The random pore model seems to work fairly well on porous solids made by compressing porous alumina powders, which is often done commercially. This model should not be extrapolated and used as yet on other radically different types of porous solids.
T V b-* X A
a E XA
PB
Acknowledgment
The Shell Oil Co. and Procter and Gamble Co. provided fellowships for this study. The Sinclair Research Co. provided measurements of the porous solids. The financial assistance provided is gratefully acknowledged. Nomenclature
a, b, c
A, B
i, k 1, n S
sion, sq. cm./sec. De = diffusion coefficient defined by Equation 11, sq. cm.,/sec. DKA = Knudsen diffusion coefficient defined by Equation 1. sq. cm./'sec. ( D K A ) e f= l effective diffusion coefficient for Knudsen diffusion, sq. cm./sec. mean Knudsen diffusion coefficient in pores, sq. cm./sec. diffusion coefficient defined by Equation 2 in a tube, sq. cm./sec. diffusion coefficient defined by Equation 6, sq. cm./sec. Knudsen tortuosity molecular tortuosity actual length of porous solid, cm. molecular weight, g./g. mole diffusion flux, g. mole A/sec. sq. cm. Knudsen number defined by Equation 24 pressure, mm. H g gas constant pore radius, cm. pore radius defined by Equation 7 , cm. mean pore radius, cm. total surface area by BET. sq. m./g.
temperature, O K. pore volume, cc./g. total pore volume, cc./g. mole fraction A flux ratio, I AVB/NA void fraction mean free path of A, cm. bulk density, g./cc.
+
SUBSCRIPTS
0,L D = diameter of pellet, cm. = molecular diffusivity for system A-B, sq. cm./sec. DAB ( D A B ) e l=f effective diffusion coefficient for molecular diffu-
= = = = = = = =
t
= three types of macropores = nitrogen and helium, respectively
= two types of micropores = indices defined in Equation 20 = nitrogen and helium sides of solid, respectively
= solid = total
literature Cited
Cunningham, R. S., Ph.D. thesis, Ohio State University, Columbus, 1966. Evans, R. B., Watson, G. M., Mason, E. A., J . Chem. Phys. 35, 2076 (1961). Flood, E. A , , Tomlinson, R. H., Leger, A. E., Can. J . Chem. 30, 372 (1952). Foster, R. N., Butt, J. B., A.Z.CI2.E. J . 12,180 (1966). Hedley, IV. H., Lavacot, F. J., IYank, S. L., .4rmstrong, W. P., A.I.CI2.E. J . 12, 321 (1966). Henry, J. P., Jr., Cunningham, R. S., Geankoplis, C. J., Chem. Eng. Sci. 22, 11 (1967). Hoogschagen, J., Znd. Eng. Chem. 47, 906 (1955). Johnson, M. F. L., Stewart, FV. E., J . Calalysis 4, 248 (1965). Michaels, A. S., A.Z.CI1.E. J . 5 , 270 (1959). Petersen, E. E., A.Z.Ch.E. J . 4, 343 (1958). Rothfeld, L. B., A.1.Ch.E. J . 9, 19 (1963). Scott, D. S., Dullien, F. A . L., A.Z.Ch.E. J . 8, 113 (1962a). Scott, D. S., Dullien, F. A. L., Chem. Eng. Sci. 17,771 (1962b). Spiegler, K. S., IND.ENG.CHEM.FUNDAMENTALS 5 , 529 (1966). i'v'akao, N., Smith, J. M., Chem. Eng. Sci. 17, 825 (1962). IYeisz, P. B., Schwartz, A. B., J . Catalysis 1, 399 (1962). Wheeler, A,, "Catalysis," Vol. 2, p. 105, Reinhold, New York, 1955. RECEIVED for review October 9, 1967 ACCEPTED February 9, 1968 \Vork supported in part by U. S. Atomic Energy Commission Contract ATf11-1)-1675.
VISCOUS FLOW THROUGH PARTICLE ASSEMBLAGES A T INTERMEDIATE REYNOLDS NUMBERS Stea4-State Solutions f o r Flow through Assemblages of Spheres B. P. L E C L A I R A N D A.
E. H A M I E L E C
Chemical Engineering Department, McMaster University, Hamilton, Ontario, Canada
deal of experimental drag data for particle assemblages in packed and fluidized beds exist in the literature (Leva, 1959; Orr, 1966; Zenz and Othmer, 1960). There is, however, a need for precise information on local velocity distributions about spheres in such assemblages. For example, local velocity distributions could be used with the equations of continuity and energy to predict heat, mass, and chemical reaction rates. This information would allow chemical reaction rate data obtained in packed or fluidized beds to be properly interpreted.
A
542
GREAT
l&EC FUNDAMENTALS
The only theoretical information on local velocities about a particle available in the literature was developed by Happel (1958) and Kuwabara (1959). Happel employed a freewrface model and Kuwabara a zero-vorticity model (two of tile many possible surface-interaction models). They obtained solutions for creeping flow through sphere assemblages, but limited to ,VRe 1. Happel calculated forces at the particle surface, while Kuwabara integrated the viscous dissipation function to predict the particle drag coefficients. Had Kuwabara used surface forces, his predicted drag would
\
STREAM FUNCTION
>"
',, ,
> k 0
'\
I
sw > cn cn w
J
,,' POROSITY
,,' ,'
z 0 cn
0.909
z W
NRE = 500 VORTICITY
z0
I DIMENSIONLESS RADIUS
r/a
Figure 4. Dimensionless angular velocity variation with radial position at N R e = 20 and 500
Figure 3.
Stream function and vorticity at NRe = 500 A. 6.
e e
= 0.741 = 0.909
angle. The flow separation angle (measured from the rear stagnation point) and the size of the standing vortex ring decrease with decreasing porosity. The magnitude of the vorticity increases significantly as porosity decreases, resulting in a rather large increase in friction drag. Surface Pressure Distribution. The surface pressure distributions for N E e = 500 and for a range of porosities (e = 0.4 to 1.0) are plotted in Figure 1, B . Again these distributions are typical and similar results have been predicted for Reynoldsnumbersof0.1, 1, 10,20, 50, 100, 500, and 1000. The pressure at the rear stagnation point is strongly dependent on porosity. This suggests that for experimental verification, pressure distributions should be measured as a function of porosity (perhaps with a test sphere in a packed bed). Frontal stagnation point pressure is virtually independent of porosity a t iVRe = 500. At lower Reynolds numbers, however, the dependence is clearly evident, the stagnation pressure increasing as the porosity decreases. The effect of porosity on frontal stagnation point pressure is shown in Figure 2, A , for a range of Reynolds numbers and porosities. Pressure, Friction, and Total Drag Coefficients. Pressure (form) drag coefficients for a range of porosities (0.4 to 1.0) and a range of Reynolds numbers are plotted in Figure 2, B. 546
I&EC FUNDAMENTALS
The values for E = 1.0 were estimated by extrapolation. First of all, the ratio of predicted pressure drag coefficient to predicted friction drag coefficient was extrapolated to e = 1. This ratio was then used with experimental total drag coefficient data (Lapple et al., 1951) to estimate the pressure and friction drag coefficients individually. These are shown as broken lines in Figure 2, B and C. Total drag coefficients (experimental) (Lapple et al., 1951) are shown as a broken line in Figure 2, D. The extrapolation procedure used involved the assumption that a t e = 1, the first derivative of the extrapolated quantity (in this case CDP/CDF)with respect to e was zero. Two points at E = 0.909 and e = 0.835 were used to evaluate the constants in the polynomial CDp/CDp
=
A,
+ A2 (1 - el2
A, equals the ratio CDP/CDF a t e = 1. Figure 2, B , C, and D ,clearly indicates the very strong dependence of flow behavior on porosity. Dimensions of Standing Vortex Ring. Lines of equivorticity and streamlines within the standing vortex ring have been plotted in Figure 3, A and B , for one Reynolds number (NRe= 500) and two porosities (E = 0.909 and 0.741). The effect of reduction in porosity is to reduce the dimensions of the vortex ring-viz., the flow separation angle and the length of the vortex ring as measured along the axis 0 = T become smaller. The critical Reynolds number for flow separation is increased as the porosity is reduced-viz., at NRe= 500, and E = 1 there exists a large vortex ring-however, a t NRe= 500, and a t E = 0.645 the vortex ring does not exist. The lines of equivorticity are swept downstream, but this convection appears to be abnormally interrupted by the posi-
10
,'
tI
4
I
I
I
I
I
1
'"'A
j 0 6-
L t
1
I
NRES I
L
0.4
.
PRESENT THEORETICAL
A
NRE. N R E . --l 1 0 Oo
8
N
I
~ -100 ~ .0
CARMPN- KOZENY (1939) FAIR- H A T C H ( I 9 3 3 )
c
0.2
L
0.0
L
0.4 POROSITY
000'02
03
04
05
07
06 I
I
08 I
(D)
IO----1 ---
1
04-
W
LT
0 4
08
06
IO
POROSITY
POROSITY
Figure 5.
Comparison of theoretical relationship with data for flow through beds
