Heat Transfer Viva

Heat Transfer VivaChemical Engineering PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Sun, 11 Nov 2012 05:57:08 UTC

ContentsArticles Absorptivity 1 Advection 1 Air well (condenser) 4 Baffle (in vessel) 15 Biot number 16 Black body 18 Black-body radiation 27 Boiling 41 Centrifugal evaporator 44 Churchill–Bernstein equation 45 Condensation 46 Condenser (heat transfer) 48 Convection–diffusion equation 49 Convective heat transfer 53 Countercurrent exchange 57 Critical heat flux 67 Dew point 69 Dynamic scraped surface heat exchanger 74 Electromagnetic radiation 76 Emissivity 92 Enthalpy of vaporization 94 Evaporative cooler 98 Evaporator 106 Evaporator (marine) 110 Falling film evaporator 115 Ficks laws of diffusion 116 First law of thermodynamics 122 Flash evaporation 134 Forced convection 137 Fouling 138 Fourier number 150 Heat 151 Heat capacity 160 Heat equation 180

Heat exchanger 194Heat flux 208Heat pipe 209Heat transfer 216Heat transfer coefficient 227Heisler Chart 234HVAC 239Interfacial thermal resistance 249Kirchhoffs law of thermal radiation 253Laser cooling 257List of thermal conductivities 258Log mean temperature difference 264Lumped capacitance model 266Moving bed heat exchanger 271Multiple-effect evaporator 272Natural convection 273NTU method 277Nucleation 278Packed bed 285Plancks law 287Plate fin heat exchanger 313Plate heat exchanger 314R-value (insulation) 317Radiant barrier 326Radiative cooling 331Radiosity (heat transfer) 332Rayleigh number 335Reboiler 336Recuperator 339Reflectivity 341Regenerative heat exchanger 343Relativistic heat conduction 346Retrograde condensation 351Reynolds number 352Rotary evaporator 359Shell and tube heat exchanger 361Steady state 365Stefan–Boltzmann constant 367

Stefan–Boltzmann law 368 Surface condenser 375 Temperature 379 Temperature gradient 393 Thermal conduction 395 Thermal conductivity 402 Thermal contact 412 Thermal contact conductance 412 Thermal diffusivity 415 Thermal energy 418 Thermal energy storage 422 Thermal equilibrium 425 Thermal expansion 428 Thermal fluids 436 Thermal hydraulics 438 Thermal insulation 438 Thermal radiation 442 Thermal science 450 Thermal transmittance 451 Thermodynamic system 453 Thermomass Theory 459 Time constant 461 Vacuum evaporation 467 Vapor-compression evaporation 468 View factor 470 Viscosity 473 Wiens displacement law 490References Article Sources and Contributors 494 Image Sources, Licenses and Contributors 503Article Licenses License 508

Absorptivity 1 Absorptivity In science, the term absorptivity (or absorptance) has various meanings: • In chemistry, it usually refers to Molar absorptivity: the constant used in the Beer-Lambert law, , where is the absorbance, is the concentration of the solution, and is the path length. • In physics, the word is used interchangeably with absorbance, and in this context it means the fraction of radiation absorbed at a given wavelength. Advection In chemistry, engineering and earth sciences, advection is a transport mechanism of a substance or conserved property by a fluid due to the fluids bulk motion. An example of advection is the transport of pollutants or silt in a river by bulk water flow downstream. Another commonly advected quantity is energy or enthalpy. Here the fluid may be any material that contains thermal energy, such as water or air. In general, any substance or conserved, extensive quantity can be advected by a fluid that can hold or contain the quantity or substance. In advection, a fluid transports some conserved quantity or material via bulk motion. The fluids motion is described mathematically as a vector field, and the transported material is described by a scalar field showing its distribution over space. Advection requires currents in the fluid, and so cannot happen in rigid solids. It does not include transport of substances by simple diffusion. Advection is sometimes confused with the more encompassing process of convection. In fact, convective transport is the sum of advective transport and diffusive transport. In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean, such as heat, humidity (see moisture) or salinity. Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle. Distinction between advection and convection The term advection is sometimes used as a synonym for convection. Technically, convection is the sum of transport by diffusion and advection. Advective transport describes the movement of some quantity via the bulk flow of a fluid (as in a river or pipeline).[1][2] Meteorology In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean, such as heat, humidity or salinity. Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle. Other quantities The advection equation also applies if the quantity being advected is represented by a probability density function at each point, although accounting for diffusion is more difficult. Mathematics of advection The advection equation is the partial differential equation that governs the motion of a conserved scalar field as it is advected by a known velocity vector field. It is derived using the scalar fields conservation law, together with Gausss theorem, and taking the infinitesimal limit.

Advection 2 One easily-visualized example of advection is the transport of ink dumped into a river. As the river flows, ink will move downstream in a "pulse" via advection, as the waters movement itself transports the ink. If added to a lake without significant bulk water flow, the ink would simply disperse outwards from its source in a diffusive manner, which is not advection. Note that as it moves downstream, the "pulse" of ink will also spread via diffusion. The sum of these processes is called convection. The advection equation In Cartesian coordinates the advection operator is . where u = (ux, uy, uz) is the velocity field, and ∇ is the del operator (note that Cartesian coordinates are used here). The advection equation for a conserved quantity described by a scalar field ψ is expressed mathematically by a continuity equation: where ∇∙ is the divergence operator and again u is the velocity vector field. Frequently, it is assumed that the flow is incompressible, that is, the velocity field satisfies and u is said to be solenoidal. If this is so, the above equation can be rewritten as In particular, if the flow is steady, then which shows that ψ is constant along a streamline. Hence, so ψ doesnt vary in time. If a vector quantity a (such as a magnetic field) is being advected by the solenoidal velocity field u, the advection equation above becomes: Here, a is a vector field instead of the scalar field ψ. Solving the equation The advection equation is not simple to solve numerically: the system is a hyperbolic partial differential equation, and interest typically centers on discontinuous "shock" solutions (which are notoriously difficult for numerical schemes to handle). Even with one space dimension and a constant velocity field, the system remains difficult to simulate. The equation becomes where ψ = ψ(x, t) is the scalar field being advected and ux is the x component of the vector u = (ux,0,0). According to Zang,[3] numerical simulation can be aided by considering the skew symmetric form for the advection operator.

Advection 3 where and u is the same as above. Since skew symmetry implies only imaginary eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities (see Boyd[4]). Using vector calculus identities, these operators can also be expressed in other ways, available in more software packages for more coordinate systems. This form also makes visible that the skew symmetric operator introduces error when the velocity field diverges. Solving the advection equation by numerical methods is very challenging and there is a large scientific literature about this. References [1] Suthan S. Suthersan, "Remediation engineering: design concepts", CRC Press, 1996. (Google books) (http:/ / books. google. ca/ books?id=Rd2QIcNKl1UC& pg=PA13& dq=advection+ and+ convection#v=onepage& q=advection and convection& f=false) [2] Jacques Willy Delleur, "The handbook of groundwater engineering", CRC Press, 2006. (Google books) (http:/ / books. google. ca/ books?id=EiXxaGH3CzcC& pg=PT485& dq=advection+ versus+ convection#v=onepage& q=advection versus convection& f=false) [3] Zang, Thomas (1991). "On the rotation and skew-symmetric forms for incompressible flow simulations". Applied Numerical Mathematics 7: 27–40. doi:10.1016/0168-9274(91)90102-6. [4] Boyd, John P. (2000). Chebyshev and Fourier Spectral Methods 2nd edition (http:/ / www-personal. engin. umich. edu/ ~jpboyd/ BOOK_Spectral2000. html). Dover. pp. 213. .

Air well (condenser) 4 Air well (condenser) An air well or aerial well is a structure or device that collects water by promoting the condensation of moisture from air.[2] Designs for air wells are many and varied, but the simplest designs are completely passive, require no external energy source and have few, if any, moving parts. Three principal designs are used for air wells: high mass, radiative and active. High-mass air wells were used in the early 20th century, but the approach failed.[3] From the late 20th century onwards, low-mass, radiative collectors proved to be much more successful.[3] Active collectors collect water in the same way as a dehumidifier; although the designs work well, they require an energy source, making them uneconomical except in special circumstances. New, innovative designs seek to minimise the energy requirements of active condensers or make use of renewable energy resources. High-mass air well of Belgian engineer Achile Knapen in Trans-en-Provence. A 550 square metres (660 sq yd) radiative [1] condenser in northwest India. Background All air well designs incorporate a substrate with a temperature sufficiently low so that dew forms. Dew is a form of precipitation that occurs naturally when atmospheric water vapour condenses onto a substrate. It is distinct from fog, in that fog is made of droplets of water that condense around particles in the air.[4] Condensation releases latent heat which must be dissipated in order for water collection to continue.[5] Global atmospheric water vapor for January 30, 2005. Northern hemisphere winter and southern An air well requires moisture from the air. Everywhere on Earth, even hemisphere summer. in the hottest climates, the surrounding atmosphere contains at least some water. According to Beysens and Milimouk: "The atmosphere contains 12,900 km3 (3,000 cubic miles) of fresh water, composed of 98 percent water vapour and 2 percent condensed water (clouds): a figure comparable to the renewable liquid water resources of inhabited lands

Air well (condenser) 5 (12,500 km3)."[4] The quantity of water vapour contained within the air is commonly reported as a relative humidity, and this depends on temperature—warmer air can contain more water vapour than cooler air. When air is cooled to the dew point, it becomes saturated, and moisture will condense on a suitable surface.[6] For instance, the dew temperature of air at 20 °C (68 °F) and 80 percent relative humidity is 18 °C (64 °F). The dew temperature falls to 10 °C (50 °F) if the relative humidity is only 25 percent.[4] A related, but quite distinct, technique of obtaining atmospheric moisture is the fog fence. An air well should not be confused with a dew pond. A dew pond is an artificial pond intended for watering livestock. The name dew pond (sometimes cloud pond or mist pond) derives from the widely held belief that the pond was filled by moisture from the air.[7] In fact, dew ponds are primarily filled by rainwater.[8] A stone mulch can significantly increase crop yields in arid areas. This is most notably the case in the Canary Islands: on the island of Lanzarote there is about 140 millimetres (5.5 in) of rain each year and there are no permanent rivers. Despite this, substantial crops can be grown by using a mulch of volcanic stones, a trick discovered after volcanic eruptions in 1730. Some credit the stone mulch with promoting dew; although the idea has inspired some thinkers, it seems unlikely that the effect is significant. Rather, plants are able to absorb dew directly from their leaves, and the main benefit of a stone mulch is to reduce water loss from the soil and to eliminate competition from weeds.[9] History Beginning in the early 20th century, a number of inventors experimented with high-mass collectors. Notable investigators were the Russian engineer Friedrich Zibold (sometimes given as Friedrich Siebold[10]), the French bioclimatologist Leon Chaptal, the German-Australian researcher Wolf Klaphake and the Belgian inventor Achille Knapen. Zibolds collector In 1900, near the site of the ancient Byzantine city of Theodosia, thirteen large piles of stones were discovered by Zibold who was a forester and engineer in charge of this area.[12] Each stone pile covered just over 900 square metres (9,700 sq ft) and was about 10 metres (33 ft) tall. The finds were associated with the A section through Zibolds dew condenser. (a) is a truncated cone of remains of 75-millimetre diameter (3 in) terracotta beach pebbles 20 metres (66 ft) in diameter at the base and 8 metres pipes that apparently led to wells and fountains in the (26 ft) in diameter at the top. (b) is a concrete bowl; a pipe (not city. Zibold concluded that the stacks of stone were shown) leads away from the base of the bowl to a collecting point. [11] (c) is ground level and (d) is the natural limestone base. condensers that supplied Theodosia with water; and calculated that each air well produced more than 55,400 litres (12,200 imp gal; 14,600 US gal) each day.[10] To verify his hypothesis Zibold constructed a stone-pile condenser at an altitude of 288 metres (945 ft) on mount Tepe-Oba near the ancient site of Theodosia. Zibold’s condenser was surrounded by a wall 1 metre (3 ft 3 in) high, 20 metres (66 ft) wide, around a bowl-shaped collection area with drainage. He used sea stones 10–40 centimetres (3.9–16 in) in diameter piled 6 metres (20 ft) high in a truncated cone that was 8 metres (26 ft) in diameter across the top. The shape of the stone pile allowed a good air flow with only minimal thermal contact between the stones.[13] Zibolds condenser began to operate in 1912 with a maximum daily production that was later estimated to have been 360 litres (79 imp gal; 95 US gal) – Zibold made no public record of his results at the time.[10] The base developed leaks that forced the experiment to end in 1915 and the site was partially dismantled before being abandoned. (The site was rediscovered in 1993 and cleaned up.)[13] Zibolds condenser was approximately the same size as the ancient

Air well (condenser) 6 stone piles that had been found,[13] and although the yield was very much less than the yield Zibold had calculated for the original structures, the experiment was an inspiration for later developers. Chaptals collector Inspired by Zibolds work, Chaptal built a small air well near Montpellier in 1929. Chaptals condenser was a pyramidal concrete structure 3 metres (9.8 ft) square and 2.5 metres (8 ft 2 in) high, it was filled with 8 cubic metres (280 cu ft) of limestone pieces being about 7.5 centimetres (3 in) in diameter. Small vent holes ringed the top and bottom of the pyramid. These holes could be closed or opened as required to control the flow of air. The structure was allowed to cool during the night, and then warm moist air was let in during the day. Dew formed on the limestone pieces and collected in a reservoir below ground level. The amount of water obtained varied from 1 litre (0.22 imp gal; 0.26 US gal) to 2.5 litres (0.55 imp gal; 0.66 US gal) per day depending on the atmospheric conditions.[14] Chaptal did not consider his experiment a success. When he retired in 1946, he put the condenser out of order, possibly because he did not want to leave an improper installation to mislead those who might later continue studies on air wells.[3] Klaphakes collectors Wolf Klaphake was a successful chemist working in Berlin during the 1920s and 30s. During that time, he tested several forms of air wells in Yugoslavia and on Vis Island in the Adriatic Sea. Klaphakes work was inspired by Zibold[15] and by the works of Maimonides, a known Jewish scholar who wrote in Arabic about 1,000 years ago and who mentioned the use of water condensers in Palestine.[13] Klaphake experimented with a very simple design: an area of mountain slope was cleared and smoothed with a watertight surface. It was shaded by a simple canopy supported by pillars or ridges. The sides of the structure were closed, but the top and bottom edges were left open. At night the mountain slope would cool, and in the day moisture would collect on and run down the smoothed surface. Although the system apparently worked, it was expensive, and Klaphake finally adopted a more compact design based on a masonry structure. This design was a sugarloaf-shaped building, about 15 metres (49 ft) high, with walls at least 2 metres (6 ft 7 in) thick, with holes on the top and at the bottom. The outer wall is made of concrete to give a high thermal capacity, and the inner surface was made of a porous material such as sandstone.[16] According to Klaphake: “ The building produces water during the day and cools itself during the night; when the sun rises, the warm air is drawn through the upper holes into the building by the out-flowing cooler air, becomes cooled on the cold surface, deposits its water, which then oozes down and is collected somewhere underneath. It is wrong to think that this process works only on days with dew, as the inner surface becomes much cooler than one should expect. In Dalmatia, that day was a rare exception which failed to produce water. [15] ” Traces of Klaphakes condensers have been tentatively identified.[17] In 1935, Wolf Klaphake and his wife Maria emigrated to Australia. The Klaphakes decision to emigrate was probably primarily the result of Marias encounters with Nazi authorities;[18][19] their decision to settle in Australia (rather than, say, in Britain) was influenced by Wolfs desire to develop a dew condenser.[19] As a dry continent, Australia was likely to need alternative sources of fresh water, and the Premier of South Australia, whom he had met in London, had expressed an interest. Klaphake made a specific proposal for a condenser at the small town of Cook, where there was no supply of potable water. At Cook, the railway company had previously installed a large coal-powered active condenser,[20] but it was prohibitively expensive to run, and it was cheaper to simply transport water. However, the Australian government turned down Klaphakes proposal, and he lost interest in the project.[21][15]

Air well (condenser) 7 Knapens aerial well Exterior Interior. Knapen, who had previously worked on systems for removing moisture from buildings,[22][23][24] was in turn inspired by Chaptals work and he set about building an ambitiously large puits aerien (aerial well) on a 180 metres (590 ft) high hill at Trans-en-Provence in France.[2][25] Beginning in 1930, Knapens dew tower took 18 months to build; it still stands today, albeit in dilapidated condition. At the time of its construction, the condenser excited some public interest.[26] The tower is 14 metres (46 ft) high and has massive masonry walls about 3 metres (9.8 ft) thick with a number of apertures to let in air. Inside there is a massive column made of concrete. At night, the whole structure is allowed to cool, and during the day warm moist air enters the structure via the high apertures, cools, descends, and leaves the building by the lower apertures.[27] Knapen’s intention was that water should condense on the cool inner column. In keeping with Chaptal’s finding that the condensing surface must be rough and the surface tension must be sufficiently low that the condensed water can drip, the central columns outer surface was studded with projecting plates of slate. The slates were placed nearly vertically to encourage dripping down to a collecting basin at the bottom of the structure.[13] Unfortunately, the aerial well never achieved anything like its hoped-for performance and produced no more than a few litres of water each day.[28] International Organisation for Dew Utilization By the end of the twentieth century, the details of how dew condenses were much better understood, the key insight being that low-mass collectors that rapidly lose heat by radiation performed best. A number of researchers worked on this method.[29] In the early 1960s, simple dew condensers made from sheets of polyethylene supported on a simple frame resembling a ridge tent were used in Israel to irrigate plants. Saplings supplied with dew and very slight rainfall from these collectors survived much better than the control group planted without Big OPUR Dew Condenser in Corsica such aids – they all dried up over the summer.[30] And in 1986 in New Mexico condensers made of a special foil produced sufficient water to supply young saplings.[4] In 1992 a party of French academics attended a condensed matter conference in Ukraine where physicist Daniel Beysens introduced

Air well (condenser) 8 them to the story of how ancient Theodosia was supplied with water from dew condensers. They were sufficiently intrigued that in 1993 they went to see for themselves. They concluded that the mounds that Zibold identified as dew condensers were in fact ancient burial mounds (a part of the necropolis of antic Theodosia) and that the pipes were medieval in origin and not associated with the mounds. They found the remains of Zibolds condenser, which they tidied up and examined closely. Zibolds condenser had apparently performed reasonably well, but in fact his exact results are not at all clear, and it is possible that the A radiative dew condenser test site at village collector was intercepting fog, which added significantly to the Kothar in the north-west India near Arabian sea coast. yield.[10] If Zibolds condenser worked at all, this was probably due to fact that a few stones near the surface of the mound were able to lose heat at night while being thermally isolated from the ground; however, it could never have produced the yield that Zibold envisaged.[3][31] Fired with enthusiasm, the party returned to France and set up the International Organisation for Dew Utilization (OPUR), with the specific objective of making dew available as an alternative source of water.[32] OPUR began a study of dew condensation under laboratory conditions; they developed a special hydrophilic film and experimented with trial installations, including a 30 square metres (320 sq ft) collector in Corsica.[33] Vital insights included the idea that the mass of the condensing surface should be as low as possible so that it cannot easily retain heat, that it should be protected from unwanted thermal radiation by a layer of insulation, and that it should be hydrophilic, so as to shed condensed moisture readily.[34] By the time they were ready for their first practical installation, they heard that one of their members, Girja Sharan, had obtained a grant to construct a dew condenser in Kothara, India. In April 2001, Sharan had incidentally noticed substantial condensation on the roof of a cottage at Toran Beach Resort in the arid coastal region of Kutch, where he was briefly staying. The following year, he investigated the phenomenon more closely and interviewed local people. Financed by the Gujarat Energy Development Agency and the World Bank, Sharan and his team went on to develop passive, radiative condensers for use in the arid coastal region of Kutch.[35] Active commercialisation began in 2006.[36] Sharan tested a wide range of materials and got good results from galvanised iron and aluminium sheets, but found that sheets of the special plastic developed by the OPUR just 400 micrometres (0.016 in) thick generally worked even better than the metal sheets and were less expensive.[37] The plastic film, known as OPUR foil, is hydrophilic and is made from polyethylene mixed with titanium dioxide and barium sulphate. Types There are three principal approaches to the design of the heat sinks that collect the moisture in air wells: high mass, radiative and active. Early in the twentieth century, there was interest in high-mass air wells, but despite much experimentation including the construction of massive structures, this approach proved to be a failure.[38] From the late twentieth century onwards, there has been much investigation of low-mass, radiative collectors; these have proved to be much more successful.

Air well (condenser) 9 High-mass The high-mass air well design attempts to cool a large mass of masonry with cool nighttime air entering the structure due to breezes or natural convection. In the day, the warmth of the sun results in increased atmospheric humidity. When moist daytime air enters the air well, it condenses on the presumably cool masonry. None of the high-mass collectors performed well, Knapens aerial well being a particularly conspicuous example. The problem with the high-mass collectors was that they could not get rid of sufficient heat during the night – despite design features intended to ensure that this would happen.[13] While some thinkers have believed that Zibold might have been correct after all,[39][40] an article in Journal of Arid Environments discusses why high-mass condenser designs of this type cannot yield useful amounts of water: “ We would like to stress the following point. To obtain condensation, the condenser temperature of the stones must be lower than the dew point temperature. When there is no fog, the dew point temperature is always lower than the air temperature. Meteorological data shows that the dew point temperature (an indicator of the water content of the air) does not change appreciably when the weather is stable. Thus wind, which ultimately imposes air temperature to the condenser, cannot cool the condenser to ensure its functioning. Another cooling phenomenon — radiative cooling — must operate. It is therefore at night-time, when the condenser cools by radiation, that liquid water can be extracted from air. It is very rare that the dew point temperature would increase significantly so as to exceed the stone temperature inside the stone heap. Occasionally, when this does happen, dew can be abundant during a short period of time. This is why subsequent attempts by L. Chaptal and A. Knapen to build massive dew condensers only rarely resulted in significant yields. [Emphasis as in original] [3] ” Although ancient air wells are mentioned in some sources, there is scant evidence for them, and persistent belief in their existence has the character of a modern myth.[3] Radiative A radiative air well is designed to cool a substrate by radiating heat to the night sky. The substrate has a low mass so that it cannot hold onto heat, and it is thermally isolated from any mass, including the ground.[41] A typical radiative collector presents a condensing surface at an angle of 30° from the horizontal. The condensing surface is backed by a thick layer of insulating material such as polystyrene foam and supported 2–3 metres (7–10 ft) above ground level. Such condensers may be conveniently installed on the ridge roofs of low buildings or supported by a simple frame.[42] Although other heights Diagram of a radiative collector. (a) radiating/condensing surface, (b) collecting do not typically work quite so well, it may be less expensive or more gutter, (c) backing insulation, (d) stand. convenient to mount a collector near to ground level or on a two-storey building.[43] The 600 square metres (6,500 sq ft) radiative condenser illustrated near the start of this article is built near the ground. In the area of north-west India where it is installed dew occurs for 8 months a year, and the installation collects about 15 millimetres (0.59 in) of dew water over the season with nearly 100 dew-nights. In a year it provides a total of about 9,000 litres (2,000 imp gal; 2,400 US gal) of potable water for the school which owns and operates the site.[1] Although flat designs have the benefit of simplicity, other designs such as inverted pyramids and cones can be significantly more effective. This is probably because the designs shield the condensing surfaces from unwanted heat radiated by the lower atmosphere, and, being symmetrical, they are not sensitive to wind direction.[44]

Air well (condenser) 10 New materials may make even better collectors.[45] One such material is inspired by the Namib Desert beetle, which survives only on the moisture it extracts from the atmosphere. It has been found that its back is coated with microscopic projections: the peaks are hydrophilic and the troughs are hydrophobic.[46][47][48] Researchers at the Massachusetts Institute of Technology have emulated this capability by creating a textured surface that combines alternating hydrophobic and hydrophilic materials. Structures with metal roofing, such as this one, can be used to harvest dew water simply by adding gutters and, for increased output, a layer of insulation underside. Without the insulation the output is nearly half of that from plastic condensers. An example of a condenser-on-roof installation, condenser made of plastic film with special properties, with insulation layer between film and concrete roof surface. This installation is on school buildings at Sayara (Kutch, India). Unlike metal roofs, concrete roofs do not attract condensation without any treatment, hence the need for an external condenser. The output from such condensers is nearly two times higher than from a bare metal roof, all else remaining constant.

Air well (condenser) 11 Active Active atmospheric water collectors have been in use since the commercialisation of mechanical refrigeration. Essentially, all that is required is to cool a heat exchanger below the dew point, and water will be produced. Such water production may take place as a by-product, possibly unwanted, of dehumidification.[13] The air conditioning system of the Burj Khalifa in Dubai, for example, produces an estimated 15 million US gallons (57,000 m3) of water each year that is used for irrigating the towers landscape plantings.[50] Because mechanical refrigeration is energy intensive, active collectors are typically restricted to places where there is no supply of water that can be desalinated or purified at a lower cost and that are sufficiently far from a supply of fresh water to make transport uneconomical. Such circumstances are uncommon, and even then large installations such as that tried in the 1930s at Cook in South Australia failed because of the cost of running the installation – it was cheaper to transport water over large distances.[21] In the case of small installations, convenience may outweigh cost. There is a wide range of small machines designed to be used in offices that produce a few litres of drinking water from the atmosphere. However, there are circumstances where there really is no source of water other than the A commercially produced atmospheric atmosphere. For example, in the 1930s, American designers added condenser water generator intended for residential [49] systems to airships – in this case the air was that emitted by the exhaust of the use. engines, and so it contained additional water as a product of combustion. The moisture was collected and used as additional ballast to compensate for the loss of weight as fuel was consumed. By collecting ballast in this way, the airships buoyancy could be kept relatively constant without having to release helium gas, which was both expensive and in limited supply.[51] More recently, on the International Space Station, the Zvezda module includes a humidity control system, the water it collects is usually used to supply the Elektron system that electrolyses water into hydrogen and oxygen, but it can be used for drinking in an emergency.[52] There are a number of designs that minimise the energy requirements of active condensers: • One method is to use the ground as a heat sink by drawing air through underground pipes.[53] This is often done to provide a source of cool air for a building by means of a ground-coupled heat exchanger (also known as Earth tubes), wherein condensation is typically regarded as a significant problem.[54] A major problem with such designs is that the underground tubes are subject to contamination and difficult to keep clean. Designs of this type require air to be drawn through the pipes by a fan, but the power required may be provided (or supplemented) by a wind turbine.[55] • Cold seawater is used in the Seawater Greenhouse to both cool and humidify the interior of greenhouse-like structure. The cooling can be so effective that not only do the plants inside benefit from reduced transpiration, but dew collects on the outside of the structure and can easily be collected by gutters.[4] • Another type of atmospheric water collector makes use of desiccants which adsorb atmospheric water at ambient temperature, this makes it possible to extract moisture even when the relative humidity is as low as 14 percent.[56] Systems of this sort have proved to be very useful as emergency supplies of safe drinking water.[57][58] For regeneration, the desiccant needs to be heated.[59] In some designs regeneration energy is supplied by the sun; air is ventilated at night over a bed of desiccants that adsorb the water vapour. During the day, the premises are closed, the greenhouse effect increases the temperature, and, as in solar desalination pools, the water vapour is

Air well (condenser) 12 partially desorbed, condenses on a cold part and is collected.[4] • A French company has recently designed a small wind turbine that uses a 30 kW electric generator to power an onboard mechanical refrigeration system to condense water.[60] References Notes [1] Sharan 2007 [2] Popular Science 1933 [3] Beysens et al. 2006 [4] Beysens & Milimouk 2000 [5] Nikolayev et al. 1996, pp. 23–26 [6] "What Exactly Is The Dew Point?" (http:/ / weathersavvy. com/ Q-Science_Dewpoint. html). Weather Savvy. . Retrieved 10 September 2010. [7] Oxford English Dictionary: "dew-pond" [8] Pugsley, 1939 [9] Pearce, Fred (9 September 2006). "The Miracle of the Stones". New Scientist: 50–51. [10] Nikolayev et al. 1996, p. 4 [11] Based on diagram by Nikolayev et all, 1996 [12] Nikolayev et al. 1996, pp. 20–23 [13] Nelson 2003 [14] Hills 1966, p. 232 [15] Klaphake 1936 [16] Sharan 2006, p. 72 [17] "In Croatia" (http:/ / opur. fr/ angl/ Bulletin7. pdf). OPUR Newletter. OPUR. April 2003. Archived (http:/ / web. archive. org/ web/ 20100911081957/ http:/ / www. opur. fr/ angl/ Bulletin7. pdf) from the original on 11 September 2010. . Retrieved 10 September 2010. [18] Neumann 2002, p. 7 [19] Klaus Neumann. "Wolf Klaphake – Immigrant or refugee" (http:/ / www. uncommonlives. naa. gov. au/ wolf-klaphake/ life/ immigrant-or-refugee. aspx). Uncommon Lives (National Archives of Australia). . Retrieved 10 September 2010. [20] Klaus Neumann. "Trans-Australian Railway photograph of a condenser cooler at Cook, 10 December 1917" (http:/ / www. uncommonlives. naa. gov. au/ wolf-klaphake/ enlargement/ trans-australian-railway-photograph-of-a-condenser-cooler-at-cook. aspx). Uncommon Lives (National Archives of Australia). . Retrieved 10 September 2010. [21] Klaus Neumann. "Wolf Klaphake – A rainmaker?" (http:/ / www. uncommonlives. naa. gov. au/ wolf-klaphake/ life/ a-rainmaker. aspx). Uncommon Lives (National Archives of Australia). . Retrieved 10 September 2010. [22] "British Knapen – The Early Years" (http:/ / www. protenservices. co. uk/ pdf/ name/ 28/ ProTen_Services_History_in_Pictures. pdf). ProTen Services. . Retrieved 10 September 2010. [23] Prevention Of Damp In Buildings. The Manchester Guardian, 27 February 1930 p. 6 column F. [24] "ProTen Services Celebrates 80 Years of Service" (http:/ / www. protenservices. co. uk/ pdf/ name/ 27/ ProTen_Services_celebrates_80_Years_of_Service. pdf). ProTen Services. . Retrieved 10 September 2010. [25] "Well Like Gigantic Ant Hill Gathers Water from Air" (http:/ / books. google. com/ ?id=fvEDAAAAMBAJ& printsec=frontcover#PPA868,M1). Popular Mechanics (Hearst Magazines) 58 (6): 868. December 1932. . Retrieved 10 September 2010. [26] "Air Well Waters Parched Farms" (http:/ / books. google. com/ books?id=oygDAAAAMBAJ& pg=PA45& dq=Popular+ Science+ 1933+ plane+ "Popular+ Science"& hl=en& ei=_P1UTf6QMYH-8Abd79mOBw& sa=X& oi=book_result& ct=result& resnum=7& ved=0CEgQ6AEwBg#v=onepage& q=Popular Science 1933 plane "Popular Science"& f=true) Popular Science, March 1933 [27] Achile Knappen. "Improved means for collecting moisture from the atmosphere" (http:/ / v3. espacenet. com/ publicationDetails/ biblio?adjacent=true& KC=A& date=19300522& NR=319778A& DB=EPODOC& locale=en_V3& CC=GB& FT=D). European Patent Office. . Retrieved 10 September 2010. [28] Sharan 2006, p. 70 [29] Sharan 2006, p. 22 [30] Gindel 1965 [31] Nikolayev et al. 1996 [32] "OPUR Ou la Conquete de la Rosee – OPUR or The Conquest of Dew" (http:/ / www. opur. fr/ index. htm) (in French with English subtitles). OPUR. Archived (http:/ / web. archive. org/ web/ 20100907070526/ http:/ / www. opur. fr/ index. htm) from the original on 7 September 2010. . Retrieved 10 September 2010. [33] Muselli, Beysens & Milimouk 2006 [34] Sharan 2006, pp. 20–28 [35] Sharan 2006, Acknowledgement section

Air well (condenser) 13 [36] Mukund, Dixit; Sharan, Girha (1 April 2007). "Leveraged Innovation Management: Key Themes from the Journey of Dewrain Harvest Systems" (http:/ / www. iimahd. ernet. in/ publications/ data/ 2007-01-04_mrdixit. pdf). Indian Institute of Management Ahmedabad, India. . Retrieved 10 September 2010. [37] Sharan 2006, p. 27 [38] Alton Stewart & Howell 2003, p. 1014 [39] "Pyramid Power" (http:/ / books. google. com/ ?id=nAEAAAAAMBAJ& printsec=frontcover#v=onepage& q). Popular Science (Bonnier Corporation) 241 (4): 30–31. October 1992. . Retrieved 10 September 2010. [40] Pearce, Fred (16 April 2005). "Pyramids of dew". New Scientist (2495). [41] Sharan, Girja. "Dew Yield From Passive Condensers in a Coastal Arid Area – Kutch" (http:/ / www. iimahd. ernet. in/ publications/ data/ 2005-01-05gsharan. pdf). p. 2. . Retrieved 10 September 2010. [42] Sharan 2006, pp. 20–39 [43] Sharan 2006, pp. 40–59 [44] Clus et al. 2006 [45] Sharan 2006, p. 20 [46] Parker, A. R. & C. R. Lawrence (2001). "Water capture by a desert beetle". Nature 414 (6859): 33–34. doi:10.1038/35102108. PMID 11689930. [47] Harries-Rees, Karen (August 31, 2005). "Desert beetle provides model for fog-free nanocoating" (http:/ / www. rsc. org/ chemistryworld/ News/ 2005/ August/ 31080502. asp). Chemistry World News (Royal Society of Chemistry). . Retrieved 10 September 2010. [48] Pawlyn, Michael (November 2010). "Using natures genius in architecture (at 7:45)" (http:/ / www. ted. com/ talks/ michael_pawlyn_using_nature_s_genius_in_architecture. html). TED. p. 2. Archived (http:/ / web. archive. org/ web/ 20110211140648/ http:/ / www. ted. com/ talks/ michael_pawlyn_using_nature_s_genius_in_architecture. html) from the original on 11 February 2011. . Retrieved 14 February 2011. [49] "Yeti Air-Conditioning-12" (http:/ / www. everestwater. com/ yetiac12. htm). Everest. . Retrieved 15 March 2011. [50] "Burj Khalifa: Towering challenge for builders" (http:/ / gulfnews. com/ business/ property/ burj-khalifa-towering-challenge-for-builders-1. 561802). GulfNews.com. January 4, 2010. Archived (http:/ / web. archive. org/ web/ 20110125112740/ http:/ / gulfnews. com/ business/ property/ burj-khalifa-towering-challenge-for-builders-1. 561802) from the original on 25 January 2011. . Retrieved 2011-01-12. [51] Allen 1931, p. 37 [52] "Zvezda" (http:/ / www. shuttlepresskit. com/ ISS_OVR/ assembly2_overview. htm). The ISS: Continued Assembly and Performance. NASA. Archived (http:/ / web. archive. org/ web/ 20100825152815/ http:/ / www. shuttlepresskit. com/ iss_ovr/ assembly2_overview. htm) from the original on 25 August 2010. . Retrieved 10 September 2010. [53] Lindsley, E.F. (January 1984). "Airwell extracts Pure Water From the Air" (http:/ / books. google. com/ ?id=eAAAAAAAMBAJ& printsec=frontcover). Popular Science (Bonnier Corporation) 224 (1). . Retrieved 10 September 2010. [54] David Darling. "Earth Cooling Tube" (http:/ / www. daviddarling. info/ encyclopedia/ E/ AE_earth_cooling_tube. html). The Encyclopedia of Alternative Energy and Sustainable Living. . Retrieved 10 September 2010. [55] US patent 4351651 (http:/ / worldwide. espacenet. com/ textdoc?DB=EPODOC& IDX=US4351651), Courneya, Calice, G., "Apparatus for extracting potable water", issued 1980-12-06 [56] Audrey Hudson (6 October 2006). "Making Water From Thin Air" (http:/ / www. wired. com/ science/ discoveries/ news/ 2006/ 10/ 71898). Wired. Archived (http:/ / web. archive. org/ web/ 20100731054540/ http:/ / www. wired. com/ science/ discoveries/ news/ 2006/ 10/ 71898) from the original on 31 July 2010. . Retrieved 10 September 2010. [57] Sher, Abe M.. "Advanced Water Technologies" (http:/ / www. aquasciences. com). Aqua Sciences. Archived (http:/ / web. archive. org/ web/ 20100917070458/ http:/ / aquasciences. com/ ) from the original on 17 September 2010. . Retrieved 10 September 2010. [58] Cartlidge 2009, pp. 26–27 [59] Cartlidge 2009, p. 16 [60] "Eolewater" (http:/ / www. eolewater. com/ gb/ home. html). . Retrieved 7 October 2011. Sources • Allen, Hugh (1931). The Story of the Airship. Goodyear Tire and Rubber Company. • Alton Stewart, Bobby; Howell, Terry A. (2003). Encyclopedia of water science. Marcel Dekker. ISBN 978-0-8247-0948-8. • Beysens, D.; Milimouk, I. (2000). "The Case For Alternative Fresh Water Sources" (http://www.opur.fr/angl/ Secheresse-angl.pdf) (pdf). International Organization For Dew Utilization. Retrieved 10 September 2010. • Beysens, D.; Milimouk, I.; Nikolayev, V.S.; Berkowicz, S.; Muselli, M.; Heusinkveld, B.; Jacobs, A.F.G. (2006). "Comment on "The moisture from the air as water resource in arid region: Hopes, doubt and facts" by Kogan and Trahtman" (http://u.cs.biu.ac.il/~trakht/JAE06.pdf). Journal of Arid Environments (Elsevier) 67 (2): 343. doi:10.1016/j.jaridenv.2006.01.011. Retrieved 10 September 2010.

Air well (condenser) 14 • Cartlidge, Cherese (2009). Water from Air: Water-Harvesting Machines. Norwood House. ISBN 978-1-59953-196-0. • Clus, Owen; Ouazzani, Jalil; Muselli, Marc; Nikolayev, Vadim; Sharan, Girja; Beysens, Daniel (2006). "Radiation-cooled Dew Water Condensers Studied by Computational Fluid Dynamic (CFD)". European PHOENICS User Meeting, //12 , Wimbledon, London, UK (CD-ROM) 1 2006 (30): 11–1. arXiv:0707.2514. Bibcode 2007arXiv0707.2514C. • Gindel, I. (11 September 1965). "Irrigation of Plants with Atmospheric Water Within the Desert". Nature 207 (5002): 1173–1175. Bibcode 1965Natur.207.1173G. doi:10.1038/2071173a0. • Hills, Edwin Sherbon (1966). Arid Lands: A Geographical Appraisal. Methuen. • Klaphake, Wolf (1936). "Practical Methods for Condensation of Water from the Atmosphere" (http://www. rexresearch.com/airwells/klaphake.htm). Proceeding of the Society of Chemical Industry of Victoria 36: 1093–1103. Retrieved 12 September 2010. • Muselli, M.; Beysens, D.; Milimouk, I. (January 2006). "Comparative Dew Yields From Two Large Planar Dew Condensers" (http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WH9-4GGXXCG-1& _user=10&_coverDate=01/31/2006&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d& _docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10& md5=129f69503ff40e1fbff76cd10ec77ef4&searchtype=a). Journal of Arid Environments 64 (1): 54–76. doi:10.1016/j.jaridenv.2005.04.007. Retrieved 6 April 2011. • Nelson, Robert A. (2003). "Air Wells, Fog Fences & Dew Ponds – Methods for Recovery of Atmospheric Humidity" (http://www.rexresearch.com/airwells/airwells.htm). Rex Research. Retrieved 10 September 2010. "This article has been widely reproduced, including extracts in Sharan, 2006." • Neumann, Klaus (2002). "Fifth Columnists? German and Austrian Refugees in Australian Internment Camps" (http://www.sisr.net/publications/0204neumann.pdf). National Archives of Australia. Retrieved 18 March 2011. • Nikolayev, V.S.; Beysens, D.; Gioda, A.; Milimouk, I.; Katiushin, E.; Morel, J. P. (1996). "Water Recovery from Dew". Journal of Hydrology (Elsevier) (182). • Pugsley, Alfred J (1939). Dewponds in Fable and Fact. Country Life Ltd. • Sharan, Girja (2006). Dew Harvest. Foundation Books. ISBN 81-7596-326-3. • Sharan, Girja (2007). "Harvesting dew to supplement drinking water supply in arid coastal villages of Gujarat" (http://www.iimahd.ernet.in/publications/data/2007-08-05Gsharan.pdf) (pdf). Indian Institute of Management. Retrieved 10 September 2010. • Anonymous (March 1933). "Air Well Waters Parched Farms" (http://books.google.com/ ?id=oygDAAAAMBAJ&printsec=frontcover#v=onepage&q). Popular Science (Bonnier Corporation) 122 (3). Retrieved 10 September 2010. External links • OPUR (International Organisation for Dew Utilization) (http://www.opur.fr/) • "Fog Quest Official Site" (http://www.fogquest.org/). Archived (http://web.archive.org/web/ 20090223071843/http://www.fogquest.org/) from the original on 23 February 2009. Retrieved 19 March 2009. • "Dew Harvesting" (http://www.rain-barrel.net/dew-harvesting.html). Rainwater Harvesting Guide. Retrieved 19 March 2009. • Pure water ... from thin Air! (https://www.youtube.com/watch?v=C57jiwUTSQE) on YouTube – KBTV: Aqua Sciences Emergency Water Station, that produces pure water out of the humidity-saturated air in the wake of a hurricane. • Compound Parabolic Concentrator for Passive Radiative Cooling (http://people.csail.mit.edu/jaffer/cool/ Aperture)

Baffle (in vessel) 15 Baffle (in vessel) Baffles are flow-directing or obstructing vanes or panels used in some industrial process vessels (tanks), such as shell and tube heat exchangers, chemical reactors, and static mixers. Baffles are an integral part of the shell and tube heat exchanger design. A baffle is designed to support tube bundles and direct the flow of fluids for maximum efficiency. Use of baffles The main roles of a baffle in a shell and tube heat exchanger are to: • Hold tubes in position (preventing sagging), both in production and operation • Prevent the effects of vibration, which is increased with both fluid velocity and the length of the exchanger • Direct shell-side fluid flow along tube field. This increases fluid velocity and the effective heat transfer co-efficient of the exchanger In a static mixer, baffles are used to promote mixing.[1] In a chemical reactor, baffles are often attached to the interior walls to promote mixing[2] and thus increase heat transfer and possibly chemical reaction rates. Cross-sectional diagram of stirred tank reactor Types of baffles Implementation of baffles is decided on the basis of size, cost and their ability to lend support to the tube bundles and direct • Longitudinal Flow Baffles (used in a two-pass shell) • Impingement Baffles (used for protecting bundle when entrance velocity is high) • Orifice Baffles. • single segmental • double segmental

Baffle (in vessel) 16 Installation of baffles As mentioned, baffles deal with the concern of support and fluid direction in heat exchangers. In this way it is vital that they are spaced correctly at installation. The minimum baffle spacing is the greater of 50.8mm or one fifth of the inner shell diameter. The maximum baffle spacing is dependent on material and size of tubes. The Tubular Exchanger Manufacturers Association [3] sets out guidelines. There are also segments with a "no tubes in window" design that affects the acceptable spacing within the design. An important design consideration is that no recirculation zones or dead spots form – both of which are counterproductive to effective heat transfer. References [1] SPE/ANTEC 1999 Proceedings edited by SPE Staff pp. 163–164 [2] Chemical reactor modeling: multiphase reactive flows by Hugo A. Jakobsen, published in 2008 by Springer-Verlag in Berlin Heidelberg, pp. 681–683 [3] Tubular Exchanger Manufacturers Association, Inc. (http:/ / www. tema. org/ ) • (Editors)Perry, R.H. and Green, D.W. (Oct,2007) Perrys Chemical Engineers Handbook (8th ed.) McGraw-Hill ISBN 0-07-142294-3 • Wolverine Tube Inc, (2008) Heat Transfer Data Book Available (http://www.wlv.com/products/databook/ ch1_4.pdf) • Professor J. Kavanagh (2009) Heat Transfer Lectures 4&5 Usyd Chemical Engineering Department Biot number The Biot number (Bi) is a dimensionless number used in heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot (1774–1862), and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body. This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, from a thermal gradient applied to its surface. In general, problems involving small Biot numbers (much smaller than 1) are thermally simple, due to uniform temperature fields inside the body. Biot numbers much larger than 1 signal more difficult problems due to non-uniformity of temperature fields within the object. The Biot number has a variety of applications, including transient heat transfer and use in extended surface heat transfer calculations. Definition The Biot number is defined as: where: • h = film coefficient or heat transfer coefficient or convective heat transfer coefficient • LC = characteristic length, which is commonly defined as the volume of the body divided by the surface area of the body, such that • kb = Thermal conductivity of the body The physical significance of Biot number can be understood by imagining the heat flow from a small hot metal sphere suddenly immersed in a pool, to the surrounding fluid. The heat flow experiences two resistances: the first within the solid metal (which is influenced by both the size and composition of the sphere), and the second at the surface of the sphere. If the thermal resistance of the fluid/sphere interface exceeds that thermal resistance offered by

Biot number 17 the interior of the metal sphere, the Biot number will be less than one. For systems where it is much less than one, the interior of the sphere may be presumed always to have the same temperature, although this temperature may be changing, as heat passes into the sphere from the surface. The equation to describe this change in (relatively uniform) temperature inside the object, is simple exponential one described in Newtons law of cooling. In contrast, the metal sphere may be large, causing the characteristic length to increase to the point that the Biot number is larger than one. Now, thermal gradients within the sphere become important, even though the sphere material is a good conductor. Equivalently, if the sphere is made of a thermally insulating (poorly conductive) material, such as wood or styrofoam, the interior resistance to heat flow will exceed that of the fluid/sphere boundary, even with a much smaller sphere. In this case, again, the Biot number will be greater than one. Applications Values of the Biot number smaller than 0.1 imply that the heat conduction inside the body is much faster than the heat convection away from its surface, and temperature gradients are negligible inside of it. This can indicate the applicability (or inapplicability) of certain methods of solving transient heat transfer problems. For example, a Biot number less than 0.1 typically indicates less than 5% error will be present when assuming a lumped-capacitance model of transient heat transfer (also called lumped system analysis).[1] Typically this type of analysis leads to simple exponential heating or cooling behavior ("Newtonian" cooling or heating) since the amount of thermal energy (loosely, amount of "heat") in the body is directly proportional to its temperature, which in turn determines the rate of heat transfer into or out of it. This leads to a simple first-order differential equation which describes heat transfer in these systems. Having a Biot number smaller than 0.1 labels a substance as thermally thin, and temperature can be assumed to be constant throughout the materials volume. The opposite is also true: A Biot number greater than 0.1 (a "thermally thick" substance) indicates that one cannot make this assumption, and more complicated heat transfer equations for "transient heat conduction" will be required to describe the time-varying and non-spatially-uniform temperature field within the material body. Together with the Fourier number, the Biot number can be used in transient conduction problems in a lumped parameter solution which can be written as, Mass transfer analogue An analogous version of the Biot number (usually called the "mass transfer Biot number", or ) is also used in mass diffusion processes: where: • hm - film mass transfer coefficient • LC - characteristic length • DAB - mass diffusivity.

Biot number 18 References [1] Incropera; DeWitt, Bergman, Lavine (2007). Fundamentals of Heat and Mass Transfer (6th edition ed.). John Wiley & Sons. pp. 260–261. ISBN 978-0-471-45728-2. Black body A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. A black body in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation. The radiation is emitted according to Plancks law, meaning that it has a spectrum that is determined by the temperature alone (see figure at right), not by the bodys shape or composition. A black body in thermal equilibrium has two notable properties:[1] 1. It is an ideal emitter: it emits as much or As the temperature decreases, the peak of the black-body radiation curve moves to more energy at every frequency than any lower intensities and longer wavelengths. The black-body radiation graph is also compared with the classical model of Rayleigh and Jeans. other body at the same temperature. 2. It is a diffuse emitter: the energy is radiated isotropically, independent of direction. An approximate realization of a black surface is a hole in the wall of a large enclosure (see below). Any light entering the hole is reflected indefinitely or absorbed inside and is unlikely to re-emerge, making the hole a nearly perfect absorber. The radiation confined in such an enclosure may or may not be in thermal equilibrium, depending upon the nature of the walls and the other contents of the enclosure.[2][3] Real materials emit energy at a fraction—called the emissivity—of black-body energy levels. By definition, a black body in thermal equilibrium has an emissivity of ε = 1.0. A source with lower emissivity independent of frequency often is referred to as a gray body.[4][5] Construction of black bodies with emissivity as close to one as possible remains a topic of current interest.[6] A white body is one with a "rough surface [that] reflects all incident rays completely and uniformly in all directions."[7] In astronomy, the radiation from stars and planets is sometimes characterized in terms of an effective temperature, the temperature of a black body that would emit the same total flux of electromagnetic energy.

Black body 19 Definition The idea of a black body originally was introduced by Gustav Kirchhoff in 1860 as follows: ...the supposition that bodies can be imagined which, for infinitely small thicknesses, completely absorb all incident rays, and neither reflect nor [8] transmit any. I shall call such bodies perfectly black, or, more briefly, black bodies. A more modern definition drops the reference to "infinitely small thicknesses":[9] An ideal body is now defined, called a blackbody. A blackbody allows all incident radiation to pass into it (no reflected energy) and internally absorbs all the incident radiation (no energy transmitted through the body). This is true of radiation for all wavelengths and for all angles of [10] incidence. Hence the blackbody is a perfect absorber for all incident radiation. Idealizations This section describes some concepts developed in connection with black bodies. Cavity with a hole A widely used model of a black surface is a small hole in a cavity with walls that are opaque to radiation.[10] Radiation incident on the hole will pass into the cavity, and is very unlikely to be re-emitted if the cavity is large. The hole is not quite a perfect black surface — in particular, if the wavelength of the incident radiation is longer than the diameter of the hole, part will be reflected. Similarly, even in perfect thermal equilibrium, the radiation inside a finite-sized cavity will not have an ideal Planck spectrum for wavelengths comparable to or larger than the size of the cavity.[11] Suppose the cavity is held at a fixed temperature T and the radiation An approximate realization of a black body as a trapped inside the enclosure is at thermal equilibrium with the tiny hole in an insulated enclosure. enclosure. The hole in the enclosure will allow some radiation to escape. If the hole is small, radiation passing in and out of the hole has negligible effect upon the equilibrium of the radiation inside the cavity. This escaping radiation will approximate black-body radiation that exhibits a distribution in energy characteristic of the temperature T and does not depend upon the properties of the cavity or the hole, at least for wavelengths smaller than the size of the hole.[11] See the figure in the Introduction for the spectrum as a function of the frequency of the radiation, which is related to the energy of the radiation by the equation E=hf, with E = energy, h = Plancks constant, f = frequency. At any given time the radiation in the cavity may not be in thermal equilibrium, but the second law of thermodynamics states that if left undisturbed it will eventually reach equilibrium,[12] although the time it takes to do so may be very long.[13] Typically, equilibrium is reached by continual absorption and re-emission of radiation by material in the cavity or its walls.[2][3][14][15] Radiation entering the cavity will be "thermalized"; by this mechanism: the energy will be redistributed until the ensemble of photons achieves a Planck distribution. The time taken for thermalization is much faster with condensed matter present than with rarefied matter such as a dilute gas. At temperatures below billions of Kelvin, direct photon–photon interactions[16] are usually negligible compared to interactions with matter.[17] Photons are an example of an interacting boson gas,[18] and as described by the H-theorem,[19] under very general conditions any interacting boson gas will approach thermal equilibrium.

Black body 20 Transmission, absorption, and reflection A bodys behavior with regard to thermal radiation is characterized by its transmission τ, absorption α, and reflection ρ. The boundary of a body forms an interface with its surroundings, and this interface may be rough or smooth. A nonreflecting interface separating regions with different refractive indices must be rough, because the laws of reflection and refraction governed by the Fresnel equations for a smooth interface require a reflected ray when the refractive indices of the material and its surroundings differ.[20] A few idealized types of behavior are given particular names: An opaque body is one that transmits none of the radiation that reaches it, although some may be reflected.[21][22] That is, τ=0 and α+ρ=1 A transparent body is one that transmits all the radiation that reaches it. That is, τ=1 and α=ρ=0. A gray body is one where α, ρ and τ are uniform for all wavelengths. This term also is used to mean a body for which α is temperature and wavelength independent. A white body is one for which all incident radiation is reflected uniformly in all directions: τ=0, α=0, and ρ=1. For a black body, τ=0, α=1, and ρ=0. Planck offers a theoretical model for perfectly black bodies, which he noted do not exist in nature: besides their opaque interior, they have interfaces that are perfectly transmitting and non-reflective.[23] Kirchhoffs perfect black bodies Kirchhoff in 1860 introduced the theoretical concept of a perfect black body with a completely absorbing surface layer of infinitely small thickness, but Planck noted some severe restrictions upon this idea. Planck noted three requirements upon a black body: the body must (i) allow radiation to enter but not reflect; (ii) possess a minimum thickness adequate to absorb the incident radiation and prevent its re-emission; (iii) satisfy severe limitations upon scattering to prevent radiation from entering and bouncing back out. As a consequence, Kirchhoffs perfect black bodies that absorb all the radiation that falls on them, cannot be realized in an infinitely thin surface layer, and impose conditions upon scattering of the light within the black body that are difficult to satisfy.[24][25] Realizations A realization of a black body is a real world, physical embodiment. Here are a few. Cavity with a hole In 1898, Otto Lummer and Ferdinand Kurlbaum published an account of their cavity radiation source.[26] Their design has been used largely unchanged for radiation measurements to the present day. It was a hole in the wall of a platinum box, divided by diaphragms, with its interior blackened with iron oxide. It was an important ingredient for the progressively improved measurements that led to the discovery of Plancks law.[27][28] A version described in 1901 had its interior blackened with a mixture of chromium, nickel, and cobalt oxides.[29] Near-black materials There is interest in blackbody-like materials for camouflage and radar-absorbent materials for radar invisibility.[30][31] They also have application as solar energy collectors, and infrared thermal detectors. As a perfect emitter of radiation, a hot material with black body behavior would create an efficient infrared heater, particularly in space or in a vacuum where convective heating is unavailable.[32] They also are useful in telescopes and cameras as anti-reflection surfaces to reduce stray light, and to gather information about objects in high-contrast areas (for example, observation of planets in orbit around their stars), where blackbody-like materials absorb light that comes

Black body 21 from the wrong sources. It has long been known that a lamp-black coating will make a body nearly black. An improvement on lamp-black is found in manufactured carbon nanotubes. Nano-porous materials can achieve refractive indices nearly that of vacuum, in one case obtaining average reflectance of 0.045%.[6][33] In 2009, a team of Japanese scientists created a material close to an ideal black body, based on vertically aligned single-walled carbon nanotubes. This absorbs between 98% and 99% of the incoming light in the spectral range from the ultra-violet to the far-infrared regions.[32] Another example of a nearly perfect black material is super black, made by chemically etching a nickel–phosphorus alloy.[34] Stars and planets A star or planet often is modeled as a black body, and electromagnetic radiation emitted from these bodies as black-body radiation. The figure shows a highly schematic cross-section to illustrate the idea. The photosphere of the star where the emitted light is generated is idealized as a layer within which the photons of light interact with the material in the photosphere and achieve a common temperature T that is maintained over a long period of time. Some photons escape and are emitted into space, but the energy they carry away is replaced by energy from within the star, so that the temperature of the photosphere is nearly steady. Changes in the core lead to changes in the supply of energy to the photosphere, but such changes are slow on the time scale of interest here. Assuming these circumstances can be realized, the An idealized view of the cross-section of a star. The photosphere contains photons of light nearly outer layer of the star is somewhat analogous to the example of an in thermal equilibrium, and some escape into enclosure with a small hole in it, with the hole replaced by the limited space as near-black-body radiation. transmission into space at the outside of the photosphere. With all these assumptions in place, the star emits black-body radiation at the temperature of the photosphere.[35] Using this model the effective temperature of stars is estimated, defined as the temperature of a black body that yields the same surface flux of energy as the star. If a star were a black body, the same effective temperature would result from any region of the spectrum. For example, comparisons in the B (blue) or V (visible) range lead to the so-called B-V color index, which increases the redder the star,[37] with the Sun having an index of +0.648 ± 0.006.[38] Combining the U (ultraviolet) and the B indices leads to the U-B index, which becomes more negative the hotter the star and the more the UV radiation. Assuming the Sun is a type G2 V star, its U-B index is +0.12.[39] The Effective temperature of a black body compared two indices for two types of stars are compared in the figure with the with the B-V and U-B color index of main effective surface temperature of the stars assuming they are black sequence and super giant stars in what is called a [36] bodies. It can be seen that there is only a rough correlation. For color-color diagram. example, for a given B-V index from the blue-visible region of the spectrum., the curves for both types of star lie below the corresponding black-body U-B index that includes the ultraviolet spectrum, showing that both types of star emit less ultraviolet light than a black body with the same B-V index. It is perhaps surprising that they fit a black body curve as well as they do, considering that stars have greatly different temperatures at different depths.[40] For example, the Sun has an effective temperature of 5780 K,[41] which

Black body 22 can be compared to the temperature of the photosphere of the Sun (the region generating the light), which ranges from about 5000 K at its outer boundary with the chromosphere to about 9500 K at its inner boundary with the convection zone approximately 500 km deep.[42] Black holes A black hole is a region of spacetime from which nothing escapes. Around a black hole there is a mathematically defined surface called an event horizon that marks the point of no return. It is called "black" because it absorbs all the light that hits the horizon, reflecting nothing, making it almost an ideal black body[43] (radiation with a wavelength equal to or larger than the radius of the hole may not be absorbed, so black holes are not perfect black bodies).[44] Physicists believe that to an outside observer, black holes have a non-zero temperature and emit radiation with a nearly perfect black-body spectrum, ultimately evaporating.[45] The mechanism for this emission is related to vacuum fluctuations in which a virtual pair of particles is separated by the gravity of the hole, one member being sucked into the hole, and the other being emitted.[46] The energy distribution of emission is described by Plancks law with a temperature T: where c is the speed of light, ℏ is the reduced Planck constant, kB is Boltzmann constant, G is the gravitational constant and M is the mass of the black hole.[47] These predictions have not yet been tested either observationally or experimentally.[48] Cosmic microwave background radiation The big bang theory is based upon the cosmological principle, which states that on large scales the Universe is homogeneous and isotropic. According to theory, the Universe approximately a second after its formation was a near-ideal black body in thermal equilibrium at a temperature above 1010 K. The temperature decreased as the Universe expanded and the matter and radiation in it cooled. The cosmic microwave background radiation observed today is "the most perfect black body ever measured in nature".[49] It has a nearly ideal Planck spectrum at a temperature of about 2.7K. It departs from the perfect isotropy of true black-body radiation by an observed anisotropy that varies with angle on the sky only to about one part in 100,000. Radiative cooling The integration of Plancks law over all frequencies provides the total energy per unit of time per unit of surface area radiated by a black body maintained at a temperature T, and is known as the Stefan–Boltzmann law: where σ is the Stefan–Boltzmann constant, σ ≈ 5.67 × 10−8 W/(m2K4).[50] To remain in thermal equilibrium at constant temperature T, the black body must absorb or internally generate this amount of power P over the given area A. The cooling of a body due to thermal radiation is often approximated using the Stefan–Boltzmann law supplemented with a "gray body" emissivity ε ≤ 1 (P/A = εσT4). The rate of decrease of the temperature of the emitting body can be estimated from the power radiated and the bodys heat capacity.[51] This approach is a simplification that ignores details of the mechanisms behind heat redistribution (which may include changing composition, phase transitions or restructuring of the body) that occur within the body while it cools, and assumes that at each moment in time the body is characterized by a single temperature. It also ignores other possible complications, such as changes in the emissivity with temperature,[52][53] and the role of other accompanying forms of energy emission, for example, emission of particles like neutrinos.[54]

Black body 23 If a hot emitting body is assumed to follow the Stefan–Boltzmann law and its power emission P and temperature T is known, this law can be used to estimate the dimensions of the emitting object, because the total emitted power is proportional to the area of the emitting surface. In this way it was found that X-ray bursts observed by astronomers originated in neutron stars with a radius of about 10 km, rather than black holes as originally conjectured.[55] It should be noted that an accurate estimate of size requires some knowledge of the emissivity, particularly its spectral and angular dependence.[56] References Citations [1] Mahmoud Massoud (2005). "§2.1 Blackbody radiation" (http:/ / books. google. com/ books?id=9KIp_fmC9A0C& pg=PA568). Engineering thermofluids: thermodynamics, fluid mechanics, and heat transfer. Springer. p. 568. ISBN 3-540-22292-8. . [2] The approach to thermal equilibrium of the radiation in the cavity can be catalyzed by adding a small piece of matter capable of radiating and absorbing at all frequencies. See Peter Theodore Landsberg. Thermodynamics and statistical mechanics (http:/ / books. google. com/ books?id=0gnWL7tmxm0C& pg=PA209) (Reprint of Oxford University Press 1978 ed.). Courier Dover Publications. p. 209. ISBN 0-486-66493-7. . [3] Planck 1914, p. 44, §52 [4] The emissivity of a surface in principle depends upon frequency, angle of view, and temperature. However, by definition, the radiation from a gray body is simply proportional to that of a black body at the same temperature, so its emissivity does not depend upon frequency (or, equivalently, wavelength). See Massoud Kaviany (2002). "Figure 4.3(b): Behaviors of a gray (no wavelength dependence), diffuse (no directional dependence) and opaque (no transmission) surface" (http:/ / books. google. com/ books?id=dKI4k-9jK88C& pg=PA381). Principles of heat transfer. Wiley-IEEE. p. 381. ISBN 0-471-43463-9. . and Ronald G. Driggers (2003). Encyclopedia of optical engineering, Volume 3 (http:/ / books. google. com/ books?id=9ExHkgDv2z0C& pg=PA2303). CRC Press. p. 2303. ISBN 0-8247-4252-4. . [5] Some authors describe sources of infrared radiation with emissivity greater than approximately 0.99 as a black body. See "What is a Blackbody and Infrared Radiation?" (http:/ / www. electro-optical. com/ eoi_page. asp?h=What is a Blackbody and Infrared Radiation?). Education/Reference tab. Electro Optical Industries, Inc.. 2008. . [6] Ai Lin Chun (25 Jan 2008). "Carbon nanotubes: Blacker than black" (http:/ / www. nature. com/ nnano/ reshigh/ 2008/ 0108/ full/ nnano. 2008. 29. html). Nature Nanotechnology. doi:10.1038/nnano.2008.29. . [7] Planck 1914, pp. 9–10 [8] Translated by F. Guthrie from Annalen der Physik: 109, 275-301 (1860): G. Kirchhoff (July, 1860). "On the relation between the radiating and absorbing powers of different bodies for light and heat" (http:/ / books. google. com/ books?id=RVYEAAAAYAAJ& pg=PA1& lpg=PA1). The London, Edinburgh and Dublin philosophical magazine and journal of science (Taylor & Francis) 20 (130). . [9] The notion of an infinitely thin layer was dropped by Planck. See Planck 1914, p. 10, footnote 2, . [10] Siegel, Robert; Howell, John R. (2002). Thermal Radiation Heat Transfer; Volume 1 (http:/ / books. google. com/ books?id=O389yQ0-fecC& pg=PA7) (4th ed.). Taylor & Francis. p. 7. ISBN 1-56032-839-8. . [11] Corrections to the spectrum do arise related to boundary conditions at the walls, curvature, and topology, particularly for wavelengths comparable to the cavity dimensions; see Roger Dale Van Zee, J. Patrick Looney (2002). Cavity-enhanced spectroscopies (http:/ / books. google. com/ books?id=OuLILePVdJQC& pg=PA202). Academic Press. p. 202. ISBN 0-12-475987-4. . [12] Clement John Adkins (1983). "§4.1 The function of the second law" (http:/ / books. google. com/ books?id=FW4Oz48TWwQC& pg=PA50). Equilibrium thermodynamics (3rd ed.). Cambridge University Press. p. 50. ISBN 0-521-27456-7. . [13] In simple cases the approach to equilibrium is governed by a relaxation time. In others, the system may hang up in a metastable state, as stated by Adkins (1983) on page 10. For another example, see Michel Le Bellac, Fabrice Mortessagne, Ghassan George Batrouni (2004). Equilibrium and non-equilibrium statistical thermodynamics (http:/ / books. google. com/ books?id=LZcdi4uzaWIC& pg=PA8). Cambridge University Press. p. 8. ISBN 0521821436. . [14] Loudon 2000, Chapter 1 [15] Mandel & Wolf 1995, Chapter 13 [16] Robert Karplus* and Maurice Neuman ,"The Scattering of Light by Light", Phys. Rev. 83, 776–784 (1951) [17] Ludwig Bergmann, Clemens Schaefer, Heinz Niedrig (1999). Optics of waves and particles (http:/ / books. google. com/ books?id=sJxv-yCSNEAC& pg=PA595). Walter de Gruyter. p. 595. ISBN 3-11-014318-6. . "Because the interaction of the photons with each other is negligible, a small amount of matter is necessary to establish thermodynamic equilibrium of heat radiation." [18] The fundamental bosons are the photon, the vector bosons of the weak interaction, the gluon, and the graviton. See Allan Griffin, D. W. Snoke, S. Stringari (1996). Bose-Einstein condensation (http:/ / books. google. com/ books?id=suqJdr2pPIsC& pg=PA4). Cambridge University Press. p. 4. ISBN 0-521-58990-8. . [19] Richard Chace Tolman (2010). "§103: Change of H with time as a result of collisions" (http:/ / books. google. com/ books?id=4TqQZo962s0C& printsec=frontcover). The principles of statistical mechanics (Reprint of 1938 Oxford University Press ed.). Dover Publications. pp. 455 ff. ISBN 0-486-63896-0. . "...we can define a suitable quantity H to characterize the condition of a gas which [will

Black body 24 exhibit] a tendency to decrease with time as a result of collisions, unless the distribution of the molecules [is already that of] equilibrium. (p. 458)" [20] Paul A. Tipler (1999). "Relative intensity of reflected and transmitted light" (http:/ / books. google. com/ books?id=Gp2tzUhbqjMC& pg=PA1044). Physics for Scientists and Engineers, Parts 1-35; Part 39 (4th ed.). Macmillan. p. 1044. ISBN 0-7167-3821-X. . [21] Massoud Kaviany (2002). "Figure 4.3(b) Radiation properties of an opaque surface" (http:/ / books. google. com/ books?id=dKI4k-9jK88C& pg=PA381#v=onepage& q& f=false). Principles of heat transfer. Wiley-IEEE. p. 381. ISBN 0-471-43463-9. . [22] BA Venkanna (2010). "§10.3.4 Absorptivity, reflectivity, and transmissivity" (http:/ / books. google. com/ books?id=IIIVHRirRgEC& pg=PA386). Fundamentals of heat and mass transfer. PHI Learning Pvt. Ltd.. pp. 385–386. ISBN 81-203-4031-0. . [23] Planck 1914, p. 10 [24] Planck 1914, pp. 9–10, §10 [25] Kirchhoff 1860c [26] Lummer & Kurlbaum 1898 [27] An extensive historical discussion is found in Jagdish Mehra, Helmut Rechenberg (2000). The historical development of quantum theory (http:/ / books. google. com/ books?id=W5kyppVPyesC& pg=PA39). Springer. pp. 39 ff. ISBN 0-387-95174-1. . [28] Kangro 1976, p. 159 [29] Lummer & Kurlbaum 1901 [30] CF Lewis (June 1988). "Materials keep a low profile" (http:/ / www. cewriters. com/ CEWritersSample_MaterialsFeature. pdf). Mech. Eng.: 37–41. . [31] Bradley Quinn (2010). Textile Futures (http:/ / books. google. com/ books?id=e_wnynE954MC& pg=PA68). Berg. p. 68. ISBN 1-84520-807-2. . [32] K. Mizuno et al. (2009). "A black body absorber from vertically aligned single-walled carbon nanotubes" (free download). Proceedings of the National Academy of Sciences 106 (15): 6044–6077. Bibcode 2009PNAS..106.6044M. doi:10.1073/pnas.0900155106. PMC 2669394. PMID 19339498. [33] Zu-Po Yang et al. (2008). "Experimental observation of an extremely dark material made by a low-density nanotube array" (http:/ / pubs. acs. org/ doi/ abs/ 10. 1021/ nl072369t). Nano Letters (American Chemical Society) 8: 446–451. Bibcode 2008NanoL...8..446Y. doi:10.1021/nl072369t. PMID 18181658. . [34] See description of work by Richard Brown and his colleagues at the UKs National Physical Laboratory: Mick Hamer (correspondent) (6 February 2003). "Mini craters key to blackest ever black" (http:/ / www. newscientist. com/ article/ dn3356-mini-craters-key-to-blackest-ever-black. html). New Scientist Magazine online. . [35] Simon F. Green, Mark H. Jones, S. Jocelyn Burnell (2004). An introduction to the sun and stars (http:/ / books. google. com/ books?id=lb5owLGIQGsC& pg=PA53). Cambridge University Press. pp. 21–22, 53. ISBN 0-521-54622-2. . "A source in which photons are much more likely to interact with the material within the source than to escape is a condition for the formation of a black-body spectrum" [36] Figure modeled after E. Böhm-Vitense (1989). "Figure 4.9" (http:/ / books. google. com/ books?id=JWrtilsCycQC& pg=PA26). Introduction to Stellar Astrophysics: Basic stellar observations and data. Cambridge University Press. p. 26. ISBN 0-521-34869-2. . [37] David H. Kelley, Eugene F. Milone, Anthony F. (FRW) Aveni (2011). Exploring Ancient Skies: A Survey of Ancient and Cultural Astronomy (http:/ / books. google. com/ books?id=ILBuYcGASxcC& pg=PA52) (2nd ed.). Springer. p. 52. ISBN 1-4419-7623-X. . [38] David F Gray (February 1995). "Comparing the sun with other stars along the temperature coordinate" (http:/ / adsabs. harvard. edu/ full/ 1995PASP. . 107. . 120G). Publications of the astronomical society of the Pacific 107: 120–123. . Retrieved 2012-01-26. [39] M Golay (1974). "Table IX: U-B Indices" (http:/ / books. google. com/ books?id=OmNuvvt31BkC& pg=PA82). Introduction to astronomical photometry. Springer. p. 82. ISBN 90-277-0428-7. . [40] Lawrence Hugh Aller (1991). Atoms, stars, and nebulae (http:/ / books. google. com/ books?id=HupvoeJDCGoC& pg=PA61) (3rd ed.). Cambridge University Press. p. 61. ISBN 0-521-31040-7. . [41] Kenneth R. Lang (2006). Astrophysical formulae, Volume 1 (http:/ / books. google. com/ books?id=HlGIXqzVEAgC& pg=PA23) (3rd ed.). Birkhäuser. p. 23. ISBN 3-540-29692-1. . [42] B. Bertotti, Paolo Farinella, David Vokrouhlický (2003). "Figure 9.2: The temperature profile in the solar atmosphere" (http:/ / books. google. com/ books?id=i-YvHNPEqAIC& pg=PA248). New Views of the Solar System. Springer. p. 248. ISBN 1-4020-1428-7. . [43] Schutz, Bernard (2004). Gravity From the Group Up: An Introductory Guide to Gravity and General Relativity (http:/ / www. amazon. com/ Gravity-Ground-Up-Introductory-Relativity/ dp/ 0521455065) (1st ed.). Cambridge University Press. p. 304. ISBN 0-521-45506-5. . [44] PCW Davies (1978). "Thermodynamics of black holes" (http:/ / cosmos. asu. edu/ publications/ papers/ ThermodynamicTheoryofBlackHoles 34. pdf). Rep Prog Phys 41 (8): 1313 ff. Bibcode 1978RPPh...41.1313D. doi:10.1088/0034-4885/41/8/004. . [45] Robert M Wald (2005). "The thermodynamics of black holes" (http:/ / books. google. com/ books?id=ZDtxnw-b1xEC& pg=PA1). In Andrés Gomberoff, Donald Marolf, eds. Lectures on quantum gravity. Springer. pp. 1 ff. ISBN 0-387-23995-2. . [46] Bernard J Carr and Steven B Giddings (2008). "Chapter 6: Quantum black holes" (http:/ / books. google. com/ books?id=N-K-q0_TexAC& pg=PA30). Beyond Extreme Physics: Cutting-edge science. Rosen Publishing Group, Scientific American (COR). p. 30. ISBN 1-4042-1402-X. . [47] Valeri P. Frolov, Andrei Zelnikov (2011). "Equation 9.7.1" (http:/ / books. google. com/ books?id=r_l5AK9DdXsC& pg=PA321). Introduction to Black Hole Physics. Oxford University Press. p. 321. ISBN 0-19-969229-7. .

Black body 25 [48] Robert M Wald. "The thermodynamics of black holes" (http:/ / books. google. com/ books?id=ZDtxnw-b1xEC& pg=PA28). cited work. p. 28. ISBN 0-387-23995-2. . "... no results on black hole thermodynamics have been subject to any experimental or observational tests, ..." [49] White, M. (1999). "Anisotropies in the CMB" (http:/ / www. dpf99. library. ucla. edu/ session9/ white0910. pdf). Proceedings of the Los Angeles Meeting, DPF 99. UCLA.. . See also arXive.org (http:/ / arxiv. org/ abs/ astro-ph/ 9903232v1). [50] "Stefan–Boltzmann constant" (http:/ / physics. nist. gov/ cgi-bin/ cuu/ Value?sigma). NIST reference on constants, units, and uncertainty. . Retrieved 2012-02-02. [51] A simple example is provided by Srivastava M. K. (2011). "Cooling by radiation" (http:/ / books. google. com/ books?id=FTpK4a4FTLQC& pg=PA610#v=onepage& q& f=false). The Person Guide to Objective Physics for the IIT-JEE. Pearson Education India. p. 610. ISBN 81-317-5513-4. . [52] M Vollmer, K-P Mõllmann (2011). "Figure 1.38: Some examples for temperature dependence of emissivity for different materials" (http:/ / books. google. com/ books?id=nAIZb8-05IwC& pg=PA45). Infrared Thermal Imaging: Fundamentals, Research and Applications. John Wiley & Sons. p. 45. ISBN 3-527-63087-2. . [53] Robert Osiander, M. Ann Garrison Darrin, John Champion (2006). MEMS and Microstructures in aerospace applications (http:/ / books. google. com/ books?id=DaqsgZ2HtIwC& pg=PA187). CRC Press. p. 187. ISBN 0-8247-2637-5. . [54] Neutrino emission is a mechanism of cooling in neutron stars, for example; see Mikhail A. Shifman (2001). "Cooling by neutrino emission" (http:/ / books. google. com/ books?id=2yhBnW_CtLIC& pg=PA2135). In B. L. Ioffe, Mikhail A. Shifman, eds. World Scientific. p. 2135. ISBN 981-02-4969-1. . [55] Walter Lewin with Warren Goldstein (2011). "X-ray bursters!" (http:/ / books. google. com/ books?id=ukItNVRRSp4C& pg=PA250#v=onepage& q& f=false). For the love of physics. Simon and Schuster. pp. 251 ff. ISBN 1-4391-0827-7. . [56] TE Strohmayer (2006). "Neutron star structure and fundamental physics" (http:/ / books. google. com/ books?id=SOzEHo1RteMC& pg=PA41). In John W. Mason, ed. Astrophysics update, Volume 2. Birkhäuser. p. 41. ISBN 3-540-30312-X. . Bibliography • Chandrasekhar, S. (1950). Radiative Transfer. Oxford University Press. • Goody, R. M.; Yung, Y. L. (1989). Atmospheric Radiation: Theoretical Basis (2nd ed.). Oxford University Press. ISBN 978-0-19-510291-8. • Hermann, A. (1971). The Genesis of Quantum Theory. Nash, C.W. (transl.). MIT Press. ISBN 0-262-08047-8. a translation of Frühgeschichte der Quantentheorie (1899–1913), Physik Verlag, Mosbach/Baden. • Kangro, H. (1976). Early History of Plancks Radiation Law. Taylor and Francis. ISBN 0-85066-063-7. • Kirchhoff, G.; [27 October 1859] (1860a). "Über die Fraunhoferschen Linien". Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 662–665. • Kirchhoff, G.; [11 December 1859] (1860b). "Über den Zusammenhang zwischen Emission und Absorption von Licht und Wärme". Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 783–787. • Kirchhoff, G. (1860c). "Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht" (http://books.google.com/books?id=RVYEAAAAYAAJ&pg=PA1& lpg=PA1). Annalen der Physik und Chemie (Leipzig) 109: 275–301. Translated by Guthrie, F. as Kirchhoff, G. (1860). "On the relation between the radiating and absorbing powers of different bodies for light and heat" (http:/ /books.google.com/books?id=RVYEAAAAYAAJ&pg=PA1&lpg=PA1). Philosophical Magazine Series 4, volume 20: 1–21. • Kirchhoff, G. (1882) [1862], "Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht", Gessamelte Abhandlungen, Leipzig: Johann Ambrosius Barth, pp. 571–598 • Kondepudi, D.; Prigogine, I. (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures. John Wiley & Sons. ISBN 0–471–97393–9. • Kragh, H. (1999). Quantum Generations: a History of Physics in the Twentieth Century. Princeton University Press. ISBN 0-691-01206-7. • Kuhn, T. S. (1978). Black–Body Theory and the Quantum Discontinuity. Oxford University Press. ISBN 0-19-502383-8. • Loudon, R. (2000) [1973]. The Quantum Theory of Light (third ed.). Cambridge University Press. ISBN 0–19–850177–3.

Black body 26 • Lummer, O.; Kurlbaum, F. (1898). "Der electrisch geglühte "absolut schwarze" Körper und seine Temperaturmessung". Verhandlungen der Deutschen Physikalischen Gesellschaft 17: 106–111. • Lummer, O.; Kurlbaum, F. (1901). "Der elektrisch geglühte „schwarze” Körper". Annalen der Physik 310 (8): 829–836. Bibcode 1901AnP...310..829L. doi:10.1002/andp.19013100809. • Mandel, L.; Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge University Press. ISBN 0–521–41711–2. • Mehra, J.; Rechenberg, H. (1982). The Historical Development of Quantum Theory. volume 1, part 1. Springer-Verlag. ISBN 0-387-90642-8. • Mihalas, D.; Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics. Oxford University Press. ISBN 0-19-503437-6. • Milne, E.A. (1930). "Thermodynamics of the Stars". Handbuch der Astrophysik 3, part 1: 63–255. • Planck, M. (1914). The Theory of Heat Radiation (http://ia600201.us.archive.org/16/items/ theheatradiation00planrich/theheatradiation00planrich.pdf). Masius, M. (transl.) (2nd ed.). P. Blakistons Son & Co. OL{{{1}}}. • Rybicki, G. B.; Lightman, A. P. (1979). Radiative Processes in Astrophysics (http://books.google.com/ books?id=LtdEjNABMlsC&printsec=frontcover). John Wiley & Sons. ISBN 0-471-82759-2. • Schirrmacher, A. (2001). Experimenting theory: the proofs of Kirchhoffs radiation law before and after Planck. Münchner Zentrum für Wissenschafts und Technikgeschichte. • Stewart, B. (1858). "An account of some experiments on radiant heat". Transactions of the Royal Society of Edinburgh 22: 1–20. External links • Lori J. Keesey. "Blacker than black" (http://www.nasa.gov/topics/technology/features/new-nano.html). NASA. Retrieved 2012-02-01. A description and video of work done making "blacker-than-pitch" materials using carbon nanotubes at the NASA Goddard Space Flight Center.

Black-body radiation 27 Black-body radiation Black-body radiation is the type of electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body (an opaque and non-reflective body) held at constant, uniform temperature. The radiation has a specific spectrum and intensity that depends only on the temperature of the body.[1][2][3][4] A perfectly insulated enclosure that is in thermal equilibrium internally contains black body radiation and will emit it through a hole made in its wall, provided the hole is small enough to have negligible As the temperature decreases, the peak of the black-body radiation curve moves to effect upon the equilibrium. lower intensities and longer wavelengths. The black-body radiation graph is also A black body at room temperature appears compared with the classical model of Rayleigh and Jeans. black, as most of the energy it radiates is infra-red and cannot be perceived by the human eye. At higher temperatures, black bodies glow with increasing intensity and colors that range from dull red to blindingly brilliant blue-white as the temperature increases. Although planets and stars are neither in thermal equilibrium with their surroundings nor perfect black bodies, black body radiation is used as a first approximation for the energy they emit.[5] Black holes are near-perfect black bodies, and it is believed that they emit black-body radiation (called Hawking radiation), with a temperature that depends on the mass of the hole.[6] The term black body was introduced by Gustav Kirchhoff in 1860. When used as a compound adjective, the term is typically written as hyphenated, for example, black-body radiation, but sometimes also as one word, as in blackbody radiation. The color (chromaticity) of black-body radiation depends on the temperature of the Black-body radiation also is called black body; the locus of such colors, shown here in CIE 1931 x,y space, is known as complete radiation or temperature the Planckian locus. radiation or thermal radiation.

Black-body radiation 28 Spectrum Black body radiation has a characteristic, continuous frequency spectrum that depends only on the bodys temperature,[7] called the Planck spectrum or Plancks law. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at room temperature most of the emission is in the infrared region of the electromagnetic spectrum.[8][9][10] As the temperature increases past about 500 degrees Celsius, black bodies start to emit significant amounts of visible light. Viewed in the dark, the first faint glow appears as a "ghostly" grey. With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises.[11][12] When the body appears white, it is emitting a substantial fraction of its energy as ultraviolet radiation. The sun, with an effective temperature of approximately 5800 °C,[13] is an approximately black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well. Black-body radiation provides insight into the thermodynamic equilibrium state of cavity radiation. If each Fourier mode of the equilibrium radiation in an otherwise empty cavity with perfectly reflective walls is considered as a degree of freedom capable of exchanging energy, then, according to the equipartition theorem of classical physics, there would be an equal amount of energy in each mode. Since there are an infinite number of modes this implies infinite heat capacity (infinite energy at any non-zero temperature), as well as an unphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the ultraviolet catastrophe. Instead, in quantum theory the occupation numbers of the modes are quantized, cutting off the spectrum at high frequency in agreement with experiment and resolving the catastrophe. The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of quantum mechanics. Explanation All matter emits electromagnetic radiation when it has a temperature above absolute zero. The radiation represents a conversion of a bodys thermal energy into electromagnetic energy, and is therefore called thermal radiation. It is a spontaneous process of radiative distribution of entropy. Conversely all matter absorbs electromagnetic radiation to some degree. An object that absorbs all radiation falling on it, at all wavelengths, is called a black body. When a black body is at a uniform temperature, its emission has a characteristic frequency distribution that depends on the temperature. Its emission is called black-body radiation. The concept of the black body is an idealization, as perfect black bodies do not exist in nature.[14] Graphite and lamp black, with emissivities greater than 0.95, however, are good approximations to a black material. Experimentally, black-body radiation may be established best as the ultimately stable steady state equilibrium radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective.[14] A closed box of graphite walls at a constant temperature with a small hole on one side produces a good approximation to ideal black-body radiation emanating from the opening.[15][16] Black-body radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity.[14] In equilibrium, for each frequency the total intensity of radiation that is emitted and reflected from a body (that is, the net amount of radiation leaving its surface, called the spectral radiance) is determined solely by the equilibrium temperature, and does not depend upon the shape, material or structure of the body.[17] For a black body (a perfect absorber) there is no reflected radiation, and so the spectral radiance is due entirely to emission. In addition, a black body is a diffuse emitter (its emission is independent of direction). Consequently, black-body radiation may be viewed as the radiation from a black body at thermal equilibrium. Black-body radiation becomes a visible glow of light if the temperature of the object is high enough. The Draper point is the temperature at which all solids glow a dim red, about 798 K.[18][19] At 1000 K, the opening in the oven

Black-body radiation 29 looks red; at 6000 K, it looks white. No matter how the oven is constructed, or of what material, as long as it is built such that almost all light entering is absorbed, it will be a good approximation to a black-body, so the spectrum, and therefore color, of the light that comes out will be a function of the cavity temperature alone. A graph of the amount of energy inside the oven per unit volume and per unit frequency interval plotted versus frequency, is called the black-body curve. Different curves are obtained by varying the temperature. Two bodies that are at the same temperature stay in thermal equilibrium, so a body at temperature T surrounded by a cloud of light at temperature T on average will emit as much light into the cloud as it absorbs, following Prevosts exchange principle, which refers to radiative equilibrium. The principle of detailed balance says that in thermodynamic equilibrium every elementary process works equally in its forward and backward sense.[20][21] Prevost also showed that the emission from a body is logically determined solely by The temperature of a Pāhoehoe lava flow can be estimated by its own internal state. The causal effect of observing its color. The result agrees well with measured thermodynamic absorption on thermodynamic temperatures of lava flows at about 1000 to 1200 °C (1832 to 2192 °F). (spontaneous) emission is not direct, but is only indirect as it affects the internal state of the body. This means that at thermodynamic equilibrium the amount of every wavelength in every direction of thermal radiation emitted by a body at temperature T, black or not, is equal to the corresponding amount that the body absorbs because it is surrounded by light at temperature T.[22] When the body is black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength λ per unit time is strictly proportional to the black-body curve. This means that the black-body curve is the amount of light energy emitted by a black body, which justifies the name. This is the condition for the applicability of Kirchhoffs law of thermal radiation: the black-body curve is characteristic of thermal light, which depends only on the temperature of the walls of the cavity, provided that the walls of the cavity are completely opaque and are not very reflective, and that the cavity is in thermodynamic equilibrium.[23] When the black body is small, so that its size is comparable to the wavelength of light, the absorption is modified, because a small object is not an efficient absorber of light of long wavelength, but the principle of strict equality of emission and absorption is always upheld in a condition of thermodynamic equilibrium. In the laboratory, black-body radiation is approximated by the radiation from a small hole in a large cavity, a hohlraum, in an entirely opaque body that is only partly reflective, that is maintained at a constant temperature. (This technique leads to the alternative term cavity radiation.) Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. Absorption occurs regardless of the wavelength of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum of the holes radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will depend only on the opacity and partial reflectivity of the walls, but not on the particular material of which they are built nor on the material in the cavity (compare with emission spectrum). Calculating the black-body curve was a major challenge in theoretical physics during the late nineteenth century. The problem was solved in 1901 by Max Planck in the formalism now known as Plancks law of black-body radiation.[24] By making changes to Wiens radiation law (not to be confused with Wiens displacement law) consistent with thermodynamics and electromagnetism, he found a mathematical expression fitting the experimental data satisfactorily. Planck had to assume that the energy of the oscillators in the cavity was quantized, i.e., it existed in

Black-body radiation 30 integer multiples of some quantity. Einstein built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the photoelectric effect. These theoretical advances eventually resulted in the superseding of classical electromagnetism by quantum electrodynamics. These quanta were called photons and the black-body cavity was thought of as containing a gas of photons. In addition, it led to the development of quantum probability distributions, called Fermi–Dirac statistics and Bose–Einstein statistics, each applicable to a different class of particles, fermions and bosons. The wavelength at which the radiation is strongest is given by Wiens displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible—indeed, the radiation of visible light increases monotonically with temperature.[25] The radiance or observed intensity is not a function of direction. Therefore a black body is a perfect Lambertian radiator. Real objects never behave as full-ideal black bodies, and instead the emitted radiation at a given frequency is a fraction of what the ideal emission would be. The emissivity of a material specifies how well a real body radiates energy as compared with a black body. This emissivity depends on factors such as temperature, emission angle, and wavelength. However, it is typical in engineering to assume that a surfaces spectral emissivity and absorptivity do not depend on wavelength, so that the emissivity is a constant. This is known as the gray body assumption. With non-black surfaces, the deviations from ideal black-body behavior are determined by both the surface structure, such as roughness or granularity, and the chemical composition. On a "per wavelength" basis, real objects in states of local thermodynamic equilibrium still follow Kirchhoffs Law: emissivity equals absorptivity, so that an object that does not absorb all incident light will also emit less The distribution of the cosmic microwave background radiation across the universe radiation than an ideal black body; the as measured by the Wilkinson Microwave Anisotropy Probe. It is the most incomplete absorption can be due to some of precisely known thermal emission spectrum and corresponds to a temperature of the incident light being transmitted through 2.725 K and an emission peak frequency of 160.2 GHz. the body or to some of it being reflected at the surface of the body. In astronomy, objects such as stars are frequently regarded as black bodies, though this is often a poor approximation. An almost perfect black-body spectrum is exhibited by the cosmic microwave background radiation. Hawking radiation is the hypothetical black-body radiation emitted by black holes, at a temperature that depends on the mass, charge, and spin of the hole. If this prediction is correct, black holes will very gradually shrink and evaporate over time as they lose mass by the emission of photons and other particles. A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (short wavelengths). For example, a black body at room temperature (300 K) with one square meter of surface area will emit a photon in the visible range (390–750 nm) at an average rate of one photon every 41 seconds, meaning that for most practical purposes, such a black body does not emit in the visible range.[26]

Black-body radiation 31 Equations Plancks law of black-body radiation Plancks law states that[27] where I(ν,T) is the energy per unit time (or the power) radiated per unit area of emitting surface in the normal direction per unit solid angle per unit frequency by a black body at temperature T; h is the Planck constant; c is the speed of light in a vacuum; k is the Boltzmann constant; ν is the frequency of the electromagnetic radiation; and T is the temperature of the body in kelvins. Wiens displacement law Wiens displacement law shows how the spectrum of black body radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. Spectral intensity can be expressed as a function of wavelength or of frequency. A consequence of Wiens displacement law is that the wavelength at which the intensity, as a function of wavelength, of the radiation produced by a black body is at a maximum, , is a function only of the temperature where the constant, b, known as Wiens displacement constant, is equal to 28977721(26) × 10−3 K m.[28] Plancks Law was also stated above as a function of frequency. The intensity maximum for this is given by .[29] Stefan–Boltzmann law The Stefan–Boltzmann law states that the power emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature: where j*is the total power radiated per unit area, T is the absolute temperature and σ = 567 × 10−8 W m−2 K−4 is the Stefan–Boltzmann constant. Human body emission

Black-body radiation 32 Much of a persons energy is radiated away in the form of infrared light. Some materials are transparent in the infrared, while opaque to visible light, as is the plastic bag in this infrared image (bottom). Other materials are transparent to visible light, while opaque or reflective in the infrared, noticeable by darkness of the mans glasses. As all matter, the human body radiates some of a persons energy away as infrared light. The net power radiated is the difference between the power emitted and the power absorbed: Applying the Stefan–Boltzmann law, The total surface area of an adult is about 2 m2, and the mid- and far-infrared emissivity of skin and most clothing is near unity, as it is for most nonmetallic surfaces.[30][31] Skin temperature is about 33 °C,[32] but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C.[33] Hence, the net radiative heat loss is about The total energy radiated in one day is about 9 MJ (megajoules), or 2000 kcal (food calories). Basal metabolic rate for a 40-year-old male is about 35 kcal/(m2·h),[34] which is equivalent to 1700 kcal per day assuming the same 2 m2 area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.[35] There are other important thermal loss mechanisms, including convection and evaporation. Conduction is negligible – the Nusselt number is much greater than unity. Evaporation via perspiration is only required if radiation and convection are insufficient to maintain a steady state temperature (but evaporation from the lungs occurs regardless). Free convection rates are comparable, albeit somewhat lower, than radiative rates.[36] Thus, radiation accounts for about two-thirds of thermal energy loss in cool, still air. Given the approximate nature of many of the assumptions, this can only be taken as a crude estimate. Ambient air motion, causing forced convection, or evaporation reduces the relative importance of radiation as a thermal loss mechanism. Application of Wiens Law to human body emission results in a peak wavelength of

Black-body radiation 33 For this reason, thermal imaging devices for human subjects are most sensitive in the 7–14 micron range. Temperature relation between a planet and its star The black-body law may be used to estimate the temperature of a planet orbiting the Sun. The temperature of a planet depends on several factors: • Incident radiation from its sun • Emitted radiation of the planet, e.g., Earths infrared glow • The albedo effect causing a fraction of light to be reflected by the planet • The greenhouse effect for planets with an atmosphere • Energy generated internally by a planet itself due to radioactive decay, tidal heating, and adiabatic contraction due by cooling. The analysis only considers the suns heat Earths longwave thermal radiation intensity, from clouds, atmosphere and ground for a planet in a Solar System. The Stefan–Boltzmann law gives the total power (energy/second) the Sun is emitting: where is the Stefan–Boltzmann constant, is the effective temperature of the Sun, and is the radius of the Sun. The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is: The Earth only has an absorbing area equal to a two dimensional disk, rather than the surface of a sphere. where is the radius of the planet and is the astronomical unit, the distance between the Sun and the planet. Because of its high temperature, the sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction of this energy where is the albedo or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction of the suns light, and reflects the rest. The power absorbed by the planet and its atmosphere is then: Even though the planet only absorbs as a circular area , it emits equally in all directions as a sphere. If the planet were a perfect black body, it would emit according to the Stefan–Boltzmann law

Black-body radiation 34 where is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting , is known as the effective temperature. The actual temperature of the planet will likely be different, depending on its surface and atmospheric properties. Ignoring the atmosphere and greenhouse effect, the planet, since it is at a much lower temperature than the sun, emits mostly in the infrared (IR) portion of the spectrum. In this frequency range, it emits of the radiation that a black body would emit where is the average emissivity in the IR range. The power emitted by the planet is then: For a body in radiative exchange equilibrium with its surroundings, the rate at which it emits radiant energy is equal to the rate at which it absorbs it:[37][38] Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, TP: In other words, given the assumptions made, the temperature of a planet depends only on the surface temperature of the Sun, the radius of the Sun, the distance between the planet and the Sun, the albedo and the IR emissivity of the planet. Temperature of Earth Substituting the measured values for the Sun and Earth yields: [39] [39] [39] [40] With the average emissivity set to unity, the effective temperature of the Earth is: or −18.8 °C. This is the temperature of the Earth if it radiated as a perfect black body in the infrared, ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrums, with an albedo of zero[41]), and assuming an unchanging albedo. The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. If we wish to estimate what the temperature of the Earth would be if it had no atmosphere, then we could take the albedo and emissivity of the moon as a good estimate. The albedo and emissivity of the moon are about 0.1054[42] and 0.95[43] respectively, yielding an estimated temperature of about 1.36 °C. Estimates of the Earths average albedo vary in the range 0.3–0.4, resulting in different estimated effective temperatures. Estimates are often based on the solar constant (total insolation power density) rather than the temperature, size, and distance of the sun. For example, using 0.4 for albedo, and an insolation of 1400 W m−2, one obtains an effective temperature of about 245 K.[44] Similarly using albedo 0.3 and solar constant of 1372 W m−2, one obtains an effective temperature of 255 K.[45][46][47]

Black-body radiation 35 Cosmology The cosmic microwave background radiation observed today is the most perfect black body radiation ever observed in nature, with a temperature of about 2.7K.[48] It is a "snapshot" of the radiation at the time of decoupling between matter and radiation in the early universe. Prior to this time, most matter in the universe was in the form of an ionized plasma in thermal equilibrium with radiation. According to Kondepudi and Prigogine, at very high temperatures (above 1010K; such temperatures existed in the very early universe), where the thermal motion separates protons and neutrons in spite of the strong nuclear forces, electron-positron pairs appear and disappear spontanteously and are in thermal equilibrium with electromagnetic radiation. These particles form a part of the black body spectrum, in addition to the electromagnetic radiation.[49] Doppler effect for a moving black body The relativistic Doppler effect causes a shift in the frequency f of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency f: where v is the velocity of the source in the observers rest frame, θ is the angle between the velocity vector and the observer-source direction measured in the reference frame of the source, and c is the speed of light.[50] This can be simplified for the special cases of objects moving directly towards (θ = π) or away (θ = 0) from the observer, and for speeds much less than c. Through Plancks law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (T) for the frequency in this equation. For the case of a source moving directly towards or away from the observer, this reduces to Here v > 0 indicates a receding source, and v < 0 indicates an approaching source. This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of c. An example is found in the cosmic microwave background radiation, which exhibits a dipole anisotropy from the Earths motion relative to this black body radiation field. History Balfour Stewart In 1858, Balfour Stewart described his experiments on the thermal radiative emissive and absorptive powers of polished plates of various substances, compared with the powers of lamp-black surfaces, at the same temperature.[22] Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of Pierre Prevost and of John Leslie. He wrote "Lamp-black, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power." More an experimenter than a logician, Stewart failed to point out that his statement presupposed an abstract general principle, that there exist either ideally in theory or really in nature bodies or surfaces that respectively have one and the same unique universal greatest possible absorbing power, likewise for radiating power, for every wavelength and equilibrium temperature. Stewart measured radiated power with a thermo-pile and sensitive galvanometer read with a microscope. He was concerned with selective thermal radiation, which he investigated with plates of substances that radiated and absorbed selectively for different qualities of radiation rather than maximally for all qualities of radiation. He

Black-body radiation 36 discussed the experiments in terms of rays which could be reflected and refracted, and which obeyed the Stokes-Helmholtz reciprocity principle (though he did not use an eponym for it). He did not in this paper mention that the qualities of the rays might be described by their wavelengths, nor did he use spectrally resolving apparatus such as prisms or diffraction gratings. His work was quantitative within these constraints. He made his measurements in a room temperature environment, and quickly so as to catch his bodies in a condition near the thermal equilibrium in which they had been prepared by heating to equilibrium with boiling water. His measurements confirmed that substances that emit and absorb selectively respect the principle of selective equality of emission and absorption at thermal equilibrium. Stewart offered a theoretical proof that this should be the case separately for every selected quality of thermal radiation, but his mathematics was not rigorously valid.[51] He made no mention of thermodynamics in this paper, though he did refer to conservation of vis viva. He proposed that his measurements implied that radiation was both absorbed and emitted by particles of matter throughout depths of the media in which it propagated. He applied the Helmholtz reciprocity principle to account for the material interface processes as distinct from the processes in the interior material. He did not postulate unrealizable perfectly black surfaces. He concluded that his experiments showed that in a cavity in thermal equilibrium, the heat radiated from any part of the interior bounding surface, no matter of what material it might be composed, was the same as would have been emitted from a surface of the same shape and position that would have been composed of lamp-black. He did not state explicitly that the lamp-black-coated bodies that he used as reference must have had a unique common spectral emittance function that depended on temperature in a unique way. Gustav Kirchhoff In 1859, not knowing of Stewarts work, Gustav Robert Kirchhoff reported the coincidence of the wavelengths of spectrally resolved lines of absorption and of emission of visible light. Importantly for thermal physics, he also observed that bright lines or dark lines were apparent depending on the temperature difference between emitter and absorber.[52] Kirchhoff then went on to consider bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at temperature T. Here is used a notation different from Kirchhoffs. Here, the emitting power E(T, i) denotes a dimensioned quantity, the total radiation emitted by a body labeled by index i at temperature T. The total absorption ratio a(T, i) of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature T . (In contrast with Balfour Stewarts, Kirchhoffs definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio E(T, i) / a(T, i) of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power, because a(T, i) is dimensionless. Also here the wavelength-specific emitting power of the body at temperature T is denoted by E(λ, T, i) and the wavelength-specific absorption ratio by a(λ, T, i) . Again, the ratio E(λ, T, i) / a(λ, T, i) of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power. In a second report made in 1859, Kirchhoff announced a new general principle or law for which he offered a theoretical and mathematical proof, though he did not offer quantitative measurements of radiation powers.[53] His theoretical proof was and still is considered by some writers to be invalid.[51][54] His principle, however, has endured: it was that for heat rays of the same wavelength, in equilibrium at a given temperature, the wavelength-specific ratio of emitting power to absorption ratio has one and the same common value for all bodies that emit and absorb at that wavelength. In symbols, the law stated that the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) has one and the same value for all bodies, that is for all values of index i . In this report there was no mention of black bodies. In 1860, still not knowing of Stewarts measurements for selected qualities of radiation, Kirchhoff pointed out that it was long established experimentally that for total heat radiation, of unselected quality, emitted and absorbed by a

Black-body radiation 37 body in equilibrium, the dimensioned total radiation ratio E(T, i) / a(T, i), has one and the same value common to all bodies, that is, for every value of the material index i.[55] Again without measurements of radiative powers or other new experimental data, Kirchhoff then offered a fresh theoretical proof of his new principle of the universality of the value of the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) at thermal equilibrium. His fresh theoretical proof was and still is considered by some writers to be invalid.[51][54] But more importantly, it relied on a new theoretical postulate of "perfectly black bodies," which is the reason why one speaks of Kirchhoffs law. Such black bodies showed complete absorption in their infinitely thin most superficial surface. They correspond to Balfour Stewarts reference bodies, with internal radiation, coated with lamp-black. They were not the more realistic perfectly black bodies later considered by Planck. Plancks black bodies radiated and absorbed only by the material in their interiors; their interfaces with contiguous media were only mathematical surfaces, capable neither of absorption nor emission, but only of reflecting and transmitting with refraction.[56] Kirchhoffs proof considered an arbitrary non-ideal body labeled i as well as various perfect black bodies labeled BB . It required that the bodies be kept in a cavity in thermal equilibrium at temperature T . His proof intended to show that the ratio E(λ, T, i) / a(λ, T, i) was independent of the nature i of the non-ideal body, however partly transparent or partly reflective it was. His proof first argued that for wavelength λ and at temperature T, at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power E(λ, T, BB), with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorption ratio a(λ, T, BB) of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorption ratio E(λ, T, BB) / a(λ, T, BB) is again just E(λ, T, BB), with the dimensions of power. Kirchhoff considered, successively, thermal equilibrium with the arbitrary non-ideal body, and with a perfectly black body of the same size and shape, in place in his cavity in equilibrium at temperature T . He argued that the flows of heat radiation must be the same in each case. Thus he argued that at thermal equilibrium the ratio E(λ, T, i) / a(λ, T, i) was equal to E(λ, T, BB), which may now be denoted Bλ (λ, T), a continuous function, dependent only on λ at fixed temperature T, and an increasing function of T at fixed wavelength λ, at low temperatures vanishing for visible but not for longer wavelengths, with positive values for visible wavelengths at higher temperatures, which does not depend on the nature i of the arbitrary non-ideal body. (Geometrical factors, taken into detailed account by Kirchhoff, have been ignored in the foregoing.) Thus Kirchhoffs law of thermal radiation can be stated: For any material at all, radiating and absorbing in thermodynamic equilibrium at any given temperature T, for every wavelength λ, the ratio of emissive power to absorptive ratio has one universal value, which is characteristic of a perfect black body, and is an emissive power which we here represent by Bλ (λ, T) . (For our notation Bλ (λ, T), Kirchhoffs original notation was simply e.)[55][57][58][59][60][61] Kirchhoff announced that the determination of the function Bλ (λ, T) was a problem of the highest importance, though he recognized that there would be experimental difficulties to be overcome. He supposed that like other functions that do not depend on the properties of individual bodies, it would be a simple function. Occasionally by historians that function Bλ (λ, T) has been called "Kirchhoffs (emission, universal) function,"[62][63][64][65] though its precise mathematical form would not be known for another forty years, till it was discovered by Planck in 1900. The theoretical proof for Kirchhoffs universality principle was worked on and debated by various physicists over the same time, and later.[54] Kirchhoff stated later in 1860 that his theoretical proof was better than Balfour Stewarts, and in some respects it was so.[51] Kirchhoffs 1860 paper did not mention the second law of thermodynamics, and of course did not mention the concept of entropy which had not at that time been established. In a more considered account in a book in 1862, Kirchhoff mentioned the connection of his law with Carnots principle, which is a form of the second law.[66] According to Helge Kragh, "Quantum theory owes its origin to the study of thermal radiation, in particular to the "black-body" radiation that Robert Kirchhoff had first defined in 1859–1860."[67]

Black-body radiation 38 References [1] Loudon 2000, Chapter 1. [2] Mandel & Wolf 1995, Chapter 13. [3] Kondepudi & Prigogine 1998, Chapter 11. [4] Peter Theodore Landsberg (1990). "Chapter 13: Bosons: black-body radiation" (http:/ / books. google. com/ books?id=0gnWL7tmxm0C& pg=PA208). Thermodynamics and statistical mechanics (Reprint of Oxford University Press 1978 ed.). Courier Dover Publications. pp. 208 ff. ISBN 0-486-66493-7. . [5] Ian Morison (2008). Introduction to Astronomy and Cosmology (http:/ / books. google. com/ books?id=yrV8vvJzgWkC& pg=PA48). J Wiley & Sons. p. 48. ISBN 0-470-03333-9. . [6] Alessandro Fabbri, José Navarro-Salas (2005). "Chapter 1: Introduction" (http:/ / books. google. com/ books?id=gUhZZtb6yA8C& pg=PA1). Modeling black hole evaporation. Imperial College Press. ISBN 1-86094-527-9. . [7] Tomokazu Kogure, Kam-Ching Leung (2007). "§2.3: Thermodynamic equilibrium and black-body radiation" (http:/ / books. google. com/ books?id=qt5sueHmtR4C& pg=PA41). The astrophysics of emission-line stars. Springer. p. 41. ISBN 0-387-34500-0. . [8] Wien, W. (1893). Eine neue Beziehung der Strahlung schwarzer Körper zum zweiten Hauptsatz der Wärmetheorie, Sitzungberichte der Königlich-Preußischen Akademie der Wissenschaften (Berlin), 1893, 1: 55–62. [9] Lummer, O., Pringsheim, E. (1899). Die Vertheilung der Energie im Spectrum des schwarzen Körpers, Verhandlungen der Deutschen Physikalischen Gessellschaft (Leipzig), 1899, 1: 23–41. [10] Planck 1914 [11] Draper, J.W. (1847). On the production of light by heat, London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, series 3, 30: 345–360. (http:/ / www. archive. org/ stream/ londonedinburghp30lond#page/ 344/ mode/ 2up) [12] Partington 1949, pp. 466–467, 478. [13] Goody & Yung 1989, pp. 482, 484 [14] Planck 1914, p. 42 [15] Wien 1894 [16] Planck 1914, p. 43 [17] Joseph Caniou (1999). "§4.2.2: Calculation of Plancks law" (http:/ / books. google. com/ books?id=X-aFGcf6pOEC& pg=PA107). Passive infrared detection: theory and applications. Springer. p. 107. ISBN 0-7923-8532-2. . [18] "Science: Drapers Memoirs" (http:/ / www. archive. org/ details/ scientificmemoi00drapgoog). The Academy (London: Robert Scott Walker) XIV (338): 408. Oct. 26, 1878. . [19] J. R. Mahan (2002). Radiation heat transfer: a statistical approach (http:/ / books. google. com/ ?id=y9zUEzA7iN0C& pg=PA58& dq=draper-point+ red) (3rd ed.). Wiley-IEEE. p. 58. ISBN 978-0-471-21270-6. . [20] de Groot, SR., Mazur, P. (1962). Non-equilibrium Thermodynamics, North-Holland, Amsterdam. [21] Kondepudi & Prigogine 1998, Section 9.4. [22] Stewart 1858 [23] Huang, Kerson (1967). Statistical Mechanics. New York: John Wiley & Sons. ISBN 0-471-81518-7. [24] Planck, Max (1901). "On the Law of Distribution of Energy in the Normal Spectrum" (http:/ / dbhs. wvusd. k12. ca. us/ webdocs/ Chem-History/ Planck-1901/ Planck-1901. html). Annalen der Physik 4: 553. Bibcode 1901AnP...309..553P. doi:10.1002/andp.19013090310. . [25] Landau, L. D.; E. M. Lifshitz (1996). Statistical Physics (3rd Edition Part 1 ed.). Oxford: Butterworth–Heinemann. ISBN 0-521-65314-2. [26] Mathematica:Planck intensity (energy/sec/area/solid angle/wavelength) is: i[w_, t_] = 2*h*c^2/(w^5*(Exp[h*c/(w*k*t)] - 1)) The number of photons/sec/area is: NIntegrate[2*Pi*i[w, 300]/(h*c/w), {w, 390*10^(-9), 750*10^(-9)}] = 0.0244173... [27] Rybicki & Lightman 1979, p. 22 [28] http:/ / physics. nist. gov/ cgi-bin/ cuu/ Value?bwien [29] Nave, Dr. Rod. "Wiens Displacement Law and Other Ways to Characterize the Peak of Blackbody Radiation" (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ quantum/ wien3. html#c1). HyperPhysics. . Provides 5 variations of Wiens Displacement Law [30] Infrared Services. "Emissivity Values for Common Materials" (http:/ / infrared-thermography. com/ material-1. htm). . Retrieved 2007-06-24. [31] Omega Engineering. "Emissivity of Common Materials" (http:/ / www. omega. com/ literature/ transactions/ volume1/ emissivityb. html). . Retrieved 2007-06-24. [32] Farzana, Abanty (2001). "Temperature of a Healthy Human (Skin Temperature)" (http:/ / hypertextbook. com/ facts/ 2001/ AbantyFarzana. shtml). The Physics Factbook. . Retrieved 2007-06-24.

Black-body radiation 39 [33] Lee, B.. "Theoretical Prediction and Measurement of the Fabric Surface Apparent Temperature in a Simulated Man/Fabric/Environment System" (http:/ / www. dsto. defence. gov. au/ publications/ 2135/ DSTO-TR-0849. pdf). . Retrieved 2007-06-24. [34] Harris J, Benedict F (1918). "A Biometric Study of Human Basal Metabolism.". Proc Natl Acad Sci USA 4 (12): 370–3. Bibcode 1918PNAS....4..370H. doi:10.1073/pnas.4.12.370. PMC 1091498. PMID 16576330. [35] Levine, J (2004). "Nonexercise activity thermogenesis (NEAT): environment and biology" (http:/ / ajpendo. physiology. org/ cgi/ content/ full/ 286/ 5/ E675). Am J Physiol Endocrinol Metab 286 (5): E675–E685. doi:10.1152/ajpendo.00562.2003. PMID 15102614. . [36] DrPhysics.com. "Heat Transfer and the Human Body" (http:/ / www. drphysics. com/ convection/ convection. html). . Retrieved 2007-06-24. [37] Prevost, P. (1791). Mémoire sur lequilibre du feu. Journal de Physique (Paris), vol 38 pp. 314-322. [38] Iribarne, J.V., Godson, W.L. (1981). Atmospheric Thermodynamics, second edition, D. Reidel Publishing, Dordrecht, ISBN 90-277-1296-4, page 227. [39] NASA Sun Fact Sheet (http:/ / nssdc. gsfc. nasa. gov/ planetary/ factsheet/ sunfact. html) [40] Cole, George H. A.; Woolfson, Michael M. (2002). Planetary Science: The Science of Planets Around Stars (1st ed.) (http:/ / books. google. com/ ?id=Bgsy66mJ5mYC& pg=RA3-PA382& dq=black-body+ emissivity+ greenhouse+ intitle:Planetary-Science+ inauthor:cole). Institute of Physics Publishing. pp. 36–37, 380–382. ISBN 0-7503-0815-X. . [41] Principles of Planetary Climate by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. From Chapter 3 which is available online here (http:/ / www-das. uwyo. edu/ ~deshler/ Atsc4400_5400_Climate/ PierreHumbert_Climate_Ch3. pdf), p. 12 mentions that Venus black-body temperature would be 330 K "in the zero albedo case", but that due to atmospheric warming, its actual surface temperature is 740 K. [42] Saari, J. M.; Shorthill, R. W. (1972). "The Sunlit Lunar Surface. I. Albedo Studies and Full Moon". The Moon 5 (1-2): 161–178. Bibcode 1972Moon....5..161S. doi:10.1007/BF00562111. [43] Lunar and Planetary Science XXXVII (2006) 2406 [44] Michael D. Papagiannis (1972). Space physics and space astronomy (http:/ / books. google. com/ ?id=SpgOAAAAQAAJ& pg=PA10). Taylor & Francis. pp. 10–11. ISBN 978-0-677-04000-4. . [45] Willem Jozef Meine Martens and Jan Rotmans (1999). Climate Change an Integrated Perspective (http:/ / books. google. com/ ?id=o1SELkgK6PcC& pg=RA1-PA53& dq=Earth+ effective-temperature+ albedo+ black-body+ 0. 3). Springer. pp. 52–55. ISBN 978-0-7923-5996-8. . [46] F. Selsis (2004). "The Prebiotic Atmosphere of the Earth" (http:/ / books. google. com/ ?id=bA_uR3iwzQUC& pg=PA279& dq=Earth+ effective-temperature+ albedo+ black-body+ 0. 3). In Pascale Ehrenfreund et al.. Astrobiology: Future Perspectives. Springer. pp. 279–280. ISBN 978-1-4020-2587-7. . [47] Wallace, J.M., Hobbs, P.V. (2006). Atmospheric Science. An Introductory Survey, second edition, Elsevier, Amsterdam, ISBN 978-0-12-732951-2, exercise 4.6, pages 119-120. [48] White, M. (1999). "Anisotropies in the CMB". Proceedings of the Los Angeles Meeting, DPF 99. UCLA. http:/ / arxiv. org/ pdf/ 1106. 2188v2. pdf. [49] Kondepudi & Prigogine 1998, pp. 227–228; also Section 11.6, pages 294–296. [50] The Doppler Effect, T. P. Gill, Logos Press, 1965 [51] Siegel 1976 [52] Kirchhoff 1860a [53] Kirchhoff 1860b [54] Schirrmacher 2001 [55] Kirchhoff 1860c [56] Planck 1914, p. 11 [57] Chandrasekhar 1950, p. 8 [58] Milne 1930, p. 80 [59] Rybicki & Lightman 1979, pp. 16–17 [60] Mihalas & Weibel-Mihalas 1984, p. 328 [61] Goody & Yung 1989, pp. 27–28 [62] Paschen, F. (1896), personal letter cited by Hermann 1971, p. 6 [63] Hermann 1971, p. 7 [64] Kuhn 1978, pp. 8, 29 [65] Mehra and Rechenberg 1982, pp. 26, 28, 31, 39 [66] Kirchhoff 1862/1882, p. 573 [67] Kragh 1999, p. 58

Black-body radiation 40 Bibliography • Chandrasekhar, S. (1950). Radiative Transfer. Oxford University Press. • Goody, R. M.; Yung, Y. L. (1989). Atmospheric Radiation: Theoretical Basis (2nd ed.). Oxford University Press. ISBN 978-0-19-510291-8. • Hermann, A. (1971). The Genesis of Quantum Theory. Nash, C.W. (transl.). MIT Press. ISBN 0-262-08047-8. a translation of Frühgeschichte der Quantentheorie (1899–1913), Physik Verlag, Mosbach/Baden. • Kirchhoff, G.; [27 October 1859] (1860a). "Über die Fraunhoferschen Linien". Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 662–665. • Kirchhoff, G.; [11 December 1859] (1860b). "Über den Zusammenhang zwischen Emission und Absorption von Licht und Wärme". Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 783–787. • Kirchhoff, G. (1860c). "Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht". Annalen der Physik und Chemie (Leipzig) 109: 275–301. Translated by Guthrie, F. as Kirchhoff, G. (1860). "On the relation between the radiating and absorbing powers of different bodies for light and heat". Philosophical Magazine Series 4, volume 20: 1–21. • Kirchhoff, G. (1882) [1862], "Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht", Gessamelte Abhandlungen, Leipzig: Johann Ambrosius Barth, pp. 571–598 • Kondepudi, D.; Prigogine, I. (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures. John Wiley & Sons. ISBN 0–471–97393–9. • Kragh, H. (1999). Quantum Generations: a History of Physics in the Twentieth Century. Princeton University Press. ISBN 0-691-01206-7. • Kuhn, T. S. (1978). Black–Body Theory and the Quantum Discontinuity. Oxford University Press. ISBN 0-19-502383-8. • Loudon, R. (2000) [1973]. The Quantum Theory of Light (third ed.). Cambridge University Press. ISBN 0–19–850177–3. • Mandel, L.; Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge University Press. ISBN 0–521–41711–2. • Mehra, J.; Rechenberg, H. (1982). The Historical Development of Quantum Theory. volume 1, part 1. Springer-Verlag. ISBN 0-387-90642-8. • Mihalas, D.; Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics. Oxford University Press. ISBN 0-19-503437-6. • Milne, E.A. (1930). "Thermodynamics of the Stars". Handbuch der Astrophysik 3, part 1: 63–255. • Partington, J.R. (1949). An Advanced Treatise on Physical Chemistry. Volume 1. Fundamental Principles. The Properties of Gases. Longmans, Green and Co.. • Planck, M. (1914) [1912]. The Theory of Heat Radiation. translated by Masius, M.. P. Blakistons Sons & Co.. • Rybicki, G. B.; Lightman, A. P. (1979). Radiative Processes in Astrophysics (http://books.google.com/ books?id=LtdEjNABMlsC&printsec=frontcover). John Wiley & Sons. ISBN 0-471-82759-2. • Schirrmacher, A. (2001). Experimenting theory: the proofs of Kirchhoffs radiation law before and after Planck. Münchner Zentrum für Wissenschafts und Technikgeschichte. • Siegel, D.M. (1976). "Balfour Stewart and Gustav Robert Kirchhoff: two independent approaches to "Kirchhoffs radiation law"". Isis 67: 565–600. • Stewart, B. (1858). "An account of some experiments on radiant heat". Transactions of the Royal Society of Edinburgh 22: 1–20. • Wien, W. (1894). "Temperatur und Entropie der Strahlung". Annalen der Physik (Leipzig) 288: 132–165.

Black-body radiation 41 Further reading • Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 0-7167-1088-9. • Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0-7167-4345-0. External links • Calculating Black-body Radiation (http://www.spectralcalc.com/blackbody/blackbody.html) Interactive calculator with Doppler Effect. Includes most systems of units. • Cooling Mechanisms for Human Body (http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/coobod. html#c1) – From Hyperphysics • Descriptions of radiation emitted by many different objects (http://www.x20.org/library/thermal/blackbody. htm) • Black-Body Emission Applet (http://webphysics.davidson.edu/Applets/java11_Archive.html) • "Blackbody Spectrum" (http://demonstrations.wolfram.com/BlackbodySpectrum/) by Jeff Bryant, Wolfram Demonstrations Project, 2007. Boiling Boiling is the rapid vaporization of a liquid, which occurs when a liquid is heated to its boiling point, the temperature at which the vapor pressure of the liquid is equal to the pressure exerted on the liquid by the surrounding environmental pressure. Nucleate boiling Nucleate boiling is characterized by the growth of bubbles or pops on a heated surface, which rise from discrete points on a surface, whose temperature is only slightly above the liquid’s. In general, the number of nucleation sites are increased by an increasing surface temperature. An irregular surface of the boiling vessel (i.e. increased surface roughness) can create additional nucleation sites, while an exceptionally smooth surface, such as plastic, lends itself to superheating. Under these conditions, a heated liquid may show boiling delay and the temperature may go somewhat above the boiling point without boiling. Transition boiling Transition boiling may be defined as the unstable boiling, which occurs at surface temperatures between the maximum attainable in nucleate and the minimum attainable in film boiling. The formation of bubbles in a heated liquid is a complex physical process which often involves cavitation and acoustic effects, such as the broad-spectrum hiss one hears in a kettle not yet heated to the point where bubbles boil to the surface.

Boiling 42 Film boiling If a surface heating the liquid is significantly hotter than the liquid then film boiling will occur, where a thin layer of vapor, which has low thermal conductivity, insulates the surface. This condition of a vapor film insulating the surface from the liquid characterizes film boiling. In distillation, boiling is used in separating mixtures. This is possible because the vapor rising from a boiling fluid generally has a ratio of components different from that in the liquid. Boiling for water sterilization Boiling can be used as a method of water disinfection but is only advocated as an emergency water treatment method, or as a method of portable water purification in rural or wilderness settings without access to a potable water infrastructure. Bringing water to the boil is effective in killing or inactivating most bacteria, viruses and pathogens. Boiling is the most certain way of killing nearly all microorganisms. According to the Wilderness Medical Society,[1] water temperatures above 160°F (70°C) kill all pathogens within 30 minutes and above 185°F (85°C) within a few minutes. So in the time it takes for the water to reach the boiling point (212°F or 100°C) from 160°F (70°C), all pathogens will be killed, even at high altitude. To be extra safe, let the water boil rapidly for one minute, especially at higher altitudes since water boils at a lower temperature.[1] Boiling in cooking In cooking, boiling is the method of cooking food in boiling water, or other water-based liquid such as stock or milk. Simmering is gentle boiling, while in poaching the cooking liquid moves but scarcely bubbles. Boiling is a very harsh technique of cooking. Delicate foods such as fish cannot be cooked in this fashion because the bubbles can damage the food. Foods such as red meat, chicken, and root vegetables can be cooked with this technique because of their tough texture. The boiling point of water is typically considered to be 100 °C or 212 °F. Pressure and a change in composition of the liquid may alter the boiling point of the liquid. For this reason, high elevation cooking generally takes longer since boiling point is a function of atmospheric Transition boiling of water pressure. In Denver, Colorado, USA, which is at an elevation of about one mile, water boils at approximately 95 °C or 203 °F.[2] Depending on the type of food and the elevation, the boiling water may not be hot enough to cook the food properly. Similarly, increasing the pressure as in a pressure cooker raises the temperature of the contents above the open air boiling point. Adding a water soluble substance, such as salt or sugar also increases the boiling point. This is called boiling-point elevation. At palatable concentrations of salt, the effect is very small, and the boiling point elevation is difficult to notice. However, while making thick sugar syrup, such as for Gulab Jamun, one will notice boiling point elevation. Due to variations in composition and pressure, the boiling point of water is almost never exactly 100 °C, but rather close enough for cooking. Bringing water to a boil is generally done by applying maximal heat, then shutting off when the water has come to a boil, which is known as bang–bang control. Keeping water at or below a boil requires more careful control of temperature, particularly by using feedback. In places where the available water supply is contaminated with disease-causing bacteria, boiling water and allowing it to cool before drinking it is practiced as a valuable health measure. Boiling is the most certain way of killing all

Boiling 43 microorganisms in emergency situations. Foods suitable for boiling include vegetables, starchy foods such as rice, noodles and potatoes, eggs, meats, sauces, stocks and soups. Boiling has several advantages. It is safe and simple, and it is appropriate for large-scale cookery. Older, tougher, cheaper cuts of meat and poultry can be made digestible. Nutritious, well flavoured stock is produced. Also, maximum color and nutritive value is retained when cooking green vegetables, provided boiling time is kept to the minimum. On the other hand, there are several disadvantages. There is a loss of soluble vitamins from foods to the water (if the water is discarded), and some boiled foods can look unattractive. Boiling can also be a slow method of cooking food. Boiling can be done in several ways: The food can be placed into already rapidly boiling water and left to cook, the heat can be turned down and the food can be simmered; or the food can also be placed into the pot, and cold water may be added to the pot. This may then be boiled until the food is satisfactory. Water on the outside of a pot, i.e. a wet pot, increases the time it takes the pot of water to boil. The pot will heat at a normal rate once all excess water on the outside of the pot evaporates. Levels of boiling In Chinese cuisine, particularly tea brewing, one distinguishes five stages of boiling:[3] "shrimp eyes, the first tiny bubbles that start to appear on the surface of the kettle water, crab eyes, the secondary, larger bubbles, then fish eyes, followed by rope of pearls, and finally raging torrent [rolling boil]". In detail:[4] shrimp eyes about 70-80 °C (155–175 °F) – separate bubbles, rising to top crab eyes about 80 °C (175 °F) – streams of bubbles fish eyes about 80-90 °C (175–195 °F) – larger bubbles rope of pearls about 90-95 °C (195–205 °F) – steady streams of large bubbles raging torrent rolling boil, swirling and roiling Effect of altitude Altitude affects the boiling point of water. As you go higher, atmosphere pressure decreases which makes it easier to boil water.[5] Also see Atmospheric pressure boiling point. References [1] OA Guide to Water Purification, The Backpackers Field Manual, by Rick Curtis, first edition published by Random House March, 1998 (http:/ / www. princeton. edu/ ~oa/ manual/ water. shtml) [2] IAPWS. "What is the effect of pressure on the boiling of water? Why does water boil at a lower temperature at high altitudes?" (http:/ / www. iapws. org/ faq1/ boil. htm). FAQs About Water and Steam (http:/ / www. iapws. org/ faq1/ ). . Retrieved 2009-12-05. [3] James Norwood Pratt, The New Tea Lovers Treasury [4] Boiling Water for Tea: Bubbles and Steam (http:/ / blog. mightyleaf. com/ bubbles-and-steam-boiling-water-for-tea/ ), by Bliss Dake, October 13th, 2009, Mighty Leaf Tea blog [5] "Is it true that you cant make a decent cup of tea up a mountain?" (http:/ / www. physics. org/ facts/ air-tea. asp). physics.org. . Retrieved 2 November 2012.

Centrifugal evaporator 44 Centrifugal evaporator A centrifugal evaporator is a device used in chemical and biochemical laboratories for the efficient and gentle evaporation of solvents from many samples at the same time, and samples contained in microtitre plates. If only one sample required evaporation then a rotary evaporator is most often used. The most advanced modern centrifugal evaporators not only concentrate many samples at the same time, they eliminate solvent bumping and can handle solvents with boiling points of up to 220 °C. This is more than adequate for the modern high throughput laboratory. Design Cutaway image of HT-4X Centrifugal Evaporator A centrifugal evaporator often comprises a vacuum pump connected to a centrifuge chamber in which the samples are placed. Many systems also have a cold trap or solvent condenser placed in line between the vacuum pump and the centrifuge chamber to collect the evaporated solvents. The most efficient systems also have a cold trap on the pump exhaust. There are many further developments available from manufacturers to speed up the process, and to provide protection for delicate samples. The system works by lowering the pressure in the centrifuge system - as the pressure drops so does the boiling point of the solvent(s) in the system. When the pressure is sufficiently low that the boiling points of the solvents are below the temperature of the sample holder, then they will boil. This enables solvent to be rapidly removed while the samples themselves are not heated to damaging temperatures. High performance systems can remove very high boiling solvents such as DMSO or NMP while keeping sample temperatures below 40 oC at all times. The centrifugal force generated by spinning the centrifuge rotor creates a pressure gradient within the solvent contained in the tubes or vials, this means that the samples boil from the top down, helping to prevent "bumping". The most advanced systems apply the vacuum slowly and run the rotor at speeds of 500 x gravity - this system is proven to prevent bumping and was patented by Genevac in the late 1990s.[1] Manufacturers Centrifugal evaporators were invented in 1960s by Savant Inc of USA, with their market leading SpeedVac brand. Other well known manufacturers of centrifugal evaporators are the German company Martin Christ, LabConco of USA and Genevac of UK. References [1] See also http:/ / www. pld. fi/ supp/ Sampleprep/ Genevac%20files/ DRI. PDF

ChurchillBernstein equation 45 Churchill–Bernstein equation In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities.[1] The need for the equation arises from the inability to solve the Navier–Stokes equations in the turbulent flow regime, even for a Newtonian fluid. When the concentration and temperature profiles are independent of one another, the mass-heat transfer analogy can be employed. In the mass-heat transfer analogy, heat transfer dimensionless quantities are replaced with analogous mass transfer dimensionless quantities. This equation is named after Stuart W. Churchill and M. Bernstein, who introduced it in 1977. This equation is also called the Churchill–Bernstein correlation. Heat transfer definition where: • is the surface averaged Nusselt number with characteristic length of diameter; • is the Reynolds number with the cylinder diameter as its characteristic length; • is the Prandtl number. The Churchill–Bernstein equation is valid for a wide range of Reynolds numbers and Prandtl numbers, as long as the product of the two is greater than or equal to 0.2, as defined above. The Churchill–Bernstein equation can be used for any object of cylindrical geometry in which boundary layers develop freely, without constraints imposed by other surfaces. Properties of the external free stream fluid are to be evaluated at the film temperature in order to account for the variation of the fluid properties at different temperatures. One should not expect much more than 20% accuracy from the above equation due to the wide range of flow conditions that the equation encompasses. The Churchill–Bernstein equation is a correlation and cannot be derived from principles of fluid dynamics. The equation yields the surface averaged Nusselt number, which is used to determine the average convective heat transfer coefficient. Newtons law of cooling can then be invoked to determine the heat loss or gain from the object, fluid and/or surface temperatures, and the area of the object, depending on what information is known. Mass transfer definition where: • is the Sherwood number • is the Schmidt number Using the mass-heat transfer analogy, the Nusselt number is replaced by the Sherwood number, and the Prandtl number is replaced by the Schmidt number. The same restrictions described in the heat transfer definition are applied to the mass transfer definition. The Sherwood number can be used to find an overall mass transfer coefficient and applied to Ficks law of diffusion to find concentration profiles and mass transfer fluxes.

ChurchillBernstein equation 46 Notes [1] Some examples of said system can be seen here: "Cylinder in Cross Flow at Various Velocities" (http:/ / www. flometrics. com/ services/ cylinder/ cylinder. html). Flometrics. 1997. . Retrieved 10/07/2007. References • Churchill, S. W.; Bernstein, M. (1977), "A Correlating Equation for Forced Convection From Gases and Liquids to a Circular Cylinder in Crossflow" (http://dx.doi.org/10.1115/1.3450685), J. Heat Transfer, Trans. ASME 99: 300–306 • Incropera, F.P.,; DeWitt, D.P., Bergman, T.L., and Lavine, A.S. (2006). Fundamentals of Heat and Mass Transfer, 6th Ed. (http://www.amazon.com/dp/0470055545). Wiley. ISBN 978-0-471-45728-2. • Tammet, Hannes; Kulmala, Markku (June), Simulating aerosol nucleation bursts in a coniferous forest (http:// www.borenv.net/BER/pdfs/ber12/ber12-421.pdf), retrieved 10/07/2007 • Ramachandran Venkatesan; Scott Fogler (2004). "Comments on Analogies for Correlated Heat and Mass Transfer in Turbulent Flow". AIChE Journal 50 (7): 1623–1626. doi:10.1002/aic.10146. • Martínez (http://webserver.dmt.upm.es/~isidoro/), Isidoro, Forced and Natural Convection (http:// webserver.dmt.upm.es/~isidoro/bk3/c12/Forced and natural convection.pdf), retrieved 2011-11-30 Condensation Condensation is the change of the physical state of matter from gaseous phase into liquid phase, and is the reverse of vaporization.[1] When the transition happens from the gaseous phase into the solid phase directly, the change is called deposition. Initiation Condensation is initiated by the formation of atomic/molecular clusters of that species within its gaseous volume—like rain drop or snow-flake formation within clouds—or at the contact between such gaseous phase Water vapor condenses into a liquid after making contact with the surface of a cold bottle. and a (solvent) liquid or solid surface. Reversibility scenarios A few distinct reversibility scenarios emerge here with respect to the nature of the surface. • absorption into the surface of a liquid (either of the same species or one of its solvents)—is reversible as evaporation.[1] • adsorption (as dew droplets) onto solid surface at pressures and temperatures higher than the species triple point—also reversible as evaporation. Condensation forming over the wing of an • adsorption onto solid surface (as supplemental layers of solid) at aircraft in a low pressure zone pressures and temperatures lower than the species triple point—is reversible as sublimation.

Condensation 47 Most common scenarios Condensation commonly occurs when a vapor is cooled and/or compressed to its saturation limit when the molecular density in the gas phase reaches its maximal threshold. Vapor cooling and compressing equipment that collects condensed liquids is called a "condenser". How condensation is measured Psychrometry measures the rates of condensation from and evaporation into the air moisture at various atmospheric pressures and temperatures. Water is the product of its vapor condensation—condensation is the process of such phase conversion. Applications of condensation Condensation is a crucial component of distillation, an important laboratory and industrial chemistry application. Because condensation is a naturally occurring phenomenon, it can often be used to generate water in large quantities for human use. Many structures are made solely for the purpose of collecting water from condensation, such as air wells and fog fences. Such systems can often be used to retain soil moisture in areas where active desertification is occurring—so much so that some organizations educate people living in affected areas about water condensers to help them deal effectively with the situation.[2] It is also a crucial process in forming particle tracks in a cloud chamber. In this case, ions produced by an incident particle act as nucleation centers for the condensation of the vapor producing the visible "cloud" trails. Biological adaptation Numerous living beings use water made accessible by condensation. A few examples of these are the Australian Thorny Devil, the darkling beetles of the Namibian coast, and the Coast Redwoods of the West Coast of the United States. Condensation in building construction Condensation in building construction is an unwanted phenomenon as it may cause dampness, mold health issues, wood rot, corrosion and energy loss due to increased heat transfer. Interstructure condensation may be caused by thermal bridges, insufficient or lacking damp proofing or insulated glazing.[3] References [1] Nic, M.; Jirat, J.; Kosata, B., eds. (2006–). "condensation in atmospheric chemistry" (http:/ / goldbook. iupac. org/ C01235. html). IUPAC Compendium of Chemical Condensation on a window during a rain shower. Terminology (Online ed.). doi:10.1351/goldbook.C01235. ISBN 0-9678550-9-8. . [2] FogQuest - Fog Collection / Water Harvesting Projects - Welcome (http:/ / www. fogquest. org/ ) [3] diydata.com (http:/ / www. diydata. com/ problem/ condensation/ condensation. php)

Condensation 48 References • "Condensation principles (Picture)" (http://www.ces.purdue.edu/pork/images/jones/condense.jpg). Condenser (heat transfer) For other Condensers not involving heat transfer, see Condenser (disambiguation). In systems involving heat transfer, a condenser is a device or unit used to condense a substance from its gaseous to its liquid state, typically by cooling it. In so doing, the latent heat is given up by the substance, and will transfer to the condenser coolant. Condensers are typically heat exchangers which have various designs and come in many sizes ranging from rather small (hand-held) to very large industrial-scale units used in plant processes. For example, a refrigerator uses a condenser to get rid of heat extracted from the interior of the unit to the outside air. Condensers are used in air conditioning, industrial chemical processes such as distillation, steam power plants and other heat-exchange systems. Use of cooling water or surrounding air as the coolant is common in many condensers.[1] Example types of condensers • A surface condenser is an example of such a heat-exchange system. The condenser coil of a refrigerator It is a shell and tube heat exchanger installed at the outlet of every steam turbine in thermal power stations. Commonly, the cooling water flows through the tube side and the steam enters the shell side where the condensation occurs on the outside of the heat transfer tubes. The condensate drips down and collects at the bottom, often in a built-in pan called a hotwell. The shell side often operates at a vacuum or partial vacuum, often produced by attached air ejectors. Conversely, the vapour can be fed through the tubes with the coolant water or air flowing around the outside. • In chemistry, a condenser is the apparatus which cools hot vapors, causing them to condense into a liquid. See "Condenser (laboratory)" for laboratory-scale condensers, as opposed to industrial-scale condensers. Examples include the Liebig condenser, Graham condenser, and Allihn condenser. This is not to be confused with a condensation reaction which links two fragments into a single molecule by an addition reaction and an elimination reaction. In laboratory distillation, reflux, and rotary evaporators, several types of condensers are commonly used. The Liebig condenser is simply a straight tube within a cooling water jacket, and is the simplest (and relatively least expensive) form of condenser. The Graham condenser is a spiral tube within a water jacket, and the Allihn condenser has a series of large and small constrictions on the inside tube, each increasing the surface area upon which the vapor constituents may condense. Being more complex shapes to manufacture, these latter types are also more expensive to purchase. These three types of condensers are laboratory glassware items since they are typically made of glass. Commercially available condensers usually are fitted with ground glass joints and come in standard lengths of 100, 200, and 400 mm. Air-cooled condensers are unjacketed, while water-cooled condensers contain a jacket for the water. • Larger condensers are also used in industrial-scale distillation processes to cool distilled vapor into liquid distillate. Commonly, the coolant flows through the tube side and distilled vapor through the shell side with distillate collecting at or flowing out the bottom.

Condenser (heat transfer) 49 • A condenser unit used in central air conditioning systems typically has a heat exchanger section to cool down and condense incoming refrigerant vapor into liquid, a compressor to raise the pressure of the refrigerant and move it along, and a fan for blowing outside air through the heat exchanger section to cool the refrigerant inside. A typical configuration of such a condenser unit is as follows: The heat exchanger section wraps around the sides of the unit with the compressor inside. In this heat exchanger section, the refrigerant goes through multiple tube passes, which are surrounded by heat transfer fins through which cooling air can move from outside to inside the unit. There is a motorized fan inside the condenser unit near the top, which is covered by some grating to keep any objects from accidentally falling inside on the fan. The fan is used to blow the outside cooling air in through the heat exchange section at the sides and out the top through the grating. These condenser units are located on the outside of the building they are trying to cool, with tubing between the unit and building, one for vapor refrigerant entering and another for liquid refrigerant leaving the unit. Of course, an electric power supply is needed for the compressor and fan inside the unit. • Direct contact condenser In this type of condenser, vapors are poured into the liquid directly. The vapors lose their latent heat of vaporization; hence, vapors transfer their heat into liquid and the liquid becomes hot. In this type of condensation, the vapor and liquid are of same type of substance. In another type of direct contact condenser, cold water is sprayed into the vapour to be condensed. References [1] Hindelang, Maureen; Palazzolo, Joseph; Robertson, Matthew, "Condensers" (http:/ / encyclopedia. che. engin. umich. edu/ Pages/ HeatTransfer/ Condensers/ Condensers. html), Encyclopedia of CHemical Engineering Equipment, University of Michigan, , retrieved May 13, 2012 Convection–diffusion equation The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation, Smoluchowski equation (after Marian Smoluchowski),[1] or (generic) scalar transport equation.[2] Equation General The general equation is[3][4] where • c is the variable of interest (species concentration for mass transfer, temperature for heat transfer), • D is the diffusivity (also called diffusion coefficient), such as mass diffusivity for particle motion or thermal diffusivity for heat transport, • is the average velocity that the quantity is moving. For example, in advection, c might be the concentration of salt in a river, and then would be the velocity of the water flow. As another example, c might be the concentration of small bubbles in a calm lake, and then would be the average velocity of bubbles rising towards the surface by buoyancy (see below).

Convectiondiffusion equation 50 • R describes "sources" or "sinks" of the quantity c. For example, for a chemical species, R>0 means that a chemical reaction is creating more of the species, and R<0 means that a chemical reaction is destroying the species. For heat transport, R>0 might occur if thermal energy is being generated by friction. • represents gradient and represents divergence. Common simplifications In a common situation, the diffusion coefficient is constant, there are no sources or sinks, and the velocity field describes an incompressible flow (i.e., it has zero divergence). Then the formula simplifies to:[5][6][7] In this form, the convection–diffusion equation combines both parabolic and hyperbolic partial differential equations. Stationary version The stationary convection–diffusion equation describes the steady-state behavior of a convective-diffusive system. In steady-state, , so the formula is: Derivation The convection–diffusion equation can be derived in a straightforward way[4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: where is the total flux and s is a net volumetric source for c. There are two sources of flux in this situation. First, diffusive flux arises due to diffusion. This is typically approximated by Ficks first law: i.e., the flux of the diffusing material (relative to the bulk motion) in any part of the system is proportional to the local concentration gradient. Second, when there is overall convection or flow, there is an associated flux called advective flux: The total flux (in a stationary coordinate system) is given by the sum of these two: Plugging into the continuity equation:

Convectiondiffusion equation 51 Complex mixing phenomena In general, D, , and s may vary with space and time. In cases in which they depend on concentration as well, the equation becomes nonlinear, giving rise to many distinctive mixing phenomena such as Rayleigh–Bénard convection when depends on temperature in the heat transfer formulation and reaction-diffusion pattern formation when s depends on concentration in the mass transfer formulation. Velocity in response to a force In some cases, the average velocity field exists because of a force; for example, the equation might describe the flow of ions dissolved in a liquid, with an electric field pulling the ions in some direction (as in gel electrophoresis). In this situation, it is usually called the drift-diffusion equation or the Smoluchowski equation,[1] after Marian Smoluchowski who described it in 1915[8] (not to be confused with the Einstein–Smoluchowski relation or Smoluchowski coagulation equation). Typically, the average velocity is directly proportional to the applied force, giving the equation:[9][10] where is the force, and characterizes the friction or viscous drag. (The inverse is called mobility.) Derivation of Einstein relation When the force is associated with a potential energy (see conservative force), a steady-state solution to the above equation (i.e. 0 = R = ∂c/∂t) is: (assuming D and are constant). In other words, there are more particles where the energy is lower. This concentration profile is expected to agree with the Boltzmann distribution (more precisely, the Gibbs measure). From this assumption, the Einstein relation can be proven: .[10] As a stochastic differential equation The convection–diffusion equation (with no sources or drains, R=0) can be viewed as a stochastic differential equation, describing random motion with diffusivity D and bias . For example, the equation can describe the Brownian motion of a single particle, where the variable c describes the probability distribution for the particle to be in a given position at a given time. The reason the equation can be used that way is because there is no mathematical difference between the probability distribution of a single particle, and the concentration profile of a collection of infinitely many particles (as long as the particles do not interact with each other). The Langevin equation describes advection, diffusion, and other phenomena in an explicitly stochastic way. One of the simplest forms of the Langevin equation is when its "noise term" is Gaussian; in this case, the Langevin equation is exactly equivalent to the convection–diffusion equation.[10] However, the Langevin equation is more general.[10]

Convectiondiffusion equation 52 Similar equations in other contexts The convection–diffusion equation is a relatively simple equation describing flows, or alternatively, describing a stochastically-changing system. Therefore, the same or similar equation arises in many contexts unrelated to flows through space. • It is formally identical to the Fokker–Planck equation for the velocity of a particle. • It is closely related to the Black–Scholes equation and other equations in financial mathematics. • It is closely related to the Navier–Stokes equations, because the flow of momentum in a fluid is mathematically similar to the flow of mass or energy. The correspondence is clearest in the case of an incompressible Newtonian fluid, in which case the Navier–Stokes equation is: where M is the momentum of the fluid (per unit volume) at each point (equal to the density multiplied by the velocity v), is viscosity, P is fluid pressure, and f is any other body force such as gravity. In this equation, the term on the left-hand side describes the change in momentum at a given point; the first term on the right describes viscosity, which is really the diffusion of momentum; the second term on the right describes the advective flow of momentum; and the last two terms on the right describes the external and internal forces which can act as sources or sinks of momentum. In semiconductor physics In semiconductor physics, this equation is called the drift–diffusion equation. The word "drift" is related to drift current and drift velocity. The equation is normally written: where • n and p are the concentrations (densities) of electrons and holes, respectively, • q>0 is the elementary charge, • Jn and Jp are the electric currents due to electrons and holes respectively, • Jn/-q and Jp/q are the corresponding "particle currents" of electrons and holes respectively, • R represents carrier generation and recombination (R>0 for generation of electron-hole pairs, R<0 for recombination.) • E is the electric field vector • and are electron and hole mobility. The diffusion coefficient and mobility are related by the Einstein relation as above: where kB is Boltzmann constant and T is absolute temperature. The drift current and diffusion current refer separately to the two terms in the expressions for J, i.e.:

Convectiondiffusion equation 53 References [1] Chandrasekhar (1943). Rev. Mod. Phys. 15: 1. Bibcode 1943RvMP...15....1C. doi:10.1103/RevModPhys.15.1. See equation (312) [2] Computational Fluid Dynamics in Industrial Combustion by Baukal and Gershtein, p67, google books link (http:/ / books. google. com/ books?id=YlSbCgjjLrcC& pg=PA67). [3] Introduction to Climate Modelling, by Thomas Stocker, p57, google books link (http:/ / books. google. com/ books?id=D4zulgFb5JwC& pg=PA57) [4] Advective Diffusion Equation, lecture notes by Scott A. Socolofsky and Gerhard H. Jirka, web link (https:/ / ceprofs. civil. tamu. edu/ ssocolofsky/ cven489/ downloads/ book/ ch2. pdf) [5] Bejan A (2004). Convection Heat Transfer. [6] Bird, Stewart, Lightfoot (1960). Transport Phenomena. [7] Probstein R (1994). Physicochemical Hydrodynamics. [8] M. v. Smoluchowski, Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und den Zusammenhang mit der verallgemeinerten Diffusionsgleichung, Ann. Phys. 353 (4. Folge 48), 1103–1112 (1915), PDF link (http:/ / matwbn. icm. edu. pl/ ksiazki/ pms/ pms2/ pms2132. pdf) [9] http:/ / www. ks. uiuc. edu/ ~kosztin/ PHYCS498NSM/ LectureNotes/ chp4. pdf [10] The Theory of Polymer Dynamics by Doi and Edwards, p46-52, google books link (http:/ / books. google. com/ books?id=dMzGyWs3GKcC& pg=PA46) • Granville Sewell, The Numerical Solution of Ordinary and Partial Differential Equations, Academic Press (1988). ISBN 0-12-637475-9 Convective heat transfer Convective heat transfer, often referred to simply as convection, is the transfer of heat from one place to another by the movement of fluids. Convection is usually the dominant form of heat transfer in liquids and gases. Although often discussed as a distinct method of heat transfer, convective heat transfer involves the combined processes of conduction (heat diffusion) and heat transfer by bulk fluid flow, a Simulation of thermal convection. Red hues designate hot areas, while regions with blue hues process technically called heat advection. are cold. A hot, less-dense lower boundary layer The term convection can refer to transfer of heat with any fluid sends plumes of hot material upwards, and likewise, cold material from the top moves movement, but advection is the more precise term for the transfer due downwards. This illustration is taken from a only to bulk fluid flow. The process of transfer of heat from a solid to a model of convection in the Earths mantle. fluid, or the reverse, requires not only transfer of heat by bulk motion of the fluid, but also diffusion/conduction of heat through the still boundary layer next to the solid. Thus, this process with a moving fluid requires both diffusion and advection of heat, a summed process that is generally called convection. Convection can be "forced" by movement of a fluid by means other than buoyancy forces (for example, a water pump in an automobile engine). In some cases, natural buoyancy forces alone are entirely responsible for fluid motion when the fluid is heated, and this process is called "natural convection." An example is the draft in a chimney or around any fire. In natural convection, an increase in temperature produces a reduction in density, which causes fluid motion due to pressures and forces when fluids of different densities are affected by gravity (or any g-force). For example, when water is heated on a stove, hot water from the bottom of the pan rises, displacing the colder denser liquid which falls. After heating has stopped, mixing and conduction from this natural convection eventually result in a nearly homogeneous density, and even temperature. The convection heat transfer mode is comprised to two mechanisms. In addition to energy transfer due to random molecular motion (diffusion), energy is also transferred by bulk, or macroscopic, motion of the fluid. This motion is associated with the fact that, at any instant, large numbers of molecules are moving collectively or as aggregates. Such motion, in the presence of a temperature gradient, contributes to heat transfer. Because the molecules in

Convective heat transfer 54 aggregate retain their random motion, the total heat transfer is then due to the superposition of energy transport by random motion of the molecules and by the bulk motion of the fluid. It is customary to use the term convection when referring to this cumulative transport and the term advection when referring to the transport due to bulk fluid motion.[1] Overview Convection is the transfer of thermal energy from one place to another by the movement of fluids or gases. Convection is usually the dominant form of heat transfer in liquids and gases. Although often discussed as a distinct method of heat transfer, convection describes the combined effects of conduction and fluid flow or mass exchange. Two types of convective heat transfer may be distinguished: • Free or natural convection: when fluid motion is caused by buoyancy forces that result from the density variations due to variations of temperature in the fluid. In the absence of an external source, when the fluid is in contact with a hot surface, its molecules separate and scatter, causing the fluid to be less dense. As a consequence, the fluid is displaced while the cooler fluid gets denser and the fluid sinks. Thus, the hotter volume transfers heat towards the cooler volume of that fluid.[2] Familiar examples are the upward flow of air due to a fire or hot object and the circulation of water in a pot that is heated from below. • Forced convection: when a fluid is forced to flow over the surface by an external source such as fans, by stirring, and pumps, creating an artificially induced convection current.[3] Internal and external flow can also classify convection. Internal flow occurs when a fluid is enclosed by a solid boundary such when flowing through a pipe. An external flow occurs when a fluid extends indefinitely without encountering a solid surface. Both of these types of convection, Papers lifted on rising convective air current either natural or forced, can be internal or external because they are from warm radiator independent of each other. The bulk temperature, or the average fluid temperature, is a convenient reference point for evaluating properties related to convective heat transfer, particularly in applications related to flow in pipes and ducts. For a visual experience of natural convection, a glass filled with hot water and some red food dye may be placed inside a fish tank with cold, clear water. The convection currents of the red liquid may be seen to rise and fall in different regions, then eventually settle, illustrating the process as heat gradients are dissipated.

Convective heat transfer 55 Newtons law of cooling Convection-cooling can sometimes be described by Newtons law of cooling in cases where the heat transfer coefficient is independent or relatively independent of the temperature difference between object and environment. This is sometimes true, but is not guaranteed to be the case (see other situations below where the transfer coefficient is temperature dependent). Newtons law, which requires a constant heat transfer coefficient, states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. The rate of heat transfer in such circumstances is derived below:[4] Newtons cooling law is a solution of the differential equation given by Fouriers law: where is the thermal energy in joules is the heat transfer coefficient (assumed independent of T here) (W/m2 K) is the surface area of the heat being transferred (m2) is the temperature of the objects surface and interior (since these are the same in this approximation) is the temperature of the environment; i.e. the temperature suitably far from the surface is the time-dependent thermal gradient between environment and object The heat transfer coefficient h depends upon physical properties of the fluid and the physical situation in which convection occurs. Therefore, a single usable heat transfer coefficient (one that does not vary significantly across the temperature-difference ranges covered during cooling and heating) must be derived or found experimentally for every system analyzed. Formulas and correlations are available in many references to calculate heat transfer coefficients for typical configurations and fluids. For laminar flows, the heat transfer coefficient is rather low compared to the turbulent flows; this is due to turbulent flows having a thinner stagnant fluid film layer on heat transfer surface.[5] However, note that Newtons law breaks down if the flows should transition between laminar or turbulent flow, since this will change the heat transfer coefficient h which is assumed constant in solving the equation. Newtons law requires that internal heat conduction within the object be large in comparison to the loss/gain of heat by convection (lumped capacitance model), and this may not be true (see heat transfer). Also, an accurate formulation for temperatures may require analysis based on changing heat transfer coefficients at different temperatures, a situation frequently found in free-convection situations, and which precludes accurate use of Newtons law. Assuming these are not problems, then the solution can be given if heat transfer within the object is considered to be far more rapid than heat transfer at the boundary (so that there are small thermal gradients within the object). This condition, in turn, allows the heat in the object to be expressed as a simple product of the objects mass, its heat capacity, and its temperature, as in the following section:

Convective heat transfer 56 Solution in terms of object heat capacity If the entire body is treated as lumped capacitance thermal energy reservoir, with a total thermal energy content which is proportional to simple total heat capacity , and , the temperature of the body, or , it is expected that the system will experience exponential decay with time in the temperature of a body. From the definition of heat capacity comes the relation . Differentiating this equation with regard to time gives the identity (valid so long as temperatures in the object are uniform at any given time): . This expression may be used to replace in the first equation which begins this section, above. Then, if is the temperature of such a body at time , and is the temperature of the environment around the body: where is a positive constant characteristic of the system, which must be in units of , and is therefore sometimes expressed in terms of a characteristic time constant given by: . Thus, in thermal systems, . (The total heat capacity of a system may be further represented by its mass-specific heat capacity multiplied by its mass , so that the time constant is also given by ). The solution of this differential equation, by standard methods of integration and substitution of boundary conditions, gives: If: is defined as : where is the initial temperature difference at time 0, then the Newtonian solution is written as: This same solution is more immediately apparent if the initial differential equation is written in terms of , as a single function of time to be found, or "solved for." References [1] Incropera DeWitt VBergham Lavine 2007, Introduction to Heat Transfer, 5th ed., pg. 6 ISBN 978-0-471-45727-5 [2] http:/ / biocab. org/ Heat_Transfer. html Biology Cabinet organization, April 2006, “Heat Transfer”, Accessed 20/04/09 [3] http:/ / www. engineersedge. com/ heat_transfer/ convection. htm Engineers Edge, 2009, “Convection Heat Transfer”,Accessed 20/04/09 [4] Louis C. Burmeister, (1993) “Convective Heat Transfer”, 2nd ed. Publisher Wiley-Interscience, p 107 ISBN 0-471-57709-X, 9780471577096, Google Book Search. Accessed 20-03-09 [5] http:/ / www. engineersedge. com/ heat_transfer/ convection. htm Engineers Edge, 2009, “Convection Heat Transfer”,Accessed 20/03/09

Countercurrent exchange 57 Countercurrent exchange Countercurrent exchange is a mechanism occurring in nature and mimicked in industry and engineering, in which there is a crossover of some property, usually heat or some component, between two flowing bodies flowing in opposite directions to each other. The flowing bodies can be liquids, gases, or even solid powders, or any combination of those. For example, in a distillation column, the vapors bubble up through the downward flowing liquid while exchanging both heat and mass. The maximum amount of heat or mass transfer that can be obtained is higher with countercurrent than cocurrent (parallel) exchange because countercurrent maintains a slowly declining difference or gradient (usually temperature or concentration difference). In concurrent exchange the initial gradient is higher but falls off quickly, leading to wasted potential. For example, in the diagram at the right, the fluid being heated (exiting top) has a higher exiting temperature than the cooled fluid (exiting bottom) that was used for heating. With cocurrent or parallel exchange the heated and cooled fluids can only approach one another. The result is that countercurrent exchange can achieve a greater amount of heat or mass transfer than parallel under otherwise similar conditions. See: flow arrangement. Counter heat current Countercurrent exchange when set up in a circuit or loop can be used for building up exchange: Note the concentrations, heat, or other properties of flowing liquids. Specifically when set up in a gradually declining loop with a buffering liquid between the incoming and outgoing fluid running in a differential and that the once hot and cold streams circuit, and with active transport pumps on the outgoing fluids tubes, the system is called exit at the reversed a Countercurrent multiplier, enabling a multiplied effect of many small pumps to temperature difference; the gradually build up a large concentration in the buffer liquid. hotter entering stream becomes the exiting cooler stream and vice versa. Other countercurrent exchange circuits where the incoming and outgoing fluids touch each other are used for retaining a high concentration of a dissolved substance or for retaining heat, or for allowing the external buildup of the heat or concentration at one point in the system. Countercurrent exchange circuits or loops are found extensively in nature, specifically in biologic systems. In vertebrates, they are called a Rete mirabile, originally the name of an organ in fish gills for absorbing oxygen from the water. It is mimicked in industrial systems. Countercurrent exchange is a key concept in chemical engineering thermodynamics and manufacturing processes, for example in extracting sucrose from sugar beet roots. Countercurrent multiplication is a similar but different concept where liquid moves in a loop followed by a long length of movement in opposite directions with an intermediate zone. The tube leading to the loop passively building up a gradient of heat (or cooling) or solvent concentration while the returning tube has a constant small pumping action all along it, so that a gradual intensification of the heat or concentration is created towards the loop. Countercurrent multiplication has been found in the kidneys[1] as well as in many other biological organs.

Countercurrent exchange 58 Three current exchange systems Countercurrent exchange along with concurrent exchange and contra-current exchange comprise the mechanisms used to transfer some property of a fluid from one flowing current of fluid to another across a barrier allowing one way flow of the property between them. The property transferred could be heat, concentration of a chemical substance, or other properties of the flow. When heat is transferred, a thermally-conductive membrane is used Three topologies of countercurrent exchange systems between the two tubes, and when the concentration of a chemical substance is transferred a semipermeable membrane is used. Concurrent flow - half transfer In the concurrent flow exchange mechanism, the two fluids flow in the same direction. As the Concurrent and countercurrent exchange mechanisms diagram shows, a concurrent exchange system has a variable gradient over the length of the exchanger. With equal flows in the two tubes, this method of exchange is only capable of moving half of the property from one flow to the other, no matter how long the exchanger is. If each stream changes its property to be 50% closer to that of the opposite Concurrent and countercurrent exchange mechanisms streams inlet condition, exchange will The magnitude of the property to be exchanged, is represented by shading. The direction of transfer across the barrier is from the greater to the lesser magnitude.. stop when the point of equilibrium is reached, and the gradient has declined to zero. In the case of unequal flows, the equilibrium condition will occur somewhat closer to the conditions of the stream with the higher flow.

Countercurrent exchange 59 Concurrent flow examples A concurrent heat exchanger is an example of a concurrent flow exchange mechanism. Two tubes have a liquid flowing in the same direction. One starts off hot at 60 °C, the second cold at 20 °C. A thermoconductive membrane or an open section allows heat transfer between the two flows. The hot fluid heats the cold one, and the cold fluid cools down the warm one. The result is thermal equilibrium: Both fluids end up at around the same temperature: 40 °C, almost exactly between the two original temperatures (20 and 60 °C). At the input end, there is a large temperature difference of 40 °C and much heat transfer; at the output end, there is a very small temperature difference (both Concurrent and Countercurrent heat exchange are at the same temperature of 40 °C or close to it), and very little heat transfer if any at all. If the equilibrium - where both tubes are at the same temperature - is reached before the exit of the liquid from the tubes, no further heat transfer will be achieved along the remaining length of the tubes. A similar example is the concurrent concentration exchange. The system consists of two tubes, one with brine (concentrated saltwater), the other with freshwater (which has a low concentration of salt in it), and a semi permeable membrane which allows only water to pass between the two, in an osmotic process. Many of the water molecules pass from the freshwater flow in order to dilute the brine, while the concentration of salt in the freshwater constantly grows (since the salt is not leaving this flow, while water is). This will continue, until both flows reach a similar dilution, with a concentration somewhere close to midway between the two original dilutions. Once that happens, there will be no more flow between the two tubes, since both are at a similar dilution and there is no more osmotic pressure. Countercurrent flow - almost full transfer In countercurrent flow, the two flows move in opposite directions. Two tubes have a liquid flowing in opposite directions. The top tube starts off hot, the bottom cold. The system can maintain a nearly constant gradient between the two flows over their entire length. With a sufficiently long length and a sufficiently low flow rate this can result in almost all of the property heat transferred. Spiral counter-current heat Countercurrent flow examples exchange schematic In a countercurrent heat exchanger, the hot fluid becomes cold, and the cold fluid becomes hot. In this example, hot water at 60 °C enters the top pipe. It warms water in the bottom pipe which has been warmed up along the way, to almost 60 °C. A minute but existing heat difference still exists, and a small amount of heat is transferred, so that the water leaving the bottom pipe is at close to 60 °C. Because the hot input is at its maximum temperature of 60 °C, and the exiting water at the bottom pipe is nearly at that temperature but not quite, the water in the top pipe can warm the one in the bottom pipe to nearly its own temperature. At the cold end - the water exit from the top pipe, because the cold water entering the bottom pipe is still cold at 20 °C, it can extract the last of the heat from the now-cooled hot water in the top pipe, bringing its temperature down nearly to the level of the cold input fluid (21 °C).

Countercurrent exchange 60 The result is that the top pipe which received hot water, now has cold water leaving it at 20 °C, while the bottom pipe which received cold water, is now emitting hot water at close to 60 °C. In effect, most of the heat was transferred. Conditions for higher transfer results It should be noted that nearly complete transfer in systems implementing countercurrent exchange, is only possible if the two flows are, in some sense, "equal". For a maximum transfer of substance concentration, an equal flowrate of solvents and solutions is required. For maximum heat transfer, the average specific heat capacity and the mass flow rate must be the same for each stream. If the two flows are not equal, for example if heat is being transferred from water to air or vice-versa, then, similar to concurrent exchange systems, a variation in the gradient is expected because of a buildup of the property not being transferred properly.[2] Countercurrent exchange in biological systems Countercurrent exchange in biological systems was discovered and studied following the discovery of Countercurrent multiplication systems by Werner Kuhn. Countercurrent exchange is used extensively in biological systems for a wide variety of purposes. For example, fish use it in their gills to transfer oxygen from the surrounding water into their blood, and birds use a countercurrent heat exchanger between blood vessels in their legs to keep heat concentrated within their bodies. In vertebrates this type of organ is referred to as a Rete mirabile (originally the name of the organ in the fish gills). Mammalian kidneys use countercurrent exchange to Rete mirabile = RM remove water from urine so the body can retain water used to move the nitrogenous waste products (see Countercurrent multiplier). Countercurrent multiplier A countercurrent multiplier is a system where fluid flows in a loop so that the entrance and exit are at similar low concentration of a dissolved substance but at the tip of the loop there is a very high concentration of that substance. A buffer liquid between the incoming and outgoing tubes receives the concentrated substance. The incoming and outgoing tubes do not touch each other. The system allows the buildup of a high concentration Counter current multiplier diagram gradually, with the use of many active transport pumps each pumping only against a very small gradient. Theoretically a similar system could exist or be constructed for heat exchange. The incoming flow starting at a low concentration has a semipermeable membrane with water passing to the buffer liquid via osmosis at a small gradient. There is a gradual buildup of concentration inside the loop until the loop tip where it reaches its maximum. In the example image water enters at 299 mg/L (NaCL / H2O). Water passes because of a small osmotic pressure to the buffer liquid in this example at 300 mg/L (NaCL / H2O). Further up the loop there is a continued flow of water out of the tube and into the buffer, gradually raising the concentration of NaCL in the tube until it reaches 1199 mg/L

Countercurrent exchange 61 at the tip. The buffer liquid between the two tubes is at a gradually rising concentration, always a bit over the incoming fluid, in our example reaching 1200 mg/L. This is regulated by the pumping action on the returning tube as explained immediately. The tip of the loop has the highest concentration of salt (NaCL) in the incoming tube - in the example 1199 mg/L, and in the buffer 1200 mg/L. The returning tube has active transport pumps, pumping salt out to the buffer liquid at a low difference of concentrations of up to 200 mg/L more than in the tube. Thus when opposite the 1000 mg/L in the buffer liquid, the concentration in the tube is 800 and only 200 mg/L are needed to be pumped out. But the same is true anywhere along the line, so that at exit of the loop also only 200 mg/L need to be pumped. In effect, this can be seen as a gradually multiplying effect - hence the name of the phenomena: a countercurrent multiplier or the mechanism: Countercurrent multiplication. In the kidney A circuit of fluid in the Loop of Henle - an important part of the kidneys allows for gradual buildup of the concentration of urine in the kidneys, by using active transport on the exiting nephrons (tubules carrying liquid in the process of gradually concentrating the urea). The active transport pumps need only to overcome a constant and low gradient of concentration, because of the countercurrent multiplier mechanism[3] Various substances are passed from the liquid entering the Nephrons until exiting the loop (See the Nephron flow diagram). The sequence of flow is as follows: • Renal corpuscle: Liquid enters the nephron system at the Bowmans capsule.[4] • Proximal convoluted tubule: It then may reabsorb urea in the thick descending limb.[5] Water is removed from the nephrons by osmosis (and Glucose and other ions are pumped out with active transport), gradually raising the concentration in the nephrons.[6] • Loop of Henle Descending: The liquid passes from the thick descending limb to the thin one.

Countercurrent exchange 62 Water is constantly released via osmosis.[7] Gradually there is a buildup of osmotic concentration, until 1200 mOsm is reached at the loop tip, but the difference across the membrane is kept small and constant. For example, the liquid at one section inside the thin descending limb is at 400 mOsm while outside its 401. Further down the descending limb, the inside concentration is 500 while outside it is 501, so a constant difference of 1 mOsm is kept all across the membrane, although the concentration inside and outside are gradually increasing. Loop of Henle (Greys Anatomy book) • Loop of Henle Ascending: after the tip (or bend) of the loop, the liquid flows in the thin ascending limb.[8] Salt - Sodium Na+ and Chlorine Cl- ions are pumped out of the liquid[9] gradually lowering the concentration in the exiting liquid, but, using the countercurrent multiplier mechanism, always pumping against a constant and small osmotic difference. For example, the pumps at a section close to the bend, pump out from 1000 mOsm inside the ascending limb to 1200 mOsm outside it, with a 200 mOsm across. Pumps further up the thin ascending limb, pump out from 400 mOsm into liquid at 600 mOsm, so again the difference is retained at 200 mOsm from the inside to the outside, while the concentration both inside and outside are gradually decreasing as the liquid flow advances. The liquid finally reaches a low concentration of 100 mOsm when leaving the thin ascending limb and passing through the thick one[10] • Distal convoluted tubule: Once leaving the loop of Henle the thick ascending limb can optionally reabsorb and re increase the concentration in the nephrons.[11] • Collecting duct: The collecting duct receives liquid between 100 mOsm if no re-absorption is done, to 300 or above if re-absorption was used. The collecting duct may continue raising the concentration if required, by gradually pumping out the same ions as the Distal convoluted tubule, using the same gradient as the ascending limbs in the loop of Henle, and reaching the same concentration.[12] • Ureter: The liquid urine leaves to the Ureter.

Countercurrent exchange 63 History Initially the countercurrent exchange mechanism and its properties were proposed in 1951 by professor Werner Kuhn and two of his former students who called the mechanism found in the Loop of Henle in mammalian kidneys a Countercurrent multiplier[13] and confirmed by laboratory findings in 1958 by Professor Carl W. Gottschalk.[14] The theory was acknowledged a year later after a meticulous study showed that there is almost no osmotic difference between liquids on both sides of nephrons.[15] Homer Smith, a considerable contemporary authority on renal physiology, opposed the model countercurrent concentration for 8 years, until conceding ground in 1959. [16] Ever since, many similar mechanisms have been found in biologic systems, the most notable of these: the Rete mirabile in fish. Countercurrent exchange of heat in organisms Countercurrent heat exchange (CCHE) is a highly efficient means of minimizing heat loss through the skins surface because heat is recycled instead of being dissipated. This way, the heart does not have to pump blood as rapidly in order to maintain a constant body core temperature and thus, metabolic rate. CCHE is used in animals living in extreme conditions of cold or hot weather as a mechanism for retaining the heat in (or out of) the body. These are countercurrent exchange systems with the same fluid, usually blood, in a circuit, used for both directions of flow. When animals like the leatherback turtle and dolphins are in colder water to which they are not acclimatized, they use this CCHE mechanism. Such CCHE systems are made up of a complex network of peri-arterial venous plexuses that run from the heart and through the blubber to peripheral sites (i.e. the tail flukes, dorsal fin and pectoral fins). Each plexus consists of a singular artery containing warm blood from the heart surrounded by a bundle of veins containing cool blood from the body surface. As these fluids flow past each other, they create a heat gradient in which heat is transferred and retained inside the body. The warm arterial blood transfers most of its heat to the cool venous blood now coming in from the outside. This conserves heat by recirculating it back to the body core. Since the arteries give up a good deal of their heat in this exchange, there is less heat lost through convection at the periphery surface.[17] Another example is found in the legs of an arctic fox treading on snow. The paws are necessarily cold, but blood can circulate to bring nutrients to the paws without losing much heat from the body. Proximity of arteries and veins in the leg results in heat exchange, so that as the blood flows down it becomes cooler, and doesnt lose much heat to the snow. As the (cold) blood flows back up from the paws through the veins, it picks up heat from the blood flowing in the opposite direction, so that it returns to the torso in a warm state, allowing the fox to maintain a comfortable temperature, without losing it to the snow. Countercurrent exchange in sea and desert birds to conserve water Sea and desert birds have been found to have a salt gland near the nostrils which concentrates brine, later to be "sneezed" out to the sea, in effect allowing these birds to drink seawater without the need to find freshwater resources. It also enables the seabirds to remove the excess salt entering the body when eating, swimming or diving in the sea for food. The kidney cannot remove these quantities and concentrations of salt. The salt secreting gland has been found in seabirds like pelicans, petrels, albatrosses, gulls, and terns. It has also been found in Namibian ostriches and other desert birds, where a buildup of salt concentration is due to dehydration and scarcity of drink water. In seabirds the salt gland with its countercurrent exchange mechanism work as follows: a. Salt enters the blood plasma from the intestines, so it is initially at a high concentration (but less than the concentration in the sea water)

Countercurrent exchange 64 b. Osmosis causes the extra-cellular fluid volume (ECFV) to rise, with large amounts of water exit the cells, while lowering the salt concentration of the extra-cellular fluid. c. A section of the salt gland further raises concentration of salt in the blood plasma. This is accomplished in a small area of the salt gland, by Active transport (moving the salt against its concentration gradient, using energy from ATP breakdown). d. In order to preserve the extremely high concentration of salt in the gland, blood plasma entering the gland and exiting it is passed through a counter current-exchange circuit, so that the high concentration of salt in plasma leaving the gland is returned to it in the incoming plasma. This system preserves the high concentration in the gland, without losing it to the bodys blood system, allowing the gland to build up the salt concentration to ever higher concentrations, by pumping salt molecules in with active transport. The anatomy of the countercurrent exchange mechanism in the salt gland is as follows: The gland has an area where salt is pumped into its tubules using active transport reaching extremely high concentrations. at the end of the gland there is an area with tubules with low salt concentration. The fluid moves in a counter-current exchange circuit so that although high concentrations leave the gland, they are constantly returned to it. Tiny arteries carrying salty blood enter the gland and are very closely juxtaposed to the tubules with low concentration, slightly adding to its concentration in the exiting tubules. This extra concentration is retrieved to the returning tubules, so that there is a continued buildup of concentration at the gland top, but a constant low concentration at the gland end, where the arteries are passing their salt to the gland. Veinules (small veins) leave the gland with a continued decreasing concentration of salt in the blood. The salty liquid thus collected into the glands is periodically sneezed out from the nostrils. The glands remove the salt efficiently and thus allow the birds to drink the salty water from their environment while they are hundreds of miles away from land.[18][19] Countercurrent exchange in industrial and scientific systems Countercurrent Chromatography is a method of separation, that is based on the differential partitioning of analytes between two immiscible liquids using countercurrent or cocurrent flow.[20] Evolving from Craigs Countercurrent Distribution (CCD), the most widely used term and abbreviation is CounterCurrent Chromatography or CCC,[21] in particular when using hydrodynamic CCC instruments. The term partition chromatography is largely a synonymous and predominantly used for hydrostatic CCC instruments. Hardendale Lime Works in the UK using countercurrent kilns to reach high temperatures

Countercurrent exchange 65 • Distillation of chemicals such as in petroleum refining is done towers or columns with perforated trays. Vapor from the low boiling fractions bubbles upward through the holes in the trays in contact with the down flowing high boiling fractions. The concentration of low boiling fraction increases in each tray up the tower as it is "stripped". The low boiling fraction is drawn off the top of the tower and the high boiling fraction drawn from the bottom. The process in the trays is a combination of heat transfer and mass transfer. Heat is supplied at the bottom, known as a "reboiler" and cooling is done at the with a condenser at the top. • Liquid-liquid extraction (also called solvent extraction or partitioning is a common method for extracting a substance from one liquid into another liquid at Counter flow in liquid-liquid a different phase (such as "slurry"). This method, which implements a extraction countercurrent mechanism, is used in nuclear reprocessing, ore processing, the production of fine organic compounds, the processing of perfumes, the production of vegetable oils and biodiesel, and other industries. • Gold can be separated from a cyanide solution with the Merrill-Crowe process using Counter Current Decantation (CCD). In some mines, Nickel and Cobalt are treated with CCD, after the original ore was treated with concentrated Sulfuric acid and steam in Titanium covered autoclaves, producing nickel cobalt slurry. The nickel and cobalt in the slurry are removed from it almost completely using a CCD system exchanging the cobalt and nickel with flash steam heated water. • Lime can be manufactured in countercurrent furnaces allowing the heat to reach high temperatures using low cost, low temperature burning fuel. Historically this was developed by the Japanese in certain types of the Anagama Countercurrent furnace (kiln) heat exchange kiln. The kiln is built in stages, where fresh air coming to the fuel is passed downwards while the smoke and heat is pushed up and out. The heat does not leave the kiln, but is transferred back to the incoming air, and thus slowly builds up to 3000 °C and more. • Cement may be created using a countercurrent kiln where the heat is passed in the cement and the exhaust combined, while the incoming air draft is passed along the two, absorbing the heat and retaining it inside the furnace, finally reaching high temperatures. Cement counter-current rotary kiln • Gasification - the process of creating methane and carbon monoxide from organic or fossil matter, can be done using a Counter-current fixed bed ("up draft") gasifier which is built in a similar way to the Anagama kiln, and must therefore withstand more harsh conditions, but reaches better efficiency. • In nuclear power plants, water leaving the plant must not contain even trace particles of Uranium. Counter Current Decantation (CCD) is used in some facilities to extract water, totally clear of Uranium.

Countercurrent exchange 66 • Some Centrifugal extractors use counter current exchange mechanisms for extracting high rates of the desired material. • Some protein skimmers - a device to clean saltwater pools and fish ponds of organic mater - use counter current technologies. External links • Countercurrent multiplier animation Exchange current decantation depicted in centrifugal extractors as 1st stage [22] from Colorado University. • Research about elephant seals [23] using countercurrent heat exchange to keep heat from leaving their body while breathing out, during hibernation. • Patent for a snow mask with a removable countercurrent exchange module [24] which keeps the warmth from leaving the mask when breathing out. • An industrial system for aerating waste water [25] and sewage which works on the countercurrent exchange principle, without pipes. Air bubbles floating upwards meet water in a down current, causing more of the air to dissolve.[26] References [1] Both countercurrent exchange and countercurrent multiplication systems have been found in the kidneys. The latter in the loop of Henle, the first in the vasa recta [2] The specific heat capacity should be calculated on a mass basis, averaged over the temperature range involved. This is in keeping with the second law of thermodynamics [3] See the countercurrent multiplier animation (http:/ / www. colorado. edu/ intphys/ Class/ IPHY3430-200/ countercurrent_ct. swf) at the Colorado University website. [4] Beginning with the Afferent arteriole, a blood vessel leading to the Glomerulus, filtered blood is passed to the nephrons in the Bowmans capsule which surrounds the Glomerulus. (The blood leaves the Glomerulus in the Efferent arteriole). [5] The liquid from the Bowmans capsule reaches the thick descending limb. Urea may be reabsorbed into the low (300 mOsm) osmotic concentration in the limb nephrons. The urea absorption in the thick descending limb is inhibited by Sartans and catalyzed by lactates and ketones. [6] Glucose, Amino acids, various ions and organic material leave the limb, gradually raising the concentration in the nephrons. Dopamin inhibits the secretion from the thick descending limb, and Angiotensin II catalyzes it [7] The semipermeable membrane of the thin descending limb does not permit passage of ions or large dissolved molecules [8] The thin ascending limbs membrane does not permit free passage of any substance including water. [9] Furosemide inhibits salt secretion from the thin ascending limb, while Aldosterone catalyzes the secretion. [10] Water or liquid with very low osmotic concentration leaving the nephrons is reabsorbed in the Peritubular capillaries and returned to the blood. [11] Reabsorbing and increasing the concentration is done by optionally absorbing Potassium (K+) and Hydrogen (H+) anions, while releasing water and the continued pumping out of Calcium (Ca+) and salt (Na+ and Chlorine Cl- ions). The repeated concentration by secretion of Calcium and salt ions is inhibited by Thiazides and catalyzed by Anti diuretic hormone and Aldosterone [12] Atial natiuretic peptide and Urodiatin inhibit water salt and calcium secretion from the collecting duct, while Antiduretic hormone and Aldosterone catalyze it. [13] The original lecture was published in 1951 in German. According to a book on Jewish scientists under the Reich (http:/ / www. isbnlib. com/ preview/ 1859734219/ ) Kuhn theorized and studied this mechanism already in the early 1940s. This was confirmed in 2001 in the translation to the original lecture (http:/ / jasn. asnjournals. org/ content/ 12/ 7/ 1566. full. pdf) published with remarks by Professor Bart Hargitay, then one of the two former student aids. Harbitay says: Before settling in Basel, Kuhn did some very fundamental work in Kiel, separating isotopes in a centrifuge. This caused him to be fascinated with the effect of countercurrents in multiplying a very small single effect to significant separations. (Journal of the American Society of Nephrology website)

Countercurrent exchange 67 [14] Gottschalk, C. W.; Mylle, M. (1958), "Evidence that the mammalian nephron functions as a countercurrent multiplier system", Science 128 (3324): 594, doi:10.1126/science.128.3324.594, PMID 13580223. [15] Gottschalk, C. W.; Mylle, M. (1959), "Micropuncture study of the mammalian urinary concentrating mechanism: evidence for the countercurrent hypothesis" (http:/ / ajplegacy. physiology. org/ cgi/ content/ abstract/ 196/ 4/ 927), American Journal of Physiology 196 (4): 927–936, PMID 13637248, . See also History of the urinary concentrating mechanism (http:/ / www. nature. com/ ki/ journal/ v31/ n2/ abs/ ki198729a. html) an article in Kidney - the Journal of International Society of Nephrology, where Prof. Gottschalk points to the heated debate prior to the acceptance of the theory of the countercurrent multiplier action of the kidney [16] Smith, Homer W., The fate of sodium and water in the renal tubules, Bull. NY Acad. Med. 35:293–316, 1959. [17] Animal physiology at Davidson (http:/ / www. bio. davidson. edu/ people/ midorcas/ animalphysiology/ websites/ 2004/ Harden/ page3. htm) [18] Proctor, Noble S.; Lynch, Patrick J. (1993). Manual of Ornithology. Yale University Press. [19] Ritchison, Gary. "Avian osmoregulation" (http:/ / people. eku. edu/ ritchisong/ bird_excretion. htm). . Retrieved 16 April 2011. [20] "TheLiquidPhase" (http:/ / www. theliquidphase. org/ ). . Retrieved 16 April 2011. [21] "Countercurrent Chromatography" (http:/ / tigger. uic. edu/ ~gfp/ countercurrent/ index2. htm). University of Illinois at Chicago. . Retrieved 16 April 2011. [22] http:/ / www. colorado. edu/ intphys/ Class/ IPHY3430-200/ countercurrent_ct. swf [23] http:/ / jeb. biologists. org/ content/ 113/ 1/ 447. full. pdf [24] http:/ / www. google. com/ patents?id=O5ktAAAAEBAJ& printsec=frontcover& dq=Mask+ with+ removable+ countercurrent+ exchange+ module& hl=en& ei=48oUTvzVLMTRhAe9n9HGDQ& sa=X& oi=book_result& ct=result& resnum=1& ved=0CCgQ6AEwAA [25] http:/ / www. diffusaire. com [26] According to the company, almost half of the electricity in the US is used to aerate sewage and wastewater. The countercurrent exchange method saves up to 50% of the electricity Critical heat flux Critical heat flux describes the thermal limit of a phenomenon where a phase change occurs during heating (such as bubbles forming on a metal surface used to heat water), which suddenly decreases the efficiency of heat transfer, thus causing localised overheating of the heating surface. The Critical heat flux for ignition is the lowest thermal load per unit area capable of initiating a combustion reaction on a given material (either flame or smoulder ignition). Description When liquid coolant undergoes a change in phase due to the absorption of heat from a heated solid surface, a higher transfer rate occurs. The more efficient heat transfer from the heated surface (in the form of heat of vaporization plus sensible heat) and the motions of the bubbles (bubble-driven turbulence and convection) leads to rapid mixing of the fluid. Therefore, boiling heat transfer has played an important role in industrial heat transfer processes such as macroscopic heat transfer exchangers in nuclear and fossil power plants, and in microscopic heat transfer devices such as heat pipes and microchannels for cooling electronic chips. The use of boiling is limited by a condition called critical heat flux (CHF), which is also called a boiling crisis or departure from nucleate boiling (DNB). The most serious problem is that the boiling limitation can be directly related to the physical burnout of the materials of a heated surface due to the suddenly inefficient heat transfer through a vapor film formed across the surface resulting from the replacement of liquid by vapor adjacent to the heated surface. Consequently, the occurrence of CHF is accompanied by an inordinate increase in the surface temperature for a surface-heat-flux-controlled system. Otherwise, an inordinate decrease of the heat transfer rate occurs for a surface-temperature-controlled system. This can be explained with Newtons law of cooling: where represents the heat flux, represents the heat transfer coefficient, represents the wall temperature and represents the fluid temperature. If decreases significantly due to the occurrence of the CHF condition,

Critical heat flux 68 will increase for fixed and while will decrease for fixed . Correlations for critical heat flux The critical heat flux is an important point on the boiling curve and it may be desirable to operate a boiling process near this point. However, one could become cautious of dissipating heat in excess of this amount. Zuber, through a hydrodynamic stability analysis of the problem has developed an expression to approximate this point. It is independent of the surface material and is weakly dependent upon the heated surface geometry described by the constant C. For large horizontal cylinders, spheres and large finite heated surfaces, the value of the Zuber constant . For large horizontal plates, a value of is more suitable. The critical heat flux depends strongly on pressure, mainly through the pressure dependence of surface tension and the heat of vaporization.[1] Applications in heat transfer The understanding of CHF phenomenon and an accurate prediction of the CHF condition are important for safe and economic design of many heat transfer units including nuclear reactors, fossil fuel boilers, fusion reactors, electronic chips, etc. Therefore, the phenomenon has been investigated extensively over the world since Nukiyama first characterized it.[2] In 1950 Kutateladze suggested the hydrodynamical theory of the burnout crisis. [3] Much of significant work has been done during the last decades with the development of water-cooled nuclear reactors. Now many aspects of the phenomenon are well understood and several reliable prediction models are available for conditions of common interests. Terminology A number of different terms are used to denote the CHF condition: departure from nucleate boiling (DNB), liquid film dryout (LFD), annular film dryout (AFD), dryout (DO), burnout (BO), boiling crisis (BC), boiling transition (BT), etc. DNB, LFD and AFD represent specific mechanisms which will be introduced later. DO means the disappearance of liquid on the heat transfer surface which properly describes the CHF condition; however, it is usually used to indicate the liquid film dryout from annular flow. BO, BC and BT are phenomenon-oriented names and are used as general terms. The CHF condition (or simply the CHF) is the most widely used today, though it may mislead one to think that there exists a criticality in the heat flux. The terms denoting the value of heat flux at the CHF occurrence are CHF, dryout heat flux, burnout heat flux, maximum heat flux, DNB heat flux, etc. The term peak pool boiling heat flux is also used to denote the CHF in pool boiling.

Critical heat flux 69 References [1] Fundamentals of Heat and Mass Transfer 6th Edition by Incropera. [2] Nukiyama, S. (1934). "Film boiling water on thin wires". Soc. Mech. Engng., Japan 37. [3] Kutateladze, S.S. (1950). "Hydromechanical model of the crisis of boiling under conditions of free convection". Journal of Technical Physics, USSR 20 (11): 1389–1392. External links • Modeling of the boiling crisis (http://www.pmmh.espci.fr/~vnikol/boiling_crisis.html) Dew point The dew point is the temperature below which the water vapor in a volume of humid air at a constant barometric pressure will condense into liquid water. Condensed water is called dew when it forms on a solid surface. The dew point is a water-to-air saturation temperature. The dew point is associated with relative humidity. A high relative humidity indicates that the dew point is closer to the current air temperature. Relative humidity of 100% indicates the dew point is equal to the current temperature and that the air is maximally saturated with water. When the dew point remains constant and temperature increases, relative humidity decreases.[1] General aviation pilots use dew-point data to calculate the likelihood of carburetor icing and fog, and to estimate the height of the cloud base. At a given temperature but independent of barometric pressure, the dew point is a consequence of the absolute humidity, the mass of water per unit volume of air. If both the temperature and pressure rise, however, the dew point will rise and the relative humidity will lower accordingly. Reducing the absolute humidity without changing other variables will bring the dew point back down to its initial value. In the same way, increasing the absolute humidity after a temperature drop brings the dew point back down to its initial level. If the temperature rises in conditions of constant pressure, then the dew point will remain constant but the relative humidity will drop. For this reason, a constant relative humidity (%) with different temperatures implies that when its hotter, a higher fraction of the air is water This graph shows the maximum percentage, by mass, of water vapor that air at vapor than when its cooler. sea-level across a range of temperatures can contain. At a given barometric pressure but independent of temperature, the dew point indicates the mole fraction of water vapor in the air, or, put differently, determines the specific humidity of the air. If the pressure rises without changing this mole fraction, the dew point will rise accordingly; Reducing the mole fraction, i.e., making the air less humid, would bring the dew point back down to its initial value. In the same way, increasing the mole fraction after a pressure drop brings the relative humidity back up to its initial level. Considering New York (33 ft elevation) and Denver (5,280 ft elevation),[2] for

Dew point 70 example, this means that if the dew point and temperature in both cities are the same, then the mass of water vapor per cubic meter of air will be the same, but the mole fraction of water vapor in the air will be greater in Denver. Relationship to human comfort When the air temperature is high, the bodys thermoregulation uses evaporation of perspiration to cool down, with the cooling effect directly related to how fast the perspiration evaporates. The rate at which perspiration can evaporate depends on how much moisture is in the air and how much moisture the air can hold. If the air is already saturated with moisture, perspiration will not evaporate. The bodys cooling system will produce perspiration in an effort to keep the body at its normal temperature even when the rate it is producing sweat exceeds the evaporation rate. So even without generating additional body heat by exercising, one can become coated with sweat on humid days. It is the unevaporated sweat that tends to make one feel uncomfortable in humid weather. As the air surrounding ones body is warmed by body heat, it will rise and be replaced with other air. If air is moved away from ones body with a natural breeze or a fan, sweat will evaporate faster, making perspiration more effective at cooling the body. The more unevaporated perspiration, the greater the discomfort. A wet bulb thermometer also uses evaporative cooling, so it provides a good analog for use in evaluating comfort level. Discomfort also exists when the dew point is low (below around −30 °C (−22 °F)). The drier air can cause skin to crack and become irritated more easily. It will also dry out the respiratory paths. OSHA recommends indoor air be maintained at 68 to 76°F (20 to 24.5°C) with a 20-60% relative humidity (a dew point of 24 to 60°F).[3] Lower dew points, less than 10 °C (50 °F), correlate with lower ambient temperatures and the body requires less cooling. A lower dew point can go along with a high temperature only at extremely low relative humidity (see graph below), allowing for relative effective cooling. Those accustomed to continental climates often begin to feel uncomfortable when the dew point reaches between 15 and 20 °C (59 and 68 °F). Most inhabitants of these areas will consider dew points above 21 °C (70 °F) oppressive. Dew point °C Dew point °F [1] Rel. humidity at 32 °C (90 °F) Human perception > Higher than 26 °C > Higher than 80 °F Severely high. Even deadly for asthma related illnesses 65% and higher 24–26 °C 75–80 °F Extremely uncomfortable, fairly oppressive 62% 21–24 °C 70–74 °F Very humid, quite uncomfortable 52–60% 18–21 °C 65–69 °F Somewhat uncomfortable for most people at upper edge 44–52% 16–18 °C 60–64 °F OK for most, but all perceive the humidity at upper edge 37–46% 13–16 °C 55–59 °F Comfortable 38–41% 10–12 °C 50–54 °F Very comfortable 31–37% < 10 °C < 49 °F A bit dry for some 30% Measurement Devices called dew point meters are used to measure dew point over a wide range of temperatures. These devices consist of a polished metal mirror which is cooled as air is passed over it. The temperature at which dew forms is, by definition, the dew point. Manual devices of this sort can be used to calibrate other types of humidity sensors, and automatic sensors may be used in a control loop with a humidifier or dehumidifier to control the dew point of the air in a building or in a smaller space for a manufacturing process.

Dew point 71 Calculating the dew point A well-known approximation used to calculate the dew point, Tdp, given just the actual ("dry bulb") air temperature, T and relative humidity (in percent), RH, is the Magnus formula: The more complete formulation and origin of this approximation involves the interrelated saturated water vapor pressure (in units of millibar, which is also hPa) at T, Ps(T), and the actual water vapor pressure (also in units of millibar), Pa(T), which can be either found with RH or approximated with the barometric pressure (in millibar units), BPmb, and "wet-bulb" temperature, Tw is: Note: unless declared otherwise, all temperatures are expressed and worked in degrees Celsius For greater accuracy, Ps(T) (and, therefore, γ(T,RH)) can be enhanced, using part of the Bögel modification, also known as the Arden Buck equation, which adds a fourth, d constant:

Dew point 72 (where ) There are several different constant sets in use, the ones used in NOAAs presentation [4] are taken from a 1980 paper by David Bolton in the Monthly Weather Review[5]: These valuations provide a minimum accuracy of 0.1%, for -30°C ≤ T ≤ +35°C; 1% < RH < 100%; [6] Also noteworthy is the Sonntag1990, Another common set of values originates from the 1974 Psychrometry and Psychrometric Charts, as presented by Paroscientific,[7] Also, in the Journal of Applied Meteorology and Climatology,[8] Arden Buck presents several different valuation sets, with different minimum accuracies for different temperature ranges. Two particular sets provide a range of -40°C → +50°C between the two, with even greater minimum accuracy than all of the other, above sets (maximum error at given |C°| extreme): Simple approximation There is also a very simple approximation that allows conversion between the dew point, temperature and relative humidity. This approach is accurate to within about ±1°C as long as the relative humidity is above 50%: and This can be expressed as a simple rule of thumb: For every 1°C difference in the dew point and dry bulb temperatures, the relative humidity decreases by 5%, starting with RH = 100% when the dew point equals the dry bulb temperature. The derivation of this approach, a discussion of its accuracy, comparisons to other approximations, and more information on the history and applications of the dew point are given in the Bulletin of the American Meteorological Society.[9] For temperatures in degrees Fahrenheit, these approximations work out to and

Dew point 73 For example, a relative humidity of 100% means dew point is the same as air temp. For 90% RH, dew point is 3 degrees Fahrenheit lower than air temp. For every 10 percent lower, dew point drops 3 °F. Frost point The frost point is similar to the dew point, in that it is the temperature to which a given parcel of humid air must be cooled, at constant barometric pressure, for water vapor to be deposited on a surface as ice without going through the liquid phase. (Compare with sublimation.) The frost point for a given parcel of air is always higher than the dew point, as the stronger bonding between water molecules on the surface of ice requires higher temperature to break.[10] References [1] Horstmeyer, Steve (2006-08-15). "Relative Humidity....Relative to What? The Dew Point Temperature...a better approach" (http:/ / www. shorstmeyer. com/ wxfaqs/ humidity/ humidity. html). Steve Horstmeyer, Meteorologist, WKRC TV, Cincinnati, Ohio, USA. . Retrieved 2009-08-20. [2] "Denver Facts Guide – Today" (http:/ / www. denvergov. org/ AboutDenver/ today_factsguide. asp). The City and County of Denver. . Retrieved March 19, 2007. [3] http:/ / www. osha. gov/ pls/ oshaweb/ owadisp. show_document?p_table=INTERPRETATIONS& p_id=24602 [4] http:/ / www. srh. noaa. gov/ images/ epz/ wxcalc/ rhTdFromWetBulb. pdf Relative Humidity and Dewpoint Temperature from Temperature and Wet-Bulb Temperature [5] "The computation of equivalent potential temperature", Monthly Weather Review (https:/ / www. rsmas. miami. edu/ users/ pzuidema/ Bolton. pdf), vol.108, pg.1047, Eq.10 [6] SHTxx Application Note Dew-point Calculation (http:/ / irtfweb. ifa. hawaii. edu/ ~tcs3/ tcs3/ Misc/ Dewpoint_Calculation_Humidity_Sensor_E. pdf) [7] MET4 AND MET4A CALCULATION OF DEW POINT (http:/ / www. paroscientific. com/ dewpoint. htm) [8] Buck, A. L. (1981), "New equations for computing vapor pressure and enhancement factor" (http:/ / www. public. iastate. edu/ ~bkh/ teaching/ 505/ arden_buck_sat. pdf), J. Appl. Meteorol. 20: 1527–1532 [9] M. G. Lawrence, "The relationship between relative humidity and the dew point temperature in moist air: A simple conversion and applications", Bull. Am. Meteorol. Soc., 86, 225–233, 2005 [10] Haby, Jeff. "Frost point and dew point" (http:/ / www. theweatherprediction. com/ habyhints/ 347/ ). . Retrieved September 30, 2011. External links • What is the dew point? (http://weathersavvy.com/Q-dew_point1.html) • Dew point definition (http://weather.gov/glossary/index.php?word=dew+point) NOAA Glossary • Dew point formula (http://www.paroscientific.com/dewpoint.htm) • Often Needed Answers about Temp, Humidity & Dew Point (http://www.faqs.org/faqs/meteorology/ temp-dewpoint/) from the sci.geo.meteorology • Humidity calculator (http://www.humidity-calculator.com/index.php) • Dew point calculator in construction (http://www.dew-point.us)

Dynamic scraped surface heat exchanger 74 Dynamic scraped surface heat exchanger The dynamic scraped surface heat exchanger (DSSHE) was designed to face some problems found in other types of heat exchangers. They increase heat transfer by: • removing the fouling layers, • increasing turbulence in case of high viscosity flow, • avoiding the generation of ice and other process by-products. DSSHEs incorporate an internal mechanism which periodically removes the product from the heat transfer wall. Introduction The most important technologies for indirect heat transfer use tubes (shell-and-tube exchangers) or flat surfaces (plate exchangers). Their goal is to exchange the maximum amount of heat per unit area by generating as much turbulence as possible below given pumping power limits. Typical approaches to achieve this consist of corrugating the tubes or plates or extending their surface with fins. However, these geometry conformation technologies, the calculation of optimum mass flows and other turbulence related factors become diminished when fouling appears, obliging designers to fit significantly larger heat transfer areas. There are several types of fouling, including particulate accumulation, precipitation (crystallization), sedimentation, generation of ice layers, etc. Another factor posing difficulties to heat transfer is viscosity. Highly viscous fluids tend to generate deep laminar flow, a condition with very poor heat transfer rates and high pressure losses involving a considerable pumping power, often exceeding the exchanger design limits. This problem becomes worsened frequently when processing non-newtonian fluids. The dynamic scraped surface heat exchangers (DSSHE) have been designed to face the above mentioned problems. They increase heat transfer by: • removing the fouling layers, • increasing turbulence in case of high viscosity flow, • avoiding the generation of ice and other process by-products. Basic description The dynamic scraped surface heat exchangers incorporate an internal mechanism which periodically removes the product from the heat transfer wall. The product side is scraped by blades attached to a moving shaft or frame. The blades are made of a rigid plastic material to prevent damage to the scraped surface. This material is FDA approved in the case of food applications. Types There are basically three types of DSSHEs depending on the arrangement of the blades: 1. Rotating, tubular DSSHEs. The shaft is placed parallel to the tube axis, not necessarily coincident, and spins at various frequencies, from a few dozen rpm to more than 1000 rpm. The number of blades oscillates between 1 and 4 and may take advantage of centrifugal forces to scrape the inner surface of the tube. Examples are the Waukesha Cherry-Burrell Votator II and the Alfa Laval Contherm. 2. Reciprocating, tubular DSSHEs. The shaft is concentric to the tube and moves longitudinally without rotating. The frequency spans between 10 and 60 strokes per minute. The blades may vary in number and shape, from baffle-like arrangements to perforated disk configurations. An example is the HRS Heat Exchangers [1] Unicus.

Dynamic scraped surface heat exchanger 75 3. Rotating, plate DSSHEs. The blades wipe the external surface of circular plates arranged in series inside a shell. The heating/cooling fluid runs inside the plates. The frequency is about several dozen rpm. An example is the HRS Spiratube T-Sensation. Evaluation Computational fluid dynamics (CFD) techniques are the standard tools to analyse and evaluate heat exchangers and similar equipment. However, for quick calculation purposes, the evaluation of DSSHEs are usually carried out with the help of ad-hoc (semi)empirical correlations based on the Buckingham π theorem: Fa = Fa(Re, Re, n, ...) for pressure loss and Nu = Nu(Re, Re, Pr, Fa, L/D, N, ...) for heat transfer, where Nu is the Nusselt number, Re is the standard Reynolds number based on the inner diameter of the tube, Re is the specific Reynolds number based on the wiping frequency, Pr is the Prandtl number, Fa is the Fanning friction factor, L is the length of the tube, D is the inner diameter of the tube, n is the number of blades and the dots account for any other relevant dimensionless parameters. Applications The range of applications covers a number of industries, including food, chemical, petrochemical and pharmaceutical. The DSSHEs are appropriate whenever products are prone to fouling, very viscous, particulate, heat sensitive or crystallizing. References • Bott, T. R. (May 1966). Design of Scraped Surface Heat Exchangers. II, No.5. British Chemical Engineering. pp. 338–339. • Bott, T. R. (November 2001). To Foul or not to Foul. CEP Magazine. pp. 30–37. • Bott, T. R.; Romero, J. J. B. (October 1963). Heat Transfer Across a Scraped Surface. The Canadian Journal of Chemical Engineering. pp. 213–219. • Chong, A. (2001). A Study of Scraped-Surface Heat Exchanger in Ice-Making Applications, M. Sc. Thesis. University of Toronto. • "Scraped surface heat exchangers project webpage" [2]. Smith Institute. • Tähti, T. (2004). Suspension Melt Crystallization in Tubular and Scraped Surface Heat Exchangers, Ph. D. Thesis. Martin-Luther-Universität. • "Scraped-Surface Heat-Exchanger Project page" [3]. University of Southampton.

Dynamic scraped surface heat exchanger 76 External links • Fouling factors in heat exchangers [4] References [1] http:/ / www. hrs-he. com [2] http:/ / www. smithinst. ac. uk/ Projects/ PD/ RA-Chemtech/ [3] http:/ / www. personal. soton. ac. uk/ cpp/ sshe_proj/ homepage. html [4] http:/ / www. hrs-heatexchangers. com/ en/ resources/ fouling-factors-in-heat-exchangers. aspx Electromagnetic radiation Electromagnetic radiation (EM radiation or EMR) is a form of energy emitted and absorbed by charged particles, which exhibits wave-like behavior as it travels through space. EMR has both electric and magnetic The electromagnetic waves that compose electromagnetic radiation can be imagined as a field components, which stand in a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram fixed ratio of intensity to each other, shows a plane linearly polarized EMR wave propagating from left to right. The electric and which oscillate in phase field is in a vertical plane and the magnetic field in a horizontal plane. The two types of fields in EMR waves are always in phase with each other, and no matter how powerful, perpendicular to each other and have a ratio of electric to magnetic intensity which is fixed and never varies. perpendicular to the direction of energy and wave propagation. In a vacuum, electromagnetic radiation propagates at a characteristic speed, the speed of light. Electromagnetic radiation is a particular form of the more general electromagnetic field (EM field), which is produced by moving charges. Electromagnetic radiation is associated with EM fields that are far enough away from the moving charges that produced them, that absorption of the EM radiation no longer affects the behavior of these moving charges. These two types or behaviors of EM field are sometimes referred to as the near and far field. In this language, EMR is merely another name for the far-field. Charges and currents directly produce the near-field. However, charges and currents produce EMR only indirectly—rather, in EMR, both the magnetic and electric fields are produced by changes in the other type of field, not directly by charges and currents. This close relationship causes the electric and magnetic fields in EMR to stand in a fixed ratio of strengths to each other, and to be found in phase, with maxima and nodes in each found at the same places in space. EMR carries energy—sometimes called radiant energy—through space continuously away from the source (this is not true of the near-field part of the EM field). EMR also carries both momentum and angular momentum. These properties may all be imparted to matter with which it interacts. EMR is produced from other types of energy when created, and it is converted to other types of energy when it is destroyed. The photon is the quantum of the electromagnetic interaction, and is the basic "unit" or constituent of all forms of EMR. The quantum nature of light becomes more apparent at high frequencies (or high photon energy). Such photons behave more like particles than lower-frequency photons do. In classical physics, EMR is considered to be produced when charged particles are accelerated by forces acting on them. Electrons are responsible for emission of most EMR because they have low mass, and therefore are easily accelerated by a variety of mechanisms. Rapidly moving electrons are most sharply accelerated when they encounter a region of force, so they are responsible for producing much of the highest frequency electromagnetic radiation observed in nature. Quantum processes can also produce EMR, such as when atomic nuclei undergo gamma decay,

Electromagnetic radiation 77 and processes such as neutral pion decay. EMR is classified according to the frequency of its wave. The electromagnetic spectrum, in order of increasing frequency and decreasing wavelength, consists of radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays. The eyes of various organisms sense a small and somewhat variable but relatively small range of frequencies of EMR called the visible spectrum or light. The effects of EMR upon biological systems (and also to many other chemical systems, under standard conditions) depends both upon the radiations power and frequency. For lower frequencies of EMR up to those of visible light (i.e., radio, microwave, infrared), the damage done to cells and also to many ordinary materials under such conditions is determined mainly by heating effects, and thus by the radiation power. By contrast, for higher frequency radiations at ultraviolet frequencies and above (i.e., X-rays and gamma rays) the damage to chemical materials and living cells by EMR is far larger than that done by simple heating, due to the ability of single photons in such high frequency EMR to damage individual molecules chemically. Physics Theory Maxwell’s equations for EM fields far from sources James Clerk Maxwell first formally postulated electromagnetic waves. These were subsequently confirmed by Heinrich Hertz. Maxwell derived a wave form of the electric and magnetic equations, thus uncovering the wave-like nature of electric and magnetic fields, and their symmetry. Because the speed of EM waves predicted by the wave equation coincided with the measured speed of light, Maxwell concluded that light itself is an EM wave. According to Maxwells equations, a spatially varying electric field causes the magnetic field to change over time. Likewise, a spatially Shows the relative wavelengths of the varying magnetic field causes changes over time in the electric field. In electromagnetic waves of three different colors of an electromagnetic wave, the changes induced by the electric field shift light (blue, green, and red) with a distance scale the wave in the magnetic field in one direction; the action of the in micrometers along the x-axis. magnetic field shifts the electric field in the same direction. Together, these fields form a propagating electromagnetic wave, which moves out into space and never again affects the source. The EM field formed by this mechanism "radiates," hence the term for it.

Electromagnetic radiation 78 Near and far fields Maxwells equations established that some charges and currents ("sources") produce a local type of electromagnetic field near them that does not have the behavior of EMR. In particular, according to Maxwell, currents directly produce a magnetic field, but it is of a magnetic dipole type which dies out rapidly with distance from the current. In a similar manner, moving charges being separated from each other in a conductor by a changing electrical potential (such as in an In electromagnetic radiation (such as microwaves from an antenna, shown here) antenna) produce an electric dipole type the term applies only to the parts of the electromagnetic field that radiate into infinite space and decrease in intensity by an inverse-square law of power, so that electrical field, but this also dies away very the total radiation energy that crosses through an imaginary spherical surface is the quickly with distance. Both of these fields same, no matter how far away from the antenna the spherical surface is drawn. make up the near-field near the EMR Electromagnetic radiation thus includes the far field part of the electromagnetic source. Neither of these behaviors are field around a transmitter. A part of the "near-field" close to the transmitter, forms part of the changing electromagnetic field, but does not count as electromagnetic responsible for EM radiation. Instead, they radiation. cause electromagnetic field behavior that only efficiently transfers power to a receiver very close to the source, such as the magnetic induction inside an electrical transformer, or the feedback behavior that happens close to the coil of a metal detector. Typically, near-fields have a powerful effect on their own sources, causing an increased “load” (decreased electrical reactance) in the source or transmitter, whenever energy is withdrawn from the EM field by a receiver. Otherwise, these fields do not “propagate,” freely out into space, carrying their energy away without distance-limit, but rather oscillate back and forth, returning their energy to the transmitter if it is not received by a receiver. By contrast, the EM far-field is composed of radiation that is free of the transmitter in the sense that (unlike the case in an electrical transformer) the transmitter requires the same power to send these changes in the fields out, whether the signal is immediately picked up, or not. This distant part of the electromagnetic field is "electromagnetic radiation" (also called the far-field). The far-fields propagate without ability for the transmitter to affect them, and this causes them to be independent in the sense that their existence and their energy, after they have left the transmitter, is completely independent of both transmitter and receiver. Because such waves conserve the amount of energy they transmit through any spherical boundary surface drawn around their source, and because such surfaces have an area that is defined by the square of the distance from the source, the power of EM radiation always varies according to an inverse-square law. This is in contrast to dipole parts of the EM field close to the source (the near-field), which varies in power according to an inverse cube power law, and thus does not transport a conserved amount of energy over distances, but instead dies away rapidly with distance, with its energy (as noted) either rapidly returning to the transmitter, or else absorbed by a nearby receiver (such as a transformer secondary coil). The far-field (EMR) depends on a different mechanism for its production than the near-field, and upon different terms in Maxwell’s equations. Whereas the magnetic part of the near-field is due to currents in the source, the magnetic field in EMR is due only to the local change in the electric field. In a similar way, while the electric field in the near-field is due directly to the charges and charge-separation in the source, the electric field in EMR is due to a change in the local magnetic field. Both of these processes for producing electric and magnetic EMR fields have a different dependence on distance than do near-field dipole electric and magnetic fields, and that is why the EMR type of EM field becomes dominant in power “far” from sources. The term “far from sources” refers to how far from the source (moving at the speed of light) any portion of the outward-moving EM field is located, by the time that

Electromagnetic radiation 79 source currents are changed by the varying source potential, and the source has therefore begun to generate an outwardly moving EM field of a different phase. A more compact view of EMR is that the far-field that composes EMR is generally that part of the EM field that has traveled sufficient distance from the source, that it has become completely disconnected from any feedback to the charges and currents that were originally responsible for it. Now independent of the source charges, the EM field, as it moves farther away, generates and regenerates itself only as a result of its own changing fields. Properties of EM radiation The physics of electromagnetic radiation is electrodynamics. Electromagnetism is the physical phenomenon associated with the theory of electrodynamics. Electric and magnetic fields obey the properties of superposition. Thus, a field due to any particular particle or time-varying electric or magnetic field contributes to the fields present in the same space due to other causes. Further, as they are vector fields, all magnetic and electric field vectors add together according to vector addition. For example, in optics two or more coherent lightwaves may interact and by constructive or destructive interference yield a resultant irradiance deviating from the sum of the component irradiances of the individual lightwaves. Since light is an oscillation it is not affected by travelling through static Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric or magnetic fields in a linear medium such as a vacuum. electric and magnetic fields. This 3D diagram However in nonlinear media, such as some crystals, interactions can shows a plane linearly polarized wave occur between light and static electric and magnetic fields — these propagating from left to right interactions include the Faraday effect and the Kerr effect. In refraction, a wave crossing from one medium to another of different density alters its speed and direction upon entering the new medium. The ratio of the refractive indices of the media determines the degree of refraction, and is summarized by Snells law. Light of composite wavelengths (natural sunlight) disperses into a visible spectrum passing through a prism, because of the wavelength dependent refractive index of the prism material (dispersion); that is, each component wave within the composite light is bent a different amount. EM radiation exhibits both wave properties and particle properties at the same time (see wave-particle duality). Both wave and particle characteristics have been confirmed in a large number of experiments. Wave characteristics are more apparent when EM radiation is This 3D diagram shows a plane linearly polarized measured over relatively large timescales and over large distances wave propagating from left to right. Note that the electric and magnetic fields in such a wave are while particle characteristics are more evident when measuring small in-phase with each other, reaching minima and timescales and distances. For example, when electromagnetic radiation maxima together is absorbed by matter, particle-like properties will be more obvious when the average number of photons in the cube of the relevant wavelength is much smaller than 1. It is not too difficult to experimentally observe non-uniform deposition of energy when light is absorbed, however this alone is not evidence of "particulate" behavior of light. Rather, it reflects the quantum nature of matter.[1] Demonstrating that the light itself is quantized, not merely its interaction with matter, is a more subtle problem.

Electromagnetic radiation 80 There are experiments in which the wave and particle natures of electromagnetic waves appear in the same experiment, such as the self-interference of a single photon. True single-photon experiments (in a quantum optical sense) can be done today in undergraduate-level labs.[2] When a single photon is sent through an interferometer, it passes through both paths, interfering with itself, as waves do, yet is detected by a photomultiplier or other sensitive detector only once. A quantum theory of the interaction between electromagnetic radiation and matter such as electrons is described by the theory of quantum electrodynamics. Wave model Electromagnetic radiation is a transverse wave, meaning that the oscillations of the waves are perpendicular to the direction of energy transfer and travel. The electric and magnetic parts of the field stand in a fixed ratio of strengths in order to satisfy the two Maxwell equations that specify how one is produced from the other. These E and B fields are also in phase, with both reaching maxima and minima at the same points in space (see illustrations). A common misconception is that the E and B fields in electromagnetic radiation are out of phase because a change in one produces the other, and this would produce a phase difference between them as sinusoidal functions (as indeed happens in electromagnetic induction, and in the near-field close to antennas). However, in the far-field EM radiation which is described by the two source-free Maxwell curl operator equations, a more correct description is that a time-change in one type of field is proportional to a space-change in the other. These derivatives require that the E and B fields in EMR are in-phase (see math section below). An important aspect of the nature of light is frequency. The frequency of a wave is its rate of oscillation and is measured in hertz, the SI unit of frequency, where one hertz is equal to one oscillation per second. Light usually has a spectrum of frequencies that sum to form the resultant wave. Different frequencies undergo different angles of refraction. A wave consists of successive troughs and crests, and the distance between two adjacent crests or troughs is called the wavelength. Waves of the electromagnetic spectrum vary in size, from very long radio waves the size of buildings to very short gamma rays smaller than atom nuclei. Frequency is inversely proportional to wavelength, according to the equation: where v is the speed of the wave (c in a vacuum, or less in other media), f is the frequency and λ is the wavelength. As waves cross boundaries between different media, their speeds change but their frequencies remain constant. Interference is the superposition of two or more waves resulting in a new wave pattern. If the fields have components in the same direction, they constructively interfere, while opposite directions cause destructive interference. The energy in electromagnetic waves is sometimes called radiant energy. Particle model and quantum theory An anomaly arose in the late 19th century involving a contradiction between the wave theory of light on the one hand, and on the other, observers actual measurements of the electromagnetic spectrum that was being emitted by thermal radiators known as black bodies. Physicists struggled with this problem, which later became known as the ultraviolet catastrophe, unsuccessfully for many years. In 1900, Max Planck developed a new theory of black-body radiation that explained the observed spectrum. Plancks theory was based on the idea that black bodies emit light (and other electromagnetic radiation) only as discrete bundles or packets of energy. These packets were called quanta. Later, Albert Einstein proposed that the quanta of light might be regarded as real particles, and (still later) the particle of light was given the name photon, to correspond with other particles being described around this time, such as the electron and proton. A photon has an energy, E, proportional to its frequency, f, by

Electromagnetic radiation 81 where h is Plancks constant, is the wavelength and c is the speed of light. This is sometimes known as the [3] Planck–Einstein equation. In quantum theory (see first quantization) the energy of the photons is thus directly proportional to the frequency of the EMR wave.[4] Likewise, the momentum p of a photon is also proportional to its frequency and inversely proportional to its wavelength: The source of Einsteins proposal that light was composed of particles (or could act as particles in some circumstances) was an experimental anomaly not explained by the wave theory: the photoelectric effect, in which light striking a metal surface ejected electrons from the surface, causing an electric current to flow across an applied voltage. Experimental measurements demonstrated that the energy of individual ejected electrons was proportional to the frequency, rather than the intensity, of the light. Furthermore, below a certain minimum frequency, which depended on the particular metal, no current would flow regardless of the intensity. These observations appeared to contradict the wave theory, and for years physicists tried in vain to find an explanation. In 1905, Einstein explained this puzzle by resurrecting the particle theory of light to explain the observed effect. Because of the preponderance of evidence in favor of the wave theory, however, Einsteins ideas were met initially with great skepticism among established physicists. Eventually Einsteins explanation was accepted as new particle-like behavior of light was observed, such as the Compton effect. As a photon is absorbed by an atom, it excites the atom, elevating an electron to a higher energy level (on average, one that is farther from the nucleus). When an electron in an excited molecule or atom descends to a lower energy level, it emits a photon of light equal to the energy difference. Since the energy levels of electrons in atoms are discrete, each element and each molecule emits and absorbs its own characteristic frequencies. When the emission of the photon is immediate, this phenomenon is called fluorescence, a type of photoluminescence. An example is visible light emitted from fluorescent paints, in response to ultraviolet (blacklight). Many other fluorescent emissions are known in spectral bands other than visible light. When the emission of the photon is delayed, the phenomenon is called phosphorescence. Wave–particle duality The modern theory that explains the nature of light includes the notion of wave–particle duality. More generally, the theory states that everything has both a particle nature and a wave nature, and various experiments can be done to bring out one or the other. The particle nature is more easily discerned if an object has a large mass, and it was not until a bold proposition by Louis de Broglie in 1924 that the scientific community realised that electrons also exhibited wave–particle duality. Wave and particle effects of electromagnetic radiation Together, wave and particle effects explain the emission and absorption spectra of EM radiation, wherever it is seen. The matter-composition of the medium through which the light travels determines the nature of the absorption and emission spectrum. These bands correspond to the allowed energy levels in the atoms. Dark bands in the absorption spectrum are due to the atoms in an intervening medium between source and observer, absorbing certain frequencies of the light between emitter and detector/eye, then emitting them in all directions, so that a dark band appears to the detector, due to the radiation scattered out of the beam. For instance, dark bands in the light emitted by a distant star are due to the atoms in the stars atmosphere. A similar phenomenon occurs for emission, which is seen when the emitting gas is glowing due to excitation of the atoms from any mechanism, including heat. As electrons descend to lower energy levels, a spectrum is emitted that represents the jumps between the energy levels of the electrons, but

Electromagnetic radiation 82 lines are seen because again emission happens only at particular energies after excitation. An example is the emission spectrum of nebulae. Today, scientists use these phenomena to perform various chemical determinations for the composition of gases lit from behind (absorption spectra) and for glowing gases (emission spectra). Spectroscopy (for example) determines what chemical elements a star is composed of. Spectroscopy is also used in the determination of the distance of a star, using the red shift. Speed of propagation Any electric charge that accelerates, or any changing magnetic field, produces electromagnetic radiation. Electromagnetic information about the charge travels at the speed of light. Accurate treatment thus incorporates a concept known as retarded time (as opposed to advanced time, which is not physically possible in light of causality), which adds to the expressions for the electrodynamic electric field and magnetic field. These extra terms are responsible for electromagnetic radiation. When any wire (or other conducting object such as an antenna) conducts alternating current, electromagnetic radiation is propagated at the same frequency as the electric current. In many such situations it is possible to identify an electrical dipole moment that arises from separation of charges due to the exciting electrical potential, and this dipole moment oscillates in time, as the charges move back and forth. This oscillation at a given frequency gives rise to changing electric and magnetic fields, which then set the electromagnetic radiation in motion. At the quantum level, electromagnetic radiation is produced when the wavepacket of a charged particle oscillates or otherwise accelerates. Charged particles in a stationary state do not move, but a superposition of such states may result in transition state which has an electric dipole moment that oscillates in time. This oscillating dipole moment is responsible for the phenomenon of radiative transition between quantum states of a charged particle. Such states occur (for example) in atoms when photons are radiated as the atom shifts from one stationary state to another. Depending on the circumstances, electromagnetic radiation may behave as a wave or as particles. As a wave, it is characterized by a velocity (the speed of light), wavelength, and frequency. When considered as particles, they are known as photons, and each has an energy related to the frequency of the wave given by Plancks relation E = hν, where E is the energy of the photon, h = 6.626 × 10−34 J·s is Plancks constant, and ν is the frequency of the wave. One rule is always obeyed regardless of the circumstances: EM radiation in a vacuum always travels at the speed of light, relative to the observer, regardless of the observers velocity. (This observation led to Albert Einsteins development of the theory of special relativity.) In a medium (other than vacuum), velocity factor or refractive index are considered, depending on frequency and application. Both of these are ratios of the speed in a medium to speed in a vacuum. Special theory of relativity By the late nineteenth century, however, a handful of experimental anomalies remained that could not be explained by the simple wave theory. One of these anomalies involved a controversy over the speed of light. The speed of light and other EMR predicted by Maxwells equations did not appear unless the equations were modified in a way first suggested by FitzGerald and Lorentz (see history of special relativity), or else otherwise it would depend on the speed of observer relative to the "medium" (called luminiferous aether) which supposedly "carried" the electromagnetic wave (in a manner analogous to the way air carries sound waves). Experiments failed to find any observer effect, however. In 1905, Albert Einstein proposed that space and time appeared to be velocity-changeable entities, not only for light propagation, but all other processes and laws as well. These changes then automatically accounted for the constancy of the speed of light and all electromagnetic radiation, from the viewpoints of all observers—even those in relative motion.

Electromagnetic radiation 83 History of discovery Electromagnetic radiation of wavelengths other than those of visible light were discovered in the early 19th century. The discovery of infrared radiation is ascribed to William Herschel, the astronomer. Herschel published his results in 1800 before the Royal Society of London. Herschel used a glass prism to refract light from the Sun and detected invisible rays that caused heating beyond the red part of the spectrum, through an increase in the temperature recorded with a thermometer. These "calorific rays" were later termed infrared. In 1801, the German physicist Johann Wilhelm Ritter made the discovery of ultraviolet in an experiment similar to Hershels, using sunlight and a glass prism. Ritter noted that invisible rays near the violet edge of a solar spectrum dispersed by a triangular prism darkened silver chloride preparations more quickly than did the nearby violet light. Ritters experiments were an early precursor to what would become photography. Ritter noted that the rays (which at first were called "chemical rays") were capable of causing chemical reactions. In 1862-4 James Clerk Maxwell developed equations for the electromagnetic field which suggested that waves in the field would travel with a speed that was very close to the known speed of light. Maxwell therefore suggested that visible light (as well as invisible infrared and ultraviolet rays by inference) all consisted of propagating disturbances (or radiation) in the electromagnetic field. Radio waves were not first detected from a natural source, but were rather produced deliberately and artificially by the German scientist Heinrich Hertz in 1887, using electrical circuits calculated to produce oscillations at a much lower frequency than that of visible light, following recipes for producing oscillating charges and currents suggested by Maxwells equations. Hertz also developed ways to detect these waves, and produced and characterized what were later termed radio waves and microwaves. Wilhelm Röntgen discovered and named X-rays. After experimenting with high voltages applied to an evaccuated tube on 8 November 1895, he noticed a fluorescence on a nearby plate of coated glass. In one month, he discovered the main properties of X-rays that we understand to this day. The last portion of the EM spectrum was discovered associated with radioactivity. Henri Becquerel found that uranium salts caused fogging of an unexposed photographic plate through a covering paper in a manner similar to X-rays, and Marie Curie discovered that only certain elements gave off these rays of energy, soon discovering the intense radiation of radium. The radiation from pitchblende was differentiated into alpha rays (alpha particles) and beta rays (beta particles) by Ernest Rutherford through simple experimentation in 1899, but these proved to be charged particulate types of radiation. However, in 1900 the French scientist Paul Villard discovered a third neutrally charged and especially penetrating type of radiation from radium, and after he described it, Rutherford realized it must be yet a third type of radiation, which in 1903 Rutherford named gamma rays. In 1910 British physicist William Henry Bragg demonstrated that gamma rays are electromagnetic radiation, not particles, and in 1914 Rutherford and Edward Andrade measured their wavelengths, and found that they were similar to X-rays but with shorter wavelengths and higher frequency.

Electromagnetic radiation 84 Electromagnetic spectrum In general, EM radiation (the designation radiation excludes static electric and magnetic and near fields) is classified by wavelength into radio, microwave, infrared, the visible spectrum we perceive as visible light, ultraviolet, X-rays, and gamma rays. Arbitrary electromagnetic waves can always be expressed by Fourier analysis in terms of sinusoidal monochromatic waves, which in turn can each be classified into these Electromagnetic spectrum with light highlighted regions of the EMR spectrum. The behavior of EM radiation depends on its frequency. Lower frequencies have longer wavelengths, and higher frequencies have shorter wavelengths, and are associated with photons of higher energy. There is no fundamental limit known to these wavelengths or energies, at either end of the spectrum, although photons with energies near the Planck energy or exceeding it (far too high to have ever been observed) will required new physical theories to describe. Soundwaves are not electromagnetic radiation. At the lower end of the Legend: γ = Gamma rays HX = Hard X-rays SX electromagnetic spectrum, about 20 Hz = Soft X-Rays EUV = Extreme-ultraviolet NUV to about 20 kHz, are frequencies that = Near-ultravioletVisible light (colored bands) might be considered in the audio range. NIR = Near-infrared MIR = Moderate-infrared However, electromagnetic waves FIR = Far-infrared EHF = Extremely high frequency (microwaves) SHF = Super-high cannot be directly perceived by human frequency (microwaves) UHF = Ultrahigh ears. Sound waves are the oscillating frequency (radio waves) VHF = Very high compression of molecules. To be frequency (radio) HF = High frequency (radio) heard, electromagnetic radiation must MF = Medium frequency (radio) LF = Low frequency (radio) VLF = Very low frequency be converted to pressure waves of the (radio) VF = Voice frequency ULF = Ultra-low fluid in which the ear is located frequency (radio) SLF = Super-low frequency (whether the fluid is air, water or (radio) ELF = Extremely low frequency(radio) something else). Radio and microwave heating and currents, and infrared heating When EM radiation interacts with matter, its behavior changes qualitatively as its frequency changes. At radio and microwave frequencies, EMR interacts with matter largely as a bulk collection of charges which are spread out over

Electromagnetic radiation 85 large numbers of affected atoms. In electrical conductors, such induced bulk movement of charges (electric currents) results in absorption of the EMR, or else separations of charges that cause generation of new EMR (effective reflection of the EMR). An example is absorption or emission of radio waves by antennas, or absorption of microwaves by water or other molecules with an electric dipole moment, as for example inside a microwave oven. These interactions produce either electric currents or heat, or both. Infrared EMR interacts with dipoles present in single molecules, which change as atoms vibrate at the ends of a single chemical bond. For this reason, infrared is reflected by metals (as is most EMR into the ultraviolet) but is absorbed by a wide range of substances, causing them to increase in temperature as the vibrations dissipate as heat. In the same process, bulk substances radiate in the infrared spontaneously (see thermal radiation section below). Reversible and nonreversible molecular changes from visible light As frequency increases into the visible range, photons of EMR have enough energy to change the bond structure of some individual molecules. It is not a coincidence that this happens in the "visible range," as the mechanism of vision involves the change in bonding of a single molecule (retinal) which absorbs light in the rhodopsin the retina of the human eye. Photosynthesis becomes possible in this range as well, for similar reasons, as a single molecule of chlorophyll is excited by a single photon. Animals which detect infrared do not use such single molecule processes, but are forced to make use of small packets of water which change temperature, in an essentially thermal process that involves many photons (see infrared sensing in snakes). For this reason, infrared, microwaves, and radio waves are thought to damage molecules and biological tissue only by bulk heating, not excitation from single photons of the radiation (however, there does remain controversy about possible non-thermal biological damage from low frequency EM radiation, see below). Visible light is able to affect a few molecules with single photons, but usually not in a permanent or damaging way, in the absence of power high enough to increase temperature to damaging levels. However, in plant tissues that carry on photosynthesis, carotenoids act to quench electronically excited chlorophyll produced by visible light in a process called non-photochemical quenching, in order to prevent reactions which would otherwise infere with photosynthesis at high light levels. There is also some limited evidence that some reactive oxygen species are created by visible light in skin, and that these may have some role in photoaging, in the same manner as ultraviolet A does.[5] Molecular damage from ultraviolet As a photon interacts with single atoms and molecules, the effect depends on the amount of energy the photon carries. As frequency increases beyond visible into the ultraviolet, photons now carry enough energy (about three electron volts or more) to excite certain doubly bonded molecules into permanent chemical rearrangement. If these molecules are biological molecules in DNA, this causes lasting damage. DNA is also indirectly damaged by reactive oxygen species produced by ultraviolet A, which has energy too low to damage DNA directly. This is why ultraviolet at all wavelengths can damage DNA, and is capable of causing cancer, and (for UVB) skin burns (sunburn) which are far worse than would be produced by simple heating (temperature increase) effects. This property of causing molecular damage that is far out of proportion to all temperature-changing (i.e., heating) effects, is characteristic of all EMR with frequencies at the visible light range and above. These properties of high-frequency EMR are due to quantum effects which cause permanent damage to materials and tissues at the single molecular level.

Electromagnetic radiation 86 Ionization and extreme types of molecular damage from X-rays and gamma rays At the higher end of the ultraviolet range, the energy of photons becomes large enough to impart enough energy to electrons to cause them to be liberated from the atom, in a process called photoionisation. The energy required for this is always larger than about 10 electron volts (eV) corresponding with wavelengths smaller than 124 nm (some sources suggest a more realistic cutoff of 33 eV, which is the energy required to ionize water). This high end of the ultraviolet spectrum with energies in the approximate ionization range, is sometimes called "extreme UV." (Most of this is filtered by the Earths atmosphere). Electromagnetic radiation composed of photons that carry minimum-ionization energy, or more, (which includes the entire spectrum with shorter wavelengths), is therefore termed ionizing radiation. (There are also many other kinds of ionizing radiation made of non-EM particles). Electromagnetic-type ionizing radiation extends from the extreme ultraviolet to all higher frequencies and shorter wavelengths, which means that all X-rays and gamma rays are ionizing radiation. These are capable of the most severe types of molecular damage, which can happen in biology to any type of biomolecule, including mutation and cancer, and often at great depths from the skin, since the higher end of the X-ray spectrum, and all of the gamma ray spectrum, are penetrating to matter. It is this type of damage which causes these types of radiation to be especially carefully monitored, due to their hazard, even at comparatively low-energies, to all living organisms. Propagation and absorption of EMR in the Earths atmosphere Most electromagnetic waves of higher frequency than visible light (UV and X-rays) are blocked by absorption from electronic excitation in ozone and dioxygen (for UV), and by ionization of air for energies in the extreme UV and above. Visible light is well transmitted in air, as it is not energetic enough to excite oxygen, but too energetic to excite molecular Rough plot of Earths atmospheric transmittance (or opacity) to various wavelengths of vibrational frequencies of molecules in electromagnetic radiation air. Below visible light, a number of absorption bands in the infrared are due to modes of vibrational excitation in water vapor. However, at energies too low to excite water vapor the atmosphere becomes transparent again, allowing free transmission of most microwave and radio waves. Finally, at radio wavelengths longer than 10 meters or so (about 30 MHz), the air in the lower atmosphere remains transparent to radio, but plasma in certain layers of the ionosphere of upper Earth atmosphere begins to interact with radio waves (see skywave). This property allows some longer wavelengths (100 meters or 3 MHz) to be reflected and results in farther shortwave radio than can be obtained by line-of-sight. However, certain ionospheric effects begin to block incoming radiowaves from space, when their frequency is less than about 10 MHz (wavelength longer than about 30 meters).

Electromagnetic radiation 87 Types and sources of electromagnetic waves, classed by spectral band (frequency) See electromagnetic spectrum for detail Radio waves When EM radiation at the frequencies for which it is referred to as "radio waves" impinges upon a conductor, it couples to the conductor, travels along it, and induces an electric current on the surface of the conductor by moving the electrons of the conducting material in correlated bunches of charge. Such effects can cover macroscopic distances in conductors (including as radio antennas), since the wavelength of radiowaves is long, by human scales. Radio waves thus have the most overtly "wave-like" characteristics of all the types of EMR, since their waves are so long. Visible light Natural sources produce EM radiation across the spectrum. EM radiation with a wavelength between approximately 400 nm and 700 nm is directly detected by the human eye and perceived as visible light. Other wavelengths, especially nearby infrared (longer than 700 nm) and ultraviolet (shorter than 400 nm) are also sometimes referred to as light, especially when visibility to humans is not relevant. Thermal radiation and electromagnetic radiation as a form of heat The basic structure of matter involves charged particles bound together in many different ways. When electromagnetic radiation is incident on matter, it causes the charged particles to oscillate and gain energy. The ultimate fate of this energy depends on the situation. It could be immediately re-radiated and appear as scattered, reflected, or transmitted radiation. It may also get dissipated into other microscopic motions within the matter, coming to thermal equilibrium and manifesting itself as thermal energy in the material. With a few exceptions related to high-energy photons (such as fluorescence, harmonic generation, photochemical reactions, the photovoltaic effect for ionizing radiations at far ultraviolet, X-ray, and gamma radiation), absorbed electromagnetic radiation simply deposits its energy by heating the material. This happens both for infrared, microwave, and radio wave radiation. Intense radio waves can thermally burn living tissue and can cook food. In addition to infrared lasers, sufficiently intense visible and ultraviolet lasers can also easily set paper afire. Ionizing electromagnetic radiation creates high-speed electrons in a material and breaks chemical bonds, but after these electrons collide many times with other atoms in the material eventually most of the energy is downgraded to thermal energy; this whole process happens in a tiny fraction of a second. This process makes ionizing radiation far more dangerous per unit of energy than non-ionizing radiation. This caveat also applies to the ultraviolet (UV) spectrum, even though almost all of it is not ionizing, because UV can damage molecules due to electronic excitation which is far greater per unit energy than heating effects produce. Infrared radiation in the spectral distribution of a black body is usually considered a form of heat, since it has an equivalent temperature, and is associated with an entropy change per unit of thermal energy. However, the word "heat" is a highly technical term in physics and thermodynamics, and is often confused with thermal energy. Any type of electromagnetic energy can be transformed into thermal energy in interaction with matter. Thus, any electromagnetic radiation can "heat" (in the sense of increase the thermal energy termperature of) a material, when it is absorbed. The inverse or time-reversed process of absorption is responsible for thermal radiation. Much of the thermal energy in matter consists of random motion of charged particles, and this energy can be radiated away from the matter. The resulting radiation may subsequently be absorbed by another piece of matter, with the deposited energy heating the material. Thermal radiation is an important mechanism of heat transfer.

Electromagnetic radiation 88 The electromagnetic radiation in an opaque cavity at thermal equilibrium is effectively a form of thermal energy, having maximum radiation entropy. Biological effects The effects of electromagnetic radiation upon living cells, including those in humans, depends upon the power and the frequency of the radiation. For low-frequency radiation (radio waves to visible light) the best-understood effects are those due to radiation power alone, acting through the effect of simple heating when the radiation is absorbed by the cell. For these thermal effects, the frequency of the radiation is important only as it affects radiation penetration into the organism (for example microwaves penetrate better than infrared). Initially, it was believed that low frequency fields that were too weak to cause significant heating could not possibly have any biological effect.[6] Despite this opinion among researchers, evidence has accumulated that supports the existence of complex biological effects of weaker non-thermal electromagnetic fields, (including weak ELF magnetic fields, although the latter does not strictly qualify as EM radiation[6][7][8]), and modulated RF and microwave fields.[9][10][11] Fundamental mechanisms of the interaction between biological material and electromagnetic fields at non-thermal levels are not fully understood.[6] Bioelectromagnetics is the study of these interactions and effects. The World Health Organization has classified radiofrequency electromagnetic radiation as a possible group 2b carcinogen.[12][13] This group contains possible carcinogens with weaker evidence, at the same level as coffee and automobile exhaust. For example, there have been a number of epidemiological studies of looking for a relationship between cell phone use and brain cancer development, which have been largely inconclusive, save to demonstrate that the effect, if it exists, cannot be a large one. See the main article referenced above. At higher frequencies (visible and beyond), the effects of individual photons of the radiation begin to become important, as these now have enough energy individually directly or indirectly to damage biological molecules.[14] All frequences of UV radiation have been classed as Group 1 carcinogens by the World Health Organization. Ultraviolet radiation from sun exposure is the primary cause of skin cancer.[15][16] Thus, at UV frequencies and higher (and probably somewhat also in the visible range),[17] electromagnetic radiation does far more damage to biological systems than simple heating predicts. This is most obvious in the "far" (or "extreme") ultraviolet, and also X-ray and gamma radiation, are referred to as ionizing radiation due to the ability of photons of this radiation to produce ions and free radicals in materials (including living tissue). Since such radiation can produce severe damage to life at powers that produce very little heating, it is considered far more dangerous (in terms of damage-produced per unit of energy, or power) than the rest of the electromagnetic spectrum. Derivation from electromagnetic theory Electromagnetic waves as a general phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwells equations. Inspection of Maxwells equations without sources (charges or currents) results in, along with the possibility of nothing happening, nontrivial solutions of changing electric and magnetic fields. Beginning with Maxwells equations in free space: where is a vector differential operator (see Del). One solution,

Electromagnetic radiation 89 is trivial. For a more useful solution, we utilize vector identities, which work for any vector, as follows: To see how we can use this, take the curl of equation (2): Evaluating the left hand side: where we simplified the above by using equation (1). Evaluate the right hand side: Equations (6) and (7) are equal, so this results in a vector-valued differential equation for the electric field, namely Applying a similar pattern results in similar differential equation for the magnetic field: These differential equations are equivalent to the wave equation: where c0 is the speed of the wave in free space and f describes a displacement Or more simply: where is dAlembertian: Notice that, in the case of the electric and magnetic fields, the speed is: This is the speed of light in vacuum. Maxwells equations have unified the vacuum permittivity , the vacuum permeability , and the speed of light itself, c0. Before this derivation it was not known that there was such a strong relationship between light and electricity and magnetism. But these are only two equations and we started with four, so there is still more information pertaining to these waves hidden within Maxwells equations. Lets consider a generic vector wave for the electric field.

Electromagnetic radiation 90 Here, is the constant amplitude, is any second differentiable function, is a unit vector in the direction of propagation, and is a position vector. We observe that is a generic solution to the wave equation. In other words for a generic wave traveling in the direction. This form will satisfy the wave equation, but will it satisfy all of Maxwells equations, and with what corresponding magnetic field? The first of Maxwells equations implies that electric field is orthogonal to the direction the wave propagates. The second of Maxwells equations yields the magnetic field. The remaining equations will be satisfied by this choice of . Not only are the electric and magnetic field waves in the far-field traveling at the speed of light, but they always have a special restricted orientation and proportional magnitudes, , which can be seen immediately from the Poynting vector. The electric field, magnetic field, and direction of wave propagation are all orthogonal, and the wave propagates in the same direction as . Also, E and B far-fields in free space, which as wave solutions depend primarily on these two Maxwell equations, are always in-phase with each other. This is guaranteed since the generic wave solution is first order in both space and time, and the curl operator on one side of these equations results in first-order spacial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time, resulting in the same phase shift for both fields in each mathematical operation. From the viewpoint of an electromagnetic wave traveling forward, the electric field might be oscillating up and down, while the magnetic field oscillates right and left; but this picture can be rotated with the electric field oscillating right and left and the magnetic field oscillating down and up. This is a different solution that is traveling in the same direction. This arbitrariness in the orientation with respect to propagation direction is known as polarization. On a quantum level, it is described as photon polarization. The direction of the polarization is defined as the direction of the electric field. More general forms of the second-order wave equations given above are available, allowing for both non-vacuum propagation media and sources. A great many competing derivations exist, all with varying levels of approximation and intended applications. One very general example is a form of the electric field equation,[18] which was factorized into a pair of explicitly directional wave equations, and then efficiently reduced into a single uni-directional wave equation by means of a simple slow-evolution approximation.

Electromagnetic radiation 91 References [1] Carmichael, H. J.. "Einstein and the Photoelectric Effect" (http:/ / www. qo. phy. auckland. ac. nz/ talks/ photoelectric. pdf). Quantum Optics Theory Group, University of Auckland. . Retrieved 22 December 2009. [2] http:/ / people. whitman. edu/ ~beckmk/ QM/ grangier/ Thorn_ajp. pdf [3] Paul M. S. Monk (2004). Physical Chemistry. John Wiley and Sons. p. 435. ISBN 978-0-471-49180-4. [4] Weinberg, S. (1995). The Quantum Theory of Fields. 1. Cambridge University Press. pp. 15–17. ISBN 0-521-55001-7. [5] J Invest Dermatol. 2012 Feb 9. doi: 10.1038/jid.2011.476. Irradiation of Skin with Visible Light Induces Reactive Oxygen Species and Matrix-Degrading Enzymes. Liebel F, et al. PMID 22318388 [6] Binhi, Vladimir N; Repiev, A & Edelev, M (translators from Russian) (2002). Magnetobiology: Underlying Physical Problems. San Diego: Academic Press. pp. 1–16. ISBN 978-0-12-100071-4. OCLC 49700531. [7] Delgado JM, Leal J, Monteagudo JL, Gracia MG (May 1982). "Embryological changes induced by weak, extremely low frequency electromagnetic fields". Journal of Anatomy 134 (3): 533–51. PMC 1167891. PMID 7107514. [8] Harland JD, Liburdy RP (1997). "Environmental magnetic fields inhibit the antiproliferative action of tamoxifen and melatonin in a human breast cancer cell line". Bioelectromagnetics 18 (8): 555–62. doi:10.1002/(SICI)1521-186X(1997)18:8<555::AID-BEM4>3.0.CO;2-1. PMID 9383244. [9] Aalto S, Haarala C, Brück A, Sipilä H, Hämäläinen H, Rinne JO (July 2006). "Mobile phone affects cerebral blood flow in humans". Journal of Cerebral Blood Flow and Metabolism 26 (7): 885–90. doi:10.1038/sj.jcbfm.9600279. PMID 16495939. [10] Cleary, SF, Liu LM, Merchant RE 1990. In vitro lyphocyte proliferation induced by radio-frequency electromagnetic radiation under isothermal conditions. Bioelectromagnetics 11(1):47-56 [11] Czerska EM, Elson EC, Davis CC 1992 Swicord ML, Czerski P, Effects of continuous and pulsed 2450-MHz radiation on spontaneous lymphoblastoid transformation of human lymphocytes in vitro. Bioelectromagnetics 13(4):247-259. [12] http:/ / www. iarc. fr/ en/ media-centre/ pr/ 2011/ pdfs/ pr208_E. pdf [13] "Trouble with cell phone radiation standard" (http:/ / www. cbsnews. com/ 2100-503063_162-20068246. html). CBS News. . [14] See PMID 22318388 for evidence of quantum damage from visible light via reactive oxygen species generated in skin. This happens also with UVA. With UVB, the damge to DNA becomes direct, with photochemical formation of pyrimidine dimers. [15] Narayanan, DL; Saladi, RN, Fox, JL (2010 Sep). "Ultraviolet radiation and skin cancer.". International Journal of Dermatology 49 (9): 978–86. doi:10.1111/j.1365-4632.2010.04474.x. PMID 20883261. [16] Saladi, RN; Persaud, AN (2005 Jan). "The causes of skin cancer: a comprehensive review.". Drugs of today (Barcelona, Spain : 1998) 41 (1): 37–53. doi:10.1358/dot.2005.41.1.875777. PMID 15753968. [17] See PMID 22318388 already quoted [18] Kinsler, P. (2010). "Optical pulse propagation with minimal approximations". Phys. Rev. A 81: 013819. arXiv:0810.5689. Bibcode 2010PhRvA..81a3819K. doi:10.1103/PhysRevA.81.013819. • Hecht, Eugene (2001). Optics (4th ed.). Pearson Education. ISBN 0-8053-8566-5. • Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks Cole. ISBN 0-534-40842-7. • Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8. • Reitz, John; Milford, Frederick; Christy, Robert (1992). Foundations of Electromagnetic Theory (4th ed.). Addison Wesley. ISBN 0-201-52624-7. • Jackson, John David (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons. ISBN 0-471-30932-X. • Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Artech House Publishers. ISBN 1-58053-832-0.

Electromagnetic radiation 92 External links • Electromagnetism (http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html) - a chapter from an online textbook • Electromagnetic Radiation (http://www.learnemc.com/tutorials/Radiation/EM_Radiation.html) - an introduction for electrical engineers • Electromagnetic Waves from Maxwells Equations (http://www.physnet.org/modules/pdf_modules/m210. pdf) on Project PHYSNET (http://www.physnet.org). • Radiation of atoms? e-m wave, Polarisation, ... (http://www.hydrogenlab.de/elektronium/HTML/ einleitung_hauptseite_uk.html) • An Introduction to The Wigner Distribution in Geometric Optics (http://scripts.mit.edu/~raskar/lightfields/ index.php?title=An_Introduction_to_The_Wigner_Distribution_in_Geometric_Optics) • The windows of the electromagnetic spectrum, on Astronoo (http://www.astronoo.com/articles/ electromagneticSpectrum-en.html) Emissivity The emissivity of a material (usually written ε or e) is the relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature. A true black body would have an ε = 1 while any real object would have ε < 1. Emissivity is a dimensionless quantity. In general, the duller and blacker a material is, the closer its emissivity is to 1. The more reflective a material is, the lower its emissivity. Highly polished silver has an emissivity of about 0.02.[1] Explanation Emissivity depends on factors such as temperature, emission angle, and wavelength. A typical physical assumption is that a surfaces spectral emissivity and absorptivity do not depend on wavelength, so that the emissivity is a constant. This is known as the "gray body assumption". Although it is common to discuss the "emissivity of a material" (such as the emissivity of highly polished silver), the emissivity of a material does in general depend on its thickness. The emissivities quoted for materials are for samples of infinite thickness (which, in practice, means samples which are optically thick) — thinner samples of material will have reduced emissivity. When dealing with non-black surfaces, the deviations from ideal black body behavior are determined by both the geometrical structure and the chemical composition, and follow Kirchhoffs law of thermal radiation: emissivity equals absorptivity (for an object in thermal equilibrium), so that an object that does not absorb all incident light will also emit less radiation than an ideal black body. Most emissivities found in handbooks and on websites of many infrared imaging and temperature sensor companies are the type discussed here, total emissivity. However, the distinction needs to be made that the wavelength-dependent or spectral emissivity is the more significant parameter to be used when one is seeking an emissivity correction for a temperature measurement device. Thus, it is important to understand the distinction between total and spectral emissivity and where they apply.

Emissivity 93 Emissivity of Earths atmosphere The emissivity of Earths atmosphere varies according to cloud cover and the concentration of gases that absorb and emit energy in the thermal infrared (i.e., wavelengths around 8 to 14 micrometres). These gases are often called greenhouse gases, from their role in the greenhouse effect. The main naturally-occurring greenhouse gases are water vapor, carbon dioxide, methane, and ozone. The major constituents of the atmosphere, N2 and O2, do not absorb or emit in the thermal infrared. Astrophysical graybody The monochromatic flux density radiated by a greybody at frequency through solid angle is given by where is the Planck function for a blackbody at temperature T and emissivity . For a uniform medium of optical depth radiative transfer means that the radiation will be reduced by a factor . The optical depth is often approximated by the ratio of the emitting frequency to the frequency where all raised to an exponent β. For cold dust clouds in the interstellar medium β is approximately two. Therefore Q becomes, .( , is the frequency where ) Emissivity between two walls Given two parallel walls whose facing surfaces have respective emissivities and at a given wavelength, a certain fraction of the radiation of that wavelength just inside one wall will leave that wall and enter the other. By Kirchhoffs law of thermal radiation for a given wavelength, whatever portion of the radiation incident on a surface, from either side, that does not pass through the surface as emission to the other side, is reflected. When this reflected radiation is neglected, the proportion of radiation emitted from the first wall is , and the proportion of that entering the second wall is therefore . When reflection is taken into account, what does not enter the second wall is reflected back to the first wall, initially in an amount . A fraction of this is then reflected back to the second wall, thereby augmenting the original emission from the first wall. These reflections bounce back and forth in diminishing quantity. Solving for the steady state then gives as the total proportion of radiation entering the second wall This formula is symmetric, and the proportion of radiation just inside the second wall that enters the first wall is the same. This is true regardless of what reflections and absorptions take place inside the two walls away from their facing surfaces, since the formula only concerns the radiation leaving one wall for the other. The quantities in these formulas are intensities rather than amplitudes, the appropriate choice when the walls are many wavelengths apart as the reflected and transmitted beams will then combine incoherently. When the walls are only a few wavelengths apart, as arises for example with the thin films used in the manufacture of optical coatings, the reflections tend to combine coherently, resulting in interference. In such a situation the above formula becomes invalid, and one must then add amplitudes instead of intensities, taking into account the phase shift as the gap is traversed and the phase reversal that occurs with reflection, concerns that did not arise in the incoherent large-gap or thick-film case.

Emissivity 94 References [1] http:/ / www. monarchserver. com/ TableofEmissivity. pdf Table of Total Emissivity External links • Emissivity of some common materials (http://www.engineeringtoolbox.com/emissivity-coefficients-d_447. html) • An open community-focused website & directory with resources related to spectral emissivity and emittance. On this site, the focus is on available data, references and links to resources related to spectral emissivity as it is measured & used in thermal radiation thermometry and thermography (thermal imaging) (http://www. spectralemissivity.com) Enthalpy of vaporization The enthalpy of vaporization, (symbol ), also known as the heat of vaporization or heat of evaporation, is the energy required to transform a given quantity of a substance from a liquid into a gas at a given pressure (often atmospheric pressure). It is often measured at the normal boiling point of a substance; although tabulated values are usually corrected to 298 K, the correction is often smaller than the uncertainty in the measured value. The heat of vaporization is temperature-dependent, though a constant heat of vaporization can be assumed for small temperature ranges and for Tr<<1.0. The heat of Temperature-dependency of the heats of vaporization for water, methanol, vaporization diminishes with increasing benzene, and acetone temperature and it vanishes completely at the critical temperature (Tr=1) because above the critical temperature the liquid and vapor phases no longer co-exist.

Enthalpy of vaporization 95 Units Values are usually quoted in J/mol or kJ/mol (molar enthalpy of vaporization), although kJ/kg or J/g (specific heat of vaporization), and older units like kcal/mol, cal/g and Btu/lb are sometimes still used, among others. Physical model for vaporization A simple physical model for the liquid-gas phase transformation has been proposed recently.[1] It is suggested that the energy required to free an atom from the liquid is equivalent to the energy needed to Fig. 1 Schematic cross section of the proposed vaporization model for monoatomic overcome the surface resistance of the liquids with one atomic surface layer. liquid. The model allows calculating the latent heat by multiplying the maximum surface area covering an atom (Fig. 1) with the surface tension and the number of atoms in the liquid. The calculated latent heat of vaporization values for the investigated 45 elements agrees well with experiments. Enthalpy of condensation The enthalpy of condensation (or heat of condensation) is by definition equal to the enthalpy of vaporization with the opposite sign: enthalpy changes of vaporization are always positive (heat is absorbed by the substance), whereas enthalpy changes of condensation are always negative (heat is released by the substance). Thermodynamic background The enthalpy of vaporization can be viewed as the energy required to overcome the intermolecular interactions in the liquid (or solid, in the case of sublimation). Hence helium has a particularly low enthalpy of vaporization, 0.0845 kJ/mol, as the van der Waals forces between helium atoms are particularly weak. On the other hand, the molecules in liquid water are held together by relatively strong hydrogen bonds, and its enthalpy of vaporization, 40.65 kJ/mol, is more than five times the energy required to heat the same quantity of water from 0 °C to 100 °C (cp = 75.3 J K−1 mol−1). Care must be taken, however, when using enthalpies of vaporization to measure the strength of intermolecular forces, as these forces may persist to an extent in the gas phase (as is the case with hydrogen fluoride), and so the calculated value of the bond strength will be too low. This is particularly true of metals, which often form covalently bonded molecules in the gas phase: in these cases, the enthalpy of atomization must be used to obtain a true value of the bond energy. An alternative description is to view the enthalpy of condensation as the heat which must be released to the surroundings to compensate for the drop in entropy when a gas condenses to a liquid. As the liquid and gas are in

Enthalpy of vaporization 96 equilibrium at the boiling point (Tb), ΔvG = 0, which leads to: As neither entropy nor enthalpy vary greatly with temperature, it is normal to use the tabulated standard values without any correction for the difference in temperature from 298 K. A correction must be made if the pressure is different from 100 kPa, as the entropy of a gas is proportional to its pressure (or, more precisely, to its fugacity): the entropies of liquids vary little with pressure, as the compressibility of a liquid is small. These two definitions are equivalent: the boiling point is the temperature at which the increased entropy of the gas phase overcomes the intermolecular forces. As a given quantity of matter always has a higher entropy in the gas phase than in a condensed phase ( is always positive), and from , the Gibbs free energy change falls with increasing temperature: gases are favored at higher temperatures, as is observed in practice. Vaporization enthalpy of electrolyte solutions Estimation of the enthalpy of vaporization of electrolyte solutions can be simply carried out using equations based on the chemical thermodynamic models, such as Pitzer model[2] or TCPC model.[3] Selected values Elements Enthalpies of vaporization of the elements (in kJ/mol, measured at their respective normal boiling points) Group → 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ↓ Period 1 H He 0.44936 0.0845 2 Li Be B C N O F Ne 145.92 292.40 489.7 355.8 2.7928 3.4099 3.2698 1.7326 3 Na Mg Al Si P S Cl Ar 96.96 127.4 293.4 300 12.129 1.7175 10.2 6.447 4 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 79.87 153.6 314.2 421 452 344.3 226 349.6 376.5 370.4 300.3 115.3 258.7 330.9 34.76 26.3 15.438 9.029 5 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe 72.216 144 363 581.6 696.6 598 660 595 493 357 250.58 100 231.5 295.8 77.14 52.55 20.752 12.636 6 Cs Ba * Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn 67.74 142 575 743 824 715 627.6 604 510 334.4 59.229 164.1 177.7 104.8 60.1 27.2 16.4 7 Fr Ra ** Rf Db Sg Bh Hs Mt Ds Rg Cn Uut Fl Uup Lv Uus Uuo n/a 37 n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a * Lanthanides La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 414 414 n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a ** Actinides Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr n/a 514.4 n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a 0–10 kJ/mol 10–100 kJ/mol 100–300 kJ/mol >300 kJ/mol

Enthalpy of vaporization 97 Other common substances Enthalpies of vaporization of common substances, measured at their respective standard boiling points: Compound Heat of Heat of vaporization vaporization (kJ mol-1) (kJ kg−1) Ammonia 23.35 1371 Butane 21.0 320 Ethanol 38.6 841 Hydrogen 0.46 451.9 Methane 8.17 480.6 Methanol 35.3 1104 Propane 15.7 356 Phosphine 14.6 429.4 Water 40.68 2257 Aluminium 294.0 10897 References [1] Garai, J. (2009). "Physical model for vaporization". Fluid Phase Equilibria 283: 89–77. doi:10.1016/j.fluid.2009.06.005. [2] X. Ge, X. Wang. Estimation of Freezing Point Depression, Boiling Point Elevation and Vaporization enthalpies of electrolyte solutions. Ind. Eng. Chem. Res. 48(2009)2229-2235. http:/ / pubs. acs. org/ doi/ abs/ 10. 1021/ ie801348c (Correction: 2009, 48, 5123)http:/ / pubs. acs. org/ doi/ abs/ 10. 1021/ ie900434h [3] X. Ge, X. Wang. Calculations of Freezing Point Depression, Boiling Point Elevation, Vapor Pressure and Enthalpies of Vaporization of Electrolyte Solutions by a Modified Three-Characteristic Parameter Correlation Model. J. Sol. Chem. 38(2009)1097-1117.http:/ / www. springerlink. com/ content/ 21670685448p5145/ • CODATA Key Values for Thermodynamics (http://www.codata.org/resources/databases/key1.html) • Kugler HK & Keller C (eds) 1985, Gmelin handbook of inorganic and organometallic chemistry, 8th ed., At, Astatine, system no. 8a, Springer-Verlag, Berlin, ISBN 3-540-93516-9, pp. 116–117 • NIST Chemistry WebBook (http://webbook.nist.gov/chemistry/) • Sears, Zemansky et al., University Physics, Addison-Wesley Publishing Company, Sixth ed., 1982, ISBN 0-201-07199-1

Evaporative cooler 98 Evaporative cooler An evaporative cooler (also swamp cooler, desert cooler, and wet air cooler) is a device that cools air through the evaporation of water. Evaporative cooling differs from typical air conditioning systems which use vapor-compression or absorption refrigeration cycles. Evaporative cooling works by employing waters large enthalpy of vaporization. The temperature of dry air can be dropped significantly through the phase transition of liquid water to water vapor (evaporation), which can cool air using much less energy than refrigeration. In extremely dry climates, evaporative cooling of air has An evaporative cooler, photographed in Rocky the added benefit of conditioning the air with more moisture for the Ford, Colorado, used in the drier parts of the comfort of building occupants. Unlike closed-cycle refrigeration, American West to provide economical cooling evaporative cooling requires a water source, and must continually consume water to operate. Air washers and wet cooling towers use the same principles as evaporative coolers but are designed for purposes other than directly cooling the air inside a building. For example, an evaporative cooler may be designed to cool the coils of a large air conditioning or refrigeration system to increase its efficiency. History Civilizations throughout the ages have found ingenious ways to combat the heat in their region. An earlier form of air cooling, the windcatcher (Bâd gir), was invented in Persia (Iran) thousands of years ago in the form of wind shafts on the roof, which caught the wind, passed it over subterranean water in a qanat and discharged the cooled air into the building. Nowadays Iranians have changed the windcatcher into an evaporative cooler (Coolere Âbi) and use it widely.[1] There are 9 million evaporative coolers in central Iran, and in just the first two months of year 1385 in the Persian/Iranian calendar (April–May 2006) 130,000 evaporative coolers were sold in Iran.[2] Schematic diagram of an ancient Iranian windcatcher and qanat, used for evaporative The evaporative cooler was the subject of numerous US patents in the cooling of buildings (click image to enlarge) 20th century; many of these, starting in 1906,[3] suggested or assumed the use of excelsior (wood wool) pads as the elements to bring a large volume of water in contact with moving air to allow evaporation to occur. A typical design, as shown in a 1945 patent, includes a water reservoir (usually with level controlled by a float valve), a pump to circulate water over the excelsior pads and a squirrel-cage fan to draw air through the pads and into the house.[4] This design and this material remain dominant in evaporative coolers in the American Southwest, where they are also used to increase humidity.[5] In the United States, the use of the term swamp cooler may be due to the odor of algae produced by early units.[6] Evaporative cooling was in vogue for aircraft engines in the 1930s, for example with the Beardmore Tornado airship engine. Here the system was used to reduce, or eliminate completely, the radiator which would otherwise create considerable drag. In these systems the water in the engine was kept under pressure with pumps, allowing it to heat to temperatures above 100°C, as the actual boiling point is a function of the pressure. The superheated water was then sprayed through a nozzle into an open tube, where it flashed into steam, releasing its heat. The tubes could be placed under the skin of the aircraft, resulting in a zero-drag cooling system.

Evaporative cooler 99 However these systems also had serious disadvantages. Since the amount of tubing needed to cool the water was large, the cooling system covered a significant portion of the plane even though it was hidden. This added complexity and reliability issues. In addition this large size meant it was very easy for it to be hit by enemy fire, and practically impossible to armor. British and U.S. developers used ethylene glycol instead, cooling the liquid in radiators. The Germans instead used streamlining and positioning of traditional radiators. Even the methods most ardent supporters, Heinkels Günter brothers, eventually gave up on it in 1940. Externally-mounted evaporative[7] cooling devices to cool interior air were used in some automobiles, often as aftermarket accessories, until modern vapor-compression air conditioning became widely available. Physical principles Evaporative cooling is a physical phenomenon in which evaporation of a liquid, typically into surrounding air, cools an object or a liquid in contact with it. Latent heat, the amount of heat that is needed to evaporate the liquid, is drawn from the air. When considering water evaporating into air, the wet-bulb temperature which takes both temperature and humidity into account, as compared to the actual air temperature (dry-bulb temperature), is a measure of the potential for evaporative cooling. The greater the difference between the two temperatures, the greater the evaporative cooling effect. When the temperatures are the same, no net evaporation of water in air occurs, thus there is no cooling effect. The wet-bulb temperature is essentially the lowest temperature which can be attained by evaporative cooling at a given temperature and humidity. A simple example of natural evaporative cooling is perspiration, or sweat, secreted by the body, evaporation of which cools the body. The amount of heat transfer depends on the evaporation rate, however for each kilogram of water vaporized 2,257 kJ of energy (about 890 BTU per pound of pure water, at 95°F) are transferred. The evaporation rate depends on the temperature and humidity of the air, which is why sweat accumulates more on hot, humid days, as it does not evaporate fast enough. Vapor-compression refrigeration uses evaporative cooling, but the evaporated vapor is within a sealed system, and is then compressed ready to evaporate again, using energy to do so. A simple evaporative coolers water is evaporated into the environment, and not recovered. In an interior space cooling unit, the evaporated water is introduced into the space along with the now-cooled air; in an evaporative tower the evaporated water is carried off in the airflow exhaust. Other types of phase-change cooling A closely related process, sublimation cooling differs from evaporative cooling in that a phase transition from solid to vapor, rather than liquid to vapor occurs. Sublimation cooling has been observed to operate on a planetary scale on the planetoid Pluto, where it has been called an anti-greenhouse effect. Another application of a phase change to cooling is the "self-refrigerating" beverage can. A separate compartment inside the can contains a desiccant and a liquid. Just before drinking, a tab is pulled so that the desiccant comes into contact with the liquid and dissolves. As it does so it absorbs an amount of heat energy called the latent heat of fusion. Evaporative cooling works with the phase change of liquid into vapor and the latent heat of vaporization, but the self-cooling can uses a change from solid to liquid, and the latent heat of fusion to achieve the same result.

Evaporative cooler 100 Applications Before the advent of refrigeration, evaporative cooling was used for millennia. A porous earthenware vessel would cool water by evaporation through its walls; frescoes from about 2500 BC show slaves fanning jars of water to cool rooms.[8] A vessel could also be placed in a bowl of water, covered with a wet cloth dipping into the water, to keep milk or butter as fresh as possible.[9] Evaporative cooling is a common form of cooling buildings for thermal comfort since it is relatively cheap and requires less energy than other forms of cooling. However, evaporative cooling requires an abundant water source as an evaporate, and is only efficient when the relative humidity is low, restricting its effective use to dry climates. Evaporative cooling also raises the internal humidity level significantly, which can cause problems such as lumpy table salt; swelling and warping of wood paneling, doors and trim; pianos going out of tune or suffering internal rusting, etc. California ranch house with evaporative cooler Evaporative cooling is especially well suited for climates where the air box on roof ridgeline (click image to enlarge) is hot and humidity is low. In the United States, the western/mountain states are good locations, with evaporative coolers prevalent in cities like Denver, Salt Lake City, Albuquerque, El Paso, Tucson, and Fresno where sufficient water is available. Evaporative air conditioning is also popular and well-suited to the southern (temperate) part of Australia. In dry, arid climates, the installation and operating cost of an evaporative cooler can be much lower than that of refrigerative air conditioning, often by 80% or so. However, evaporative cooling and vapor-compression air conditioning are sometimes used in combination to yield optimal cooling results. Some evaporative coolers may also serve as humidifiers in the heating season. In locations with moderate humidity there are many cost-effective uses for evaporative cooling, in addition to their widespread use in dry climates. For example, industrial plants, commercial kitchens, laundries, dry cleaners, greenhouses, spot cooling (loading docks, warehouses, factories, construction sites, athletic events, workshops, garages, and kennels) and confinement farming (poultry ranches, hog, and dairy) often employ evaporative cooling. In highly humid climates, evaporative cooling may have little thermal comfort benefit beyond the increased ventilation and air movement it provides. Other examples On Earth, trees transpire large amounts of water through pores in their leaves called stomata, and through this process of evaporative cooling, forests interact with climate at local and global scales.[10] Evaporative cooling is commonly used in cryogenic applications. The vapor above a reservoir of cryogenic liquid is pumped away, and the liquid continuously evaporates as long as the liquids vapor pressure is significant. Evaporative cooling of ordinary helium forms a 1-K pot, which can cool to at least 1.2 K. Evaporative cooling of helium-3 can provide temperatures below 300 mK. These techniques can be used to make cryocoolers, or as components of lower-temperature cryostats such as dilution refrigerators. As the temperature decreases, the vapor pressure of the liquid also falls, and cooling becomes less effective. This sets a lower limit to the temperature attainable with a given liquid. Evaporative cooling is also the last cooling step in order to reach the ultra-low temperatures required for Bose–Einstein condensation (BEC). Here, so-called forced evaporative cooling is used to selectively remove high-energetic ("hot") atoms from an atom cloud until the remaining cloud is cooled below the BEC transition temperature. For a cloud of 1 million alkali atoms, this temperature is about 1μK. Although robotic spacecraft use thermal radiation almost exclusively, many manned spacecraft have short missions that permit open-cycle evaporative cooling. Examples include the Space Shuttle, the Apollo Command/Service

Evaporative cooler 101 Module (CSM), Lunar Module and Portable Life Support System. The Apollo CSM and the Space Shuttle also had radiators, and the Shuttle could evaporate ammonia as well as water. The Apollo spacecraft used sublimators, compact and largely passive devices that dump waste heat in water vapor (steam) that is vented to space. When liquid water is exposed to vacuum it boils vigorously, carrying away enough heat to freeze the remainder to ice that covers the sublimator and automatically regulates the feedwater flow depending on the heat load. The water expended is often available in surplus from the fuel cells used by many manned spacecraft to produce electricity. Evaporative cooler designs Most designs take advantage of the fact that water has one of the highest known enthalpy of vaporization (latent heat of vaporization) values of any common substance. Because of this evaporative coolers use only a fraction of the energy of vapor-compression or absorption air conditioning systems. Unfortunately, except in very dry climates, the single-stage (direct) Evaporative cooler illustration cooler can increase relative humidity to a level that makes occupants uncomfortable. Indirect and Two-stage evaporative coolers keep the RH lower. Direct evaporative cooling (open circuit) is used to lower the temperature of air by using latent heat of evaporation, changing liquid water to water vapor. In this process, the energy in the air does not change. Warm dry air is changed to cool moist air. The heat of the outside air is used to evaporate water. The Relative Humidity increases to 70 to 90% which reduces the cooling effect of human perspiration. The moist air has to be continually released to outside or else the air becomes saturated and evaporation stops. Indirect evaporative cooling (closed circuit) is similar to direct evaporative cooling but uses some type of heat exchanger. The cooled moist air never comes in direct contact with the conditioned air. The moist air stream is released outside or used to cool other external devices such as solar cells which are more efficient if kept cool. One indirect cooler manufacturer uses the so-called Maisotsenko cycle which employs an iterative (multi-step) heat exchanger that can reduce the temperature to below the wet-bulb temperature.[11] While no moisture is added to the incoming air the RH does rise a little according to the Temperature-RH formula. Conditioned air without added moisture increases the evaporation of perspiration improving the cooling effect of Indirect compared to Direct. Two-stage evaporative cooling, or indirect-direct. In the first stage of a two-stage cooler, warm air is pre-cooled indirectly without adding humidity (by passing inside a heat exchanger that is cooled by evaporation on the outside). In the direct stage, the pre-cooled air passes through a water-soaked pad and picks up humidity as it cools. Since the air supply is pre-cooled in the first stage, less humidity is transferred in the direct stage, to reach the desired cooling temperatures. The result, according to manufacturers, is cooler air with a relative humidity between 50-70%, depending on the climate, compared to a traditional system that produces about 70–80% relative humidity in the conditioned air. Hybrid. Direct or Indirect cooling has been combined with vapor-compression or absorption air conditioning to increase the overall efficiency and /or to reduce the temperature below the wet-bulb limit. Materials. Traditionally, evaporative cooler pads consist of excelsior (wood wool) (aspen wood fiber) inside a containment net, but more modern materials, such as some plastics and melamine paper, are entering use as cooler-pad media. Wood absorbs some of the water and has a larger surface area which allows the wood fibers to cool passing air to a lower temperature than some synthetic materials, but natural fibers also can pose a problem with

Evaporative cooler 102 harboring or supporting mildew growth. Typical installations Typically, residential and industrial evaporative coolers use direct evaporation, and can be described as an enclosed metal or plastic box with vented sides. Air is moved by a centrifugal fan or blower, (usually driven by an electric motor with pulleys known as "sheaves" in HVAC terminology, or a direct-driven axial fan), and a water pump is used to wet the evaporative cooling pads. The cooling units can be mounted on the roof (down draft, or downflow), or exterior walls or windows (side draft, or horizontal flow) of buildings. To cool, the fan draws ambient air through vents on the units sides and through the damp pads. Heat in the air evaporates water from the pads which are constantly re-dampened to continue the cooling process. Then cooled, moist air is delivered into the building via a vent in the roof or wall. Because the cooling air originates outside the building, one or more large vents must exist to allow air to move from inside to outside. Air should only be allowed to pass once through the system, or the cooling effect will decrease. This is due to the air reaching the saturation point. Often 15 or so air changes per hour (ACHs) occur in spaces served by evaporative coolers, a relatively high rate of air exchange. Evaporative (wet) cooling towers Cooling towers are structures for cooling water or other heat transfer media to near-ambient wet-bulb temperature. Wet cooling towers operate on the evaporative cooling principle, but are optimized to cool the water rather than the air. Cooling towers can often be found on large buildings or on industrial sites. They transfer heat to the environment from chillers, industrial processes, or the Rankine power cycle, for example. Large hyperboloid cooling towers made of structural steel for a power plant in Kharkov (Ukraine) Misting systems Misting systems work by forcing water via a high pressure pump and tubing through a brass and stainless steel mist nozzle that has an orifice of about 5 micrometres, thereby producing a micro-fine mist. The water droplets that create the mist are so small that they instantly flash evaporate. Flash evaporation can reduce the surrounding air temperature by as much as 35 F° (20 C°) in just seconds.[12] For patio systems, it is ideal to mount the mist line approximately 8 to 10 feet (2.4 to 3.0 m) above the ground for optimum cooling. Misting is used for applications such as flowerbeds, pets, livestock, kennels, insect Mist spraying system with water pump beneath control, odor control, zoos, veterinary clinics, cooling of produce, and greenhouses.

Evaporative cooler 103 Misting fans A misting fan is similar to a humidifier. A fan blows a fine mist of water into the air. If the air is not too humid, the water evaporates, absorbing heat from the air, allowing the misting fan to also work as an air cooler. A misting fan may be used outdoors, especially in a dry climate. Small portable battery-powered misting fans, consisting of an electric fan and a hand-operated water spray pump, are sold as novelty items. Their effectiveness in everyday use is unclear. Performance Understanding evaporative cooling performance requires an understanding of psychrometrics. Evaporative cooling performance is variable due to changes in external temperature and humidity level. A residential cooler should be able to cool air to within 3–4 °C (5–7 °F) of the corresponding wet-bulb temperature. It is simple to predict cooler performance from standard weather report information. Because weather reports usually contain the dewpoint and relative humidity, but not the wet-bulb temperature, a psychrometric chart or a simple computer program must be used to compute the wet bulb temperature. Once the wet bulb temperature and the dry bulb temperature are identified, the cooling performance or leaving air temperature of the cooler may be determined: TLA = TDB – ((TDB – TWB) x E) TLA = Leaving Air Temp TDB = Dry Bulb Temp TWB = Wet Bulb Temp E = Efficiency of the evaporative media. Evaporative media efficiency usually runs between 80% to 90%, and the evaporation efficiency drops very little over time. Typical aspen pads used in residential evaporative coolers offer around 85% efficiency while CELdek type of evaporative media offer efficiencies of >90% depending on air velocity. The CELdek media is more often used in large commercial and industrial installations. As an example, in Las Vegas, Nevada, with a typical summer design day of 108°F DB/66°F WB or about 8% relative humidity, the leaving air temperature of a residential cooler would be: TLA = 108° – ((108° – 66°) x 85% efficiency) TLA = 72.3°F However, either of two methods can be used to estimate performance: • Use a psychrometric chart to calculate wet bulb temperature, and then add 6–8 °F as described above. • Use a rule of thumb which estimates that the wet bulb temperature is approximately equal to the ambient temperature, minus one third of the difference between the ambient temperature and the dew point. As before, add 6–8 °F as described above. Some examples clarify this relationship: • At 32 °C (90 °F) and 15% relative humidity, air may be cooled to nearly 16 °C (61 °F). The dew point for these conditions is 2 °C (36 °F). • At 32 °C (90 °F) and 50% relative humidity, air may be cooled to about 24 °C (75 °F). The dew point for these conditions is 20 °C (68 °F). • At 40 °C (104 °F) and 15% relative humidity, air may be cooled to nearly 21 °C (70 °F). The dew point for these conditions is 8 °C (46 °F). (Cooling examples extracted from the June 25, 2000 University of Idaho publication, " Homewise [13]"). Because evaporative coolers perform best in dry conditions, they are widely used and most effective in arid, desert regions such as the southwestern USA and northern Mexico.

Evaporative cooler 104 The same equation indicates why evaporative coolers are of limited use in highly humid environments: for example, a hot August day in Tokyo may be 30 °C (86 °F), 85% relative humidity, 1,005 hPa pressure. This gives dew point 27.2 °C (81 °F) and wet-bulb temperature 27.88 °C (82.18 °F). According to the formula above, at 85% efficiency air may be cooled only down to 28.2 °C (82.8 °F) which makes it quite impractical. Comparison to air conditioning Comparison of evaporative cooling to phase-change air conditioning: Advantages Less expensive to install • Estimated cost for installation is about half that of central refrigerated air conditioning.[14] Less expensive to operate • Estimated cost of operation is 1/4 that of refrigerated air. A misting fan • Power consumption is limited to the fan and water pump. Because the water vapor is not recycled, there is no compressor that consumes most of the power in closed-cycle refrigeration. • The refrigerant is water. No special refrigerants, such as ammonia, sulfur dioxide or CFCs, are used that could be toxic, expensive to replace, contribute to ozone depletion and/or be subject to stringent licensing and environmental regulations. Ease of maintenance • The only two mechanical parts in most basic evaporative coolers are the fan motor and the water pump, both of which can be repaired at low cost and often by a mechanically inclined homeowner. Ventilation air • The constant and high volumetric flow rate of air through the building reduces the "age-of-air" in the building dramatically. • Evaporative cooling increases humidity. In dry climates, this may improve comfort and decrease static electricity problems. • The pad itself acts as a rather effective air filter when properly maintained; it is capable of removing a variety of contaminants in air, including urban ozone caused by pollution, regardless of very dry weather. Refrigeration-based cooling systems lose this ability whenever there is not enough humidity in the air to keep the evaporator wet while providing a constant trickle of condensate that washes out dissolved impurities removed from the air. Disadvantages Performance • High dewpoint (humidity) conditions decrease the cooling capability of the evaporative cooler. • No dehumidification. Traditional air conditioners remove moisture from the air, except in very dry locations where recirculation can lead to a buildup of humidity. Evaporative cooling adds moisture, but in dry climates, dryness may improve thermal comfort at higher temperatures. Comfort • The air supplied by the evaporative cooler is typically 80–90% relative humidity; very humid air reduces the evaporation rate of moisture from the skin, nose, lungs, and eyes.

Evaporative cooler 105 • High humidity in air accelerates corrosion, particularly in the presence of dust. This can considerably shorten the life of electronic and other equipment. • High humidity in air may cause condensation of water. This can be a problem for some situations (e.g., electrical equipment, computers, paper, books, old wood). Water • Evaporative coolers require a constant supply of water to wet the pads. • Water high in mineral content will leave mineral deposits on the pads and interior of the cooler. Depending on the type and concentration of minerals, possible safety hazards during the replacement and waste removal of the pads could be present. Bleed-off and refill (purge pump) systems may reduce this problem. • The water supply line may need protection against freeze bursting during off-season, winter temperatures. The cooler itself needs to be drained too, as well as cleaned periodically and the pads replaced. Mosquitoes In India, an evaporative cooler is a common place for mosquito breeding. Indian authorities consider a poorly maintained cooler to be a big threat to health. See Dengue and Coolers [15] Miscellaneous • Odors and other outdoor contaminants may be blown into the building unless sufficient filtering is in place. • Mold and bacteria may be dispersed into interior air from poorly maintained or defective systems, causing Sick Building Syndrome • Asthma patients may need to avoid poorly maintained evaporatively cooled environments. • A sacrificial anode may be required to prevent excessive evaporative cooler corrosion. • Wood wool of dry cooler pads can catch fire even by small sparks. References [1] Kheirabadi, Masoud (1991). Iranian cities: formation and development. Autin, TX: University of Texas Press. p. 36. ISBN 978-0-292-72468-6. [2] "Statistical Center of Iran > Home" (http:/ / www. amar. org. ir/ Default. aspx?tabid=133) (in Farsi). Teheran: Statistical Centere of Iran. . Retrieved 25 February 2012. [3] John Zellweger (1906). "Air filter and cooler" (http:/ / www. google. com/ patents?=04pHAAAAEBAJ& printsec=abstract& zoom=4& dq=cooling+ excelsior& as_drrb_is=b& as_minm_is=1& as_miny_is=1799& as_maxm_is=1& as_maxy_is=1910& num=30#PPA2,M1). U.S. patent 838602. . [4] Bryant Essick (1945). "Pad for evaporative coolers" (http:/ / www. google. com/ patents?id=Z2BKAAAAEBAJ& pg=PA1& dq=excelsior+ evaporative-cooler& as_drrb_is=b& as_minm_is=1& as_miny_is=1900& as_maxm_is=1& as_maxy_is=1950& num=30& rview=1& source=gbs_selected_pages& cad=0_1#PPA1,M1). U.S. patent 2391558. . [5] Scott Landis (1998). The Workshop Book (http:/ / books. google. com/ books?id=bs7I7qf5cUQC& pg=PA120& dq=evaporative+ cooler+ "squirrel+ cage"+ southwest+ popular). Taunton Press. p. 120. ISBN 978-1-56158-271-6. . [6] Arthur William Gutenberg (1955). The Economics of the Evaporative Cooler Industry in the Southwestern United States (http:/ / books. google. com/ books?id=uq1EAAAAIAAJ). Stanford University Graduate School of Business. p. 167. . [7] Such units were mounted on the passenger-side window of the vehicle; the window was rolled nearly all the way up, leaving only enough space for the vent which carried the cool air into the vehicle. [8] AZEVAP: History of Evaporative Cooling Technology (http:/ / www. azevap. com/ EvaporativeCooling/ historytechnology. php) [9] www.1900s.org.uk: Food storage in a working class London household in the 1900s (http:/ / www. 1900s. org. uk/ 1900s-food-storage. htm) [10] Gordon B. Bonan. Forests and Climate Change: Forcings, Feedbacks, and the Climate Benefits of Forests. 13 June 2008 Vol. 320 Science [11] see Independent Testing tab, Thermodynamic performance assessment of a novel air cooling cycle and other papers http:/ / www. coolerado. com/ products/ material-resource-center/ [12] http:/ / www. cool-off. com/ faqs. html [13] http:/ / info. ag. uidaho. edu/ homewise/ homewise_062500. htm [14] John Krigger and Chris Dorsi (2004). Residential Energy: Cost Savings and Comfort for Existing Buildings (http:/ / books. google. com/ books?id=7HlKF4trR-YC& pg=PA207& dq=evaporative+ coolers+ cost+ install#v=onepage& q=evaporative coolers cost install& f=false) (4th ed.). Saturn Resource Management. p. 207. ISBN 978-1-880120-12-5. . [15] http:/ / www. thenewstribe. com/ 2011/ 08/ 20/ dengue-can-spread-from-unclean-coolers/

Evaporative cooler 106 External links • Holladay, April (2001). "A swamp cooler cools air by evaporation" (http://www.wonderquest.com/ swamp-coolers.htm). WonderQuest Weekly Q&A science column. USAToday.com. Retrieved 2006–07–14. • PATH Tech Inventory: Two Stage Evaporative Cooler (http://www.toolbase.org/Techinventory/TechDetails. aspx?ContentDetailID=789&BucketID=6&CategoryID=6) • PATH Tech Inventory: Evaporative Cooler (http://www.toolbase.org/TechInventory/TechDetails. aspx?ContentDetailID=749&BucketID=1&CategoryID=6) • Evaporative cooling simulation (http://www.colorado.edu/physics/2000/bec/evap_cool.html) • Coolerado indirect evaporative cooling (http://www1.eere.energy.gov/femp/pdfs/tir_coolerado.pdf) • Air Conditioning Repair (http://dallas-air-conditioning-repair.net) Self Help Air Conditioning Repair • Innovative Evaporative and Thermally Activated Technologies Improve Air Conditioning (http://www.nrel. gov/innovation/pdfs/47566.pdf) Evaporator An evaporator is a device used to turn the liquid form of a chemical into its gaseous form. The liquid is evaporated, or vaporized, into a gas. Uses An evaporator is used in an air-conditioning system to allow a compressed cooling chemical, such as Freon or R-410A, to evaporate from liquid to gas while absorbing heat in the process.[1] It can also be used to remove water or other liquids from mixtures. The process of evaporation is widely used to concentrate foods and chemicals as well as salvage solvents. In the concentration process, the goal of evaporation is to vaporize most of the water from a solution which contains the desired product. In the case of desalination of sea water, the reverse purpose applies; evaporation removes the desirable drinking water from the undesired product, salt. One of the most important applications of evaporation is in the food and beverage industry. Foods or beverages that need to last for a considerable amount of time or need to have certain consistency, like coffee, go through an evaporation step during processing. In the pharmaceutical industry, the evaporation process is used to eliminate excess moisture, providing an easily handled product and improving product stability. Preservation of long-term activity or stabilization of enzymes in laboratories are greatly assisted by the evaporation process. Another example of evaporation is in the recovery of sodium hydroxide in kraft pulping.[2] Cutting down waste-handling cost is another major reason for large companies to use evaporation applications. Legally, all producers of waste must dispose of waste using methods compatible with environmental guidelines; these methods are costly. By removing moisture through vaporization, industry can greatly reduce the amount of waste product that must be processed. Energetics Water can be removed from solutions in ways other than evaporation, including membrane processes, liquid-liquid extractions, crystallization, and precipitation. Evaporation can be distinguished from some other drying methods in that the final product of evaporation is a concentrated liquid, not a solid. It is also relatively simple to use and understand since it has been widely used on a large scale, and many techniques are generally well known. In order to concentrate a product by water removal, an auxiliary phase is used which allows for easy transport of the solvent (water) rather than the solute. Water vapor is used as the auxiliary phase when concentrating non-volatile

Evaporator 107 components, such as proteins and sugars. Heat is added to the solution, and part of the solvent is converted into vapor. Heat is the main tool in evaporation, and the process occurs more readily at high temperature and low pressures. Heat is needed to provide enough energy for the molecules of the solvent to leave the solution and move into the air surrounding the solution. The energy needed can be expressed as an excess thermodynamic potential of the water in the solution. Leading to one of the biggest problems in industrial evaporation, the process requires enough energy to remove the water from the solution and to supply the heat of evaporation. When removing the water, more than 99% of the energy needed goes towards supplying the heat of evaporation. The need to overcome the surface tension of the solution also requires energy. The energy requirement of this process is very high because a phase transition must be caused; the water must go from a liquid to a vapor. When designing evaporators, engineers must quantify the amount of steam needed for every mass unit of water removed when a concentration is given. An energy balance must be used based on an assumption that a negligible amount of heat is lost to the systems surroundings. The heat that needs to be supplied by the condensing steam will approximately equal the heat needed to vaporize the water. Another consideration is the size of the heat exchanger which affects the heat transfer rate. Some common terms: A = heat transfer area, and q = overall heat transfer rate. How an evaporator works The solution containing the desired product is fed into the evaporator and passes across a heat source. The applied heat converts the water in the solution into vapor. The vapor is removed from the rest of the solution and is condensed while the now-concentrated solution is either fed into a second evaporator or is removed. The evaporator, as a machine, generally consists of four sections. The heating section contains the heating medium, which can vary. Steam is fed into this section. The most common medium consists of parallel tubes but others have plates or coils. The concentrating and separating section removes the vapor being produced from the solution. The condenser condenses the separated vapor, then the vacuum or pump provides pressure to increase circulation. Types of evaporators used today Natural/forced circulation evaporator Natural circulation evaporators are based on the natural circulation of the product caused by the density differences that arise from heating. In an evaporator using tubing, after the water begins to boil, bubbles will rise and cause circulation, facilitating the separation of the liquid and the vapor at the top of the heating tubes. The amount of evaporation that takes place depends on the temperature difference between the steam and the solution. Problems can arise if the tubes are not well-immersed in the solution. If this occurs, the system will be dried out and circulation compromised. In order to avoid this, forced circulation can be used by inserting a pump to increase pressure and circulation. Forced circulation occurs when hydrostatic head prevents boiling at the heating surface. A pump can also be used to avoid fouling that is caused by the boiling of liquid on the tubes; the pump suppresses bubble formation. Other problems are that the residing time is undefined and the consumption of steam is very high, but at high temperatures, good circulation is easily achieved.

Evaporator 108 Falling film evaporator This type of evaporator is generally made of long tubes (4–8 meters or 13–26 ft in length) which are surrounded by steam jackets. The uniform distribution of the solution is important when using this type of evaporator. The solution enters and gains velocity as it flows downward. This gain in velocity is attributed to the vapor being evolved against the heating medium, which flows downward as well. This evaporator is usually applied to highly viscous solutions, so it is frequently used in the chemical, food, and fermentation industries. Rising film (Long Tube Vertical) evaporator In this type of evaporator, boiling takes place inside the tubes, due to heating made (usually by steam) outside the same. Submergence is therefore not desired; the creation of water vapor bubbles inside the tube creates an ascensional flow enhancing the heat transfer coefficient. This type of evaporator is therefore quite efficient, the disadvantage being to be prone to quick scaling of the internal surface of the tubes. This design is then usually applied to clear, non-salting solutions. Tubes are usually quite long, typically 4+ meters (13+ ft). Sometimes a small recycle is provided. Sizing this type of evaporator is usually a delicate task, since it requires a precise evaluation of the actual level of the process liquor inside the tubes. Recent applications tend to favor the falling-film pattern rather than rising-film. Climbing and falling-film plate evaporator Climbing and falling-film plate evaporators have a relatively large surface area. The plates are usually corrugated and are supported by frame. During evaporation, steam flows through the channels formed by the free spaces between the plates. The steam alternately climbs and falls parallel to the concentrated liquid. The steam follows a co-current, A rising film evaporator counter-current path in relation to the liquid. The concentrate and the vapor are both fed into the separation stage where the vapor is sent to a condenser. This type of plate evaporator is frequently applied in the dairy and fermentation industries since they have spatial flexibility. A negative point of this type of evaporator is that it is limited in its ability to treat viscous or solid-containing products. There are other types of plate evaporators, which work with only climbing film. In the climbing film type much higher viscosities can be used compared to the climbing and falling film described above. Multiple-effect evaporators Unlike single-stage evaporators, these evaporators can be made of up to seven evaporator stages or effects. The energy consumption for single-effect evaporators is very high and makes up most of the cost for an evaporation system. Putting together evaporators saves heat and thus requires less energy. Adding one evaporator to the original decreases the energy consumption to 50% of the original amount. Adding another effect reduces it to 33% and so on. A heat-saving-percent equation can be used to estimate how much one will save by adding a certain amount of effects. The number of effects in a multiple-effect evaporator is usually restricted to seven because after that, the equipment cost starts catching up to the money saved from the energy-requirement drop. There are two types of feeding that can be used when dealing with multiple-effect evaporators. Forward feeding takes place when the product enters the system through the first effect, which is at the highest temperature. The

Evaporator 109 product is then partially concentrated as some of the water is transformed into vapor and carried away. It is then fed into the second effect which is a little lower in temperature. The second effect uses the heated vapor created in the first stage as its heating source (hence the saving in energy expenditure). The combination of lower temperatures and higher viscosities in subsequent effects provides good conditions for treating heat-sensitive products, such as enzymes and proteins. In using this system, an increase in the heating surface area of subsequent effects is required. Another way to proceed is by using backward feeding. In this process, the dilute products are fed into the last effect which has the lowest temperature and are transferred from effect to effect, with the temperature increasing. The final concentrate is collected in the hottest effect, which provides an advantage in that the product is highly viscous in the last stages, and so the heat transfer is considerably better. Problems Technical problems can arise during evaporations, especially when the process is applied to the food industry. Some evaporators are sensitive to differences in viscosity and consistency of the dilute solution. These evaporators could work inefficiently because of a loss of circulation. The pump of an evaporator may need to be changed if the evaporator needs to be used to concentrate a highly viscous solution. Fouling also occurs when hard deposits form on the surfaces of the heating mediums in the evaporators. In foods, proteins and polysaccharides can create such deposits that reduce the efficiency of heat transfer. Foaming can also create a problem since dealing with the excess foam can be costly in time and efficiency. Antifoam agents are to be used, but only a few can be used when food is being processed. Corrosion can also occur when acidic solutions such as citrus juices are concentrated. The surface damage caused can shorten the long-life of evaporators. Quality and flavor of food can also suffer during evaporation. Overall, when choosing an evaporator, the qualities of the product solution need to be taken into heavy consideration. Marine Use Large ships usually carry evaporating plants to produce fresh water, thus reducing their reliance on shore-based supplies. Steam ships must be able to produce high-quality distillate in order to maintain boiler-water levels. Diesel-engined ships often utilise waste heat as an energy source for producing fresh water. In this system, the engine-cooling water is passed through a heat exchanger, where it is cooled by concentrated seawater (brine). Because the cooling water (which is chemically treated fresh water) is at a temperature of 70–80 °C (158–176 °F), it would not be possible to flash off any water vapour unless the pressure in the heat exhanger vessel was dropped. To alleviate this problem, a brine-air ejector venturi pump is used to create a vacuum inside the vessel. Partial evaporation is achieved, and the vapour passes through a demister before reaching the condenser section. Seawater is pumped through the condenser section to cool the vapour sufficiently to precipitate it. The distillate gathers in a tray, from where it is pumped to the storage tanks. A salinometer monitors salt content and diverts the flow of distillate from the storage tanks if the salt content exceeds the alarm limit. Sterilisation is carried out after the evaporator. Evaporators are usually of the shell-and-tube type (known as an Atlas Plant) or of the plate type (such as the type designed by Alfa Laval). Temperature, production and vacuum are controlled by regulating the system valves.

Evaporator 110 Seawater temperature can interfere with production, as can fluctuations in engine load. For this reason, the evaporator is adjusted as seawater temperature changes, and shut down altogether when the ship is manoeuvring. An alternative in some vessels, such as naval ships and passenger ships, is the use of the reverse osmosis principle for fresh-water production, instead of using evaporators. References [1] "How Does an Air Conditioner Work?" (http:/ / www. energyquest. ca. gov/ how_it_works/ air_conditioner. html). . Retrieved 27 April 2012. [2] Smook, G.A. (1990), Handbook of pulp and paper technology. Angus Wilde Publications. Sources • Fennema, Owen R., Marcus Karel, and Daryl B. Lund. Physical Principles of Food Preservation. Marcel Deker, Inc. New York and Basel, 1975. • Krijgsman, Ir J., Principal Scientist and Research Project Manager, Gist-brocades, Delft and Delft University of Technology, Delft, and The Netherlands. Product Recovery in Bioprocess Technology. Butterworth-Heinemann, 1992. Evaporator (marine) An evaporator, distiller or distilling apparatus is a piece of ships equipment used to produce fresh drinking water from sea water by distillation. As fresh water is bulky, may spoil in storage, and is an essential supply for any long voyage, the ability to produce more in mid-ocean is clearly important for any ship. Although distillers are often associated with steam ships, their use pre-dates this. Nelsons HMS Victory was fitted with distilling apparatus in her galley.[1] Distilling apparatus was only fitted to larger warships and some exploratory ships at this time: a warships large crew needed more water and they could ill afford the space to carry enough. Cargo ships, and their smaller crews, merely carried their supplies with them. Boiler feedwater With the development of the marine steam engine, their boilers also required a continual supply of feedwater. This steam evaporator aboard HMS Belfast Early boilers used seawater directly, but this gave problems with the distilled up to six tons of fresh water per hour for build-up of brine and scale.[2] For efficiency, as well as conserving the boiler and for drinking. feedwater, marine engines have usually been condensing engines. By 1865, the use of an improved surface condenser permitted the use of fresh water feed,[3] as the additional feedwater now required was only the small amount required to make up for losses, rather than the total passed through the boiler. Despite this, a large warship could still require up to 100 tons of fresh water makeup to the feedwater system, when under full power.[4] Attention was also paid to de-aereating feedwater, to further reduce boiler corrosion.[3] The distillation system for boiler feedwater at this time was usually termed an evaporator, partly to distinguish it from a separate system or distiller used for drinking water. Separate systems were often used, especially in early systems, owing to the problem of contamination from oily lubricants in the feedwater system and because of the greatly different capacities required in larger ships. In time, the two functions became combined and the two terms

Evaporator (marine) 111 were applied to the separate components of the system. Potable water distillers The first water supply by distillation of boiler steam appeared on early paddle steamers and used a simple iron box in the paddle boxes, cooled by water splash. A steam supply direct from the boiler, avoiding the engine and its lubricants, was led to them.[1] With the development of steam heating jackets around the cylinders of engines such as the trunk engine, the exhaust from this source, again unlubricated, could be condensed.[1] Evaporators Combined supply The first distilling plants that boiled a separate water supply from that of the main boiler, appeared around 1867.[1] These are not directly heated by a flame, but have a primary steam circuit using main boiler steam through coils within a steam drum or evaporator.[5] The distillate from this vessel then passes to an adjacent vessel, the distilling condenser.[5] As these evaporators used a clean seawater supply directly, rather than contaminated water from the boiler circuit, they could be used to supply both feedwater and drinking water. These double distillers appeared around 1884.[1] For security against failure, ships except the smallest were fitted with two sets.[5] Vacuum evaporators Evaporators consume a great deal of steam, and thus fuel, in relation to the quantity of fresh water produced. Their efficiency is improved by working them at a partial vacuum, supplied by the main engine condensers.[5][6][7] On modern diesel-powered ships, this vacuum can instead be produced by an ejector, usually worked by the output from the brine pump. Working under vacuum also reduces the temperature required to boil seawater and thus permits evaporators to be used with lower-temperature waste heat from the diesel cooling system. Scale One of the greatest operational problems with an evaporator is the build-up of scale. Its design is tailored to reduce this, and to make its cleaning as effective as possible. The usual design, as developed by Weir and the Admiralty, is for a vertical cylindrical drum, heated by steam-carrying drowned coils in the lower portion.[6] As they are entirely submerged, they avoid the most active region for the deposition of scale, around the waterline. Each coil consists of one or two spirals in a flat plane. Each coil is easily removed for cleaning, being fastened by individual pipe unions through the side of the evaporator. A large door is also provided, allowing the coils to be removed or replaced. Cleaning may be carried out mechanically, with a manual scaling hammer.[7] This also has a risk of mechanical damage to the tubes, as the slightest pitting tends to act as a nucleus for scale or corrosion.[7] It is also common practice to break light scaling free by thermal shock, by passing steam through the coils without cooling water present[5][7] or by heating the coils, then introducing cold seawater.[8] Despite the obvious salinity of seawater, salt is not a problem for deposition until it reaches the saturation concentration.[2] As this is around seven times that of seawater and evaporators are only operated to a concentration of two and a half times,[9] this is not a problem in service. A greater problem for scaling is the deposition of calcium sulphate.[6] The saturation point for this compound decreases with temperature above 60°C, so that beginning from around 90°C a hard and tenuous deposit is formed. To further control scale formation, equipment may be provided to automatically inject a weak citric acid solution into the seawater feed. The ratio is 1:1350, by weight of seawater.[10]

Evaporator (marine) 112 Compound evaporators Operation of an evaporator represents a costly consumption of main boiler steam, thus fuel. Evaporators for a warship must also be adequate to supply the boilers at continuous full power when required, even though this is rarely required. Varying the vacuum under which the evaporator works, and thus the boiling point of the feedwater, may optimise production for either maximum output, or better efficiency. Greatest output is achieved when the evaporator operates at near atmospheric pressure and a high temperature (for saturated steam this will be at a limit of 100°C), which may then have an efficiency of 0.87 kg of feedwater produced for each kg of steam supplied.[6] If condenser vacuum is increased to its maximum, evaporator temperature may be reduced to around 72°C. Efficiency increases until the mass of feedwater produced almost equals that of the supplied steam, although production is now restricted to 0.86 of the previous maximum.[6] Evaporators are generally installed as a set, where two evaporators are coupled to a single distiller.[11] For reliability, large ships will then have a pair of these sets.[11] It is possible to arrange these sets of evaporators in either parallel or in series, for either maximum or most efficient production.[6] This arranges the two evaporators so that the first operates at atmospheric pressure and high temperature (the maximum output case), but then uses the resultant hot output from the first evaporator to drive a second, running at maximum vacuum and low temperature (the maximum efficiency case).[11] The overall output of feedwater may exceed the weight of steam first supplied, as up to 1.6 times as much. Capacity is however reduced, to 0.72 times the maximum.[6] Evaporator pumps The unevaporated seawater in an evaporator gradually becomes a concentrated brine and, like the early steam boilers with seawater feed, this brine must be intermittently blown down every six to eight hours and dumped overboard.[5] Early evaporators were simply mounted high-up and dumped their brine by gravity.[1] As the increasing complexity of surface condensers demanded better feedwater quality, a pump became part of the evaporator equipment.[5] This pump had three combined functions as a seawater feed pump, a fresh water delivery pump and a brine extraction pump, each of progressively smaller capacity.[4] The brine salinity was an important factor in evaporator efficiency: too dense encouraged scale formation, but too little represented a waste of heated seawater. The optimum operating salinity was thus fixed at three times that of seawater, and so the brine pump had to remove at least one third of the total feedwater supply rate.[12] These pumps resembled the steam-powered reciprocating feedwater pumps already in service. They were usually produced by the well-known makers, such as G & J Weir. Vertical and horizontal pumps were used, although horizontal pumps were favoured as they encouraged the de-aeration of feedwater. Electrically powered rotary centrifugal pumps were later adopted, as more efficient and more reliable. There were initial concerns whether these would be capable of pumping brine against the vacuum of the evaporator and so there was also a transitional type where a worm gear-driven plunger pump for brine was driven from the rotary shaft.[4] Flash distillers A later form of marine evaporator is the flash distiller.[13] Heated seawater is pumped into a vacuum chamber, where it flashes into pure water vapour. This is then condensed for further use. As the use of vacuum reduces the vapour pressure, the seawater need only be raised to a temperature of 77 °C (171 °F).[14] Both evaporator and distiller are combined into a single chamber, although most plants use two joined chambers, worked in series. The first chamber is worked at 23.5 inHg vacuum, the second at 26-27 inHg.[13] Seawater is supplied to the distiller by a pump at around 20 psi. The cold seawater passes through a condenser coil in the upper part of each chamber before being heated by steam in an external feedwater heater. The heated seawater enters the lower part of the first chamber, then drains over a weir and passes to the second chamber, encouraged by the differential vacuum between them. The brine produced by a flash distiller is only slightly concentrated and is pumped overboard continuously.[13]

Evaporator (marine) 113 Fresh water vapour rises through the chambers and is condensed by the seawater coils. Baffles and catchment trays capture this water in the upper part of the chamber. Vacuum itself is maintained by steam ejectors.[13] The advantage of the flash distiller over the compound evaporator is its greater operating efficiency, in terms of heat supplied. This is due to working under vacuum, thus low temperature, and also the regenerative use of the condenser coils to pre-heat the seawater feed.[13] A limitation of the flash distiller is its sensitivity to seawater inlet temperature, as this affects the efficiency of the condenser coils. In tropical waters, the distiller flowrate must be throttled to maintain effective condensation.[13] As these systems are more modern, they are generally fitted with an electric salinometer and some degree of automatic control.[13] Vapour-compression distillers Diesel-powered motorships do not use steam boilers as part of their main propulsion system and so may not have steam supplies available to drive evaporators. Some do, as they use auxiliary boilers for non-propulsion tasks such as this. Such boilers may even be heat-recovery boilers that are heated by the engine exhaust.[15] Where no adequate steam supply is available, a vapour-compression distiller is used instead. This is driven mechanically, either electrically or by its own diesel engine.[16] Seawater is pumped into an evaporator, where it is boiled by a heating coil. Vapour produced is then compressed, raising its temperature. This heated vapour is used to heat the evaporator coils. Condensate from the coil outlet provides the One of two vapour-compression fresh water supply. To start the cycle, an electric pre-heater is used to heat the distillers in the engine room of WW2 first water supply. The main energy input to the plant is in mechanically driving submarine USS Pampanito (SS-383) the compressor, not as heat energy.[16] Both the fresh water production and the waste brine from the evaporator are led through an output cooler. This acts as a heat exchanger with the inlet seawater, pre-heating it to improve efficiency. The plant may operate at either a low pressure or slight vacuum, according to design. As the evaporator works at pressure, not under vacuum, boiling may be violent. To avoid the risk of priming and a carry over of saltwater into the vapour, the evaporator is divided by a bubble cap separator.[16] Submarines Vapour-compression distillers were installed on US submarines shortly before World War 2.[17] Early attempts had been made with evaporators running from diesel engine exhaust heat, but these could only be used when the submarine was running at speed on the surface. A further difficulty with submarines was the need to produce high-quality water for topping up their large storage batteries. Typical consumption on a war patrol was around 500 US gallons per day for hotel services, drinking, cooking, washing[18] etc. and also for topping up the diesel engine cooling system. A further 500 gallons per week was required for the batteries.[17] The standard Badger model X-1 for diesel submarines could produce 1,000 gallons per day. Tank capacity of 5,600 gallons (1,200 of which was battery water) was provided, around 10 days reserve.[17] With the appearance of nuclear submarines and their plentiful electricity supply, even larger plant could be installed. The X-1 plant was designed so that it could be operated when snorkelling, or even when completely submerged. As the ambient pressure increased when submerged, and thus the boiling point, additional heat was required in these submarine distillers and so they were designed to run with electric heat continuously.[17]

Evaporator (marine) 114 Notes [1] Rippon 1998, pp.78-79 [2] Rippon 1998, p.30 [3] Rippon 1998, p.60 [4] Rippon 1998, p.164 [5] Stokers Manual (1912 ed.). Admiralty, via HMSO, via Eyre & Spottiswoode. 1901. pp. 42–45. [6] Rippon 1998, pp.160-164 [7] Drover, Engineer-Captain F.J., RN (1925). Marine Engineering Repairs. Chapman & Hall. pp. 105–106. [8] Naval Marine Engineering Practice 1971, pp. 227 [9] Machinery Handbook 1941, pp. 156-166 [10] Naval Marine Engineering Practice 1971, pp. 225-226 [11] Machinery Handbook 1941, pp. 159-160 [12] Rippon 1998, p.161 [13] Naval Marine Engineering Practice 1971, pp. 212-215 [14] A temperature of at least 71 °C (160 °F) is required, for sterilisation purposes. [15] Milton, J. H. (1961) [1953]. Marine Steam Boilers (2nd ed.). Newnes. pp. 119–137. [16] Naval Marine Engineering Practice 1971, pp. 230-232 [17] Fleet Submarine, Distilling Systems [18] Although German U-boats relied on saltwater soap, US practice was to fit adequate distilling plant. References Bibliography • Rippon, Commander P.M., RN (1998). The evolution of engineering in the Royal Navy. Vol 1: 1827-1939. Spellmount. ISBN 0-946771-55-3. • BR 77 Machinery Handbook. later replacement for the Stokers Manual. Admiralty, via HMSO. 1941. • Naval Marine Engineering Practice. later replacement for the Stokers Manual. Vol 1. Royal Navy, via HMSO. 1971 [1959]. ISBN 11-770223-4. • Submarine Distilling Systems (http://www.maritime.org/fleetsub/still/chap1.htm). The Fleet Type Submarine. 5. Bureau of Naval Personnel. January 1955. Navpers 16170.

Falling film evaporator 115 Falling film evaporator A falling film evaporator is an industrial device to concentrate solutions, especially with heat sensitive components. The evaporator is a special type of heat exchanger. In general evaporation takes place inside vertical tubes, but there are also applications where the process fluid evaporates on the outside of horizontal or vertical tubes. In all cases, the process fluid to be evaporated flows downwards by gravity as a continuous film. The fluid will create a film along the tube walls, progressing downwards (falling) - hence the name. The fluid distributor has to be designed carefully in order to maintain an even liquid distribution for all tubes along which the solution falls. In the majority of applications the heating medium is placed on the outside of the tubes. High heat transfer coefficients are required in order to achieve equally balanced heat transfer resistances. Therefore, condensing steam is commonly used as a heating medium. For internally evaporating fluids, separation between the liquid phase A falling film triple effect evaporator for caustic soda (the solution) and the gaseous phase takes place inside the tubes. In order to maintain conservation of mass as this process proceeds, the downward vapor velocity increases, increasing the shear force acting on the liquid film and therefore also the velocity of the solution. The result can be a high film velocity of a progressively thinner film resulting in increasingly turbulent flow. The combination of these effects allows very high heat transfer coefficients. The heat transfer coefficient on the evaporating side of the tube is mostly determined by the hydrodynamic flow conditions of the film. For low massflows or high viscosities the film flow can be laminar, in which case heat transfer is controlled purely by conduction through the film. Therefore in this condition the heat transfer coefficient decreases with increased mass flow. With increased mass flow the film becomes wavy laminar and then turbulent. Under turbulent conditions the heat transfer coefficient increases with increased flow. Evaporation takes place at very low mean temperature differences between heating medium and process stream, typically between 3 - 6K, therefore these devices are ideal for heat recovery in multi stage processes[1] [2] A further advantage of the falling film evaporator is the very short residence time of the liquid and the absence of superheating of the same. Not considering the vapour separator, the residence time inside the tubes is measured in seconds, making it ideal for heat-sensitive product such as milk, fruit juice, pharmaceuticals and many others. Falling Film Evaporators are also characterised by very low pressure drops. Therefore they are often used in deep vacuum applications.

Falling film evaporator 116 References [1] Chun, K. R.; Seban, R. A. (1971). "Heat transfer to evapaporating liquid films". Journal of Heat Transfer 93 (197): 391–396. doi:10.1115/1.3449836. [2] Alhusseini, A; Tuzla, K; Chen, J (1998). "Falling film evaporation of single component liquids". International Journal of Heat and Mass Transfer 41 (12): 1623–1632. doi:10.1016/S0017-9310(97)00308-6. External links • Falling Film Evaporators (http://www.fallfilmverdampfer.info/english/applications) • Wolverine Tube Heat Transfer Databooks (http://www.wlv.com/products/thermal-management-databooks. html) Ficks laws of diffusion Ficks laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855. Ficks first law Ficks first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative). In one (spatial) dimension, the law is Molecular diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the where solute diffuses to fill the whole container. Top: A single molecule • is the "diffusion flux" [(amount of substance) per moves around randomly. Middle: With more molecules, there is a clear trend where the solute fills the container more and more unit area per unit time], example . uniformly. Bottom: With an enormous number of solute molecules, measures the amount of substance that will flow all randomness is gone: The solute appears to move smoothly and through a small area during a small time interval. systematically from high-concentration areas to low-concentration areas. This smooth flow is described by Ficks laws. • is the diffusion coefficient or diffusivity in dimensions of [length2 time−1], example • (for ideal mixtures) is the concentration in dimensions of [amount of substance per unit volume], example • is the position [length], example is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes-Einstein relation. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0.6x10−9 to 2x10−9 m2/s. For biological molecules the diffusion coefficients normally range from 10−11 to 10−10 m2/s. In two or more dimensions we must use , the del or gradient operator, which generalises the first derivative, obtaining

Ficks laws of diffusion 117 . The driving force for the one-dimensional diffusion is the quantity which for ideal mixtures is the concentration gradient. In chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of chemical potential of this species. Then Ficks first law (one-dimensional case) can be written as: where the index i denotes the ith species, c is the concentration (mol/m3), R is the universal gas constant (J/(K mol)), T is the absolute temperature (K), and μ is the chemical potential (J/mol). If the primary variable is mass fraction ( , given, for example, in ), then the equation changes to: where is the fluid density (for example, in ). Note that the density is outside the gradient operator. Ficks second law Ficks second law predicts how diffusion causes the concentration to change with time: where • is the concentration in dimensions of [(amount of substance) length−3], example • is time [s] • is the diffusion coefficient in dimensions of [length2 time−1], example • is the position [length], example It can be derived from Ficks First law and the mass conservation in absence of any chemical reactions: Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiation and multiply by the constant: and, thus, receive the form of the Ficks equations as was stated above. For the case of diffusion in two or more dimensions Ficks Second Law becomes , which is analogous to the heat equation. If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, Ficks Second Law yields An important example is the case where is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant , the solution for the concentration will be a linear change of concentrations along . In two or more dimensions we obtain

Ficks laws of diffusion 118 which is Laplaces equation, the solutions to which are called harmonic functions by mathematicians. Example solution in one dimension: diffusion length A simple case of diffusion with time t in one dimension (taken as the x-axis) from a boundary located at position , where the concentration is maintained at a value is . where erfc is the complementary error function. The length is called the diffusion length and provides a measure of how far the concentration has propagated in the x-direction by diffusion in time t (Bird, 1976). As a quick approximation of the error function, the first 2 terms of the Taylor series can be used: If is time-dependent, the diffusion length becomes . This idea is useful for estimating a diffusion length over a heating and cooling cycle, where D varies with temperature. Generalizations 1. In the inhomogeneous media, the diffusion coefficient varies in space, . This dependence does not affect Ficks first law but the second law changes: 2. In the anisotropic media, the diffusion coefficient depends on the direction. It is a symmetric tensor . Ficks first law changes to For the diffusion equation this formula gives The symmetric matrix of diffusion coefficients should be positive definite. It is needed to make the right hand side operator elliptic. 3. For the inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in 4. The approach based on the Einsteins mobility and Teorell formula gives the following generalization of Ficks equation for the multicomponent diffusion of the perfect components: where are concentrations of the components and is the matrix of coefficients. Here, indexes i,j are related to the various components and not to the space coordinates. The Chapman-Enskog formulas for diffusion in gases include exactly the same terms. It should be stressed that these physical models of diffusion are different from the toy-models which are valid for very small

Ficks laws of diffusion 119 deviations from the uniform equilibrium. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation. For anisotropic multicomponent diffusion coefficients one needs 4-index quantities, for example, , where i, j are related to the components and α, β=1,2,3 correspond to the space coordinates. History In 1855, physiologist Adolf Fick first reported[1][2] his now-well-known laws governing the transport of mass through diffusive means. Ficks work was inspired by the earlier experiments of Thomas Graham, which fell short of proposing the fundamental laws for which Fick would become famous. The Ficks law is analogous to the relationships discovered at the same epoch by other eminent scientists: Darcys law (hydraulic flow), Ohms law (charge transport), and Fouriers Law (heat transport). Ficks experiments (modeled on Grahams) dealt with measuring the concentrations and fluxes of salt, diffusing between two reservoirs through tubes of water. It is notable that Ficks work primarily concerned diffusion in fluids, because at the time, diffusion in solids was not considered generally possible.[3] Today, Ficks Laws form the core of our understanding of diffusion in solids, liquids, and gases (in the absence of bulk fluid motion in the latter two cases). When a diffusion process does not follow Ficks laws (which does happen),[4][5] we refer to such processes as non-Fickian, in that they are exceptions that "prove" the importance of the general rules that Fick outlined in 1855. Applications Equations based on Ficks law have been commonly used to model transport processes in foods, neurons, biopolymers, pharmaceuticals, porous soils, population dynamics, nuclear materials, semiconductor doping process, etc. Theory of all voltammetric methods is based on solutions of Ficks equation. A large amount of experimental research in polymer science and food science has shown that a more general approach is required to describe transport of components in materials undergoing glass transition. In the vicinity of glass transition the flow behavior becomes "non-Fickian". It can be shown that the Ficks law can be obtained from the Maxwell-Stefan equations[6] of multi-component mass transfer. The Ficks law is limiting case of the Maxwell-Stefan equations, when the mixture is extremely dilute and every chemical species is interacting only with the bulk mixture and not with other species. To account for the presence of multiple species in a non-dilute mixture, several variations of the Maxwell-Stefan equations are used. See also non-diagonal coupled transport processes (Onsager relationship). Biological perspective The first law gives rise to the following formula:[7] in which, • is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature. • is the difference in concentration of the gas across the membrane for the direction of flow (from to ). Ficks first law is also important in radiation transfer equations. However, in this context it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this situation, one can use a flux limiter. The exchange rate of a gas across a fluid membrane can be determined by using this law together with Grahams law.

Ficks laws of diffusion 120 Ficks flow in liquids When two miscible liquids are brought into contact, and diffusion takes place, the macroscopic (or average) concentration evolves following Ficks law. On a mesoscopic scale, that is, between the macroscopic scale described by Ficks law and molecular scale, where molecular random walks take place, fluctuations cannot be neglected. Such situations can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale. [8] In particular, fluctuating hydrodynamic equations include a Ficks flow term, with a given diffusion coefficient, along with hydrodynamics equations and stochastic terms describing fluctuations. When calculating the fluctuations with a perturbative approach, the zero order approximation is Ficks law. The first order gives the fluctuations, and it comes out that fluctuations contribute to diffusion. This represents somehow a tautology, since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by renormalizing fluctuating hydrodynamics equations. Semiconductor fabrication applications IC Fabrication technologies, model processes like CVD, Thermal Oxidation, and Wet Oxidation, doping, etc. use diffusion equations obtained from Ficks law. In certain cases, the solutions are obtained for boundary conditions such as constant source concentration diffusion, limited source concentration, or moving boundary diffusion (where junction depth keeps moving into the substrate). Derivation of Ficks 1st law in 1 dimension The following derivation is based on a similar argument made in Berg 1977 (see references). Consider a collection of particles performing a random walk in one dimension with length scale and time scale . Let be the number of particles at position at time . At a given time step, half of the particles would move left and half would move right. Since half of the particles at point move right and half of the particles at point move left, the net movement to the right is: The flux, J, is this net movement of particles across some area element of area a, normal to the random walk during a time interval . Hence we may write: Multiplying the top and bottom of the righthand side by and rewriting, we obtain: We note that concentration is defined as particles per unit volume, and hence . In addition, is the definition of the diffusion constant in one dimension, . Thus our expression simplifies to: In the limit where is infinitesimal, the righthand side becomes a space derivative:

Ficks laws of diffusion 121 Notes [1] A. Fick, Ann. der. Physik (1855), 94, 59, doi:10.1002/andp.18551700105 (in German). [2] A. Fick, Phil. Mag. (1855), 10, 30. (in English) [3] Jean Philibert, One and a Half Century of Diffusion: Fick, Einstein, before and beyond, Diffusion Fundamentals 2, 2005 1.1–1.10 (http:/ / www. uni-leipzig. de/ diffusion/ journal/ pdf/ volume2/ diff_fund_2(2005)1. pdf) [4] J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press. [5] A.N. Gorban, H.P. Sargsyan and H.A. Wahab (2011), Quasichemical Models of Multicomponent Nonlinear Diffusion (http:/ / arxiv. org/ pdf/ 1012. 2908v4. pdf), Mathematical Modelling of Natural Phenomena (http:/ / journals. cambridge. org/ action/ displayJournal?jid=MNP), Volume 6 / Issue 05, 184−262. [6] Taylor, Ross; R Krishna (1993). Multicomponent mass transfer. Wiley. [7] Physiology at MCG 3/3ch9/s3ch9_2 (http:/ / web. archive. org/ web/ 20080401093403/ http:/ / www. lib. mcg. edu/ edu/ eshuphysio/ program/ section3/ 3ch9/ s3ch9_2. htm) [8] D. Brogioli and A. Vailati, Diffusive mass transfer by nonequilibrium fluctuations: Ficks law revisited, Phys. Rev. E 63, 012105/1-4 (2001) (http:/ / arxiv. org/ abs/ cond-mat/ 0006163) References • W.F. Smith, Foundations of Materials Science and Engineering 3rd ed., McGraw-Hill (2004) • H.C. Berg, Random Walks in Biology, Princeton (1977) • R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley & sons, (1976) External links • Diffusion fundamentals (http://www.timedomaincvd.com/CVD_Fundamentals/xprt/intro_diffusion.html) • Diffusion in Polymer based Materials (http://www.composite-agency.com/messages/3875.html) • Ficks equations, Boltzmanns transformation, etc. (with figures and animations) (http://dragon.unideb.hu/ ~zerdelyi/Diffusion-on-the-nanoscale/node2.html) • Wilson, Bill. Ficks Second Law. Connexions. 21 Aug. 2007 (http://cnx.org/content/m1036/2.11/) • (http://webserver.dmt.upm.es/~isidoro/bk3/c11/Mass Transfer.htm)

First law of thermodynamics 122 First law of thermodynamics The first law of thermodynamics is a version of the law of conservation of energy, specialized for thermodynamical systems. It is usually formulated by stating that the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the amount of work done by the system on its surroundings. The law of conservation of energy can be stated: The energy of an isolated system is constant. Original statements The first explicit statement of the first law of thermodynamics, by Rudolf Clausius in 1850, referred to cyclic thermodynamic processes. "In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to the work done; and conversely, by the expenditure of an equal quantity of work an equal quantity of heat is produced."[1] Clausius stated the law also in another form, this time referring to the existence of a function of state of the system called the internal energy, and expressing himself in terms of a differential equation for the increments of a thermodynamic process. This equation may be translated into words as follows: In a thermodynamic process of a closed system, the increment in the internal energy is equal to the difference between the increment of heat accumulated by the system and the increment of work done by it.[2] Description The first law of thermodynamics was expressed in two ways by Clausius. One way referred to cyclic processes and the inputs and outputs of the system, but did not refer to increments in the internal state of the system. The other way referred to any incremental change in the internal state of the system, and did not expect the process to be cyclic. A cyclic process is one which can be repeated indefinitely often and still eventually leave the system in its original state. In each repetition of a cyclic process, the work done by the system is proportional to the heat consumed by the system. In a cyclic process in which the system does work on its surroundings, it is necessary that some heat be taken in by the system and some be put out, and the difference is the heat consumed by the system in the process. The constant of proportionality is universal and independent of the system and was measured by James Joule in 1845 and 1847, who described it as the mechanical equivalent of heat. In any incremental process, the change in the internal energy is considered due to a combination of heat added to the system and work done by the system. Taking as an infinitesimal (differential) change in internal energy, one writes where and are infinitesimal amounts of heat supplied to the system by its surroundings and work done by the system on its surroundings, respectively. This sign convention is implicit in Clausius statement of the law given above, and is consistent with the use of thermodynamics to study heat engines which provide useful work, which is regarded as positive. In chemistry, however, it is conventional to use the IUPAC convention where the first law is formulated in terms of the work done on the system. With this alternate sign convention for work, the first law for a closed system may be written: [3] This convention follows physicists such as Max Planck[4], and considers all net energy transfers to the system as positive and all net energy transfers from the system as negative, independently of any use for the system as an

First law of thermodynamics 123 engine or otherwise. When a system expands in a quasistatic process, the work done by the system on the environment is the product, P dV, of pressure, P, and volume change, dV, whereas the work done on the system is -P dV. Using either sign convention for work, the change in internal energy of the system is: Work and heat are expressions of actual physical processes which supply or remove energy, while is a mathematical abstraction that keeps account of the exchanges of energy that befall the system. Thus the term heat for means that amount of energy added or removed by conduction of heat or by thermal radiation, rather than referring to a form of energy within the system. Likewise, work energy for means "that amount of energy gained or lost as the result of work". Internal energy is a property of the system whereas work done and heat supplied are not. A significant result of this distinction is that a given internal energy change can be achieved by, in principle, many combinations of heat and work. The internal energy of a system is not uniquely defined. It is defined only up to an arbitrary additive constant of integration, which can be adjusted to give arbitrary reference zero levels. This non-uniqueness is in keeping with the abstract mathematical nature of the internal energy. The internal energy is stated relative to a conventionally chosen standard reference state of the system. Various statements of the law for closed systems The law is of very great importance and generality and is consequently thought of from several points of view. Most careful textbook statements of the law express it for closed systems. It is stated in several ways, sometimes even by the same author.[5][6] For the thermodynamics of closed systems, the distinction between transfers of energy as work and as heat is central and is within the scope of the present article. For the thermodynamics of open systems, such a distinction is beyond the scope of the present article, but some limited comments are made on it in the section below headed First law of thermodynamics for open systems. There are two main ways of stating a law of thermodynamics, physically or mathematically. They should be logically coherent and consistent with one another.[7] An example of a physical statement is that of Planck (1897/1903): It is in no way possible, either by mechanical, thermal, chemical, or other devices, to obtain perpetual motion, i.e. it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing."[8] This physical statement is restricted neither to closed systems nor to systems with states that are strictly defined only for thermodynamic equilibrium; it has meaning also for open systems and for systems with states that are not in thermodynamic equilibrium. An example of a mathematical statement is that of Crawford (1963): For a given system we let ΔE kin = large-scale mechanical energy, ΔE pot = large-scale potential energy, and ΔE tot = total energy. The first two quantities are specifiable in terms of appropriate mechanical variables, and by definition For any finite process, whether reversible or irreversible, The first law in a form that involves the principle of conservation of energy more generally is

First law of thermodynamics 124 Here Q and W are heat and work added, with no restrictions as to whether the process is reversible, quasistatic, or irreversible.[Warner, Am. J. Phys., 29, 124 (1961)][9] This statement by Crawford, for W, uses the sign convention of IUPAC, not that of Clausius. Though it does not explicitly say so, this statement refers to closed systems, and to internal energy U defined for bodies in states of thermodynamic equilibrium, which possess well-defined temperatures. The history of statements of the law for closed systems has two main periods, before and after the work of Bryan (1907),[10] of Carathéodory (1909),[11] and the approval of Carathéodorys work given by Born (1921).[12] The earlier traditional versions of the law for closed systems are nowadays often considered to be out of date. Carathéodorys celebrated presentation of equilibrium thermodynamics[11] refers to closed systems, which are allowed to contain several phases connected by internal walls of various kinds of impermeability and permeability, explicitly including walls that are permeable only to heat. Carathéodorys version of the first law of thermodynamics was stated in an axiom which refrained from defining or mentioning temperature or quantity of heat transferred. That axiom stated that the internal energy of a phase in equilibrium is a function of state, that the sum of the internal energies of the phases is the total internal energy of the system, and that the value of the total internal energy of the system is changed by the amount of work done adiabatically on it, considering work as a form of energy. That article considered this statement to be an expression of the law of conservation of energy for such systems. This version is nowadays widely accepted as authoritative, but is stated in slightly varied ways by different authors. The Carathéodory statement of the law in axiomatic form does not mention heat or temperature, but the equilibrium states to which it refers are explicitly defined by variable sets that necessarily include "non-deformation variables", such as pressures, which, within reasonable restrictions, can be rightly interpreted as empirical temperatures, and the walls connecting the phases of the system are explicitly defined as possibly impermeable to heat or permeable only to heat. According to Münster (1970), "A somewhat unsatisfactory aspect of Carathéodorys theory is that a consequence of the Second Law must be considered at this point [in the statement of the first law], i.e. that it is not always possible to reach any state 2 from any other state 1 by means of an adiabatic process." Münster instances that no adiabatic process can reduce the internal energy of a system at constant volume.[13] Carathéodorys paper asserts that its statement of the first law corresponds exactly to Joules experimental arrangement, regarded as an instance of adiabatic work. It does not point out that Joules experimental arrangement performed essentially irreversible work, through friction of paddles in a liquid, or passage of electric current through a resistance inside the system, driven by motion of a coil and inductive heating, or by an external current source, which can access the system only by the passage of electrons, and so is not strictly adiabatic, because electrons are a form of matter, which cannot penetrate adiabatic walls. The paper goes on to base its main argument on the possibility of quasi-static adiabatic work, which is essentially reversible. The paper asserts that it will avoid reference to Carnot cycles, and then proceeds to base its argument on cycles of forward and backward quasi-static adiabatic stages, with isothermal stages of zero magnitude. Some respected modern statements of the first law for closed systems assert the existence of internal energy as a function of state defined in terms of adiabatic work and accept the Carathéodory idea that heat is not defined in its own right, that is to say calorimetrically or as due to temperature difference; they define heat as a residual difference between change of internal energy and work done on the system, when that work does not account for the whole of the change of internal energy and the system is not adiabatically isolated.[14][13][15] Sometimes the concept of internal energy is not made explicit in the statement.[16][17][18] Sometimes the existence of the internal energy is made explicit but work is not explicitly mentioned in the statement of the first postulate of thermodynamics. Heat supplied is then defined as the residual change in internal energy after work has been taken into account, in a non-adiabatic process.[19] A respected modern author states the first law of thermodynamics as "Heat is a form of energy", which explicitly mentions neither internal energy nor adiabatic work. Heat is defined as energy transferred by thermal contact with a

First law of thermodynamics 125 reservoir, which has a temperature, and is generally so large that addition and removal of heat do not alter its temperature.[20] A current student text on chemistry defines heat thus: "heat is the exchange of thermal energy between a system and its surroundings caused by a temperature difference." The author then explains how heat is defined or measured by calorimetry, in terms of heat capacity, specific heat capacity, molar heat capacity, and temperature.[21] A respected text disregards the Carathéodorys exclusion of mention of heat from the statement of the first law for closed systems, and admits heat calorimetrically defined along with work and internal energy.[22] Another respected text defines heat exchange as determined by temperature difference, but also mentions that the Born (1921) version is "completely rigorous".[23] These versions follow the traditional approach that is now considered out of date, exemplified by that of Planck (1897/1903).[24] Evidence for the first law of thermodynamics for closed systems The first law of thermodynamics for closed systems was originally induced from empirically observed evidence, however, it is now taken to be the definition of heat via the law of conservation of energy and the definition of work in terms of changes in the external parameters of a system. The original discovery of the law was gradual over a period of perhaps half a century or more, and some early studies were in terms of cyclic processes.[25] The following is an account in terms of changes of state through compound processes that are not necessarily cyclic. This account first considers processes for which the first law is easily verified because of their simplicity, namely adiabatic processes (in which no heat is transferred) and adynamic processes (in which no work is transferred). Adiabatic processes Given a closed system in an initial state, if work is done on the system in an adiabatic (i.e. no heat transfer) way, and given the final state after a process, the amount of work required to be transferred to the system is the same, irrespective of how this work is performed. The work done on the system is defined and measured by changes in mechanical or quasi-mechanical variables external to the system. Physically, adiabatic transfer of energy as work requires the existence of adiabatic enclosures. For instance, in Joules experiment, the initial system is a tank of water with a paddle wheel inside. If we isolate thermally the tank and move the paddle wheel with a pulley and a weight we can relate the increase in temperature with the height descended by the mass. Now the system is returned to its initial state, isolated again, and the same amount of work is done on the tank using different devices (an electric motor, a chemical battery, a spring,...). In every case, the amount of work can be measured independently. The return to the initial state is not conducted by doing adiabatic work on the system. The evidence shows that the final state of the water (in particular, its temperature and volume) is the same in every case. It is irrelevant if the work is electrical, mechanical, chemical,... or if done suddenly or slowly, as long as it is performed in an adiabatic way, that is to say, without heat transfer into or out of the system. Evidence of this kind shows that to increase the temperature of the water in the tank, the qualitative kind of adiabatically performed work does not matter. No qualitative kind of adiabatic work has ever been observed to decrease the temperature of the water in the tank. A change from one state to another, for example an increase of both temperature and volume, may be conducted in several stages, for example by externally supplied electrical work on a resistor in the body, and adiabatic expansion allowing the body to do work on the surroundings. It needs to be shown that the time order of the stages, and their relative magnitudes, does not affect the amount of adiabatic work that needs to be done for the change of state. According to one respected scholar: "Unfortunately, it does not seem that experiments of this kind have ever been carried out carefully. ... We must therefore admit that the statement which we have enunciated here, and which is equivalent to the first law of thermodynamics, is not well founded on direct experimental evidence."[26]

First law of thermodynamics 126 This kind of evidence, of independence of sequence of stages, combined with the above-mentioned evidence, of independence of qualitative kind of work, would show the existence of a very important state variable that corresponds with adiabatic work, but not that such a state variable represented a conserved quantity. For the latter, another step of evidence is needed, which may be related to the concept of reversibility, as mentioned below. That very important state variable was first recognized and denoted by Clausius in 1850, but he did not then name it, and he defined it in terms not only of work but also of heat transfer in the same process. It was also independently recognized in 1850 by Rankine, who also denoted it ; and in 1851 by Kelvin who then called it "mechanical energy", and later "intrinsic energy". In 1865, after some hestitation, Clausius began calling his state function "energy". In 1882 it was named as the internal energy by Helmholtz.[27] If only adiabatic processes were of interest, and heat could be ignored, the concept of internal energy would hardly arise or be needed. The relevant physics would be largely covered by the concept of potential energy, as was intended in the 1847 paper of Helmholtz on the principle of conservation of energy, though that did not deal with forces that cannot be described by a potential, and thus did not fully justify the principle; moreover that paper was very critical of the early work of Joule which had by then been performed.[28] A great merit of the internal energy concept is that it frees thermodynamics from a restriction to cyclic processes, and allows a treatment in terms of thermodynamic states. In an adiabatic process, adiabatic work takes the system either from a reference state with internal energy to an arbitrary one with internal energy , or from the state to the state : Except under the special, and strictly speaking, fictional, condition of reversibility, only one of the processes or is empirically feasible by a simple application of externally supplied work. The reason for this is given as the second law of thermodynamics and is not considered in the present article. The fact of such irreversibility may be dealt with in two main ways, according to different points of view. since the work of Bryan (1907), • To deal with it nowadays, the most accepted way, followed by Carathéodory,[11][29][15] is to rely on the previously established concept of quasi-static processes,[30][31][32] as follows. Actual physical processes of transfer of energy as work are always at least to some degree irreversible. The irreversibility is often due to mechanisms known as dissipative, that transform bulk kinetic energy into internal energy. Examples are friction and viscosity. If the process is performed more slowly, the frictional or viscous dissipation is less. In the limit of infinitely slow performance, the dissipation tends to zero and then the limiting process, though fictional rather than actual, is notionally reversible, and is called quasi-static. Throughout the course of the fictional limiting quasi-static process, the internal intensive variables of the system are equal to the external intensive variables, those which describe the reactive forces exerted by the surroundings.[33] This can be taken to justify the formula • Another way to deal with it is to allow that experiments with processes of heat transfer to or from the system may be used to justify the formula just above. Moreover, it deals to some extent with the problem of lack of direct experimental evidence that the time order of stages of a process does not matter in the determination of internal energy. This way does not provide theoretical purity in terms of adiabatic work processes, but is empirically feasible, and is in accord with experiments actually done, such as the Joule experiments mentioned just above, and with older traditions. The formula above allows that to go by processes of quasi-static adiabatic work from the state to the state we can take a path that goes through the reference state , since the quasi-static adiabatic work is independent of the path This kind of empirical evidence, coupled with theory of this kind, largely justifies the following statement:

First law of thermodynamics 127 For all adiabatic processes between two specified states of a closed system of any nature, the net work done is the same regardless the details of the process, and determines a state function called internal energy, ." Adynamic processes A complementary observable aspect of the first law is about heat transfer. Adynamic transfer of energy as heat can be measured empirically by changes in the surroundings of the system of interest by calorimetry. This again requires the existence of adiabatic enclosure of the entire process, system and surroundings, though the separating wall between the surroundings and the system is thermally conductive or radiatively permeable, not adiabatic. A calorimeter can rely on measurement of sensible heat, which requires the existence of thermometers and measurement of temperature change in bodies of known sensible heat capacity under specified conditions; or it can rely on the measurement of latent heat, through measurement of masses of material which change phase, at temperatures fixed by the occurrence of phase changes under specified conditions in bodies of known latent heat of phase change. The calorimeter can be calibrated by adiabatically doing externally determined work on it. The most accurate method is by passing an electric current from outside through a resistance inside the calorimeter. The calibration allows comparison of calorimetric measurement of quantity of heat transferred with quantity of energy transferred as work. According to one textbook, "The most common device for measuring is an adiabatic bomb calorimeter."[34] According to another textbook, "Calorimetry is widely used in present day laboratories."[35] According to one opinion, "Most thermodynamic data come from calorimetry..."[36] According to another opinion, "The most common method of measuring “heat” is with a calorimeter."[37] When the system evolves with transfer of energy as heat, without energy being transferred as work, in an adynamic process, the heat transferred to the system is equal to the increase in its internal energy: General case for reversible processes Heat transfer is practically reversible when it is driven by practically negligibly small temperature gradients. Work transfer is practically reversible when it occurs so slowly that there are no frictional effects within the system; frictional effects outside the system should also be zero if the process is to be globally reversible. For a particular reversible process in general, the work done reversibly on the system, , and the heat transferred reversibly to the system, are not required to occur respectively adiabatically or adynamically, but they must belong to the same particular process defined by its particular reversible path, , through the space of thermodynamic states. Then the work and heat transfers can occur and be calculated simultaneously. Putting the two complementary aspects together, the first law for a particular reversible process can be written This combined statement is the expression the first law of thermodynamics for reversible processes for closed systems. In particular, if no work is done on a thermally isolated closed system we have . This is one aspect of the law of conservation of energy and can be stated: The internal energy of an isolated system remains constant.

First law of thermodynamics 128 General case for irreversible processes If, in a process of change of state of a closed system, the energy transfer is not under a practically zero temperature gradient and practically frictionless, then the process is irreversible. Then the heat and work transfers may be difficult to calculate, and irreversible thermodynamics is called for. Nevertheless, the first law still holds and provides a check on the measurements and calculations of the work done irreversibly on the system, , and the heat transferred irreversibly to the system, , which belong to the same particular process defined by its particular irreversible path, , through the space of thermodynamic states. This means that the internal energy is a function of state and that the internal energy change between two states is a function only of the two states. Overview of the weight of evidence for the law The first law of thermdynamics is very general and makes so many predictions that they can hardly all be directly tested by experiment. Nevertheless, very very many of its predictions have been found empirically accurate. And very importantly, no accurately and properly conducted experiment has ever detected a violation of the law. Consequently, within its scope of applicability, the law is so reliably established, that, nowadays, rather than experiment being considered as testing the accuracy of the law, it is far more practical and realistic to think of the law as testing the accuracy of experiment. An experimental result that seems to violate the law may be assumed to be inaccurate or wrongly conceived, for example due to failure to consider an important physical factor. State functional formulation for infinitesimal processes When the heat and work transfers in the equations above are infinitesimal in magnitude, they are often denoted by δ, rather than exact differentials denoted by "d", as a reminder that heat and work do not describe the state of any system. The integral of an inexact differential depends upon the particular path taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states. If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential will always be zero. The path taken by a thermodynamic system through a chemical or physical change is known as a thermodynamic process. For a homogeneous system, with a well-defined temperature and pressure, the expression for dU can be written in terms of exact differentials, if the work that the system does is equal to its pressure times the infinitesimal increase in its volume. Here one assumes that the changes are quasistatic, so slow that there is at each instant negligible departure from thermodynamic equilibrium within the system. In other words, δW = -PdV where P is pressure and V is volume. As such a quasistatic process in a homogeneous system is reversible, the total amount of heat added to a closed system can be expressed as δQ =TdS where T is the temperature and S the entropy of the system. Therefore, for closed, homogeneous systems: The above equation is known as the fundamental thermodynamic relation, for which the independent variables are taken as S and V, with respect to which T and P are partial derivatives of U. While this has been derived for quasistatic changes, it is valid in general, as U can be considered as a thermodynamic state function of the independent variables S and V. E.g., suppose that the system is initially in a state of thermal equilibrium defined by S and V, and then the system is suddenly perturbed so that thermal equilibrium breaks down and no temperature and pressure can be defined. Then the system settles down again to a state of thermal equilibrium, defined by an entropy and a volume which differ infinitesimally from the initial values. The infinitesimal difference in internal energy between the initial and final state will then satisfy the above equation. The work done and heat added to the system will then not satisfy the above

First law of thermodynamics 129 expressions, they will instead satisfy the inequalities: δQ < TdS and δW < PdV. In the case of a closed system in which the particles of the system are of different types and, because chemical reactions may occur, their respective numbers are not necessarily constant, the expression for dU becomes: where dNi is the (small) increase in amount of type-i particles in the reaction, and μi is known as the chemical potential of the type-i particles in the system. If dNi is expressed in kg then μi is expressed in J/kg. The statement of the first law, using exact differentials is now: If the system has more external mechanical variables than just the volume that can change, the fundamental thermodynamic relation generalizes to: Here the Xi are the generalized forces corresponding to the external variables xi. The parameters Xi are independent of the size of the system and are called intensive parameters and the xi are proportional to the size and called extensive parameters. For an open system, there can be transfers of particles as well as energy into or out of the system during a process. For this case, the first law of thermodynamics still holds, in the form that the internal energy is a function of state and the change of internal energy in a process is a function only of its initial and final states, as noted in the section below headed First law of thermodynamics for open systems. A useful idea from mechanics is that the energy gained by a particle is equal to the force applied to the particle multiplied by the displacement of the particle while that force is applied. Now consider the first law without the heating term: dU = -PdV. The pressure P can be viewed as a force (and in fact has units of force per unit area) while dVis the displacement (with units of distance times area). We may say, with respect to this work term, that a pressure difference forces a transfer of volume, and that the product of the two (work) is the amount of energy transferred out of the system as a result of the process. If one were to make this term negative then this would be the work done on the system. It is useful to view the TdS term in the same light: here the temperature is known as a "generalized" force (rather than an actual mechanical force) and the entropy is a generalized displacement. Similarly, a difference in chemical potential between groups of particles in the system drives a chemical reaction that changes the numbers of particles, and the corresponding product is the amount of chemical potential energy transformed in process. For example, consider a system consisting of two phases: liquid water and water vapor. There is a generalized "force" of evaporation which drives water molecules out of the liquid. There is a generalized "force" of condensation which drives vapor molecules out of the vapor. Only when these two "forces" (or chemical potentials) are equal will there be equilibrium, and the net rate of transfer will be zero. The two thermodynamic parameters which form a generalized force-displacement pair are termed "conjugate variables". The two most familiar pairs are, of course, pressure-volume, and temperature-entropy. Spatially inhomogeneous systems Classical thermodynamics is initially focused on closed homogeneous systems (e.g. Planck 1897/1903[24]), which might be regarded as zero-dimensional in the sense that they have no spatial variation. But it is desired to study also systems with distinct internal motion and spatial inhomogeneity. For such systems, the principle of conservation of energy is expressed in terms not only of internal energy as defined for homogeneous systems, but also in terms of kinetic energy and potential energies of parts of the inhomogeneous system with respect to each other and with

First law of thermodynamics 130 respect to long-range external forces.[38] How the total energy of a system is allocated between these three more specific kinds of energy varies according to the purposes of different writers; this is because these components of energy are to some extent mathematical artefacts rather than actually measured physical quantities. For any closed homogeneous component of an inhomogeneous closed system, if denotes the total energy of that component system, one may write where and denote respectively the total kinetic energy and the total potential energy of the component closed homogeneous system, and denotes its internal energy.[9][39] Potential energy can be exchanged with the surroundings of the system when the surroundings impose a force field, such as gravitational or electromagnetic, on the system. A compound system consisting of two interacting closed homogeneous component subsystems has a potential energy of interaction between the subsystems. Thus, in an obvious notation, one may write The distinction between internal and kinetic energy is hard to make in the presence of turbulent motion within the system, as friction gradually dissipates macroscopic kinetic energy of localised bulk flow into molecular random motion of molecules that is classified as internal energy. The rate of dissipation by friction of kinetic energy of localised bulk flow into internal energy,[40][41][42] whether in turbulent or in streamlined flow, is an important quantity in non-equilibrium thermodynamics. This is a serious difficulty for attempts to define entropy for time-varying spatially inhomogeneous systems. First law of thermodynamics for open systems For the first law of thermodynamics, there is no trivial passage of physical conception from the closed system view to an open system view.[43] For closed systems, the concepts of an adiabatic enclosure and of an adiabatic wall are fundamental. Matter and internal energy cannot permeate or penetrate such a wall. For an open system, there is a wall that allows penetration by matter. In general, matter in motion will carry with it some internal energy, and some potential energy changes will accompany the motion. An open system is not adiabatically enclosed. By definition therefore, adiabatic work cannot be done on an open system.[44] In contrast to the case of closed systems, for open systems, in the presence of diffusion, there is no unconstrained and unconditional physical distinction between convective transfer of internal energy by bulk flow of matter, the transfer of internal energy without transfer of matter (usually called heat conduction), and change of various potential energies. The older traditional way and the Carathéodory way agree that there is no physically unique definition of heat and work transfer processes between open systems.[45][46][47][48] The ideas of heat and work transfer for closed systems are superseded for open systems by the ideas of transfer of kinetic energy of bulk flow, of bulk potential and of internal energies, and of entropy. An example is evaporation. One may consider an open system consisting of a collection of vapour in a controlled volume, enclosed except where it is allowed to receive more vapour from or to condense into its parent liquid, which may be considered as another open system in open contact with the vapour system. The process might be a mechanical increase in the controlled volume of the vapour. Some work will be done by the vapour in the mechanical part of the process, but also some of the parent liquid will evaporate and enter the vapour collection which is the system. Some internal energy will accompany the vapour that enters the system, but it will not make sense to try to uniquely identify part of that internal energy as heat and part of it as work. Consequently, the energy transfer of the process as a whole, though having a component of mechanical work, cannot be uniquely split into heat and work transfers to or from the open system. The component of total energy transfer that accompanies the transfer of vapour into the system is customarily called latent heat of evaporation, but this is a quirk of historical language usage, not in strict compliance with the thermodynamic definition of transfer of energy as heat. In this example, kinetic energy of bulk flow and potential energy with respect to external long range forces such as gravity are both

First law of thermodynamics 131 considered to be zero. The first law of thermodynamics refers to the change of internal energy of the open system. History The discovery of the first law of thermodynamics was by way of many tries and mistakes of investigation, over a period of about half a century. The first full statements of the law were made by Clausius in 1850 as noted above, and by Rankine also in 1850; Rankines statement was perhaps not quite as clear and distinct as was Clausius.[25] A main aspect of the struggle was to deal with the previously proposed caloric theory of heat. Germain Hess in 1840 stated a conservation law for the so-called heat of reaction for chemical reactions.[49] His law was later recognized as a consequence of the first law of thermodynamics, but Hesss statement was not explicitly concerned with the relation between energy exchanges by heat and work. Julius Robert von Mayer According to Truesdell (1980), Julius Robert von Mayer in 1841 made a statement that meant that "in a process at constant pressure, the heat used to produce expansion is universally interconvertible with work", but this is not a general statement of the first law.[50][51] References [1] Clausius, R. (1850). Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen, Annalen der Physik und Chemie (Poggendorff, Leipzig), 155 (3): 368-394, particularly on page 373 (http:/ / gallica. bnf. fr/ ark:/ 12148/ bpt6k15164w/ f389. image), translation here taken from Truesdell, C.A. (1980), pp. 188-189. [2] Clausius, R. (1850). Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen, Annalen der Physik und Chemie (Poggendorff, Leipzig), 155 (3): 368-394, page 384 (http:/ / gallica. bnf. fr/ ark:/ 12148/ bpt6k15164w/ f400. image). [3] Quantities, Units and Symbols in Physical Chemistry (IUPAC Green Book) (http:/ / media. iupac. org/ publications/ books/ gbook/ IUPAC-GB3-2ndPrinting-Online-22apr2011. pdf) See Sec. 2.11 Chemical Thermodynamics [4] Planck, M.(1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans, Green & Co., London. (https:/ / ia700200. us. archive. org/ 15/ items/ treatiseonthermo00planrich/ treatiseonthermo00planrich. pdf), p. 43 [5] Münster, A. (1970). [6] Bailyn, M. (1994), p. 79. [7] Kirkwood, J.G., Oppenheim, I. (1961), pp. 31–33. [8] Planck, M.(1897/1903), p. 40. [9] Crawford, F.H. (1963), pp. 106–107. [10] Bryan, G.H. (1907), p. 47. [11] C. Carathéodory (1909). "Untersuchungen über die Grundlagen der Thermodynamik". Mathematische Annalen 67: 355–386. A partly reliable translation is to be found at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.. doi:10.1007/BF01450409. [12] Born, M. (1921). Kritische Betrachtungen zur traditionellen Darstellung der Thermodynamik, Physik. Zeitschr. 22: 218–224. [13] Münster, A. (1970), pp. 23–24. [14] Reif, F. (1965), p. 122. [15] Haase, R. (1971), pp. 24–25. [16] Pippard, A.B. (1957/1966), p. 14. [17] Reif, F. (1965), p. 82. [18] Adkins, C.J. (1968/1983), p. 31. [19] Callen, H.B. (1960/1985), pp. 13, 17. [20] Kittel, C. Kroemer, H. (1980). Thermal Physics, (first edition by Kittel alone 1969), second edition, W.H. Freeman, San Francisco, ISBN 0-7167-1088-9, pp. 49, 227. [21] Tro, N.J. (2008). Chemistry. A Molecular Approach, Pearson/Prentice Hall, Upper Saddle River NJ, ISBN 0-13-100065-9, p. 246. [22] Kirkwood, J.G., Oppenheim, I. (1961), pp. 17–18. Kirkwood & Oppenheim 1961 is recommended by Münster, A. (1970), p. 376. It is also cited by Eu, B.C. (2002), Generalized Thermodynamics, the Thermodynamics of Irreversible Processes and Generalized Hydrodynamics, Kluwer Academic Publishers, Dordrecht, ISBN 1-4020-0788-4, pp. 18, 29, 66.

First law of thermodynamics 132 [23] Guggenheim, E.A. (1949/1967). Thermodynamics. An Advanced Treatment for Chemists and Physicists, (first edition 1949), fifth edition 1967, North-Holland, Amsterdam, pp. 9–10. Guggenheim 1949/1965 is recommended by Buchdahl, H.A. (1966), p. 218. It is also recommended by Münster, A. (1970), p. 376. [24] Planck, M.(1897/1903). [25] Truesdell, C.A. (1980). [26] Pippard, A.B. (1957/1966), p. 15. According to Herbert Callen, in his most widely cited text, Pippards text gives a "scholarly and rigorous treatment"; see Callen, H.B. (1960/1985), p. 485. It is also recommended by Münster, A. (1970), p. 376. [27] Cropper, W.H. (1986). Rudolf Clausius and the road to entropy, Am. J. Phys., 54: 1068–1074. [28] Truesdell, C.A. (1980), pp. 161–162. [29] Buchdahl, H.A. (1966), p. 43. [30] Maxwell, J. C. (1871). Theory of Heat, Longmans, Green, and Co., London, p. 150. [31] Planck, M. (1897/1903), Section 71, p. 52. [32] Bailyn, M. (1994), p. 95. [33] Adkins, C.J. (1968/1983), p. 35. [34] Atkins, P., de Paula, J. (1978/2010). Physical Chemistry, (first edition 1978), ninth edition 2010, Oxford University Press, Oxford UK, ISBN 978-0-19-954337-3, p. 54. [35] Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester, ISBN 978-0-470-01598-8, p. 63. [36] Gislason, E.A., Craig, N.C. (2005). Cementing the foundations of thermodynamics:comparison of system-based and surroundings-based definitions of work and heat, J. Chem. Thermodynamics 37: 954–966. [37] Rosenberg, R.M. (2010). From Joule to Caratheodory and Born: A conceptual evolution of the first law of thermodynamics, J. Chem. Edu., 87: 691–693. [38] Bailyn, M. (1994), 254-256. [39] Glansdorff, P., Prigogine, I. (1971), page 8. [40] Thomson, William (1852 a). " On a Universal Tendency in Nature to the Dissipation of Mechanical Energy (http:/ / zapatopi. net/ kelvin/ papers/ on_a_universal_tendency. html)" Proceedings of the Royal Society of Edinburgh for April 19, 1852 [This version from Mathematical and Physical Papers, vol. i, art. 59, pp. 511.] [41] Thomson, W. (1852 b). On a universal tendency in nature to the dissipation of mechanical energy, Philosophical Magazine 4: 304-306. [42] Helmholtz, H. (1869/1871). Zur Theorie der stationären Ströme in reibenden Flüssigkeiten, Verhandlungen des naturhistorisch-medizinischen Vereins zu Heidelberg, Band V: 1-7. Reprinted in Helmholtz, H. (1882), Wissenschaftliche Abhandlungen, volume 1, Johann Ambrosius Barth, Leipzig, pages 223-230 (http:/ / echo. mpiwg-berlin. mpg. de/ ECHOdocuViewfull?url=/ mpiwg/ online/ permanent/ einstein_exhibition/ sources/ QWH2FNX8/ index. meta& start=231& viewMode=images& pn=237& mode=texttool) [43] Münster A. (1970), Sections 14, 15, pp. 45–51. [44] Münster, A. (1970), p. 46. [45] Münster, A. (1970), p. 50. [46] Haase, R. (1963/1969), p. 15. [47] Haase, R. (1971), p. 20. [48] Smith, D.A. (1980). Definition of heat in open systems, Aust. J. Phys., 33: 95–105. (http:/ / www. publish. csiro. au/ paper/ PH800095. htm) [49] Hess, H. (1840). Thermochemische Untersuchungen, Annalen der Physik und Chemie (Poggendorff, Leipzig) 126(6): 385-404 (http:/ / gallica. bnf. fr/ ark:/ 12148/ bpt6k151359/ f397. image. r=Annalen der Physik (Leipzig) 125. langEN). [50] Truesdell, C.A. (1980), pp. 157-158. [51] Mayer, Robert (1841). Paper: Remarks on the Forces of Nature"; as quoted in: Lehninger, A. (1971). Bioenergetics - the Molecular Basis of Biological Energy Transformations, 2nd. Ed. London: The Benjamin/Cummings Publishing Company. Cited sources • Adkins, C.J. (1968/1983). Equilibrium Thermodynamics, (first edition 1968), third edition 1983, Cambridge University Press, ISBN 0-521-25445-0. • Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York, ISBN 0-88318-797-3. • Bryan, G.H. (1907). Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications, B.G. Teubner, Leipzig (https://ia700208.us.archive.org/6/items/Thermodynamics/ Thermodynamics.tif). • Buchdahl, H.A. (1966), The Concepts of Classical Thermodynamics, Cambridge University Press, London. • Callen, H.B. (1960/1985), Thermodynamics and an Introduction to Thermostatistics, (first edition 1960), second edition 1985, John Wiley & Sons, New York, ISBN 0–471–86256–8.

First law of thermodynamics 133 • Crawford, F.H. (1963). Heat, Thermodynamics, and Statistical Physics, Rupert Hart-Davis, London, Harcourt, Brace & World, Inc.. • Glansdorff, P., Prigogine, I., (1971). Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley, London, ISBN 0-471-30280-5. • Haase, R. (1963/1969). Thermodynamics of Irreversible Processes, English translation, Addison-Wesley Publishing, Reading MA. • Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of Thermodynamics, pages 1–97 of volume 1, ed. W. Jost, of Physical Chemistry. An Advanced Treatise, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081. • Kirkwood, J.G., Oppenheim, I. (1961). Chemical Thermodynamics, McGraw-Hill Book Company, New York. • Münster, A. (1970), Classical Thermodynamics, translated by E.S. Halberstadt, Wiley–Interscience, London, ISBN 0-471-62430-6. • Pippard, A.B. (1957/1966). Elements of Classical Thermodynamics for Advanced Students of Physics, original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK. • Planck, M.(1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans, Green & Co., London. (https://ia700200.us.archive.org/15/items/treatiseonthermo00planrich/treatiseonthermo00planrich.pdf) • Reif, F. (1965). Fundamentals of Statistical and Thermal Physics, McGraw-Hill Book Company, New York. • Truesdell, C.A. (1980). The Tragicomical History of Thermodynamics, 1822-1854, Springer, New York, ISBN 0-387-90403-4. Further reading • Goldstein, Martin, and Inge F. (1993). The Refrigerator and the Universe. Harvard University Press. ISBN 0-674-75325-9. OCLC 32826343. Chpts. 2 and 3 contain a nontechnical treatment of the first law. • Çengel Y.A. and Boles M. (2007). Thermodynamics: an engineering approach. McGraw-Hill Higher Education. ISBN 0-07-125771-3. Chapter 2. • Atkins P. (2007). Four Laws that drive the Universe. OUP Oxford. ISBN 0-19-923236-9. External links • MISN-0-158, The First Law of Thermodynamics (http://35.9.69.219/home/modules/pdf_modules/m158. pdf) (PDF file) by Jerzy Borysowicz for Project PHYSNET (http://www.physnet.org). • First law of thermodynamics (http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node8. html) in the MIT Course Unified Thermodynamics and Propulsion (http://web.mit.edu/16.unified/www/ FALL/thermodynamics/notes/notes.html) from Prof. Z. S. Spakovszky

Flash evaporation 134 Flash evaporation Flash (or partial) evaporation is the partial vapor that occurs when a saturated liquid stream undergoes a reduction in pressure by passing through a throttling valve or other throttling device. This process is one of the simplest unit operations. If the throttling valve or device is located at the entry into a pressure vessel so that the flash evaporation occurs within the vessel, then the vessel is often referred to as a flash drum.[1][2] If the saturated liquid is a single-component liquid (for example, liquid propane or liquid ammonia), a part of the liquid immediately "flashes" into vapor. Both the vapor and the residual liquid are cooled to the saturation temperature of the liquid at the reduced pressure. This is often referred to as "auto-refrigeration" and is the basis of most conventional vapor compression refrigeration systems. If the saturated liquid is a multi-component liquid (for example, a mixture of propane, isobutane and normal butane), the flashed vapor is richer in the more volatile components than is the remaining liquid. Uncontrolled flash evaporation can result in a boiling A typical flash drum liquid expanding vapor explosion (BLEVE). Flash evaporation of a single-component liquid The flash evaporation of a single-component liquid is an isenthalpic process and is often referred to as an adiabatic flash. The following equation, derived from a simple heat balance around the throttling valve or device, is used to predict how much of a single-component liquid is vaporized. where: = weight fraction vapourized = upstream liquid enthalpy at upstream temperature and pressure, J/kg = flashed vapor enthalpy at downstream pressure and corresponding saturation temperature, J/kg = residual liquid enthalpy at downstream pressure and corresponding saturation temperature, J/kg If the enthalpy data required for the above equation is unavailable, then the following equation may be used.

Flash evaporation 135 where: = weight fraction vapourized = liquid specific heat at upstream temperature and pressure, J/(kg °C) = upstream liquid temperature, °C = liquid saturation temperature corresponding to the downstream pressure, °C = liquid heat of vaporization at downstream pressure and corresponding saturation temperature, J/kg Here, the words "upstream" and "downstream" refer to before and after the liquid passes through the throttling valve or device. This type of flash evaporation is used in the desalination of brackish water or ocean water by "Multi-Stage Flash Distillation." The water is heated and then routed into a reduced-pressure flash evaporation "stage" where some of the water flashes into steam. This steam is subsequently condensed into salt-free water. The residual salty liquid from that first stage is introduced into a second flash evaporation stage at a pressure lower than the first stage pressure. More water is flashed into steam which is also subsequently condensed into more salt-free water. This sequential use of multiple flash evaporation stages is continued until the design objectives of the system are met. A large part of the worlds installed desalination capacity uses multi-stage flash distillation. Typically such plants have 24 or more sequential stages of flash evaporation. Equilibrium flash of a multi-component liquid The equilibrium flash of a multi-component liquid may be visualized as a simple distillation process using a single equilibrium stage. It is very different and more complex than the flash evaporation of single-component liquid. For a multi-component liquid, calculating the amounts of flashed vapor and residual liquid in equilibrium with each other at a given temperature and pressure requires a trial-and-error iterative solution. Such a calculation is commonly referred to as an equilibrium flash calculation. It involves solving the Rachford-Rice equation:[3][4][5][6] where: • zi is the mole fraction of component i in the feed liquid (assumed to be known); • β is the fraction of feed that is vaporised; • Ki is the equilibrium constant of component i. The equilibrium constants Ki are in general functions of many parameters, though the most important is arguably temperature; they are defined as: where: • xi is the mole fraction of component i in liquid phase; • yi is the mole fraction of component i in gas phase. Once the Rachford-Rice equation has been solved for β, the compositions xi and yi can be immediately calculated as: The Rachford-Rice equation can have multiple solutions for β, at most one of which guarantees that all xi and yi will be positive. In particular, if there is only one β for which:

Flash evaporation 136 then that β is the solution; if there are multiple such βs, it means that either Kmax<1 or Kmin>1, indicating respectively that no gas phase can be sustained (and therefore β=0) or conversely that no liquid phase can exist (and therefore β=1). It is possible to use Newtons method for solving the above water equation, but there is a risk of converging to the wrong value of β; it is important to initialise the solver to a sensible initial value, such as (βmax+βmin)/2 (which is however not sufficient: Newtons method makes no guarantees on stability), or, alternatively, use a bracketing solver such as the bisection method or the Brent method, which are guaranteed to converge but can be slower. The equilibrium flash of multi-component liquids is very widely utilized in petroleum refineries, petrochemical and chemical plants and natural gas processing plants. Contrast with spray drying Spray drying is sometimes seen a form of flash evaporation. However, although it is a form of liquid evaporation, it is quite different from flash evaporation. In spray drying, a slurry of very small solids is rapidly dried by suspension in a hot gas. The slurry is first atomized into very small liquid droplets which are then sprayed into a stream of hot dry air. The liquid rapidly evaporates leaving behind dry powder or dry solid granules. The dry powder or solid granules are recovered from the exhaust air by using cyclones, bag filters or electrostatic precipitators. References [1] Stanley M. Walas (1988). Chemical Process Equipment:Selection and Design. Butterworth- Heinemann. ISBN 0-409-90131-8. [2] Gas Processing Suppliers Association (GPSA) (1987). Engineering Data Book (10th Edition, Vol. 1 ed.). Gas Processing Suppliers Association, Tulsa, Oklahoma. [3] Harry Kooijman and Ross Taylor (2000). The ChemSep Book (http:/ / www. chemsep. com/ downloads/ docs/ book2. pdf) (2nd ed.). ISBN 3-8311-1068-9. See page 186. [4] Analysis of Objective Functions (https:/ / www. e-education. psu. edu/ png520/ m13_p2. html) (Pennsylvania State University) [5] Flash Calculations using the Soave-Redlich-Kwong equation of state (http:/ / www. mathworks. com/ matlabcentral/ fileexchange/ loadFile. do?objectId=9341& objectType=file) (view full-size image) [6] Curtis H. Whitson, Michael L. Michelsen, The Negative Flash, Fluid Phase Equilibria, 53 (1989) 51–71. External links • Vapor and Flash Steam (http://www.tlv.com/global/US/steam-theory/flash-steam-and-vapor.html) Animation, photos and technical explanation of the difference between Flash Steam and Vaporized fraction. • Flash Steam Tutorial (http://www.spiraxsarco.com/resources/steam-engineering-tutorials/ condensate-recovery/flash-steam.asp) The benefits of recovering flash steam, how it is done and typical applications. • Water Desalination Technologies (http://www.escwa.org.lb/information/publications/edit/upload/ tech-01-3-e.pdf) in the Middle East and Western Asia • Discussion of spray drying (http://www.niro.com/ndk_website/niro/cmsresources.nsf/filenames/ spray-drying.pdf/$file/spray-drying.pdf) • Flash evaporation program online (http://petrochemical.gronerth.com/flash_distillation_program_explorer. htm) Flash distillation of the hydrocarbon compounds.

Forced convection 137 Forced convection Forced convection is a mechanism, or type of transport in which fluid motion is generated by an external source (like a pump, fan, suction device, etc.). It should be considered as one of the main methods of useful heat transfer as significant amounts of heat energy can be transported very efficiently and this mechanism is found very commonly in everyday life, including central heating, air conditioning, steam turbines and in many other machines. Forced convection is often encountered by engineers designing or analyzing heat exchangers, pipe flow, and flow over a plate at a different temperature than the stream (the case of a shuttle wing during re-entry, for example). However, in any forced convection situation, some amount of natural convection is always present whenever there are g-forces present (i.e., unless the system is in free fall). When the natural convection is not negligible, such flows are typically referred to as mixed convection. When analyzing potentially mixed convection, a parameter called the Archimedes number (Ar) parametrizes the relative strength of free and forced convection. The Archimedes number is the ratio of Grashof Forced convection by a fan in a snow machine. number and the square of Reynolds number, which represents the ratio of buoyancy force and inertia force, and which stands in for the contribution of natural convection. When Ar >> 1, natural convection dominates and when Ar << 1, forced convection dominates. [1] When natural convection isnt a significant factor, mathematical analysis with forced convection theories typically yields accurate results. The parameter of importance in forced convection is the Peclet number, which is the ratio of advection (movement by currents) and diffusion (movement from high to low concentrations) of heat. When the Peclet number is much greater than unity (1), advection dominates diffusion. Similarly, much smaller ratios indicate a higher rate of diffusion relative to advection. References [1] Incropera, F. P. (2001). Fundamentals of Heat and Mass Transfer, 5th Ed.. Wiley. ISBN 978-0471386506. Bibliography • Cebeci, Tuncer (2002). Convective Heat Transfer. Springer. ISBN 096684615X. • Burmeister, Louis (1993). Convective Heat Transfer, 2E. Wiley-Interscience. ISBN 047157709X. • Hewitt, G.F (1994). Process Heat Transfer. CRC Press Inc. ISBN 0-8493-9918-1.

Fouling 138 Fouling Fouling is the accumulation of unwanted material on solid surfaces to the detriment of function. The fouling material can consist of either living organisms (biofouling) or a non-living substance (inorganic or organic). Fouling is usually distinguished from other surface-growth phenomena in that it occurs on a surface of a component, system or plant performing a defined and useful function, and that the fouling process impedes or interferes with this function. Other terms used in the literature to describe fouling include: deposit formation, encrustation, crudding, deposition, scaling, scale formation, slagging, and sludge formation. The last six terms have a more narrow meaning than fouling within the scope of the fouling science and technology, and they also have meanings outside of this Heat exchanger in a steam power plant, fouled by macro fouling scope; therefore, they should be used with caution. Fouling phenomena are common and diverse, ranging from fouling of ship hulls, natural surfaces in the marine environment (marine fouling), fouling of heat-transfer components through ingredients contained in the cooling water or gases, and even the development of plaque or calculus on teeth, or deposits on solar panels on Mars, among other examples. Condenser tube with residues of biofouling (cut open) This article is primarily devoted to the fouling of industrial heat exchangers, although the same theory is generally applicable to other varieties of fouling. In the cooling technology and other technical fields, a distinction is made between macro fouling and micro fouling. Of the two, micro fouling is the one which is usually more difficult to prevent and therefore more important. Condenser tube with calcium carbonate scaling (cut open)

Fouling 139 Components subject to fouling Following are examples of components that may be subject to fouling and the corresponding effects of fouling: • Heat exchanger surfaces – reduces thermal efficiency, decreases heat flux, increases temperature on the hot side, Brass tube with corrosion traces (cut open) decreases temperature on the cold side, induces under-deposit corrosion, increases use of cooling water; • Piping, flow channels – reduces flow, increases pressure drop, increases upstream pressure, increases energy expenditure, may cause flow oscillations, slugging in two-phase flow, cavitation; may increase flow velocity elsewhere, may induce vibrations, may cause flow blockage; • Ship hulls – creates additional drag, increases fuel usage, reduces maximum speed;[1] • Turbines – reduces efficiency, increases probability of failure; Cost relations between the individual types of fouling • Solar panels – decreases the electrical power generated; • Reverse osmosis membranes – increases pressure drop, increases energy expenditure, reduces flux, membrane [2] failure (in severe cases); • Electrical heating elements – increases temperature of the element, increases corrosion, reduces lifespan; • Nuclear fuel in pressurized water reactors – axial offset anomaly,[3] may need to de-rate the power plant; • Injection/spray nozzles (e.g., a nozzle spraying a fuel into a furnace) – incorrect amount injected, malformed jet, component inefficiency, component failure; • Venturi tubes, orifice plates – inaccurate or incorrect measurement of flow rate; • Pitot tubes in airplanes – inaccurate or incorrect indication of airplane speed; • Spark plug electrodes in cars – engine misfiring;[4] • Production zone of petroleum reservoirs and oil wells – decreased petroleum production with time; plugging; in some cases complete stoppage of flow in a matter of days;[5] • Teeth – promotes tooth or gum disease, decreases aesthetics; • Living organisms – deposition of excess minerals (e.g., calcium, iron, copper) in tissues is (sometimes controversially) linked to aging/senescence.

Fouling 140 Macro fouling Macro fouling is caused by coarse matter of either biological or inorganic origin, for example industrially produced refuse. Such matter enters into the cooling water circuit through the cooling water pumps from sources like the open sea, rivers or lakes. In closed circuits, like cooling towers, the ingress of macro fouling into the cooling tower basin is possible through open canals or by the wind. Sometimes, parts of the cooling tower internals detach themselves and are carried into the cooling water circuit. Such substances can foul the surfaces of heat exchangers and may cause deterioration of the relevant heat transfer coefficient. They may also create flow blockages, redistribute the flow inside the components, or cause fretting damage. Examples • Manmade refuse; • Detached internal parts of components; • Tools and other "foreign objects" accidentally left after maintenance; • Algae; • Mussels; • Leaves, parts of plants up to entire trunks. Micro fouling As to micro fouling, distinctions are made between:[6] • Scaling or precipitation fouling, as crystallization of solid salts, oxides and hydroxides from water solutions, for example, calcium carbonate or calcium sulfate; • Particulate fouling, i.e., accumulation of particles, typically colloidal particles, on a surface; • Corrosion fouling, i.e., in-situ growth of corrosion deposits, for example, magnetite on carbon steel surfaces; • Chemical reaction fouling, for example, decomposition or polymerization of organic matter on heating surfaces; • Solidification fouling - when components of the flowing fluid with a high-melting point freeze onto a subcooled surface; • Biofouling, like settlements of bacteria and algae; • Composite fouling, whereby fouling involves more than one foulant or fouling mechanism. Precipitation fouling Scaling or precipitation fouling involves crystallization of solid salts, oxides and hydroxides from solutions. These are most often water solutions, but non-aqueous precipitation fouling is also known. Precipitation fouling is a very common problem in boilers and heat exchangers operating with hard water and often results in limescale. Through changes in temperature, or solvent evaporation or degasification, the concentration of salts may exceed the saturation, leading to a precipitation of solids (usually crystals). Limescale buildup inside a pipe both reduces liquid flow through the pipe, as well as reduces thermal conduction from the liquid to the outer pipe shell. Both effects will reduce the pipes overall thermal efficiency when used as a heat exchanger.

Fouling 141 As an example, the equilibrium between the readily soluble calcium bicarbonate - always prevailing in natural water - and the poorly soluble calcium carbonate, the following chemical equation may be written: The calcium carbonate that forms through this reaction precipitates. Due to the temperature dependence of the reaction, and increasing volatility of CO2 with increasing temperature, the scaling is higher at the hotter outlet of the heat exchanger than at the cooler inlet. Temperature dependence of the solubility of calcium sulfate (3 phases) in pure water. The In general, the dependence of the salt water is pressurized so that it can be maintained in the liquid state at the elevated solubility on temperature or presence temperatures. of evaporation will often be the driving force for precipitation fouling. The important distinction is between salts with "normal" or "retrograde" dependence of solubility on temperature. The salts with the "normal" solubility increase their solubility with increasing temperature and thus will foul the cooling surfaces. The salts with "inverse" or "retrograde" solubility will foul the heating surfaces. An example of the temperature dependence of solubility is shown in the figure. Calcium sulfate is a common precipitation foulant of heating surfaces due to its retrograde solubility. Precipitation fouling can also occur in the absence of heating or vaporization. For example, calcium sulfate decreases it solubility with decreasing pressure. This can lead to precipitation fouling of reservoirs and wells in oil fields, decreasing their productivity with time.[7] Fouling of membranes in reverse osmosis systems can occur due to differential solubility of barium sulfate in solutions of different ionic strength.[2] Similarly, precipitation fouling can occur because of solubility changes induced by other factors, e.g., liquid flashing, liquid degassing, redox potential changes, or mixing of incompatible fluid streams. The following lists some of the industrially common phases of precipitation fouling deposits observed in practice to form from aqueous solutions: • Calcium carbonate (calcite, aragonite usually at t > ~50 °C, or rarely vaterite); • Calcium sulfate (anhydrite, hemihydrate, gypsum); • Calcium oxalate (e.g., beerstone); • Barium sulfate (barite); • Magnesium hydroxide (brucite); magnesium oxide (periclase); • Silicates (serpentine, acmite, gyrolite, gehlenite, amorphous silica, quartz, cristobalite, pectolite, xonotlite); • Aluminium oxide hydroxides (boehmite, gibbsite, diaspore, corundum); • Aluminosilicates (analcite, cancrinite, noselite); • Copper (metallic copper, cuprite, tenorite); • Phosphates (hydroxyapatite); • Magnetite or nickel ferrite (NiFe2O4) from extremely pure, low-iron water.[8] The deposition rate by precipitation is often described by the following equations: Transport:

Fouling 142 Surface crystallisation: Overall: where: m - mass of the material (per unit surface area), kg/m2 t - time, s Cb - concentration of the substance in the bulk of the fluid, kg/m3 Ci - concentration of the substance at the interface, kg/m3 Ce - equilibrium concentration of the substance at the conditions of the interface, kg/m3 n1, n2 - order of reaction for the crystallisation reaction and the overall deposition process, respectively, dimensionless kt, kr, kd - kinetic rate constants for the transport, the surface reaction, and the overall deposition reaction, respectively; with the dimension of m/s (when n1 and n2 = 1) Particulate fouling Fouling by particles suspended in water ("crud") or in gas progresses by a mechanism different than precipitation fouling. This process is usually most important for colloidal particles, i.e., particles smaller than about 1 μm in at least one dimension (but which are much larger than atomic dimensions). Particles are transported to the surface by a number of mechanisms and there they can attach themselves, e.g., by flocculation or coagulation. Note that the attachment of colloidal particles typically involves electrical forces and thus the particle behaviour defies the experience from the macroscopic world. The probability of attachment is sometimes referred to as "sticking probability", P:[6] where kd and kt are the kinetic rate constants for deposition and transport, respectively. The value of P for colloidal particles is a function of both the surface chemistry, geometry, and the local thermohydraulic conditions. An alternative to using the sticking probability is to use a kinetic attachment rate constant, assuming the first order reaction[9][10]: and then the transport and attachment kinetic coefficients are combined as two processes occurring is series: where: • dm/dt is the rate of the deposition by particles, kg m−2 s−1, • ka, kt and kd are the kinetic rate constants for deposition, m/s, • Ci and Cb are the concentration of the particle foulant at the interface and in the bulk fluid, respectively; kg m3. Being essentially a surface chemistry phenomenon, this fouling mechanism can be very sensitive to factors that affect colloidal stability, e.g., zeta potential. A maximum fouling rate is usually observed when the fouling particles and the substrate exhibit opposite electrical charge, or near the point of zero charge of either of them. Particles larger than those of colloidal dimensions may also foul e.g., by sedimentation ("sedimentation fouling") or straining in small-size openings.

Fouling 143 With time, the resulting surface deposit may harden through processes collectively known as "deposit consolidation" or, colloquially, "aging". The common particulate fouling deposits formed from aqueous suspensions include: • iron oxides and iron oxyhydroxides (magnetite, hematite, lepidocrocite, maghemite, goethite); • Sedimentation fouling by silt and other relatively coarse suspended matter. Fouling by particles from gas aerosols is also of industrial significance. The particles can be either solid or liquid. The common examples can be fouling by flue gases, or fouling of air-cooled components by dust in air. The mechanisms are discussed in article on aerosol deposition. Corrosion fouling Corrosion deposits are created in-situ by the corrosion of the substrate. They are distinguished from fouling deposits, which form from material originating ex-situ. Corrosion deposits should not be confused with fouling deposits formed by ex-situ generated corrosion products. Corrosion deposits will normally have composition related to the composition of the substrate. Also, the geometry of the metal-oxide and oxide-fluid interfaces may allow practical distinction between the corrosion and fouling deposits. An example of corrosion fouling can be formation of an iron oxide or oxyhydroxide deposit from corrosion of the carbon steel underneath. Corrosion fouling should not be confused with fouling corrosion, i.e., any of the types of corrosion that may be induced by fouling. Chemical reaction fouling Chemical reactions may occur on contact of the chemical species in the process fluid with heat transfer surfaces. In such cases, the metallic surface sometimes acts as a catalyst. For example, corrosion and polymerization occurs in cooling water for the chemical industry which has a minor content of hydrocarbons. Systems in petroleum processing are prone to polymerization of olefins or deposition of heavy fractions (asphaltenes, waxes, etc.). High tube wall temperatures may lead to carbonizing of organic matter. Food industry, for example milk processing, also experiences fouling problems by chemical reactions. Fouling through an ionic reaction with an evolution of an inorganic solid is commonly classified as precipitation fouling (not chemical reaction fouling).

Fouling 144 Solidification fouling Solidification fouling occurs when a component of the flowing fluid "freezes" onto a surface forming a solid fouling deposit. Examples may include solidification of wax (with a high melting point) from a hydrocarbon solution, or of molten ash (carried in a furnace exhaust gas) onto a heat exchanger surface. The surface needs to have a temperature below a certain threshold; therefore, it is said to be subcooled in respect to the solidification point of the foulant. Biofouling Biofouling or biological fouling is the undesirable accumulation of micro-organisms, algae and diatoms, plants, and animals on surfaces, for example ships hulls, or piping and reservoirs with untreated water. This can be accompanied by microbiologically influenced corrosion (MIC). Bacteria can form biofilms or slimes. Thus the organisms can aggregate on surfaces using colloidal hydrogels of water and extracellular polymeric substances (EPS) (polysaccharides, lipids, nucleic acids, etc.). The biofilm structure is usually complex. Bacterial fouling can occur under either aerobic (with oxygen dissolved in water) or anaerobic (no oxygen) conditions. In practice, A fragment of a canal lock in Northern France, aerobic bacteria prefer open systems, when both oxygen and nutrients covered with zebra mussels are constantly delivered, often in warm and sunlit environments. Anaerobic fouling more often occurs in closed systems when sufficient nutrients are present. Examples may include sulfate-reducing bacteria (or sulfur-reducing bacteria), which produce sulfide and often cause corrosion of ferrous metals (and other alloys). Sulfide-oxidizing bacteria (e.g., Acidithiobacillus), on the other hand, can produce sulfuric acid, and can be involved in corrosion of concrete. Zebra mussels serve as an example of larger animals that have caused widespread fouling in North America. Composite fouling Composite fouling is common. This type of fouling involves more than one foulant or more than one fouling mechanism[11] working simultaneously. The multiple foulants or mechanisms may interact with each other resulting in a synergistic fouling which is not a simple arithmetic sum of the individual components. Fouling on Mars NASA Mars Exploration Rovers (Spirit and Opportunity) experienced (presumably) abiotic fouling of solar panels by dust particles from the Martian atmosphere.[12] Some of the deposits subsequently spontaneously cleaned off. This illustrates the universal nature of the fouling phenomena. Quantification of fouling The most straightforward way to quantify fairly uniform fouling is by stating the average deposit surface loading, i.e., kg of deposit per m² of surface area. The fouling rate will then be expressed in kg/m²s, and it is obtained by dividing the deposit surface loading by the effective operating time. The normalized fouling rate (also in kg/m²s) will additionally account for the concentration of the foulant in the process fluid (kg/kg) during preceding operations, and is useful for comparison of fouling rates between different systems. It is obtained by dividing the fouling rate by the foulant concentration. The fouling rate constant (m/s) can be obtained by dividing the normalized fouling rate by the mass density of the process fluid (kg/m³).

Fouling 145 Deposit thickness (μm) and porosity (%) are also often used for description of fouling amount. The relative reduction of diameter of piping or increase of the surface roughness can be of particular interest when the impact of fouling on pressure drop is of interest. In heat transfer equipment, where the primary concern is often the effect of fouling on heat transfer, fouling can be quantified by the increase of the resistance to the flow of heat (m²K/W) due to fouling (termed "fouling resistance"), or by development of heat transfer coefficient (W/m²K) with time. If under-deposit or crevice corrosion is of primary concern, it is important to note non-uniformity of deposit thickness (e.g., deposit waviness), localized fouling, packing of confined regions with deposits, creation of occlusions, "crevices", "deposit tubercles",[13] or sludge piles. Such deposit structures can create environment for underdeposit corrosion of the substrate material, e.g., intergranular attack, pitting, stress corrosion cracking, or localized wastage. Porosity and permeability of the deposits will likely influence the probability of underdeposit corrosion. Deposit composition can also be important - even minor components of the deposits can sometimes cause severe corrosion of the underlying metal (e.g., vanadium in deposits of fired boilers causing hot corrosion). There is no general rule on how much deposit can be tolerated, it depends on the system. In many cases, a deposit even a few micrometers thick can be troublesome. A deposit in a millimeter-range thickness will be of concern in almost any application. Progress of fouling with time Deposit on a surface does not always develop steadily with time. The following fouling scenarios can be distinguished, depending on the nature of the system and the local thermohydraulic conditions at the surface: • Induction period. Sometimes, a near-nil fouling rate is observed when the surface is new or very clean. This is often observed in biofouling and precipitation fouling. After the "induction period", the fouling rate increases. • "Negative" fouling. This can occur when fouling rate is quantified by monitoring heat transfer. Relatively small amounts of deposit can improve heat transfer, relative to clean surface, and give an appearance of "negative" fouling rate and negative total fouling amount. Negative fouling is often observed under nucleate-boiling heat-transfer conditions (deposit improves bubble nucleation) or forced-convection (if the deposit increases the surface roughness and the surface is no longer "hydraulically smooth"). After the initial period of "surface roughness control", the fouling rate usually becomes strongly positive. • Linear fouling. The fouling rate can be steady with time. This is a common case. • Falling fouling. Under this scenario, the fouling rate decreases with time, but never drops to zero. The deposit thickness does not achieves a constant value. The progress of fouling can be often described by two numbers: the initial fouling rate (a tangent to the fouling curve at zero deposit loading or zero time) and the fouling rate after a long period of time (an oblique asymptote to the fouling curve). • Asymptotic fouling. Here, the fouling rate decreases with time, until it finally reaches zero. At this point, the deposit thickness remains constant with time (a horizontal asymptote). This is often the case for relatively soft or poorly adherent deposits in areas of fast flow. The asymptote is usually interpreted as the deposit loading at which the deposition rate equals the deposit removal rate. • Accelerating fouling. Under this scenario, the fouling rate increases with time; the rate of deposit buildup accelerates with time (perhaps until it becomes transport limited). Mechanistically, this scenario can develop when fouling increases the surface roughness, or when the deposit surface exhibits higher chemical propensity to fouling than the pure underlying metal. • Seesaw fouling. Here, fouling loading generally increases with time (often assuming a generally linear or falling rate), but, when looked at in more detail, the fouling progress is periodically interrupted and takes the form of sawtooth curve. The periodic sharp variations in the apparent fouling amount often correspond to the moments of system shutdowns, startups or other transients in operation. The periodic variations are often interpreted as periodic removal of some of the deposit (perhaps deposit re-suspension due to pressure pulses, spalling due

Fouling 146 thermal stresses, or exfoliation due to redox transients). Steam blanketing has been postulated to occur between the partially spalled deposits and the heat transfer surface. However, other reasons are possible, e.g., trapping of air inside the surface deposits during shutdowns, or inaccuracy of temperature measurements during transients ("temperature streaming").[14] Fouling modelling Fouling of a system can be modelled as consisting of several steps: • Generation or ingress of the species that causes fouling ("foulant sourcing"); • Foulant transport with the stream of the process fluid (most often by advection); • Foulant transport from the bulk of the process fluid to the fouling surface. This transport is often by molecular or turbulent-eddy diffusion, but may also occur by inertial coasting/impaction, particle interception by the surface (for particles with finite sizes), electrophoresis, thermophoresis, diffusiophoresis, Stefan flow (in condensation and evaporation), sedimentation, Magnus force (acting on rotating particles), thermoelectric effect,[15][16] and other Schematics of the fouling process consisting of mechanisms. simultaneous foulant deposition and deposit • Induction period, i.e., a near-nil fouling rate at the initial period of removal. fouling (observed only for some fouling mechanisms); • Foulant crystallisation on the surface (or attachment of the colloidal particle, or chemical reaction, or bacterial growth); • Sometimes fouling autoretardation, i.e., reduction (or potentially enhancement) of crystallisation/attachment rate due to changes in the surface conditions caused by the fouling deposit; • Deposit dissolution (or re-entrainment of loosely attached particles); • Deposit consolidation on the surface (e.g., through Oswald ripening or differential solubility in temperature gradient) or cementation, which account for deposit losing its porosity and becoming more tenacious with time; • Deposit spalling, erosion wear, or exfoiliation. Deposition consists of transport to the surface and subsequent attachment. Deposit removal is either through deposit dissolution, particle re-entrainment, or deposit spalling, erosive wear, or exfoliation. Fouling results from foulant generation, foulant deposition, deposit removal, and deposit consolidation. For the modern model of fouling involving deposition with simultaneous deposit re-entrainment and consolidation,[17] the fouling process can be represented by the following scheme: Following the above scheme, the basic fouling equations can be written as follows (for steady-state conditions with flow, when concentration remains constant with time): where:

Fouling 147 • m is the mass loading of the deposit (consolidated and unconsolidated) on the surface (kg/m2); • t is time (s); • kd is the deposition rate constant (m/s); • ρ is the fluid density (kg/m3); • Cm - mass fraction of foulant in the fluid (kg/kg); • λr is the re-entrainment rate constant (1/s); • mr is the mass loading of the removable (i.e., unconsolidated) fraction of the surface deposit (kg/m2); and • λc is the consolidation rate constant (1/s). This system of equations can be integrated (taking that m = 0 and mr = 0 at t = 0) to the form: where λ = λr + λc. This model reproduces either linear, falling, or asymptotic fouling, depending on the relative values of k, λr, and λc. The underlying physical picture for this model is that of a two-layer deposit consisting of consolidated inner layer and loose unconsolidated outer layer. Such a bi-layer deposit is often observed in practice. The above model simplifies readily to the older model of simultaneous deposition and re-entrainment[18] (which neglects consolidation) when λc=0. In the absence of consolidation, the asymptotic fouling is always anticipated by this older model and the fouling progress can be described as: where m* is the maximum (asymptotic) mass loading of the deposit on the surface (kg/m2). Economic and environmental importance of fouling Fouling is ubiquitous and generates tremendous operational losses, not unlike corrosion. For example, one estimate puts the losses due to fouling of heat exchangers in industrialized nations to be about 0.25% of their GDP.[19] Another analysis[20] estimated (for 2006) the economical loss due to boiler and turbine fouling in China utilities at 4.68 billion dollars, which is about 0.169% the country GDP . The losses initially result from impaired heat transfer, corrosion damage (in particular under-deposit and crevice corrosion), increased pressure drop, flow blockages, flow redistribution inside components, flow instabilities, induced vibrations (possibly leading to other problems, e.g., fatigue[21]), fretting, premature failure of electrical heating elements, and a large number of other often unanticipated problems. In addition, the ecological costs should be (but typically are not) considered. The ecological costs arise from the use of biocides for the avoidance of biofouling, from the increased fuel input to compensate for the reduced output caused by fouling, and an increased use of cooling water in once-through cooling systems. For example, "normal" fouling at a conventionally fired 500 MW (net electrical power) power station unit accounts for output losses of the steam turbine of 5 MW and more. In a 1,300 MW nuclear power station, typical losses could be 20 MW and up (up to 100% if the station shuts down due to fouling-induced component degradation). In seawater desalination plants, fouling may reduce the gained output ratio by two-digit percentages (the gained output ratio is an equivalent that puts the mass of generated distillate in relation to the steam used in the process). The extra electrical consumption in compressor-operated coolers is also easily in the two-digit area. In addition to the operational costs, also the capital cost increases because the heat exchangers have to be designed in larger sizes to compensate for the heat-transfer loss due to fouling. To the output losses listed above, one needs to add the cost of down-time required to inspect, clean, and repair the components (millions of dollars per day of shutdown in lost revenue in a typical power plant), and the cost of actually doing this maintenance. Finally, fouling is often a root cause of serious degradation problems that may limit the life of components or entire plants.

Fouling 148 Fouling control The most fundamental and usually preferred method of controlling fouling is to prevent the ingress of the fouling species into the cooling water circuit. In steam power stations and other major industrial installations of water technology, macro fouling is avoided by way of pre-filtration and cooling water debris filters. Some plants employ foreign-object exclusion program (to eliminate the possibility of salient introduction of unwanted materials, e.g., forgetting tools during maintenance). Acoustic monitoring is sometimes employed to monitor for fretting by detached parts. In the case of micro fouling, water purification is achieved with extensive methods of water treatment, microfiltration, membrane technology (reverse osmosis, electrodeionization) or ion-exchange resins. The generation of the corrosion products in the water piping systems is often minimized by controlling the pH of the process fluid (typically alkanization with ammonia, morpholine, ethanolamine or sodium phosphate), control of oxygen dissolved in water (for example, by addition of hydrazine), or addition of corrosion inhibitors. For water systems at relatively low temperatures, the applied biocides may be classified as follows: inorganic chlorine and bromide compounds, chlorine and bromide cleavers, ozone and oxygen cleavers, unoxidizable biocides. One of the most important unoxidizable biocides is a mixture of chloromethyl-isothiazolinone and methyl-isothiazolinone. Also applied are dibrom nitrilopropionamide and quaternary ammonium compounds. For underwater ship hulls bottom paints are applied. Chemical fouling inhibitors[22] can reduce fouling in many systems, mainly by interfering with the crystallization, attachment, or consolidation steps of the fouling process. Examples for water systems are: chelating agents (for example, EDTA), long-chain aliphatic amines or polyamines (for example, octadecylamine, helamin, and other "film-forming" amines), organic phosphonic acids (for example, etidronic acid), or polyelectrolytes (for example, polyacrylic acid,[23] polymethacrylic acid, usually with a molecular weight lower than 10000). For fired boilers, aluminum or magnesium additives can lower the melting point of ash and promote creation of deposits which are easier to remove. See also process chemicals. Magnetic water treatment has been a subject of controversy as to its effectiveness for fouling control since 1950s. The prevailing opinion is that it simply "does not work".[24] Nevertheless, some studies suggest that it may be effective under some conditions to reduce buildup of calcium carbonate deposits.[25] On the component design level, fouling can often (but not always) be minimized by maintaining a relatively high (for example, 2 m/s) and uniform fluid velocity throughout the component. Stagnant regions need to be eliminated. Components are normally overdesigned to accommodate the fouling anticipated between cleanings. However, a significant overdesign can be a design error because it may lead to increased fouling due to reduced velocities. Periodic on-line pressure pulses or backflow can be effective if the capability is carefully incorporated at the design time. Blowdown capability is always incorporated into steam generators or evaporators to control the accumulation of non-volatile impurities that cause or aggravate fouling. Low-fouling surfaces (for example, very smooth, implanted with ions, or of low surface energy like Teflon) are an option for some applications. Modern components are typically required to be designed for ease of inspection of internals and periodic cleaning. On-line fouling monitoring systems are designed for some application so that blowing or cleaning can be applied before unpredictable shutdown is necessary or damage occurs. Chemical or mechanical cleaning processes for the removal of deposits and scales are recommended when fouling reaches the point of impacting the system performance or an onset of significant fouling-induced degradation (e.g., by corrosion). These processes comprise pickling with acids and complexing agents, cleaning with high-velocity water jets ("water lancing"), recirculating ("blasting") with metal, sponge or other balls, or propelling offline mechanical "bullet-type" tube cleaners. Whereas chemical cleaning causes environmental problems through the handling, application, storage and disposal of chemicals, the mechanical cleaning by means of circulating cleaning balls or offline "bullet-type" cleaning can be an environmentally friendlier alternative. In some heat-transfer applications, mechanical mitigation with dynamic scraped surface heat exchangers is an option. Also ultrasonic or abrasive cleaning methods are available for many specific applications.

Fouling 149 References [1] "Marine fouling and its prevention"; prepared for Bureau of Ships, Navy Dept, Woods Hole Oceanographic Institution, United States, Navy Dept. Bureau of Ship, 1952. (pdf) (https:/ / darchive. mblwhoilibrary. org/ handle/ 1912/ 191) [2] Siobhán Francesca E. Boerlage, "Scaling and Particulate Fouling in Membrane Filtration Systems", Taylor & Francis; 2001, ISBN 90-5809-242-9 (Google books) (http:/ / books. google. com/ books?id=SqHUSd3vk1oC& lpg=PP1& pg=PP1#v=onepage& q=& f=false) [3] Joshua M. Hawkes, "The Simulation and Study of Conditions Leading to Axial Offset Anomaly in Pressurized Water Reactors", Georgia Institute of Technology Master of Science Thesis, December 2004. (pdf) (http:/ / etd. gatech. edu/ theses/ available/ etd-11162004-143152/ unrestricted/ hawkes_joshua_m_200412_mast. pdf) [4] "Spark Plug Faces", brochure "Bosch Spark Plugs 0307", Part 1 (pdf) (http:/ / www. boschautoparts. com/ Technical Resources/ Spark Plugs/ SparkPlugFaces. pdf) [5] G.A. Mansoori "Physicochemical Basis of Arterial Blockage / Fouling. Prediction and Prevention." Department of Chemical Engineering, University of Illinois at Chicago, on-line publication, September 2001 (pdf) (http:/ / www. uic. edu/ labs/ trl/ PhysicochemicalBasisOfFouling. pdf) [6] T.R. Bott, "Fouling of Heat Exchangers (Chemical Engineering Monographs)", Elsevier Science, 1995. [7] J. Moghadasi, H. Müller-Steinhagen, M. Jamialahmadi, and A. Sharif, "Scale Deposition in Porous Media and their Removal by EDTA Injection ", ECI Engineering Conferences International Symposium Series, Heat Exchanger Fouling and Cleaning VII, July 1–6, 2007 - Tomar, Portugal. (pdf) (http:/ / services. bepress. com/ cgi/ viewcontent. cgi?article=1009& context=eci/ heatexchanger2007) [8] "Modeling PWR Fuel Corrosion Product Deposition and Growth Processes (5)", Technical Report 1009734, Electric Power Research Institute, Palo Alto, California, USA, 2004. [9] Eli Ruckenstein and Dennis C. Prieve, "Rate of deposition of Brownian particles under the action of London and double-layer forces", J. Chem. Soc., Faraday Trans. 2, 1973, 69, 1522-1536 (abstract) (http:/ / pubs. rsc. org/ en/ Content/ ArticleLanding/ 1973/ F2/ F29736901522). [10] Bruce D. Bowen and Norman Epstein, "Fine particle deposition in smooth parallel-plate channels", Journal of Colloid and Interface Science, Volume 72, Issue 1,15 October 1979, Pages 81-97 (abstract) (http:/ / www. sciencedirect. com/ science/ article/ pii/ 002197977990184X) [11] Hong Lu, "Composite Fouling of Heat Exchanger Surfaces", Nova Science Books, New York, 2007. [12] Mars Pathfinder - Dust Settling (MAE) (http:/ / mars. jpl. nasa. gov/ MPF/ ops/ mae. html) [13] H. M. Herro (Nalco Chemical Company), "Deposit-Related Corrosion in Industrial Cooling Water Systems", Presented at the National Association of Corrosion Engineers Corrosion ’89 meeting, New Orleans, Louisiana, April 17–21, 1989 ( (pdf) (http:/ / www. onlinewatertreatment. com/ literature/ Nalco/ docs/ R-482. pdf). [14] "Steam Generator Thermal Performance Degradation Case Studies", Report TR-110018, Electric Power Research Institute, Palo Alto, California, USA, 1998 (abstract) (http:/ / my. epri. com/ portal/ server. pt?Abstract_id=TR-110018). [15] V.P. Brusakov, "Law for the Deposition of Materials on Heat-Transmitting Surfaces under the Action of Thermoelectric Effects", Atomnaya Energiya, Vol.30, No.1, pp.10-14, January 1971. [16] D.H. Lister, ""Corrosion products in power generating systems". In: Fouling of Heat Exchanger Equipment", E.F. Somerscales and J.G. Knudsen (eds.), Hemisphere Pub. Corp., Washington, DC, USA, 1981, pp.135-200. [17] C.W. Turner, S.J. Klimas, "Modelling the Effect of Surface Chemistry on Particle Fouling Under Flow-Boiling Conditions", Proceeding of Heat Exchanger Fouling: Fundamental Approaches and Technical Solutions, 2001, July 8–13, Davos, Switzerland, AECL Report 12171. [18] D.O.Kern and R.E. Seaton, "A theoretical analysis of thermal surface fouling", Brit. Chem. Eng., 14, 5, 258, 1959. [19] H. Mueller-Steinhagen, M.R. Malayeri and A.P. Watkinson, "Fouling of Heat Exchanger--New Approaches to Solve Old Problem", Heat Transfer Engineering, 26(2), 2005. [20] Xu Zhi-Ming, ZHANG Zhong-Bin, and YANG Shan-Rang, "Costs due to utility fouling in China", ECI Engineering Conferences International Symposium Series, Heat Exchanger Fouling and Cleaning VII, July 1–6, 2007 - Tomar, Portugal. (pdf) (http:/ / services. bepress. com/ cgi/ viewcontent. cgi?article=1015& context=eci/ heatexchanger2007) [21] Herve BODINEAU and Thierry SOLLIER, "Tube support plate clogging up of French steam generators", Eurosafe webpage (http:/ / www. eurosafe-forum. org/ files/ Presentations2008/ Seminar 1/ Abstracts/ 1. 5 Tubes support plates clogging up of French PWR steam generators V5 correct figures Bodineau_20080923. pdf) [22] J.C. Cowan and D.J. Weintritt, "Water-Formed Scale Deposits. A Comprehensive Study of the Prevention, Control, Removal and Use of Mineral Scale", Gulf Publishing Company, Houston, Texas, 1976. [23] "Dispersants for Tube Fouling Control: Volume 2: Short-Term Trial at ANO-2", Report 1003144, Electric Power Research Institute, Palo Alto, California, USA, 2001 (abstract) (http:/ / my. epri. com/ portal/ server. pt?space=CommunityPage& cached=true& parentname=ObjMgr& parentid=2& control=SetCommunity& CommunityID=404& RaiseDocID=000000000001003144& RaiseDocType=Abstract_id) [24] " Magnetic Water Treatment (http:/ / www. wbdg. org/ ccb/ ARMYCOE/ PWTB/ pwtb_420_49_34. pdf)", Public Works Technical Bulletin 420-49-34, U.S. Army Corps of Engineers, 15 June 2001. [25] A. Szkatula, M. Balanda, M. Kopec, "Magnetic treatment of industrial water. Silica activation". Eur. Phys. J.Applied Physics, 1, vol. 18, p. 41-49, 2002 (abstract) (http:/ / www. epjap. org/ index. php?option=article& access=doi& doi=10. 1051/ epjap:2002025)

Fouling 150 External links • Crude Oil Fouling research (http://www.imperial.ac.uk/crudeoilfouling) • On-line & In-situ fouling/biofilm/scales/slime monitoring (http://www.neo-sens.com) • A green Antifouling alternative to Chlorine (http://www.anti-fouling.eu) • fouling monitor instrumentation to quantify problems and solutions (http://www.bridgersci.com) • http://economictimes.indiatimes.com/articleshow/msid-3720536,flstry-1.cms The Economic Times: Cleantech firms gearing up for bigger opportunities in biofouling Fourier number In physics and engineering, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes heat conduction. Conceptually, it is the ratio of the heat conduction rate to the rate of thermal energy storage. Together with the Biot number, it characterizes transient conduction problems. It is defined as: where: • α is the thermal diffusivity [m2/s] • t is the characteristic time [s] • R is the length through which conduction occurs [m] For transient mass transfer by diffusion, there is an analogous mass Fourier Number (also denoted Fo) defined as: where: • "D" is the Diffusivity • "t" is the characteristic timescale • "L" is the length scale of interest Using Fourier number Together with the Biot number, the Fourier number can be used to solve unsteady state conduction problems. The Fourier number is frequently used as a nondimensional time parameter. If the Biot number is less than 0.1, the following equation derived from the Biot and Fourier numbers can be used to find the time. Where T0 is the initial temperature and T is the temperature at the center. References • Incropera, Frank P.; DeWitt, David P. Fundamentals of Heat and Mass Transfer (5th Edition ed.). Wiley.

Heat 151 Heat In the natural sciences, heat is energy transferred from one body to another by thermal interaction.[1][2] In contrast to work, heat is always accompanied by a transfer of entropy. Heat flow is characteristic of macroscopic bodies. Its mechanisms and properties can be understood in terms of the microscopic constituents of bodies. Heat flow from a high to a low temperature body occurs spontaneously. This flow of energy can be harnessed and partly converted into useful work by means of a heat engine. The second law of thermodynamics prohibits heat flow directly Nuclear fusion in the Sun converts nuclear potential energy into available internal energy and keeps the temperature of the Sun very high. Consequently, heat is from a low to a high temperature body, but transported to Earth as electromagnetic radiation. This is the main source of energy with the aid of a heat pump external work for life on Earth. can be used to transport internal energy indirectly from a low to a high temperature body. In ordinary language, heat has a diversity of meanings, including temperature.[3] The present article is about natural science, especially physics, for which heat by strict definition always refers to a process of transfer of energy. In many writings, the ordinary language usage remains alongside the strict physical definition, and this can lead unwary readers to become confused. In physics, there are two kinds of thermal interaction that account for heat transfer: conduction,[4] and electromagnetic radiation.[5] The SI unit of heat is the joule. Heat can be measured by calorimetry,[6] or determined indirectly by calculations based on other quantities, relying for instance on the first law of thermodynamics. In physics, especially in calorimetry, and in meteorology, the concepts of latent heat and of sensible heat are used. Latent heat produces changes of state without temperature change, while sensible heat produces temperature change. Overview

Heat 152 This article is about heat in physics, where heat is defined as energy transferred to the system by thermal interactions. Heat flows spontaneously from systems of higher temperature to systems of lower temperature. When two systems come into thermal contact, they exchange energy through the microscopic interactions of their particles. When the systems are at different temperatures, this entails spontaneous net flow of energy from the hotter to the cooler, so that the hotter decreases in temperature and the cooler increases in temperature. This will continue until their temperatures are equal. Then the net flow of energy has settled to zero, and the systems are said to be in a relation of thermal equilibrium. Spontaneous heat transfer is an irreversible Heat may flow across the boundary of the system and thus change its internal energy. process. The first law of thermodynamics requires that the internal energy of an isolated system is conserved. To change the internal energy of a system, energy must be transferred to or from the system. For a closed system, heat and work are the mechanisms by which energy can be transferred. For an open system, internal energy can be changed also by transfer of matter.[7] Work performed by a body is, by definition, an energy transfer from the body that is due to a change to external or mechanical parameters of the body, such as the volume, magnetization, and location of center of mass in a gravitational field.[2][8][9][10][11][12] Heat is not regarded as being stored within a body. Like work, it exists only as energy in transit from one body to another or between a body and its surroundings. When energy in the form of heat is added to a body, it is stored as kinetic and potential energy of the atoms and molecules in the body, which are jointly totalled as internal energy.[13] Definitions Scottish physicist James Clerk Maxwell, in his 1871 classic Theory of Heat, was one of many who began to build on the already established idea that heat has something to do with matter in motion. This was the same idea put forth by Sir Benjamin Thompson in 1798, who said he was only following up on the work of many others. One of Maxwells recommended books was Heat as a Mode of Motion, by John Tyndall. Maxwell outlined four stipulations for the definition of heat: • It is something which may be transferred from one body to another, according to the second law of thermodynamics. • It is a measurable quantity, and thus treated mathematically. • It cannot be treated as a substance, because it may be transformed into something that is not a substance, e.g., mechanical work. • Heat is one of the forms of energy. Mechanisms of heat transfer Referring to conduction, Partington writes: "If a hot body is brought in conducting contact with a cold body, the temperature of the hot body falls and that of the cold body rises, and it is said that a quantity of heat has passed from the hot body to the cold body."[14] Referring to radiation, Maxwell writes: "In Radiation, the hotter body loses heat, and the colder body receives heat by means of a process occurring in some intervening medium which does not itself thereby become hot."[15] From these empirically based ideas of heat, and from other empirical observations, the notions of internal energy and of entropy can be derived, so as to lead to the recognition of the first and second laws of thermodynamics.[16] This was the way of the historical pioneers of thermodynamics.[17][18]

Heat 153 Notation and units As a form of energy heat has the unit joule (J) in the International System of Units (SI). However, in many applied fields in engineering the British Thermal Unit (BTU) and the calorie are often used. The standard unit for the rate of heat transferred is the watt (W), defined as joules per second. The total amount of energy transferred as heat is conventionally written as Q for algebraic purposes. Heat released by a system into its surroundings is by convention a negative quantity (Q < 0); when a system absorbs heat from its surroundings, it is positive (Q > 0). Heat transfer rate, or heat flow per unit time, is denoted by . This should not be confused with a time derivative of a function of state (which can also be written with the dot notation) since heat is not a function of state. Heat flux is defined as rate of heat transfer per unit cross-sectional area, resulting in the unit watts per square metre. Estimation of quantity of heat The quantity of heat transferred by some process can either be directly measured, or determined indirectly through calculations based on other quantities. Direct measurement is by calorimetry and is the primary empirical basis of the idea of quantity of heat transferred in a process. The transferred heat is measured by changes in a body of known properties, for example, temperature rise, change in volume or length, or phase change, such as melting of ice.[19][20] Indirect estimations of quantity of heat transferred rely on the law of conservation of energy, and, in particular cases, on the first law of thermodynamics. Indirect estimation is the primary approach of many theoretical studies of quantity of heat transferred.[21][22][23] Internal energy and enthalpy In the case where the number of particles in the system is constant (closed systems), the first law of thermodynamics states that the differential change in internal energy dU of a system is given by an infinitesimal amount of heat δQ supplied to the system minus the infinitesimal amount of work δW exerted by the system:[24] This can also be interpreted as that δQ makes contributions to the internal energy and to the work done by the system: The work done by the system includes boundary work (when the system increases its volume against an external force, such as that exerted by a piston) and other work (e.g. shaft work performed by a compressor fan): In this Section we will neglect the "other-work" contribution. The internal energy, U, is a state function. In cyclical processes, such as the operation of a heat engine, state functions return to their initial values after completing one cycle. Thus, the differential for the internal energy is an exact differential dU. The symbol for exact differentials is the lowercase letter d. In contrast, neither Q nor W represents the state of the system. Thus, infinitesimal amounts of heat and work are inexact differentials, denoted by δQ and δW, respectively. The lowercase Greek letter delta, δ, is the symbol for inexact differentials. The integral of any inexact differential over the time it takes for a system to leave and return to the same thermodynamic state does not necessarily equal zero. However, if heat is supplied to a system in which no irreversible processes take place and which has a well-defined temperature T, the heat δQ and the temperature T form the exact differential

Heat 154 with S the entropy of the system. Likewise, with a well-defined pressure p behind the moving boundary, the work δW and the pressure p form the exact differential with V the volume of the system. In general, for homogeneous systems, Associated with this differential equation is that the internal energy may be considered to be a function U (S,V) of its natural variables S and V. Thus it is said that the internal energy representation of the fundamental thermodynamic relation is written [25][26] If V is constant and if p is constant with H the enthalpy given by The enthalpy may be considered to be a function H (S,p) of its natural variables S and p. Thus it is said that the enthalpy representation of the fundamental thermodynamic relation is written [26][27] The internal energy representation and the enthalpy representation are partial Legendre transforms of one another. They contain the same physical information, written in different ways.[27][28] Path-independent examples for an ideal gas For a simple compressible system such as an ideal gas inside a piston, the internal energy change ΔU at constant volume and the enthalpy change ΔH at constant pressure are modeled by separate heat capacity values, which are CV and Cp, respectively. Constrained to have constant volume, the heat, Q, required to change its temperature from an initial temperature, T0, to a final temperature, Tf, is given by Allowing the system to expand or contract at constant pressure, the heat, Q, required to change its temperature from an initial temperature, T0, to a final temperature, Tf, is given by Here we used the definition of the enthalpy and the fact that p is constant. When integrating an exact differential (e.g. dU), the lowercase letter d is substituted for Δ (e.g. ΔU). Note that the symbol Δ is convenient since it is compact, but it can lead to sign errors. So it may be better to write Uf - U0 instead of ΔU. When integrating an inexact differential (e.g. δQ), the lowercase Greek letter δ is removed with no replacement (e.g. Q).

Heat 155 Chemical reactions For a closed system in which a chemical reaction is of interest, the extent of reaction, denoted by ξ, states the degree of advancement of the reaction and is included as a further natural variable for internal energy and for enthalpy. This is written In practice, chemists often use tables of a special but unnamed thermodynamic potential that is not the enthalpy expressed in its natural variables; instead they use the enthalpy expressed as a function of temperature instead of entropy. This special potential is related to the natural form of the enthalpy H (S,p,ξ) by another partial Legendre transform, that makes its natural variables T, p, and ξ. The special unnamed potential is still usually called the enthalpy. It can be written This enthalpy is used to report the enthalpy change of reaction, also called the heat of reaction.[29][30] Latent and sensible heat In an 1847 lecture entitled On Matter, Living Force, and Heat, James Prescott Joule characterized the terms latent heat and sensible heat as components of heat each affecting distinct physical phenomena, namely the potential and kinetic energy of particles, respectively.[31] He described latent energy as the energy possessed via a distancing of particles where attraction was over a greater distance, i.e. a form of potential energy, and the sensible heat as an energy involving the motion of particles or what was known as a living force. At the time of Joule kinetic energy either held invisibly internally or held visibly externally was known as a living force. Latent heat is the heat released or absorbed by a chemical substance or a thermodynamic system during a change of state that occurs without a change in Joseph Black temperature. Such a process may be a phase transition, such as the melting of ice or the boiling of water.[32][33] The term was introduced around 1750 by Joseph Black as derived from the Latin latere (to lie hidden), characterizing its effect as not being directly measurable with a thermometer. Sensible heat, in contrast to latent heat, is the heat exchanged by a thermodynamic system that has as its sole effect a change of temperature.[34] Sensible heat therefore only increases the thermal energy of a system. Consequences of Blacks distinction between sensible and latent heat are examined in the Wikipedia article on calorimetry. Specific heat Specific heat, also called specific heat capacity, is defined as the amount of energy that has to be transferred to or from one unit of mass (kilogram) or amount of substance (mole) to change the system temperature by one degree. Specific heat is a physical property, which means that it depends on the substance under consideration and its state as specified by its properties. The specific heats of monatomic gases (e.g., helium) are nearly constant with temperature. Diatomic gases such as hydrogen display some temperature dependence, and triatomic gases (e.g., carbon dioxide) still more.

Heat 156 Microscopic explanation of conduction of heat Heat is a macroscopic characteristic of systems, but like other thermodynamic quantities it has a microscopic explanation given by statistical mechanics. Temperature in ideal gases is directly proportional to the kinetic energy of translational motion of microscopic particles, and heat conduction is the net effect of two-way exchange of such energy through molecular collisions. An early and vague expression of this was by Francis Bacon.[35][36] Precise and detailed versions of it were developed in the nineteenth century.[37] Conduction is conceptually and physically distinct but physically inseparable from radiative transfer of heat within a body. For solids, conduction of heat is through collective motions of microscopic particles, including phonons.[38] Entropy In 1856, German physicist Rudolf Clausius defined the second fundamental theorem (the second law of thermodynamics) in the mechanical theory of heat (thermodynamics): "if two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat Q from work at the temperature T, has the equivalence-value:"[39][40] In 1865, he came to define the entropy symbolized by S, such that, due to the supply of the amount of heat Q at temperature T the entropy of the system is increased by Rudolf Clausius and thus, for small changes, quantities of heat δQ (an inexact differential) are defined as quantities of TdS, with dS an exact differential: This equality is only valid for a closed system and if no irreversible processes take place inside the system while the heat δQ is applied. If, in contrast, irreversible processes are involved, e.g. some sort of friction, then there is entropy production and, instead of the above equation, one has This is the second law of thermodynamics for closed systems.

Heat 157 Heat transfer in engineering The discipline of heat transfer, typically considered an aspect of mechanical engineering and chemical engineering, deals with specific applied methods by which thermal energy in a system is generated, or converted, or transferred to another system. Although the definition of heat implicitly means the transfer of energy, the term heat transfer encompasses this traditional usage in many engineering disciplines and laymen language. Heat transfer includes the mechanisms of heat conduction, thermal radiation, and mass transfer. In engineering, the term convective heat transfer is used to describe the combined A red-hot iron rod from which heat transfer to the surrounding environment will be primarily through radiation. effects of conduction and fluid flow and is often regarded as an additional mechanism of heat transfer. Although distinct physical laws may describe the behavior of each of these methods, real systems often exhibit a complicated combination which are often described by a variety of complex mathematical methods. Practical applications In accordance with the first law for closed systems, energy transferred as heat enters one body and leaves another, changing the internal energies of each. Transfer, between bodies, of energy as work is a complementary way of changing internal energies. Though it is not logically rigorous from the viewpoint of strict physical concepts, a common form of words that expresses this is to say that heat and work are interconvertible. Heat engines operate by converting heat flow from a high temperature reservoir to a low temperature reservoir into work. One example are steam engines, where the high temperature reservoir is steam generated by boiling water. The flow of heat from the hot steam to water is converted into mechanical work via a turbine or piston. Heat engines achieve high efficiency when the difference between initial and final temperature is high. Heat pumps, by contrast, use work to cause thermal energy to flow from low to high temperature, the opposite direction heat would flow spontaneously. An example is a refrigerator or air conditioner, where electric power is used to cool a low temperature system (the interior of the refrigerator) while heating a higher temperature environment (the exterior). High efficiency is achieved when the temperature difference is small. Usage of words The strictly defined physical term quantity of energy transferred as heat has a resonance with the ordinary language noun heat and the ordinary language verb heat. This can lead to confusion if ordinary language is muddled with strictly defined physical language. In the strict terminology of physics, heat is defined as a word that refers to a process, not to a state of a system. In ordinary language one can speak of a process that increases the temperature of a body as heating it, ignoring the nature of the process, which could be one of adiabatic transfer of energy as work. But in strict physical terms, a process is admitted as heating only when what is meant is transfer of energy as heat. Such a process does not necessarily increase the temperature of the heated body, which may instead change its phase, for example by melting. In the strict physical sense, heat cannot be produced, because the usage production of heat misleadingly seems to refer to a state variable. Thus, it would be physically improper to speak of heat production by friction, or of heating by adiabatic compression on descent of an air parcel or of heat production by chemical reaction; instead, proper physical usage speaks of conversion of kinetic energy of bulk flow, or of potential energy of bulk matter,[41] or of chemical potential energy, into internal energy, and of transfer of energy as heat. Occasionally a present-day author, especially when referring to history, writes of "adiabatic heating", though this is a

Heat 158 contradiction in terms of present day physics.[42] Historically, before the concept of internal energy became clear over the period 1850 to 1869, physicists spoke of "heat production" where nowadays one speaks of conversion of other forms of energy into internal energy.[43] Notes [1] Reif, F. (1965), pp. 67, 73. [2] Kittel, C. Kroemer, H. (1980). Thermal Physics, second edition, W.H. Freeman, San Francisco, ISBN 0-7167-1088-9, p. 227. [3] Oxford English Dictionary, second edition, Oxford University Press, Oxford, UK. [4] Guggenheim, E.A. (1949/1967), p. 8. [5] Planck. M. (1914). The Theory of Heat Radiation, a translation by Masius, M. of the second German edition, P. Blakistons Son & Co., Philadelphia. [6] Maxwell 1871, Chapter III. [7] Kondepudi, D. (2008), p. 59. [8] Planck, M. (1927), p. 4. [9] Crawford, F.H. (1963). Heat, Thermodynamics, and Statistical Physics, Rupert Hart-Davis, Harcourt, Brace & World, p. 98. [10] Reif, F. (1965), p. 73. [11] Gislason, E.A., Craig, N.C. (2005). Cementing the foundations of thermodynamics: Comparison of system-based and surroundings-based definitions of heat and work, J. Chem. Thermodynamics, 37: 954–966. [12] Anacleto, J., Ferreira, J.M. (2008). Surroundings-based and system-based heat and work definitions: Which one is most suitable?, J. Chem. Thermodynamics, 40: 134–135. [13] Smith, J.M., Van Ness, H.C., Abbot, M.M. (2005). Introduction to Chemical Engineering Thermodynamics. McGraw-Hill. ISBN 0073104450. [14] Partington, J.R. (1949), p. 118. [15] Maxwell, J.C. (1871), p. 10. [16] Planck, M. (1903). [17] Partington, J.R. (1949). [18] Truesdell, C. (1980), page 15. [19] Maxwell J.C. (1872), p. 54. [20] Planck (1927), Chapter 3. [21] Bryan, G.H. (1907), p. 47. [22] C. Carathéodory (1909). "Untersuchungen über die Grundlagen der Thermodynamik". Mathematische Annalen 67: 355–386. doi:10.1007/BF01450409. A partly reliable translation is to be found at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA. [23] Callen, H.B. (1985), Section 1-8. [24] An alternate convention is to consider the work performed on the system by its surroundings. This leads to a change in sign of the work. This is the convention adopted by many modern textbooks of physical chemistry, such as those by Peter Atkins and Ira Levine, but many textbooks on physics define work as work done by the system. [25] Callen, H.B., (1985), Section 2-3, pp. 40–42. [26] Adkins, C.J. (1983), p. 101. [27] Callen, H.B. (1985), p. 147. [28] Adkins, C.J. (1983), pp. 100–104. [29] Prigogine, I.; Defay, R. (1954). Chemical Thermodynamics. translated by D.H. Everett. Longmans, Green & Co., London, Section 2-6, pp. 29–31. [30] Callen, H.B., (1985), Section 5-3, pp. 146–148, Section 6-6, pp. 173–179. [31] J. P. Joule (1884), The Scientific Paper of James Prescott Joule, The Physical Society of London, p. 274, "Heat must therefore consist of either living force or of attraction through space. In the former case we can conceive the constituent particles of heated bodies to be, either in whole or in part, in a state of motion. In the latter we may suppose the particles to be removed by the process of heating, so as to exert attraction through greater space. I am inclined to believe that both of these hypotheses will be found to hold good,—that in some instances, particularly in the case of sensible heat, or such as is indicated by the thermometer, heat will be found to consist in the living force of the particles of the bodies in which it is induced; whilst in others, particularly in the case of latent heat, the phenomena are produced by the separation of particle from particle, so as to cause them to attract one another through a greater space.", Lecture on Matter, Living Force, and Heat. May 5 and 12, 1847 [32] Perrot, Pierre (1998). A to Z of Thermodynamics. Oxford University Press. ISBN 0-19-856552-6. [33] Clark, John, O.E. (2004). The Essential Dictionary of Science. Barnes & Noble Books. ISBN 0-7607-4616-8. [34] Ritter, Michael E. (2006). "The Physical Environment: an Introduction to Physical Geography" (http:/ / www. uwsp. edu/ geo/ faculty/ ritter/ geog101/ textbook/ energy/ energy_balance. html). . [35] Bacon, F. (1620). Novum Organum Scientiarum, translated by Devey, J., P.F. Collier & Son, New York, 1902.

Heat 159 [36] Partington, J.R. (1949), page 131. [37] Partington, J.R. (1949), pages 132–136. [38] Kittel, C. (1953/1980). Introduction to Solid State Physics, (first edition 1953), fifth edition 1980, John Wiley & Sons, New York, ISBN 0-471-49024-5, Chapters 4, 5. [39] Published in Poggendoff’s Annalen, Dec. 1854, vol. xciii. p. 481; translated in the Journal de Mathematiques, vol. xx. Paris, 1855, and in the Philosophical Magazine, August 1856, s. 4. vol. xii, p. 81 [40] Clausius, R. (1865). The Mechanical Theory of Heat –with its Applications to the Steam Engine and to Physical Properties of Bodies. London: John van Voorst, 1 Paternoster Row. MDCCCLXVII. [41] Iribarne, J.V., Godson, W.L. (1973/1989). Atmospheric thermodynamics, second edition, reprinted 1989, Kluwer Academic Publishers, Dordrecht, ISBN 90-277-1296-4, p. 136. [42] Bailyn, M. (1994), Part D, pp. 50–56. [43] For example, Clausius, R. (1857). Über die Art der Bewegung, welche wir Wärme nennen, Annalen der Physik und Chemie, 100: 353–380. (http:/ / gallica. bnf. fr/ ark:/ 12148/ bpt6k15185v/ f371. table) Translated as On the nature of the motion which we call heat, Phil. Mag. series 4, 14: 108–128. (http:/ / books. google. com. au/ books?id=B5QOAAAAIAAJ& pg=RA1-PA108& dq=On+ the+ Nature+ of+ the+ Motion+ which+ we+ call+ Heat,+ clausius,+ philosophical+ magazine,+ 1857& hl=en& ei=FH84TJvOEIKknQfXuoznAw& sa=X& oi=book_result& ct=result& redir_esc=y#v=onepage& q& f=false) References Bibliography • Adkins, C.J. (1968/1983). Equilibrium Thermodynamics, (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, ISBN 0-521-25445-0. • Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York, ISBN 0-88318-797-3. • Beattie, J.A., Oppenheim, I. (1979). Principles of Thermodynamics, Elsevier, Amsterdam, ISBN 0–444–41806–7. • Bryan, G.H. (1907). Thermodynamics. An Introductory Treatise dealing mainly with First Principles and their Direct Applications, B.G. Teubner, Leipzig. • Callen, H.B. (1960/1985). Thermodynamics and an Introduction to Thermostatistics, (1st edition 1960) 2nd edition 1985, Wiley, New York, ISBN 0-471-86256-8. • Guggenheim, E.A. (1967) [1949], Thermodynamics. An Advanced Treatment for Chemists and Physicists (fifth ed.), Amsterdam: North-Holland Publishing Company. • Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of Thermodynamics, pages 1–97 of volume 1, ed. W. Jost, of Physical Chemistry. An Advanced Treatise, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081. • Kondepudi, D. (2008), Introduction to Modern Thermodynamics, Chichester UK: Wiley, ISBN 978–0–470–01598–8. • Kondepudi, D., Prigogine, I. (1998). Modern Thermodynamics: From Heat Engines to Dissipative Structures, John Wiley & Sons, Chichester, ISBN 0–471–97393–9. • Maxwell, J.C. (1871), Theory of Heat (first ed.), London: Longmans, Green and Co. • Partington, J.R. (1949), An Advanced Treatise on Physical Chemistry., volume 1, Fundamental Principles. The Properties of Gases, London: Longmans, Green and Co. • Planck, M. (1903) [1897], Treatise on Thermodynamics (http://www.onread.com/reader/145819) (first ed.), London: Longmans, Green and Co. • Planck, M. (1927) [1923], Treatise on Thermodynamics (third ed.), London: Longmans, Green and Co. • Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. New York: McGraw-Hll, Inc.. • Truesdell, C. (1980). The Tragicomical History of Thermodynamics 1822-1854, Springer, New York, ISBN 0–387–90403–4.

Heat 160 External links • Heat (http://www.bbc.co.uk/programmes/b00fq3d4) on In Our Time at the BBC. ( listen now (http://www. bbc.co.uk/iplayer/console/b00fq3d4/In_Our_Time_Heat)) • Plasma heat at 2 gigakelvins (http://www.foxnews.com/story/0,2933,187464,00.html) - Article about extremely high temperature generated by scientists (Foxnews.com) • Correlations for Convective Heat Transfer (http://www.cheresources.com/convection.shtml) - ChE Online Resources • An Introduction to the Quantitative Definition and Analysis of Heat written for High School Students (http:// canadaconnects.ca/chemistry/10114/) Heat capacity Heat capacity (usually denoted by a capital C, often with subscripts), or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substances temperature by a given amount. In the International System of Units (SI), heat capacity is expressed in units of joule(s) (J) per kelvin (K). Derived quantities that specify heat capacity as an intensive property, i.e., independent of the size of a sample, are the molar heat capacity, which is the heat capacity per mole of a pure substance, and the specific heat capacity, often simply called specific heat, which is the heat capacity per unit mass of a material. Occasionally, in engineering contexts, a volumetric heat capacity is used. Because heat capacities of materials tend to mirror the number of atoms or particles they contain, when intensive heat capacities of various substances are expressed directly or indirectly per particle number, they tend to vary within a much more narrow range. Temperature reflects the average kinetic energy of particles in matter while heat is the transfer of thermal energy from high to low temperature regions. Thermal energy transmitted by heat is stored as kinetic energy of atoms as they move, and in molecules as they rotate. Additionally, some thermal energy may be stored as the potential energy associated with higher-energy modes of vibration, whenever they occur in interatomic bonds in any substance. Translation, rotation, and a combination of the two types of energy in vibration (kinetic and potential) of atoms represent the degrees of freedom of motion which classically contribute to the heat capacity of atomic matter (loosely bound electrons occasionally also participate). On a microscopic scale, each system particle absorbs thermal energy among the few degrees of freedom available to it, and at high enough temperatures, this process contributes to a specific heat capacity that classically approaches a value per mole of particles that is set by the Dulong-Petit law. This limit, which is about 25 joules per kelvin for each mole of atoms, is achieved by many solid substances at room temperature (see table below). For quantum mechanical reasons, at any given temperature, some of these degrees of freedom may be unavailable, or only partially available, to store thermal energy. In such cases, the specific heat capacity will be a fraction of the maximum. As the temperature approaches absolute zero, the specific heat capacity of a system also approaches zero, due to loss of available degrees of freedom. Quantum theory can be used to quantitatively predict specific heat capacities in simple systems. Background Before the development of modern thermodynamics, it was thought that heat was a fluid, the so-called caloric. Bodies were capable of holding a certain amount of this fluid, hence the term heat capacity, named and first investigated by Joseph Black in the 1750s.[1] Today one instead discusses the internal energy of a system. This is made up of its microscopic kinetic and potential energy. Heat is no longer considered a fluid. Rather, it is a transfer of disordered energy at the microscopic level. Nevertheless, at least in English, the term "heat capacity" survives. Some other languages prefer the term thermal capacity, which is also sometimes used in English.

Heat capacity 161 Older units and English units An older unit of heat is the kilogram-calorie (Cal), originally defined as the energy required to raise the temperature of one kilogram of water by one degree Celsius, typically from 15 to 16 °C. The specific heat capacity of water on this scale would therefore be exactly 1 Cal/(°C·kg). However, due to the temperature-dependence of the specific heat, a large number of different definitions of the calorie came into being. Whilst once it was very prevalent, especially its smaller cgs variant the gram-calorie (cal), defined so that the specific heat of water would be 1 cal/(K·g), in most fields the use of the calorie is now archaic. In the United States other units of measure for heat capacity may be quoted in disciplines such as construction, civil engineering, and chemical engineering. A still common system is the English Engineering Units in which the mass reference is pound mass and the temperature is specified in degrees Fahrenheit or Rankine. One (rare) unit of heat is the pound calorie (lb-cal), defined as the amount of heat required to raise the temperature of one pound of water by one degree Celsius. On this scale the specific heat of water would be 1 lb-cal/(K·lb). More common is the British thermal unit, the standard unit of heat in the U.S. construction industry. This is defined such that the specific heat of water is 1 BTU/(°F·lb). Extensive and intensive quantities An objects heat capacity (symbol C) is defined as the ratio of the amount of heat energy transferred to an object to the resulting increase in temperature of the object, In the International System of Units, heat capacity has the unit joules per kelvin. Heat capacity is an extensive property, meaning it is a physical property that scales with the size of a physical system. A sample containing twice the amount of substance as another sample requires the transfer of twice the amount of heat ( ) to achieve the same change in temperature ( ). For many experimental and theoretical purposes it is more convenient to report heat capacity as an intensive property - an intrinsic characteristic of a particular substance. This is most often accomplished by expressing the property in relation to a unit of mass. In science and engineering, such properties are often prefixed with the term specific.[2] International standards now recommend that specific heat capacity always refer to division by mass.[3] The units for the specific heat capacity are . In chemistry, heat capacity is often specified relative to one mole, the unit of amount of substance, and is called the molar heat capacity. It has the unit . For some considerations it is useful to specify the volume-specific heat capacity, commonly called volumetric heat capacity, which is the heat capacity per unit volume and has SI units . This is used almost exclusively for liquids and solids, since for gases it may be confused with specific heat capacity at constant volume. Metrology, the measurement of heat capacity The heat capacity of most systems is not a constant. Rather, it depends on the state variables of the thermodynamic system under study. In particular it is dependent on temperature itself, as well as on the pressure and the volume of the system, and the ways in which pressures and volumes have been allowed to change while the system has passed from one temperature to another. The reason for this is that pressure-volume work done to the system raises its temperature by a mechanism other than heating, while pressure-volume work done by the system absorbs heat without raising the systems temperature. Different measurements of heat capacity can therefore be performed, most commonly at constant pressure and constant volume. The values thus measured are usually subscripted (by p and V, respectively) to indicate the

Heat capacity 162 definition. Gases and liquids are typically also measured at constant volume. Measurements under constant pressure produce larger values than those at constant volume because the constant pressure values also include heat energy that is used to do work to expand the substance against the constant pressure as its temperature increases. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. The specific heat capacities of substances comprising molecules (as distinct from monatomic gases) are not fixed constants and vary somewhat depending on temperature. Accordingly, the temperature at which the measurement is made is usually also specified. Examples of two common ways to cite the specific heat of a substance are as follows: • Water (liquid): cp = 4.1855 [J/(g·K)] (15 °C, 101.325 kPa) • Water (liquid): CvH = 74.539 J/(mol·K) (25 °C) For liquids and gases, it is important to know the pressure to which given heat-capacity data refer. Most published data are given for standard pressure. However, quite different standard conditions for temperature and pressure have been defined by different organizations. The International Union of Pure and Applied Chemistry (IUPAC) changed its recommendation from one atmosphere to the round value 100 kPa (≈750.062 Torr).[4] Calculation from first principles The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below. Thermodynamic relations and definition of heat capacity The internal energy of a closed system changes either by adding heat to the system or by the system performing work. Written mathematically we have For work as a result of an increase of the system volume we may write, If the heat is added at constant volume, then the second term of this relation vanishes and one readily obtains This defines the heat capacity at constant volume, CV. Another useful quantity is the heat capacity at constant pressure, CP. With the enthalpy of the system given by our equation for dU changes to and therefore, at constant pressure, we have

Heat capacity 163 Relation between heat capacities Measuring the heat capacity, sometimes referred to as specific heat, at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume implying the containing vessel must be nearly rigid or at least very strong (see coefficient of thermal expansion and compressibility). Instead it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws. Starting from the fundamental Thermodynamic Relation one can show where the partial derivatives are taken at constant volume and constant number of particles, and constant pressure and constant number of particles, respectively. This can also be rewritten where is the coefficient of thermal expansion, is the isothermal compressibility. The heat capacity ratio or adiabatic index is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor. Ideal gas [5] For an ideal gas, evaluating the partial derivatives above according to the equation of state where R is the gas constant for an ideal gas → = → substituting = this equation reduces simply to Mayers relation,

Heat capacity 164 Specific heat capacity The specific heat capacity of a material on a per mass basis is which in the absence of phase transitions is equivalent to where is the heat capacity of a body made of the material in question, is the mass of the body, is the volume of the body, and is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include isobaric (constant pressure, ) or isochoric (constant volume, ) processes. The corresponding specific heat capacities are expressed as From the results of the previous section, dividing through by the mass gives the relation A related parameter to is , the volumetric heat capacity. In engineering practice, for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the mass-specific heat capacity (specific heat) is often explicitly written with the subscript , as . Of course, from the above relationships, for solids one writes For pure homogeneous chemical compounds with established molecular or molar mass or a molar quantity is established, heat capacity as an intensive property can be expressed on a per mole basis instead of a per mass basis by the following equations analogous to the per mass equations: = molar heat capacity at constant pressure = molar heat capacity at constant volume where n = number of moles in the body or thermodynamic system. One may refer to such a per mole quantity as molar heat capacity to distinguish it from specific heat capacity on a per mass basis.

Heat capacity 165 Polytropic heat capacity The polytropic heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change = molar heat capacity at polytropic process The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (γ or κ) Dimensionless heat capacity The dimensionless heat capacity of a material is where C is the heat capacity of a body made of the material in question (J/K) n is the amount of substance in the body (mol) R is the gas constant (J/(K·mol)) N is the number of molecules in the body. (dimensionless) k is Boltzmann’s constant (J/(K·molecule)) In the ideal gas article, dimensionless heat capacity is expressed as , and is related there directly to half the number of degrees of freedom per particle. This holds true for quadratic degrees of freedom, a consequence of the equipartition theorem. More generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the dimensionless entropy per particle , measured in nats. Alternatively, using base 2 logarithms, C* relates the base-2 logarithmic increase in temperature to the increase in the dimensionless entropy measured in bits.[6] Heat capacity at absolute zero From the definition of entropy the absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperature Tf The heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, which would violate the third law of thermodynamics. One of the strengths of the Debye model is that (unlike the preceding Einstein model) it predicts the proper mathematical form of the approach of heat capacity toward zero, as absolute zero temperature is approached.

Heat capacity 166 Negative heat capacity (stars) Most physical systems exhibit a positive heat capacity. However, even though it can seem paradoxical at first,[7][8] there are some systems for which the heat capacity is negative. These include gravitating objects such as stars; and also sometimes some nano-scale clusters of a few tens of atoms, close to a phase transition.[9] A negative heat capacity can result in a negative temperature. According to the virial theorem, for a self-gravitating body like a star or an interstellar gas cloud, the average potential energy UPot and the average kinetic energy UKin are locked together in the relation The total energy U (= UPot + UKin) therefore obeys If the system loses energy, for example by radiating energy away into space, the average kinetic energy and with it the average temperature actually increases. The system therefore can be said to have a negative heat capacity.[10] A more extreme version of this occurs with black holes. According to black hole thermodynamics, the more mass and energy a black hole absorbs, the colder it becomes. In contrast, if it is a net emitter of energy, through Hawking radiation, it will become hotter and hotter until it boils away. Theory of heat capacity Factors that affect specific heat capacity For any given substance, the heat capacity of a body is directly proportional to the amount of substance it contains (measured in terms of mass or moles or volume). Doubling the amount of substance in a body doubles its heat capacity, etc. However, when this effect has been corrected for, by dividing the heat capacity by the quantity of substance in a body, the resulting specific heat capacity is a function of the structure of the substance itself. In particular, it depends on the number of degrees of freedom that are available to the particles in the substance, each of which type of freedom allows substance particles to store energy. The kinetic energy of substance particles is the only one of the many possible degrees of freedom which manifests as temperature change, and thus the larger the number of degrees of freedom available Molecules undergo many characteristic internal vibrations. Potential to the particles of a substance other than kinetic energy stored in these internal degrees of freedom contributes to a [11] [12] energy, the larger will be the specific heat capacity sample’s energy content, but not to its temperature. More internal degrees of freedom tend to increase a substances specific heat for the substance. capacity, so long as temperatures are high enough to overcome quantum effects. In addition, quantum effects require that whenever energy be stored in any mechanism associated with a bound system which confers a degree of freedom, it must be stored in certain minimal-sized deposits (quanta) of energy, or else not stored at all. Such effects limit the full ability of some degrees of freedom to store energy when their lowest energy storage quantum amount is not easily supplied at the average energy of particles at a given

Heat capacity 167 temperature. In general, for this reason, specific heat capacities tend to fall at lower temperatures where the average thermal energy available to each particle degree of freedom is smaller, and thermal energy storage begins to be limited by these quantum effects. Due to this process, as temperature falls toward absolute zero, so also does heat capacity. Degrees of freedom Molecules are quite different from the monatomic gases like helium and argon. With monatomic gases, thermal energy comprises only translational motions. Translational motions are ordinary, whole-body movements in 3D space whereby particles move about and exchange energy in collisions—like rubber balls in a vigorously shaken container (see animation here [13]). These simple movements in the three dimensions of space mean individual atoms have three translational degrees of freedom. A degree of freedom is any form of energy in which heat transferred into an object can be stored. This can be in translational kinetic energy, rotational kinetic energy, or other forms such as potential energy in vibrational modes. Only three translational degrees of freedom (corresponding to the three independent directions in space) are available for any individual atom, whether it is free, as a monatomic molecule, or bound into a polyatomic molecule. As to rotation about an atoms axis (again, whether the atom is bound or free), its energy of rotation is proportional to the moment of inertia for the atom, which is extremely small compared to moments of inertia of collections of atoms. This is because almost all of the mass of a single atom is concentrated in its nucleus, which has a radius too small to give a significant moment of inertia. In contrast, the spacing of quantum energy levels for a rotating object is inversely proportional to its moment of inertia, and so this spacing becomes very large for objects with very small moments of inertia. For these reasons, the contribution from rotation of atoms on their axes is essentially zero in monatomic gases, because the energy spacing of the associated quantum levels is too large for significant thermal energy to be stored in rotation of systems with such small moments of inertia. For similar reasons, axial rotation around bonds joining atoms in diatomic gases (or along the linear axis in a linear molecule of any length) can also be neglected as a possible "degree of freedom" as well, since such rotation is similar to rotation of monatomic atoms, and so occurs about an axis with a moment of inertia too small to be able to store significant heat energy. In polyatomic molecules, other rotational modes may become active, due to the much higher moments of inertia about certain axes which do not coincide with the linear axis of a linear molecule. These modes take the place of some translational degrees of freedom for individual atoms, since the atoms are moving in 3-D space, as the molecule rotates. The narrowing of quantum mechanically determined energy spacing between rotational states results from situations where atoms are rotating around an axis that does not connect them, and thus form an assembly that has a large moment of inertia. This small difference between energy states allows the kinetic energy of this type of rotational motion to store heat energy at ambient temperatures. Furthermore (although usually at higher temperatures than are able to store heat in rotational motion) internal vibrational degrees of freedom also may become active (these are also a type of translation, as seen from the view of each atom). In summary, molecules are complex objects with a population of atoms that may move about within the molecule in a number of different ways (see animation at right), and each of these ways of moving is capable of storing energy if the temperature is sufficient. The heat capacity of molecular substances (on a per-atom, or atom-molar, basis) does not exceed the heat capacity of monatomic gases, unless vibrational modes are brought into play. The reason for this is that vibrational modes allow energy to be stored as potential energy in intra-atomic bonds in a molecule, which are not available to atoms in monatomic gases. Up to about twice as much energy (on a per-atom basis) per unit of temperature increase can be stored in a solid as in a monatomic gas, by this mechanism of storing energy in the potentials of interatomic bonds. This gives many solids about twice the atom-molar heat capacity at room temperature of monatomic gases. However, quantum effects heavily affect the actual ratio at lower temperatures (i.e., much lower than the melting temperature of the solid), especially in solids with light and tightly bound atoms (e.g., beryllium metal or diamond).

Heat capacity 168 Smaller polyatomic gases store intermediate amounts of energy, giving them a per-molecule heat capacity that is between that of monatomic gases (3⁄2 R per mole of atoms or molecules, where R is the ideal gas constant), and the maximum of fully excited warmer solids (3 R per mole of atoms). For gases, heat capacity never falls below the minimum of 3⁄2 R per mole (of molecules), since the kinetic energy of gas molecules is always available to store at least this much thermal energy. However, at cryogenic temperatures in solids, heat capacity falls toward zero, as temperature approaches absolute zero. Example of temperature-dependent specific heat capacity, in a diatomic gas To illustrate the role of various degrees of freedom in storing heat, we may consider nitrogen, a diatomic molecule that has five active degrees of freedom at room temperature: the three comprising translational motion plus two rotational degrees of freedom internally. Although the constant-volume molar heat capacity of nitrogen at this temperature is five-thirds that of monatomic gases, on a per-mole of atoms basis, it is five-sixths that of a monatomic gas. The reason for this is the loss of a degree of freedom due to the bond when it does not allow storage of thermal energy. Two separate nitrogen atoms would have a total of six degrees of freedom—the three translational degrees of freedom of each atom. When the atoms are bonded the molecule will still only have three translational degrees of freedom, as the two atoms in the molecule move as one. However, the molecule cannot be treated as a point object, and the moment of inertia has increased sufficiently about two axes to allow two rotational degrees of freedom to be active at room temperature to give five degrees of freedom. The moment of inertia about the third axis remains small, as this is the axis passing through the centres of the two atoms, and so is similar to the small moment of inertia for atoms of a monatomic gas. Thus, this degree of freedom does not act to store heat, and does not contribute to the heat capacity of nitrogen. The heat capacity per atom for nitrogen (5/2 per mole molecules = 5/4 per mole atoms) is therefore less than for a monatomic gas (3/2 per mole molecules or atoms), so long as the temperature remains low enough that no vibrational degrees of freedom are activated.[14] At higher temperatures, however, nitrogen gas gains two more degrees of internal freedom, as the molecule is excited into higher vibrational modes that store thermal energy. Now the bond is contributing heat capacity, and is contributing more than if the atoms were not bonded. With full thermal excitation of bond vibration, the heat capacity per volume, or per mole of gas molecules approaches seven-thirds that of monatomic gases. Significantly, this is seven-sixths of the monatomic gas value on a mole-of-atoms basis, so this is now a higher heat capacity per atom than the monatomic figure, because the vibrational mode enables for diatomic gases allows an extra degree of potential energy freedom per pair of atoms, which monatomic gases cannot possess.[15] See thermodynamic temperature for more information on translational motions, kinetic (heat) energy, and their relationship to temperature. However, even at these large temperatures where gaseous nitrogen is able to store 7/6ths of the energy per atom of a monatomic gas (making it more efficient at storing energy on an atomic basis), it still only stores 7/12 ths of the maximal per-atom heat capacity of a solid, meaning it is not nearly as efficient at storing thermal energy on an atomic basis, as solid substances can be. This is typical of gases, and results because many of the potential bonds which might be storing potential energy in gaseous nitrogen (as opposed to solid nitrogen) are lacking, because only one of the spatial dimensions for each nitrogen atom offers a bond into which potential energy can be stored without increasing the kinetic energy of the atom. In general, solids are most efficient, on an atomic basis, at storing thermal energy (that is, they have the highest per-atom or per-mole-of-atoms heat capacity).

Heat capacity 169 Per mole of... Per mole of molecules When the specific heat capacity, c, of a material is measured (lowercase c means the unit quantity is in terms of mass), different values arise because different substances have different molar masses (essentially, the weight of the individual atoms or molecules). In solids, thermal energy arises due to the number of atoms that are vibrating. "Molar" heat capacity per mole of molecules, for both gases and solids, offer figures which are arbitrarily large, since molecules may be arbitrarily large. Such heat capacities are thus not intensive quantities for this reason, since the quantity of mass being considered can be increased without limit. Per mole of atoms Conversely, for molecular-based substances (which also absorb heat into their internal degrees of freedom), massive, complex molecules with high atomic count—like octane—can store a great deal of energy per mole and yet are quite unremarkable on a mass basis, or on a per-atom basis. This is because, in fully excited systems, heat is stored independently by each atom in a substance, not primarily by the bulk motion of molecules. Thus, it is the heat capacity per-mole-of-atoms, not per-mole-of-molecules, which is the intensive quantity, and which comes closest to being a constant for all substances at high temperatures. This relationship was noticed empirically in 1819, and is called the Dulong-Petit law, after its two discoverers.[16] Historically, the fact that specific heat capacities are approximately equal when corrected by the presumed weight of the atoms of solids, was an important piece of data in favor of the atomic theory of matter. Because of the connection of heat capacity to the number of atoms, some care should be taken to specify a mole-of-molecules basis vs. a mole-of-atoms basis, when comparing specific heat capacities of molecular solids and gases. Ideal gases have the same numbers of molecules per volume, so increasing molecular complexity adds heat capacity on a per-volume and per-mole-of-molecules basis, but may lower or raise heat capacity on a per-atom basis, depending on whether the temperature is sufficient to store energy as atomic vibration. In solids, the quantitative limit of heat capacity in general is about 3 R per mole of atoms, where R is the ideal gas constant. This 3 R value is about 24.9 J/mole.K. Six degrees of freedom (three kinetic and three potential) are available to each atom. Each of these six contributes 1⁄2R specific heat capacity per mole of atoms.[17] This limit of 3 R per mole specific heat capacity is approached at room temperature for most solids, with significant departures at this temperature only for solids composed of the lightest atoms which are bound very strongly, such as beryllium (where the value is only of 66% of 3 R), or diamond (where it is only 24% of 3 R). These large departures are due to quantum effects which prevent full distribution of heat into all vibrational modes, when the energy difference between vibrational quantum states is very large compared to the average energy available to each atom from the ambient temperature. For monatomic gases, the specific heat is only half of 3 R per mole, i.e. (3⁄2R per mole) due to loss of all potential energy degrees of freedom in these gases. For polyatomic gases, the heat capacity will be intermediate between these values on a per-mole-of-atoms basis, and (for heat-stable molecules) would approach the limit of 3 R per mole of atoms, for gases composed of complex molecules, and at higher temperatures at which all vibrational modes accept excitational energy. This is because very large and complex gas molecules may be thought of as relatively large blocks of solid matter which have lost only a relatively small fraction of degrees of freedom, as compared to a fully integrated solid. For a list of heat capacities per atom-mole of various substances, in terms of R, see the last column of the table of heat capacities below.

Heat capacity 170 Corollaries of these considerations for solids (volume-specific heat capacity) Since the bulk density of a solid chemical element is strongly related to its molar mass (usually about 3 R per mole, as noted above), there exists a noticeable inverse correlation between a solid’s density and its specific heat capacity on a per-mass basis. This is due to a very approximate tendency of atoms of most elements to be about the same size, despite much wider variations in density and atomic weight. These two factors (constancy of atomic volume and constancy of mole-specific heat capacity) result in a good correlation between the volume of any given solid chemical element and its total heat capacity. Another way of stating this, is that the volume-specific heat capacity (volumetric heat capacity) of solid elements is roughly a constant. The molar volume of solid elements is very roughly constant, and (even more reliably) so also is the molar heat capacity for most solid substances. These two factors determine the volumetric heat capacity, which as a bulk property may be striking in consistency. For example, the element uranium is a metal which has a density almost 36 times that of the metal lithium, but uraniums specific heat capacity on a volumetric basis (i.e. per given volume of metal) is only 18% larger than lithiums. Since the volume-specific corollary of the Dulong-Petit specific heat capacity relationship requires that atoms of all elements take up (on average) the same volume in solids, there are many departures from it, with most of these due to variations in atomic size. For instance, arsenic, which is only 14.5% less dense than antimony, has nearly 59% more specific heat capacity on a mass basis. In other words; even though an ingot of arsenic is only about 17% larger than an antimony one of the same mass, it absorbs about 59% more heat for a given temperature rise. The heat capacity ratios of the two substances closely follows the ratios of their molar volumes (the ratios of numbers of atoms in the same volume of each substance); the departure from the correlation to simple volumes in this case is due to lighter arsenic atoms being significantly more closely packed than antimony atoms, instead of similar size. In other words, similar-sized atoms would cause a mole of arsenic to be 63% larger than a mole of antimony, with a correspondingly lower density, allowing its volume to more closely mirror its heat capacity behavior. Other factors Hydrogen bonds Hydrogen-containing polar molecules like ethanol, ammonia, and water have powerful, intermolecular hydrogen bonds when in their liquid phase. These bonds provide another place where heat may be stored as potential energy of vibration, even at comparatively low temperatures. Hydrogen bonds account for the fact that liquid water stores nearly the theoretical limit of 3 R per mole of atoms, even at relatively low temperatures (i.e. near the freezing point of water). Impurities In the case of alloys, there are several conditions in which small impurity concentrations can greatly affect the specific heat. Alloys may exhibit marked difference in behaviour even in the case of small amounts of impurities being one element of the alloy; for example impurities in semiconducting ferromagnetic alloys may lead to quite different specific heat properties.[18] The simple case of the monatomic gas In the case of a monatomic gas such as helium under constant volume, if it is assumed that no electronic or nuclear quantum excitations occur, each atom in the gas has only 3 degrees of freedom, all of a translational type. No energy dependence is associated with the degrees of freedom which define the position of the atoms. While, in fact, the degrees of freedom corresponding to the momenta of the atoms are quadratic, and thus contribute to the heat capacity. There are N atoms, each of which has 3 components of momentum, which leads to 3N total degrees of freedom. This gives:

Heat capacity 171 where is the heat capacity at constant volume of the gas is the molar heat capacity at constant volume of the gas N is the total number of atoms present in the container n is the number of moles of atoms present in the container (n is the ratio of N and Avogadro’s number) R is the ideal gas constant, (8.3144621[75] J/(mol·K). R is equal to the product of Boltzmann’s constant and Avogadro’s number The following table shows experimental molar constant volume heat capacity measurements taken for each noble monatomic gas (at 1 atm and 25 °C): Monatomic gas CV, m (J/(mol·K)) CV, m/R He 12.5 1.50 Ne 12.5 1.50 Ar 12.5 1.50 Kr 12.5 1.50 Xe 12.5 1.50 It is apparent from the table that the experimental heat capacities of the monatomic noble gases agrees with this simple application of statistical mechanics to a very high degree. The molar heat capacity of a monatomic gas at constant pressure is then Diatomic gas In the somewhat more complex case of an ideal gas of diatomic molecules, the presence of internal degrees of freedom are apparent. In addition to the three translational degrees of freedom, there are rotational and vibrational degrees of freedom. In general, the number of degrees of freedom, f, in a molecule with na atoms is 3na: Mathematically, there are a total of three rotational degrees of freedom, one corresponding to rotation about each of the axes of three dimensional space. However, in practice only the existence of two degrees of rotational freedom for linear molecules will be considered. This approximation is valid because the moment of inertia about the internuclear axis is vanishingly small with respect to other moments of inertia in the molecule (this is due to the extremely small radii of the atomic nuclei, compared to the distance between them in a molecule). Quantum mechanically, it can be shown that the interval between successive rotational energy eigenstates is inversely proportional to the moment of inertia about that axis. Because the moment of inertia about the internuclear axis is vanishingly small relative to the other two rotational axes, the energy spacing can be considered so high that no excitations of the rotational state can possibly occur unless the temperature is extremely high. It is easy to calculate the expected number of vibrational degrees of freedom (or vibrational modes). There are three degrees of translational freedom, and two degrees of rotational freedom, therefore Each rotational and translational degree of freedom will contribute R/2 in the total molar heat capacity of the gas. Each vibrational mode will contribute to the total molar heat capacity, however. This is because for each vibrational mode, there is a potential and kinetic energy component. Both the potential and kinetic components will

Heat capacity 172 contribute R/2 to the total molar heat capacity of the gas. Therefore, a diatomic molecule would be expected to have a molar constant-volume heat capacity of where the terms originate from the translational, rotational, and vibrational degrees of freedom, respectively. The following is a table of some molar constant-volume heat capacities of various diatomic gases at standard temperature (25 oC = 298 K) Diatomic gas CV, m (J/(mol·K)) CV, m / R H2 20.18 2.427 CO 20.2 2.43 N2 19.9 2.39 Cl2 24.1 3.06 Br2 (vapour) 28.2 3.39 From the above table, clearly there is a problem with the above theory. All of the diatomics examined have heat capacities that are lower than those predicted by the equipartition theorem, except Br2. However, as the atoms composing the molecules become heavier, the heat capacities move closer to their expected values. One of the reasons for this phenomenon is the quantization of vibrational, and to a lesser extent, rotational states. In fact, if it is assumed that the molecules remain in their lowest energy vibrational state because the inter-level energy spacings for vibration-energies are large, the predicted molar constant volume heat capacity for a diatomic molecule becomes just that from the contributions of translation and rotation: which is a fairly close approximation of the heat capacities of the lighter molecules in the above table. If the quantum harmonic oscillator approximation is made, it turns out that the quantum vibrational energy level spacings are actually inversely proportional to the square root of the reduced mass of the atoms composing the diatomic molecule. Therefore, in the case of the heavier diatomic molecules such as chlorine or bromine, the quantum vibrational energy level spacings become finer, which allows more excitations into higher vibrational levels at lower temperatures. This limit for storing heat capacity in vibrational modes, as discussed above, becomes 7R/2 = 3.5 R per mole of gas molecules, which is fairly consistent with the measured value for Br2 at room temperature. As temperatures rise, all diatomic gases approach this value. General gas phase The specific heat of the gas is best conceptualized in terms of the degrees of freedom of an individual molecule. The different degrees of freedom correspond to the different ways in which the molecule may store energy. The molecule may store energy in its translational motion according to the formula: where m is the mass of the molecule and is velocity of the center of mass of the molecule. Each direction of motion constitutes a degree of freedom, so that there are three translational degrees of freedom. In addition, a molecule may have rotational motion. The kinetic energy of rotational motion is generally expressed as where I is the moment of inertia tensor of the molecule, and is the angular velocity pseudo-vector (in a coordinate system aligned with the principle axes of the molecule). In general, then, there will be three additional

Heat capacity 173 degrees of freedom corresponding to the rotational motion of the molecule, (For linear molecules one of the inertia tensor terms vanishes and there are only two rotational degrees of freedom). The degrees of freedom corresponding to translations and rotations are called the rigid degrees of freedom, since they do not involve any deformation of the molecule. The motions of the atoms in a molecule which are not part of its gross translational motion or rotation may be classified as vibrational motions. It can be shown that if there are n atoms in the molecule, there will be as many as vibrational degrees of freedom, where is the number of rotational degrees of freedom. A vibrational degree of freedom corresponds to a specific way in which all the atoms of a molecule can vibrate. The actual number of possible vibrations may be less than this maximal one, due to various symmetries. For example, triatomic nitrous oxide N2O will have only 2 degrees of rotational freedom (since it is a linear molecule) and contains n=3 atoms: thus the number of possible vibrational degrees of freedom will be v = (3*3)-3-2 = 4. There are four ways or "modes" in which the three atoms can vibrate, corresponding to 1) A mode in which an atom at each end of the molecule moves away from, or towards, the center atom at the same time, 2) a mode in which either end atom moves asynchronously with regard to the other two, and 3) and 4) two modes in which the molecule bends out of line, from the center, in the two possible planar directions that are orthogonal to its axis. Each vibrational degree of freedom confers TWO total degrees of freedom, since vibrational energy mode partitions into 1 kinetic and 1 potential mode. This would give nitrous oxide 3 translational, 2 rotational, and 4 vibrational modes (but these last giving 8 vibrational degrees of freedom), for storing energy. This is a total of f = 3+2+8 = 13 total energy-storing degrees of freedom, for N2O. For a bent molecule like water H2O, a similar calculation gives 9-3-3 = 3 modes of vibration, and 3 (translational) + 3 (rotational) + 6 (vibrational) = 12 degrees of freedom. The storage of energy into degrees of freedom If the molecule could be entirely described using classical mechanics, then the theorem of equipartition of energy could be used to predict that each degree of freedom would have an average energy in the amount of (1/2)kT where k is Boltzmann’s constant and T is the temperature. Our calculation of the constant-volume heat capacity would be straightforward. Each molecule would be holding, on average, an energy of (f/2)kT where f is the total number of degrees of freedom in the molecule. Note that Nk = R if N is Avogadros number, which is the case in considering the heat capacity of a mole of molecules. Thus, the total internal energy of the gas would be (f/2)NkT where N is the total number of molecules. The heat capacity (at constant volume) would then be a constant (f/2)Nk the mole-specific heat capacity would be (f/2)R the molecule-specific heat capacity would be (f/2)k and the dimensionless heat capacity would be just f/2. Here again, each vibrational degree of freedom contributes 2f. Thus, a mole of nitrous oxide would have a total constant-volume heat capacity (including vibration) of (13/2)R by this calculation. In summary, the molar heat capacity (mole-specific heat capacity) of an ideal gas with f degrees of freedom is given by This equation applies to all polyatomic gases, if the degrees of freedom are known.[19] The constant-pressure heat capacity for any gas would exceed this by an extra factor of R (see Mayers relation, above). As example Cp would be a total of (15/2)R/mole for nitrous oxide.

Heat capacity 174 The effect of quantum energy levels in storing energy in degrees of freedom The various degrees of freedom cannot generally be considered to obey classical mechanics, however. Classically, the energy residing in each degree of freedom is assumed to be continuous—it can take on any positive value, depending on the temperature. In reality, the amount of energy that may reside in a particular degree of freedom is quantized: It may only be increased and decreased in finite amounts. A good estimate of the size of this minimum amount is the energy of the first excited state of that degree of freedom above its ground state. For example, the first vibrational state of the hydrogen chloride (HCl) molecule has an energy of about 5.74 × 10−20 joule. If this amount of energy were deposited in a classical degree of freedom, it would correspond to a temperature of about 4156 K. If the temperature of the substance is so low that the equipartition energy of (1/2)kT is much smaller than this excitation energy, then there will be little or no energy in this degree of freedom. This degree of freedom is then said to be “frozen out". As mentioned above, the temperature corresponding to the first excited vibrational state of HCl is about 4156 K. For temperatures well below this value, the vibrational degrees of freedom of the HCl molecule will be frozen out. They will contain little energy and will not contribute to the thermal energy or the heat capacity of HCl gas. Energy storage mode "freeze-out" temperatures It can be seen that for each degree of freedom there is a critical temperature at which the degree of freedom “unfreezes” and begins to accept energy in a classical way. In the case of translational degrees of freedom, this temperature is that temperature at which the thermal wavelength of the molecules is roughly equal to the size of the container. For a container of macroscopic size (e.g. 10 cm) this temperature is extremely small and has no significance, since the gas will certainly liquify or freeze before this low temperature is reached. For any real gas translational degrees of freedom may be considered to always be classical and contain an average energy of (3/2)kT per molecule. The rotational degrees of freedom are the next to “unfreeze". In a diatomic gas, for example, the critical temperature for this transition is usually a few tens of kelvins, although with a very light molecule such as hydrogen the rotational energy levels will be spaced so widely that rotational heat capacity may not completely "unfreeze" until considerably higher temperatures are reached. Finally, the vibrational degrees of freedom are generally the last to unfreeze. As an example, for diatomic gases, the critical temperature for the vibrational motion is usually a few thousands of kelvins, and thus for the nitrogen in our example at room temperature, no vibration modes would be excited, and the constant-volume heat capacity at room temperature is (5/2)R/mole, not (7/2)R/mole. As seen above, with some unusually heavy gases such as iodine gas I2, or bromine gas Br2, some vibrational heat capacity may be observed even at room temperatures. It should be noted that it has been assumed that atoms have no rotational or internal degrees of freedom. This is in fact untrue. For example, atomic electrons can exist in excited states and even the atomic nucleus can have excited states as well. Each of these internal degrees of freedom are assumed to be frozen out due to their relatively high excitation energy. Nevertheless, for sufficiently high temperatures, these degrees of freedom cannot be ignored. In a few exceptional cases, such molecular electronic transitions are of sufficiently low energy that they contribute to heat capacity at room temperature, or even at cryogenic temperatures. One example of an electronic transition degree of freedom which contributes heat capacity at standard temperature is that of nitric oxide (NO), in which the single electron in an anti-bonding molecular orbital has energy transitions which contribute to the heat capacity of the gas even at room temperature. An example of a nuclear magnetic transition degree of freedom which is of importance to heat capacity, is the transition which converts the spin isomers of hydrogen gas (H2) into each other. At room temperature, the proton spins of hydrogen gas are aligned 75% of the time, resulting in orthohydrogen when they are. Thus, some thermal energy has been stored in the degree of freedom available when parahydrogen (in which spins are anti-aligned) absorbs energy, and is converted to the higher energy ortho form. However, at the temperature of liquid hydrogen,

Heat capacity 175 not enough heat energy is available to produce orthohydrogen (that is, the transition energy between forms is large enough to "freeze out" at this low temperature), and thus the parahydrogen form predominates. The heat capacity of the transition is sufficient to release enough heat, as orthohydrogen converts to the lower-energy parahydrogen, to boil the hydrogen liquid to gas again, if this evolved heat is not removed with a catalyst after the gas has been cooled and condensed. This example also illustrates the fact that some modes of storage of heat may not be in constant equilibrium with each other in substances, and heat absorbed or released from such phase changes may "catch up" with temperature changes of substances, only after a certain time. In other words, the heat evolved and absorbed from the ortho-para isomeric transition contributes to the heat capacity of hydrogen on long time-scales, but not on short time-scales. These time scales may also depend on the presence of a catalyst. Less exotic phase-changes may contribute to the heat-capacity of substances and systems, as well, as (for example) when water is converted back and forth from solid to liquid or gas form. Phase changes store heat energy entirely in breaking the bonds of the potential energy interactions between molecules of a substance. As in the case of hydrogen, it is also possible for phase changes to be hindered as the temperature drops, so that they do not catch up and become apparent, without a catalyst. For example, it is possible to supercool liquid water to below the freezing point, and not observe the heat evolved when the water changes to ice, so long as the water remains liquid. This heat appears instantly when the water freezes. Solid phase For matter in a crystalline solid phase, the Dulong-Petit law, which was discovered empirically, states that the mole-specific heat capacity assumes the value 3 R. Indeed, for solid metallic chemical elements at room temperature, molar heat capacities range from about 2.8 R to 3.4 R. Large exceptions at the lower end involve solids composed of relatively low-mass, tightly bonded atoms, such as beryllium at 2.0 R, and diamond at only 0.735 R. The latter conditions create larger quantum vibrational energy-spacing, so that many vibrational modes have energies too high to be populated (and thus are "frozen out") at room temperature. At the higher end of possible heat capacities, heat capacity may exceed R by modest The dimensionless heat capacity divided by three, as a function of temperature as predicted by the Debye model and by Einstein’s earlier model. The horizontal axis amounts, due to contributions from is the temperature divided by the Debye temperature. Note that, as expected, the anharmonic vibrations in solids, and dimensionless heat capacity is zero at absolute zero, and rises to a value of three as sometimes a modest contribution from the temperature becomes much larger than the Debye temperature. The red line conduction electrons in metals. These are corresponds to the classical limit of the Dulong-Petit law not degrees of freedom treated in the Einstein or Debye theories. The theoretical maximum heat capacity for multi-atomic gases at higher temperatures, as the molecules become larger, also approaches the Dulong-Petit limit of 3 R, so long as this is calculated per mole of atoms, not molecules. The reason for this behavior is that, in theory, gases with very large molecules have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas.

Heat capacity 176 The Dulong-Petit limit results from the equipartition theorem, and as such is only valid in the classical limit of a microstate continuum, which is a high temperature limit. For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at standard ambient temperature, quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3 R per mole of atoms in the solid, although in molecular solids, heat capacities calculated per mole of molecules in molecular solids may be more than 3 R. For example, the heat capacity of water ice at the melting point is about 4.6 R per mole of molecules, but only 1.5 R per mole of atoms. As noted, heat capacity values far lower than 3 R "per atom" (as is the case with diamond and beryllium) result from “freezing out” of possible vibration modes for light atoms at suitably low temperatures, just as happens in many low-mass-atom gases at room temperatures (where vibrational modes are all frozen out). Because of high crystal binding energies, the effects of vibrational mode freezing are observed in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3 R per mole of atoms of the Dulong-Petit theoretical maximum. For a more modern and precise analysis of the heat capacities of solids, especially at low temperatures, it is useful to use the idea of phonons. See Debye model. Phonons can also be applied to the heat capacity of liquids[20] The specific heat of amorphous materials has characteristic discontinuities at the glass transition temperature due to rearrangements that occur in the distribution of atoms.[21] These discontinuities are frequently used to detect the glass transition temperature where a supercooled liquid transforms to a glass.[22] Table of specific heat capacities Note that the especially high molar values, as for paraffin, gasoline, water and ammonia, result from calculating specific heats in terms of moles of molecules. If specific heat is expressed per mole of atoms for these substances, none of the constant-volume values exceed, to any large extent, the theoretical Dulong-Petit limit of 25 J/(mol·K) = 3 R per mole of atoms (see the last column of this table). Paraffin, for example, has very large molecules and thus a high heat capacity per mole, but as a substance it does not have remarkable heat capacity in terms of volume, mass, or atom-mol (which is just 1.41 R per mole of atoms, or less than half of most solids, in terms of heat capacity per atom). In the last column, major departures of solids at standard temperatures from the Dulong-Petit law value of 3R, are usually due to low atomic weight plus high bond strength (as in diamond) causing some vibration modes to have too much energy to be available to store thermal energy at the measured temperature. For gases, departure from 3R per mole of atoms in this table is generally due to two factors: (1) failure of the higher quantum-energy-spaced vibration modes in gas molecules to be excited at room temperature, and (2) loss of potential energy degree of freedom for small gas molecules, simply because most of their atoms are not bonded maximally in space to other atoms, as happens in many solids. Table of specific heat capacities at 25 °C (298 K) unless otherwise noted Substance Phase (mass) Constant Constant Volumetric Constant specific pressure volume heat vol. heat capacity molar molar capacity atom-molar cp or cm heat capacity heat capacity Cv heat J·g−1·K−1 Cp,m Cv,m J·cm−3·K−1 capacity J·mol−1·K−1 J·mol−1·K−1 in units of R Cv,m(atom) atom-mol−1 Air (Sea level, dry, gas 1.0035 29.07 20.7643 0.001297 ~ 1.25 R 0 °C (273.15 K))

Heat capacity 177 gas 1.012 29.19 20.85 0.00121 ~ 1.25 R Air (typical room conditionsA) Aluminium solid 0.897 24.2 2.422 2.91 R Ammonia liquid 4.700 80.08 3.263 3.21 R mixed 3.5 3.7* Animal tissue [23] (incl. human) Antimony solid 0.207 25.2 1.386 3.03 R Argon gas 0.5203 20.7862 12.4717 1.50 R Arsenic solid 0.328 24.6 1.878 2.96 R Beryllium solid 1.82 16.4 3.367 1.97 R [24] solid 0.123 25.7 1.20 3.09 R Bismuth Cadmium solid 0.231 26.02 3.13 R [19] gas 0.839* 36.94 28.46 1.14 R Carbon dioxide CO2 Chromium solid 0.449 23.35 2.81 R Copper solid 0.385 24.47 3.45 2.94 R Diamond solid 0.5091 6.115 1.782 0.74 R Ethanol liquid 2.44 112 1.925 1.50 R Gasoline (octane) liquid 2.22 228 1.64 1.05 R [24] solid 0.84 Glass Gold solid 0.129 25.42 2.492 3.05 R [24] solid 0.790 2.17 Granite Graphite solid 0.710 8.53 1.534 1.03 R Helium gas 5.1932 20.7862 12.4717 1.50 R Hydrogen gas 14.30 28.82 1.23 R [19] gas 1.015* 34.60 1.05 R Hydrogen sulfide H2S Iron solid 0.450 25.1 3.537 3.02 R Lead solid 0.129 26.4 1.44 3.18 R Lithium solid 3.58 24.8 1.912 2.98 R [25] liquid 4.379 30.33 2.242 3.65 R Lithium at 181 °C Magnesium solid 1.02 24.9 1.773 2.99 R Mercury liquid 0.1395 27.98 1.888 3.36 R Methane at 2 °C gas 2.191 35.69 0.66 R [26] liquid 2.14 68.62 1.38 R Methanol (298 K) Nitrogen gas 1.040 29.12 20.8 1.25 R Neon gas 1.0301 20.7862 12.4717 1.50 R Oxygen gas 0.918 29.38 21.0 1.26 R Paraffin wax solid 2.5 (ave) 900 2.325 1.41 R C25H52

Heat capacity 178 solid 2.3027 Polyethylene [27][28] (rotomolding grade) liquid 2.9308 Polyethylene [27][28] (rotomolding grade) Silica (fused) solid 0.703 42.2 1.547 1.69 R [24] solid 0.233 24.9 2.44 2.99 R Silver Sodium solid 1.230 28.23 3.39 R Steel solid 0.466 Tin solid 0.227 27.112 3.26 R Titanium solid 0.523 26.060 3.13 R [24] solid 0.134 24.8 2.58 2.98 R Tungsten Uranium solid 0.116 27.7 2.216 3.33 R Water at 100 °C (steam) gas 2.080 37.47 28.03 1.12 R Water at 25 °C liquid 4.1813 75.327 74.53 4.1796 3.02 R Water at 100 °C liquid 4.1813 75.327 74.53 4.2160 3.02 R [24] solid 2.11 38.09 1.938 1.53 R Water at −10 °C (ice) [24] solid 0.387 25.2 2.76 3.03 R Zinc Substance Phase Cp Cp,m Cv,m Volumetric J/(g·K) J/(mol·K) J/(mol·K) heat capacity J/(cm3·K) A Assuming an altitude of 194 metres above mean sea level (the world–wide median altitude of human habitation), an indoor temperature of 23 °C, a dewpoint of 9 °C (40.85% relative humidity), and 760 mm–Hg sea level–corrected barometric pressure (molar water vapor content = 1.16%). *Derived data by calculation. This is for water-rich tissues such as brain. The whole-body average figure for mammals is approximately 2.9 [29] J/(cm3·K) Specific heat capacity of building materials (Usually of interest to builders and solar designers) Specific heat capacity of building materials Substance Phase cp J/(g·K) Asphalt solid 0.920 Brick solid 0.840 Concrete solid 0.880 Glass, silica solid 0.840 Glass, crown solid 0.670 Glass, flint solid 0.503 Glass, pyrex solid 0.753

Heat capacity 179 Granite solid 0.790 Gypsum solid 1.090 Marble, mica solid 0.880 Sand solid 0.835 Soil solid 0.800 Sulphur Hexafluoride gas 0.664 Wood solid 1.7 (1.2 to 2.3) Substance Phase cp J/(g·K) Notes [1] Laider, Keith J. (1993). The World of Physical Chemistry (http:/ / books. google. com/ ?id=01LRlPbH80cC& printsec=frontcover). Oxford University Press. ISBN 0-19-855919-4. . [2] International Union of Pure and Applied Chemistry, Physical Chemistry Division. "Quantities, Units and Symbols in Physical Chemistry" (http:/ / old. iupac. org/ publications/ books/ gbook/ green_book_2ed. pdf). Blackwell Sciences. p. 7. . "The adjective specific before the name of an extensive quantity is often used to mean divided by mass." [3] International Bureau of Weights and Measures (2006), The International System of Units (SI) (http:/ / www. bipm. org/ utils/ common/ pdf/ si_brochure_8_en. pdf) (8th ed.), ISBN 92-822-2213-6, [4] Nic, M.; Jirat, J.; Kosata, B., eds. (2006–). "Standard Pressure" (http:/ / goldbook. iupac. org/ S05921. html). IUPAC Compendium of Chemical Terminology (Online ed.). doi:10.1351/goldbook.S05921. ISBN 0-9678550-9-8. .. Besides being a round number, this had a very practical effect: relatively few people live and work at precisely sea level; 100 kPa equates to the mean pressure at an altitude of about 112 metres (which is closer to the 194–metre, world–wide median altitude of human habitation). [5] Yunus A. Cengel and Michael A. Boles,Thermodynamics: An Engineering Approach 7th Edition, , McGraw-Hill, 2010,ISBN: 007-352932-X [6] Fraundorf, P. (2003). "Heat capacity in bits". American Journal of Physics 71 (11): 1142. arXiv:cond-mat/9711074. Bibcode 2003AmJPh..71.1142F. doi:10.1119/1.1593658. [7] D. Lynden-Bell & R. M. Lynden-Bell; Lynden-Bell (Nov. 1977). "On the negative specific heat paradox". Monthly Notices of the Royal Astronomical Society 181: 405–419. Bibcode 1977MNRAS.181..405L. [8] Lynden-Bell, D. (Dec. 1998). "Negative Specific Heat in Astronomy, Physics and Chemistry". Physica A 263: 293–304. arXiv:cond-mat/9812172v1. Bibcode 1999PhyA..263..293L. doi:10.1016/S0378-4371(98)00518-4. [9] Schmidt, Martin; Kusche, Robert; Hippler, Thomas; Donges, Jörn; Kronmüller, Werner; Von Issendorff, Bernd; Haberland, Hellmut (2001). "Negative Heat Capacity for a Cluster of 147 Sodium Atoms". Physical Review Letters 86 (7): 1191–4. Bibcode 2001PhRvL..86.1191S. doi:10.1103/PhysRevLett.86.1191. PMID 11178041. [10] See eg, Wallace, David (2010). "Gravity, entropy, and cosmology: in search of clarity" (http:/ / philsci-archive. pitt. edu/ archive/ 00004744/ 01/ gravent_archive. pdf). British Journal for the Philosophy of Science 61 (3): 513. arXiv:0907.0659. Bibcode 2010BJPS...61..513W. doi:10.1093/bjps/axp048. . Section 4 and onwards. [11] Reif, F. (1965). Fundamentals of statistical and thermal physics. McGraw-Hill. pp. 253–254. ISBN 07-051800-9. [12] Charles Kittel; Herbert Kroemer (2000). Thermal physics. Freeman. pp. 78. ISBN 0-7167-1088-9. [13] http:/ / upload. wikimedia. org/ wikipedia/ commons/ 6/ 6d/ Translational_motion. gif [14] Smith, C. G. (2008). Quantum Physics and the Physics of large systems, Part 1A Physics. University of Cambridge. [15] The comparison must be made under constant-volume conditions—CvH—so that no work is performed. Nitrogen’s CvH (100 kPa, 20 °C) = 20.8 J mol–1 K–1 vs. the monatomic gases which equal 12.4717 J mol–1 K–1. Citations: Freeman’s, W. H.. "Physical Chemistry Part 3: Change Exercise 21.20b, Pg. 787" (http:/ / www. whfreeman. com/ college/ pdfs/ pchem8e/ PC8eC21. pdf). .. Also Georgia State University. "Molar Specific Heats of Gases" (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ kinetic/ shegas. html). . [16] Petit A.-T., Dulong P.-L. (1819). Translation "Recherches sur quelques points importants de la Théorie de la Chaleur" (http:/ / web. lemoyne. edu/ ~giunta/ PETIT. html). Annales de Chimie et de Physique 10: 395–413. Translation. [17] "Discussion of heat capacity of solids in terms of degrees of freedom" (http:/ / ruelle. phys. unsw. edu. au/ ~gary/ Site/ PHYS3020_files/ SM3_6. pdf) (PDF). . [18] Hogan, C. (1969). "Density of States of an Insulating Ferromagnetic Alloy". Physical Review 188 (2): 870. Bibcode 1969PhRv..188..870H. doi:10.1103/PhysRev.188.870. [19] Young; Geller (2008). Young and Geller College Physics (8th ed.). Pearson Education. ISBN 0-8053-9218-1. [20] Bolmatov, D.; Brazhkin, V. V.; Trachenko, K. (2012). "The phonon theory of liquid thermodynamics". Scientific Reports 2. arXiv:1202.0459. Bibcode 2012NatSR...2E.421B. doi:10.1038/srep00421. Lay summary (http:/ / physicsworld. com/ cws/ article/ news/ 2012/ jun/ 13/ phonon-theory-sheds-light-on-liquid-thermodynamics) – Physics World.

Heat capacity 180 [21] Ojovan, M. I. (2008). "Configurons: thermodynamic parameters and symmetry changes at glass transition" (http:/ / www. mdpi. org/ entropy/ papers/ e10030334. pdf) (PDF). Entropy 10 (3): 334–364. Bibcode 2008Entrp..10..334O. doi:10.3390/e10030334. . [22] Ojovan, Michael I. (2008). "Viscosity and Glass Transition in Amorphous Oxides". Advances in Condensed Matter Physics 2008: 1. Bibcode 2008AdCMP2008....1O. doi:10.1155/2008/817829. [23] Page 183 in: Cornelius, Flemming (2008). Medical biophysics (6th ed.). ISBN 1-4020-7110-8. (also giving a density of 1.06 kg/L) [24] "Table of Specific Heats" (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ tables/ sphtt. html#c1). . [25] "Materials Properties Handbook" (http:/ / fusionnet. seas. ucla. edu/ input/ PDF/ 1997 - Iter Material Properties Handbook - volAR01-3108 - no1 - p1-4. pdf). UCLA. . [26] "HCV (Molar Heat Capacity (cV)) Data for Methanol" (http:/ / ddbonline. ddbst. de/ EE/ 110 HCV (Molar Heat Capacity (cV)). shtml). Dortmund Data Bank Software and Separation Technology. . [27] Crawford, R. J.. Rotational molding of plastics. ISBN 1-59124-192-8. [28] Gaur, Umesh; Wunderlich, Bernhard (1981). "Heat capacity and other thermodynamic properties of linear macromolecules. II. Polyethylene" (http:/ / www. nist. gov/ data/ PDFfiles/ jpcrd178. pdf). Journal of Physical and Chemical Reference Data 10: 119. Bibcode 1981JPCRD..10..119G. doi:10.1063/1.555636. . [29] Faber, P.; Garby, L. (1995). "Fat content affects heat capacity: a study in mice". Acta Physiologica Scandinavica 153 (2): 185–7. doi:10.1111/j.1748-1716.1995.tb09850.x. PMID 7778459. References External links • Air Specific Heat Capacity Calculator (http://www.enggcyclopedia.com/calculators/physical-properties/ air-specific-heat-calculator/) Heat equation The heat equation is a parabolic partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. Statement of the equation For a function u(x,y,z,t) of three spatial variables (x,y,z) (see cartesian coordinates) and the time variable t, the heat equation is In this example, the heat equation in two dimensions predicts that if one area of an otherwise cool metal plate has been heated, say More generally in any coordinate system: with a torch, over time the temperature of that area will gradually decrease, starting at the edge and moving inward. Meanwhile the part of the plate outside that region will be getting warmer. Eventually the entire plate will reach a uniform intermediate temperature. In this animation, both height and color are used to show temperature. where α is a positive constant, and Δ or ∇2 denotes the Laplace operator. In the physical problem of temperature variation, u(x,y,z,t) is the temperature and α is the thermal diffusivity. For the mathematical treatment it is sufficient to consider the case α = 1.

Heat equation 181 The heat equation is of fundamental importance in diverse scientific fields. In mathematics, it is the prototypical parabolic partial differential equation. In probability theory, the heat equation is connected with the study of Brownian motion via the Fokker–Planck equation. In financial mathematics it is used to solve the Black–Scholes partial differential equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes. General description Suppose one has a function u which describes the temperature at a given location (x, y, z). This function will change over time as heat spreads throughout space. The heat equation is used to determine the change in the function u over time. The image to the right is animated and describes the way heat changes in time along a metal bar. One of the interesting properties of the heat equation is the maximum principle which says that the maximum value of u is either earlier in time than the region of concern or on the edge of the region of concern. This is Solution of a 1D heat equation PDE. The essentially saying that temperature comes either from some source or temperature (u) is initially distributed over a from earlier in time because heat permeates but is not created from one-dimensional, one-unit-long interval (x=[0,1]) with insulated endpoints. The distribution nothingness. This is a property of parabolic partial differential approaches equilibrium over time. equations and is not difficult to prove mathematically (see below). Another interesting property is that even if u has a discontinuity at an initial time t = t0, the temperature becomes smooth as soon as t > t0. For example, if a bar of metal has temperature 0 and another has temperature 100 and they are stuck together end to end, then very quickly the temperature at the point of connection will become 50 and the graph of the temperature will run smoothly from 0 to 100. The heat equation is used in probability and describes random walks. It is also applied in financial mathematics for this reason. It is also important in Riemannian geometry and thus topology: it was adapted by Richard Hamilton when he defined the Ricci flow that was later used by Grigori Perelman to solve the topological Poincaré conjecture. The physical problem and the equation Derivation in one dimension The heat equation is derived from Fouriers law and conservation of energy (Cannon 1984). By Fouriers law, the flow rate of heat energy through a surface is proportional to the negative temperature gradient across the surface, where k is the thermal conductivity and u is the temperature. In one dimension, the gradient is an ordinary spatial derivative, and so Fouriers law is where In the absence of work done, a change in internal energy per unit volume in the material, ΔQ, is proportional to the change in temperature, Δu. (In this section only, Δ is the ordinary difference operator, not the Laplacian.) That is,

Heat equation 182 where cp is the specific heat capacity and ρ is the mass density of the material. Choosing zero energy at absolute zero temperature, this can be rewritten as . The increase in internal energy in a small spatial region of the material over the time period is given by[1] where the fundamental theorem of calculus was used. If no work is done and there are neither heat sources nor sinks, the change in internal energy in the interval [x-Δx, x+Δx] is accounted for entirely by the flux of heat across the boundaries. By Fouriers law, this is again by the fundamental theorem of calculus.[2] By conservation of energy, This is true for any rectangle [t−Δt, t+Δt] × [x−Δx, x+Δx]. By the fundamental lemma of the calculus of variations, the integrand must vanish identically: Which can be rewritten as: or: which is the heat equation, where the coefficient (often denoted α) is called the thermal diffusivity. An additional term may be introduced into the equation to account for radiative loss of heat, which depends upon the excess temperature at a given point compared with the surroundings. At low excess temperatures, the radiative loss is approximately , giving a one-dimensional heat-transfer equation of the form . At high excess temperatures, however, the Stefan-Boltzmann law gives a net radiative heat-loss proportional to , and the above equation is inaccurate. For large excess temperatures, , giving a high-temperature heat-transfer equation of the form

Heat equation 183 where . Here, is Stefans constant, is a characteristic constant of the material, is the sectional perimeter of the bar and is its cross-sectional area. However, using instead of gives a better approximation in this case. Three-dimensional problem In the special cases of wave propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is where: • u = u(x, y, z, t) is temperature as a function of space and time; • is the rate of change of temperature at a point over time; • uxx, uyy, and uzz are the second spatial derivatives (thermal conductions) of temperature in the x, y, and z directions, respectively; • is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity k, the mass density ρ, and the specific heat capacity cp. The heat equation is a consequence of Fouriers law of cooling (see heat conduction). If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify boundary conditions for u. To determine uniqueness of solutions in the whole space it is necessary to assume an exponential bound on the growth of solutions; this assumption is consistent with observed experiments. Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object. Generally, many different states and starting conditions will tend toward the same stable equilibrium. As a consequence, to reverse the solution and conclude something about earlier times or initial conditions from the present heat distribution is very inaccurate except over the shortest of time periods. The heat equation is the prototypical example of a parabolic partial differential equation. Using the Laplace operator, the heat equation can be simplified, and generalized to similar equations over spaces of arbitrary number of dimensions, as where the Laplace operator, Δ or ∇2, the divergence of the gradient, is taken in the spatial variables. The heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion or the propagation of action potential in nerve cells. Although they are not diffusive in nature, some quantum mechanics problems are also governed by a mathematical analog of the heat equation (see below). It also can be used to model some phenomena arising in finance, like the Black–Scholes or the Ornstein-Uhlenbeck processes. The equation, and various non-linear analogues, has also been used in image analysis. The heat equation is, technically, in violation of special relativity, because its solutions involve instantaneous propagation of a disturbance. The part of the disturbance outside the forward light cone can usually be safely neglected, but if it is necessary to develop a reasonable speed for the transmission of heat, a hyperbolic problem should be considered instead – like a partial differential equation involving a second-order time derivative. Some models of nonlinear heat conduction (which are also parabolic equations) have solutions with finite heat transmission speed.[3][4]

Heat equation 184 Internal heat generation The function u above represents temperature of a body. Alternatively, it is sometimes convenient to change units and represent u as the heat density of a medium. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units. Suppose that a body obeys the heat equation and, in addition, generates its own heat per unit volume (e.g., in watts/litre - W/L) at a rate given by a known function q varying in space and time.[5] Then the heat per unit volume u satisfies an equation For example, a tungsten light bulb filament generates heat, so it would have a positive nonzero value for q when turned on. While the light is turned off, the value of q for the tungsten filament would be zero. Solving the heat equation using Fourier series The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822. Let us consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions. (1) where u = u(x, t) is a function of two variables x and t. Here • x is the space variable, so x ∈ [0,L], where L is the length of the rod. • t is the time variable, so t ≥ 0. We assume the initial condition (2) where the function f is given, and the boundary conditions . (3) Let us attempt to find a solution of (1) which is not identically zero satisfying the boundary conditions (3) but with the following property: u is a product in which the dependence of u on x, t is separated, that is: (4) This solution technique is called separation of variables. Substituting u back into equation (1),

Heat equation 185 Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value −λ. Thus: (5) and (6) We will now show that nontrivial solutions for (6) for values of λ ≤ 0 cannot occur: 1. Suppose that λ < 0. Then there exist real numbers B, C such that From (3) we get and therefore B = 0 = C which implies u is identically 0. 2. Suppose that λ = 0. Then there exist real numbers B, C such that From equation (3) we conclude in the same manner as in 1 that u is identically 0. 3. Therefore, it must be the case that λ > 0. Then there exist real numbers A, B, C such that and From (3) we get C = 0 and that for some positive integer n, This solves the heat equation in the special case that the dependence of u has the special form (4). In general, the sum of solutions to (1) which satisfy the boundary conditions (3) also satisfies (1) and (3). We can show that the solution to (1), (2) and (3) is given by where Generalizing the solution technique The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator uxx with the zero boundary conditions can be represented in terms of its eigenvectors. This leads naturally to one of the basic ideas of the spectral theory of linear self-adjoint operators. Consider the linear operator Δu = uxx. The infinite sequence of functions for n ≥ 1 are eigenvectors of Δ. Indeed

Heat equation 186 Moreover, any eigenvector f of Δ with the boundary conditions f(0)=f(L)=0 is of the form en for some n ≥ 1. The functions en for n ≥ 1 form an orthonormal sequence with respect to a certain inner product on the space of real-valued functions on [0, L]. This means Finally, the sequence {en}n ∈ N spans a dense linear subspace of L2(0, L). This shows that in effect we have diagonalized the operator Δ. Heat conduction in non-homogeneous anisotropic media In general, the study of heat conduction is based on several principles. Heat flow is a form of energy flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space. • The time rate of heat flow into a region V is given by a time-dependent quantity qt(V). We assume q has a density, so that • Heat flow is a time-dependent vector function H(x) characterized as follows: the time rate of heat flowing through an infinitesimal surface element with area dS and with unit normal vector n is Thus the rate of heat flow into V is also given by the surface integral where n(x) is the outward pointing normal vector at x. • The Fourier law states that heat energy flow has the following linear dependence on the temperature gradient where A(x) is a 3 × 3 real matrix that is symmetric and positive definite. By Greens theorem, the previous surface integral for heat flow into V can be transformed into the volume integral • The time rate of temperature change at x is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constant κ Putting these equations together gives the general equation of heat flow: Remarks. • The coefficient κ(x) is the inverse of specific heat of the substance at x × density of the substance at x. • In the case of an isotropic medium, the matrix A is a scalar matrix equal to thermal conductivity.

Heat equation 187 • In the anisotropic case where the coefficient matrix A is not scalar (i.e., if it depends on x), then an explicit formula for the solution of the heat equation can seldom be written down. Though, it is usually possible to consider the associated abstract Cauchy problem and show that it is a well-posed problem and/or to show some qualitative properties (like preservation of positive initial data, infinite speed of propagation, convergence toward an equilibrium, smoothing properties). This is usually done by one-parameter semigroups theory: for instance, if A is a symmetric matrix, then the elliptic operator defined by is self-adjoint and dissipative, thus by the spectral theorem it generates a one-parameter semigroup. Fundamental solutions A fundamental solution, also called a heat kernel, is a solution of the heat equation corresponding to the initial condition of an initial point source of heat at a known position. These can be used to find a general solution of the heat equation over certain domains; see, for instance, (Evans 1998) for an introductory treatment. In one variable, the Greens function is a solution of the initial value problem where δ is the Dirac delta function. The solution to this problem is the fundamental solution One can obtain the general solution of the one variable heat equation with initial condition u(x, 0) = g(x) for -∞ < x < ∞ and 0 < t < ∞ by applying a convolution: In several spatial variables, the fundamental solution solves the analogous problem in -∞ < x i < ∞, i = 1,...,n, and 0 < t < ∞. The n-variable fundamental solution is the product of the fundamental solutions in each variable; i.e., The general solution of the heat equation on Rn is then obtained by a convolution, so that to solve the initial value problem with u(x, t = 0) = g(x), one has The general problem on a domain Ω in Rn is with either Dirichlet or Neumann boundary data. A Greens function always exists, but unless the domain Ω can be readily decomposed into one-variable problems (see below), it may not be possible to write it down explicitly. The method of images provides one additional technique for obtaining Greens functions for non-trivial domains.

Heat equation 188 Some Greens function solutions in 1D A variety of elementary Greens function solutions in one-dimension are recorded here. In some of these, the spatial domain is the entire real line (-∞,∞). In others, it is the semi-infinite interval (0,∞) with either Neumann or Dirichlet boundary conditions. One further variation is that some of these solve the inhomogeneous equation where f is some given function of x and t. Homogeneous heat equation Initial value problem on (-∞,∞) Comment. This solution is the convolution with respect to the variable x of the fundamental solution and the function g(x). Therefore, according to the general properties of the convolution with respect to differentiation, is a solution of the same heat equation, for Moreover, and so that, by general facts about approximation to the identity, as t → 0 in various senses, according to the specific g. For instance, if g is assumed bounded and continuous on R then converges uniformly to g as t → 0, meaning that u(x, t) is continuous on R × [0, ∞) with u(x,0) = g(x). Initial value problem on (0,∞) with homogeneous Dirichlet boundary conditions Comment. This solution is obtained from the preceding formula as applied to the data g(x) suitably extended to R, so as to be an odd function, that is, letting g(−x) := −g(x) for all x. Correspondingly, the solution of the initial value problem on (−∞,+∞) is an odd function with respect to the variable x for all values of t, and in particular it satisfies the homogeneous Dirichlet boundary conditions Initial value problem on (0,∞) with homogeneous Neumann boundary conditions Comment. This solution is obtained from the first solution formula as applied to the data g(x) suitably extended to R so as to be an even function, that is, letting g(−x) := +g(x) for all x. Correspondingly, the solution of the initial value problem on (−∞,+∞) is an even function with respect to the variable x for all values of t > 0, and in particular, being smooth, it satisfies the homogeneous Neumann boundary conditions ux(0, t) = 0. Problem on (0,∞) with homogeneous initial conditions and non-homogeneous Dirichlet boundary conditions

Heat equation 189 Comment. This solution is the convolution with respect to the variable t of and the function h(t). Since Φ(x, t) is the fundamental solution of the function ψ(x, t) is also a solution of the same heat equation, and so is u := ψ ∗ h, thanks to general properties of the convolution with respect to differentiation. Moreover, and so that, by general facts about approximation to the identity, ψ(x, ⋅) ∗ h → h as x → 0 in various senses, according to the specific h. For instance, if h is assumed continuous on R with support in [0, ∞) then ψ(x, ⋅) ∗ h converges uniformly on compacta to h as x → 0, meaning that u(x, t) is continuous on [0, ∞) × [0, ∞) with u(0, t) = h(t). Inhomogeneous heat equation Problem on (-∞,∞) homogeneous initial conditions Comment. This solution is the convolution in R2, that is with respect to both the variables x and t, of the fundamental solution and the function f(x, t), both meant as defined on the whole R2 and identically 0 for all t → 0. One verifies that which is expressed in the language of distributions as where the distribution δ is the Diracs delta function, that is the evaluation at 0. Problem on (0,∞) with homogeneous Dirichlet boundary conditions and initial conditions Comment. This solution is obtained from the preceding formula as applied to the data f(x, t) suitably extended to R × [0,∞), so as to be an odd function of the variable x, that is, letting f(−x, t) := −f(x, t) for all x and t. Correspondingly, the solution of the inhomogeneous problem on (-∞,+∞) is an odd function with respect to the variable x for all values of t, and in particular it satisfies the homogeneous Dirichlet boundary conditions u(0, t) = 0. Problem on (0,∞) with homogeneous Neumann boundary conditions and initial conditions

Heat equation 190 Comment. This solution is obtained from the first formula as applied to the data f(x, t) suitably extended to R × [0,∞), so as to be an even function of the variable x, that is, letting f(−x,t) := f(x, t) for all x and t. Correspondingly, the solution of the inhomogeneous problem on (−∞,+∞) is an even function with respect to the variable x for all values of t, and in particular, being a smooth function, it satisfies the homogeneous Neumann boundary conditions Examples Since the heat equation is linear, solutions of other combinations of boundary conditions, inhomogeneous term, and initial conditions can be found by taking an appropriate linear combination of the above Greens function solutions. For example, to solve let where u and v solve the problems Similarly, to solve let where w, v, and r solve the problems Mean-value property for the heat equation Solutions of the heat equations satisfy a mean-value property analogous to the mean-value properties of harmonic functions, solutions of , though a bit more complicated. Precisely, if u solves and then where Eλ is a "heat-ball", that is a super-level set of the fundamental solution of the heat equation:

Heat equation 191 Notice that as λ → ∞ so the above formula holds for any (x, t) in the (open) set dom(u) for λ large enough. Conversely, any function u satisfying the above mean-value property on an open domain of Rn × R is a solution of the heat equation. This can be shown by an argument similar to the analogous one for harmonic functions. Stationary Heat Equation The (time) stationary heat equation is not dependent on time. In other words, it is assumed conditions exist such that: This condition depends on the time constant and the amount of time passed since boundary conditions have been imposed. Thus, the condition is fulfilled in situations in which the time equilibrium constant is fast enough that the more complex time-dependent heat equation can be approximated by the stationary case. Equivalently, the stationary condition exists for all cases in which enough time has passed that the thermal field u no longer evolves in time. In the stationary case, a spacial thermal gradient may (or may not) exist, but if it does, it does not change in time. This equation therefore describes the end result in all thermal problems in which a source is switched on (for example, an engine started in an automobile), and enough time has passed for all permanent temperature gradients to establish themselves in space, after which these spacial gradients no longer change in time (as again, with an automobile in which the engine has been running for long enough). The other (trivial) solution is for all spacial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well. The equation is much simpler and can help to understand better the physics of the materials without focusing on the dynamic of the heat transport process. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time. Stationary condition: The stationary heat equation for a volume that contains a heat source (the inhomogeneous case), is the Poissons equation: In electrostatics, this is equivalent to the case where the space under consideration contains an electrical charge. The stationary heat equation without a heat source within the volume (the homogeneous case) is the equation in electrostatics for a volume of free space that does not contain a charge. It is described by Laplaces equation: where u is the temperature, k is the thermal conductivity and q the heat source density.

Heat equation 192 Applications Particle diffusion One can model particle diffusion by an equation involving either: • the volumetric concentration of particles, denoted c, in the case of collective diffusion of a large number of particles, or • the probability density function associated with the position of a single particle, denoted P. In either case, one uses the heat equation or Both c and P are functions of position and time. D is the diffusion coefficient that controls the speed of the diffusive process, and is typically expressed in meters squared over second. If the diffusion coefficient D is not constant, but depends on the concentration c (or P in the second case), then one gets the nonlinear diffusion equation. Brownian motion The random trajectory of a single particle subject to the particle diffusion equation (or heat equation) is a Brownian motion. If a particle is placed at R = 0 at time t = 0, then the probability density function associated with the position vector of the particle R will be the following: which is a (multivariate) normal distribution evolving in time. Schrödinger equation for a free particle With a simple division, the Schrödinger equation for a single particle of mass m in the absence of any applied force field can be rewritten in the following way: , where i is the imaginary unit, ħ is the reduced Plancks constant, and ψ is the wavefunction of the particle. This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation: Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of the Schrödinger equation, which in turn can be used to obtain the wavefunction at any time through an integral on the wavefunction at t = 0: , with Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the wavefunction satisfying Schrödingers equation might have an origin other than diffusion.

Heat equation 193 Thermal diffusivity in polymers A direct practical application of the heat equation, in conjunction with Fourier theory, in spherical coordinates, is the measurement of the thermal diffusivity in polymers (Unsworth and Duarte). The dual theoretical-experimental method demonstrated by these authors is applicable to rubber and various other materials of practical interest. Further applications The heat equation arises in the modeling of a number of phenomena and is often used in financial mathematics in the modeling of options. The famous Black–Scholes option pricing models differential equation can be transformed into the heat equation allowing relatively easy solutions from a familiar body of mathematics. Many of the extensions to the simple option models do not have closed form solutions and thus must be solved numerically to obtain a modeled option price. The equation describing pressure diffusion in an porous medium is identical in form with the heat equation. Diffusion problems dealing with Dirichlet, Neumann and Robin boundary conditions have closed form analytic solutions (Thambynayagam 2011). The heat equation is also widely used in image analysis (Perona & Malik 1990) and in machine-learning as the driving theory behind scale-space or graph Laplacian methods. The heat equation can be efficiently solved numerically using the Crank–Nicolson method of (Crank & Nicolson 1947). This method can be extended to many of the models with no closed form solution, see for instance (Wilmott, Howison & Dewynne 1995). An abstract form of heat equation on manifolds provides a major approach to the Atiyah–Singer index theorem, and has led to much further work on heat equations in Riemannian geometry. Notes [1] Here we are assuming that the material has constant mass density and heat capacity through space as well as time, although generalizations are given below. [2] In higher dimensions, the divergence theorem is used instead. [3] The Mathworld: Porous Medium Equation (http:/ / mathworld. wolfram. com/ PorousMediumEquation. html) and the other related models have solutions with finite wave propagation speed. [4] Juan Luis Vazquez (2006-12-28), The Porous Medium Equation: Mathematical Theory, Oxford University Press, USA, ISBN 0-19-856903-3 [5] Note that the units of u must be selected in a manner compatible with those of q. Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. References • Cannon, John Rozier (1984), The One–Dimensional Heat Equation (http://books.google.com/ ?id=XWSnBZxbz2oC&printsec=frontcover#v=onepage&q=), Encyclopedia of Mathematics and Its Applications, 23 (1st ed.), Reading-Menlo Park–London–Don Mills–Sidney–Tokyo/ Cambridge–New York–New Rochelle–Melbourne–Sidney: Addison-Wesley Publishing Company/Cambridge University Press, pp. XXV+483, ISBN 978-0-521-30243-2, MR0747979, Zbl 0567.35001. • Crank, J.; Nicolson, P.; Hartree, D. R. (1947), "A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type", Proceedings of the Cambridge Philosophical Society 43: 50–67, Bibcode 1947PCPS...43...50C, doi:10.1017/S0305004100023197 • Einstein, Albert (1905), "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen", Ann. Phys. Leipzig 17 322 (8): 549–560, Bibcode 1905AnP...322..549E, doi:10.1002/andp.19053220806 • Evans, L.C. (1998), Partial Differential Equations, American Mathematical Society, ISBN 0-8218-0772-2 • John, Fritz (1991), Partial Differential Equations (4th ed.), Springer, ISBN 978-0-387-90609-6 • Wilmott, P.; Howison, S.; Dewynne, J. (1995), The Mathematics of Financial Derivatives:A Student Introduction, Cambridge University Press

Heat equation 194 • Carslaw, H. S.; Jaeger, J. C. (1959), Conduction of Heat in Solids (2nd ed.), Oxford University Press, ISBN 978-0-19-853368-9 • Thambynayagam, R. K. M. (2011), The Diffusion Handbook: Applied Solutions for Engineers, McGraw-Hill Professional, ISBN 978-0-07-175184-1 • Perona, P; Malik, J. (1990), "Scale-Space and Edge Detection Using Anisotropic Diffusion", IEEE Transactions on Pattern Analysis and Machine Intelligence 12 (7): 629–639 • Unsworth, J.; Duarte, F. J. (1979), "Heat diffusion in a solid sphere and Fourier Theory", Am. J. Phys. 47 (11): 891–893, Bibcode 1979AmJPh..47..981U, doi:10.1119/1.11601 External links • Derivation of the heat equation (http://www.mathphysics.com/pde/HEderiv.html) • Linear heat equations (http://eqworld.ipmnet.ru/en/solutions/lpde/heat-toc.pdf): Particular solutions and boundary value problems - from EqWorld Heat exchanger A heat exchanger is a piece of equipment built for efficient heat transfer from one medium to another. The media may be separated by a solid wall, so that they never mix, or they may be in direct contact.[1] They are widely used in space heating, refrigeration, air conditioning, power plants, chemical plants, petrochemical plants, petroleum refineries, natural gas processing, and sewage treatment. The classic example of a heat exchanger is found in an internal combustion engine in which a circulating fluid known as engine coolant flows through radiator coils and air flows past the coils, which cools the coolant and heats the incoming air. An interchangeable plate heat exchanger Tubular heat exchanger.

Heat exchanger 195 Flow arrangement Countercurrent (A) and parallel (B) flows Fig. 1: Shell and tube heat Fig. 2: Shell and tube heat Fig. 3: Shell and tube heat exchanger, single pass (1–1 exchanger, 2-pass tube side exchanger, 2-pass shell side, parallel flow) (1–2 crossflow) 2-pass tube side (2-2 countercurrent) There are two primary classifications of heat exchangers according to their flow arrangement. In parallel-flow heat exchangers, the two fluids enter the exchanger at the same end, and travel in parallel to one another to the other side. In counter-flow heat exchangers the fluids enter the exchanger from opposite ends. The counter current design is the most efficient, in that it can transfer the most heat from the heat (transfer) medium due to the fact that the average temperature difference along any unit length is greater. See countercurrent exchange. In a cross-flow heat exchanger, the fluids travel roughly perpendicular to one another through the exchanger. For efficiency, heat exchangers are designed to maximize the surface area of the wall between the two fluids, while minimizing resistance to fluid flow through the exchanger. The exchangers performance can also be affected by the addition of fins or corrugations in one or both directions, which increase surface area and may channel fluid flow or induce turbulence. The driving temperature across the heat transfer surface varies with position, but an appropriate mean temperature can be defined. In most simple systems this is the "log mean temperature difference" (LMTD). Sometimes direct

Heat exchanger 196 knowledge of the LMTD is not available and the NTU method is used. Types of heat exchangers Shell and tube heat exchanger Shell and tube heat exchangers consist of a series of tubes. One set of these tubes contains the fluid that must be either heated or cooled. The second fluid runs over the tubes that are being heated or cooled so that it can either provide the heat or absorb the heat required. A set of tubes is called the tube bundle and can be made up of several types of tubes: plain, longitudinally finned, etc. Shell and tube heat exchangers are typically used for high-pressure applications (with pressures greater than 30 bar and temperatures greater than 260 °C).[2] This is because the shell and tube heat exchangers are robust due to their shape. A Shell and Tube heat exchanger Several thermal design features must be considered when designing the tubes in the shell and tube heat exchangers: • Tube diameter: Using a small tube diameter makes the heat exchanger both economical and compact. However, it is more likely for the heat exchanger to foul up faster and the small size makes mechanical cleaning of the fouling difficult. To prevail over the fouling and cleaning problems, larger tube diameters can be used. Thus to determine the tube diameter, the available space, cost and the fouling nature of the fluids must be considered. • Tube thickness: The thickness of the wall of the tubes is usually determined to ensure: • There is enough room for corrosion • That flow-induced vibration has resistance • Axial strength • Availability of spare parts • Hoop strength (to withstand internal tube pressure) • Buckling strength (to withstand overpressure in the shell) • Tube length: heat exchangers are usually cheaper when they have a smaller shell diameter and a long tube length. Thus, typically there is an aim to make the heat exchanger as long as physically possible whilst not exceeding production capabilities. However, there are many limitations for this, including space available at the installation site and the need to ensure tubes are available in lengths that are twice the required length (so they can be withdrawn and replaced). Also, long, thin tubes are difficult to take out and replace. • Tube pitch: when designing the tubes, it is practical to ensure that the tube pitch (i.e., the centre-centre distance of adjoining tubes) is not less than 1.25 times the tubes outside diameter. A larger tube pitch leads to a larger overall shell diameter, which leads to a more expensive heat exchanger. • Tube corrugation: this type of tubes, mainly used for the inner tubes, increases the turbulence of the fluids and the effect is very important in the heat transfer giving a better performance. • Tube Layout: refers to how tubes are positioned within the shell. There are four main types of tube layout, which are, triangular (30°), rotated triangular (60°), square (90°) and rotated square (45°). The triangular patterns are employed to give greater heat transfer as they force the fluid to flow in a more turbulent fashion around the piping. Square patterns are employed where high fouling is experienced and cleaning is more regular. • Baffle Design: baffles are used in shell and tube heat exchangers to direct fluid across the tube bundle. They run perpendicularly to the shell and hold the bundle, preventing the tubes from sagging over a long length. They can also prevent the tubes from vibrating. The most common type of baffle is the segmental baffle. The semicircular segmental baffles are oriented at 180 degrees to the adjacent baffles forcing the fluid to flow upward and downwards between the tube bundle. Baffle spacing is of large thermodynamic concern when designing shell and

Heat exchanger 197 tube heat exchangers. Baffles must be spaced with consideration for the conversion of pressure drop and heat transfer. For thermo economic optimization it is suggested that the baffles be spaced no closer than 20% of the shell’s inner diameter. Having baffles spaced too closely causes a greater pressure drop because of flow redirection. Consequently having the baffles spaced too far apart means that there may be cooler spots in the corners between baffles. It is also important to ensure the baffles are spaced close enough that the tubes do not sag. The other main type of baffle is the disc and donut baffle, which consists of two concentric baffles. An outer, wider baffle looks like a donut, whilst the inner baffle is shaped like a disk. This type of baffle forces the fluid to pass around each side of the disk then through the donut baffle generating a different type of fluid flow. Plate heat exchanger Another type of heat exchanger is the plate heat exchanger. One is composed of multiple, thin, slightly separated plates that have very large surface areas and fluid flow passages for heat transfer. This stacked-plate arrangement can be more effective, in a given space, than the shell and tube heat exchanger. Advances in gasket and brazing technology have made the plate-type heat exchanger increasingly practical. In HVAC applications, large heat exchangers of this type are called plate-and-frame; when used in open loops, these heat exchangers are normally of the gasket type to allow periodic disassembly, cleaning, and inspection. There are many types of permanently bonded plate heat exchangers, such as dip-brazed, vacuum-brazed, and welded plate varieties, and they are often specified for closed-loop applications such as refrigeration. Plate heat exchangers also differ in the types of plates that are used, and in the Conceptual diagram of a plate and frame heat exchanger. configurations of those plates. Some plates may be stamped with "chevron", dimpled, or other patterns, where others may have machined fins and/or grooves. Plate and shell heat exchanger A third type of heat exchanger is a plate and shell heat exchanger, which combines plate heat exchanger with shell and tube heat exchanger technologies. The heart of the heat exchanger contains a fully welded circular plate pack made by pressing and cutting round plates and welding them together. Nozzles carry flow in and out of the A single plate heat exchanger platepack (the Plate side flowpath).The fully welded platepack is assembled into an outer shell that creates a second flowpath ( the Shell side). Plate and shell technology offers high heat transfer, high pressure, high operating temperature, compact size, low fouling and close approach temperature. In particular, it does completely without gaskets, which provides security against leakage at high pressures and temperatures.

Heat exchanger 198 Adiabatic wheel heat exchanger A fourth type of heat exchanger uses an intermediate fluid or solid store to hold heat, which is then moved to the other side of the heat exchanger to be released. Two examples of this are adiabatic wheels, which consist of a large wheel with fine threads rotating through the hot and cold fluids, and fluid heat exchangers. Plate fin heat exchanger This type of heat exchanger uses "sandwiched" passages containing fins to increase the effectivity of the unit. The designs include crossflow and counterflow coupled with various fin configurations such as straight fins, offset fins and wavy fins. An interchangeable plate heat exchanger applied to the system of a swimming pool Plate and fin heat exchangers are usually made of aluminium alloys, which provide high heat transfer efficiency. The material enables the system to operate at a lower temperature and reduce the weight of the equipment. Plate and fin heat exchangers are mostly used for low temperature services such as natural gas, helium and oxygen liquefaction plants, air separation plants and transport industries such as motor and aircraft engines. Advantages of plate and fin heat exchangers: • High heat transfer efficiency especially in gas treatment • Larger heat transfer area • Approximately 5 times lighter in weight than that of shell and tube heat exchanger. • Able to withstand high pressure Disadvantages of plate and fin heat exchangers: • Might cause clogging as the pathways are very narrow • Difficult to clean the pathways • Aluminum alloys are susceptible to Mercury Liquid Embrittlement Failure Pillow plate heat exchanger A pillow plate exchanger is commonly used in the dairy industry for cooling milk in large direct-expansion stainless steel bulk tanks. The pillow plate allows for cooling across nearly the entire surface area of the tank, without gaps that would occur between pipes welded to the exterior of the tank. The pillow plate is constructed using a thin sheet of metal spot-welded to the surface of another thicker sheet of metal. The thin plate is welded in a regular pattern of dots or with a serpentine pattern of weld lines. After welding the enclosed space is pressurized with sufficient force to cause the thin metal to bulge out around the welds, providing a space for heat exchanger liquids to flow, and creating a characteristic appearance of a swelled pillow formed out of metal.

Heat exchanger 199 Fluid heat exchangers This is a heat exchanger with a gas passing upwards through a shower of fluid (often water), and the fluid is then taken elsewhere before being cooled. This is commonly used for cooling gases whilst also removing certain impurities, thus solving two problems at once. It is widely used in espresso machines as an energy-saving method of cooling super-heated water to use in the extraction of espresso. Waste heat recovery units A Waste Heat Recovery Unit (WHRU) is a heat exchanger that recovers heat from a hot gas stream while transferring it to a working medium, typically water or oils. The hot gas stream can be the exhaust gas from a gas turbine or a diesel engine or a waste gas from industry or refinery. Dynamic scraped surface heat exchanger Another type of heat exchanger is called "(dynamic) scraped surface heat exchanger". This is mainly used for heating or cooling with high-viscosity products, crystallization processes, evaporation and high-fouling applications. Long running times are achieved due to the continuous scraping of the surface, thus avoiding fouling and achieving a sustainable heat transfer rate during the process. Phase-change heat exchangers In addition to heating up or cooling down fluids in just a single phase, heat exchangers can be used either to heat a liquid to evaporate (or boil) it or used as condensers to cool a vapor and condense it to a liquid. In chemical plants and refineries, reboilers used to heat incoming feed for distillation towers are often heat exchangers.[3][4] Distillation set-ups typically use condensers to condense distillate vapors back into liquid. Power plants that use steam-driven turbines commonly use heat exchangers to boil water into steam. Heat exchangers or similar units for producing steam from Typical kettle reboiler used for industrial distillation towers water are often called boilers or steam generators. In the nuclear power plants called pressurized water reactors, special large heat exchangers pass heat from the primary (reactor plant) system to the secondary (steam plant) system, producing steam from water in the process. These are called steam generators. All fossil-fueled and nuclear power plants using steam-driven turbines have surface condensers to convert the exhaust steam from the turbines into condensate (water) for re-use.[5][6] To conserve energy and cooling capacity in chemical and other plants, regenerative heat exchangers can transfer heat from a stream that must be cooled to Typical water-cooled surface condenser another stream that must be heated, such as distillate cooling and reboiler feed pre-heating.

Heat exchanger 200 This term can also refer to heat exchangers that contain a material within their structure that has a change of phase. This is usually a solid to liquid phase due to the small volume difference between these states. This change of phase effectively acts as a buffer because it occurs at a constant temperature but still allows for the heat exchanger to accept additional heat. One example where this has been investigated is for use in high power aircraft electronics. Direct contact heat exchangers Direct contact heat exchangers involve heat transfer between hot and cold streams of two phases in the absence of a separating wall.[7] Thus such heat exchangers can be classified as: • Gas – liquid • Immiscible liquid – liquid • Solid-liquid or solid – gas Most direct contact heat exchangers fall under the Gas- Liquid category, where heat is transferred between a gas and liquid in the form of drops, films or sprays. [2] Such types of heat exchangers are used predominantly in air conditioning, humidification, industrial hot water heating, water cooling and condensing plants.[8] [9] Continuous phase Driving force Change of phase Examples Phases Gas – Liquid Gas Gravity No Spray columns, packed columns Yes Cooling towers, falling droplet evaporators Forced No Spray coolers/quenchers Liquid flow Yes Spray condensers/evaporation, jet condensers Liquid Gravity No Bubble columns, perforated tray columns Yes Bubble column condensers Forced No Gas spargers Gas flow Yes Direct contact evaporators, submerged combustion HVAC air coils One of the widest uses of heat exchangers is for air conditioning of buildings and vehicles. This class of heat exchangers is commonly called air coils, or just coils due to their often-serpentine internal tubing. Liquid-to-air, or air-to-liquid HVAC coils are typically of modified crossflow arrangement. In vehicles, heat coils are often called heater cores. On the liquid side of these heat exchangers, the common fluids are water, a water-glycol solution, steam, or a refrigerant. For heating coils, hot water and steam are the most common, and this heated fluid is supplied by boilers, for example. For cooling coils, chilled water and refrigerant are most common. Chilled water is supplied from a chiller that is potentially located very far away, but refrigerant must come from a nearby condensing unit. When a refrigerant is used, the cooling coil is the evaporator in the vapor-compression refrigeration cycle. HVAC coils that use this direct-expansion of refrigerants are commonly called DX coils. On the air side of HVAC coils a significant difference exists between those used for heating, and those for cooling. Due to psychrometrics, air that is cooled often has moisture condensing out of it, except with extremely dry air flows. Heating some air increases that airflows capacity to hold water. So heating coils need not consider moisture condensation on their air-side, but cooling coils must be adequately designed and selected to handle their particular latent (moisture) as well as the sensible (cooling) loads. The water that is removed is called condensate.

Heat exchanger 201 For many climates, water or steam HVAC coils can be exposed to freezing conditions. Because water expands upon freezing, these somewhat expensive and difficult to replace thin-walled heat exchangers can easily be damaged or destroyed by just one freeze. As such, freeze protection of coils is a major concern of HVAC designers, installers, and operators. The introduction of indentations placed within the heat exchange fins controlled condensation, allowing water molecules to remain in the cooled air. This invention allowed for refrigeration without icing of the cooling mechanism.[10] The heat exchangers in direct-combustion furnaces, typical in many residences, are not coils. They are, instead, gas-to-air heat exchangers that are typically made of stamped steel sheet metal. The combustion products pass on one side of these heat exchangers, and air to heat on the other. A cracked heat exchanger is therefore a dangerous situation that requires immediate attention because combustion products may enter living space. Spiral heat exchangers A spiral heat exchanger (SHE), may refer to a helical (coiled) tube configuration, more generally, the term refers to a pair of flat surfaces that are coiled to form the two channels in a counter-flow arrangement. Each of the two channels has one long curved path. A pair of fluid ports are connected tangentially to the outer arms of the spiral, and axial ports are common, but optional.[11] The main advantage of the SHE is its highly efficient use of space. This attribute is often leveraged and partially reallocated to gain other improvements in performance, according to well known tradeoffs in heat Schematic drawing of a spiral heat exchanger. exchanger design. (A notable tradeoff is capital cost vs operating cost.) A compact SHE may be used to have a smaller footprint and thus lower all-around capital costs, or an over-sized SHE may be used to have less pressure drop, less pumping energy, higher thermal efficiency, and lower energy costs. Construction The distance between the sheets in the spiral channels are maintained by using spacer studs that were welded prior to rolling. Once the main spiral pack has been rolled, alternate top and bottom edges are welded and each end closed by a gasketed flat or conical cover bolted to the body. This ensures no mixing of the two fluids occurs. Any leakage is from the periphery cover to the atmosphere, or to a passage that contains the same fluid.[12] Self cleaning SHEs are often used in the heating of fluids that contain solids and thus tend to foul the inside of the heat exchanger. The low pressure drop lets the SHE handle fouling more easily. The SHE uses a “self cleaning” mechanism, whereby fouled surfaces cause a localized increase in fluid velocity, thus increasing the drag (or fluid friction) on the fouled surface, thus helping to dislodge the blockage and keep the heat exchanger clean. "The internal walls that make up the heat transfer surface are often rather thick, which makes the SHE very robust, and able to last a long time in demanding environments." They are also easily cleaned, opening out like an oven where any build up of foulant can be removed by pressure washing.

Heat exchanger 202 Self-Cleaning Water filters are used to keep the system clean and running without the need to shut down or replace cartridges and bags. Flow arrangements There are three main types of flows in a spiral heat exchanger: 1. Counter-current Flow: Fluids flow in opposite directions. These are used for liquid-liquid, condensing and gas cooling applications. Units are usually mounted vertically when condensing vapour and mounted horizontally when handling high concentrations of solids. 2. Spiral Flow/Cross Flow: One fluid is in spiral flow and the other in a cross flow. Spiral flow passages are Concurrent and countercurrent flow. welded at each side for this type of spiral heat exchanger. This type of flow is suitable for handling low density gas, which passes through the cross flow, avoiding pressure loss. It can be used for liquid-liquid applications if one liquid has a considerably greater flow rate than the other. 3. Distributed Vapour/Spiral flow: This design is that of a condenser, and is usually mounted vertically. It is designed to cater for the sub-cooling of both condensate and non-condensables. The coolant moves in a spiral and leaves via the top. Hot gases that enter leave as condensate via the bottom outlet. Applications The SHE is good for applications such as pasteurization, digester heating, heat recovery, pre-heating (see: recuperator), and effluent cooling. For sludge treatment, SHEs are generally smaller than other types of heat exchangers. Selection Due to the many variables involved, selecting optimal heat exchangers is challenging. Hand calculations are possible, but many iterations are typically needed. As such, heat exchangers are most often selected via computer programs, either by system designers, who are typically engineers, or by equipment vendors. To select an appropriate heat exchanger, the system designers (or equipment vendors) would firstly consider the design limitations for each heat exchanger type. Though cost is often the primary criterion, several other selection criteria are important: • High/low pressure limits • Thermal performance • Temperature ranges • Product mix (liquid/liquid, particulates or high-solids liquid) • Pressure drops across the exchanger • Fluid flow capacity • Cleanability, maintenance and repair

Heat exchanger 203 • Materials required for construction • Ability and ease of future expansion Choosing the right heat exchanger (HX) requires some knowledge of the different heat exchanger types, as well as the environment where the unit must operate. Typically in the manufacturing industry, several differing types of heat exchangers are used for just the one process or system to derive the final product. For example, a kettle HX for pre-heating, a double pipe HX for the ‘carrier’ fluid and a plate and frame HX for final cooling. With sufficient knowledge of heat exchanger types and operating requirements, an appropriate selection can be made to optimise the process.[13] Monitoring and maintenance Online monitoring of commercial heat exchangers is done by tracking the overall heat transfer coefficient. The overall heat transfer coefficient tends to decline over time due to fouling. U=Q/AΔTlm By periodically calculating the overall heat transfer coefficient from exchanger flow rates and temperatures, the owner of the heat exchanger can estimate when cleaning the heat exchanger is economically attractive. Integrity inspection of plate and tubular heat exchanger can be tested in situ by the conductivity or helium gas methods. These methods confirm the integrity of the plates or tubes to prevent any cross contamination and the condition of the gaskets. Mechanical integrity monitoring of heat exchanger tubes may be conducted through Nondestructive methods such as eddy current testing. Fouling Fouling occurs when impurities deposit on the heat exchange surface. Deposition of these impurities can be caused by: • Low wall shear stress • Low fluid velocities • High fluid velocities • Reaction product solid precipitation • Precipitation of dissolved impurities due to elevated wall temperatures The rate of heat exchanger fouling is determined by the rate of particle deposition less re-entrainment/suppression. This model was originally proposed in 1959 by Kern and Seaton. Crude Oil Exchanger Fouling. In commercial crude oil refining, crude oil is heated from 21 °C to 343 °C prior to entering the distillation column. A series of A heat exchanger in a steam power station contaminated with shell and tube heat exchangers typically exchange heat macrofouling. between crude oil and other oil streams to heat the crude to 260 °C prior to heating in a furnace. Fouling occurs on the crude side of these exchangers due to asphaltene insolubility. The nature of asphaltene solubility in crude oil was successfully modeled by Wiehe and Kennedy.[14] The precipitation of insoluble asphaltenes in crude preheat trains has been successfully modeled as a first order reaction by Ebert and Panchal[15] who expanded on the work of Kern and Seaton.

Heat exchanger 204 Cooling Water Fouling. Cooling water systems are susceptible to fouling. Cooling water typically has a high total dissolved solids content and suspended colloidal solids. Localized precipitation of dissolved solids occurs at the heat exchange surface due to wall temperatures higher than bulk fluid temperature. Low fluid velocities (less than 3 ft/s) allow suspended solids to settle on the heat exchange surface. Cooling water is typically on the tube side of a shell and tube exchanger because its easy to clean. To prevent fouling, designers typically ensure that cooling water velocity is greater than 0.9 m/s and bulk fluid temperature is maintained less than 60 °C. Other approaches to control fouling control combine the “blind” application of biocides and anti-scale chemicals with periodic lab testing. Maintenance Plate heat exchangers must be disassembled and cleaned periodically. Tubular heat exchangers can be cleaned by such methods as acid cleaning, sandblasting, high-pressure water jet, bullet cleaning, or drill rods. In large-scale cooling water systems for heat exchangers, water treatment such as purification, addition of chemicals, and testing, is used to minimize fouling of the heat exchange equipment. Other water treatment is also used in steam systems for power plants, etc. to minimize fouling and corrosion of the heat exchange and other equipment. A variety of companies have started using water borne oscillations technology to prevent biofouling. Without the use of chemicals, this type of technology has helped in providing a low-pressure drop in heat exchangers. In nature Humans The human nasal passages serve as a heat exchanger, which warms air being inhaled and cools air being exhaled. You can demonstrate its effectiveness by putting your hand in front of your face and exhaling, first through your nose and then through your mouth. Air exhaled through your nose is substantially cooler.[16][17] In species that have external testes (such as humans), the artery to the testis is surrounded by a mesh of veins called the pampiniform plexus. This cools the blood heading to the testis, while reheating the returning blood. Birds, fish, marine mammals Further information: Countercurrent exchange in biological systems "Countercurrent" heat exchangers occur naturally in the circulation system of fish, whales and other marine mammals. Arteries to the skin carrying warm blood are intertwined with veins from the skin Countercurrent exchange conservation circuit carrying cold blood, causing the warm arterial blood to exchange heat with the cold venous blood. This reduces the overall heat loss in cold waters. Heat exchangers are also present in the tongue of baleen whales as large volumes of water flow through their mouths.[18][19] Wading birds use a similar system to limit heat losses from their body through their legs into the water.

Heat exchanger 205 Carotid rete The carotid rete is a counter-current heat exchanging organ in some ungulates. The blood ascending the carotid arteries on its way to the brain, flows via a network of vessels where heat is discharged to the veins of cooler blood descending from the nasal passages. The carotid rete allows Thomsons gazelle to maintain its brain almost 3°C cooler than the rest of the body, and therefore aids in tolerating bursts in metabolic heat production such as associated with outrunning cheetahs (during which the body temperature exceeds the maximum temperature at which the brain could function).[20] In industry Heat exchangers are widely used in industry both for cooling and heating large scale industrial processes. The type and size of heat exchanger used can be tailored to suit a process depending on the type of fluid, its phase, temperature, density, viscosity, pressures, chemical composition and various other thermodynamic properties. In many industrial processes there is waste of energy or a heat stream that is being exhausted, heat exchangers can be used to recover this heat and put it to use by heating a different stream in the process. This practice saves a lot of money in industry, as the heat supplied to other streams from the heat exchangers would otherwise come from an external source that is more expensive and more harmful to the environment. Heat exchangers are used in many industries, indluding: • Waste water treatment • Refrigeration • Wine and beer making • Petroleum refining In waste water treatment, heat exchangers play a vital role in maintaining optimal temperatures within anaerobic digesters to promote the growth of microbes that remove pollutants. Common types of heat exchangers used in this application are the double pipe heat exchanger as well as the plate and frame heat exchanger. In aircraft In commercial aircraft heat exchangers are used to take heat from the engines oil system to heat cold fuel.[21] This improves fuel efficiency, as well as reduces the possibility of water entrapped in the fuel freezing in components.[22] Early 2008, a Boeing 777 flying as British Airways Flight 38 crashed just short of the runway. In an early-2009 Boeing-update sent to aircraft operators, the problem was identified as specific to the Rolls-Royce engine oil-fuel flow heat exchangers.[22] Other heat exchangers, or Boeing 777 aircraft powered by GE or Pratt and Whitney engines, were not affected by the problem.[22] A model of a simple heat exchanger A simple heat exchanger[23] might be thought of as two straight pipes with fluid flow, which are thermally connected. Let the pipes be of equal length L, carrying fluids with heat capacity (energy per unit mass per unit change in temperature) and let the mass flow rate of the fluids through the pipes be (mass per unit time), where the subscript i applies to pipe 1 or pipe 2. Temperature profiles for the pipes are and where x is the distance along the pipe. Assume a steady state, so that the temperature profiles are not functions of time. Assume also that the only transfer of heat from a small volume of fluid in one pipe is to the fluid element in the other pipe at the same position. There is no transfer of heat along a pipe due to temperature differences in that pipe. By Newtons law of cooling the rate of change in energy of a small volume of fluid is proportional to the difference in temperatures between it and the corresponding element in the other pipe:

Heat exchanger 206 where is the thermal energy per unit length and γ is the thermal connection constant per unit length between the two pipes. This change in internal energy results in a change in the temperature of the fluid element. The time rate of change for the fluid element being carried along by the flow is: where is the "thermal mass flow rate". The differential equations governing the heat exchanger may now be written as: Note that, since the system is in a steady state, there are no partial derivatives of temperature with respect to time, and since there is no heat transfer along the pipe, there are no second derivatives in x as is found in the heat equation. These two coupled first-order differential equations may be solved to yield: where , , and A and B are two as yet undetermined constants of integration. Let and be the temperatures at x=0 and let and be the temperatures at the end of the pipe at x=L. Define the average temperatures in each pipe as: Using the solutions above, these temperatures are: Choosing any two of the temperatures above eliminates the constants of integration, letting us find the other four temperatures. We find the total energy transferred by integrating the expressions for the time rate of change of internal energy per unit length:

Heat exchanger 207 By the conservation of energy, the sum of the two energies is zero. The quantity is known as the Log mean temperature difference, and is a measure of the effectiveness of the heat exchanger in transferring heat energy. References [1] Sadik Kakaç and Hongtan Liu (2002). Heat Exchangers: Selection, Rating and Thermal Design (2nd ed.). CRC Press. ISBN 0-8493-0902-6. [2] Saunders, E. A. (1988). Heat Exchanges: Selection, Design and Construction. New York: Longman Scientific and Technical. [3] Kister, Henry Z. (1992). Distillation Design (1st ed.). McGraw-Hill. ISBN 0-07-034909-6. [4] Perry, Robert H. and Green, Don W. (1984). Perrys Chemical Engineers Handbook (6th ed.). McGraw-Hill. ISBN 0-07-049479-7. [5] Air Pollution Control Orientation Course (http:/ / www. epa. gov/ oar/ oaqps/ eog/ course422/ ce6b3. html) from website of the Air Pollution Training Institute [6] Energy savings in steam systems (http:/ / kolmetz. com/ pdf/ ENERGY EFFICIENCY IMPROVEMENT. pdf) Figure 3a, Layout of surface condenser (scroll to page 11 of 34 PDF pages) [7] Coulson, J. & Richardson, J. (1983), Chemical Engineering – Design (SI Units), Volume 6, Pergamon Press, Oxford. [8] Hewitt G, Shires G, Bott T (1994), Process Heat Transfer, CRC Press Inc, Florida. [9] Table: Various Types of Gas – Liquid Direct Contact Heat Exchangers (Hewitt G, Shires G & Bott T, 1994) [10] Patent 2,046,968 John C Raisley (http:/ / www. google. com/ patents/ pdf/ _RAISLEY_X. pdf?id=zwxKAAAAEBAJ& output=pdf& sig=63-9nepnkjAp6ngTtM6iTQCAca4) issued July 7, 1936; filed Jan. 8, 1934 (http:/ / www. google. com/ patents?id=zwxKAAAAEBAJ& dq=2046968) [11] Cooling Text (http:/ / www. gc3. com/ techdb/ manual/ cooltext. htm#Cooling Water Heat Exchangers) [12] E.A.D.Saunders (1988). Heat Exchangers:Selection Design And Construction Longman Scientific and Technical ISBN 0-582-49491-5 • White, F.M. ‘Heat and Mass Transfer’ © 1988 Addison-Wesley Publishing Co. pp. 602–604 • Heat Exchangers (http:/ / www. geothermie. de/ egec-geothernet/ prof/ heat_exchangers. htm) Kevin D. Rafferty, Gene Culver Geo-Heat Center, 1996–2001 Last Accessed 17/3/08 • For manufacturing engineers who use heat processing equipment- Heat exchanger basics (http:/ / www. process-heating. com), BNP Media, 2007 Last Accessed 17/3/08 [14] I. A. Wiehe and R. J. Kennedy, Energy & Fuels, 14, 56 – 63 (2000). [15] Panchal C;B; and Ebert W., Analysis of Exxon Crude-Oil-Slip-Stream Coking Data, Proc of Fouling Mitigation of Industrial Heat-Exchanger Equipment, San Luis Obispo, California, USA, p 451, June 1995 [16] Heat Loss from the Respiratory Tract in Cold (http:/ / www. dtic. mil/ srch/ doc?collection=t3& id=AD0293194), Defense Technical Information Center, April 1955 [17] Randall, David J.; Warren W. Burggren, Kathleen French, Roger Eckert (2002). Eckert animal physiology: mechanisms and adaptations. Macmillan. p. 587. ISBN 0-7167-3863-5. [18] (http:/ / www. nhm. org/ research/ mammals/ jj/ ) [19] Heyning and Mead; Mead, JG (November 1997). "Thermoregulation in the Mouths of Feeding Gray Whales" (http:/ / www. sciencemag. org/ cgi/ content/ abstract/ 278/ 5340/ 1138). Science 278 (5340): 1138–1140. doi:10.1126/science.278.5340.1138. PMID 9353198. . Retrieved 2010-01-19. [20] Carotid rete cools brain: Thomsons gazelle -- AskNature (http:/ / www. asknature. org/ strategy/ 10e254e7306d0e0ac711e77df408bd0d) [21] "United States Patent 4498525, Fuel/oil heat exchange system for an engine" (http:/ / www. freepatentsonline. com/ 4498525. html). United States Patent and Trademark Office. . Retrieved 3 February 2009. [22] "Boeing links Heathrow, Atlanta Trent 895 engine rollbacks" (http:/ / www. flightglobal. com/ articles/ 2009/ 02/ 03/ 322023/ boeing-links-heathrow-atlanta-trent-895-engine-rollbacks. html). FlightGlobal.com. . Retrieved 3 February 2009. [23] Kay J M & Nedderman R M (1985) Fluid Mechanics and Transfer Processes, Cambridge University Press • Coulson, J. and Richardson, J (1999). Chemical Engineering- Fluid Flow. Heat Transfer and Mass Transfer- Volume 1; Reed Educational & Professional Publishing LTD • Dogan Eryener (2005), ‘Thermoeconomic optimization of baffle spacing for shell and tube heat exchangers’, Energy Conservation and Management, Volume 47, Issue 11–12, Pages 1478–1489. • G.F.Hewitt, G.L.Shires, T.R.Bott (1994)Process Heat Transfer, CRC Press, Inc, United States Of America.

Heat exchanger 208 External links • Heat Exchangers (http://www.dmoz.org/Business/Industrial_Goods_and_Services/Machinery_and_Tools/ Thermal_Process/Heat_Exchangers/) at the Open Directory Project • Shell and Tube Heat Exchanger Design Software for Educational Applications (http://www.ijee.dit.ie/articles/ Vol14-3/ijee1015.pdf) (PDF) • EU Pressure Equipment Guideline (http://ec.europa.eu/enterprise/pressure_equipment/ped/guidelines/ guideline2-4_en.html) • A Thermal Management Concept For More Electric Aircraft Power System Application (http://www.dtic.mil/ dticasd/sbir/sttr04/m010b.pdf) (PDF) • Heat exchanger shellside and tubeside pressure drop calculator (http://www.enggcyclopedia.com/ welcome-to-enggcyclopedia/calculators/heat-exchanger-pressure-drop) Calculate shellside / tubeside pressure drop for an exchanger. Heat flux Heat flux or thermal flux is the rate of heat energy transfer through a given surface. The SI derived unit of heat rate is joule per second, or watt. Heat flux is the heat rate per unit area. In SI units, heat flux is measured in [W/m2][1]. Heat rate is a scalar quantity, while heat flux is a vectorial quantity. To define the heat flux at a certain point in space, one takes the limiting case where the size of the surface becomes infinitesimally small. Heat flux is often denoted , the subscript q specifying heat rate, as opposed to mass or momentum rate. Fouriers law is an important application of these concepts. Measuring heat flux The measurement of heat flux is most often done by measuring a temperature difference over a piece of material with known thermal conductivity. This method is analogous to a standard way to measure an electric current, where one measures the voltage drop over a known resistor. Relevance to science and engineering One of the tools in a scientists or engineers toolbox is the energy balance. Such a balance can be set up for any physical Heat flux through a surface. system, from chemical reactors to living organisms, and generally takes the following form where the three terms stand for the time rate of change of respectively the total amount of incoming energy, the total amount of outgoing energy and the total amount of accumulated energy.

Heat flux 209 Now, if the only way the system exchanges energy with its surroundings is through heat transfer, the heat rate can be used to calculate the energy balance, since where we have integrated the heat flux density over the surface of the system. In real-world applications one cannot know the exact heat flux at every point on the surface, but approximation schemes can be used to calculate the integral, for example Monte Carlo integration. References [1] The NIST Reference on Constants, Units, and Uncertainty http:/ / physics. nist. gov/ cuu/ Units/ units. html Heat pipe A heat pipe or heat pin is a heat-transfer device that combines the principles of both thermal conductivity and phase transition to efficiently manage the transfer of heat between two solid interfaces. At the hot interface within a heat pipe, which is typically at a very low pressure, a liquid in contact with a thermally conductive solid surface turns into a vapor by absorbing heat from that surface. The vapor condenses back into a liquid at the cold interface, releasing the latent heat. The liquid then returns to the hot interface through either capillary action or gravity A laptop heat pipe system action where it evaporates once more and repeats the cycle. In addition, the internal pressure of the heat pipe can be set or adjusted to facilitate the phase change depending on the demands of the working conditions of the thermally managed system. Structure, design and construction A typical heat pipe consists of a sealed pipe or tube made of a material with high thermal conductivity such as copper or aluminium at both hot and cold ends. A vacuum pump is used to remove all air from the empty heat pipe, and then the pipe is filled with a fraction of a percent by volume of working fluid (or coolant) chosen to match the operating temperature. Alternatively, the pipe is heated until the fluid boils, and sealed while hot. Examples of such fluids include water, ethanol, acetone, sodium, or mercury. Due to the partial vacuum that is near or below the vapor pressure of the fluid, some of the fluid will be in the liquid phase and some Diagram showing components and mechanism for a heat pipe will be in the gas phase. The use of a vacuum containing a wick

Heat pipe 210 eliminates the need for the working gas to diffuse through any other gas and so the bulk transfer of the vapor to the cold end of the heat pipe is at the speed of the moving molecules. In this sense, the only practical limit to the rate of heat transfer is the speed with which the gas can be condensed to a liquid at the cold end.[1] Inside the pipes walls, an optional wick structure exerts a capillary pressure on the liquid phase of the working fluid. This is typically a sintered metal powder or a series of grooves parallel to the pipe axis, but it may be any material capable of exerting capillary pressure on the condensed liquid to wick it back to the heated end. Cut-away view of a 500 µm thick flat heat pipe, with a thin planar The heat pipe may not need a wick structure if gravity capillary (aqua colored) or some other source of acceleration is sufficient to overcome surface tension and cause the condensed liquid to flow back to the heated end. A heat pipe is not a thermosiphon, because there is no siphon. Thermosiphons transfer heat by single-phase convection. (See also: Perkins tube, after Jacob Perkins.) Heat pipes contain no mechanical moving parts and typically require no maintenance, though non-condensing gases (that diffuse through the pipes walls, result from breakdown of the working fluid, or exist as impurities in the materials) may eventually reduce the pipes effectiveness at transferring heat. This Thin flat heat pipe (heat spreader) with remote heat sink and fan is significant when the working fluids vapour pressure is low. The materials chosen depend on the temperature conditions in which the heat pipe must operate, with coolants ranging from liquid helium for extremely low temperature applications (2–4 K) to mercury (523–923 K) & sodium (873–1473 K) and even indium (2000–3000 K) for extremely high temperatures. The vast majority of heat pipes for low temperature applications use some combination of ammonia (213–373 K), alcohol (methanol (283–403 K) or ethanol (273–403 K)) or water (303–473 K) as working fluid. Water, for instance, at low pressure will boil at just above 273 K (0 degrees Celsius) and so can start to effectively transfer latent heat at this low temperature. The advantage of heat pipes over many other heat-dissipation mechanisms is their great efficiency in transferring heat. They are fundamentally better at heat conduction over a distance than an equivalent cross-section of solid copper (a heat sink alone, though simpler in design and construction, does not take advantage of the principle of matter phase transition). Some heat pipes have demonstrated a heat flux of more than 230 MW/m².[2] Active control of heat flux can be effected by adding a variable volume liquid reservoir to the evaporator section. Variable conductance heat pipes employ a large reservoir of inert immiscible gas attached to the condensing section. Varying the gas reservoir pressure changes the volume of gas charged to the condenser which in turn limits the area available for vapor condensation. Thus a wider range of heat fluxes and temperature gradients can be accommodated with a single design. A modified heat pipe with a reservoir having no capillary connection to the heat pipe wick at the evaporator end can also be used as a thermal diode. This heat pipe will transfer heat in one direction, acting as an insulator in the other.

Heat pipe 211 Vapor Chamber or Flat heat pipes Thin planar heat pipes (heat spreaders) have the same primary components as tubular heat pipes. These components are a hermetically sealed hollow vessel, a working fluid, and a closed-loop capillary recirculation system. Compared to a one-dimensional tubular heat pipe, the width of a two-dimensional heat pipe allows an adequate cross section for heat flow even with a very thin device. These thin planar heat pipes are finding their way into “height sensitive” applications, such as notebook computers, and surface mount circuit board cores. It is possible to produce flat heat pipes as thin as 0.5 mm (thinner than a credit card). Loop heat pipe A loop heat pipe (LHP) is a two-phase heat transfer device that uses capillary action to remove heat from a source and passively move it to a condenser or radiator. LHPs are similar to heat pipes but have the advantage of being able to provide reliable operation over long distance and the ability to operate against gravity. They can transport a large heat load over a long distance with a small temperature difference.[3][4] Different designs of LHPs ranging from powerful, large size LHPs to miniature LHPs (micro loop heat pipe) have been developed and successfully employed in a wide sphere of applications both ground based as well as space applications. Heat transfer Heat pipes employ evaporative cooling to transfer thermal energy from one point to another by the evaporation and condensation of a working fluid or coolant. Heat pipes rely on a temperature difference between the ends of the pipe, and cannot lower temperatures at either end beyond the ambient temperature (hence they tend to equalise the temperature within the pipe). When one end of the heat pipe is heated the working fluid inside the pipe at that end evaporates and increases the vapour pressure inside the cavity of the heat pipe. The latent heat of evaporation absorbed by the vaporisation of the working fluid reduces the temperature at the hot end of the pipe. The vapour pressure over the hot liquid working fluid at the hot end of the pipe is higher than the equilibrium vapour pressure over A heat sink (aluminium) with heat pipe (copper) condensing working fluid at the cooler end of the pipe, and this pressure difference drives a rapid mass transfer to the condensing end where the excess vapour condenses, releases its latent heat, and warms the cool end of the pipe. Non-condensing gases (caused by contamination for instance) in the vapour impede the gas flow and reduce the effectiveness of the heat pipe, particularly at low temperatures, where vapour pressures are low. The speed of molecules in a gas is approximately the speed of sound, and in the absence of noncondensing gases (i.e., if there is only a gas phase present) this is the upper limit to the velocity with which they could travel in the heat pipe. In practice, the speed of the vapour through the heat pipe is limited by the rate of condensation at the cold end and far lower than the molecular speed. The condensed working fluid then flows back to the hot end of the pipe. In the case of vertically-oriented heat pipes the fluid may be moved by the force of gravity. In the case of heat pipes containing wicks, the fluid is returned by capillary action. When making heat pipes, there is no need to create a vacuum in the pipe. One simply boils the working fluid in the heat pipe until the resulting vapour has purged the non condensing gases from the pipe and then seals the end.

Heat pipe 212 An interesting property of heat pipes is the temperature over which they are effective. Initially, it might be suspected that a water charged heat pipe would only work when the hot end reached the boiling point (100 °C) and steam was transferred to the cold end. However, the boiling point of water is dependent on absolute pressure inside the pipe. In an evacuated pipe, water will boil just slightly above its melting point (0 °C). Thus the heat pipe can operate at hot-end temperatures as low as just slightly warmer than the melting point of the working fluid. Similarly, a heat pipe with water as a working fluid can work well above the boiling point (100 °C), if the cold end is low enough in temperature to condense the fluid. The main reason for the effectiveness of heat pipes is the evaporation and condensation of the working fluid. The heat of vaporization greatly exceeds the sensible heat capacity. Using water as an example, the energy needed to evaporate one gram of water is 540 times the amount of energy needed to raise the temperature of that same one gram of water by 1 °C. Almost all of that energy is rapidly transferred to the "cold" end when the fluid condenses there, making a very effective heat transfer system with no moving parts. Origins and research in the United States The general principle of heat pipes using gravity (commonly classified as two phase thermosiphons) dates back to the steam age. The modern concept for a capillary driven heat pipe was first suggested by R.S. Gaugler of General Motors in 1942 who patented the idea.[5] The benefits of employing capillary action were independently developed and first demonstrated by George Grover at Los Alamos National Laboratory in 1963 and subsequently published in the Journal of Applied Physics in 1964.[6] Grover noted in his notebook:[7] "Heat transfer via capillary movement of fluids. The "pumping" action of surface tension forces may be sufficient to move liquids from a cold temperature zone to a high temperature zone (with subsequent return in vapor form using as the driving force, the difference in vapor pressure at the two temperatures) to be of interest in transferring heat from the hot to the cold zone. Such a closed system, requiring no external pumps, may be of particular interest in space reactors in moving heat from the reactor core to a radiating system. In the absence of gravity, the forces must only be such as to overcome the capillary and the drag of the returning vapor through its channels." Between 1964 and 1966, RCA was the first corporation to undertake research and development of heat pipes for commercial applications (though their work was mostly funded by the US government). During the late 1960s NASA played a large role in heat pipe development by funding a significant amount of research on their applications and reliability in space flight following from Grovers suggestion. NASA’s attraction to heat pipe cooling systems was understandable given their low weight, high heat flux, and zero power draw. Their primary interest however was based on the fact that the system wouldn’t be adversely affected by operating in a zero gravity environment. The first application of heat pipes in the space program was in thermal equilibration of satellite transponders. As satellites orbit, one side is exposed to the direct radiation of the sun while the opposite side is completely dark and exposed to the deep cold of outer space. This causes severe discrepancies in the temperature (and thus reliability and accuracy) of the transponders. The heat pipe cooling system designed for this purpose managed the high heat fluxes and demonstrated flawless operation with and without the influence of gravity. The developed cooling system was the first description and usage of variable conductance heat pipes to actively regulate heat flow or evaporator temperature. Corporate R&D Publications in 1967 and 1968 by Feldman, Eastman, & Katzoff first discussed applications of heat pipes to areas outside of government concern and that did not fall under the high temperature classification such as: air conditioning, engine cooling, and electronics cooling. These papers also made the first mentions of flexible, arterial, and flat plate heat pipes. 1969 publications introduced the concepts of the rotational heat pipe with its applications to turbine blade cooling and the first discussions of heat pipe applications to cryogenic processes.

Heat pipe 213 Starting in the 1980s Sony began incorporating heat pipes into the cooling schemes for some of its commercial electronic products in place of both forced convection and passive finned heat sinks. Initially they were used in tuners & amplifiers, soon spreading to other high heat flux electronics applications. During the late 1990s increasingly hot microcomputer CPUs spurred a threefold increase in the number of U.S. heat pipe patent applications. As heat pipes transferred from a specialized industrial heat transfer component to a consumer commodity most development and production moved from the U.S. to Asia. Modern CPU heat pipes are typically made from copper and use water as the working fluid.[8] Applications Grover and his colleagues were working on cooling systems for nuclear power cells for space craft, where extreme thermal conditions are found. Heat pipes have since been used extensively in spacecraft as a means for managing internal temperature conditions. Heat pipes are extensively used in many modern computer systems, where increased power requirements and subsequent increases in heat emission have resulted in greater demands on cooling systems. Heat pipes are typically used to move heat away from components such as CPUs and GPUs to heat sinks where thermal energy may be dissipated into the environment. Solar Thermal Heat pipes are also being widely used in solar thermal water heating applications in combination with evacuated tube solar collector arrays. In these applications, distilled water is commonly used as the heat transfer fluid inside a sealed length of copper tubing that is Alaska pipeline support legs cooled by heat pipes to keep permafrost frozen. located within an evacuated glass tube and oriented towards the sun. In solar thermal water heating applications, an individual absorber tube of an evacuated tube collector can deliver up to 40% more efficiency when compared to more traditional "flat plate" solar water collectors. This is largely due to the vacuum that exists within the tube which slows down convective and conductive heat loss. Relative efficiencies of the evacuated tube system are reduced however, when compared to "flat plate" collectors because "flat plate" collectors have a larger aperture size and can absorb more solar energy per unit area. This means that while an individual evacuated tube has better insulation (lower conductive and convective losses) due to the vacuum created inside the tube, an array of tubes found in a completed solar assembly absorbs less energy per unit area due to there being less absorber surface area pointed toward the sun because of the rounded design of an evacuated tube collector. Therefore, real world efficiencies of both designs are about the same. Evacuated tube collectors reduce the need for anti-freeze additives to be added as the vacuum helps slow heat loss. However, under prolonged exposure to freezing temperatures the heat transfer fluid can still freeze and precautions must be taken in the design to ensure that the freezing liquid does not damage the evacuated tube. Properly designed solar thermal water heaters can be frost protected down to more than -3 °C with special additives and are being used in Antarctica to heat water.

Heat pipe 214 Permafrost cooling Building on permafrost is difficult because heat from the structure can thaw the permafrost. To avoid the risk of destabilization, heat pipes are used in some cases. For example, on the Trans-Alaska Pipeline System residual ground heat remaining in the oil, as well as that produced by friction and turbulence in the moving oil could conduct down the pipes support legs and melt the permafrost on which the supports are anchored. This would cause the pipeline to sink and possibly sustain damage. To prevent this each vertical support member has been mounted with 4 vertical heat pipes.[9] Heat pipes are also used to dissipate heat alongside parts of the Qinghai–Tibet Railway. The embankment and track absorb the Suns heat. Vertical heat pipes either side of the formation prevent that heat spreading any further into the surrounding ground. Cooking Heat pipes have been designed to speed the cooking of roasts. The pipe is poked through the roast. One end of the pipe extends into the oven where it draws heat to the middle of the roast.[10] The principle has also been applied to camping stoves, transferring a large volume of heat at low temperature to allow baking of goods in camping-type situations, as well as cooking other dishes. An example of this is the Bakepacker system.[11] Ventilation heat recovery In heating, ventilation and air-conditioning systems, HVAC, heat pipes are positioned within the supply and exhaust air streams of an air handling system, or in the exhaust gases of an industrial process, in order to recover the heat energy. The device consists of a battery of multi-row finned heat pipe tubes located within both the supply and exhaust air streams. Within the exhaust air side of the heat pipe, the refrigerant evaporates, taking its heat from the extract air. The refrigerant vapour moves towards the cooler end of the tube, within the supply air side of the device, where it condenses and gives up its heat. The condensed refrigerant returns by a combination of gravity and capillary action in the wick. Thus heat is transferred from the exhaust air stream through the tube wall to the refrigerant, and then from the refrigerant through the tube wall to the supply air stream. Because of the characteristics of the device, better efficiencies are obtained when the unit is positioned upright with the supply air side mounted over the exhaust air side, this allows the liquid refrigerant to flow quickly under gravity back to the evaporator. Generally, gross heat transfer efficiencies of up to 75% are claimed by manufacturers. Nuclear Reactor Cooling Since the early 1990s, numerous nuclear reactor power systems have been proposed using heat pipes for transporting heat between the reactor core and power conversion system.[12] The first nuclear reactor to produce electricity using heat pipes, Demonstration Using Flattop Fission was first operated September 13, 2012. Limitations Heat pipes must be tuned to particular cooling conditions. The choice of pipe material, size and coolant all have an effect on the optimal temperatures in which heat pipes work. When heated above a certain temperature, all of the working fluid in the heat pipe will vaporize and the condensation process will cease to occur; in such conditions, the heat pipes thermal conductivity is effectively reduced to the heat conduction properties of its solid metal casing alone. As most heat pipes are constructed of copper (a metal with high heat conductivity), an overheated heatpipe will generally continue to conduct heat at around 1/80 of the original conductivity.

Heat pipe 215 In addition, below a certain temperature, the working fluid will not undergo phase change, and the thermal conductivity will be reduced to that of the solid metal casing. One of the key criteria for the selection of a working fluid is the desired operational temperature range of the application. The lower temperature limit typically occurs a few degrees above the freezing point of the working fluid. Most manufacturers cannot make a traditional heat pipe smaller than 3mm in diameter due to material limitations (though 1.6mm thin sheets can be fabricated). Experiments have been conducted with micro heat pipes, which use piping with sharp edges, such as triangular or rhombus-like tubing. In these cases, the sharp edges transfer the fluid through capillary action, and no wick is necessary. References [1] Faghri, Amir (1995). Heat Pipe Science And Technology. Taylor & Francis Group. ISBN 978-1-56032-383-9. [2] Jim Danneskiold, Los Alamos-developed heat pipes ease space flight (http:/ / www. lanl. gov/ news/ releases/ archive/ 00-064. shtml). Los Alamos News Release, April 26, 2000. [3] Ku, Jentung; Ottenstein, Laura; Douglas, Donya; Hoang, Triem. "Multi-Evaporator Miniature Loop Heat Pipe for Small Spacecraft Thermal Control" (http:/ / hdl. handle. net/ 2060/ 20110015223). American Institute of Aeronautics and Astronomics. Goddard Space Flight Center. . Retrieved 14 September 2011. [4] Ku, Jentung; Paiva, Kleber; Mantelli, Marcia. "Loop Heat Pipe Transient Behavior Using Heat Source Temperature for Set Point Control with Thermoelectric Converter on Reservoir" (http:/ / hdl. handle. net/ 2060/ 20110015224). NASA. Goddard Space Flight Center. . Retrieved 14 September 2011. [5] Gaugler, Richard (1944). Heat Transfer Devices. Dayton, Ohio: U.S. Patent Office. pp. 4. 2350348 [6] Grover, G.M., T. P. Cotter, and G. F. Erickson (1964). "Structures of Very High Thermal Conductance". Journal of Applied Physics 35 (6): 1990–1991.. Bibcode 1964JAP....35.1990G. doi:10.1063/1.1713792. [7] Heat Pipe research at LANL (http:/ / www. lanl. gov/ orgs/ esa/ epe/ Heat_Pipe_Site/ ancient. shtml) [8] Jansson, Dick (2010). "Heat Pipes" (http:/ / www. arrl. org/ files/ file/ QEX_Next_Issue/ Jul-Aug_2010/ Jansson. pdf). QEX (ARRL) (Jul-Aug2010): 3–9. . Retrieved November 14, 2011. [9] http:/ / www. alyeska-pipe. com/ InTheNews/ MonthlyNews/ 2004/ December/ dec2004_featurestory. asp [10] Kew, David Anthony Reay ; Peter. A. (2006). Heat pipes (5th ed. ed.). Oxford: Butterworth-Heinemann. p. 309. ISBN 978-0-7506-6754-8. [11] "Bakepacker Product Details" (http:/ / www. bakepacker. com/ bak. html). bakepacker.com. . [12] "Nuclear Reactors for Space" (http:/ / www. world-nuclear. org/ info/ inf82. html). World Nuclear Association. . Retrieved 21 September 2012. External links • Frontiers in Heat Pipes (FHP) - An International Journal (http://www.thermalfluidscentral.org/e-journals/ index.php/Heat_Pipes) • House_N Research (mit.edu) (http://architecture.mit.edu/house_n/web/resources/tutorials/House_N Tutorial Heat Pipes.htm) • What is a Heat Pipe? (http://www.cheresources.com/htpipes.shtml) • Heat pipe selection guide (pdf) (http://www.enertron-inc.com/enertron-resources/PDF/ How-to-select-a-heat-pipe.pdf)

Heat transfer 216 Heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy and heat between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, thermal convection, thermal radiation, and transfer of energy by phase changes. Engineers also consider the transfer of mass of differing chemical species, either cold or hot, to achieve heat transfer. While these mechanisms have distinct characteristics, they often occur simultaneously in the same system. Heat conduction, also called diffusion, is the direct microscopic exchange of kinetic energy of particles through the boundary between two systems. When an object is at a different temperature from another body or its surroundings, heat flows so that the body and the surroundings reach the same temperature, at which point they are in thermal equilibrium. Such spontaneous heat transfer always occurs from a region of high temperature to another region of lower temperature, as described by the second law of thermodynamics. Heat convection occurs when bulk flow of a fluid (gas or liquid) carries heat along with the flow of matter in the fluid. The flow of fluid may be forced by external processes, or sometimes (in gravitational fields) by buoyancy forces caused when thermal energy expands the fluid (for example in a fire plume), thus influencing its own transfer. The latter process is often called "natural convection". All convective processes also Heat diffusion move heat partly by diffusion, as well. Another form of convection is forced convection. In this case the fluid is forced to flow by use of a pump, fan or other mechanical means. The final major form of heat transfer is by radiation, which occurs in any transparent medium (solid or fluid) but may also even occur across vacuum (as when the Sun heats the Earth). Radiation is the transfer of energy through space by means of electromagnetic waves in much the same way as electromagnetic light waves transfer light. The same laws that govern the transfer of light govern the radiant transfer of heat.[1] Heat radiation

Heat transfer 217 Overview Heat is defined in physics as the transfer of thermal energy across a well-defined boundary around a thermodynamic system. It is a characteristic of a process and is not statically contained in matter. In engineering contexts, however, the term heat transfer has acquired a specific usage, despite its literal redundancy of the characterization of transfer. In these contexts, heat is taken as synonymous to thermal energy. This usage has its origin in the historical interpretation of heat as a fluid (caloric) that can be transferred by various causes,[2] and that is also common in the language of laymen and everyday life. Fundamental methods of heat transfer in engineering include conduction, convection, and radiation. Physical laws describe the Linear Heat flow behavior and characteristics of each of these methods. Real systems often exhibit a complicated combination of them. Heat transfer methods are used in numerous disciplines, such as automotive engineering, thermal management of electronic devices and systems, climate control, insulation, materials processing, and power plant engineering. Various mathematical methods have been developed to solve or approximate the results of heat transfer in systems. Heat transfer is a path function (or process quantity), as opposed to a state quantity; therefore, the amount of heat transferred in a thermodynamic process that changes the state of a system depends on how that process occurs, not only the net difference between the initial and final states of the process. Heat flux is a quantitative, vectorial representation of the heat flow through a surface.[3] Heat transfer is typically studied as part of a general chemical engineering or mechanical engineering curriculum. Typically, thermodynamics is a prerequisite for heat transfer courses, as the laws of thermodynamics are essential to the mechanism of heat transfer.[3] Other courses related to heat transfer include energy conversion, thermofluids, and mass transfer. The transport equations for thermal energy (Fouriers law), mechanical momentum (Newtons law for fluids), and mass transfer (Ficks laws of diffusion) are similar[4][5] and analogies among these three transport processes have been developed to facilitate prediction of conversion from any one to the others.[5] Mechanisms The fundamental modes of heat transfer are: Conduction or diffusion The transfer of energy between objects that are in physical contact Convection The transfer of energy between an object and its environment, due to fluid motion Radiation The transfer of energy to or from a body by means of the emission or absorption of electromagnetic radiation Advection The transfer of energy from one location to another as a side effect of physically moving an object containing that energy

Heat transfer 218 Conduction On a microscopic scale, heat conduction occurs as hot, rapidly moving or vibrating atoms and molecules interact with neighboring atoms and molecules, transferring some of their energy (heat) to these neighboring particles. In other words, heat is transferred by conduction when adjacent atoms vibrate against one another, or as electrons move from one atom to another. Conduction is the most significant means of heat transfer within a solid or between solid objects in thermal contact. Fluids—especially gases—are less conductive. Thermal contact conductance is the study of heat conduction between solid bodies in contact.[6] Steady state conduction (see Fouriers law) is a form of conduction that happens when the temperature difference driving the conduction is constant, so that after an equilibration time, the spatial distribution of temperatures in the conducting object does not change any further.[7] In steady state conduction, the amount of heat entering a section is equal to amount of heat coming out.[6] Transient conduction (see Heat equation) occurs when the temperature within an object changes as a function of time. Analysis of transient systems is more complex and often calls for the application of approximation theories or numerical analysis by computer.[6] Convection Convective heat transfer, or convection, is the transfer of heat from one place to another by the movement of fluids, a process that is essentially the transfer of heat via mass transfer. (In physics, the term fluid means any substance that deforms under shear stress; it includes liquids, gases, plasmas, and some plastic solids.) Bulk motion of fluid enhances heat transfer in many physical situations, such as (for example) between a solid surface and the fluid.[8] Convection is usually the dominant form of heat transfer in liquids and gases. Although sometimes discussed as a third method of heat transfer, convection is usually used to describe the combined effects of heat conduction within the fluid (diffusion) and heat transference by bulk fluid flow streaming.[9] The process of transport by fluid streaming is known as advection, but pure advection is a term that is generally associated only with mass transport in fluids, such as advection of pebbles in a river. In the case of heat transfer in fluids, where transport by advection in a fluid is always also accompanied by transport via heat diffusion (also known as heat conduction) the process of heat convection is understood to refer to the sum of heat transport by advection and diffusion/conduction. Free, or natural, convection occurs when bulk fluid motion (steams and currents) are caused by buoyancy forces that result from density variations due to variations of temperature in the fluid. Forced convection is a term used when the streams and currents in the fluid are induced by external means—such as fans, stirrers, and pumps—creating an artificially induced convection current.[10] Convective heating or cooling in some circumstances may be described by Newtons law of cooling: "The rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings." However, by definition, the validity of Newtons law of cooling requires that the rate of heat loss from convection be a linear function of ("proportional to") the temperature difference that drives heat transfer, and in convective cooling this is sometimes not the case. In general, convection is not linearly dependent on temperature gradients, and in some cases is strongly nonlinear. In these cases, Newtons law does not apply.

Heat transfer 219 Radiation Thermal radiation is energy emitted by matter as electromagnetic waves due to the pool of thermal energy that all matter possesses that has a temperature above absolute zero. Thermal radiation propagates without the presence of matter through the vacuum of space.[11] Thermal radiation is a direct result of the random movements of atoms and molecules in matter. Since these atoms and molecules are composed of charged particles (protons and electrons), their movement results in the emission of electromagnetic radiation, which carries energy away from A red-hot iron object, transferring heat to the surrounding the surface. environment primarily through thermal radiation. Unlike conductive and convective forms of heat transfer, thermal radiation can be concentrated in a small spot by using reflecting mirrors, which is exploited in concentrating solar power generation. For example, the sunlight reflected from mirrors heats the PS10 solar power tower and during the day it can heat water to 285 °C (545 °F). Advection By transferring matter, energy—including thermal energy—is moved by the physical transfer of a hot or cold object from one place to another.[12] This can be as simple as placing hot water in a bottle and heating a bed, or the movement of an iceberg in changing ocean currents. A practical example is thermal hydraulics. This can be described by the formula Convection vs. conduction In a body of fluid that is heated from underneath its container, conduction and convection can be considered to compete for dominance. If heat conduction is too great, fluid moving down by convection is heated by conduction so fast that its downward movement will be stopped due to its buoyancy, while fluid moving up by convection is cooled by conduction so fast that its driving buoyancy will diminish. On the other hand, if heat conduction is very low, a large temperature gradient may be formed and convection might be very strong. The Rayleigh number ( ) is a measure determining the result of this competition. where • g is acceleration due to gravity • ρ is the density with being the density difference between the lower and upper ends • μ is the dynamic viscosity • α is the Thermal diffusivity • β is the volume thermal expansivity (sometimes denoted α elsewhere) • T is the temperature and • ν is the kinematic viscosity. The Rayleigh number can be understood as the ratio between the rate of heat transfer by convection to the rate of heat transfer by conduction; or, equivalently, the ratio between the corresponding timescales (i.e. conduction timescale divided by convection timescale), up to a numerical factor. This can be seen as follows, where all

Heat transfer 220 calculations are up to numerical factors depending on the geometry of the system. The buoyancy force driving the convection is roughly , so the corresponding pressure is roughly . In steady state, this is canceled by the shear stress due to viscosity, and therefore roughly equals , where V is the typical fluid velocity due to convection and the order of its timescale. The conduction timescale, on the other hand, is of the order of . Convection occurs when the Rayleigh number is above 1,000–2,000. Phase changes Transfer of heat through a phase transition in the medium—such as water-to-ice, water-to-steam, steam-to-water, or ice-to-water—involves significant energy and is exploited in many ways: steam engines, refrigerators, etc.[13] For example, the Mason equation is an approximate analytical expression for the growth of a water droplet based on the effects of heat transport on evaporation and condensation. Boiling Heat transfer in boiling fluids is complex, but of considerable technical importance. It is characterized by an S-shaped curve relating heat flux to surface temperature difference.[14] At low driving temperatures, no boiling occurs and the heat transfer rate is controlled by the usual single-phase mechanisms. As the surface temperature is increased, local boiling occurs and vapor bubbles nucleate, grow into the surrounding cooler fluid, and collapse. This is sub-cooled nucleate boiling, and is a very efficient heat transfer mechanism. At high bubble generation rates, the bubbles begin to interfere and the heat flux no longer increases rapidly with surface temperature (this is the departure from nucleate boiling, or DNB). At higher temperatures still, a maximum in the heat flux is reached (the critical heat flux, or CHF). The regime of falling heat transfer that follows is not easy to study, but is believed to be characterized by alternate periods of nucleate and film boiling. Nucleate boiling slows the heat transfer due to gas bubbles on the heaters surface; as mentioned, gas-phase thermal conductivity is much lower than liquid-phase thermal conductivity, so the outcome is a kind of "gas thermal barrier". At higher temperatures still, the hydrodynamically-quieter regime of film boiling is reached. Heat fluxes across the stable vapor layers are low, but rise slowly with temperature. Any contact between fluid and the surface that may be seen probably leads to the extremely rapid nucleation of a fresh vapor layer ("spontaneous nucleation"). Condensation Condensation occurs when a vapor is cooled and changes its phase to a liquid. Condensation heat transfer, like boiling, is of great significance in industry. During condensation, the latent heat of vaporization must be released. The amount of the heat is the same as that absorbed during vaporization at the same fluid pressure. There are several types of condensation: • Homogeneous condensation, as during a formation of fog. • Condensation in direct contact with subcooled liquid. • Condensation on direct contact with a cooling wall of a heat exchanger: This is the most common mode used in industry: • Filmwise condensation is when a liquid film is formed on the subcooled surface, and usually occurs when the liquid wets the surface. • Dropwise condensation is when liquid drops are formed on the subcooled surface, and usually occurs when the liquid does not wet the surface. Dropwise condensation is difficult to sustain reliably; therefore, industrial equipment is normally designed to operate in filmwise condensation mode.

Heat transfer 221 Modeling approaches Complex heat transfer phenomena can be modeled in different ways. Heat equation The heat equation is an important partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. In some cases, exact solutions of the equation are available; in other cases the equation must be solved numerically using computational methods. For example, simplified climate models may use Newtonian cooling, instead of a full (and computationally expensive) radiation code, to maintain atmospheric temperatures. Lumped system analysis System analysis by the lumped capacitance model is a common approximation in transient conduction that may be used whenever heat conduction within an object is much faster than heat conduction across the boundary of the object. This is a method of approximation that reduces one aspect of the transient conduction system—that within the object—to an equivalent steady state system. That is, the method assumes that the temperature within the object is completely uniform, although its value may be changing in time. In this method, the ratio of the conductive heat resistance within the object to the convective heat transfer resistance across the objects boundary, known as the Biot number, is calculated. For small Biot numbers, the approximation of spatially uniform temperature within the object can be used: it can be presumed that heat transferred into the object has time to uniformly distribute itself, due to the lower resistance to doing so, as compared with the resistance to heat entering the object. Lumped system analysis often reduces the complexity of the equations to one first-order linear differential equation, in which case heating and cooling are described by a simple exponential solution, often referred to as Newtons law of cooling. Applications and techniques Heat transfer has broad application to the functioning of numerous devices and systems. Heat-transfer principles may be used to preserve, increase, or decrease temperature in a wide variety of circumstances. Insulation and radiant barriers Thermal insulators are materials specifically designed to reduce the flow of heat by limiting conduction, convection, or both. Radiant barriers are materials that reflect radiation, and therefore reduce the flow of heat from radiation sources. Good insulators are not necessarily good radiant barriers, and vice versa. Metal, for instance, is an excellent reflector and a poor insulator. The effectiveness of an insulator is indicated by its R-value, or resistance value. The R-value of a material is the inverse of the conduction coefficient (k) multiplied by the thickness Car exhausts usually require some form of heat barrier, especially high performance exhausts where a ceramic coating (d) of the insulator. In most of the world, R-values are is often applied.

Heat transfer 222 measured in SI units: square-meter kelvins per watt (m²·K/W). In the United States, R-values are customarily given in units of British thermal units per hour per square-foot degrees Fahrenheit (Btu/h·ft²·°F). Rigid fiberglass, a common insulation material, has an R-value of four per inch, while poured concrete, a poor insulator, has an R-value of 0.08 per inch.[15] Heat exposure as part of a fire test for firestop products. The tog is a measure of thermal resistance, commonly used in the textile industry, and often seen quoted on, for example, duvets and carpet underlay. The effectiveness of a radiant barrier is indicated by its reflectivity, which is the fraction of radiation reflected. A material with a high reflectivity (at a given wavelength) has a low emissivity (at that same wavelength), and vice versa. At any specific wavelength, reflectivity = 1 - emissivity. An ideal radiant barrier would have a reflectivity of 1, and would therefore reflect 100 percent of incoming radiation. Vacuum flasks, or Dewars, are silvered to approach this ideal. In the vacuum of space, satellites use multi-layer insulation, which consists of many layers of aluminized (shiny) Mylar to greatly reduce radiation heat transfer and control satellite temperature. Critical insulation thickness Low thermal conductivity (k) materials reduce heat fluxes. The smaller the k value, the larger the corresponding thermal resistance (R) value. Thermal conductivity is measured in watts-per-meter per kelvin (W·m−1·K−1), represented as k. As the thickness of insulating material increases, the thermal resistance—or R-value—also increases. However, adding layers of insulation has the potential of increasing the surface area, and hence the thermal convection area. For example, as thicker insulation is added to a cylindrical pipe, the outer radius of the pipe-and-insulation system increases, and therefore surface area increases. The point where the added resistance of increasing insulation thickness becomes overshadowed by the effect of increased surface area is called the critical insulation thickness. In simple cylindrical pipes, this is calculated as a radius:[16] Heat exchangers A heat exchanger is a tool built for efficient heat transfer from one fluid to another, whether the fluids are separated by a solid wall so that they never mix, or the fluids are in direct contact. Heat exchangers are widely used in refrigeration, air conditioning, space heating, power generation, and chemical processing. One common example of a heat exchanger is a cars radiator, in which the hot coolant fluid is cooled by the flow of air over the radiators surface. Common types of heat exchanger flows include parallel flow, counter flow, and cross flow. In parallel flow, both fluids move in the same direction while transferring heat; in counter flow, the fluids move in opposite directions; and in cross flow, the fluids move at right angles to each other. Common constructions for heat exchanger include shell and tube, double pipe, extruded finned pipe, spiral fin pipe, u-tube, and stacked plate.

Heat transfer 223 When engineers calculate the theoretical heat transfer in a heat exchanger, they must contend with the fact that the driving temperature difference between the two fluids varies with position. To account for this in simple systems, the log mean temperature difference (LMTD) is often used as an "average" temperature. In more complex systems, direct knowledge of the LMTD is not available, and the number of transfer units (NTU) method can be used instead. Heat dissipation A heat sink is a component that transfers heat generated within a solid material to a fluid medium, such as air or a liquid. Examples of heat sinks are the heat exchangers used in refrigeration and air conditioning systems, and the radiator in a car (which is also a heat exchanger). Heat sinks also help to cool electronic and optoelectronic devices such as CPUs, higher-power lasers, and light-emitting diodes (LEDs). A heat sink uses its extended surfaces to increase the surface area in contact with the cooling fluid. Buildings In cold climates, houses with their heating systems form dissipative systems, often resulting in a loss of energy (known colloquially as "Heat Bleed") that makes home interiors uncomfortably cool or cold. For the comfort of the inhabitants, the interiors must be maintained out of thermal equilibrium with the external surroundings. In effect, these domestic residences are islands of warmth in a sea of cold, and the thermal gradient between the inside and outside is often quite steep. This can lead to problems such as condensation and uncomfortable air currents, which—if left unaddressed—can cause cosmetic or structural damage to the property. Such issues can be prevented through the execution of an energy audit, and the implementation of recommended corrective procedures (such as the installation of adequate insulation, the air sealing of structural leaks, and the addition of energy-efficient windows and doors.[17] Thermal transmittance is the rate of transfer of heat through a structure divided by the difference in temperature across the structure. It is expressed in watts per square meter per kelvin, or W/m²K. Well-insulated parts of a building have a low thermal transmittance, whereas poorly-insulated parts of a building have a high thermal transmittance. A thermostat is a device capable of starting the heating system when the houses interior falls below a set temperature, and of stopping that same system when another (higher) set temperature has been achieved. Thus, the thermostat controls the flow of energy into the house, that energy eventually being dissipated to the exterior. Thermal energy storage Thermal energy storage refers to technologies that store energy in a thermal reservoir for later use. They can be employed to balance energy demand between daytime and nighttime. The thermal reservoir may be maintained at a temperature above (hotter) or below (colder) than that of the ambient environment. Applications include later use in space heating, domestic or process hot water, or to generate electricity. Most practical active solar heating systems have storage for a few hours to a days worth of heat collected in insulated hot water tanks, but this can be extended to interseasonal thermal storage using underground thermal energy storage.[18] Evaporative cooling Evaporative cooling is a physical phenomenon in which evaporation of a liquid, typically into surrounding air, cools an object or a liquid in contact with it. Latent heat describes the amount of heat that is needed to evaporate the liquid; this heat comes from the liquid itself and the surrounding gas and surfaces. The greater the difference between the two temperatures, the greater the evaporative cooling effect. When the temperatures are the same, no net evaporation of water in air occurs; thus, there is no cooling effect. A simple example of natural evaporative cooling is perspiration, or sweat, which the body secretes in order to cool itself. An evaporative cooler is a device that cools air through the simple evaporation of water.

Heat transfer 224 Radiative cooling Radiative cooling is the process by which a body loses heat by radiation. It is an important effect in the Earths atmosphere. In the case of the Earth-atmosphere system, it refers to the process by which long-wave (infrared) radiation is emitted to balance the absorption of short-wave (visible) energy from the Sun. Convective transport of heat and evaporative transport of latent heat both remove heat from the surface and redistribute it in the atmosphere, making it available for radiative transport at higher altitudes. Laser cooling Laser cooling refers to techniques in which atomic and molecular samples are cooled through the interaction with one or more laser light fields. The most common method of laser cooling is Doppler cooling. In Doppler cooling, the frequency of the laser light is tuned slightly below an electronic transition in the atom. Thus, the atoms would absorb more photons if they moved towards the light source, due to the Doppler effect. If an excited atom then emits a photon spontaneously, it will be accelerated. The result of the absorption and emission process is to reduce the speed of the atom. Eventually the mean velocity, and therefore the kinetic energy of the atoms, will be reduced. Since the temperature of an ensemble of atoms is a measure of the random internal kinetic energy, this is equivalent to cooling the atoms. Sympathetic cooling is a process in which particles of one type cool particles of another type. Typically, atomic ions that can be directly laser-cooled are used to cool nearby ions or atoms. This technique allows cooling of ions and atoms that cannot be laser cooled directly. Magnetic cooling Magnetic evaporative cooling is a technique for lowering the temperature of a group of atoms. The process confines atoms using a magnetic field. Over time, individual atoms will become much more energetic than the others due to random collisions, and will escape—removing energy from the system and reducing the temperature of the remaining group. This process is similar to the familiar process by which standing water becomes water vapor. Heat transfer in the human body The principles of heat transfer in engineering systems can be applied to the human body in order to determine how the body transfers heat. Heat is produced in the body by the continuous metabolism of nutrients which provides energy for the systems of the body.[19] The human body must maintain a consistent internal temperature in order to maintain healthy bodily functions. Therefore, excess heat must be dissipated from the body to keep it from overheating. When a person engages in elevated levels of physical activity, the body requires additional fuel which increases the metabolic rate and the rate of heat production. The body must then use additional methods to remove the additional heat produced in order to keep the internal temperature at a healthy level. Heat transfer by convection is driven by the movement of fluids over the surface of the body. This convective fluid can be either a liquid or a gas. For heat transfer from the outer surface of the body, the convection mechanism is dependent on the surface area of the body, the velocity of the air, and the temperature gradient between the surface of the skin and the ambient air.[20] The normal temperature of the body is approximately 37°C. Heat transfer occurs more readily when the temperature of the surroundings is significantly less than the normal body temperature. This concept explains why a person feels “cold” when not enough covering is worn when exposed to a cold environment. Clothing can be considered an insulator which provides thermal resistance to heat flow over the covered portion of the body.[21] This thermal resistance causes the temperature on the surface of the clothing to be less than the temperature on the surface of the skin. This smaller temperature gradient between the surface temperature and the ambient temperature will cause a lower rate of heat transfer than if the skin were not covered. In order to ensure that one portion of the body is not significantly hotter than another portion, heat must be distributed evenly through the bodily tissues. Blood flowing through blood vessels acts as a convective fluid and

Heat transfer 225 helps to prevent any buildup of excess heat inside the tissues of the body. This flow of blood through the vessels can be modeled as pipe flow in an engineering system. The heat carried by the blood is determined by the temperature of the surrounding tissue, the diameter of the blood vessel, the thickness of the fluid, velocity of the flow, and the heat transfer coefficient of the blood. The velocity, blood vessel diameter, and the fluid thickness can all be related with the Reynolds Number, a dimensionless number used in fluid mechanics to characterize the flow of fluids. Latent heat loss, also known as evaporative heat loss, accounts for a large fraction of heat loss from the body. When the core temperature of the body increases, the body triggers sweat glands in the skin to bring additional moisture to the surface of the skin. The liquid is then transformed into vapor which removes heat from the surface of the body.[22] The rate of evaporation heat loss is directly related to the vapor pressure at the skin surface and the amount of moisture present on the skin.[20] Therefore, the maximum of heat transfer will occur when the skin is completely wet. The body continuously loses water by evaporation but the most significant amount of heat loss occurs during periods of increased physical activity. Other A heat pipe is a passive device constructed in such a way that it acts as though it has extremely high thermal conductivity. Heat pipes use latent heat and capillary action to move heat, and can carry many times as much heat as a similar-sized copper rod. Originally invented for use in satellites, they have applications in personal computers. A thermocouple is a junction between two different metals that produces a voltage related to a temperature difference. Thermocouples are a widely used type of temperature sensor for measurement and control, and can also be used to convert heat into electric power. A thermopile is an electronic device that converts thermal energy into electrical energy. It is composed of thermocouples. Thermopiles do not measure the absolute temperature, but generate an output voltage proportional to a temperature difference. Thermopiles are widely used, e.g., they are the key component of infrared thermometers, such as those used to measure body temperature via the ear. A thermal diode or thermal rectifier is a device that preferentially passes heat in one direction: a "one-way valve" for heat. References [1] Geankoplis, Christie John (2003). Transport processes and separation process principles : (includes unit operations) (4th ed. ed.). Upper Saddle River, NJ: Prentice Hall Professional Technical Reference. ISBN 0-13-101367-X. [2] Lienhard, John H.,V; Lienhard, John H., V (2008). A Heat Transfer Textbook (3rd ed.). Cambridge, Massachusetts: Phlogiston Press. ISBN 978-0-9713835-3-1. OCLC 230956959. [3] New Jersey Institute of Technology, Chemical Engineering Dept. "B.S. Chemical Engineering" (http:/ / catalog. njit. edu/ undergraduate/ programs/ chemicalengineering. php). NJIT. . Retrieved 9 April 2011. [4] Welty, James R.; Wicks, Charles E.; Wilson, Robert Elliott (1976). Fundamentals of momentum, heat, and mass transfer (http:/ / books. google. be/ books?cd=3& hl=en& id=hZxRAAAAMAAJ) (2 ed.). New York: Wiley. ISBN 978-0-471-93354-0. OCLC 2213384. . [5] Faghri, Amir; Zhang, Yuwen; Howell, John (2010). Advanced Heat and Mass Transfer. Columbia, MO: Global Digital Press. ISBN 978-0-9842760-0-4. [6] Abbott, J.M. Smith, H.C. Van Ness, M.M. (2005). Introduction to chemical engineering thermodynamics (7th ed. ed.). Boston ; Montreal: McGraw-Hill. ISBN 0-07-310445-0. [7] "Thermal-FluidsPedia | Heat conduction" (https:/ / www. thermalfluidscentral. org/ encyclopedia/ index. php/ Heat_Conduction). [8] Çengel, Yunus A. (2003). Heat Transfer: a practical approach (http:/ / books. google. be/ books?id=nrbfpSZTwskC& dq=9780072458930& hl=en& ei=1q7YTNKMJ8OqlAe25_CSCQ& sa=X& oi=book_result& ct=result& resnum=1& ved=0CCkQ6AEwAA). McGraw-Hill series in mechanical engineering (2nd ed.). Boston: McGraw-Hill. ISBN 978-0-07-245893-0. OCLC 300472921. . Retrieved 2009-04-20. [9] "Thermal-FluidsPedia | Convective heat transfer" (https:/ / www. thermalfluidscentral. org/ encyclopedia/ index. php/ Convective_Heat_Transfer) [10] "Convection — Heat Transfer" (http:/ / www. engineersedge. com/ heat_transfer/ convection. htm). Engineers Edge. Engineers Edge. . Retrieved 2009-04-20. [11] "Thermal-FluidsPedia | Radiation" (https:/ / www. thermalfluidscentral. org/ encyclopedia/ index. php/ Radiation) [12] "Thermal-FluidsPedia | Mass transfer" (https:/ / www. thermalfluidscentral. org/ encyclopedia/ index. php/ Mass_Transfer)

Heat transfer 226 [13] "Thermal-FluidsPedia | Multiphase systems" (https:/ / www. thermalfluidscentral. org/ encyclopedia/ index. php/ Multiphase_systems) [14] Kay, John Menzies; Nedderman, R. M. (1985). Fluid Mechanics and Transfer Processes (2nd ed.). CUP Archive. p. 529. ISBN 978-0-521-31624-8. [15] Martin, Randy L. (2008-07-29). "R-Value Table" (http:/ / coloradoenergy. org/ procorner/ stuff/ r-values. htm). ColoradoENERGY.org. Windsor, Colorado: R. L. Martin & Associates, Inc.. . Retrieved 2010-11-08. [16] Balku, Şaziye (2007-05-22). "Steady Heat Transfer and Thermal Resistance Networks" (http:/ / mechatronics. atilim. edu. tr/ courses/ mece310/ ch9mechatronics. ppt) PPT Course notes: Mece 310: Thermodynamics and Heat Transfer. p. 20. Ankara, Turkey: Atılım University. Retrieved 2010-11-08. [17] "EnergySavers: Tips on Saving Money & Energy at Home" (http:/ / www. energysavers. gov/ pdfs/ energy_savers. pdf). U.S. Department of Energy. . Retrieved March 2, 2012. [18] "Underground Thermal Energy Storage" (http:/ / www. icax. co. uk/ Underground_Thermal_Energy_Storage. html). Icax.co.uk. . Retrieved 2012-30-6. [19] Hartman,Carl and Bibb, Lewis. "The Human body and its enemies: a textbook of physiology hygiene and sanitation", World Book Co., 1913, p.232. [20] Cengel, Yunus A. and Ghajar, Afshin J. "Heat and Mass Transfer: Fundamentals and Applications." , McGraw-Hill, 4th Edition, 2010. [21] Tao, Xiaoming. "Smart fibres, fabrics, and clothing" , Woodhead Publishing, 2001 [22] Wilmore, Jack H., Costill, David L., Kenney, Larry, "Physiology of sport and exercise", Human Kinetics, 2008, p.256. Further reading • Frontiers in Heat and Mass Transfer (https://www.thermalfluidscentral.org/e-journals/index.php/ Heat_Mass_Transfer) • Heat Transfer Engineering (http://www.tandf.co.uk/journals/titles/01457632.asp) • Experimental Heat Transfer (http://www.tandf.co.uk/journals/titles/08916152.asp) • International Journal of Heat and Mass Transfer (http://www.sciencedirect.com/science/journal/00179310) • ASME Journal of Heat Transfer (http://scitation.aip.org/dbt/dbt.jsp?KEY=JHTRAO) • Numerical Heat Transfer Part A (http://www.tandf.co.uk/journals/titles/10407782.asp) • Numerical Heat Transfer Part B (http://www.tandf.co.uk/journals/titles/10407790.asp) • Nanoscale and Microscale Thermophysical Engineering (http://www.tandf.co.uk/journals/titles/15567265. asp) • Journal of Enhanced Heat Transfer (http://www.begellhouse.com/journals/4c8f5faa331b09ea.html) External links • Advanced Heat and Mass Transfer (https://www.thermalfluidscentral.org/e-books/book-intro.php?b=37) - A textbook for free online reading. • Thermal-FluidsPedia (https://www.thermalfluidscentral.org/encyclopedia/index.php/Main_Page) - An online thermal fluids encyclopedia. • COMSOL Heat Transfer Module (http://www.comsol.com/products/ht/) - CAD software that models and simulates heat transfer problems. • Heat Transfer Tutorial (http://www.spiraxsarco.com/resources/steam-engineering-tutorials/ steam-engineering-principles-and-heat-transfer/heat-transfer.asp) Modes of heat transfer (conduction, convection, radiation) within or between media are explained, together with calculations and other issues such as heat transfer barriers - Spirax Sarco • Heat Transfer Podcast - Arun Majumdar - Department of Mechanical Engineering - University of California, Berkeley (http://webcast.berkeley.edu/courses/archive.php?seriesid=1906978353) • Heat Transfer Basics (http://www.cheresources.com/heat_transfer_basics.shtml) - Overview • A Heat Transfer Textbook (http://web.mit.edu/lienhard/www/ahtt.html) - Downloadable textbook (free) • Thermal Resistance Circuits (http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node118. html) - Overview • Hyperphysics Article on Heat Transfer (http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/heatra.html) - Overview

Heat transfer 227 • Interseasonal Heat Transfer (http://www.icax.co.uk/thermalbank.html) - a practical example of how heat transfer is used to heat buildings without burning fossil fuels. • Heat transfer fundamentals (http://www.hrs-heatexchangers.com/en/resources/ heat-transfer-fundamentals-01-05.aspx) • Aspects of Heat Transfer, Cambridge University (http://www.msm.cam.ac.uk/phase-trans/2007/HT/ heat_transfer.html) • Thermal-Fluids Central (https://www.thermalfluidscentral.org/) • Energy2D: Interactive Heat Transfer Simulations for Everyone (http://energy.concord.org/energy2d/) Heat transfer coefficient The heat transfer coefficient, in thermodynamics and in mechanical and chemical engineering, is used in calculating the heat transfer, typically by convection or phase transition between a fluid and a solid: where Q = heat flow in input or lost heat flow , J/s = W h = heat transfer coefficient, W/(m2K) A = heat transfer surface area, m2 = difference in temperature between the solid surface and surrounding fluid area, K From the above equation, the heat transfer coefficient is the proportionality coefficient between the heat flux, that is heat flow per unit area, q/A, and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ΔT). The heat transfer coefficient has SI units in watts per squared meter -kelvin: W/(m2K). Heat transfer coefficient is the inverse of thermal insulance. This is used for building materials (R-value) and for clothing insulation. There are numerous methods for calculating the heat transfer coefficient in different heat transfer modes, different fluids, flow regimes, and under different thermohydraulic conditions. Often it can be estimated by dividing the thermal conductivity of the convection fluid by a length scale. The heat transfer coefficient is often calculated from the Nusselt number (a dimensionless number). There are also online calculators [1] available specifically for heat transfer fluid applications. An understanding of convection boundary layers is necessary to understanding convective heat transfer between a surface and a fluid flowing past it. A thermal boundary layer develops if the fluid free stream temperature and the surface temperatures differ. A temperature profile exists due to the energy exchange resulting from this temperature difference. The heat transfer rate can then be written as, And because heat transfer at the surface is by conduction, These two terms are equal; thus Thermal Boundary Layer

Heat transfer coefficient 228 Rearranging, Making it dimensionless by multiplying by representative length L, The right hand side is now the ratio of the temperature gradient at the surface to the reference temperature gradient. While the left hand side is similar to the Biot modulus. This becomes the ratio of conductive thermal resistance to the convective thermal resistance of the fluid, otherwise known as the Nusselt number, Nu. Alternative Method (A simple method for determining the overall heat transfer coefficient) A simple method for determining an overall heat transfer coefficient that is useful to find the heat transfer between simple elements such as walls in buildings or across heat exchangers is shown below. Note that this method only accounts for conduction within materials, it does not take into account heat transfer through methods such as radiation. The method is as follows: Where: • = the overall heat transfer coefficient (W/m2 K) • = the contact area for each fluid side (m2) (with A_1 and A_2 expressing either surface) • = the thermal conductivity of the material (W/mK) • = the individual convection heat transfer coefficient for each fluid (W/m2 K) • = the wall thickness (m) As the areas for each surface approach being equal the equation can be written as the transfer coefficient per unit area as shown below: or NOTE: Often the value for is referred to as the difference of two radii where the inner and outer radii are used to define the thickness of a pipe carrying a fluid, however, this figure may also be considered as a wall thickness in a flat plate transfer mechanism or other common flat surfaces such as a wall in a building when the area difference between each edge of the transmission surface approaches zero. In the walls of buildings the above formula can be used to derive the formula commonly used to calculate the heat through building components. Architects and engineers call the resulting values either the U-Value or the R-Value of a construction assembly like a wall. Each type of value (R or U) are related as the inverse of each other such that R-Value = 1/U-Value and both are more fully understood through the concept of an overall heat transfer coefficient described in lower section of this document. Heb ik ook gezien. Citaat van Peter van Deurzen

Heat transfer coefficient 229 Convective heat transfer Correlations Although convective heat transfer can be derived analytically through dimensional analysis, exact analysis of the boundary layer, approximate integral analysis of the boundary layer and analogies between energy and momentum transfer, these analytic approaches may not offer practical solutions to all problems when there are no mathematical models applicable. As such, many correlations were developed by various authors to estimate the convective heat transfer coefficient in various cases including natural convection, forced convection for internal flow and forced convection for external flow. These empirical correlations are presented for their particular geometry and flow conditions. As the fluid properties are temperature dependent, they are evaluated at the film temperature , which is the average of the surface and the surrounding bulk temperature, . Natural convection External flow, Vertical plane Churchill and Chu correlation[2] for natural convection adjacent to vertical planes. NuL applies to all fluids for both laminar and turbulent flows. L is the characteristic length with respect to the direction of gravity, and RaL is the Rayleigh Number with respect to this length. For laminar flows in the range of , the following equation can be further improved. External flow, Vertical cylinders For cylinders with their axes vertical, the expressions for plane surfaces can be used provided the curvature effect is not too significant. This represents the limit where boundary layer thickness is small relative to cylinder diameter D. The correlations for vertical plane walls can be used when External flow, Horizontal plates W.H. McAdams [2] suggested the following correlations. The induced buoyancy will be different depending upon whether the hot surface is facing up or down. For a hot surface facing up or a cold surface facing down, For a hot surface facing down or a cold surface facing up, The length is the ratio of the plate surface area to perimeter. If the plane surface is inclined at an angle θ, the equations for vertical plane by Churchill and Chu may be used for θ up to . When boundary layer flow is laminar, the gravitational constant g is replaced with g cosθ for calculating the Ra in the equation for laminar flow

Heat transfer coefficient 230 External flow, Horizontal cylinder For cylinders of sufficient length and negligible end effects, Churchill and Chu has the following correlation for External flow, Spheres For spheres, T. Yuge [2] has the following correlation. for Pr≃1 and Forced convection Internal flow, Laminar flow Sieder and Tate [2] has the following correlation for laminar flow in tubes where D is the internal diameter, μ_b is the fluid viscosity at the bulk mean temperature, μ_w is the viscosity at the tube wall surface temperature. Internal flow, Turbulent flow The Dittus-Boelter correlation (1930) is a common and particularly simple correlation useful for many applications. This correlation is applicable when forced convection is the only mode of heat transfer; i.e., there is no boiling, condensation, significant radiation, etc. The accuracy of this correlation is anticipated to be ±15%. For a fluid flowing in a straight circular pipe with a Reynolds number between 10 000 and 120 000 (in the turbulent pipe flow range), when the fluids Prandtl number is between 0.7 and 120, for a location far from the pipe entrance (more than 10 pipe diameters; more than 50 diameters according to many authors[3]) or other flow disturbances, and when the pipe surface is hydraulically smooth, the heat transfer coefficient between the bulk of the fluid and the pipe surface can be expressed as: where - thermal conductivity of the bulk fluid - - Hydraulic diameter Nu - Nusselt number (Dittus-Boelter correlation) Pr - Prandtl number Re - Reynolds number n = 0.4 for heating (wall hotter than the bulk fluid) and 0.33 for cooling (wall cooler than the bulk fluid).[4] The fluid properties necessary for the application of this equation are evaluated at the bulk temperature thus avoiding iteration

Heat transfer coefficient 231 Forced convection, External flow In analyzing the heat transfer associated with the flow past the exterior surface of a solid, the situation is complicated by phenomena such as boundary layer separation. Various authors have correlated charts and graphs for different geometries and flow conditions. For Flow parallel to a Plane Surface, where x is the distance from the edge and L is the height of the boundary layer, a mean Nusselt number can be calculated using the Colburn analogy[2]. Thom correlation There exist simple fluid-specific correlations for heat transfer coefficient in boiling. The Thom correlation is for flow boiling of water (subcooled or saturated at pressures up to about 20 MPa) under conditions where the nucleate boiling contribution predominates over forced convection. This correlation is useful for rough estimation of expected temperature difference given the heat flux:[5] where: is the wall temperature elevation above the saturation temperature, K q is the heat flux, MW/m2 P is the pressure of water, MPa Note that this empirical correlation is specific to the units given. Heat transfer coefficient of pipe wall The resistance to the flow of heat by the material of pipe wall can be expressed as a "heat transfer coefficient of the pipe wall". However, one needs to select if the heat flux is based on the pipe inner or the outer diameter. where k is the effective thermal conductivity of the wall material and x is the wall thickness. If the above assumption does not hold, then the wall heat transfer coefficient can be calculated using the following expression: where di and do are the inner and outer diameters of the pipe, respectively. The thermal conductivity of the tube material usually depends on temperature; the mean thermal conductivity is often used. Combining heat transfer coefficients For two or more heat transfer processes acting in parallel, heat transfer coefficients simply add: For two or more heat transfer processes connected in series, heat transfer coefficients add inversely:[6] For example, consider a pipe with a fluid flowing inside. The rate of heat transfer between the bulk of the fluid inside the pipe and the pipe external surface is: where q = heat transfer rate (W)

Heat transfer coefficient 232 h = heat transfer coefficient (W/(m2·K)) t = wall thickness (m) k = wall thermal conductivity (W/m·K) A = area (m2) = difference in temperature. Overall heat transfer coefficient The overall heat transfer coefficient is a measure of the overall ability of a series of conductive and convective barriers to transfer heat. It is commonly applied to the calculation of heat transfer in heat exchangers, but can be applied equally well to other problems. For the case of a heat exchanger, can be used to determine the total heat transfer between the two streams in the heat exchanger by the following relationship: where = heat transfer rate (W) = overall heat transfer coefficient (W/(m²·K)) = heat transfer surface area (m2) = log mean temperature difference (K) The overall heat transfer coefficient takes into account the individual heat transfer coefficients of each stream and the resistance of the pipe material. It can be calculated as the reciprocal of the sum of a series of thermal resistances (but more complex relationships exist, for example when heat transfer takes place by different routes in parallel): where R = Resistance(s) to heat flow in pipe wall (K/W) Other parameters are as above.[7] The heat transfer coefficient is the heat transferred per unit area per kelvin. Thus area is included in the equation as it represents the area over which the transfer of heat takes place. The areas for each flow will be different as they represent the contact area for each fluid side. The thermal resistance due to the pipe wall is calculated by the following relationship: where x = the wall thickness (m) k = the thermal conductivity of the material (W/(m·K)) A = the total area of the heat exchanger (m2) This represents the heat transfer by conduction in the pipe. The thermal conductivity is a characteristic of the particular material. Values of thermal conductivities for various materials are listed in the list of thermal conductivities. As mentioned earlier in the article the convection heat transfer coefficient for each stream depends on the type of fluid, flow properties and temperature properties. Some typical heat transfer coefficients include:

Heat transfer coefficient 233 • Air - h = 10 to 100 W/(m2K) • Water - h = 500 to 10,000 W/(m2K) Thermal resistance due to fouling deposits Surface coatings can build on heat transfer surfaces during heat exchanger operation due to fouling. These add extra thermal resistance to the wall and may noticeably decrease the overall heat transfer coefficient and thus performance. (Fouling can also cause other problems.) The additional thermal resistance due to fouling can be found by comparing the overall heat transfer coefficient determined from laboratory readings with calculations based on theoretical correlations. They can also be evaluated from the development of the overall heat transfer coefficient with time (assuming the heat exchanger operates under otherwise identical conditions). This is commonly applied in practice, e.g.[8] The following relationship is often used: = where = overall heat transfer coefficient based on experimental data for the heat exchanger in the "fouled" state, = overall heat transfer coefficient based on calculated or measured ("clean heat exchanger") data, = thermal resistance due to fouling, References [1] http:/ / www. heat-transfer-fluid. com/ resources/ heat-transfer-coefficient-calculator. php [2] James R. Welty; Charles E. Wicks; Robert E. Wilson; Gregory L. Rorrer., "Fundamentals of Momentum, Heat and Mass transfer" 5th edition, John Wiley and Sons [3] S.S. Kutateladze and V.M. Borishanskii, A Concise Encyclopedia of Heat Transfer, Pergamon Press, 1966. [4] F. Kreith (editor), "The CRC Handbook of Thermal Engineering", CRC Press, 2000. [5] W.Rohsenow, J.Hartnet, Y.Cho, "Handbook of Heat Transfer", 3rd edition, McGraw-Hill, 1998. [6] This relationship is similar to the harmonic mean; however, note that it is not multiplied with the number n of terms. [7] Coulson and Richardson, "Chemical Engineering", Volume 1,Elsevier, 2000 [8] Turner C.W.; Klimas S.J.; Bbrideau M.G., "Thermal resistance of steam-generator tube deposits under single-phase forced convection and flow-boiling heat transfer", Canadian Journal of Chemical Engineering, 2000, vol. 78, No 1, pp. 53-60 External links • Overall Heat Transfer Coefficients (http://www.cheresources.com/uexchangers.shtml) • Correlations for Convective Heat Transfer (http://www.cheresources.com/convection.shtml) • Heat Transfer fundamentals (http://www.hrs-heatexchangers.com/en/resources/ heat-transfer-fundamentals-01-05.aspx) • Heat Transfer Coefficient Calculator (http://www.heat-transfer-fluid.com/resources/ heat-transfer-coefficient-calculator.php)

Heisler Chart 234 Heisler Chart Heisler Charts are a graphical analysis tool for the evaluation of heat transfer in thermal engineering. They are a set of two charts per included geometry introduced in 1947 by M. P. Heisler[1] which were supplemented by a third chart per geometry in 1961 by H. Gröber. Heisler Charts permit evaluation of the central temperature for transient heat conduction through an infinitely long plane wall of thickness 2L, an infinitely long cylinder of radius ro, and a sphere of radius ro. Although Heisler-Gröber Charts are a faster and simpler alternative to the exact solutions of these problems, there are some limitations. First, the body must be at uniform temperature initially. Additionally, the temperature of the surroundings and the convective heat transfer coefficient must remain constant and uniform. Also, there must be no heat generation from the body itself.[2][3][4] Infinitely long plane wall These first Heisler-Gröber Charts were based upon the first term of the exact Fourier Series solution for an infinite plane wall: ,[2] where Ti is the initial temperature of the slab, T∞ is the constant temperature imposed at the boundary, x is the location in the plane wall, λn is π(n+1/2), and α is thermal diffusivity. The position x=0 represents the center of the slab. The first chart for the plane wall is plotted using 3 different variables. Plotted along the vertical axis of the chart is dimensionless temperature at the midplane, θo* . Plotted along the horizontal axis is the Fourier Number, Fo=αt/L2 . The curves within the graph are a selection of values for the inverse of the Biot Number, where "Bi = hL/k. k is the thermal conductivity of the material and h is the heat transfer coefficient."[2] [5] The second chart is used to determine the variation of temperature within the plane wall for different Biot Numbers. The vertical axis is the ratio of a given temperature to that at the centerline θ/θo where the x/L curve is the position at which T is taken. The horizontal axis is the value of Bi−1.

Heisler Chart 235 [5] The third chart in each set was supplemented by Gröber in 1961 and this particular one shows the dimensionless heat transferred from the wall as a function of a dimensionless time variable. The vertical axis is a plot of Q/Qo, the ratio of actual heat transfer to the amount of total possible heat transfer before T=T∞ . On the horizontal axis is the plot of (Bi2)(Fo), a dimensionless time variable. [5]

Heisler Chart 236 Infinitely long cylinder For the infinitely long cylinder, the Heisler chart is based on the first term in an exact solution to a Bessel function.[2] Each chart plots similar curves to the previous examples, and on each axis is plotted a similar variable. [5] [5]

Heisler Chart 237 [5] Sphere (of radius ro) The Heisler Chart for a sphere is based on the first term in the exact Fourier series solution: [2] These charts can be used similar to the first two sets and are plots of similar variables. [5]

Heisler Chart 238 [5] [5]

Heisler Chart 239 Modern alternatives Currently there are programs that provide numerical solutions to the same problems, without using transcendental functions or infinite series. Examples of these programs can be found here [6].[7] References [1] Transactions ASME, 69, 227-236, 1947 [2] Cengel, Yunus A. (2007). Heat and Mass Transfer: A Practical Approach (3rd edition ed.). McGraw Hill. pp. 231-236. ISBN 978-0-07-312930-3. [3] http:/ / www. slideshare. net/ erlaurito/ unsteady-state-basics-presentation [4] http:/ / www. scribd. com/ doc/ 17462198/ Heat-conduction-in-cylinder [5] Lee Ho Sung, http:/ / www. mae. wmich. edu/ faculty/ Lee/ me431/ ch05_supp_heisler. pdf [6] http:/ / www. che. utexas. edu/ cache/ newsletters/ summer2006_JAVA. pdf [7] http:/ / www. che. utexas. edu/ cache/ newsletters/ summer2006_JAVA. pdf HVAC HVAC (heating, ventilation, and air conditioning) is the technology of indoor and automotive environmental comfort. HVAC system design is a major subdiscipline of mechanical engineering, based on the principles of thermodynamics, fluid mechanics, and heat transfer. Refrigeration is sometimes added to the fields abbreviation as HVAC&R or HVACR, or ventilating is dropped as in HACR (such as the designation of HACR-rated circuit breakers). HVAC is important in the design of medium to large industrial and office buildings such as skyscrapers and HVAC systems use ventilation air ducts installed throughout a in marine environments such as aquariums, where safe building that supply conditioned air to a room through rectangular or round outlet vents, called diffusers; and ducts that remove air through and healthy building conditions are regulated with return-air grilles respect to temperature and humidity, using fresh air from outdoors.

HVAC 240 Background Heating, ventilating, and air conditioning[1] is based on inventions and discoveries made by Nikolay Lvov, Michael Faraday, Willis Carrier, Reuben Trane, James Joule, William Rankine, Sadi Carnot, and many others. The invention of the components of HVAC systems went hand-in-hand with the industrial revolution, and new methods of modernization, higher efficiency, and system control are constantly introduced by companies and inventors all over the world. The three central functions of heating, ventilating, and air-conditioning are interrelated, especially with the need to provide thermal comfort and acceptable indoor air quality within reasonable installation, operation, and maintenance Ventilation (architecture) on the downdraught costs. HVAC systems can provide ventilation, reduce air infiltration, system, by impulsion, or the plenum principle, and maintain pressure relationships between spaces. How air is applied to schoolrooms (1899) delivered to, and removed from spaces is known as room air distribution.[2] The starting point in carrying out a heat estimate both for cooling and heating will depend on the ambient and inside conditions specified. However before taking up the heat load calculation, it is necessary to find fresh air requirements for each area in detail, as pressurization is an important consideration. In modern buildings the design, installation, and control systems of these functions are integrated into one or more HVAC systems. For very small buildings, contractors normally "size" and select HVAC systems and equipment. For larger buildings, building services designers and engineers, such as mechanical, architectural, or building services engineers analyze, design, and specify the HVAC systems, and specialty mechanical contractors build and commission them. Building permits and code-compliance inspections of the installations are normally required for all sizes of buildings. The HVAC industry is a worldwide enterprise, with roles including operation and maintenance, system design and construction, equipment manufacturing and sales, and in education and research. The HVAC industry was historically regulated by the manufacturers of HVAC equipment, but Regulating and Standards organizations such as HARDI, ASHRAE, SMACNA, ACCA, Uniform Mechanical Code, International Mechanical Code, and AMCA have been established to support the industry and encourage high standards and achievement.

HVAC 241 Heating There are many different types of heating systems. Central heating is often used in cold climates to heat private houses and public buildings. Such a system contains a boiler, furnace, or heat pump to heat water, steam, or air in a central location such as a furnace room in a home or a mechanical room in a large building. The use of water as the heat transfer medium is known as hydronics. These systems also contain either ductwork for forced air systems or piping to distribute a heated fluid and radiators to transfer this heat to the air. The term radiator in this context is misleading since most heat transfer from the heat exchanger is by convection, not radiation. The radiators may be mounted on walls or buried in the floor to give under-floor heat. All but the simplest boiler-fed or radiant heating systems have a pump to circulate the water and ensure an equal supply of heat to all the radiators. The heated water can also be fed through another (secondary) heat exchanger inside a storage cylinder to provide hot running water. Forced-air systems send heated air through ductwork. During warm weather the same ductwork can be used for air conditioning. The forced air can also be filtered or passed through air cleaners. Central heating unit Heat can also be provided electrically by resistive heating, in which conductive filaments are heated by the passage of electricity. This is used in baseboard heaters, portable heaters, and as backup or supplemental heating for heat pump (or reverse heating) systems. The heat pump is a form of heating that gained popularity in the 1950’s. Heat pumps can extract heat from the air or suck heat from the ground. Heat pumps work well in moderate climates, where summers are long and winters are mild. However, they tend to be more expensive than conventional heating systems and although more energy efficient, a ground extraction system is more costly.[3] The heating elements (radiators or vents) should be located in the coldest part of the room, typically next to the windows, to minimize condensation and offset the convective air current formed in the room due to the air next to the window becoming negatively buoyant due to the cold glass. Devices that direct vents away from windows to prevent "wasted" heat defeat this design intent. Cold air drafts can contribute significantly to subjectively feeling colder than the average room temperature, and for this reason it is important to control air leaks from outside in addition to properly designing the heating system. The invention of central heating is often credited to the ancient Romans, who installed systems of air ducts called hypocausts in the walls and floors of public baths and private villas.[4] The use of furnaces, space heaters and boilers as means of indoor heating may result in incomplete combustion and the emission of carbon monoxide, NOx, formaldehyde, VOC’s and other combustion by-products. Incomplete combustion occurs when there is insufficient oxygen; the inputs are fuels containing various contaminants and the outputs are the harmful by-products, most dangerously carbon monoxide which is a tasteless and odorless gas that has serious adverse health effects when inhaled. [5] Without proper ventilation, carbon monoxide can be extremely dangerous and can vary from a small, limited amount to a lethal amount. Carbon monoxide can be lethal at high concentration, usually less than 1000 ppmv. However, at several hundred ppmv, carbon monoxide exposure can induce headaches, fatigue, nausea and vomiting. Carbon monoxide binds with hemoglobin in the blood, forming carboxyhemoglobin, reducing the blood’s ability to transport oxygen. The primary health concerns associated with carbon monoxide exposure are its cardiovascular and neurobehavioral effects. Carbon monoxide can cause atherosclerosis; the hardening of arteries, and can also trigger heart attacks. Neurologically, carbon monoxide exposure reduces hand to eye coordination, vigilance and continuous

HVAC 242 performance. It can also affect your time discrimination. [6] Ventilation Ventilation is the process of "changing" or replacing air in any space to control temperature or remove any combination of moisture, odors, smoke, heat, dust, airborne bacteria or carbon dioxide, and to replenish oxygen. Ventilation includes both the exchange of air with the outside as well as circulation of air within the building. It is one of the most important factors for maintaining acceptable indoor air quality in buildings. Methods for ventilating a building may be divided into mechanical/forced and natural types.[7] An air handling unit is used for the heating and cooling of air in a central location (click on image for legend). Mechanical or forced ventilation "Mechanical" or "forced" ventilation is provided by an air handler and used to control indoor air quality. Excess humidity, odors, and contaminants can often be controlled via dilution or replacement with outside air. However, in humid climates much energy is required to remove excess moisture from ventilation air. Kitchens and bathrooms typically have mechanical exhausts to control odors and sometimes humidity. Factors in the design of such systems include the flow rate (which is a function of the fan speed and exhaust vent size) and noise level. Direct drive fans are available for many applications, and can reduce maintenance needs. Ceiling fans and table/floor fans circulate air within a room for the purpose of reducing the perceived temperature by increasing evaporation of perspiration on the skin of the occupants. Because hot air rises, ceiling fans may be used to keep a room warmer in the winter by circulating the warm stratified air from the ceiling to the floor. Natural ventilation Natural ventilation is the ventilation of a building with outside air without the use of fans or other mechanical systems. It can be achieved with openable windows or trickle vents when the spaces to ventilate are small and the architecture permits. In more complex systems warm air in the building can be allowed to rise and flow out upper openings to the outside (stack effect) thus forcing cool outside air to be drawn into the building naturally through openings in the lower areas. These systems use very little energy but care must be taken to ensure the occupants comfort. In warm or humid months in many climates maintaining thermal comfort solely via natural ventilation may not be possible so conventional air conditioning systems are used as backups. Air-side economizers perform the same function as natural ventilation, but use mechanical systems fans, ducts, dampers, and control systems to introduce and distribute cool outdoor air when appropriate. An important component of natural ventilation is the concept of air changes per hour. Air changes per hour is a rate used to describe the amount of ventilation moving through an area with respect to the size of the space. AC/hr is used to determine room pressure, whether it is positive or negative. Positive pressure occurs when there is more air being supplied than exhausted and conversely, negative pressure occurs when more air is being exhausted than supplied. When contaminants are being kept out, positive pressure is occurring and when things are being kept in, negative pressure is occurring. [8] To establish AC/hr the room volume must be calculated first. Next, identify all supply and exhaust points in the room, these are where air enters and leaves the space. Determine the area and cross sectional velocity for each supply or exhaust grill in the room. Then, calculate the flow rate for each and sum the flows. Finally, use the calculation

HVAC 243 –AC/hr= Q x60/volume [9] Airborne Illnesses Natural ventilation is a key factor in reducing the spread of airborne illnesses such as tuberculosis, the common cold, influenza and meningitis. Opening doors, windows and using ceiling fans are all ways to maximize natural ventilation and reduce the risk of airborne contagion. Natural ventilation requires no maintenance and is not costly. [10] Air conditioning Air conditioning and refrigeration are provided through the removal of heat. Heat can be removed through radiation, convection, and by heat pump systems through a process called the refrigeration cycle. Refrigeration conduction media such as water, air, ice, and chemicals are referred to as refrigerants. An air conditioning system, or a standalone air conditioner, provides cooling, ventilation, and humidity control for all or part of a house or building. HVAC Ventilation Exhaust for a 12-story The refrigeration cycle uses four essential elements to create a cooling building effect. The system refrigerant starts its cycle in a gaseous state. The compressor pumps the refrigerant gas up to a high pressure and temperature. From there it enters a heat exchanger (sometimes called a "condensing coil" or condenser) where it loses energy (heat) to the outside. In the process the refrigerant condenses into a liquid. The liquid refrigerant is returned indoors to another heat exchanger ("evaporating coil" or evaporator). A metering device allows the liquid to flow in at a low pressure at the proper rate. As the liquid refrigerant evaporates it absorbs energy (heat) from the inside air, returns to the compressor, and repeats the cycle. In the process heat is absorbed from indoors and transferred outdoors, resulting in cooling of the building. In variable climates, the system may include a reversing valve that automatically switches from heating in winter to cooling in summer. By reversing the flow of refrigerant, the heat pump refrigeration cycle is changed from cooling to heating or vice versa. This allows a residence or facility to be heated and cooled by a single piece of equipment, by the same means, and with the same hardware. Central, all-air air conditioning systems (or package systems) with a combined outdoor condenser/evaporator unit are often installed in modern residences, offices, and public buildings, but are difficult to retrofit (install in a building that was not designed to receive it) because of the bulky air ducts required to carry the needed air to heat or cool an area. The duct system must be carefully maintained to prevent the growth of pathogenic bacteria such as legionella in the ducts. An alternative to central systems is the use of separate indoor and outdoor coils in split systems. These systems, although most often seen in residential applications, are gaining popularity in small commercial buildings. The evaporator coil is connected to a remote condenser unit using refrigerant piping between an indoor and outdoor unit instead of ducting air directly from the outdoor unit. Indoor units with directional vents mount onto walls, suspend from ceilings, or fit into the ceiling. Other indoor units mount inside the ceiling cavity, so that short lengths of duct handle air from the indoor unit to vents or diffusers around the room or rooms. Dehumidification in an air conditioning system is provided by the evaporator. Since the evaporator operates at a temperature below dew point, moisture in the air condenses on the evaporator coil tubes. This moisture is collected at the bottom of the evaporator in a pan and removed by piping to a central drain or onto the ground outside. A dehumidifier is an air-conditioner-like device that controls the humidity of a room or building. It is often employed in basements which have a higher relative humidity because of their lower temperature (and propensity for damp

HVAC 244 floors and walls). In food retailing establishments, large open chiller cabinets are highly effective at dehumidifying the internal air. Conversely, a humidifier increases the humidity of a building. Air-conditioned buildings often have sealed windows, because open windows would work against an HVAC system intended to maintain constant indoor air conditions. All modern air conditioning systems, down to small "window" package units, are equipped with internal air filters. These are generally of a lightweight gauzy material, and must be replaced as conditions warrant (some models may be washable). For example, a building in a high-dust environment, or a home with furry pets, will need to have the filters changed more often than buildings without these dirt loads. Failure to replace these filters as needed will contribute to a lower heat-exchange rate, resulting in wasted energy, shortened equipment life, and higher energy bills; low air flow can result in "iced-up" or "iced-over" evaporator coils, which can completely stop air flow. Additionally, very dirty or plugged filters can cause overheating during a heating cycle, and can result in damage to the system or even fire. It is important to keep in mind that because an air conditioner moves heat between the indoor coil and the outdoor coil, both must be kept just as clean. This means that, in addition to replacing the air filter at the evaporator coil, it is also necessary to regularly clean the condenser coil. Failure to keep the condenser clean will eventually result in harm to the compressor, because the condenser coil is responsible for discharging both the indoor heat (as picked up by the evaporator) and the heat generated by the electric motor driving the compressor. Outside, "fresh" air is generally drawn into the system by a vent into the indoor heat exchanger section, creating positive air pressure. The percentage of return air made up of fresh air can usually be manipulated by adjusting the opening of this vent. Energy efficiency For the last 20 to 30 years, manufacturers of HVAC equipment have been making an effort to make the systems they manufacture more efficient. This was originally driven by rising energy costs, and has more recently been driven by increased awareness of environmental issues. In the USA, the EPA has also imposed tighter restrictions. There are several methods for making HVAC systems more efficient. Heating energy Water heating is more efficient for heating buildings and was the standard many years ago. Today forced air systems can double for air conditioning and are more popular. Some benefits of forced air systems, which are now widely used in churches, schools and high-end residences, are • Better air conditioning effects • Energy savings of up to 15-20% • Even conditioning. A drawback is the installation cost, which can be slightly higher than traditional HVAC systems. Energy efficiency can be improved even more in central heating systems by introducing zoned heating. This allows a more granular application of heat, similar to non-central heating systems. Zones are controlled by multiple thermostats. In water heating systems the thermostats control zone valves, and in forced air systems they control zone dampers inside the vents which selectively block the flow of air. In this case, the control system is very critical to maintaining a proper temperature.

HVAC 245 Geothermal Heat Pump Geothermal heat pumps are similar to ordinary heat pumps, but instead of using heat found in outside air, they rely on the stable, even heat of the earth to provide heating, air conditioning and, in most cases, hot water. The heat extracted through a geothermal heat pump can come from any source, despite the temperature. However, the warmer the source of heat, the more energy efficient it will be. [11] From Montanas −70 °F (−57 °C) temperature to the highest temperature ever recorded in the U.S.—134 °F (56.7 °C) in Death Valley, California, in 1913—many parts of the country experience seasonal temperature extremes. A few feet below the earths surface, however, the ground remains at a relatively constant temperature. Although the temperatures vary according to latitude, at 6 feet (1.83 m) underground, temperatures only range from 45 to 75 °F (7.2 to 23.9 °C). While they may be more costly to install than regular heat pumps, they can produce markedly lower energy bills—30 to 40 percent lower, according to estimates from the U.S. Environmental Protection Agency. Ventilation energy recovery Energy recovery systems sometimes utilize heat recovery ventilation or energy recovery ventilation systems that employ heat exchangers or enthalpy wheels to recover sensible or latent heat from exhausted air. This is done by transfer of energy to the incoming outside fresh air. Air conditioning energy The performance of vapor compression refrigeration cycles is limited by thermodynamics. These air conditioning and heat pump devices move heat rather than convert it from one form to another, so thermal efficiencies do not appropriately describe the performance of these devices. The Coefficient-of-Performance (COP) measures performance, but this dimensionless measure has not been adopted, but rather the Energy Efficiency Ratio (EER). EER is the Energy Efficiency Ratio based on a 35 °C (95 °F) outdoor temperature. To more accurately describe the performance of air conditioning equipment over a typical cooling season a modified version of the EER is used, the Seasonal Energy Efficiency Ratio (SEER), or in Europe the ESEER. SEER ratings are based on seasonal temperature averages instead of a constant 35 °C outdoor temperature. The current industry minimum SEER rating is 13 SEER. Engineers have pointed out some areas where efficiency of the existing hardware could be improved. For example, the fan blades used to move the air are usually stamped from sheet metal, an economical method of manufacture, but as a result they are not aerodynamically efficient. A well-designed blade could reduce electrical power required to move the air by a third.[12] Air Filtration and Cleaning Air cleaning and filtration is an important factor of our indoor environment because cleaning the air filters out what the lungs cannot by removing particles, contaminants, vapors and gases from the air. The filtered and cleaned air then is used in heating, ventilation and air conditioning. Air cleaning and filtration should be taken in account when protecting our building environments.[13] Clean Air Delivery Rate and Filter Performance Clean air delivery rate is the amount of clean air an air cleaner provides to a room or space. When determining CADR, the amount of airflow in a space is taken into account. For example, an air cleaner with a flow rate of 100 cfm (cubic feet per minute) and an efficiency of 50% has a CADR of 50 cfm. Along with CADR, filtration performance is very important when it comes to the air in our indoor environment. Filter performance depends on the size of the particle or fiber, the filter packing density and depth and also the air flow rate.[14] http://www.cdc.gov/niosh/docs/2003-136/2003-136b.html[15]

HVAC 246 HVAC industry and standards International ISO16813:2006 is one of the ISO building environment standards.[16] It establishes the general principles of building environment design. It takes into account the need to provide a healthy indoor environment for the occupants as well as the need to protect the environment for future generations and promote collaboration among the various parties involved in building environmental design for sustainability. ISO16813 is applicable to new construction and the retrofit of existing buildings.[17] The building environmental design standard aims to[17]: • provide the constraints concerning sustainability issues from the initial stage of the design process, with building and plant life cycle to be considered together with owning and operating costs from the beginning of the design process; • assess the proposed design with rational criteria for indoor air quality, thermal comfort, acoustical comfort, visual comfort, energy efficiency and HVAC system controls at every stage of the design process; • iterate decisions and evaluations of the design throughout the design process. North America USA In the United States, HVAC engineers generally are members of the American Society of Heating, Refrigerating, and Air-Conditioning Engineers (ASHRAE), EPA Universal CFC certified, or locally engineer certified such as a Special to Chief Boilers License issued by the state or, in some jurisdictions, the city. ASHRAE is an international technical society for all individuals and organizations interested in HVAC. The Society, organized into regions, chapters, and student branches, allows exchange of HVAC knowledge and experiences for the benefit of the fields practitioners and the public. ASHRAE provides many opportunities to participate in the development of new knowledge via, for example, research and its many technical committees. These committees typically meet twice per year at the ASHRAE Annual and Winter Meetings. A popular product show, the AHR Expo, is held in conjunction with each winter meeting. The Society has approximately 50,000 members and has headquarters in Atlanta, Georgia, USA. The most recognized standards for HVAC design are based on ASHRAE data. The most general of four volumes of the ASHRAE Handbook is Fundamentals; it includes heating and cooling calculations. Each volume of the ASHRAE Handbook is updated every four years. The design professional must consult ASHRAE data for the standards of design and care as the typical building codes provide little to no information on HVAC design practices; codes such as the UMC and IMC do include much detail on installation requirements, however. Other useful reference materials include items from SMACNA, ACCA, and technical trade journals. American design standards are legislated in the Uniform Mechanical Code or International Mechanical Code. In certain states, counties, or cities, either of these codes may be adopted and amended via various legislative processes. These codes are updated and published by the International Association of Plumbing and Mechanical Officials (IAPMO) or the International Code Council (ICC) respectively, on a 3-year code development cycle. Typically, local building permit departments are charged with enforcement of these standards on private and certain public properties. In the United States, as well as throughout the world, HVAC contractors and companies are members of NADCA, the National Air Duct Cleaners Association. NADCA was formed in 1989 as a non-profit association of companies engaged in the cleaning of HVAC systems. Its mission was to promote source removal as the only acceptable method of cleaning and to establish industry standards for the association. NADCA has expanded its mission to include the representation of qualified companies engaged in the assessment, cleaning, and restoration of HVAC

HVAC 247 systems, and to assist its members in providing high quality service to their customers. The goal of the association is to be the number one source for the HVAC cleaning and restoration services. NADCA has experienced large membership growth and has been extremely successful with the training and certification of air systems cleaning specialists, mold remediators, and HVAC inspectors. The association has also published important standards and guidelines, educational materials, and other useful information for the consumers and members of NADCA. Their headquarters are located in Washington, D.C. Europe United Kingdom The Chartered Institution of Building Services Engineers is a body that covers the essential Service (systems architecture) that allow buildings to operate. It includes the electrotechnical, heating, ventilating, air conditioning, refrigeration and plumbing industries. To train as a building services engineer, the academic requirements are GCSEs (A-C) / Standard Grades (1-3) in Maths and Science, which are important in measurements, planning and theory. Employers will often want a degree in a branch of engineering, such as building environment engineering, electrical engineering or mechanical engineering. To become a full member of CIBSE, and so also to be registered by the Engineering Council UK as a chartered engineer, engineers must also attain an Honours Degree and a Masters Degree in a relevant engineering subject. CIBSE publishes several guides to HVAC design relevant to the UK market, and also the Republic of Ireland, Australia, New Zealand and Hong Kong. These guides include various recommended design criteria and standards, some of which are cited within the UK building regulations, and therefore form a legislative requirement for major building services works. The main guides are: • Guide A: Environmental Design • Guide B: Heating, Ventilating, Air Conditioning and Refrigeration • Guide C: Reference Data • Guide D: Transportation systems in Buildings • Guide E: Fire Safety Engineering • Guide F: Energy Efficiency in Buildings • Guide G: Public Health Engineering • Guide H: Building Control Systems • Guide J: Weather, Solar and Illuminance Data • Guide K: Electricity in Buildings • Guide L: Sustainability • Guide M: Maintenance Engineering and Management Within the construction sector, it is the job of the building services engineer to design and oversee the installation and maintenance of the essential services such as gas, electricity, water, heating and lighting, as well as many others. These all help to make buildings comfortable and healthy places to live and work in. Building Services is part of a sector that has over 51,000 businesses and employs represents 2%-3% of the GDP.

HVAC 248 Australia The Air Conditioning and Mechanical Contractors Association of Australia (AMCA), Australian Institute of Refrigeration, Air Conditioning and Heating (AIRAH), and CIBSE are responsible. Philippines The Philippine Society of Ventilating, Air Conditioning and Refrigerating Engineers (PSVARE) along with Philippine Society of Mechanical Engineers (PSME) govern on the codes and standards for HVAC / MVAC in the Philippines. India The Indian Society of Heating, Refrigerating and Air Conditioning Engineers (ISHRAE) was established to promote the HVAC industry in India. ISHRAE is an associate of ASHRAE. ISHRAE was started at Delhi in 1981 and a chapter was started in Bangalore in 1989. Between 1989 & 1993, ISHRAE chapters were formed in all major cities in India and also in the Middle East. References [1] Swenson, S. Don (1995). HVAC: heating, ventilating, and air conditioning. Homewood, Ill.: American Technical Publishers. ISBN 978-0-8269-0675-5. [2] Designers Guide to Ceiling-Based Air Diffusion, Rock and Zhu, ASHRAE, Inc., New York, GA, USA, 2002 [3] Ierley, M (1999). The Comforts of Home: The American House and the Evolution of Modern Convenience. New York: Clarkson Potter Publishers. pp. 186–200. [4] "Hypocaust" (http:/ / www. britannica. com/ EBchecked/ topic/ 279869/ hypocaust). Encyclopedic. Britannica Online. 2009. . Retrieved 2009-01-29. [5] Bearg, David W (1993). Indoor Air Quality and HVAC Systems. New York: Lewis Publishers. pp. 107–112. [6] Dianat, Nazari, I,I. "Characteristic of unintentional carbon monoxide poisoning in Northwest Iran- Tabriz" (http:/ / www. pubmed. com). International Journal of Injury Control and Promotion. . Retrieved 11/15/2011. [7] Ventilation and Infiltration chapter, Fundamentals volume of the ASHRAE Handbook, ASHRAE, Inc., Atlanta, GA, 2005 [8] Bell, Geoffrey. "Room Air Change Rate" (http:/ / ateam. lbl. gov/ Design-Guide/ DGHtm/ roomairchangerates. htm). A Design Guide for Energy-Efficient Research Laboratories. . Retrieved 11/15/2011. [9] None, None. "Open Channel Flow- Velocity Flow" (http:/ / www. flowtechnology. co. uk/ freevelocityarea. htm). Davenport Flowtechnology Simple Calculations. . Retrieved 11/15/2011. [10] Escombe, AR; Oeser CC, Gilman RH, et al. (2007). "Natural ventilation for the prevention of airborne contagion". PLoS Med 4 (68). [11] Egg, J (2011). Geothermal HVAC: Heating and Cooling. United States of America: McGraw-Hill Companies. pp. 45–62. [12] Keeping cool and green, The Economist 17 July 2010, p. 83 [13] Howard, J (2003). "Guidance for Filtration and Air-Cleaning Systems to Protect Building Environments". Department of Health and Human Services: Centers for Disease Control and Prevention, National Institute for Occupational Safety and Health 2003 (136): 1–63. [14] Howard, J (2003). "Guidance for Filtration and Air-Cleaning Systems to Protect Building Environments". Department of Health and Human Services: Centers for Disease Control and Prevention, National Institute for Occupational Safety and Health 2003 (136): 1–63. [15] Howard, J (2003). "Guidance for Filtration and Air-Cleaning Systems to Protect Building Environments". Department of Health and Human Services: Centers for Disease Control and Prevention, National Institute for Occupational Safety and Health 2003 (136): 1–63. [16] ISO. "Building environment standards" (http:/ / www. iso. org/ iso/ iso_catalogue/ catalogue_tc/ catalogue_tc_browse. htm?commid=54740). www.iso.org. . Retrieved 2011-05-14. [17] ISO. "Building environment design -- Indoor environment -- General principles" (http:/ / www. iso. org/ iso/ iso_catalogue/ catalogue_tc/ catalogue_detail. htm?csnumber=41300). . Retrieved 14 May 2011.

HVAC 249 Further reading • International Mechanical Code (March 6, 2006) by the International Code Council, Thomson Delmar Learning; 1st edition • Modern Refrigeration and Air Conditioning (August 2003) by Althouse, Turnquist, and Bracciano, Goodheart-Wilcox Publisher; 18th edition Interfacial thermal resistance Interfacial thermal resistance, also known as thermal boundary resistance, or Kapitza resistance, is a measure of an interfaces resistance to thermal flow. This thermal resistance differs from contact resistance, as it exists even at atomically perfect interfaces. Due to the differences in electronic and vibrational properties in different materials, when an energy carrier (phonon or electron, depending on the material) attempts to traverse the interface, it will scatter at the interface. The probability of transmission after scattering will depend on the available energy states on side 1 and side 2 of the interface. Assuming a constant thermal flux is applied across an interface, this interfacial thermal resistance will lead to a finite temperature discontinuity at the interface. From an extension of Fouriers law, we can write where is the applied flux, is the observed temperature drop, is the thermal boundary resistance, and is its inverse, or thermal boundary conductance. Understanding the thermal resistance at the interface between two materials is of primary significance in the study of its thermal properties. Interfaces often contribute significantly to the observed properties of the materials. This is even more critical for nanoscale systems where interfaces could significantly affect the properties relative to bulk materials. Low thermal resistance at interfaces is technologically important for applications where very high heat dissipation is necessary. This is of particular concern to the development of microelectronic semiconductor devices as defined by the International Technology Roadmap for Semiconductors in 2004 where an 8 nm feature size device is projected to generate up to 100000 W/cm3 and would need efficient heat dissipation of an anticipated die level heat flux of 1000 W/cm3 which is an order of magnitude higher than current devices.[1] On the other hand, applications requiring good thermal isolation such as jet engine turbines would benefit from interfaces with high thermal resistance. This would also require material interfaces which are stable at very high temperature. Examples are metal-ceramic composites which are currently used for these applications. High thermal resistance can also be achieved with multilayer systems. As stated above, thermal boundary resistance is due to carrier scattering at an interface. The type of carrier scattered will depend on the materials governing the interfaces. For example, at a metal-metal interface, electron scattering effects will dominate thermal boundary resistance, as electrons are the primary thermal energy carriers in metals. Two widely used predictive models are the acoustic mismatch model (AMM) and the diffuse mismatch model (DMM). The AMM assumes a geometrically perfect interface and phonon transport across it is entirely elastic, treating phonons as waves in a continuum. On the other hand, the DMM assumes scattering at the interface is diffusive, which is accurate for interfaces with characteristic roughness at elevated temperatures.

Interfacial thermal resistance 250 Theoretical Models There are two primary models that are used to understand the thermal resistance of interfaces, the acoustic mismatch and diffuse mismatch models (AMM and DMM respectively). Both models are based only on phonon transport, ignoring electrical contributions. Thus it should apply for interfaces where at least one of the materials is electrically insulating. For both models the interface is assumed to behave exactly as the bulk on either side of the interface (e.g. bulk phonon dispersions, velocities, etc.). The thermal resistance then results from the transfer of phonons across the interface. Energy is transferred when higher energy phonons which exist in higher density in the hotter material propagate to the cooler materials, which in turn transmits lower energy phonons, creating a net energy flux.[2] A crucial factor in determining the thermal resistance at an interface is the overlap of phonon states. Given two materials, A and B, if material A has a low population (or no population) of phonons with certain k value, there will be very few phonons of that wavevector to propagate from A to B. Further, due to the detailed balance, very few phonons of that wavevector will propagate the opposite direction, from B to A, even if material B has a large population of phonons with that wavevector. Thus as the overlap between phonon dispersions is small, there are less modes to allow for heat transfer in the material, giving at a high thermal interfacial resistance relative to materials with a high degree of overlap.[3] Both AMM and DMM reflect this principle, but differ in the conditions they require for propagation across the interface. Neither model is universally effective for predicting the thermal interface resistance (with the exception of very low temperature), but rather for most materials they act as upper and lower limits for real behavior. Both models differ greatly in their treatment of scattering at the interface. In AMM the interface is assumed to be perfect, resulting in no scattering, thus phonons propagate elastically across the interface. The wavevectors that propagate across the interface are determined by conservation of momentum. In DMM, the opposite extreme is assumed, a perfectly scattering interface. In this case the wavevectors that propagate across the interface are random and independent of incident phonons on the interface. For both models the detailed balance must still be obeyed. For both models some basic equations apply. The flux of energy from one material to the other is just: where n is the number of phonons at a given wavevector and momentum, E is the energy, and α is the probability of transmission across the interface. The net flux is thus the difference of the energy fluxes: Since both fluxes are dependent on T1 and T2, the relationship between the flux and the temperature difference can be used to determine the thermal interface resistance based on: where A is the area of the interface. These basic equations form the basis for both models. n is determined based on the Debye model and Bose-Einstein statistics. Energy is given simply by: where ν is the speed of sound in the material. The main difference between the two models is the transmission probability, whose determination is more complicated. In each case it is determined by the basic assumptions that form the respective models. The assumption of elastic scattering makes it more difficult for phonons to transmit across the interface, resulting in lower probabilities. As a result, the acoustic mismatch model typically represents an upper limit for thermal interface resistance, while the diffuse mismatch model represents the lower limit.[4]

Interfacial thermal resistance 251 Examples Liquid Helium Interfaces The presence of thermal interface resistance, corresponding to a discontinuous temperature across an interface was first proposed from studies of liquid helium in 1936. While this idea was first proposed in 1936,[3] it wasn’t until 1941 when Kapitza carried out the first systematic study of thermal interface behavior in liquid helium.[6] The first major model for heat transfer at interfaces was the acoustic mismatch model which predicted a T−3 temperature dependence on the interfacial resistance, but this failed to properly model the thermal conductance of helium interfaces by as much as two orders of Typical Interfacial Resistance of Liquid Helium with metals. Resistance has been magnitude. Another surprising behavior of [5] multiplied by T3 to remove the expected T-3 dependence. Adapted from the thermal resistance was observed in the pressure dependence. Since the speed of sound is a strong function of temperature in liquid helium, the acoustic mismatch model predicts a strong pressure dependence of the interfacial resistance. Studies around 1960 surprising showed that the interfacial resistance was nearly independent of pressure, suggesting that other mechanisms were dominant. The acoustic mismatch theory predicted a very high thermal resistance (low thermal conductance) at solid-helium interfaces. This was potentially disastrous to researchers working at ultra-cold temperatures because it greatly impedes cooling rates at low temperatures. Fortunately such a large thermal resistance was not observed due to many mechanisms which promoted phonon transport. In liquid helium, Van der Waals forces actually work to solidify the first few monolayers against a solid. This boundary layer functions much like an anti-reflection coating in optics, so that phonons which would typically be reflected from the interface actually would transmit across the interface. This also helps to understand the pressure independence of the thermal conductance. The final dominant mechanism to anomalously low thermal resistance of liquid helium interfaces is the effect of surface roughness, which is not accounted for in the acoustic mismatch model. For a more detailed theoretical model of this aspect see the paper by A. Khater and J. Szeftel.[7]. Like electromagnetic waves which produce surface plasmons on rough surfaces, phonons can also induce surface waves. When these waves eventually scatter, they provide another mechanism for heat to transfer across the interface. Similarly, phonons are also capable of producing evanescent waves in a total internal reflection geometry. As a result, when these waves are scattered in the solid, additional heat is transferred from the helium beyond the prediction of the acoustic mismatch theory. For a more complete review on this topic see the review by Swartz.[8] Notable Room Temperature Thermal Conductance In general there are two types of heat carriers in materials: phonons and electrons. The free electron gas found in metals is a very good conductor of heat and dominates thermal conductivity. All materials though exhibit heat transfer by phonon transport so heat flows even in dielectric materials such as silica. Interfacial thermal conductance is a measure of how efficiently heat carriers flow from one material to another. The lowest room temperature thermal conductance measurement to date is the Bi/Hydrogen-terminated diamond with a thermal conductance of 8.5 MW

Interfacial thermal resistance 252 m−2 K−1. As a metal, bismuth contains many electrons which serve as the primary heat carriers. Diamond on the other hand is a very good electrical insulator (although it has a very high thermal conductivity) and so electron transport between the materials is minimized. Further, these materials have very different lattice parameters so phonons do not efficiently couple across the interface. Finally, the Debye temperature between the materials is significantly different. As a result, bismuth, which has a low Debye temperature, has many phonons at low frequencies. Diamond on the other hand has a very high Debye temperature and most of its heat-carrying phonons are at frequencies much higher than are present in bismuth.[9] Increasing in thermal conductance, most phonon mediated interfaces (dielectric-dielectric and metal-dielectric) have thermal conductances between 80 and 300 MW m−2 K−1. The largest phonon mediated thermal conductance measured to date is between TiN (Titanium Nitride) and MgO. These systems have very similar lattice structures and Debye temperatures. While there are no free electrons to enhance the thermal conductance of the interface, the similar physical properties of the two crystals facilitate a very efficient phonon transmission between the two materials.[6] [8][9][10] At the highest end of the spectrum, one of Thermal Conductance Data adapted from, the highest thermal conductances measured is between aluminum and copper. At room temperature, the Al-Cu interface has a conductance of 4 GW m−2 K−1. The high thermal conductance of the interface should not be unexpected given the high electrical conductivity of both materials.[11] Interfacial Resistance in Carbon Nanotubes The superior thermal conductivity of Carbon nanotubes makes it an excellent candidate for making composite materials. But interfacial resistance impacts the effective thermal conductivity. This area is not well studied and only a few studies have been done to understand the basic mechanism of this resistance.[12] References [1] Hu, M., Keblinski, P., Wang, JS., and Raravikar, N., Journal of Applied Physics 104 (2008) [2] Swartz, E.T, Solid-solid Boundary Resistance, PhD Dissertation, Cornell University 1987 [3] Swartz, E.T., Pohl, R.O., Rev. Mod. Phys. 61 605 (1989) [4] Zeng, T., and Chen, G., Transactions of the ASME, 123, (2001) [5] Swartz, E.T., Pohl, R.O., Rev. Mod. Phys. 61 605 (1989). [6] Kapitza, P.L., J. Phys (USSR) 4 (1941) [7] A. Khater and J. Szeftel, Phys. Rev. B 35, 6749 (1987) [8] H. Lyeo, D. G. Cahill, Phys. Rev. B. 73 144301 (2006) [9] R. M. Costescu, M. A. Wall, D. G. Cahill, Phys Rev. B. 67 054302 (2003) [10] B. C. Grundum, D. G. Cahill, R. S. Averback, Phys Rev B. 72 245426 (2005) [11] B. C. Grundum, D. G. Cahill, R. S. Averback, Phys Rev B. 72 245426 (2005) [12] Zhong, H. and Lukes, J.R., Phys. Rev. B,vol. 74, 125403 (2006)

Kirchhoffs law of thermal radiation 253 Kirchhoffs law of thermal radiation See also Kirchhoffs laws for other laws named after Kirchhoff. In thermodynamics, Kirchhoffs law of thermal radiation refers to wavelength-specific radiative emission and absorption by a material body in thermodynamic equilibrium, including radiative exchange equilibrium. A body at temperature T radiates electromagnetic energy. A perfect black body in thermodynamic equilibrium absorbs all light that strikes it, and radiates energy according to a unique law of radiative emissive power for temperature T, universal for all perfect black bodies. Kirchhoffs law states that: For a body of any arbitrary material, emitting and absorbing thermal electromagnetic radiation at every wavelength in thermodynamic equilibrium, the ratio of its emissive power to its dimensionless coefficient of absorption is equal to a universal function only of radiative wavelength and temperature, the perfect black-body emissive power.[1][2][3][4][5][6] Here, the dimensionless coefficient of absorption (or the absorptivity) is the fraction of incident light (power) that is absorbed by the body when it is radiating and absorbing in thermodynamic equilibrium. In slightly different terms, the emissive power of an arbitrary opaque body of fixed size and shape at a definite temperature can be described by a dimensionless ratio, sometimes called the emissivity, the ratio of the emissive power of the body to the emissive power of a black body of the same size and shape at the same fixed temperature. With this definition, a corollary of Kirchhoffs law is that for an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity. In some cases, emissive power and absorptivity may be defined to depend on angle, as described below. Kirchhoffs Law has another corollary: the emissivity cannot exceed one (because the absorptivity cannot, by conservation of energy), so it is not possible to thermally radiate more energy than a black body, at equilibrium. In negative luminescence the angle and wavelength integrated absorption exceeds the materials emission, however, such systems are powered by an external source and are therefore not in thermodynamic equilibrium. Before Kirchhoffs law was recognized, it had been experimentally established that a good absorber is a good emitter, and a poor absorber is a poor emitter. Naturally, a good reflector must be a poor absorber. This is why, for example, lightweight emergency thermal blankets are based on reflective metallic coatings: they lose little heat by radiation. Theory In a blackbody enclosure that contains electromagnetic radiation with a certain amount of energy at thermodynamic equilibrium, this "photon gas" will have a Planck distribution of energies.[7] One may suppose a second system, a cavity with walls that are opaque, rigid, and not perfectly reflective to any wavelength, to be brought into connection, through an optical filter, with the blackbody enclosure, both at the same temperature. Radiation can pass from one system to the other. For example, suppose in the second system, the density of photons at narrow frequency band around wavelength were higher than that of a the first system. If the optical filter passed only that frequency band, then there would be a net transfer of photons, and their energy, from the second system to the first. This is in violation of the second law of thermodynamics, which requires that there can be no net transfer of heat between two bodies at the same temperature. In the second system, therefore, at each frequency, the walls must absorb and emit energy in such a way as to maintain the black body distribution.[8] The absorptivity is the ratio of the energy absorbed by the wall to the energy incident on the wall, for a particular wavelength. This will be proportional to where is the intensity of black body radiation at wavelength and temperature . The emissivity of the wall is defined as the ratio of emitted energy to the amount that would be radiated if the wall were a perfect black body. That will be where is the emissivity at wavelength . These two quantities must be equal,

Kirchhoffs law of thermal radiation 254 or else the distribution of photon energies in the cavity will deviate from that of a black body. This yields Kirchoffs law: By a similar, but more complicated argument, it can be shown that, since black body radiation is equal in every direction (isotropic), the emissivity and the absorptivity, if they happen to be dependent on direction, must again be equal for any given direction.[8] Average and overall absorptivity and emissivity data are often given for materials with values which differ from each other. For example, white paint is quoted as having an absorptivity of 0.16, while having an emissivity of 0.93.[9] This is because the absorptivity is averaged with weighting for the solar spectrum, while the emissivity is weighted for the emission of the paint itself at normal ambient temperatures. The absorptivity quoted in such cases is being calculated by: while the average emissivity is given by: Where is the emission spectrum of the sun, and is the emission spectrum of the paint. Although, by Kirchoffs law, in the above equations, the above averages and are not generally equal to each other. The white paint will serve as a very good insulator against solar radiation, because it is very reflective of the solar radiation, and although it therefore emits poorly in the solar band, its temperature will be around room temperature, and it will emit whatever radiation it has absorbed in the infrared, where its emission coefficient is high. Black bodies Near-black materials It has long been known that a lamp-black coating will make a body nearly black. Some other materials are nearly black in particular wavelength bands. Such materials do not survive all the very high temperatures that are of interest. An improvement on lamp-black is found in manufactured carbon nanotubes. Nano-porous materials can achieve refractive indices nearly that of vacuum, in one case obtaining average reflectance of 0.045%.[10][11] Opaque bodies Bodies that are opaque to thermal radiation that falls on them are valuable in the study of heat radiation. Planck analyzed such bodies with the approximation that they be considered topologically to have an interior and to share an interface. They share the interface with their contiguous medium, which may be rarefied material such as air, or transparent material, through which observations can be made. The interface is not a material body and can neither emit nor absorb. It is a mathematical surface belonging jointly to the two media that touch it. It is the site of refraction of radiation that penetrates it and of reflection of radiation that does not. As such it obeys the Helmholtz reciprocity principle. The opaque body is considered to have a material interior that absorbs all and scatters or transmits none of the radiation that reaches it through refraction at the interface. In this sense the material of the opaque body is black to radiation that reaches it, while the whole phenomenon, including the interior and the interface, does not show perfect blackness. In Plancks model, perfectly black bodies, which he noted do not exist in nature, besides their opaque interior, have interfaces that are perfectly transmitting and non-reflective.[2]

Kirchhoffs law of thermal radiation 255 Cavity radiation The walls of a cavity can be made of opaque materials that absorb significant amounts of radiation at all wavelengths. It is not necessary that every part of the interior walls be a good absorber at every wavelength. The effective range of absorbing wavelengths can be extended by the use of patches of several differently absorbing materials in parts of the interior walls of the cavity. In thermodynamic equilibrium the cavity radiation will precisely obey Plancks law. In this sense, thermodynamic equilibrium cavity radiation may be regarded as thermodynamic equilibrium black-body radiation to which Kirchhoffs law applies exactly, though no perfectly black body in Kirchhoffs sense is present. A theoretical model considered by Planck consists of a cavity with perfectly reflecting walls, initially with no material contents, into which is then put a small piece of carbon. Without the small piece of carbon, there is no way for non-equilibrium radiation initially in the cavity to drift towards thermodynamic equilibrium. When the small piece of carbon is put in, it transduces amongst radiation frequencies so that the cavity radiation comes to thermodynamic equilibrium.[2] A hole in the wall of a cavity For experimental purposes, a hole in a cavity can be devised to provide a good approximation to a black surface, but will not be perfectly Lambertian, and must be viewed from nearly right angles to get the best properties. The construction of such devices was an important step in the empirical measurements that led to the precise mathematical identification of Kirchhoffs universal function, now known as Plancks law. Kirchhoffs perfect black bodies Planck also noted that the perfect black bodies of Kirchhoff do not occur in physical reality. They are theoretical fictions. Kirchhoffs perfect black bodies absorb all the radiation that falls on them, right in an infinitely thin surface layer, with no reflection and no scattering. They emit radiation in perfect accord with Lamberts cosine law.[1][2] Original statements Gustav Kirchhoff stated his law in several papers in 1859 and 1860, and then in 1862 in an appendix to his collected reprints of those and some related papers.[12] Prior to Kirchhoffs studies, it was known that for total heat radiation, the ratio of emissive power to absorptive ratio was the same for all bodies emitting and absorbing thermal radiation in thermodynamic equilibrium. This means that a good absorber is a good emitter. Naturally, a good reflector is a poor absorber. For wavelength specificity, prior to Kirchhoff, the ratio was shown experimentally by Balfour Stewart to be the same for all bodies, but the universal value of the ratio had not been explicitly considered in its own right as a function of wavelength and temperature. Kirchhoffs original contribution to the physics of thermal radiation was his postulate of a perfect black body radiating and absorbing thermal radiation in an enclosure opaque to thermal radiation and with walls that absorb at all wavelengths. Kirchhoffs perfect black body absorbs all the radiation that falls upon it. Every such black body emits from its surface with a spectral radiance that Kirchhoff labeled I (for specific intensity, the traditional name for spectral radiance). Kirchhoffs postulated spectral radiance I was a universal function, one and the same for all black bodies, only of wavelength and temperature. The precise mathematical expression for that universal function I was very much unknown to Kirchhoff, and it was just postulated to exist, until its precise mathematical expression was found in 1900 by Max Planck. It is nowadays referred to as Plancks law.

Kirchhoffs law of thermal radiation 256 Then, at each wavelength, for thermodynamic equilibrium in an enclosure, opaque to heat rays, with walls that absorb some radiation at every wavelength: For an arbitrary body radiating and emitting thermal radiation, the ratio E / A between the emissive spectral radiance, E, and the dimensionless absorptive ratio, A, is one and the same for all bodies at a given temperature. That ratio E / A is equal to the emissive spectral radiance I of a perfect black body, a universal function only of wavelength and temperature. Cited references [1] Kirchhoff 1860 [2] Planck 1914 [3] Milne 1930, p. 80 [4] Chandrasekhar 1950, p. 8 [5] Mihalas & Weibel-Mihalas 1984, p. 328 [6] Goody & Yung 1989, pp. 27–28 [7] Rybicki & Lightman, pp. 15–20. [8] Rybicki, George B. and Lightman, Alan P., "Radiative Processes in Astrophysics" (John Wiley and Sons, New York, 1979) [9] http:/ / www. solarmirror. com/ fom/ fom-serve/ cache/ 43. html [10] Ai Lin Chun, " Carbon nanotubes: Blacker than black (http:/ / www. nature. com/ nnano/ reshigh/ 2008/ 0108/ full/ nnano. 2008. 29. html)," Nature Nanotechnology (25 Jan 2008), doi:10.1038/nnano.2008.29. [11] Z.-P. Yang et al., " Experimental Observation of an Extremely Dark Material Made By a Low-Density Nanotube Array (http:/ / pubs. acs. org/ doi/ abs/ 10. 1021/ nl072369t)," Nano Letters vol. 8, pp. 446-451 (2008). [12] Kirchhoff, G. (1862). Appendix, Über das Verhältniß zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht, to Untersuchungen über das Sonnenspectrum und die Spectren der chemischen Elemente, Ferd. Dümmlers Verlagsbuchhandlung, Berlin, pages 22-39. Reprinted with the same title in Kangro, H. (1972), Otto Zeller Verlag, Osnabrück, ISBN 3–535–00820–4, pages 45–64. Bibliography • Chandrasekhar, S. (1960) [1950]. Radiative Transfer (Revised reprint ed.). Dover Publications. ISBN 978-0-486-60590-6. • Goody, R. M.; Yung, Y. L. (1989). Atmospheric Radiation: Theoretical Basis (2nd ed.). Oxford University Press. ISBN 978-0-19-510291-8. • Kirchhoff, G. (1860). "Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht". Annalen der Physik und Chemie (Leipzig) 109: 275–301. Translated by Guthrie, F. as Kirchhoff, G. (1860). "On the relation between the radiating and absorbing powers of different bodies for light and heat". Philosophical Magazine Series 4, volume 20: 1–21. • Mihalas, D.; Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics. Oxford University Press. ISBN 0-19-503437-6. • Milne, E.A. (1930). "Thermodynamics of the Stars". Handbuch der Astrophysik 3, part 1: 63–255. • Planck, M. (1914). The Theory of Heat Radiation. Masius, M. (transl.) (2nd ed.). P. Blakistons Son & Co. OL{{{1}}}. General references • Evgeny Lifshitz and L. P. Pitaevskii, Statistical Physics: Part 2, 3rd edition (Elsevier, 1980). • F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill: Boston, 1965).

Laser cooling 257 Laser cooling Laser cooling refers to a number of techniques in which atomic and molecular samples are cooled down to near absolute zero through the interaction with one or more laser light fields. The first example of laser cooling, and also still the most common method (so much so that it is still often referred to simply as laser cooling) is Doppler cooling. Other methods of laser cooling include: • Sisyphus cooling • Resolved sideband cooling • Velocity selective coherent population trapping (VSCPT) • Anti-Stokes inelastic light scattering (typically in the form of fluorescence or Raman scattering) • Cavity mediated cooling • Sympathetic cooling • Use of a Zeeman slower Doppler cooling Doppler cooling, which is usually accompanied by a magnetic trapping force to give a magneto-optical trap, is by far the most common method of laser cooling. It is used to cool low density gasses down to the Doppler cooling limit, which for Rubidium 85 is around 150 microkelvin. As Doppler cooling requires a very particular energy level structure, known as a closed optical loop, the method is limited to a small handful of elements. In Doppler cooling, the frequency of light is tuned slightly below an electronic transition in the atom. Because the light is detuned to the "red" (i.e. at lower frequency) of the transition, the atoms will absorb more photons if they move towards the light source, due to the Doppler effect. Thus if one applies light from two opposite directions, the atoms will always scatter more photons from the laser beam pointing opposite to their direction of motion. In each scattering event the atom loses a momentum equal to the momentum of the photon. If the atom, which is now in the excited state, then emits a photon spontaneously, it will be kicked by the same amount of momentum, but in a random direction. Since the initial momentum loss was opposite to the direction of motion, while the subsequent momentum gain was in a random direction, the overall result of the absorption and emission process is to reduce the speed of the atom (provided its initial speed was larger than the recoil speed from scattering a single photon). If the absorption and emission are repeated many times, the average speed, and therefore the kinetic energy of the atom will be reduced. Since the temperature of a group of atoms is a measure of the average random internal kinetic energy, this is equivalent to cooling the atoms. Other methods of laser cooling Several somewhat similar processes are also referred to as laser cooling, in which photons are used to pump heat away from a material and thus cool it. The phenomenon has been demonstrated via anti-Stokes fluorescence, and both electroluminescent upconversion and photoluminescent upconversion have been studied as means to achieve the same effects. In many of these, the coherence of the laser light is not essential to the process, but lasers are typically used to achieve a high irradiance.

Laser cooling 258 Uses Laser cooling is primarily used for experiments in Quantum Physics to achieve temperatures of near absolute zero (−273.15°C, −459.67°F). This is done to observe the unique quantum effects that can only occur at this heat level. Generally, laser cooling has been only used on the atomic level to cool down elements. This may soon change, as a new breakthrough in the technology has successfully cooled a macro-scale object to near absolute zero.[1] References [1] Massachusetts Institute of Technology (2007, April 8). Laser-cooling Brings Large Object Near Absolute Zero. ScienceDaily. Retrieved January 14, 2011, from http:/ / www. sciencedaily. com/ releases/ 2007/ 04/ 070406171036. htm 2^D.J. Wineland, R.E. Drullinger and F.L. Walls (1978). "Radiation-pressure cooling of bound resonant absorbers" (http:/ / tf. nist. gov/ general/ pdf/ 128. pdf). Phys. Rev. Lett. 40 (25): 1639. Bibcode 1978PhRvL..40.1639W. doi:10.1103/PhysRevLett.40.1639. 3^W. Neuhauser, M. Hohenstatt, P. Toschek and H. Dehmelt (1978). "Optical-sideband cooling of visible atom cloud confined in parabolic well". Phys. Rev. Lett. 41 (4): 233. Bibcode 1978PhRvL..41..233N. doi:10.1103/PhysRevLett.41.233. 4^ Nobel Lecture by William D. Phillips, Dec 8, 1997. (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1997/phillips-lecture.pdf) 5^ Foot, C.J. Atomic Physics. Oxford University Press (2005). (http:/ / www. oup. com/ uk/ catalogue/ ?ci=9780198506959) 6^Cohen-Tanoudji, Claude (2011). Advances in Atomic Physics (http:/ / www. worldscibooks. com/ physics/ 6631. html). World Scientific. pp. 791. ISBN 978-981-277-496-5. List of thermal conductivities In heat transfer, the thermal conductivity of a substance, k, is an intensive property that indicates its ability to conduct heat. Thermal conductivity is often measured with laser flash analysis. Alternative measurements are also established. Mixtures may have variable thermal conductivities due to composition. Material Temperature [K] Notes Thermal conductivity [W·m−1·K−1] Electrical conductivity @ 293 K −1 −1 [Ω ·m ]Acrylic Glass [1] [1] [2] [1] [1] [1] 0.17 -0.19 -0.2 296 7.143E-15 - 5.0E-14(Plexiglas V045i) [3][4][5] [6] [3][4] [6] [5] [8] [8] (N,21%O+0.93%Ar+0.04%CO2) (1 atm) 0.024 -0.025 273 -293 -298 hiAerosols2.95 -loAerosols7.83 ×10−15Air [7] [7] 0.0262 (1 bar) 300 [7] [7] 0.0457 (1 bar) 600Alcohols OR Oils [5][6] [9] [5][6] [9] [6] [5] [9] 0.1 -0.110 -0.21 -0.212 293 -298 -300 [10] [3] [11] [6][12][13][14] [5] [6][10] [5][13][14] [13] [15] 204.3 -205 -220 -237 -250 293 -298 37,450,000 - 37,740,000Aluminium, pure [10] [10] 214.6 366 [10] [10] 249.3 478Aluminium nitride [12] [16] [16] [16] [16] 170 -175 -190 293 0×10−11Aluminium oxide, [17] [6] [17] [12] [18] [6][17][18] [17][18] 26 -30 -35 -39 -40 293 0×10−12 -pure

List of thermal conductivities 259Ammonia, [9] [9] 0.507 300saturatedArgon [5] [14] [14][19] [5][14] [14][19] 0.016 -0.01772 -0.0179 298 -300Beryllium oxide [12] [20] [20] [20] −12 [20] 218 -260 -300 293 0×10Bismuth [14] [14] 7.97 300Brass Cu63% [21] [21] [21] [21] (Cu63%, Zn37%) 125 296 15,150,000 - 16,130,000Brass Cu70% [3][22] [22] [3] [22] [22] [22] (Cu70%, Zn30%) 109 - 121 293 -296 12,820,000 - 16,130,000Brick [3] [3] [5] [5] [3] [5] 0.15 -0.6 -0.69 -1.31 293 -298 [11] [10] [23] [23] [23] [11]Bronze 26 293 -296 5,882,000 - 7,143,000 Sn25% [23] [10][23] [23] 42 -50 (Cu89%, Sn11%)Calcium silicate [24] [24]

Heat Transfer Viva

by Chaitanya Bharathi Institute of Technology on Nov 11, 2012

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Heat Transfer Viva Document Transcript