2226
Ind. Eng. Chem. Res. 1993,32,2226-2235
A Shortcut Computational Method for Designing Na PSA Adsorbents Joan M. Schork,’ R. Srinivasan, and Steven R. Auvil Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195-1501
A computationally efficient procedure has been developed for determining the influences of various adsorbent properties and operating conditions on Nz pressure swing adsorption (PSA) process performance. This procedure is unconventional in that it employs a detailed model of mass transfer and limits computational time by assuming linear concentration profiles along the column axis. It provides a facile means of ranking kinetics-based adsorbents. Parametric studies conducted using this procedure show, quite unexpectedly, that the Nz PSA process performance of carbon molecular sieves is more sensitive t o equilibrium capacity and total void volume than t o kinetic selectivity or gas uptake rates.
Introduction In the development of adsorption process simulators, three important elements are often “traded-off” against one another. These are (1) the precision to which convection, dispersion,and accumulation in the bulk fluid phase are modeled, (2) the detail employed in modeling mass transfer into and out of the adsorbent particles, and (3) computational efficiency. Increasing the precision in modeling the bulk phase generally translates into the use of more nodes or elements along the axial column dimension. The modeling of processes with very sharp mass transfer zones (MTZ) requires a large number of nodes or elements, or the use of a more advanced technique such as moving grids. Detailed modeling of mass transfer can require accounting for several resistances, e.g., fluid film, macropore, and micropore. Inclusion of a resistance that is distributed throughout the adsorbent particle necessitates the use of multiple radial nodes or elements. Obviously, increasing precision or detail in the modeling of either the bulk or the solid phase increases computational time. A compromise must generally be made between precision/detail and computational efficiency. Such a compromise forms the basis of the computational method described in this paper, a computationally efficient procedure for predicting N2 pressure swing adsorption (PSA) process performance. Nz PSA is a kinetics-based process for the production of 97-99.7 % N2. Because of its relatively small size and operational simplicity, Nz PSA has found a niche (the 141200 standard cubic meters per hour market) in meeting the growing demand for Nz. Carbon molecular sieve (CMS),the adsorbent employed in Nz PSA, does not offer any significant 0dNz selectivity at equilibrium. Instead, the separation is based on a kinetic (i.e., diffusion rate) selectivity; typical commercial CMSs uptake 02 30-40 times faster than N2. An accurate representation of diffusion should be a key element in modeling this unusual diffusion-based process. Literature models have addressed this requirement to varying degrees. They have employed variations of the linear driving force (LDF) model (Chu, 1991; Hassan et al., 1986; Raghavan and Ruthven, 1985) and the more appropriate Langmuir kinetics model (La Cava et al., 1988, 1989; Chen et al., 1992). Both can be seen as lumpedparameter models in which the mass transfer resistance
* Author to whom correspondence should be addressed.
+ The authors
take pleasure in dedicating this paper to J. R. Fair, who delights and excels in “making the right approximations”.
of the CMS is taken to be concentrated in a “surface barrier” at the entrance to the micropores (Srinivasan et al., 1992). A distributed parameter model in which the resistance is taken to be homogeneously distributed throughout the CMS micropores has also been used (Farooq and Ruthven, 1991). In contrast to either of these extremes, CMSs can be modeled more generally with two intraparticle resistances in series: a “surface barrier” and a homogeneously distributed micropore resistance (Chihara et al., 1983; Loughlin et al., 1992). As will be demonstrated in this paper, important information about the adsorbent can be lost if only a single resistance is considered; adsorbents that would perform quite differently in the Nz PSA process cannot be distinguished by a single resistance model. This paper describes a computational procedure that accounts for both “surface barrier” and interior resistances to mass transfer yet is highly computationally efficient. This procedure is unconventional in that it controls computational time by treating the axial bed profiles in a greatly simplified manner. It was developed for use in the evaluation of new kinetics-based adsorbents and is intended to provide a means of predicting Nz PSA process performance given fundamental adsorbent properties.
Carbon Molecular Sieves CMSs are a modified form of activated carbon, designed to have both high internal surface area/equilibrium capacity and partial or total molecular sieving capability. While the particulars of CMS synthesis are largely proprietary, an outline based on accounts in the literature can be given: The starting materials are unselective, highsurface-area( 200-600 m2/g) carbon pellets-assemblages of microporous particles with macropores amid the particles. Additional carbon is deposited in the micropores by means such as hydrocarbon cracking (Juntgen et al., 19811, or (thermoset) polymer coating followed by carbonization (Walker, 1966; Walker et al., 1966; J h t g e n et al., 1976). The carbon deposits bestow kinetic selectivity (i.e., OdNz separation by partial molecular sieving) by acting as constrictions in the micropores of the base carbon. If the constrictions are present mostly in a “shell” at the periphery of the (micro)particles, then the CMS can be visualized as a collection of ink-bottle-shaped spaces, with transport to and from the wide sorption spaces occurring through the narrow bottlenecks. If, on the other hand, the constrictions are distributed throughout the microparticles, then the CMS can be seen as zeolite-like, the constrictions acting as windows amid sorption cavities. Corresponding to either of these CMS structures, models
-
Q888-5885/93/2632-2226$04.00/00 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993 2227 Adsorption of 02 and N2 mixtures was assumed to obey the binary version of the pure gas isotherm; e.g., nl* =
- --
0.0 0
SHELL MODEL TWO RESISTANCES MODEL CORE MODEL
I
I
I
I
I
I
200
400
600
800
1000
1200
TIME (SEC)
Figure 1. Nz uptake by a commercial CMS.
of uptake transients are available. The ink-bottle system is amenable f 6 analysis by a “surface barrier” model (Kocirik et al., 1986),in which the mass-transfer resistance of the entire CMS microparticle is lumped into a single discrete intraparticle “shell” or “pore mouth” resistance. The zeolite-like system can be adequately described by a single distributed “core”or “poreinterior”resistancemodel (Crank, 1956). None of the N2 uptake transients in our fairly extensive data base of gas uptake by commercial CMSs can be satisfactorily fitted by the core resistance model. The shell resistance model adequately fits some of the data but a combined two-resistances model provides an excellent fit of all of the data. Nz uptake data for a commercial CMS are shown in Figure 1along with “best” fits using all three models. In light of the improved fits obtained with the two-resistances model and given the knowledge of how CMSs are produced, it can be postulated that at least the CMSs that we have studied have inkbottle structures and hence the pore mouth resistancemust be considered; however, the “bottle” itself is neither wide nor short and hence it is desirableto account for the interior pore resistance as well. It is pertinent to note that Chihara et al. (1983) and Loughlin et al. (1992)have come to similar conclusions. It could be argued that the failure of the shell model to adequately fit the data is due to a significant variation in micropore size. This, however, is unlikely for the commercial air separation CMSs as it has been shown that the micropore size distributions in these materials are quite narrow (Gaffney et al., 1992). A general model should also account for the macropore resistance, but this resistance is typically insignificant in comparison with the shell and core resistances. Adsorption Equilibrium. The 0 2 and N2 pure gas isotherms display considerable curvature in the pressure range of the PSA process. Further, we have found significant heterogeneityin N2 sorption (Auvilet al., 1993). Accordingly, we employed a two site Langmuir isotherm equation; Le.,
*Pi +- ‘Pi nj* = l+B,.Pj l + D , P j This “dual site Langmuir (DSL) equation”-unlike the single site Langmuir isotherm-is able to account for the heterogeneity in the energetics of adsorption. 0 2 and Nz single component equilibrium data can be fitted excellently with the saturation capacity of each site the same for both gases (i.e., AllBl = AdB2 and C1/D1 = CdD2).
+
Alp1
ClPl
+
1 BIP, BZp2 + 1 + D,Pl
+ DZp2
(2)
The Intraparticle Mass-TransferModel. The main purpose of this paper is to describe a new way of analyzing the column mass balances in kinetics-basedair separation. It is appropriate, however, to outline the intraparticle mass transfer model used. The model is similar to the classic “bidispersepore model” (Ruckenstein et al., 1971;Chihara et al., 1978) in which the microparticles are taken to be uniformly distributed point sinks, with their external concentrations in equilibrium with the local macroparticle (or pellet) concentrations. The present model differs from the classic model in the following ways. (1)It includes a surface barrier resistance at the entrance to the microparticles; i.e., while the adsorbate concentration on the external surface of the microparticle (at P = r,) is in equilibrium with the gas in the macropores, the concentration underneath the barrier (at r = rc - 6) is not. (2) A DSL isotherm is used instead of a linear isotherm. (3) The model is adiabatic rather than isothermal. (4) All diffusion fluxes are based on chemical potential rather than concentration driving forces. Macropore fluxes are described by the “dustygas model” for a binary mixture in a tortuous capillary (e.g., Jackson, 1977; Sotirchos, 1989).
-q + RT
= pZhTl--lNz
V P , -VP) BPl
PDlZ
flK1
+-Nl 4
(3)
1
(5) 2
B v = -‘ma -
“ma
‘ma
8
As described in a recent work on a related model (Farooq and Ruthven, 19911, the micropore fluxes are (7)
The derivation of these flux expressions using irreversible thermodynamics can be found in the literature (e.g., Round et al.,1966; Kiirger and Biilow, 1978). It should be noted that these expressions were derived on the basis of the single site Langmuir isotherm and hence are not exact when adsorption equilibria are described by DSL isotherms. We have derived exact expressions for the latter case but did not use them in the present study; instead, the above expressions were used, with the modification that the 8’s were evaluated using the DSL isotherm coefficients; e.g., ‘1
+
BIPl
= 1 BIPl
+ B2p2
DIP,
+
1
+ D,Pl + DZp2
(9)
The differential mass balance equations-incorporating
2228 Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993
the flux expressions detailed above-are
N2PRODU,CT
as follows:
,
macroparticle:
microparticle: AIR FEED NF
The following boundary conditions are enforced.
Pj
0,F
Figure 2. Schematic representation of the basic Na PSA process.
macroparticle:
at r, = rp,
t
unequally distributed among the grids; the grid beneath the micropore entrance corresponds to 10% , the grid at the micropore core to 65% ,and the intervening grid to the remaining 25 % . The equations were linearized (as in Newman (1991)) and the resulting system of equations was solved by Gaussian elimination. The results from the isothermal case were used as initial guesses for the adiabatic case.
= Pj bulk
microparticle:
The Column Model Alternative Expressions for Recovery and Productivity. N2 PSA cycles are all variations upon a relatively simple, two bed, three step cycle. The process steps are (1)a high-pressure feed step during which highpurity N2 is produced, (2) pressure equalization of the beds, and (3) column depressurization by venting to the atmosphere. This cycle and variations upon it have been described elsewhere (Hassan et al., 1987;Ng et al., 1993). The basic cycle is represented by the diagram in Figure
at ri = rc,
and
2.
where the quantities with the overbars are evaluated at the local pressures in the macropores, and ksj = DsjfS
(17)
This “shell” mass transfer coefficient (i.e., ksj or rather ksj/rc)is estimated from pure gas uptake transients like the ones in Figure 1,as is the interior/core mass transfer coefficient:
Itcj = Dcjfrc2 These mass balance equations are coupled with a straightforward heat balance in which the intrapellet gas and solid phases are assumed to be at the same temperature. Enthalpy changes due to adsorption/desorption as well as the enthalpy content of the incoming/outgoing gas streams are accounted for. The DSL isotherm coefficients and the micropore mass-transfer coefficients depend on temperature according to the Clausius-Clapeyron and Arrhenius equations, respectively. The corresponding heats of sorption and diffusion activation energies were estimated from pure gas isotherm and uptake transient data at two temperatures. The differential mass balance equations-in spherical coordinates-were converted to nonlinear algebraic equations using finite differences. To reduce computational time, the number of spatial grid points was minimized. One grid was used for the macropores and three were used for the microparticle interior. The micropore volume was
N2 PSA performance is commonly assessed in terms of process productivity and air recovery at a given product purity. Productivity, the product flow rate per volume of bed, reflects the size of the adsorbent beds required; the higher the productivity, the lower the capital cost of a system. Air recovery is the amount of N2 product generated per volume of feed air compressed. Higher recovery translates into lower power costsand lower capital cost for the compressor. Using the symbols shown in Figure 2, the standard definitions of productivity and air recovery yield the following equations.
P + OP productivity = N -
(19)
tC
P + 0, air recovery = N Nf + Of These equations do not make explicit the relationships between process performance and fundamental adsorbent properties. It would be desirable to have recovery and productivity in terms of 0, and N, since these variables can be related to adsorbent properties through the adsorbent working capacities. 0, (Nd, referred to as the overall 0 2 (N2) working capacity, is the sum of the adsorbent’s 0 2 (N2) working capacity plus the quantity of 0 2 (N2) in the void spaces at the end of the pressure equalization step. Defining p and f as the mole fractions of N2 in the product and feed streams respectively, and applying component mass balances around the system shown in
Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993 2229 Figure 2, the equations for recovery and productivity can be rewritten as productivity =
0,f
- N,(1 t,O,
-f,
-f ,
W v)
a
2
tc
*e----
(21)
It is particularly instructive to look at simplified expressions for the case in which the N2 product purity approaches 100'36. Assuming that p = 1and the feed gas is air, eqs 21 and 22 can be simplified to productivity =
____-------
1.2
v)
' 3.760, - N,
1.4
P
2
s
0.6
0.2
o
0
(23)
air recovery = 0.79 - 0.21NW/0, (24) It is clear from these equations that recovery is a function of the working selectivity of the adsorbent while productivity is primarily a function of the 02 working capacity. The N2 working capacity also affects productivity but with a gain factor that is only about a fourth of that on the 0 2 working capacity. These expressions make explicit the intuitively-derived and experimentally-supported premises that recovery should increase with kinetic selectivity and productivity should increase with uptake rates. Approximate Representationof Bed Concentration Profiles. The above expressions for recovery and productivity are exact if integral values of 0, and N, are used. Obtaining accurate values of these parameters, however, would require the use of a full scale simulator that accounts for convection and accumulation in the bed voids as well as diffusion and accumulation in the CMS. Significant compuation time can be saved by using a simpler masstransfer model, e.g., a LDF or Langmuir kinetics model. Alternatively, the method described here models mass transfer within the CMS pellets in detail and limits computational time by approximating the solid phase concentration profiles in the column as straight lines. In order to make this approximation,it must be assumed that both the 0 2 and N2 MTZs span the entire length of an N2 PSA column. In light of the sluggish rate of N2 diffusion in an air separation CMS, one would expect the N2 MTZ to span the entire column; the feed step in a normal cycle is about an order of magnitude shorter than the time required for a CMS to reach even 67 '36 of its N2 equilibrium loading. It has been shown experimentally that the 0 2 MTZ also spans a large portion of the bed under commercially important conditions, i.e., conditions under which recovery is maximized (Ng et al., 1993). Simulated 0 2 and N2 concentration profiles are shown in Figures 3 and 4. These results, generated using a PSA simulator based on the Langmuir kinetics model (Chen et al., 1992), indicate that straight lines are reasonable approximations of the solid phase concentration profiles. Procedure Implementation. In order to determine the working capacity of a CMS, the gas-phase conditions under which it is cycled must be specified. If one assumes linear gas phase concentration profiles in a bed of CMS, the working capacity of the bed could be approximated by determining the performance of a CMS cycling between the average adsorption conditions and the average desorption conditions. Since the solid-phase concentration profile in a kinetics-based adsorptive separation is always less well developed (i.e., less sharp) than the gas-phase profile, a slightly better approximation is made if the solid phase concentration profiles are assumed to be linear. In
N, LOADING 02 LOADING GAS PHASE 0,
.2
.4
.6
.a
.IO
DIMENSIONLESS COLUMN LENGTH
Figure 3. Concentration profiles at the end of the feed step.
:
1.2
1
I
cn
C C C _ _ _ - _ - - - - - - - - *
- - - N2 LOADING - 02 LOADING
gr 5%
0.4
ga
0.2
-
c!J
s
o
-
I
0
.2
.4
.6
GAS PHASE 02
.8
.IO
DIMENSIONLESS COLUMN LENGTH
Figure 4. Concentration profiles at the end of the blowdown step.
this case, the working capacities at the two ends of the bed are calculated and the column working capacities are assumed to be the arithmetric averages of the end point values. The procedure described here assumes linear solid-phase profiles. Working capacities at the end points are determined by using the mass-transfer model described above to calculate the changes in solid- and gas-phase loadings of an adsorbent cycling between two sets of bulk gas conditions. Even at the two end points, however, the precise, time-dependent bulk gas conditions during adsorption and desorption cannot be specified a priori. Thus, some representative, time-averaged, bulk gas conditions for the adsorption and desorption steps must be defined. In this context, the following approximations have been employed. Adsorption: 1. The bed pressure is assumed to be constant and equal to the final adsorption operating pressure. 2. The bulk gas composition at the feed end of the bed is fixed at the feed composition. 3. The bulk gas composition at the product end of the bed is set as a constant value equal to the user-specified purity of the N2 product. This is actually the timeaveraged value of the gas flowing out of the bed. Desorption: 1. The average bed pressure during the desorption step is a function of cycle time. A time-averaged value, calculated from experimental depressurization data, is used. Use of this approximation can be shown to be superior to the use of a constant, cycle time-independent value.
2230 Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993 A 0
-*
4-."5
h
ae
K
1
E r 1
0 0 0 0
Y 5-
120
-
100
-
42
E * t 2
39
36
---
CMS-D DATA CMS-D CALCULATED CMS-N DATA = CMS- N CALCULATED
CMS-D CALCULATED CMS-N DATA = CMS N CALCULATED
100
150
200
250
300
-
80 -
0 60
-
n 1
40 50
......-
.
W
g
- - -CMS-D DATA
140
50
I
I
100
CYCLE TIME (SEC)
I 150
I
I 200
I
I
I
250
3 0
CYCLE TIME (SEC)
Figure 5. Measured and calculated values of air recovery.
Figure 6. Measured and calculated values of productivity.
2. The bulk gas composition at the product end of the bed is that of the instantaneous flux out of the adsorbent pellets, a time-dependent value. At this end of the bed, desorption occurs without the benefit of purge gas. 3. The bulk gas composition at the feed end of the bed varies as a function of the volume and composition of the gas desorbed from the rest of the bed. To account for this, the bulk gas composition is calculated as the arithmetic average of the composition of the instantaneous flux out of the pellets and the average composition of the gas desorbed from the product end of the bed during the desorption step. This is also a time-dependent value since the composition of the flux out of the pellets is timedependent. In summary, the computational algorithm is as follows. First, the performance of the adsorbent at the product end of the bed is approximated by determining the overall working capacities, 0, and N,, of an adsorbent cycled between (1) product gas at feed pressure and (2) the adsorbent's own desorptive flux at a calculated, average desorption pressure (close to atmospheric). The overall 02 and N2 working capacities are stored, and the composition of the resultant blowdown gas is calculated. The performance of the adsorbent at the feed end of the bed is then approximated by determining the overall working capacities of an adsorbent cycled between (1) the feed gas at feed pressure and (2) a 50/50mixture of the adsorbent's own time-dependent desorptive flux and the blowdown gas from the product end at the calculated desorption pressure. Finally, arithmetic averages of the values of 0, and N , calculated for the bed end points are used in eqs 21 and 22 to approximate recovery and productivity.
This consistency allows the method to be used to predict process performance relativeto a base case. The procedure has also been shown to differentiate between different lots of CMS-D, materials that differ by relatively little. The cycle time range of maximum recovery is correctly predicted for CMS-N. The prediction that recovery will level off at a much shorter cycle time with CMS-D is consistent with the lab data. Calculated results also show excellentagreement with lab and process data on the effects of temperature and pressure on the performances of these two CMSs. The assumption that the 02MTZ spans nearly the entire bed would of course be invalid for cases in which the cycle time is long enough to allow 0 2 saturation of a significant portion of the bed. This could explain the increasing departure of the calculated productivity curve from experimental data at long cycle times. The match between the calculated and experimental results at shorter times suggests that the assumptions made are reasonableat near optimum cycle times. A fundamental limitation of the procedure is a lack of sensitivityto product purity. Calculated values of recovery and productivity both decrease with increased purity, as expected, but not to the extent that experimental data do. This is probably due to the procedure's inability to account for changes in the shape of the 02MTZ. This fundamental limitation, however, should not affect the evaluation of adsorbents on a relative basis. Since the effect of purity cannot be examined using this procedure, a constant value of 99% N2 was used for all calculations.
Parametric Studies Procedure Verification In order to ratify the computational procedure, the predicted process performance was compared to laboratory and process data for two commercial CMSs, CMS-D and CMS-N, that differ by over a factor of 2 in 02 diffusion rate, about 10% in kinetic selectivity, and about 10% in equlibrium capacity. While the procedure does not accurately predict absolute values of recovery and productivity,it is quite sensitive to variations in both operating conditions and adsorbent properties and tracks experimental data quite well. As shown in Figures 5 and 6, recovery is consistently overpredicted by 10-20 % while productivity is consistently underpredicted by about the same amount. The performances of these two very different CMSs are approximated with a similar degree of accuracy without the use of any adjustable parameters.
The computational procedure described above was used to study the sensitivity of process performance to various adsorbent properties. The effects of certain process variables were also studied since the fundamental properties of an adsorbent affect its response to changes in the operating conditions. Recoveries and productivities are reported relative to those of a base case. The operating conditions for the base case are air feed at 20 OC and 825 kPa, a 0.35-column void fraction, and full pressure equalization. A typical commercial CMS was used as the base case adsorbent. Cycle Time. The performance of a kinetics-based adsorbent is a strong function of cycle time. As demonstrated by the curves in Figure 7, there are maxima in both the recovery and the productivity versus cycle time curves. The existence of a maximum in the recovery curve
37
Ind. Eng. Chem. Res., Vol. 32,No. 10,1993 2231 Table I. Effects of Varying 01and NIUDtake Rates
120
performance relative to base case (%)110
36
100 35
CI
u)
90 0
?
0
34
80
?
factor 1.0 1.5 2.0 2.5 3.0 4.0
feed step(s) 120 90 70 60 50 45
recovery
productivity
-2 -5 -7 -10 -11
+30 +58 +77 +lo0 +118
Figure 7. Effect of cycle time on productivity and air recovery.
Table 11. Location of the Mass-Transfer Resistance performance relative to base case (%) recovery productivity all resistance interior -28 -25 all resistance at pore mouth +7 +60
is due to the presence of nonselective (macropore and interparticle) voids. If a bed contained no such voids, the overall working capacities would be simply those of the sorbed phase. Recovery would decrease monotonically as feed time was increased because the ratio of 0 2 to Nz in the sorbed phase decreases with time. Recovery would approach its maximum value of 79% as cycle time approached zero. In a system with nonselective voids, the ratio of the 02 to N2 overall working capacities is a weighted average of the ratios in the sorbed and void gas phases. The ratio in the void spaces is close to that of the feed and thus represents zero recovery. As cycle time approaches zero, the overall working capacities approach those of the void spaces and recovery goes to zero. To maximize recovery, the feed time must be sufficientlylong to allow the quantity of gas sorbed to "outweigh" that contained in the voids and thus bring the overall ratio up. As time approaches infinity, however, the adsorbent becomes fully saturated with both components a t the feed conditions and recovery again approaches a value very close to zero. Productivity goes to zero as cycle time approaches infinity because the rate of gas uptake decreases as saturation is approached. Productivity is low at short cycle times as well because of the effect of nonproductive cycle time. With very short cycle times, the nonproductive time required for pressure equalization, bed repressurization, and valve switching is a large enough portion of the total cycle time that its effect overwhelms that of the fast uptake rates at short feed times. The cycle times at which the recovery and productivity maxima occur are functions of adsorbent properties and operating conditions. For each case discussed below, process performance was determined at several different cycle times. The results reported are those for the cycle time at which recovery is maximized. Gas Uptake Rates. To determine the effect of overall gas uptake rates on the process performance of a kineticsbased adsorbent, calculationswere made with all four masstransfer coefficients(surfacebarrier and micropore interior for both 0 2 and Nz) multiplied by a single factor. In all cases, the kinetic selectivity was the same as that of the base case. The results of this study are listed in Table I. As shown by the feed times listed, the optimum cycle time decreases as the rates are increased. This is because the time required for a given quantity of gas to be sorbed (enough to "outweigh" the nonselective capacity of the bed) decreases. The performance predictions listed in Table I show that productivity will increase as the uptake rates are increased but that the maximum obtainable recovery will decrease. The reduction in recovery is surprising; in the difference
equations describing mass transfer in a particle, the transfer coefficients always appear multiplied by time. Thus, one would expect that if cycle time were varied inversely with the mass transfer coefficients, the amounts of each component adsorbed would be the same in each case and recovery would remain constant. Productivity would increase proportionally with the mass transfer coefficient because a given quantity of gas would be adsorbed in proportionally less time. The expected results are obtained if a single,cycle-time-independent desorption pressure is used for all cases. For these parametric studies, however, the desorption pressure is a time-averaged value that increases as cycle time is decreased. This explains why the computational procedure, as normally implemented, predicts that recovery will drop as rates are increased. Location of the Mass-Transfer Resistance. As discussed above, in our model, mass transfer into a CMS particle is controlled by two resistances in series. For convenience, the overall resistance to a given component can be expressed in terms of the characteristic time, the time required for an initially clean adsorbent to reach 67 % of saturation when exposed to the given gas at 20 "C and 1 atm. The characteristic time can be shown to be approximately equal to the reciprocal of the LDF masstransfer coefficient. It is possible for two CMSs with the same 0 2 and N2 characteristic times to have very different values of the surface barrier and micropore interior mass transfer coefficients. As shown by the results in Table 11, the location of the resistance has a strong effect on CMS performance. To obtain these results, one of the mass transfer coefficients, either the barrier or interior, was set to a very large value for both components. The remaining two coefficientswere then adjusted until the characteristic times were the same as in the base case. It is clear from these results that the location of the mass transfer resistance is very important. Since more 02 than N2 enters the micropores, interior resistance has a larger effect upon the 0 2 uptake rate than upon the N2 rate. (Taking this to the limit, if no Nz passed through the surface barrier, interior resistance would slow the 02 uptake rate without contributing at all to the separation.) For a given selectivity, it is desirable to have as much of the resistance as possible at the pore entrance. Kinetic Selectivity. The kinetic selectivity of a CMS can be increased by either increasing the 0 2 uptake rate or decreasing the N2 rate. Increasing selectivity by either means increases the ratio of the OdNz working capacities and hence recovery. As shown in Table 111, a larger recovery increase is obtained by decreasing the N2 rate. Unfortunately, productivity in this case decreases.
-n
I -
30
35
40
45
50
55
60
70
FEED STEP (SEC)
2232 Ind. Eng.Chem. Res., Vol. 32, No. 10,1993
Table 111. Variation of the Kinetic Selectivity performance relative to baaewe(%) recoverv orcductivitv
ealectivitv feed steofa) 37 120
base w e
01 rate doubled
Table V. Effects of SOW Less Macropore Volnme performancerelative to
m
+7
halved
74 18.5
110
-14
+44 -39
halved doubled
74 18.5
140 90
+11 -19
-5 +5
Nz rate
Table IV. Effects of Increawd Saturation Capacity performance relative to base caae (%) factor 1.00 1.25 1.50 2.00
f e d step(s) 120 110 100
3.00
IO
m
recovery
productivity
+7
+33 +I4 +146 +296
+11 +18 +25
3
I
I
I
I
I
Figure 8. Selective and nonselective working Capacities.
WhenselectivityisincreasedhyincreasingtheOzuptake rate, the absolute value of the 02 working capacity goes up and with it productivity. When the Nz rate is decreased, the NZworking capacity for a given cycle time goes down. This causes the 02 working capacity to decrease as well because there is less Nz available to purge 02 out of the system during the blowdown step. Thus, while the ratio of the OdNz working capacities is increased, the absolute values of the working capacities for a given cycle time are lower, causing productivity to be lower. Equilibrium Capacity. To determine the effect of the adsorbent equilibrium capacity on NZ PSA performance, the saturation capacities in the dual-Langmuir isotherm of the base case CMS were multiplied by a factor greater than 1. This does not affect the equilibrium selectivity of the CMS. The results in Table IV indicate that increasing the saturation capacity increases both recovery and productivity. The recovery increase occurs because the ratios of selective to nonselective capacity for the two components are different. This is illustrated by Figure 8. In the base case, the nonselective voids account for over half of the total Nz working capacity but only about 10%ofthe total02 workingcapacity. Doublingthe saturation capacity increases the sorbed phase working capacities of both components by about 60%. This bas amucbgreater effectupontheoverall02working capacity than it does upon the overall NZworking capacity. As a result, the ratioofthe02 to Nz working capacities increases and with it recovery. Three fadors account for the higher productivity. First, with increased equilibriumcapacity, the working capacities of both components are larger for a given pressure swing.
additive
feed step(s)
inert faer additional CMS
110 90
baaewe(%) prcduetivity +I4
mvery +9 +14
+68
Second, the larger difference in equilibrium loading for a given pressure swing increases the driving force for mass transfer, reducing the time required to obtain a given loading and thus allowing a shorter cycle time. Finally, the higher 02 to NZ overall working capacity ratio also favors a shorter cycle time. As discussed above, in the majority of cases, a shorter cycle time provides higher productivity. Void Volume. The nonselective capacity of a column obviously plays an important detrimental role in the performanceofanNzPSAsystem. Themacroporevolume of a typical commercial CMS accounts for a major portion of this capacity (almost 60 % in the base case). T w omeans of reducing the macropore volume can be envisioned; replacing a portion of the void space with an inert material or replacing it withadditionalCMS. The resultspresented in Table V show that reducing macroporosity by either method increases both recovery and productivity. Reducing the macropore volume increases the ratio of the selective to nonselective capacities of the system by reducingthe nonselective component. As discussed above, increasing this ratio increases both recovery and productivity. Replacing the void space with additional CMS raises the ratio further by also increasing the selective capacity per unit volume of bed. Decreasing the interparticle and extra-bed void volumes in a column is another way of eliminating nonselective capacity. Benefits similar to those derived by reducing macroporosity are obtained. Again, void volume can he reduced by replacing it with inerta or additionaladsorbent. The benefits of reducing bed void fraction have been recognized by others (Kaplan et al., 1989). Combined Capacity and Void Volume Improvements. Since NZPSA performance appears to he most sensitive to changes in equilibrium capacity and void fraction, it is of interest to look at the effects of simultaneous improvements in these two areas. To evaluate this, the effects of increasing the equilibrium capacity of a material with 50% of the macropore volume filled with additional CMS were compared to the effects on the base case CMS. As illustrated by Figure 9, the recovery benefita are not completely additive. The slope of the recovery versus capacity curve decreases when the void fraction is reduced because the importance of the nonselective capacity has been reduced; when there is less nonselective capacity in the system, increasing the sorbed-phase capacity has a smaller effectupontheratiooftheOztoNzoverallworking capacities. Taking this to the limit, if there were no nonselective capacity a t all, increasing the equilibrium capacity of the CMS would have no effect upon recovery. As shown in Figure 10, the slope of the productivity versus capacity curve increases as the void fraction is reduced. This is because productivity is primarily a function of overall 02 working capacity. For a given percentage change in sorbed phase capacity (on a per gram of CMS basis), the increase in the capacity of a unit volume of bed is greater for the case in which the solid makes up a larger portion of the bed. Temperature. The computationalprocedurewasused to explore two aspects of the effects of temperature on
Ind. Eng. Chem. Res., Vol. 32, No. 10,1993 2233 I C h
c
30
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a 0
25 100
20
-I I I
15 10
BED VOID FRACTION = 0.175 BED VOID FRACTION = 0.35
5
0
10
0
1.0
2.0
1.5
2.5
Figure 9. Effect of combined capacity and void volume improvementa on air recovery. finn "VU
c
400
- - BED VOID FRACTION = 0.35
/
30
35
40 I
h
8
Y
/ / / &
25
40
Figure 12. Effect of temperature on the performance of CMS-D with a 60-8 feed step.
46
--- BED VOID FRACTION = 0.175
20
TEMPERATURE ("C)
3.0
CAPACITY MULTIPLIER
15
/'
1
L
80
75
c"
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-I
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kl K
44 105
42
35
-i
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10 (70 SEC)
8
h
95
15
20 ( 5 0 SEC)
25
30 ( 4 0 SEC)
35
40 (35 SEC)
TEMPERATURE ("C) (FEED STEP)
Figure 14. Performance of CMS-Dwith cycle time adjusted for temperature. 1
90 1
E r
damental properties of the adsorbents; no adjustable parameters are employed. The procedure was designed to be used to study the relationships between fundamental adsorbent properties and process performance. With its detailed model of mass transfer in a CMS, it is well suited for this task. While it has found use as a tool for evaluating the effects of certain process variables, the limitations discussed above restrict its usefulness in this arena. A full scale simulator, albeit with a simpler model of mass transfer within the CMS, should generally prove more useful for the analysis of process cycle variations. Parametric studies conducted using this procedure show, quite unexpectedly, that the process performance of CMSs in the kinetics-based NZPSA process is more sensitive to equilibrium capacity and total void volume than to kinetic selectivityor gas uptake rates. Broadly, it can be concluded that increasing the ratio of the selective to nonselective capacities in a system increases the ratio of the 02 to NZ overall working capacities and hence the recovery. Progress is being made in realizing the improved adsorbents proposed in this work (Farris et al., 1992).
Acknowledgment The authors wish to thank Air Products and Chemicals, Inc., for permission to publish this work. PRODUCTIVITY AIR RECOVERY 40
80 (653)
100 (790)
120 140 160 (928) (1066) (1204) FEEDPRESSURE
180 (1342)
200
PSlG
(1480)
Figure 15. Effects of feed pressure on process performance.
becomes so short that productivity begins to decrease as temperature is increased further. In this case, the benefit of increased rates is outweighed by the detrimental effect of lower capacity. Feed Pressure. Increasing operating pressure has several effects: it increases the mass transfer rates, the equilibrium capacities at the feed conditions, and the quantity of gas in the void spaces. The computational procedure should account for these changes. Increasing pressure also slows the volumetric flow rate, increasing gas contact time, and thus sharpens the MTZs. The procedure does not account for changes in the shape of the MTZs. The calculated values plotted in Figure 15 show productivity increasing and recovery decreasing slightly as the adsorption pressure is increased from 650to 1475 kPa. Productivity increases with pressure because the 02 working capacity is larger and the mass-transfer rates are faster. Recovery decreases slightly because the void space capacity increase is larger than that of the sorbed phase (i.e., the slope of the isotherm between 650 and 1475 kPa is less than 1IRT) and hence the ratio of the selective to nonselective capacities decreases.
Conclusions A computationally efficient procedure for determining the effects of various adsorbent properties and operating conditions on Nz PSA process performance has been developed. This procedure tracks experimental data quite well and provides a facile means of ranking kinetics-based adsorbents. The relative performances of materials can be determined given only independently measured, fun-
Nomenclature A, C: product of the Langmuir saturation capacity and the Langmuir affinity constant in the dual site Langmuir isotherm (mol cm-3 atm-l) B, D: affinity constant in the dual site Langmuir isotherm (atm-1) B,: effective Darcy permeability of macropores (cm2) Dj: effective micropore diffusion coefficient of component j (cm2 s-1) Dcj: value of Dj in the microparticle interiodcore (cmz 8-1) DK:effective Knudsen diffusion coefficient in macropores (cmz 8-1) D~z:effective binary bulk diffusion coefficient in macropores (cm2 8-1) Dsj: value of Dj in the micropore entrance/shell (cm2 8-1) f : mole fraction of N2 in the feed J: micropore diffusion flux (mol cm-2 8-1) kc: microparticle interior/core mass transfer coefficient (cm 9-11
ks: microparticle entrance/shellmass-transfer coefficient (5-1) Mj: molecular weight of component j (g mol-') n: concentration in the adsorbed phase (mol cm-3) n*: value of n at equilibrium (mol cm-3) I t macropore flux (mol cm2 8-11 Nf: moles of NZin the feed/bed volume/cycle N,,: moles of N2 in the product/bed volume/cycle N,: moles of Nz in waste stream/bed volume/cycle Of: moles of 0 2 in the feed/bed volume/cycle 0,: moles of 02 in the product/bed volume/cycle 0,: moles of 0 2 in the waste stream/bed volume/cycle Pj: partial pressure of component j (atrn) I? total pressure (atrn) (dyn cm-2in the VPterm of the dusty gas model) p: mole fraction of Nz in the product R: gas constant (cm3 atm mol-' K-l) r,: radial coordinate in the macroparticle (cm) r,: spherical-equivalentradius of microparticle (cm) ri: radial coordinate in the microparticle (cm) r,: volume-average radius of macropores (cm) rp: radius of the macroparticle (cm) t: time (8) t,: cycle time (e)
Ind. Eng. Chem. Res., Vol. 32, No. 10,1993 2235
T: temperature (K) 6: thickness of micropore mouth barrier or "shell" (cm) em: macropore volume per unit volume of pellet fraction of the microparticle's external area that is cut through by micropores; also the volume of micropores per unit volume of microparticle p: viscosity of gas phase (g cm-1 8-1) 8: total degree of saturation over both sites of the dual site Langmuir isotherm T:
tortuosity factor
V: divergencelgradient operator Subscripts 1 and 2: components 1 and 2 (02and Nz) ma: macropores mi: micropores
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* Abstract published in Advance ACS Abstracts, September 15, 1993.
