Standing Wave Annealing Technique - American Chemical Society

Ind. Eng. Chem. Res. 2006, 45, 8697-8712

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Standing Wave Annealing Technique: For the Design and Optimization of Nonlinear Simulated Moving Bed Systems with Significant Mass-Transfer Effects Fattaneh G. Cauley SeaVer College, Pepperdine UniVersity, Malibu, California 90263

Stephen F. Cauley School of Electrical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907

Ki Bong Lee Department of Chemical Engineering, Lehigh UniVersity, Bethlehem, PennsylVania 18015

Yi Xie Eli Lilly and Company, Lilly Corporate Center, Indianapolis, Indiana 46285

Nien-Hwa Linda Wang* School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907

This paper introduces a flexible and computationally efficient technique for the optimization of nonlinear simulated moving bed (SMB) systems with significant mass-transfer effects. The efficiency results from a combination of standing wave design equations (SWD), with a stochastic optimization algorithm, simulated annealing. Standing wave annealing technique (SWAT) extends the applicability of the SWD to the simultaneous optimization of a large number of variables that include material parameters. Several interrelated issues regarding the design of an SMB system are addressed through an example, the resolution of racemic mixtures of FTC-esters. Models containing 16, 18, and 19 decision variables are considered in terms of two alternative objectives: maximum productivity or minimum purification cost. SWAT’s computational efficiency (each optimization takes minutes rather than hours or days) helps identify the important role that maximum operating pressure plays in determining the economical design of an SMB system. Among the key findings are the following: (1) The costs of building a high-pressure SMB system that maximizes productivity can be nearly 60% greater than those of a cost-minimizing medium-pressure system. (2) For optimal separation, the maximal binding capacity should be as high as possible, and the adsorption affinity for the low-affinity solute should be as low as possible. The affinity for the high-affinity solute should be moderate in order to maximize productivity while keeping the solvent cost low. (3) Optimizing the material parameters has the potential of increasing productivity by 7-10-fold, reducing the solvent cost by 60%, and reducing the average purification cost by 60-70%. 1. Introduction Simulated moving bed (SMB) is a flexible and powerful separation tool that has been used for the separation of petrochemicals since the early 1970s and of sugars since the 1980s.1,2 The application of SMB has been extended into recovery and purification of biochemicals and pharmaceuticals since the early 1990s.3-7 A typical four-zone SMB system for binary separation consists of four or more chromatographic columns that are connected to form a continuous circuit (Figure 1). The circuit is divided into four regions (zones) of different flow rates by four ports. Two of these ports are for inlet streams, one is for a feed solution, and the other is for an eulent (or desorbent). The other two are for outlet streams, one for the slow-migrating solute, which has a higher affinity to the stationary phase (the extract), and the other for the fast-migrating solute, which has a lower affinity (the raffinate). The ports for the inlet and outlet streams move periodically in the direction * To whom correspondence concerning this article should be addressed. E-mail: [email protected]. Tel.: +1-765-494-4081. Fax: +1-765-494-0805.

of the fluid flow to follow the migrating solute bands. When the average velocity of port movement is greater than the migration velocity of the slow solute, but is less than the migration velocity of the fast solute, the slow solute will lag behind and the fast solute will move past the feed port. This difference in migration velocities provides the mechanism to separate the two components and to collect pure products from the outlet ports. In an ideal SMB system, where there are no mass-transfer effects, local equilibrium analysis guarantees complete separation, achieving 100% purity and yield.8,9 However, in nonideal systems, where mass-transfer effects are prevalent, high purity and yield in separation is not easily achieved (Figure 2). Thus, to minimize mass-transfer effects, practitioners have turned to the use of fine adsorbent particles and high-pressure equipment that are costly. Early research in this area focused on the optimal operation of an existing SMB unit. Here, optimal design refers to the determination of the values of the five operating variables (the four zone flow rates and the average velocity of port movement) that minimizes the eulent consumption or maximizes throughput.

10.1021/ie060300a CCC: $33.50 © 2006 American Chemical Society Published on Web 11/11/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006

Figure 1. Schematics of a conventional four-zone simulated moving bed: (a) step N; (b) step N + 1.

In 1997, Ma and Wang10 proposed the standing wave design (SWD) for determining the optimal operating variables of nonideal SMB systems with linear isotherms. SWD involves solving a system of algebraic equations that link product purity and yield to zone lengths, zone interstitial velocities, port switching time, isotherms, and mass-transfer parameters (axial dispersion and lumped mass-transfer coefficients). This method provides a simple and systematic procedure for achieving the desired purity and yield in the presence of mass-transfer effects.11-13 Later, SWD was modified for nonideal, nonlinear SMB systems.14-18 Furthermore, unlike methods that rely on the equilibrium theory, SWD does not require time-consuming simulations of SMB processes to ensure that the purity and yield requirements are satisfied. Many experimental studies and computer simulations based on a detailed rate model (VERSE) have verified that SWD guarantees the desired purity and yield for nonideal SMB systems.19-32 Recent research has considered optimizing additional variables that are thought to have a large impact on the economic efficiency of an SMB unit. Procedures based on SWD,33-36 and other procedures including the triangle theory,9,37-44 have been used to determine the optimal particle size, total number of columns, column lengths, and the feed concentrations when the objective is to maximize throughput, minimize eulent consumption, or maximize product purity. In 2004, Cauley et al.35 developed the optimization tool, standing wave annealing technique (SWAT), to optimize the design of nonideal, linear SMB systems with linear isotherms. This tool, which is based on the SWD, employs a stochastic optimization algorithm, simulated annealing (SA), for simultaneous optimization of 12 variables: 5 operating variables (the four zone flow rates and the average velocity of port movement), and 6 system variables (particle size, column length, column

Figure 2. Schematics of the SMB column profiles for a binary nonlinear SMB system with mass-transfer effects.

diameter, and number of columns in each zone). The authors showed that SWAT was computationally at least an order of magnitude faster than a grid search method.34 This paper extends and expands SWAT so that it is applicable to the design of nonlinear SMB systems. Specifically, the SWD equations are reformulated and augmented with additional variables and constraints. The resulting mathematical programming model includes all variables that control the cost of the SMB design, construction, and operation. The model, which is highly nonconvex,45 is solved using a SA algorithm; the integration of SA with SWD provides an efficient tool for solving any single or multiobjective optimization of nonideal, nonlinear SMB systems. In this paper, we will illustrate SWAT through an example, the resolution of racemic mixtures of FTC-esters, to address several interrelated issues regarding the design of an economical SMB system. In large-scale production, both capital and operation costs must be considered; in the first part of this paper, we will determine the design that minimizes the major cost of building and operating an SMB unit. A total of 16 decision variables will be simultaneously optimized. In addition, our analysis highlights the key role that the maximum operating pressure plays in the optimal design of an SMB system when the objective is to minimize the purification cost. Our findings include the surprising result that, if constructed optimally, a medium-pressure system may be able to reduce total purification costs to half that of a high-pressure system. In the second part of the paper, we will expand the set of decision variables to some of the material parameters, for example, isotherm and

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Figure 3. Diagram of SWAT optimization tool.

mass-transfer parameters. The goal is to determine whether optimizing the parameters can affect the productivity of the adsorbent or the separation costs. The analysis is similar to a “thought experiment”, because optimizing the material parameters would involve invention of new adsorbents. The result of this analysis suggests that, for optimal separation, the maximal binding capacity should be as high as possible and the adsorption affinity for the low-affinity solute should be as low as possible. The affinity for the high-affinity solute should be moderate in order to maximize productivity while keeping the solvent cost low. Furthermore, optimizing the material parameters has the potential of increasing productivity by 7-10-fold, reducing solvent cost by 60%, and reducing the average purification cost by 60-70%. 2. Model and Optimization Technique This section introduces the mathematical programming model that defines SWAT. It includes descriptions of the input parameters, decision variables, constraints, and alternative objective functions for the mathematical program. The input parameters, with known values, are associated either with the separation problem, such as the intrinsic adsorption parameters, or represent characteristics of the SMB unit. The decision variables represent aspects of the design that are not locked in place and will be determined by the optimization process. The constraints are a reformulation of the SWD equations to extend their applicability to solve efficiently any single or multiobjective optimization problem for a nonideal, nonlinear SMB system. Figure 3 shows a schematic diagram for inputs and outputs of SWAT. The mathematical programming models for nonideal, linear SWAT, as described in Cauley et al.,35 and nonideal, nonlinear SWAT, as described in this section, are similar with respect to some of their inputs, variables, and constraints. However, there are important differences between the models, mainly as a result of competitive adsorption effects in nonlinear systems.15,18 The model introduced in this paper is discussed in detail in the next section. With respect to the mathematical model, different SMB systems are represented by different input parameters. What is

unique to this model is the flexibility to expand the set of decision variables to include a number of material parameters that are normally considered as input parameters with fixed values. This flexibility will be illustrated in Section 4, where the effects of variations in these parameters on the optimal design of an SMB system are investigated. 2.1. SWAT. The model’s decision variables were clustered into four tiers to facilitate the explanation of various facets that determine the optimal design of an SMB unit. In Section 3, it will be shown that previous papers on SMB systems can be considered as special cases of this model, where the decision variables in some of the tiers are regarded as fixed. Furthermore, the four-tiered design provides an intuitive setting for describing and analyzing the different components of purification cost. 2.1.1. Decision Variables. Tier0, which will be referred to as the material tier, is composed of two variables: feed concentrations (CF,i) for component i, i ) 1, 2. In Section 4, Tier0 variables are expanded to include decision variables that represent variations in the material parameters. Tier1 is comprised of four system variables: column length (Lc), column cross-sectional area (S), total number of columns (Ncol), and particle diameter (dp). These variables determine the fixed cost and physical capacity of an SMB system; once purchased and placed in use, the values of these variables cannot be changed without incurring substantial additional costs. The three Tier2 variables that determine the system’s configuration are the number of columns (Njcol) in zone j, j ) 1, 2, 3. Note that, in this model, the number of columns in zone four, N4col, is not a variable but is uniquely determined by the other variables (Ncol and the Tier2 variables). The seven Tier3 variables include the five variables that are generally considered as operating variables: the interstitial velocities (uj0) in zone j, j ) 1, 2, 3, 4, and the average velocity of port movement (V). The additional two variables, which are unique to nonlinear systems, are the diluted plateau concentrations near the feed port15 (Cs,i) for component i, i ) 1, 2 (Figure 2). All variables except for those of Tier1 (system variables) will have an impact on the variable costs of production and can be altered and optimized in response to changes in demand.

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u10 ) (1 + Pδ12)V + ∆12

variables in the calculation of plateau concentrations Cp,i.14,15 Additionally, Yi and Purreq represent the required yield and purity, respectively. Appendix A contains a detailed description of all derived variables and terms as functions of the decision variables and input parameters.

u20 ) (1 + Pδ21)V + ∆21

u30 - u20 e uF,max

2.1.2. Constraints. The model contains five distinct groups of constraints. As was previously mentioned, the SWD equations, given below, constitute the core group of constraints:

u30 ) (1 + Pδ32)V - ∆32

(1)

u30 < u10

u40 ) (1 + Pδ41)V - ∆41

u20 < u30

Lc - Vts,min g 0

u40 < u10

The mass-transfer correction terms in eq 1 are defined as

∆ji )

βji NjcolLc

(

j Eb,i +

)

P(δji)2V2 Kji

, for j ) 1, 2, 3, 4

u20 < u10

where the subscript i that designates the component is equal to 1 in zones 2 and 4 and equal to 2 in zones 1 and 3. The phase ratio P is defined as (1 - b)/b, where b is the interparticle void fraction. In the above equation, β is the natural logarithm of the ratio of the highest concentration to the lowest concentration of a standing wave in a particular zone. Thus, β is, in effect, an index of product purity and yield; the greater the value of β, the higher are the product purity and yield. The axial dispersion coefficient is Eb, and K is the overall mass-transfer coefficient as defined by Ma and Wang.10 An important difference between linear and nonlinear SWAT is evident in the next equation. In nonlinear systems, the effective retention factors (δ) vary with the concentrations of both components and are complex functions of several of the decision variables. This difference adds substantially to the computational complexity of the optimization algorithm, which is discussed in the next section.

δ12

DV ) p + (1 - p)a2 + PLcSb

a1 DV δ21 ) p + (1 - p ) + 1 + b2Cp,2 PLcSb δ32

a2

(3)

DV ) p + (1 - p) + 1 + b1Cs,1 + b2Cs,2 PLcSb δ41 ) p + (1 - p)

u40 < u30

(2)

a1 DV + 1 + b1Cp,1 PLcSb

In the above, ai and bi are the Langmuir adsorption isotherm parameters and Cp,i, a derived variable of the model, represents the plateau concentration of component i. In addition, p is the intraparticle void fraction, and DV is the extra-column dead volume. The next group of constraints is primarily dictated by the SWD. However, the difference between the models for linear and nonlinear isotherms is observed in this group as well. The first eight constraints were introduced in Cauley et al.,35 for linear systems. The first constraint denotes the requirement of mass balance at the feed port and a physically feasible solution of eq 1. The maximum feed uF,max in the right-hand side of the first constraint is a nonlinear function of the decision variables. The remaining constraints are dictated by SWD for separations with nonlinear isotherms.15,18 The derived variable, γ+,F, is unique to nonlinear design; it and γ- are intermediate

2 (a2b1γ+,F + a1b2γ+,F)Cs,2 - (a2b1γ+,F + a1b2)Cs,1 + (a1 - a2)γ+,F ) 0 (4)

u30 - u20 u30 - u40

CF,1Y1 ) Cp,1

u30Cs,1 - (u30 - u20)CF,1 > 0 i ) 1, 2

Cs,i e CF,i

CF,1 ) CF,2 (chiral models only) PurE ) PurR )

CF,2Y2 CF,2Y2 + CF,1(1 - Y1) CF,1Y1 CF,1Y1 + CF,2(1 - Y2)

g PurEreq g PurRreq

The feed and zone flow rates of an SMB system are limited by the maximum operating pressure of the system, ∆pmax. The pressure drop in an SMB system depends on its pump configuration. In this study, the desorbent pump is assumed to control the flow rates from zones 1 to 4. For such a system, the pressure can be calculated by summing the pressure drop of the four zones. The Ergun equation46 is commonly used to relate the pressure drop in each zone, ∆pj, to particle size, zone length, and other parameters. The following equation represents the pressure drop constraint. 4

∆pj e ∆pmax ∑ j)1 where

∆p ) j

[

(150 × 108)µuj0NjcolLcP2 6dp2

+

]

1.75F(uj0)2NjcolLc 1.47 , j ) 1, 2, 3, 4 (5) 3.6dp (1.013 × 103)

In the above, µ and F represent the viscosity and mobile phase density, respectively. As will be seen in the next section, while determining the optimal values of the decision variables, SWAT consistently forces the above constraint to its maximum value. Thus, the system’s maximum operating pressure will play a vital role in determining the optimal design of an SMB system.


The desired annual production of the product Q is a nonlinear function of the decision variables. This function is entered as a constraint with upper and lower bounds. By varying Q, SWAT is able to analyze the impact of differences in the desired levels of output on the optimal system design. This provides flexibility in the analysis of the economic viability (e.g., determining long run costs) of a nonlinear SMB system,

Qmin e Q e Qmax for Q ) Sb(u30 - u20)CF,pYpt

(6)

where t, measured in min, is the available time for annual production. In addition to the constraints, there are upper and lower bounds for all variables. The bounds are determined by physical restrictions, validity of the SWD, maximum concentration limits, and solubility constraints. We will refer to the model described above as the benchmark model in the remainder of the paper. 2.1.3. Objective Function. SWAT is capable of optimizing any objective function that is expressed as a function of the decisions variables, including multiobjective optimization. The early literature on SMB optimization was mainly concerned with maximizing performance measures such as productivity (PR), yield, or product purity.33,37-39 This focus resulted from the desire to optimize the production of an SMB unit that is already built and is in operation. The following is a well-established definition of PR.

PR )

Q FBV(365)

(7)

In eq 7, Q is the annual production as given in eq 6, FB is the bulk adsorbent packing density, and V is the total bed volume of the system. The factor 365 is used to convert the annual production rate Q to a daily production rate, based on the assumption of no down time. PR is expressed in (kg of product)/ (kg of CSP)/day. Noting that SWAT is based on SWD, the desired levels of purity and yield are guaranteed in the optimization process. In recent literature, attention has shifted to cost minimization and optimizing additional variables that are thought to have a large impact on the economic efficiency of an SMB system.34,35,41,42 The annual purification cost (PC) can be separated into the fixed costs and the variable costs of purification, where

annual fixed cost ) annual equipment cost + annual CSP cost AnnualEqCost )

CCSPSLcFBNcol CE , AnnualCSPCost ) rE rCSP (8)

The numerator in the first term of eq 8 is the acquisition price of SMB equipment, and the denominator is the depreciation period of the equipment. In the second term, the numerator represents the CSP cost for the SMB system, where CCSP is the price/(kg of CSP). The denominator is the depreciation period of CSP. Similarly,

annual variable cost ) annual solvent cost + annual feed cost AnnualSolCost ) Sbt[(u10 - u40) + (u30 - u20)] × [Csol(1 - SRcy) + CRcy sol SRcy] (9) AnnualFeedCost ) Cfeed

Q Yp

In the first term of eq 9,Csol, CRcy sol , and SRcy represent, respectively, the price/L of solvent, the price/L of solvent recovery, and the solvent recycle ratio. In the next term, Cfeed represents the price/(kg of feed). Throughout this analysis, all three terms for annual feed cost remain fixed and have no effect on the optimization result. For this reason, annual feed cost is ignored in this analysis. It should be noted that the above definition for purification cost ignores facility, labor, and other operating costs. In the analysis that follows, the results will be couched in terms of average cost (AC) as defined in the next equation. This step is taken for ease of presentation, on the grounds that, for any given level of output (25 000 kg/year in this analysis), operations that minimize purification cost will also minimize average cost. Combining the definitions in eqs 8-10, the

AC )

PC Q

(10)

following equation represents the breakdown of average cost where all terms are expressed in $/(kg of product).

AC ) EqC + CSPC + SolC

(11)

2.2. Optimization Technique. The SWAT model is composed of nonlinear and nonconvex constraints and objective function.45 This produces a feasible set with many local optima and a substantial computational challenge to locate the global optimum. The options available for successfully optimizing nonconvex problems (i.e., not getting stuck in local “optima”) are limited. During the last two decades, stochastic optimization techniques, where sampling for the global optimum follows a probability distribution, have been developed to solve such problems. At the heart of any stochastic optimization technique is the opportunity for a jump to another part of the feasible set that is independent of the current point (i.e., acceptable moves do not always have to be associated with better values of the objective function). Among these algorithms, two stand out: genetic algorithm (GA) and simulated annealing (SA). The former works on the principle of mimicking a biological system, and the latter mimics a physical system. GA generally does well with unconstrained problems regardless of the complexity of the objective function, but its efficiency drops substantially in the presence of a large number of constraints,47 such as in our problem. In contrast, SA algorithms are well-suited for problems with a large number of constraints. Furthermore, it has also been shown that SA has a strong record of finding a near-optimal solution for most nonconvex optimization problems.48,49 SA is a technique that is widely used in physics and electrical engineering for the design of computer chips.50 Cauley et al.35 developed an SA algorithm for solving the model associated with the SW equations for SMB systems with linear isotherms. The choice of the move class, acceptance rule, and annealing schedule in implementation of SWAT was given in detail by Cauley et al.35 Our experience in solving nonideal, linear SMB systems by SWAT was favorable. We found the optimization results obtained by SWAT to be similar or superior to those previously reported by a grid search method,34 and the computation time was at least an order of magnitude faster. In Section 2.1 of this manuscript, the model associated with the SW equations for SMB systems with nonlinear isotherms is presented, and the differences between the current model and the model by Cauley et al.35 are described. Although there are important differences between the two models, only the implementation of the move class in SWAT

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Table 1. Input Parameters for the FTC Separation System minimum step time ts,min (min) yield, Y (%) purity, Pr (%) interparticle void fraction, b intraparticle void fraction, p extra-column dead volume (% of bed volume) viscosity µ (g/cm/min) mobile phase density F (g/cm3) packing density FB (g/cm3) Brownian diffusivity D∞ (cm2/min) intraparticle diffusivity Dp (cm2/min) kf (cm/min) Eb (cm2/min) isotherms (+)-FTC (-)-FTC

0.5 99 99 0.24 0.60 5.73 0.32 0.79 0.60 6.12 × 10-4 2.0 × 10-5 Wilson and Geankoplis51 correlation Chung & Wen correlation52 a ) 1.12, b ) 0.0168 a ) 2.12, b ) 0.0383

required significant change. Specifically, the additional complexities associated with the current model required additional effort in determining a feasible initial guess prior to the basinhopping stage. In this procedure, a local search was preformed around the stochastically perturbed minimum35 in order to determine a valid starting point before initiating the basin hopping. Additional detail regarding the implementation of SA in SWAT for solving the SMB optimization problem can be found in the paper by Cauley et al.35 3. Results and Discussion of Application of SWAT to Chiral Separations To illustrate the efficiency of SWAT, a separation problem from previous literature, the resolution of racemic mixtures of FTC-esters, is analyzed. The results of simultaneous optimization, with respect to all tiers (16 variables), are presented below. As will be shown, the flexibility and computational tractability of SWAT allows us to answer design questions that have not been addressed previously. For example, we will analyze the relationship between the maximum operating pressure of an SMB system and system productivity and costs. These analyses will lead to an interesting finding; that medium- or low-pressure systems, when designed optimally, can reduce the costs of purification substantially as compared to high-pressure systems. 3.1. Separation Problem of FTC-ester. In 2003, Xie et al.16 examined the resolution of racemic mixtures of FTC-esters, which is a precursor of a potential anti-HIV drug. Details regarding stationary phase, mobile phase, isotherm parameters, and mass-transfer parameters were reported by Xie et al. These parameters are used as inputs in this analysis, and they are given in Table 1. The problem that Xie et al.16 solved is equivalent to fixing material (Tier0), system (Tier1), and configuration (Tier2) variables for the benchmark model and using the objective of maximum productivity to determine the operating (Tier3) variables. In the next section we expand on the work of Xie et al.16 That is, we do not fix any of the tiers and determine simultaneously all 16 variables of the benchmark model, first for the objective of maximum productivity and then for the objective of minimum purification cost. By determining all 16 decisions variables for a given objective, we are providing answers to the design questions that arise when building a new SMB unit (i.e., before the design is locked in). The bounds for the decision variables for the benchmark model are given in Table 2. One of the most important considerations in this analysis was the acquisition price of an SMB unit. In general, this price is a

Table 2. Bounds of Variables for Optimizing Benchmark Model variable

bounds

feed concentration ((g of racemate)/L) particle diameter dp (µm) column diameter dc (cm) column length Lc (cm) total number of columns Ncol number of columns in each zone Nj zone velocity uj0 (cm/min) average port moving velocity V (cm/min)

10-80 5-200 1-300 10- 100 8-20 2 - (Ncol - 6) 0.01-200 0.01-20

complex function of many parameters and is difficult to estimate precisely. In practice, large systems are generally designed for a specific separation, and the price can change depending on the detailed requirements of the application. A high-pressure SMB unit requires high-pressure parts (column, pumps, valves, sensors, tubing, and other accessories), which are much more costly than the corresponding parts in a low- or medium-pressure SMB. High-pressure SMB columns, for example, are constructed by drilling out the center portion from a single solid piece of stainless steel. By contrast, medium- and low-pressure columns are constructed differently and cost much less. As a result, the costs of columns are not major factors in the acquisitions prices of low- and medium-pressure SMB units. In the analysis that follows, three pressure classes are considered: low-pressure systems (e1.0 MPa), medium-pressure systems (e2.4 MPa), and high-pressure systems (e13.8 MPa). The acquisitions prices for the three systems are based on estimates provided by Lee et al.36 As is seen in Table 3, the three classes can differ substantially in acquisition price of equipment. The acquisition price of SMB equipment for highpressure systems is based on the data from Novasep and is given as a function of column diameter. The acquisition price of SMB equipment for low- and medium-pressure systems are based on estimates from U.S. Filter for the production of high fructose corn syrup upgraded for pharmaceutical production. Excluding acquisition price of equipment, all other costs and inputs are the same for the three pressure classes throughout this paper. 3.2. Optimization of Tiers0-3 Variables for Determining the Optimal SMB Design. This section begins by using SWAT to determine the optimal SMB system for maximum productivity (eq 7). To clarify the ensuing results, we examine the optimization for the medium-pressure class in detail. For this pressure class, eight values (equidistant from 1.2 to 2.4 MPa) within the feasible range of pressures for this class (e2.4 MPa) are chosen. Each value signifies the maximum operating pressure for an SMB system and would appear as the limit in the pressuredrop constraint (eq 5). Values below 1.0 MPa are not considered, because it would be inefficient to incur the fixed cost of a medium-pressure SMB system and operate it at a pressure below 1.0 MPa. Likewise, values below 2.4 MPa for the high-pressure class are not considered. The 7 values for the low-pressure class and 33 values for the high-pressure class are all chosen equidistant within the feasible range. For each case, SWAT is run for the objective of maximum productivity for an annual production of 25 000 kg. That is, for each case, SWAT simultaneously determines the optimal values of the material (Tier0); the system, including CSP particle size (Tier1); the configuration (Tier2); and the operation of the SMB unit (Tier3) variables. By simultaneously optimizing the 16 design variables, we find that the maximum productivity for each pressure class is realized at the maximum operating pressure for the class in question. A triangle plot of all SWAT solutions is shown in Figure 4. In this plot, the vertex shows the point where the highest productivity can be attained, that is, the point where there is no


Figure 4. Diagram of triangle plot with SWAT solutions. Table 3. Costs and Other Information Used in All Optimizations production rate (kg/year) CSP (Chiralpak AD) price ($/kg) (CSP price is independent of the value of material parameters) CSP depreciation period (years) solvent (methanol) price ($/L) solvent recycle ratio (%) solvent recovery cost ($/L) equipment depreciation period (years) acquisition price of SMB equipment low-pressure system (e1.0 MPa) medium-pressure system (e2.4 MPa) high-pressure system (e13.8 MPa)

25 000 15 000 4 0.20 95 0.10 7 $700 000 $1 000 000 is a function of column diameter dc (cm),(-927.67dc2+ 260 600dc + 96 292)

pressure limit on the system. The movement of the solutions toward the vertex is easily explained. By optimizing the material, and the system variables within their bounds, the mass-transfer effect is largely removed and the productivity of the SMB system is primarily limited by its maximum operating pressure. The conclusion of this analysis is that, to maximize productivity, the optimal SMB system is a high-pressure system that can be operated at 13.8 MPa, the largest maximum operating pressure considered in this paper. The SWAT result for this system is presented in Table 4. Additionally, from the information given in Table 3, the acquisition price for this high-pressure system based on a column diameter of 46.5 cm is estimated at $10 207 000. However, a firm’s objective should be to maximize profits, or to minimize the costs associated with production. Thus, the objective of maximum productivity and its conclusions would be appropriate if, and only if, changes in the decision variables that result in an increase in productivity also result in a decrease in purification cost. To further examine the optimal solutions for each of the (productivity-maximized) SMB systems, the associated average purification costs are estimated. Figure 5 shows the maximum productivity and the associated average cost for each system. The x-axis in this graph, as well as the other graphs in this paper, denotes the maximum operating pressure for an SMB system. Notice that both the productivity and average cost relationships are broken into three segments, where the first segment is data for the 7 cases for the lowpressure systems, the second segment is data for the 8 mediumpressure systems, and the last segment is data for 33 highpressure systems. Small gaps observed between pressure classes on the x-axis are intentional, to draw attention to distinctive trends in the three pressure classes. From this figure, we see that, as expected, productivity increases monotonically, both within a given pressure class (low, medium, or high) and

between pressure classes. However, the same monotonicity is not observed for average costs. Specifically, within a pressure class, increases in productivity are associated with decreases in average cost, but between pressure classes, we see that this relationship does not appear to hold because of the large increase in the acquisition price of equipment for high-pressure systems. To further explore the relationship between maximum operating pressure and purification cost, the above analysis is extended. SWAT is run for the objective of minimum average purification cost (eq 10) for annual production of 25 000 kg. As before, SWAT simultaneously determines the values of all 16 decision variables for each of the cases of the three pressure classes. Similar to the previous results, in all optimizations, the minimum average cost for each case is obtained at the maximum operating pressure for the case. The conclusion of this analysis is that, to minimize average cost, the optimal SMB system is a mediumpressure system that can be operated at 2.4 MPa, the highest maximum operating pressure for the class (full data for this analysis will be presented in the next section). The average cost for such a system is estimated to be $64.50/kg. From Table 4, the average cost for the optimal high-pressure SMB system operated at 13.8 MPa is estimated at $103.10. Our analysis suggests that, by building the SMB system for maximum productivity, the average cost will be almost 60% higher than necessary. To further clarify this important result, Table 5 presents an analysis of the sensitivity of the above conclusion to variations in the acquisition price of medium-pressure SMB systems. All entries in the table are for the cost-minimizing 2.4 MPa medium-pressure system. The first row of this table shows cost estimates based upon the previously described $1 000 000 acquisition price of a medium-pressure system. Subsequent rows present the same information for alternative (higher) estimates of acquisition price of a medium-pressure system up to $7 000 000. From this table, it can be seen that acquisition price for medium-pressure equipment can be more than 7× the U.S. Filter estimates, and it is still more economical to adopt a medium-pressure SMB system. Two conclusions can be drawn from the above analysis concerning the design of an SMB system; first if low- and medium-pressure systems were available as viable alternatives, the optimal SMB system as determined by SWAT for the objective of minimum average cost is a medium-pressure system. Second and most importantly, the price of upgrading medium-pressure equipment for pharmaceutical production can be substantially higher, and these systems remain competitive compared to high-pressure systems. The above analysis illustrates why it is essential to consider a large set of decision variables when designing SMB systems. The flexibility and computational efficiency of SWAT was an invaluable tool in this analysis, where approximately 100 optimizations were performed, each taking SWAT a few minutes (rather than hours or days) on a 2.2gh Pentium4. In the next section, some of the results obtained in the analysis of minimum average cost and their implications for the optimal design are discussed. 3.3. Discussion of SWAT Results for Minimum Average Cost. As stated earlier, SWAT was run for each of the values of the three pressure classes (48 cases). In each case, the 16 variables of the benchmark model are determined for the objective of minimum average cost. The optimal value for the Tier0 variable feed concentration corresponds to its upper bound value of 80 ((g of racemate)/L), as predicted by the equilibrium model. The optimal value for the variable column length is its lower bound value of 10 cm. The optimal number of columns is its lowest value of eight columns with the corresponding

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Table 4. SWAT Result for Optimal SMB System, Objective Maximum PR variables PR, costs PR costs

Tier 0 Tier 1

Tier 2

Tier 3

13.8 MPa SMB system

SWAT result

(kg of product)/(kg of CSP)/day AC ($/kg) EqC ($/kg) CSPC ($/kg) SolC ($/kg) feed concentrationCF,1 ((g of racemate)/L) feed concentration CF,2 ((g of racemate)/L) column cross section S (cm2) (column diameter dc (cm)) column length Lc (cm) particle diameter dP (µm) total number of columns Ncol zone 1 number of columns N1col zone 2 number of columns N2col zone 3 number of columns N3col zone 1 interstitial velocity u10 (cm/min) zone 1 interstitial velocity u20 (cm/min) zone 1 interstitial velocity u30 (cm/min) zone 1 interstitial velocity u40 (cm/min) average port moving velocity V (cm/min) diluted plateau concentration of component 1 CS,1 ((g of racemate)/L) diluted plateau concentration of component 2 CS,2 ((g of racemate)/L)

0.84 $103.1 $58.3 $12.3 $32.5 40.0 40.0 1 697.7 (46.5) 10.0 25.0 8 2 2 2 103.0 73.1 76.0 69.6 17.1 10.0 18.1

the above conclusion and result in the medium-pressure system becoming more favorable. In Figure 6b, the rapid decrease of column diameter with the increase in system pressure results in the increase in productivity that was reported earlier. In Figure 6c, the interstitial velocities of the four zones increase as the system pressure increases, as expected by the Ergun equation,46

∆p ∝

Figure 5. PR and corresponding AC for benchmark model; objective maximum PR. Table 5. Sensitivity Analysis AC Breakdown for 2.4 MPa SMB System Benchmark Model; Objective Minimum AC alternative acquisition price for 2.4 MPa SMB equipment

AC ($/kg)

equipment cost ($/kg)

CSP cost ($/kg)

solvent cost ($/kg)

$1 000 000 $2 000 000 $3 000 000 $4 000 000 $5 000 000 $6 000 000 $7 000 000

$64.50 $70.22 $75.94 $81.65 $87.37 $93.08 $98.8

$5.71 $11.43 $17.14 $22.86 $28.57 $34.29 $40.00

$31.28 $31.28 $31.28 $31.28 $31.28 $31.28 $31.28

$27.51 $27.51 $27.51 $27.51 $27.51 $27.51 $27.51

optimized configuration of two columns in each zone (or 2, 2, 2, 2). This result can be explained by eq 7, where we observe that the column length and the number of columns are inversely related to productivity. Therefore, holding all other factors constant, the lower bounds of the two variables provide the highest productivity. As was previously the case, the data graphed in the following figures is broken into segments for low-, medium-, and high-pressure systems. Figure 6a shows the optimal particle size, which decreases steadily as the maximum operating pressure increases. The particle size for the optimal SMB system, shown on the mediumpressure curve at 2.4 MPa, is estimated at 34 µm. It should be noted that, in our optimizations, the CSP price was held constant for all particle sizes. If CSP price is lower for a larger particle, the purification cost for the medium-pressure system will be lower. Thus, the price differential will add to the validity of

Luj0 dp2

(12)

In Figure 6d, the normalized product concentrations CP,i/CF,i for both extract and raffinate increase within a given pressure class, and between low- and medium-pressure classes. The extract product concentration, which is ∼0.10-0.115, is more diluted, and raffinate product concentration, 0.51-0.56, is less diluted. The key observation here is that there is very little difference between low-, medium-, and high-pressure systems. The above provides insight into the minimized cost breakdown. Table 6 shows the cost breakdown for the three conditionally optimal systems for the three pressure classes. The table shows that, once the systems are built optimally, there is very little difference in the solvent cost. Although, the CSP cost is higher for the medium-pressure system, the large difference in AC is due to the high equipment cost for the high-pressure systems. Figure 6e is designed to demonstrate this point for all pressure classes. The curves in this figure show the three cost components as they are added to arrive at the average cost. As an example, consider the optimal SMB system, the mediumpressure unit with the maximum operating pressure of 2.4 MPa. The equipment cost for this system is estimated at $5.71/kg, as shown in Table 6 and on the Eq/kg curve. The next curve is the fixed cost relationship which is the sum of the equipment and CSP costs, and it shows the value of $36.99 on the (Eq + CSP)/kg curve for this system. Thus, we can observe visually the CSP cost as the difference between these two curves. Similarly, the solvent cost is the difference between the (Eq + CSP)/kg curve and the AC curve. The rapid decrease in CSP cost observed as the system pressure increases can be attributed to the fact that, as system pressure increases and productivity increases, less CSP is required per unit of output. The solvent cost is ∼$28 for all pressure classes. The relatively high average

Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006 8705 Table 6. Cost Breakdown for Optimal System of Each Pressure Class, Benchmark Model; Objective Minimum AC

AC ($/kg) SolC ($/kg) CSPC ($/kg) EqC ($/kg)

1.0 MPa

2.4 MPa

13.8 MPa

79.31 28.54 46.76 4.00

64.50 27.51 31.28 5.71

100.17 27.82 12.85 59.50

Table 7. Bounds of Variables for Optimizing Material Parameters Models variable

bounds

Adsorption Isotherm Parameters Model saturation capacity qmax (g/L) 0.0011-100 isotherm parameter b (L/g) 0.0011-1000 Intraparticle Diffusivity Model intraparticle diffusivity Dp (cm2/min) 6.12 × 10-6 to 5.508 ×10-4

cost for the high-pressure systems clearly can be attributed to the high acquisition price of the SMB equipment. 4. Expanding SWAT for Optimization of Material Parameters

Figure 6. (a) Particle size for benchmark model; objective minimum AC. (b) Column diameter and column length for benchmark model; objective minimum AC. (c) Zone velocities for benchmark model; objective minimum AC. (d) Normalized extract and raffinate product concentrations for benchmark model; objective minimum AC. (e) AC breakdown for benchmark model; objective minimum AC.

The result presented above provides direction for how, given a set of input parameters, to construct and operate an SMB unit. At the core of these results are the SWD equations. Given the validity of the SWD equations, there is little room for any further reduction in costs. However, an area that could impact the efficiency of production is the potential for making changes in the material parameters. We will now use SWAT to analyze the potential impact of changes in these parameters on the optimal design and average cost of SMB systems. SWAT is easily modified to include the material parameters in the set of decision variables. By releasing these parameters from their previous role, as an input parameter with fixed values, we are able to explore the potential impact of new adsorbents on system productivity and average cost. Specifically, the main change occurs when the set of material variables (Tier0) is expanded to include additional material parameters as variables. In the remainder of this section, we investigate the impact of changing material parameters on the separation of FTC enantiomers. 4.1. SWAT Models for Analysis of Material Parameters. To investigate the impact of changes in material parameters, SWAT is modified, and two separate models are considered: the first explores variations in adsorption isotherm parameters, and the second explores variations in intraparticle diffusivity. For each case, the set of material variables (Tier0) is expanded. The other tiers and the variables they comprise will remain as described in Section 2. The five groups of constraints presented in Section 2 will also hold in their entirety, but with added bounds corresponding to the new variables. The bounds, as given in Table 7 represent an interval, dictated by physical considerations, around the original values of the parameters given by Xie et al.16 The model for the analysis of adsorption isotherm parameters includes 19 decision variables. Three new variables are added to the 16 variables of the benchmark model; they are qmax and the two isotherm parameters bi. The upper bound for qmax is chosen at 100, to provide a high number of adsorption sites per CSP volume, which should result in high productivity for the system. The model for the analysis of mass-transfer parameters intraparticle diffusivities contains 18 decision variables. The two intraparticle diffusivities Dp,i are added to the benchmark model. For chiral systems, the two intraparticle diffusivities should be

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Figure 7. Ratio of PR of material parameters models to benchmark model; objective maximum PR.

the same. This is dealt with by adding a constraint in a general model, where the two diffusivities can be different. The bounds for the new variables are based on the value of the Brownian diffusivity, D∞. The lower and upper bounds represent 1% and 90% of the value of the Brownian diffusivity, respectively. The two models described above will be referred to by the name of the material parameters that are considered as variables for the model. 4.2. Optimization of Material Parameters for Maximum Productivity. We will first investigate the impact of changes in material parameters on productivity. SWAT is run for each of the material parameters models for the objective of maximum productivity. The figures presented in this section should be interpreted in the same way as those for Section 3, where the x-axis represents the maximum allowable pressure drop for the SMB system. Figure 7 shows the ratio of the optimal productivity of each of the two material parameters models to the benchmark. The highest proportional gain in productivity was achieved through optimizing the adsorption isotherm parameters. As this figure shows, optimizing these three parameters results in a 12-fold increase in productivity for the low- and mediumpressure systems, and on the average an 11-fold increase in productivity for the high-pressure systems. For all pressure systems, SWAT’s optimal value for qmax is the upper bound, where the highest productivity can be obtained. The optimal value for b1, for all pressure systems, is the lower bound, so the interference with adsorption of the high-affinity component can be minimized. More interesting are the optimal values of isotherm parameter b2, which vary from 0.07 (L/g) in the low- and medium-pressure systems to 0.05 (L/g) in the highpressure systems. All are interior values are fairly small but larger than the actual b2 value of the existing chiral sorbent, 0.038 reported by Xie et al.16 The optimal values of b2 clearly demonstrate the presence of two competing factors. On one hand, b2 should be sufficiently large to have the needed selectivity to separate the two components. On the other hand, a large value of b2 will result in a large ∑ibiCF,i value or strong competitive adsorption, which in turn will reduce productivity. Noting that SWAT optimized b1 at the lower bound value, which is close to zero, and CF,i at the upper bound value of 40, it falls on b2 to keep the values of ∑ibiCF,i small. Thus, the optimal values of b2, although interior points, are moderately small and result in values of 2-3 for ∑ibiCF,i. Optimizing the intraparticle diffusivities also increases productivity, where the increase for low- and medium-pressure systems is 3-4-fold and for the high-pressure systems is 1-3fold. For all pressure systems, SWAT’s optimal value for intraparticle diffusivity is close to the upper bound value. This value, as shown in Table 7, represents 90% of Brownian

Figure 8. AC of the material parameters models and benchmark model; objective minimum AC.

diffusivity. This result is consistent with the fact that the SMB system attains high productivity as the intraparticle diffusivity approaches Brownian diffusivity as its limiting value.The above analysis strongly suggests that the low- and medium-pressure systems can gain from optimizing adsorption isotherms and intraparticle diffusivity. More notably, the fact that optimizing Dp,i in high-pressure systems show only a slight improvement in productivity compared to the benchmark model leads us to conclude that the existing CSP is designed to maximize productivity in high-pressure systems. Our results suggest that designing CSP that is optimal for low- and medium-pressure systems can eliminate the gap in productivity that is currently observed between these systems and high-pressure systems. 4.3. Optimization of Material Parameters for Minimum Average Cost. Because the ultimate goal is to design an SMB unit that has minimum average cost, SWAT is run for the two material parameters models using this objective. For this analysis, it is assumed that the material parameters within the bounds given in Table 7 are aVailable at the same cost as existing materials. As will be shown, even with substantially larger costs for “new” materials, there is the potential for large cost savings through the development of these materials. Figure 8 shows the minimum average cost for the two material parameters models as well as the benchmark model. Optimizing the adsorption isotherm parameters has the most impact on cost reduction for all pressure classes. The average cost of this model shows reductions of 70% to almost 80% for low- and mediumpressure systems and 60% for high-pressure systems. The optimal values for the adsorption isotherm parameters and other results for this model will be given in detail in the next section. Optimizing the intraparticle diffusivities also results in a large reduction in costs, where low- and medium-pressure systems show reductions of 40% to mid 50% and high-pressure systems show reductions in the range of 30% to 40% compared to the benchmark model. Similar to the results of Section 4.2, the optimal value for the intraparticle diffusivity is the upper bound value. It can be concluded from Figure 8 that optimizing the material parameters for minimum average cost has the potential of resulting in a substantial reduction in cost, where the lowand medium-pressure systems show the largest reduction. For both parameters models, similar to the analysis of benchmark, the optimal system as determined by SWAT for minimum average cost is a medium-pressure system with maximum operating pressure of 2.4 MPa, the highest pressure for the class. The volume of data that resulted from running SWAT for two objectives and two parameters models is too large to fully present here. However, the results of the last two sections clearly show that, when compared to the benchmark model, optimizing the adsorption isotherm parameters results in the largest increase in productivity and the largest decrease


in average cost. Therefore, in the next section, some important results from the analysis of the isotherm parameter model will be discussed and compared with the benchmark model. 4.4. Discussion of SWAT Results: Optimization of Isotherm Parameters for Minimum Average Cost. Figure 9a shows the three optimized isotherm parameters. Similar to the results presented in Section 4.2, SWAT’s optimal value for qmax is the upper bound, and b1 is optimized at the lower bound of nearly zero for all pressure classes. All optimized b2 values are fairly small interior points: around 0.02 for low- and mediumpressure systems and 0.03 for high-pressure systems. The optimized values of both isotherm parameters are smaller than the actual b1 (0.016) and b2 (0.038) of the existing chiral sorbent reported by Xie et al.16 To achieve lower purification cost, smaller values of ∑ibiCF,i are needed to reduce the solvent cost. Note from eq 9 that solvent cost is proportional to (u10 - u40), which in turn, from eq 1, is proportional to (δ12 - δ41). According to eq 3, a large b2 (or b1) will result in a large (δ12 δ41) and higher solvent cost. SWAT’s optimized values for b1 and b2 result in values of 0.8-1.4 for ∑ibiCF,i, which are ∼50% of the 2.2 value of the existing chiral sorbent for the above product. The effect of this result is observed when comparing Figures 6e and 9f. In Figure 6e, the solvent cost for the benchmark model is seen to be ∼$20, while in Figure 9f, the solvent cost is ∼$10. The impact of optimizing the adsorption isotherm parameters on productivity and adsorption consumption is further evident in parts b-d of Figure 9. The figures show the optimal values of the variables for this model and benchmark model. In Figure 9b, it is seen that optimizing the isotherm parameters has reduced the need for fine particle size, especially in high-pressure systems. The same impact is evident in Figure 9c, where the need for large column diameters is substantially reduced, especially for lower-pressure systems. Figure 9d shows the feed velocity uF, the difference of interstitial velocities in zone 3 and zone 2. A higher feed velocity will result in a higher productivity (eq 7). The two curves are wide apart, especially for high-pressure systems. This result is consistent with the results of Figure 7, where a large increase in productivity was observed when the adsorption isotherm parameters were optimized. The normalized product concentrations CP,i/CF,i for extract and raffinate are shown in Figure 9e. Although the trend is similar to the benchmark model, both product concentrations are higher than the concentrations observed in the benchmark. Higher concentration implies less dilution, which will result in lower cost for downstream evaporation or crystallization. Optimizing the isotherm parameters has resulted in a normalized extract product concentration of ∼0.3-0.4, which represents a 2-3-fold increase when compared to the benchmark. Normalized raffinate product concentration is ∼0.9, which is almost twice of that of the benchmark. Again, it should be noted that there is very little difference between the concentrations of low-, medium-, and high-pressure systems. Figure 9f shows the breakdown of the three costs that determine the average cost of purification. This figure is similar to Figure 6e, where the curves in the figure show the three cost components as they are added to arrive at the average cost. Optimizing the isotherm parameters has resulted in an efficient CSP, where its cost represents a small fraction of average cost, particularly for high-pressure systems. It is interesting that there is very little difference in the solvent cost for different pressures. This is similar to what was observed earlier in Figure 6e for the benchmark. However, as stated earlier, optimizing the isotherm parameters can reduce the solvent cost of an SMB

Figure 9. (a) Isotherm parameters for isotherm parameters model; objective minimum AC. (b) Particle size for isotherm parameters model and benchmark model; objective minimum AC. (c) Column diameter for isotherm parameters model and benchmark model; objective minimum AC. (d) Feed velocity for isotherm parameters model and benchmark model; objective minimum AC. (e) Normalized extract and raffinate product concentrations for isotherm parameters model; objective minimum AC. (f) AC breakdown for isotherm parameters model; objective minimum AC.

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unit by ∼60% for all pressures. Combining this information with the fact that equipment cost represents a large percentage of the average cost in high-pressure systems, we can conclude that optimally designed medium-pressure systems are likely to be the most economical SMB systems.

analysis of variations in material parameters suggests that substantial efficiency gains may be possible through optimizing material parameters. Designing CSP that is optimal for low- or medium-pressure systems can assist in eliminating the gap in productivity that is currently observed between these systems and high-pressure systems.

5. Summary and Conclusions This paper introduces a flexible and computationally efficient optimization tool for nonlinear SMB systems with significant mass-transfer effects. The tool, standing wave annealing technique (SWAT), is based on the SWD equations and employs a stochastic optimization algorithm, simulated annealing. SWAT can simultaneously optimize all variables that control the cost of the SMB design, construction, and operation. At the core of SWAT are the SWD equations, which are algebraic relationships that offer significant modeling advantage in optimizing SMB systems. Specifically, by solving these equations, the desired purity and yield for nonideal systems is guaranteed, and the need for numerical solutions of ordinary or partial differential equations is eliminated. In this paper, the flexibility and computational efficiency of SWAT is illustrated through an example, the resolution of racemic mixtures of FTC-esters. Several interrelated issues regarding the design of an economically efficient SMB system are addressed. SWAT identifies the important role that the maximum operating pressure plays in determining the economical design of a SMB system. In Section 3, 16 variables were simultaneously determined for either maximum productivity or minimum purification cost. Among our key findings are the following: (1) The total cost of building a high-pressure SMB system that maximizes productivity can be nearly 60% greater than those of a cost-minimized medium-pressure SMB. (2) Once the SMB system is built optimally given its maximum operating pressure, there is very little difference in the solvent cost between low-, medium-, and high-pressure systems. (3) The most economical SMB system is a medium-pressure unit, using 34 µm particle size and 10 cm column length, with eight columns and a configuration of 2, 2, 2, 2. The optimal operating conditions for this system are determined from the SWD. In Section 4, SWAT was modified to include a number of material parameters in the set of decision variables. Two separate models were considered: the first explores variations in adsorption isotherm parameters, and the second explores variations in intraparticle diffusivity. Among the key findings are the following: (1) For both parameters models, the optimal system as determined by SWAT for minimum average costs is a medium-pressure system. (2) For optimal separation, the maximal binding capacity qmax should be as high as possible, and the b for the low-affinity solute should be as low as possible. The b for the high-affinity solute should be an interior value, where the product of ∑ibiCF,i is small, in order to maximize productivity while keeping the solvent cost low. In the separation of racemic mixtures of FTC-esters, the value of ∑ibiCF,i for minimum purification cost is 0.8-1.2. (3) Optimizing the intrinsic adsorption parameters has the potential of increasing productivity by 7-10-fold, reducing the solvent cost by 60%, and reducing the average purification cost by 60-70%. The above findings illustrate why it is essential to consider a large set of decision variables when designing SMB systems and that SWAT is a valuable tool in such an analysis. Combining the above findings, we can conclude that medium-pressure systems are likely to be the most economical SMB systems in the future and that serious attention regarding research and development of these systems is warranted. In addition, SWAT’s

Acknowledgment Support from Seaver Research Council Grant at Pepperdine University, Indiana 21st Century Research and Technology Fund, and Chiral Technologies are gratefully acknowledged. The authors are thankful to Mr. Geoffrey B. Cox for insightful discussions and comments, and to valuable suggestions by two anonymous reviewers. Appendix A: Model of Nonideal Nonlinear SMB System for Binary Separation Input Parameters

( )

(

)

1 - b DV , ts,min, ∆pmax, b P ) , Lc S b p, F, FB, µ, and ai, bi, qmax, Dp,i, D∞,i, Ke,i, Yi for i ) 1, 2

PurEreq, PurRreq, DV'

Decision Variables

Tier0 variables: CF,1, CF,2 Tier1 variables: S, dp, Lc, Ncol Tier2 variables: Njcol, j ) 1, 2, 3 Tier3 variables: V, uj0, j ) 1, 2, 3, 4, Cs,i, i ) 1, 2 Derived Variables

λF ) 4a1a2b1b2CF,1CF,2 χF ) -a1(1 + b2CF,2) + a2(1 + b1CF,1)

γ+,F )

χF + xχF2 + λF 2a2b1CF,2

Cp,2 )

a2 - a1 a1b2 + a2b1γ+,F

λ ) 4a1a2b1b2Cs,1Cs,2 χ ) -a1(1 + b2Cs,2) + a2(1 + b1Cs,1) δ12 ) p + (1 - p)a2 +

γ- )

DV' 1 - b

χ - xχ2 + λ 2a2b1Cs,2


Cp,1 )

a1 - a 2 a1b2 a2b1 + γ-

(dp × 10-4)2 dp × 10-4 1 ) + K32 60Ke2pDp2 6k3f,2

(

(dp × 10-4)2 dp × 10-4 1 ) + K41 60Ke1pDp1 6k4f,1

)

a1 DV' δ21 ) p + (1 - p) + 1 + b2Cp,2 1 - b δ32 ) p + (1 - p)

(

a2

1 + b1Cs,1 + b2Cs,2

(

)

)

+

DV' 1 - b

Ratio of Highest Concentration to the Lowest Concentration of the Standing Wave in Each Zone

a1 DV' δ41 ) p + (1 - p) + 1 + b1Cp,1 1 - b Axial Dispersion Coefficients

β21 ) ln

u10b(dP × 10-4)

(

E1b,2 )

)

Fu10bdP × 10-4 0.2 + 0.011 60µ u20b(dP × 10-4)

E2b,1 )

0.2 + 0.011

(

β12

)

Fu20bdP × 10-4 60µ

[

) ln

]

u20(u30 - u20)CF,1(1 - Y1) ) ln

[ [

(u30 - u40)Cs,2

] ]

(u30 - u20)CF,2(1 - Y2)

β41 ) ln

0.48

]

u40(u10 - u20)(1 - Y2)

(u10 - u20)(u30Cs,1 - (u30 - u20)CF,1)

0.48

β32

[

u10(u30 - u40)Y2

u40(u10 - u20)Y1

u10(u30 - u40)(1 - Y1)

Maximum Feed Interstitial Velocity

u30b(dP × 10-4)

E3b,1 )

0.2 + 0.011

(

Fu30bdP

× 10 60µ

)

-4 0.48

u40b(dP × 10-4)

(

E4b,2 )

)

Fu40bdP × 10-4 0.2 + 0.011 60µ

4

0.48

)

k2f,1

)

1.09(u10)1/3

( ( ( (

1.09(u20)1/3

) ) ) )

bdp × 10-4 D∞,2

-2/3

bdp × 10-4 D∞,1

-2/3

Pβ32(δ32)2 k32N3colLc

+

( )

Pβ21(δ21)2 k21N2colLc

-

β32E3b2

N3colLc

Subject to Constraints 1. Bounds on Variables:

Smin e S e Smax Lc,min e Lc e Lc.max Njcol g 2, j ) 1, 2, 3 Ncol,min e Ncol e Ncol,max

-4 -2/3

k3f,2 ) 1.09(u30)1/3

bdp × 10 D∞,2

k4f,1 ) 1.09(u40)1/3

bdp × 10-4 D∞,1

-2/3

Overall Mass-Transfer Coefficients

(dp × 10-4)2 dp × 10-4 1 + ) K1 60Ke2pDp2 6k1 2

(

P2(δ32 - δ21)2

dp,min e dp e dp,max

Film Mass-Transfer Coefficients

k1f,2

uF,max )

CF,min e CF,1 + CF,2 e CF,max ujmin e uj0 e ujmax , j ) 1, 2, 3, 4 Ve

Lc,min ts,min

2. Linear System Constraints:

Cs,i e CF,i, i ) 1, 2

f,2

(dp × 10-4)2 dp × 10-4 1 ) + K21 60Ke1pDp1 6k2f,1

CF,1 ) CF,2 u30 < u10

+

β21E2b1 N2colLc

)

8710


u20 < u30

∆p1 )

u40 < u10 u20 < u10 u40 < u30

∆p2 )

3. Nonlinear System Constraints:

u30 - u20 e uF,max (a2b1γ+,F2 + a1b2γ+,F)Cs,2 - (a2b1γ+,F + a1b2)Cs,1 + (a1 - a2)γ+,F ) 0

∆p3 )

u30Cs,1 - (u30 - u20)CF,1 > 0 u30 - u20 u30 - u40 PurE )

PurR )

CF,1Y1 ) Cp,1

CF,2Y2 CF,2Y2 + CF,1(1 - Y1) CF,1Y1 CF,1Y1 + CF,2(1 - Y2)

4. SW Equations:

ν-

ν-

u10 1 + Pδ12 u20 1 + Pδ21

ν-

ν-

)-

)-

u30 1 + Pδ32 u40 1+

Pδ41

)

)

β12

β21 Pδ21)N2colLc β32

(1 + Pδ32)N3colLc

(1 +

β41 Pδ41)N4colLc

g PurEreq

( (

( (

(

p

(

p

(

p

( )

+

( )

150µu20N2colLc 1 - b 2 106 + b 6 d2

E4b1 +

)

1.75F(u20)2N2colLc 1 - b 1 14.7 dp b 3.6 1.013 × 105

( )

150µu30N3colLc 1 - b 2 106 + b 6 d2

)

1.75F(u30)2N3colLc 1 - b 1 14.7 dp b 3.6 1.013 × 105

( )

150µu40N4colLc 1 - b 2 106 + b 6 d2

)

1.75F(u40)2N4colLc 1 - b 1 14.7 dp b 3.6 1.013 × 105

Objectives for Maximization

) )

Pν2(δ12)2 K12 2

+

)

1.75F(u10)2N1colLc 1 - b 1 14.7 dp b 3.6 1.013 × 105

Productivity (PR), PR )

E2b1 +

E3b2

p

Appendix B: Objective Functions

g PurRreq

E1b2

(1 + Pδ12)N1colLc

(1 +

∆p4 )

(

150µu10N1colLc 1 - b 2 106 + b 6 d2

Pν (δ21)2 K21

) )

Pν2(δ32)2 K32 2

Pν (δ41)2 K41

Q FBV(365)

Objectives for Minimization

(1) PC ) FC + VC where

annual fixed cost ) annual equipment cost + annual CSP cost AnnualEqCost )

CCSPSLcFBNcol CE , AnnualCSPCost ) rE rCSP

annual variable cost ) annual solvent cost + annual feed cost AnnualSolCost ) Sbt[(u10 - u40) + (u30 - u20)][Csol(1 - SRcy) + CRcy sol SRcy]

Lc - Vts,min g 0 AnnualFeedCost ) Cfeed

5. Demand Constraint:

Qmin e Q e Qmax where

PC , or AC ) EqC + CSPC + SolC ($/kg) Q

Nomenclature

Q)

Sb(u30

-

u20)CF,pYpt,

t ) 525 600 min

6. Pressure Drop Constraint:

∑j ∆pj ) ∆pmax, j ) 1, 2, 3, 4 where

(2) AC )

Q Yp

ai ) Langmuir isotherm parameter of component i based on solid volume, L/L S. V. bi ) Langmuir isotherm parameter of component i, L/g CSPC) average CSP cost, $/kg CCSP ) CSP price, $/g CE ) acquisition price of SMB equipment, $ Cfeed ) feed price, $/g CF,i ) feed concentration of component i, g/L


Cp,i ) plateau concentration of component i, g/L Cs,i ) diluted plateau concentration of component i at the feed port, g/L Csol ) solvent price, $/L CRcy sol ) solvent recovery price, $/L dc ) column diameter, cm dP ) particle diameter, µm DP,i ) intraparticle diffusivity of component i, cm2/min D∞,i ) Brownian diffusivity of component i, cm2/min DV ) extra-column dead volume, cm3 EqC ) average equipment cost, $/kg Eb,ji ) axial dispersion coefficient of component i in zone j, cm2/min FeedC ) feed cost, $/kg kf,ji ) film mass-transfer coefficient of component i in zone j, cm/min Ke,i ) size exclusion factor for component i Kij ) lumped mass-transfer coefficient of component i in zone j, min-1 Lc ) single column length, cm Ncol ) total number of columns in SMB j Ncol )number of columns in zone j P ) (1 - b)/b ) phase ratio PC )purification cost, $ PR ) productivity, (kg of product)/(kg of CSP)/day PurEreq ) purity required for extract PurRreq ) purity required for raffinate qmax ) saturation capacity of the stationary phase, g/L S. V. Q ) annual production, kg rE ) depreciation period of equipment, year rCSP ) depreciation period of CSP, year S ) column cross-sectional area, cm2 SolC ) average solvent cost, $/kg SRcy ) solvent recycle ratio, % TBV ) throughput per bed volume per day, (kg of product)/ (L of bed volume)/(day) t ) duration of production, min ts,min ) minimum step time, min uF,max ) maximum feed interstitial velocity, cm/min j u0 ) interstitial velocity in zone j, cm/min V ) average port moving velocity, cm/min Yi ) yield of component i Greek Letters βji ) decay factor of standing component i in zone j δi ) retention factor of component i ∆pj ) pressure drop for zone j, psi ∆pmax ) maximum pressure drop, psi b ) interparticle void fraction p ) intraparticle void fraction (or particle porosity) F ) mobile phase density, g/mL FB ) bulk packing density, kg/m3 µ ) viscosity, g/cm/s Subscripts and Superscripts i ) component index j ) zone number index Literature Cited (1) Broughton, D. B. Production-Scale Adsorptive Separations of Liquid Mixtures by Simulated Moving-Bed Technology. Sep. Sci. Prog. 1961, 66, 70.

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ReceiVed for reView March 12, 2006 ReVised manuscript receiVed August 14, 2006 Accepted September 5, 2006 IE060300A

Standing Wave Annealing Technique - American Chemical Society

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