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Aof

General Expression for the CorrelationRates of Transfer and Other Phenomena

The expression Y = (1 + Zn)l;n where Y and Z are expressed in termsof the solutions for asymptotically large and small values of the independentvariable is shown to be remarkably successful in correlating rates of transferfor processes which vary uniformly between these limiting cases. The arbi-trary exponent n can be evaluated simply from plots of Y versus Z and Y/Zversus 1/Z. The expression is quite insensitive to the choice of n and theclosest integral value can be chosen for simplicity. The process of correla-tion can be repeated for additional variables in series. Illustrative applica-tions are presented only for flow, conduction, forced convection, and freeconvection, but the expression and procedure are applicable to any phe-nomenon which varies uniformly between known, limiting solutions.

S. W. CHURCHILLand R. USAGISchool of Chemical EngineeringUniversity of PennsylvaniaPhiladelphia, Pennsylvania 19104

. The limiting solutions for large and small values of th eindependent variables, and parameters such as the Reyn-olds number, the Prandtl number, and time are known formany transfer processes such as laminar and turbulentflow through porous media, laminar forced convection,and thermal conduction. Solutions for intermediate casesare generally not in closed form.The use of a simple equation containing one arbitraryconstant is proposed for interpolation between these lim-iting solutions. The constant can be evaluated from oneor more experimental or theoretical values. The equationitself has been used previously for correlation, but itsgeneral utility has not been recognized and a systematic

method of evaluating the constant has not been proposed.The general applicability of the equation and the pro-cedure of correlation are illustrated for a number of proc-esses of flow and heat transfer. The expression and pro-cedure appear however to be applicable to any phenom-enon which varies uniformly between known, limitingsolutions.

The proposed expression is useful for evaluat ing aswell a; summarizing experimental data and values ob-tained from computer solutions. It is convenient for de-sign purposes, particularly since all of the correlationshave the same form.

CONCLUSIONS AND SIGNIFICANCEThe equation Y = (1 + Zn)l/n,where n is an arbitraryconstant and Y and Z are defined in terms of the limiting

solutions for large and small values of t he independentvariable, is remarkably successful in representing data forwhich the dependent variable varies uniformly from smallto large values of the independent variable. The majorityof transfer processes fall in this category. Examples arethe dependence of the pressure drop on the rate of flowthrough porous media and the dependence of the Nusseltnumber on the Prandtl number in laminar free convection.Processes which have an irregular dependence such as thepressure drop through a pipe on the Reynolds number inthe region of transition from laminar to turbulent flowcannot of course be successfully represented.For processes in which the dependence on the inde-pendent variable increases as the independent variableincreases, Y is chosen as y(z)/y(z + 0) and 2 as y(z 4 w)/y( z -+ 0) where y(z + 0) and y(z -+ w ) are the asymp-totic solutions for small and large z , respectively. Forprocesses in which the dependent variable decreases asthe independent variable increases, Y is chosen instead asy(z -+ O)/y(z) and Z as y(z + O)/y(z -+ w) . Sincey(z + 0) and y(z + w ) cannot both be constants Z is afunction of z. The correlation is equivalent to the nth-ordersum of the asymptotic solutions and is proportional tothe nth root mean value. For example, if n = 2 the corre-lation is equivalent to dx imes the root mean square of

the asymptotic values.Experimental data and computer solutions for repre-

sentative processes appear to follow the functional rela-tionship provided by this equation very closely eventhough there is no apparent theoretical justification forits use. This success apparently is a result of the greatsimilarity of the different rate processes when expressedin this canonical form and of the limited variance fromthe asymptotic solutions. The equation or a plot of Yversus Z can thus be considered to be a general representa-tion of all rate processes with uniform behavior, individualprocesses being characterized by n alone.

The arbitrary exponent n can be evaluated by compar-ing experimental data or computed solutions with curvesof Y versus Z or Y/Z versus 1 /Z for fixed values of n.The correlations for typical processes prove to be rela-tively insensitive to the choice of n and in many instancesan integral value can be chosen without serious error.Converiely such plots provide a very sensitive test of theconsistency, precision, and accuracy of experimental dataand computer solutions. In some cases the equation canbe applied in series (with different Y, 2, and n) for two ormore independent variables or parameters. In other casesn can be correlated as a function of the additional varia-bles.

The intersection of the two limiting solutions, whichoccurs at 2 = 1, defines a central value for the inde-

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pe nd en t variable. T his cen1:al value is different fo r differ-en t processes. F or example, the central value of the P randtlnumber is 0.492 for free convection from a vertical iso-thermal plate, but is 0.0467 for forced convection from anisothermal plate. This central value is optimal for experi-mentation or numerical solution insofar as correlation isconcerned since the related deviation of Y from the l im-iting solutions is greatest. That is, the value of n is mostsensitive to experimental or computed values at or nearz = 1.Flow through porous media, the free fall of a sphere,laminar free convection from an isothermal vertical plate,laminar forced convection from a flat plate, and transientthermal conduction to an insulated semi-infinite regionare examples of applications in which the supporting data,series solutions, and computer solutions are quite con-sistent and dependable and in which precise correlationsare obtained by this procedure. Laminar forced convec-tion in a tube is illustrative of applications in which thereis a choice between different asymptotic solutions and inwhich the correlation is not quite so successful. This latterexample is included to indicate that the development ofthe best correlation is not necessarily routine and may re-quire ingenuity and judgement despite the simplicity ofthe gene ral expression.Although these examples are for simple processes offlow and heat transfer, the equation and procedure appear

to be applic able to mass transfer, to m ore complex proc-esses of transfer, and even to physical relationships otherthan transfer.Tabulations of experimental or theoretical values areessential for the development of correlations. The com-mon practice of presenting such results only in graphicalform is unfortunate in this respect . On the other hand,tabulations as well as theoretical solutions in series andintegral form are generally inconvenient for direct use indesign and analysis. Log-log or semilog plots generallyshow the forest but not the trees; deviations are com-pressed and effectively disguised. In printed form suchplots can seldom be read with sufficient accuracy forapplications. Arithmetric plots are usually limited in utilityto a narrow range of one of the variables. By contrast, thespecial graphical representations advocated herein dis-play only deviations from the limiting solutions and em-phasize the scatter of individual points. Hence they pro-vide a better basis for critical analysis of the individualpoints and of the success of proposed correlations. Eventhese plots are not very convenient for direct use. Em-pirical equations are generally the most convenient forthe application of experimental and complex theoreticalresults. The form of the empirical equation proposedherein has the advantage of simplicity, generality, inherentaccuracy, and convergence to theoretical solutions in thelimits.

Solutions in closed form have been developed for manytransfer processes for asymptotically larg e an d small valuesof time, Pr, Sc, Re, Gz, Gr, etc. Solutions for intermediatevalues have generally been accomplished only in the formof infinite series, definite integrals, and tabulations ofnumerical integrations. These intermediate solutions areoften inconvenient to use. For example, the solution de-rived by Graetz (1885) for convective heat transfer infully developed laminar flow in a tube following a step-chan ge in wall-temperature converges very slowly for largeGz. In most cases such solutions for intermediate valuescan be e val uate d only at a series of discre te values of theindependent variable, and a graphical representation oran empirical correlation of these values is desirable forinterpolation. Many different expressions, including powerseries, have been used for this purpose, but apparentlyno general expression has been proposed.Empirical expressions for the entire range have beenconstructed in some cases from the limiting solutions forlarge and small values alone. For example, the equationdeveloped by Ergun (1952) for the pressure drop in flowthrough randomly packed beds of uniformly sized spheresconsists of th e simple sum of t he emp irical corre lations forthe two limiting cases of purely laminar flow and purelyturbulent flow. In most cases additional terms and arbitraryconstants have been found necessary to fi t the intermediatevalues. For example, Le Fevre (1956) proposed the fol-lowing empirical equ ation for laminar free convection froma vertical, isothermal p late:

GrPr22.435 + 4.884 Pr'lz + 4.952 PrNu, =3 [4

where the first and third terms in the denominator of thebracket correspond to the asymptotic solutions fo r smalland large Pr, respectively, and the coefficient of Pr"2 waschosen to fit th e interm ediate values of the numerical solu-tion of Ostrach (1953).

Acrivos ( 1961) uti l ized Equation (7 ) below with thevariables interpreted as in Equat ion (16) fo r representa-tion of the effect of Pr and Sc on convection in severallaminar boundary layer flows. He recommended specific,integral values of n for these different situations but didnot discuss the method of evaluation of the arbitrary ex-ponent n or the accuracy of the representation. Manyothers have utilized this expression but generally in a morerestricted sense.The development of a general expression and procedurefor correlation in terms of asym ptotic solutions for largeand small values of the independent variable and onearbitrary constant is outlined in the following paragraphs.The general expression is the same as that used b y Acrivosbut its application is extended to other variables and proc-esses and to increasing as well as decreasing dependenceon the independent variable. Principle attention is givento the evaluation of the arbitrary exponent and to thechoice of the most appropriate asymptotic solutions.DEVELOPMENTCase A. Increasing Dependenceat the higher l imit, that is, whe nWhen the power of the independent variable is greater

y+AzP as z+O (2)y+Bzq as z+co (3 )

with q > p, the expressiony = [ Azp)" + (Bz~ ) " ] ' " ' (4 )

is usually suitable for interpolation. (T he variables y andz and hence the coefficients A an d B will be considereddimensionless for convenience although this restriction isnot necessary.) The right side of Equation ( 4 ) can be in -terpreted as the nth-orde r sum of th e two asymptotic solu-AlChE Journal (Vol. 18, No. 6 )

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108

6

4

y 3

2

I01 0 2 0 4 06 08 10 2 4 6 8 1 0z

Fig. 1. Gener al functi onal relationship for transfer processes.

\ 4 .6

0 0 2 0 4 0 6 08 I 0 0 8 0 6 0 4 0 2 0Z I / Z

Fig. 2. Worki ng plot for correlat ion.

tions or as 2 lIn times the nth-root mean value, for example,if n = 2 as v'z times the root mean square value. Equa-tion ( 4 ) can be rearranged in the two more convenientformsand

both of which have the genera1 form

,$ y/BzQ = [l + ( A / B z ~ - P ) ~ ] ~ / ~y/Azp = [1+ ( B z q - p / A ) n]l/n

Y = [l + Zn-jl'n( 5 )( 8 )(7)

Evaluation of n. The exponent n can be evaluated froma single intermediate value of y ( z ) or Y ( 2 ) .f Y is knownfor Z = 1For any other value of 2, n must be calculated indirectly.The best value of n for a series of values of Y (2) couldbe calculated by nonlinear regression. However, Equation(7) is relatively insensitive to n as indicated by Figure 1in which log Y is plotted versus log 2 for a series of in-tegral values of n. The known experimental or theoreticalvalues can therefore be plotted in Figure 1 and a rea-sonable value of n chosen by inspection. Alternatively theplotted values can be extrapolated or interpolated to yieldY ( 1) so that n can be calculated from Equation (8) . Aplot in the form of Figure 2 which has Y/Z as an ordinateand 1/Z as an abscissa for Z > 1 s more convenient andaccurate for the determination of n since it permits expan-sion of the scale of the ordinate. Such a working plot canbe readily constructed for just the observed range of val-ues of Y and 2. Figure 2 also demonstrates the reciprocalsymmetry of Equations ( 5 ) and (6) about Z = Bzq-p/A= 1. In both Figures 1 and 2, the point (Y = 1,2 = I )corresponds to the intersection of the lines pepresenting thetwo limiting solutions. Y ( l ) - 1 = 21/n - 1 representsthe maximum fractional deviation of Y from the limitingsolutions. 2 = 1 represents the dividing line between largeand small values of the independent variable. Hence if aAlChE Journal (Vol. 18, No. 6)

n = l n ( 2 ) / l n Y ( l ) ( 8 )

single calculation or experiment is to be carried out, itshould be at or near 2 = 1 since n is most sensitive toY ( 1 ) . Even if Equation (7) were not used for correlationthis value of Y would provide a useful bound for theprocess.

The experimental data for laminar %owthrough packed beds of spheres are well represented bythe Kozeny-Carman equation

Example 1.

cp = 150 Re, (9)(10)

and the data for turbulent flow by the Burke-Plummerequation@ = 1.75 Re,2

The corresponding value of Y is then @/150Re, and of Zis 1.75 Re,/150 = Re , /87 .5 . The central value of Re, at2 = 1 s thus 87.5. Figure 3 is a plot in the form of Figure2 of a sampling of the data used by Ergun (1952) to de-velop his correlation. These points scatter widely but arerepresented on the mean by the straight lines correspond-ing to n = 1.This value yields the Ergun equation

cp = 150 Re, + 1.75Re,2 (11)These same experimental values are plotted in Figure4 in the form of Figure 1. This plot would have exactly

the same form as the original plot prepared by Ergun toillustrate the success of his correlation if the ordinate weremultiplied by 150 and the abscissa by 87.5. Comparisonof Figures 3 and 4 confirms the greater sensitivity of theformer. Ergun arrived at Equation (11) by heuristicreasoning and then tested his postulate in the form ofFigure 4.On the other hand, the data suggest an exponentof n without any postulation when plotted in the form

R e p / 8 5 7 8 5 7 / R e p

Fig. 3. Pressure drop in flow through a packed bed of spheres.

I 0 0

10

150 R e p

I .O

001 0 10 1.0 10 100R e , / 8 7 5

Fig. 4. Loga rithmi c correlation for pressure drop through a packedbe d o f spheres.

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of Figure 3. In this instance the exponent n was knownfrom the prior work of Ergun. In the several dozen otherprocesses which have been examined the best value of theexponent was not known in advance. This process alsohappens to be the only one examined so far in which thebest exponent is unity.

The drag coefficient for a solid spherefalling at low velocity in an incompressible fluid is repre-sented by Stokes' law

Example 2.

CD= 24/Red (12)As Red approaches 3000 the drag coefficient approachesa constant value of approximately 0.40. (At still highervelocities the drag coefficient has a more complex depen-dence on velocity.) The experimental data of Christiansen(1943) which are plotted in Figure 5 with Y = CDRed/24and z = 0.4 Red/% = Red/60 are reasonably well rep-resented by Equation ( 7 ) with n = 0.56 or say 5/9. It isnot certain whether the values for 60/Re < 0.4 are inslight error or whether the physical behavior cannot berepresented any more closely by such a simple relationshipwith constant n. The correlation

CD= (24/Red) (1+ [Red/60]5/9)9/5 (13)may be used for 0 < Red < 3000. The central value ofRed is 60, Drag coefficients for particles of different shapes,both supported in a fluid stream and falling, have a quali-tatively similar dependence on Re and could obviouslybe correlated in the form of Equation (13) with slightlydifferent values of n and the appropriate coefficients sub-stituted for 24 and 60.Case 8. Decreosing DependenceWhen the power of the dependent variable is lower atthe higher limit, that is, when p > q in Equations (2)and (3)

y = [(AzP)--n+ (Bzq)--n]--lln (14)1can be used instead of Equation (4) for interpolation.Equation (14) can be rearranged as

or

which both have the same form as Equation (7 ) . Henceplots such as Figures 1 and 2 can be used for the deter-mination of n in this case as well. (Alternately, Case Bcan be considered the same as Case A but with a negativevalue of n.)

The processes considered by Acrivos (1961) were allin this category and he noted that both asymptotic solu-tions were upper bounds for the dependent variable y.

The limiting solutions obtained by LeFevre (1956) for laminar free convection from an iso-thermal vertical plate are, as indicated by Equation (1)

Bzq/y = [1 + (B /Azp -q )n]l/nAzp/y = [1 + ( A z p - q / B )nl1l-n (15)(16)

Example 3.

N u, = 0.6004 Gr*PrZ for Pr+ 0 (17)N u , = 0.5027 Gr'hPr'A for P r + 00 (18)

Hence in the form of Equation (15) Y = 0.5027 Gr%Pr%/Nu,, Z = (0.492/Pr)'A and the central value of Pr =0.492. Values obtained from the numerical solutions ofOstrach ( 1953), Sparrow and Gregg ( 1959), Sugawaraand Michiyoshi (1953) and Saunders (1939) are plottedin Figure 6. A value of n = 9/4 represents these valuesremarkably well yielding

N u z = 0.503 Gr%Pr'h/[ 1 + (0.492/Pr)9/16]4/9 (19)The agreement of the values of Ostrach and of SparrowPage 11 24 November, 19 72

and Gregg with each other and with the curve for la =9/4 over the entire range suggests that it is the isolatedvalues of Saunders and of Sugawara and Michiyoshi whichare in error. The fact that these latter deviations are lessthan 1% emphasizes the extreme sensitivity of this plot.The better representation is at least in part due to thegreater precision of these computed values as contrastedwith the use of experimental data in Figures 3 and 5. Sur-prisingly, Equations (1) and (19) agree to within about0.1% over the entire range. Ede (1967) indicates thatEquation (1 ) is central to the widely scattered experi-mental data. Equation (19) can, therefore, also be consid-ered to represent the experimental data.

This was one of the processes considered by Acrivos(1961). He proposed n = 2 for representation of thecomputed values of Ostrach (1953), presumably on thebasis of a log-log plot of 1/Y versus 2. The superiority ofthe coordinates of Figure 2 in general and of Figure 6 spe-cifically is thus substantiated in that n = 9/5 is showndecisively to be a better choice than n = 2, even thoughthe latter value is acceptable for practical purposes.Cuse C. N o n p o w e r D e p e n de n c e

The above derivations are all for processes in which therate depends on the independent variable to a fixed powerin the limits. Many transfer processes fall in this category.However, the usefulness of Equat ion (7) for interpolationis not limited to such functional relationships. Equation( 5 ) can be generalized to the fo?

.

I

y(z)/y(z-, m )

R e d / 6 0 6 0 / R e dFig. 5. Drag coefficient for a freely falling sphere.

I I I I I I , I I

( Pr /0.4 92 '"0 4 9 2 / P r 1 " 4Fig. 6. Laminar free convection from a vertical isothermal plate.

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= [ l + ( y ( z + O)/y(z+ w))"]ll" (20)where y ( z -+ 0) and y ( z+ O ) are the asymptotic solu-tions for z -+ 0 and z -+ 00, respectively. Equations ( 6 ) ,(15), and (16) can be generalized similarly.

Transient conduction through a finite layerof insulation to a semi-infinite region of lesser conductivityfollowing a step change in the external surface tempera-ture with both the insulation and the region at a uniforminitial temperature, as discussed by Churchill ( 1962), canbe used to illustrate this more general case. The asymptoticsolutions for the heat flux density at the external surfaceare

i ( n ~ t ) ' / z / k ( T , T , ) = 1 as t+ 0 (21)and

Example 4 .

j 6/k Tt , .- T,) = eat/@@rfc ( Y t/(rW)/i s t+ M(22)

When plotted in the form suggested by Equations (20),(21) , and (22) the intermediate values computed fromthe series solution for u = 49 suggest a value of n = 8,yieldingj ( n d ) " Z / k ( T , - T,)

(23)Equation (23) agrees with the computed values to foursignificant figures over the entire range as indicated inFigure 7. Analysis suggests that n will become a significantfunction of u for c < 20. Hence a plot of n versus uwould complete the correlation. The central value of timecorresponding to the intersection of the dashed line anddashed curve in Figure 7 or to the simultaneous solutionof Equations (21) and (22) is approximately S ' / ~ (Y .Case D. Ambiguity in the Choice of the Asymptotic Solution

Churchill and Ozoe (1972) developed correlations forlaminar forced convection in flow over flat plates andthrough tubes. Their work for fully developed flowthrough tubes illustrates the process of analysis whichmay be required to choose the appropriate asymptoticsolutions.

Following a step function in wall tempera-ture in fully developed laminar flow in a tube N u D ap-proaches 3.657 as Gz + 0. For the other limit Lkv&que(1928) derived a solution

= ( 1 + [ (n t / P ) eat/@*" erfc d / u 2 S 2 ) '/218)

Example 5 .

X U D= 1.167Gz ' /~ for Gz+ 0 (24)Lipkis (1935) re-expressed Equation (24) in terms of themixed-mean temperature rather than the inlet tempera-ture. The resulting equation

N U D= 1.167 Gz113/[l - 5 . 5 / G ~ ~ / ~ ]would be expected to be applicable to lower values of Gzthan Equation (24). However, a correlation based onEquation (25) would retain the singularity at Gz r 3.0.Hence, Churchill and Ozoe (1972) used Equation (24)rather than Equation (25) to derive the following expres-sion for interpolation

(25)

Nu,/3.657 = [l + ( G ~ / 3 0 . 8 ) ~ / ~ ] ' / ~ 26)Equation (26) represents the values obtained from theGraetz series and by direct numerical integration of thedifferential energy balance reasonably well. However, thecomputed points fall slightly below unity for large Gzindicating that the computations are in error or that theLCv&que solution is not a true lower bound.The solution obtained by Worsge-Schmidt (1967) asan expansion about the LCv&quesolution confirms that theAlChE Journal (Vol. 18, No. 6 )

108 I I

06 .0 4 I I i10 2 10 ' ,

a t / 6 'Fig. 7. Conduction to an insulated, semi-infinite region ofter a stepin surface temperature u = 49

LCvBque solution is not a lower bound for the general solu-tion and hence is not satisfactory on theoretical groundsfor the construction of a correlation. The results of Worspre-Schmidt for large Gz can be approximated by the expres-sion

N u D = 1.167 Gz113- 1.7as suggested by Lipkis (1955). If the right side of Equa-tion (27) is used as the asymptotic solution for large Gzthe general correlation takes the form

(27)

Nu~/3.657= (1+ [ (Gd30.8)'I3 - 0.4651")""(28)

Equation (28) is clearly limited to Gz > 3.10 in order toavoid negative values for the term in square brackets. Thisis not a serious limitation since N uD can simply be takenas 3.657 without significant error for Gz < 3.10. A valueof n = 2 represents the computed values well for all Gz> 3.10; but some dissymmetry is apparent.

Because of these two discrepancies in the representationprovided by Equation (28) an alternative correlation wasconstructed as follows. Equation (27) was rearranged as

N u D + 1.7= 1.167 Gz "~ (29)The corresponding expression for correlation is then( N u + 1.7)/5.357 = [ l + (G~/97)"/~] ' /" (30)

Again in this form the computed values do not demon-strate the symmetry found previously in Examples 1through 4. Apparently the appearance of an additive termin one of the asymptotic solutions precludes the symmetryimplied by Equation (7) . However n = 8/3 representsthe computed values well for all Gz.

These examples illustrate a limitation on the ability ofEquation (7) to represent this particular transfer process.However, the limitation is of greater intrinsic than practi-cal interest. Equation (26) and Equations (28) and (30)with a single compromise value of n provide reasonablygood correlations (within 5% ) over the entire range ofGz from 0 to co.Correlations within the range of certaintyof the most reliable sets of computed values ( - 1%) areprovided by both Equations (28) and (30) by using dif-ferent values of n for large and small Gz. Equation (26)is even more accurate for all but the short range of Gz inwhich the additive term in Equation (27) is influential.All of these correlations have a greater range of applica-bility than the Wors@e-Schmidt series solution and aremore convenient to use than the Graetz series solution orthe various sets of tabulated values. The sensitivity of thismethod of correlation is again confirmed in that the failureof the LCv&que solution by a very slight amount to pro-vide a lower bound was revealed in the plot which led toEquation (26).A similar discrepancy was revealed in the development

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of a correlation for steady heat and component transferfrom a sphere in Stokes flow by Brian and Hales (1969).They correlated their values with an equation equivalent

presumed to be a lower bound for developing flow. Hence,the reduced form of Equation (20) was combined insteadwith Equation (26) to yield

)"]"" (33)Nu D (Gz/33 )3.657 [ l + (Gz/30.8)8/311/8 [ + ( [1 + (Pr/0.0468)2/3]1/4[ 1 + ( ~ 3 0 . 8 )/3 ] l l 8to Equation (7 ) but determined the coefficient in theasymptotic solution for large flow rates empirically. Thiscoefficient was observed to be 10% higher than that ofthe asymptotic solution obtained analytically after severalapproximations. Their calculations were sufficiently pre-cise to reveal this discrepancy in a log-log plot of theoriginal variables. A plot in the form of Figure 2 wouldprobably have revealed the discrepancy even with less pre-cise values.Case E. Multiple Variables

The effect of secondary variables can be taken into ac-count by determining n for a series of values of the secondvariable and developing a graphical correlation for n assuggested in Example 4. In some instances the use ofEquation (20) can be repeated in series for two or morevariables. This process is illustrated below.Churchill and Ozoe (1972) obtained anxample 6.

Equation (33) with n = 4 was found to represent thecomputed values equally well and behaves correctly in allof the chosen limiting cases. For example for Pr + 0 andGz + 03 Equation (33) appears to converge to the Graetzsolution for plug flow which is presumed to be an upperbound for developing flow. Again the previously men-tioned anomalies in the computed values provide someuncertainty in the choice of n. Furthermore, the com-puted values for very large Gz are consistently less thanpredicted by Equation (33) and indeed by Equation(32). This discrepancy indicates that Equation (32) isnot truly a lower bound for developing flow in tube justas Equation (25) is not for developing flow.

The value of 1.7 was therefore arbitrarily added to theleft side of Equation (32). This expression without the4Gz-1/2-term was then combined with Equation (30)with 8/3 to yield

N U D+ 1.7 (Gz/71) l j 25.357 [1 + (Gz/97) 8/9]3/8 = ( [ l + (Pr/0.0468)2/3]1/* [ l + (Gz/97)*/9]3/8 (34)

excellent representation for their computer values andthose of prior workers for laminar forced convection inflow over an isothermal plate with the expression

Nu, = 0.564 Re,1'2Pr1/2/[ 1 + (Pr/0.0468)2/3]1/4A value of n less than 4 would represent the computedvalues slightly better for large Pr and a value greater than4 for small Pr, but Equation (31) agrees to approximately1% with all of the computed values.

Acrivos (1961) recommended n = 3 for all wedge flowsincluding the flat plate. Analysis of his computed values(which he graciously supplied) in the form of Figure 2indicates that n = 3 provides an excellent fit for flow nor-mal to the plate but reveals that the best value of n in-creases to 4 with decreasing wedge angles.Equation (31) was reinterpreted and rearranged as anasymptotic solution for Gz + 00 for forced convection indeveloping laminar flow in an isothermal tube in the fol-lowing form:NuD = 0.637 GZ ' / ~ / [1 + (Pd0.0468) /3]1/4Equation (32) might be combined in turn with N U D =3.657 for fully developed heat transfer to yield an expres-sion for all Gz and all Pr in developing flow. However, itis apparent that the resulting expression would blow upfor Cz L 16/[1 + (Pr/0.0468)2/3]1/2.Hence the term4Gz-'l2 in Equation (32) was dropped for this applica-tion just as was the comparable term in Equation (25)since these terms become negligible anyway as Gz + 00 .The resulting combined expression with a value of n =9/5 was found to yield a reasonable representation for theseveral sets of computed values which are available forPr 6 2.0. The choice of n and the success of the form ofthe correlation is subject to considerable uncertainty be-cause the computed values scatter widely, and it is notapparent which are the most reliable and to what degree.However, this correlation yields values for U U D for Pr+ a3 below those given by the Graetz solution, which is

(31)

- 4Gz-'/') (32)

Page 11 26 November, 19 72

Equation (34) appears to behave correctly in all of thelimiting cases of large and small Gz and Pr. n = 8/3represents the computed values for all Gz reasonably well.The additive term does not appear to introduce as muchasymmetry as it did in the case of Equation (30). Someof the computed Nusselt numbers are less than the valuesgiven by Equation (30) which certainly should be alower bound. However, these deviations and other widedivergences are probably due to computing error ratherthan to the inadequacy of the correlation.The discrepancies in these several correlations for forcedconvection in developing flow arise not from the applica-tion of Equation (20) in series for two variables butrather from the limitations in the component equations.Equations (33) and (34) appear somewhat ungainly butare probably the simplest possible expressions that con-verge to all four of the chosen asymptotic solutions.INTERPRETATION

In all of the above processes a uniform transition occursbetween the limiting values. Clearly Equation (7) cannotrepresent processes which have an irregular transition suchas from laminar to turbulent flow in pipes. A sufficient re-striction to permit the use of Equation (7) is that the firstderivative of the functional relationship be continuous andthat the second derivative not change sign. This restric-tion can be relaxed somewhat as indicated by the successachieved in example 4 in which the second derivativechanges sign, but slowly.

Symmetry about Z = 1 is not necessarily to be expectedfor all transfer processes, that is, the same value of nwould not be expected to represent best both the data forZ greater and less than unity. Indeed the functional rela-tionship itself provided by Equation (7) cannot be justi-fied theoretically for either 2 greater or less than unityfor the inherently complex transfer processes. The remark-able agreement in functional form in most of the aboveexamples and the relative insensitivity to the value of ndemonstrated in some of the examples above must there-

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fore be considered somewhat fortuitous. The maximumdeviation of the correlation from the asymptotic solutionsis 2lln - 1. Hence the larger the value of n the smallerthe range of the correlation. The success of Equation (7)in correlating diverse processes is related to this constraint.Some care must be taken in the choice of the properasymptotic solution. A singularity which is not of concernin an asymptotic solution may be unacceptable in thecombined expression. The chosen asymptotic solutionsshould be lower bounds in Case A (increasing depen-dence) and upper bounds in Case B (decreasing depen-dence) to avoid anomalies in the combined expression.Additive terms in the asymptotic solutions appear to pro-duce asymmetry in the correlation and it may be necessaryto choose different exponents for large and small valuesof 2 to provide a precise correlation in such cases. Theasymptotic solutions yielding the largest value of n shouldbe chosen if possible since this reduces the range of Y (2 ) .Thus the use of Equation (26) instead of the limitingvalue of 3.657 yielded n = 4 as opposed to n = 1.8 inExample 6. Equation (26) with n = 8 is competitivewith Equation (28) with n = 2 and with Equation (30)with n = 813 for this reason despite its theoretical short-comings. The previously discussed experience of Brianand Hales (1969) suggests that the coefficient in one orboth of the asymptotic solutions might be determined bytrial and error if the values are uncertain.

The central value of the independent variable is definedby the limiting solutions and is not a function of n. It isnoteworthy that the central value of Pr is 0.492 in Exam-ple 3 but 0.0468 in Example 6. Similarly the central valueof Gz is 30.8 in Equation (26) but depends on Pr inEquation ( 3 3 ) .Thus the central value of the independentvariable depends on the process itself. This central valueshould be given priority in experimentation or in a com-putational program since it leads most directly and mostaccurately to the best value of n.

Expressions4,having a form equivalent to Equation (7)have been us68 for correlation in many instances in thepast, for example by Sparrow (1955) and Brian and Hales(1969). The generality acquired by the definition of Yand 2 in terms of the appropriate asymptotic solutions hasreceived more limited attention, notably from Acrivos(1961) for the effect of Pr and Sc on convection in bound-ary layer flows. However, he did not discuss the evaluationof n or the applicability of the expression to other proc-esses.Equation (7) and Figures 1 and 2 can be consideredto be approximate representations for all phenomena inwhich the dependent variable varies uniformly from smallto large values of the independent variable. The variablesY and Z are defined by the limiting solutions for largeand small values of the independent variable and the in-dividual processes are characterized by the choice of nalone, Indeed a value for n is a sufficient documentationfor the correlation if the choice of the limiting solutionsis unambiguous.

In summary Equation (7) and the procedure describedabove appear to have wide applicability for correlatingor testing experimental or theoretical values for processesin which the limiting behavior is known. The examplesabove were for simple processes of flow and heat transferbut the procedure is equally applicable to componenttransfer, to more complex processes of transfer and indeedto the correlation of thermodynamic data, etc. Processeswhich produce anomalies were deliberately chosen for il-lustration to indicate the limitations as well as the SUC-cesses of the equation and procedure.AlChE Journal (Vol. 18, No. 6)

ACKNOWLEDGMENT

gratefully acknowledged.NOTATIONA Equation (2)]B

Equation (3)3c = heat capacity, J/kg.KcD = drag coefficient for sphered p = diameter of sphere, mD = diameter of pipe, mDf = diffusivity, m2/sg = gravitational acceleration, m/s2Gr = gB (T , - T , ) x 3 / v 2 = Grashof numberGz = wc/kx = Graetz numberh = local heat transfer coefficient, W/m2*Ki = local heat flux density, W/m2k = thermal conductivity of fluid or insulation, W/k = thermal conductivity of semi-infinite region, W/L = distance through packed bed, mn = exponent in Equation (7)N u , = hx/k = local Nusselt number for plateN U D = h D / k = local Nusselt number for tubeP = pressure, kg/m.s2 \p = exponent in asymptotic solution for z 4 [seeq = exponent in asymptotic solution for z + co [seeR e , = XUJV local Reynolds number for plateRed = d,uO/v = Reynolds number for falling sphereRe, = d P u g / v (1- ) = Reynolds number for packedbed of spheresSc = v / D f = Schmidt numberT, = temperature of wall, KT, = ambient temperature, Kuo = superficial velocity through packed bed or veloc-ity of sphere, m/su, = free-stream velocity, m/swxy ( z ) = dependent variabley ( z3 0) = asymptotic behavior of dependent variable

for z + 0y ( z + w ) = asymptotic behavior of dependent variablefor z + c,

Y (2) = normalized dependent variable [see Equationz = independent variable2

The suggestions of J. D. Hellums and of the reviewers are

= coefficient in asymptotic solution for z 4 [see= coefficient in asymptotic solution for z + 00 [see

m*Km*K

Equation (2)Equation ( 3 )1

= mass flow rate in pipe, kg/s= distance from start of plate, m

(7)1(7 ) 1= normalized independent variable [see Equation

Greek LettersCY

aBS6p = viscosity, kg/m*sv = kinematic viscosity, m2/sp = density, kg/m3u = ( k / k ) ( C U / ~ ) ~ ~@

-- thermal diffusivity of fluid or insulation, m2/sI hermal diffusivity of semi-infinite region, m2/s5 volumetric coefficient of expansion with tempera-= thickness of insulation, m= void fraction in packed bed

ture, 1 /K

= p e 3 d P 3 - d P / d L ) /p2 (1 - ) ~November, 19 72 Page 11 27

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LITERATURE CITEDAcrivos, A., On the Solution of the Convection Equation inLaminar Boundary Layer Flows, Chem. Eng. Sci., 17, 457(1962).Brian, P. L. T., and H. B. Hales, Effects of Transpiration andChanging Diameter on Heat and Mass Transfer to Spheres,AIChE J . , 15,419 (1969).Christiansen, E. B., Effect of Particle Shape on Free SettlingRates, Ph.D. thesis, Univ. Michigan, Ann Arbor (1943).Churchill, S. W., Heat Transfer Rates and Temperature Fieldsfo r Underground Storage Tanks, SOC . Pet. Engr. J . , 2, 28

(1962).., and H. Ozoe, Correlations for Laminar ForcedConvection in Flow over an Isothermal Plate and in De-veloping and Fully Developed Flow in an Isothermal Tube,in review.Ede, A. J., Advances in Free Convection, Advances in HeatTransfer, J. P. Hartnett and T. F. Irvine, Jr., eds., AcademicPress, N . Y. (1967).Ergun, S., Fluid Flow through Packed Columns, Chern. Eng.Progr., 48, 89 ( 1952).

Graetz, L., Uber die Warmeleitungsfahigkeit von Fliissig-keiten, Ann. Phys. Che m., 25, 337 ( 1885).Le Fevre, E. J., Laminar Free Convection from a VerticalPlane Surface, Proc. 9th Intern. Congr. Appl. Mech., Brus-sels, 4, 168 ( 1956).

LQvQque,M. A., Les lois de la transmission de la chaleur parconvection, Ann. mines, 13, 201, 305, 381 ( 1928).Lipkis, R., Discussion of Numerical Solutions for LaminarFlow Heat Transfer in Circular Tubes by W. M. Kays,Trans. A S M E , 77,1272 ( 1955).Ostrach, S., An Analysis of Laminar Free-Convection Flowand Heat Transfer about a Flat Plate Parallel to the Directionof the Generating Body Force, Nut. A&. Comm. Aero-nautics, Report 1111,Washington, D. C. ( 1953).Saunders, 0. A., Natural Convection in Liquids, Proc. Ray.SOC . (London),A172,55 ( 1939).Sparrow, E. M., Analysis of Laminar Forced-Convection Hea;Transfer in Entrance Region of Flat Rectangular Ducts,Nat. Advisory Comm. Aeronautics, TN 3331, WashingtonD. C. (1955).., and J. L. Gregg, Details of Low Prandtl NumberBoundary-Layer Solutions for Forced and for Free Convec-tion, Nat. Aeronautics Space Adm., Memo 2-27-59 E,Washington, D. C. ( 1959).Sugawara, S., and I. Michiyoshi, Effects of Prandtl Number onHeat Transfer by Natural Convection, Proc. 3rd JapanNatl. Cong. Appl. Mech., 315 (1953).Worsoe-Schmidt, P. M., Heat Transfer in the Thermal En-trance Region of Circular Tubes and Annular Passages withFully Developed Laminar Flow, Intern. J . Heat Mass Trans-fer, 10,541 ( 1967).Manuscript received March 31, 1972. revision received May 30, 1972;paper accepted June 2, 1972.

Modal AnalysisAxial Diffusion

of Convect onThe one-dimensional model for convection with diffusion and with a

source term for mass or energy generation or interchange is analyzed forthe eigenvalues and the corresponding spatial eigenmodes as a function ofthe Peclet number. It is shown how the modal analysis of the source case,when the source coefficients for heat exchange and chemical reaction arespatially independent, is directly related to the no source solution. Numericalexamples of determining the source term coefficient are included. Thesesolutions form a base to discuss dynamic characteristics and stability andto which solutions for spatially dependent coefficients can be compared.

with

W IL LIA M C. COHENALFRED M. LOEB andKENNETH WESTONSchool of Chem ical EngineeringUniversity of PennsylvaniaPhiladelphia, Pennsylvania 19104

SCOPEEigenvalues and corresponding eigenfunctions arise

naturally and are useful in connection with systems de-scribed by linear partial differential equations. They areuseful in the solution of the equations with given initialconditions and in the determination of system stability,measurement, and control. Linear partial differentialequations arise when small perturbations about someoperating state are considered for a linear or nonlinearsystem. In this paper we obtain the eigenvalues and asso-ciated eigenfunctions for the one dimensional linearized

model for convection with axial diffusion and with asource term for mass or energy generation or interchangesubject to the widely used Wehner and Wilhelm (1956)boundary conditions.

a4+a+=--a4 a4Pe ax2 ax a t----

Correspondence concerning this paper should be addressed to W. C.Cohen, Department of Chemical Engineering, Northwestern University,Evanston, Illinois. A. M. Loeb is with NAVSEC, Philadelphia Division,U. S. Naval Base, Philadelphia, Pennsylvania 191 2.

-Y (1, t) = 0ax

This model has wide applicability arising in connectionPage 1128 November, 1972 AlChE Journal (Vol. 18, NO. 6)