2740
Ind. Eng. Chem. Res. 1993,32, 274Q-2751
Pressure Swing Adsorption Processes: Intraparticle Diffusion/ Convection Models Zuping Lu, Jose M. Loureiro, M. Douglas Levan,+and Alfrio E. Rodrigues' Laboratory of Separation and Reaction Engineering, School of Engineering, University of Porto, 4099 Porto Codex, Portugal
The dynamic behavior of a three-step one-column isothermal PSA process is assessed by simulation; three models are used: equilibrium, intraparticle diffusion, and intraparticle diffusion/ convection. Two process performance parameters, product enrichment and recovery of the light component, are used t o measure the separation performance. The effects of several operating variables on the process performance are addressed. Higher pressure ratios (%/PI) and higher adsorption capacities increase the process performance. The process performance is practically independent of the step rates, although it can decrease if high feed rates are used together with large feed duration times. The limiting performance of the system is found in the absence of mass-transfer resistances. Intraparticle convection, enhancing mass transfer inside particles, increases the process performance relative to intraparticle diffusion alone; the improvement is limited by the equilibrium situation.
Introduction The growing application of pressure swing adsorption
(PSA)processes calls for a better understanding of process dynamics. An interesting topic in PSA is rapid cycling with large pressure gradients in the bed (Jones and Keller, 1981). However,difficulties arise in simulating the process. Frozen solid phase or instantaneous gas/solid equilibrium assumptions have been made for mathematical simulation of the pressure-changing steps, i.e., pressurization and blowdown (Chihara and Suzuki, 1983; Fernandez and Kenney, 1983; Richter et al., 1982). Unfortunately, these assumptions do not provide a good understanding of PSA processes. Preliminary theoretical and experimental studies on the effect of pressure gradients inside the bed on process dynamics in pressurization and blowdown have been carried out with the assumption of isothermal operation and local gas/solid adsorption equilibrium (Sundaram and Wankat, 1988;Hart et al., 1990;Rodrigues et al., 1991a; Scott, 1991; Lu et al., 1993a) and with consideration of intraparticle mass-transfer resistances in simulating the bed dynamics (Zhong et al., 1992;Lu et al., 1992a,b). The temperature variations due to the exothermic/endothermic nature of adsorption/desorption in pressurization and blowdown have also been taken into account in the simulations (Zhong et al., 1992; Lu et al., 1992~). The one-column pressure swing adsorption process was invented by Turnock and Kadlec (1971) and Kowler and Kadlec (1972) for the separation of nitrogen and methane mixtures. A substantial refinement of this process was made by Jones and Keller (1981) for commercial oxygen production from air. Cheng and Hill (1985) developed a modified well-stirred cell local equilibrium model to simulate the separation performance of a helium and methane mixture by a three-step one-column pressure swing adsorption process, and they concluded that the discrepancy between model predictions and experimental results was due to flow resistances and heat release that were not taken into account in their model. In one of our previous works (Lu et al., 1993b), separation of a binary mixture (one inert and one adsorbable species) by this process had been simulated with the consideration of
* To whom correspondence should be addressed.
University of Virginia, Department of Chemical Engineering, Charlottesville,VA 22903-2442. t
0888-5885/93/2632-2740$04.00/0
pressure drop and temperature variations inside the bed by use of the local gas/solid equilibrium model. The effects of the nature of the adsorption isotherm, adsorption capacity of the adsorbent, feed composition, incomplete pressurization and blowdown,"dead" volumes at both sides of the bed, and temperature variations on the separation performance have been studied. The bed dynamics in the transient regime and at the cyclic steady state in the three steps have also been addressed. As noted by Farooq et al. (19881, it was found that many cycles were needed to build up the temperature profiles inside an adiabatic bed, as a result of the high ratio between the heat capacities for the solid and fluid phases. The local equilibrium model is widely used to gain insight into PSA processes due to ita simplicity. However, the mass-transfer resistances must be considered due to the high fluid velocity with the need of high mass transport between adsorbents and fluid in many PSA processes, particularly when very short cycle times are used. The pore/surface diffusion and linear driving force models have been developed and used to simulate the PSA processes (Yang and Doong, 1985; Doong and Yang, 1986; Hassan et al., 1987; Shin and Knaebel, 1987; Farooq et al., 1989; Farooq and Ruthven, 1991). But in all of these works, the pressure gradient inside the bed was assumed to be negligible. In some PSA processes, mass transport through the macropores is the controlling step (Doongand Yang, 1986; Cen and Yang, 1986;Yang, 1987;Rodrigues et al., 1992a,b), particularly when a high-pressure ratio (%/PI)is used (Ruthven, 1992). In these cases the enhancement of intraparticle mass transport due to intraparticle forced convective flow should be considered when "large-pore" adsorbents are used in the presence of large pressure gradients inside the bed (Rodrigues et al., 1991b). The improved bed dynamics have been shown by Lu et al. (1992a-c) when consideringintraparticle forced convective flow, and the enhanced separation performance of a threestep one-column pressure swing adsorption process due to intraparticle convection has been assessed by Rodrigues et al. (1992a,b). Intensification of sorption processes obtained by either reducing particle sizes or introducing large pore packings has been discussed by Lu and Rodrigues (1993). In this paper three different models, equilibrium, intraparticle diffusion and intraparticle diffusion/convection, will be used to address the following 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2741 aspects: (1)simulation of a three-step one-column pressure swing adsorption process in order to assess the improvement of process separation performance due to intraparticle forced convective flow; (2) effects of pressurization, feed and blowdown rates, pressure ratios, and adsorption capacities on process separation performance; (3) bed dynamics in the transient regime and a t the cyclic steady State.
Process Description The PSA process to be modeled is the same as that used in a previous paper (Lu et al., 1993b) in which an equilibrium model was developed for a three-step onecolumn pressure swing adsorption process. A binary mixture to be separated consists of one inert and one adsorbable species. A cycle of the process is composed of pressurization from low pressure to high pressure with feed, feed at high pressure with withdrawal of a light product (inert) from the product end, and blowdown of the column to low pressure a t the feed end of the bed. An enriched inert species is obtained as a product during the high-pressure feed step at the product end of the bed, and an enriched adsorbable species is obtained during the blowdown step at the feed end of the bed. Pressurization, feed, and blowdown rates of the bed are controlled by valves and measured by the parameters Mp,ur*, and hfb in the model equations, respectively. Generally, for a given separation system and a given packed adsorbent, the process operation can be selected in the local gaslsolid equilibrium region by using very small fluid velocity, Le., having a very small CY (reciprocal of the number of intraparticle diffusion units; Rodrigues et al., 1991b), which results in a better separation performance but lower adsorbent productivity. However, PSA processes strive for higher adsorbent productivity. High fluid velocity in the bed has an upper limit for the given PSA system, beyond which the requested product purity cannot be obtained for a given recovery. This limit is dependent on the parameter CY, which means the limit is related to the particle diameter and intraparticle effective diffusivity of the components. There is an optimization problem in choosing the operating parameters in each process step to reach the best combination of the adsorbent productivity and product purity for a requested product recovery in a given system. This must be based on the good understanding of the process dynamics. Jones and Keller (1981) have pointed out that optimum bed design and process cycle occur just where intraparticle mass transfer begins to be significant, and a predictive mathematical model must include mass transfer.
Mathematical Models An intraparticle diffusionlconvection model, which includes the adsorbable species and overall mass balances, and momentum balances inside particles and in the bed, was constructed for an isothermal bed with the following assumptions: (i) the flow pattern in the bed can be described by the axial dispersion plug flow model; (ii) the bulk fluid superficial velocity is described locally by Ergun's relation and the intraparticle forced convective flow follows Darcy's law; (iii) intraparticle mass transfer is assumed to be due to intraparticle diffusion and convection in the macropores; therefore, micropore and external film mass-transfer resistances are assumed to be negligible; (iv) the adsorption equilibrium is described by a Langmuir isotherm.
We introduce the variables f = tic,, x = Z I L , in the bed and
e = tiT0, u*
= Uiu,
= zfii, e = tiTo, u* = vivo inside particles, where uo is the reference superficial intraparticle velocity, uois the reference superficial fluid velocity, and r0 is the reference space time. The mathematical model is then as follows: (i) Mass balances inside adsorbent particles
p = C'ic,,
species mass balance
overall mass balance
where the adsorption particle capacity factor [ is
(ii) Momentum equation inside particles (Darcy's law)
u x
u* = U, = (iii) Mass balances in the bed
overall mass balance
where the dimensionless fluxes of species i and overall from bulk fluid to the particle are given by the following relations.
(iv) Momentum equation in the bulk fluid (Ergun equation)
2742 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 Table I. Dimensionless Parameters of the Models
- 2 = bSu*+ b&*’ ax where the coefficients of the Ergun equation are respectively 1.75pg,(l - c)L 150p(1- e)’L b, = u, and b, = UO d,2e3P0 P,d,e3 (1la) (v) Adsorption isotherm
cp12
%=--
Demo
uo
J, Dmo
-
b,
Dko
k’ = 1 + kc,
The dimensionless parameters in the above equations are listed in Table I. The PSA cycle usually starts with the pressurization step, and two initial conditions, feed-saturated and clean bed, were used in the simulations. e=o, y i = y f ; f = f l ; o < x i i (13a) and e=o, y i = o ; f = f , ; o < x i i (13b) Boundary conditions of the bulk fluid in the bed for cyclic operation are (a) pressurization
Demo
[ = -1 - BP tp
_-
1 --=J), Pe Lu
k’q* [1+ (k’- l)@f’y{l2 b,
+
b2
Po L fi, = -P o 1 fiR
u*f
--!&!E
1
= U,
(b) the dimensionless productivity normalized by the reference mole flux u,,Po
Nprd= LJefu*f(outlet) d8
4
O
where et = 8, + 8f + ob, et, 8,, Of, and 8b are cycle, pressurization, feed, and blowdown times, respectively. (c) the recovery of the light component in the product stream RE = EN X CUT (22) where the product cut (CUT) is the total amount of product stream during the feed step over the total amount input into the bed during pressurization and feed steps in a cycle.
(b) feed
Simulation Results and Discussion x=l, f=fe;
aY.
’=o ax
(17)
(c) blowdown
where the parameters Mp and Mb determine the pressurization and blowdown rates. The limiting cases are as follows: (a) If X,= 0, then the intraparticle diffusion/convection model reduces to an intraparticle diffusion model. (b) If there are no masstransfer resistances inside particles, i.e., a, = 0, the model equations reduce to the equilibrium model.
Process Performance The process performance under cyclic steady state can be described by the following performance parameters as defined in a previous paper (Lu et al., 1993b): (a) the average enrichment of the light component in the product stream EN=-
v)p
1-9, 1-Yf
(20)
where is the average mole fraction of the adsorbable species in the product stream.
Model equations inside the particle were first reduced to first-order partial differential equations on the independent variables x and 8 using orthogonal collocation on finite elements in the p coordinate; four elements were used with two collocation points in each element. The partial differential equations inside particles coupled with the partial differential equations in the bulk fluid were then solved by the commercial PDECOL package; 10 elements were used in the axial direction. The maximum CPU time for one cycle under steady state (three-step) is around 10 min on an IBM RS/6000-530 computer. The effect of particle permeability, particle size, pressurization, feed and blowdown rates, adsorption capacity of adsorbent, and pressure ratio on the process performance will be studied, as well as the bed dynamics using the three models: intraparticle diffusion/convection, intraparticle diffusion, and equilibrium. The reference case presented in this paper is based on the followingparameter values: q* = 20, k’ 2, f h = 5, f e = 4.5, fi = 1, Mp = 5, Mb = 5, u,* = 0.43, T , = 0.225 s, d , = 0.1 em, B = 0 (for intraparticle diffusion alone) and 1.25 X 10-B cm2 (for intraparticle diffusionlconvection) 8, = 3, 8b = 10, and Of varies with the recovery RE. Some values of system properties, operating variables, and constants used in simulations are listed in Table 11. Intraparticle Diffusion/Convection and Equilibrium Models. In modeling adsorption bed operations, the equilibrium model is a limiting case corresponding to the absence of intraparticle mass-transfer resistances;when intraparticle mass-transfer resistances are present, the
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2743 Table 11. Values of System Properties, Operating Variables, and Constants Used in Simulations hiah-Dressure P h = 5 and 10 bar gas mixture density pe0 = 10-9 g/cm3 low-pressure PI 1bar molecular diffusivity D,, = 0.625 cm2/s column length L = 100 cm gas viscosity p = 1O-Cg/s.cm interparticle porosity t = 0.43 particle tortuosity T,* = 10 intraparticle porosity e, = 0.595 particle permeability B, = 0,1.25 X lW, and 5 X 1W cm2 particle diameter d , = 0.05 and 0.1 cm reference pressure Po = 1bar gaa mixture composition yf = 0.5 constants y1= 20 and yz = 0.5 initial mole fraction in the bed yo = 0 and 0.5
0.43, T~ = 0.225s, d, = 0.1 cm, andB, = 0,1.25 X 10-9, and 5X cm2. As expected, the separation performance of '0 the diffusion-controlled and the equilibrium-dominated *.. '\. 1.9 ,d = 0.05 cm cases locate the two limits, and intraparticle convection improves the performance from the diffusion to the ....._t, 0. .......... 1.8 equilibrium limits with increasing particle permeability. The simulation results with a smaller particle size, d, = EN . 0.05 cm, by using the intraparticle diffusion model, are 1.7 also shown in the figure for the above values except T~ = 5 . 0 e - 9 J/ 0.335 s and ur* = 0.4. Higher product purity can be ..... \ 1.25e-9 .... 1.6 obtained using a smaller particle size; however, a decrease of the productivity of the adsorbent is observed due to the increase of the cycle time. It must be pointed out that the 1.5 reduction of intraparticle mass-transfer resistances due 0.2 0.4 0.6 0.8 to intraparticle forced convective flow is based on two RE necessary conditions: large pore materials and large Figure 1. Effects of particle size and particle permeability on the pressure gradients across them. Due to the coexistence enrichment/recovery relation for the intraparticle diffusion and equiintraparticle diffusion/convection models, respectively: 0, of small pressure drop regions (the closed end of the bed) , large pressure drop regions (the open end of the bed) libriummodel; -O-, intraparticle diffusion model (d, = 0.1 cm); -.and intraparticle diffusion model ( d , = 0.05 cm); -0-, intraparticle in the PSA process, the simulation results of the PSA diffusion/convection model (d, = 0.1 cm). cycles lead to the conclusion that the advantages due to intraparticle forced convective flow cannot be explained intraparticle diffusion model is normally used to simulate only on the basis of the enhancement of intraparticle mass the behavior of the system. The importance of the transport by convection as was the case in a classic intraparticle mass-transfer resistance is measured by the adsorption bed operation (Rodrigues et al., 1991b). reciprocal of the number of intraparticle diffusion units Moreover, the global results of the intraparticle diffusion/ (Rodrigues et al., 1991b). If large pressure gradients exist convection model cannot be simulated simply by the along the bed, when large-pore adsorbents are used, a Giffusion model with an apparent augmentedjiffusivity convective flow, measured by the intraparticle Peclet De,or by the introduction of a new parameter cyo, as when number A, develops inside the particles which is superthe particle size is changed. However, the approach of imposed to the diffusive flow, enhancing the intraparticle the equilibrium separation performance either by increasmass transport relative to diffusion alone (Rodrigues et ing the particle permeability or by reducing the particle al., 1991b). size can still be clearly seen. A better efficiency for the adsorption (pressurization) (ii) Effect of Pressurization, Feed, and Blowdown and desorption (blowdown)due to intraparticle convection Rates. Different pressurization, feed, and blowdown rates was assessed by using the intraparticle diffusion/convecmean different fluid velocities which nearly do not tion models in the isothermal and adiabatic cases (Lu et influence the separation performance, i.e., the enrichment/ al., 1992a-c). The numerical results also show that an recovery (EN-RE) relation, of the PSA cycle when local increase in particle permeability (increase of intraparticle gadsolid equilibrium dominates (Cheng and Hill, 1985; forced convective flow) can be equivalent to a decrease in Lu et al., 1993b). I t is known that the fluid velocity does particle size in reducing intraparticle mass-transfer renot change the nature of the breakthrough curve in an sistances (Lu et al., 1992b). The improvement of the adsorption fixed-bed operation if mass-transfer resistances separation performance of a PSA process due to intraand axial dispersion are not important. However, when particle convection has been studied briefly by Rodrigues intraparticle mass-transfer resistances are present, the et al. (1992a,b). reciprocal of the number of intraparticle diffusion units (i) Effect of the Particle Permeability Bp and is proportional to the fluid velocity, which directly affects Particle Size dp. The comparisons of the separation the output of an adsorption bed (Rodrigues et al., 1991b). performance using the three models are based on the same The bed dynamics in the cyclic steady-state operation pressurization and blowdown rates and times, and comof PSA processes are different from that of a classic plete pressurization and blowdown. In previous work (Lu adsorption fixed bed. The concentration front of the et al., 1993b), it has been shown that the enrichment/ adsorbable species is substantially far from the production product cut relation (EN-CUT) is almost linear in the end of the bed; the fluid velocity not only changes its operating region of interest when interphase equilibrium magnitude along the bed but it also changes its direction dominates; i.e., the equilibrium model is used; it is a little in the pressurization and blowdown steps. This implies concave for small cut when the intraparticle diffusion/ that conclusions reached for a traditional adsorption fixed convection models are used (Rodrigues et al., 1992a).The bed may not be applied to the cyclic operation of a PSA results obtained in this work, depicted in Figure 1, show process, at least to some extent. Simulation results shown a similar behavior for the enrichment/recovery relation of in Figures 2 and 3 illustrated this point for the intraparticle a three-step one-column PSA process using the three diffusion and intraparticle diffusion/convection models, models for the following values: q* = 20, k' = 2, f h = 5, respectively. The parameters values used in the simufe = 4.5, ep = 3, 6b = 10, of 2-10, Mp = 5, Mb = 5, Ur* =
*....,. ... "
....-.. -. O-.
I
2744 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
a a/
-
1.9
t 1.5' u.2
.*
4I f i
0.4
RE
u.2
0.8
0.6
0.4
RE
0.6
0.8
4 0 ,
"*I
b
b /
.-.*. ... .
RE
1.8
-
1.7.
'.
1.6 -
1
0.4
-
EN
1.5 .-
u.2
1.9
1.51 ._
u.2
0.8
0.6
1.9,
0.4
RE
0.8
0.6
I
C 1.9 -
c /
1.8
.
EN
I
I
'19. .'*y 'D
1.5' u.2
I 0.4
RE
0.6
0.8
-
1.7
-
1.6
-
t
1.5'
u.2
0.4
RE
0.6
I
0.8
Figure 2. Effects of pressurization (a), feedb), and blowdown (c) rates on the enrichment/recovery relation when the process is simulated by the intraparticle diffusion model.
Figure 3. Effects of pressurization (a), feed (b), and blowdown (c) rates on the enrichment/recovery relation when the process is simulated by the intraparticle diffusion/ convection model.
lations are those of the reference case unless noted. In both cases, the enrichment/recovery relation is almost independent on the velocities used in the feed and blowdown steps. A possible explanation is that, although the feed step operation is very similar to the classic operation of an adsorption fixed bed, the concentration front of the adsorbable species never reaches the outlet of the bed. A higher fluid velocity means mass-transfer resistances are more important, which leads to more dispersive concentration fronts; this effect increases with the increase of feed duration, i.e., recovery. However, this effect is very small for the interesting operating range of a PSA process. It is expected that the effect of the feed rate on separation performance will increase with the adsorption capacity of the adsorbent since the penetration distance for pressurization is smaller for higher adsorption capacity (Lu et al., 1993a), and then the feed duration is longer for a given recovery. In the blowdown step, we think that the important region for mass transfer is not the outlet of the bed, but the region where the concentration front is located. The fluid velocity at the outlet changes significantly when the blowdown rate changes;however, the fluid velocity in the region where
the mass-transfer resistances are important does not change much, which can be seen from the bed dynamics shown by Lu et al. (1993b). It can be expected that by increasing the product purity and decreasing the product recovery, the concentration front will be closer to the feed end of the bed and the importance of the blowdown rate will increase. For pressurization, results inconsistent with the adsorption fixed-bed classic operation were obtained: the separation performance is improved with increasing pressurization rate by using the two models. However, the reasons are different in the two cases. When dealing with the intraparticle diffusion model, mass-transfer resistances are more important near the bed inlet; as a matter of fact, it was observed that the concentration of the adsorbable species inside the particles at the open end of the bed is not equilibrated with that of the bulk fluid phase, even at the end of pressurization (Lu et al., 1992a). This allows adsorption to occur at the bed inlet during the feed step, which follows the pressurization step. As the pressurization rate increases, a higher amount of adsorbable species is retained in the bed after the feed step (as checked
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2745 2.0
q*=lO
a
a
..."...
1.81
1.5'
0.4
u.2
I
0.8
0.6
RE
u.4
q*=20 1.9 2*o*
I
-'O
u.2
0.4
RE
0.8
-
2.0
c
k''
1.7
I
0.6
C
'.
-
1.6.
I. . I
u.2
q*=60 0.4
0.6
I
'
0.5
0.6
R
0.8
0.9
Figure 5. Enrichment/recovery relation when @ = 0.1: -0-, equilibrium model; -O-, intraparticle diffusion model; -0-,intraparticle diffusion/convection model.
\
1.5'
..,
17L
0.8
RE
Figure 4. Effect of the adsorption capacity q* on the enrichment/ equilibriummodel;-o-, intraparticle recoveryrelation (@ = 0.2): -0-, diffusion model; +, intraparticle diffueion/convection model.
by global mass balances in each step), then improving the separation performance. When the intraparticle diffusion/ convection model is used, higher fluid velocities lead to higher pressure drops across the particles which, in turn, yield higher intraparticle convection, improving the intraparticle mass transfer, the result being again a less dispersive concentration front (now already obtained at the end of pressurization) and a better separation performance. The discussion above can lead to the conclusion that high pressurization, feed, and blowdown rates should be used when a three-step one-column pressure swing adsorption process is used for the operation region of interest. However, a high pressurization rate has a mechanical limit, a high feed rate has a high-pressure limit, and a high blowdown rate has an intrinsic limit. The proper selection of pressurization, feed, and blowdown rates should be based on the understanding of the bed dynamics. Nevertheless, it should be pointed out that this conclusion is only valid in the operation region of interest for the three-step onecolumn PSA process for the system under study. If very small fluid velocities are used in the three steps, the results
of the diffusion model should approach those of the equilibrium model. (iii)Effect of Adsorption Capacity q*and Pressure Ratio a. The separation performance of a three-step onecolumn PSA process is significantly dependent on the adsorption capacity of the adsorbents when the local interphase equilibrium dominates (Lu et al., 1993b). By analyzing chromatography of a linearly adsorbable species, it is known that the importance of macropore mass-transfer resistance increases with the adsorption capacity when it is small and nearly does not depend on it when it is large; however, the importance of micropore mass-transfer and adsorption kinetic resistances first increases and then decreases with the increase of the adsorption capacity when it is small (Rodrigues et al., 1992c; Ruthven, 1984). For a Langmuir adsorption isotherm, the effect of the adsorption capacity of the adsorbent on the separation performance of the PSA process by using the three models is shown in Figure 4. The product enrichment decreases 11.9, 10.5, and 9.7% at RE = 0.5 and 8.7, 8.4, and 7.8% at RE = 0.6 for q* = 10,20, and 60, respectively, from the equilibrium dominated to the intraparticle diffusioncontrolled cases due to the macropore mass-transfer resistance; however, it decreases 7.6,7.5, and 7.7 % at RE = 0.5, and 6.1,6.5, and 6.0% at RE = 0.6 for q* = 10, 20, and 60, respectively, from the equilibrium dominated to the diffusion/convection-controlledcases. The reason for the decrease of the importance of the macropore masstransfer resistance with the increase of the adsorption capacity is related to what we mentioned above: the extent of saturation of the particles at the bed inlet during pressurization decreaseswith the increase in the adsorption capacity, which leads to a little better separation performance for high adsorption capacity when the diffusion model is used. In general, it can be concluded that the importance of the macropore mass-transfer resistance is nearly independent on the adsorption capacity as is the case in a chromatographic column. The importance of mass-transfer resistances also decreases with the increase in product recovery. In order to assess the effect of the pressure ratio a, simulations were done with a = 0.1, i.e., doubling the highpressure value relative to the base case. Figure 5 shows the results as represented by the relation enrichment/ recovery for the three models. Whichever model is used, an important increase of the enrichment is seen for the same value of recovery when decreases. The decrease of enrichment when recovery increases is slower; Le., it is possible to maintain high values of enrichment with higher values of recovery. For these new conditions, the product
2746 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
enrichment decreases 6.8% at RE = 0.6 when one goes from the equilibrium to the diffusion-controlled cases. Finally, for the cases simulated, the presence of intraparticle convection improves the behavior of the system relative to the diffusion case; the relative importance of this improvement increases with recovery, that is, although the curve for the diffusion/convection model is always between the other two curves, the equilibrium curve approaches the curves of the diffusion and diffusion/ convection cases when recovery increases,while the relative distance of these last two curves scarcely changes. Evolution of the Separation Performance and Bed Dynamics. The evolution of the performance parameters and of the bed dynamics of a PSA process starting from different initial conditions was studied in the isothermal and adiabatic cases when mass- and heat-transfer resistances are absent (Lu et al., 1993b). No multiple steady states or quasi-steady states were found in the simulations. Hundreds of cycles are needed to reach the cyclic steady state in the adiabatic case due to the difficulty of building up the temperature profile in the bed, although the changes of the performance parameters from cycle to cycle are very small at large cycle number. In this and the next sections, simulation results are presented for the reference cases unlessnoted. The process performance parameters enrichment (EN), recovery (RE), and mole fraction (yfe)of the adsorbable species at x = 1 in the end of the feed step as a function of the number of cycles for initial conditions yo = 0 and yo = 0.5 are shown in Figure 6. The effect of the initial condition on the evolution of the performance parameters for various operating conditions was checked. At the cyclic steady state, the mole fraction axial profiles are exactly recovered whatever initial condition is used. For the equilibrium model, 100 finite elements were used in the bed to avoid numerical oscillations in simulations. The number of cycles needed to reach the cyclic steady state is usually between 20 and 50, the number increasing with the decrease of recovery or the increase of enrichment. For the intraparticle diffusion/convection models, the number of cycles to reach the cyclic steady state is lower than that for the equilibrium model for relative low product recovery and larger for relative high product recovery. For example, 33,25, and 22 cycles are needed to reach the cyclic steady state for 8f = 3 (RE -0.44)as shown in Figure 6 for the equilibrium, diffusion/ convection, and diffusion models, respectively; and about 9, 11, and 12 cycles, respectively, are needed for 6f = 10 (RE -0.72). For very high product enrichment, i.e., very low recovery, hundreds of cycles are needed to reach the cyclic steady state when intraparticle mass transfer resistances are present (-95 cycles are needed to reach the cyclic steady state closely for RE = 0.22, Le., Of = 0.8, when the diffusion model is used with the reference case parameter values), and the number is nearly independent of the initial condition used, i.e., starting from a clean bed (yo = 0) or starting from a feed saturated bed (yo = 0.5). However, very low recovery is an operating condition seldom used in these rapid PSA processes. The development of axial mole fraction profiles of the adsorbable species from a uniform initial mole fraction in the bed (yo = 0.5) to the cyclic steady state has been studied by Lu et al. (199313) when the equilibrium model is used; changes occur mainly in the region of the bed near the production end where a concentration plateau develops, with a mole fraction value different from the initial and feed values. A shock front of mole fraction in the bed was observed in pressurization and feed steps during evolution,
2.1
-
L o
2.0
I.,
a
0
20
10
30
40
cycles
0.48
bl
-
RE 0.46 -
0.44
0.42
-
t' L A
3
10
J
20
30
cycles
40
nic.
"I
c l
%.
,'
\
: \ :
10'2
0.00 3
10
/--
. - - - r c* 20
cycles
30
1 40
Figure 6. Evolution of the process performance parameters enrichment (a) and recovery (b) and of the mole fraction of the adsorbable species in the product at the end of the feed step (c) from the initial to the cyclic steady-state values, as calculated with the three models: - - -, equilibrium model; -., intraparticle diffusion model; -, intraparticle diffusion/convection model.
as shown in Figures 7-9. These runs correspond to the filled circles in Figure 4b (RE =0.44). When the intraparticle diffusion/convection models are used, a dispersive mole fraction front is formed in the pressurization and feed steps due to mass-transfer resistances during evolution. However, the main changes of the axial mole fraction profiles still occur in the downstream region of the bed to form a mole fraction plateau in the cyclic steady state. From the axial mole fraction profiles at the end of the steps shown in Figures 7-9 by using the three models for cycles 1and 5 and at the cyclic steady state, respectively, it seems that the greater influence of mass-transfer resistances on the bed dynamics occurs in pressurization and blowdown steps. In pressurization, bed efficiency decreases when going from the equilibrium to the intraparticle diffusion/ convection to the diffusion-alone cases. The importance of mass-transfer resistances is larger in the region near the feed end of the bed. Intraparticle convection improves the bed efficiency from the diffusion
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2747 ne.
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Figure 7. Mole fraction profiles at the end of pressurization (a), feed (b), and blowdown (c) for the fiist cycle; initial condition yo = 0.5 in the bed, @ = 0.2: - - -, equilibrium model; intraparticle diffusion model; -, intraparticle diffusion/convection model.
Figure 8. Mole fraction profiles at the end of pressurization (a), feed (b), and blowdown (c) for cycle 5; initial condition yo = 0.5 in the bed, @ = 0 . 2 - - -,equilibriummodel; -., intraparticlediffusion model; -, intraparticle diffusion/convection model.
alone to the equilibrium cases, since it reduces the masstransfer resistance, as observed before (Lu et al., 1992a,b). In the feed step, the difference among the axial mole fraction profiles for the three models nearly does not change and all their shapes show less dispersive fronts due to the favorable adsorption isotherm used. In blowdown, the region where mass transfer resistances are most important is near the mole fraction front, not at the bed outlet, as we pointed out before. Big differences in mole fraction near the open end of the bed were observed between the diffusion and equilibrium models for the first several cycles. These differences propagate (and even increase) to the rest of the column as the number of cycles increases, the final profiles being almost parallel to each other with a difference of more than 0.2 mole fraction (0.4 in the downstream side of the bed). The axial mole fraction profiie for the diffusion/convection model is located somewhere between the other two, and it is almost parallel to them, as well; thie again confirms that intraparticle convection improves mass transport relative to the diffusion-alone case, the limit being the equilibrium case.
Relative large fluid velocities can also be found around the region where the mole fraction front is located (Lu et al., 1993b) which yield an intraparticle forced convective flow and lead to a more efficient desorption. During evolution, the important downstream mole fraction plateaus are formed in pressurization and feed steps for all the three models in the magnitude order diffusion > diffusion/convection > equilibrium, which leads to the enrichment of the product in the magnitude order equilibrium > diffusion/convection > diffusion. Figure 10 shows the steady-state mole fraction profiles at the end of the three steps, obtained with the three models when CP = 0.1. These profiles correspond to the runs represented by filled circles in Figure 5 (RE = 0.6). If Figure 10 is compared with Figure 9, it can be seen that the transition between the feed and the product plateaus in the profiles of mole fraction a t the end of the feed step is displaced to the downstream side of the column when a higher pressurization (and feed) pressure is used because more adsorbable species is being fed (ita concentration is doubled). When the equilibrium model is used,
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2748 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 0.6 I
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X Figure 10. Mole fraction profides at the end of pressurization (a), feed (b),and blowdown (e)in the cyclic steady state, initial condition yo = 0.5 in the bed, Q, = 0.1: -,equilibrium model; -.,intraparticle diffusion model; -, intraparticle diffueion/convection model.
the mole fraction of the adsorbable component is scarcely altered, coherently with the fact that the enrichment is almost the same in both situations. However, both the intraparticle diffusion and diffusion/convection models give a more pure product (loweryi) when the high pressure is increased; i.e., the product enrichment increases when (cdcf) is decreased. It should be noted that we are comparing situations with different recoveries (RE = 0.44 for @ = 0.2, RE = 0.6 for @ = 0.1) and that the conclusions drawn above should be reinforced if the recovery were the same for both situations. Bed Dynamics at the Cyclic Steady State. It is very interesting to studythe histories of the process parameters at both ends of the bed at the cyclic steady state since they are easily monitored and controlled in practical operations. The bed dynamics of the PSA process at the cyclic steady state have been studied by using the equilibrium model in isothermalandadiabaticcases (Luetal., 1993b). Shocks in fluid velocity and temperature were observed which coincide with the mole fraction shock front of the adsorbable species.
For an isothermal three-step PSA process, Figures 1113show the reduced pressure, reduced fluid velocity, and mole fraction of the adsorbable species in the fluid and inside the particle (at p = 0.4155) at x = 0 and x = 1by using the intraparticle diffusion, intraparticle diffusion/ convection, and equilibrium models, respectively. When the diffusion model is used, large differences are observed in the values of pressure and mole fraction in the bulk fluid and inside the particles at the feed end of the bed during pressurization and blowdown. The differencesarise due to intraparticle mass-transfer resistances, which control the process during these two steps near the feed end of the bed. When intraparticle convection is present, the reduction of those resistances becomes apparent. The total pressure (total concentration) differences are eliminated; moreover, although at the early times the differences in mole fraction of the active component increase, the steps (pressurization and blowdown) are much faster and the differences are eliminated sooner than when only diffusion accounts for mass transfer. The contribution of intraparticle convection to mass transport is so important
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Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2749 6
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Figure 11. Histories of reduced pressure (a), reduced fluid velocity (b), and mole fraction of the adsorbable species (c) in the fluid and inside particles ( p = 0.4155)at the axial positions x = 0 (feed end) and x = 1(production end) of the bed, as calculated with the diffusion model, during a complete cycle at the cyclic steady state: -, in the fluid; 0,inside particles at x = 0; 0 , inside particles at r = 1.
that even the mole fraction in the bulk fluid is seen to pass through a maximum at the early times during blowdown. During the production (feed) step, the behavior is very similar for the three models at this feed end of the column; i.e., mass-transfer resistances do not play an important role in this step. Considering now what is happening in the production end of the bed, it can be seen that mass-transfer resistances are not important in this region, since no differences are observed between the pressures and mole fractions in the bulk fluid and particles, even when the diffusion model is used. Nevertheless, the presence of mass-transfer resistances during the pressure-changing steps (pressurization and blowdown) influences the mole fraction observed at the production end of the bed, contributing decisively to the increase of its value and, consequently, to the decrease of the enrichment performance parameter. And again, the presence of intraparticle convection improves the behavior of the system, as measured by the
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Figure 12. Histories of reduced pressure (a), reduced fluid velocity (b), and mole fraction of the adsorbable species (c) in the fluid and inside particles ( p = 0.4155)at the axial poeitions x = 0 (feed end) andx = 1 (productionend)ofthebed,ascalculat.edwiththediffusion/ convection model, during a complete cycle at the cyclic steady state -, in the fluid;O, inside particles at x = 0; 0 , inside particles at x = 1.
performance parameters, from the diffusion-alone to the equilibrium cases. The histories also show that the blowdown time could be reduced, increasing the productivity without significantly reducing the separation performance, since none of the represented variables change during the final part of the blowdown time used (ob = 10).
Conclusions A three-step one-column isothermal PSA system was studied by simulation with three models: one is the equilibrium model, while the other two differ in the mechanism of intraparticle mass-transfer diffusion or diffusion/convection. The effects of several operating variables on the process separation performance, measured by product enrichment and recovery, as well as the dynamics of the bed in the transient regime and at the cyclic steady state were assessed.
2750 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 6
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Both a decrease in particle size and an increase in particle permeability improve the process performance, since the results of the intraparticle diffusion and diffusion/convection models then approach the results of the equilibrium model (the optimum theoretical limit). For the process studied, it was found that high pressurization and blowdown rates should be used, since they have no apparent effect on the process separation performance. As a rule, when intraparticle resistances are present, the increase in the feed rate leads to more dispersive concentration fronts; however, as these fronts are normally far from the production end of the bed, a noticeable decrease in the separation performance would be seen only if large feed duration times are used. However, if a large pore adsorbent is used, this effect can be overcome, since higher intraparticle convective velocities arise when larger feed rates are used and the consequent reduction of mass-transfer resistances can lead to less dispersive fronts.
The performance of the process can also be increased if larger pressure ratios (PdP1)and adsorbents with larger adsorption capacities are used. Mass-transfer resistances during the pressure-changing steps (pressurization and blowdown) are directly observed only near the feed end and the concentration front, where differences in pressure and mole fraction are observed between the bulk fluid and the solid phases. Nevertheless, when they are present, their effects are propagated to the production end of the bed, increasing the mole fraction of the adsorbable component in this region and decreasing the product enrichment in the light component. The main changes in the axial mole fraction profiles during evolution occur in the downstream side of the bed, where a mole fraction plateau develops. The transition between the feed and the product end mole fraction plateaus is a dispersive wave for all cases studied, except for the pressurization and feed steps when the equilibrium model is used. The product mole fraction plateau increases when one goes from the equilibrium to the intraparticle diffusion models and, accordingly,the product enrichment decreases.
Acknowledgment Financial support from FUNDACAO ORIENTE, JNICT, NATO CRG 890600, and EEC JOULE 0052 is gratefully acknowledged. Nomenclature bl, ...,b6 = dimensionlessquantities, stated in Table I and eq lla B, = permeability of the adsorbent, cm2 c = total concentration in the bulk fluid phase, moVcm3 c' = total concentration in the fluid inside the adsorbent, mol/cm3 c, = total concentration in the bulk fluid phase under the reference conditions, mol/cm3 cf = total concentration of the feed in pressurization, mol/ cm3 d, = adsorbent particle diameter, cm D, = axial mass dispersion coefficient, cm2/s De = effective diffusivity, cm2/s De, = effectivemolecular diffusivity at reference conditions, cm2h D, = molecular diffusivity, cm2/s D,, = molecular diffusivity at reference conditions, cm2/s Dko = Knudsen diffusivity at reference conditions, cm2/s f = dimensionlesstotal concentration in the bulk fluid phase fh = dimensionless high pressure fi = dimensionless low pressure f' = dimensionless total concentration in the fluid inside the particle k = parameter of the Langmuir equilibrium relation, cm3/ mol k' = constant for the normalized equilibrium relation L = bed length, cm 1 = slab thickness, cm Mb = parameter that controls the blowdown rate, eq 18 M, = parameter that controls the pressurization rate, eq 14 N = dimensionless total mole flux from the bulk fluid to the adsorbent Ni = dimensionlessmole flux of species i from the bulk fluid to the adsorbent P = pressure in the bulk fluid, Pa p = pressure inside the particle, Pa Po= atmospheric pressure, Pa AP, = pressure drop across the bed under the reference conditions, Pa Pe = Peclet number in the bulk fluid
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2751 q* = constant for the normalized equilibrium relation qm = parameter of the Langmuir equilibrium relation, mol/ cm3 t = time, s u = superficial velocity in the bulk fluid, cm/s uo= superficialvelocity under the reference conditions,cm/s u* = dimensionless velocity in the bulk fluid u = intraparticle velocity, cm/s uo = reference intraparticle velocity, cm/s u* = dimensionless intraparticle velocity x = dimensionless axial coordinate in the bed yf = mole fraction of the adsorbable species in the feed yfe = mole fraction of the adsorbable species at x = 1, in the end of the feed step yi = mole fraction of adsorbable species in the bed yo = initial mole fraction in the bed yi’ = mole fraction of adsorbable species in the fluid inside the adsorbent z = axial coordinate in the bed, cm z’ = space coordinate in the adsorbent, cm Greek Letters cto= ratio between the referencetime constant for intraparticle mass diffusion and reference space time 81 = dimensionless parameter defined in Table I 8~ = ratio between the half-thickness of the slab and bed length y1,yz = constants t = bed porosity tp = adsorbent porosity CP = concentration ratio (cdcf) ho = intraparticle mass Peclet number under the reference conditions p = fluid viscosity, g/cms p = dimensionless space coordinate in the adsorbent p o = fluid density, g/cm3 8 = time reduced by the reference space time T~ = tortuosity factor for the particle T~ = reference space time, s 4 = adsorption capacity factor, stated in Table I
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Investigation of a Modified Cycle with Pressure Equalization and No Purge. Chem. Eng. Sci. 1987,42,2037. Jones, R. L.; Keller, G. E. Pressure Swing Parametric Pumping-A New Adsorption Process. J. Sep. Proc. Technol. 1981,2,17. Kowler, D. E.; Kadlec, R. H. The Optimal Control of a Periodic Adsorber. AIChE J . 1972,18,1207. Lu, Z. P.; Rodrigues,A. E. Intensificationof Sorption ProcessesUsing Large-Pore Adsorbent. Id.Eng. Chem. Res. 1993,32,230. Lu, Z.P.; Loureiro,J. M.; LeVan,M. D.; Rodrigues,A. E. Intraparticle Convection Effect on Pressurization and Blowdown of Adsorbers. AIChE J. 1992a,38,857-867. Lu, Z. P.; Loureiro,J. M.; LeVan,M. D.; Rodrigues,A. E. Intraparticle Diffusion/Convection Models in Pressurization and Blowdown: Nonlinear Equilibrium. Sep. Sci. Technol. 1992b,272 (14),1857. Lu, Z.P.; Loureiro, J. M.; LeVan, M. D.; Rodrigues, A. E. Pressurization and Blowdown of an Adiabatic Adsorption Bed. IV, Diffusion/Convection Model. G m Sep. Purif. 1992c, 6, 84-100. Lu, Z.P.; Loureiro, J. M.; LeVan, M. D.; Rodrigues, A. E. Pressurization and Blowdown of Adsorption Beds. 11. Effect of the Momentum and Equilibrium Relations on the Isothermal Operation. Chem. Eng. Sci. 1993a,48,1699. Lu, Z.P.; Loureiro,J. M.; LeVan, M. D.; Rodrigues,A. E. Simulation of a Three Step One Column Pressure Swing Adsorption Process. AIChE J. 1993b,in press. Ricther, E.; Strunk, J.; Knoblauch, K.; Juntgen, H. Modeling of Desorption by Depressurization aa Partial Step in Gas Separation by Pressure Swing Adsorption. Ger. Chem. Eng. 1982,5,147. Rodrigues, A. E.; Loureiro, J. M.; LeVan, M. D. Simulated Pressurization of Adsorption Beds. Gas Sep. Purif. 1991a,5,115. Rodrigues, A. E.; Lu, Z. P.; Loureiro, J. M. Residence Time Distribution of Inert and Linearly Adsorbed Speciesin Fixed-Bed Containing “Large-Pore” Supports Applications in Separation Engineering. Chem. Eng. Sci. 1991b,46,2765. Rodrigues, A. E.; Lu, Z. P.; Loureiro, J. M.; LeVan, M. D. Diffusion and Convectionin Pressure SwingAdsorption Processes. Presented at the Fourth International Conference on Fundamentals of Adsorption, Kyoto, Japan, May 1992a. Rodrigues, A. E.; Lu, Z. P.; LeVan, M. D. Diffueion and Convection in Pressure Swing Adsorption Processes. Presented at the AIChE Meeting, Miami Beach, Nov 1992b. Rodrigue8,A.E.; Lopes, J. C.; Lu, Z. P.; Loureiro, J. M.; Dim, M. M. Importance of Intraparticle Convection on the Performance of Chromatographic Processes. Int. Symp. Prep. Chromatogr., PREP’92, Arlington, VA (1991);J. Chromatogr. 1992c,590,93. Ruthven, D. M. Principles of Adsorption and Adsorption Process, Wiley: New York, 1984. Ruthven, D. M. Sorption Kineticsin PSA Syateme. In Gas Separation by PSA; Symposium a t the Twente University, Netherlands, Feb 1992. Scott, D. M. Similarity Solutions for Pressurization and Depressurization witha Two-ComponentGas in an Adsorbing Bed. Chem Eng. Sci. 1991,46,2977. Shin, H. S.;Knaebel, K. S. Pressure Swing Adsorption: A Theoretical Study of Diffusion-Induced Separation. AIChE J . 1987,33,654. Sundaram,N.; Wankat, P. Pressure Drop Effecte in the Pmurization and Blowdown Steps of Pressure Swing Adsorption. Chem. Eng. Sci. 1988,43,123. Turnock, P. H.; Kadlec, R. H. Separation of Nitrogen and Methane via Periodic Adsorption. AIChE J. 1971,17,335. Yang, R. T. Gas Separation by Adsorption Processes; Butterworthe: Boston, 1987. Yang, R. T.; Doong, S. J. Gas Separation by Pressure Swing Adsorption: A Pore-DiffusionModel for Bulk Separation. MChE J . 1985,31, 1829. Zhong, G. M.; Meunier,F.; Huberson, S.; Chalfen,J. B. Pressurization of a SingleComponent Gas in an Adsorption Column. Chem. Eng. Sci. 1992,47,543.
Received for reuiew March 25, 1993 Accepted July 20, 1993. 0 Abstract published in Aduance ACS Abstracts, October 1, 1993.
