literature Cited
Brown, L. E., Haynes, H. W., Ind. Eng. Fundamentals 8, 600 (1969). Hwang, S-T., Kammermeyer, K., Can. J . Chem. Eng. 44, 82 (1966a). Hwang, S-T., Kammermeyer, K., Separation Sci. 1 (5), 629 (1966b).
Hwang, S.-T., Kammermeyer, K., Separation Sci. 2 (4), 555 (1967). Tock, R. W., Kammermeyer, K., A.I.Ch.E.J., in press (1969). Sun-Talc Hwang Karl Kammermeyer University of Iowa Iowa City, Iowa 62240
VISCOUS FLOW THROUGH PARTICLE ASSEMBLAGES AT INTERMEDIATE REYNOLDS NUMBERS SIR: LeClair and Hamielec (1968) estimated the nondimensionalized frontal stagnation pressure, Po, for flow past an isolated sphere by extrapolating the values of Po for porosities of 0.909 and less to a porosity of unity. The validity of this large extrapolation is negated by our current work, using similar boundary conditions and a similar numerical technique, on axisymmetric flow past spheroids. The limiting case of a sphere is plotted for Re = 1.0 in Figure 1, which includes the relatively low porosity data of LeClair and Hamielec, our own low and high porosity data, and an intermediate point by Rhodes (1967). An interruptedly smooth curve connects all these data points, and Po is an even stronger function of E a t high than a t lower porosities. According to Jenson (1959), the surface pressure distribution for creeping flow past an isolated sphere is given by
15
i \re
=LO (LeClair 8 Hamielec extrapolation)
(Masliyah & Epstein) I
1
I
, 1 , 1
lo
Re
0
Figure 2. Variation of dimensionless stagnation pressure with superficial particle Reynolds number
Therefore a t the front of the sphere, where 0 = 0, Po = P
=
7
3025-
+
Rhodes
20-
-e-
LeClair & Harnielec
+ I
Masliyah & Epstein
e IS--
,-LeClair .----
8 Harnielec extrapolation
for a particle Reynolds number of unity. Our own value of 7.9 compares favorably with this Stokes value, considering that the latter is strictly valid only for creeping flow (Re 7 0.1). On the other hand, the corresponding LeClairHamielec extrapolated value of 11.5 is considerably in error. Figure 2 shows that the LeClair-Hamielec extrapolation from E = 0.909 to E = 1.0 is also in error at Reynolds numbers considerably in excess of unity. It may be concluded that such extrapolations of Po to E = 1.0 are permissible only from porosities which are already very close to unity. literature Cited
16'
lo"
10-
16'
l a
162
IO'
IO0
I-•
Figure 1. Dependence of dimensionless frontal stagnation pressure on volumetric concentration
Jenson, V. G., Proc. Roy. SOC.(London) 249A, 346 (1959). LeClair, B. P., Hamielec, A. E.. IND.ENG.CHEM.FUNDAMENTALS 7, 542 (1968). Rhodes, J. M., Ph.D. thesis, University of Tennessee, Knoxville, Tenn., 1967. Jacob H . Masliyah Norman Epstein University of British Columbia Vancouver, B. C., Canada
VISCOUS FLOW THROUGH PARTICLE ASSEMBLAGES AT INTERMEDIATE REYNOLDS NUMBERS SIR: Masliyah and Epstein are absolutely correct in pointing out that frontal stagnation pressures for a single sphere in an infinite found by extrapolation using porosities of 0.909 and less are inaccurate a t low Reynolds numbers. The accuracies of frontal stagnation pressures found by extrapolation for Reynolds numbers greater than about 20 are, however, adequate. This suggests that such extrapolation 602
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FUNDAMENTALS
procedures may be very useful for these larger Reynolds numbers where computation times become excessive, particularly when solutions are attempted for porosities as large as 0.999963. We are well aware of the accuracy of the extrapolation technique since (Hamielec et al., 1967) we have published accurately calculated frontal stagnation pressure obtained for
