Vol. 23, KO.10
INDUSTRIAL A N D ENGI-VEERI,VG CHEMISTRY
1180
NOTES AND CORRESPONDENCE Heat Transfer to Liquids in Viscous Flow Editor of Industrial and Engineering Chemistry: In the article under this title by C. G. Kirkbride and W. L. McCabe [IND.ENG. CHEM.,23, 625 (1931)], certain numerical errors have been copied from the foreign literature. As a consequence, if the authors’ graphical integrations are accurate, a considerable discrepancy has been introduced in the location of the theoretical curve in Figure 4 of their article. The correct expressions for the first three terms of the two infinite series are:
x
+
+
( x ) = 2.995e-14.6272 2,18e-82.222 l.OO6e-2125
and @
From this it follows that the correct values of the coefficients in Series B are 0.819, 0.0976, and 0.01897, respectively. This relation is not satisfied by the coefficients of the series given by Grober. The graphical integrations to obtain the thoretical curve of Kirkbride and McCabe’s Figure 4 are unnecessary. From Equation 6 of their article,
At the end of the pipe, tau = t 2 , and therefore the heat transferred in unit time over the entire tube area is:
(A)
+
( x ) = 0.819e-14.6272 4-0.0976e-sQ~22z0.01897e-2122 (B)
By using this equation with Equation 13 of the article and
A of this note, it follows that
wherex = PeL/D The corresponding series given by Kirkbride and McCabe are: 2,996e-14.6342 + 2 228e-88.72 + 1,006-212.180 4 (x) = 0.821e-14.634z 0.0987e-88.72 0.0135e-212.18z
x (2)
+
+
They occur in the equations numbered 5 and 6, respectively, in the article, and were correctly quoted from Grober “Die Grundgesetze der Warmeleitung und des Wdrmeiiberganges,” Springer, 1921, who based his development on Nusselt’s work [ Z . Vtr. deut. Ing., 54, 1154 (1910)l. Actually, the theory under discussion was originated by Graetz [Ann. Physik, 18, 79 (1883); 25, 337 (1885)l and rediscovered by Nusselt some years later.
Nusselt gave inaccurate values for the coefficient and exponent of the second term of Series A. The present writer has quite closely checked Nusselt’s values for the first term of Series A, has corrected the values for the second term, and has found that the values for the third term of this series are of the correct order of magnitude. It can be shown mathematically that the coefficients of the terms of Series B are definitely related to those of Series A. If the n’th term of Series A is
a
--bz
t h e n’th term of Series B must be
4 ane-*Z b
x ( x )ds
= 1- 4(x)
Since the series @ is integrable term by term, the denominator of Kirkbride and McCabe’s Equation 15 is readily evaluated. The resulting expression for ha,D / k is:
h& k
-
+
+
1 - 8(0.10238e-14.6272 0.01220e-sQ.222 k[0.01430 - 0.013999e-14.6272 - 0.000273e-89.221: 0.00237e-2120 , , , ) 0.0000224e-212z- , .
+
The curve of Figure 1marked ‘’ T . M.”is a graph of Equation C. (Figure 1 is a copy of Kirkbride and McCabe’s Figure 4). If one accepts this (ha* D / k ) as the proper quantity to compare with the authors’ ( h D / k ) , i t is apparent that the discrepancy between theory and experiment is much less than Kirkbride and McCabe suppose. The asymptote of Equation C for large values of x is 4.37. The above curve is based on the true mean temperature difference, as defined by Kirkbride and McCahe’s Equation 11. Since the authors used substantially an arithmetic mean of the temperature differences in computing from their data the points shown on Figure 4, it is of interest, and quite as logical, to compare the points with the theoretical curve for (ha.m.D)/k, where ha.m. is the coefficient based on an arithmetic mean of
INDUSTRIAL AND ENGINEERING CHEhfISTRY
October, 1931
the terminal temperature differences. The latter curve is marked
A . M . in Figure 1. The right-hand part is computed from the equation
ha.m. - = -Dk
Z’ D 2 [ 1 - 41 2kL l + @
c
which, like the other equations derived from Graetz’s theory, presupposes constancy of the wall temperature. The left-hand part of the curve has been extended by the approximation of LkvCque [Ann. mines, [12] 13, 201, 305, 381 (1928)1]. At the right-hand end, the curve A . M . is asymptotic to the 45-degree line, (ha.,,,. D / k ) = ‘/2 (D/Pe L ) , as may be seen from Equation D. Of course, this latter fact becomes obvious, independently of the theory, if the definition of ha.,,,. is substituted in a heat balance. Naturally, in the case of constant wall temperature, no correct experimental point can lie to the right of the 45-degree line. At the left, the curves A . M. and T. M . are necessarily the same, since for small values of the abscissa the terminal values of At are nearly equal and any reasonable method of averaging gives the same result as the arithmetic mean. I t is easily possible, and even probable, that in the authors’ case of constant rate of heat flow per unit length, the theoretical value of (ha.m. D / k ) would approach a horizontal asymptote a t the right. However, it does not seem quite clear either from theory, or from their own data, why 3.65 should be chosen by them as the asymptote of their empirical formula. I n so far as the writer is aware, there is no theoretically calculable (ha” D / k ) having the limiting value of 3.65, which could conceivably be represented by the quantity that the authors have computed. In themselves, the data plotted in the right-hand half of Figure 1 seem to call for a n empirical curve that is concave downwards. With a slide rule, the figures in column 6 of Table I1 in the article under discussion have been computed from the data given in Table 111. For runs 1, 10, 17, 28, 29, 30, 38, 41, 44, 52, 53, and 54, there appear to be rather large discrepancies. Since the data help to fill what has been a serious gap in published measurements, i t would be advantageous if the authors would repeat the computations and publish any necessary corrections for these runs. THOMAS 13. DREW MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASS. June 24, 1931
., . , ., . . . . .. . . Editor of Industrial and Engineering Chemistry: Basing the correction on his extensive theoretical study of heat transfer to liquids in viscous flow, Mr. Drew shows that the theoretical integrated curve given by the writers in Figure 4 of their paper [IND.ENG. CHEM.,23, 625 (1931)] is too low, and replaces it with a more accurate plot. Since the primary purpose of the article under discussion was to present new data which could be used for design purposes in a field of heat transfer where previously published data were lacking, the theory was used mainly to show how the data might be correlated. Accordingly, no attempt was made to check the very lengthy calculations necessary to evaluate the constants involved in the Nusselt-Grober equation, and the errors in the coefficients and exponents were carried over. There is no doubt that the Drew curve is the more correct. I t still falls short of the experimental curve, as is to be expected, but the difference between theory and experiment is a much more reasonable one than that shown by the writers’ Figure 4. I n the writers’ calculations, the arithmetic mean temperature head was not used in computing the average coefficient, although in many of the runs the difference between an arithmetic
1181
mean and a true mean is small. The scattering of the experimental points a t the higher values of P e L / D is attributed by the writers to the fact that the velocities were so low in these runs that some natural convection took place. The asymptote of 3.65 implied in the empirical Equation 12 of the article has no significance. Equation 12 fits only the experimental data that were obtained. It is hardly safe to extrapolate it beyond a value of PeL/D = 0.15. The writers greatly regret the fact that certain numerical errors, mainly typographical and arising in an early stenographic copy of the data, appeared in the tables. Although the errors have no effect on the final experimental curve, they are corrected as follows: Table I RUN 1 27 61 64
CORRECTAs PRINTED F. F. ~. 110.2 115.2 85.5 89.5 100.1 102.1 103.5 106.5 103.0 103.9
ITEM Thermocouple Thermocouple Thermocouple Thermocouple Thermocouple
5 6 6 5
6
Table I1
B. 1. u./hr. B . t. u./hr. 1 9 22 38 39 41 44 53 54 66
From temp. rise Gross heat flow From temp. rise From temp. rise From temp. rise Radiation
of fluid
of fluid of fluid of fluid
From temp. rise of fluid From temp. rise of fluid From temp. rise of fluid Radiation Table I11
3 4
ha. hav hosD/k
10 12
uov
14 17 22
has I1
29 30 38 40
ha. hovD/k 4 hav hauD/k 11
%/D has fl 12
44 %ID
52
12
53 62
hov hov hauD/k
C. G. KIRKBRIDE W. L. MCCABE DEPARTMENT OF CHEMICAL ENGINEERING OF MICHIGAN UNIVERSITY ANN ARBOR,MICH. August 10, 1931
Corrections In the letter to th; editor entitled “Burning Characteristics of Smokeless Powder, by C. G. Dunkle [IND.ENG. CHEM.,23, 1076 ( 1931)], the abbreviation “cal.” was erroneously printed for “Cal.,” meaning kilogram calories. This is important in connection with the author’s mention of the “zero heat effect, at room temperature, of the water-gas reaction.” Measured in small calories, this will, of course, be a considerable number, depending on the particular value of heats of formation used in calculating it. Measured in large calories, it is considerably less than one, for the reaction as written. An error occurs in literature reference (1) in the article entitled “Composition of Kapok Seed,” by E. P. Griffing and C. L. Alsberg, IND.ENG.CHEM.,23,909 (1931). This reference should 40, 647 (1918). read: Carruth, F. E., J . A m . Chem. SOC.,
