Ind. Eng. Chern. Fundarn., Vol. 19, No. 2, 1980
stants are given with five digits accuracy) with a maximal error of 0.4651% for 4 X lo8 5 R e 5 4000 and 5 X loe7I €105 0.05. Since this fit is better than the fit obtained when using other explicit methods, the author concludes that his equation is the best one and it should be used. I would like to show that it is quite easy to develop “explicit” equations which have similar or better accuracy than Chen’s eq 2. After that I would like to discuss what a reasonable accuracy is and whether an explicit equation is really needed. It has been shown by Shacham (1976) that the successive substitution method always converges for eq 1 from a reasonable starting point and its convergence is usually very fast. Since the value of f D should be between 0.005 < f D < 0.08 for turbulent flow [see the Moody (1944) chart] we can select an initial guess approximately in the middle of this range, like f D = 0.03. Substituting this initial value for f D into the right-hand side of eq 1 and applying the successive substitution method twice, the following “explicit” equation is obtained 1
fi
t
5.02 Re
t
7- - log
3.10
(3) (3.70 + E)] Re
This equation is much simpler than eq 2 and still the results obtained from it are within 1% of agreement with the correct solution of eq 1, for the same region which was indicated by Chen. But we can do even better than that. Denoting by x the expression that appears inside the square brackets in eq 3 and applying a single iteration of the Newton-Raphson method we get 1 =
v%
[x(l-Inx)-3.70
]/[
+
1.15129~ Re
229
This equation correlates the correct solution of eq 1within 0.02% relative error! This result seems to be very impressive when comparing with the error obtained when using Chen’s equation, so the conclusion should be that eq 4 is the best “explicit” method for friction factor calculation. Unfortunately, such an accuracy is not needed. According to Moody (1944), the error in predicting f D using eq 1 is f 5 % for a smooth pipe, f 1 0 % for commercial steel, and can be much more than that when uncertainty regarding the pipe roughness is involved. That means than an explicit equation which correlates eq 1with a maximal error of f l % is absolutely adequate and no better accuracy is needed. It can be concluded that the attempt to develop an explicit correlation for an implicit equation when it is not really needed may lead to the strange result that the best “explicit” correlation is not more than an iterative solution of the implicit equation, started from a good initial guess and stopped after two iterations. Comparing correlations according to their accuracy may be misleading, if the physical phenomena behind the equation is forgotten. L i t e r a t u r e Cited Chen, N. H., Ind. Eng. Chem. Fundam., 18, 296 (1979) Colebrook, C. F., J . Inst. Civil Eng., 133 (1939). Moody, L. F., Trans. ASME. 66, 641 (1944). Shacham, M., Isr. Chem. Eng., 8, 7E (1976).
Department of Chemical Engineering University of Connecticut Storrs, Connecticut 06268
Mordechai Shacham*
(4)
On leave from the Department of Chemical Engineering, Ben Gurion University of the Negev, Beer-Sheva, Israel.
Sir: The main objective of my article is t o propose an explicit equation for friction factor in pipe to replace the implicit equation of Colebrook. Hence the article is solely responsible for the accuracy of the equation against the modified equation of Colebrook, eq 4 in the article which is used for comparison, so as t o improve the accuracy of my proposed equation. The original equation of Colebrook has the constants 3.7 and 2.51 instead of 3.7065 and 2.5226. The discrepancy between these two equations is extremely small so that it can be neglected. Except the accuracy, all other problems are beyond the scope of the article. Thus, what conditions the Colebrook equation holds are also good for the proposed equation. For instance, the Colebrook equation has been verified (Schlichting, 1968) to be good in the transition region. Therefore, my proposed equation is also applied in this region but not only the turbulent region as indicated by Churchill. Since a t high Reynolds number the Colebrook equation becomes the Von Karman equation, which applies in the turbulent region, the proposed equation is also valid in this region. Then automatically it is understood that the proposed equation is good for all values of Reynolds number and €10within these two regions. This range of application of the Colebrook and the proposed equations coincides with Figure 14.1 of Welty et al. (1969) and many others. Hence Churchill’s comment that the Colebrook and Chen equations are limited to the turbulent regime is incorrect. Furthermore, we should be concerned not only with the accuracy of the proposed equation, which can be best done by finding its deviation from the Colebrook equation, but also with the readers’ query on the quality of the existing
explicit equations. In order to compare the results a t the same level, the same basis should be chosen by comparing the proposed, Churchill, and Wood equations with the Colebrook equation. How well the Colebrook equation fits the experimental data is again beyond the scope of this article. Schlichting (1968) indicated the law of friction for pipes roughened with sand in his Figure 20.18. He stated that in the region of laminar flow, all rough pipes have the same resistances as a smooth pipe. This implies that the Hagen-Poiseuille equation holds also for rough pipe. This is the reason why the Colebrook equation does not include this region of laminar flow. So do the Moody friction factor chart and my proposed equation. Now Churchill introduces a term into his equation without sufficient proof to account for both the laminar and the critical regions. The inclusion of this term is redundant and doubtful. The validity criteria for the Colebrook and the Von Karman equations are the same as those of Welty e t al. (1969). The constant 1.74 in the Von Karman equation is correct and is verified by Schlichting (1968). Because of a typographical error, the diameter D in eq 3 should be the radius R , or the constant 1.74 should be 1.14, as indicated by Schorle. Moreover, if the pressure gradient is specified, use of the Colebrook equation is preferred because it does not require a trial-and-error method. (Note the difference of this and the Churchill’s statements). On the contrary, if the pressure gradient is not specified, use of the Colebrook equation would require laborious trial and error. Hence for the later case, use of my proposed equation is recommended.
230
Ind. Eng. Chem. Fundam. 1980, 19, 230-231
It appears that Shacham's eq 3 is modified from my proposed equation by changing only the numerical values. Hence Shacham's eq 3 should be actually called the Chen-Shacham equation or the modified Chen equation, just like the eq 4 in my article which is different from the Colebrook equation on the numerical constants only should be called the modified Colebrook equation. Furthermore, because of different numerical values, this Chen-Shacham equation is therefore not as accurate as my proposed equation. Then Shacham attempted to improve accuracy. Consequently, eq 4 in his comments was obtained in which he claims that its accuracy is better than my proposed equation. Taking for granted that his eq 4 is good, I feel that it is so complicated that its practical application is extremely limited. At last, Shacham, failing to obtain a simpler and more accurate correlation, indicated unreasonably that an accurate equation is not needed. However,
it is obvious that good correlation should have the qualifications of simplicity, applicability, and accuracy. Frequently an equation may be accurate but not simple so that its use is limited or vice versa. From this point of view, my proposed equation is simpler than Shacham's eq 4 and is more accurate than the Chen-Shacham equation. In conclusion, my proposed equation is still recommended. Literature Cited Schlichting, H., "Boundary-LayerTheory", 6th ed,McGraw-Hill, New York, N.Y., 1968. Welty, J. R., Wicks, C. E., Wilson, R. E., "Fundamentals of Momentum, Heat, and Mass Transfer", Wiley, New York, N.Y., 1969.
D e p a r t m e n t of Chemical E n g i n e e r i n g University of Lowell Lowell, M a s s a c h u s e t t s 01854
Ning Hsing Chen
On the Effectiveness Factor of Partially Wetted Catalysts in Trickle-Bed Reactors
Sir: In a recent article Mills and Dudukovic (1979) analyze the effect of fractional pore fill-up on the effectiveness factor of partially wetted catalysts in trickle-bed reactors. From their Figure 10, where the effectiveness factor for slab geometry at an external contacting efficiency of 0.5 is represented, they conclude that "incomplete internal wetting is seen to have a major role over all ranges of (Thiele) moduli". However, if we consider the expression for the Thiele modulus modified to take into account both internal and external wetting (Dudukovic and Mills, 1978) the effectiveness factor is given approximately by
I 710 I
I 7 , :10 oa
I
06 OL
l
02
where 7 is the effectiveness factor of a totally wetted pellet (notation identical with that used by Mills and Dudukovic, 1979). It follows now at low values of 4 that OTB = Ti
while for high values of 4 OTB
=
OCE/~
The asymptotes given by eq 2 and 3 do not coincide, except for complete internal wetting, with the curves represented in the mentioned Figure 10. At low Thiele modulus, since the reactant concentration is uniform, the effectiveness factor is directly proportional to the internal wetted volume (eq 2) and does not depend on the external wetting. On the other hand, at sufficiently high Thiele modulus, the partial internal wetting does not affect the effectiveness factor, since a sharp drop in reactant concentration near the wetted surface of the catalyst occurs, and the limiting reactant does not reach the dry zone (eq 3). A very simple model for which eq 1represents the exact solution can be proposed. It consists of an infinite catalyst slab, one of its faces wet and the other dry, that is vcE = 0.5. Partial internal wetting is taken into account by considering that the region near the dry surface is not wetted. The mass balance equation describing this model is d2u - = 42u; @2 = L 2 k / D , (4) dt2 0196-4313/80/1019-0230$01 .OO/O
Figure 1. Effect of internal wetting o n the catalyst effectiveness factor a t external contacting efficiency, 'ICE = 0.5.
with the boundary conditions [ = O
u = l ; [=27,
-du= o
a
(5)
where L is half the width of the slab, where it has been assumed that there is no diffusional resistance in the liquid film, and the liquid has a negligible vapor pressure. The solution of eq 4 leads to tanh (2477,) (6) OTB = 24 It can be seen that it is equivalent to eq 1 for vCE = 0.5. The effectiveness factor is represented in Figure 1 as a function of the Thiele modulus. The curve for complete internal wetting coincides at high and low values of 4 with the numerical solution of the Dudukovic-Mills model. At intermediate values, lower effectiveness factors are obtained, which can be attributed to the difference in the models (Figures 1 and 3 of Dudukovic and Mills, 1978). For incomplete internal wetting, the curves behave as 0 1980 American Chemical Society
