Comprehensive Correlating Equations For Heat, Mass and Momentum Transfer in Fully Developed Flow in Smooth Tubes Stuart W. Churchill Department of Chemical and Biochemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19 174
A single correlating equation is constructed for the mean Nusselt (or Sherwood) number for all Reynolds and Prandtl (or Schmidt) numbers, and for either uniform wall temperature or a uniform heat flux density. The applicability of this equation is limited to fully developed flow in smooth tubes. However, developing as well as fully developed convection is considered. A corresponding equation is constructed for the friction factor for all Reynolds numbers. These expressions are based on interpolation between the various limiting cases, using the model of Churchill and Usagi. The correlating equations appear to represent available experimental and theoretical values within their uncertainty and to be at least as accurate as prior expressions for restricted ranges of Re and Pr or Sc. The equations are suitable for hand-held computers as well as for incorporation in algorithms for design and optimization.
Introduction Heat, mass, and momentum transfer to fluids passing through tubes attracted the attention of Thomas K. Sherwood throughout his career as indicated by references ( I ) through (21). (References to the publications of T. K. Sherwood are herein distinguished by the use of a reference number rather than a year.) His continued interest is a testimony to the importance of the problem and to his focus on practical matters. His contributions generally combined theory and experiment and ranged from overall rates to eddy diffusivities. The designation of the dimensionless rate of mass transfer as the Sherwood n u m b e r is seen to be most appropriate. However, despite the extensive efforts of Sherwood and many others, the existing correlations for Nu and Share highly empirical, generally quite limited in scope, and subject to disagreement. The objective of this work has been to construct simple correlating equations with as sound a theoretical basis and as much generality as possible. The construction is based on the model of Churchill and Usagi (1972,1973) which interpolates between the solutions or correlations for limiting cases. Attention is first focused on fully developed flow and transfer in smooth pipe and on the idealized case of invariant physical properties. The complications of developing convection and variable physical properties are subsequently considered. The analogy between heat and mass transfer in flow through tubes has been well confirmed and will be assumed throughout. The equations are constructed in terms of heat transfer but are implied to be applicable to mass transfer with Sh and Sc substituted for Nu and Pr. The extensive literature on this subject has recently been reviewed by Petukhov (1970), Kays and Perkins (1973), and Gnielinski (1975). Therefore attention will be restricted to those theories, data and correlations which are directly relevant to the analysis herein. General Development of the Correlating Equations Churchill and Usagi ( 1972,1973) have demonstrated that highly successful correlations for most phenomena can be constructed by taking the p t h root of the sum of the p t h power of the limiting solutions or correlations for large and small values of the independent variable. Intermediate values are then utilized to choose the best value for the arbitrary exponent p . This procedure is first used herein to construct an expression for the effect of P r on Nu at large Re. This corre-
lating equation for large Re is then combined with a value of Nu for the limiting case of turbulent flow as RePr decreases, to yield a correlating equation for all Pr over the entire regime of fully developed turbulent flow. This second correlating equation is in turn combined with an asymptotic expression for transition to laminar flow. Finally this third correlating equation is combined with the appropriate expression for Nu in the laminar regime to give an overall correlating equation for all Re and Pr. The above procedure can be expressed symbolically as follows
+ (NummIa NU^')^ +
(Nu,)" = (Nu,')" =
(1)
(2)
+ [(NU,')" + NU,")"]^" (3) + (NU,)' (4) NU^)' + ((Nuo')~+ [(NU,')" + NU^")"]^'^)^'^ (5) (NU) = (NU,) + (NUT) (6) + N NU^)' + + NU,^)" + NU^")^]^/^)^/^]^/^ (7a) =
(NUT)' = NU^)'
=
=
or NU =
+
NU^)'
+
[(Nu,')"
NU^')^
+
+
NU,^)^]^/")^/^]^/(.)^/^
(7b)
Equation 7 contains five functions Nul, Nui, Nu'', Nu,", and Nu,' which must be specified and four exponents, a , b, c, and d , which must be determined. Fortunately, these functions and exponents can be determined serially and independently, insofar as sufficient experimental data or theoretical solutions are available for the different regimes. The regime of greatest practical interest is that of moderately high Re and intermediate Pr. Hence the development herein will proceed in the reverse direction of the derivation of eq 7 with particular attention to this regime. Although the functions and coefficients in eq 2 were actually chosen and evaluated by reference to experimental and computed values at each step, the evaluation herein will be deferred until the completion of the derivation in the interest of economy of presentation. The development of a correlating equation for the friction Ind. Eng. Chern., Fundam., Vol. 16,No. 1, 1977
109
factor follows the same course with expressions for the laminar and transition regime first combined, then this expression combined with an expression for the turbulent regime. Thus
+ (fm)g)'/g
f = ([VI)' + (fi"]'/'
-.
(8)
Re 03 and Pr + a.A number of expressions have been postulated or derived for Nu for the limiting case of both large Re and large Pr. The assumption that the resistance to transfer is limited to the region near the wall where U + = fly+], and the further assumption that d v is proportional to (y+)" in this region lead to Nu," = ARe 2 X los. Such values are not available. Extrapolation of the computed values of Notter and Sleicher (1972) suggest that the same value of 0.079 may be used for B as for A, with considerable advantage in simplicity. Interpolating Equation for all Pr at Large Re. An interpolating equation for all Pr a t large Re is obtained by substituting eq 11 and 12 in 1,resulting in ( N u m ) == ( 0 . 0 7 9 R e C P r ) a 110
+ (0.079RecPr1/3)a
Ind. Eng. Chem., Fundam.,Vol. 16, No. 1, 1977
(13)
The value of a is most sensitive to data for P r 2 1.0. For Pr = 1.0 a = In {2)/ln(N~,~/0.079Re*l
(14)
The value of Nu1 computed by Notter and Sleicher (1972) for UWT a t Re = 500 000 gives a = -1.2. This value of a was found to predict values of Nu, in good agreement with experimental and computed values for other Pr as subsequently demonstrated. The proposed correlating equation for all P r a t large Re is thus
[
Nu, = 0 . 0 7 9 R e C P 1 - ~ ' ~1/
+pr:5 ' ]516
or
(It should be noted that values for B as low as 0.06 and values for -a from 1to $ are suggested by some sets of self-consistent data. Hence the chosen values must be considered tentative pending further experimental or theoretical resolution of the behavior for small Pr.) Turbulent Regime, RePr -+ 0. Equation 15 implies that Nu 0 as RePr 0. However, Lyon (1951) showed that for turbulent flow Nu approaches a finite value, here designated as Nuo", as RePr 0. For plug flow NuoO has a theoretical value of 8.0 for a uniform heat flux density (UHF) and 5.76 for a uniform wall temperature (UWT). Values determined experimentally or theoretically for more realistic velocity distributions are somewhat lower, and the values of 6.3 and 4.8 obtained from the computations of Notter and Sleicher (1972) appear to be satisfactory for use as asymptotes in eq 2. Interpolating Equation For Entire Turbulent Regime. The experimental data for intermediate values of Re suggest a value of approximately unity for b resulting in the expression
-
-
-
Nut =
+
0.079ReqPr [I + pr4/5]5/6
Equation 16 is proposed for all Pr with Re I10 000. The subscript has been dropped from f in eq 16 since this expression is ultimately incorporated in expressions for the transition and laminar regimes. Transition Regime. Petersen and Christiansen (1966) found that the transition from completely turbulent convection a t Re 2 10 000 to completely laminar convection a t Re 5 2100 was the same function of Re for all conditions and fluids, including finite L / D and non-Newtonian liquids. The interpolating equation which they propose for the transition is inconvenient for incorporation in eq 4. Instead the arbitrary expression Nui = NU,c,(Re-2200)/730
(17)
is proposed as the asymptote for Re 5 2100. A value of the exponent c = -2 appears to approximate the observed behavior in the transition regime. Hence the proposed expression for the entire transition and turbulent region (Re > 2100) is
with Nut and Nui given by eq 16 and 17. Laminar Regime. In the laminar regime (Re < 2100) for fully developed conditions Nul attains a constant value of 3.657 for UWT and 4.364 for UHF, and Nulc = Nul. Overall Equation for Fully Developed Conditions. The shift from laminar to transitional convection a t Re = 2100 is quite abrupt. This behavior can be simulated by choosing a large value of say 10 for the exponent d . The behavior for all
Re and all Pr with fully developed conditions is then represented by (NU)"' = (Nul)'"
e (L'"OO-Ke.i/:365
+
[
+
(
1 (j
Nuo
)2,-5
(19)
+ 0.079ReqPr (1 + Pr4/6)5/fi
The appropriate values of Nul, Nul,, and Nu()') for UWT or UHF are to be substitut.ed. Equation 19 appears to be complicated, but only superficially. I t obviously reduces to Nu = Nul for Re < 2100, to eq 18 for Re > 2100, to eq 16 for Re > 10 000 and further to eq 11 for P r > 100. Friction Factor Correlation. A graphical correlation for the friction factor in eq 19 is inconvenient if a computer is to be used. Empirical and theoretical equations for the friction factor are generally restricted in range and, in particular, omit the transition regime from 2100 < Re < 3000. The desired comprehensive correlating equation can be constructed by specifying the functions and choosing the exponents in eq 8. For Re < 2100, Poiseuille's law f 1 = 8/Re
(20)
is applicable. The empirical equation
provides a good representation for the data of Pate1 and Head (1969) and the recent computed values of Wilson and Azad (1 975) for Re in the range from 2200 to 2700. For the abrupt shift from the laminar to1 the transition zone an exponent e of 10 appears to be appropriate. Churchill (1972) has asserted that the empirical equation of Colebrook (1938) (22)
provides a satisfactory representation for the entire turbulent regime. An exponent g of 5 appears to f i t the experimental data for the shift from transitional to turbulent flow. The resulting expression for all Re is \ i/i
(23) It is probable that the transition is effected by secondary variables. Some of the earlier data would be better represented by a linear dependence on Re in the transition regime and a lower pair of exponents In any event eq 2 1 should be considered somewhat tentative. Equation 20 may be used in lieu of eq 23 for Re < 2000 and must be used for Re < 7 owing to the change of sign in the logarithmic term in eq 23.
Evaluation for Fully Developed Convection The prediction provided by eq 19 and 23 is plotted in Figure 1for P r (or Sc) = 0.02,0.03,0.72,3.0,6.0,8.0,48,75,930,and 9810. These values of Pr (or Sc) are those for which experimental data exist for small temperature differences, fixed P r (or Sc) and a reasonable range of Re. The computed values of Notter and Sleicher (1972) based on eddy diffusivities are included in the plot. Dashed curves representing UHF are included for Pr = 0.02, 0.03, 0.72, and 8.0 since the data for
these cases extend to conditions such that the results for the two boundary conditions differ significantly. The overall success of eq 19 and 23 in representing the effects of Re and P r (or Sc) is apparent. The results for each chosen value of Pr (or Sc) will next be examined in detail. The data of Johnson et al. (1954) for molten lead-bismuth eutectic (Pr 2 0.029) and UHF agree with the dashed curve for Pr = 0.03 within their scatter. The data of Sleicher et al. (1973) for molten sodium-potassium eutectic a t UWT, with Pr ranging upward with Re from 0.0203 to 0.0245, are in even better agreement. The computed values of Notter and Sleicher (1972) are also in excellent agreement with the predicted curves. The experimental values of Kolar (1965) for air (Pr 2 0.71) and UWT are in reasonable agreement with the curve for P r = 0.72 although slightly higher for intermediate Re and slightly lower for low Re. The values of Cholette (1948) for air (Pr z 0 . 7 ) and UWT are in excellent agreement with the predicted curve for Pr = 0.72 in the transition regime, The deviations a t Re = 1650 and 2190 are undoubtedly due to insufficient L / D to attain complete thermal development. The solid circles represent the data of Delpont (1973) extrapolated to zero temperature differences. The indicated dependence on Re is somewhat different but is well within the scatter of the raw data. The computed values of Notter and Sleicher for Pr = 0.72 are i n good agreement at high Re but fall above the predictions for intermediate Re. The curves representing eq 19 and 23 are in excellent accord with points representing the measured values of Malina and Sparrow (1964) for UWT and water and oils (Pr = 3,48 and 75) and with those of Allen and Eckert (1964) for UHF and water ( P r = 8).These two sets of values have been given considerable credence by Nijsing (1969), Notter and Sleicher (1971),and others because they represent extrapolations to zero temperature difference. (The values for Pr = 3 are mistabulated 1 i ~Xijsing (1969).) The measured v a l u a of McCarthy et al. (1964) for 50% hydrazine and 50% unsymmetrical dimethyl hydrazine are included in Figure 1 despite a variation in Pr because they exceed most other experiments in the magnitude of Re and hence test the dependence on Re. Their values with 7 < P r < 8 and those with 5.2 < Pr < 7 are separately coded. The values of Stone et al. (1962) for a number of organic liquids with 8.17 < Pr < 8.83 are also seen to scatter about the predicted curve for Pr = 8. The agreement of the predicted curves and computed values for Pr = :3 and 8 is excellent except for Re = 5 X 10' and 10'' where the computed values are significantly lower. The agreement between the curve for Pr = 48 and the computed values for P r = 50 is excellent for all Re. The predicted curves for P r = 930 and 9810 are in almost exact accord with the measurements of Harriott and Hamilton (1965) for mass transfer and also with the computed values for Pr = 1000 and 10 000. The experimental data of Hubbard and Lightfoot (1966)and of Meyerink and Friedlander (1962) for mass transfer at high Pr are significantly lower, and Notter and Sleicher (1Y71)question their accuracy. Data do not exist to test the predictions of transition at high P r or Sc because the requirements of tube length for the complete development of laminar convection are excessive for experimentation. The correlating equations of Petukhov and Popov (1963), Notter and Sleicher (1972) and Gnielinski (1976) are of comparable accuracy, but have more restricted ranges of applicability. The correlating equation of Hughmark (1975) for turbulent flow gives predictions of comparable accuracy for low and moderate Pr (or Sc). For high Sc it fits the electrochemical data of Hubbard and Lightfoot (1966),Meyerink and Friedlander (19621,and Mizushina et al. (1971), upon which it is based, much better than eq 16 but grossly underpredicts Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
111
10:
I
I
2200-Re
I
I'
Pr or Sc
gT930
,/
10'
10'
r
cn L
0 3
Z
10;
10
1
I
I
the solution data of Harriott and Hamilton (1965) and the computed values of Notter and Sleicher (1972).
Developing Convection in Fully Developed Flow Laminar Flow. Analytical solutions in the form of infinite series have been derived for developing heat transfer in fully developed laminar flow. These solutions are rather inconvenient for calculations. The following semiempirical equation proposed by Seider and Tate (1936) has proven reasonably satisfactory large RePrD/L: Nu = 1.86(RePrD/L)lI3
(24)
This may be combined with the constant value for fully developed convection to give the following approximate relationship for the whole laminar regime with UWT 112
Ind. Eng. Chem., Fundam., Vol. 16, No. 1. 1977
I
I
[ (RePrDIL 7.60 ) ]
Nul = 3.657 1 +
8/3
118
(25)
Based on this expression and the correlating equations of Churchill and Ozoe (1973) for the local coefficient the following expression is proposed for UHF Nul = 4.364
[ 1+ (RePrDIL 7.3 ) ] 2
116
(26)
For Re < 2100 eq 25 and 26 give the value of Nul for use in eq 19, or can be used directly. For Re > 2100, Nulc for substitution in eq 19 is obtained by substituting Re = 2100 in eq 25 and 26 giving for UWT
and for UHF
A complication arises in developing laminar convection in that the arithmetic-mean temperature difference rather than the log-mean temperature difference is sometimes used in evaluating the heat transfer coefficient. However, the latter choice is implied herein even though the former was originally used in constructing eq 24. Turbulent Flow. The situation is even more poorly defined for developing convection in fully developed turbulent flow. The correction for LID is a function of Re, Pr, and the thermal boundary conditions. Sleicher et al. (1973) proposed Nu 80 -=l+-+-ln Nut L
20 L
I42 -
(29)
for liquid metals. Deissler (1961) has prepared graphical correlations for Nu/Nut for moderate and large P r and for a wide range of Re, based on theoretical calculations. His results indicate that the correction is ordinarily negligible for practical conditions. For very low P r or very small LID, such that convective development. is significant, Nut should be multiplied by the factor obtained from eq 29 or from these graphs. Evaluation. The experimental data of Friend and Metzner (1958) for corn syrup with LID = 161 are compared with the predictions of eq 19,23,25, and 27 in Figure 2. The correction for LID is negligible for the completely turbulent regime but is significant for the laminar and transition regimes. The experimental values are in general agreement with the predicted curves but are slightly high a t high Re and indicate some anomalies a t low Re. The experimental values of Sherwood et al. (2) for a hydrocarbon oil with LID = 234 are included despite a wide variation in viscosity within the fluid and from run-to-run, since they are one of the few sets of data which encompass both the laminar and the transition regimes. The properties were evaluated as indicated in the next section, and the dashed curve, which becomes essentially coincident with the curve for P r = 93 for Re > 4000, was drawn for the varying Pr corresponding to the individual data points. The agreement is fair for Re > 1200. The deviations are probably due to the use of large temperature differences in the experiments.
Variable Physical Properties The effect of the variation of physical properties with temperature cannot in principle be generalized for all fluids by the choice of particular properties and/or the use of property or temperature ratios. Nevertheless many such methods have been used with some success. The suggestion of Sleicher and Rouse (1975) to correlate the data for completely turbulent flow in terms of NUh, Ref, and Pr, has the merit of simplicity and appears to be as successful as any other choice. The use of these properties in eq 16 and in the corresponding term in eq 19 is recommended. (Nub is of course an anomaly since h is based on the rate of conduction a t the surface. Hence it must be presumed that the use of Nub is merely an empirical artifact.) Colburn (1933) noted that the onset of turbulent motion occurred at Reb z 2100 for fixed Tb and varying T,. Hence the use of Reb in eq 17 and in the corresponding term of eq 19 seems appropriate. Allen and Eckert (1964) indicate that the effect of physical property variation on the friction factor for turbulent flow can be taken into account approximately merely by evaluating the viscosity in the Reynolds number at the wall temperature. The seriously nonisothermal data in Figure 1 were actually plotted in terms of Ref, Pr, and Nub.
Figure 2. Correlation for developing convection Pr
93 185 340 530
95-181
L/D
Prediction
161 161 161 161
-
234
------
Data
V\
”I
Friend and Metzner
AI J
Sherwood, Kiley, and Mangsen
Laminar Flow. The problem of property variation is much more serious in laminar flow than in turbulent flow. The velocity profile across the tube may change and in turn change the rate of heat transfer significantly. At low flow rates the change in density may superimpose natural convection. It is tempting to suggest the use of the same choice of properties in laminar as in turbulent flow. However the appropriate data to evaluate this proposal do not appear to exist. The generally accepted procedure is to substitute Reb, Prb, ~ ~ Re, Pr, and Nu in eq 20 and 25-28. This and Nub ( p , / ~ t , ) ” .for procedure, as well as others which have been proposed, is not entirely successful and the relatively poor correlations for laminar convection have generally been attributed to property variations. Critical Overall Evaluation The correlating equations are tested more critically and thoroughly in Figures 3 and 4 in which the fractional deviations of the measured (and computed) values from the predicted values are plotted vs. P r (or Sc) and Re, respectively. These plots permit inclusion and testing of the experimental data obtained at odd and varying values of Pr and Sc. Only essentially isothermal and reasonably precise data are plotted, except as noted below. Figure 3 tests the dependence on Pr at Re = 10 000 only. A single Re was chosen to avoid confusion, and this value because of more measurements than at any other. The deviations of the computed values of Notter and Sleicher (1972) are less than 3% except for 0.2 < Pr < 2 for which the computed values are much higher than those predicted. The deviations of the experimental values of Friend and Metzner (1958) and of Harriott and Hamilton (1965) are seen to be quite random in sign and quite small except again near P r = 0.5. The deviations of the data of Hubbard and Lightfoot (1966) and of Mizushina et al. (1971) fall consistently 20% below the predictions. The deviations of the experimental data plotted in Figure 4 appear to scatter rather randomly and hence to confirm the predicted dependence on Re. The principal exceptions are the sets of data of Kolar (1965) and of Delpont (1964) for air, which display a somewhat different trend even though the Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
113
* ++
+0.21 $? * O . l C
e
e
L
-0.3-
I
,
I
I
I
I
I
lo-’
1
10
to2
1o3
1 oL
I 0’
I
IO
Pr or Sc
Figure 3 . Test of correlation for Pr and Sc at Re = 1 0 000: 0 , Friend and Metzner-heat transfer; 0,Harriott and Hamiltonmass transfer; A; Hubbard and Lightfoot-mass transfer; v, Mizushina et a1.-mass transfer; +, Notter and Sleicher (1972)computed.
+0.4
I
I
I
I
I
I
I
I
I
I
,
I
I
I
I
I
I
l
l
)
I
I
I
I T
I
+0.3+0.2-
z 3
A
+0.1 -
-z Z
\
0
3
-0.1 - v
z
V
I
3
z
-0.21r
I
-0.3-
V
V
0
-0.4
I
I
I
I
I
l
l
l
1o3
l
I
I
I
I
I
1 o4
I
I
I
I
I
I
I
I
I
1o5
I
1o6
Re Figure 4 . Test of correlation for all Re and Pr or Sc Symbol m
N
Heat Transfer ____Lit. source Sleicher et al. Johnson et al.
___-____
Pr 0.022
Ko 1ar
0.070
____
Del p on t
0.070
0
Lawrence and Sherwood McCarthy et al. McCarthy et al. Allen and Eckert
1.75-3.4
e
AI
Malina and Sparrow
Lit. source
0
Meyerink and Friedlander
n}
Linton and Sherwood
8 13
Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
~
sc 900
1700 Hubbard and Lightfoot
5-7 7 -8
absolute deviations are not serious. The data of Hubbard and Lightfoot deviate even more negatively a t higher Re than in Figure 3 and are joined by the data of Meyerink and Friedlander (1962). The data of Mizushina e t al. (1971) (not shown) would lie even lower. One is forced to the conclusion that these three sets of measurements are not consistent with those upon which this correlation was indirectly based. The 1950 data of Linton and Sherwood (12) scatter more widely than more modern values but show overall agreement even into the laminar regime. The 1931 data of Lawrence and Sherwood (1) also stand up well with modern valuesagreeing with the predictions in the transition region but showing a downward trend with increasing Re, probably due to insufficient correction for property variations. Comparison of Figures 1with 2 and 3 indicates how much 114
Mass Transfer __
Symbol
0.029
0
+ ‘1+
-----______-----
1 0 000 30 000
m
+I
X
Harriott and Hamilton
930 9810
more sensitive the latter plots are in revealing discrepancies and trends.
Conclusions (1)Equation 19 provides a prediction for Nu or Sh for all Re and all Pr or Sc in fully developed, near-isothermal flow in smooth pipes. Equation 23 provides the corresponding prediction for f for all Re > 7. For Re < 7 eq 23 should be replaced by eq 20. Equations 25 and 26 provide the indicated values of Nul, and eq 27 and 28 the corresponding values of Nulc for UWT and UHF. Nu00 = 4.8 for UWT and 6.3 for UHF. A correction factor for developing turbulent convection is provided by eq 29 for low Pr and by the graphical correlations of Deissler (1961) for moderate or large Pr.
For nonisothermal conditions the use of Nub, Ref, and Pr, or sew, except for Reb in the exponential, is suggested for eq 19. The use of Re, is suggested in eq 23 and the use of NU~(/.L~//.L Reb, ~ ) ~and . ' ~ Prb , or SCb in eq 25-28. (2) These expressions appear to represent the experimental data within their uncertainty and as well as prior correlating equations for more limited ranges. It may be desirable to modify some of the coefficients and exponents and property choices in the future when measurements of improved accuracy for the critical situations are attained. In particular, resolution of the current discrepancy in the measured rates of mass transfer in favor of the electrochemical data would require revision of the arbitrary exponent and coefficient in eq 15. (3) Equations 19, 23, and 25-28 are convenient for calculations with a hand-held computer and also for incorporation in algorithms for the design and optimization of heat exchangers. (4) Equations 19 and 23 obviously reduce to one or more of their component parts for various ranges of the independent variables. For example, eq 19 reduces to eq 16 for Re > lo4 and further to eq 11 for P r > 30. (5) The difference in Nu for uniform heating and uniform wall temperature increases as Re and P r decrease and is significant only insofar as Nu00 and/or Nul or Nulc contribute significantly to eq 19. (6) Despite the worthy efforts of Sherwood and his coworkers to develop a rational basis for the direct dependence of Nu and Sh on the friction factor as in the original ChiltonColburn analogy, a dependence on appears to have greater experimental support. ( 7 ) The failure of authors to include and journals to insist on tabulations of experimental values in completely accessible form, i.e., not in combined groups such as N~,l(Prnf)O.~ is a serious impediment to the subsequent use of this information. The early papers of Sherwood are paragons in this respect. Acknowledgment The provision of a tabulation of the experimental values of McCarthy et al. (1964) by Professor J. D. Seader is greatly appreciated. This work has been greatly enhanced by the several significant contributions of Professor C. A. Sleicher and his coworkers as noted. Nomenclature A = coefficient in eq 9 A' = coefficient in eq 10 a = exponent in eq 1 B = coefficient in eq 12 b = exponent i n e q 2 c = exponentineq4 D = tubediameter,D d = exponent in eq 6 B = diffusivity, L2/8 e = exponentineq8 f = rwp/G2 = friction factor g = exponentineq8 C = mass velocity, M/L28 h = heat transfer coefficient based on log-mean temperature difference, Q/L20T k= =thermal conductivity, QIL0T k' = mass transfer coefficient based on log-mean concentration difference, LIB L = length of heated section, L n = power-dependence of t / v on y + Nu = h D / k p = exponent in Churchill-Usagi model P r = v/ct = Prandtl number Re = D G / p = Reynoldsnumber Sc = u/B = Schmidtnumber Sh = k'DIa) = Sherwoodnumber
T = temperature, T u = local velocity, L/0 u + = uy = distance from wall, L y+ = y ~ l v y++ = Y+g
Greek Letters = thermal diffusivity, L2/8 t = eddy diffusivity for heat or mass transfer, L2/8 /.L = viscosity, MIL% u = kinematic viscosity, L2/0 p = density, MIL3 T, = shear stress at wall, MILO2 cy
Subscripts b = properties at mixed-mean temperature c = atReb = 2100 f = properties at average of mixed-mean and surface temperatures i = asymptotic behavior in transition to laminar regime 1 = laminar regime 0 = asymptotic behavior as Re 2100 T = turbulent plus transition regime t = turbulent regime w = a t surface = asymptotic behavior as Re m Superscripts 1 = at P r of 1.0 0 = asymptotic behavior as Pr m = asymptotic behavior as P r
--
0 03
L i t e r a t u r e Cited Sherwood a n d Coworkers (1)Lawrence, A. E., Sherwood, T. K., ind. Eng. Chem., 23, 301 (1931). (2)Sherwood, T. K., Kiley, D. D., Mangsen, G. E., ind. Eng. Chem., 24, 273 (1932). (3)Sherwood, T. K., Petrie, J. M., Ind. Eng. Chem., 24, 736 (1932). (4)Gilliland, E. R., Sherwood, T. K., Ind. Eng. Chem., 26, 516 (1934). (5) Towle, W. L.. Sherwood, T. K.. Seder, L. A,, Ind. Eng. Chem., 31, 462 (1939). (6)Towle, W. L., Sherwood, T. K., Ind. Eng. Chem., 31, 457 (1939). (7)Sherwood, T. K., Woertz, B. B., lnd. Eng. Chem., 31, 1034 (1939). ( 8 ) Sherwood, T. K., Woertz, B. B., Trans. Am. Inst. Chem. Eng., 35, 517 (1939). (9)Sherwood, T. K., Trans. Am. inst. Chem. Eng., 36, 817 (1940). (10)Sherwood, T. K.. in "Fluid Machanics: Statistical Methods", p 55, University of Pennsylvania Press, Philadelphia, Pa., 1941. (11)Sherwood, T. K., Gilliland, E. R., ind. Eng. Chem., 26, 1093 (1943). (1 2) Linton, W. H., Jr., Sherwood, T. K., Chem. Eng. Prog., 46, 258 (1950). (13)Sherwood, T. K., ind. Eng. Chem., 42, 2077 (1950). (14)Sherwood, T. K., Pigford, R . L., "Absorption and Extraction," 2d ed, McGraw-Hill, New York, N.Y., 1952. (15) Sherwood, T. K., Ryan, J. M., Chem. Eng. Sci., 11, 82 (1959). (16)Sherwood, T. K., Chem. Eng. frog. Symp. Ser., 55, No. 25,71 (1959). (17)Sherwood, T. K., "Mass Transfer Between Phases," Pennsylvania State University Press, State College, Pa., 1959. (18)Sherwood, T. K., Ryan, J. M., in "Recent Advances in Heat and Mass Transfer." T. P. Hartnett, Ed., p 208,McGraw-Hill, New York, N.Y., 1961. (19)Vieth, W. R., Porter, J. H., Sherwood, T. K., ind. Eng. Chem., Fundam., 2, l(1963). (20)Sherwood, T. K., Smith, K. A., Fowles, P. E.,Chem. Eng. Sci., 23, 1225 (1968). (21)Sherwood, T. K., Pigford, R. L., Wilke, C. R., "Mass Transfer," McGraw-Hill, New York, N.Y., 1975.
Others Allen, R. W., Eckert, E. R . G., J. Heat Transfer, 64C, 301 (1964). Chilton, T. A., Colburn, A. P. lnd. Eng. Chem., 26, 1183 (1934). Cholette, A., Chem. Eng. frog., 44, 81 (1948). Churchill, S.W., AIChEJ., 19, 375 (1973). Churchill, S.W., Ozoe, H., J. Heat Transfer, 95C, 78 (1973). Churchill, S. W., Usagi, R., AIChEJ.. 18, 1121 (1972). Churchill. S.W., Usagi, R., ind. Eng. Chem., Fundam., 13, 39 (1973). Colburn. A. P., Trans. Am. Inst. Chem. Eng., 29, 174 (1933). Colbrook, C.F., J. Inst. CiviIEng., 11, 133 (1938). Deissler, R. G.,in ''Recent Advances in Heat and Mass Transfer," J. P. Hartnett, Ed., p 253,McGraw-Hill, New York, N.Y., 1961. Delpont, J. P., lnt. J. Heat Mass Transfer, 7,517 (1964). Friend, W. L., Metzner, A. B., AlChEJ., 4, 393 (1958). Gnielinski, V., Forsch. Ingenieurwes., 41, 6 (1975); English trans., Int. Chem. Eng., 16, 359 (1976). Hanna, 0. T.. Sandall, 0. C., AIChEJ., 18, 527 (1972). Harriott, P., Hamilton, R. M., Chem. Eng. Sci., 20, 1073 (1965). Hubbard. D. W., Lightfoot. E. N., Ind. Eng. Chem., Fundam., 5,371 (1966). Hughmark, G.A.. AIChEJ., 18, 1072 (1972).
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Hughmark, G. A,, AlChEJ., 21, 1031 (1975). Johnson, H. A., Hartnett, J. P., Clabaugh, W. T.. Trans. ASME, 75, 1197 (1953). Kays, W. M., Perkins, H. C., in “Handbook of Heat Transfer.” W. M. Rohsenow, J. P. Hartnett, Ed., Sect. 7, pp 7-1/7-193, McGraw-Hill, New York, N.Y., 1973. Kolar, V., Int. J. Heat Mass Transfer, 8, 639 (1965). Lyon, R . N. Chem. Eng. Prog., 47, 75 (1951). Malina, J. A,, Sparrow, E. M., Chem. Eng. Sci., 19, 953 (1964). McCarthy, J. R.. Hines, W. S., Seader, J. D., Trebes. D. M., Bull. 6th Liquid Propulsion Symposium, pp 73-100, Chemical Propulsion Information Agency, Pub. No. 56, Silver Spring, Md., Aug 1964. Meyerink, E. S.C., Friedlander, S. K., Chem. Eng. Sci., 17, 121 (1962). Mizushina, T., Ogino, F., Oka, Y., Fukuda, H., lnt. J. HeatMss Transfer, 14, 1705 (1971). Nijsing, R., Warme- Stoffubertragung, 2, 65 (1969). Notter, R. N., Sleicher, C. A,, AlChf J., 15, 936 (1969).
Notter, R . N., Sleicher. C. A.. Chem. Eng. Sci., 26, 161 (1971). Notter, C. A., Sleicher, C. A., Chem. fng. fng. Sci., 27, 2073 (1972). Patel, V. C., Head, M. R., J. NuidMech., 38, 181 (1969). Petersen, A. W., Christiansen, E. B., AlChEJ., 12, 221 (1966). Petukhov, B. S., Adv. Heat Transfer, 6, 503 (1970). Petukhov, B. S., Popov, V. N., Teplofiz. Vysok. Temperatur, 1, 69 (1963). English trans., High Temperature, 1, 69 (1963). Seider, E. N., Tate. G. E., lnd. Eng. Chem., 28, 1429 (1936). Sleicher, C. A., Awad, A. S., Notter, R. H., lnt. J. HeatMass Transfer, 16, 1565 (1973). Sleicher, C. A., Rouse, M. W., lnt. J. Heat Mass Transfer, 18, 677 (1975). Wilson, N. W., Azad, R. S., J. Appi. Mech., 42, 51 (1975).
Received for review September 3 , 1976 Accepted October 29,1976
Longitudinal Dispersion in Packed Gas-Absorption Columns William E. Dunn, Theodore Vermeulen,’ Charles R. Wilke,. and Tracy T. Word Chemical Engineering Department and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720
Axial-mixing experiments were conducted in a 2 4 diameter gas-absorption column by using tracer-injection techniques. The carrier streams were air and water. Empirical Peclet number correlations are presented for both phases, with I-in. Berl saddles, I-in. Raschig rings, and 2-in. Raschig rings used as packing materials. The extent of mixing was much greater in the liquid phase than the gas phase. Gas-phase Peclet numbers decreased with both liquid and gas flow rates. On the other hand, liquid-state Peclet numbers were found to increase with liquid flow rates, and no quantitative effect of gas rate was observed. It is found that mixing can contribute significantly to absorption-column design requirements, especially if liquid-phase diffusion controls.
Introduction In the design of packed columns for gas-absorption or stripping operations, the tower height is often estimated by multiplying the total number of transfer units (NTU) by a correlational height of a transfer unit. Previous work has shown that axial mixing, a deviation from piston flow, may be a significant design factor for liquid-liquid extraction systems in packed columns. Axial mixing arises from the fact that “packets” of fluid do not all move through a packed bed at a constant and uniform velocity, either because of either velocity gradients in the fluid, or eddy motion in the packing voids. Axial mixing tends to reduce the concentration driving force for mass transfer that which would exist for piston flow, as illustrated in Figure 1; the concentration profiles for piston flow are dotted lines, and solid lines represent the axial-mixing case (Miyauchi et al., 1963). To achieve a given separation, more transfer units are required for the axial-mixing case owing to the reduced driving force. Likewise, for a given column under conditions of axial mixing, HTU’s (heights of a transfer unit) estimated with plug flow assumed are higher than the true HTU’s that would be calculated from the actual mass-transfer coefficients. The object of this study has been to determine whether axial mixing is an important factor in gas-absorption column design, and to provide correlations of dispersion coefficients in the form of Peclet numbers for both liquid and gas phases. Axial mixing was measured for each phase separately, in the absence of interphase transfer of any solute. It appears that gas holdup is not affected in a major way by mass transfer, but may be different for liquids with lower surface tension than that of water. The flow patterns for liquid are also expected to be independent of mass transfer unless surface-active solutes are involved, so that the liquid-phase results obtained here may 116
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prove to be rather generally applicable. The calculation of axial-dispersion effects in actual column separations has been discussed by Miyauchi et al. (1963), and Vermeulen et al. (1966), among others. A numerical example of the effect in absorber performance, based on results obtained in the present study, is given by Sherwood et al. (1975). A general discussion of axial dispersion has been given by Mecklenburgh and Hartland (1975). Theoretical The extent of axial mixing may be evaluated quantitatively, independently of any mass transfer, by tracer-injection techniques. A tracer amount of a component is injected into one of the bulk phases in the form of a step, impulse, or sinewave input. A step input was used as a basis for this study. The buildup of tracer at a fixed distance downstream from the injection point is measured as a concentration-time relation which is called a breakthrough curve. The characteristics of the experimental breakthrough curve may then be compared with the forms predicted by a mathematical mixing model; the value of the mixing parameter in the mathematical model that gives the best fit to the experimental curve is designated as the mixing parameter characteristic of the experimental system. Two such models were found to be pertinent to the case at hand, and their results are briefly described. Random-Walk Model. The random-walk model is characterized by “packets” of fluid moving through the packing in a series of discrete jumps corresponding to a certain mean free path, 1 , and with a characteristic velocity, u . If the total bed length is h , and the time that a packet of fluid has been in the bed is t , then the number of mixing lengths N (or column Peclet number) is defined as h/l , and the dimensionless time scale. T , as ut/l. If a step input of tracer with magnitude
