Ind. Eng. Chem. Res. 1991,30, 585-589 Vogel, A. I. Aliphatic Compounds. In Elementary Practical Organic Chemistry, 2nd ed.; Wiley: New York, 1966; Part 1, Chapter 3, pp 204-205. Weininger, S. J.; Stermitz, F. R Organic Chemistry; Academic Press: New York, 1984; Chapter 17, pp 664-666. Weissermel, K.; Arpe, H.-J. Components for Polyamides. In Zndustrial Organic Chemistry; Verlag Chemie: Weinheim, New York, 1978 Chapter 10, pp 222-230. Willem, M.; Bruylants, A. Etude cinetique de l'hydrolyae des amides aliphatiques. Bull. SOC.Chim. Belg. 1961,60, 191-207. Zuman, P.; Patel, R.C. Techniques of Reaction Kinetics. In Tech-
585
niques in Organic Reaction Kinetice; Wiley: New York, 1984; Chapter 2.
* To whom correspondenceshould be addressed. Tarakad 5.Devarajan, Peter N.Pintauro' Department of Chemical Engineering, Tulane University New Orleans, Louieiana 70118 Receioed for reoiew March 21, 1990 Accepted September 7,1990
The Water Gas Shift Reaction Assisted by a Palladium Membrane Reactor The water gas shift reaction w& operated at 673 K by use of a double tubular type membrane reactor, the inner tube of which was palladium membrane (thickness, 20 pm) supported on a porous glass cylinder. The membrane reactor provided higher levels of carbon monoxide conversion beyond equilibrium attainable in a closed system resulting from shift of thermodynamic equilibrium toward the product side, which was caused by selective and rapid removal of hydrogen from the reaction system. Further, it was found that the amount of steam required to achieve reasonable levels of conversion could be reduced, and on the basis of computer simulation, the high reaction efficiency attained by this membrane reactor is credited to the thinness of the palladium film. Introduction Recently, there has been rising interest in a reactor embodying inorganic membrane through which a gaseous component can selectively permeate (e.g., Armor, 1989). The usage of such a reactor improves reaction efficiency through shifting the chemical equilibrium toward the product side, as the products are selectively removed from the reaction zone through the membrane. Most researches in this field have dealt with hydrogen-separating-type membrane reactors, where investigations were made on dehydrogenation of cyclohexane (Wood, 1968, Shinji et al., 1982; Mohan and Govind, 1986, Itoh, 1987; Itoh et al., 1988, 1989; Sun and Khang, 19881, ethylbenzene (Mohan and Govind, 1988), and butene (Itoh and Govind, 1989); steam reforming of methane (Oertel et al., 1987; Uemiya et al., 1991); decomposition of hydrogen sulfide (Raymont, 1975; Dokiya et al., 1977) and hydrogen iodide (Itoh et al., 1984); and aromatization of propane (Clayson and Howard, 1987; Uemiya et al., 1990a). A commercial application of the water gas shift (WGS) reaction is in raising the concentration of hydrogen in gas mixtures produced via steam reforming or partial oxidation of hydrocarbons. Copper-based catalysts are extremely active in this reaction, but these cannot be used at high concentration of reactant because activity would be deteriorated by sintering, caused by evolved heat of reaction. As the reaction is exothermic, thermodynamically, low reaction temperatures are desirable. Thus, commercial processes of the WGS reaction usually consist of two converters: the first being operated at high temperatures, between 623 and 673 K, with an iron-based catalyst having high thermal stability to suppress the concentration of carbon monoxide, and the second being operated at comparatively low temperatures, ranging from 473 to 523 K, using a copper-based catalyst. Complete conversion, however, is possible even at high temperatures, around 673 K, in case product hydrogen is separated from the reaction system by use of palladium membrane, which has been briefly reported previously (Kikuchi et al., 1989). Further, ultra-high-purity hydrogen is produced directly, since palladium membrane is permeable only to hydrogen. In this work, we additionally 0888-S885/91/2630-0585$02.60/0
deliberate on the influence of palladium membrane on the WGS reaction, and results obtained experimentally and those obtained from simulation were compared. Experimental Section Experimental apparatus and procedures were the same as those shown in the previous report (Kikuchi et aL, 1989). A double tubular type reactor was employed in this study. It was fabricated of palladium membrane ( 1 0 " 0.d.) and a quartz tube (1%" i.d.), which formed the inner and outer tubes, respectively. The palladium membrane was of a composite structure consisting of thin palladium film (palladium thickness, 20 pm) supported on the outer surface of a porous-glass cylinder (mean pore size, 300 nm), supplied by Ise Chemical Industry Co. The complete procedure for preparation of this composite membrane was described in a previous paper (Uemiya et al., 1988). A weighed amount (12.1 g) of a commercial iron-chromium oxide catalyst, designed as Girdler G-3,was packed uniformly in the reaction side (outside the membrane). The height of the catalyst bed was 8 cm, and the effective area of the palladium membrane for hydrogen separation was 25.1 cm2. The catalyst was heated to 673 K in a stream of nitrogen, as a precaution, to prevent the membrane from rupturing, as pinholes and cracks were formed in contact with hydrogen below 573 K, probably resulting from hydrogen embrittlement (Uemiya et al. 1988). The catalyst was, then, reduced at 673 K for 2 h in a stream of hydrogen dyuted with steam. Reactions were operated at 673 K, under atmospheric pressure. A mixture of carbon monoxide and steam was quantitatively fed to the reaction side so as to flow downward. Inert gas, argon, was supplied to the permeation side (inside the membrane) in a concurrent manner to sweep the permeated hydrogen. Compoeitione of effluent gases, from both the reaction and permeation sides, were determined by means of gas chromatography. Flow rates were measured volumetrically. Computer simulation of reaction kinetics was conducted to understand the function of palladium membrane reactor. In order to determine the rate equation for the iron-chromium oxide Catalyst UBBd in thia study, reactions were also carried out at 673 K and under atmospheric (8
1991 American Chemical Society
586 Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991
K 0
1
I
I
1
2
(W/F)/lO3g-cat min CO-mol-'
Figure 2. Comparison of experimental levels of carbon monoxide conversion with those calculated, represented by solid curves. H,O/CO molar ratio: 0, 1; A, 2; 0 , 3. 0
Pressure Difference/atm 1.0 1,s 2,O 2,s
0,s
3,O
Figure 1. Flow model of reaction and permeation in palladium membrane reactor.
pressure in a conventional flow reactor. The equation for hydrogen permeation rate through palladium membrane was determined with the use of pure hydrogen. The pressure drop between the two sides of the membrane was varied from 0.48 to 2.9 atm, maintaining the atmospheric pressure at the permeation side.
Results and Discussion The model of flow in the palladium membrane reactor is illustrated in Figure 1. Material balance, for both reaction and permeation sides, leads'to eqs 1-6. reaction side dF, = -r dl dFb
= -r dl
(1)
(2)
dF,=rdl dFd = r dl - j dl
(3)
dfd = j dl
(5)
(4)
permeation side df, = 0 (6) where 1 is the length of catalyst bed; Fi and fi are the flow rates of component i in the reaction and permeation sides, respectively; and r and j are the rates of the WGS reaction and hydrogen permeation, respectively. Many rate expressions have been proposed for the WGS reaction on iron-based catalysts (Morita and Tsuchimoto, 1967; Newsome, 1980). These were put to test, to judge how closely the experimental results could be reproduced. Among them, the rate equation reported by Kodama et al. (1956) most closely correlated with experimentally obtained data, as shown typically in Figure 2. The rate equation determined for reaction at 673 K is expressed as follows: r=
P.P, - Kf'PJ'd k l + 4.4Pb + 13Pc
(7)
Here Kp is the equilibrium constant, 11.92 at 673 K,and the rate constant k is determined to be 5.4 X lo3 [cm3(STP) cm-I min-l atm-l] per unit length of catalyst bed,
0
0,5 1,O 1,s ( ~ ~ 0 . 7 6- 0.76)/,tm0.76
2,O
pa
Figure 3. Pressure dependence of rate of hydrogen permeation through palladium/porous-glasscomposite membrane having palladium thickness of 20 um.
while Pi is the partial pressure of component i. The mechanism of hydrogen permeation through palladium membrane has been studied widely. It has been found that hydrogen is permeated through palladium membrane via a solution diffusion transport mechanism, and the rate of hydrogen permeation, J, per unit area of membrane, is written in terms of Fick's first law as follows (Lewis, 1967): J = (Q/t)(Pdn - Pd")
(8) where Q is the hydrogen permeation coefficient, t is the thickness of the palladium film, and n is a constant indicating pressure dependency. p d and & are the partial pressures of hydrogen in the high- and low-pressuresides, respectively. In a permeation process, if diffusion through the bulk of palladium is slow, relative to hydrogen dissociation, the concentration of dissolved hydrogen is proportional to the square root of hydrogen pressure, in which condition the constant n is 0.5. In Figure 3 is shown the pressure dependence of hydrogen permeation rate through the palladium/ porousglass composite membrane having a thickness of 20 pm at 673 K, pressure difference of 0.48-2.9 atm, and atmospheric pressure at the permeation side. The rate of hydrogen permeation was found to be proportional to the difference between high and low pressures to their 0.76 power. It has been pointed out by Lewis (1967) that the
Ind. Eng. Chem. Res., Vol. 30, No.3, 1991 587 100
90 0 L
,080 VI
L
4
4
0 0 H
w 70
.
25
60
I 0
I
2.5
I
I
1
5.0 7,5 10,o (W/F)/1O3q-cat mln CO-mol-'
1 12,5
Figure 4. Effect of time factor (W/F) on conversion of carbon monoxide. 0 ,experimental reaults; solid curve represents calculated results. Experimental conditions: temperature, 673 K; H20/C0 molar ratio, 1; flow rate of sweep argon, 400 cm3(STP) min-l.
rate of hydrogen permeation is rather better correlated to higher orders of hydrogen pressure than 0.5: for instance, 0.8 (DeRosset, 1960) or 0.68 (Hurlbert, 1961), comparable to the value observed in our experiments. We observed that the concentration of dissolved hydrogen was proportional to the square root of hydrogen pressure. Therefore, we have to take another factor into consideration in order to explain the order of permeation rate higher than 0.5. As the hydrogen permeation coefficient is the product of the hydrogen diffusion coefficient and solubility constant, we consider that the former may vary with the concentration of dissolved hydrogen. Suzuki and Kimura (1984) also pointed out that the diffusion coefficient of hydrogen increased with the concentration of dissolved hydrogen. We have also shown in a previous paper (Uemiya et al., 1990b) that the rate of hydrogen permeation through the composite membrane is inversely proportional to the thickness of the palladium film. The rate of hydrogen permeation j is expressed as follows: j = (q/ t )(Pdo"'- pd0*'6) (9) Here q is determined to be 5.9 X 102 [cm3(STP)pm-' cm-' min-' atm4*76]per unit length of catalyst bed. Numerical simulation was conducted by using rate equations for reaction and hydrogen permeation on the basis of the model shown in Figure 1. Equations 1-7 and 9 were solved on the presumption that plug flow, isobaric, and isothermal conditions existed. Moreover, it was assumed that the rate of hydrogen permeation was not affected by any of the coexisting gases and that the reaction occurred only on the iron-chromium oxide catalyst, and not on the palladium membrane. In order to ascertain that the proposed flow model is logical, experimental results were compared with those calculated. Figures 4 and 5 show the levels of carbon monoxide conversion as a function of the time factor W/F, which was defined by the weight of catalyst Wand the feed rate of carbon monoxide F, and as a function of the molar ratio of steam to carbon monoxide, respectively. The experimental and the calculated results agreed quite well. As shown in Figure 4, the level of carbon monoxide conversion increased as W/F increased. Further, at high W/F, higher levels of conversion beyond equilibrium attainable in a closed system were obtained, by shift of the thermodynamic equilibrium toward the product side, as a result of selective and rapid removal of produced hydrogen.
0
I
2
3
4
5
6
H20/C0 Molar Ratio
Figure 5. Effect of molar ratio of steam to carbon monoxide on conversion of carbon monoxide. 0 , experimental results; solid curve represents cdculated results. Experimental conditions: temmrat&e, 673 K; H,O/CO molar ratio; 1;flow rate of sweep argon, 400 cm3(STP) min-'. 100
90 0 L
c
80 L
4 0
equilibrium
w
i I I
I I I I
0,001 0.01 0,05 0 , l P a r t i a l Pressure of Hydrogen/atm
0.5
Figure 6. Conversion of carbon monoxide as a function of partial pressure of hydrogen. 0,experimental results; solid curve repreaenta calculated results on the assumption of chemical equilibrium.
From a thermodynamics viewpoint, commercial processes of the WGS reaction are operated in the presence of excess steam. As shown in Figure 5, a membrane reactor gave the same level of carbon monoxide conversion, at an equimolar ratio of steam to carbon monoxide, as obtained at a molar ratio of 2 in a conventional reactor. It is obvious, therefore, that a membrane reactor serves to reduce the amount of steam needed to achieve reasonable levels of conversion. Experimental results shown in Figures 4 and 5 were reproduced in Figure 6 as a function of the partial pressure of hydrogen in the reaction system. It was found that the experimental resulta almost coincided on the calculated curve, which was determined by assuming the thermodynamic equilibrium of the reaction. These results lead us to conclude that the hydrogen production rate is sufficiently high relative to the hydrogen permeation rate through the membrane. The dependence of the level of carbon monoxide conversion on palladium thickness was investigated with computer simulation. The calculated results are shown in Figure 7. The level of carbon monoxide conversion increased with decreasing thickness of palladium, as a result of improved rata of hydrogen permeation. Complete conversion of carbon monoxide, however, was not attained. When the level of carbon monoxide conversion was stabilized, the partial pressure of hydrogen remaining in the
588 Ind. Eng. Chem. Res., Vol. 30,No. 3, 1991 100
1
I
50
0
100
150
200
250
Thickness/um
Figure 7. Conversion of carbon monoxide as a function of palladium thickness. Feed rate of CO: (a) 25, (b) 25, and (c) 100 cmS(STP) mi&. Flow rate of sweep argon: (a) 3200, (b) 400, and (c) 400 cma(STP) min-'.
reaction side was at the same level as that in the permeation side. The level of conversion was raised by an increase in the flow rate of argon, resulting from reduction of the partial pressure of hydrogen in the permeation side. These results indicate that carbon monoxide with steam could be completely converted to hydrogen and carbon dioxide by use of a vacuum pump. The palladium membrane used in this study had the least thickness so far reported with 100% hydrogen selectivity remaining. Thus,it is concluded that a membrane reactor constructed with composite palladium membrane provides a significantlyhigh reaction efficiency attributed to its excellent hydrogen permeation performance. Conclusion It is shown that, by use of a palladium membrane reactor, carbon monoxide with steam was converted to carbon dioxide and hydrogen in high proportions beyond equilibrium. The level of carbon monoxide conversion is dependent on the thickness of palladium film, as the rate of hydrogen production is higher than that of hydrogen permeation through the membrane. The high level of conversion obtained by use of a composite membrane consisting of thin palladium f i i supported on porous cylindrical glass is credited to the thinness of the palladium film. Acknowledgment This work was partly supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture. Nomenclature F = feed rate of carbon monoxide, mol min-' Fi 5 flow rate of component i in the reaction side (outaide the membrane), cm3(STP) min-' f i = flow rate of component i in the permeation side (inside the membrane), cm3(STP) min-' J = rate of hydro en permeation per unit area through the membrane, cm (STP) cmm2min-' J' = rate of hydrogen permeation per unit length of catalyst bed through the membrane, cma(STP) cm-' min-' Kp = equilibrium constant of the WGS reaction, dimensionless k = rate constant of the WGS reaction per unit length of catalyst bed, cm3(STP) cm-' min-' atm-' 1 = length of the catalyst bed, cm n = constant indicating partial pressure dependency, dimensionless
f
P = partial pressure of component i in the reaction side (outside the membrane), atm p = partial pressure of component i in the permeation side (inside the membrane), atm P,, = partial pressure of hydrogen in the high-pressure side, atm P d = partial pressure of hydrogen in the low-pressure side, atm Q = hydrogen permeation coefficient per unit area, cm3(SW) gm-' cm-2 min-' atm-" q = hydrogen permeation coefficient per unit length of catalyst bed, cm3(STP)Fm-' cm-' min-' atm-'".76 r = rate of the WGS reaction per unit length of catalyst bed, cm3(STP) cm-' min-' t = palladium thickness, Fm W = catalyst weight, g Su bscriptjj a = carbon monoxide b = steam c = carbon dioxide d = hydrogen e = inert gas, argon Abbreviation
WGS reaction
= water gas shift reaction Registry No. COz, 124-38-9; HP,1333-74-0; Pd, 7440-05-3.
Literature Cited Armor, J. N. Catalysis with Permselective Inorganic Membranes. Appl. Catal. 1989,49, 1-25. Clayson, D. M.; Howard, P. UK Patent Application, 2190397A, 1987. DeRosset, A. J. Diffusion of Hydrogen Through Palladium Membranes. Znd. Eng. Chem. 1960,52,525-528. Dokiya, M.; Kameyama, T.; Fukuda, K. The Application of the Effusion on the Thermochemically Limited Reaction. Denki Kagaku 1977,45, 701-703. Hurlbert, R. C.; Konecny, J. 0. Diffusion of Hydrogen through Palladium. J. Chem. Phys. 1961,34,655-658. Itoh, N. A Membrane Reactor Using Palladium. AZChE J. 1987,33, 1576-1578.
Itoh, N.; Govind, R. Combined Oxidation and Dehydrogenation in a Palladium Membrane Reactor. Znd. Eng. Chem. Res. 1989,28, 1554-1557.
Itoh, N.; Shindo, T.; Hakuta, T.; Yoshitome, Y. Enhanced Catalytic Decomposition of HI by Using a Microporous Membrane. Znt. J. Hydrogen Energy 1984,9, 835-839. Itoh, N.; Shindo, Y.; Haraya, K.; Hakuta, T. A Membrane Reactor Using Microporous Glass for Shifting Equilibrium of Cyclohexane Dehydrogenation. J . Chem. Eng. Jpn. 1988,21, 399-404. Itoh, N.; Miura, K.; Shindo, Y.; Haraya, K.; Obata, K.; W a k a b a y d , K. A Novel Method for Dehydrogenation of Cyclohexane with a Palladium Membrane. Sekiyu Gakkaishi 1989,32,47-50. Kikuchi, E.; Uemiya, S.; Sato, N.; Inoue, H.; Ando, H.; Matauda, T. Membrane Reactor Using Microporous Gh-supported Thin Film of Palladium. Application to the Water Gas Shift Reaction. Chem. Lett. 1989, 489-492. Kodama, S.; Mazume, A.; Fukuba, K.; Fukui, K. Reaction Rate of Water-Gas Shift Reaction. Bull. Chem. SOC.Jpn. 1955, 28, 318-324.
Lewis, F. A. The Palladium Hydrogen System; Academic Prese; London, 1967; Chapter 7, pp 94-117. Mohan, K.; Govind, R. Analysis of a Cocurrent Membrane Reactor. AIChE J . 1986,32, 2083-2086. Mohan, K.; Govind, R.Effect of Temperature on Equilibrium Shift in Reactors with a Permselective Wall. Znd. Eng. Chem. Res. 1988,27, 2064-2070.
Morita, Y.; Tsuchimoto, K. Conversion of Carbon Monoxide and ita Catalysts. Nenryo Kyokai Shi 1967,46,802-818. Newsome, D. S. The Water-Gas Shift Reaction. Catal. Reo. Sci. Eng. 1980,21,275-318. Oertel, M.; Schmitz, J.; Weirich, W.; Jendryssek-Neumann, D.; Schulten, R. Steam Reforming of Natural Gas with Integrated Hydrogen Separation for Hydrogen Production. Chem. Eng. Technol. 1987, 10, 248-255. Raymont, M.E. D. Make Hydrogen from Hydrogen Sulfide. Hydrocarbon Process. 1975, July, 139-142.
Ind. Eng. Chem. Res. 1991,30, 689-691 Shinji, 0.;Misono, M.; Yoneda, Y.The Dehydrogenation of Cyclohexane by the Use of a Porous-glaee Reactor. Bull. Chem. SOC. Jpn. 1982,56,2760-2764. Sun, Y. M.; Khang, S. J. Catalytic Membrane for Simultaneous Chemical Reaction and Separation Applied to a Dehydrogenation Reaction. Znd. Eng. Chem. Res. 1988,27,1136-1142. Suzuki, Y.;Kimura, S. Separation and Concentration of Hydrogen Isotopea by Palladium-Alloy Membrane (I). Nippon Genshiryoku Gakkai Shi 1984,26,802-810. Uemiya, S.; Kude, Y.;Sugino, K.; Sato, N.; Matauda, T.; Kikuchi, E. A Palladium/Porous-Glass Composite Membrane for Hydrogen Separation. Chem. Lett. 1988,1687-1690. Uemiya, S.;Matauda, T.; Kikuchi, E. Aromatization of Propane Assisted by Palladium Membrane Reactor. Chem. Lett. 1990a,
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on a Porous Glass Tube. J . Membr. Sci. 1990b,in press. Uemiya, 5.;Sato, N.;Ando, H.; Matsuda, T.; Kikuchi, E. Steam Reforming of Methane in a Hydrogen-permeable Membrane Reactor. Appl. Catal. 1991,67,223-230. Wood,B. J. Dehydrogenationof Cyclohexane on a Hydrogen-Porous Membrane. J. Catal. 1968,11,30-34.
Shigeyuki Uemiya,* Noboru Sato Hiroehi Ando, Eiichi Eikuchi* Department of Applied Chemistry
School of Science and Engineering, Waseda University 3-4-1Okubo, Shinjuku-ku, Tokyo 169,Japan Received for review June 25, 1990 Revised manuscript received December 4,1990 Accepted December 17, 1990
1335-1338. Uemiya, S.;Sato, N.; Ando, H.; Kude, Y.; Matauda, T.; Kikuchi, E. Separation of Hydrogen through Palladium Thin Film Supportad
Mass Transport in Finite Baths: Effect of Surface Barriers Mass transfer of diffusants to the surface of solid polymeric material from some external medium can occur through a diffusional boundary layer. Equations exist that permit the description of such mass-transfer phenomena, but such equations are valid only for infinite bath systems, i.e., systems in which the concentration of diffusant in the external medium is constant. A new technique ia given that permits the mathematical description of mass transfer through a diffusional boundary layer for finite bath systems, i.e., systems in which the concentration of diffusant in the external medium changes during the sorption process. Diffusion-controlled mass transfer of diffusanta to (or from) the surface of porous solids of various geometrical shapes is strongly influenced by the thickness of diffusional boundary layer barriers at the solid surface (Levich, 1962). Newman was the first to describe in mathematical terms the effect of the diffusional boundary layer on rates of sorption by the slab, cylinder, and sphere (Newman, 1931). Newman’s equations are, however, only applicable to infinite bath sorption systems, i.e., systems in which the concentration of diffusant in the external medium is constant during the sorption process. No equation exists that describes the effect of the diffusional boundary layer on mass transport in finite bath systems, Le., systems in which the concentration of diffusant in the external medium is not constant during the sorption process. Since many real sorption systems are finite bath systems in which a significant boundary layer exists, it is useful to be able to model such systems mathematically. The purpose of the present work is to provide a new technique that can be effective in describing sorption rates of diffusanta from finite bath systems in which a diffusional barrier exists at the solid surface. The technique will be illustrated only for the case of diffusant uptake by a morphologically stable, homogeneous, endless cylinder. However, it should be understood that the technique also is applicable to the case of the plane sheet (slab) and the sphere.
where the P i s are the roots of the transcendental equation: OnJl(On) - LJo(Bn) = 0 (3) in which Joand J1are zero- and first-order Bessel functions. Equation 2 reveals that the rate of sorption is strongly influenced by the numerical value of the dimensionleas parameter, L. As L decreases, the rate of sorption decreases. Equations similar to eq 2 also are given by Crank for the case of the plane sheet and the sphere (Crank, 1975).
Finite Bath Systems When no diffusional boundary layer exists at the solid surface, i.e., when the dimensionless parameter, L, is equal to infinity, Wilson’s equation (Wilson, 1948)can be used to describe the relationship between fractional sorption of diffusant, M t / M m dimensionless , time, Dt/a2,and the dimensionless bath exhaustion parameter, a. In functional form:
M , / M , = Ff= f(Dt/a2,.)
(4)
Wilson’s equation for diffusant uptake by a cylinder from f i i t e baths in which no boundary layer exists at the solid surface is given by
Infinite Bath Systems The infinite bath, surface barrier equations of Newman are notationally encumbered and have been rewritten in a more straightforward manner by Crank (Crank, 1976). The functional relationship between fractional sorption of diffusant, Mt/M,, dimensionless time, D t / a 2 ,and dimensionless boundary layer, L, is given by M t / M - = Fi = f(Dt/UZ,L) (1) For the case of diffusant sorption by a cylinder surrounded by a diffusional boundary layer, Crank’s computational solution is given by
-Mt =
M,
F,= 1 -
-
c nil
4a(1 + a) exp(-qn2(Dt/a2)) 4 + 4a + d q , 2
(5)
where the qn’s are the positive, nonzero roota of W n J o ( q n ) + 2Jl(Qn) = 0 (6) in which Jo and Jl again are zero- and first-order Bessel functions. Equation 5 reveals that the rate of sorption is strongly influenced by the value of the dimensionless bath exhaustion parameter, a. As a increases, the rate of sorption decreases. Finite bath equations for the case of (8
1991 American Chemical Society
