Ind. Eng. Chem. Res. 2006, 45, 6289-6301
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Design of Simulated-Moving-Bed Chromatography with Enriched Extract Operation (EE-SMB): Langmuir Isotherms Galatea Paredes, Hyun-Ku Rhee,* and Marco Mazzotti† Institute of Process Engineering, ETH Zurich, Sonneggstrasse 3, CH-8092 Zurich, Switzerland
A variation of the simulated-moving-bed (SMB) operation, called Enriched Extract SMB (EE-SMB), is investigated, in which a portion of the extract product is concentrated and the resulting enriched stream is re-injected into the SMB at that same point, i.e., at the inlet of section 2. This operation has been recently patented. [Bailly et al., U.S. Patent Application No. WO2004039468, 2004.] Equilibrium Theory is used (i) to obtain the constraints on the operating parameters and (ii) to identify the operating regimes for the EESMB operation. This analysis is performed for a binary mixture whose adsorption behavior is described by a Langmuir isotherm. The operating regimes are translated into regions on the (m2, m3) plane representation, which can be easily compared with the corresponding operating regimes of the standard SMB given by the Triangle Theory. In particular, for the EE-SMB operation, the region of complete separation turns out to be a line segment, i.e., a one-dimensional locus. The theoretical findings are confirmed through detailed simulations of the process, and the value of the new EE-SMB operation mode is assessed. following two equivalent forms:
1. Introduction The simulated-moving-bed (SMB)1 process is a multicolumn chromatographic technique that allows for the continuous separation of a mixture into two fractions. The solid phase is contacted counter-currently with the fluid phase by switching the inlet and outlet ports in the direction of the fluid flow, as a way of simulating the true-moving-bed (TMB) process, which is unfeasible in practice. To guarantee the same separation performance both in the TMB and in the SMB, the following equivalence relationships must be fulfilled: namely, that between the flow rates of the two units (i.e., QSMB and QTMB ) j j and that between the switch time of the SMB (t*) and the volumetric solid flow rate of the TMB unit (Qs):
QSMB ) QTMB + j j Qs V ) t* 1 -
Qs 1-
(1) (2)
with V being the volume of the column of the SMB unit and being its bed void fraction. With respect to batch chromatography, the SMB technique allows one to achieve higher productivities and lower solvent consumptions, mainly because of the countercurrent flow configuration.2,3 SMB has been proven to be successful for several applications that include fine chemicals and pharmaceuticals, particularly the separation of enantiomers.4-6 The design of an SMB unit for a given separation performance consists of selecting the net flow-rate ratios (mj) in the different sections (typically, j ) 1, ..., 4), where a section is considered to be each group of chromatographic columns located between two inlet or outlet ports. The flow-rate ratio mj is the ratio between the net fluid flow rate and the solid flow rate in section j, which, using eqs 2 and 1, can be written in the * To whom correspondence should be addressed. Tel.: +41-446322456. Fax: +41-44-6321141. E-mail: marco.mazzotti@ ipe.mavt.ethz.ch. † Currently with School of Chemical and Biological Engineering, Seoul National University, Seoul 151-744, Korea.
mj )
QTMB - Qsp j Qs(1 - p)
)
t* -V* QSMB j V(1 - *)
(j ) 1, ..., 4)
(3)
where p is the intraparticle void fraction and * is the overall bed void fraction (* ) + (1 - )p). To design an SMB operation, it is useful to exploit the equivalence between the SMB and the TMB units. A simplified model is used where mass-transfer resistance and axial dispersion are neglected (i.e., the Equilibrium Theory model). This approach yields the so-called Triangle Theory, which provides a good prediction of the optimal operating conditions and the purity trends, with respect to the operating parameters.7-9 More precisely, it gives the necessary and sufficient conditions for complete separation of the two components in the feed mixture that correspond to an operating region in the operating parameter space, which is usually projected onto the (m2, m3) plane. The boundaries of this region are explicit functions of the isotherm parameters and of the composition of the mixture to be separated, in the case of Langmuirian-type isotherms.7,10 This approach is used both in academia and in industry to obtain preliminary operating conditions that can then be refined and optimized with the help of detailed simulations.11 The objective of the present work is to investigate a modification of the SMB technique, where a portion of the extract stream is concentrated continuously and re-injected at the same point of the unit, i.e., at the inlet of section 2, as shown in Figure 1. This idea has been recently patented;12,13 we call the corresponding process Enriched Extract SMB, or EE-SMB. The advantage of this approach, with respect to the standard SMB, stems in principle from the possibility of changing the nature of the composition fronts or transitions occurring in section 2. In the case of a Langmuir isotherm, which will be considered in this work, or any otherwise favorable isotherm, the simple wave transition (present in section 2) of a standard SMB, is replaced in the EE-SMB by a shock transition, thanks to the enriched extract inlet. Similar to that observed in the standard SMB case, the use of Equilibrium Theory yields constraints on the flow-rate ratios
10.1021/ie060256z CCC: $33.50 © 2006 American Chemical Society Published on Web 08/03/2006
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Figure 1. Scheme of a four-section simulated moving bed (SMB) unit used for the continuous separation of species 1 and 2. Table 1. Adsorption Isotherm Parameters and Transport Parameters for the Model System under Investigation in This Study component
Hi
Ki [mL/mg]
ki [1/s]
DLi [cm/s2]
1 2
4.7025 6.0075
0.0209 0.0267
0.45 0.45
4.0 × 10-3u(cm/s) 4.0 × 10-3u(cm/s)
that must be fulfilled to obtain complete separation. In this work, the separation of two enantiomers is considered; the parameters that describe the retention behavior of this binary system are given in Table 1 (axial dispersion and mass-transfer resistance parameters are also shown in Table 1). Moreover, the validity of the theoretical conclusions of this study is assessed through detailed simulations. Finally, the potential of the EE-SMB operation, as compared to the standard SMB, is analyzed and discussed. 2. Design of the Operating Conditions of EE-SMBs In this section, the constraints on the operating conditions of an EE-SMB unit that lead to complete separation are obtained in section 2.2. This requires that the corresponding derivation for a standard SMB be summarized, as shown in the following section (section 2.1). 2.1. Design Criteria for SMB Separation: Triangle Theory. Let us consider the two-section TMB unit shown in Figure 2a to be working under complete separation conditions. The feed mixture (F) comprises species 1 (the less-retained component, or weak) and species 2 (the more-retained component, or strong) in an inert solvent, and the interaction with the chromatographic stationary phase is described by a binary Langmuir isotherm:
Hici Hici ) ni ) δ 1 + K1c1 + K2c2
(for i ) 1, 2)
(4)
where Hi ) KiNi is the Henry constant of the species i (with H1 < H2), and Ki and Ni are its equilibrium constant and saturation loading capacity, respectively. According to their retention properties, species 2 is recovered in the extract (E), whereas species 1 is recovered in the raffinate (R), as shown in Figure 1. It has been proven elsewhere14 that the results obtained for this simplified two-section TMB unit, in terms of complete separation region in the (m2, m3) plane,
Figure 2. Scheme of a two-section countercurrent SMB unit with indication of the states associated to each stream and yielding complete separation. In panel a, the states are characterized by their composition, whereas in panel b, the corresponding ω values are indicated.
also are valid for the general configuration that includes sections 1 and 4, provided sections 1 and 4 are properly operated to regenerate the stationary phase and the mobile phase, respectively. For the sake of simplicity, the value of p will be considered to be zero throughout this study; however, the results may be extended easily to account for the case where p * 0, as presented in previous publications.14 At complete separation, species 1 and 2 cannot be present in sections 2 and 3, respectively; hence, the following steady-state mass-balance equations must be fulfilled: (1) Mass-balance equations through the cross section labeled A in Figure 2a, which are given as
m2cR1 ) nγ1
(5)
m2cR2 ) nγ2 + (m2c22 - n22)
(6)
where cji and nji are, respectively, the fluid and solid-phase concentrations of species i in the corresponding stream (labeled R, β, or γ, as in the figure) or section (j). Equation 5 implies that the net flux of species 1 through section 2, (m2c12 - n12), is equal to zero, which is a necessary condition for the complete separation. (2) Mass-balance equations through the cross section labeled B in Figure 2a, which are given as
m3cβ1 ) nγ1 + (m3c13 - n13)
(7)
m3cβ2 ) nγ2
(8)
Equation 8 indicates that the net flux of species 2 through section 3, (m3c23 - n23), is equal to zero. (3) Mass balances at the feed node for species 1 and 2, which are given as
m3cβ1 ) m2cR1 + (m3 - m2)cF1
(9)
m3cβ2 ) m2cR2 + (m3 - m2)cF2
(10)
For a given set of operating conditions (m2 and m3), and after the states that prevail at steady state in sections 2 and 3 are defined (e.g., complete separation), the system of eqs 5-10 is
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a linear system that involves the following six unknowns: cβ1 , cβ2 , nγ1 , nγ2 , cR1 , and cR2 . The following two overall mass-balance equations around the control volume defined by the rectangular dotted envelope in Figure 2a can be obtained as a linear combination of eqs 5-10:
(m3 - m2)cF1 ) m3c13 - n13
(11)
(m3 - m2)cF2 ) -(m2c22 - n22)
(12)
These two balances may replace two equations, e.g., eqs 6 and 7. The following additional mass-balance equations may be written at the raffinate and the extract nodes to calculate the R product concentrations (i.e., nES 2 and c1 ):
m3c13 - n13 ) m3cR1
(13)
-(m2c22 - n22) ) nES 2
(14)
Under complete separation conditions, the steady-state concentrations in sections 2 and 3 are univocally defined and may be calculated using Equilibrium Theory.15,16 It can be demonstrated that a one-to-one mapping exists between the space of fluid or adsorbed phase concentrations and the vectors Ω in a related space (a two-dimensional (2D) plane in the case of a binary system). The two components of this vector associated with a given equilibrium composition of solid and fluid phase (ci, ni for i ) 1, 2) are called (ω1,ω2). These fulfill the following constraints:
0 e ω 1 e H1 e ω2 e H2
(15)
A summary of the required background information is given in Appendixes A and B. In particular, eq 41 is needed to calculate ω1 and ω2 for a given composition state, whereas eqs 42 and 43 are used to calculate the composition that corresponds to a given pair of ω values. In Figure 2b, the states that correspond to the inlet and outlet streams are indicated as E(ωE1 , ωE2 ) for the extract outlet, F( ω1F, ω2F) for the feed inlet, and R(ωR1 , ωR2 ) for the raffinate outlet. In a complete separation regime, the extract stream is free of species 1; it can be proven that, in this case, one of the ω values that characterize the extract state must be equal to H1. Similarly, a pure raffinate stream (no species 2 is present) requires H2 to be one of the corresponding ω values. As a consequence, a pure solvent stream is associated to the ω pair (H1, H2). The problem can be illustrated in the hodograph plane, (c1, c2), as shown in Figure 3. The desorbent stream as well as the pure solid-phase stream correspond to the origin; states β and γ are both binary mixtures and correspond to points within the first quadrant. It has been proven elsewhere that, under optimum operating conditions, states γ and β coincide,14 as it is the case in the example of Figure 3. The intermediate states that connect the origin with state β ) γ (which are labeled P and Q) are necessarily located on one of the two axes, because only one species is present. The character of the transition paths that connect the states mentioned previously, i.e., shocks or waves (see Appendix B), is indicated in Figure 3 by the symbols Σ and Γ, respectively. Connecting the inlet fluid state to the inlet solid state in section 2 yields the following sequence of states:
(H1, H2) f (H1, ωγ2 ) f (ωγ1 , ωγ2 )
Figure 3. Transition paths in the (c2, c1) plane (hodograph plane) for the two-section TCC unit shown in Figure 2, working under optimum operating conditions. The model system parameters are reported in Table 1.
According to Appendix B, both transitions are simple waves, because ωi < Hi. The intermediate state P(H1, ωγ2 ) contains the more-retained component exclusively; hence, it is compatible with complete separation. This prevails at steady state if
H1 e m2δ2 e ωγ2
(16)
Accordingly, in section 3, we shall consider the sequence of states
(ωβ1 , ωβ2 ) f (ωβ1 , H2) f (H1, H2) where the corresponding transitions are shocks. Again, the intermediate state Q(ωβ1 , H2) shall prevail at steady state for complete separation, and this requires that
H1 e m3δ3 e ωβ2
(17)
In the aforementioned equations, δj is the denominator of the Langmuir isotherm to be calculated at the corresponding steadystate composition in section j. It follows that δ2 ) H2/ωγ2 and δ3 ) H1/ωγ1 , because of eq 44. Also note that the transition state (H1, ω), with ωγ2 < ω < H2, which corresponds to one of the composition states along the segment between the state O and the intermediate state P in the hodograph plane (see Figure 3), would fulfill the complete separation condition. This is obtained at steady state if and only if m2δ2 ) ω, i.e., if and only if m2 ) ω2/H2. As discussed elsewhere,7,17 this condition does not add any additional set of operating points to those determined through eq 16 and, therefore, can be ignored. The six equations that have been identified as eqs 5, 8, 9, 10, 11, and 12 may also be recast in terms of the ω values, and, for a given set of operating conditions (m2 and m3) and given states in the columns (e.g., those which correspond to complete separation), one obtains a nonlinear system in the unknowns ωγ1 , ωγ2 , ωβ1 , ωβ2 , ωR1 , and ωR2 . In particular, the overall mass balances 11 and 12 would read as follows:
(ωβ1 )2 - ωβ1 [m3 + H1 + (m3 - m2)K1cF1 ] + m3H1 ) 0
(18)
(ωγ2 )2 - ωγ2 [m2 + H2 - (m3 - m2)K2cF2 ] + m2H2 ) 0 (19) Two of the six equations can be used to demonstrate that the relationships ωγ1 ) ωβ1 and ωR2 ) ωγ2 hold, thus reducing the number of equations and unknowns to four. It can also be proven
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point belong to a Γ- characteristic, whereas all points located to the right belong to a Γ+ characteristic.16 When the final enriched extract concentration (cE2 ) is smaller than the watershed concentration (cw2 ), the inlet state to section 2 of the EE-TMB corresponds to the pair (H1, ω2E). Using Equilibrium Theory, the steady-state S2 to guarantee complete separation operation can be defined; this corresponds to the intermediate state (H1, ωγ2 ) (also, state E guarantees complete separation; however, this case does not add any operating points to the region of complete separation obtained when the intermediate state is considered). Connecting the inlet fluid state to the inlet solid state in section 2 yields the following sequence of states:
(H1, ωE2 ) f (H1, ωγ2 ) f (ωγ1 , ωγ2 )
Figure 4. Scheme of a two-section countercurrent enriched extract simulated-moving-bed (EE-SMB) unit with indication of the states associated to each stream and yielding complete separation. In panel a, the states are characterized by their composition, whereas in panel b, the corresponding ω values are indicated.
that the relevant solution of eq 18 is the smaller root, whereas that of eq 19 is the larger root. In previous publications, the Equilibrium Theory solution of this problem has been presented in detail.9,18 Usually it is solved by enforcing the upper and lower constraints that are given by eqs 16 and 17, and by obtaining the boundary of the complete separation region in the form of constraints that involve m2 and m3. Another approach consists of selecting a pair (m2, m3) and solving the mass-balance equations in sequence: eq 18 gives ωγ1 ) ωβ1 , eq 19 serves to calculate ωγ2 ) ωR2 , and eqs 5 and 8 are used to calculate the last two unknowns: ωR1 and ωβ2 . Finally, the constrains 16 and 17 are tested to conclude whether the selected pair (m2, m3) belongs to the solution, i.e., to the complete separation region. Therefore, once the problem is defined in the manner previously described, all unknowns may be calculated and the system is perfectly defined for a given pair (m2, m3). 2.2. Design Criteria for EE-SMBs. Let us now consider the EE-TMB unit shown in Figure 4a. This is similar to Figure 2a, except for the concentration of the fluid stream entering section 2, which is now the enriched extract, E, containing only species 2, but at higher concentration than that of the stream leaving section 1. The mass-balance equations written in the previous section hence apply, but the steady-state concentration that prevails in section 2 of the unit achieving complete separation is different, because of the enriched extract inlet. The enriched extract stream does not contain species 1, and, therefore, one of the ω values that characterize this stream must be equal to H1. The second component, ωEi , is dependent on the level of enrichment, with respect to the watershed concentration. This corresponds to a particular point of the hodograph plane for Langmuir isotherms in which both families of characteristics (usually called Γ+ and Γ-) have a slope of zero. Indeed, the envelope of the characteristics is tangent to the c2 axes at the watershed point of coordinates:
cw1 ) 0 cw2 )
( )
H2 1 -1 H1 K2
(20a) (20b)
All points of the c2 axis located to the left of the watershed
Because the conditions in section 3 do not change, with respect to a standard SMB, the same sequence of states is observed:
(ωβ1 , ωβ2 ) f (ωβ1 , H2) f (H1, H2) where the corresponding transitions are shocks. It is readily observed that this case (cE2 < cw2 ) is applicable for when both sections are identical to that of the standard SMB and, therefore, leads to the same final solution, namely, the complete separation criteria of the Triangle Theory for a standard SMB. On the other hand, let us assume that cE2 is larger than cw2 , which is the case that is considered in Figure 4b. In this case, the fluid inlet state to section 2 corresponds to the pair (ωE1 , H1); connecting this state to the inlet solid state (in section 2) yields the following sequence of states:
(ωE1 , H1) f (ωE1 , ωγ2 ) f (ωγ1 , ωγ2 ) According to Appendix B, the first transition is a shock, as expected, whereas the second can be either a shock or a simple wave, depending on the concentration achieved in state γ. The fluid inlet state (ωE1 , H1) contains only the more-retained species (species 2), but not the less-retained species (species 1); hence, this is the state that must prevail in section 2 to obtain complete separation (see Figure 4b). Note that the behavior of section 3 is the same as that in the standard SMB also in this case. It follows that the conditions to achieve complete separation in an EE-SMB unit where cE2 > cw2 are
ωγ2 e m2δ2 < ∞
(21)
H1 e m3δ3 e ωβ2
(22)
where δ2 ) H2/ωE1 and δ3 ) H1/ωγ1 (eq 22 is identical to eq 17). The mapping of eqs 42 and 43 yields the relationship between cE2 and ωE1 in the enriched extract stream, and cβ1 and ωβ1 in the state prevailing in section 3:
ωE1 )
H2 1 + K2cE2
K1cβ1 )
H1 - ωβ1 ωβ1
(23)
(24)
The problem is illustrated in the hodograph plane in Figure 5, where (as was the case for SMB in Figure 3) the transition paths are those which correspond to the optimum conditions of the
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Figure 5. Transition paths in the (c2, c1) plane (hodograph plane) for the two-section TCC unit shown in Figure 4, assuming m3 ) mmax 3 . The model system parameters are reported in Table 1.
particular example studied and defined in the caption, for which states γ and β coincide. As already mentioned, eqs 5-12 also apply in the case of the EE-SMB unit. Equations 5 and 8-12 are independent and contain all the unknowns of the design problem. This set of equations may be transformed into the equivalent form that involves the ω values, i.e., the unknowns of the problem (for a given pair m2 and m3, and after the prevailing states in the sections are defined) are as follows: ωγ1 , ωγ2 , ωβ1 , ωβ2 , ωR1 , and ωR2 . When doing so, the result of the overall mass balance of component 1 (eq 11) is the same quadratic expression in ωβ1 that is obtained in the case of the SMB and is given by eq 18, whereas the overall mass balance of component 2 (eq 12) takes the following form:
m3 )
H2cE2 cF2 (1 + K2cE2 )
-
( ) cE2 cF2
- 1 m2
(25)
or, in terms of ω values,
m3 )
(H2 - ωE1 )(H2 - H1)ωF1 ωF2 (H2 - ωF2 )(H2 - ωF1 )H1
[
- m2
(H2 - ωE1 )(H2 - H1)ωF1 ωF2 (H2 - ωF2 )(H2 - ωF1 )H1ωE1
]
- 1 (26)
Neither of these equations involves any of the unknowns, but rather, they represent a linear relationship between m2 and m3 that constrains the choice of operating conditions and confines the region of complete separation in the (m2, m3) plane onto a line, i.e., a segment of the line of operation given by eqs 25 or 26. As a consequence, the system to be solved is constituted of five equations in the six unknowns already mentioned. One of the five equations may be used to prove that ωγ1 ) ωβ1 , reducing the number of equations to four and the number of unknowns to five. Because the number of unknowns is larger than the number of equations, there exists an infinite number of solutions that fulfills the system of eqs 5 and 8-12, for a given (m2, m3) pair selected according to eq 26 and for the selected steady states in the sections. This is a remarkable result, indicating that, under the conditions to achieve complete separation in an EE-SMB unit, the composition states inside the unit are not univocally defined, although the outlet compositions are fixed, being defined by the overall material balances.
Figure 6. Regions of complete separation in the (m2, m3) plane obtained through equilibrium theory, for the EE-SMB operation (segment ey) and the standard SMB operation (triangular region delimited by the lines connecting points a, b, and w), for the model system parameters and the conditions specified in Tables 1 and 2. In the case of the standard SMB, the region above the curve aw leads to pure extract, whereas the region on the left of line bw leads to pure raffinate; for any other operating point neither outlet stream is pure. In the case of the EE-SMB, the region located on the right of segment ey leads to pure extract, whereas the region on the left of this same segment leads to pure raffinate. The square triangle shows, for comparison, the region of complete separation in the case of infinite dilution of the feed mixture (linear range) for the standard SMB.
2.2.1. Region of Complete Separation. Even if all the unknowns cannot be calculated using Equilibrium Theory, the region of complete separation for the EE-SMB operation is welldetermined. Figure 6 shows an example that corresponds to the case study that will be analyzed below. Equation 25 is a straight line with negative slope in the (m2, m3) plane. The segment ey that belongs to this line is the locus of operating points leading to complete separation. The triangle-shaped complete separation region, bwa, which corresponds to the same separation performed in a standard SMB unit, is also shown. To better specify the position of the line of eq 25 where complete separation is achieved, let us consider the constraints defined by eqs 21 and 22 and replace the expressions for δ2 and δ3:
ωγ2 ωE1 m2 g H2
(27)
ωβ2 ωγ1 m3 e H1
(28)
These two equations, together with the mass balance of species 1, through cross section A (eq 5), can be combined to prove that ωβ2 e ωγ2 . Equations 27 and 28 then may be recast as
m 3H 1 ωβ1
< ωβ2 e ωγ2
cw2 is slightly more pronounced. Considering Equilibrium Theory, the optimal operating conditions are always the same below the watershed concentration, regardless of the value of cE2 , because the complete separation region does not change; these conditions are identical to the optimal operating conditions for the SMB. Above the watershed, the optimal operating conditions change with cE2 , as the segment of complete separation moves on the (m2, m3) plane, and the productivity increases, as explained previously (see Figures 8 and 9). Figure 17 shows the recovery of species 2 in the extract as a function of cE2 , which has been calculated using the following expression:
ReE2
) 100
cE2 QE cF2 QF
[
) 100 1 -
]
cR2 (m3 - m4) cF2 (m3 - m2)
(40)
Above the watershed, the recovery of species 2 in the extract increases with cE2 and, at the same time, the raffinate purity improves. The minimum recovery (97.9%) is, in fact, achieved at the watershed. Let us compare this with the standard SMB separation performed under the same operating conditions (mj and feed concentration) as all the EE-SMB separations at cE2 < cw2 . Under these conditions, the SMB operation achieves the following values of purity and recovery: PuE2 ) 96.7%, PuR1 ) 96.3%, ReE2 ) 96.3%, and ReR1 ) 96.7%. These values are worse than those obtained in all the EE-SMB cases, both below and above the watershed concentration. The detailed simulation results presented in Figure 16 show that even below the watershed, where the nominal productivities of the SMB and EE-SMB processes given by eq 34 under optimal conditions are the same (P* ) 1), the SMB leads to
Equilibrium Theory has been used to derive complete separation criteria for the enriched extract simulated-movingbed (EE-SMB) operation.12 It was determined that these correspond to a one-dimensional locus in the (m2, m3) plane, i.e., as compared to the two-dimensional triangle-shaped region of the standard SMB operation. However, Equilibrium Theory does not allow one to fully resolve the problem, in the sense that this has multiple solutions for a given pair of (m2, m3) values. However, knowing the complete separation region, it is easy to select operating points in a systematic way in which detailed simulations can be run. The introduction of the dispersive effects defines the solution in a univocal way, so that the internal concentration profiles can be determined. The detailed simulations allow to validate the model and study the performance of the EE-SMB operation. From this study, it follows that the performance of the EE-SMB operation is always at least as good as that of SMB when the enriched extract concentration is equal to the watershed point, and it is better for any enriched extract concentration above this point. Because of the asymmetric character of the extract and raffinate purity trends (shown in Figures 11 and 13), the operation is particularly advantageous to recover species 2 in the extract. Figure 9 shows how the productivity of EE-SMB may be 1.5 times larger than that of SMB, for a sufficient enrichment of the extract stream. Furthermore, at these convenient larger enrichments, the raffinate purity increases, as shown in Figure 16, at the same time as the recovery of component 2 in the extract increases (see Figure 17). Comparing the SMB purity values with those shown in Figure 16 for the EE-SMB operation at cE2 > cw2 , it results that the extract purity in the last case is significantly higher (up to 99.9%). In terms of recovery of species 2, the minimum value achieved in the EE-SMB, i.e., the one corresponding to cE2 ) cw2 , is larger than that which corresponds to the standard SMB operation. This confirms the attractiveness of the EE-SMB operation whenever the target compound in the mixture to be separated is the more-retained species and it must be recovered at high purity. Acknowledgment The support of the Swiss Innovation Promotion Agency (KTI, Project No. 6138.1 KTS) is gratefully acknowledged. Appendix A. One-to-One Mapping between the (c1, c2) Plane and the Hodograph Plane, Namely the (ω1,ω2) Plane It can be demonstrated, using Equilibrium Theory, that a oneto-one mapping exists between the space of fluid or adsorbed phase concentrations and the ω components in a related twodimensional space.15,16 The two ω values associated with a given equilibrium composition of solid and fluid phase (ni and ci for i ) 1, 2) are the roots of the following equation:
K2n2 K1n1 + )1 H1 - ω H2 - ω
(41)
The two real and positive roots of these equations fulfill the fundamental rule given by eq 15. The mapping of eq 41 may be inverted to get the concentrations as a function of the ω
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values (ω1, ω2):
K1c1 )
(H1 - ω1)(ω2 - H1) ω1ω2 (H2 - H1)
(42)
K2c2 )
(H2 - ω2)(H2 - ω1) ω1ω2 (H2 - H1)
(43)
H2 H1
From the aforementioned equations, it can be readily shown that the denominator of the Langmuir isotherm may be written as
δ ) 1 + K1c1 + K2c2 )
H 1H 2 ω1ω2
(44)
and that the solid-phase concentrations are given by
K1n1 )
(H1 - ω1)(ω2 - H1) H 2 - H1
(45)
K2n2 )
(H2 - ω2)(H2 - ω1) H 2 - H1
(46)
When one of the two species i is absent from a specific stream j, i.e., cji ) nji ) 0, then one of the ω values is given by eq 41, whereas the other is equal to Hi. Appendix B. Relationship between the Steady State and the Operating Conditions in a Given Simulated-Moving-Bed (SMB) Section Let us consider a generic section of the countercurrent unit with inlet states a and b, which correspond to the fluid entering at the left end of the section and the solid entering at the right end of the section, respectively. At steady state, three states may prevail in the column: state a, state b, and the intermediate state defined by the two (state I). Every possible steady state is separated from the others by transitions that are either a continuous simple wave when ωbi < ωai , or a shock when the opposite occurs. Which state prevails in the section is dependent on the operating conditions (i.e., on the flow-rate ratio m) and can be expressed in the following manner:
State a:
(ωa1,ωa2) if max{ωa2,ωb2} e mδa
(47)
State I: (ωa1,ωb2) if max{ωa1,ωb1} e mδI e min{ωa2,ωb2} (48) State b:
(ωb1,ωb2) if mδb e min{ωa1,ωb1}
(49)
When the composition of the constant state attained at steady state in the column is known, based on the composition of the inlet streams and on the value of m, the composition of the outlet streams can be calculated through mass balances at the column ends. Nomenclature cji ) concentration of species i cwi ) watershed concentration
in the stream j [mol/L, mg/mL] of species i [mol/L, mg/mL] DLi ) axial dispersion coefficient of species i [cm2/s] Hi ) Henry’s constant of species i ki ) mass-transfer coefficient of species i [cm/s] Ki ) Langmuir equilibrium constant of species i [mL/mg]
mj ) flow-rate ratio in section j n ) total number of columns nji ) adsorbed phase concentration of species i in the stream j [mol/L] Ni ) saturation loading capacity of species i [mg/mL] Pji ) productivity based on species i in the product stream j [mg/(h mL)] P* ) relative productivity of the EE-SMB to the SMB Puji ) purity of the product stream j with respect to species i [%] Qj ) volumetric fluid flow-rate in section j [mL/min] Qs ) volumetric solid flow-rate [mL solid/min] Reji ) recovery of species i in the stream j [%] S ) column section [cm2] SC ) solvent consumption per unit mass of product [mL/(h mg)] t* ) switch time [s] V ) column volume [mL] Greek Letters Γ ) characteristic in the hodograph plane, i.e., image of a simple wave δj ) denominator of the Langmuir isotherm in section j ) bed void fraction * ) overall bed void fraction p ) intraparticle void fraction Σ ) locus of the state on the other side of a shock in the hodograph plane ωj ) characteristic parameter corresponding to the state j Subscripts and Superscripts 1 ) less-retained species 2 ) more-retained species D ) desorbent E ) extract ES ) index related to the solid phase of the extract state F ) feed i ) species; i )1, 2, ..., c I, P, Q ) intermediate state j ) stream/section index; j )1, 2, ... opt ) optimum R ) raffinate S2 ) steady state in section 2 of the SMB T ) total TMB ) true moving bed x, Q ) related to the larger and smaller root Literature Cited (1) Ruthven, D. M.; Ching, C. B., Counter current and simulated counter current adsorption separation processes. Chem. Eng. Sci. 1989, 44, 1011. (2) Nicolaos, A.; Muhr, L.; Gotteland, P.; Nicoud, R. M.; Bailly, M. J. Chromatogr., A 2001, 908, 71. (3) Nicolaos, A.; Muhr, L.; Gotteland, P.; Nicoud, R. M.; Bailly, M. J. Chromatogr., A 2001, 908, 87. (4) Nicoud, R.-M. The separation of optical isomers by simulated moving bed chromatography (Part II). Pharm. Technol. Eur. 1999, 11, 28. (5) Juza, M.; Mazzotti, M.; Morbidelli, M. Simulated moving-bed chromatography and its application to chirotechnology. Trends Biotechnol. 2000, 18, 108. (6) Schulte, M.; Strube, J. Preparative enantioseparations by simulated moving bed chromatography. J. Chromatogr., A 2001, 906, 399. (7) Mazzotti, M.; Storti, G.; Morbidelli, M. Optimal operation of Simulated Moving Bed units for nonlinear chromatographic separations. J. Chromatogr., A 1997, 769, 3. (8) Migliorini, C.; Mazzotti, M.; Morbidelli, M. Simulated moving bed units with extracolumn dead volume. AIChE J. 1999, 45, 1411.
Ind. Eng. Chem. Res., Vol. 45, No. 18, 2006 6301 (9) Migliorini, C.; Mazzotti, M.; Morbidelli, M. Continuous chromatographic separation through simulated moving beds under linear and nonlinear conditions. J. Chromatogr., A 1998, 827, 161. (10) Mazzotti, M. Local Equilibrium Theory for the Binary Chromatography of Species Subject to a Generalized Langmuir Isotherm. Ind. Eng. Chem. Res. 2006, 45 (15), 5332-5350. (11) Zhang, Z.; Mazzotti, M.; Morbidelli, M. Multiobjctive optimization of simulated moving bed and Varicol processes using a genetic algorithm. J. Chromatogr., A 2003, 989, 95. (12) Bailly, M.; Nicoud, R.-M.; Philippe, A.; Ludemann-Hombourger, O. Method and device for chromatography comprising a concentration step, U.S. Patent Application No. WO2004039468, 2004. (13) Abdelmoumen, S. Etude thorique et exprimentale d’un nouveau procd multi-colonnes continu de chromatographie prparative intgrant une tape de concentration (05INPL003N). Ph.D. Thesis, INPL-Nancy, Nancy, France, 2005. (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.
(15) Rhee, H.-K.; Aris, R.; Amundson, N. R. On the theory of multicomponent chromatography. Philos. Trans. R. Soc. London A 1970, A267, 419. (16) Rhee, H.-K.; Aris, R.; Amundson, N. R. First-Order Partial Differential Equations; Prentice-Hall: Englewood Cliffs, NJ, 1989; Vol. II. (17) Mazzotti, M. Design of Simulated Moving Bed Separations: Generalized Langmuir Isotherm. Ind. Eng. Chem. Res. 2006, 45, 63116324. (18) Migliorini, C.; Mazzotti, M.; Morbidelli, M. Design of simulated moving bed multicomponent separations: Langmuir systems. Sep. Purif. Technol. 2000, 20, 79. (19) Migliorini, C.; Gentilini, A.; Mazzotti, M.; Morbidelli, M. Design of simulated moving bed units under nonideal conditions. Ind. Eng. Chem. Res. 1999, 38, 2400.
ReceiVed for reView March 2, 2006 ReVised manuscript receiVed June 27, 2006 Accepted July 5, 2006 IE060256Z
