Ind. Eng. Chem. Res. 1990,29, 625-631
625
GENERALRESEARCH Multiple Periodic States for Thermal Swing Adsorption of Gas Mixtures M. Douglas L e V a n Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442
This paper reports the results of a theoretical study. The adsorption of two dilute components from inert gas in a n adiabatic fixed bed is considered. The adsorption cycle is operated for recovery of both components rather than to separate them from one another. Cycles with two and three steps are considered. Multiple, stable, periodic states are found. For a single heating time, two different adsorption times may exist, corresponding to different depths of penetration of the heavy component into the bed at breakthrough of the light component. The system studied is benzene and cyclohexane adsorbed from nitrogen onto activated carbon. This paper elaborates on a previous article published in this journal. In Davis et al. (1988), we referred to the existence of multiple periodic states in discussing an example of the periodic states (cyclic steady states) that develop in a thermal swing adsorption bed over repeated cycling. This paper examines multiplicity of periodic states for thermal swing adsorption in greater detail by means of mathematical modeling. The goals are to consider theoretically the existence of multiple periodic states for thermal swing adsorption and to examine the structure of the solution to identify periodic states as either stable or unstable. The cycle considered is precisely that of Davis et al. (1988). Two adsorbed components are present, both of which are recovered in the bed; thus, the bed is not used to separate the components from one another. The cycle has two or three steps: (1)adsorbing to breakthrough, (2) heating with a noncondensable purge gas for a fixed time with the flow direction reversed, and (3) possibly cooling with the noncondensable purge gas for a fixed time with the flow in the same direction as for heating. The system considered is benzene and cyclohexane adsorbed from nitrogen onto activated carbon. There have been numerous studies of nonisothermal adsorption and single steps of the thermal swing cycle in which it is assumed that the bed initially has a uniform loading and temperature. We have recently discussed much of this research (Davis et al., 1988; Davis and LeVan, 1989). It should be noted that, depending on system parameters, particularly pressure, it is often not necessary to follow the heating step with a cooling step to prevent premature breakthrough during the adsorption step (Basmadjian, 1975). Far less research has been carried out on the entire thermal swing adsorption cycle. For short heating times, such as those close to optimal for minimizing heating costs (Davis and LeVan, 1989), the bed is not at a uniform condition at the start of the various cycle steps. Davis and LeVan (1987) have carried out an equilibrium theory analysis of complete two- and three-step adiabatic adsorption cycles. The results show that, by proper timing of a cooling step, heating requirements can be reduced significantly. Davis et al. (1988) considered the periodic states that develop in a bed for two-component adsorption
with both components recovered. It was shown that at modest and high regeneration pressures the heavy component does not accumulate in the bed over repeated cycles but rather the cycle operates with the bed enriched in the light component. Davis and LeVan (1989) report results of an experimental study of two- and three-step cycles with n-hexane adsorbed on activated carbon. Optimal cycles, as measured in terms of energy requirements, were found to correspond to short heating times. Proper timing of the cooling step was found to lead to significant energy savings. The literature on multiple periodic states in adsorption beds is particularly scant. For thermal swing adsorption, Davis et al. (1988) mention the existence of two periodic states for a fixed heating time. For pressure swing adsorption, which differs considerably from the thermal swing adsorption cycle considered here, F a r m et al. (1988) and Suh and Wankat (1989) have recently demonstrated the existence of multiple periodic states. The results of two analyses are reported below. First, a preliminary analysis is carried out to identify regions in the parameter space near which multiple periodic states may occur; this preliminary analysis follows the method of Davis et al. (1988). Then, repetitive calculations of cycles are performed to show the approach to the various periodic states. Mathematical Model The mathematical model consists of material and energy balances, adsorption equilibria relations, and physical properties. The system of equations is cast in the form of the equilibrium stage model of Davis et al. (1988). This is a simple finite difference model which is entirely adequate to achieve our goals. It should be recognized that local equilibrium models give results that are asymptotically correct for high rates of mass and heat transfer. Conservation Equations. Material balances on adsorbable components are of the form
The energy balance for an adiabatic column is
0888-5885/90/2629-0625$02.50/0 C 1990 American Chemical Society
626
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990
where
u, = (c, + C p l 4 1 + cp242)(T- Tref)- A141 - A242 (3)
u f = hf - P / c
(4)
(5) The balances were simplified by introducing the following dimensionless variables tuot 7=-
L {=z/L u* =
(7)
u/vo
(8)
where uo is the velocity at the bed inlet and T is the number of superficial column volumes of gas passed into bed. The interstitial velocity, u, was calculated assuming a constant flux of inert gas in a dilute system. The axial derivatives appearing in the balances were written by using backward differences to give the equilibrium stage model. This gives equations for dqi/dr and dT/dT for each stage, which are integrated to obtain new values of qi and T. The results reported here were obtained using 50 stages. Adsorption Equilibrium. Adsorption equilibria for benzene and cyclohexane adsorbed on activated carbon were described by using Langmuir isotherms and the parameters of Rhee et al. (1972), which are based on the data of James and Phillips (1954). This is the most extensively studied system for nonisothermal adsorption; LeVan (1989) cites 19 previous studies that have made use of these adsorption equilibrium relations. Single-component Langmuir isotherms were written in the form
QiKici qi =
i = 1,2
with
Ki = KOiT112 exp(Ai/RT)
(10)
This temperature dependence for Ki has been discussed in considerable detail by de Boer (1968). Binary equilibria were constructed by using the ideal adsorbed solution theory (Myers and Prausnitz, 1965); thus, partial pressures are given by
Pi = XiPiO(?r,T) (11) where n is the spreading pressure of the mixed adsorbate and Pio(a,T)is the pure component reference state. The spreading pressure must be determined by iteration. Initial estimates of ?r for isotherms (used to load the bed initially, when T and ci are given) were determined from the two-term Taylor series expansion of LeVan and Vermeulen (19811, which is
Table I. System Parameters Physical Properties Pb, kg/m3 CPl 480 c, kJ/(kg K) c 2 1.05 K cpI, kJ/(mol K) 0.0291
Adsorption Equilibrium 4.4 KO2 Qz 3.0 XI, kJ/mol K o l ,m3/(mol K1/2) 3.88 X Xz 61, m
oPg
0.0936 0.124 298 1.04 43.5 32.6
X
lo*
point for iterative improvement by using Newton's method. System. Numerical values of parameters used in the simulations are given in Table I. Benzene is component 1, and cyclohexane is component 2. Preliminary Analysis As mentioned above, a preliminary analysis was carried out to identify regions in the parameter space that might yield multiple periodic states. From the results of Davis et al. (1988), it was apparent that multiple periodic states could occur only for fairly short heating times. The method used for the preliminary analysis is as follows and pertains only to the two-step cycle. Further details are provided by Davis et al. (1988). First, a value is chosen for {1, the depth of penetration of the heavy component into the bed, and the bed is loaded at the end of the adsorption step with shocks obtained from the equilibrium theory. It is assumed that subsequent adsorption steps return the bed to this assumed initial loading and that switching from an adsorption step to a heating step occurs at breakthrough of one of the two components. Second, the heating step is carried out (with 25 stages on each side of {J, and the effluent ratio, E , is followed. This ratio is defined by mol of 2/mol of 1 removed during heating E= (14) c2/c1 in feed for adsorption
At a periodic state, we have E = 1. For E < 1, an excess of component 1 has been removed during heating. Similarly, for E > 1, an excess of component 2 has been removed. Third, the stability of the periodic states is examined. After examining system behavior for several pressures, adsorption feeds, and heating temperatures, cycle step feeds for more detailed analysis were set as follows. For the adsorption step, the feed is at Pa = 1.0 MPa, T , = 298 K, @1 = 0.02, and @2 = 0.04. (& is the fractional saturation of the feed with component i; thus, the feed was nitrogen containing benzene and cyclohexane a t partial pressures of 2% and 4% of their pure component saturation vapor pressures). For the countercurrent heating step, the feed is pure nitrogen at Ph= 1.0 MPa and Th = 398 K. For the cooling step, when used, the feed is pure nitrogen at P, = 1.0 MPa and T , = 298 K, fed to the bed in the same direction as the feed for the heating step. An example of the assumed bed loading at the end of the adsorption step is shown in Figure 1. There, the depth of penetration of the heavy component into the bed is = 0.323, which is the value obtained for an adsorption step beginning with an initially clean bed. Immediately downstream of the bed prior to breakthrough is pure nitrogen at a temperature of 298.96 K. In Figure 1and in similar figures that follow, flow is from left to right for adsorption and from right to left for heating and cooling. As the cycle develops and the bed is not completely regenerated, the depth of penetration of the heavy component into the bed at the end of the adsorption step will be deeper or shallower, and Cl will move toward
rl
For isosteres (used during the remainder of the simulation, when T and qi are known), an equally accurate initial estimate of R was developed and is
These estimates are accurate to two or more significant digits for the system considered and provided the starting
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 627 21
1
Y
I
I
/
1
I
I
I-
: ' Is,:
[,
I
ik
01 0
,I I
3 Figure 1. Loading of a clean bed at the end of the adsorption step following equilibrium theory. Benzene (l),cyclohexane (2), and temperature profiles are shown at the start of the heating step. Flow for the previous adsorption step is from left to right. Flow for heating (and cooling) is from right to left. A t the end of the adsorption step, cyclohexane has just reached the end of the bed at { = 1. The two-component shock is located at [ = = 0.323. Pure nitrogen left the bed prior to breakthrough at a temperature of 298.96 K.
- c 8
/
01 0
I
IO0
I
200
1
I
300
400
I
500
r h
Figure 3. Effluent ratio as a function of heating time for various depths of penetration of benzene into the bed. E = 1 is not reached for > 0.323 at long times.
4 360
m
E
V
4
80
90
I10
I00
120
I30
r h
0 r h
Figure 2. Breakthrough curves for benzene (l),cyclohexane (2), and temperature for the heating step. At the end of the previous adsorption step, benzene penetrated into the bed to = 0.323, as shown in Figure 1.
larger or smaller values. According to equilibrium theory, the concentrations and temperatures for the plateaus at the end of the adsorption step remain the same; i.e., they are unaffected by the value of ll. Figure 2 shows the heating breakthrough curves for the initial bed loading of Figure 1. Breakthrough begins after about 45 column volumes of hot nitrogen have been passed into the bed. The concentration of cyclohexane in the effluent peaks early, while that for benzene passes through a more gradual maximum. Benzene is completely removed from the bed before cyclohexane. The effluent ratio for various initial values of 5; is shown in Figure 3. After the initial breakthrough, we find E > 1,indicating that more of component 2 has been removed from the bed relative to its concentration in the feed. At least for a while thereafter, for t1 > 0.127, we have the opposite imbalance, with more of component 1 removed. The curve labeled Cl = 0.323 finally returns to the E = 1 line at large 7h, after benzene and cyclohexane have been completely removed from the bed. Figure 4 is a sketch of the detailed behavior found near Th = 100 in Figure 3. The stability criterion proposed by Davis et al. (1988) can be used to determine the stability of the different families of curves sweeping through this region. According to this criterion, a periodic state is stable
Figure 4. Detail of Figure 3. Four families of curves sweep through the region as described in the text.
if in the vicinity of the periodic state 5; increases in passing vertically downward across the E = 1line. There are four families of curves sweeping through the region shown in Figure 4. First, there is an unstable family passing sharply downward across the E = 1 line from 7h = 85 to 103 with values of 3; increasing from 0.16 to 0.80 (0.16,0.2,0.3,0.6, and 0.8 shown) with increasing Th. Second, rotating backward from the first family at i h = 103 is a stable family with Cl increasing from 0.8 to unity (0.8, 0.95, and 0.99 shown) with decreasing 7h. Third, rotating forward from the first family is a stable family for which ll decreases from 0.16 to 0.127 (0.16, 0.15, 0.14, and 0.13 shown) as sh increases from 85 to 93. Fourth, continuing forward from the third family is a stable family with increasing from 0.127 to 0.323 (0.13,0.14, and 0.15 shown) as 7 h increases from 93 to large values. In Figure 4, three families of curves cross E = 1 at Th = 100. Thus, based on the preliminary analysis, three periodic states exist for a heating time of 7 h = 100. They correspond to depths of penetration of benzene into the bed of roughly 5; = 0.13 (stable), 5; = 0.60 (unstable), and = 0.95 (stable). Figure 5 shows the periodic states that arise from the four families of curves sweeping through Figure 4. Corresponding values of 5; and 7 h along the E = 1 line in Figure 4 have been plotted versus one another. The curve for the unstable family is dashed. The lower branch corresponds to the third and fourth families (joining where tl is a minimum) with values of increasing to 0.323 at
cl
628 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 or--
--- y-
-----1
,
1
08-
I
/
I
i 706 e
u-
0 i____L-d--,____I 8C 10s
,.-
'r'O
tk
Figure 5. Periodic states predicted by preliminary analysis. The dashed and solid curves indicate unstable and stable periodic states, respectively. Between Th = 85 and 103,two stable periodic states are predicted.
large 7 h . The multiple periodic states a t readily apparent.
7h
= 100 are
Analysis of Cycles An important assumption was made above as part of the preliminary analysis. Specifically, it was assumed that, after heating steps, adsorption steps would return the bed to loading profiles closely resembling those shown in Figure 1. Repeated calculations are now performed for the twostep cycle to check this assumption and to further investigate the approach to the periodic states. Also, the development of periodic states for the three-step cycle is considered. The approach taken is to determine the periodic state approached from various assumed initial conditions for the bed at the start of cycling. Cycling begins with an assumed initial condition for the bed a t the end of an adsorption step. The parameter llis varied in choosing various initial conditions. However, the initial plateau adsorbed-phase concentrations and temperatures upstream and downstream of are always as shown in Figure 1. The stage model is used for all cycle steps with the bed discretized into 50 equal sized stages. Switching from the adsorption step to the heating step is arbitrarily taken to occur when one of the components has broken through at the bed outlet to 5% of its feed concentration. The heating and cooling steps are carried out for fixed times. Feeds are those identified in the preliminary analysis above. Thus, a heating time 7 h (and a cooling time rc for the three-step cycle) is fixed, and the approach to the periodic state is calculated for various initial bed loadings. The adsorption time, 7,, is allowed to float and is determined at 5 % breakthrough for each adsorption step. The adsorption time varies during the approach to the periodic state according to how well the bed was regenerated in the previous heating (and cooling) step. A periodic state is not reached until the amounts of the adsorbates removed from the bed during a heating (and cooling) step become equal to the amounts taken up during an adsorption step. Two-step Cycle. For the two-step cycle, we first consider the approach to a periodic state for a moderately long heating time. A heating time of 7 h = 200 is chosen arbitrarily. (This is longer than the optimum, as discussed below.) From Figure 3, a single periodic state is expected with slightly greater than 0.2; thus, benzene is expected to penetrate to a depth slightly greater than 20% of the way through the bed. Figure 6 shows the results of repetitive calculations of the cycle for this case. The figure pertains to the condition
Cycle N u m b e r Figure 6. Development of the periodic state determined by repetitive calculation of cycle steps for the two-step cycle with ~h = 200.
I
2
' I - 2 9 8 5\ I \
1T -
400
b
Figure 7. Periodic state for two-step cycle with T~ = 200. Bed profiles are shown at the end of the adsorption step (top) and heating step (bottom). The adsorption time is T. = 2480.
of the bed at the end of adsorption steps. The average adsorbed-phase concentration of benzene in the bed is plotted versus cycle number. The scale on the right side of the figure gives the effective depth of penetration of benzene into the bed, obtained by dividing the average adsorbed-phase concentration by the concentration existing in equilibrium with the feed. There are five different curves on the figure, each corresponding to a different initial condition at the start of cycling. These initial conditions have = 0.1,0.3,0.5, 0.7, and 0.9. It is readily apparent that a single periodic state is approached in the bed and that it is approached rapidly. After roughly eight cycles, the periodic state has been reached from all initial conditions. At this point, all dependence on the initial condition of the bed at the start of cycling has been lost. The periodic state has a value of 3; that is slightly greater than 0.2, in good agreement with the prediction of Figure 3. Figure 7 shows bed profiles for this periodic state. The bed profiles at the end of the adsorption step resemble the assumed profiles shown in Figure 1. Note that considerable regeneration takes place during the heating step for 7 h = 200; a large fraction of each adsorbate has been removed from the bed during heating, giving a subsequent adsorption time of 7, = 2480 to the 5% breakthrough. Multiple periodic states are expected to exist at shorter heating times. A stability analysis was performed using the feeds above to determine a heating time that would
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 629
0.6
-
c 0
w0.4
0.2
0.5
I
0 0
2
4
6
8
1
0 0
Cycle Number Figure 8. Development of periodic states determined by repetitive calculation of cycle steps for the two-step cycle with 7 h = 75. An unstable periodic state exists near the center of the bed. Stable periodic states exist near ilea = 0.2 and a t large TleF The two dashed lines on the right margin show the conditions of the bed after 30 cycles for beds beginning with Cl = 0.4 and 0.7.
0
1
' 1
'298 I
Figure 9. Periodic state for the two-step cycle with 7 h = 75 for shallow penetration of the heavy component into the bed. Bed profiles are shown at the end of the adsorption step (top) and heating step (bottom). The adsorption time is 7, = 550. 299
5
298
locate an unstable periodic state near the center of the bed. This was done simply by varying heating times until a periodic state was located near the center of the bed, from which the state of the bed would diverge. The heating time 731 = 75 was chosen for the two-step cycle. Alternating cycle steps were then calculated repeatedly. The results are shown in Figure 8. Nine initial conditions are shown, with fl varying from 0.1 to 0.9 by increments of 0.1. Initial decreases in the average q1 and the effective ll are observed for all initial conditions. (This is due to a short first adsorption step, for which a minimum T, of 100 was set. The first adsorption step is short because the initial loading of the bed, as in Figure 1, has more of the adsorbates than the bed can accommodate at 5% breakthrough, a t least at this early phase of the development of a periodic state. For example, compare Figure 7 (top) with Figure 1. Thus, the bed breaks through quickly during adsorption, and heating is begun with the bed containing less of the adsorbates than a t the beginning of the previous heating step.) As the cycle continues, states approached by the system become clear. An unstable periodic state is located near the center of the bed. Two stable periodic states are being approached. For a bed initially loaded at a small value of 11,the system is approaching a periodic state near an effective of about 0.2. For a large initial value of tl,however, benzene establishes itself deep within the bed. The two dashed lines on the right margin of the figure show the conditions of the bed after 30 cycles for the cases beginning with 1; = 0.4 and 0.7. Both are steadily moving away from the unstable periodic state in the center of the bed. Bed profiles for the stable periodic states with shallow and deep penetration of benzene into the bed are shown in Figures 9 and 10 , respectively. At the end of the adsorption steps, neither of the periodic states has a bed loading that closely resembles Figure 1, although the profile for benzene shown in Figure 9 is remotely similar. Instead, incomplete wave interactions remain in the bed, with Figure 10 even indicating some subcooling as benzene is desorbed. Adsorption times are about T* = 550 for the shallow periodic state and T, = 750 for the deep one. During heating, the beds get reasonably hot. In Figure 10, extensive "roll-up" (Basmadjian et al., 1975) occurs for benzene, and the temperature profile passes through a
m
0 0
5r
1
I - ,
I
297
400
Figure 10. Periodic state for the two-step cycle with rh = 75 for deep penetration of the heavy component into the bed. Bed profiles are shown a t the end of the adsorption step (top) and heating step (bottom). The adsorption time is T, = 750.
minimum near the bed outlet. This minimum is actually due to the readsorption of benzene near the bed outlet, increasing the temperature there. Three-Step Cycle. By reasoning similar to that used to choose the heating time for the two-step cycle, heating and cooling times of Th = 22 and T~ = 100 were chosen for the three-step cycle. With this short heating time and heating and cooling flows in the same direction, more adsorbate is removed from the bed during cooling than during heating, leading to an energetically efficient cycle. Figure 11shows the approach to the periodic states. An unstable periodic state is located near the center of the bed. Stable periodic states are located at about Cleff = 0.2 and > 1, because the average value at large Cleff (actually at lIeff of q1 in the bed exceeds that in equilibrium with the feed for the adsorption step). The dashed lines indicate the state of the system after 30 cycles for initial conditions of = 0.4 and 0.8. Bed profiles are shown in Figures 12 and 13 for the periodic states with shallow and deep penetration of benzene, respectively. These are similar in many respects to Figures 9 and 10 for the two-step cycle. For the adsorption steps, incomplete wave interactions remain in the beds. The benzene profile for the shallow periodic state is remotely similar to that in Figure 1, and the deep periodic state again shows some subcooling. During heating, the principal thermal wave advances only about one-third
630 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990
Y 5
\ -
0
E
5r
Y
0 -
400
U
0
0
2
6
4
8
0
IO
C y c l e Number
I
I
-
Figure 11. Development of periodic states determined by repetitive
m
cdculation of the cycle steps for the three-step cycle with T h = 22 and 7, = 100. An unstable periodic state exists near the center of the bed. = 0.2 and a t large The two Stable periodic states exist near dashed lines on the right margin show the conditions of the bed after 30 cycles for beds beginning with = 0.4 and 0.8.
rl
5r
A 299
Figure 13. Periodic state for the three-step cycle with
7 h = 22 and = 100 for deep penetration of the heavy component into the bed. Bed profiles are shown a t the end of the adsorption step (top), heating step (middle), and cooling step (bottom). The adsorption time is T , = 520. T~
0
f
I
3
f
I
i 300
1
0
296 0
z
Figure 12. Periodic state for the three-step cycle with
7h = 22 and = 100 for shallow penetration of the heavy component into the bed, Bed profiles are shown a t the end of the adsorption step (top), heating step (middle), and cooling step (bottom). The adsorption time is T~ = 470. T,
of the way through the bed, but it is then followed by the cooling wave, which sweeps the heated zone on through the bed. Again, for the deep periodic state, extensive roll-up occurs for benzene during regeneration. It is interesting to note that, for the deep periodic state, the bed contains very little cyclohexane. Adsorption times for the shallow and deep periodic states are T~ = 470 and 520, respectively. Discussion The location of multiple periodic states has involved a two-step process: a preliminary analysis followed by calculation of repetitive cycles. The preliminary analysis was found to give an accurate prediction of the behavior
of cycles with moderately long heating times, for which only a single periodic state exists. For relatively short heating times, the manner in which the bed would load during the adsorption step was not known a priori, and it was necessary to carry out calculations of repetitive cycles to predict performance. Nevertheless, the knowledge of the solution structure gained from the preliminary analysis and the feeds identified there were very important in identifying the multiple periodic states. Furthermore, the structure of the solution found by calculation of repetitive cycles is qualitatively that obtained by the preliminary analysis and shown in Figure 5. To determine the conditions for the existence of multiple periodic states in the calculation of repetitive cycles, it was necessary to decrease the heating time relative to that in the preliminary analysis primarily because the bed was not loading on adsorption as shown in Figure 1 and secondarily because some material was lost during the 5% breakthroughs. With a long heating time, periodic states were shown to be approached quickly, whereas for a short heating time they were approached more slowly. This should be expected-the more material removed from the bed during regeneration, the more the system responds to the adsorption feed in a single cycle. If regeneration were carried out to completion, then the periodic state would be approached in a single cycle, regardless of the initial condition of the bed. Above, the concept of an economic optimum has been alluded to. We have previously considered the minimization of the ratio T ~ / T ,to determine the optimal way in which to operate a cycle to minimize the cost of heating purge gas (Davis et al., 1988; Davis and LeVan, 1989). For the system considered here, Davis et al. found a minimum a t ?h = 120 for a pressure of 1.0 MPa but different feed conditions. Based on our preliminary analysis, roughly equal minima, with ?h/T, = 0.7, occur a t Th = 100 (for the deep periodic state) and new = 150. These pertain only to the two-step cycle. An optimization based on analysis
Ind. Eng. Chem. Res., Vol. 29, No. 4,1990 631 of repetitive cycles would involve an extensive study of the behavior of the system for many different heating times, and this has not been done. It is interesting, however, that the preliminary analysis gave an optimum not only in the region of multiplicity but for the deep penetration case. Note that in Figure 5 the deep penetration branch is not the branch of stable solutions connected to the periodic states for long regeneration times. Thus, it is surprising that an optimum could be associated with this fairly difficult to identify and perhaps difficult to reach periodic state. The calculations for repetitive cycles lead to the same conclusion. Based on the adsorption times cited above in association with Figures 9 and 10 for the two-step cycle and Figures 12 and 13 for the three-step cycle, the periodic state with deep penetration is preferable. This paper has been concerned with multiple periodic states for recovery of two components present in a feed, rather than for separation of the two components. It is expected that multiple periodic states exist for other applications of thermal swing adsorption. For example, a direct extension of this work would involve the separation of a three-component mixture, with two components recovered in the bed. Conclusions Multiple, stable, periodic states have been predicted for the thermal swing adsorption cycle for adsorptive recovery of two components. For a single heating time, two different adsorption times may exist, corresponding to different depths of penetration of the heavy component into the bed a t breakthrough of the light component. The state approached will be determined by the condition of the bed a t the start of the cycling. Acknowledgment This research was supported by the National Science Foundation under Grant CBT-8417673. Nomenclature A = surface area of adsorbent, m2/kg c = vapor-phase concentration, mol/m3 c p = vapor-phase heat capacity, kJ/(mol K) c, = heat capacity of adsorbent, kJ/(kg K) hf = enthalpy of gas phase, kJ/mol K = constant in Langmuir isotherm P = total pressure, MPa Pi = partial pressure of component i, MPa q = adsorbed-phase concentration, mol/kg Q = Langmuir monolayer capacity, mol/kg R = gas constant t = time, s T = temperature, K Trer= reference temperature, K uf = internal energy of gas phase, kJ/mol u, = internal energy of stationary phase, kJ/kg
u = interstitial velocity, m/s x = mole fraction in adsorbed z = axial coordinate, m
phase
Greek Letters t = void fraction of packing t’ = volume fraction occupied by vapor { = dimensionless axial coordinate
= depth of penetration of heavy component into bed X = heat of desorption, kJ/mol A
Pb
= spreading pressure, kJ/m2 = bulk density of packing, kg/m3
T = dimensionless time 4 = fractional saturation of vapor phase
Subscripts a = adsorption step c = cooling step h = heating step I = inert carrier gas (nitrogen)
1 = heavy component (benzene) 2 = light component (cyclohexane)
Literature Cited Basmadjian, D. On the Possibility of Omitting the Cooling Step in Thermal Gas Adsorption Cycles. Can. J. Chem. Eng. 1975,53, 234. Basmadjian, D.; Ha, D.; Pan, C. Y. Nonisothermal Desorption by Gas Purge of Single Solutes in Fixed-Bed Adsorbers. Ind. Eng. Chem. Process Des. Deu. 1975, 14, 328. Davis, M. M.; LeVan, M. D. Equilibrium Theory for Complete Adiabatic Adsorption Cycles. AIChE J . 1987, 33, 470. Davis, M. M.; LeVan, M. D. Experiments on Optimization of Thermal Swing Adsorption. Ind. Eng. Chem. Res. 1989, 28, 778. Davis, M. M.; McAvoy, R. L., Jr.; LeVan, M. D. Periodic States for Thermal Swing Adsorption of Gas Mixtures. Ind. Eng. Chem. Res. 1988, 27, 1229. de Boer, J. H. The Dynamical Character of Adsorption, 2nd ed.; Oxford: London, 1968. Farooq, S.; Hassan, M. M.; Ruthven, D. M. Heat Effects in Pressure Swing Adsorption Systems. Chem. Eng. Sci. 1988, 43, 1017. James, D. H.; Phillips, C. S. G. The Chromatography of Gases and Vapors. 111: The Determination of Adsorption Isotherms. J . Chem. SOC.1954, 1066. LeVan, M. D. Thermal Swing Adsorption: Regeneration, Cyclic Behavior, and Optimization. In Adsorption: Science and Technology;Rodrigues, A. E., LeVan, M. D., Tondeur, D., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1989; pp 339-355. LeVan, M. D.; Vermeulen, T. Binary Langmuir and Freundlich Isotherms for Ideal Adsorbed Solutions. J . Phys. Chem. 1981,85, 3247. Myers, A. L.; Prausnitz, J. M. Thermodynamics of Mixed-Gas Adsorption. AIChE J . 1965, 11, 121. Rhee, H. K.; Heerdt, E. D.; Amundson, N. R. An Analysis of an Adiabatic Adsorption Column. Chem. Eng. J . 1972, 3, 22. Suh, S. S.; Wankat, P. C. Combined Cocurrent-Countercurrent Blowdown Cycle in Pressure Swing Adsorption. AIChE J . 1989, 35, 523.
Received for review August 14, 1989 Accepted December 18, 1989
