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fit their equations with a linear rate function. “Higher order kinetic regimes” are invoked to explain the departure. But no independent considerations are given for such a rejection. Even if the authors’ rate model and integrated column equation were correct, they err in using only bed exit conditions to judge application of their model. The breakthrough or bed exit concentration is a function of all previous fluid and adsorbent concentrations for the entire elapsed time. This situation is true regardless of whether the rate (dispersion) processes are of the point or distributed type. Whatever rate model is used must apply to the whole range of column conditions. Finally, the dimensions of the carbon bed are too small for the particle size used. Cohen and Metzner (10) have recently shown that the column to particle diameter (CPD) ratio must be at least 30 to reduce flow nonuniformity errors to less than other errors. For Newtonian fluids, a CPD ratio of less than 30 results in appreciably greater flow near the column walls. This flow maldistribution is due to the small variations in bed porosity across the column. From the authors’ own figures of volumetric flow rate of 285 cm3/min, superficial linear velocity of 323 cm/min, and mean particle diameter of 0.268 cm, a CPD ratio of 3.95 is found. The flow maldistribution error for a CPD value of 4 is too large to estimate from ref 10. Of course, there may be some compensation of errors in some circumstances. All such potential problems can be avoided by using adequate adsorbent bed diameters. For gas-phase work with commercially used sizes of activated carbon, this means column diameters of 8-10 cm. Literature Cited Jonas, L. A.; Sansone, E. B. Enuiron. Sci. Technol. 1981, 15, 1367. Wheeler, A. “Collective Protection Against CB Agents”; Makowski, J., Ed.; 6th Bimonthly Progress Report, Army Contract DA 18-035-AMC-279(A), Sept 1965. Wheeler, A.; Robell, A. J. J. Catal. 1969, 13, 299. Miura, K.; Hashimoto, K. J. Chem. Eng. Jpn. 1977,10,490. Hall, K. R.; Eagleton, L. C.; Acrivos, A.; Vermeulen, T. Ind. Eng. Chem. Fund. 1966,5, 212. Cooney, D. 0.;Lightfoot, E. N. Ind. Eng. Chem. Fundam. 1966, 4, 233. Masamune, S.; Smith, J. M. Am. Inst. Chem. Eng. J. 1965, 11, 41. Rehrmann, J. A.; Jonas, L. A. Carbon 1978,16,47. Wilde, K. A. In “Activated Carbon Adsorption of Organics from the Aqueous Phase”; Suffet, I. H., McGuire, M. J., Eds.; Ann Arbor Science: Ann Arbor, MI, 1980; p 251. Cohen, Y.; Metzner, A. D. Am. Inst. Chem. Eng. J. 1981, 27, 705.
Kenneth Wllde 3604 Laurel Ledge Lane Austin, Texas 78731
SIR: Wilde suggests that our desorption paper (1) is based on an incorrectly derived equation, an erroneous rate model, and a sample size inadequate to eliminate flow maldistribution. We believe he is mistaken on all counts. In 1964, Wheeler derived an equation for adsorption kinetics from a continuity equation of mass balance per unit area (2). The derivation appears elsewhere (3’4) and 732
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is also available from us. The form was similar to that obtained by Bohart and Adams (5), Short and Pierce ( 6 ) , and Klotz (7). The innovative feature of Wheeler’s equation was the inclusion of a pseudo-first-order rate constant for adsorption, with units of reciprocal time, in the critical bed portion of the equation. Wheeler’s derivation begins with four differential equations that require simultaneous solution and includes the concept of a constant velocity wave in the carbon bed. By selection of particular adsorption conditions, viz., on the plateau of a type 1 isotherm, past the linear portion, where the adsorption capacity is no longer dependent upon inlet concentration and exit concentration is much smaller than inlet concentration, the four differential equations were reduced to one that could be solved analytically. Without imposing these conditions, asymptotic solutions for the entire exit concentration vs. time curve have been obtained by Rosen (8), Masamune and Smith (9, IO),and Schneider and Smith (11). In studying the kinetics of desorption, Wheeler (2) considered the desorption curve (desorbing concentration as a function of time) to be a mirror image of the sigmoidally shaped adsorption curve and used four transformation equations to relate the adsorption and desorption equations. The equation Wheeler derived was eq 1 in our paper (1). The derivation is available from us. The work of Cooney and Lightfoot (12) cited by Wilde is not applicable to the Wheeler adsorption and desorption equations, which are kinetic and in which exit concentration depends upon bed depth and time. Cooney and Lightfoot state that “ X and y [dimensionless concentrations] can be expressed entirely as functions of the single distance variable z* and independently of time, provided that t is very large”. The mirror image concept of adsorption-desorption can be visualized by the respective sigmoidal-antisigmoidal shapes of the concentration vs. time curves. In the adsorption cycle, the adsorption rate constant displays pseudo-first-order kinetics with respect to the decrease in gas molecules; in the desorption cycle, the desorption rate constant displays pseudo-first-order kinetics with respect to the reappearance (or regeneration) of active sites. In our experimental desorption tests we determined the maximum desorption rate from the carbon by waiting until the adsorption-desorption-adsorption wave had concentrated the adsorbed vapor at the exit of the bed. Thus, comments regarding elution processes or nonuniform initial bed loading are not applicable. Physical adsorption can be considered to be second order in kinetics, involving the reaction between an active site and a free vapor molecule (9,13). However, when the rate of change of gas molecules is much greater than that of active sites (active sites >> gas molecules), the reaction is pseudo first order with respect to gas molecules. Likewise, when the rate of change (appearance or regeneration) of active sites is much greater than that of gas molecules (gas molecules >> active sites), the reaction is pseudo first order with respect to active sites. A rigorous derivation of pseudo-first-order kinetics reactions for generalized gas adsorption is available from us. Wilde’s statement that “Mass transfer is a distributive process and must be dependent upon particle size” is not applicable to our conditions. Mass transfer would affect the overall rate only if external diffusion itself, or in combination with another mechanism, were rate controlling. In our studies we have found internal or intraparticle diffusion to be rate controlling. Thus, if the total adsorption process were conceived to consist of the sequence
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(1) external diffusion (mass transfer), (2) internal diffusion, and (3) adsorption at the site (9,11,14),then the internal diffusion rate would control the overall adsorption rate constant. Our finding that internal diffusion controls the overall adsorption rate constant is in consonance with Masamune and Smith’s conclusion, from their study of nitrogen adsorption rates, that surface adsorption was very rapid compared with intraparticle diffusion (internal diffusion) for spherical particles larger than 0.02 cm diameter (9). In dynamic flow testing of packed beds, maintenance of the proper column to particle diameter ratio is not the only requirement for satisfactory contact of the gas-air stream with the gross surfaces of the particles. Other important considerations in eliminating flow maldistribution are the porosity and tortuosity of the bed, the bed depth or volume, and the mean residence time of the gas in the bed. In our studies, the mean residence time ranged from 0.35 to 0.79 s, 900-2000 times longer than has been found necessary for this type of activated carbon to adsorb organic vapors in the molecular weight range 140-160 (15). The true measure, however, of a flow maldistribution can be seen in deviations from the expected linear dependence of breakthrough time t b (in the range 0 < C,/Co 5 0.04) on carbon weight W. If flow maldistribution exists, deviations in the t b vs. W plot would occur mostly at the low W values, tending to show more rapid bed breakthrough, resulting in shorter tbvalues and causing deviations from linearity. No deviations from linearity were observed in our tests. Literature Cited Jonas, L. A.; Sansone, E. B. Enuiron. Sci. Technol. 1981, 15, 1367. Wheeler, A. “Collective Protection Against CB Agents”; Makowski, J., Ed.; 6th Bimonthly Progress Report, Army Contract DA 18-035-AMC-279(A), Sept 1965. Wheeler, A.; Robell, A. J. J. Catal. 1969, 13, 299 Jonas, L. A. Ph.D. Thesis, University of Maryland, College Park, MD, 1970; Appendix E. Bohart, G.; Adams, E. J. Am. Chem. SOC.1920,42, 523. Short, 0. A.; Pierce, F. G. MIT Memorandum Report No. 114, Massachusetts Institute of Technology, Cambridge, MA, 1946. Klotz, I. M. Chem. Rev. 1946, 39, 241. Rosen, J. B. J. Chem. Phys. 1952,20,387. Masamune, S.; Smith, J. M. AIChE J. 1964, 10, 246. Masamune, S.; Smith, J. M. AIChE J. 1965, 11, 34. Schneider, P.; Smith, J. M. AIChE J. 1968, 14, 762. Cooney, D. 0.;Lightfoot, E. N. Ind. Eng. Chem. Fundam. 1965, 4 , 233. Hiester, N. K.; Vermeulen, T. Chem. Eng. Progr. 1952,48, 505. Tien, C.; Thodos, G. AIChE J. 1959, 5, 373. Rehrmann, J. A.; Jonas, L. A. Carbon 1978,16,47.
Eric B. Sansone,* Leonard A. Jonas Environmental Control and Research Laboratory National Cancer Institute Frederick Cancer Research Facility Frederick, Maryland 21 701
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Another Comment on “Desorption Klnetlcs of Carbon Tetrachloride from Activated Carbon” SIR: In a recent paper on ”Desorption Kinetics of Carbon Tetrachloride from Activated Carbon” (Enuiron. Sci. Technol. 1981,15, 1367-1369), the authors claim to have demonstrated “a more fundamental relationship between the theoretical equations of adsorption and desorption kinetics than had previously been supposed”. This demonstration rests on the observation that the “log of the desorption rate constant was a linear function of the percentage saturation of the bed ....” The purpose of this letter is to question how fundamental and generally significant the observed relationship is. In this reader’s opinion, any truly fundamental relationship between desorption rate and adsorbate loading ought to account explicitly for the adsorption equilibrium relation, as done by Grubner and Burgess in a paper in the same issue (Ibid. 1981,15,1346-1351). The paper under discussion does not treat the equilibrium relation in any way, nor are the properties of the equilibrium isotherm of the system studied (CC14/BC-ACactivated carbon) mentioned in the paper. This omission seems a serious shortcoming. Moreover, the reference cited (ref 3) for the derivation of the kinetic expression is an unpublished progress report, which is scarcely available to the scientific community. I invite the authors to comment on the sensitivity of their model to the form of the equilibrium isotherm (i.e., favorable or linear). Paul V. Roberts Environmental Engineering and Science Department of Civil Engineering Terman Engineering Center Stanford University, Stanford, California
S I R Roberta has misinterpreted us. He states that “the authors claim to have demonstrated ‘a more fundamental relationship’ and understands us to have put forward eq 6 as that fundamental relationship. However, we said that the result (i.e., the linearity of the relationship between log kd and percent CCll saturation) “suggests that a more fundamental relationship exists”. We did not claim to have found it. We do not plan to respond to Roberts’ other points concerning adsorption equilibrium and derivation of the kinetic expression at this time, because we believe that they are made moot by the preceding paragraph. Eric B. Sansone,* Leonard A. Jonas Environmental Control and Research Laboratory National Cancer Institute Frederick Cancer Research Facility Frederick, Maryland 21701
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