288
Ind. Eng. Chem. Res. 1995,34,288-301
Design of Optimal Operating Conditions of Simulated Moving Bed Adsorptive Separation Units Giuseppe Storti? Renato Baciocchi, Marco Mazzotti, and Massimo Morbidelli' Dipartimento di Chimica Fisica Applicata, Politecnico di Milano, Piazza Leonard0 da Vinci 32, 20133 Milano, Italy
The design of the optimal operating conditions for simulated moving bed (SMB) adsorptive separation units is considered. A procedure for the a priori selection of the operating conditions to achieve an assigned separation requirement is developed in the frame of equilibrium theory for the equivalent four section countercurrent unit, using a model where the adsorption equilibria are described through the constant selectivity stoichiometric model, while both mass transfer resistance and axial dispersion are neglected. The space of the operating parameters, i.e. the mass flow rate ratios mj,is divided in regions with different separation regimes. Curves a t constant outlets purity and recovery are drawn in the (m2,m3)plane. The introduction of three performance parameters, desorbent requirement, adsorbent requirement, and productivity, allows the development of a procedure for the design of optimal operating conditions. This procedure is completed, accounting for the effect of the switching time on the separation performances, with a detailed model of the SMB unit, considering both axial dispersion and mass transfer resistance. This result constitutes a useful tool for determining the range of operating conditions to achieve an assigned separation requirement and then for selecting the optimal operating condition within this range. 1. Introduction
Adsorption separation continuous processes are usually preferred to batch processes, because in general they allow one to achieve better performances (Ruthven, 1984; Ruthven and Ching, 1989; Storti et al., 1993a). The simulated moving bed (SMB) technology makes it possible to achieve the separation performance of a true continuous countercurrent unit (TCC), while avoiding the difficulties in the movement of the solid phase. In SMB units the solid phase is stationary and the movement of the solid is simulated by moving periodically the inlet and outlet ports in the same direction as the fluid flow. Let us consider the TCC unit reported in Figure 1. This is divided in four sections, each one having a specific task. Thus, with reference to a binary mixture, in sections 2 and 3 the two components are separated from each other, whereas in sections 1 and 4 the adsorbent and desorbent, respectively, are regenerated and the enrichment of the extract and raffinate streams is also performed. The scheme of an equivalent SMB unit is shown in Figure 2. Each section of the unit is divided in subsections in order to simulate the movement of the solid stream in the true countercurrent unit. In particular, the case illustrated in the figure involves a 5-1-3-3 configuration,with a total of 12 subsections. The geometric and kinematic equivalence between TCC and SMB units is given by
Lj = L?lj us = -(1 L t*
* To whom
(1)
-E)
correspondence should be addressed. E-mail:
[email protected]. Dipartimento di Chimica Inorganica, Metallorganica e Analitica. Universita degli Studi di Padova. Via Marzolo, 1. 35131 Padova, Italy.
SECTION
ADSORPTION OF 0
4
rl A;-.To +B GI1, SECTION
FEED
1 1 SECTION
,
ADSORPTION OF A
: 2
;
DESORPTION OF B
EXTRACT
SECT I O N
1
DESORPTION OF A
Figure 1. Scheme of a four section TCC separation unit.
uj
E +us = lPJ I--€
(3)
where u sis the solid velocity and Lj and uj are the length and the fluid superficialvelocity, respectively, in the j t h section of the TCC unit; L is the length of each subsection; t* is the switching time (i.e. the time between two instantaneous displacements of the feed and withdrawal locations); and nj and u*j are the
0888-5885/95/2634-0288$09.00/00 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 289
S
r
T
1
i
n
o
T
-->X
r
,
i
Figure 3. Scheme of the single section of a countercurrent adsorptive unit.
I"'
t
Figure 2. Scheme of an SMB separation unit with 5-1-3-3 configuration.
number of subsections and the fluid superficial velocity, respectively, in thejth section of the SMB unit. Both SMB and TCC units can be simulated with two different approaches. The first one involves a detailed model, accounting for dispersion phenomena and mass transfer resistances, which gives an accurate simulation of the unit behavior. The second approach, based on the equilibrium theory (i.e. neglecting both masstransport resistance and axial dispersion), provides an explicit solution which allows one to fully characterize the qualitative behavior of the separation. Unfortunately, such a solution is available only for the TCC configuration (c.f. Storti et al., 1988). Thus, in general, it is convenient to analyze the TCC unit behavior using the equilibrium theory solution and then to transfer the obtained results to the corresponding SMB unit, using the equivalence conditions 1-3. For a more accurate analysis of the SMB performance we need to use the detailed model (i.e. the first approach) and to solve it numerically. In previous papers (Storti et al., 1993b; Mazzotti et al., 1994)we have addressed the problem of determining the optimal operating conditions for a TCC unit or for the equivalent SMB one. It was shown that the key operating parameters for the design of these units are the ratios between the net fluid mass flow rate and the adsorbed phase mass flow rate, which, with reference to thejth section, are given by
for the TCC unit (4) for the SMB unit
m.=
The optimal design problem was solved in the case of complete separation, which implies both absolute purity and complete recovery of the desired components, i.e. the weak components appear only in the raffinate, while the strong components appear only in the extract. In particular, the boundaries of the region in the operating parameter space mj,j = 1-4, which assures the achievement of complete separation were determined. However, in applications we are not interested in complete separation per se, but rather in those conditions which are economically most convenient. Often these happen to include situations where the aim of the separation is to recover as much as possible, but not all, of a given component with very high purity, i.e. we have a strong purity requirement only in the raffinate or in the extract, not in both. Since in applications many such situations arise, in this work we analyze the performance of the separation unit in all possible operating conditions. The aim is to cover all possible situations which may arise in applications. In particular, using equilibrium theory, we characterize the separation performances of a wide range of operating conditions in terms of two parameters: purity and recovery of the outlet streams. This allows us to select a priori the range of operating conditions which makes it possible to meet certain given separation requirements, only on the basis of the knowledge of the equilibrium properties and the feed composition of the system under examination. The choice of the optimal operating condition among those compatible with the assigned separation requirement can be performed on the basis of economic considerations. For this, three performance parameters are introduced accounting for desorbent requirement, adsorbent requirement, and productivity of the unit. These parameters are evaluated for each operating condition so that the optimal one can be selected by finding the best compromise between them. This is done using a detailed model, which accounts for the effect of axial dispersion and mass transfer resistance. The final result is a procedure that makes it possible to select the optimal operating conditions for SMB units, through the characterization of the separation performances (purity and recovery) and of the process economics (desorbent requirement, adsorbent requirement, and productivity of the unit) for each operating condition.
2. The Region of Complete Separation
Gjt* Q&(I
(5)
- E*)r-
where uj and us are the fluid and solid superficial velocities in the TCC unit; Gj and A, the fluid flow rate and the column section in the SMB unit; ef and es,the fluid and solid density, respectively; E* = [ E (1- E ) E ~ I , being E and ep the inter- and intraparticle void fractions, respectively; and r" is the adsorbed phase saturation concentration.
With reference to the genericj t h section of a TCC unit shown in Figure 3, the mass balance according to equilibrium theory is given by (Storti et al., 1989)
+
with the following boundary and initial conditions
290 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995
yf(z,O)=
(z = 1, ...,NC)
(7)
$(z,l) =
(i = 1, ..., NC)
(8)
$(O,x) = ($)'
(z = 1,...,NC)
(9)
and the nonlinear Langmuir constant selectivity equilibrium model (10) where yi and bi are the mole fractions of component i in the fluid phase and adsorbed phase, respectively; Ki is the corresponding equilibrium factor; d= esP/ef)is a phase capacity ratio and NC the number of components, including the desorbent (i.e. NC = N 1). In the frame of equilibrium theory, a set of N characteristic parameters wk(k = 1, ..., N) corresponds t o each set of compositions in the fluid and adsorbed phases. The steady state solution for a single countercurrent section is given by a constant state with a discontinuity a t either one or both ends of the column. Let Qa and Qb be the sets of characteristic parameters associated with the compositions of the incoming fluid and solid streams, respectively, i.e.:
+
(11) b b b ab= (wI,w2,...,wN)
(12)
Then, the constant state prevailing in the column at steady state is given by
(13) Depending upon the ratio between the fluid and adsorbed phase flow rates (i.e. the parameter mj given by eq 41, the index k can take one of the following values, thus identifying the constant state prevailing at steady state conditions: (1)k = 0. The state prevailing in the column is that of the incoming solid stream. (2) 0 < k < N . The state prevailing in the column is intermediate between those of the two incoming streams. (3) k = N . The state prevailing in the column is that of the incoming fluid stream. In general, when dealing with multicomponent mixtures, the separation processes are aimed a t dividing the feed stream in two fractions, so as to recover a component, i.e. the strong key (sk), in the extract stream and another component, i.e. the weak key (wk), in the raffinate. The adsorptivities of the key components with respect to that of the other components of the feed mixture can be ordered as follows:
The aforementioned separation is achieved through the following process requirements: the concentrations of component sk in the raffinate and of component wk in the extract must be smaller than a prescribed maximum value, say 0.1% weight fraction. Thus, we obtain an extract stream which contains component wk and the more adsorbable ones, i.e. E = {wk,sk,...,IT), whereas the raflnate stream contains component sk and the less adsorbable ones, i.e. R = {1,2,...,wk,sk). In
the frame of equilibrium theory, it is possible to reduce to zero the maximum allowable concentration of components wk and sk in the extract and raffinate, respectively. This corresponds to complete separation described above, i.e. E = {sk,...JV) and R = {1,2,...,wk}. The region of operating conditions for which complete separation is achieved can be computed in terms of the mass flow rate ratios mj in the four sections of the SMB unit, in the case of both binary (Storti et al., 1993b) and multicomponent (Mazzotti et al., 1994) separations.
3. Performance Parameters In this section, using the equilibrium theory model, we investigate the performance of SMB units in the case where only one of the outlet streams (raffinate or extract) has to satisfy a strict purity specification. This requirement, which is less stringent than complete separation examined earlier (Storti et al., 1993b; Mazzotti et al., 1994), is indeed the most common one in applications. The aim is to fully characterize the unit operating conditions in terms of two performance parameters: purity and recovery, so as to allow for the a priori selection of the range of operating conditions feasible for any given separation requirement. Once this range has been determined, the choice of the specific optimal operating conditions can be performed on the basis of economic considerations. In particular, we have assumed the separation costs to be a linear function of both the adsorbent and the desorbent requirements, as well as of the ratio between the feed flow rate and the volume of the unit, defined as the productivity of the process. The separation performance has been characterized in terms of these parameters so as t o select the optimal operating conditions among those feasible with assigned purity and recovery of the outgoing streams. 3.1. Purity. Let us consider the case of an N component mixture plus the desorbent, where the equilibrium constants of the components of the feed mixture are ordered as in eq 14. We assume that the separation requirement is to recover component sk in the extract and to separate it from component wk and the weaker ones. This is equivalent to minimizing the concentration of component wk in the extract, i.e. to maximize the purity of the extract, defined as follows: V N
-.E
(15) Similarly, if the separation requirement is to recover component wk in the raffinate, we are interested in maximizing the purity of the raffinate defined by
(16) According to these definitions, the extract stream is pure when PE= loo%, Le., if it does not contain components less adsorbable than the strong key one, whereas the raffinate is pure when PR= loo%, i.e., if it does not contain components more adsorbable than the weak key one. In the case of a binary mixture, with KA > KB,if A has to be recovered in the extract stream and B in the raffinate, the two purity parameters defined above
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 291 2.0
1.5
E"
1.0
COMPLETE SEPARATION
n NO 0.5
1.0
SEPARATION
1.5
"0
2.0
0.5
1.0
1.5
2.0
"2
m2
Figure 4. Regions of the (mz,m3)plane with different separation regimes in terms of purity of the outlet streams: y: = y i = 0.5, KA = 1.95, KB = 1,KD = 1.5.
Figure 5. Regions with different separation regimes and curves at constant extract (-1 and raffinate (- -1 purity in the plane: yfk = yg = 0.5, KA = 1.95, Kg = 1,KD = 1.5.
(m2,m3)
is described by a characteristic parameter, QF, defined by
become
(17)
(18) The extract stream is pure (i.e., PE = loo%), if it does not contain component B, whereas the raffinate is pure (Le., PR = loo%), if it does not contain component A. In general, in a separation process, a desorbent with any adsorptivity with respect to those of the components to be separated can be used. Depending on the relative adsorptivity of the desorbent and the feed components, the following situations may occur: (1)weak desorbent, i.e. KD < K I ; (2) weak intermediate desorbent, i.e. K1 KD Kwk; ( 3 )intermediate desorbent, i.e. Kwk KD Ksk; (4)strong intermediate desorbent, i.e. Ksk KD KN; (5) strong desorbent, i.e. KN KD. As mentioned in the previous section, using equilibrium theory we can determine the operating conditions in the flow rate ratio parameter space mj,j = 1, ..., 4,which lead to complete separation of an N component mixture, using desorbents with various adsorptivity values (Mazzotti et al., 1994). Such conditions, in the case of sections 1and 4 reduce to a lower bound for ml and an upper bound for m4, which can be evaluated explicitly. These assure that both the fluid leaving section 4 as well as the solid leaving section 1are fully regenerated, i.e., they do not contain components from the feed stream. Accordingly, in the following we analyze the effect of m2 and m3 on the process performance, whereas both ml and m4 are kept constant such as to satisfy the above-mentioned explicit constraints. Let us first consider the separation of a binary mixture with intermediate desorbent, i.e., KB < KD KA where A and B are the feed components and D is the desorbent. Using equilibrium theory we obtain the results shown in Figures 4 and 5, in terms of purity of the outlet streams. It is worth noticing that in the context of the equilibrium theory the feed composition
(19) The equilibrium theory results shown in Figure 4 identify a specific region in the (m2,m3)plane where the separation is possible. In particular, there exists an upper limit for m2, given by m2 = KA/KD and a lower limit for m3, given by m3 = K$KD. For m2 values larger than KA/KD, the state prevailing in sections 1 and 2 is the pure desorbent, and similarly, for m3 values lower than K$KD the state in sections 3 and 4 is again the pure desorbent. In both cases no separation takes place. Therefore, the region in the (m2,m3)plane, where the separation can be performed is constrained by m2 KA/ KD and m3 > KB/KD,together with the condition m3 > m2, corresponding to feed flow rates greter than zero. Moreover, we find that the region in the (m2,m3)plane where the separation can be performed, can be further divided in four regions, depending on the purity values of the outlet streams. With reference to Figure 4, the following regions can be identified: (1) Complete separation region. The extract and raffinate streams are pure. The boundaries of this region can be evaluated analitically (Storti et al., 1993b). (2) Pure extract region. This region includes the complete separation region and is limited by the vertical lines m2 = on the left and m2 = KAIKD,on the right hand side. For all the operating conditions corresponding to points inside this region, pure extract is obtained. (3) Pure raffnate region. This region includes the complete separation region and is bounded by the horizontal lines m3 = K$KD from below and m3 = QF/ KB from above. All the operating points inside this region lead to pure raffinate. (4) No purity region. It is a rectangular region defined by m3 > QF/KB and m2 < QF/KA;in this region neither the extract nor the raffinate streams are pure. A better understanding of the unit behavior can be achieved by considering the effect of the operating parameters m2 and m3 on the purity of the outlet streams. For this the curves corresponding to constant
292 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995
extract (continuous curves) and raffinate purity (broken curves) have been reported in Figure 5. Let us first consider the pure extract region. We see that by decreasing m2 toward the vertical line through point W, while keeping m3 constant and larger than QFIKB, the raffinate purity increases. In this region the constant state prevailing in section 2 is that intermediate between the states corresponding to the inlet fluid and solid stream. When this vertical line is crossed, such constant state becomes that of the inlet solid stream so that raffinate purity becomes independent of mz. A similar conclusion can be drawn by examining the behavior of constant extract purity curves in the pure raffinate region, with m2 smaller than QFIKA, The extract purity increases for constant mz and increasing values of m3 up to the horizontal line through point W. When this horizontal line is crossed, the constant state in section 3 shifts from the intermediate one t o that of the fluid inlet stream and the extract purity becomes independent of m3. 3.2. Recovery. The second feature, beside purity, which characterizes the performance of a separation process is the amount of desired component which is recovered in each outlet stream. Let us consider, in the frame of equilibrium theory, the three following possible objectives of a multicomponent separation process: (1)Pure extract. Purity of the extract is equal to 100%. (2) Pure raffinate. Purity of the raffinate is equal to 100%. (3) Complete separation. Purity of both extract and raffinate is equal to 100%. When the objective is pure extract, this must contain only components whose adsorptivity is Ki 1 K s k . The question is what is the fraction of these components which is recovered in the extract and how much is left in the rafiinate. For this we introduce the parameter recovery in the extract stream, RE,which is defined with reference to the strong key component, as follows:
where yfk is the mole fraction of the strong key component in the extract stream and E is the extract flow rate, whereas "&is the mole fraction of the same component in the feed stream and F is the feed flow rate. Note that this definition has been preferred to an alternative one, which accounts for all components recovered in the extract stream, as it is the case for the definition of purity (eq 15). The reason is that by monitoring only the less adsorbable component, i.e., the strong key, we obtain a more sensitive index of the separation performance.
If pure raffinate is the objective of the separation, then it will contain only the components whose equilibrium constants are Ki IKwk, i.e. the weak key is the most adsorbable component recovered in the raffinate stream. In this case, the parameter recovery in the raffinate stream, RR, is defined with reference t o the weak key component, as follows:
(21) where the variables have a similar meaning as in the case of RE defined by eq 20.
O'T/
"0
0.5
1.0
1.5
2.0
"2
Figure 6. Regions with different separation regimes and curves at constant extract (-1 and raflinate (- -) recovery in the (??22,??23) plane: y T ; = y ~ = 0 . 5 , K ~ = 1 . 9 5 , K g = l , K ~ = 1 . 5 .
Let us consider as an example the separation of a binary mixture (components A and B) with intermediate desorbent (componentD). Using equilibrium theory, the recovery of A in the extract and that of B in the r f i n a t e have been calculated for various operating conditions, i.e. by changing the flow rate ratios in sections 2 and 3, while keeping the flow rate ratios in sections 1 and 4 constant and fulfilling the explicit constraints mentioned above. The obtained results are shown in Figure 6 in terms of curves with constant extract (continuous curves) or rafinate (broken curves) recovery in the (m2,m3)plane. Let us first consider the region of pure extract. It can be seen that, as expected, RE is larger when the operating conditions are close to the complete separation region. When moving away from this region, for example by increasing m3 while keeping m2 constant, the recovery of component A in the extract stream decreases. A similar behavior is exhibited by the recovery of component B in the raffinate stream in the region of pure r a n a t e . The largest recovery values are obtained for operating conditions close to the complete separation region. For decreasing values of m2, with constant m3, the recovery of component B in the raffinate stream decreases. 3.3. Role of the Desorbent. All the results presented in the previous sections refer to separations involving an intermediate desorbent. In general, this situation is optimal, since the desorbent has to perform the followingtwo tasks (Johnson and Kabza, 1993): (1) to be adsorbed in section 1 (see Figure 1) so as to facilitate the desorption of the strong components and then convey them all t o the extract port (for this aim, a strong desorbent is the most efficient); and (2) to be desorbed in section 4 so as t o facilitate the adsorption of the light components and then convey them all to the raffinate port (for this purpose, a weak desorbent is recommended). Therefore, the choice of an intermediate desorbent represents a good compromise between these two requirements. However, in some cases it may be convenient to use a desorbent whose adsorptivity is not intermediate between that of the two components to be separated. For example, larger recovery values in the extract stream are obtained using a strong desorbent, while a weak desorbent increases the recovery in the raffinate stream (Storti et al., 1989).
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 293
' i
0 . 5 K 0.5
, 1.0
the complete separation region: (intermediate desorbent) yF - 0.5,KA = 1.95,KB = 1, KD = 1.5;(strong desorbent) y41 4: YB 0.5,KA - 1.50,KB = 1, KD = 1.95;(weak desorbent)yA 0.5,K A - 1.95,KB = 1.5,KD = 1.
1.5
2.0
2.5
m2
m2
Figure 7. Effect of the desorbent on the shape and position of
l
Figure 8. Curves at constant extract and raffinate recovery, in the case of weak desorbent: y: = yi = 0.5,KA = 1.95,KB = 1.5, KD = 1.
51 z
In order to properly define the role of the desorbent in determining the separation performance, let us consider two different cases: one involving a strong desorbent, i.e. a desorbent whose adsorptivity is larger than that of all the feed components, and one involving a weak desorbent, i.e. a desorbent whose adsorptivity is lower than that of all the feed components. In both cases, we characterize the binary separation performance by comparison with the case of intermediate desorbent. We first note that a change in the adsorptivity of the desorbent strongly affects both the shape and the position of the complete separation region, as shown in Figure 7. By taking the intermediate desorbent case as a reference, we see that, if a weak desorbent is used, the complete separation region is shifted to the right in the (m2,m3)plane, so that larger values of 771.2 and m3 are needed to obtain complete separation. On the other hand, if a strong desorbent is used, the opposite behavior is observed, i.e. smaller values of m2 and m3 are needed to achieve complete separation. The constant recovery curves in the (m2,m3)plane in the cases of weak and strong desorbent have been reported in Figures 8 and 9, respectively. It can be seen that the shape of these curves changes in a similar fashion as the shape of the corresponding complete separation region. Besides, we note that when a weak desorbent is used (Figure 81, the curves characterized by large extract recovery values are very close to the complete separation region. Thus, in this region very small changes in the flow ratios m2 and m3 may produce large variations of the extract recovery and then of the separation performances. On the other hand, if a strong desorbent (Figure 9) is used, the curves corresponding to large values of the extract recovery are far from the complete separation region and well separated from each other. Accordingly, the separation performance is less sensitive to disturbances in the flow-rate ratios, and then we have, at least from this point of view, the most robust operation. Thus summarizing, if a component has t o be recovered in the extract stream, it is possible to increase the robustness of the operation using a strong desorbent. Following similar arguments, we can
L
OO
0.5
1.0
1.5
1
m2
Figure 9. Curves at constant extract and raffinate recovery, in the case of strong desorbent: y: = yi = 0.5,K A= 1.50,KB= 1,KD = 1.95.
show that if the desired component has to be collected in the raffinate stream, the most robust operation is achieved when a weak desorbent is used. 3.4. Multicomponent Separation. In the previous sections we have always considered binary separations. These allow a better understanding of the main parameters affecting the separation performance and provide a valid reference case for the characterization of more complex systems. However, in applications high purity separation of mixtures consisting of more than two components is often of interest. Thus, we now consider the case of multicomponent mixtures using the same numerical procedure previously adopted for the binary case. Using the equilibrium theory solution for multicomponent mixture, we can evaluate the separation performance corresponding to each operating condition. As for the separation of a binary mixture, various operating conditions have been analyzed, corresponding to different values of the mass flow rate ratios in sections 2 and 3, while ml and m4 are kept constant and fulfilling the explicit constraints for pure outlet adsorbent and desorbent, respectively (Mazzotti et al., 1994). This allows
294 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995
Curve b. If the operating conditions are chosen below this curve, the component C alone is recovered in the railhate, whereas both components B and C are recovered in the raffinate if the operating conditions are chosen above it. Curve c . For operating conditions lying on the left hand side of this curve, all three components are recovered in the extract stream, whereas component C is fully recovered in the raffinate stream if the operating conditions are chosen on its right hand side. Curve d . On the right hand side we have component A alone recovered in the extract stream, whereas on the left hand side we have both components A and B in the extract. Through the determination of the curves above, we can identify the following six separation regimes in Figure 10. Region I (pure extract). In this region, whose boundaries are provided by curve d on the left and m2 = &/ KD on the right, component A alone is collected in the extract stream, while all three components are present in the raffinate. Region 2 (pure raffinate). In this region, bounded above by curve b and below by m3 = K&D, component C alone is collected in the raffinate, while all three components are present in the extract. Moving away from the region of pure extract, by crossing line d, the purity is lost and some of component B is collected in the extract stream. In a similar way, moving away from the region of pure raffinate, by crossing line b, purity of the raffinate is lost and some of component B is recovered in the raffinate. Thus, we can characterize the separation performance of the unit in regions 3 and 4 as follows: Region 3. All three components are present in the raffinate, while the extract stream contains only components A and B. Region 4 . The extract stream contains all the feed stream components, while the raffinate contains only B and C. By further analyzing the relative position of the four curves defined above, we define the two remaining regions which complete the possible separation regimes of the unit. Region 5. The extract stream contains components A and B and the raffinate stream components B and C . Region 6 . When line a is crossed upward, component A shows up in the raffinate and similarly when line c is crossed leftward, component C shows up in the extract. Thus, in this region all three components are collected in both the outlet streams. From the analysis above, it is clear that the operating conditions of interest in applications are usually those corresponding to regions 1and 2 where one component, A or C respectively, is produced pure with a more or less high percentage of recovery from the feed mixture. However in some applications we may be interested in splitting the original mixture in different ways, thus asking for the unit t o be operated under conditions corresponding to another one of the regions defined above. Note that the analysis above can be made quantitatively complete by adding the curves with constant purity and recovery. However, the obtained results are in full similarity with those described above in the case of binary separations. Thus here, we do not discuss this aspect further.
I
OO
0.5
1.0
1.5
2 .o
m2
Figure 10. Predicted regions with different separation regimes for a three component mixture. Coordinates of the points: A (KA/ KD,K ~ K DB) ,VWKD,KBIKD),C (KJKD,KJKD). y i = 0.253;yi = 0.248; = 0.499; KA = 2.2; KB = 1.34; Kc = 1; KD = 1.50.
one to identify in the (m2,m3)plane various regions corresponding to different separation requirements. ks an example we consider the case of a ternary mixture. It is worth noticing that the results of the analysis reported by Mazzotti et al. (1994) allow one t o consider more complex situations. However the representation of the results becomes difficult without a corresponding conceptual gain. In Figure 10 the results of the characterization of the separation performance for a mixture of three components (A, B, and C) with desorbent D and the following equilibrium constants
(22) are shown. Note that we are interested only in the region in the (m2,m3) plane, where separations are possible, i.e. that delimited by m2 < KP/KD,m3 > Kd KD,and m3 > m2. Moreover, in this case (Mazzotti et al., 1994)we see that two regions of complete separation are present, corresponding to the two possible separations that can be performed. The first one corresponds to the separation of the most adsorbable component (A), recovered in the extract stream, from the other ones (B and C), which are left in the raffinate stream. The second one leads to the separation of the less adsorbable component (C) in the raffinate stream from the other ones (A and B), which are left in the extract stream. Obviously, there are no operating conditions which allow one to obtain pure component B. Using the equilibrium theory model, we can divide the (m2,ms) plane into six regions corresponding to different separation regimes for the unit. In particular, these are defined as follows repeatedly solving the equilibrium model with various m2 and m3 in order to identify the four characteristic curves which represent the boundaries of such regions. Without reporting the computational details, in the following we discuss only the final results. Curve a. For operating conditions corresponding to points lying above this curve, component A is present both in the raffinate and in the extract streams, whereas it is fully recovered in the extract if the operating conditions fall below the curve.
Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 295 in the (m2,m3)plane the curves corresponding to constant desorbent requirement (broken curves) are straight lines with unitary slope. Since Dr increases as (m3-m2) decreases, the minimum desorbent requirement is obtained for the maximum (m3-m2) value. Thus, recalling the shape of the constant recovery curves shown in Figure 6, it is clear that maximization of recovery and minimization of desorbent requirement cannot be achieved simultaneously. On the contrary, an increase of recovery implies an increase of desorbent requirement, while a lower D, value is obtained only at the expense of a lower recovery. Let us assume that the extract must be obtained pure with a given recovery, say RE, of the strong-key component. With reference to Figure 11, for a binary mixture, this means that the operating conditions compatible with this requirement lie on the curve at constant recovery RE, which connects A and W1 and belongs to the pure extract region. Among these operating conditions, the one with the lowest desorbent requirement is given by point WI. It can be readily seen that for any recovery requirement, the optimal operating conditions, with respect t o the desorbent requirement, lie on the vertical straight line originating from the point W. Then, this is the locus of the optimal operating points when pure extract is required. Finally, it is worthwhile noting that all these optimal operating conditions are sensitive to modifications of the flow rates inside the unit but, depending on which one of the flow rates changes, the consequences may be different. In particular, if m3 changes, the separation performance changes in terms of recovery, whereas the purity of the extract is maintained. On the other hand, if m2 changes, the purity of extract stream or recovery can be reduced depending upon the sign of the flow rate variation, negative or positive, respectively. Similar arguments can be applied to the case in wNch pure raffinate is required with a given recovery, say RR, of the weak key component. Again, with reference to Figure 11,the optimal operating point, selected Cmong those lying on the constant recovery curve RR,is represented by point W2. For any recovery requirement, the optimal operating point lies on the horizontal straight line originating from the point W. Thus, in this case, the purity of the raffinate may be lost if a variation of the flow rate in section 3 occurs, whereas a modification of m2 affects only the value of the recovery. 3.6. Adsorbent Requirement. In an SMB separation process the adsorbent inventory is an important factor in determining the economics of the process. The operating conditions affect the solid requirement as well as the deactivation process of the solid adsorbent which, once deactivated, has to be replaced. Accordingly, operating conditions have to be chosen in order to minimize the deactivation phenomena. Often, as for example in the case of the separation of hydrocarbons through adsorption on molecular sieves, the deactivation process is mainly due t o the formation of coke on the adsorbent due to cracking of hydrocarbon molecules. If a time on stream deactivation mechanism (Wojciechowsky, 1968) is assumed, deactivation is strongly affected by temperature and fluid flow. Thus, the key factor in the deactivation process is represented by the fluid flow rate inside the column: the larger this is, the more coke is produced and deposited on the adsorbent particles, thus resulting in larger deactivation rates. When the adsorbent undergoes a deactivation process, the required separation performance can still m4,
I/
I
I
I
where U F and U D are the fluid superficial velocities of the feed stream and of the fresh desorbent stream, respectively. With reference to Figure 1,the following relationships hold: U J J = u1 UF
u4
(24)
= u3 - u2
(25)
Substituting the definition of mj (eq 4) in eqs 24 and 25 and then in eq 23, the following expression for D, is obtained:
Dr = m1m3 -
m4
m2
(26)
As shown in Figure 11,for constant values of ml and
296 Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995
be achieved, by suitably changing the operating conditions. This "compensation" can be applied, until the decrease of the adsorbent performance is so large as to require the replacement of the adsorbent. The time between two successive replacements of the zeolite is defined as the lifetime of the adsorbent. In the frame of the deactivation mechanism described above, the lifetime of the adsorbent increases for decreasing values of the internal flow rates. On the other hand, by increasing the solid inventory, which is proportional to the unit volume V, a larger lifetime is obtained. Accordingly, the lifetime, tv, depends on both the internal fluid flow rate and the volume of the unit, as follows: (27) where G is the fluid flow rate inside the unit and the simplest functional relationship has been chosen, i.e. t, proportional to V and to the inverse of G . The latter can be evaluated for an SMB unit as a mean value among the flow rates inside the four sections of the unit. In order t o keep only the two independent variables, mg and m3, it is convenient to neglect the contribution of sections 1and 4,since the flow rates in these sections are the largest and the smallest, respectively, and t o adopt the following definition of G:
The mean fluid flow rate inside the unit can be written as a function of the mean fluid superficial velocity, as follows:
G = efAu
(29)
where the mean fluid superficial velocity can be evaluated as the mean arithmetic value among the superficial velocities in sections 2 and 3. Since the solid adsorbent has to be replaced after a time equal to its mean lifetime, t,, we can introduce a theoretical solid flow, 2, which is proportional t o the ratio between the quantity of adsorbent in the unit and its lifetime as follows:
(30) By substituting eq 27 in the above equation, the theoretical solid flow specific to the feed flow rate, Z,, is given by (31) where F and U F are the feed flow rate and the superficial velocity of the feed, respectively. We define the ratio in the right hand side of eq 31 as adsorbent requirement, A,. By expressing the average superficial velocity in terms of ug and u3,the A, parameter becomes
A, =
+
+
VE* (V/2)dl - c*)(m2 m3) Vutl - c*)(m3 - m,)
(32)
and the curves corresponding to constant solid requirement are represented in the (m2,m3) plane by the
201
I
I '
I
I'
,"/
I
0.5
''
I
/
'
I ,
/i
A
I
I
I
1.0
1.5
2.0
m2 Figure 12. Curves at constant adsorbent requirement in the (m2,m3) operating parameter plane, with physical parameters typical of vapor phase operations.
following equation:
M,+l 2E* m -3 -24, - i m 2 + 4 1 - E * X ~ A , - 1)
(33)
These curves are shown in Figure 12. It can be seen that these are straight lines whose slope and intercept on the m3 axis both depend upon the A, value. In particular, A, is lower for lines with larger slope, whereas the intercept on the m3 axis is very close to zero, when u is sufficiently large, which is the case for vapor phase operations. As in the case of the desorbent requirement, it is clear that recovery and adsorbent requirement cannot be optimized bimultaneously, i.e. increasing recovery can be achieved only with larger adsorbent requirement. Finally, with reference to the requirement of pure outlet stream (extract or raffinate) and given recovery of a key component (strong or weak), the procedure for the selection of the optimal operating condition with respect to A, is the same as that for D,, discussed in the previous section, since the adsorbent requirement increases as (m3 - m2) decreases as the desorbent requirement does. So, the locus of the optimal operating points with respect to A,, in the case of partial recovery of raffinate or extract, is given by the same two straight lines, originating from the point W and parallel to the m2 or the m3 axis, respectively. 3.7. Productivity. In general, we may need to solve two design problems: (1)design a separation unit for given separation requirements and feed flow rate; and (2) design the operating conditions for a given separation unit and process specifications. In both cases an optimal design procedure must account for the maximization of a productivity parameter, defined as the ratio between the feed flow rate and the volume of the unit:
p = -QF =- UF ' V L
(34)
where QFis the volumetric feed flow rate. Once Pr has been maximized, it is possible to solve both the design problems defined above, using eq 34. In particular, if the flow rate of the feed to be separated is known, then from eq 34 we can evaluate the volume of the unit,
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 297 whereas if a unit of given volume is available, then we can compute the flow rate of the feed to be separated. In order to evaluate the maximum productivity, it is useful t o express this parameter for the SMB unit as a function of mass flow rate ratios and switching time as follows:
P, = a(l - €*)
material balance of the ith component in the solid phase
equation for fluid velocity
boundary and initial conditions
>
0
(40)
t = 0; 0 < x < L (43) =y'$x) where yi and ypi are the fluid phase concentrations in the bulk and in the macropores, respectively, Pe and St y'pi
1
2
3
4
Pe St
2.13 33.4
2.33 69.0
2.17
2.70 176
60.1
(35)
material balance of the ith component in the fluid phase
x = 1;t
section
adimensional number
'A = 1.767 cm2; L = 100 cm; t* = 300 s.
m3 - m2
t* According to this equation and in the frame of equilibrium theory, the maximum value of P, is obtained by minimizing the switching time, i.e. for t* = 0. In practice, such a condition cannot be attained. This can be readily understood by recalling that the solid velocity of the equivalent TCC unit is linked to the SMB parameters through eq 2. Hence, if the switching time approaches zero, then the length of the unit reduces to zero, as well. This unrealistic result is due to the simplifying assumptions on which equilibrium theory is based. Actually, both mass transfer resistances and axial dispersion phenomena cannot be neglected when considering the effect of the switching time on the process performance, in particular for t* approaching zero. In order t o proceed with our analysis, we have to consider a detailed model of the unit. In particular, a lumped pore diffusion model of the SMB unit, accounting for both axial dispersion and intra- and interphase mass transfer resistances, has been adopted (Storti et aZ., 1988). The model equations can be written with reference to the generic j t h section of the SMB unit, as follows:
WiIaX = 0
Table 1. Values of the Dimensionless Peclet and Stanton in Each Section of the SMB Pilot PlanP
I
~
7
I -
\7'°0
{
995
\1 g61 I
98 k
I
94
1
i
92
,
1
90 0
I
1
2
3
4
Productivity, P,
Figure 13. Influence of the productivity on the separation performances for an operating point lying inside the pure extract region.
are the Peclet and Stanton numbers, and Ti is the adsorbed phase concentration. Note that equilibrium conditions between the fluid in the macropores and the adsorbed phase are assumed everywhere along the unit, i.e. rL= Eq(y,). The system of eqs 36-43 has been solved using suitable numerical techniques discussed elsewhere (Storti et al., 1988). Let us now illustrate the procedure for determining the optimal productivity, P,, using as example a binary separation with intermediate desorbent. The operating conditions have been selected so as to operate the unit in the region of complete separation, as described in the previous section in the context of equilibrium theory. As expected, using the detailed model above, which includes transport resistance and axial dispersion according to the parameter values Pe and St given in Table 1,the obtained extract and raffmate purity values are lower than loo%, i.e. than the value predicted by the equilibrium theory model. For example, in Figure 13 the values of the extract purity predicted by the detailed model are shown as a function of the process productivity, which from eq 35 implies changing values of the switching time, t*, with all the other parameters kept constant. It is seen that as the productivity value decreases to zero, the detailed model predicts a unit behavior in good agreement with the equilibrium theory model and the purity in the extract approaches 100%. However, for increasing productivity values the effect of transport resistance and axial dispersion becomes significant and the extract purity decreases. A similar behavior is observed, although not shown, for the raffinate purity. Thus, if we now select a desired value for the extract purity, say 99.5%, we can readily determine from the results shown in Figure 13 the maximum possible productivity value, and the corresponding minimum value for the switching time.
298 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 2
1 1
Pure Extract
1
,/
1
mw
E
G'
.
0
0.5
1.5
1
2
m2
Figure 14. Relative positions of the experimental operating pointa (0)with respect to the predicted regions corresponding to different separation performances for an equimolar &we of 49% p-xylene (A) and 51% m-xylene (B),using isopropylbenzene (D) as desorbent. Equilibrium parameters: KA = 1.95, KB = 1,KD = 1.5, and P = 0.135 g/g.
By substituting this productivity value in eq 34,the design problems defined at the beginning of this section are readily solved. Before developing the procedure for determining the optimal operating conditions, corresponding to minimum desorbent and adsorbent requirements, as well as to maximum productivity, it is convenient to discuss the reliability of the results of the analysis above by comparison with appropriate experimental data.
4. Comparison with Experimental Data In the previous sections we have characterized the separation performance of an SMB unit depending upon the location of the corresponding operating point in various regions of the parameter space (m2,m3).We now compare the results of this analysis with the actual separation performance of an SMB pilot plant with a 2-1-2-1 configuration operating in the vapor phase for the separation of mixtures of m-xylene,p-xylene,and ethylbenzene with various compositions, using isopropylbenzene as desorbent (Storti et al., 1992;Storti et al., 1993~). Let us first consider the separation of an equimolar mixture of m-xylene andp-xylene (m-Xy 51%, p-Xy 49%, which corresponds t o QF = 1.33). Using the results of the analysis above, we can compute for the system under examination the boundaries of the complete separation region, of the pure extract region and of the pure rafYinate one, as shown in Figure 14 using adsorption equilibrium parameters estimated through breakthrough experiments: KA= 1.95,KB = 1,and KD= 1.50, where A is p-xylene and B is m-xylene. For the same system, a set of experimental runs has been performed using the values of the operating parameters m2 and m3 shown in Figure 14 which lead to the separation performances reported in Table 2. Note that in all the experimental runs the adopted values for the parameters ml and m4 fulfill the corresponding explicit bounds given in the previous sections. It can be seen that complete separation has been achieved in runs F and H, where both p-xylene in the
Table 2. Experimental Performances of the SMB Pilot PlanP run B C F G H J
PE(%) 81.0 &(%)
69.0
>99.6 84.6
299.7 >99.9
89.5 >99.9
>99.8 299.9
>99.9 99.7
Separation of p-xylene from m-xylene: yFx = 0.49, y k = 0.51. Desorbent isopropylbenaene. T = 258 "C. P = 3 atm (Storti et al., 1993a).
raffinate and m-xylene in the extract are below the analytical detection limit. The separation performance measured in runs F and H is in good agreement with the position of the two corresponding operating points in the (m2,m3)plane, since these are both well inside the complete separation region. On the other hand, the points corresponding to runs B, C, G, and J lie outside this region, again in agreement with the experimental data in Table 2. In all these cases some of the undesired component is present in a t least one of the two outlet streams. In particular, inside the region of extract purity, the separation is not complete in the raffinate stream, as is the case for runs C and J. Similarly, the extract stream is not pure for operating conditions represented by points inside the region of pure r a f i a t e , as indicated by the separation performance measured in run G. Finally, the rather poor separation performance obtained in run B also agrees well with the model predictions, since the corresponding operating points lie inside the region of no purity for both the outlet streams. Let us now compare the predicted separation performances for a multicomponent mixture constituted of 49.9% m-xylene (B), 25.3% p-xylene (A),and ethylbenzene (C),using isopropylbenzene (D) as desorbent, with the actual performances obtained in the pilot plant. Using the following equilibrium parameters, i.e. KA = 2.2, KB = 1, KD = 1.5, and Kc = 1.34,we obtain, through the same analysis illustrated above for the case of Figure 10,the boundaries of the various separation regions in the (m2,m3)plane. The differences between the values of these equilibrium parameters from those of the binary case arise mainly because of the deviation that this system exhibits with respect to the constant selectivity equilibrium model adopted in this work. Moreover, in Figure 15 the two regions of complete separation and the regions where pure p-xylene and pure m-xylene are collected are shown together with the region where ethylbenzene is collected in both outlets. In the same figure the curves at constant extract and raffinate recovery are also shown, together with the points representing the operating conditions adopted in each experimental run. It can be seen that the experimental separation performances reported in Table 3 are in good agreement with those expected by considering the relative positions of the experimental points in the h 2 , m 3 ) plane. Small differences between predicted and experimental values are due to uncertainties in the estimation of the equilibrium parameters. However, at least in a qualitative way, the same arguments reported above in the case of the binary separation illustrated in Figure 14 can be repeated in this case. In particular, recalling that operating conditions corresponding to equal desorbent requirement are represented by straight lines with unitary slope in the (m2,m3)plane (see Figure 111, it can be seen that when complete separation is desired, the desorbent requirement is lower bounded by the value corresponding to point W, which at least from this point of view is then the optimal operating point.
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 299
0.5
1.0
1.5
2.0
2.5
"2
Figure 15. Relative positions of the experimental operating points (01,with respect to the predicted regions corresponding to different separation performances and the contour lines at constant extract recovery for a ternary mixture made up of 25.3% p-xylene (A), 49.9% m-xylene (C), and ethylbenzene (B),using isopropylbenzene as desorbent. Equilibrium parameters: KA= 2.20,KB = 1,KC = 1.34,KD = 1.5,and r" = 0.13g/g.
Table 3. Experimental Performances of the SMB Pilot Plant: (a) Separation of p-Xylene from m-Xylene and Ethylbenzene and (b) Separation of m-Xylene from p-Xylene and Ethylbenzene run
A
B1 B2
c1 c2 D
H A E F G2 H
PE (a)
RE (%)
a 54.0 96.7 99.5 98.0 98.7 100 62.2 b
100 100 80.8 57.4 31.0 41.4 100
89.5 100 99.6 99.8 82.3
95.6 17.8 29.4 83.1 98.6
a y L = 0.253,yEh = 0.248,y& = 0.499. Desorbent: isopropylbenzene. T = 250 "C. P = 3 atm (Storti et al., 1993~).
In practical applications, it may be convenient to decrease the desorbent requirement a t the expense of leaving some p-xylene in the raffinate, thus yielding values of p-xylene recovery in the extract less than 100%. This kind of performance is provided by operating conditions represented by points which lie inside the region of pure extract in Figure 15. Let us consider in particular the experimental runs B2, C1, C2, and D, which all achieve rather high p-xylene purity in the extract. It can be seen that by going from B2 to C1 and t o C2 and D the recovery of p-xylene in the extract substantially decreases, in good agreement with the position of the constant recovery curves, shown in Figure 15. On the other hand for the pairs of experimental runs Bl-B2 and Cl-C2, where the desorbent requirement is always constant, we see that larger recovery values correspond to lower p-xylene purities in the extract stream. It is worth pointing out that similar conclusions are
"2
Figure 16. Complete separation region and constant extract recovery curve in the (m2,m3)plane for a binary mixture (A B)
+
with intermediate desorbent. Relative position of the operating points corresponding to the calculated performances, shown in Figure 17. Optimal operating point (W3) with respect to D,, A,, and P,.
obtained by considering the adsorbent requirement that changes in the same direction as the desorbent requirement. 5. Design of Optimal Operating Conditions
The results obtained in the previous sections make it possible to characterize each operating condition, in terms of the corresponding performance parameters. Thus, if a separation requirement is assigned (i.e. purity and recovery of either extract or rafinate), it is possible to identify a range of operating conditions, which allows one to achieve such a specification. The optimal operating condition is then selected among these as that which represents the best compromise among the desorbent and adsorbent requirements and productivity. Let us consider, as an example a binary separation (A B), with intermediate desorbent (D).In this section we discuss the procedure for the selection of the optimal operating conditions in the case where purity is required only in the extract stream with partial recovery, say 80%, of component A. In Figure 16 the complete separation region for the system under examination is shown in the ( m ~ , ? ? Z 3 ) plane, together with the pure extract region and the curve AW1 corresponding to constant recovery in the extract, RE = 80%. Recalling from sections 3.5 and 3.6 that both the desorbent requirement, Dr, and the adsorbent requirement, A,, decrease as (m3 - mz) increases, we conclude that at least with respect to these performance parameters, the optimal operating conditions correspond to the point W1. In order to complete the analysis, we have to account for the unit productivity, i.e. to select the value of the switching time t*. In section 3.7 we have seen that when the effect of transport resistances and axial dispersion is properly accounted for, there exists a minimum value oft* (or a maximum productivity) above (below)which the process performance predicted by the equilibrium theory, in terms of the extract purity, can in fact be achieved. If t* is chosen below this limit, purity of the extract stream drops below the equilibrium theory value. In order to investigate this aspect, we have examined for
+
300 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995
,
_____I
Afterward, each set of operating conditions of the SMB unit has been characterized in terms of the abovementioned performance parameters. The characterization has been performed in the frame of the equilibrium theory and with reference to the flow rate ratios m,,j = 1, ..., 4, which are the key operating parameters in determining the separation performances (Storti et al., 1993b; Mazzotti et al., 1994). The results obtained using the equilibrium theory model have been compared with a set of experimental data obtained in a six port SMB unit, operated in the vapor phase for the separation of various mixtures of the alkylaromatic c8 fraction. This comparison shows a good qualitative agreement between theory and experiments. In the frame of the equilibrium theory an important limitation arises when looking for a criterion to select the optimal value of the switching time, t*. Since, the productivity of the unit is proportional to the inverse of t*, equilibrium theory requires that t* goes t o zero to obtain the maximum productivity. This phisically unrealistic result is a consequence of the approximations implied by equilibrium theory, which neglects mass transfer resistance and axial dispersion. These phenomena play an important role when t* decreases and the productivity increases, since in this case high flow rates are required inside the columns of the SMB unit. It follows that the choice oft* must be done using a detailed model, which accounts for all the dispersion phenomena. In the last section of this paper a procedure for selecting the optimal t* is described and applied to an example, which is particularly interesting in applications.
Acknowledgment various operating conditions, corresponding to extract recovery RE = 80%, the effect oft* on the extract purity. The obtained results are shown in Figure 17 where each curve corresponds to a different operating point along the AW1 curve, whose position is indicated in Figure 16. It is found that the minimum t* value for which an extract purity equal to, say 99%, is obtained, decreases as the operating point moves from W1 to A. On the other hand, since (m3 - m2) decreases when going from W1 to A, the value of productivity, Pr, defined by eq 35, exhibits a maximum. This is shown by the productivity value reported in Table 4, where it appears that P, is maximum in some point between W2 and W4. Hence, considering that W1 represents the operating point where Dr and A, are minimized, the optimal operating point can be selected somewhere on the curve w1w4, as the best compromise between maximum productivity on one hand and minimum desorbent and adsorbent requirements on the other hand. 6. Conclusions
In this work the issue of the optimal design of simulated moving bed adsorption separation units has been addressed. First, several performance parameters have been introduced and discussed. These include the purity of the outlet streams, extract and r f f i a t e ; the recovery of the key components in the outlet streams; the desorbent and adsorbent requirements, which account for the consumption of desorbent and solid sorbent with respect to the feed flow rate; and the productivity of the process, which is defined as the ratio between the feed flow rate and the size of the separation unit.
The financial support of the Consiglio Nazionale delle Ricerche-Progetto Finalizzato Chimica Fine, is gratefully acknowledged.
Nomenclature A = cross section of the adsorption column a, = pellet specific external area A, = adsorbent requirement, defined by eq 32 d, = particle diameter DL= axial dispersion coefficient D,= desorbent requirement, defined by eq 23 E = set of strong components collected in the extract stream GJ= fluid mass flow rate in sectionj of a SMB unit k, = global mass transfer coefficient of component i K,= equilibrium adsorption constant of component i L = length of SMB subsection LJ= length of sectionj of TCC unit m, = mass flow rate ratio in sectionj nJ = number of columns in the jth section of the SMB unit N = total number of components except for the desorbent NC = total number of components P = relative pressure PE = extract purity, defined by eq 17 PR = raffinate purity, defined by eq 18 P, = productivity, defined by eq 35 Pe = Peclet number, Pe = U&DLE St = Stanton number, S( = k,a&JElu, R = set of weak components collected in the raffinate stream RE = extract recovery, defined by eq 20 RR = raf€inate recovery, defined by eq 21 t = time T = temperature
Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 301 t* = switching time in a SMB unit t, = lifetime of the adsorbent U j = superficial fluid phase velocity in sectionj of a TCC unit U*j = superficial fluid phase velocity in sectionj of a SMB unit ur = reference superficial fluid velocity us = superficial solid phase velocity x = dimensionless axial coordinate, x = zlLj y'i = fluid phase dimensionless concentration of component i in section j z = axial coordinate Greek Letters
P = adsorbed phase saturation concentration = interparticle void fraction cP = intraparticle void fraction E* = overall void fraction, E* = E E
+ ep(l - E)
= solid phase dimensionless concentration of component i in sectionj p = volumetric flow rate ratio, p = u$ur Qi = sifer ef = fluid phase density eS= bulk solid mass density u = capacity ratio, u = esPlef t = dimensionless
time, t = tur/(LE)
w = equilibrium theory parameter 5j = SMB dimensionless fluid velocity in thej t h section,
= Uj/Ur Subscripts and Superscripts a = incoming fluid state A, B = components to be separated b = incoming solid state D = desorbent E = extract f = fluid state F = feed i = component index j = section index o = initial conditions r = reference conditions
R = raffinate s = solid state
e
sk = strong key component wk = weak key component
Literature Cited Johnson, J. A.; Kabza, R. G. Sorbex: Industrial-Scale Adsorptive Separation. In Preparative and Production Scale Chromatog raphy; Ganetsos, G., Barker, P. E., Eds.; Marcel Dekker: New York, 1993; p 257. Mazzotti, M.; Storti, G.; Morbidelli, M. Robust Design of Countercurrent Adsorption Separation Processes. 2. Multicomponent Systems. AIChE J. 1994, in press. Ruthven, D. M. Principles of adsorption and adsorption processes. John Wiley: New York, 1984. Ruthven, D. M.; Ching, C. B. Counter-current and simulated counter-current adsorption separation processes. Chem. Eng. Sci. 1989,44, 1011. Storti,G.; Masi, M.; Carrl,S.; Morbidelli, M. Modeling and Design of Simulated Moving-Bed - Adsorption Separation Units. Prep. Chrom. 1988,1,1. Storti, G.; Masi, M.; Cad,S.; Morbidelli, M. Optimal design of multicomponent adsorption separation processes involving nonlinear equilibria. Chem. Eng. Sci. 1989,44,1329. Storti, G.; Mazzotti, M.; F'urlan, L. T.; Morbidelli, M., Carrl, S. Performance of a six port Simulated Moving Bed pilot plant for vapor-phase adsorption separations. Sep. Sci. Technol. 1992, 27, 1889. Storti, G.; Masi, M.; Morbidelli, M. Modeling of Countercurrent Adsorption Processes. In Preparative and Production Scale Chromatography; Ganetsos, G., Barker, P. E., Eds.; Marcel Dekker: New York, 1993a; p 673. Storti,G.; Mazzotti, M.; Morbidelli, M.; Carrl,S. Robust Design of Binary Countercurrent Adsorption Separation Processes. AIChE J. 1@93b, 39,471. Storti,G.; Mazzotti, M.; Furlan, L. T.; Morbidelli, M. Analysis of a six port Simulated Moving Bed separation unit. In Proceedings of the Fourth International Conferenceon Fundamentals ofAdsorption; Suzuki, M., Ed.; Kodansha: Tokyo,Japan, 1993c; p 607. Wojciechowsky, B. W. Theoretical treatment of catalyst decay. Can. J. Chem. Eng. 1968,46,48.
Received for review December 27, 1993 Revised manuscript received August 8, 1994 Accepted August 24, 1994"
* Abstract published in Advance ACS Abstracts, November 1, 1994.
