The document discusses Newton's law of cooling, which states that the rate of change of an object's temperature is proportional to the temperature difference between the object and its surroundings. It also discusses the heat transfer version of the law. The law has applications in predicting cooling rates and requires the temperature difference and heat transfer mechanism to remain constant. Kirchhoff's law of thermal radiation states that an object's emissivity and absorptivity are equal when the object is in thermal equilibrium. The law explains that good absorbers are also good emitters. The document also discusses critical thickness of insulation and its relationship to heat transfer through cylindrical and spherical objects.
3. Newtons Law of Cooling states that
THE RATE OF CHANGE OF THE TEMPERATURE OF AN OBJECT IS PROPORTIONAL TO THE DIFFERENCE
BETWEEN ITS OWN TEMPERATURE AND THE AMBIENT TEMPERATURE I.E. THE TEMPERATURE OF ITS
SURROUNDINGS.
What it means is that the rate at which an object loose or gain heat depends on the difference between its own
temperature and the surrounding’s temperature
The law is always tied with the condition that
o The temperature difference is small between object and surroundings
o The nature of heat transfer mechanism remains the same
conduction, convection etc.
temperature
time
4. Rate of temperature change
Temperature changing coefficient Initial temperature of object
Temperature of surroundings
(ambient temperature)
Temperature at time t
Temperature of object at time t
5. HEAT TRANSFER VERSION OF THE LAW
The heat-transfer version of Newton's 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.
Thermal energy
Heat transfer coefficient
Heat transfer surface area
Temperature of the object's surface and interior
Temperature of the environment
6. APPLICATIONS
o To predict how long it takes for a hot object to cool down at a certain temperature.
o To find the temperature of a soda placed in a refrigerator by a certain amount of time.
o It helps to indicate the time of death given the probable body temperature at the time of death and current body
temperature.
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 that can be analysed using the
presumption that Newton's law will hold.
LIMITATIONS OF NEWTONS LAW OF COOLING
o The difference in temperature between the body and surroundings must be small,
o The loss of heat from the body should be by radiation only,
o The temperature of surroundings must remain constant during the cooling of the body.
Newton's Law is more closely followed for forced air or pumped fluid cooling, where the velocity of the fluid does not vary with
temperature. In the case of heat transfer by thermal radiation, Newton's law of cooling holds only for very small temperature
changes, and a more accurate description is given by Planck's Law.
8. In plane walls heat transfer through the walls can be decreased by adding insulation on the surface of the wall.
But in the case of cylindrical or spherical objects this may not be the case
This is due to the fact that Heat transfer from an object depends on the surface area of the object
By adding insulation to a plane wall, surface area is not changed, hence heat transfer is effectively dampened by insulation provided.
But in the case of spherical or cylindrical or any other curved surface, adding insulation excessively could increase the surface area of
the exterior enough so that insulation cannot provide heat protection similar to plane walls
Adding insulation in this case decreases the convection resistance at the outer surface.
The thickness upto which heat flow increases and after which heat flow decreases is termed as critical thickness.
In the case of cylinders and spheres it is called CRITICAL RADIUS.
The critical radius of insulation for optimum heat transfer
rc= k
ho
Conductivity of material
Heat transfer coefficient
9. APPLICATIONS
o While deciding the thickness of insulation for an electric wire, from which you want to release the heat generated because
of the passage of current, the thickness of insulation should be lower than the critical thickness value.
o Finding best insulation thickness for HVAC shafts, vents, pipes carrying heated or cooled water
o While designing insulation thickness for a thermos flask from which you don't want heat leakage, the thickness should be
more than the critical thickness.
ASSUMPTIONS FOR CALCULATING CRITICAL RADIUS
o All air gaps are neglected.
o Heat is assumed to be uniformly distributed
o Convection between material and the covering is neglected.
o Radiations losses are neglected.
o Clamps used for holding the insulated rod do not absorb any heat during the process and hence aren't
considered in calculations.
10. Example
Assume a steel pipe of r1 = 10 mm, which is exposed to natural convection at h = 50 W/m2.K. This pipe is insulated by
material of thermal conductivity k = 0.5 W/m.K. Determine the critical thickness of this combination:
Hence rcr > r1 and heat transfer will increase with the addition of insulation up to a thickness of rcr – r1 = (0.010 –
0.005)m = 0.005 m
Critical Thickness of Insulation – Spherical Coordinates
It can be shown in a similar manner that the critical radius of insulation for a spherical shell is:
12. Kirchhoff’s Law of thermal radiation:
FOR AN ARBITRARY BODY EMITTING AND ABSORBING THERMAL RADIATION IN
THERMODYNAMIC EQUILIBRIUM, THE EMISSIVITY IS EQUAL TO THE ABSORPTIVITY.
emissivity ε = absorptivity α
all bodies above absolute zero temperature radiate some heat. Two objects radiate heat toward each other. But what if a
colder object with high emissivity radiates toward a hotter object with very low emissivity? This is resolved by the fact that
each body must be in direct line of sight of the other to receive radiation from it. Therefore, whenever the cool body is
radiating heat to the hot body, the hot body must also be radiating heat to the cool body. Moreover, the hot body will radiate
more energy than cold body. The case of different emissivities is solved by the Kirchhoff’s Law of thermal radiation, which
states that object with low emissivity have also low absorptivity. As a result, heat cannot spontaneously flow from cold system
to hot system
13. The emissivity, ε, of the surface of a material is its effectiveness in emitting energy as thermal radiation and varies between 0.0
and 1.0.
By definition, a blackbody in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate
as much heat as a perfect black body. They radiate less heat than a black body and therefore are
called gray bodies.
Quantitatively, emissivity is the ratio of the thermal radiation from a surface to the radiation from an
ideal black surface at the same temperature as given by the Stefan–Boltzmann law. Emissivity is simply a
factor by which we multiply the black body heat transfer to take into account that the black body is the
ideal case.
The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square
metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper
wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m2). Emissivity plays
important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces
that have very low emissivities. These collectors waste very little of the solar energy through emission of
thermal radiation.
EMISSIVITY
14. Another important radiation property of a surface is its absorptivity, α, which is the fraction of the radiation energy incident on a
surface that is absorbed by the surface.
Like emissivity, value of absorptivity is in the range 0 < α < 1.From its definition, a blackbody, which is an idealized physical body,
absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. That is, a blackbody is a perfect
absorber. Since for real objects the absorptivity is less than 1, a real object can not absorb all incident light. The incomplete
absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the
surface of the body.
In general, the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation
visible radiation occupies a very narrow band of the spectrum from 0.4 to 0.76 nm. hence, we cannot make any judgments about
the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus
appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-
wavelength radiation.
ABSORPTIVITY
15. APPLICATIONS
Sand is rough black, so it is a good absorber and hence in deserts, days (when radiation from the sun is incident on sand) will be very
hot. Now in accordance with Kirchhoff's law, good absorber is a good emitter so nights (when sand emits radiation) will be cold. This is
why days are hot and nights are cold in desert.
When a shining metal ball having some black spots on its surface is heated to a high temperature and is seen in dark, the black spots
shine brightly and the shining ball becomes dull or invisible. The reason is that the black spots on heating absorb radiation and so emit
these in dark while the polished shining part reflects radiations and absorb nothing and so does not emit radiations and becomes
invisible in the dark.
When a green glass is heated in furnace and taken out, it is found to glow with red light. This is because red and green are
complimentary colours. At ordinary temperatures, a green glass appears green, because it transmits green colour and absorb red
colour strongly. According to Kirchhoff's law, this green glass, on heating must emit the red colour, which is absorbed strongly.
Similarly when a red glass is heated to a high temperature it will glow with green light.
A person with black skin experiences more heat and more cold as compared to a person of white skin because when the outside
temperature is greater, the person with black skin absorbs more heat and when the outside temperature is less the person with black
skin radiates more energy.
16. Kirchoff’s law also explains the Fraunhofer lines :
o Sun's innermost part (photosphere) emits radiation of all wavelength at high temperature.
o When these radiation enters in outer part (chromosphere) of sun, few wavelength are absorbed by some terrestrial elements
(present in vapor form at lower temperature)
o These absorbed wavelengths, which are missing appear as dark lines in the spectrum of the sun called Fraunhofer lines.
o During total solar eclipse these lines appear bright because the gases and vapour present in the chromosphere start emitting
those radiation which they had absorbed
