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1BASIC CONCEPTS1.1 Convection Heat TransferIn general, convection heat transfer deals with thermal interaction betweena surface and an adjacent moving fluid. Examples include the flow of fluidover a cylinder, inside a tube and between parallel plates. Convection alsoincludes the study of thermal interaction between fluids. An example is ajet issuing into a medium of the same or a different fluid.1.2 Important Factors in Convection Heat TransferConsider the case of the electric bulb shown inFig. 1.1. Surface temperature and heat fluxare Ts and q csc , respectively. The ambient fluidtemperature is Tf . Electrical energy is dissipated into heat at a fixed rate determined by thecapacity of the bulb. Neglecting radiation, thedissipated energy is transferred by convectionfrom the surface to the ambient fluid. Supposethat the resulting surface temperature is too highand that we wish to lower it. What are ouroptions?qcscTsxVfTf Fig. 1.1(1) Place a fan in front of the bulb and force theambient fluid to flow over the bulb.(2) Change the fluid, say, from air to a non-conducting liquid.(3) Increase the surface area by redesigning the bulb geometry.We conclude that three factors play major roles in convection heat transfer:(i) fluid motion, (ii) fluid nature, and (iii) surface geometry.Other common examples of the role of fluid motion in convection are:

21 Basic Conceptsx x x x Fanning to feel cool.Stirring a mixture of ice and water.Blowing on the surface of coffee in a cup.Orienting a car radiator to face air flow.Common to all these examples is a moving fluid which is exchanging heatwith an adjacent surface.1.3 Focal Point in Convection Heat TransferOf interest in convection heat transfer problems is the determination ofsurface heat transfer rate and/or surface temperature. These importantengineering factors are established once the temperature distribution in themoving fluid is determined. Thus the focal point in convection heat transferis the determination of the temperature distribution in a moving fluid. InCartesian coordinates this is expressed asTT ( x, y , z , t ) .(1.1)1.4 The Continuum and Thermodynamic Equilibrium ConceptsIn the previous sections we have invoked the concept of temperature andfluid velocity. The study of convection heat transfer depends on materialproperties such as density, pressure, thermal conductivity, and specificheat. These familiar properties which we can quantify and measure are infact manifestation of the molecular nature and activity of material. Allmatter is composed of molecules which are in a continuous state of randommotion and collisions. In the continuum model we ignore the characteristicsof individual molecules and instead deal with their average or macroscopiceffect. Thus, a continuum is assumed to be composed of continuous matter.This enables us to use the powerful tools of calculus to model and analyzephysical phenomena. However, there are conditions under which thecontinuum assumption breaks down. It is valid as long as there issufficiently large number of molecules in a given volume to make thestatistical average of their activities meaningful. A measure of the validityof the continuum assumption is the molecular-mean-free path O relative tothe characteristic dimension of the system under consideration. The meanfree-path is the average distance traveled by molecules before they collide.The ratio of these two length scales is called the Knudson number, Kn,defined as

1.5 Fourier’s Law of ConductionKnODe,3(1.2)where De is the characteristic length, such as the equivalent diameter orthe spacing between parallel plates. The criterion for the validity of thecontinuum assumption is [1]Kn 10 1 .(1.3a)Thus this assumption begins to break down, for example, in modelingconvection heat transfer in very small channels.Thermodynamic equilibrium depends on the collisions frequency ofmolecules with an adjacent surface. At thermodynamic equilibrium thefluid and the adjacent surface have the same velocity and temperature.This is called the no-velocity slip and no-temperature jump, respectively.The condition for thermodynamic equilibrium isKn 10 3 .(1.3b)The continuum and thermodynamic equilibrium assumptions will beinvoked throughout Chapters 1-8. Chapter 9, Convection in Microchannels,deals with applications where the assumption of thermodynamicequilibrium breaks down.1.5 Fourier’s Law of ConductionOur experience shows that if one end of a metal bar is heated, itstemperature at the other end will eventually begin to rise. This transfer ofenergy is due to molecular activity. Molecules at the hot end exchange theirkinetic and vibrational energies with neighboring layers through randommotion and collisions. A temperature gradient, or slope, is established withenergy continuously being transported in the direction of decreasingtemperature. This mode of energy transfer is called conduction. The samemechanism takes place in fluids, whether they are stationary or moving. Itis important to recognize that the mechanism for energy interchange at theinterface between a fluid and a surface is conduction. However, energytransport throughout a moving fluid is by conduction and convection.

41 Basic ConceptsWe now turn our attention toLformulating a law that will help usdetermine the rate of heat transfer byAconduction. Consider the wall shownin Fig.1.2 . The temperature of onesurface (x 0) is Tsi and of the othersurface (x L) is Tso . The wallthickness is L and its surface area isA. The remaining four surfaces areTsiTsowell insulated and thus heat isx0transferred in the x-direction only.dxAssume steady state and let q x bethe rate of heat transfer in the xFig. 1.2direction. Experiments have shownthat q x is directly proportional to Aand (Tsi Tso ) and inversely proportional to L. That isqx vqxA (Tsi Tso ).LIntroducing a proportionality constant k, we obtainqxkA (Tsi Tso ),L(1.4)where k is a property of material called thermal conductivity. We mustkeep in mind that (1.4) is valid for: (i) steady state, (ii) constant k and (iii)one-dimensional conduction.These limitations suggest that a reformulation is in order. Applying (1.4) to the element dx shown inFig.1.2 and noting that Tsi o T ( x), Tso o T ( x dx), and L is replacedby dx, we obtainqx k AT ( x dx) T ( x)T ( x) T ( x dx). k AdxdxSince T(x dx) T(x) dT, the above givesqx k AdT.dx(1.5)It is useful to introduce the term heat flux q cxc , which is defined as theheat flow rate per unit surface area normal to x. Thus,

1.6 Newton's Law of Coolingq cxcqx.A5(1.6)Therefore, in terms of heat flux, (1.5) becomesq cxc kdT.dx(1.7)Although (1.7) is based on one-dimensional conduction, it can begeneralized to three-dimensional and transient conditions by noting thatheat flow is a vector quantity. Thus, the temperature derivative in (1.7) ischanged to partial derivative and adjusted to reflect the direction of heatflow as follows:q cxc kwT,wxq cyc kwT,wyq czc kwT,wz(1.8)where x, y, and z are the rectangular coordinates. Equation (1.8) is knownas Fourier's law of conduction. Four observations are worth making: (i)The negative sign means that when the gradient is negative, heat flow is inthe positive direction, i.e., towards the direction of decreasing temperature,as dictated by the second law of thermodynamics. (ii) The conductivity kneed not be uniform since (1.8) applies at a point in the material and not toa finite region. In reality thermal conductivity varies with temperature.However, (1.8) is limited to isotropic material, i.e., k is invariant withdirection. (iii) Returning to our previous observation that the focal point inheat transfer is the determination of temperature distribution, we nowrecognize that once T(x,y,z,t) is known, the heat flux in any direction can beeasily determined by simply differentiating the function T and using (1.8).(iv) By manipulating fluid motion, temperature distribution can be altered.This results in a change in heat transfer rate, as indicated in (1.8).1.6 Newton's Law of CoolingAn alternate approach to determining heat transfer rate between a surfaceand an adjacent fluid in motion is based on Newton’s law of cooling. Usingexperimental observations by Isaac Newton, it is postulated that surfaceflux in convection is directly proportional to the difference in temperaturebetween the surface and the streaming fluid. That isq csc v Ts Tf ,

61 Basic Conceptswhere q csc is surface flux, Ts is surface temperature and Tf is the fluidtemperature far away from the surface. Introducing a proportionalityconstant to express this relationship as equality, we obtainq csch (Ts Tf ) .(1.9)This result is known as Newton's law of cooling. The constant ofproportionality h is called the heat transfer coefficient. This simple result isvery important, deserving special attention and will be examined in moredetail in the following section.1.7 The Heat Transfer Coefficient hThe heat transfer coefficient plays a major role in convection heat transfer.We make the following observations regarding h:(1) Equation (1.9) is a definition of h and not a phenomenological law.(2) Unlike thermal conductivity k, the heat transfer coefficient is not amaterial property. Rather it depends on geometry, fluid properties, motion,and in some cases temperature difference, 'T (Ts Tf ) . That ishf (geometry, fluid motion, fluid properties, 'T ) .(1.10)(3) Although no temperature distribution is explicitly indicated in (1.9), theanalytical determination of h requires knowledge of temperaturedistribution in a moving fluid.This becomes evident when bothFourier’s law and Newton’s laware combined. Application ofFourier’s law in the y-directionfor the surface shown in Fig. 1.3givesq csc kwT ( x,0, z ),wy(1.11)Fig. 1.3where y is normal to the surface, wT ( x,0, z ) / w y is temperature gradient inthe fluid at the interface, and k is the thermal conductivity of the fluid.Combining (1.9) and (1.11) and solving for h, gives

1.7 The Heat Transfer Coefficient hhwT ( x,0, z )wy. kTs Tf7(1.12)This result shows that to determine h analytically one must determinetemperature distribution.(4) Since both Fourier’s law and Newton’s law give surface heatflux, what is the advantage of introducing Newton’s law? In someapplications the analytical determination of the temperature distributionmay not be a simple task, for example, turbulent flow over a complexgeometry. In such cases one uses equation (1.9) to determine hexperimentally by measuring q csc , Ts and Tf and constructing an empiricalequation to correlate experimental data. This eliminates the need for thedetermination of temperature distribution.(5) We return now to the bulb shown in Fig. 1.1. Applying Newton’s law(1.9) and solving for surface temperature Ts , we obtainTsTf q csc.h(1.13)For specified q csc and Tf , surface temperature Ts can be altered bychanging h. This can be done by changing the fluid, surface geometryand/or fluid motion. On the other hand, for specified surface temperatureTs and ambient temperature Tf ,Table 1.1equation (1.9) shows that surfaceTypical values of hflux can be altered by changing h.2 o(6) One of the major objectives ofconvection is the determination of h.(7) Since h is not a property, itsvalues cannot be tabulated as is thecase with thermal conductivity,enthalpy, density, etc. Nevertheless,it is useful to have a rough idea ofits magnitude for common processesand fluids. Table 1.1 gives theapproximate range of h for variousconditions.ProcessFree convectionGasesLiquidsh ( W/m C)5-3020-1000Forced convectionGasesLiquidsLiquid metals20-30050-20,0005,000-50,000Phase 0

81 Basic Concepts1.8 Radiation: Stefan-Boltzmann LawRadiation energy exchange between two surfaces depends on the geometry,shape, area, orientation, and emissivity of the two surfaces. In addition, itdepends on the absorptivity D of each surface. Absorptivity is a surfaceproperty defined as the fraction of radiation energy incident on a surfacewhich is absorbed by the surface. Although the determination of the netheat exchange by radiation between two surfaces, q12 , can be complex, theanalysis is simplified for an ideal model for which the absorptivity D isequal to the emissivity H . Such an ideal surface is called a gray surface.For the special case of a gray surface which is completely enclosed by amuch larger surface, q12 is given by Stefan-Boltzmann radiation lawq12H 1V A1 (T14 T24 ) ,(1.14)where H 1 is the emissivity of the small surface, A1 its area, T1 its absolutetemperature, and T2 is the absolute temperature of the surrounding surface.Note that for this special case neither the area A2 of the large surface norits emissivity H 2 affect the result.1.9 Differential Formulation of Basic LawsThe analysis of convection heat transfer relies on the application of thethree basic laws: conservation of mass, momentum, and energy. Inaddition, Fourier’s conduction law and Newton’s law of cooling are alsoapplied. Since the focal point is the determination of temperaturedistribution, the three basic laws must be cast in an appropriate form thatlends itself to the determination of temperature distribution. This castingprocess is called formulation. Various formulation procedures areavailable. They include differential, integral, variational, and finitedifference formulation. This section deals with differential formulation.Integral formulation is presented in Chapter 5.Differential formulation is based on the key assumption of continuum.This assumption ignores the molecular structure of material and focuses onthe gross effect of molecular activity. Based on this assumption, fluids aremodeled as continuous matter. This makes it possible to treat variablessuch as temperature, pressure, and velocity as continuous function in thedomain of interest.

1.10 Mathematical Background1.10 Mathematical BackgroundyWe review the following mathematicaldefinitions which are needed in the differentialformulation of the basic laws.&Vu i v j wk .vuw&(a) Velocity Vector V . Let u, v, and w be thevelocity components in the& x, y and z directions,respectively. The vector V is given by9xz(1.15a)Fig. 1.4(b) Velocity Derivative. The derivative of the velocity vector with respectto any one of the three independent variables is given by&wVwxwuwvwwi j k.wxwxwx(1.15b)(c) The Operator . In Cartesian coordinates the operator is a vectordefined as {wwwi j k.wxwywz(1.16)In cylindrical coordinates this operator takes the following form {ww1 wir iT i z .wrr wTwz(1.17)Similarly, the form in spherical coordinate is {1 w1wwir iT iI .r wTr sin T wIwr(1.18)&(d) Divergence of a Vector. The divergence of a vector V is a scalardefined as&&div.V { Vwu wv ww. wx wy wz(1.19)

101 Basic Concepts(e) Derivative of the Divergence. The derivative of the divergence withrespect to any one of the three independent variables is given by&w Vwxw § wu wv ww · . wx wx wy wz ¹(1.20)The right hand side of (1.20) represents the divergence of the derivative of&the vector V . Thus (1.20) can be rewritten as&w Vwx wui v j wk ,wxor&w Vwx&wV .wx(1.21)(f) Gradient of Scalar. The gradient of a scalar, such as temperature T, is avector given byGrad T { TwTwTwTi j k.wzwxwy(1.22)(g) Total Differential and Total Derivative. We consider a variable ofthe flow field designated by the symbol f. This is a scalar quantity such astemperature T, pressure p, density U , or velocity component u. In generalthis quantity is a function of the four independent variables x, y, z and t.Thus in Cartesian coordinates we writeff ( x, y , z , t ) .(a)The total differential of f is the total change in f resulting from changes inx, y, z and t. Thus, using (a)dfDividing through by dtwfwfwfwfdx dy dz dt .wtwzwywx

1.10 Mathematical Backgroundwf dx wf dy wf dz wf .wx dt wy dt wz dt tuting (c) into (b)dfdtDfDtuwf wfwfwf. w vwz wtwywx(1.23)df / dt in the above is called the total derivative. It is also written asDf / Dt , to emphasize that it represents the change in f which resultsfrom changes in the four independent variables. It is also referred to as thesubstantial derivative. Note that the first three terms on the right hand sideare associated with motion and are referred to as the convective derivative.The last term represents changes in f with respect to time and is called thelocal derivative. Thusuwfwfwf w vwzwywxwfwtconvective derivative,(d)local derivative.(e)To appreciate the physical significance of (1.23), we apply it to the velocitycomponent u. Setting f u in (1.23) givesdudtDuDtuwu wuwuwu. w vwz wtwywx(1.24)Following (d) and (e) format, (1.24) representsuwuwuww v wwxwywzconvective acceleration in the x-direction,(f)

121 Basic Conceptswulocal acceleration .wt(g)Similarly, (1.23) can be applied to the y and z directions to obtain thecorresponding total acceleration in these directions.The three components of the total acceleration in the cylindricalcoordinates r ,T , z aredv rdtdvTdtDv rDtDvTDtdv zdtwvwvwv r vT wv r vT2 v z r r , (1.25a) r wTrwtwzwrvrvrDv zDtwvwvwvT vT wvT v r vT v z T T , (1.25b) wzwrr wTrwtvrwv z vT wv zwvwv vz z z .wrwzr wTwt(1.25c)Another example of total derivative is obtained by setting f T in(1.23) to obtain the total temperature derivativedTdtDTDtuwTwTwT wT v w. wxwywz wt(1.26)1.11 UnitsSI units are used throughout this text. The basic units in this system are:Length (L): meter (m).Time (t): second (s).Mass (m): kilogram (kg).Temperature (T): kelvin (K).Temperature on the Celsius scale is related to the kelvin scale byT(oC) T(K) - 273.15.(1.27)Note that temperature difference on the two scales is identical. Thus, achange of one kelvin is equal to a change of one Celsius. This means thatquantities that are expressed per unit kelvin, such as thermal conductivity,

1.12 Problem Solving Format13heat transfer coefficient, and specific heat, are numerically the same as perdegree Celsius. That is, W/m2-K W/ m2-oC.The basic units are used

1.1 Convection Heat Transfer 1 1.2 Important Factors in Convection Heat Transfer 1 1.3 Focal Point in Convection Heat Transfer 2 1.4 The Continuum and Thermodynamic Equilibrium Concepts 2 1.5 Fourier’s Law of Conduction 3 1.6 Newton