Critique of the Classical Algebraic Analogies between Heat, Mass

3866

Ind. Eng. Chem. Res. 1997, 36, 3866-3878

Critique of the Classical Algebraic Analogies between Heat, Mass, and Momentum Transfer Stuart W. Churchill† Department of Chemical Engineering, University of Pennsylvania, 311A Towne Building, 220 South 33rd Street, Philadelphia, Pennsylvania 19104-6393

New exact formulations for the limiting case of Pr f 0, the particular case of Pr ) Prt ) PrT, and the asymptotic case of Pr f ∞ reveal and define functional and numerical errors in all of the classical algebraic analogies between heat, mass, and momentum transfer in fully developed turbulent convection in a round tube and along a flat plate. The neglect or misrepresentation of the variation of the heat (or mass) flux density within the fluid and the misrepresentation of the molecular and turbulent transport near the wall, in the turbulent core, or in both regions are the principal sources of error in the analogies. These deficiencies are not generally identified or emphasized in our current textbooks. The continued use of some of these analogies directly or as components of a correlating equation contributes uncertainty to the design of equipment for heat and mass transfersan uncertainty that is counterbalanced by the use of large safety factors. Introduction The new, simplified differential models proposed by Churchill (1996a) for the turbulent transport of momentum and energy lead to new simplified integral formulations for the friction factor and drag coefficient and for the Nusselt number. These new integral formulations provide a basis for identification of the functional and numerical errors in the classical algebraic analogies for heat, mass, and momentum transfer that appear in all of our current textbooks and that are still widely used in practice. The objective of the work reported herein was to evaluate these classical analogies one by one. Analogies have evolved as a mainstay in engineering practice for prediction of the rates of heat and mass transfer for three primary reasons. One reason is the greater ease and accuracy of measurements of the local time-averaged velocity and shear stress relative to those of the local time-averaged temperature, composition, heat flux density, and component flux densities. A second reason is the theoretical or semitheoretical form of the analogies, which inspires confidence in their generality as contrasted with purely empirical expressions. A third reason, historically, has been to reduce the tedium of numerical calculations by means of shortcuts, simplifications, and generalizations. The speed, storage capacity, and ready availability of computing machinery, together with the development of user-friendly interfaces and efficient algorithms for numerical integration, have now eliminated or greatly minimized the need for simplicity at the expense of accuracy. However, the use of such simplistic expressions persists because of tradition, habit, and a widespread lack of confidence in the validity and accuracy of purely empirical correlations on the one hand and of the heuristic differential models for turbulent transport in terms of the eddy viscosity, eddy conductivity, turbulent Prandtl number, and mixing length on the other. In the past decade a completely new approach has emerged for the prediction of turbulent flow and convec†

Tel: 215-898-5579. Fax: 215-572-2093. Email: churchil@ cheme.seas.upenn.edu. S0888-5885(96)00750-6 CCC: $14.00

tion. Essentially exact solutions have been accomplished by the method of direct numerical simulation. Such results are currently very limited in scope, namely, to the lowest fringe of fully turbulent flow along a semiinfinite flat plate, in a round tube, and between parallel plates and to fully developed heat transfer from one parallel plate to the other. With one exception the results for heat transfer are limited to Prandtl numbers of 0.7 and 1.0. These limitations may persist for some time owing to the truly great computational requirements and inherent characteristics of the modeling itself (the postulated absence of large eddies). In addition the solutions by direct numerical simulation consist of instantaneous velocities and temperatures and hence are not convenient for direct application. Despite these limitations in scope and form, the solutions that have been accomplished to date by this methodology are invaluable criteria for testing speculative asymptotes and empirical models and for evaluating the unknown coefficients and exponents thereof. Churchill and Chan (1995b) and Churchill (1997a) have proposed to avoid the shortcomings of the eddy viscosity, eddy conductivity, and mixing length by modeling directly in terms of the time-averaged local turbulent shear stress and heat flux density within the fluid. The latter two quantities as contrasted with the eddy viscosity, eddy conductivity, and mixing length do not invoke any mechanistic postulates and are continuous, finite, and experimentally defined for all geometries and conditions. For a round tube, the time-averaged differential force-momentum balance may be solved formally to obtain an exact integral expression for the velocity distribution and in turn an exact integral expression for the mixed-mean velocity, and thereby for the friction factor, in terms of the time-averaged turbulent shear stress only. The time-averaged differential energy balance may similarly be solved formally to obtain exact integrals for the temperature distribution and the mixed-mean temperature, and thereby for the Nusselt number, in terms of the turbulent and total heat flux densities. For uniform heating on the wall, the total heat flux density within the fluid may be expressed in terms of an exact integral of the local turbulent shear stress only. For a uniform wall-temperature, which is © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3867

implied in the derivation of most of the classical algebraic analogies, the integral equations for the temperature distribution, the mixed-mean temperature, and the Nusselt number must be solved iteratively. For unconfined flow along a semiinifinite flat plate, the general integral solutions for the velocity distribution, free-stream velocity, and drag coefficient incorporate the total local shear stress, which is not known a priori, as a parameter. The general integral solutions for the temperature distribution, the free-stream temperature, and the Nusselt number similarly incorporate the total local heat flux density as a parameter. This complication reduces the utility of these solutions as quantitative but not as qualitative criteria for evaluation of the classical algebraic analogies. Churchill and Chan (1994) developed a very accurate correlating equation for the dimensionless local shear stress within the fluid in a round tube based on a speculative asymptotic structure, selected experimental data, and the available results from direct numerical simulations. Churchill (1997a) has also demonstrated that a correlating equation for either the turbulent Prandtl number Prt or the total Prandtl number PrT, both of which are independent of geometry and of the thermal boundary conditions, may be utilized in lieu of a correlating equation for the turbulent heat flux density without any loss of generality. The principal uncertainty in numerical evaluations of the Nusselt number from the new integral formulations arises from the uncertainty of the currently available correlations and predictive equations for Prt and PrT. Fortunately for the present investigation, this uncertainty may be avoided for several special conditions. The time-averaged, differential force-momentum and energy balances in terms of the new variables are essentially the same for all one-dimensional, fully developed turbulent flows. The integral formulations for the velocity distributions, the friction factor or drag coefficient, the temperature distribution, and the Nusselt number are, however, geometry-dependent. Since the classical algebraic analogies were generally derived for round tubes or flat plates, the integral formulations examined herein will be limited to these two geometries. Also, only such detail as is necessary will be presented for the models and integral formulations. The details of the derivations as well as of the formulations for other geometries may be found in Churchill (1997a). Numerical values of the friction factor based on such formulations are given by Churchill and Chan (1994) and of the Nusselt number and related quantities for the asymptotic conditions by Heng et al. (1997). Invariant physical properties are postulated throughout. Mass transfer is not considered explicitly since the results for heat transfer are presumed to be directly adaptable insofar as the net mass flux density normal to the wall does not have a significant effect. Differential Momentum and Energy Balance The differential, time-averaged force-momentum balance for fully developed one-dimensional or pseudoone-dimensional (unconfined) flow may be expressed in general as

τ)µ

du - Fu′v′ dy

(1)

Here, Fu′v′ is the shear stress in the x-direction on the y-plane (or the momentum flux density in the y-

direction) that is produced by the turbulent fluctuations in the velocity. Churchill (1997a) has recently pointed out the advantages of the following new dimensionless form of eq 1:

τ du+ [1 - (u′v′)++] ) + τw dy

(2)

where (u′v′)++ ≡ -Fu′v′/τ is seen to represent the fraction of the total shear stress due to the turbulence. Before proceeding with the integral formulations, it is constructive to compare eq 2 with its more conventional counterpart in terms of the eddy viscosity, namely,

(

)

µt du+ τ ) 1+ τw µ dy+

(3)

Elimination of du+/dy+ between eqs 2 and 3 results in the following rather surprising relationship:

µt (u′v′)++ ) µ 1 - (u′v′)++

(4)

The principal merit of eq 4 is the revelation that the eddy viscosity is independent of its heuristic diffusional origin. For flow in a round tube and along a flat plate the only advantage of (u′v′)++ over µt/µ in terms of correlation or integral formulations is one of simplicity. However, Churchill and Chan (1995b) have pointed out that µt/µ is unbounded at one or more points and negative over an adjacent range in all geometries, including, for example, circular concentric annuli, in which the shear stress is not equal at opposing points on the walls. This failure in applicability extends to the mixing length and to the κ- model, which functions by predicting the eddy viscosity or the mixing length. The differential, time-averaged energy balance corresponding to eq 1 may be expressed as

j ) -k

dT + FcpT′v′ dy

(5)

Here, FcpT′v′ is the heat flux density in the y-direction due to the turbulent fluctuations in temperature and velocity. The new dimensionless form corresponding to eq 2 is

dT+ j [1 - (T′v′)++] ) + jw dy

(6)

where here T+ ≡ k(τwF)1/2(Tw - T)/µjw and (T′v′)++ ≡ FcpT′v′/j. The latter quantity is seen to represent the fraction of the local heat flux density due to the turbulence. The conventional counterpart of eq 6 in terms of the eddy conductivity is

(

)

kt dT+ j ) 1+ jw k dy+

(7)

Elimination of dT+/dy+ between eqs 6 and 7 yields

kt (T′v′)++ ) k 1 - (T′v′)++

(8)

3868 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997

Equation 8 reveals that the eddy conductivity is also independent of its heuristic diffusional origin. Equation 7 is, however, usually expressed in the following expanded form:

( )]

[

[ ( )]

cpµ kt µt dT+ j Pr µt dT+ ) 1+ ) 1+ (9) + jw k cpµt µ dy Prt µ dy+ which replaces the requirement of a correlation for kt/k by one for Prt/Pr. The turbulent Prandtl number Prt ≡ cpµt/kt has often been disparaged as an artificial quantity, even if a useful one because of its very constrained variation for moderate and large values of Pr. However, taking the ratio of eqs 4 and 8 yields

(

(u′v′)++

1 - (u′v′)

)(

) ( )( ) ( )( )

1 - (T′v′)++

++

)

(T′v′)

++

µt k ) µ kt

cpµt k Prt (10) ) kt cpµ Pr

Equation 10 indicates that Prt/Pr may be interpreted as simply a symbol for the ratio of the shear stress due to turbulence to that due to the molecular motion, divided by the corresponding ratio for the heat flux density. Prt is thus independent of its heuristic diffusional origin just as are µt and kt. Substitution for µt/µ in eq 6 from eq 4 results in

[ (

)]

++ j Pr (u′v′) ) 1+ jw Prt 1 - (u′v′)++

Integral Formulations for a Round Tube

dT+ dy+

(11)

Equation 11 is perfectly general and will be utilized to some extent to evaluate the classical algebraic analogies, but an even simpler formulation is possible. From eqs 4 and 8 it follows that

µt 1+ cp(µ + µt) k PrT 1 - (T′v′)++ µ ) ) ) ++ k + kt cpµ Pr kt 1 - (u′v′) 1+ k

(

)

(12)

where the total Prandtl number PrT ≡ cp(µ + µt)/(k + kt) is seen to represent the ratio of the fraction of the local heat flux density due to thermal conduction to the fraction of the local shear stress due to molecular motion. PrT is also independent of its heuristic diffusional origin. Substitution for 1 - (T′v′)++ in eq 6 from eq 12 results in

PrT dT+ j ) + [1 - (u′v′)++] jw Pr dy

++

+

1 - (u′v′)++ Pr

Momentum Transfer. A force-momentum balance on an annular element of fluid adjacent to the wall of a round tube may be expressed in the following forms, which thereby define the relationship between the several coordinates utilized herein:

y+ r τ )1- +) )R τw a a

(15)

Introducing R for τ/τw and a+ dR for -dy+ in eq 2 followed by formal integration from T+ ) 0 at R ) 1 results in

u+ )

∫R1 [1 - (u′v′)++] dR2

a+ 2

2

(16)

Specializing eq 16 for R ) 0 gives the following integral expression for the dimensionless velocity at the centerline

u+ c )

(13)

a+ 2

∫01[1 - (u′v′)++] dR2

(17)

while integrating eq 16 by parts yields

Equation 13 is the starting point for most of the evaluations herein of the classical analogies. Before proceeding, it is useful to note the following relationship between the quantities PrT, Prt, and Pr:

(u′v′) 1 ) PrT Prt

0. Equation 11 is thereby more convenient than eq 13 in terms of integration despite its greater superficial complexity. Although eqs 1-14 are all independent of geometry, the parameters (u′v′)++, (T′v′)++, µt/µ, kt/k, τ/τw, and j/jw are not. It may be inferred speculatively that Prt and PrT are geometry-independent and depend only on (u′v′)++ and Pr. This postulate has been made explicitly or implicitly in most correlating equations for Prt or PrT and is inherent in the recent expression for PrT derived by Yahkot et al. (1987) using renormalization group theory. It follows that Prt and PrT are also independent of the temperature field and hence of the thermal boundary condition or conditions. It further follows that (T′v′)++ and kt/k share that independence. The most critical experimental test of these speculations appears to be that of Abbrecht and Churchill (1960) who determined longitudinal as well as radial gradients of the time-averaged temperature in fully developed turbulent flow of air in a round tube following a step in the wall temperature. They found the values of Prt derived from these measurements to be independent of axial distance within their uncertainty and to be in good agreement with the values of Prt determined by Page et al. (1952) for fully developed heat transfer from one parallel plate to another at the equivalent rate of flow (a+ ) b+).

(14)

As contrasted with µt and kt, Prt and PrT are positive and finite for all geometries and physical conditions. For Pr f ∞, Prt has been found experimentally to approach ∼0.85 for all y+. It follows from eq 14 that PrT varies strongly very near the wall where (u′v′)++ f

(2f )

1/2

∫01u+ dR2 ) a4 ∫01[1 - (u′v′)++] dR4 +

≡ u+ m )

(18)

Since a+ ) (f/2)1/2ReD/2, eq 18 may also be expressed as

f)

16

∫0 [1 - (u′v′)++] dR4

ReD

1

(19)

Equation 19 may be seen to reduce to Poiseuille’s law for (u′v′)++ ) 0.


Churchill and Chan (1994) devised a generalized and almost exact correlating equation for (u′v′)+ ) (1 - (y+/a+))(u′v′)++ in round tubes and utilized that + expression to evaluate u+, u+ c , um, and f with great accuracy by carrying out the integrations of eqs 16-19 numerically. Such values for the centerline and mixedmean velocities were in turn found to be represented very closely by + u+ c ) 5.92 + 2.5 ln{a }

(20)

and

u+ m

( )

161.2 47.6 ) 1.99 + a+ a+

2

NuD )

2a+ ) T+ m 4

(

∫0 ∫ 1

1

(1 + γ)[1 - (u′v′)++] R2

)( )

( )

PrT u+ dR2 + dR2 Pr um

(26)

It would be consistent with the prior formulations herein to substitute for u+/u+ m in eqs 25 and 26 from eq 16. This substitution will not be made, however, for reasons that will eventually become apparent. Instead, eq 24 will be specialized for R ) 0, resulting in

T+ c )

+ 2.5 ln{a } (21) +

∫01(1 + γ)[1 - (u′v′)++]

a+ 2

( )

PrT dR2 Pr

(27)

from which it follows that Equations 20 and 21 serve as predictors of the rate of momentum transfer to which the rates of heat and mass transfer are related by the analogies to follow. The extended expression for f according to eq 18 reveals its direct dependence on a+ and implies that it cannot be a linear function of Ren as is postulated in most purely empirical correlating equations. The recognition that u+ f y+ near the wall requires the presence of the terms in (a+)-1 and (a+)-2. The experimental data for f do confirm a simpler dependence on a+ than on ReD but are too imprecise to confirm the existence of the terms in (a+)-1 and (a+)-2. The three significant figures of some of the constants in eqs 20 and 21 are not justified by the data for f but are by the experimental data and computed values of u and u′v′. Energy Transfer. The variation of the heat flux density ratio j/jw with R differs from that of the shear stress ratio τ/τw, but only moderately. Hence it is convenient, before integrating eq 11 or 13, to introduce a new variable γ defined by

j/jw ) (1 + γ)(τ/τw)

(22)

For a round tube, eq 22 simplifies to

j/jw ) (1 + γ)R

(23)

Reichardt (1951) was apparently the first to utilize the formulation represented by eq 22. The quantity γ is essentially a perturbation with respect to unity, but a numerically significant one in terms of NuD. Substituting for j/jw in eq 13 from eq 23 and then integrating results in

T+ )

∫R1 (1 + γ)[1 - (u′v′)++]

a+ 2

2

( )

PrT dR2 Pr

()

()

+ 2a+ 2a+ Tc ) ) + T+ T+ m c Tm + 4(T+ c /Tm)

( )

∫

1

PrT dR2 Pr

(1 + γ)[1 - (u′v′)++] 0

(28)

For Pr f 0, for which, according to eq 14, Pr/PrT f 1 - (u′v′)++, eq 28 reduces to

NuD{Pr ) 0} )

+ 4(T+ c /Tm)

∫01(1 + γ) dR2

)

+ 4(T+ c /Tm) (29) 1 + 1

where

1 ≡

∫01γ dR2

(30)

is the integrated mean value of γ over the cross section of the tube. The designation NuD{Pr ) 0} rather than NuD{Pr f 0} is used here in that the asymptotic dependence on Pr has been lost and eq 29 is merely an expression for the lower limiting Nusselt number. Insofar as the variation of PrT with y+ may be neglected, and by virtue of eq 17, eq 28 may be expressed as

( )( ) ( )

+ + f Pr Tc um ReD PrT T+ u+ 2 m c NuD ) 1 + 2

(31)

where

(24)

It follows that

∫01T+ uu+

NuD )

()

∫01γ[1 - (u′v′)++] dR2 1 u+ ≡∫ γd 2 ≡ 1 ∫0 [1 - (u′v′)++] dR2 0 u+c

(32)

+

T+ m ≡

a+ 2

(

dR2 )

m

∫0 ∫R (1 + γ)[1 - (u′v′)++] 1

1

2

and thereby that

( )

)( )

PrT u+ dR2 + dR2 Pr um (25)

Thus 2 may be interpreted either as the integrated mean value of γ, weighted by 1 - (u′v′)++, over the cross section or as the integrated mean value of γ with respect to u+ over the radius. According to the semitheoretical expression of Yahkot et al. (1987) for PrT{Pr,(u′v′)++}, PrT is truly independent of y+ only for Pr ) PrT ) Prt ) 0.848. Thereby, eq 31 is exact only for that value of Pr, for which it reduces to


( ) ()

+ T+ c um

ReD

+ T+ m uc NuD{Pr ) PrT ) 0.848} ) 1 + 2

f 2

(33)

It may be inferred from eqs 16 and 24 that even for this + + + condition, T+ c /Tm differs from uc /um by virtue of the finite values of γ. Experimental data indicate that Pr ) Prt ) PrT for Pr = 0.86, which differs negligibly from the prediction of 0.848 by Yahkot et al. Despite the presence of the unknown factors of + + + + + (T+ c /Tm)/(1 + 1) in eq 29 and (Tc /Tm)(um/uc )/(1 + 2) in eq 33, these two expressions prove to be very useful in evaluating the classical algebraic analogies. The effects of the thermal boundary conditions are exerted wholly through these two groupings. Since somewhat simpler expressions are obtained below for a uniformly heated wall, the principal utility of eqs 29 and 33 is for a uniform wall-temperature. For Pr . 1, eq 11 is a more convenient starting point than eq 13 or 27 for the derivation of an expression for NuD because of the aforementioned lesser variation of Prt than of PrT with y+ very near the wall where the entire development of the temperature profile occurs. In this region j/jw = 1 and there is general agreement that

(34)

These postulates allow the analytical integration of eq 11 and the specialization of that result to obtain

( ) ( ) ( ) ()

Prt 3 NuD{Pr . 1} ) 12π Pr

4/3

1/3

RPr Prt

j 1 ) jw R

∫0R

2

( )( )

T+ u+ dR2 + T+ u m m

T+ c T+ m

while for a uniform heat flux density on the wall, the corresponding expression is

∫0R

1 j ) jw R

2

()

u+ dR2 u+ m

(38)

The substitution of γ corresponding to eq 37 in eq 24 results in an integral equation that must be solved in conjunction with eq 25. An iterative numerical solution is feasible but the results are not of much help for the functional evaluation of the classical algebraic analogies. On the other hand the relationship represented by eq 38 is remarkable in its simplicity and in its independence from the temperature distribution and the Prandtl number. It allows great simplification for uniform heating in the integral formulations for NuD.

f ReD 2

By virtue of eq 38, the denominator of eq 26 may be integrated by parts to obtain

NuD )

1/2

(35)

For the generally accepted values of R ) 7 × 10-4 and Prt ) 0.85 for Pr f ∞ and the recognition that (1 - (Prt/ + Pr))4/3 and T+ c /Tm both approach unity, eq 35 may be reduced to

8 2

where, by means of eqs 23, 26, and 38 and again integration by parts, γ) 1 - R2 R2

(

)∫

R4

0

[1 - (u′v′)++] dR4 +

∫ (1 -R R )[1 - (u′v′) R4

∫ [1 - (u′v′)

(36)

The equivalent of eq 35 was apparently first derived by Churchill (1997b), but the equivalent of eq 36, usually with Prt postulated to be unity, has been derived previously by Petukhov (1970) and others. The utility of the term (1 - (Prt/Pr))4/3 is in providing a first-order correction for the effect of a finite value of Pr and conversely of defining the limit of applicability of eq 36 in terms of a finite value of Pr. As contrasted with eq 29, eq 36 is an asymptotic expression in that it retains a dependence on Pr. The shifting proportionality of NuD from ReD(f/2) to ReD(f/2)1/2 as Pr increases reflects the + decreasing dependence of NuD on u+ c and um. Some uncertainty exists with respect to the reliability of eqs 35 and 36 owing to the variation of Prt near the wall for very large values of Pr . 100 as found by Papavassiliou and Hanratty (1997) by Lagrangean direct numerical simulations. However, most of the experimental data for heat transfer, which are limited to Pr < 100, appear to be in accord with eq 36. Heat Flux Density Ratio General and specific expressions for the heat flux density ratio j/jw in fully developed convection may be

2

1

++

NuD{Pr f ∞} = 0.078Pr ReD(f/2)

1/2

(39)

∫0 (1 + γ) [1 - (u′v′)++](PrT/Pr) dR4 1

1

1/3

(37)

Integral Formulations for Uniform Heating

(u′v′)++ f R(y+)3

3/2

derived by means of an integral energy balance over an arbitrary central core of the fluid stream in a round tube. For a uniform wall-temperature the result may be expressed as

0

2

++

] dR4

] dR4

(40)

The quantity γ, which may readily be evaluated numerically by means of a correlating equation for (u′v′)++ such as that of Chan and Churchill (1994), rises monotonically from zero at the wall (R ) 1) to a maximum value at the centerline. It decreases with increasing ReD for all y+. The analogue of eq 39 in terms of Prt is, by virtue of eq 11,

NuD )

∫0

1

1+

8 (1 + γ)2 dR4

(

(41)

)

++ Pr (u′v′) Prt 1 - (u′v′)++

Lyon (1951) was apparently the first investigator to reduce eq 26 for uniform heating by means of integration by parts. His final formulation is equivalent to 2 2 eq 41 but with (1/R2) ∫R0 (u+/u+ m) dR substituted for 1 + ++ γ and kt/k for (Pr/Prt)((u′v′) /[1 - (u′v′)++]). For Pr f 0, eq 41 and eq 39, by virtue of eq 15, both reduce to


NuD{Pr ) 0} )

8

∫0 (1 + γ)2 dR4 1

)

8 1 + 3

(42)

where

1 + 3 ≡

∫01(1 + γ)2 dR4

(43)

is the integrated mean value of (1 + γ)2 with respect to R4. Equation 42 represents a lower bound for NuD in the turbulent regime for uniform heating, while 3 characterizes the dependence of this lower bound on ReD. Since eq 29 is applicable for uniform heating as well as for other thermal boundary conditions, it may be combined with eq 42 to obtain

∫ ∫

1

2 2(1 + 1) 2 0 (1 + γ) dR {Pr ) 0} ) ) 1 (44) 1 + 3 T+ (1 + γ)2 dR4 m 0

T+ c

with eq 40 applicable for γ. Equation 4 may of course be derived directly from the individual exact integral + expressions for T+ c and Tm for uniform heating. For + plug flow, uniform heating, and Pr f 0, T+ c /Tm f 2 and NuD f 8. Equation 44 provides a measure of the + deviation of T+ c /Tm from 2.0 due to the actual velocity distribution in turbulent flow, again for uniform heating and Pr f 0. For Pr ) PrT ) Prt = 0.86, eqs 39 and 41 both reduce, by virtue of eq 14, to

NuD{Pr ) PrT} )

8

(45)

∫0 (1 + γ) [1 + (u′v′)++] dR4 1

2

which, by virtue of eq 18, may also be expressed as

NuD{Pr ) PrT} )

ReD(f/2) 1 + 4

(46)

where

∫01(1 + γ)2[1 - (u′v′)++] dR4 1 + 4 ≡ ∫01[1 - (u′v′)++] dR4

(47)

is the integrated-mean value of (1 + γ)2 weighted by 1 - (u′v′)++, over R4. For uniform heating, the quantity 4 characterizes completely the deviation of NuD{Pr ) PrT}/ReD(f/2) from unity. From eqs 33 and 46 or from + the individual expressions for u+ c and um, as well as for + + Tc and Tm for uniform heating, it follows that

1 + 2 {Pr ) PrT} ) ) + + 1 + 4 T u

(

)

∫0 (1 + γ)[1 - (u′v′) ] dR × ∫01(1 + γ)2[1 - (u′v′)++] dR4 ∫01[1 - (u′v′)++] dR4 ∫01[1 - (u′v′)++] dR2 1

++

(

Integral Formulations for a Flat Plate Many of the classical algebraic analogies were originally derived specifically for a flat plate. Hence the integral formulations will also be examined for this geometry. Momentum Transfer. In the regime of fully developed turbulence in pseudo-fully-developed unconfined flow along a semiinfinite flat plate, it follows from eq 2 that

u+ )

2

)

∫0y

+

()

τ [1 - (u′v′)++] dy+ τw

(49)

Specialization of eq 49 for the free-stream velocity that is approached as y+ f ∞ results in

()

1/2

2 Cf

≡ u∞+ )

∫0δ

()

+

()

τ y+ [1 - (u′v′)++] d + τw δ

(50)

where δ+ is arbitrarily defined as the dimensionless boundary-layer thickness beyond which u+ = u+ ∞ and τ/τw is effectively equal to zero. Equation 50 differs fundamentally from eq 18, its counterpart for a round tube, in that τ/τw and δ+ are dependent variables and thus not known a priori. Rather than using eq 50 with correlating equations for τ/τw and (u′v′)++, Churchill (1993) used a correlating equation for the velocity distribution and an integral momentum balance to derive the equivalent of

() 2 Cf

1/2

) 2.53 +

{

CfRex 1 ln 0.41 1 - 8C1/2 f + 22Cf

}

(51)

which serves the same role for a flat plate as eq 21 for a round tube. The remarks concerning the structure of eq 21 are applicable to eq 51. Heat Transfer. For fully developed convection from the plate, it follows from eq 6 that

T+ )

+ T+ c um m c

to be congruent because of the identical mechanisms for the transport of momentum and energy. Equation 48 is an exact integral measure of the failure of that congruence owing to the deviation of j/jw, as represented by γ, from τ/τw. Despite their restriction to uniform heating, eqs 42 and 45 are a great improvement on eqs 29 and 33 for the evaluation of the classical algebraic analogies by virtue of their more explicit form. It may be inferred + from eqs 44 and 48 that T+ c /Tm is greater for a uniform wall-temperature than for uniform heating for the same values of ReD and Pr.

∫0y

()

+

( )

PrT j [1 - (u′v′)++] dy+ jw Pr

(52)

Specialization of eq 52 for y+ f ∞ results in

T∞+ ) (48)

For Pr ) PrT the dimensionless temperature and velocity distributions might erroneously be presumed

()

( )

∫0∞ jjw [1 - (u′v′)++]

) δ+ E

()

∫01 jjw

PrT dy+ Pr

( )()

[1 - (u′v′)++]

PrT y+ d + Pr δE

(53)

where δ+ E is an arbitrarily defined dimensionless thickness of the thermal boundary layer beyond which


T+ = T+ ∞ and j/jw is effectively equal to zero. It follows from eq 53 that

Nux ≡

Rex(Cf/2)1/2

Rex(Cf/2)1/2

()

)

∫0∞ jjw [1 - (u′v′)++]

T∞+

x+/δ+ E

()

)

∫01 jjw [1 - (u′v′)++]

( )

PrT dy+ Pr

( )() PrT y+ d + Pr δE

(54) For Pr f 0, for which Pr/PrT f 1 - (u′v′)++, eq 54 reduces to

Nux{Pr ) 0} )

Rex(Cf/2)

()

∫0∞ jjw

x+/δ+ E

()()

)

∫01 jjw

dy+

d

)

y+ δ+ E

x+/δ+ E 1 - 5 (55)

where

1 - 5 ≡

()()

∫01 jjw

d

y+ δ+ E

(56)

is the integrated-mean value of j/jw over the thermal boundary layer. Nux{0} is observed to be finite for a flat plate just as for a round tube. + For Pr ) PrT ) Prt = 0.86, for which δ+ E = δ , eq 54 reduces to

Nux{Pr ) PrT} )

Rex(Cf/2)1/2

∫0∞(j/jw)[1 - (u′v′)++] dy+

)

∫0 (j/jw)[1 - (u′v′)++] d(y+/δ+)

(57)

Introducing γ, per eq 22, and utilizing eqs 49 and 50 allow eq 57 to be reexpressed as

Rex(Cf/2) 1 + 6

(58)

where

6 ≡

() ∫( )

() ()

∫01γ ττw [1 - (u′v′)++] d δy+ 1

0

+

τ y+ [1 - (u′v′)++] d + τw δ

()

∫01γ d uu+

+

)

Generalities Concerning the Integral Formulations Although numerical computations and comparisons are considered to beyond the scope of this presentation, several generalities concerning the relative behavior for uniform wall-temperature and uniform heating are perhaps appropriate before comparing the classical algebraic analogies with the exact integral formulations. From the illustrative graphical results of Churchill and Balzhiser (1959), it is apparent that γ is much greater for uniform wall-temperature than for uniform heating. Their results are limited to round tubes, but a similar effect may be inferred for unconfined flow along a flat plate. From the Graetz-type series solutions of Sleicher and Tribus (1957) and Notter and Sleicher (1972), it is apparent that the Nusselt number is much greater for uniform heating than for uniform wall-temperature at very small values of the Prandtl number but is only slightly greater at moderate and large values. It follows, as was deduced on other grounds, that + T+ c /Tm is greater for a uniform wall-temperature than for uniform heating. These series solutions are exact except for the uncertainty introduced by the empirical correlations that were utilized for u+{y+,a+} and thereby µt, and particularly for Prt. Since the same expressions were used for both modes of heating, the predicted relative behavior should be reliable, at least semiquantatively. Classical Algebraic Analogies

x+/δ+ 1

Nux{Pr ) PrT} )

Equations 54, 55, 58, and 60 are applicable for any thermal boundary condition that produces fully developed convection. The thermal boundary condition influences eqs 54, 55, and 58 only by virtue of j/jw and γ. Equations 55, 58, and 60 constitute the primary criteria for evaluation of the classical algebraic analogies for unconfined flow along a semiinfinite flat plate.

(59)

∞

The quantity 6 may thus be interpreted as the integrated mean value of γ, weighted by (τ/τw)[1 (u′v′)++] over the boundary layer, or more simply as the integrated mean value of γ with respect to u+/u+ ∞. For Pr f ∞, eq 36, by virtue of its inherent independence from a characteristic length and a characteristic velocity as well as from j/jw, is directly adaptable for a flat plate in the form

Nux{Pr f ∞} ) 0.078Pr1/3Rex(Cf/2)1/2

(60)

A representative set of the classical algebraic analogies, including the most widely known ones, will now be evaluated one-by-one in terms of the essentially exact integral formulations above for Pr ) 0, Pr ) Prt ) PrT, and Pr f ∞. SInce derivations of most of these analogies are readily available in the standard textbooks on heat transfer, mass transfer, and transport phenomena or in the recent archival literature, they will in most cases simply be presented here with a brief identification of their origin and explicit idealizations. Mechanistically and Heuristically Based Analogies. The earliest and best known analogies were derived on mechanistic or heuristic grounds rather than from the partial differential equations of conservation. As a consequence the inherent idealizations and approximations are not necessarily evident. The Reynolds, Prandtl-Taylor, Thomas and Fan, and Colburn analogies may be considered somewhat arbitrarily to fall in this class. The Reynolds Analogy. The first and most widely known analogy is that of Reynolds (1874), which may be expressed as follows for a round tube:

NuD ) PrReDf/2

(61)

The original derivation of eq 61 was based on the postulate that momentum and energy are transported at equal mass rates between the bulk of the fluid and


the wall of the tube by the oscillatory radial motion of the eddies. Comparison of eq 61 with the above exact integral solutions indicates that it is valid only for Pr = 0.86, for which Pr ) PrT ) Prt, and even then is in error by a factor of PrT(1 + 4) for uniform heating and + + + of PrT(1 + 2)(T+ m/Tc )(uc /um) for a uniform wall-tem+ + + perature. Since Tc /Tm, u+ c /um, and 1 + 2 each exceed unity, some numerical compensation may be expected from their combination. All three of these factors decrease as ReD increases but not in the same proportion. The predicted deviation of experimental data from eq 61 for Pr = 1 may be inferred from Figure 13 of Friend and Metzner (1957). For unconfined flow along a flat plate, the Reynolds analogy may be expressed as

Nux ) PrRexCf/2

(62)

Comparison of eq 62 with the above exact integral solutions for a flat plate indicates that it too is valid only for Pr = 0.86 and even then is in error by a factor of PrT(1 + 6). It is interesting to note as an aside that eq 62 holds exactly for the thin-laminar-boundary-layer regime of flow along an isothermal plate for Pr ) 1 but fails numerically by a factor of 1.382 17 for a uniformly heated plate and also that NuD ) AReDf/2 for fully developed laminar convection in a round tube for all Pr, with A ) 0.2727 for uniform heating and A ) 0.2286 for a uniform wall-temperature. The Prandtl-Taylor Analogy. Prandtl (1910) and Taylor (1916) independently attempted to correct for one of the conceptional shortcomings of the Reynolds analogy by postulating that the eddies penetrate only to a finite distance δs from the wall and that transport of momentum and energy through the remaining distance occurs by a linear process of molecular diffusion. Their resulting expression for unconfined flow along a flat plate may be written in the form

Nux )

PrRexCf/2 1+

δ+ s (Pr

- 1)(Cf/2)1/2

(63)

The additive form of the denominator of eq 63 arises from the two resistances in series of the models for momentum and energy transfer. On the basis of “the law of the wall”, which implies that u+ is a function only of y+ and hence is independent of Rex, δ+ s may be inferred to be a constant with an empirically determined numerical value of about 11. Although eq 63 has the comparative merit with respect to eq 62 of predicting, in accord with eqs 58 and 60, a shifting proportionality of Nux from ReCf/2 to Re(Cf/2)1/2 as Pr increases, it shares the shortcomings of the Reynolds analogy for Pr = 1 and erroneously predicts the independence of Nux from Pr for Pr f ∞. The expression corresponding to eq 63 for a round tube is further deficient, for example, at Pr ) PrT ) Prt = 0.86 by a factor of 1 + 4 for uniform + + + heating and by a factor of (1 + 2)(T+ m/Tc )(uc /um) for a uniform wall-temperature. The Analogy of Thomas and Fan. The model of penetration and surface renewal, as originated by Higbie (1935) for bubbles and extended by Danckwerts (1957) for liquid films, has been utilized by Thomas and Fan (1971) to attempt to correct for another implicit idealization of the analogy of Reynolds, namely, the instantaneous interchange of all of the momentum and energy of each eddy with the wall. In their model, this

interchange occurs by transient one-dimensional molecular diffusion to and from eddies of effectively semiinfinite thickness for a fixed time of contact. The resulting expression for convection from a flat plate as derived by eliminating the time of contact between the solutions of the momentum and energy balances is

Nux ) Pr1/2RexCf/2

(64)

Equation 64 is equivalent to the Reynolds analogy with Pr replaced by Pr1/2. The predicted proportionality of Nux to Pr1/2 is an artifact of the models for transient molecular diffusion and has no experimental or theoretical support for the turbulent regime for any value of Pr. In every other respect, eq 64 shares the shortcomings of the Reynolds analogy. The Colburn Analogy. The most widely used analogy even today is that of Colburn (1933), which has no mechanistic or theoretical basis whatsoever. He simply observed that some empirical correlating equations for f/2 and NuD/ReDPrm appeared to have a very similar dependence, both numerically and functionally, on ReD. Accordingly he proposed the expression

NuD ReDPr

1/3

)

f 2

(65)

His choice of m ) 1/3 was simply a convenient compromise for empirically determined values ranging from 0.3 to 0.4. The degeneration of eq 65 to coincidence with the Reynolds analogy for Pr ) 1 was misinterpreted as some justification for this relationship. Actually, except for the proportionality of NuD to Pr1/3 for Pr f ∞, and of NuD to ReDf for Pr = 1, eq 65 is in error functionally in every respect. For example, the correct proportionality of NuD is to ReD(f/2)1/2 rather than to ReDf/2 for Pr f ∞ and to Pr rather than to Pr1/3 for Pr ) PrT ) Pr = 0.86. For this latter condition the factors 1 + 4 and + + + (1 + 2)(T+ m/Tc )(uc /um) are again missing for uniform heating and uniform wall-temperature, respectively. Equation 65 necessarily predicts reasonable numerical values for moderate values of ReD and Pr, since it is directly based on empirical correlations for both NuD and f for such fluids and conditions. However, the previously mentioned Figure 13 of Friend and Metzner (1957) clearly demonstrates the numerical failure of the Colburn analogy for large values of Pr as well as for Pr = 1. How has the Colburn analogy retained such great credibility despite its numerical and functional failures? The answer is in part simply the acceptance of significant error in the predictions of the rate of convective heat and mass transfer and its compensation in the form of excessive safety factors and in part the excellent reputation of Colburn himself and the aura he cast by the designation of the grouping on the left-hand side of eq 65 as the “j-factor for heat transfer”. Indeed, in spite of all of the theoretical considerations and experimental evidence refuting the equality of NuD to Pr1/3ReDf/2 for all Pr, several attempts have been made to rationalize this dependence. For example, Sherwood and Ryan 3 (1959) noted that a proportionality of µt to (y+/u+ m) would lead to the Colburn analogy for Pr f ∞, while Bejan (1984) used dimensional arguments and the concept of a laminar sublayer of time-varying depth and length to justify the proportionality of NuD to Pr1/3ReDf for all Pr.


Analogies Based on Velocity and Temperature Distributions. The analogies of Reynolds, PrandtlTaylor, and Thomas and Fan are based on heuristic and simplistic models for turbulent transport, while that of Colburn is simply based on an algebraic ratio of empirical correlating equations for the Nusselt number and the friction factor. The following two analogies are instead based on correlating equations for the velocity distribution, the temperature distribution, and directly related quantities. Greater accuracy, both functionally and numerically, would be anticipated from such usage of local quantities. The Analogy of Kadar and Yaglom. Empirical correlating equations for the velocity distribution and the boundary layer thickness, together with experimental data for the temperature distribution and the rates of heat and mass transfer from a flat plate, as well as some conjecture, were utilized by Kadar and Yaglom (1972) to devise the expression

Nux )

PrRex(Cf/2)1/2 12.5Pr2/3 + 2.12 ln{PrRexCf/2} - 5.73

(66)

The additive form of the denominator of eq 66 is reminiscent of that of eq 63 but reflects a more sophisticated representation for the diffusional regime. The coefficient of 2.12, which was based on experimental data for the temperature distribution, implies that Prt ) 0.85. The factor Pr in the argument of the logarithm was introduced to assure the explicit independence of Nux from µ in the limit of Pr f 0, while the term 12.5Pr2/3 was introduced to force conformity to experimental data for large values of Pr and Sc. Finally, the constant -5.73 was determined from experimental data for Nux for 0.7 e Pr e 64. Kadar and Yaglom proposed the following expression for the drag coefficient in eq 66:

(2/Cf)1/2 ) 2.4 + 2.5 ln{RexCf}

(67)

The combination of eqs 66 and 67 results in a proportionality of Nux to ReCf/2 for Pr ) 0.711 rather than for Pr ) 0.86, with a correction factor corresponding to 1 + 6 ) 1.193. Equation 66 fails to reduce to a finite value for Pr f 0. Using an equivalent procedure, Kadar and Yaglom devised the following expression for a round tube:

NuD )

PrReD(f/2)1/2 12.5Pr2/3 + 2.12 ln{PrReD(f/2)1/2} - 9.36

(68)

They found eq 68 to be in good agreement with experimental data for Pr g 0.7 and Sc e 106 but to fail + seriously for liquid metals. Introducing T+ c /Tm as a factor in the numerator of eq 68 and substituting +3.84 for -9.36 in the denominator was found to improve the representation for 0.006 e Pr e 0.026 somewhat. They proposed the following expression for the friction factor in eq 64:

(2/f)1/2 ) -0.85 + 2.5 ln{ReDf1/2}

The Analogy of Churchill. An analogy for convection from an isothermal flat plate was derived by Churchill (1997b) by a procedure somewhat similar to that of Kadar and Yaglom, but using more exact and detailed expressions for the velocity and temperature distributions. His resulting expression for a flat plate is

Nux )

[(

( ) () Cf Pr Rex Prt 2

)

Pr 13.62 Prt

2/3

] ()

2 -1 + Cf

{( ) }

+ 1 Pr δE ln + 0.41 Prt δ+

(70)

The coefficient 13.62 corresponds to 0.078 of eq 36. Equation 51 is a compatible expression for Cf. The + parameters Prt and δ+ E /δ of eq 70 replace the fixed, mean numerical values of eq 66. Equation 70 reduces to the exact expression, eq 58, for Pr ) Prt ) 0.86, + insofar as the terms (1/0.41)(Cf/2)1/2 ln{δ+ E /δ } and 6 are equal or both negligible with respect to unity. It avoids an explicit dependence on µ as Pr f 0 but does not converge to a finite value in that limit. The corresponding expression for an isothermal round tube is

( ) ()

+ f Pr Tc ReD Prt T+ 2 m

NuD ) 13.62

[( ) Pr Prt

2/3

1/2

] [( )

-1 +

{ }]

u+ c 2 1/2 Pr + 2.5 ln + f Prt u m

(71) Equation ?? is a compatible expression for f. Equation 71 appears to be superior to eq 68 of Kadar and Yaglom + by virtue of the explicit appearance of Prt, T+ c /Tm, and + + uc /um as parameters, its exact reduction to eq 36 for Pr f ∞, and its reduction to eq 33, except for factor of 1 + 2, for Pr ) Prt = 0.86. Analogies Based on a Differential Energy Balance. Several approximate analogies have been derived starting from a differential energy balance. Those of von Ka´rmań, Martinelli, Reichardt, and Friend and Metzner will be examined as representative. The von Ka´ rma´ n Analogy. von Ka´rmań (1939) integrated eq 9 for a round tube in three steps by postulating u+ ) y+ for y+ e 5, u+ ) -3.05 + 5.0 ln{y+} for 5 e y+ e 30, and the Reynolds analogy for 30 e y+ e a+, as well as τ/τw ) j/jω ) 1 for the entire cross + section. He also postulated implicitly that T+ c /Tm, + + uc /um, and Prt equal unity. His result may be expressed as

NuD )

PrReD(f/2)1/2

{

}

2 1/2 1 + 5Pr + 5(Pr - 1) + 5 ln f 6

()

(72)

For |Pr - 1| , 1, eq 72 may be reduced to

(69)

Combination of eqs 68 and 69 indicates a proportionality of NuD to ReDf/2 for Pr ) 0.7224 with a correction factor + of 1.174 corresponding to (1 + 2)u+ c /um of eq 31. Equation 68 obviously fails to approach a finite limit as Pr f 0.

1/2

1/2

NuD )

PrReDf/2 55 f 1/2 1+ (Pr - 1) 6 2

()

(73)

which is equivalent to the Prandtl-Taylor analogy for δ+ s ) 55/6 ) 9.17. Equation 72 reduces to the Reynolds


analogy for Pr ) 1 and becomes independent of Pr for Pr f ∞, just as does the Prandtl-Taylor analogy, but proves to be more accurate for small and moderate values of Pr by virtue of the replacement of δ+ s (Pr 1) by 5(Pr - 1) + 5 ln{(1 + 5Pr)/6}, as illustrated in Figure 15.12 of Knudsen and Katz (1958). The Martinelli Analogy. Martinelli (1947) improved upon the von Ka´rmań analogy by retaining Prt from eq 9 and postulating that j/jw ) τ/τw ) 1 - (y+/a+) and u+ ) 5.5 + 2.5 ln{a+} over the turbulent core (30 e y+ e a+). He thereby obtained

NuD )

( ) () { ( )}] { () } T+ c Pr f ReD + Pr 2 T t

1/2

m

[

ReD f 1/2 + 2.5F ln 60 2

Pr Pr 5 + ln 1 + 5 Prt Prt

(74)

Here, F is an integrated mean value over the turbulent core of kt/(k + kt), that is of the fraction of the transport of energy due to turbulence. If molecular transport is neglected in the turbulent core, F ) 1. His own slightly + corrected values for F and T+ c /Tm were subsequently reported in tabular form by McAdams (1954). Equation 74 erroneously predicts independence of NuD from Pr for Pr f ∞ and fails to reduce to the Reynolds analogy for Pr ) Prt. It is, however, decisively superior to all of the other classical algebraic analogies for small Pr in that NuD approaches a finite value by virtue of the factor F. The Reichardt Analogy. Reichardt (1951) developed a formal analogy for moderate and large values of Pr and a uniform wall-temperature. He started by taking the ratio of the equivalent of eqs 3 and 9 to obtain

(

) [ ( )]

µt 1 + du+ j/jw µ ) τ/τw Pr µt 1+ dT+ Prt µ

T+ c )

∫0

(75)

( ( ))

µt µ (1 + γ) du+ Pr µt 1+ Prt µ 1+

(76)

For Pr ) Prt, eq 76 may be recognized as exactly equivalent to eq 27. Reichardt expedited the approximate evaluation of the integral of eq 76 for moderate and large values of Pr by expanding the integrand to obtain

T+ c

∫0

u+ c

)

[

(

) ( )

]

Prt µt 1Prt µ Pr + du+ (77) + Pr µt Pr Pr µt 1+ 1+ Prt µ Prt µ γ 1+

( )

NuD )

( ) () ( )( )( ) ( )∫ T+ c

He then asserted that the left-most term of the integrand is negligible for small u+ owing to γ being small with respect to unity and is approximately equal to γPrt/ Pr for large u+ owing to µt/µ being large with respect to

ReD

T+ m

f 2

1/2

1/2

Prt u+ c 2 + Pr u f m

(1 + 2)

+ 1-

Prt Pr

du+ Pr µt 1+ Prt µ (78)

u+ c 0

( )

For moderate and large values of Pr the value of the remaining integral in eq 78 is very small except near the wall where du+ ) (du+/dy+)dy+ = dy+. Although Reichardt at first considered an expression for µt/µ equivalent to eq 34, he ultimately and erroneously utilized an expression proportional to (y+)5. Had he used eq 34 with R ) 7 × 10-4 he would have obtained

NuD )

( ) () ( )( )( ) ( )( ) T+ c

T+ m

Prt u+ c 2 (1 + 2) Pr u+ f m

ReD

1/2

f 2

1/2

Prt Prt + 13.62 1 Pr Pr

1/3

(79)

+ Reichardt evaluated 1 + 2 and T+ c /Tm as functions of ReD and Pr and presented these results graphically. He + also evaluated u+ c /um as a function of ReD. The analogue of eq 79 for unconfined flow along a flat plate is

Rex(Cf/2)1/2

NuD )

It may be noted that integrating eq 75 for j/jw ) τ/τw and Prt ) Pr yields the Reynolds analogy. Instead, introducing the factor γ from eq 22 and integrating formally results in

u+ c

unity. Then postulating Prt to be invariant over the radius allows eq 77 to be integrated in part to obtain

(1 + 6)

( )( ) Prt 2 Pr Cf

1/2

(

+ 13.62 1 -

)( )

Prt Prt Pr Pr

1/3

(80)

Reichardt asserted that 6 was negligible with respect to unity and hence could be dropped. Equations 79 and 80 reduce exactly to eqs 33 and 58, respectively, for Pr ) Prt and to eqs 36 and 60 for Pr f ∞ and Prt ) 0.85 but are of course inapplicable for Pr , 1. Analogy of Friend and Metzner. Friend and Metzner (1958) [also see Metzner and Friend (1958)] evaluated the integral of eq 78 indirectly, using experimental data for heat and mass transfer as follows. They + + + postulated 2 ) 0, T+ c ) Tm, uc /um ) 1.20 and Prt ) 1 and then rearranged the resulting reduced form of eq 78 as

∫0u

+ c

PrReD(f/2)/NuD - 1.20

du+

()

µt 1 + Pr µ

()

)

f (Pr - 1) 2

1/2

(81)

From a logarithmic plot of the right-hand side of eq 81 versus Pr, they concluded that the integral could be represented on the mean by 11.8/Pr1/3, resulting in

NuD )

PrReDf/2 f 1/2(Pr - 1) 1.2 + 11.8 2 Pr1/3

()

(82)

Their previously mentioned plot of experimental data


for gases and ordinary liquids demonstrated good agreement with eq 82 (as was to be expected since the coefficient of 11.8 was chosen to fit these same values). Correlating Equations in the Form of Analogies. Many correlating equations for heat and mass transfer in the form of analogies (that is, incorporating the friction factor) have been proposed. Equation 65 of Colburn, eqs 66 and 68 of Kadar and Yaglom, and eq 82 of Friend and Metzner could actually be placed in that category. Two additional ones, both based on numerically predicted values from eddy diffusional models, will now be examined as representative. Petukhov (1970) correlated values of NuD predicted by Petukhov and Popov (1963) for uniform heating, 5 × 104 e ReD e 106 and 0.5 e Pr e 2000, with the expression

NuD )

PrReDf/2 f 1/2 1/3 1.07 + 12.7 (Pr - 1) 2

()

(83)

The predicted values were obtained by numerical evaluation of the equivalent of eq 41 using an eddy diffusional model of Reichardt (1951) with Prt postulated to be equal to unity. The predictions of the expression used for the friction factor in eq 83 differ negligibly from those of eq 21 for ReD > 5 × 104. Equation 83 reduces to eq 46 with 1 + 4 ) 1.07 for Pr ) 1 (rather than for Pr ) 0.86) and essentially to eq 36 for Pr f ∞. It is of course not applicable for small Pr. Churchill (1977) utilized asymptotic expressions to develop the following correlating equation for the numerically predicted values of NuD by Notter and Sleicher (1972) for a uniform wall-temperature, 104 e ReD e 106 and 0.004 e Pr e 104:

0.079ReD NuD ) 4.8 +

(2f )

1/2

Pr

(1 + Pr4/5)5/6

(84)

With 6.3 substituted for 4.8, the same expression was recommended for uniform heating. The numerical values of NuD were computed by Notter and Sleicher using a Graetz-type series solutions together with their own expressions for the eddy viscosity and the turbulent Prandtl number. For Pr ) 0, eq 84 reduces to a fixed value of 4.8 as compared to the slightly variable, exact + value of 4(T+ c /Tm)/(1 + 1) for a uniform wall-temperature, while for uniform heating, the fixed value of 6.3 is to be compared with 8/(1 + 3). Equation 84 does not reduce to eq 33 for Pr ) 0.86, but its numerical and functional predictions differs only slightly. It does reduce almost exactly to eq 36 for Pr f ∞. Equation 21 is applicable for the prediction of the friction factor in eq 84. The numerical predictions of NuD upon which eqs 83 and 84 are based take into account the radial variations of τ/τw and j/jw more or less exactly. The principal uncertainty of these values of NuD arises from the arbitrary expressions used for the eddy viscosity and the turbulent Prandtl number. The essentially exact expressions for (u′v′)+, and hence for (u′v′)++, of Churchill and Chan (1994) would result in some improvement in the predictions of NuD vis-a´-vis the values of the eddy viscosity that led to eqs 83 and 84, but such a revision is not justified until significant improvements are made in the generality and reliability of the expressions for the prediction of the turbulent Prandtl number.

Conclusions (i) The new, simplified models of Churchill (1997a) for the local turbulent shear stress and the local turbulent heat flux density lead to simple but exact expressions for the Nusselt number for a round tube and a flat plate for the limiting case of Pr ) 0, the particular case of Pr ) Prt ) PrT = 0.86, and possibly for the asymptotic case of Pr f ∞. These explicit expressions for Pr ) 0 for any thermal boundary condition (eq 29 for a round tube and eq 55 for a flat plate) and for Pr ) Prt for any thermal boundary condition (eq 33 for a round tube and eq 58 for a flat plate) have apparently not appeared previously in the literature. The corresponding expressions for a uniformly heated round tube (eq 42 for Pr ) 0 and eq 46 for Pr ) Prt) were apparently first presented by Churchill (1997b), although such an expression may be inferred from the integral analogy of Lyon (1951), and numerical values of NuD{0} have been computed by a number of investigators. (ii) The exact solutions for Pr ) 0 and Pr ) Prt ) PrT = 0.86 are not particularly useful as components of an overall correlating equation because they incorporate integrals that depend slightly on the Reynolds number. However, together with the possibly exact solution for Pr f ∞, they have been shown herein to be invaluable in identifying functional and numerical errors in the classical algebraic analogies for heat, mass, and momentum transfer that still grace our textbooks and design algorithms. (iii) In particular, all of the classical algebraic analogies are found in to be in significant numerical error due to the neglect or misrepresentation of the spatial variation of the heat flux density in their derivation. (iv) The analogies of Rayleigh, Prandtl-Taylor, von Ka´rmań, and Martinelli are found to be in significant functional and numerical error for large values of Pr because of the neglect in their derivation of combined molecular and turbulent transport near the wall. (v) All of the algebraic analogies except that of Martinelli are in serious error, both functionally and numerically, for Pr f 0 because of the neglect in their derivation of molecular transport in the turbulent core. (vi) The analogy of Thomas and Fan has the distinction of being in functional error for all values of Pr because of the use of transient molecular diffusion as the mechanism of turbulent transport near the wall. (vii) The analogy of Colburn is in functional error with respect to Pr for all but very large values because of the incorporation of an empirical correlation for that regime only. (viii) The analogies of Rayleigh and of Thomas and Fan are in error with respect to the friction factor or drag coefficient for all values of Pr other than those near unity owing to their simplistic models for the transport of momentum and energy in the turbulent core. The Colburn analogy is in error in the same regard but owing instead to the incorporation of an empirical correlating equation of limited range with respect to the Reynolds number. Since the variation of the friction factor is quite limited, this functional error has generally been overlooked. (ix) All of the algebraic analogies except those of Churchill (1997b) and Martinelli neglect the explicit dependence of the Nusselt number on Prt. On the other hand, Prt varies only slightly for Pr > 0.7; hence a mean value buried in some constant does not introduce great error.


(x) All of the analogies for a round tube with a uniform wall-temperature, except those of Churchill (1997b) and Reichardt, neglect the explicit dependence of the Nusselt + + + number on T+ c /Tm and uc /um. (xi) Those analogies that are empirical in the sense that one or more coefficients are evaluated from experimental data for the rate of heat and/or mass transfer, including those of Colburn, Kadar and Yaglom, and Friend and Metzner, thereby compensate on-the-mean to some extent for their idealizations with respect to + + + τ/τw, j/jw, T+ c /Tm, uc /um, and Prt or PrT. Such compensation is also inherent in the correlating equations of Churchill (1977) and Petukhov. (xii) The absence of an additive term in the denominator of the analogies of Reynolds, Thomas and Fan, and Colburn may be seen in retrospect to be a signal of their simplistic functional origin. (xiii) Because of their erroneous functional and numerical predictions, as identified herein, the time appears to have come to abandon all of the classical algebraic analogies in favor of correlating equations based on theoretical asymptotes, such as those of Churchill (1977) and Petukhov and improvements thereon insofar as numerical predictions are concerned. (xiv) On the other hand, the historical role of the classical analogies in the development of an understanding of turbulent convection should not be forgotten or underestimated. The more sophistocated ones have provided a rational, even if incomplete, explanation for the varying dependence of NuD on Pr and ReD, whereas the purely empirical correlating equations in use even today neglect or obscure these variations. In many instances the “scatter” in graphical correlations for convection is not due wholly to experimental error but may to a considerable degree be due to a parametric variation that is incompletely represented by the combination of variables chosen for the absciessa and ordinate. An example is the startling discovery in the 1940’s that the empirical correlating equations that were reasonably successful for both gases and ordinary liquids failed seriously to predict NuD for liquid metals. The reason for this failure was the postulate of a fixed power of Pr. In that instance the classical analogies also failed, although to a lesser extent, but those of Martinelli (1947) and Lyon (1951) provided not only good predictions but also explanations for the failures. All in all, the classical analogies provide excellent examples for the classroom in terms of both their failures and partial successes. (xv) The new models described herein for turbulent transport, together with the almost exact expressions for the local turbulent shear stress, might be expected to produce improved numerical predictions of the Nusselt number and hence improved overall correlating equations. However, a significant improvement in this respect must await the establishment of more reliable and general expressions for Prt or PrT than are yet available. The critique of the analogies was limited to Pr ) 0, Pr ) Prt ) PrT, and Pr f ∞, for which such a general expression is not required. Nomenclature a ) radius of a round tube (m) a+ ) a(τwF)1/2/µ cp ) specific heat capacity at constant pressure (J/kg‚K) Cf ) 2τw/Fu2∞, drag coefficient D ) diameter of a round tube (m) f ) 2τw/Fu2m, Fanning friction factor

j ) heat flux density (W/m2) jw ) heat flux density at the wall (W/m2) k ) thermal conductivity (W/m‚K) kt ) eddy thermal conductivity (W/m‚K) m ) arbitrary exponent NuD ) jwD/k(Tw - Tm), Nusselt number for a round tube Nux ) jwx/k(Tw - T∞), Nusselt number for a flat plate Pr ) cpµ/k, Prandtl number Prt ) cpµt/kt, turbulent Prandtl number PrT ) cp(µt + µ)/(kt + k), total Prandtl number r ) radial coordinate (m) R ) r/a ReD ) DumF/µ, Reynolds number for a round pipe Rex ) xu∞F/µ, Reynolds number for a flat plate T ) time-mean temperature of fluid (K) Tc ) time-mean temperature of the fluid at centerline of a round pipe (K) Tm ) mixed-mean, time-mean temperature of the fluid (K) Tw ) temperature of the wall (K) T∞ ) free-stream temperature of the fluid (K) T+ ) k(τwF)1/2(Tw - T)/µjw T′ ) turbulent fluctuation in temperature about the timemean value (K) (T′v′)++ ) FcpT′v′/j, fraction of the heat flux density due to turbulence u ) time-mean component of the velocity in the x-direction (m/s) uc ) time-mean velocity at the centerline of a round pipe (m/s) um ) time-mean, mixed-mean velocity (m/s) u∞ ) free-stream velocity (m/s) u+ ) u(F/τw)1/2 u′ ) turbulent fluctuation in velocity about the time-mean value (m/s) (u′v′)+ ) (-Fu′v′)/τw (u′v′)++ ) (-Fu′v′)/τ, fraction of the momentum flux density due to turbulence v′ ) turbulent fluction in velocity normal to the wall (m/s) x ) coordinate along the wall (m) x+ ) x(τwF)1/2/µ y ) coordinate normal to the wall (m) y+ ) y(τwF)1/2/µ Greek Symbols R ) arbitrary coefficient γ ) (j/jw)(τw/τ) - 1 δ ) total thickness of the boundary layer on a flat plate (m) δE ) total thickness of the thermal boundary layer on a flat plate (m) δs ) thickness of the laminar sublayer (m) δ+ ) δ+(τwF)1/2/µ 1 ) see eq 24 2 ) see eq 32 3 ) see eq 43 4 ) see eq 53 5 ) see eq 56 6 ) see eq 59 µ ) dynamic viscosity (Pa‚s) µt ) eddy dynamic viscosity (Pa‚s) F ) specific density of the fluid (kg/m3) τ ) time-mean shear stress in the fluid (Pa) τw ) time-mean shear stress on the wall (Pa)

Literature Cited Abbrecht, P. H.; Churchill, S. W. The Turbulent Entrance Region in Fully Developed Turbulent Flow. AIChE J. 1960, 6, 268. Bejan, A. Convection Heat Transfer; John Wiley and Sons: New York, 1984.

3878 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Churchill, S. W. Comprehensive Correlating Equations for Heat, Mass and Momentum Transfer in Fully Developed Flow in Smooth Tubes. Ind. Eng. Chem. Fundam. 1977, 6, 109. Churchill, S. W. Theoretically Based Expressions in Closed Form for the Local and Mean Coefficients of Skin Friction in Fully Turbulent Flow along Smooth and Rough Plates. Int. J. Heat Fluid Flow 1993, 14, 231. Churchill, S. W. New, Simplified Models and Formulations for Turbulent Flow and Convection. AIChE J. 1997a, 43, 1125. Churchill, S. W. The Prediction of Turbulent Convection without Explicit Empiricism. Thermal Science and Engineering, in press (1997b). Churchill, S. W.; Balzhiser, R. E. The Radial Heat Flux. Chem. Eng. Prog. Symp. Ser. 1959, 55 (29), 127. Churchill, S. W.; Chan, C. Improved Correlating Equations for the Friction Factor for Fully Turbulent Flow in Round Tubes and between Identical Parallel Plates, both Smooth and Naturally Rough. Ind. Eng. Chem. Res. 1994, 33, 2016. Churchill, S. W.; Chan, C. Theoretically Based Correlating Equations for the Local Characteristics of Fully Turbulent Flow in Round Tubes and between Parallel Plates. Ind. Eng. Chem. Res. 1995a, 33, 1332. Churchill, S. W.; Chan, C. Representation of the Time-Averaged Characteristics of Turbulent Flow in Channels in Terms of the Local Turbulent Shear and Normal Stresses. AIChE J. 1995b, 41, 2513. Colburn, A. P. A Method of Correlating Forced Convection Heat Transfer Data and a Comparison with Fluid Friction. Trans. Am. Inst. Chem. Eng. 1933, 29, 174. Danckwerts, P. V. Significance of Liquid Film Coefficients in Gas Absorption. AIChE J. 1957, 3, 1460. Friend, W. L.; Metzner, A. B. Turbulent Heat Transfer Inside Tubes and the Analogy Among Heat, Mass and Momentum Transfer. AIChE J. 1958, 4, 393. Heng, Ly; Chan, C.; Churchill, S. W. Exact Characteristics of Turbulent Convection. In preparation. Higbie, R. The Rate of Absorption of Pure Gas into a Still Liquid During Short Periods of Exposure. Trans. Am. Inst. Chem. Eng. 1935, 31, 365. Kadar, B. A.; Yaglom, A. M. Heat and Mass Transfer Laws for Fully Turbulent Wall Flows. Int. J. Heat Mass Transfer 1972, 15, 2229. Knudsen, J. G.; Katz, D. L. Fluid Dynamics and Heat Transfer; McGraw-Hill Book Co.: New York, 1958; p 425. Lyson, R. E. Liquid Metal Heat Transfer Coefficients. Chem. Eng. Prog. 1951, 47, 75. Martinelli, R. C. Heat Transfer of Molten Metals. Trans. ASME 1947, 69, 947. McAdams, W. H. Heat Transmission, 3rd ed.; McGraw-Hill Book Co.: New York, 1954; p 212. Metzner, A. B.; Friend, W. L. Theoretical Analogies between Heat, Mass, and Momentum Transfer and Modifications for Fluids of High Prandtl or Schmidt Numbers. Can. J. Chem. Eng. 1958, 36, 235.

Notter, R. H.; Sleicher, C. A. A Solution to the Turbulent Graetz Problem-III. Fully Developed and Entry Region Heat Transfer Rates. Chem. Eng. Sci. 1972, 27, 2073. Page, F., Jr.; Schlinger, W. G.; Breaux, D. K.; Sage, B. H. Point Values of Eddy Conductivity and Eddy Viscosity in Uniform Flow between Parallel Plates. Ind. Eng. Chem. 1952, 44, 424. Papavassiliou, D. V.; Hanratty, T. J. Transport of a Passive Scalar in a Turbulent Channel Flow. Int. J. Heat Mass Transfer 1997, 40, 1303. 196. Petukhov, B. S. Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties. Adv. Heat Transfer 1970, 6, 503. Petukhov, B. S.; Popov, V. N. Theoretical Calculations of Heat Exchange and Frictional Resistance in Tubes of an Incompressible Fluid with Variable Physical Properties. High Temperature 1963, 1, 69. Prandtl, L. Eine Beziehung zwischen Wa¨rmeaustausch und Stro¨mungswiderstand der Flu¨ssigkeit. Phys. Z 1910, 11, 1072. Reichardt, H. Die Grundlagen des turbulenten Wa¨rmeu¨bertraganges. Archiv. Wa¨rmetechnik 1951, 2, 129; English transl. The Principles of Turbulent Heat Transfer. Nat. Adv. Comm. Aeronaut. TM1408, Washington, D.C., Sept. 1957. Reynolds, O. On the Extent and Action of the Heating Surface on Steam Boilers. Proc. Lit. Philos. Soc. Manchester 1874, 14, 7; Papers on Mechanical and Physical Subjects; Cambridge University Press: Cambridge, 1901; Vol. I, Paper 14, p 81. Sherwood, T. K.; Ryan, J. M. Mass Transfer to a Turbulent Fluid with and without Chemical Reaction. Chem. Eng. Sci. 1959, 11, 81. Sleicher, C. A.; Tribus, M. Heat Transfer in a Pipe with Turbulent Flow and Arbitrary Wall-Temperature Distribution. Trans. ASME 1957, 79, 789. Taylor, G. I. Conditions at the Surface of a Hot Body Exposed to the Wind. British Adivsory Committee Aeronautical Reports and Memoranda, No. 272, 1916. Thomas, L. C.; Fan, L. T. Heat and Momentum Transfer Analogy for Incompressible Boundary Layer Flow. Int. J. Heat Mass Transfer 1971, 14, 715. von Ka´rmań, Th. The Analogy Between Fluid Friction and Heat Transfer. Trans. ASME 1939, 61, 705. Yahkot, V.; Orszag, S. A.; Yahkot, A. Heat Transfer in Turbulent Fluids-1. Pipe Flow. Int. J. Heat Mass Transfer 1987, 30, 15.

Received for review November 26, 1996 Revised manuscript received May 5, 1997 Accepted May 12, 1997X IE960750A

X Abstract published in Advance ACS Abstracts, July 15, 1997.

Critique of the Classical Algebraic Analogies between Heat, Mass

Recommend Documents