Znd. Eng. Chem. Res. 1996,34, 2762-2768
2762
Modeling and Simulation of Carbon Mask Adsorptive Reactors Rui C. Soares, Jose M. Loureiro, Carlos Sereno: and Alirio E. Rodrigues* Laboratory of Separation and Reaction Engineering, School of Engineering, University of Porto, Rua dos Bragas, 4099 Porto Codex, Portugal, and Department of Mathematics, University of Beira Interior, Rua Marqu&sde Avila e Bolama, 6200 Covilhd, Portugal
A carbon gas mask is considered in this paper for the elimination of CNCl by adsorption in the activated carbon support and simultaneous second order chemical reaction with the impregnant metal. A model, taking into account nonlinear adsorption, chemical reaction, film and intraparticle mass transfer resistances, and axial dispersion is written and solved by the moving finite elements method. In the very short beds used i n carbon gas masks, axial dispersion is probably important. Simulations are carried out for different values of the parameters in order to assess the influence of the intraparticle and film mass transfer resistances and of the rate of reaction on the system behavior in the presence of axial dispersion. It is shown that, although axial dispersion does not alter the general behavior of the system, its presence can reduce dramatically the usefulness of these respirators. On the other hand, two different correlations give more than 1 order of magnitude differences in the value of axial dispersion to be expected in the bed.
Introduction The removal of toxic and irritating gases or vapors from a stream is commonly achieved by the use of carbon filters. There are several applications in the civilian and military areas. Carbon filters can be found in personal respirator cartridges or in big ventilation systems (Balieu, 1990). The first observations of the ability of carbons t o retain gases date back to 1777 and 1780 (Greggand Sing, 19821,but only during World War I they were used for that purpose. Physical adsorption and capillary condensation are two of the processes by which carbons manage to trap gases. Therefore a large specific surface area and well developed meso and micropore structures are essential to effective gas removal by carbon filters. Activated carbons used in gas filters usually have specific surfaces in the range 900-1300 m2/g. Nevertheless, only organic vapors with boiling points above 65 "C are efficiently removed by activated carbons. Gases and organic vapors with boiling points below 65 "C often are not irreversibly adsorbed and pure activated carbons have low retention efficiencies for such gases. The performance of the gas mask can be strongly enhanced by impregnating the activated carbon with metallic ions, N€&+,co32-,and organic compounds. The impregnants interact with the toxic agent either by stoichiometric chemical reaction or by catalysis. For both situations an improvement in removal efficiency is achieved by irreversible conversion of the toxic agent to another species (Balieu, 1990). The products of the reactions should remain, if still harmful, irreversibly adsorbed, at normal conditions of use. During World War I, the Germans pioneered the use of activated and impregnated carbons. Their early masks contained activated carbon impregnated with either alkali or hexamethylenetetramine. The use of impregnated carbons by the German army stimulated the Allies research of the subject. One of the most important developments in that period was the use of copper as an impregnant. This kind of activated and impregnated carbons was called whetlerite by J . C. Whetzel and E. W. Fuller (Noyes, 1946).
* Author to whom correspondence should be addressed. +
University of Beira Interior.
0888-5885l95/2634-2762$09.00/0
In the last years of World War I, adsorbents based on copper impregnation of activated carbons were manufactured. In the beginning of World War I1 some progress was achieved, but it was not too significant. In the following years research was intensified and many new impregnants such as silver, thyocyanate, molybdenum, vanadium, tartaric acid, chromium, and zinc were tested. All these efforts led t o the development of the ASC whetlerite. ASC whetlerite is an activated carbon impregnated with Cu2+, Cr6+, Ag+, NH4+, and co32-. This adsorbent has good efficiency for the removal of many toxic agents including warfare gases and vapors. It is still in use today for military and civilian purposes with very little change from the original formulation. The first mathematical approach to the adsorption phenomena in packed columns was carried out by Bohart and Adams in 1920 (Balieu, 1990). In their work they developed an equation for the shape of the wave front of the nonadsorbed gas in a fixed-bed adsorber (Danby et al., 1946). According to Danby et al. (1946), in 1925 Mecklenburg and Kubelka presented the first predictive equation for a system where only physical adsorption is present. Using the same approach, Danby et al. (1946) developed similar equations for systems where physical adsorption and chemical reaction are coupled and tested their validity with experimental results. The equations of the latter researchers differ essentially from the ones of Mecklenburg and Kubelka only in the physical interpretation of the constants (Danby et al., 1946). Since then more predictive equations have been developed such as the ones by Klotz (19461, WheelerJonas (Jonas and Rehrmann, 19731, and Yoon-Nelson (1984). We must stress that predictive equations are simple mathematical expressions, with more or less empirical content, whose sole purpose is t o give an approximate estimate of the breakthrough time of the toxic gas emerging from the packed bed. These equations can be applied only if temperature, flow, and inlet concentration of the toxic sorbate are kept constants. If any of these variables is not constant, a more complex analysis is required. In the past 40 years many papers have appeared on the modeling of physical adsorption systems. 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 8,1995 2763
A mathematical approach to systems where physical adsorption and chemical reaction are both present was suggested by Magee (1963), for the case of a chromatographic reactor. He considered a reversible reaction in terms of gas phase concentrations and linear adsorption equilibrium. This model was later improved by Gore (19671, who considered the reaction rate in terms of concentrations of the adsorbed species. Chu and Tsang (1971) continued these studies applying the LangmuirHinshelwood model. By using an equilibrium model, i.e., neglecting all kinds of dispersive effects and mass transfer resistances, Schweich et al. (1980) developed a method where linear equilibrium isotherms with equilibrated reversible reaction mechanisms were analyzed. Using the same approach, Bunge and Radke (1982) and Loureiro et al. (1983, 1990) studied the case of Langmuir equilibrium isotherms and irreversible reaction mechanisms of several orders. None of these studies considered the elimination of toxic gases by activated and impregnated carbon filters, except the Danby et al. (1946) research. In 1988 Friday proposed a model for the retention of CNCl with an ASC filter. A n irreversible reaction was considered between one of the impregnants and adsorbed CNC1. The reaction rate was assumed to be first order relative to both species. Mass transfer was considered between the moving phase and the particle surface. Intraparticle mass transfer was described in terms of a lumped model. A very similar study was carried out by Graceffo et al. (1989). The main differences from Friday's approach were that reaction rate was expressed in terms of the gas phase and the use of a diffusion model to describe the intraparticle mass transfer resistances. A different approach was taken in the analysis by Chatterjee and Tien (1990);in this work the reaction between the impregnant and the toxic agent was assumed to be instantaneous. This work dealt only with the particle side modeling. A more complex methodology validated by comparison with experimental data was developed by Gail et al. (1989)for SO2 elimination. They proposed a Langmuir-Hinshelwood model where simultaneous adsorption and reaction occur. The recently developed moving finite elements method (Sereno et. al., 1992) is a numerical method appropriate for the solution of problems involving wave propagation phenomena, namely when the propagation velocities are not constant; this is the case in the carbon mask adsorptive reactor, where the toxic component concentration decreases, the same happening to its velocity, as it moves along the filter. The objectives of this paper are (i) to develop a mathematical model for carbon mask adsorptive reactors, (ii)to use a new numerical method-moving finite elements method-for the solution of the model equations, and (iii) to study the influence of model and operating parameters on the carbon mask behavior, as measured by the breakthrough (useful) time.
Conservation equations for both components (solute
A and metal N) are mass balance for species A i n the fluid phase: a2c
-gDax aZ2
a c a c + (1- e)kpp (C - c*) = 0 + az + E at EU~
(1)
mass balances for species A in the solid phase:
(3) mass balance for metal N: (4)
In these equations a is the stoichiometric coefficient of the metal species with concentration N (moles of impregnant consumed in the reaction with 1 mol of solute), c is the concentration of species A in the gas phase, q is the average adsorbed phase concentration of A, q* is the adsorbed concentration of A at the particle surface in equilibrium with c*, and r ( q N is the reaction rate given by
r(qJV)= kqN
(5)
The concentrations of the adsorbed and gas phases in equilibrium are expressed by the Langmuir adsorption isotherm: q" = qsatKLC"
+
1 KLc* The following initial and boundary conditions hold for the above model equations: t=O
Vz>O
c=q=c*=q*=O,
z=L VtrO
-ac =o aZ
N = N o (7)
(9)
Introducing dimensionless variables,
Model Equations Consider a fixed bed of impregnated and activated carbon where the impregnant is consumed in a reaction with the solute. The reaction takes place in the adsorbed phase and is of global second order (first order in each of the two reactants), where the solute is nonlinearly adsorbed on the carbon. We will consider a film resistance between the fluid and solid and an intraparticle resistance, described by a lumped model of the linear driving force (LDF) type.
and the dimensionless parameters,
UiL
Nd= kpapt, Pe = -, K = 1+ KLcE Dax eqs 1-9 become
2764 Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995
ae = -aDaxyM
(11)
ae = Nd(y*- y ) - DayM
(12)
In the Lth finite element of the mth equation, Y, is fitted by a polynomial y”, with degree N - 1. The degree of the fitting polynomial can be different for another finite element. This approximation is defined by the Lagrange interpolating polynomial: N
(22) i=l
(13)
y* = 1
+
kk* (K- l)x*
where umpL v is the independent variable in the Lth I is the ith Lagrange finite element [ u r , U ~ + ~Zy8L(umrL) is the value of in interpolating coeficient, and the ith interpolating point of the Lth finite element, UFL.
t=o
For the mth differential equation the residual is defined by
ve10
{=1
(16)
v010
& -O
(17)
at
The model parameters are the following: a = stoichiometric coefficient of the impregnant metal x = ratio between the adsorbed solute concentration QE in equilibrium with CE and the initial metal concentration in the solid No 5 = adsorption capacity factor, i.e., ratio between the amounts of solute in the solid and in the fluid phases in equilibrium with each other K = modified constant of the adsorption isotherm Da = Damkohler number Nd = number of mass transfer units by intraparticle diffusion Nf= number of mass transfer units by film diffusion Pe = Peclet number
The Moving Finite Elements Method Model equations were numerically solved by using the moving finite elements method (Sereno et al., 1992).In this method the node mesh is automatically adjusted to the current solution a t each moment. Let us consider the mth generic equation of a system of partial differential equations (PDE):
where 9, is the approximation of Y, for every point u E [a,bl in such a way that 9, = for every u E
[v,
VY+J
The possibility of expressing the approximations of two consecutive finite elements by the same polynomial o r the possibility of two separation nodes to collapse in only one point must be avoided. This is prevented by penalizing the movement of the nodes. Then, the Gallerkin method for the development of the moving finite elements method must be reformulated. We will use the formulation proposed by Miller and Miller (1981, 19831,defining the function
v+l
where z r = is the length of the finite element L of the finite element system associated to the mth equation and E? and Sr are terms corresponding to “viscosity” and “spring” forces, respectively, penalizing the movement of the separation nodes. In order t o obtain the system of ordinary differential equations which is an approximation of the actual system of partial differential equations, we have t o minimize the function F, in order t o all its parameters
where k = {Yl, Yz,..., YN}*is the vector of dependent variables and where Y, = Ym(v,t)
when t L 0 and a 5 u Ib, f o r m = 1, ..., n (19)
with the following initial and boundary conditions Y, = gm(v) aYm Ym-+a,Y,=/3,
av
when t = 0 and a
Iu 5
b (20)
fort > O a n d v = a and/or u = b (21)
The domain of the system of equations, [a,bl, is divided, for the mth equation, in Q finite elements, separated by Q + 1 separation nodes, the first being = a and the last T+l b.
The system of ordinary differential equations is integrated by using the LSODI integrator, developed in the Livermore National Laboratory (Hindmarsh, 1986).
Variables and Parameters Used in the Simulations In this work the removal of cyanogen chloride from an air stream using ASC whetlerite contained in a gas mask canister is considered. The canister and adsorbent bed characteristics are summarized in Table 1.The equilibrium and kinetic parameters as well as the operating conditions, taken from Friday (19881,are listed in Table 2. The film mass transfer coefficient, under the reference case operating conditions, was estimated by the cor-
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2765 Table 1. Canister and Adsorbent Bed Characteristics bed length bed diameter bed porosity bulk density particle size particle porosity initial metal (impregnant) concentration
0.02 m 0.1 m 0.4 600 kg/m3 0.001m 0.63 0.38 moVkg
Table 2. Operating Conditions, Equilibrium Parameters, and Kinetic Parameters operating conditions ASC whetlerite equilibrated initial conditions with clean air at 80% relative humidity feed conditions air at 80% relative humidity with 0.065 mol/m3CNCl 5.0 x 10-4m3/s flow rate 294-298 K temperature adsorption equilibrium parameters 1.8m3/m01 equilibrium constant 3.4 moVkg saturation capacity kinetic parameters 0.2 mol (metal)/mol (CNCl) stoichiometric coefficient 4.33 x 10-6 mV(mo1.s) reaction rate constant Table 3. Dimensionless Parameters Calculated from Physical Constants and Variables for the Reference Case Da 1.241 x Nd 9.35 x 10-1 Pe 4.42 a 0.200 K 1.12 E 8.22 103 Nf 67.7 X 0.937
relation proposed by Wakao and Funazkri (1978):
Sh = 2.0
+ l . l S ~ ” ~ R (e3~ 30 it can be neglected; i.e., for the reference
2766 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
Bp2.0 2‘5 104 l
o
4
7
e
101
1
e
a I
00
Figure 4. Effect of axial dispersion, measured by the axial Peclet number, Pe, on breakthrough time of the bed for two different reaction rates, as measured by the Damkohler number, Da; 0 ,Da = 1.241 x 0,Da = 1.241 x loy3.
I:
e
eBp 1400;
t(o)----j 2000
e
1100
1000
O 0
2
4
6
I SDa’
”
Figure 3. Effect of number of reaction units in the fluid phase, tDa, on breakthrough time of the bed.
“1
0
. 0.5
1.0
1.5
2.0 2.5
3.0 3.5 4.0
5
L(cm) Figure 5. Effect of bed length on breakthrough time of the bed.
case (Nf = 67.7) the film mass transfer resistances can be neglected. The Damkohler number, Da, measures the amount of reaction occurring in the bed; since the reaction is taking place in the solid phase, it translates the conversion of the reactants CNCl and metal impregnant in the solid. Ultimately, to evaluate filter performance, one must consider design in terms of the conversion of CNCl in the fluid phase. Since the equilibrium amounts of CNCl in the fluid and solid phases are related through the capacity factor 5, the use of EDa as a number of reaction units is preferable (Loureiro and Rodrigues, 1991). Figure 3 shows the influence of this number of reaction units, fDa, on the breakthrough time for the bed, &pa It clearly shows that as the reaction amount increases, so does the useful time of the filter; this is a consequence of the lower concentration propagated (Loureiro et al., 1983) with a nonlinear adsorption equilibrium isotherm; the effect should be more pronounced with a more nonlinear isotherm. Axial dispersion, measured by the Peclet number Pe, has an enormous influence on the breakthrough time as Figure 4 shows. As a matter of fact, the reference case operates in a region of no intraparticle or film mass transfer resistances, but the estimated axial dispersion coefficient results in a very low Peclet number (Pe = 4.42) conducive to a breakthrough time that is only about 115 of the OBp that should be observed in the absence of dispersion. This means that although the presence of axial dispersion does not alter the general behavior of the system, it certainly reduces drastically its usefulness. Since the axial Peclet number is proportional to the bed length, some simulations were done with different bed lengths (and the respective estimated axial dispersion coefficients); the results are shown in Figure 5 where the breakthrough time, in seconds, is plotted
against the bed length for various bed lengths. In this figure, real time is used, since the space time varies with the length of the bed. One important conclusion to be drawn based on the simulation results is that if a different adsorbent with a more nonlinear isotherm and/or an impregnant conducive to a faster chemical reaction and/or a bed filling process giving a higher Peclet number cannot be found, the usefulness of the filter is far from being good; as a matter of fact, for the reference case the space time is z = 0.126 s, and for the observed 6Bp of about 1100 this means about 2.3 min of protection. The comparison of these calculated breakthrough times with the experimental results published by Friday (1988) shows that the prediction is not good; since the protection achieved by this kind of systems is accomplished because of the combination of nonlinear adsorption and reaction, some runs were done with a Damkohler number 10 times larger than the one shown in Table 3, i.e., with Da = 1.241 x the results are plotted, against the Peclet number, in Figure 4. The improvement is dramatic, namely, for the high Peclet region. For Pe RZ 80, the calculated breakthrough time compares well with the experimental one and with the simulated by Friday (1988) using 40 cells in a cells in series model, roughly equivalent to Pe = 80. The Damkohler number for the reference case was calculated with the reaction kinetic constant published by Friday (1988); however, the correct value should be K = 4.33 x m3/(mol.s) (Friday, 1995). The fact that the comparison between the experimental (Friday, 1988) and the simulated results points to a higher value of the Peclet number than the one estimated by eq 26 seems t o indicate that this correlation is not appropriate, at least for the reported experimental conditions. As a matter of fact, the Peclet value should
Ind. Eng. Chem. Res., Vol. 34, No. 8,1995 2767 be 58.2 for the reference case, if the followingcorrelation (Yang, 1987) were used instead
L - 0.3 --2R$e Re Sc
+
0.5 1 -k 3.8l(Re Sc)
(29)
It should be noted that it is not necessary to estimate a different film mass transfer coefficient, since the film mass transfer resistance is negligible anyway. The observed discrepancy in the estimated Peclet values points to the care that should be taken whenever one needs to sort to an estimation of axial dispersion based on the available correlations.
Conclusions In this paper a carbon gas mask in which CNCl is removed by simultaneous nonlinear adsorption and second order reaction is considered. A model including film and intraparticle (with a LDF approximation) mass transfer resistances, a global second order chemical reaction (first order in the solute CNCl and first order in the metal impregnant), and axial dispersion is built and solved with the moving finite elements method. The effects of the film and intraparticle resistances and of the chemical reaction rate on the breakthrough (useful) time are assessed through simulations for different values of the respective numbers of mass transfer or reaction units. The simulated results agree with the experimental and simulated data published by Friday (1988), when a reaction rate constant of K = 4.33 x 10-5 m3/(mol*s)is used. It is shown that the presence of axial dispersion does not alter the general behavior of the system; nevertheless the useful time of the carbon mask can be drastically increased when axial dispersion is reduced (the Peclet number is increased). Two correlations for the estimation of the axial dispersion coefficient are used in this paper, and the resulting Peclet numbers differ by more than 1order of magnitude. Roughly the same difference is observed on the useful time for the moderate to high Peclet region. Given the discrepancy between the two estimated axial dispersion coefficients, care must be taken when using this kind of correlation; moreover and surprisingly, when the results are compared with experimental data, it seems that the recommended correlation to be used in the operating conditions of the filter fails more than the other one used.
Acknowledgment Financial support from the Portuguese Ministry of Defense and JNICT is gratefully acknowledged.
Nomenclature a = left extremum for the independent variable u up = particle specific surface area b = right extremum for the independent variable u c = solute concentration in the gas phase CE = inlet solute concentration in the gas phase c* = solute concentration in the gas phase at the interface gadsolid Du = Damkohler number D,, = axial dispersion coefficient D m = diffision coefficient De = effective diffusivity D, = molecular diffisivity
F, = penalizing function in the moving finite elements method g, = initial value of Y m k = reaction rate constant k, = intraparticle mass transfer coefficient (LDF approximation) k, = film mass transfer coefficient K = dimensionless adsorption constant KL = Langmuir adsorption constant IFL = ith Lagrange interpolating coefficient L = bed length M = dimensionless concentration of the metal impregnant n = number of equations in the system of PDE N = concentration of the metal impregnant Nd = number of mass transfer units by intraparticle diffusion N f = number of film mass transfer units NO= initial concentration of the metal impregnant P = pressure Pe = bed Peclet number q = solute concentration in the adsorbed phase q E = solute concentration in the adsorbed phase in equilibrium with CE qsat= saturation concentrationof the solute in the adsorbed phase q* = solute concentration in the adsorbed phase in equilibrium with c* Q = number of finite elements r = reaction rate R , = residual for the mth differential equation Re = Reynolds number R, = particle radius SF = “spring” forces in the penalizing function of the moving finite elements method Sc = Schmidt number Sh = Sherwood number t = time u1 = interstitial velocity umaL= independent variable in the Lth finite element u = space-independent variable in the PDE system u L = atomic diffusion volume of species i x = dimensionless concentration in the gas phase x* = dimensionless concentration in the gas phase at the gadsolid interface y = dimensionless concentration in the adsorbed phase y* = dimensionless concentration in the adsorbed phase in equilibrium with x* 9 = vector of dependent variables in the PDE system ym= generic dependent variable in the PDE system Y, = approximation of Y, y“, = Lagrange interpolating polynomial z = bed axial coordinate ZF = length of the finite element L associated to the mth equation Greek Letters
a = stoichiometric coefficient of the metal impregnant a, = constant in the boundary condition for the general PDE 3/, = constant in the boundary condition for the general PDE y m = constant in the boundary condition for the general PDE E = bed porosity E; = “viscosity” forces in the penalizing function of the moving finite elements method E , = particle porosity 5 = dimensionless axial coordinate of the bed 8 = time reduced by the space time OBp = breakthrough time reduced by the space time ij = bed capacity factor
2768 Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 = bulk density of the bed z = space time p, = tortuosity factor for the particle x = ratio between the adsorbed phase concentration qE and the metal concentration No @b
Literature Cited Balieu, E. Fundamental Aspects in Air Filtration and Purification by Means of Activated Carbon Filters. In Gas Separation Technology; Vansant, E. F., Dewolfs, R., Eds.; Elsevier Sci. Publ.: Amsterdam, 1990. Bunge, A. L.; Radke, C. J. Pulse Chromatography of Adsorbing and Reacting Chemicals. In Recent Advances in Adsorption and Ion Exchange; Ma, Y. H., Ed.; AIChE Symposium Series 219; AIChE: New York, 1982;pp 3-18. Chatterjee, S. G.; Tien, C. Retention of Toxic Gases by Modified Carbon in Fixed Beds. Carbon 1990,28(6),839-848. Chu, C.;Tsang, L. Behavior of a Chromatographic Reactor. Znd. Eng. Chem. Process Des. Dev. 1971,10,47-53. Danby, C. J.; Davoud, J . G.; Everett, D. H.; Hinshelwood, C. N.; Lodge, R. M. The Kinetics of Absorption of Gases from an Air Stream by Granular Reagents. J. Chem. SOC. 1946,115,918934. Friday, D. K. The Breakthrough Behavior of a Light Gas in a Fixed-Bed Adsorptive Reactor. AZChE Symp. Ser. 1988, 84 (2641,89-93. Friday, D. K. Private communication, 1995. Gail, E.; Richter, H.; Kast, W. On the Kinetics of Adsorption and Reaction of Sulphur Dioxide in a Fixed Bed of Activated Carbon. Presented at the Third International Conference on Fundamentals of Adsorption, Sonthofen, Germany, 1989. Glueckauf, E. Theory of Chromatography, Part 10-Formulae for Diffusion into Spheres and their Application to Chromatography. Trans. Faraday SOC.1955,51,1540-1551. Gore, F. E. Performance of Chromatographic Reactors in Cyclic Operation. Ind. Eng. Chem. Process Dev. 1967,6 (11, 10-16. Graceffo, L.A,; Chatterjee, S. G.; Moon, H.; Tien, C. A Model for the Retention of Toxic Gases by Impregnated Carbon. Carbon 1989,27 (31, 441-456. Hindmarsh, A. C. LSODE and LSODI, Two New Initial Value Ordinary Differential Equation Solvers. ACM-SIGNUM Newsl. 1986,15,10-11. Jonas, L. A,; Rehrmann, J. A. Predictive Equations in Gas Adsorption Kinetics. Carbon 1973,11, 59-64. Klotz, I. M. The Adsorption Wave. Chem. Rev. 1946,39, 241268.
Loureiro, J. M.; Costa, C. A.; Rodrigues, A. E. Dynamics of Adsorptive Reactors. I-Instantaneous Nonlinear Adsorption and Finite Zero Order Irreversible Reaction. Can. J. Chem. Eng. 1990, 68,127-138. Loureiro, J. M.; Costa, C. A.; Rodrigues, A. E. Propagation of Concentration Waves in Fixed Bed Adsorptive Reactors. Chem. Eng. J. 1983,27,135-148. Loureiro, J. M.; Rodrigues, A. E. Adsorptive Reactors. In Fundamentals of Adsorption; Mersmann, A. B., Scholl, S. E., Eds.; AIChE: New York, 1991. Magee, E. M. The Course of a Reaction in a Chromatographic Column. Ind. Eng. Chem. Fundam. 1963,2,32-36. Miller, K.Alternate Modes to Control the Nodes in the Moving Finite Element Method. In Adaptive Computational Methods for Partial Differential Equations; Babuska, I., Chandra, J., Flaherty, J. E., Eds.; SIAM: Philadelphia, 1983;pp 165-182. Miller, K.; Miller, R. N. Moving Finite Elements-Parts I and 11. SIAM J. Numer. Anal. 1981,18,1019-1057. Noyes, W. A. Summary Technical Report ofDivision 10;National Defense Research Committee: Washington, DC, 1946;Vol. 1. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids, 4th ed.; McGraw-Hill Book Company: New York, 1987. Gregg, S.J.; Sing, K. S. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. Schweich, D.; Villermaux, J.; Sardin, M. An Introduction to the Theory of Adsorptive Reactors. AZChE J. 1980,26,477-486. Sereno, C. A.; Villadsen, J.; Rodrigues, A. E. Solution of PDEs Systems by the Moving Finite Elements Method. Comput. Chem. Eng. 1992,6,583-592. Wakao, N.; Funazkri, T. Effect of Fluid dispersion Coefficientsand Particle-to-Fluid Mass Transfer Coefficients in Packed Beds. Chem. Eng. Sci. 1978,33,1375-1384. Yang, R. T. Gas Separation by Adsorption Processes; Butterworths: Boston, 1987. Yoon, Y. H.; Nelson, J. H. Applicationof Gas Adsorption Kinetics-I. A Theoretical Model for Respirator Cartridge Service Life. Amer. Znd. Hyg. Assoc. J. 1984,45,509-516.
Received for review November 15,1994 Revised manuscript received April 18,1995 Accepted April 26, 1995@
IE9406730
* Abstract published in Advance A C S Abstracts, July 1, 1995.
