Temperature Gradient Operation of a Simulated Moving Bed Unit

Further improvements of separation performance are expected when each section of the unit is optimized independently by applying a gradient of tempera...
0 downloads 0 Views 126KB Size
Download PDF
2606

Ind. Eng. Chem. Res. 2001, 40, 2606-2617

Temperature Gradient Operation of a Simulated Moving Bed Unit Cristiano Migliorini, Michael Wendlinger, and Marco Mazzotti Institut fu¨ r Verfahrenstechnik, ETH Zu¨ rich, Sonneggstrasse 3, CH-8092 Zu¨ rich, Switzerland

Massimo Morbidelli* Laboratorium fu¨ r Technische Chemie, ETH Zu¨ rich, Universita¨ tstrasse 6, CH-8092 Zu¨ rich, Switzerland

The simulated moving bed (SMB) technology has shown great potential for fine chemical separations, particularly for the resolution of the enantiomers of chiral compounds. Further improvements of separation performance are expected when each section of the unit is optimized independently by applying a gradient of temperature, pressure, or solvent composition along the unit. The aim of this work is to extend the design criteria for nonlinear SMBs to the case where a temperature gradient mode is adopted. It is shown how beneficial this can be in terms of productivity and solvent consumption. Finally, the temperature transient in the columns is studied. It is shown that temperature changes in the column of the unit yield a constraint on the maximum fluid velocity. This is analogous to the constraints due to column efficiency and packing stability requirements. The results show that the temperature gradient operation of an SMB unit is feasible and may have significant advantages over the traditional isothermal mode. 1. Introduction The increasing interest for the separation of enantiomers and natural and biological products and the development of new products and new drugs requires preparative techniques to realize more and more complex separations. The simulated moving bed (SMB) technology (usually followed by other purification steps, such as crystallization) allows one to carry out chromatographic separations at a production scale, reducing the high costs typical of analytical-scale chromatographic processes.1,2 SMBs, which are now established in the fine chemicals and pharmaceuticals industries, particularly for the resolution of racemates,3 are expected to find new applications in bioseparations, such as purification of enzymes, peptides, antibiotics, and natural extracts.4 In the true countercurrent (TCC) chromatography, the solid and liquid phases move in opposite directions. Under complete separation conditions, the ratio between the stationary and mobile phase flow rates is adjusted so that the more retained component is carried in the direction of the solid phase while the less retained one moves with the mobile phase, as shown in Figure 1. The regeneration of the solid and fluid phases required for a continuous process is carried out in sections 1 and 4, respectively. In practical applications, the continuous countercurrent movement of the solid in the TCC unit is simulated by the periodic column switching in the SMB unit. The performance of the TCC and SMB units is equal if equivalence relationships between the two units are fulfilled,5 and we will exploit this property to transfer the results that will be obtained by analyzing the simpler TCC unit to the SMB unit. Each of the four sections of the equivalent TCC unit in Figure 1 plays a different role in the separation. For instance, in section 1 the desorption of the more retained * Corresponding author. Tel.: +41-1-6323034. Fax: +411-6321082. E-mail: [email protected].

Figure 1. Four-section true countercurrent (TCC) unit. The specific role of each section is highlighted.

compound must be favored (hence, low retention times are required), whereas in section 4 also the less retained compound B must be adsorbed (hence, large retention times are required). It is clear that the separation performance of the unit can be optimized by adjusting the adsorption behavior in accordance to the task of each section. On the other hand, it is known that the adsorption behavior of the system is affected by changes of several operating conditions such as temperature, solvent composition, and pressure. In the case of enantiomer separations, the interaction energies between the chiral active sites and the enantiomers are rather large, thus making adsorption for these systems particularly sensitive to temperature changes.6 This behavior can be exploited to improve the separation performance by properly changing the temperature in the different sections of a SMB unit. Similarly, pH or eluent composi-

10.1021/ie000825h CCC: $20.00 © 2001 American Chemical Society Published on Web 05/15/2001

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2607

tion gradients can be established along a SMB unit in order to make the task of each section easier, thus improving the overall performance. In this same direction goes the idea of applying a pressure gradient in a SMB using a supercritical eluent; experimental results7 and theoretical analysis8 proved that the separation performance can be significantly improved with respect to the case of constant-pressure operation, particularly in the case of low selectivity. The objective of this work is to derive and apply a design technique for binary SMB separations with gradients of some operating conditions through the different sections. Although the analysis could be, in principle, extended to gradients of pressure or eluent composition, for the sake of simplicity in the following we will focus on temperature gradients. In addition, we will refer to nonlinear adsorption equilibria, because the advantages of the SMB technology are best exploited by operating the unit under overload conditions, where higher productivities are reached.9 The work is organized as follows: the equilibrium theory model is briefly introduced together with the detailed model which allows a direct numerical simulation of the behavior of the SMB unit. The temperature gradient is first applied under linear conditions, and the results are extended to the nonlinear case where the effect of the feed concentration is studied. With reference to a specific model system, a general strategy for the optimization of the temperature profile is devised. Finally, the design criteria are completed with the discussion of the additional constraints which arise from the need of properly changing the temperature of each chromatographic column of the SMB as soon as it goes from one section to the other. 2. TCC and SMB Modeling The equivalence between TCC and SMB units is widely exploited to design the proper operating conditions for a given separation regime of the SMB unit by solving the simpler TCC model. This approach justifies the development of detailed TCC models solved numerically,10 where mass-transfer resistance and axial dispersion are accounted for, as well as stagewise5 and continuous11 equilibrium models, where local equilibrium, plug flow, and constant fluid and solid velocities are assumed. For example, using the equilibrium model the material balance for component i in section j can be written as

∂ ∂ [*ci + (1 - *)neq (m c - neq i ] + uS(1 - p) i ) ) 0 ∂t ∂z j i (1) where neq i is the adsorbed phase concentration at equilibrium with the local fluid phase composition c. Based on this model and assuming stoichiometric Langmuir adsorption isotherm explicit criteria on the TCC operating conditions to achieve complete separation of binary mixtures have been derived.11 Extensions have included different binary equilibrium isotherms, i.e., nonstoichiometric Langmuir,9,12 bi-Langmuir,13 and IAS (Ideal Adsorbed Solution Theory) models,14 as well as some multicomponent separations.15-17

These criteria are given in terms of the flow-rate ratios

mj )

uTCC - uSp j uS(1 - p)

(j ) 1, ..., 4)

(2)

in the different sections of the unit and depend on the feed concentration and the thermodynamic parameters of the system (see Notation for symbols). These yield a lower limit on the flow-rate ratio in section 1, i.e., m1 > m1,cr, and an upper limit on the flow-rate ratio in section 4, i.e., m4 < m4,cr. The constraints on the flow rate ratios in sections 2 and 3 are more complex, because of the physical coupling between the two sections. In general, these yield a triangular-shaped region in the (m2, m3) plane, which is constituted of operating points that allow one to achieve complete separation of the feed mixture. The vertex of this region, i.e., the point furthest from the diagonal, corresponds to optimal separation performance in terms of productivity and solvent consumption.9 Thus, we first select the values of the mj parameters so as to achieve the desired separation in a TCC unit, and then we design the corresponding SMB unit using the following relationship based on the equivalence between TCC and SMB:

mj )

t* - L* uSMB j L(1 - *)

(3)

which, once the switch time t* is fixed, provides the SMB flow rates. The value of the switch time has to be selected based on considerations beyond equilibrium theory, i.e., by accounting for pressure drop and column efficiency issues.18,19 On the other hand, the detailed SMB model provides a complete picture of the performance of the unit, because it takes into account the dynamic behavior (startup) and the effect of mass transfer, of axial dispersion, and also of extracolumn dead volume whenever necessary.18,20 The detailed simulations presented in this work are based on the solution of the following material balance equations:

*

∂ci ∂2ci ∂ni ∂ci + (1 - *) + uj ) bDi 2 ∂t ∂t ∂z ∂z ∂ni ) kiap(neq i - ni) ∂t

(4)

(5)

The model system used to illustrate the results of this work is the separation of N-benzoyl-D- and L-alanine using a 0.1 M aqueous phosphate buffer solution in 3% (v/v) propanol at pH 6.8 on immobilized bovine serum albumin.6 This system exhibits a strong nonlinear behavior even at low concentrations, which significantly reduces the maximum feed concentration at which the SMB operation can be carried out in practice. In fact, we have calculated that at a feed concentration of 100 µmol/L for each enantiomer the region of separation in the (m2, m3) plane becomes so close to the diagonal that the SMB separation cannot be conveniently carried out anymore. Over this small concentration range it is possible to describe well the adsorption equilibrium data

2608

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001

with a nonstoichiometric Langmuir isotherm with an inert solvent:

neq i )

ΓiKici 1+

∑j

i ) A, B

(6) 3. Temperature Gradient Mode Operation

Kjcj

where A indicates the more retained component and the Henry constant γi is given by ΓiKi. In particular, in fitting the isotherm parameters at temperatures of 293, 303, and 313 K, the saturation capacity Γi has been assumed to remain constant, whereas the equilibrium parameter Ki follows the exponential law

Ki ) Ki0e-∆Hi /RT 0

(∆Hi0 < 0)

(7)

The fitting of the experimental data yields ΓA ) 2800 µmol/L, ΓB ) 4000 µmol/L, KA0 ) 1.62 × 10-8 L/µmol, KB0 ) 2.40 × 10-7 L/µmol, ∆HA0 ) -3.15 × 104 J/mol, and ∆HB0 ) -2.25 × 104 J/mol. In the following examples and in the optimization of the temperature gradient operation, the temperature levels of the sections in the SMB unit are kept within a 10 K range. This is meant to avoid unrealistic heat-transfer conditions and thermal and mechanical stress on the stationary phase. Within this range, physical properties such as density of the solution are assumed to be constant. It is worth noting that, although the temperature variation is small, the effect on the adsorption behavior is significant. For instance, the data above indicate that a temperature increase of 10 K leads to Henry constant values for A and B smaller by 35% and 25%, respectively. The quality of the SMB performance will be assessed with reference to three performance parameters: productivity, desorbent requirement, and enrichment. The productivity is defined as the amount of feed separated per unit time and unit mass of stationary phase:

cFT(m3 - m2) PR ) NcFst*

(8)

Note that this relationship and the following ones are obtained using the definition of the mj parameters (2) and assuming that the densities of the feed and desorbent streams are the same; this is correct as long as the feed concentration is small, as, for example, in the case of fine chemicals and enantiomer separations where the solubility is limited to a few grams per liter of solvent. The desorbent requirement is defined as the mass of desorbent per unit mass of pure product separated:

DR )

(

cD cFT

1+

)

m1 - m4 m3 - m2

(9)

The enrichment is defined as the ratio between the concentration of the products A and B in extract and raffinate, respectively, and their corresponding feed concentration. Under complete separation conditions, the following relationships are obtained:

EA )

It is worth noting that desorbent requirement and enrichment depend on the mj ratios in all sections of the unit, whereas productivity depends only on m2 and m3.

m3 - m2 m3 - m2 ; EB ) m1 - m2 m3 - m4

(10)

In this section the design of the operating conditions of a TCC unit operated in the temperature gradient mode is discussed in the frame of equilibrium theory. In the following, it is assumed that the temperature of the unit can be adjusted at the desired level instantaneously. Therefore, the design of the temperature gradient mode TCC requires that a different adsorption isotherm is used in each section following the appropriate temperature dependence. How realistic this assumption is and what process constraints the requirement of a temperature change between two adjacent columns of the TCC or SMB unit imposes will be discussed in the final part of the paper. We proceed by first considering the regeneration sections 1 and 4. Next we analyze the two central sections, under the assumption of complete regeneration for both the adsorbent and the eluent. The regeneration of the stationary phase in section 1 is fulfilled at any feed concentration when m1 > m1,cr ) γA.9 In general, increasing the temperature makes adsorption less favorable and reduces the value of m1,cr. Therefore, a temperature increase in section 1 leads to a decrease of the desorbent consumption (9) due to a smaller difference between m1 and m4 and a larger enrichment of component A according to eq 10. With reference to section 4, regeneration of the mobile phase is achieved for m4 < m4,cr. The adsorption of the less retained compound B is favored by lower temperatures, which result in larger values of m4,cr. This can be verified by substituting eq 7 in the relationships for m4,cr,9 i.e.

1 m4,cr(m2,m3) ) {γB + m3 + KBcFB(m3 - m2) 2

x[γB + m3 + KBcFB(m3 - m2)]2 - 4γBm3}

(11)

and analyzing the resulting temperature dependence. Therefore, lower temperature values allow for smaller m4 values, which lead to a decrease in the desorbent consumption (9) and to an increase in the enrichment of B (10). The effect of changing the temperatures of sections 2 and 3 is more complex, because these are coupled through the mass balances at the feed node. Therefore, we discuss the behavior of these two sections more in detail in the following, starting with the simple case of a linear adsorption isotherm. It is worth noting that the scheme of the process illustrated in Figure 1 imposes the obvious constraints m1 > m2, m3 > m4, and m3 > m2. 3.1. Linear Systems. First, let us consider the separation of two components whose adsorption behavior is described by the linear isotherm ni ) γici. As shown in Figure 2, under isothermal conditions, i.e., when T2 ) T3, a temperature increase shifts the region of complete separation along the diagonal toward the lower left corner of the diagram, because of the corresponding decrease of the Henry constants according to eq 7. For the selected model system, the increase in temperature corresponds also to a decrease in selectivity, which leads to an optimal operating point w closer

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2609

Figure 2. Regions of complete separation under linear and isothermal (T2 ) T3) conditions: broken line, T ) 303 K; solid line, T ) 313 K. The coordinates of the intersection points with the diagonal are a1 (γ3A, γ3A), a2 (γ2A, γ2A), b1 (γ3B, γ3B), and b2 (γ2B, γ2B).

Figure 3. Region of complete separation under linear conditions when the temperature gradient between sections 2 and 3 is applied. T2 ) 313 K, and T3 ) 303 K. Regions: 1, complete separation; 2, pure extract; 3, pure raffinate; 4, no pure outlet; 5, extract flooded with pure desorbent; 6, raffinate flooded with pure desorbent; 7, extract flooded with pure desorbent (B is collected in the raffinate, and A accumulates in section 2); 8, raffinate flooded with pure desorbent (A is collected in the extract, and B accumulates in section 3).

to the diagonal. Under linear conditions the feed concentration cFT does not affect the boundaries of the complete separation region. Therefore, performance can be measured by the specific productivity parameter PR/cFT, which is proportional to the difference m3 - m2. In this example the specific productivity is larger at a lower temperature, which is often the case in applications. Under linear conditions, the temperature gradient mode operation leads always to an improvement of the specific productivity independently of the relative change of the Henry constants with temperature. This is shown in Figure 3 where the region of complete separation (b1, w3, a2) for a TCC unit where T3 ) 303 K and T2 ) 313 K is shown. The boundaries of this region can easily be found by solving the flux conditions to achieve complete separation5,8,11 which in the linear case are given by

γ2B < m2 < γ2A

(12)

γ3B < m3 < γ3A

(13)

Figure 4. Regions of complete separation under nonlinear and isothermal (T2 ) T3) conditions (solid line): a1w1b1, T ) 303 K; a2w2b2, T ) 313 K. Region of complete separation under nonlinear and nonisothermal conditions (broken line): b2w3a2, cFA ) cFB ) 70 µmol/L, T2 ) 313 K, and T3 ) 303 K.

The optimal point w3, whose coordinates are (γ2B, γ3A), is further away from the diagonal than the optimal points w1 and w2 in Figure 2, i.e., those corresponding to a unit operated at constant temperature, namely, 303 and 313 K, respectively. In the temperature gradient mode the higher temperature in section 2 favors the desorption of B, while the lower temperature in section 3 favors the adsorption of A (see Figure 1), thus improving the overall performance of the separation. This situation is similar to the one found for a supercritical SMB unit, where the adsorptivity of the various components is changed through proper changes of pressure rather than of temperature as in this work.8 In this case, the region of complete separation can take various different shapes other than the triangular one, as for example that of a truncated rectangle intersecting the diagonal shown in Figure 3. The different shapes of the complete separation regions as well as the different regimes of separation, which are attained in the surrounding regions, have been discussed in detail in the context of pressure gradients elsewhere and will not be repeated.8 It is sufficient here to recall that, according to the analysis based on linear systems, in region 8 component B accumulates in the unit because it can leave neither with the extract nor with the raffinate. In reality, its holdup is upper bounded by the achievement of saturation conditions either in the liquid or in the adsorbed phase. In the latter case, the linear picture is no more valid and one has to refer to the nonlinear situation discussed in the next section. Similar considerations apply to region 7 and component A. In other words, under linear conditions only regions 1-6 in Figure 3 are consistent with a steady-state operation of the unit, whereas regions 7 and 8 correspond to physically nonrealistic operation, whereby components A and B, respectively, would indefinitely accumulate in the unit. This ambiguity will be resolved in the next section. 3.2. Nonlinear Systems. Let us now consider the separation of a binary system whose adsorption equilibrium is described by the nonlinear Langmuir isotherm (6). First, let us consider the effect of changing the temperature on the separation region when the unit is operated isothermally, i.e., when T2 ) T3. In Figure 4 it is shown that the temperature increase under nonlinear conditions shifts the region of complete sepa-

2610

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001

ration (solid boundaries) along the diagonal, similarly to the linear case (cf. Figure 2). This is consistent with the property that the isothermal region of complete separation under nonlinear conditions has the same intersections with the diagonal as the corresponding linear one, i.e., at the values corresponding to the Henry constants of A and B. This can be explained by considering that when the operating point is close to the diagonal, the feed stream is so small compared to the flow rates inside the unit that at the high dilution conditions achieved linear adsorption equilibria prevail.9 Because in our example the Henry constants and the selectivity decrease with temperature, the region of separation moves downward along the diagonal and its basis shrinks, while the shape remains similar. The corresponding shift of the optimal operating point leads to smaller productivity at higher temperature. The region of complete separation when sections 2 and 3 are operated at different temperatures, i.e., when T2 > T3, is represented by the broken line b2w3a2 in Figure 4. The solution of the equilibrium theory model in this case is derived in Appendix A. As a general result, we may note that when a temperature gradient is applied, the optimal operating point is located along the line w2b2 (or its prolongation). In this example the distance between the optimal operating point w3 and the diagonal in the temperature gradient mode is much larger than that in the high-temperature isothermal case (w2) and slightly smaller than that in the lowtemperature case (w1). Therefore, at the feed concentration considered, the temperature gradient mode does not lead to an improvement. However, this is not a general result, and the calculations at different feed concentrations presented in the next section show that under different conditions the temperature gradient operation can lead to improvements in productivity as well as in other performance parameters. The results above have been obtained in the frame of equilibrium theory and therefore need to be verified. In particular, for the sake of robustness of the separation and to account for finite column efficiency, in practice an operating point close to w3 but located inside the separation region has to be adopted. To better illustrate this point, the detailed model introduced in section 2 has been used to show the effect of the finite efficiency on the behavior of an SMB unit in the temperature gradient mode. The parameters reported in the captions of Figure 5 for the model simulations correspond to a column efficiency of about 20 plates under linear conditions; to be on the safe side, the same efficiency is assumed at different temperatures even if an increase in the temperature leads, in general, to a higher efficiency. Three operating points, i.e., A-C, have been chosen as shown in Figure 5 (point D is discussed in Appendix B), each one within a different separation region according to equilibrium theory. The purity values calculated with the detailed model are consistent with the position of the points in the (m2, m3) plane; in particular, point B, which is inside the complete separation region assuming infinite efficiency, achieves a purity larger than 99% in both product streams. This confirms that equilibrium theory provides useful information to determine feasible or even optimal operating conditions also in the temperature gradient mode. Finally it is worth noting that the region of separation in the temperature gradient case is sharper than the isothermal regions in the vicinity of the optimal vertex.

Figure 5. Region of complete separation under nonlinear and nonisothermal conditions: cFA ) cFB ) 25 µmol/L, T2 ) 313 K, and T3 ) 303 K. SMB detailed simulation parameters: eight columns, layout 2-2-2-2, length 20 cm, cross-sectional area 1 cm2, * ) 0.5, t* ) 800 s, DA,B ) 8.25 × 10-4 cm2/s, apkA,B ) 0.07 s-1, m1 ) 1.2m1,cr, m4 ) 0.8m4,cr.

Figure 6. Regions of complete separation under nonlinear and nonisothermal conditions: T2 ) 313 K, and T3 ) 303 K. cFA ) cFB ) 1.4, 14, 60, and 200 µmol/L. The broken line is the linear region of separation when the temperature gradient is applied. Operating points inside the patterned area (region 8 in Figure 3) lead under linear conditions to unrealistic accumulation of B in section 3.

This implies that the optimal operating point is less robust in the temperature gradient mode than in the isothermal mode. By comparison of Figures 3 and 4, it is observed that the linear and nonlinear regions of complete separation in the temperature gradient mode have different intersections with the diagonal, i.e., b1a2 and b2a2, respectively. This is a peculiar aspect of the combination of the temperature gradient operation and the nonlinear competitive adsorption behavior, which deserves some explanation. Under temperature gradient conditions, the nonlinear region of separation at decreasing feed concentration does not approach the region calculated with the linear model, as shown in Figure 6, even though at infinite dilution the Langmuir model approaches linear behavior. The right triangle intersecting the diagonal in b1b2, i.e., the patterned region in Figure 6 and region 8 in Figure 3, belongs, in fact, to the nonlinear complete separation region when the feed concentration approaches zero, whereas it does not belong to the complete separation region under linear conditions. This supposed inconsistency can be resolved by noting that, as mentioned above, operating conditions

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2611 Table 1. Performance Parameters at cFT ) 50 µmol/La run

T1

T2

T3

T4

m1

m2

m3

m4

PR

DR × 103

EA

EB

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

303.0 306.3 309.7 313.0 313.0 313.0 313.0 309.7 306.3 309.7 313.0 313.0 313.0 313.0 313.0 313.0 313.0 313.0

303.0 306.3 309.7 313.0 313.0 313.0 313.0 309.7 306.3 309.7 309.7 313.0 309.7 308.0 306.3 303.0 309.7 306.3

303.0 306.3 309.7 313.0 309.7 306.3 303.0 303.0 303.0 306.3 306.3 313.0 309.7 308.0 306.3 303.0 303.0 303.0

303.0 306.3 309.7 313.0 309.7 306.3 303.0 303.0 303.0 306.3 303.0 303.0 303.0 303.0 303.0 303.0 303.0 303.0

12.2 10.6 9.3 8.2 8.2 8.2 8.2 9.3 10.6 9.3 8.2 8.2 8.2 8.2 8.2 8.2 8.1 8.1

6.5 6.0 5.5 5.0 4.9 4.8 4.7 5.2 5.8 5.4 5.4 5.0 5.5 5.7 6.0 6.5 5.2 5.8

10.6 9.4 8.3 7.4 8.1 8.7 9.4 9.8 10.2 9.1 9.1 7.4 8.3 8.8 9.4 10.6 9.8 10.2

6.8 6.2 5.7 5.2 5.6 6.1 6.6 6.7 6.8 6.2 6.7 6.4 6.6 6.7 6.8 6.8 6.7 6.8

102 85 70 60 80 97 117 115 110 92 92 60 70 77 85 102 115 110

92.7 91.8 91.4 90.0 72.5 61.5 53.6 62.6 74.5 73.5 56.2 70.0 62.9 59.4 56.5 53.7 52.7 51.8

0.72 0.74 0.74 0.75 0.97 1.15 1.34 1.12 0.92 0.95 1.32 0.75 1.04 1.24 1.55 2.41 1.59 1.91

1.08 1.06 1.08 1.09 1.28 1.50 1.68 1.48 1.29 1.28 1.54 2.40 1.65 1.48 1.31 1.08 1.48 1.29

a

In eq 8 it is assumed that NcFst* ) 2 h/L, so that productivity is given in µmol/h. In eq 9 it is assumed that cD ) 2 mol/L.

4. Selection of the Temperature Profile

Figure 7. Regions of complete separation under nonlinear and nonisothermal conditions: T2 ) 313 K, and T3 ) 293 K. cFA ) cFB ) 2, 10, 50, 100, and 200 µmol/L. The broken line is the linear region of separation when the temperature gradient is applied and γ2A < γ3B. Operating points inside the patterned area lead under linear conditions to unrealistic accumulation of B in section 3.

in region 8 cannot be realistically described with a linear model. What happens in fact, at infinite dilution for an operating point in region 8, is that component A is collected in the extract, whereas component B accumulates in the unit until overload conditions are achieved and B breaks through at the raffinate outlet. This leads to a steady-state behavior achieving complete separation, which is correctly predicted by the Langmuir isotherm accounting for the saturation of the adsorbent, as shown in Figure 6, but cannot be predicted by a linear equilibrium model not accounting for saturation. Following similar arguments, it can be shown that an operating point in region 7 of Figure 3 will ultimately lead to flooding of the raffinate with the whole amount of both components fed to the unit, i.e., to no separation. Similar considerations can be repeated for the situation illustrated in Figure 7, where γA2 < γB3 and the linear complete separation region does not intersect the diagonal. The transient behavior of the unit and the development of the internal profiles are also strongly affected by the same effect, as discussed in Appendix B.

The results presented in the previous sections allow one to determine for any given temperature profile in the TCC unit and under the assumption of ideal column behavior, i.e., in the frame of equilibrium theory, the operating point achieving optimal performance, which is given by the vertex of the triangular-shaped complete separation region. Therefore, it is now possible to compare different possible temperature profiles and select the optimal one. In principle, an overall objective function should be defined and the temperatures and flow rates in the four sections of the unit should be let free to vary until the objective function is optimized. For the sake of simplicity, we limit the search for optimal temperature values to only four levels, namely, 303, 306.3, 309.6, and 313 K, and we impose that the temperature decreases from section 1 to 4 within the range 303-313 K. Moreover, four separate performance parameters are evaluated, namely, productivity, desorbent requirement, and enrichment of A in the extract and of B in the raffinate, as defined by eqs 8-10. It is worth noting that according to equilibrium theory we consider for such a temperature profile the vertex of the complete separation region as the optimal operating point, where complete separation of the two components, i.e., 100% purity, is obtained. The 18 different temperature profiles reported in Tables 1 and 2 have been analyzed, which include also 4 isothermal (isocratic) operations. Because different results may be obtained for different values of the overall feed concentration, Tables 1 and 2 correspond to a low and high overall feed concentration, respectively, i.e., 50 and 200 µmol/L. For each temperature configuration, the optimal mj values reported in the tables have been calculated using the method described in the previous section. The corresponding separation performance has been evaluated using eqs 8-10 and reported in the last four columns. By analysis of the results reported in Tables 1 and 2, a few general remarks can be made, despite the rather evident differences between the low and high feed concentrations. The desorbent requirement reduces by 20-50% when going from isocratic to temperature gradient mode, and the largest reductions are obtained when T1 ) 313 K and T4 ) 303 K, whatever the temperature values in the other two sections are. Among the four isocratic cases, productivity is best at low

2612

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001

Table 2. Performance Parameters at cFT ) 200 µmol/L run

T1

T2

T3

T4

m1

m2

m3

m4

PR

DR × 103

EA

EB

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

303.0 306.3 309.7 313.0 313.0 313.0 313.0 309.7 306.3 309.7 313.0 313.0 313.0 313.0 313.0 313.0 313.0 313.0

303.0 306.3 309.7 313.0 313.0 313.0 313.0 309.7 306.3 309.7 309.7 313.0 309.7 308.0 306.3 303.0 309.7 306.3

303.0 306.3 309.7 313.0 309.7 306.3 303.0 303.0 303.0 306.3 306.3 313.0 309.7 308.0 306.3 303.0 303.0 303.0

303.0 306.3 309.7 313.0 309.7 306.3 303.0 303.0 303.0 306.3 303.0 303.0 303.0 303.0 303.0 303.0 303.0 303.0

12.2 10.6 9.3 8.2 8.2 8.2 8.2 9.3 10.6 9.3 8.2 8.2 8.2 8.2 8.2 8.2 8.2 8.2

5.7 5.3 4.9 4.6 4.4 4.3 4.2 4.6 5.1 4.7 4.7 4.6 4.9 5.1 5.3 5.7 4.6 5.1

8.0 7.3 6.6 6.0 6.1 6.3 6.3 6.9 7.5 6.8 6.8 6.0 6.6 7.0 7.3 8.1 6.9 7.5

6.0 5.5 5.1 4.7 4.9 5.1 5.3 5.5 5.7 5.3 5.5 5.3 5.6 5.7 5.8 6.0 5.5 5.8

230 200 170 140 170 200 210 230 240 210 210 140 170 190 200 240 230 240

37.0 35.5 34.7 35.0 29.4 25.5 23.8 26.5 30.4 29.0 22.9 30.7 25.3 23.2 22.0 19.2 21.3 19.6

0.35 0.38 0.39 0.39 0.45 0.51 0.53 0.49 0.44 0.46 0.60 0.39 0.52 0.61 0.69 0.96 0.66 0.80

1.15 1.11 1.13 1.08 1.42 1.67 2.10 1.64 1.33 1.40 1.62 2.00 1.70 1.46 1.33 1.14 1.64 1.41

Figure 8. Productivity as a function of the total feed concentration, cFT. The productivity of temperature profile 16 is the same as that calculated under isothermal conditions at T ) 303 K (profile 1).

temperature because of the decrease of selectivity with the temperature characteristic of the model system under examination. The best temperature gradient operations can only slightly improve this productivity performance. Other cases achieve similar productivity values; among profiles 5-18, the ones achieving the best productivity depend strongly on the nonlinear behavior of the isotherm. Enrichments of A and B are maximum when T1 - T2 and T3 - T4 are maximum, respectively. A rather good compromise between these two is observed for runs 7 and 11. On the basis of these considerations, four temperature profiles, i.e., 1, 7, 11, and 16, have been selected for a further analysis of the effect of the overall feed concentration in the whole range up to 200 µmol/L. The calculated productivity and enrichments are plotted in Figures 8-10 . It is apparent that different profiles are optimal with respect to different performance parameters and that these change when changing the feed concentration. All of these effects can be explained based on the functional form of eqs 8-10 and on the temperature dependence of the adsorption parameters of the selected system. For instance, under strong dilution conditions the temperature gradient operation for the selected model system is always better than the isothermal mode, as proved in section 3. However, at higher concentration the nonlinearity bends the temperature gradient region of separation faster than in

Figure 9. Enrichment of component A as a function of the total feed concentration, cFT. Profile 1 is calculated for the isothermal conditions T ) 303 K.

Figure 10. Enrichment of component B as a function of the total feed concentration, cFT. The enrichment of temperature profile 16 is the same as that calculated under isothermal conditions at T ) 303 K (profile 1).

the isothermal case (the proof is given in Appendix A). This explains the crossover in productivity corresponding to different operating modes shown in Figure 8. For systems with the selectivity increasing with temperature, different behaviors are expected, which can be analyzed with the same approach followed here. This should be used at the early stage of the development of a new separation in order to determine whether the

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2613

temperature gradient mode operation promises significant improvement with respect to the isocratic mode and, if this is the case, which are the temperature profiles for which the best performance can be expected. 5. Design Criteria for the Temperature Change To realize in practice the temperature gradient mode operation discussed above, we need to change the temperature of a jacketed chromatographic column at the time of the valve actuation when inlet and outlet ports to the unit are switched. Let us assume that thermostating fluids at the different temperatures required are available and can be fed independently to each column jacket. Moreover, no heat exchangers are inserted between two successive columns of the SMB unit in order to avoid the introduction of detrimental extracolumn dead volumes. Under these assumptions the situation of interest is that of a fluid at a given temperature entering a column where the heating (cooling) fluid is at a temperature 10 degrees higher (lower). Because in the analysis above we have assumed that the column temperature switches instantaneously at the nominal operating temperature, i.e., that of the thermostating fluid, the question to be answered is how much the dynamics of column heating and cooling affects the separation performance. It is evident that the answer depends on the heat-transfer efficiency, which depends, in turn, on the physical properties of the mobile and stationary phases and on the fluid velocity. In particular, it can be expected that in order to have sufficiently fast heat transfer a lower bound on the fluid velocity will have to be imposed. It is interesting to note that this bound is analogous to the widely analyzed bounds on fluid velocity arising from masstransfer (maximum number of theoretical plates to guarantee enough column efficiency) and hydrodynamics (maximum pressure drop to avoid mechanical instability) considerations.19,21 In the following we will derive this constraint using, on the one hand, a short-cut approach based on the two-dimensional heat balance equation in the packed bed and, on the other hand, a commercial computational fluid dynamic code, Fluent version 5.3 (Fluent Inc., Lebanon, NH). For the sake of simplicity, the following assumptions are made. First, we assume that the controlling heattransfer resistance is mainly inside the column and that the inner column wall reaches immediately the new temperature after the switch, i.e., Tj. Second, the packed bed is modeled as a homogeneous medium with average physical properties given by

(Fcp)m ) (1 - *)(Fcp)s + *(Fcp)f

(14)

km ) (1 - *)ks + *kf

(15)

These are kept constant because temperature changes over a rather small range. Finally, a flat velocity profile is assumed, and radiative transport effects and viscous dissipation are neglected. The following two-dimensional heat balance equation can be used to study the development of the temperature profile, T(z,r,t):

(Fcp)m

∂T km ∂ ∂T ∂T r + (Fcp)fu ) ∂t ∂z r ∂r ∂r

( )

(16)

Because the column is slender, the conductive term in the axial direction has been neglected with respect

Figure 11. Ratio of the retention time for the temperature profile given by eq 16, tx, and the column at the uniform wall temperature tj as a function of the superficial velocity. Column 25 cm × 0.46 cm i.d., and * ) 0.5. Packing: microcrystalline cellulose triacetate. Eluent: ethanol. (Fcp)m ) 2.06 J/(m3 K), and km ) 0.2 W/(m K). For the more retained enantiomer γA ) γ0Ae-∆HA0/RT, where γ0A ) 1.6 × 10-3 and ∆H0A ) -5079 J/mol.

to the radial one. In the following we analyze the effect of temperature dynamics on the retention time in the column by assuming linear adsorption equilibrium, with Henry constants depending on temperature according to eq 7. Let us analyze eq 16 in the two limiting cases where the accumulation or the convective term are neglected, respectively. The former case occurs when thermal steady state is achieved, the holdup term is zero, and eq 16 can be solved analytically as a sum of Bessel’s functions, thus obtaining Tss(z,r). Because the cooling and heating processes occur through the wall of the column, the largest difference in retention behavior with respect to the nominal one is attained at the column axis, i.e., at r ) 0. The nominal retention time, i.e., tj, is given by

L tj ) (* + (1 - *)γ(Tj)) u

(17)

whereas the one attained along the column axis is

tx )

∫0L[* + (1 - *)γ(Tss(z,0))] dz

1 u

(18)

where the steady-state solution Tss depends on the fluid velocity u. The ratio tx/tj for a model system is shown as a function of fluid velocity u in Figure 11. It is apparent that the two limits at vanishing u, where tx ) tj obviously, and at infinite velocity, where tx ) (* + (1 - *)γ(T0))/u and T0 equals the inlet temperature of the fluid, are satisfied. Enforcing the constraint |tx - tj|/tj e R < 1 yields an upper bound uH on the fluid velocity in the column, as indicated, for example, by the open circle in Figure 11 in the case of R ) 0.05. Let us now consider the temperature profiles along the column axis for u ) uH and for velocity values larger and smaller than uH. The profiles illustrated in Figure 12 have been calculated with Fluent in the laminar regime with no shear at the column wall. It is seen that for the critical velocity value the wall temperature is achieved only at the column outlet. The second limiting case is where the longitudinal convection term in eq 16 is neglected, i.e., at small times.

2614

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001

Figure 12. Axial temperature profile along the column at different velocities: u1 ) 0.10 cm/s, uH ) 0.43 cm/s, and u2 ) 0.92 cm/s. Data are as in Figure 11.

that the restricting velocity bound is uS ) 0.15 cm/s. This constraint must be compared with the constraints that allow a stable hydrodynamic operation of the unit and good column efficiency mentioned above. The former, from our experience, corresponds to a maximum pressure drop per column of 20 bar, which leads to a superficial velocity u ) 0.05 cm/s. The latter yields for this model system a less stringent constraint. Therefore, we can conclude that the temperature gradient operation can be applied to this unit without changing the flow rates and the switch time. We believe that this conclusion actually bears some generality and that many small-scale fine chemical separations operated isothermally could be operated in the temperature gradient mode because they happen to be under conditions where thermal transients are not important, while exploiting an improvement in separation performance. 6. Conclusion

Table 3. Values of the Ratio uS/uH for Different Fluid/ Stationary Phase Systemsa uS/uH

water

ethanol

hexane

nitrogen

sc-CO2

silica activated carbon polymeric resin

7.97 7.64 8.45

5.96 5.56 6.58

5.29 4.89 5.92

0.22 0.19 0.27

5.91 5.51 6.54

a A representative m value for each system has been used in the calculations. For illustrative purposes β is set equal to 1.

An estimate of the characteristic time of this transient period is given by the time necessary for the wall temperature to reach that of the column axis, i.e.

t0 ) (Fcp)md2/(4km)

(19)

Because during this time the temperature profile is not the steady-state one, Tss, the corresponding retention time will also be different. This is acceptable as far as this time is short enough compared to the nominal retention time, which then leads to the constraint t0/tj e β < 1. Using eq 17, this can be cast in terms of fluid velocity as follows:

u e uS )

4L km (* + (1 - *)γ(Tj))β d2 (Fcp)m

(20)

We can then conclude that the upper bound on the fluid velocity coming from the heat-transfer limitation can be taken as the minimum between the two derived above, i.e.

u e min {uH, uS}

(21)

Some illustrative calculations of the ratio uS/uH for systems of practical interest are reported in Table 3. It is seen that, in general, for liquid systems and supercritical CO2 the velocity bound uH is more restrictive. In this case the energy transferred into the system is carried away by the fluid because of its heat capacity. On the contrary, for gases the velocity bound uS is limiting. In fact, the balance between conductionconvection is easily achieved because of the low heat capacity of the fluid, and therefore the balance between thermal inertia and conduction is controlling. With illustrative purposes let us apply the results of the analysis above to the case of the separation of Tro¨ger’s base enantiomers on microcrystalline cellulose triacetate (CTA).22 Assuming R ) 0.05 and β ) 0.1 gives

The results of this work provide guidelines to design SMB separations when a temperature gradient is applied. The problem of optimizing the temperature profiles and the design criteria to account for the temperature gradient has been studied. The equilibrium theory model allows one to study the effect of the temperature gradient on the different performance parameters. The analysis is carried out under linear and nonlinear conditions by introducing a novel and general technique, which can be extended to tackle the design problem also with pressure and solvent composition gradients. These predictions can be refined using a detailed model where nonideal effects are accounted for. The conclusion of this analysis is that the temperature gradient operation allows one to improve productivity with respect to the isothermal mode to an extent which depends on the system parameters and the feed concentration. Moreover, it is possible to tune the enrichment by selecting a proper temperature profile. This choice may have an important effect on the unit operation that follows SMB. Finally the application of a temperature gradient leads to a remarkable reduction of the solvent consumption. Enforcing a temperature gradient imposes that the temperature of the chromatographic columns constituting the SMB unit as well as of the fluid flowing therein should be controlled precisely. In other words, temperature gradients must be fast enough with respect to the characteristic retention times of the system. It has been shown that this requirement yields a constraint on the maximum fluid velocity in the columns. This establishes an analogous situation where mass transport, which determines column efficiency, momentum transfer, which controls the mechanical stability by determining pressure drop in the columns, and heat transfer, each impose an upper bound on the fluid velocity. We have shown that among these three, the limiting one is different depending on the physicochemical properties of the system under investigation. It can be concluded that the temperature gradient operation of an SMB is a feasible and realistic operating mode that can have a positive impact on process performance. This might have a significant impact in the case of SMB reactors,23 where not only selective adsorption but also reaction kinetics are temperature dependent.

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2615

Acknowledgment We kindly acknowledge the support of F. HoffmannLa Roche AG, Basel. Appendix A The procedure developed to calculate the region of complete separation of a TCC unit when the adsorption behavior is described with nonconstant selectivity isotherms14 can be applied also to solve the design problem when different isotherms are used in each section of the TCC unit because of temperature, pressure, or solvent composition gradients. The aim of this appendix is to show how this extension can be made. Under the assumption of complete separation, the states in sections 2 and 3 are given by the compositions M2 ) (c2A, 0) and M3 ) (0, c3B). (i) Optimal Point w. The calculation of the optimal operating point requires the solution of the system of equations requiring (a) the fulfillment of the mass balances for A and B under the assumptions of complete separation and (b) the optimal conditions, i.e., m2 ) m2,min and m3 ) m3,max. The mass balances for A and B are

m2c2A

(22)

(m3 - m2)cFB ) m3c3B - n3B

(23)

(m3 -

m2)cFA

)

n2A

-

The system of eqs 22-26 can be solved in the W unknowns c2A, c3B, cβA, cβB, mW 2 , and m3 . (ii) Curve wb. The solution of eqs 22 and 25 using c2A as the running parameter determines the line wb. The equation of this line in the case of temperature gradient is the same as that under isothermal conditions if the thermodynamic parameters of section 2 are used, that is

(γ2A - γ2B(1 + K2AcFA))m2 + K2AcFAγ2Bm3 ) γ2B(γ2A - γ2B) (30) At constant temperature in section 2, the effect of the temperature gradient between sections 2 and 3 shifts the coordinate of the optimal point w along this line, without changing its slope. (iii) Curve wr. Solving the system of eqs 22-24 and 26 in the unknowns c2A, cβA, cβB, m2, and m3 using c3B as the running parameter yields the boundary wr. (iv) Curve ra. The last part of the region of separation is calculated by solving eq 22 and requiring that m2 ) m2,max, where

m2,max )

|

∂nA ∂cA

(31) M2

This leads to the following equation for ra:

m3 ) m2 +

The condition m3 ) m3,max can be written as

m3 ) m3,max )

nβB - n3B cβB - c3B

)

nβA cβA

(24)

where β indicates the state of the fluid stream entering section 3 of the TCC unit. The last condition that must be satisfied in the optimal operating point is m2 ) m2,min which is given by

m2 ) m2,min )

|

∂nB ∂cB

(25) M2

For a Langmuir isotherm, the intermediate state M2 can be calculated from

-b - xb2 - 4ac 2a 2

c2A )

(26)

where

a)

2 nβ,2 B KA

cβB

nβ,2 B β 2 b ) β (1 - cβAK2A) - γ2A + nβ,2 A cBKA cB c ) nβ,2 A -

cβA β,2 nB cβB

(xγ2A - xm2)2 K2A cFA

(32)

Again, the equation for ra is the same as that obtained for the isothermal case, because only the mass balance of A and its thermodynamic parameters in section 2 are involved. Let us now prove that, if the selectivity is decreasing with the temperature as in eq 7, then the nonlinear competitive adsorption equilibria described by eq 6 make the region of separation sharper at high concentrations when the temperature increases. The proof applies to the isothermal cases as well as when temperature gradients are applied. As the feed concentration increases at constant composition, the shape of the region of separation differs from the one calculated under linear conditions. When the feed concentration of A reaches the watershed value (cFA ) cWS A ) defined as

1 cWS A ) (S - 1) 2 KA

(33)

where the selectivity S is defined as

(27)

(28)

(29)

Equation 26 specializes the procedure to find the optimal point outlined in section 3.3 of ref 14 to the case of a Langmuir isotherm.

S)

ΓAK2A ΓBK2B

)

γ2A γ2B

(34)

then the curve wb becomes horizontal. It is worth noticing that the effect of the feed concentration on the sharpness of the region of separation depends on the selectivity, S. The values S and K2A determine through eq 33 how strong the effect of the feed concentration is on the shape of the region. If S is very large or K2A is small, then large feed concentrations are required to significantly change the shape of the separation region from the linear one. It is worth noting that this effect depends only on the feed concentration of A.18 Let us

2616

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001

Figure 13. Fluid-phase concentration profile in the unit for point D (see Figure 5) when cyclic steady-state conditions have been attained. The feed composition for A and B is 25 µmol/L (broken line).

take cWS A as the reference concentration for the analysis. Because we have assumed that ΓA and ΓB are constant, if S is decreasing with the temperature, then K2A is necessarily decreasing faster than K2B. This imis decreasing with the temperature. plies that cWS A Therefore, in the temperature gradient mode, a smaller feed concentration will lead to a region of separation with a horizontal wb curve. Appendix B The aim of this appendix is to investigate through the results of Appendix A the behavior of the unit under nonlinear conditions when the operating point belongs to the intersection between the nonlinear region of separation and the region of accumulation of B under linear conditions, i.e., region 8 in Figure 3. Here, the concentration of B in section 3 in the limit where m3 m2 f 0, i.e., close to the diagonal, is given by

c3B )

γ3B - m3 K Bm 3

(35)

Therefore, regardless of the feed concentration, the concentration of B for points in region 8 and close to the diagonal can achieve high values. To further clarify this point and to understand its consequences on the dynamics of the SMB unit, let us analyze the results of the simulation (D) in Figure 5. As an example, the region of complete separation is drawn at a feed concentration of cFT ) 50 µmol/L. From Eq. (35) it is found that c3B ) 184 µmol/L. In this example, a remarkable accumulation of B in section 3 of the unit is expected. This can be explained by considering that points belonging to region 8 (see Figure 3) lead to accumulation of B under linear conditions until the nonlinear regime of separation is achieved. This is shown in Figure 13 where the internal profiles in the mobile phase at steady state after the switch has occurred are depicted. The internal concentration of B largely exceeds the feed concentration of B (broken line), as predicted by the equilibrium theory model. This effect must be accounted for when nonlinear separations under gradient mode are carried out. The accumulation of B could in fact lead to an unstable operation of the unit if the solubility limit is exceeded. Let us remark

Figure 14. Average fluid-phase concentration at the outlet ports in the transient startup for point D (see Figure 5). The raffinate and the extract achieve 100% purity. The asymmetric behavior of the two species is evidenced. Note that, even if the internal concentration of B largely exceeds its feed concentration (broken line), the enrichment of B in the raffinate is low.

that the enrichment in point D is lower than that in point w; in other words, the concentration averaged over a switch of B leaving the unit is lower with respect to the optimal operating conditions. The accumulation of B has a strong impact on the startup dynamics of the SMB unit. In Figure 14 the effluent average concentrations of B in the raffinate and A in the extract are shown. The behavior of A and B is strongly asymmetric. In fact, B leaves the unit after the system deviates from the linear regime of separation. This is achieved only when c3B is high enough. This example shows that the prediction of the equilibrium theory model can also be used to understand better the complex dynamics of the SMB unit. Notation ap ) specific mass-transfer area ci ) fluid phase concentration of component i cp ) specific heat d ) column diameter Di ) axial dispersion coefficient of component i DR ) desorbent requirement Ei ) enrichment of component i k ) thermal conductivity ki ) mass-transfer coefficient of component i Ki ) equilibrium constant of component i in the Langmuir isotherm L ) column length mj ) flow-rate ratio in section j Nc ) number of columns in the SMB unit ni ) adsorbed phase concentration of component i PR ) productivity r ) radial coordinate R ) ideal gas constant S ) selectivity t ) time t* ) switching time T ) temperature uj ) superficial velocity in section j uS ) solid velocity z ) axial coordinate Greek Letters γi ) Henry constant of component i Γi ) saturation capacity of component i b ) bed void fraction

Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2617 p ) particle porosity * ) overall bed void fraction F ) density Superscripts and Subscripts A ) more retained component B ) less retained component D ) desorbent E ) extract eq ) equilibrium isotherm f ) fluid phase F ) feed i ) component index, i ) A, B m ) average property j ) section index, j ) 1, ..., 4 R ) raffinate s ) solid phase WS ) watershed point

Literature Cited (1) Francotte, E.; Richert, J. Application of Simulated Moving Bed chromatography to the separation of the enantiomers of chiral drugs. J. Chromatogr. A 1997, 769, 101-107. (2) Pynnonen, B. Simulated moving bed processing: escape from the high cost box. J. Chromatogr. A 1998, 827, 143-160. (3) Juza, M.; Mazzotti, M.; Morbidelli, M. Simulated movingbed chromatography and its application to chirotechnology. Trends Biotechnol. 2000, 18, 108-118. (4) Nicoud, R. M. Simulated moving bed: some possible applications of biotechnology. In Bioseparation and bioprocessing; Subramanian, G., Ed.; Wiley-VCH: New York, 1998. (5) Ruthven, D. M.; Ching, C. B. Counter current and simulated counter current adsorption separation processes. Chem. Eng. Sci. 1989, 44, 1011-1038. (6) Jacobson, S.; Golshan-Shirazi, S.; Guiochon, G. Chromatographic band profile and band separation of enantiomers at high concentration. J. Am. Chem. Soc. 1990, 112, 6492-6498. (7) Clavier, J. Y.; Perrut, M.; Nicoud, R. M. A new efficient fractionation process: the simulated moving bed with supercritical eluent. In High-Pressure Chemical Engineering; Rudolf von Rohr, Ph., Trepp, Ch., Eds.; Elsevier Science: London, 1996. (8) Mazzotti, M.; Storti, G.; Morbidelli, M. Supercritical fluid simulated moving bed chromatography. J. Chromatogr. A 1997, 786, 309-320. (9) Mazzotti, M.; Storti, G.; Morbidelli, M. Optimal operation of Simulated Moving Bed units for nonlinear chromatographic separations. J. Chromatogr. A 1997, 769, 3-24.

(10) Storti, G.; Masi, M.; Paludetto, R.; Morbidelli, M.; Carra´, S. Adsorption separation processes: countercurrent and simulated countercurrent operations. Comput. Chem. Eng. 1988, 12, 475482. (11) Storti, G.; Mazzotti, M.; Morbidelli, M.; Carra´, S. Robust design of binary countercurrent separation processes. AIChE J. 1993, 39, 471-492. (12) Mazzotti, M.; Storti, G.; Morbidelli, M. Robust design of binary countercurrent separation processes. 3. Nonstoichiometric systems. AIChE J. 1996, 42, 2784-2796. (13) Gentilini, A.; Migliorini, C.; Mazzotti, M.; Morbidelli, M. Optimal operation of Simulated Moving Bed units for nonlinear chromatographic separations. II. Bi-Langmuir isotherms. J. Chromatogr. A 1998, 805, 37-44. (14) Migliorini, C.; Mazzotti, M.; Morbidelli, M. Robust design of binary countercurrent separation processes. 5. Non constant selectivity binary systems. AIChE J. 2000, 46, 1384-1399. (15) Chiang, A. S. T. Complete separation conditions for a local equilibrium TCC adsorption unit. AIChE J. 1998, 44, 332-340. (16) Chiang, A. S. T. Equilibrium theory for simulated moving bed adsorption processes. AIChE J. 1998, 44, 2431-2441. (17) Migliorini, C.; Mazzotti, M.; Morbidelli, M. Design of simulated moving bed multicomponent separations: Langmuir systems. Sep Purif. Technol. 2000, 20, 79-96. (18) Migliorini, C.; Gentilini, A.; Mazzotti, M.; Morbidelli, M. Design of simulated moving bed units under nonideal conditions. Ind. Eng. Chem. Res. 1999, 38, 2400-2410. (19) Biressi, G.; Ludemann-Hombourger, O.; Mazzotti, M.; Nicoud, R. M.; Morbidelli, M. Design and Optimisation of a SMB Unit: role of deviations from equilibrium theory. J. Chromatogr. A 2000, 876, 3-15. (20) Migliorini, C.; Mazzotti, M.; Morbidelli, M. Simulated moving bed units with extracolumn dead volume. AIChE J. 1999, 45, 1411-1422. (21) Charton, F.; Nicoud, R. M. Complete design of a Simulated Moving Bed. J. Chromatogr. A 1995, 702, 97-112. (22) Pedeferri, M. P.; Zenoni, G.; Mazzotti, M.; Morbidelli, M. Experimental analysis of a chiral separation through simulated moving bed chromatography. Chem. Eng. Sci. 1999, 54, 37353748. (23) Lode, F.; Houmard, M.; Migliorini, C.; Mazzotti, M.; Morbidelli, M. Continuous reactive chromatography. Chem. Eng. Sci. 2001, 56, 269-291.

Received for review September 18, 2000 Accepted February 28, 2001 IE000825H