Design of Simulated Moving Bed Units under Nonideal Conditions

More recently, the design problem of SMB units has been solved for .... second term on the right-hand side of eq 9) corresponds to the axial dispersio...
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Ind. Eng. Chem. Res. 1999, 38, 2400-2410

Design of Simulated Moving Bed Units under Nonideal Conditions Cristiano Migliorini,† Andrea Gentilini,‡ Marco Mazzotti,† and Massimo Morbidelli* Laboratorium fu¨ r Technische Chemie, ETH Zu¨ rich, Universita¨ tstrasse 6, CH-8092 Zu¨ rich, Switzerland

The influence of dispersive effects (axial mixing and mass-transport resistance) on the performance of simulated moving bed (SMB) units is studied through detailed modeling. The results are represented in the space of the dimensionless operating parameters defined as the ratios between the fluid and the simulated stationary phase flow rate in the different sections of the SMB unit. The complete separation region calculated by numerical simulations is compared to the one obtained through equilibrium theory (i.e., by assuming ideal behavior and neglecting dispersive effects). The differences are shown to be relatively small when realistic values of the process parameters are used; however, they must be taken into account for fine-tuned process optimization. This proves the usefulness of the equilibrium theory approach to provide a first solution that can then be refined through detailed model simulations. A thorough analysis of the optimal design of the operating conditions when only one compound is required pure is carried out and a rather interesting asymmetric behavior is evidenced and explained. Finally, the usefulness of the theoretical findings is assessed by using them to discuss and explain a set of experimental data reported in the literature. 1. Introduction Continuous large-scale chromatographic separations using the simulated moving bed (SMB) technology1 are well-known processes in the petrochemical and food industries and are now extended to pharmaceutical, fine chemical, and biological separations.2-16 In particular, the separation of enantiomeric mixtures on chiral stationary phases has received recently great interest because of its potential in achieving high performances and in reducing the time from discovery to production of new products. The development of new separations and the need to optimize their performances motivate the use of SMB models. The first studies of the behavior of SMB units through detailed models date back to the ‘80s.17-19 The effects of changes in the operating parameters and in the layout of the unit have been analyzed by modeling SMB units directly or by exploiting the equivalence between SMB and true countercurrent (TCC) units. However, the great number of parameters involved and the computational burden needed for the solution of these models prevent the use of this approach to the design of SMB units. More recently, the design problem of SMB units has been solved for Langmuir adsorption isotherms in the framework of equilibrium theory, that is, using a model where axial mixing and mass-transfer resistances are neglected.20-22 It was demonstrated that the flow rate ratios in the four sections of the unit, that is,

mj )

Qjt* - V V(1 - )

(j ) 1, ..., 4)

(1)

are the key parameters in determining the behavior of SMBs and allow one to define rigorous criteria for the design of optimal and robust operating conditions yield* To whom correspondence should be addressed. Tel.: +411-6323034. E-mail: [email protected]. † Institut fu ¨ r Verfahrenstechnik, ETH Zu¨rich, Sonneggstrasse 3, CH-8092, Switzerland. ‡ Institut fu ¨ r Automatik, ETH Zu¨rich, Physikstrasse 3, CH8092. Switzerland.

ing desired separation performances. This approach provides a rather useful geometrical representation in the (m2, m3) plane, that is, the plane spanned by the flow rate ratios in the two central sections of the SMB unit, and allows a thorough understanding of the SMB behavior under linear and nonlinear conditions. The illustration of the application of this method and its validation by comparison with experimental data are reported in previous works.21,23-25 It is worth noting that the equilibrium theory approach is based on the equivalence between TCC and SMB configurations, whose reliability has been already investigated in the past.18 On the contrary, the quantitative understanding of the effect of axial dispersion and mass-transfer resistances on the separation regions in the operating parameter space identified through the equilibrium theory is still lacking. The aim of this work is to address this issue, by determining the region of complete separation in the real case and by comparing it with the solution provided by the ideal equilibrium theory model. The study case considered is the separation of enantiomers in a nonadsorbable solvent, where the system is described by a bi-Langmuir isotherm. This is a rather general case, representative also of the Langmuir and modified Langmuir isotherms,21 which can be regarded as a special case of the bi-Langmuir isotherm.22 The region of complete separation has been determined by performing simulations using a detailed model on a fine square grid laid on the (m2, m3) plane, while keeping the operating parameters for the first and last section of the SMB unit far away from their critical values. Since this procedure is computationally intensive and time-consuming, an equilibrium-dispersive model of the SMB, which accounts for dispersive effects, has been used.26,27 This is a rather realistic model, since axial dispersion and mass-transfer resistances play a qualitatively similar role in determining the performances of chromatographic columns.28 It is worth noting that in this work the analytical results of the equilibrium theory, which are based on the TCC/SMB equivalence,18 have been compared di-

10.1021/ie980262y CCC: $18.00 © 1999 American Chemical Society Published on Web 04/27/1999

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2401

rectly with the cyclic steady-state results of the SMB model. This is the only approach which can assess the deviation of equilibrium theory predictions from the performances of the real SMB model. A comparison with the TCC model with axial dispersion does not provide reliable answers in this regard. In fact, it is difficult to give a clear physical interpretation of the parameters in the TCC model; for instance, the HETP values for a fixed bed and a countercurrent one are different even though the relative velocity between the fluid and the solid phase is the same, that is, even when the two units are kinetically equivalent.1 Moreover, experimental concentration profiles along the SMB columns should be compared with results obtained using an SMB model27 and not with simulated TCC profiles,29 particularly when the overall number of columns is equal or smaller than eight, as in most practical small-scale HPLC-SMB applications. As a matter of fact, it has been demonstrated that SMB profiles approach TCC ones for a large number of subsections per section18 and that for a small number of columns the two profiles can be significantly different.18,30 In the following the equilibrium-dispersive model is first presented and the underlying assumptions are discussed. For reference, an analysis of literature data is reported giving the values of the number of theoretical plates for different pilot units used for the separation of enantiomers. Then, for the study case considered, the region of complete separation in the (m2, m3) plane in the presence of axial dispersion is drawn and compared with the one provided by the equilibrium theory. It is worth noting that this approach, which is based on numerical calculations. may be applied also to isotherms for which the equilibrium theory solution is not available at all. An analysis of the performances of the unit when the operating point crosses the region of separation is reported; an asymmetric behavior is observed when only one stream has to be collected pure. This finding is rather important in applications since it allows one to optimize the separation when only one of the compounds is needed pure. Finally, some experimental data are discussed in the light of the theoretical results obtained, in order to assess their reliability and practical relevance. 2. Modeling SMB Units 2.1. Single Chromatographic Column. The use of several different models for the description of the dynamic behavior of chromatographic columns has been reported in the literature. Models based on the use of the linear driving force approximation are widely applied; despite their simplicity they constitute an accurate tool for the prediction of breakthrough profiles in most situations.31 The lumped solid diffusion model is one of these and is constituted of the following set of equations (i ) A, B in the cases of interest here):

∂ci ∂2ci ∂ni ∂ci  + (1 - ) +u ) Di 2 ∂t ∂t ∂z ∂z

(2)

∂ni ) kiap(n/i - ni) ∂t

(3)

n/i ) f eq i (c)

(4)

where ki is the overall mass-transfer coefficient of

component i, ap is the specific surface of the adsorbent particles, and the other symbols are defined in the Notation section. Together with proper initial and Dankwerts boundary conditions, these equations can be solved using several different numerical techniques (e.g., orthogonal collocations32 or finite differences33). It is worth noting that chromatographic column efficiency (i.e., the capacity of the column to minimize band broadening and to achieve good separation performance under analytical conditions) depends on the masstransfer and axial dispersion coefficients (i.e., ki and Di). Column efficiency is often given in terms of the number of theoretical stages, Np. In the case where the adsorption isotherm is linear (i.e., n/i ) Hici), then the solid diffusion model, (2) to (3), is equivalent to the equilibrium dispersive model, where mass-transfer resistance and axial dispersion are lumped in an apparent axial dispersion coefficient and local equilibrium conditions are enforced:26,27,34



∂ci ∂2ci ∂n/i ∂ci + (1 - ) +u ) Dap,i 2 ∂t ∂t ∂z ∂z

(5)

The equivalence between the two models implies equal values for the dispersive parameters (i.e., the same number of theoretical stages). In the case of the lumped solid diffusion model under infinite dilution linear conditions this is given by the following relationship:35

(

2Di (1 - )Hi 1 2u + ) sol uL (1 - )LHikiap  + (1 - )Hi Np,i

)

2

(6)

sol The efficiency parameter Np,i may be different for different solutes, since the controlling parameters (i.e., ki, Di, and Hi) are solute-specific. This equation can be recast as follows:

(

)

(1 - )Hi 1 2 1 ) sol + sol sol Np,i Nmix,i Nmt,i  + (1 - )Hi

2

(7)

where each dispersive mechanism contributes independently to the overall number of theoretical stages sol through a number of mixing stages, Nmix,i , and a sol number of mass-transfer stages, Nmt,i. In the case of the equilibrium dispersive model (5), the number of theoretical stages is given by the following relationship:35

2Dap,i 1 ) ead uL Np,i

(8)

The simplified equilibrium axial dispersive model is equivalent to the solid diffusion model, provided that Dap,i is chosen so that the number of theoretical plates ead sol ) Np,i for i ) in the two models are the same (i.e., Np,i A, B). Under the assumption that the apparent axial dispersion is the same for all components (i.e., Dap,i ) Dap), the numerical solution of the equilibrium axial dispersive model can be obtained in a computing efficient way by using a finite difference scheme and letting numerical dispersion play the role of the apparent axial dispersion in eq 5. This is obtained by discretizing the first-order space derivative in eq 5 through backward

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Figure 1. Scheme of a simulated moving bed unit for continuous chromatographic separations. with column configuration 2-22-2.

differences:

|

|

the inlet and outlet streams are moved one column forward in the direction of the fluid flow. Thus, the space composition profiles at the end of the previous period (i.e. at time nt*) are used as initial conditions for the next integration from time nt* to (n + 1) t*. Whatever the model used, after space discretization, the partial differential equations are reduced to a system of ordinary differential equations, which is integrated in time using a commercial stiff integrator. Mass balances at the nodes and boundary conditions are written in terms of the concentration at the end points of the relevant columns and directly substituted in the system of ordinary differential equations of the SMB model. It is worth noting that when using the solid diffusion model, the axial dispersion coefficient is calculated in every section of the unit using the Chung and Wen equation31 and neglecting the molecular diffusion contribution; this yields the following relationship for the number of mixing stages which is independent of the fluid flow rate and is the same for all solutes:

∂2ci ∆z ci(z) - ci(z - ∆z) ∂ci ) + O(∆z2) (9) - 2 ∆z ∂z z ∂z z 2 where the neglected terms are proportional to ∆z2. If the space interval, ∆z, is chosen so that

∆z ) 2Dap/u

(10)

then the numerical error (i.e., the second term on the right-hand side of eq 9) corresponds to the axial dispersion term in eq 5, which is therefore not included in the numerical scheme. It can be observed that using this algorithm the number of grid points (i.e., NG ) L/∆z) is equal to the number of theoretical stages given by eq 8. Mass-transfer resistance and axial dispersion effects are often lumped into an apparent axial dispersion coefficient also when the adsorption equilibria (4) are nonlinear. However, this is not rigorous anymore and the equivalence between the solid diffusion model and the equilibrium axial dispersive model exploited in the linear case becomes now only an approximation. However, it has been observed that this approximation leads to satisfactory results in many cases of interest, even when the number of stages is rather small (i.e., Np,i < 100 (cf., for example, refs 36-38)). This result is confirmed in the next section with reference to the specific system considered in this work. As a conclusion, in the following we use the equilibrium axial dispersive model, not only because the solution technique described above is efficient but also because it allows one to lump all dispersive terms in a single parameter (i.e., the number of theoretical stages Np). This is particularly convenient for the analysis to be developed in this work. 2.2. Simulated Moving Bed. A schematic of the SMB unit considered in this work is shown in Figure 1: it is constituted of eight columns distributed according to a 2-2-2-2 configuration (i.e., two columns per section). Therefore, the model describing this SMB unit is constituted of one set of the equations reported in the previous section for each column (i.e., either the lumped solid diffusion equations, (2)-(4), or the equilibrium axial dispersion equations (5)) with the adsorption isotherm (4), together with the single component and overall material balances at the four inlet and outlet nodes of the unit.18 At every instant of time equal to an integer multiple of the switching time, t*, the ports of

L 10dp

Nsol mix )

(11)

Because of eq 6, a different number of theoretical stages in the different sections of the SMB unit are used. On the other hand, if the axial dispersive model is used and the algorithm described above is implemented using the same number of grid points in each section of the unit, then further approximation is introduced. This is not critical in the context of the analysis carried out in this work, provided that the number of grid points is chosen as the smallest one among those exhibited in the different sections of the unit. In order to quantitatively analyze this point, we compare the results obtained using the two models with reference to the system considered in this work. This is the separation of the enantiomers of 1-1′-bi-2-naphtol on a 3,5-dinitrobenzoyl phenylglycine bonded to a silica gel stationary phase, using a mixture of heptanehexane (78:22) as the mobile phase.13,14 The adsorption equilibrium is described by the bi-Langmuir isotherm:

ni )

γici 1+

∑jajcj

+

δici 1+

∑jbjcj

i ) A, B (12)

where γA ) 3.728, γB ) 2.688, δA ) 0.3, δB ) 0.1, aA ) 0.0466 L/g, aB ) 0.0336 L/g, bA ) 3 L/g, and bB ) 1 L/g.13,14 It follows that A denotes the more retained enantiomer and B the less retained one. The separation is carried out in an eight-column SMB unit with a 2-22-2 configuration, such as the one shown in Figure 1; the column volume is V ) 55.75 cm3 and the overall void fraction is  ) 0.4. In the case of the solid diffusion model sol the parameters used are Nsol mt ) 45, Nmix ) 1000, and sol Np ) 32 (the same values for A and B have been adopted). Note that the number of mass-transfer stages and the number of theoretical stages refer to the conditions in section 1, but they increase in the other sections where the flow rates are smaller; on the contrary, the number of mixing stages is constant, because of eq 11. This set of parameters indicates that in this case mass transfer is controlling; hence, the assumptions on which the equilibrium axial dispersive model is based are particularly critical. In the case of the approximate model a value of the apparent axial ) 32 has dispersion coefficient corresponding to Nead p

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2403

Figure 2. Separation of bi-naphtol enantiomers: composition profiles in the SMB unit immediately after the port switch (see section 2.2 for operating conditions and model parameters). Broken line, solid diffusion model; solid line, equilibrium axial dispersive model with Nead ) 32. Components A and B are collected in the p extract and raffinate, respectively.

been used for all SMB sections. The following operating conditions have been considered: Q1 ) 56.9 cm3/min; Q2 ) 40.8 cm3/min; Q3 ) 44.5 cm3/min; Q4 ) 35.4 cm3/ min; t* ) 165 s; cFA ) cFB ) 2.9 g/L. The concentration profiles in the SMB unit obtained using the two models are shown in Figure 2; only minor differences can be observed. Other simulations not reported here confirm these results and indicate that when the number of mass-transfer stages is larger than that in the previous example a better agreement between the two models is achieved, as expected. In conclusion, the axial dispersive model will be used in this work to analyze in general terms the effect of dispersive phenomena on SMB performance by investigating the effect of the single parameter Np. However, whenever a specific system has to be simulated and optimized, it is recommended that a detailed model be used, such as the lumped solid diffusion model discussed above. 3. Region of Complete Separation In the framework of equilibrium theory the optimal operating conditions for SMBs are obtained by calculating the region of complete separation in the (m2, m3) plane.21,22 This region is made of operating points leading to complete separation, that is, both components are collected pure in the product streams, provided that proper constraints on the parameters of the regenerating sections m1 and m4 are fulfilled. These are simply given by a lower bound for m1 and an upper bound for m4, corresponding to the specific tasks of section 1 and 4 (i.e., the regeneration of the eluent and of the adsorbent, respectively). The aim of this section is to determine the complete separation region in the (m2, m3) plane in the presence of dispersive effects (i.e. using the equilibrium dispersive model introduced in the previous section). Since sections 2 and 3 are the heart of the separation, in the following we concentrate on the (m2, m3) plane and select the operating conditions for parameters m1 and m4 far enough from their respective critical values. In such a way, we are guaranteed that the results obtained for sections 2 and 3 are not affected by any improper operation of the regenerating sections 1 and 4. The objective of this analysis is to draw conclusions of general validity about the effect of dispersive phenomena on SMB performances. This is conventionally done by using a case study to illustrate the various findings. In particular, we have selected the separation

Figure 3. Purity contour lines in the operating parameter plane (m2, m3). Contour lines for the raffinate intersect the diagonal at the top right corner, contour lines of the extract at the bottom left corner. The efficiency of each column is 30 theoretical stages.

of the enantiomers of bi-naphtol referred to in the previous section. In all the following simulations a few parameters have been held constant: column volume V ) 55.75 cm3; overall void fraction  ) 0.4; switch time t* ) 170 s; flow rates Q1 ) 78.7 cm3/min (i.e., m1 ) 6) and Q4 ) 19.7 cm3/min (i.e., m4 ) 1); feed concentration cFA ) cFB ) 2.9 g/L. This choice of the parameters m1 and m4 guarantees complete regeneration of stationary and mobile phases in sections 1 and 4, respectively. We will come back to these constraints in section 5 in order to analyze their effect on the separation performances. In all simulations cyclic steady-state conditions have been reached, as confirmed by overall and single-component mass balances. In Figure 3, the results obtained by assuming an apparent axial dispersion corresponding to Np ) 30 are illustrated. In particular, the contour lines of the purities in the outlet streams, obtained by performing simulations on a square grid in the (m2, m3) plane spaced by ∆m ) 0.025, are drawn. Contour lines for the purity in the raffinate, PR, intersect the diagonal on the top right corner of the diagram; the region below each specific PR contour line (i.e., between this and the diagonal) corresponds to operating conditions leading to a purity in the raffinate larger than the value labeling the line itself (e.g., 99% for the lowest thick solid curve). Contour lines for purity in the extract, PE, intersect the diagonal on the bottom left corner of the diagram; the region above and on the right-hand side of a specific PE contour line is constituted of operating points leading to a purity in the extract larger than the value corresponding to the line itself. On the basis of these definitions, it is possible to readily identify all the operating conditions leading to purity values in the extract and raffinate larger than any given pair of values. For instance, the triangleshaped region whose boundaries are the diagonal and the two thick solid contour lines corresponding to PE ) 99% and PR ) 99% identifies operating conditions achieving 99% purity or more in both product streams. The vertex of this region (i.e., the furthest point from the diagonal) corresponds to optimal operating conditions in terms of productivity and eluent consumption when dispersive effects are accounted for.21 The results illustrated in Figure 3 allow one to choose the optimal operating conditions for the SMB separation fulfilling any process requirement, in terms of purity of the outlet streams. Once the values of the mj parameters have been chosen in this way, the specific values of flow rates

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Figure 4. The region of complete separation, calculated through equilibrium theory assuming an infinite column efficiency (s) is compared with the region where the purity of both extract and raffinate is greater than 99%, assuming an efficiency of 30 (- - -) and 20 (-‚-‚) theoretical stages. The vertex, the triangle-shaped region obtained through equilibrium theory, is the ideal optimal operating point.21 The three regions corresponding to different separation regimes are also drawn, as calculated through equilibrium theory. The points A, B, C, and D represent the operating points of the experimental runs reported in Table 1 and discussed in section 5 (the operating points of runs B and B' coincide).

to be selected can be calculated using eq 1. To this aim, a further constraint to select a proper value of t* must be enforced; this may be done by requiring that either column efficiency is large enough or pressure drop is small enough, or else by imposing a given value for the overall feed flow rate. The effect of changes in column efficiency is illustrated in Figure 4. Here, the same region corresponding to PE and PR values larger than 99% as in Figure 3 is shown (broken boundaries), together with the region calculated in the same way but with a lower column efficiency, namely, with Np ) 20 (dashed-dotted boundaries). It can be readily observed that reducing column efficiency makes the region of the operating parameter plane where the desired purity values are achieved (99% in this example) smaller. For a comparison, in Figure 4 the ideal complete separation region (solid boundaries) corresponding to the isotherms (12) and calculated in the frame of equilibrium theory,22 (i.e., assuming Np f ∞) is also shown. It is seen that the separation region corresponding to finite columnn efficiency approaches the ideal region given by equilibrium theory solution when Np increases. In particular, this applies to the optimal operating points (i.e., the vertices of the regions drawn in Figure 4). Moreover, it can be noted that even with Np ) 30 (i.e., with a rather low column efficiency compared to the typical experimental) the vertex of the 99% purity region is very close to the optimal point of the ideal complete separation region calculated without dispersive effects, which leads to 100% purity. Finally, let us consider Figure 5 where the extract and raffinate purity are plotted as a function of the number of theoretical plates. The same operating conditions, corresponding to the optimal point of the ideal complete separation region in Figure 4 (i.e., m2 ) 2.49 and m3 ) 3.14) but different column efficiencies have been adopted. This corresponds to different Np values (i.e., to different number of space discretization points). It is seen that the complete separation prediction of equilibrium theory is reached very rapidly for the raffinate (at Np = 50) and a bit slower for the extract. However, it is noteworthy that in both cases very high

Figure 5. Extract and raffinate purity as a function of the number of theoretical stages. The simulations are performed in the ideal optimal operating point W shown in Figure 4; the corresponding operating conditions are m2 ) 2.49 and m3 ) 3.14.

Figure 6. Regions of separation in the (m2, m3) plane at cFA ) cFB ) 2.9 g/L. The simulations are carried out moving the operating point along the straight lines AB, CD, EF, and GE, while keeping all the other parameters constant. The corresponding flow rate values for Q2 and Q3 can be obtained from eq 1.

purity values (above 99%) in both streams are achieved at relatively low efficiency values (Np = 40). The same pattern of behavior (i.e., a first fast improvement of performances while increasing column efficiency at low values of Np followed by a slow asymptotical attainement of the ideal dispersion-free performances) is typical for SMB units under overload conditions and not only for the specific system used for the calculations above. In particular, the same qualitative behavior is exhibited also by non-Langmuirian systems for which the equilibrium theory analysis is not feasible. A similar analysis of the effect of dispersive phenomena on SMB behavior has been reported by Rodrigues and co-workers,13,14 who have also investigated experimentally the system discussed above. Their results (see Figures 2, 3, 6, and 7 in ref 13) are qualitatively similar to ours, as illustrated in Figures 3 and 4, but quantitatively different. This can be explained by observing that in our analysis the flow rates in the different sections of the unit are chosen so as to always fulfill the constraints on sections 1 and 4, independent of the values of m2 and m3. Therefore, the performance of the separation corresponding to the pairs of m2 and m3 values can be attributed entirely to the operation of sections 2 and 3. On the contrary, Rodrigues and coworkers establish a constraint among the involved flow rates, which therefore are not chosen independently. In particular, the overall product flow rate (i.e., the sum of the flow rates of the extract and raffinate) is kept constant together with the value of m4. Therefore, in

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2405

Figure 7. Extract and raffinate purity as a function of m2 in the set of simulations AB of Figure 6. The efficiency is 30 theoretical stages.

Figure 8. Extract and raffinate purity as a function of m2 in the set of simulations CD of Figure 6. The column efficiency is 30 (s) and 20 (- - -) theoretical stages.

all their calculations, m1 is a linear function of m2 and m3, namely, m1 ) 4.76 - m3 + m2, and not a constant as in ours. It follows that in this case a pair of values for m2 and m3 determines not only the operation of sections 2 and 3 but also indirectly that of section 1. This implies that performances are different for different values of the constant in the linear relationship above among m1, m2, and m3 and also different from those obtained through our approach. In conclusion, these representations cannot be compared on a quantitative basis and the boundaries separating different separation regimes illustrated in Figure 4 should not be expected to coincide with those presented in ref 13.

tion region in the (m2, m3) plane for the case study considered in the previous section as calculated through equilibrium theory is shown, together with the other regions of partial separation (i.e., only pure extract, only pure raffinate, and no pure outlet streams). In particular, four sets of operating points are considered, each corresponding to a straight segment in the (m2, m3) plane. The same parameters as those in the previous section are held constant. First, let us consider the operating points A f B on a segment parallel to the diagonal. In practice, these conditions are obtained by changing the extract and raffinate flow rates by the same quantity but in the opposite direction, so as to keep constant the feed and eluent flow rates. This strategy has been adopted, for example, by Francotte and Richert11,39 in their experimental investigation of the guaifenesine separation. A similar situation occurs when all the flow rates are kept constant and the switching time is changed. In this case the obtained set of operating points lies on a straight line which is almost parallel to the diagonal, as for the experimental results reported by Pais et al.13,14 and Pedeferri et al.16 In Figure 7 the calculated purities in the outlet streams as a function of the m2 value are shown, when the dispersive effects correspond to Np ) 30. It appears that dispersive effects reduce the complete separation region which equilibrium theory predicts for m2 values between 2.54 and 2.72, to a very small neighborhood of the value m2 ) 2.68. It is worth noting that the dispersive complete separation region is very small in this case because the segment AB crosses the ideal complete separation region at a location where this is rather narrow (i.e., very close to the ideal optimal operating point represented by the vertex of the equilibrium theory region). When considering the set of operating points C f D in Figure 6, which are further away from the optimal point, then both the ideal and the dispersive complete separation regions are larger. This is confirmed by the purity values shown in Figure 8 corresponding to operating points along the segment C f D, with Np ) 30. In the same figure the results obtained with a lower column efficiency (i.e., Np ) 20) are also shown. As expected from the results shown in Figure 4, the range of m2 values achieving complete separation is smaller than that in the case of Np ) 30. It is worth noting that on the one hand extract purity at Np ) 20 is always smaller than that at Np ) 30, while on the other hand raffinate purity exhibits a crossover behavior for m2 larger than about 3.15. This is somehow unexpected, even though its importance should not be

4. Performance Parameters The operating parameter space of SMB units is multidimensional. Experimental and numerical results of SMB performances are usually performed along a one-dimensional subset of the whole space, with the aim of locating the complete separation region or determining the optimal operating conditions with respect to a single operating parameter. In this section, we revise a number of these strategies focusing on the effect of dispersive phenomena on process optimization. In order to quantitatively determine the performance of a separation process, let us introduce the following parameters: purity of the product streams, defined as

PE )

PR )

cEA cEA + cEB cRB cRA + cRB

× 100

(13)

× 100

(14)

and specific productivity with respect to the species collected pure in the relevant outlet stream (this is of particular importance whenever only one of the components to be separated is of interest21):

PRA )

QEcEA ncV

(15)

PRB )

QRcRB ncV

(16)

The different strategies considered in the following are illustrated in Figure 6, where the complete separa-

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Figure 9. Performance parameters along the set of points EF in Figure 6 as a function of the operating parameter m3. The purity of the extract PE and of the raffinate PR are drawn as well as the normalized value of the productivity of A.

Figure 10. Performance parameters along the set of points GE in Figure 6 as a function of the operating parameter m2. The purity of the extract PE and of the raffinate PR are drawn as well as the normalized value of the productivity of B.

overestimated since this phenomenon occurs for operating conditions where raffinate purity is rather poor (less than 90%), hence, not very interesting from the application viewpoint. Finally, note that for both sets of points A f B and C f D the purity performances illustrated in Figures 6 and 7 are consistent with the position of each operating point in the (m2, m3) parameter plane shown in Figure 6. In particular, as m2 and m3 increase, first 100% purity in the raffinate and low purity in the extract are observed (pure raffinate region); then, both streams achieve 100% purity (complete separation region) and finally PE is 100% and the raffinate purity is lost (pure extract region). The third set of operating points considered (i.e., E f F in Figure 6) is obtained by increasing the feed flow rate and the raffinate flow rate by the same amount (keeping all the other flow rates constant) and was adopted by Francotte and Richert11,39 to improve the productivity of the unit with respect to the component collected pure in the extract. The purity performances in the case of Np ) 30 are reported in Figure 9 as a function of m3, together with the productivity parameter PRA defined by eq 15. The purity values are consistent with the position of the operating points in the (m2, m3) parameter plane, which are first inside the complete separation region and then move to the pure extract region. Therefore, PE ) 100% in all cases whereas PR drops below 100% for m3 larger than 3.25 (i.e., a bit earlier than that predicted by equilibrium theory (m3 ) 3.27)). The behavior exhibited by PRA is rather interesting and indicates that when entering the pure extract region, the extract purity remains at 100%, but no further improvement of the productivity of the species collected in the extract is possible. In other words, in these conditions all the additional amount of A fed to the unit is directly conveyed to the raffinate outlet. A different behavior is observed if, starting from the operating point E, the segment E f G is followed (see Figure 6). From the practical point of view this corresponds to the simultaneous increase of the feed and extract flow rates, while the other streams are left unchanged. With reference to Figure 10, where the results corresponding to Np ) 30 are illustrated, it can be seen that as m2 decreases we go from an operation where both the extract and raffinate are pure to one where the raffinate purity remains 100% while that in the extract decreases. This is consistent with the transition from complete separation to a pure raffinate region

shown in Figure 6. However, when compared to the raffinate purity behavior in Figure 9, it is found that the extract purity drops earlier with respect to the critical ideal value, m2 ) 2.57. This is consistent with the results shown in Figure 4, where it appears that the dispersive effects strongly bend the left-hand side of the complete separation region (i.e., the region of pure raffinate expands rightward with respect to the ideal case). Another difference with respect to the previous case, which has even more relevant practical implications, is that the productivity of B defined by eq 16 increases steadily while m2 decreases (i.e., the feed flow rate increases). This means that, moving leftward along segment EG or along other lines parallel to the horizontal axis allow one to increase the productivity of the species collected in the raffinate, while keeping its purity equal to 100%. This indicates that separations where we are interested in the recovery of only one pure component exhibit a rather asymmetric behavior depending on whether such a component is recovered in the extract or in the raffinate. In the first case there is an upper bound in the feed flow rate above which the productivity of the pure component does not increase. In the second case such a bound does not exist and it is possible by increasing the feed flow rate continuosly to improve the productivity of the pure component. Of course, when doing this, the process yield (i.e., the fraction of pure component in the raffinate with respect to the total feed amount) continuously decreases. The above-mentioned asymmetrical behavior is obviously quite relevant when designing a separation process. Its nature is actually not related to dispersive phenomena, but only to the countercurrent flow of the two phases and to the phase equilibria. This phenomenon can be in fact explained using equilibrium theory, as shown in detail in the Appendix. Here, it suffices to say that the bi-Langmuir isotherm, as well as the Langmuir isotherm, exhibits a watershed point in the phase plane spanned by the concentrations of the two species to be separated.40 In the frame of equilibrium theory, this is located on the cA axis and represents the unique composition state where the two characteristic lines emanating from each point in the (cA, cB) plane coincide. From the practical point of view, its cA concentration value represents the maximum value of the concentration of A achievable in the second section of the SMB unit, whatever the feed concentration cFA is. Since no similar constraint exists for the less retained species B, it follows that the cEA value, hence PRA from eq 15, is

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2407 Table 1. Values for the mj Parameters Taken from References 13 and 14 and Critical Values for m1 and m4 as Obtained through Equilibrium Theory (The Position of the Operating Points in the (m2, m3) Plane Is Shown in Figure 4) run

(m1,cr)

m1

m2

m3

m4

(m4,cr)

t* (s)

A B B′ C D

4.03 4.03 4.03 4.03 4.03

3.67 4.01 4.20 4.09 4.51

2.45 2.69 2.69 2.75 3.06

2.72 2.99 3.00 3.05 3.39

2.03 2.24 2.39 2.29 2.56

2.46 2.56 2.55 2.57 2.63

153 165 172 168 183

Table 2. Purity Values and Results of the Simulations for the Operating Conditions in Table 1 experimental purity

Np ) 20

Np ) 30

run

PE

PR

PE

PR

PE

PR

A B B′ C D

74.0 93.0 94.5 95.6 91.5

93.0 96.2 98.9 95.4 70.9

81.7 98.2 98.3 99.2 99.2

94.5 99.1 99.8 99.4 85.6

84.3 99.3 99.3 99.8 99.8

95.4 99.5 99.9 99.8 85.3

upper-bounded, while on the other hand from eq 16, can reach any value.

cRB,

hence PRB

5. Comparison with Experimental Data In this section the above theoretical findings are used to discuss and explain a set of experimental results reported by Pais et al.13,14 This is possible because the bi-Langmuir isotherm which describes the adsorption thermodynamics of this system has been used in all previous calculations. The characteristics of the experimental system and the SMB unit have already been discussed in section 3. The experimental operating conditions in terms of mj values and switching time t* are summarized in Table 1, while the corresponding operating points are shown in Figure 4, where their position can be compared with the complete separation region calculated through equilibrium theory and the 99% purity regions obtained for Np ) 20 and Np ) 30. It is worth noting that runs A, B, C, and D have been performed with the same fluid flow rates for all SMB streams, but with increasing values of the switching time t*. On the other hand, run B′, which shares the same operating point as that of run B in the (m2, m3) plane, has been performed with smaller flow rate values and a larger t* than those of run B, so as to obtain a better column efficiency. In order to understand the unit behavior in these different operating conditions, we should consider that by changing the switching time value at constant flow rates, not only m2 and m3 change in such a way that the complete separation region is crossed (see Figure 4), but also m1 and m4 change according to eq 1. Therefore, a careful check of the values of those parameters with respect to the corresponding critical values is necessary in order to guarantee that the regenerating sections of the SMB operate properly. To this aim, in Table 1 the critical values of m1 and m4 are reported, as calculated using the adsorption isotherm (12). The experimental separation performances are reported in Table 2, together with those calculated using the equilibrium-dispersive model considering rather low-efficiency columns (i.e., Np ) 20 and Np ) 30). First, let us consider runs A, B, C, and D. It is readily seen that in no experiment complete separation has been achieved even though the complete separation region is indeed crossed when going from point A to

Table 3. Values of the m1 and m4 Parameters To Guarantee Regeneration in Sections 1 and 4 (The Position of the Operating Points in the (m2, m3) Plane Is the Same as That Reported in Table 1) run

m1 ()1.2m1,cr)

m2

m3

m4 ()0.8m4,cr)

t* (s)

A B B′ C D

4.84 4.84 4.84 4.84 4.84

2.45 2.69 2.69 2.75 3.06

2.72 2.99 3.00 3.05 3.39

1.97 2.05 2.04 2.06 2.10

153 165 172 168 183

Table 4. Results of the Simulations for the Conditions Reported in Table 3 Np ) 20

Np ) 30

run

PE

PR

PE

PR

A B B′ C D

81.3 98.1 98.2 99.1 99.9

99.9 99.9 99.9 99.9 85.8

83.9 99.2 99.3 99.8 99.9

99.9 99.9 99.9 99.9 85.3

point D in Figure 4. However, this inconsistency is only apparent and can be explained by considering the performances of sections 1 and 4. Let us consider the raffinate purity. This should be the highest in run A which is in the pure raffinate region. On the contrary, experimental value is relatively low (i.e., 93%) and is qualitatively consistent with the calculated value which is about 95%. This can be explained by noting that the parameter m1 (cf. Table 1) is smaller than the critical value and therefore the position of the operating point in the (m2, m3) plane is not sufficient anymore to define the unit performances. In this case, in fact, although sections 2 and 3 properly perform their task, section 1 does not achieve complete regeneration of the stationary phase. Therefore, some of the more retained component is kept by the stationary phase when the switch occurs and it is carried to the raffinate outlet, thus spoiling its purity (see Mazzotti et al.24 for a similar effect in the case of the separation of linear and nonlinear paraffins). The poor raffinate purity in run A is not due to dispersive effects. This point can be best demonstrated by a simple numerical experiment, whose results are reported in Tables 3 and 4. The first one reports the operating conditions adopted in the simualtions and the second one the calculated purity performances, corresponding to Np ) 20 and Np ) 30. Operating conditions for the calculations are chosen in such a way that the m2 and m3 values are the same as those in the experimental runs, whereas m1 and m4 fulfill the corresponding constraint with a 20% margin (cf. Tables 1 and 3). In this case it is found for run A that, for both column efficiencies, 99.9% purity in the raffinate is achieved. The same considerations, as far as the raffinate purity is considered, apply also to runs B and C, where the experimental value of m1 is very close to the critical value. In run D the adsorbent regeneration is likely to be complete since the experimental value of m1 is about 10% larger than m1,cr. However, in this case the raffinate purity is actually the lowest because the values selected for m2 and m3 correspond to high purity only in the extract, as shown in Figure 4. Accordingly, in all simulations of run D with different m1 values (see Tables 2 and 4), but values always larger than m1,cr, the raffinate purity is very low (i.e., about 85.5%). The conclusion is that the behavior of this unit, with respect to the raffinate, is qualitatively determined by the choice

2408 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999

of the parameters m1 and m4, while dispersive effects are relevant only for the quantitative values of the performance parameters. This clearly appears from the results in Table 4, indicating that once the parameters m1 and m4 are properly selected, the calculated performances are in full agreement with the positions of the operating points in the (m2, m3) plane in Figure 4. The interpretation of the experimental results becomes less clear when we consider the extract purity achieved in runs A, B, C, and D. Indeed, the very low experimental value of PE in run A, 74%, is consistent with the position of the operating point in the pure raffinate region and with all calculated values reported in Tables 2 and 4, which are between 81% and 84%. However, in the other three operating conditions, higher purity values in the extract should be expected. This is in fact what is found with the equilibrium dispersive model simulations reported in Table 2, which predict in all cases larger purity values for the extract. One possible explanation for this disagreement is that the m4 value is smaller than the critical value m4,cr (see Table 1) but only with a very small margin (i.e., about 12%, 12%, and 3% for runs B, C, and D, respectively). Thus, small inaccuracies in the experimental variables which affect the behavior of the section could lead to an incomplete regeneration of the mobile phase which through recycle would carry some amount of compound B in section 1, thus polluting the extract. Note that this observation applies particularly to run D, which is the one where the highest extract purity should be expected. On the other hand, for runs B and C there is another aspect which is worth considering. These points are in fact close to the left boundary of the ideal complete separation region and this zone of the (m2, m3) plane is rather sensitive to the column efficiency as illustrated in Figure 4. Therefore, it is useful to consider the position of the operating points in the (m2, m3) plane with respect to the 99% regions calculated with a different number of theoretical plates and shown in Figure 4. It can be observed that experimental values of PE less than 96% in run B and C can be explained also by assuming that the column efficiency does not correspond to Np ) 30, but it is close to or even less than Np ) 20. Finally, let us consider run B′ and compare it to run B. The qualitative behavior is similar, that is, low extract purity and larger, although not very high, raffinate purity. The experimental m1 value in this run has been chosen 5% greater than the critical value and thus a better regeneration of the solid phase is fulfilled. According to the previous analysis, the raffinate purity achieved, 98.9%, is much higher than that in run B. Moreover, from the quantitative point of view, separation performances are better in the case of run B′, also because the switching time is larger and the flow rates are smaller. This implies that the improvement in terms of purity of the outlet streams is compensated by a lower productivity. The effect of the increase of the m1 value is observed by considering the calculated performances for runs B and B′ (see Table 2).

of SMB units, which is detailed enough to properly account for the mentioned effects and simple enough to allow for massive numerical computations. The outcome of this analysis demonstrates the usefulness of the ideal prediction of the complete separation region in the operating parameters plane obtained through equilibrium theory (i.e., assuming infinite column efficiency). The effect of dispersive phenomena on the shape of the complete separation region and the location of the optimal operating point have been quantitatively elucidated. It is found that when considering values of axial dispersion and transport resistances which are typical in applications, the changes of the separation regions, with respect to the predictions of equilibrium theory, are relatively small. Nevertheless, they cannot be neglected in the quantitative evaluation of process performances, where even differences of less than 1% in product purity may be quite critical. Although the analysis has been performed with reference to a specific system, as it is necessary when a numerical approach is adopted, its results bear a more general qualitative validity. In this regard, one aspect particularly relevant for applications is the behavior of SMBs in terms of purity of the outlet streams and productivity. An asymmetric behavior with respect to the more and less retained component to be separated has been evidenced. This must be taken into account and exploited in order to select not only optimal operating conditions but also stationary and mobile phases (i.e., elution order of the components to be separated). Finally, the value and usefulness of this theoretical analysis has been proved by discussing and explaining a set of experimental results in the literature. It is apparent that because of the complexity of these systems, an appropriate theoretical analysis is needed to understand the experimental behaviors and to improve the separation performances.

6. Concluding Remarks

In a countercurrent section, where a fluid stream of pure eluent and a solid stream containing the mixture of A and B considered above are fed at the two opposite ends of the column, the steady state achievable in a finite time can be either one of the feed states, that is, pure eluent or the A plus B mixture, or the pure A plateau discussed above; the specific steady state is

In this work, the design of the optimal operating conditions of simulated moving bed units in the presence of nonneglegible dispersive effects (i.e., axial mixing and mass-transfer resistances) has been considered. This has been done using an equilibrium dispersive model

Appendix The aim of this appendix is to explain the role of the watershed point in countercurrent processes as compared to its well-known features in chromatography. Let us consider a column initially saturated with a binary mixture constituted of the more retained component A and the less retained one B. Assume that pure eluent is fed. At first, the outlet stream will have the same concentration of the initial state of the column; then, through a transition the weak component B will be eluted and a plateau concentration of pure A will be reached. At the end, after a second transition, through which A is eluted, only eluent comes out, thus indicating equilibration of the column with the pure eluent feed stream. It is known that the intermediate concentration of A is upper-bounded by the watershed concentration, which is defined as41

∂nA ∂cA

|

) (cA,WS,0)

|

∂nB ∂cB

(17)

(cA,WS,0)

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2409

selected by the choice of the parameter m.20 It has been recently shown that the behavior of this countercurrent section is representative of that of section 2 of a foursection TCC unit. Moreover, the extract will contain only component A if and only if at steady state section 2 achieves the intermediate pure A plateau discussed above (cf. ref 42 for the demonstration and ref 20 for the same demonstration in a slightly different context); therefore, the steady-state concentration of A in section 2 of a TCC unit is upper-bounded by the watershed composition. Let us consider the operating points represented by line FE in Figure 6, which have a fixed m2 value. When the operating point crosses the region of separation to enter the one corresponding to pure extract, the productivity of A does not increase anymore. In fact, under these conditions the state inside section 2 achieves its upper concentration limit that depends only on m2 and is given by the implicit relationship42

m2 )

|

∂nB ∂cB

(18)

(cA2,0)

In the case of the bi-Langmuir isotherm this can be made explicit as follows:

cA,max2 )

δBaA + γBbA - m2(bA + aA) + x∆ (19) 2m2aAbA

where

∆ ) [δBaA + γBbA + m2(bA - aA)]2 + 4m2γBbA(aA - bA) Therefore, an increase in the amount of A fed, because of an increase of the distance of the operating point from the diagonal, does not correspond to an increase of the productivity of A. In other words, the amount of A added to the feed leaves the unit through the raffinate stream because section 2 is already overloaded. Since for a given overall feed concentration cFT it is possible to draw the region of complete separation for the bi-Langmuir isotherm, it follows that the range of possible values of m2 falls between the abscissa of the optimal point W and the Henry constant of A. Among these the smallest value of m2 corresponds to the abscissa of W and corresponding to this value from eq 18 the largest c2A,max value is obtained. Since the optimal point shifts leftward as the overall feed concentration increases,22 the maximum concentration of component A in section 2 increases as well, having cA,WS as a limit value for cFT f ∞. In the case of a Langmuir isotherm, for which the abscissa of point W is given explicitly, this can be proved analytically; in fact, as cFA f ∞21

m2 f cA,max2 )

(

γB2 γA

) (

)

1 γB 1 γA -1 f - 1 ) cA,WS (20) aA m2 aA γB

This proves that the steady-state concentration of component A in section 2 of a TCC unit, hence, the extract stream of a TCC unit and equivalent SMB unit, is upper-bounded by the watershed concentration. The

same constraint does not hold for the less retained compound B; this explains why its productivity may increase in the region of the pure raffinate. This result elucidates the asymmetrical behavior of SMB’s performances outside the region of separation. Notation ap ) specific surface ai ) bi-Langmuir parameter bi ) bi-Langmuir parameter ci ) fluid-phase concentration dp ) particle diameter Dap,i ) apparent axial dispersion coefficient Di ) axial dispersion coefficient f eq i ) equilibrium relationship Hi ) Henry constant HETP ) height equivalent to a theoretical plate ki ) overall mass-transfer coefficient L ) column length mj ) flow rate ratio in section j defined by eq 1 ni ) adsorbed-phase concentration NG ) number of space intervals Np ) number of overall theoretical stages Nmix ) number of mixing stages Nmt ) number of mass-transport stages Qj ) volumetric fluid flow in section j PE ) extract purity PR ) raffinate purity PRi ) productivity t ) time t* ) switching time u ) superficial velocity V ) volume of a single adsorption column z ) axial coordinate Greek Letters γi ) bi-Langmuir parameter δi ) bi-Langmuir parameter  ) overall void fraction Subscripts and Superscripts A ) more retained species B ) less retained species E ) extract ead ) equilibrium axial dispersive model F ) feed i ) component index, i ) A, B j ) section index, j ) 1, ..., 4 R ) raffinate sol ) solid diffusion model WS ) watershed point

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Received for review April 27, 1998 Revised manuscript received March 18, 1999 Accepted March 23, 1999 IE980262Y