How does Convection Heat transfer work

how to calculate convective heat transfer coefficient and convective heat transfer laminar and turbulent flow convective heat transfer lecture notes
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Chapter19 Convective Heat Transfer Heat transfer by convection is associated with energy exchange between a surface and an adjacent fluid. There are very few energy-transfer situations of practical importance in which fluid motion is not in some way involved. This effect has been eliminated as much as possible in the preceding chapters, but will now be considered in some depth. The rate equation for convection has been expressed previously as q ¼ hDT (15-11) A where the heat flux, q/A, occurs by virtue of a temperature difference. This simple equation is the defining relation for h, the convective heat-transfer coefficient. The determination of the coefficient h is, however, not at all a simple undertaking. It is related to the mechanism of fluid flow, the properties of the fluid, and the geometry of the specific system of interest. In light of the intimate involvement between the convective heat-transfer coefficient and fluid motion, we may expect many of the considerations from the momentum transfer to be of interest. In the analyses to follow, much use will be made of the developments and concepts of Chapters 4 through 14. 19.1 FUNDAMENTAL CONSIDERATIONS IN CONVECTIVE HEAT TRANSFER AsmentionedinChapter12,thefluidparticlesimmediatelyadjacenttoasolidboundaryare stationary,andathinlayeroffluidclosetothesurfacewillbeinlaminarflowregardlessof the nature of the free stream.Thus, molecular energyexchange or conduction effects will alwaysbepresent,andplayamajorroleinanyconvectionprocess.Iffluidflowislaminar, then all energy transfer between a surface and contacting fluid or between adjacent fluid layersisbymolecularmeans.If,onthecontrary,flowisturbulent,thenthereisbulkmixing of fluid particles between regions at different temperatures, and the heat transfer rate is increased. The distinction between laminar and turbulent flow will thus be a major consideration in any convective situation. There are two main classifications of convective heat transfer. These have to do with the driving force causing fluid to flow. Natural or free convection designates the type of processwhereinfluidmotionresultsfromtheheattransfer.Whenafluidisheatedorcooled, the associated density change and buoyant effect produce a natural circulation in which the affected fluid moves of its own accord past the solid surface, the fluid that replaces it is similarly affected by the energy transfer, and the process is repeated. Forced con- vection is the classification used to describe those convection situations in which fluid circulation is produced by an external agency such as a fan or a pump. 27419.2 Significant Parameters in Convective Heat Transfer 275 The hydrodynamic boundary layer, analyzed in Chapter 12, plays a major role in convectiveheattransfer,asonewouldexpect.Additionally,weshalldefineandanalyzethe thermal boundary layer, which will also be vital to the analysis of a convective energy- transfer process. Therearefourmethodsofevaluatingtheconvectiveheat-transfercoefficientthatwill be discussed in this book. These are as follows: (a) dimensional analysis, which to be useful requires experimental results; (b) exact analysis of the boundary layer; (c) approximate integral analysis of the boundary layer; and (d) analogy between energy and momentum transfer. 19.2 SIGNIFICANT PARAMETERS IN CONVECTIVE HEAT TRANSFER Certain parameters will be found useful in the correlation of convective data and in the functional relations for the convective heat-transfer coefficients. Some parameters of this type have been encountered earlier; these include the Reynolds and the Euler numbers. Several of the new parameters to be encountered in energy transfer will arise in such a mannerthattheirphysicalmeaningisunclear.Forthisreason,weshalldevoteashortsection to the physical interpretation of two such terms. Themoleculardiffusivitiesofmomentumandenergyhavebeendefinedpreviouslyas m momentum diffusivity : n  r and k thermal diffusivity : a  rc p Thatthesetwoaredesignatedsimilarlywouldindicatethattheymustalsoplaysimilarroles intheirspecifictransfermodes.Thisisindeedthecase,asweshallseeseveraltimesinthe developmentstofollow.Forthemomentweshouldnotethatbothhavethesamedimensions, 2 those of L /t; thus their ratio must be dimensionless. This ratio, that of the molecular diffusivity of momentum to the molecular diffusivity of heat, is designated the Prandtl number. mc n p Pr  ¼ (19-1) a k ThePrandtlnumberisobservedtobeacombinationoffluidproperties;thusPritselfmay be thought of as a property. The Prandtl number is primarily a function of temperature and is tabulated in Appendix I, at various temperatures for each fluid listed. ThetemperatureprofileforafluidflowingpastasurfaceisdepictedinFigure19.1.In thefigure,thesurfaceisatahighertemperaturethanthefluid.Thetemperatureprofilethat existsisduetotheenergyexchangeresultingfromthistemperaturedifference.Forsucha case the heat-transfer rate between the surface and the fluid may be written as q ¼ hA(T T ) (19-2) y s 1 and, because heat transfer at the surface is by conduction q ¼kA (TT )j (19-3) y s y¼0 y276 Chapter 19 Convective Heat Transfer v ∞ T – T s ∞ v x T – T s y Figure 19.1 Temperature and velocity profiles for a fluid flowing past a heated plate. T s These two terms must be equal; thus h(T T )¼k (TT )j s 1 s y¼0 y which may be rearranged to give (T T)/yj h s y¼0 ¼ (19-4) k T T s 1 Equation (19-4) may be made dimensionless if a length parameter is introduced. Multiplying both sides by a representative length, L, we have (T T)/yj hL s y¼0 ¼ (19-5) k (T T )/L s 1 The right-hand side of equation (19-5) is now the ratio of the temperature gradient at the surfacetoanoverallorreferencetemperaturegradient.Theleft-handsideofthisequation is written in a manner similar to that for the Biot modulus encountered in Chapter 18. It may be considered a ratio of conductive thermal resistance to the convective thermal resistance of the fluid. This ratio is referred to as the Nusselt number hL Nu (19-6) k where the thermal conductivity is that of the fluid as opposed to that of the solid, which was the case in the evaluation of the Biot modulus. These two parameters, Pr and Nu, will be encountered many times in the work to follow. 19.3 DIMENSIONAL ANALYSIS OF CONVECTIVE ENERGY TRANSFER ForcedConvection. Thespecificforced-convectionsituation,whichweshallnowconsi- der,isthatoffluidflowinginaclosedconduitatsomeaveragevelocity,v,withatemperature difference existing between the fluid and the tube wall. The important variables, their symbols, and dimensional representations are listed below.Itisnecessarytoincludetwomoredimensions—Q,heat,andT,temperature—tothe fundamental group considered in Chapter 11; thus all variables must be expressed dimensionally as some combination of M, L, t, Q, and T. The above variables include termsdescriptiveofthesystemgeometry,thermalandflowpropertiesofthefluid,andthe quantity of primary interest, h.19.3 Dimensional Analysis of Convective Energy Transfer 277 Variable Symbol Dimensions Tube diameter DL 3 Fluid density r M/L Fluid viscosity m M/Lt Fluid heat capacity c Q/MT p Fluid thermal conductivity kQ/tLT Velocity v L/t 2 Heat-transfer coefficient hQ/tL T UtilizingtheBuckinghammethodofgroupingthevariablesaspresentedinChapter 11, the required number of dimensionless groups is found to be 3. Note that the rank of thedimensionalmatrixis4,onemorethanthetotalnumberoffundamentaldimensions. ChoosingD,k,m,andvasthefourvariablescomprisingthecore,wefindthatthethree p groups to be formed are a b c d p ¼ D k m v r 1 e f g h p ¼ D km v c p 2 and i j k l p ¼ D k m v h 3 Writing p in dimensional form 1  b c d Q M L M a 1¼ðLÞ 3 LtT Lt t L andequatingtheexponentsofthefundamentaldimensionsonbothsidesofthisequation,we have for L: 0¼ abcþd3 Q: 0¼ b t: 0¼bcd T: 0¼b and M: 0¼ cþ1 Solving these equations for the four unknowns yields a¼ 1 c¼1 b¼ 0 d¼ 1 and p becomes 1 Dvr p ¼ 1 m which is the Reynolds number. Solving for p and p in the same way will give 2 3 mc hD p p ¼ ¼ Pr and p ¼ ¼ Nu 2 3 k k278 Chapter 19 Convective Heat Transfer Theresultofadimensionalanalysisofforced-convectionheattransferinacircularconduit indicates that a possible relation correlating the important variables is of the form Nu¼ f (Re, Pr) (19-7) 1 If, in the preceding case, the core group had been chosen to include r,m,c , and v, the p analysis would have yielded the groups Dvr/m,mc /k, andh/rvc : The first two of these p p we recognize as Re and Pr. The third is the Stanton number. h St (19-8) rvc p This parameter could also have been formed by taking the ratio Nu/ðRe PrÞ.An alternative correlating relation for forced convection in a closed conduit is thus St¼ f (Re, Pr) (19-9) 2 NaturalConvection. Inthecaseofnatural-convectionheattransferfromaverticalplane wall to an adjacent fluid, the variables will differ significantly from those used in the precedingcase.Thevelocitynolongerbelongsinthegroupofvariables,asitisaresultof othereffectsassociatedwiththeenergytransfer.Newvariablestobeincludedintheanalysis arethoseaccountingforfluidcirculation.Theymaybefoundbyconsideringtherelationfor buoyant force in terms of the density difference due to the energy exchange. The coefficient of thermal expansion, b, is given by r¼ r (1bDT) (19-10) 0 wherer is the bulk fluid density,r is the fluid density inside the heated layer, andDT is 0 thetemperaturedifferencebetweentheheatedfluidandthebulkvalue.Thebuoyantforce per unit volume, F ,is buoyant F ¼ (r r)g buoyant 0 which becomes, upon substituting equation (19-10) F ¼ bgr DT (19-11) buoyant 0 Equation (19-11) suggests the inclusion of thevariables b,g,andDT into the list of those important to the natural convection situation. The list of variables for the problem under consideration is given below. Variable Symbol Dimensions Significant length LL 3 Fluid density r M/L Fluid viscosity m M/Lt Fluid heat capacity c Q/MT p Fluid thermal conductivity kQ/LtT Fluid coefficient of thermal expansion b 1/T 2 Gravitational acceleration gL/t Temperature difference DTT 2 Heat-transfer coefficient hQ/L tT19.4 Exact Analysis of the Laminar Boundary Layer 279 The Buckingham p theorem indicates that the number of independent dimensionless parameters applicable to this problem is 95¼ 4. Choosing L,m,k,g,andb as the core group, we see that the p groups to be formed are a b c d e p ¼ L m k b g c 1 p f g h i j p ¼ Lm k b g r 2 k l m n o p ¼ L m k b g DT 3 and p q r s t p ¼ L m k b g h 4 Solving for the exponents in the usual way, we obtain mc p p ¼ ¼ Pr p ¼ bDT 1 3 k 3 2 L gr hL p ¼ and p ¼ ¼ Nu 2 4 2 m k 2 3 2 The product of p andp , which must be dimensionless, is (bgr L DT)/m . This para- 2 3 meter, used in correlating natural-convection data, is the Grashof number. 2 3 bgr L DT Gr (19-12) 2 m From the preceding brief dimensional-analyses considerations, we have obtained the following possible forms for correlating convection data: (a) Forced convection Nu¼ f (Re, Pr) (19-7) 1 or St¼ f (Re, Pr) (19-9) 2 (b) Natural convection Nu¼ f (Gr, Pr) (19-13) 3 The similarity between the correlations of equations (19-7) and (19-13) is apparent. In equation(19-13),GrhasreplacedReinthecorrelationindicatedbyequation(19-7).Itshould benotedthattheStantonnumbercanbeusedonlyincorrelatingforced-convectiondata.This becomes obvious when we observe the velocity, v, contained in the expression for St. 19.4 EXACTANALYSIS OF THE LAMINAR BOUNDARY LAYER An exact solution for a special case of the hydrodynamic boundary layer is discussed in Section12.5.Blasius’ssolutionforthelaminarboundarylayeronaflatplatemaybeextended to include the convective heat-transfer problem for the same geometry and laminar flow. The boundary-layer equations considered previously include the two-dimensional, incompressible continuity equation v v x y þ ¼ 0 (12-10) x y280 Chapter 19 Convective Heat Transfer and the equation of motion in the x direction 2 v v v dv v x x x 1 x þv þv ¼ v þn (12-9) x y 1 2 t x y dx y Recall that the y-directional equation of motion gave the result of constant pressure throughthe boundary layer. The proper form of theenergyequationwillthusbe equation (16-14), for isobaric flow, written in two-dimensional form as  2 2 T T T T T þv þv ¼ a þ (19-14) x y 2 2 t x y x y 2 2 With respect to the thermal boundary layer depicted in Figure 19.2, T/x is much 2 2 smaller in magnitude than T/y : T ∞ Edge of thermal boundary layer T = T(y) x y Figure 19.2 The thermal boundary layer for laminar flow past a flat T s surface. In steady, incompressible, two-dimensional, isobaric flow the energy equation that applies is now 2 T T T v þv ¼ a (19-15) x y 2 x y y From Chapter 12, the applicable equation of motion with uniform free-stream velocity is 2 v v v x x x v þv ¼ n (12-11a) x y 2 x y y and the continuity equation v v x y þ ¼ 0 (12-11b) x y The lattertwo oftheaboveequationswereoriginallysolvedbyBlasius togivetheresults discussed in Chapter 12. The solution was based upon the boundary conditions v v x y ¼ ¼0at y¼ 0 v v 1 1 and v x ¼1at y¼1 v 119.4 Exact Analysis of the Laminar Boundary Layer 281 The similarity in form between equations (19-15) and (12-11a) is obvious. This situation suggests the possibility of applying the Blasius solution to the energy equation. In order that this be possible, the following conditions must be satisfied: (1) Thecoefficients ofthesecond-ordertermsmustbeequal.Thisrequiresthatn¼ a or that Pr¼ 1. (2) The boundary conditions for temperature must be compatible with those for the velocity.ThismaybeaccomplishedbychangingthedependentvariablefromT to (TT )/(T T ). The boundary conditions now are s 1 s v v TT x y s ¼ ¼ ¼0at y¼ 0 v v T T 1 1 1 s v TT x s ¼ ¼1at y¼1 v T T 1 1 s Imposing these conditions upon the set of equations (19-15) and (12-11a), we may now writetheresultsobtainedbyBlasiusfortheenergy-transfercase.Usingthenomenclatureof Chapter 12, v TT x s 0 f ¼ 2 ¼ 2 (19-16) v T T 1 1 s rffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi y v y xv y 1 1 h¼ ¼ ¼ Re (19-17) x 2 nx 2x n 2x and applying the Blasius result, we obtain    0  df d½2(v /v ) x 1  00  pffiffiffiffiffiffiffiffi ¼ f (0)¼   dh  d½(y/2x) Re  x y¼0 y¼0   df2½(TT )/(T T )g s 1 s  ¼ pffiffiffiffiffiffiffiffi ¼ 1:328 ð19-18Þ   d½(y/2x) Re  x y¼0 It should be noted that according to equation (19-16), the dimensionless velocity profile in the laminar boundary layer is identical with the dimensionless temperature pro- file. This is a consequence of having Pr¼ 1: A logical consequence of this situation is thatthehydrodynamicandthermalboundarylayersareofequalthickness.Itissignificant that the Prandtl numbers for most gases are sufficiently close to unity that the hydro- dynamic and thermal boundary layers are of similar extent. We may now obtain the temperature gradient at the surface    T 0:332 1/2  ¼ (T T ) Re (19-19) y¼0 1 s x  y x Application of the Newton and Fourier rate equations now yields   q T y  ¼ h (T T )¼k x s 1  A y y¼0 from which   k T 0:332k  1/2 h ¼  ¼ Re (19-20) x x  T T y x s 1 y¼0282 Chapter 19 Convective Heat Transfer or h x x 1/2 ¼ Nu ¼ 0:332Re (19-21) x x k 1 Pohlhausen considered the same problem with the additional effect of a Prandtl number other than unity. He was able to show the relation between the thermal and hydrodynamic boundary layers in laminar flow to be approximately given by d 1/3 ¼ Pr (19-22) d t 1/3 The additional factor of Pr multiplied byh allows the solutionto the thermal boundary layertobeextendedtoPrvaluesotherthanunity.Aplotofthedimensionlesstemperature 1/3 vs. h Pr is shown in Figure 19.3. The temperature variation given in this form leads to an expression for the convective heat-transfer coefficient similar to equation (19-20). At y¼ 0, the gradient is    T 0:332 1/2 1/3  ¼ (T T ) Re Pr (19-23) 1 s x  y x y¼0 Slope = 1.328 Figure 19.3 Temperature variation for laminar flow over a flat plate. 1/3 (y/2x)Re Pr x which, when used with the Fourier and Newton rate equations, yields k 1/2 1/3 h ¼ 0:332 Re Pr (19-24) x x x or h x x 1/2 1/3 ¼ Nu ¼ 0:332Re Pr (19-25) x x k 1/3 The inclusion of the factor Pr in these equations extends the range of application of equations (19-20) and (19-21) to situations in which the Prandtl number differs considerably from 1. 1 E. Pohlhausen, ZAMM, 1, 115 (1921). f' = 2 (T – T ) / (T – T ) s ∞ s19.5 Approximate Integral Analysis of the Thermal Boundary Layer 283 Themeanheat-transfercoefficientapplyingoveraplateofwidthwandlengthLmaybe obtained by integration. For a plate of these dimensions Z q ¼ hA(T T )¼ h (T T )dA y s 1 x s 1 A Z L 1/2 Re 1/3 x h(wL)(T T )¼ 0:332kwPr (T T ) dx s 1 s 1 x 0   Z 1/2 L v r 1 1/3 1/2 hL¼ 0:332kPr x dx m 0  1/2 v r 1 1/3 1/2 ¼ 0:664kPr L m 1/3 1/2 ¼ 0:664kPr Re L The mean Nusselt number becomes hL 1/3 1/2 Nu ¼ ¼ 0:664 Pr Re (19-26) L L k and it is seen that Nu ¼2Nu atx¼ L (19-27) L x In applying the results of the foregoing analysis it is customary to evaluate all fluid properties at the film temperature, which is defined as T þT s 1 T ¼ (19-28) f 2 the arithmetic mean between the wall and bulk fluid temperatures. 19.5 APPROXIMATE INTEGRAL ANALYSIS OF THE THERMAL BOUNDARY LAYER The application of the Blasius solution to the thermal boundary layer in Section 19.4 was convenientalthoughverylimitedinscope.Forflowotherthanlaminarorforaconfiguration other than a flat surface, another method must be utilized to estimate the convective heat- transfer coefficient. An approximate method for analysis of the thermal boundary layer employstheintegralanalysisasusedbyvonKa´rma´nforthehydrodynamicboundarylayer. This approach is discussed in Chapter 12. ConsiderthecontrolvolumedesignatedbythedashedlinesinFigure19.4,applyingto flowparalleltoaflatsurfacewithnopressuregradient,havingwidthDx;aheightequaltothe q 3 q q 1 2 δ ∆ x th y Figure 19.4 Control volume q 4 x for integral energy analysis.284 Chapter 19 Convective Heat Transfer thicknessofthethermalboundarylayer,d ,andaunitdepth.Anapplicationofthefirstlawof t thermodynamics in integral form ZZ ZZZ dQ dW dW s m :   ¼ (eþP/r)r(v n) dAþ erdV (6-10) dt dt dt t c:s: c:v: yields the following under steady-state conditions:   dQ T  ¼kDx  dt y y¼0 dW dW s m ¼ ¼ 0 dt dt  ZZ Z d 2 t  v P x  : (eþP/r)r(v n) dA¼ þgyþuþ rv dy x  2 r c:s: 0 xþDx  Z d 2 t  v P x   þgyþuþ rv dy x  2 r 0 x   Z d 2 t  d v P x   rv þgyþuþ dy Dx x  dx 2 r 0 d t and ZZZ erdV ¼ 0 t c:v: Intheabsenceofsignificantgravitationaleffects,theconvective-energy-fluxtermsbecome 2 v P x þuþ ¼ h ’ c T 0 p 0 2 r where h is the stagnation enthalpy and c is the constant-pressure heat capacity. The 0 p stagnation temperature will now be written merely as T (without subscript) to avoid confusion. The complete energy expression is now    Z Z Z d d d t t t    T d    k Dx ¼ rv c Tdy  rv c Tdy rc Dx v T dy x p x p p x 1    y dx 0 0 0 y¼0 xþDx x ð19-29Þ Equation (19-29) can also be written as q ¼ q q q where these quantities are 4 2 1 3 shown in Figure 19.4. In equation (19-29), T represents the free-stream stagnation 1 temperature. If flow is incompressible, and an average value of c is used, the product p rc may be taken outside the integral terms in this equation. Dividing both sides of p equation (19-29) byDx and evaluating the result in the limit as byDx approaches zero, we obtain  Z  d t k T d   ¼ v (T T)dy (19-30) x 1  rc y dx p 0 y¼0 Equation(19-30)isanalogoustothemomentumintegralrelation,equation(12-37),withthe momentumtermsreplacedbytheirappropriateenergycounterparts.Thisequationmaybe solved if both velocity and temperature profile are known. Thus, for the energy equation both the variation in v and in Twith y must be assumed. This contrasts slightly with the x momentum integral solution in which the velocity profile alone was assumed.19.6 Energy- and Momentum-Transfer Analogies 285 An assumed temperature profile must satisfy the boundary conditions (1) TT ¼0at y¼ 0 s (2) TT ¼ T T at y¼ d s 1 s t (3) (TT )¼0at y¼ d s t y 2 (4) (TT )¼0at y¼ 0½see equation(19-15) s 2 y If a power-series expression for the temperature variation is assumed in the form 2 3 TT ¼ aþbyþcy þdy s the application of the boundary conditions will result in the expression for TT s   3 TT 3 y 1 y s ¼  (19-31) T T 2 d 2 d 1 s t t If the velocity profile is assumed in the same form, then the resulting expression, as obtained in Chapter 12, is  3 v 3 y 1 y ¼  (12-40) v 2 d 2 d 1 Substituting equations (19-31) and (12-40) into the integral expression and solving, we obtain the result 1/2 1/3 Nu ¼ 0:36Re Pr (19-32) x x which is approximately 8% larger than the exact result expressed in equation (19-25). Thisresult,althoughinexact,issufficientlyclosetotheknownvaluetoindicatethatthe integralmethodmaybeusedwithconfidenceinsituationsinwhichanexactsolutionisnot known.Itisinterestingtonotethatequation(19-32)againinvolvestheparameterspredicted from dimensional analysis. A condition ofconsiderableimportance is that ofanunheatedstarting length. Problem 19.17attheendofthechapterdealswiththissituationwherethewalltemperature,T,isrelated s to the distance from the leading edge, x, and the unheated starting, length X, according to T ¼ T for 0 x X s 1 and T T for X x s 1 Theintegraltechnique,aspresentedinthissection,hasprovedeffectiveingeneratinga modified solution for this situation. The result for T ¼ constant, and assuming both the s hydrodynamic and temperature profiles to be cubic, is "1 3 Pr Nu ffi0:33 Re (19-33) x x 3/4 1(X/x) Note that this expression reduces to equations (19-25) for X=0 19.6 ENERGY- AND MOMENTUM-TRANSFER ANALOGIES Manytimesinour considerationof heat transferthus farwehavenotedthesimilaritiesto momentum transfer both in the transfer mechanism itself and in the manner of its quantitative description. This section will deal with these analogies and use them to develop relations to describe energy transfer.286 Chapter 19 Convective Heat Transfer Osborne Reynolds first noted the similarities in mechanism between energy and 2 3 momentum transfer in 1874. In 1883, he presented the results of his work on frictional resistancetofluidflowinconduits,thusmakingpossiblethequantitativeanalogybetween the two transport phenomena. As we have noted in the previous sections, for flow past a solid surface with a Prandtl numberofunity,thedimensionlessvelocityandtemperaturegradientsarerelatedasfollows:     d v d TT x s   ¼ (19-34)   dy v dy T T y¼0 1 1 s y¼0 For Pr¼ mc /k¼ 1; we have mc /k and we may write equation (19-34) as p p     d v d TT x s   mc ¼ k p   dy v dy T T 1 1 s y¼0 y¼0 which may be transformed to the form     mc dv k d p x   ¼ (TT ) (19-35) s   v dy T T dy 1 s 1 y¼0 y¼0 Recalling a previous relation for the convective heat-transfer coefficient   h d (T T) s  ¼ (19-4)  k dy (T T ) s 1 y¼0 it is seen that the entire right-hand side of equation (19-34) may be replaced by h, giving   mc dv p x h¼  (19-36)  v dy 1 y¼0 Introducing next the coefficient of skin friction   t 2m dv x 0  C ffi ¼ f  2 2 rv /2 rv dy 1 1 y¼0 we may write equation (19-36) as C f h¼ (rv c ) 1 p 2 which, in dimensionless form, becomes h C f St¼ (19-37) rv c 2 1 p Equation (19-37) is the Reynolds analogy and is an excellent example of the similar nature of energy and momentum transfer. For those situations satisfying the basis for the development of equation (19-37), a knowledge of the coefficient of frictional drag will enable the convective heat-transfer coefficient to be readily evaluated. The restrictions on the use of the Reynolds analogy should be kept in mind; they are (1)Pr¼ 1and(2)noformdrag.Theformerofthesewasthestartingpointinthepreceding developmentandobviouslymustbesatisfied.Thelatterissensiblewhenoneconsidersthat, in relating two transfer mechanisms, the manner of expressing them quantitatively must 2 O. Reynolds, Proc. Manchester Lit. Phil. Soc., 14:7 (1874). 3 O. Reynolds, Trans. Roy. Soc. (London, 174A, 935 (1883).19.7 Turbulent Flow Considerations 287 remain consistent. Obviously the description of drag in terms of the coefficient of skin friction requires that the drag be wholly viscous in nature. Thus, equation (19-37) is applicableonlyforthosesituationsinwhichformdragisnotpresent.Somepossibleareasof applicationwouldbeflowparalleltoplanesurfacesorflowinconduits.Thecoefficientof skinfrictionforconduitflowhasalreadybeenshowntobeequivalenttotheFanningfiction factor, which may be evaluated by using Figure 14.1. 4 The restriction that Pr¼ 1 makes the Reynolds analogy of limited use. Colburn has suggested a simple variation of the Reynolds analogy form that allows its application to situationswherethePrandtlnumberisotherthanunity.TheColburnanalogyexpressionis C f 2/3 St Pr ¼ (19-38) 2 which obviously reduces to the Reynolds analogy when Pr¼ 1. Colburn applied this expression to a wide range of data for flow and geometries of differenttypesandfoundittobequiteaccurateforconditionswhere(1)noformdragexists, and(2)0:5 Pr 50.ThePrandtlnumberrangeisextendedtoincludegases,water,and severalotherliquidsofinterest.TheColburnanalogyisparticularlyhelpfulforevaluating heattransferininternalforcedflows.Itcanbeeasilyshownthattheexactexpressionfora laminar boundary layer on a flat plate reduces to equation (19-38). The Colburn analogy is often written as C f j ¼ (19-39) H 2 where 2/3 j ¼ St Pr (19-40) H isdesignatedtheColburnjfactorforheattransfer.Amass-transferjfactor,isdiscussedin Chapter 28. NotethatforPr¼ 1,theColburnandReynoldsanalogiesarethesame.Equation(19-38) is thus an extension of the Reynolds analogy for fluids having Prandtl numbers other than unity,withintherange0.5–50asspecifiedabove.HighandlowPrandtlnumberfluidsfalling outside this range would be heavy oils at one extreme and liquid metals at the other. 19.7 TURBULENT FLOW CONSIDERATIONS Theeffectoftheturbulentflowonenergytransferisdirectlyanalogoustothesimilareffects on momentum transfer as discussed in Chapter 12. Consider the temperature profile variationinFigure19.5toexistinturbulentflow.Thedistancemovedbyafluid‘‘packet’’ intheydirection,whichisnormaltothedirectionofbulkflow,isdenotedbyL,thePrandtl mixing length. The packet of fluid moving through the distance L retains the mean temperature from its point of origin, and upon reaching its destination, the packet will differintemperaturefromthatoftheadjacentfluidbyanamountTj Tj :Themixing yL y length is assumed small enough to permit the temperature difference to be written as   dt  Tj Tj ¼L (19-41) yL y  dy y 0 We now define the quantity T as the fluctuating temperature, synonymous with 0 the fluctuating velocity component, v , described in Chapter 12. The instantaneous x 4 A. P. Colburn, Trans. A.I.Ch.E., 29, 174 (1933).288 Chapter 19 Convective Heat Transfer y T T L T' T T t (a) (b) Figure 19.5 Turbulent-flow temperature variation. temperatureisthesumofthemeanandfluctuatingvalues,asindicatedinFigure19.5(b),or, in equation form 0 T ¼ TþT (19-42) Anysignificantamountofenergytransferintheydirection,forbulkflowoccurringinthe 0 xdirection,isaccomplishedbecauseofthefluctuatingtemperature,T ;thus,itisapparent from equations (19-41) and (19-42) that dT 0 T ¼L (19-43) dy The energy flux in the y direction may now be written as  q y 0  ¼ rc Tv (19-44) p y y A 0 wherev maybeeitherpositiveornegative.SubstitutingforTitsequivalent,accordingto y equation (19-42)  q  y 0 0  ¼ rc v (TþT ) p y y A and taking the time average, we obtain, for the y-directional energy flux due to turbulent effects  q y 0 0 ¼ rc (v T ) (19-45)  p y A turb 0 or, with T in terms of the mixing length   q dT y 0 ¼ rc v L (19-46)  p y A dy turb The total energy flux due to both microscopic and turbulent contributions may be written as q dT y 0 ¼rc ½aþjv Lj (19-47) p y A dy 0 As a is the molecular diffusivity of heat, the quantityjv Lj is the eddy diffusivity of heat, y designated as e . This quantity is exactly analogous to the eddy diffusivity of H momentum, e ; as defined in equation (12-52). In a region of turbulent flow, e a for M H all fluids except liquid metals.19.7 Turbulent Flow Considerations 289 AsthePrandtlnumberistheratioofthemoleculardiffusivitiesofmomentumandheat, an analogous term, the turbulent Prandtl number, can be formed by the ratio e /e . M H Utilizing equations (19-47) and (12-55), we have 2 2 e L jdv /dyj L jdv /dyj M x x Pr ¼ ¼ ¼ ¼ 1 (19-48) turb 0 2 e jLv j L jdv /dyj H x y Thus, in a region of fully turbulent flow the effective Prandtl number is unity, and the Reynolds analogy applies in the absence of form drag. In terms of the eddy diffusivity of heat, the heat flux can be expressed as   q DT y  ¼rc e (19-49) p H  turb A dy The total heat flux, including both molecular and turbulent contributions, thus becomes q dT y ¼rc (aþ e ) (19-50) p H A dy Equation (19-50) applies both to the region wherein flow is laminar, for which a e , and to that for which flow is turbulentand e a. It is in this latter region that H H 5 the Reynolds analogy applies.Prandtl achieveda solution that includesthe influences of both the laminar sublayer and the turbulent core. In his analysis solutions were obtained in each region and then joined at y¼ j, the hypothetical distance from the wall that is assumed to be the boundary separating the two regions. Within the laminar sublayer the momentum and heat flux equations reduce to dv x t¼ rv (a constant) dy and q dT y ¼rc a p A dy Separatingvariablesandintegratingbetweeny¼ 0andy¼ j,wehave,forthemomentum expression Z Z v j j x j t dv ¼ dy x rv 0 0 and for the heat flux Z Z T j j q y dT¼ dy Arc a p T 0 s Solving for the velocity and temperature profiles in the laminar sublayer yields tj v j ¼ (19-51) x j rv and q j y T T ¼ (19-52) s j Arc a p 5 L. Prandtl, Zeit. Physik., 11, 1072 (1910).290 Chapter 19 Convective Heat Transfer Eliminating the distance j between these two expressions gives rvv j x rAc a j p ¼ (T T ) (19-53) s j t q y Directing our attention now to the turbulent core where the Reynolds analogy applies, we may write equation (19-37) h C f ¼ (19-37) rc (v v j ) 2 p 1 x j and, expressing h and C in terms of their defining relations, we obtain f q /A t y ¼ 2 rc (v v j )(T T ) r(v v j ) p 1 x j 1 j 1 x j Simplifying and rearranging this expression, we have r(v v j ) 1 x (T T ) 1 j j ¼ rAc (19-54) p t q y which is a modified form of the Reynolds analogy applying from y¼ jtoy¼ y : max Eliminating T between equations (19-53) and (19-54), we have j hi  r v rAc p v þv j 1 ¼ (T T ) (19-55) 1 x s 1 j t a q y Introducing the coefficient of skin friction t C ¼ f 2 rv /2 1 and the convective heat-transfer coefficient q y h¼ A(T T ) s 1 we may reduce equation (19-54) to v þv j (v/a1) 1 x rc j p ¼ 2 v C /2 h f 1 Inverting both sides of this expression and making it dimensionless, we obtain C /2 h f  St¼ (19-56) rc v 1þ(v j /v )½(v/a)1 p 1 x 1 j This equation involves the ratio v/a, which has been defined previously as the Prandtl number. For a value of Pr¼ 1, equation (19-56) reduces to the Reynolds analogy. For Pr¼ 1, the Stanton number is a function of C, Pr, and the ratio v j /v . It would be x 1 f j convenient to eliminate the velocity ratio; this may be accomplished by recalling some results from Chapter 12. At the edge of the laminar sublayer þ þ v ¼ y ¼ 5 pffiffiffiffiffiffiffi þ and by definition v ¼ v /( t/r). Thus for the case at hand x pffiffiffiffiffiffiffi þ v ¼ v j /( t/r)¼ 5 x j19.7 Turbulent Flow Considerations 291 Again introducing the coefficient of skin friction in the form t C ¼ f 2 rv /2 1 we may write rffiffiffiffiffi rffiffiffi t C f ¼ v 1 r 2 which, when combined with the previous expression given for the velocity ratio, gives rffiffiffiffiffi v j C x j f ¼ 5 (19-57) v 2 1 Substitution of equation (19-57) into (19-56) gives C /2 f St¼ pffiffiffiffiffiffiffiffiffi (19-58) 1þ5 C /2(Pr1) f which is known as the Prandtl analogy. This equation is written entirely in terms of measurable quantities. 6 ´ ´ vonKarman extended Prandtl’s work to include the effect of the transition or buffer ´ ´ layer in addition to the laminar sublayer and turbulent core. His result, the von Karman analogy, is expressed as C /2 f St¼ pffiffiffiffiffiffiffiffiffi (19-59) 5 1þ5 C /2fPr1þln½1þ (Pr1)g f 6 Note that, just as for the Prandtl analogy, equation (19-59) reduces to the Reynolds analogy for a Prandtl number of unity. ´ ´ TheapplicationofthePrandtlandvonKarmananalogiesis,quitelogically,restrictedto thosecasesinwhichthereisnegligibleformdrag.Theseequationsyieldthemostaccurate results for Prandtl numbers greater than unity. Anillustrationoftheuseofthefourrelationsdevelopedinthissectionisgiveninthe example below.  EXAMPLE 1 Waterat50 Fentersaheat-exchangertubehavinganinsidediameterof1in.andalengthof10ft.  The water flows at 20 gal/min. For a constant wall temperature of 210 F, estimate the exit temperature of the water using (a) the Reynolds analogy, (b) the Colburn analogy, (c) the Prandtl analogy,and(d)thevonKa´rma´nanalogy.Entranceeffectsaretobeneglected,andthepropertiesof water may be evaluated at the arithmetic-mean bulk temperature. Considering a portion of the heat-exchanger tube shown in Figure 19.6, we see that an applicationofthefirstlawofthermodynamicstothecontrolvolumeindicatedwillyieldtheresult that 8 9 8 9 8 9 rateofheat rateofheat rateofheat = = = transferintoc:v: þ transferintoc:v: ¼ transferoutofc:v: : ; : ; : ; byfluidflow byconvection byfluidflow 6 ´ ´ T. von Karman, Trans. ASME, 61, 705 (1939).292 Chapter 19 Convective Heat Transfer q q 3 Flow 2 q D 1 ∆ x Figure 19.6 Analog analysis of T s water flowing in a circular tube. Iftheseheat-transferratesaredesignatedasq ,q ,andq ,theymaybeevaluatedasfollows: 1 2 3 2 pD q ¼ r v c Tj 1 x p x 4 q ¼ hpDD (T T) 2 x s and 2 pD q ¼ r v c Tj 3 x p xþDx 4 The substitution of these quantities into the energy balance expression gives 2 pD r v c ½Tj TjhpDDx(T T)¼ 0 x p s xþDx x 4 which may be simplified and rearranged into the form D Tj Tj h xþDx x þ (TT )¼ 0 (19-60) s 4 Dx rv c x p Evaluated in the limit asDx0, equation (19-59) reduces to dT h 4 þ (TT )¼ 0 (19-61) s dx rv c D x p Separating the variables, we have dT h 4 þ dx¼ 0 TT rv c D s x p and integrating between the limits indicated, we obtain Z Z T L L dT h 4 þ dx¼ 0 TT rv c D s x p T 0 0 T T h 4L L s ln þ ¼ 0 T T rv c D (19-62) 0 s x p Equation(19-62)maynowbesolvedfortheexittemperatureT .Observethatthecoefficientofthe L right-handterm,h/rv c ,istheStantonnumber.Thisparameterhasbeenachievedquitenaturally x p from our analysis. The coefficient of skin friction may be evaluated with the aid of Figure 14.1. The velocity is calculated as 3 2 2 v ¼ 20gal/min(ft /7:48gal)½144/(p/4)(1 )ft (min/60s)¼ 8:17fps x Initially,wewillassumethemeanbulktemperaturetobe908F.Thefilmtemperaturewillthen 5 2 be 1508F, at which n¼ 0:47410 ft /s. The Reynolds number is Dv (1/12ft)(8:17ft/s) x Re¼ ¼ ¼ 144,000 2 5 v 0:47410 ft /s At this value of Re, the friction factor, f, assuming smooth tubing, is 0.0042. For each of the four f analogies, the Stanton number is evaluated as follows:19.8 Closure 293 (a) Reynolds analogy C f St¼ ¼ 0:0021 2 (b) Colburn analogy C f 2/3 2/3 St¼ Pr ¼ 0:0021(2:72) ¼ 0:00108 2 (c) Prandtl analogy C /2 f pffiffiffiffiffiffiffiffiffi St¼ 1þ5 C /2(Pr1) f 0:0021 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:00151 1þ5 0:0021(1:72) ´ ´ (d) von Karman analogy C /2 f St¼ pffiffiffiffiffiffiffiffiffi  5 1þ5 C /2 Pr1þln 1þ (Pr1) f 6 0:0021 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  5 1þ5 0:0021 2:721þln 1þ (2:721) 6 ¼ 0:00131 Substituting these results into equation (19-62), we obtain, for T , the following results: L  (a) T ¼ 152 F L  (b) T ¼ 115 F L  (c) T ¼ 132 F L  (d) T ¼ 125 F L Some fine tuning of these results may be necessary to adjust the physical property values for the calculated film temperatures. In none of these cases is the assumed film temperature different than the calculated one by more than 68F, so the results are not going to change much. The Reynolds analogy value is much different from the other results obtained. This is not surprising, as the Prandtl number was considerably above a value of one. The last three analogies yieldedquiteconsistentresults.TheColburnanalogyisthesimplesttouseandispreferablefromthat standpoint. 19.8 CLOSURE Thefundamentalconceptsofconvectionheattransferhavebeenintroducedinthischapter. New parameters pertinent to convection are the Prandtl, Nusselt, Stanton, and Grashof numbers.