March, 1952
RATE-DETERMINING STEPSIN RADIAL ADSORPTION
373
THE RATE-DETERMINING STEPS IN RADIAL ADSORPTION ANALYSIS BY LEONLAP ID US^ AND NEALR. AMUNDSON University of Minnesota, Minneapolis] Minn. Received December 11, 1050
Equations have been derived based upon various kinetic, liquid and solid diffusion rate mechanisms for the saturation of an initially em t y radial adsor tion disc. Three experimental systems were employed in an endeavor to test the validity and usefulness of t i e equations unjer a variety of operating conditions. The results indicate that for certain of the solute concentrations studied a single rate mechanism can be chosen with a fair degree of certainty.
The kinetics of adsorption columns have received a considerable amount of attention by various authors in the past few years. However, a careful perusal of the literature reveals the fact that a vast majority of the publications have been primarily of a theoretical nature. Hiester and Vermeulen2 have summarized very adequately the various theoretical postulates presented up to 1948 and the Faraday Society3 has since published a symposium on chromatography which contains a number of pertinent papers. Application of the various kinetic and diffusion equations to actual experimental data in an endeavor to confirm or disprove the theoretical postulates has received little attention. Of the attacks made on this problem, a-11 the theoretical equations have not been subjected to sufficient variation in the operating variables to allow the authors to make any definite statements as to the applicability of the equations. The present authors, therefore, undertook the collection of data under a variety of operating conditions obtained on a radial adsorption disc. The choice of a radial disc rather than the usual vertical column was based on the number of commercial advantages that the former method presents.12 These data were then applied to theoretical equations based on kinetic and diffusional mechanisms and the results analyzed for the applicability of the equations. It was hoped that by pursuing this course a more adequate basis would be established toward understanding the phenomenon of rate-dependent adsorption analysis.
Theoretical I n a previous publicationi3 the authors have presented under very general conditions the solution of the mathematical problem which results when one considers the flow of solution containing a single solute through a radial adsorption disc. (1) Forrestal Research Center, Chemical Kinetics Division, Princeton University, Princeton, N. J. (2) N. K. Hiester and T. Vermeulen, J . Chem. Phus., 16, 1087 (1948). (3) "Chromatographio Analysis," Discussion Faraday SOC., No. 7 (1949). (4) R. H. Beaton and C. C. Furnas, Ind. Ene. Chem., 38,150 (1941). ( 5 ) G. A. Bohart and E. 0. Adams, J . A m . Chem. SOC.,48, 523 (1920). (6) C. N. Hinshelwood, et ar., J . Chem. Soc.. 918 (1946); 401 (1947). (7) E. R. Tompkins, J. X. Khym and W. E. Cohn, J . A m . Chem. Soc.. 69, 2769 (1947). (8)J. J. Kipling, J . Chem. Soc., 1487 (1948). (9) W. A. Selke, Ph.D. Thesis, Yale Univ., 1949. (10) L. G. Sillen and E. Ekedahl, Arkin. Kemi Mineral Qeol.. 828, No. 16 (1946). (1 1) F. C. Williams, N. K. Hiester and T. Vermeulen, in press. (12) H. Weil, Can. Chcm. and PTOC. Ind., 956 (1949). (13) L. Lapidua and N. R. Amundson, Tma JOVRNAL,64, 811 (1950).
For the sake of continuity this paper will be summarized first, and the equations changed into forms more desirable for actual usage. Let: T
= distance from origin to any point in the disc, cm
mass of adsorbent per unit area of disc,. g,/.cm.2 radius of circular hole into which solution is introduced] cm. c = amount of solute in solution per unit volume of solution, mmol./ml. n = amount of solute adsorbed per unit mass of adsorbent] mmol./g. = volume rate of flow of solution, ml./min. v m = porosity of adsorbent bed, ml./g. co = initial solute concentration. = total solute capacity of adsorbent. no = concentration of solute in equilibrium with adc* sorbed solute. D1 = diffusion constant for solute in solution. D. = diffusion constant for solute in solid. b = distribution coefficient. k, = mass-transfer coefficient. kll kz = kinetic velocity constants. a0 = particle radius, cm. Aa = film thickness. = mass of adsorbent/unit volume, g./ml. jPk = constants. y = shape factor. h
R
= =
The system of equations which has been shown to define the physical problem is ac 5ar
an ac = 0 + 2 ~ r hat + 2 ~ r h mat
c(rlt) = ~ ( t ) when , T = R n(r,t) = no(r), when 5t Thm(r* - R2)
