Ind. Eng. Chem. Process Des. Dev. 1984, 23,860
a60
CORRESPONDENCE Comments on “Estlmation of the Drag Coefflclent of Regularly Shaped Particles In Slow Flows from Morphological Descriptors” Sir: For circular profile, two s u m s of the shape parameters (EL, and EL3)used by Carmichael (1982) to formulate the correlation for the shape correction factor, K, are equal to zero. Therefore, Carmichael’s correlation (eq 9) gives K = 0 for any particle whose projected profile on a plane parallel or perpendicular to the flow direction is a circle. I t is surprising to note that for such cases predicted by Carmichael the values for the K factor are different from zero, for example, K values in lines 1to 3 , 7 to 12, and 25 to 38 in Table IV. An attempt to reproduce Carmichael’s results by using his own data from Table IV and omitting zero terms in eq 9 has failed. I have compared Carmichael’s correlation for the shape correction factor, K, with one obtained from exact solution for spheroids (Clift et al., 1978). When a spheroid axis of symmetry is oriented parallel or perpendicular to the flow direction, the spheroid projection on the plane normal or parallel to the flow direction is a circle. The zero terms in eq 9 were neglected. Fourier series coefficients were generated by the discrete fast Fourier transformation routine (Cooley et al., 1970) using twelve points of contour representation. A comparison between theoretical, predicted, and experimental K values as a function of aspect ratio, E, is shown in Figure 1. There is good agreement between the experimental data and the theoretical curves. There is, however, limited agreement observed between the theoretical solution and the predicted K curves. Certain agreement is observed at the aspect ratios for which the experimental data were available to develop the correlation 9, i.e., for E = 0.25,0.5, 2,3, and 4 (not for E = 1) and only for spheroids with their symmetry axis parallel to the flow direction. Two things are now clearly and distinctly visible. First, the form of Carmichael’s correlation 9 was carelessly chosen by not taking into account the possibility of existence of vanishing terms. Second, it is unsafe to use statistical correlation over a limited range of data.
0196-4305/84/ 1 123-0860$01.50/0
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E Figure 1. Shape correction factor ( K ) vs. aspect ratio (E) for spheroids having axis of symmetry parallel (11) and perpendicular (I) to the flow direction.
Finally, a small typographic error occurred in eq 6 which should be
Ro2 = Ao2 + -
1 N
A2
2n=1
in order to have the coefficients Lo, L2,L3,and K dimensionless. Literature Cited Carmichaei, G. R. Ind. Eng. Chem. Rocess Des. Dev. 1982, 21, 401. CiM, R.; &ace, J. R.; Weber, M. E. “Bubbles, Drop8 and Particlee”;Academ IC Press: New York, 1978. Cooky, J. W.; Lewis. P. A. W.; Welch, P. D. J . Sound V b . 1970, 12(3),315.
Department of Chemical Engineering The University of British Columbia Vancouver, British Columbia Canada, V6T 1 W5
0 1984 American Chemical Society
Robert M. Soszynski
