Ind. Eng. Chem. Res. 2006, 45, 739-752
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Optimal Standing-Wave Design of Nonlinear Simulated Moving Bed Systems for Enantioseparation Ki Bong Lee School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907
Sungyong Mun Department of Chemical Engineering, Hanyang UniVersity, Seoul 133-791, South Korea
Fattaneh Cauley SeaVer College, Pepperdine UniVersity, Malibu, California 92365
Geoffrey B. Cox Chiral Technologies, West Chester, PennsylVania 19380
Nien-Hwa Linda Wang* School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907
An efficient optimization tool is developed based on the standing-wave design for simulated moving bed (SMB) systems with nonlinear isotherms and significant mass-transfer effects. A maximum operating pressure is considered in the optimization. Both system parameters (particle size, column length, column diameter, total number of columns, column configuration, and feed concentration) and operating parameters (zone flow rates and switching time) are optimized to achieve the maximum productivity or the minimum separation cost. Under a pressure limit, medium particle size (10-40 µm), short columns (5-15 cm), and a longer zone II give higher productivity and lower separation cost. Nonlinear effects resulting from high feed concentration can decrease productivity and increase separation cost. High-pressure SMB systems (5.2 MPa) can have higher productivity, but low- and medium-pressure SMB systems (1.0 and 2.4 MPa, respectively) are more economical. 1. Introduction The simulated moving bed (SMB) chromatography was first patented in the early 1960s by Broughton and Gerhold to separate the isomers of xylenes.1 This technology has been used for several decades in the petrochemical and sugar industries for large-scale separation.2,3 In the past decade, SMB chromatography has gained attention for the production of fine chemicals and pharmaceuticals, particularly enantiomers.4-6 A standard SMB is particularly suited for binary separations, such as chiral separations.7-9 In the SMB process, a countercurrent flow between the mobile and stationary phases is mimicked by periodic switching of the inlet (feed and desorbent) and outlet (extract and raffinate) positions by one column in the direction of the mobile-phase flow (Figure 1). In this continuous operation, the fast-migrating component is recovered from the raffinate port while the slowmigrating component is recovered from the extract port. The two migrating bands overlap, allowing a high product concentration and a large fraction of the stationary phase to be used for separation. For this reason, high product purity and high yield can be achieved. Furthermore, SMB results in less solvent consumption and higher productivity than batch chromatography. An optimal design and operation can exploit the economic potential of the SMB system. In general, a four-zone SMB * To whom correspondence should be addressed. Tel.: +1-765-4944081. Fax: +1-765-494-0805. E-mail:
[email protected].
Figure 1. Schematics of a conventional four-zone simulated moving bed b Fast-migrating component, (gray triangle) slow-migrating component: (a) step N and (b) step N + 1.
process has six system parameters and five operating parameters. The six system parameters include particle size, column length, column diameter, total number of columns, column configuration, and feed concentration. The five operating parameters consist of four zone flow rates and port velocity (or switching time).
10.1021/ie0504248 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/21/2005
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In 1997, Ma and Wang10 developed the standing-wave design (SWD) for optimizing the operating parameters of an SMB system with linear isotherms. Analytical solutions were derived from continuous moving bed equations. The solutions were used to identify the operating conditions that give the highest productivity with the lowest desorbent requirement. SWD has been successfully used for various separations.11-13 SWD has been used for the optimization of system parameters as well as operating parameters for linear isotherm systems. Wu et al.14 used SWD to optimize throughput per bed volume and desorbent requirement for paclitaxel purification. Mun et al.15 developed a grid search method using SWD to optimize a tandem SMB design for insulin purification to achieve the lowest purification cost. Cauley et al.16 introduced a novel approach using simulated annealing, a stochastic optimization algorithm. This method, called the standing-wave annealing technique (SWAT), is based on the standing-wave equations, and it can significantly reduce the computation time for simultaneous optimization of both system and operating parameters. Mallmann et al.17 and Xie et al.18 broadened the applicability of SWD to nonlinear isotherm systems. The nonlinear SWD has been employed successfully for optimization of the operating parameters of SMB systems in chiral and organic acids separations.18-20 Recent extensions to this method include the investigation of systems with a pressure drop limit.21 In parallel, Mazzotti et al.22 developed the triangle theory for ideal systems, based on the local equilibrium theory, and showed that the theoretical optimal operating conditions for the largest productivity are on the vertex point of the triangle region. However, for nonideal systems where mass-transfer effects are important, the theoretical optimal operating conditions from the triangle theory can deviate from the actual optimum. Iterative experiments or simulations have been used to find the actual optimal operating conditions.9,23-25 For linear isotherm systems, Silva et al.26 derived an analytical solution which was used to find the actual optimal operating conditions. The approach was similar to that in Ma and Wang,10 except axial dispersion was ignored in the solution. They used the solutions to generate a separation-region plot which was similar to the triangle plot. A few studies concentrated on finding the optimal system parameters by use of the triangle theory. Biressi et al.27 developed an algorithm requiring simulations to optimize operating variables and column length. They used the algorithm to study the effect of parameters such as particle size and purity requirement. Ludemann-Hombourger et al.28 derived algebraic equations which were functions of particle size, column length, and column diameter. They used the Kozeny-Karman equation to calculate a pressure drop and the Van Deemter equation to consider mass-transfer effects. They estimated the optimal particle size for the productivity with a physically acceptable column length. However, the study was limited to the linear isotherms, and simulations were needed to estimate the required number of plates for target purity. Jupke et al.29 compared batch chromatography and SMB processes which were optimized for both the operating parameters and the system parameters. However, the only system parameter considered was the column dimension, and iterative simulations were used to satisfy both purity and yield. More recently, Zhang et al.30 presented a comparison of SMB and Varicol processes for nonlinear isotherm systems and optimized each system using a genetic algorithm. However, in their study several system parameters were fixed; only the configurations with a total of 5 or 6 columns were considered. More importantly, the procedure lacked computational efficiency, because the solutions of ordinary
differential equations were needed to ensure that the product purity and yield were satisfied. In this study, a method based on SWD is developed to optimize both the system and operating parameters for nonlinear isotherm SMB with significant mass-transfer effects. The grid search method developed by Mun et al.15 for linear isotherm systems is extended to nonlinear isotherm systems under a pressure limit. The proposed grid search method contains an inner loop and an outer loop and provides a flexible, systematic procedure for determining the optimal operating and system parameters of an SMB system. The inner loop of the optimization algorithm is based on SWD. In this loop, the five operating parameters (four zone flow rates and port velocity) satisfy both the pressure constraint and the SWD equations for a fixed set of system parameters. This solution ensures high purity, high yield, highest productivity, and lowest desorbent requirement. Also, the solution requires little computation time since the SWD equations are algebraic equations. By extending the grid search to the outer loop, the optimal values for the system parameters can also be determined. The inner loop of the grid search algorithm is used to investigate the effects of each individual parameter on the SMB performance, such as productivity, desorbent requirement, and separation cost. Additionally, each analysis is investigated under three different operating pressures: high pressure (5.2 MPa), medium pressure (2.4 MPa), and low pressure (1.0 MPa). The outer loop of the grid search algorithm provide a systematic tool for determining the overall optimal SMB design that achieves the maximum productivity or the minimum separation cost. In the overall optimal design, all 11 parameters (6 system parameters and 5 operating parameters) are optimized under the three different operating pressures. The results show that the SMB productivity can be limited by either mass-transfer efficiency or maximum operating pressure. Under a pressure limit, medium particle size (10-40 µm), short columns (5-15 cm), and long separation zones (particularly zone II) result in high productivity and low separation cost. However, nonlinear effects due to very high feed concentration can decrease productivity and increase separation cost. The high-pressure SMB generally has higher productivity than the medium- or low-pressure systems, but it also has a higher separation cost because of high equipment cost. 2. Theory 2.1. Standing-Wave Design for an Ideal System. To achieve binary separation in moving bed systems, the following constraints for concentration wave velocities in each zone have been proposed for a system without any mass-transfer effects, or an ideal system.31
uIw,2 > ν
(1a)
uIIw,1 > ν
(1b)
uIII w,2 < ν
(1c)
uIV w,1 < ν
(1d)
j uw,i )
u0j 1 + Pδij
(2)
where the subscripts 1 and 2 denote the fast-migrating solute and the slow-migrating solute, respectively; the superscripts
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For a given feed flow rate, the five operating parameters (four zone flow rates and port velocity) can be determined using eq 4 in addition to eqs 3a-3d. II Ffeed ) bS(uIII 0 - u0 )
(4)
where Ffeed is the feed flow rate and S is the cross-sectional area of the column. 2.2. Standing-Wave Design for a Nonideal System under a Pressure Constraint. If mass-transfer effects are significant (or a nonideal system), the concentration waves are spread, causing contaminated products (Figure 2b). For such a system, the interstitial velocities in zones I and II should be increased such that the key wave velocities migrate faster than the average port velocity, whereas the interstitial velocities in zones III and IV should be decreased such that the key wave velocities migrate slower than the average port velocity.10,18 The differences between the port velocity and the key wave velocities can focus the waves toward the zone boundaries to counter wave spreading and maintain high product purity and high yield in a nonideal system.
uI0 ) (1 + PδI2)ν + uII0 ) (1 + PδII1 )ν + Figure 2. Schematics of the SMB column profiles for a binary nonlinear isotherm system: (a) without mass-transfer effects and (b) with significant mass-transfer effects. j I-IV denote the four zones; uw,i is the wave velocity of j component i in zone j; u0 is the interstitial velocity in zone j; P is the phase ratio, defined as (1 - b)/b, where b is the interparticle void fraction; δ is the effective retention factor; and ν is the average port velocity. In 1997, Ma and Wang10 developed the standing-wave design (SWD) for continuous moving beds with linear isotherms. Mallmann et al.17 and Xie et al.18 expanded this method to nonlinear isotherm systems. In the SWD for a system without any mass-transfer resistances and axial dispersion (or an ideal system), the wave velocity of a key component in each zone is set to match the average port velocity. Under the standing-wave conditions, the desorption-concentration wave of the fastmigrating component stands in zone II and the adsorption wave stands in zone IV. The concentration waves of the slowmigrating component stand in zones I and III respectively (Figure 2a).
uIw,2 ) ν
(3a)
)ν
(3b)
uIII w,2 ) ν
(3c)
uIV w,1 ) ν
(3d)
uIIw,1
Note that the standing-wave conditions (eqs 3a-3d) are the boundary values of the constraints for separation in moving beds (eqs 1a-1d). Any other velocities that satisfy eqs 1a-1d have IV higher uI0 and uII0 and lower uIII 0 and u0 than those of eqs 3a3d, resulting in a lower feed flow rate and a higher desorbent flow rate. Therefore, the standing-wave conditions define simultaneously the highest productivity and the lowest desorbent requirement.
uIII 0
) (1 +
PδIII 2 )ν
uIV 0
) (1 +
PδIV 1 )ν
-
LI
βII1 LII
EIb,2 +
EIII b,2 III
βIV 1 L
EIV b,1 IV
) )
Pν2(δI2)2
EIIb,1 +
βIII 2 L
-
( ( ( (
βI2
KIf,2
(5a)
Pν2(δII1 )2
+
+
KIIf,1
(5b)
) )
2 Pν2(δIII 2 )
KIII f,2
2 Pν2(δIV 1 )
KIV f,1
(5c)
(5d)
where βji is the natural logarithm of the ratio of the highest concentration to the lowest concentration of standing-wave component i in zone j; Lj is the length of zone j; Eb is the axial dispersion coefficient; and Kf is the lumped mass-transfer coefficient, which can be estimated from the particle radius (Rp), the intraparticle diffusivity (Dp), and the film mass-transfer coefficient (kf) by10
Rp2 Rp 1 ) + Kf 15pDp 3kf
(6)
For a linear isotherm system (qi ) aici), the effective retention factor (δji) of component i in zone j is defined10
δji ) p + (1 - p)ai +
DV PLcSb
(7)
where p is the intraparticle void fraction, ai is the linear equilibrium constant, DV is the extra-column dead volume, and Lc is the single column length. In a binary competitive (or nonlinear) isotherm system, the solute concentration in the stationary phase (qi) and its bulk phase concentration (ci) can be related using the multicomponent Langmuir isotherm equations.
qi )
aici 2
1+
bjcj ∑ j)1
(where i ) 1 or 2)
(8)
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The effective retention factors for such systems are calculated from the isotherm parameters (ai and bi) and the plateau concentrations (Cs,i and Cp,i; see Figure 2).
δI2 ) p + (1 - p)a2 + δII1 ) p + (1 - p)
(
)
)
a2
1 + b1Cs,1 + b2Cs,2
δIV 1 ) p + (1 - p)
(9a)
a1 DV + 1 + b2Cp,2 PLcSb
(
δIII 2 ) p + (1 - p)
DV PLcSb
(
+
(9b)
DV PLcSb
)
a1 DV + 1 + b1Cp,1 PLcSb
(9c) (9d)
The effective retention factors for a linear isotherm system are independent of concentrations (eq 7). However, in a nonlinear isotherm system, the effective retention factors of zones II-IV are dependent on concentrations (eqs 9b-9d). If the saturation capacity (qmax) can be assumed to be the same for both enantiomers, the four parameters (a1, a2, b1, and b2) for the binary competitive Langmuir isotherm can be reduced to three parameters (qmax, b1, and b2), and the selectivity can be defined as the ratio of the Langmuir b values.
a1 a2 qmax ) ) b1 b2
[( 4
)
βIII 2 L
EIII b,2 III
-
βII1
]
EIIb,1 II
L
(12)
A maximum allowable pressure in an SMB system can limit the feed and zone flow rates; therefore, a pressure limit should be considered in the design of an SMB. The maximum pressure drop in an SMB depends on its pump configuration. In this study, the desorbent pump is assumed to control the flow rates from zones I to IV. Then the system pressure can be obtained by summing the pressure drop over each zone. The Ergun equation has been commonly used to calculate pressure drop in a packed bed.32 IV
∆P )
∆Pj ) ∑ j)I IV
L ∑ j)I
[ ( )
j
150 µuj0 1 - b 4Rp2
b
2
+
FfeedCF,iYi (14) FbedBV
DR ((L of solvent)/(kg of product)) )
Fdes (15) FfeedCF,iYi
SC ) CSP cost + solvent cost + equipment cost (16)
Ffeed,max )
-
PR [((kg of product)/day)/(kg of CSP)] )
(11)
When mass-transfer resistances and axial dispersion are significant and the operating conditions are not limited by a maximum pressure drop, the maximum feed flow rate can be determined from the SWD equations as follows:10
bS
2.3. Productivity, Desorbent Requirement, and Separation Cost. To quantify and compare the SMB performance, productivity, desorbent requirement, and separation cost have been used.14,15,33 In this study, productivity (PR) is defined as the production rate per kg of chiral stationary phase (CSP); desorbent requirement (DR) is defined as the liter of desorbent required per kg of pure enantiomer produced; and separation cost (SC) is defined as the sum of CSP cost, solvent cost, and equipment cost. Costs of labor, utilities, waste disposal, and others are excluded from the estimation of the total separation cost.
(10)
a2 b2 R(selectivity) ) ) a1 b1
II 2 P(δIII 2 - δ1 ) III 2 βII1 (δII1 )2 βIII 2 (δ2 ) + III KIII KIIf,1LII f,2L
Figure 3. SMB equipment price as a function of column diameter: highpressure system (e13.8 MPa), medium-pressure system (2.4 MPa), and lowpressure system (1.0 MPa).
]
1.75 F(uj0)2(1 - b) 2Rpb
(13)
where ∆P is the pressure drop, µ is the fluid viscosity, and F is the fluid density.
CSP cost ($/(kg of product)) ) CSP price ($/(kg of CSP))/(CSP lifetime (yr)) (17a) PR [((kg of product)/yr)/(kg of CSP)] Solvent cost ($/(kg of product)) ) solvent consumption ((L of solvent)/(kg of product)) × solvent price ($/(L of solvent)) × (1 - recycle ratio) + solvent recovery cost ($/(L of solvent)) × recycle ratio (17b)
[
]
equipment cost ($/(kg of product)) ) SMB unit price ($) × number of SMB required SMB lifetime (yr) × production rate ((kg of product)/yr) (17c) where CF,i is the feed concentration of component i, Yi is the yield of component i, Fbed is the packing density, BV is the bed volume, and Fdes is the desorbent flow rate. In the solvent cost estimation, both solvent in feed and desorbent are considered, and the fraction of solvent is assumed to be reused after recovering it from impurities. Only one SMB unit is considered in this study. The SMB unit price for high-pressure systems (13.8 MPa) is assumed to be a function of column diameter and independent of column number (Figure 3). The unit price for medium-pressure or low-pressure systems is assumed to be independent of column diameter and column number. 2.4. Maximum Productivity and Desorbent Requirement Obtained from Standing-Wave Equations. When column dimension, total number of columns, and feed concentration are
Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006 743
fixed for a given system without any pressure limit, the maximum productivity (PRmax) can be estimated from the maximum feed flow rate (eq 12).
Ffeed,maxCF,iYi Ffeed,maxCF,iYi ) FbedBV FbedSLcNc,tot
PRmax )
[(
PRmax )
bCF,iYi FbedNc,tot
II 2 P(δIII 2 - δ1 )
III 2 βIII 2 (δ2 ) III KIII f,2 Nc
4
+
)
βII1 (δII1 )2 KIIf,1 NIIc
-
βII1 EIIb,1
NIII c (Lc)
NIIc (Lc)2
]
(19)
where Nc,tot is the total number of columns and N jc is the number of columns in zone j. It is clear from eq 19 that the maximum productivity decreases as Kf decreases or Eb increases. If Kf and Eb are fixed, the maximum productivity increases with increasing column length and then reaches a limiting value (eq 19). Also, if the total number of columns is fixed, the maximum productivity increases with increasing column numbers in zone II or zone III. Since II δIII 2 is larger than δ1 , eq 19 suggests that more columns in zone III are more favorable for high productivity. In addition, the higher the product purity requirement (high β), the lower the productivity. II If λ is defined as the ratio of δIII 2 and δ1 , eq 19 is simplified as follows:
λ) PRmax ) bCF,iYi FbedNc,tot
[( 4
δIII 2
P(λ - 1)2 2 βIII 2 λ
KIII f,2
NIII c
+
βII1 KIIf,1
NIIc
(20)
δII1
)
-
III βIII 2 Eb,2 2 NIII c (Lc)
-
[
βII1 EIIb,1 NIIc (Lc)2
]
For a linear isotherm system, λ is constant and the maximum productivity linearly increases with increasing feed concentration (eq 21). If a1 is fixed, as a2 increases, the selectivity R and λ increase. As λ increases, the maximum productivity increases and eventually approaches a limiting value (eq 21). If the selectivity is fixed, as a1 and a2 increase, λ first increases and eventually approaches the selectivity. Therefore, the maximum productivity first increases with increasing a1 and a2 and eventually approaches a limiting value. For a nonlinear isotherm system, δII1 and δIII 2 are less than those of a corresponding linear system if the nonlinearity terms (bici) in eqs 9b and 9c are significant. Since δIII 2 decreases more than δII1 (eqs 9b and 9c), as nonlinearity increases, λ decreases and the maximum productivity also decreases (eq 21). Since λ decreases as feed concentration increases, this effect counters the linear CF,i term in eq 21. For this reason, the maximum productivity first increases with increasing CF,i in the linear isotherm region and eventually decreases with CF,i when the nonlinear effects are significant. If axial dispersion is not important and intraparticle diffusion is dominating (see Appendix A), the relation of mass-transfer coefficients and the equation of the maximum productivity can be further simplified and the maximum productivity is expressed as a function of particle size and intraparticle diffusivity (eq 23).
2
4Rp
(
2 βIII 2 λ
Dp,2f III c
+
]
)
βII1
Dp,1f IIc
(22)
(23)
where f jc is the fractional zone length of zone j, defined as f jc ) N jc/Nc,tot. It is clear from eq 23 that, without any pressure limit, the maximum productivity is inversely proportional to Rp2. Furthermore, the maximum productivity increases linearly with increasing intraparticle diffusivity, since both enantiomers should have the same diffusivity. It is also evident from eq 23 that increasing the fractional zone length in zone II or zone III increases the maximum productivity. Also, for the maximum feed flow rate (eq 12), desorbent requirement can be derived as follows: DR ) Pν(δI2 - δIV 1 )+
[( 4
βI2 NIcLc
P(δIII 2
{
EIb,2 +
-
III 2 βIII 2 (δ2 ) III KIII f,2 Nc
+
} {
Pν2(δI2)2
δII1 )2Lc
KIf,2
)
βII1 (δII1 )2 KIIf,1 NIIc
-
+
βIII 2
βIV 1
NIV c Lc
EIII b,2
LcNIII c
-
βII1
EIV b,1 +
]
EIIb,1
LcNIIc
}
2 Pν2(δIV 1 )
KIV f,1
CF,iYi
(24a)
If intraparticle diffusion controls,
DR ) (21)
15pP(λ - 1)2
bCF,iYi PRmax ) Fbed
(18)
III βIII 2 Eb,2
2
Rp2 Rp Rp2 1 ) + = Kf 15pDp 3kf 15pDp
Pν(δI2 - δIV 1 )+
{
}
{
}
2 2 βIV Pν2(δI2)2Rp2 Pν2(δIV 1 1 ) Rp + 15pDp,1 NIcLc 15pDp,2 NIV c Lc
βI2
II 2 15pCF,iYiP(δIII 2 - δ1 ) Lc
(
4Rp2
III 2 βIII 2 (δ2 )
Dp,2NIII c
+
)
βII1 (δII1 )2 Dp,1NIIc
(24b)
It is clear from eq 24 that desorbent requirement increases with increasing purity requirement (β), Eb, or particle size Rp. It decreases with increasing feed concentration, Dp, or zone length in zone I or zone IV. For a fixed feed flow rate, as the column length increases, the desorbent requirement decreases and eventually reaches a limiting value, which corresponds to the desorbent requirement for an ideal system. 3. Grid Search Method For the optimal performance of an SMB system, the six system parameters (particle size, column length, column diameter, total number of columns, column configuration, and feed concentration) and the five operating parameters (four zone flow rates and switching time) must be optimized. A grid search method, shown in Figure 4, is introduced in this section and illustrated in the next section. The method contains an inner loop and an outer loop. The inner loop, at the core, is based on SWD. In this loop, the five operating parameters are obtained from the SWD for a fixed set of system parameters. This solution ensures high product purity and yield, highest productivity, and lowest desorbent requirement for SMB operation.
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Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006 Table 1. Intrinsic Parameters of the Phenylpropanolamine Separation System interparticle void fraction, b intraparticle void fraction, p extra-column dead volume (% of the bed volume) D∞ (cm2/min) Dp (cm2/min) kf (cm/min) Eb (cm2/min) isotherms (+)-PPA (-)-PPA
0.32 0.55 5.73 7.47 × 10-4 7.0 × 10-5 Wilson and Geankoplis correlation34 Chung and Wen correlation35 a ) 0.448, b ) 0.0102 a ) 1.44, b ) 0.0354
Table 2. System Parameters of the Example System total number of columns column configuration feed concentration ((g of racemate)/L) column length (cm) CSP particle diameter (µm)
Figure 4. Schematic diagram of the optimization algorithm with inner loop (thick dashed line) and outer loop (thin dashed line).
More importantly, the inner loop requires little computation time because the SWD equations are algebraic equations. The procedure starts by assigning a very small initial value as the feed linear velocity. This velocity is then increased by variable step sizes until it reaches the maximum velocity, which is limited either by the mass-transfer efficiency (eq 12) or by the maximum allowable system pressure (eq 13). The discussion in the next section exemplifies this. For each set of system and operating parameters, the objective function is calculated and, if it shows an improvement compared to the incumbent, the new optimal parameters are saved. By extending the use of the grid search to the outer loop, the optimal values for the system parameters can also be determined. The four system parameters, particle size, column length, total number of columns, and feed concentration, are initialized at the lower-bound values. The initialized values are increased with a small step at each time until they reach the upper bound. The step size for the total number of columns is a discrete integer, and all possible column configurations are investigated at each step.15 For the other three parameters, dividing the range of the parameter by 50 or 100 steps determines the initial step size. After finding a bracket that includes a provisional optimal value, the search is concentrated within the bracket with a much smaller step size. The optimal column diameter is determined from the optimal linear velocities to satisfy the production rate. Both inner and outer loops are repeated until the results satisfy a given tolerance. The grid search method is coded in Fortran 90. 3.1. Optimization of an Example System Using the Grid Search Method. In the results section, use of the grid search method is illustrated through the preparative enantioseparation of phenylpropanolamine (PPA). In a previous study,21 competitive Langmuir isotherms were obtained for the enantiomers and the isotherms and the mass-transfer parameters were confirmed by experimental data and simulations. Table 1 lists these intrinsic
8 2-2-2-2 50 10 27
values. The purity and yield were set at 99% for the SWD throughout the analysis. The first part of the analysis concentrates on the results obtained from the use of the inner loop of the grid search method. The system parameters are fixed at values referred to as the example system (Table 2), and the inner loop is used to find the optimal values of the five operating parameters. The example system is based on commercially available system parameters. The example system contains eight columns with the column configuration of 2-2-2-2 (two columns in each zone). The column length is chosen as 10 cm, and column diameter is chosen to allow annual production of 25 000 kg with 20% downtime. The particle size is 27 µm, and the racemic feed concentration is fixed at the upper-bound value of 50 g/L.21 In the second part of the analysis, the outer loop is called upon to optimize both system parameters and operating parameters. Three different pressure limits (1.0 MPa, low pressure; 2.4 MPa, medium pressure; and 5.2 MPa, high pressure) were analyzed and compared to the case without any pressure limit. The comparison illustrates clearly that the operating conditions and productivity are limited either by the mass-transfer efficiency or by the maximum operating pressure limit. Overall optimizations were performed under two different objective functions, maximum productivity or minimum separation cost. Assumptions regarding the cost estimations are given in detail in Table 3. Figure 3 illustrates the substantial equipment price differences for the SMB systems with different pressure limits. 4. Results and Discussion 4.1. Sensitivity Analysis Based on the Example System. The following sections present sensitivity analyses regarding the system parameters of the example system (Table 2). Each section contains figures that correspond, respectively, to effects of change in the parameter on productivity, desorbent requirement, and separation cost for each of the three maximum pressure limits under investigation. Table 4 shows the range of system parameters studied in the sensitivity analysis. The results are based on the optimal operating parameters to achieve the highest productivity for a given set of system parameters. The switching time is assumed to have no limit. 4.1.1. Sensitivity Analysis on the Effects of CSP Particle Size. Without any pressure limit, the feed linear velocity is limited only by the mass-transfer efficiency. As expected from eq 23, smaller particles should have higher productivity and the productivity exponentially decreases with increasing particle
Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006 745 Table 3. Assumptions Used for Cost Estimation in the Sensitivity Analysis and the Optimization Studies production rate (kg/yr) system down time (%) CSP (Chiralpak AD) price ($/kg) maximum CSP operating pressure (MPa) CSP lifetime (yr) solvent (methanol) price ($/L) solvent recycle ratio (%) solvent recovery cost ($/L) SMB equipment price
25 000 20 15 000 5.2 4 0.20 95 0.10 $700 000 for a low-pressure system (1.0 MPa). $1 000 000 for a medium-pressure system (2.4 MPa). Use a function of column diameter (dc in cm) based on Novasep SMB for a high-pressure system (5.2 MPa), $(96 292 + 260 600dc - 927.67dc2) The SMB price does not depend on the number of columns 7
equipment lifetime (yr) Table 4. Range of System Parameters for Sensitivity Analysis and Optimization Studies
sensitivity analysis optimization particle diameter (µm) column length (cm) total number of columns feed concentration ((g of racemate)/L)
1-60 1-20 8-16 10-200
5-100 5-100 8-16 10-100
size (Figure 5a). Under a pressure limit, the linear velocities and, subsequently, the productivity for small particles (300 g/L), the competitive adsorption
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Figure 7. Effects of total number of columns on (a) productivity (PR); (b) desorbent requirement (DR); and separation cost (SC) at (c) 1.0 MPa, (d) 2.4 MPa, and (e) 5.2 MPa. Table 5. Optimal Column Configuration with the Highest Productivity total no. of columns
1.0 MPa
pressure limit 2.4 MPa
5.2 MPa
without pressure limit
8 9 10 11 12 13 14 15 16
2-2-2-2 2-3-2-2 2-4-2-2 2-5-2-2 2-6-2-2 2-7-2-2 2-8-2-2 2-9-2-2 2-10-2-2
2-2-2-2 2-3-2-2 2-4-2-2 2-5-2-2 2-6-2-2 2-7-2-2 2-7-3-2 2-8-3-2 2-9-3-2
2-2-2-2 2-3-2-2 2-3-3-2 2-4-3-2 2-5-3-2 2-6-3-2 2-7-3-2 2-8-3-2 2-9-3-2
2-2-2-2 2-2-3-2 2-3-3-2 2-3-4-2 2-3-5-2 2-4-5-2 2-4-6-2 2-5-6-2 2-5-7-2
effects and mass-transfer effects become dominant, and in this mass-transfer-limiting region, the maximum productivity eventually decreases with increasing feed concentration (not shown here). The viscosity of the solution can change with PPA solute concentration,21 and the increasing viscosity can reduce linear velocities and corresponding productivity to satisfy a pressure
limit. However, within the feed concentration range considered, the effect of viscosity change on the SMB performance is insignificant. As feed concentration increases, desorbent requirement decreases and approaches a limiting value (Figure 8b). Without any pressure limit, the desorbent requirement is inversely proportional to the feed concentration (eq 24). The feed linear velocity becomes smaller for a fixed production rate as feed concentration increases. Less desorbent is needed to maintain product purity and yield at a lower feed linear velocity. For these reasons, desorbent requirement decreases as feed concentration increases. These effects are also dominant under a pressure limit. For a high-pressure limit, a higher feed linear velocity is allowed. However, a higher zone I velocity is needed to maintain product purity and yield, resulting in a higher desorbent requirement (Figure 8b). The separation cost decreases as the feed concentration increases for all pressure limits (Figure 8 parts c-e). Both CSP
Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006 749
Figure 8. Effects of feed concentration on (a) productivity (PR); (b) desorbent requirement (DR); and separation cost (SC) at (c) 1.0 MPa, (d) 2.4 MPa, and (e) 5.2 MPa. Table 6. Effect of Individual Parameter Optimization on Productivity Improvement
particle diameter (µm) column length (cm) feed concentration ((g of racemate)/L) a
value in example system
optimum value for max productivity
productivity at optimum {((kg of product)/day)/(kg of CSP)} [productivity improvement]a
27 10 50
42.5 3.8 535
1.73 [76.5%] 4.38 [346.9%] 1.51 [54.1%]
The improvement is based on the productivity of 0.98 ((kg of product)/day)/(kg of CSP) in the base system at 2.4 MPa.
and solvent costs decrease because productivity increases, and desorbent requirement decreases with increasing feed concentration. However, as discussed earlier, if the upper bound of feed concentration is sufficiently large, the separation cost eventually increases with increasing feed concentration because of nonlinear effects (not shown here). 4.1.6. Effect of Individual Parameter Optimization on Productivity Improvement. Table 6 shows the productivity improvement based on the sensitivity analysis when each parameter is optimized with other system parameters fixed at
values of the example system. The maximum pressure is 2.4 MPa, and the productivity of the example system is 0.98 ((kg of product)/day)/(kg of CSP). Individual parameter optimization can increase productivity by 54.1-346.9%. Especially the optimization of column length can significantly improve productivity, since the operating conditions and productivity of the base case are limited by the maximum pressure and a short column can decrease pressure drop. 4.2. Optimization of All System and Operating Parameters. In the previous sections, the effects of a single parameter
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Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006
Table 7. Optimal System and Operating Parameters to Achieve the Maximum Productivity for PPA Separation SMB equipment
low pressure
medium pressure
high pressure
max operating pressure (MPa) particle diameter (µm) column length (cm) column diameter (cm) total no. of columns column configuration feed concentration ((g of racemate)/L) uI0 (cm/min) uII0 (cm/min) uIII 0 (cm/min) uIV 0 (cm/min) switching time (min) productivity [((kg of product)/day)/(kg of CSP)] CSP cost ($/(kg of product)) solvent cost ($/(kg of product)) equipment cost ($/(kg of product)) total separation cost ($/(kg of product))
1.0 27.6 5.0 47.1 8 2-2-2-2 100 37.7 26.1 28.3 26.0 0.5 2.05 6.3 (26.3%) 13.6 (57.0%) 4.0 (16.7%) 23.9
2.4 18.0 5.0 45.6 8 2-2-2-2 100 37.4 26.0 28.2 26.1 0.5 2.18 5.9 (24.3%) 12.6 (52.1%) 5.7 (23.6%) 24.2
5.2 12.3 5.0 45.0 8 2-2-2-2 100 37.3 25.9 28.3 26.1 0.5 2.24 5.7 (7.7%) 12.2 (16.3%) 56.9 (76.0%) 74.8
were investigated while the other parameters were fixed. For a specific application, however, all 11 system and operating parameters should be optimized. The parameters include the six system parameters (particle size, column length, column diameter, total number of columns, column configuration, and feed concentration) and the five operating parameters (four zone flow rates and switching time). Two different objective functions (maximum productivity and minimum separation cost) are considered at three different pressures (high, medium, and low pressures). In the overall optimization, the minimum switching time of 0.5 min is added as an additional constraint, since it is difficult to implement an accurate switching time that is
