1346
INDUSTRIAL AND ENGINEERING CHEMISTRY
Equation 23 gives the relation between inlet and outlet fluid concentrations but its use is not quite so simple as in the equilibrium case. The application of this formula t o a particular case is made difficult by the paucity of literature on adsorption kinetics. I n passing, i t might be noted that the w. appearing in Equation 23 are roots of the same equation occurring in the equilibrium case. The solution for the case of infinite mass transfer coefficient is obtained by letting E ~ 3 . As before wn -+ nr, so in the limit Equation 3 becomes +
m
SUMMARY AND CRITIQUE
I
Adsorption in a fluidized bed has been considered assuming uniform, porous spheres as the adsorbent medium. The spheres have been assumed to move in a completely random fashion in the bed with the supporting fluid of uniform concentration. Two cases were considered. If equilibrium is attained within the spheres, the relation between the adsorbate concentration of the fluid and of the solid must be according t o the isotherm for that system. I n this paper the isotherm had to be linear. In the second case the nonequilibrium condition was discussed and a linear, reversible kinetic relation was used. This equation is the most general relation for the problem involved which can be solved by elementary analytical methods. It was supposed there existed a resistance to mass transfer a t the fluid-sphere interface. It might be that the adsorbate concentration is not uniform throughout the fluid since there might be a continuous concentration gradient in the fluid bed from inlet to outlet. Gilliland and Mason (6)found back-mixing of gas in fluidized beds relatively low, although more data is needed for vessels of low length-diameter ratios. Perhaps a better approximation would be to assume the spheres are in contact with a fluid whose adsorbate concentration is the average of the inlet and outlet concentrations. This would have the effect of replacing the lefthand side of Equation 14 by 2(Cl 2co -
- c2) c1 - c2
D,
diffusion coefficient of adsorbate in fluid in the sphere void volume, sq. ft./hr. G = mass velocity of fluid in vessel, Ib./hr./sq. ft. kj = mass transfer film coefficient, Ib. molcs/sq. ft./hr./unit concentration difference kl, kz = constants in the kinetic equation in reciprocal hours K1, K L= constants in the adsorption isotherm n = adsorbate on adsorbent, Ib. moles/unit apparent volume of adsorbent no = initial adsorbate on adsorbent, Ib. moles/unit apparent volume of adsorbent N = Laplace transform of n p = Laplace transform parameter corresponding to t T = radius variable in sphere, ft. R = external radius of spheres, ft. S = cross-sectional area, sq. ft. t = time, hr. uQ = superficial velocity of fluid, ft./sec. v =c-c1 V = fluid feed rate to bed, cu. ft./hr. w = adsorbent feed rate, lb./hr. R’ = mass of adsorbent in fluidized bed, Ib. 01 = porosity of spheres, cubic feet of void volume per cubic foot of apparent volume of solid or square feet of open area per square foot of sphere surface Y = D,,oi/(K, N) = wR;/bF@, dimensionless e = kfR/D,, dimensionless (modified Nusselt number) p = apparent density of spheres, Ib./cu. ft. P B = bulk density of fluid bed, Ib./cu. ft. p = viscosity of fluid, Ib./ft.-hr. w = a root of the transcendental equation: w cot w = 1 - E =
+
LITERATURE CITED
Arnold, h l . H. hI., and Baxter, D., C . 6. Patent 2,476,472 (July 19, 1949). Campbell, D. L., Tyson, C. W., Martin, H. Z., and hIurphree, E. V., U. S. Patent 2,428,690 (Oct. 7, 1947); Zbid., 2,446,076 (July 27, 1948).
Carslaw, H. S., and Jaeger, J. C., “Conduction of Heat in Solids,” London, Oxford University Press, 1947 Churchill, R. V., “Modern Operational Mathematics in Engineering,” New York, McGraw-Hill Book Co., 1944. Gilliland, E. R . , and Mason, E. A , , IND.ENG.CHICM.,41, 1191 (1949).
Kalbach, J. C., Chsm. Eng., 54, No. 1, 105 (1917); 54, No. 2, 136 (1947).
Laidler, K. J., Bull. soc. chim. France, 1949, D 171-6. Leva, M., Grummer, hl.. Weintraub, M., and Pollchik, AM., Chem. Eng. Progress, 44, 511, 619 (1948).
Lewis, W. K., Gilliland, E, R., and Reed, W. A., IND.ENQ. CHEM.,
with similar changes in the other formulas. The assumption of completely random motion of the particles with the resultant use of probability to determine the residence time of spheres is probably valid since the amount of adsorbent used would be large and the particle size small. For the case of a linear equilibrium isotherm, Equation 14 reveals that for industrial adsorbents, whose K1 values run from lo3 to 104, there is some doubt about the utility of fluidized beds for adsorption purposes. This is readily apparent since the assumption of a uniform fluid concentration means that the exit fluid has the concentration of the bed fluid and hence the driving force for diffusion into the particles is small with the result that the quantity adsorbed per unit mass of adsorbent is not very great, The results of this paper, however, should be applicable to the much more important process of catalysis. NOMENCLATURE c = adsorbate concentration of fluid phase, Ib. moles/cu. f t . co = initial adsorbate concentration of spheres in fluid phase
in the void volume, Ib. moles/cu. f t .
c1
= adsorbate concentration of fluid phase in reactor vessel, Ib.
moles/cu. ft. = inlet adsorbate concentration of fluid stream, Ib. moles/cu. ft. C = Laplace transform of c
c2
Vol. 42, No. 7
41, 1227 (1949).
MacMullin, R. B., and Weber, M., Jr., Trans. Am. Znst. C h m . Engrs., 31, 409 (1935). Parent, J. D., Jagol, No,and Steiner, C. S., Chem. E ~ QProgrcea, . 43, 429 (1947).
Resniok, W., and White, R. R., Ibid.,45,377 (1949). Symposium on Fluidization, IND.ENG.CHEM.,41, 1098-1260 (1949).
Von Kfirmdn, T., and Biot, M. A., “Mathematical Methods in Engineering,” Iiew York, hIcGraw-Hill Book Co., 1940. Wilhelm, R. H., and Kwauk, M., Chem. Eng. Progress, 44, 201 (1948). RECEIVED January 16,1950.
Dust and Fume Standards-Correction I n the article on “Dust and Fume Standards” [McCabe, Rose, Hamming, and Viets, IND.ENG.CHEM.,41, 2388 (1949)l the formula on page 2390 should be corrected by changing the minua signs to plus signs. LOUISC. MCCABE
