- Email: [email protected]
Exact solution and linear driving force approximation for cyclic mass transfer in a bidisperse sorbent
Chemienl Engimming Science. Vol. 48, No. 9, pp. 1613-1618, Printed in Great Britain.
000%2509,93 $6.00 + 0.00 0 1993 Pergamon Prss Ltd
1993.
EXACT SOLUTION AND LINEAR DRIVING FORCE APPROXIMATION FOR CYCLIC MASS TRANSFER IN A BIDISPERSE SORBENT Department
GIORGIO CARTA of Chemical Engineering, University of Virginia, Charlottesville, VA 22903.2442, U.S.A. (Received
3 September
1992; accepted
forpublication
3 November
1992)
Abstract-A purely periodic analytic solution for cyclic mass transfer in an adsorbent pellet with a bidisperse pore structure is obtained. The solution is used to define the effective rate coefficient, K, for the linear driving force (LDF) approximation for cyclic adsorption and desorption. The results obtained differ from those previously derived using an equivalent solid diffusion model for the pellet with a single diffusional time constant, approaching this limit only when macropore diffusion is dominant. In fact, for rapid cycling the mass transfer rate cannot be expressed in the terms of a simple linear combination of the diffusional time constants for the bidisperse pellet independent of cycle time. Thus, in this case, the exact solution must be used to arrive at an appropriate definition of the K-values. With these K-values the LDF approximation is found to provide a reasonable representation of mass transfer rates in cyclic adsorption, which can he used for the si&lation of sorptive separations.
INTRODUCTION
of the LDF
rate coefficient to both the cycle time and the intraparticle diffusional time constant. More recently, Alpay and Scott (1992) and Carta (1993) have obtained analytical solutions that allow a direct calculation of the effective rate coefficient for cyclic mass transfer, also extending the solution to the case of unequal adsorption and desorption periods. Other approaches have also been considered. For example, Buzanowski and Yang (1991) introduced an “extended” LDF approximation for rapid cyclic operaobtained by matching the instantaneous tions, adsorption rates, rather than the total amounts
A description of mass transfer rates is often required for the modeling of sorptive separation processes. Intraparticle diffusion typically limits the overall transport rate in porous sorbents and, thus, an accurate representation requires, in principle, a description in terms of a particle diffusion model. Nevertheless, to diminish the complexity of the numerical calculations needed to simulate sorptive separation processes, simpler alternative descriptions, such as the linear driving force (LDF) approximation, are often used in practice. In this case, regardless of the exact nature of the mass transfer process, mass transfer rates are represented with an equivalent film resistance model, with an effective rate coefficient determined empirically in such a way as to provide an acceptably accurate description of the process (Ruthven, 1984; Yang, 1987). For linear adsorption equilibrium, the well-known Glueckauf LDF approximation (Glueckauf and Coates, 1947) provides an excellent description of mass transfer rates for the calculation of breakthrough curves in adsorption beds. The effective rate coefficient, in this case, can be shown to be dependent only on the diffusional time constants of the physical system. Nakao and Suzuki (1983), on the other hand, have shown that this approximation proves inadequate for the description of intraparticle mass transfer in cyclic adsorptionAesorption operations, indicating that the proper value of the effective rate coefficient should be taken as a function of the cycle time. Nakao and Suzuki solved independently the LDF and particle diffusion models for cyclic mass transfer in a single particle, and matched the two numerical solutions by determining the value of the effective rate coefficient that would give the same amount of adsorption with the two models. They presented an empirical correlation relating the value
adsorbed. These LDF approximations for cyclic mass transfer have been obtained under the assumption that intraparticle transport can be described with a solid diffusion model, with a single characteristic diffusional time constant. These solutions can be applied with a bidisperse pore structure, i.e. to those sorbents containing both macropores and microparticles such as zeolites, but only when macropore transport is the dominant mass transfer resistance. In this case, in fact, the macropore and the microparticles of which the sorbent pellet is composed, may be assumed to be in local equilibrium at each point within the pellet. As a result, intraparticle transport is again characterized by a single diffusional time constant and the LDF approximations apply. On the other hand, when the time constants for diffusion are similar for transport through the macropores and through the microparticles, the appropriate LDF approximation for cyclic adsorption may be quite different from those developed assuming solid diffusion. Since two comparable time constants are involved, we can expect that the effective rate coefficient would depend on both, as well as on the cycle time. In order to provide such a relationship, we present the time periodic solution of the complete
also to sorbents
1613
GIORGIO CARTA
1614
macropore-microparticle diffusion model for a single spherical pellet in a cyclic adsorptiondesorption operation with a linear isotherm. The purely periodic solution is then matched to the LDF approximation to obtain an analytic expression for the effective rate coefficient.
THEORETICAL
DEVELOPMENT
Cyclic adsorption and desorption is considered in a single spherical pellet with a bidisperse pore structure. The pellet is assumed to consist of spherical microparticles with interconnecting macropares. For a zeolite adsorbent, for example, the microparticle represent the zeolite crystals, and the macropores are those in the binder used in forming the pellet. We also assume that there is no external mass transfer resistance, that the pellet is isothermal, and that the adsorption equilibrium is described by a linear isotherm, 4 = mc, where m is the Henry’s law constant. Finally, we assume that the surface of the microparticles is in equilibrium with the macropore fluid at any point within the pellet. The dimensionless conservation equations and boundary conditions for the microparticles and the pellet are given by (Cen and Yang, 1986):
T2 = 3 *‘E”mT, %
(13)
In these equations, y and x are dimensionless microparticle and macropore concentrations, cr is a reference concentration, { and 5 are dimensionless radial coordinates for the microparticles and for the pellet, and 0 is a dimensionless time based on the diffusional time constant for macropore diffusion, D,/Rg. The parameters T1 and Tz depend upon the time constants for intraparticle diffusion, the pellet macroporosity, Ed, and the Henry’s law constant. x, is the dimensionless concentration at the external pellet surface. This is assumed to vary in a time periodic manner according to
xs(e) = 1 for 2jar < e c 2jkr + nr, x,(8) = 0
for 2jnr + nr, i
e < 2jzr + 2nr withj=0,1,2,...
where zr, = D&JR: and 2nr = D,t,/R$ are the dimensionless durations of the adsorption period and of the total cycle, respectively. Initial conditions are not needed since here we seek only the purely periodic solution. The Laplace transform of these equations and boundary conditions provides
For the microparticles:
p,T(g+;$)
(14)
(15)
(1) (16)
$p,O)= 0
(2) (3)
(18)
For the pellet:
g
(4)
1 1 - exp(n(1) = s 1 - exp(-
(5) (6) where
y’4 m=F
xc5 CF
nr,s) Znrs)
Solution of these ordinary differential equations yields the following expression for the intrapellet concentration profile: _
(7) (8)
(19)
(0) = 0
1 1 - exp(-
x=Sl
nr,,s) sinh (
-exp(-22rrrs)
(21)
and for the instantaneous flux at the pellet surface: 1 1 - exp(= S 1 - exp(-
(9)
nr,s) (J;; 211~s)
coth J;f
- 1)
(22) (10)
,=o,t R:
DC/r:
T, = __
WR:
(11) (12)
where rl=s+
T,(&coth&-I).
(23)
The time-domain solution is found by applying the residue theorem to the general inversion integral, as
Exact solution and linear driving force approximation shown, for example, by Carta (1988). For the dimensionless rate of adsorption or desorption, we obtain dX
de=
periodic solution of this equation yields (Carta, 1993) dJ _=__ de’
ax
3%
(>
Epf(l-E&l
ip
1615
2Kr’ s
@ sin*(knrb/2r’) c kZ, (Kr’)’ + k2
x[;cos($-$)
e=,
-Ksin(F-%)]
(24) where A
ak
=
k
sinh (2ak) + @ksin(28k) cash (2ak) -
r
k
sin
ak
=
(2ak)
-
COS (2pk) -
sinh c28k)
Bk
cash (2tck) -
1
COS (2/$‘)
1 1
1’2
a h p =
(af + b;)“* [
uk
Uk
2
k
l’*
sinh da
+ sin ,/‘m
cash ,/m
- cos ,/w
+ T,
sinh Jm
-
2’“Tl cash JklrT,
Ap _ 4Kr’ g sinz(kxri/2r’). R ,‘:I (Kr’)* + k*
(28) _ 1
sin Jw
(34)
This expression coincides with the result of Nakao and Suzuki (1983) for equal adsorption and desorption periods: 1 - exp (-
(271
(29) bk =;
where nr: and 2d are the dimensionless adsorption period and total cycle time based on the dimensionless time @. By integration one finds
(25) (26)
(33)
(A_Fhm= 1 + exp(-
Ksr’) Knr’)’
(35)
The effective mass transfer parameter, K, of the LDF approximation is found by matching eq. (34) [or eq. (35) for equal periods] to the exact solution for the macropore-microparticle diffusion model [eq. (3 l)]. In the case where macropore diffusion is dominant, the microparticles may be assumed to be in equilibrium with the macropore fluid at any point in the pellet. Then, eq. (4) may be replaced by
- cos ,/m’
(30) In eq. (24), X is the total amount of solute, adsorbed and held in the macropores, per unit pellet volume normalized with respect to the amount of solute in the pellet at equilibrium with the concentration cF prevailing during the adsorption period. The latter amount is given by [Ed + (1 - ep)m]cF. The dimensionless amount of solute adsorbed and desorbed during each cycle per unit pellet volume is found by integrating eq. (24) EP 7c &p+ (1 - EJrn
m sin’ (kxrJ2r) k=l
8’ =
+‘,I% pP + (1 - .zp)m t ’
kZ
A . ’
(31)
Next we consider the LDF approximation for cyclic mass transfer. Following Nakao and Suzuki (1983), we express the instantaneous rate of adsorption or desorption as
where J is the average concentration in the micropartitles, assumed uniform across the pellets, and 8’ = o’tfRz is a dimensionless time based on an appropriate diffusional time constant, D’/Rz. The purely
(37)
The periodic solution of eq. (36) can then be matched to eq. (34) or eq. (35), by taking the diffusional time constant DyRj, equal to q,D,/Ri [E, + (1 - &Jm]. In this case, the equivalence of the two solutions is expressed by AX=AJ
nradX -do=!? AX = 0 de s
xc
where
(38)
and the matching K-values are those given by Nakao and Suzuki (1983) for equal adsorption and desorption periods, or those calculated from the expressions given by Carta (1993) for unequal periods. In the more general case, when both macropore and microparticle diffusion are important, different values of K are obtained. As before, these may be calculated by equating the AX values obtained from the exact solution with those calculated with the LDF approximation, eq. (34) [or eq. (35)], with a diffusional time constant given by D’/Ri = E,,D,/R~ [E,, + (1 - E&I]. Since, by choosing this definition of dimensionless time, we are in fact basing the LDF approximation on the assumption of macropore control, we should find that the exact solution yields K-values that converge to those of Nakao and Suzuki (1983), when the microparticle resistance can indeed be neglected. From eqs
1616
GIORGIO CARTA
(31), (35), and (3&), for equal adsorption and desorption periods the correct K-values for a bidisperse pellet are given by K=
E,, +
(1 - ~,)m ln Ep7tl
(39)
where AX is calculated from eq. (31). For unequal periods, the K-values have to be determined by trial and error, since a compact expression for eq. (34) is not available. RESULTS
AND DISCtJSSION
The purely periodic solution of the macropore-microparticle diffusion model depends on the parameters T, and Tz. For zeolite adsorbents under practical conditions, Cen and Yang (1986) give typical values of the parameters in the ranges T, = 0.1-0.001,
m = lo-looo,
Fig. 2. Effect of cycle time on the effective rate coefficient for the LDF approximation. m = 10, e, = 0.1, xr = VW,.
Ep= 0.1-0.5.
We then obtain TZ = 0.03-3000. Calculations were carried out for parameter values covering this range. To ensure accuracy 1000 terms were used in the series solution, and some typical results are shown in Figs l-5. Figure 1 shows the calculated amount adsorbed per cycle relative to the amount adsorbed at equilibrium for m = 10 and sg = 0.1. For a given value of the ratio of diffusional time constants, T,, the amount adsorbed and desorbed per cycle increases with cycle time. For a given cycle time, however, as T, is reduced, the amount adsorbed decreases, because of the increased microparticle resistance. For very large values of T, and high cycle times, the curves calculated from the exact model approach the upper limit of macropore control. This, of course, coincides with the curve calculated from the LDF approximation assuming no microparticle resistance. However, as the cycle time is reduced while keeping T, relatively large, a deviation of the exact solution from the macropore control limit is seen. For T, = 0.01, for
4 , ,,,,,, .,,,,,, ,,,,,,, ,,,,,,,, , &ol ckcm 0.01 01 1
,,,,,”
,
10
.A
loo
O.COl
T1 Fig. 3. Effect of ratio of diffusionaltime constantson K and AX.m=10,~~=0.1,~r=~r,,=0.1.
I.
,‘:
-.
5
Y
-
e*sohlbo”
--.
Fig. 4. Effect of Henry’s law constant on K T, = 0.01, Ep = 0.1, Rr = nr, = 0.1.
Fig. 1. Effect of cycle time on the dimensionlessamount adsorbed and desorbed per cycle, normalized with the amount adsorbed at equilibrium. The line for macropore control is calculated from the LDF approximation. m = 10, Ep = 0.1, 11r = zr,.
and
AX.
example, we see that the exact solution approaches the macropore control limit at low cycling frequencies. At high cycling frequencies, on the other hand, lower AX-values are attained, corresponding to a significant microparticle resistance. For these high cycling frequencies, the depth of penetration of the concentration profiles is very small both within the pellet and the microparticles. The parameter T,, on the other hand, merely compares diffusional time con-
Exact solution and linear driving force approximation
3
0.3
I
ozO.l-
0.0-l 0
I 002
0.04
0.06
0.08
0.1
e
Fig. 5. Instantaneous rate of adsorption calculated from the diffusion model and the LDF approximation. T1 = 0.01, m = 10, ep = 0.1, nr = nr. = 0.1.
macropore-microparticle
stants using the radii of the pellet and the micropartitles as characteristic lengths. It is not surprising, then, that, for given values of m and Q,, the magnitude of T1 provides an unequivocal indication of the relative importance of macropore and microparticle resistances only in the case of low cycling frequencies, when the penetration of sorbate approaches the pellet and microparticle radii. At high cycling frequencies, T, provides inadequate information, since, as we can see from the exact solution, in this case microparticle diffusion may be important even at relatively high values of T,. Figure 1 also shows, for a comparison, the dimensionless amount adsorbed and desorbed per cycle calculated from the LDF approximation by assuming that microparticle diffusion is controlling for all cycle times. For this calculation we assumed that the dimensionless macropore concentration changes instantaneously from a value of one, during the adsorption period, to zero, during the desorption period. The average concentration within the microparticles was therrcomputed from the LDF approximation, by taking the dimensionless cycle time 2xr’ equal to ZnrT, . The K-values are again those given by Nakao and Suzuki (1983) for this dimensionless cycle time, and the dimensionless amount adsorbed in the micropartitles is given by eq. (35). Taking into account the amount of solute held in the macropores, it is easy to show that in this case AX =
Ed + (I - &P)mAB Ep + (1 - Ep)m .
(40)
The AX values calculated in this fashion are seen to approach those obtained from the exact solution, for conditions where the macropore resistance is dominant. Deviations occur, however, at high cycling frequencies. In this case, in fact, the amount of sorbate exchanged is very small and comparable to the holdup in the macropore. This was assumed to be exchanged instantaneously with the fluid outside the pellet in the calculations, resulting in an erroneous prediction when the cycle times are short.
1617
Figure 2 shows the calculated K-values obtained from the macropore-microparticle diffusion model. The calculations converge to the K-values of Nakao and Suzuki (1983) for macropore control when T1 is large and the cycling frequency is low. Lower Kvalues are obtained as T, is decreased, with an increasing departure from the upper limit of macropore control for lower cycling frequencies. The effects of T, and Henry’s law constant, m, are shown in Figs 3 and 4 for a fixed cycle time (TCT = 0.1). From Fig. 3 (m = lo), it is evident how, at high values of T1, the exact solution approaches the macropore control limit. This coincides with the curves obtained from the LDF approximation with .the K-values of Nakao and Suzuki (1983). At lower T,-values, however, microparticle diffusion becomes more important and lower values are obtained. The effects of Henry’s law constant are shown in Fig. 4 (T, = 0.01). In this case, as m is increased, K increases, remaining, however, always below the curve computed from the LDF approximation with the assumption of macropore diffusion control. The AX-values behave in a corresponding way, decreasing as m is increased, and lying below the values computed from the LDF approximation for macropore diffusion control. Figure 5 shows the instantaneous rate of adsorption computed from the exact solution for the macropore-microparticle diffusion model and from the LDF approximation with the correct K-value (K = 37.6 for this case). Since we have assumed equal adsorption and desorption periods for this calculation (Bra = ar = 0.1). the instantaneous adsorption and desorption rates are mirror images of each other in the two periods. The LDF approximation is seen to underpredict the rate for small times and to overpredict it for long times. By definition, on the other hand, the exact solution and the LDF approximation give exactly the same total amount adsorbed at the end of the period. This behavior is the same as that found with the LDF description in terms of solid diffusion, and derives from the inability of the film resistance model to correctly predict the instantaneous rate. As shown by Buzanowski and Yang (1991), the discrepancy in the instantaneous rates becomes worse as the cycle time is reduced. CONCLUSIONS We have obtained an exact purely periodic solution of a macropore-microparticle diffusion model for cyclic adsorption and desorption in a pellet with a bidisperse pore structure. Through this solution, we have obtained the effective rate coefficients for the LDF approximation, defined in such a way that the total amount adsorbed and desorbed per cycle is the same for this approximation and for the exact solution. Thus, with this definition, the LDF approximation does not provide an exact prediction of the instantaneous rates, but only of the average rate. This is often found to be acceptably accurate for the simulation of sorptive separations.
1618
GIORGIO
The K-values obtained in this work coincide with those obtained by Nakao and Suzuki (1983) when macropore diffusion controls. The conditions under which this occurs are determined not only by the magnitude of the ratio of diffusional time constants (diffusivity/radius square) for macropore and microparticle transport, but also by the cycle time. We note that, while in the simulation of breakthrough curves and chromatography the diffusion times for macropore and microparticle transport can be linearly combined into a single effective diffusional time constant, this is generally not possible for rapid cyclic adsorption. In this case, in fact the combination of the two diffusional time constants depends on the cycle time, as shown, for example, in Figs 1 and 2. NOTATION
constant defined by eq. (29) constant defined by eq. (30) fluid-phase concentration reference concentration effec_tivediffusion coefficient microparticle diffusion coefficient macropore diffusion coefficient Henry’s law adsorption constant microparticle concentration dimensionless cycle time ( = D, t,/R: 2x) dimensionless adsorption period ( = D,t,,/ R;N
microparticle radius radial coordinate in pellet pellet radius time adsorption time total cycle time ratio of diffusional time constants, eq. (12) parameter defined by eq. (13) dimensionless macropore concentration, eq. (8) value of x at pellet surface dimensionless amount in pellet per unit pellet volume dimensionless microparticle concentration, eq. (7) mean microparticle dimensionless concentration
CARTA
P Greek ak
Pk Tk
& 5 C z 8’ r P
mean microparticle concentration across pellet
averaged
letters
constant defined by eq. (27) constant defined by eq. (28) constant defined by eq. (26) constant defined by eq. (25) pellet macroporosity dimensionless radial coordinate in microparticles variable defined by eq. (23) dimensionless time based on macropore diffusion time constant, eq. (11) dimensionless time based on effective diffusion time constant, eq. (37) dimensionless radial coordinates in pellet radial coordinate in microparticles
REFERENCES
Alpay, E. and Scott, D. M., 1992, The linear driving force model for short-cycle adsorption and desorption in a spherical particle. Chem. Engng Sci. 47,499-502. Buzanowski, M. A. and Yang, R. T., 1991, Approximations for intraparticle diffusion rates in cyclic adsorption and desorption. Chem Engng Sci. 46, 2589-2598. Carta, G., 1988, Exact analytic solution of a mathematical model for chromatographic operations. Chem. Engng Sci. 43, 2877-2883. Carta, G., 1993, The linear driving force approximation for cvclic mass transfer in suherical uarticles. Chem. Enuno I I S;i. 48, 622-625. Cen, P. L. and Yang, R. T., 1986, Analytic solution for adsorber breakthr&h curves with bidisperse sorbents (zeolites). A.1.Ck.E. J. 32, 1635-1641. Glueckauf, E. and Coates, J. J., 1947, Theory of chromatography. Part IV. The influence of incomplete equilibrium on the front boundary of chromatograms and on the effectiveness of separation. J. ckem. Sot. pp. 1315-1321. Haynes, H. W., Jr. and Sarma, P. N., 1973, A model for the application of gas chromatography to measurements of diffusion in bidisperse structured catalysts. A.I.Ck.E. J. 19, 1043. Nakao, S. and Suzuki, M., 1983, Mass transfer coefficient in cyclic adsorption and desorption. J. them. Engng Japan
16, 114119. Ruthven, D. M., 1984, Principles
of Adsorption and Adsorption Processes. Wiley, New York. Yang, R. T., 1987, Gas Separation by Adsorption Processes. Butterworth, Boston, MA.
000%2509,93 $6.00 + 0.00 0 1993 Pergamon Prss Ltd
1993.
EXACT SOLUTION AND LINEAR DRIVING FORCE APPROXIMATION FOR CYCLIC MASS TRANSFER IN A BIDISPERSE SORBENT Department
GIORGIO CARTA of Chemical Engineering, University of Virginia, Charlottesville, VA 22903.2442, U.S.A. (Received
3 September
1992; accepted
forpublication
3 November
1992)
Abstract-A purely periodic analytic solution for cyclic mass transfer in an adsorbent pellet with a bidisperse pore structure is obtained. The solution is used to define the effective rate coefficient, K, for the linear driving force (LDF) approximation for cyclic adsorption and desorption. The results obtained differ from those previously derived using an equivalent solid diffusion model for the pellet with a single diffusional time constant, approaching this limit only when macropore diffusion is dominant. In fact, for rapid cycling the mass transfer rate cannot be expressed in the terms of a simple linear combination of the diffusional time constants for the bidisperse pellet independent of cycle time. Thus, in this case, the exact solution must be used to arrive at an appropriate definition of the K-values. With these K-values the LDF approximation is found to provide a reasonable representation of mass transfer rates in cyclic adsorption, which can he used for the si&lation of sorptive separations.
INTRODUCTION
of the LDF
rate coefficient to both the cycle time and the intraparticle diffusional time constant. More recently, Alpay and Scott (1992) and Carta (1993) have obtained analytical solutions that allow a direct calculation of the effective rate coefficient for cyclic mass transfer, also extending the solution to the case of unequal adsorption and desorption periods. Other approaches have also been considered. For example, Buzanowski and Yang (1991) introduced an “extended” LDF approximation for rapid cyclic operaobtained by matching the instantaneous tions, adsorption rates, rather than the total amounts
A description of mass transfer rates is often required for the modeling of sorptive separation processes. Intraparticle diffusion typically limits the overall transport rate in porous sorbents and, thus, an accurate representation requires, in principle, a description in terms of a particle diffusion model. Nevertheless, to diminish the complexity of the numerical calculations needed to simulate sorptive separation processes, simpler alternative descriptions, such as the linear driving force (LDF) approximation, are often used in practice. In this case, regardless of the exact nature of the mass transfer process, mass transfer rates are represented with an equivalent film resistance model, with an effective rate coefficient determined empirically in such a way as to provide an acceptably accurate description of the process (Ruthven, 1984; Yang, 1987). For linear adsorption equilibrium, the well-known Glueckauf LDF approximation (Glueckauf and Coates, 1947) provides an excellent description of mass transfer rates for the calculation of breakthrough curves in adsorption beds. The effective rate coefficient, in this case, can be shown to be dependent only on the diffusional time constants of the physical system. Nakao and Suzuki (1983), on the other hand, have shown that this approximation proves inadequate for the description of intraparticle mass transfer in cyclic adsorptionAesorption operations, indicating that the proper value of the effective rate coefficient should be taken as a function of the cycle time. Nakao and Suzuki solved independently the LDF and particle diffusion models for cyclic mass transfer in a single particle, and matched the two numerical solutions by determining the value of the effective rate coefficient that would give the same amount of adsorption with the two models. They presented an empirical correlation relating the value
adsorbed. These LDF approximations for cyclic mass transfer have been obtained under the assumption that intraparticle transport can be described with a solid diffusion model, with a single characteristic diffusional time constant. These solutions can be applied with a bidisperse pore structure, i.e. to those sorbents containing both macropores and microparticles such as zeolites, but only when macropore transport is the dominant mass transfer resistance. In this case, in fact, the macropore and the microparticles of which the sorbent pellet is composed, may be assumed to be in local equilibrium at each point within the pellet. As a result, intraparticle transport is again characterized by a single diffusional time constant and the LDF approximations apply. On the other hand, when the time constants for diffusion are similar for transport through the macropores and through the microparticles, the appropriate LDF approximation for cyclic adsorption may be quite different from those developed assuming solid diffusion. Since two comparable time constants are involved, we can expect that the effective rate coefficient would depend on both, as well as on the cycle time. In order to provide such a relationship, we present the time periodic solution of the complete
also to sorbents
1613
GIORGIO CARTA
1614
macropore-microparticle diffusion model for a single spherical pellet in a cyclic adsorptiondesorption operation with a linear isotherm. The purely periodic solution is then matched to the LDF approximation to obtain an analytic expression for the effective rate coefficient.
THEORETICAL
DEVELOPMENT
Cyclic adsorption and desorption is considered in a single spherical pellet with a bidisperse pore structure. The pellet is assumed to consist of spherical microparticles with interconnecting macropares. For a zeolite adsorbent, for example, the microparticle represent the zeolite crystals, and the macropores are those in the binder used in forming the pellet. We also assume that there is no external mass transfer resistance, that the pellet is isothermal, and that the adsorption equilibrium is described by a linear isotherm, 4 = mc, where m is the Henry’s law constant. Finally, we assume that the surface of the microparticles is in equilibrium with the macropore fluid at any point within the pellet. The dimensionless conservation equations and boundary conditions for the microparticles and the pellet are given by (Cen and Yang, 1986):
T2 = 3 *‘E”mT, %
(13)
In these equations, y and x are dimensionless microparticle and macropore concentrations, cr is a reference concentration, { and 5 are dimensionless radial coordinates for the microparticles and for the pellet, and 0 is a dimensionless time based on the diffusional time constant for macropore diffusion, D,/Rg. The parameters T1 and Tz depend upon the time constants for intraparticle diffusion, the pellet macroporosity, Ed, and the Henry’s law constant. x, is the dimensionless concentration at the external pellet surface. This is assumed to vary in a time periodic manner according to
xs(e) = 1 for 2jar < e c 2jkr + nr, x,(8) = 0
for 2jnr + nr, i
e < 2jzr + 2nr withj=0,1,2,...
where zr, = D&JR: and 2nr = D,t,/R$ are the dimensionless durations of the adsorption period and of the total cycle, respectively. Initial conditions are not needed since here we seek only the purely periodic solution. The Laplace transform of these equations and boundary conditions provides
For the microparticles:
p,T(g+;$)
(14)
(15)
(1) (16)
$p,O)= 0
(2) (3)
(18)
For the pellet:
g
(4)
1 1 - exp(n(1) = s 1 - exp(-
(5) (6) where
y’4 m=F
xc5 CF
nr,s) Znrs)
Solution of these ordinary differential equations yields the following expression for the intrapellet concentration profile: _
(7) (8)
(19)
(0) = 0
1 1 - exp(-
x=Sl
nr,,s) sinh (
-exp(-22rrrs)
(21)
and for the instantaneous flux at the pellet surface: 1 1 - exp(= S 1 - exp(-
(9)
nr,s) (J;; 211~s)
coth J;f
- 1)
(22) (10)
,=o,t R:
DC/r:
T, = __
WR:
(11) (12)
where rl=s+
T,(&coth&-I).
(23)
The time-domain solution is found by applying the residue theorem to the general inversion integral, as
Exact solution and linear driving force approximation shown, for example, by Carta (1988). For the dimensionless rate of adsorption or desorption, we obtain dX
de=
periodic solution of this equation yields (Carta, 1993) dJ _=__ de’
ax
3%
(>
Epf(l-E&l
ip
1615
2Kr’ s
@ sin*(knrb/2r’) c kZ, (Kr’)’ + k2
x[;cos($-$)
e=,
-Ksin(F-%)]
(24) where A
ak
=
k
sinh (2ak) + @ksin(28k) cash (2ak) -
r
k
sin
ak
=
(2ak)
-
COS (2pk) -
sinh c28k)
Bk
cash (2tck) -
1
COS (2/$‘)
1 1
1’2
a h p =
(af + b;)“* [
uk
Uk
2
k
l’*
sinh da
+ sin ,/‘m
cash ,/m
- cos ,/w
+ T,
sinh Jm
-
2’“Tl cash JklrT,
Ap _ 4Kr’ g sinz(kxri/2r’). R ,‘:I (Kr’)* + k*
(28) _ 1
sin Jw
(34)
This expression coincides with the result of Nakao and Suzuki (1983) for equal adsorption and desorption periods: 1 - exp (-
(271
(29) bk =;
where nr: and 2d are the dimensionless adsorption period and total cycle time based on the dimensionless time @. By integration one finds
(25) (26)
(33)
(A_Fhm= 1 + exp(-
Ksr’) Knr’)’
(35)
The effective mass transfer parameter, K, of the LDF approximation is found by matching eq. (34) [or eq. (35) for equal periods] to the exact solution for the macropore-microparticle diffusion model [eq. (3 l)]. In the case where macropore diffusion is dominant, the microparticles may be assumed to be in equilibrium with the macropore fluid at any point in the pellet. Then, eq. (4) may be replaced by
- cos ,/m’
(30) In eq. (24), X is the total amount of solute, adsorbed and held in the macropores, per unit pellet volume normalized with respect to the amount of solute in the pellet at equilibrium with the concentration cF prevailing during the adsorption period. The latter amount is given by [Ed + (1 - ep)m]cF. The dimensionless amount of solute adsorbed and desorbed during each cycle per unit pellet volume is found by integrating eq. (24) EP 7c &p+ (1 - EJrn
m sin’ (kxrJ2r) k=l
8’ =
+‘,I% pP + (1 - .zp)m t ’
kZ
A . ’
(31)
Next we consider the LDF approximation for cyclic mass transfer. Following Nakao and Suzuki (1983), we express the instantaneous rate of adsorption or desorption as
where J is the average concentration in the micropartitles, assumed uniform across the pellets, and 8’ = o’tfRz is a dimensionless time based on an appropriate diffusional time constant, D’/Rz. The purely
(37)
The periodic solution of eq. (36) can then be matched to eq. (34) or eq. (35), by taking the diffusional time constant DyRj, equal to q,D,/Ri [E, + (1 - &Jm]. In this case, the equivalence of the two solutions is expressed by AX=AJ
nradX -do=!? AX = 0 de s
xc
where
(38)
and the matching K-values are those given by Nakao and Suzuki (1983) for equal adsorption and desorption periods, or those calculated from the expressions given by Carta (1993) for unequal periods. In the more general case, when both macropore and microparticle diffusion are important, different values of K are obtained. As before, these may be calculated by equating the AX values obtained from the exact solution with those calculated with the LDF approximation, eq. (34) [or eq. (35)], with a diffusional time constant given by D’/Ri = E,,D,/R~ [E,, + (1 - E&I]. Since, by choosing this definition of dimensionless time, we are in fact basing the LDF approximation on the assumption of macropore control, we should find that the exact solution yields K-values that converge to those of Nakao and Suzuki (1983), when the microparticle resistance can indeed be neglected. From eqs
1616
GIORGIO CARTA
(31), (35), and (3&), for equal adsorption and desorption periods the correct K-values for a bidisperse pellet are given by K=
E,, +
(1 - ~,)m ln Ep7tl
(39)
where AX is calculated from eq. (31). For unequal periods, the K-values have to be determined by trial and error, since a compact expression for eq. (34) is not available. RESULTS
AND DISCtJSSION
The purely periodic solution of the macropore-microparticle diffusion model depends on the parameters T, and Tz. For zeolite adsorbents under practical conditions, Cen and Yang (1986) give typical values of the parameters in the ranges T, = 0.1-0.001,
m = lo-looo,
Fig. 2. Effect of cycle time on the effective rate coefficient for the LDF approximation. m = 10, e, = 0.1, xr = VW,.
Ep= 0.1-0.5.
We then obtain TZ = 0.03-3000. Calculations were carried out for parameter values covering this range. To ensure accuracy 1000 terms were used in the series solution, and some typical results are shown in Figs l-5. Figure 1 shows the calculated amount adsorbed per cycle relative to the amount adsorbed at equilibrium for m = 10 and sg = 0.1. For a given value of the ratio of diffusional time constants, T,, the amount adsorbed and desorbed per cycle increases with cycle time. For a given cycle time, however, as T, is reduced, the amount adsorbed decreases, because of the increased microparticle resistance. For very large values of T, and high cycle times, the curves calculated from the exact model approach the upper limit of macropore control. This, of course, coincides with the curve calculated from the LDF approximation assuming no microparticle resistance. However, as the cycle time is reduced while keeping T, relatively large, a deviation of the exact solution from the macropore control limit is seen. For T, = 0.01, for
4 , ,,,,,, .,,,,,, ,,,,,,, ,,,,,,,, , &ol ckcm 0.01 01 1
,,,,,”
,
10
.A
loo
O.COl
T1 Fig. 3. Effect of ratio of diffusionaltime constantson K and AX.m=10,~~=0.1,~r=~r,,=0.1.
I.
,‘:
-.
5
Y
-
e*sohlbo”
--.
Fig. 4. Effect of Henry’s law constant on K T, = 0.01, Ep = 0.1, Rr = nr, = 0.1.
Fig. 1. Effect of cycle time on the dimensionlessamount adsorbed and desorbed per cycle, normalized with the amount adsorbed at equilibrium. The line for macropore control is calculated from the LDF approximation. m = 10, Ep = 0.1, 11r = zr,.
and
AX.
example, we see that the exact solution approaches the macropore control limit at low cycling frequencies. At high cycling frequencies, on the other hand, lower AX-values are attained, corresponding to a significant microparticle resistance. For these high cycling frequencies, the depth of penetration of the concentration profiles is very small both within the pellet and the microparticles. The parameter T,, on the other hand, merely compares diffusional time con-
Exact solution and linear driving force approximation
3
0.3
I
ozO.l-
0.0-l 0
I 002
0.04
0.06
0.08
0.1
e
Fig. 5. Instantaneous rate of adsorption calculated from the diffusion model and the LDF approximation. T1 = 0.01, m = 10, ep = 0.1, nr = nr. = 0.1.
macropore-microparticle
stants using the radii of the pellet and the micropartitles as characteristic lengths. It is not surprising, then, that, for given values of m and Q,, the magnitude of T1 provides an unequivocal indication of the relative importance of macropore and microparticle resistances only in the case of low cycling frequencies, when the penetration of sorbate approaches the pellet and microparticle radii. At high cycling frequencies, T, provides inadequate information, since, as we can see from the exact solution, in this case microparticle diffusion may be important even at relatively high values of T,. Figure 1 also shows, for a comparison, the dimensionless amount adsorbed and desorbed per cycle calculated from the LDF approximation by assuming that microparticle diffusion is controlling for all cycle times. For this calculation we assumed that the dimensionless macropore concentration changes instantaneously from a value of one, during the adsorption period, to zero, during the desorption period. The average concentration within the microparticles was therrcomputed from the LDF approximation, by taking the dimensionless cycle time 2xr’ equal to ZnrT, . The K-values are again those given by Nakao and Suzuki (1983) for this dimensionless cycle time, and the dimensionless amount adsorbed in the micropartitles is given by eq. (35). Taking into account the amount of solute held in the macropores, it is easy to show that in this case AX =
Ed + (I - &P)mAB Ep + (1 - Ep)m .
(40)
The AX values calculated in this fashion are seen to approach those obtained from the exact solution, for conditions where the macropore resistance is dominant. Deviations occur, however, at high cycling frequencies. In this case, in fact, the amount of sorbate exchanged is very small and comparable to the holdup in the macropore. This was assumed to be exchanged instantaneously with the fluid outside the pellet in the calculations, resulting in an erroneous prediction when the cycle times are short.
1617
Figure 2 shows the calculated K-values obtained from the macropore-microparticle diffusion model. The calculations converge to the K-values of Nakao and Suzuki (1983) for macropore control when T1 is large and the cycling frequency is low. Lower Kvalues are obtained as T, is decreased, with an increasing departure from the upper limit of macropore control for lower cycling frequencies. The effects of T, and Henry’s law constant, m, are shown in Figs 3 and 4 for a fixed cycle time (TCT = 0.1). From Fig. 3 (m = lo), it is evident how, at high values of T1, the exact solution approaches the macropore control limit. This coincides with the curves obtained from the LDF approximation with .the K-values of Nakao and Suzuki (1983). At lower T,-values, however, microparticle diffusion becomes more important and lower values are obtained. The effects of Henry’s law constant are shown in Fig. 4 (T, = 0.01). In this case, as m is increased, K increases, remaining, however, always below the curve computed from the LDF approximation with the assumption of macropore diffusion control. The AX-values behave in a corresponding way, decreasing as m is increased, and lying below the values computed from the LDF approximation for macropore diffusion control. Figure 5 shows the instantaneous rate of adsorption computed from the exact solution for the macropore-microparticle diffusion model and from the LDF approximation with the correct K-value (K = 37.6 for this case). Since we have assumed equal adsorption and desorption periods for this calculation (Bra = ar = 0.1). the instantaneous adsorption and desorption rates are mirror images of each other in the two periods. The LDF approximation is seen to underpredict the rate for small times and to overpredict it for long times. By definition, on the other hand, the exact solution and the LDF approximation give exactly the same total amount adsorbed at the end of the period. This behavior is the same as that found with the LDF description in terms of solid diffusion, and derives from the inability of the film resistance model to correctly predict the instantaneous rate. As shown by Buzanowski and Yang (1991), the discrepancy in the instantaneous rates becomes worse as the cycle time is reduced. CONCLUSIONS We have obtained an exact purely periodic solution of a macropore-microparticle diffusion model for cyclic adsorption and desorption in a pellet with a bidisperse pore structure. Through this solution, we have obtained the effective rate coefficients for the LDF approximation, defined in such a way that the total amount adsorbed and desorbed per cycle is the same for this approximation and for the exact solution. Thus, with this definition, the LDF approximation does not provide an exact prediction of the instantaneous rates, but only of the average rate. This is often found to be acceptably accurate for the simulation of sorptive separations.
1618
GIORGIO
The K-values obtained in this work coincide with those obtained by Nakao and Suzuki (1983) when macropore diffusion controls. The conditions under which this occurs are determined not only by the magnitude of the ratio of diffusional time constants (diffusivity/radius square) for macropore and microparticle transport, but also by the cycle time. We note that, while in the simulation of breakthrough curves and chromatography the diffusion times for macropore and microparticle transport can be linearly combined into a single effective diffusional time constant, this is generally not possible for rapid cyclic adsorption. In this case, in fact the combination of the two diffusional time constants depends on the cycle time, as shown, for example, in Figs 1 and 2. NOTATION
constant defined by eq. (29) constant defined by eq. (30) fluid-phase concentration reference concentration effec_tivediffusion coefficient microparticle diffusion coefficient macropore diffusion coefficient Henry’s law adsorption constant microparticle concentration dimensionless cycle time ( = D, t,/R: 2x) dimensionless adsorption period ( = D,t,,/ R;N
microparticle radius radial coordinate in pellet pellet radius time adsorption time total cycle time ratio of diffusional time constants, eq. (12) parameter defined by eq. (13) dimensionless macropore concentration, eq. (8) value of x at pellet surface dimensionless amount in pellet per unit pellet volume dimensionless microparticle concentration, eq. (7) mean microparticle dimensionless concentration
CARTA
P Greek ak
Pk Tk
& 5 C z 8’ r P
mean microparticle concentration across pellet
averaged
letters
constant defined by eq. (27) constant defined by eq. (28) constant defined by eq. (26) constant defined by eq. (25) pellet macroporosity dimensionless radial coordinate in microparticles variable defined by eq. (23) dimensionless time based on macropore diffusion time constant, eq. (11) dimensionless time based on effective diffusion time constant, eq. (37) dimensionless radial coordinates in pellet radial coordinate in microparticles
REFERENCES
Alpay, E. and Scott, D. M., 1992, The linear driving force model for short-cycle adsorption and desorption in a spherical particle. Chem. Engng Sci. 47,499-502. Buzanowski, M. A. and Yang, R. T., 1991, Approximations for intraparticle diffusion rates in cyclic adsorption and desorption. Chem Engng Sci. 46, 2589-2598. Carta, G., 1988, Exact analytic solution of a mathematical model for chromatographic operations. Chem. Engng Sci. 43, 2877-2883. Carta, G., 1993, The linear driving force approximation for cvclic mass transfer in suherical uarticles. Chem. Enuno I I S;i. 48, 622-625. Cen, P. L. and Yang, R. T., 1986, Analytic solution for adsorber breakthr&h curves with bidisperse sorbents (zeolites). A.1.Ck.E. J. 32, 1635-1641. Glueckauf, E. and Coates, J. J., 1947, Theory of chromatography. Part IV. The influence of incomplete equilibrium on the front boundary of chromatograms and on the effectiveness of separation. J. ckem. Sot. pp. 1315-1321. Haynes, H. W., Jr. and Sarma, P. N., 1973, A model for the application of gas chromatography to measurements of diffusion in bidisperse structured catalysts. A.I.Ck.E. J. 19, 1043. Nakao, S. and Suzuki, M., 1983, Mass transfer coefficient in cyclic adsorption and desorption. J. them. Engng Japan
16, 114119. Ruthven, D. M., 1984, Principles
of Adsorption and Adsorption Processes. Wiley, New York. Yang, R. T., 1987, Gas Separation by Adsorption Processes. Butterworth, Boston, MA.