Znd. Eng. Chem. Res. 1989,28, 1677-1683 ri = inner radius of capillary, m
ro = outer radius of capillary, m R = recycle ratio T = annual operating time TI= absolute temperature of gas at the intake condition of a compressor, K v,, = intake flow rate of a compressor, std m3/h W = power requirement of a compressor, kW xp = purity of product Greek Letters
y = membrane permselectivity 0 = stage cut u1 = depreciation rate of the membrane v2
= depreciation rate of the compressor Registry No. COz, 124-38-9; 0,7782-44-7.
1677
Matson, S. L.; Ward, W. J.; Kimura, S. G.; Browall, W. R. Membrane oxygen enrichment, 11. Economics Assessment. J. Membr. Sci.
1986,29,79-96. Ohno, M.; Tetsuo, M.; Ozaki, 0.; Miyauchi, T. Comparison of Gas Membrane Separation Cascades Using Conventional Separation Cell and Two-Unit Separation Cells. J. Nuclear Sci. Technol. 1978 15 376-386. Peters, M. S.; Timmerhaus, K. D. Plant Design and Economics for Chemical Engineers; McGraw-Hill: New York, 1968; pp 106-108. Kao, Y. K.; Hwang, S. T.; Qiu, M. Critical Evaluations of Two Membrane Gas Separator Designs. Znd. Eng. Chem. Res. 1989, in press. Schell, W. J. Industrial Gas Application. In Membrane Gas Separations for Chemical Process and Energy Application; Whyte, T. E., Yon C. M., Wagenes, H., Eds.; ACS Symposium Series 223; American Chemical Society: Washington, DC, 1983;pp 125-143. Schell, W. J.; Houston, C. D. Use of Membranes for Biogas treatment. Energy Prog. 1983,3,96-100. S p r i n g ” , H. Plan Large O2and N2Plants. Hydrocarbon Process.
1977,2,97-101.
Literature Cited Kuester, J. L.; Mize, J. H. Optimization Techniques with Fortran; McGraw-Hill: New York, 1973;Chapters 7 and 9.
Received for review September 19,1988 Accepted July 28, 1989
Adsorption Column Blowdown: Adiabatic Equilibrium Model for Bulk Binary Gas Mixtures Ravi Kumar Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195
An adiabatic equilibrium model to simulate the blowdown step of a pressure swing adsorption process is presented. T h e partial differential equations describing the mass and heat balances for the desorption of a bulk binary gas mixture are reduced to simple first-order differential equations by assuming local equilibrium a t all times in the column and independence of column temperature (e), pressure (P),and gas-phase composition (y) on the column length ( x ) . The Langmuir model is used to describe the nonlinear equilibrium isotherms of the adsorbate-adsorbent systems presented in this study. However, the adiabatic equilibrium model is general in nature, and any equilibrium isotherm model with a n explicit from may be used. T h e adiabatic equilibrium model is solved to study the effect of initial column pressure, initial gas composition, selectivity, and isotherm shape on the blowdown step. The adiabatic model is also compared with the isothermal model. Pressure-swing adsorption processes for gas separation always involve a pressure reduction or a blowdown step from a high adsorption pressure to a lower desorption pressure. Mathematical models available for this step are rather limited. One approach to simulate this step is to solve the relevant partial differential equations by a numerical technique. Sebastian (1975) simulated the evacuation of a single component from an isothermal adsorption column and solved the resulting partial differential equations by a finite difference technique. This resulted in a multiplicity of solutions. Some of the solutions, therefore, had to be eliminated due to physical considerations. The model predicted the observed pressure profiles quite closely. However, details of the simulated system, such as the form of the equilibrium isotherm and the concentration of the adsorbing species, were not given. Richter et al. (1982) solved the mass balance equations for the isothermal blowdown of an adsorption column numerically and also found some numerical instability. The model was used to simulate the desorption of methane and nitrogen from a binary mixture of CH4/H2 and N2/H2, respectively, during the blowdown step from a carbon molecular sieve. The kinetic parameters were extracted by a curve-fitting technique. Another approach for simulating various adsorption process steps is the use of equilibrium theory (Amundson
et al., 1965; Pan and Basmadjian, 1971; Basmadjian et al., 1975; Sircar and Kumar, 1985). This theory assumes local equilibrium inside the adsorption column at all points and, therefore, neglects the effect of the mass-transfer rate. The partial differential equations are reduced to ordinary differential equations and then solved either analytically or by a numerical technique. Fernandez and Kenney (1983) and later Knaebel and Hill (1985) used this approach to simulate the pressure change in an isothermal adsorption column with linear equilibrium isotherms. Fernandez and Kenney (1983) also mentioned a nonlinear equilibrium isotherm modification of their isothermal model. Analytical solutions were obtained. Since, depending upon the heat of adsorption, the effects of temperature change on any one of the adsorption process steps could be very significant (Sircar and Kumar, 1985; Sircar et al., 1983),the present study was undertaken to simulate the blowdown of an adsorption column under adiabatic conditions. Equilibrium theory was used, and therefore, the rates of heat and mass transfer were assumed to be instantaneous. The partial differential equations were reduced to ordinary differential equations and solved by the numerical technique of finite differences. Bulk binary gas mixtures with nonlinear equilibrium isotherms were considered. The adsorbate-adsorbent systems (C02/N2, C02/CH4, and C02/H2 on BPL carbon and
0888-5885/89/2628-1677$01.5~~0 0 1989 American Chemical Society
1678 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989
C02/N2on 5A zeolite) simulated in this study were described by the Langmuir model. However, any equilibrium isotherm model with an explicit form can be used. Mathematical Model The mass and heat balance equations for the blowdown of an adiabatic adsorption column initially saturated and pressurized with a bulk binary gas mixture are described as follows: mass balance for component one (i = 1)
-=-( dB
dP
dy F + n y X 8 dP X 2 - n s X a
)+
E + npXa X2-nsX8
(7)
where A , B, E , F, and Xl-X8 are given by eq A-6-A-17 of the Appendix section. Substitution of eq 4 and 5 into eq 2 and integration over the entire column length give the specific gas quantity desorbed as a function of time d(t)= -
1
+ np)+ ny dp dY + ne
'%((A
*)
dt ( 8 )
dP
or the total quantity of desorbed gas as a function of
column pressure
overall mass balance
overall heat balance
The solid-phase capacity of component i is a function of p , Y , and 8, ni = ~ ; ( P , Y , @ and the total solid-phase capacity is the sum of the individual component capacities, n = Eni
Equilibrium Isotherm Equilibrium isotherms are needed to calculate nip,niy, and niB.While any equilibrium model with an explicit form may be used, the Langmuir model is used for mathematical simplicity to demonstrate the applicability of the present model:
at an _ - n p -dP +n at dt
dt
dt
-dY+ n o - d8 dt dt
(4)
1+
n2 =
mb2P(1- y f 1 + b1Py + bZP(1 - y )
and
The assumption of local equilibrium at all times in the column gives
dP dY dd -ani- - nipdt + niy + nie-
mblPY blPy + bJ'(1 - y )
n, =
where bi = iiexp (R(;;
(5)
0))
Therefore,
nlp =
mb1y z
mblP(l + b2P) n1y
=
n2y= -
and
z
+
(12)
mb2P(1 blP)
z
(13)
np = Enip ny = Eniy ne = Cnie
Equations 1-3 assume that the axial mass transport due to diffusion and axial heat transport due to conduction in the bulk gas and the adsorbed phases are negligible. It is further assumed that e, C,, C,, pa, and q iare independent of temperature. The heat capacity of the adsorbed phase is neglected. If it is further assumed that 8, y , and P are independent of column length, then, as outlined in the Appendix section, eq 1-5 can be solved to give dy - XlX2 E X , + XaX, _ (6) dP X5X2 - FXB - XaXG - X ,
+
and
where
Method of Solution For the blowdown step, eq 6 and 7 provide a set of ordinary differential equations relating y and 0 to the equilibrium parameters and other constants. The independent parameter in these equations is column pressure (P). Therefore, eq 6 and 7 can be solved for y and 0 as a function of P. The total quantity of desorbed gas (D)is
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1679 and
Table I. Blowdown from a BPL Carbon Column Saturated with a C02/N2Mixture P, atm yCol T, K d, mmol/g FL,mmol/s t,s 15.00 14.00 13.00 12.00 11.00 10.00 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.01
0.500 0.504 0.508 0.513 0.519 0.526 0.534 0.544 0.557 0.574 0.595 0.626 0.672 0.746 0.880
300.0 299.7 299.3 298.9 298.4 297.8 297.2 296.5 295.7 294.7 293.5 291.9 289.8 286.7 280.9
0.000 0.083 0.168 0.256 0.346 0.440 0.538 0.642 0.752 0.871 1.002 1.151 1.327 1.555 1.911
4.526 4.225 3.923 3.621 3.319 3.018 2.716 2.414
0.00 2.88 6.06 9.59 13.55 18.06 23.28 29.42 36.84 46.10 58.17 74.93 100.62 147.59 348.37
2.112
1.811 1.509 1.207 0.905 0.604 0.050
for Pefit> 0.53P (21)
Typical results for a sample problem are listed in Table I. The column packed with BPL carbon was initially saturated with a 50/50 COz/N2 mixture at 15 atm. Adsorbent properties and other relevant constants are listed in Tables I1 and 111.
Isothermal System For a binary isothermal system with linear equilibrium isotherms, the adiabatic model can be reduced to the isothermal model in the following manner: n, = K,Py n2 = K2P(1- y) nu = nZ8= 8 = 0
Table 11. Adsorbent Properties and Other Constants L 100 cm C,, 3.70 X i.d.
2 cm 3.14 X
C".
T.
COZ
NZ
8.86 44
c, PI c
1 atm 300 K
Pf
6.96 28 BPL carbon 0.22 0.484 0.763
CHI
Therefore, eq 6 gives the following expression:
H2
8.53 6.99 16 2 5A zeolite 0.22 0.731 0.762
where
calculated as a function of column pressure from eq 9. The blowdown step variables (y, 8, and D) can be related with time by either of the following approaches. (a) If the column pressure as a function of time is empirically known, P = fl(t) (17) then, since y, 8, and D are known functions of column pressure, eq 17 reduces y, 8, and D as functions of time,
Equation 22 is identical with the previously published eq 16 of Fernandez and Kenney (1983) and eq 1 2 of Knaebel and Hill (1985).
Parametric Study In the following study, it is assumed that the adsorption column is first saturated by flowing a binary gas mixture at pressure P, and gas composition ys. Then, the above described local equilibrium theory and the method of solution were used to evaluate the effect of initial column pressure (PJ, initial gas composition (ye),selectivity and shape of the equilibrium isotherm on the adiabatic blowdown process. Pressure increments of 0.005 atm were used to solve the ordinary differential eq 6,7, and 9 by the numerical technique of finite differences. Pressure increments of 0.005 atm were found to be sufficiently accurate after comparing blowdown profiles for the pressure increments of 0.1, 0.01, 0.005, and 0.002 atm. A typical computation time on an IBM 370/165 computer was approximately 2 s. BPL activated carbon and 5A zeolite were used as the adsorbents. C02was taken as component 1and N2, CH4, and Hzwere considered as component 2 in each of the binary cases: CO2/NZ,C02/CH4, and C02/H2. All the runs were carried out at an initial column temperature of 300 K and a final column pressure of 1 atm. Other constants used in the calculations are listed in Table 11. The pure-componentadsorption equilibriumof the gases on both the adsorbents was described by the Langmuir isotherm, and the binary equilibrium was described by the
t. (b) If the flow characteristic of the valve at the column exit is empirically known, (18) FL = f z ( p ) then the combination of eq 9 and 18 gives time as a function of column pressure:
Again, since y, 8, and D are known functions of column pressure, eq 19 gives y, 8, and D as functions of time, t. In the present study, the second approach was chosen since eq 17 has to be empirically determined by a blowdown experiment, and eq 18 is generally available from valve manufacture's technical information. The following equations were used: P FL = c"(qyMW, + (1 - Y ) M W ~ ) ) ~ / ~ for Pedt50.53P (20) Table 111. Langmuir Parameters for the Pure Adsorbate
absorbent
BPL activated carbon COZ
NZ CH4 HZ
m 3.65 x 10-3 3.65 x 10-3 3.65 x 10-3 3.65 x 10-3
b
2.8 X 13.5 x 1.5 X 0.5 X
lo4 10-4 lo4 lo4
5A zeolite Q 4900 2500 4700 3300
m 3.66 x 10-3 3.66 x 10-3
b 1.0 x 10-5 2.1 x 10-6
Q 8870 5000
1680 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989
Table IV. Summary of the Results initial gas quantity in the column, mmol/g .~
S -
1 2 3 4 5
svstem COP(l)/NP (2)-BPL carbon, SlIz= 11.7 COP (l)/NP (2)-BPL carbon, Sl12= 11.7 COz (1)/N2 (2)-BPL carbon, Sl/z= 11.7 COz (I)/Nz (2)-BPL carbon, SliZ= 11.7 COP(1)/N2 (2)-BPL carbon, S1 - 11.7 COz (l)/CH, (2)-BPL carbon, 2.6 COP(l)/HP (2)-BPL carbon, Sl12= 80.0 COP (l)/NP(2)-5A zeolite, SIIz= 314 C02 (I)/CH, (2)-BPL carbon, isothermal
P,, atm
Y,
5
0.50 0.50 0.50 0.20 0.80 0.20 0.20 0.20 0.20
15 24 15 15 15 15 15 15
&;=
6 7 8 9
Tf, K Yf 4 mmol/g 287.5 0.825 0.92 280.9 0.880 1.91 2.65 279.2 0.886 2.01 283.3 0.570 279.2 0.971 1.91 2.15 274.3 0.263 286.8 0.931 1.55 295.5 0.562 0.72 0.332 2.74 300
170
0'
I
n nil
I
m
I,
final gas quantity in the column, mmol/g 1 2 1 2 2.643 0.372 2.057 0.038 3.488 0.738 2.295 0.020 3.940 1.070 2.345 0.015 2.387 1.520 1.794 0.103 4.083 0.263 2.435 0.003 1.469 2.732 0.994 1.057 2.855 0.901 2.199 0.007 3.690 0.553 3.370 0.025 1.469 2.732 0.809 0.652
YD
f
0.637 0.624 0.602 0.295 0.864 0.221 0.420 0.269 0.241
0.982 0.991 0.994 0.946 0.999 0.485 0.997 0.993 0.554
Figure 2. Variation in the desorbed gas quantity with column pressure during the blowdown step: COz/N2on BPL carbon.
Figure 1. (a) Variation in the concentration of the effluent gas and (b) variation in column temperature with column pressure during the blowdown step: C02/N2on BPL carbon.
mixed Langmuir equations (Sircar and Kumar, 1983). Table I11 summarizes the parameters of the equilibrium isotherms.
Effect of Initial Column Pressure The effect of initial column pressure on the blowdown step was calculated for Pa= 5, 10, and 25 atm. CO2/N2 on BPL carbon with y s = 0.5 was used as the example. Figure 1plots the concentration of the effluent gas (y) and the column temperature (2') against the column pressure (P). It is observed that as the blowdown step proceeds, i.e., the column pressure (P)decreases, the column temperature decreases, and the concentration of the strongly adsorbed species in the blowdown gas increases. This implies that the blowdown step can be used to increase the concentration of the strongly adsorbed species inside the adsorption column well above its concentration in the feed gas. Table IV shows that the final concentration (yf) of COz in the blowdown gas increases with initial column pressure (Pa). The mole fraction of carbon dioxide remaining in the column at the end of the blowdown step
(f 1 f = carbon dioxide on the adsorbent and in the
voids/binary gas mixture on the adsorbent and in the voids also increases, from 98.2% to 99.4%, as the initial column pressure is increased, from 5 to 25 atm. Table IV shows that, as expected, the initial gas quantity in the column, i.e., the void and the adsorbed gas, increases with initial column pressure. However, the final gas quantity of the more strongly adsorbed species is higher for runs with higher initial column pressure, although the final column pressure for all runs is the same (Pf= 1 atm). This is explained by noticing that, for the higher initial
O ' I 0.3
Figure 3. Variation in desorbed gas concentration with desorbed gas quantity during the blowdown step: CO2/N2on BPL carbon.
column pressure runs, the final gas-phase concentration of the strongly adsorbed species is higher and the final column temperature is lower. These two factors result in a higher solid-phase loading of the strongly adsorbed species and hence a higher final gas quantity of the more strongly adsorbed species. Figure 2 plots the blowdown gas quantity as a function of column pressure for various initial pressures. As the blowdown step proceeds, the slope of this curve ( A d / @ ) increases. This increase is due to the shape of the Langmuir isotherm, which has an increasing slope (An/ A€') with decreasing pressure. Figure 3 shows the change in the blowdown gas concentration with the blowdown gas quantity for various initial column pressures. The gas-phase concentration,
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1681
/I
10
1: 0.7
-
0.a
-
0.5
-
0.4
-
\
\
I
14[ 0.3
//1
y.=0.2
0.1
0.2
I
0.1 01
1
~
-
I
.o
2.0
d. m w / g
Figure 4. Effect of initial gas composition on blowdown gas concentration vs blowdown gas quantity profiles: C02/N2 on BPL carbon.
OI
Figure 6. Effect of selectivity on gas composition during the blowdown S ~ P (1) . CO2/CHd, S i p = 2.6; (2) COa/N2, S i p = 11.7; and (3) COz/H2, S1,p = 80.0.
I 02
08
04
08
I D
I.
Figure 5. Effect of initial gas composition on yf and 9 ~ C02/N2 : on BPL carbon, Sl/2 = 11.7. (-.-) S1/2 = 1.
0
8%
especially for higher pressure runs (P,= 25 atm), first rises slowly and then more rapidly as the blowdown step proceeds. As shown in Table IV, the average concentration of the strongly adsorbed species in the blowdown gas @D) decreases with initial column pressure (Pa).
Figure 7. Effect of selectivity on yt and yD Adsorbent: BPL carbon.
Effect of Initial Gas Composition The effect of the initial gas composition on the blowdown step was calculated for y s = 0.2, 0.5, and 0.8. A mixture of C02 and N2 on BPL carbon with Ps= 15 atm was used as the example. Figure 4 plots the concentration of the blowdown gas (y) against the blowdown gas quantity for various initial gas compositions (ys). Table IV shows that, as y s increases, the final column temperature (Tf)decreases but the final gas-phase concentration of the more strongly adsorbed species (yf), the average blowdown gas concentration @D), and the mole fraction of carbon dioxide remaining in the column at the end of the blowdown step (f) increase. Figure 5 shows the variation of yf and jjD with ys. Since the C02/N2selectivity on BPL carbon is higher than 1 (Sl12= 11.7), C02 enrichment in the blowdown gas (Ep = jjD/yJ and in the gas remaining inside the column at the end of the blowdown step (Ef= yf/ys) is observed (i.e., E p > 1and Et > 1). This results in the yr and jjD curves being convex toward the vertical axis as shown in Figure 5. For a nonselective system (&I2 = I), a 45' line (Le., yf = ys = j j D ) will be observed as shown in Figure 5.
for three binary gas mixtures with different selectivities on BPL carbon: (1)C02/CH4,Slj2= 2.6; (2) C02/N2,S1/2 = 11.7, and (3) C02/H2,Sl12= 80.0. Figure 6 shows the variation of C02 concentration with column pressure for the three systems during the blowdown step. The rise in the blowdown gas concentration is sharpest for the system with the highest selectivity. As the selectivity between the components increases from 2.6 to 11.7 to 80, the temperature drop inside the column (Tf- TJ decreases from 25.7 to 19.1 to 13.2 K. This is due to the fact that, as the selectivity increases, the total gas quantity desorbed from the column, which is the difference between the initial and the final gas quantities in the column (Table IV),decreases from 2.15 to 2.01 to 1.55 mmol/g for the three systems. The effect of selectivity on yf and jjD is plotted in Figure 7. The average concentration of the blowdown gas @D) decreases with decreasing selectivity and approaches the initial gas composition (y,) as the selectivity approaches one. The gas-phase concentration of carbon dioxide inside the column at the end of the blowdown step (yf) first rises slowly, then at a steady rate, and then asymptotically approaches one as the selectivity increases. For the highest selectivity system (C02/H2/BPLcarbon, Sl12= 80),Table IV shows that, at the end of the blowdown step, essentially pure carbon dioxide is left inside the adsorption column (f = 0.997).
Effect of Selectivity The effect of selectivity on the blowdown step was demonstrated by solving the adiabatic blowdown model
Effect of Isotherm Shape The entire shape of the equilibrium isotherm is one of the key factors in determining the appropriate adsorbent
1682 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989
05
04
i
03
1
30
I
\
02-
01
10
10
d, m mol&
Figure 8. Effect of isotherm shape on gas composition profile during the blowdown step. (1) C02/N2-5A zeolite and (2) COz/ N,-BPL carbon.
for a PSA process. Selectivity and adsorption capacity, alone, cannot provide complete information for an adsorbent selection. This is illustrated by comparing the blowdown steps for two adsorbents 5A zeolite and BPL carbon separating a C02/N2mixture with ys = 0.2. Table IV demonstrates that, although the C02/N2selectivity of 5A zeolite is 314 as compared to the selectivity of 11.7 for BPL carbon and although the initial COz quantity on 5A zeolite is 3.690 mmol/g as compared to 2.387 mmol/g on BPL carbon, the working capacity (i.e., initial gas quantity - final gas quantity) of 5A zeolite is -46% lower than the working capacity of BPL carbon (0.320 vs 0.593 mmol/g). Therefore, at these operating conditions, BPL carbon is a better adsorbent for PSA application than 5A zeolite even through the carbon has a lower selectivity and a lower initial capacity than the zeolite. Table IV also demonstrates that yf and ?, for the zeolite system are lower than for the carbon system. This is due to the fact that the shape of the equilibrium isotherm for the carbon system is such that it releases more carbon dioxide than the zeolite system as the pressure is reduced during the blowdown step. As a result, the total desorbed gas quantity (Figure 8) is also higher for the carbon system than for the zeolite system.
Isothermal Blowdown The above-described adiabatic local equilibrium model could be reduced to an isothermal local equilibrium model by neglecting the heats of adsorption for both of the gases, i.e., ql = q2 = 0. The blowdown of the C02/CH4-BPL carbon system with y s = 0.2, T, = 300 K, and Ps= 15 atm was selected as the example (bco = 1.04 atm-' and bCH, = 0.40 atm-l at T, = 300 K). hgure 9 compares the blowdown gas concentration and the blowdown gas quantity predicted by the adiabatic and the isothermal models. It shows that the isothermal model predicts a higher blowdown gas concentration and higher blowdown gas quantity at all pressures than the adiabatic model during the entire step. The working capacity for carbon dioxide as predicted by the isothermal model (Table IV) is -60% higher than predicted by the adiabatic model (0.660 vs 0.412 mmol/g). Therefore, the use of isothermal simulation for blowdown step would result in significant undersizing of the adsorption column. This effect is even greater for higher initial pressure and higher initial concentration runs because of the larger column temperature drop during the blowdown step at these conditions. In addition, Table IV also shows that the isothermal model predicts -27% higher blowdown gas quantity (2.74 vs 2.15 mmol/g), higher final gas-phase concentration (yf = 33.2% vs 26.3'%), higher average blowdown gas concentration (5jD = 24.1% vs 22.1%), and higher C02 mole fraction left on
Figure 9. Comparison of isothermal (- - -) and adiabatic (-) models for C02/CH, on BPL carbon. P, = 15 atm, T , = 300 K, ya = 0.20.
the bed (f = 55.4% vs 48.5%) than the adiabatic model. These factors will result in erroneous prediction of the column performance by the isothermal model during the blowdown step.
Conclusions The adiabatic simulation of the blowdown step for bulk systems shows that an isothermal model is inadequate for design purposes. The error introduced by the isothermality assumption is higher for higher initial pressures and higher initial gas-phase concentrations. Adsorbent choice for a PSA process should take the shape of the entire equilibrium isotherm into consideration. Selectivity and initial capacity, alone, are not sufficient parameters for adsorbent selection. The model shows that the blowdown step can be used to enrich the desorbed gas as well as the gas remaining inside the adsorption column at the end of the desorption step. The "local equilibrium" assumption along with the assumption that 0, y, and P are independent of column length simplifies the adiabatic model for simulating the blowdown of a bulk binary gas mixture considerably by reducing the partial differential equations (PDE's) into simple first-order differential equations. The simplified equations can be solved in a fraction of CPU time required to solve such P D E s on a computer. Even though such a simplified model may not be accurate for process design, it is an excellent tool for predicting the column behavior and trends in a semiquantitative manner. Acknowledgment The author is grateful to Air Products and Chemicals, Inc., for permission to publish this work. Portions of this paper were presented at the AIChE Summer National Meeting, Denver, CO, Aug 21-24, 1988.
Nomenclature A = defined by eq A-6 A, = area of column cross section, cm2 B = defined by eq A-7 = Langmuir constant, atm-' b = Langmuir constant at infinite T , atm-' C, = empirical valve flow coefficient below critical pressure C,, = empirical valve flow coefficient above critical pressure C, = gas-phase heat capacity, cal/(mol K) C, = adsorbent heat capacity, cal/(g K) D = total desorbed gas quantity, mol d = specific desorbed gas quantity, mol/g E = defined by eq A-8
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1683
F = defined by eq A-9 f = mole fraction of the strongly adsorbed species left in the adsorption column at the end of the blowdown step FL = flow rate at column exit, mol/s i.d. = inside column diameter K = Henry's law constant for linear equilibrium isotherm, mol/(g atm) L = length of the column, cm MW = molecular weight of a gas species, g/mol m = monolayer capacity in the Langmuir model, mol/g n = solid-phase capacity, mol/g P = pressure, atm Pelit = pressure at the column exit after flow control valve, atm Q = gas flow rate, mol/(cm2 s) q = isosteric heat of adsorption, cal/mol R = gas constant = (nly2)/(n2yl), binary selectivity T = temperature, K t = time, s X1-X7 = defined by eq A-10-A-17 x = distance variable, cm y = gas phase mole fraction of the strongly adsorbed species (i = 1) y D = average concentration of the strongly adsorbed species in the blowdown gas Z = defined by eq 16 Greek Symbols t = adsorbent column void fraction, dimensionless 6 = T - T,, K pB = gas-phase density, mol/cm3 p, = adsorbent density, g/cm3
Eliminating d6ldt by combining eq A-2 and A-3 gives XiX2 EX3 X8X4 dy _ (-4-4) dP X5X2 + FX3 - X,XG - X7
+
+
Substitution for d y / W in eq A-2 gives (A-5)
where (A-6) (A-7) (A4 (A-9) (A-10) (A-11) (A-12) (A-13) (A-14) (A-15) (A-16)
and
Superscripts and Subscripts 1 = more strongly adsorbed species i = component i ( i = 1, 2) s = conditions at the start of the blowdown step f = conditions at the end of the blowdown step
x8
= 6Cg
(A-17)
Literature Cited Amundson, N. R.; Aria, R.; Swanson, R. On Simple Exchange Waves in Fixed Beds. Proc. R. SOC.London 1965, A286, 129. Basmadjian, D.; Ha, K. D.; Pan, C. Y. Nonisothermal Desorption by Gas Purge of Single Solutes in Fixed Bed Adsorbers. 1. Equilibrium Theory. Ind. Eng. Chem. Process Des. Deu. 1975,14(3),
Appendix Equation 3 is rewritten as
328.
(A-1)
Fernandez, G. F.; Kenney, C. N. Modelling of the Pressure Swing Air Separation Process. Chem. Eng. Sci. 1983, 38(6),827. Knaebel, K. S.; Hill,F. B. Pressure Swing Adsorption: Development of an Equilibrium Theory for Gas Separations. Chem. Eng. Sci.
Substitution for the first term on the right-hand side of eq A-1 from eq 2 and combination with eq 4 give
Pan, C. Y.; Basmadjian, D. An Analysis of Adiabatic Sorption of Single Solutes in Fixed Bed: Equilibrium Theory. Chem. Eng.
2 (c,+ C,AP - B - neeCg)= dt dP dY - ( E + npOC,) + - (F + ny6C,) (A-2) dt dt where A , B , E , and F are defined later. Substitution for the first term on the right-hand side of eq A-1 from eq 1 and combination with eq 4 give
Richter, E.; Strunk, J.; Knoblauch, K.; Juntgen, H.Modelling of Desorption by Depressurizationas Partial Step in Gas Separation by Pressure Swing Adsorption. Gem. Chem. Eng. 1982, 5 , 147. Sebastian, D. J. G. A Mathematical Model for Pressure Swing Adsorption. Proceedings of the 7th IFIP Conference on Optimization and Modelling, Nice, 1975; p 440. Sircar, S.; Kumar, R. Adiabatic Adsorption of Bulk Binary Gas Mixtures: Analysis by Constant Pattern Model. Ind. Eng. Chem. Process Des. Dev. 1983,22, 271. Sircar, S.; Kumar, R. Equilibrium Theory for Adiabatic Desorption of Bulk Binary Gas Mixtures by Purge. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 358. Sircar, S.; Kumar, R.; Anselmo, K. J. Effects of Column Nonisothermality or Nonadiabaticity on the Adsorption Breakthrough Curves. Znd. Eng. Chem. Process Des. Dev. 1983,22, 10.
(A-3)
Received f o r review August 16, 1988 Accepted June 23,1989
1985,40(2), 2351.
Sci. 1971, 26, 45.
