Food Science Text SeriesThe Food Science Text Series provides faculty with the leading teaching tools. The Editorial Boardhas outlined the most appropriate and complete content for each food science course in a typicalfood science program and has identiﬁed textbooks of the highest quality, written by the leading foodscience educators.Series EditorDennis R. HeldmanEditorial BoardDavid A. Golden, Ph.D., Professor of Food Microbiology, Department of Food Science andTechnology, University of TennesseeRichard W. Hartel, Professor of Food Engineering, Department of Food Science, University ofWisconsinHildegarde Heymann, Professor of Food Sensory Science, Department of Food Science andTechnology, University of California-DavisJoseph H. Hotchkiss, Professor, Institute of Food Science and Institute for Comparative andEnvironmental Toxicology, and Chair, Food Science Department, Cornell UniversityMichael G. Johnson, Ph.D., Professor of Food Safety and Microbiology, Department of Food Science,University of ArkansasJoseph Montecalvo, Jr., Professor, Department of Food Science and Nutrition, California Polytechnicand State University-San Luis ObispoS. Suzanne Nielsen, Professor and Chair, Department of Food Science, Purdue UniversityJuan L. Silva, Professor, Department of Food Science, Nutrition and Health Promotion, MississippiState UniversityFor further volumes:http://www.springer.com/series/5999
P.G. SmithIntroduction to Food ProcessEngineeringSecond Edition123
Preface to the First EditionThere are now a large number of food-related ﬁrst-degree courses offered at universities in Britain andelsewhere in the world which either specialise in, or contain a signiﬁcant proportion of, food technol-ogy or food engineering. This is a new book on food process engineering which treats the principlesof processing in a scientiﬁcally rigorous yet concise manner, which can be used as a lead-in to morespecialised texts for higher study and which is accessible to students who do not necessarily possess atraditional science A-level background. It is equally relevant to those in the food industry who desire agreater understanding of the principles of the processes with which they work. Food process engineer-ing is a quantitative science and this text is written from a quantitative and mathematical perspectiveand is not simply a descriptive treatment of food processing. The aim is to give readers the conﬁdenceto use mathematical and quantitative analyses of food processes and most importantly there are alarge number of worked examples and problems with solutions. The mathematics necessary to readthis book is limited to elementary differential and integral calculus and the simplest kind of differentialequation. This book is the result of 15 years experience of teaching food processing technology and foodengineering to students on a variety of diploma, ﬁrst degree and postgraduate courses. It is designed,inter alia, to• emphasise the importance of thermodynamics and heat transfer as key elements in food processing• stress the similarity of heat, mass and momentum transfer and make the fundamentals of these essential concepts readily accessible• develop the theory of mass transfer, which is underused in studies of food processing and little understood in a useful and readily applicable way• widen the usual list of unit operations treated in textbooks for undergraduates to include the use of membranes• introduce a proper treatment of the characterisation of food solids and of solids processing and handling Chapters 1 and 2 set out the background for a quantitative study of food processing by deﬁningthe objectives of process engineering, by describing the mathematical and analytical approach to thedesign and operation of processes and by establishing the use of SI units. Much of what follows inthe book is made easier by a thorough understanding of the SI system. Important thermodynamicsconcepts are introduced in Chapter 3 which underpins the sections on energy balances and heat trans-fer, itself so central to food processing. Chapter 4 is concerned with material and energy balancesand concentrates upon the techniques required to solve problems. Most of the chapter is devoted tonumerical examples drawn from a wide range of operations. In Chapter 5 the concepts of heat, mass and momentum transfer are introduced; the similaritybetween heat, mass and momentum transfer is stressed. This acts as an introduction to the material v
vi Preface to the First Editionof the following three chapters. These cover ﬁrst, the ﬂow of food ﬂuids, in which the importance oflaminar ﬂow in food processing is emphasised, and food rheology, where the objective is to enablethe reader to apply rheological models to experimental data and to understand their signiﬁcance inmechanistic and structural terms. Second, heat transfer, which is at the heart of many food processingoperations. The basic principles are covered in detail and illustrated with numerous worked examples.Third, mass transfer, which is often perceived as a difﬁcult topic and indeed is poorly treated in manyfood texts. As a consequence, mass transfer theory is underused in the analysis of food processes.Chapter 8 is intended to redress this imbalance, and the treatment of mass transfer is extended inChapter 9, where the principles of psychrometry are explained. The principal preservation operations are covered in Chapters 10, 11 and 12. These include thecommercial sterilisation of foods, where the bases of the general and mathematical models are outlinedand emphasis is given to a clear explanation of calculation procedures; low-temperature preservation,including coverage of the principles of the refrigeration cycle; evaporation and drying. The processingof food particulates is often overlooked and Chapter 13 is an attempt to address this oversight. It con-siders the characterisation of individual particles and the development of relationships for particle –ﬂuid interaction, and ﬂuidisation is included at this point because it is a fundamental processing tech-nique with wide application to many unit operations. Finally, Chapter 14 covers mixing and physicalseparation processes, including the increasingly important area of separation using ultraﬁltration andreverse osmosis.
Preface to the Second EditionIn this second edition two chapters have been added. Chapter 15 covers some of the mass transferoperations used in the food industry but which are not always considered to be core food processes:distillation (including both batch and continuous operation), leaching (or solid–liquid extraction) andsupercritical ﬂuid extraction, a process of increasing importance in the food industry. In the case ofdistillation and leaching the outline of the relevant theory is supported by detailed worked exam-ples to illustrate the common graphical methods which are used to determine the number of ideal orequilibrium stages. The growing demand for safer food of ever higher quality has led to the investigation of a rangeof techniques which may together be labelled as minimum processing technologies. The principlesof some of these techniques are outlined in Chapter 16, including ohmic heating, pulsed electric ﬁeldheating (PEF), radio frequency heating (RF), high-pressure processing, irradiation and ultrasound. The content of a number of other chapters has been updated or amended. Methods of temperaturemeasurement, especially the details of various types of thermocouples in use, have been included inChapter 7. A new section on the application of mass transfer to food packaging has been added toChapter 8; data on the permeability of packaging ﬁlms are presented. The coverage of freeze drying(Chapter 12) has been extended considerably to include the use of heat and mass transfer models inthe prediction of drying time. The section on ﬂuidisation in Chapter 13 has been rewritten to includemore information on the estimation of heat and mass transfer coefﬁcients in ﬂuidised beds used infood processes. In addition to these changes, the opportunity has been taken to review all the worked examples andproblems in the book and to correct a number of errors in the ﬁrst edition. The reading lists at the endof each chapter have been updated where appropriate.Lincoln, UK P.G. SmithJune 2010 vii
Contents xviiAppendix A List of Unit Preﬁxes; Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . 479Appendix B Fundamental and derived SI Units; Conversion Factors . . . . . . . . . . . . . . . . 481Appendix C Derivation of a Dimensionless Correlation for Film Heat Transfer Coefﬁcients . . . . 483Appendix D Properties of Saturated Water and Water Vapour . . . . . . . . . . . . . . . . . . . . 487Appendix E Derivation of Logarithmic Mean Temperature Difference . . . . . . . . . . . . . . . 489Appendix F Derivation of Fourier’s First Law of Conduction . . . . . . . . . . . . . . . . . . . . 491Answers to Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
Chapter 1An Introduction to Food Process EngineeringA process may be thought of as a sequence of operations which take place in one or more pieces ofequipment, giving rise to a series of physical, chemical or biological changes in the feed material andwhich results in a useful or desirable product. More traditional deﬁnitions of the concept of processwould not include the term biological but, because of the increasing sophistication, technologicaladvance and economic importance of, the food industry, and the rise of the biotechnology industries,it is ever more relevant to do so. Process engineering is concerned with developing an understanding of these operations and withthe prediction and quantifying of the resultant changes to feed materials (such as composition andphysical behaviour). This understanding leads in turn to the speciﬁcation of the dimensions of pro-cess equipment and the temperatures, pressures and other conditions required to achieve the necessaryoutput of product. It is a quantitative science in which accuracy and precision, measurement, math-ematical reasoning, modelling and prediction are all important. Food process engineering is aboutthe operation of processes in which food is manufactured, modiﬁed and packaged. Two major cat-egories of process might be considered; those which ensure food safety, that is the preservationtechniques such as freezing or sterilisation, which usually involve the transfer of heat and inducechanges to microbiological populations, and those which may be classiﬁed as food manufacturingsteps. Examples of the latter include the addition of components in mixing, the separation of compo-nents in ﬁltration or centrifugation or the formation of particles in spray drying. Classiﬁcation in thisway is rather artiﬁcial and by no means conclusive but serves to illustrate the variety of reasons forprocessing food materials. Although foods are always liquid or solid in form, many foods are aerated (e.g. ice cream), manyprocesses utilise gases or vapours (e.g. steam as a heat source) and many storage procedures requiregases of a particular composition. Thus it is important for the food technologist or the food engineer tounderstand in detail the properties and behaviour of gases, liquids and solids. In other words the trans-fer of heat, mass and momentum in ﬂuids and an understanding of the behaviour of solids, especiallyparticulate solids, form the basis of food processing technology. At the heart of process engineering isthe concept of the unit operation. Thus the principles which underlie drying, extraction, evaporation,mixing and sterilisation are independent of the material which is being processed. Once understood,these principles can be applied to a wide range of products. The overall purpose of food process engineering then is to design processes which result in safefood products with speciﬁc properties and structure. Foods, of course, have their own particular andpeculiar properties: most food liquids are non-Newtonian; structures are often complex and multi-phase; non-isotropic properties are common. In addition to this, hygiene is of paramount importancein all manufacturing steps. The correct design of such processes is possible only as a result of thedevelopment of mathematical models which incorporate the relevant mechanisms. Thus it is importantto understand the chemical, structural and microbiological aspects of food in so far as they contributeP.G. Smith, Introduction to Food Process Engineering, Food Science Texts Series, 1DOI 10.1007/978-1-4419-7662-8_1, C Springer Science+Business Media, LLC 2011
2 1 An Introduction to Food Process Engineeringto an understanding of the process, that is, how to develop, design, operate and improve the processto give better performance at reduced cost and, above all, improved safety and quality. The ﬁrst step in the design of a process is the conception stage. What is the product to be manufac-tured? What steps will be needed in order to manufacture it? In some cases the necessary steps may bevery well known and there is no particular innovation required. As an example take the manufactureof ice cream. Whilst individual products may be innovative to a degree, the essential production stepsare well known. There will be a mixing step in which the solid and liquid ingredients are added tothe batch, followed by pasteurisation, storage or ageing, freezing and ﬁnally ﬁlling and packaging.For many food products there is an established way of doing things and there may be no realisticalternative. In other cases it may be far less obvious what the ﬁnal process design will look like. Ineach case a simple ﬂow sheet of the process should be prepared. At this point it is likely to become apparent whether the process is to be batch or continuous.A batch process is one in which a given mass of material is subject to a series of operations in aparticular sequence. For example a batch of liquid may be heated, a second component added, themixture agitated and then the resultant liquid cooled, all within a single vessel. Alternatively thesequence of operations may involve a number of pieces of equipment. In a simple mixing operation,or where a chemical reaction occurs, the composition of the batch changes with time. If a liquid isheated in a stirred vessel the temperature of the liquid will be uniform throughout the vessel, providedthe agitation is adequate, but will change with time. Batch processes generally have two disadvantages.First they are labour intensive because of the bulk handling of material involved and the large numberof individual operations which are likely to be used. Second the quality of the product may wellvary from batch to batch. These problems are largely overcome if the process becomes continuous.Here, material ﬂows through a series of operations and individual items of equipment, undergoinga continuous change without manual handling. Once running, a continuous process should run for along period under steady-state conditions, that is the composition, ﬂow rate, temperature or any othermeasurable quantity should remain constant at any given point in the process. In this way a continuousprocess gives a more consistent product. The mathematical analysis of a process also highlights an important difference between batch andcontinuous operation. Continuous, steady-state processes are usually considerably simpler to analysethan are unsteady-state batch processes because the latter involve changes in composition or temper-ature with time. However, the difference between batch and continuous may not always be clear-cut;many individual operations in the food industry are batch (often because of the scale of operationrequired) but are placed between other continuous operations. Thus the entire process, or a majorsection of it, is then best described as either semi-batch or semi-continuous. The second stage of the design process may be called process analysis and this entails establishingboth a material balance and an energy balance. The material balance aims to answer the question:What quantities of material are involved? What ﬂow rates of ingredients are needed? In many casesthis will be simply a case of establishing the masses of components to be added to a batch mixer. Inothers it will require the determination of ﬂow rates of multi-component streams at several points ina complex process covering a large factory unit. In food processing the energy or enthalpy balanceassumes enormous signiﬁcance; sterilisation, pasteurisation, cooking, freezing, drying and evapora-tion all involve the addition of heat to, or removal of heat from, the product. Establishing the necessaryheat ﬂows with accuracy is therefore of crucial importance for reasons both of food safety and ofprocess efﬁciency. A third stage comprises the speciﬁcation of each operation and the design of individual piecesof equipment. In order to do this the prevailing physical mechanisms must be understood as well asthe nature and extent of any chemical and biochemical reaction and the kinetics of microbiologicalgrowth and death. Speciﬁcation of the size of heat transfer process equipment depends upon beingable to predict the rate at which heat is transferred to a food stream being sterilised. In turn thisrequires knowledge of the physical behaviour of the ﬂuid, in short an understanding of ﬂuid ﬂow
An Introduction to Food Process Engineering 3and rheology. This allows judgements to be made about how best to exploit the ﬂow of material, forexample whether the ﬂow should be co-current or counter-current. Crucial to any process design is knowledge of equilibrium and kinetics. Equilibrium sets the bound-aries of what is possible. For example, in operations involving heat transfer knowledge of thermalequilibrium (the heat capacity and the ﬁnal temperatures required) allows the calculation of the quan-tities of heat to be removed or added. Equipment and processes can be sized only if the rate at whichheat is transferred is known. Each rate process encountered in food engineering follows the samekind of law: where molecular diffusion is responsible for transfer, the rate of transfer of heat, massor momentum is dependent upon the product of a gradient in temperature, concentration or velocity,respectively, and a diffusivity – a physical property which characterises the particular system underinvestigation. Where artiﬁcial convection currents are introduced, by the use of deliberate agitation,then an empirical coefﬁcient must be used in conjunction with gradient term; little progress can bemade in the application of heat transfer in food processing without a knowledge of the relevant heattransfer coefﬁcient. The overall design of the food process now moves onto the speciﬁcation of instrumentation andprocess control procedures, to detailed costing and economic calculations, to detailed mechanicaldesign and to plant layout. However, all of these latter stages are beyond the scope of this book.
Chapter 2Dimensions, Quantities and UnitsNomenclaturea AccelerationA Aread DiameterF Forceg Acceleration due to gravityh Heightm MassP Pressuret Timeu Speed or velocityux Velocity in the x-directionW Workx DistanceGreek symbolsρ Density2.1 Dimensions and UnitsThe dimensions of all physical quantities can be expressed in terms of the four basic dimensions: mass,length, time and temperature. Thus velocity has the dimensions of length per unit time and densityhas the dimensions of mass per unit length cubed. A system of units is required so that the magni-tudes of physical quantities may be determined and compared one with another. The internationallyagreed system which is used for science and engineering is the Systeme International d’Unites, usu-ally abbreviated to SI. Table 2.1 lists the SI units for the four basic dimensions together with those forelectrical current and plane angle which, although strictly are derived quantities, are usually treatedas basic quantities. Also included is the unit of molar mass which somewhat illogically is the grammolecular weight or gram mole and which is usually referred to simply as a ‘mole’. However, it isoften more convenient to use the kilogram molecular weight or kmol. The SI system is based upon the general metric system of units which itself arose from the attemptsduring the French Revolution to impose a more rational order upon human affairs. Thus the metrewas originally deﬁned as one ten-millionth part of the distance from the North Pole to the equatorP.G. Smith, Introduction to Food Process Engineering, Food Science Texts Series, 5DOI 10.1007/978-1-4419-7662-8_2, C Springer Science+Business Media, LLC 2011
6 2 Dimensions, Quantities and Units Table 2.1 Dimensions and SI units of the four basic quantities and some derived quantities Dimension Symbol SI unit Symbol Mass M Kilogram kg Length L Metre m Time T Second s Temperature Degree kelvin K Plane angle – Radian rad Electrical current – Ampere A Molar mass – Gram-molecular weight molalong the meridian which passes through Paris. It was subsequently deﬁned as the length of a barof platinum–iridium maintained at a given temperature and pressure at the Bureau International desPoids et Measures (BIPM) in Paris, but is deﬁned now by the wavelength of a particular spectral lineemitted by a Krypton 86 atom. The remaining units in Table 2.1 are deﬁned as follows:kilogram: The mass of a cylinder of platinum–iridium kept under given conditions at BIPM, Paris.second: A particular fraction of a certain oscillation within a caesium 133 atom.degree kelvin: The temperature of the triple point of water, on an absolute scale, divided by 273.16. The degree kelvin is the unit of temperature difference as well as the unit of thermodynamic temperature.radian: The angle subtended at the centre of a circle by an arc equal in length to the radius.ampere: The electrical current which if maintained in two straight parallel conductors of inﬁnite length and negligible cross-section, placed 1 m apart in a vacuum, produces a force between them of 2 × 10−7 N per metre length.mol: The amount of substance containing as many elementary units (atoms or molecules) as there are in 12 g of carbon 12. The SI system is very logical and, in a scientiﬁc and industrial context, has a great many advan-tages over previous systems of units. However, it is usually criticised on two counts. First that thenames given to certain derived units, such as the pascal for the unit of pressure, of themselves meannothing and that it would be better to remain with, for example, the kilogram per square metre. Thisis erroneous; the deﬁnitions of newton, joule, watt and pascal are simple and straightforward if theunderlying principles are understood. Derived units which have their own symbols, and which areencountered in this book, are listed in Table 2.2. The second criticism concerns the magnitude of many units and the resulting numbers which areoften inconveniently large or small. This problem would occur with any system of units and is notpeculiar to SI. However, there are instances when strictly non-SI units may be preferred. For example,the wavelengths of certain kinds of electromagnetic radiation may be more conveniently written in Table 2.2 Some derived SI units Name Symbol Quantity represented Basic units newton N Force kg m s−2 joule J Energy or work Nm watt W Power J s−1 pascal Pa Pressure N m−2 hertz Hz Frequency s−1 volt V Electrical potential W A−1
2.2 Deﬁnitions of Some Basic Physical Quantities 7terms of the angstrom, 1 Å being equal to 10−10 m. Flow rates and production ﬁgures, when expressedin kg h−1 or even in t day−1 , may be more convenient than in kg s−1 . Pressures are still often quotedin bars or standard atmospheres rather than pascals simply because the pascal is a very small unit.Many of these latter disadvantages can be overcome by using preﬁxes. Thus a pressure of 105 Pamight better be expressed as 100 kPa. A list of preﬁxes is given in Appendix A. It must be stressed that whatever shorthand methods are used to present data, the strict SI unit mustbe used in calculations. Mistakes are made frequently by using, for example, kW m−2 K−1 for theunits of heat transfer coefﬁcients in place of W m−2 K−1 . Although such errors ought to be obviousit is often the case that compound errors of this kind result in plausible values based upon erroneouscalculations. A further note on presentation is appropriate at this point. I believe ﬁrmly that the use of negativeindices, as in W m−2 K−1 , avoids confusion and is to be preferred to the solidus as in W/m2 K. Theformer method is used throughout this book. Appendix B gives a list of conversion factors betweendifferent units.2.2 Deﬁnitions of Some Basic Physical Quantities2.2.1 Velocity and SpeedVelocity and speed are both deﬁned as the rate of change of distance with time. Thus, speed u is givenby dx u= (2.1) dtwhere x is distance and t is time. Average speed is then the distance covered in unit elapsed time.Velocity and speed differ in that speed is a scalar quantity (its deﬁnition requires only a magnitudetogether with the relevant units) but velocity is a vector quantity and requires a direction to be speci-ﬁed. A process engineering example would be the velocity of a ﬂuid ﬂowing in a pipeline; the velocitymust be speciﬁed as the velocity in the direction of ﬂow as opposed to, for example, the velocity per-pendicular to the direction of ﬂow. Thus velocity in the x-direction might be designated ux where dx ux = (2.2) dt In practice the term velocity is used widely without specifying direction explicitly because thedirection is obvious from the context. The SI unit of velocity is m s−1 .2.2.2 AccelerationAcceleration is the rate of change of velocity with time. Thus the acceleration at any instant, a, isgiven by du a= (2.3) dtIt should be noted that the term acceleration does not necessarily indicate an increase in velocity withtime. Acceleration may be positive or negative and this should be indicated along with the magnitude.The SI unit of acceleration is m s−2 .
8 2 Dimensions, Quantities and Units2.2.3 Force and MomentumThe concept of force can only be understood by reference to Newton’s laws of motion. First law: A body will continue in its state of rest or uniform motion in a straight line unless acted upon by an impressed force. Second law: The rate of change of momentum of the body with time is proportional to the impressed force and takes place in the direction of the force. Third law: To each force there is an equal and opposite reaction. These laws cannot be proved but they have never been disproved by any experimental observation.The momentum of an object is the product of its mass m and velocity u: momentum = mu (2.4) From Newton’s second law, the magnitude of a force F acting on a body may be expressed as d(mu) F∝ (2.5) dtIf the mass is constant, then du F∝m (2.6) dtor F ∝ ma (2.7)In other words, force is proportional to the product of mass and acceleration. A suitable deﬁnition ofthe unit of force will result in a constant of proportionality in Eq. (2.7) of unity. In the SI system theunit of force is the newton (N). One newton is that force, acting upon a body with a mass of 1 kg,which produces an acceleration of 1 m s−2 . Hence F = ma (2.8)2.2.4 WeightWeight is a term for the localised gravitational force acting upon a body. The unit of weight is thereforethe newton and not the kilogram. The acceleration produced by gravitational force varies with the dis-tance from the centre of the earth and at sea level the standard value is 9.80665 m s−2 which is usuallyapproximated to 9.81 m s−2 . The acceleration due to gravity is normally accorded the symbol g. The magnitude of a newton can be gauged by considering the apple falling from a tree which wasobserved supposedly by Isaac Newton before he formulated the theory of universal gravitation. Theforce acting upon an average-sized apple with a mass of, say, 0.10 kg, falling under gravity, would be,using Eq. (2.8) F = 0.10 × 9.81 N F = 0.981 N F ≈ 1.0 N
2.2 Deﬁnitions of Some Basic Physical Quantities 9The fact that an average-sized apple falls with a force of about 1.0 N is nothing more than an interest-ing coincidence. However, this simple illustration serves to show that the newton is a very small unit.2.2.5 PressureA force F acting over a speciﬁed surface area A gives rise to a pressure P. Thus F P= (2.9) AIn the SI system the unit of pressure is the pascal (Pa). One pascal is that pressure generated bya force of 1 N acting over an area of 1 m2 . Note that standard atmospheric pressure is 1.01325 ×105 Pa. There are many non-SI units with which it is necessary to become familiar; the bar is equalto 105 Pa and ﬁnds use particularly for pressures exceeding atmospheric pressure. A pressure can beexpressed in terms of the height of a column of liquid which it would support and this leads to manycommon pressure units, derived from the use of the simple barometer for pressure measurement. Thusatmospheric pressure is approximately 0.76 m of mercury or 10.34 m of water. Very small pressuresare sometimes expressed as mm of water or ‘mm water gauge’. Unfortunately it is still common toﬁnd imperial units of pressure, particularly pounds per square inch or psi. Consider a narrow tube held vertically so that the lower open end is below the surface of a liquid(Fig. 2.1). The pressure of the surrounding atmosphere P forces a column of liquid up the tube toa height h. The pressure at the base of the column (point B) is given by the weight of liquid in thecolumn acting over the cross-sectional area of the tube: πd 2 mass of liquid in tube = hρ (2.10) 4where ρ is the density of the liquid. The weight of liquid (force acting on the cross-section) F is πd 2 F= hρg (2.11) 4The pressure at B must equal the pressure at A, therefore πd 2 4 P= hρg 2 (2.12) 4 πd P h P A BFig. 2.1 Pressure at the base of a column of liquid
10 2 Dimensions, Quantities and Unitsor P = ρgh (2.13)Example 2.1What pressure will support a column of mercury (density = 13,600 kg m−3 ) 80 cm high? From Eq. (2.13) the pressure at the base of the column is P = 13, 600 × 9.81 × 0.80 Paand therefore P = 1.0673 × 105 Pa2.2.6 Work and EnergyWork and energy are interchangeable quantities; work may be thought of as energy in transition. Thus,for example, in an internal combustion engine chemical energy in the fuel is changed into thermalenergy which in turn produces expansion in a gas and then motion, ﬁrst of a piston within a cylinderand then of a crankshaft and of a vehicle. The SI unit of all forms of energy and of mechanical workis the joule (J). A joule is deﬁned as the work done when a force of 1 N moves through a distance of1 m. For example, if an apple of weight 1 N (see Section 2.2.4) is lifted through a vertical distanceof 1 m then the net work done on the apple is 1 J. This is the energy required simply to lift the appleagainst gravity and does not take into account any inefﬁciencies in the device – be it a mechanicaldevice or the human arm. Clearly the joule is a small quantity. A further illustration of the magnitudeof the joule is that approximately 4180 J of thermal energy is required to increase the temperature of1 kg of water by 1 K (see Section 3.5).Example 2.2A mass of 250 kg is to be raised by 5 m against gravity. What energy input is required to achieve this? From Eq. (2.8) the gravitational force acting on the mass is F = 250 × 9.81 Nor F = 2452.5 N The work done (or energy required) W in raising the mass against this force is W = F × distanceand therefore W= 2452.5 × 5 Jor W= 1.23 × 104 J
2.3 Dimensional Analysis 112.2.7 PowerPower is deﬁned as the rate of working or the rate of usage or transfer of energy and thus it involvestime. The apple may be lifted slowly or quickly. The faster it is lifted through a given distance thegreater is the power of the mechanical device. Alternatively the greater the mass lifted (hence thegreater the force overcome) within the same period, the greater will be the power. A rate of energyusage of 1 joule per second (1 J s−1 ) is deﬁned as 1 watt (W).Example 2.3The mass in Example 2.2 is lifted in 5 s. What power is required to do this? What power is requiredto lift it in 1 min? The power required is equal to the energy used (or work done) divided by the time over which theenergy is expended. Therefore 1.23 × 104 power = W 5 = 2452.5 W or 2.45 kW. If now the mass is lifted over a period of 1 min, 1.23 × 104 power = W 60 = 204.4 W Whilst most examples in this book are concerned with rates of thermal energy transfer, it is impor-tant to understand (and may be easier to visualise) the deﬁnitions of work and power in mechanicalterms. In addition, students of food technology and food engineering do require some basic knowl-edge of electrical power supply and usage, although that is outside the scope of this book. Sufﬁce itto say that the electrical power consumed (in watts) when a current ﬂows in a wire is given by theproduct of the current (in amperes) and the electrical potential (in volts).2.3 Dimensional Analysis2.3.1 Dimensional ConsistencyAll mathematical relationships which are used to describe physical phenomena should be dimension-ally consistent. That is, the dimensions (and hence the units) should be the same on each side of theequality. Take Eq. (2.13) as an example. P = ρgh (2.13)Using square brackets to denote dimensions or units, the dimensions of the terms on the right-handside are as follows:
12 2 Dimensions, Quantities and Units [ρ] = ML−3 g = LT−2 [h] = LThus the dimensions of the right-hand side of Eq. (2.13) are [ρgh] = (ML−3 )(LT−2 )(L)or [ρgh] = ML−1 T−2Pressure is a force per unit area, force is given by the product of mass and acceleration and thereforethe dimensions of the left-hand side of Eq. (2.13) are [mass] [acceleration] [P] = [area]or (M)(LT−2 ) [P] = L2or [P] = ML−1 T−2Similarly the units must be the same on each side of the equation. This is simply a warning not to mixSI and non-SI units and to be aware of preﬁxes. The units of the various quantities in Eq. (2.13) are [P] = Pa [ρ] = kg m−3 [g] = m s−2 [h] = mand therefore the right-hand side of Eq. (2.13) becomes [ρgh] = (kgm−3 ) × (ms−2 ) × (m)or [ρgh] = kg × ms−2 × m−2
2.3 Dimensional Analysis 13As a force of 1 N, acting upon a body of mass 1 kg, produces an acceleration of 1 m s−2 , the units ofthe right-hand side of Eq. (2.13) are now [ρgh] = Nm−2and thus [ρgh] = Pa2.3.2 Dimensional AnalysisThe technique of dimensional analysis is used to rearrange the variables which represent a physicalrelation so as to give a relationship that can more easily be determined by experimentation. For exam-ple, in the case of the transfer of heat to a food ﬂuid in a pipeline, the rate of heat transfer might dependupon the density and viscosity of the food, the velocity in the pipeline, the pipe diameter and othervariables. Dimensional analysis, by using the principle of dimensional consistency, suggests a moredetailed relationship which is then tested experimentally to give a working equation for predictiveor design purposes. It is important to stress that this procedure must always consist of dimensionalanalysis followed by detailed experimentation. The theoretical basis of dimensional analysis is tooinvolved for this text but an example, related to heat transfer (Chapter 7), is set out in Appendix C.Problems2.1 What is the pressure at the base of a column of water 20 m high? The density of water is 1000 kg m−3 .2.2 What density of liquid is required to give standard atmospheric pressure (101.325 kPa) at the base of a 12 m high column of that liquid?2.3 A person of mass 75 kg climbs a vertical distance of 25 m in 30 s. Ignoring inefﬁciencies and friction (a) how much energy does the person expend and (b) how much power must the person develop?2.4 Determine the dimensions of the following: (a) linear momentum (mass × velocity) (b) work (c) power (d) pressure (e) weight (f) moment of a force (force × distance) (g) angular momentum (linear momentum × distance) (h) pressure gradient (i) stress (j) velocity gradient
Chapter 3Thermodynamics and EquilibriumNomenclatureA Areacp Heat capacity at constant pressurecv Heat capacity at constant volumeC Molar concentration of gasCA Molar concentration of component A in the gas phaseh EnthalpyH Henry’s constantH Henry’s constantH Henry’s constantl LengthM Molecular weightn Molar mass of gasnA Molar mass of Ap Partial pressurepA Partial pressure of component Ap Pure component vapour pressurepA Pure component vapour pressure of AP PressurePc Critical pressurePext External pressure of surroundingsQ Heat; ﬂow rate of heatQ Heat per unit mass of ﬂuidR Universal gas constantT Absolute temperatureTc Critical temperatureu Internal energyV VolumeVo Volume at 0◦ CW Work; rate of workingW Work per unit mass of ﬂuidxA Mole fraction of A in the liquid phaseyA Mole fraction of component A in vapour phaseP.G. Smith, Introduction to Food Process Engineering, Food Science Texts Series, 15DOI 10.1007/978-1-4419-7662-8_3, C Springer Science+Business Media, LLC 2011