Optimal Design of a Size-Exclusion Tandem ... - ACS Publications

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Ind. Eng. Chem. Res. 2003, 42, 1977-1993

1977

Optimal Design of a Size-Exclusion Tandem Simulated Moving Bed for Insulin Purification Sungyong Mun,† Yi Xie,† Jin-Hyun Kim,‡ and Nien-Hwa Linda Wang*,† School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907, and Department of Chemical Engineering, KongJu National University, KongJu City, ChungNam, South Korea

A tandem simulated moving bed (SMB) was developed previously for the purification of insulin from two impurities in two sequential steps. In this study, an efficient optimization tool based on the standing wave design is developed to find the optimal tandem SMB for insulin purification. Both system parameters (total number of columns, zone configuration, column diameter, and column length for each ring) and operating parameters (zone flow rates and switching time for each ring) are optimized to achieve the lowest purification cost. In the production range of interest, equipment cost dominates. For this reason, the optimal design has eight or fewer columns in each ring. The optimal design also has a small column length and a large column diameter, as the stationary phase for insulin purification is compressible and has a zone linear velocity limit. Splitting strategies and constraints on the column diameter and column length have significant effects on the optimal design. If there is a limit on the column diameter, optimization of the splitting strategy and column length can result in 24 and 25% savings, respectively, in total purification costs. Finally, if the limit on linear velocity is removed, a lower purification cost can be achieved by using longer columns with smaller diameters. The method developed in this study can be applied to other size-exclusion systems or linear isotherm systems. 1. Introduction Simulated moving beds (SMBs) represent an efficient separation technique. They were first introduced for hydrocarbon purification.1 Recently, many SMB processes have been developed for pharmaceutical applications,2 including a tandem SMB process (two SMB units in series) for the separation of insulin from two impurities, high-molecular-weight proteins (HMWPs) and ZnCl2.3 Figure 1 shows a schematic diagram of the tandem SMB for insulin purification.3 HMWPs are removed from the raffinate port in the first ring (ring A). The effluent from the extract port of ring A is loaded into the second ring (ring B), where insulin is separated from ZnCl2. In the previous study, the operating parameters (zone flow rates and switching time) were determined from the standing wave design,4,5 whereas system parameters (total number of columns, zone configuration, column length, column diameter, and particle size) were kept the same in all experiments. However, process cost, throughput, and solvent consumption are complex functions of both the operating and system parameters that should be optimized according to a given objective. Previous studies in the literature have focused on finding the optimal operating parameters for a fixed feed flow rate in a given SMB system.4,6-10 Optimization of five operating parameters, however, does not guarantee the minimum process cost. In this study, both the system and operating parameters for the tandem SMB for insulin purification are optimized to reduce purification costs. * To whom correspondence should be addressed. Tel.: +1765-494-4081. Fax: +1-765-494-0805. E-mail: wangn@ ecn.purdue.edu. † Purdue University. ‡ KongJu National University.

Figure 1. Schematic diagram of the tandem four-zone SMB for insulin purification.

Optimization of several of the system parameters for a single four-zone SMB has been addressed in several recent publications. Charton and Nicoud11 studied the effects of column length and particle size on the optimal design of an SMB, but they determined the optimal flow rates entirely on the basis of equilibrium theory (no mass-transfer effects were considered). Biressi et al.12

10.1021/ie020680+ CCC: $25.00 © 2003 American Chemical Society Published on Web 04/05/2003

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and Jupke et al.13 improved this simplified approach further by employing simulations in the optimization. They used predictions from the equilibrium theory as an initial guess for the operating parameters. An equilibrium stage model (Biressi et al.12) or a lumped kinetic model (Jupke et al.13) was used to consider masstransfer effects in searching of the optimal operating parameters. Some trial-and-error searching is required to find the optimal operating parameters in this approach. The procedure was repeated for different column lengths. For each case, simulations based on the model were used to confirm product purity. The objective function was evaluated for each case to find the optimal column length. Biressi et al.12 fixed the flow rates in zones I and IV in their optimization procedures. By contrast, Jupke et al.13 varied all of the flow rates and reduced the number of variables by using several dimensionless parameters for optimization. Du¨nnebier et al.14 presented a modelbased optimization strategy that was used to find the optimal operating parameters and feed concentration in their examples. Optimization of the total number of columns and zone configuration was not attempted in the previous studies. Azevedo and Rodrigues15 considered the effects of zone configuration in their design method “separation volume analysis”. Pynnonen16 pointed out that zone configuration is also an important issue in the design economics of SMB separation. More importantly, only a single SMB for binary separation has been optimized. No previous studies have addressed the optimization of a tandem SMB for multicomponent separation. In this work, the standing wave design method4,5 is used as a basis for SMB optimization. The standing wave design (SWD) equations calculate the linear velocity in each zone and the average port velocity (or switching time) that can guarantee desired purity and yield for a given set of feed flow rate and zone length data. Unlike the triangle theory,17,18 which is based on an ideal system (no mass-transfer effects), the SWD considers the mass-transfer effects in the estimation of the operating parameters (zone flow rates and switching time). No iteration or trial-and-error is required in the SWD. This allows a quick determination of the SMB operating conditions for a desired purity and yield. Several experimental studies19-21 have shown that the SWD is efficient and guarantees high purity and high yield for binary separations. Recently, Hritzko et al.5 extended the SWD for multicomponent fractionation. They also proposed optimal splitting strategies to reduce solvent consumption for the recovery of a product with intermediate affinity using a tandem SMB. The goals of this paper are (1) to optimize the system and operating parameters of the tandem SMB proposed for insulin purification3 to achieve the lowest purification cost, (2) to understand how different splitting strategies and constraints on system and operating parameters affect the purification cost, (3) to investigate the difference between optimization of a single SMB and that of a tandem SMB, and (4) to develop a general method for the optimization of a single SMB and a tandem SMB for size-exclusion or other linear isotherm systems. In this study, the SWD equations for multicomponent separation5 are used to find the zone flow rates and switching time for each ring. The number of SMB units, total number of columns, zone configuration, column

length, and column diameter for each ring are optimized to achieve the lowest purification cost. Different splitting strategies, such as whether one impurity is distributed in the first ring, are also compared. One of the important results in this work is that independent optimization of the individual SMB units in a tandem SMB does not always lead to optimization of the tandem SMB as a whole. Product dilution in the first ring was found to play an important role in the overall optimization. Low dilution results in a smaller column diameter, fewer SMB units in the second ring, and therefore a lower purification cost. The rules for decreasing product dilution were derived from the SWD equations. The splitting strategies and constraints on the column diameter and column length have significant effects on the optimal design. Finally, the splitting strategies to reduce product dilution for multicomponent separation proposed by Hritzko et al.5 were found to give the lowest purification cost in this system. 2. Theory 2.1. Systematic Method of SMB Design and Optimization. A systematic method of SMB design and optimization for multicomponent separation is proposed (Figure 2). This method significantly reduces trial-anderror in the design and number of experiments for process development. In this method, a series of batch chromatography tests is first applied to obtain the intrinsic parameters that are independent of the scale or the operating conditions of the system. These parameters include bed voidage, particle porosity, isotherms, and mass-transfer parameters. These intrinsic parameters are used in the standing wave design to determine the operating parameters, which include the maximum feed flow rate, zone flow rates, and switching time. In addition to the operating parameters, system parameters can also be optimized. The mobile phase and stationary phase determine the selectivity for the separation. A high selectivity is preferred because it results in less product dilution and higher productivity of the stationary phase.5 Additionally, solubility should be considered in the selection of the mobile phase because higher solubility leads to lower solvent consumption and high productivity of the stationary phase. Particle size also has an important effect on pressure drop and stationary-phase productivity. For these reasons, the selection of mobile and stationary phases is important for optimization. For ternary separation in a tandem SMB, two separation sequences can be applied to recover a component with an intermediate affinity.5 For insulin purification, insulin can be separated either from HMWPs or from ZnCl2 in the first ring. Because HMWPs are unstable, they should be removed in the first ring. For this separation sequence, there are two splitting strategies. The first strategy is to separate HMWPs from both insulin and ZnCl2. The second strategy is to separate HMWPs from insulin and allow ZnCl2 to be distributed between the extract and the raffinate. For each splitting strategy, the optimal column length, number of columns, and zone configuration are found from the standing wave design results. In general, the product yield can also be one of the optimization variables. The reason is that the feed flow rate can be increased at the expense of the yield. In this study, the yield is kept at the highest possible value of 99.99%, because insulin is a high-value

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Figure 2. SMB design, cost analysis, and optimization based on the standing wave design. Table 1. Intrinsic Parameters Used in the SWD Equations for Optimization in This Work HMWP Ke D∞ (cm2/min) Dp (cm2/min) kf (cm/min) Eb (cm2/min) column properties

insulin

Table 2. Parameter Values Used to Evaluate Total Purification Cost ZnCl2

0.19 0.74 0.99 5.49 × 10-5 3.96 × 10-4 4.80 × 10-5 2.29 × 10-5 1.65 × 10-4 2.00 × 10-5 Wilson and Geankoplis correlation22 Chung and Wen correlation23 a b ) 0.35, p ) 0.89

a E value of insulin in zone III of ring A was chosen to be 40 b times as large as that estimated from the Chung and Wen correlation.23

product. Any gain from increased throughput cannot offset the loss resulting from a lower yield. In this study, the following 20 parameters are optimized for insulin purification: total number of columns, zone configuration, column length, column diameter, feed flow rate, zone flow rates, and switching time for each ring. Particle size, mobile phase, and stationary phase are fixed in this study. The objective function in this study is the total cost required to purify 1 kg of insulin. Total purification cost consists of resin, solvent, and equipment costs. Resin and solvent costs are related to productivity (or throughput per bed volume) and solvent consumption, respectively. Equipment cost per kilogram of product is related to the production rate and depreciation time. Input parameters needed for the optimization are divided into two groups. The first group consists of intrinsic parameters, which include column properties (b, p), size-exclusion factors, and mass-transfer parameters. The intrinsic parameter values used in this work were reported by Xie et al.3 and are listed in Table 1. Note that, in the SWD equations, the Eb value of insulin in zone III of ring A is chosen to be 40 times the value estimated from the Chung and Wen correlation.23 This is to overcome the dispersion of insulin as a result of nonuniform flow in zone III of ring A.3 The second group of parameters for optimization is a set of cost parameters, which are used to evaluate total purification cost. Table 2 lists the cost parameter values

a

parameter

value

resin price ($/L) resin lifetime (year) solvent price ($/L) solvent recycle SMB price ($/unit)a depreciation time (month) maximum equipment production rate (kg/year) equipment utilization factor down time (%)

75.0 2 0.1 0 400 000 + 10 000Ncol 70 10 000 0.6 4

Ncol is the total number of columns per SMB unit.

used in this work. The cost of the stationary phase was provided by Pharmacia. The cost of the mobile phase (1 N acetic acid) was provided by experienced engineers in pharmaceutical companies. Because SMBs have never been used in industry for insulin production, the equipment cost function is based on estimates provided by two SMB equipment vendors, U.S. Filter and Applexion. The equipment cost is controlled by the requirements of high purity, high yield, reliability, and good manufacturing practice for pharmaceutical production. The cost is a weak function of the number of columns in the system and is assumed to be unaffected by specific equipment configurations. Notice in Table 2 that a portion of the SMB equipment price increases linearly with increasing number of columns. This is because, as the number of columns increases, the associated costs of columns, valves, and controllers also increase. The following constraints are considered: First, product purity should be higher than 99.9% in both ring A and ring B. Second, the amount of output from ring A should be equal to or less than the maximum allowable input of ring B. The optimal tandem SMB should also satisfy a production rate requirement. In addition, some other issues are worthy of consideration for large-scale production. First, the stationary phase used in the previous study, and also employed here, is compressible

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Figure 3. Summary of the 10 cases proposed for the optimization of the tandem SMB for insulin purification. Table 3. All Possible Combinations of Splitting Strategies and Constraints on the Column Length and Column Diameter case

distribution in ring A

fixed column length

limit on column diameter

I-a I-b I-d I-e II-a II-b II-d II-e

no no no no yes yes yes yes

yes yes no no yes yes no no

no yes no yes no yes no yes

and has a maximum allowable pressure drop per unit length. Thus, the zone linear velocity should not exceed a certain limit for a given column length. Second, successful packing and a uniform flow distribution cannot be achieved if the column diameter exceeds a certain limit for a given column length. Third, the use of identical columns in rings A and B is preferred for the convenience of control and maintenance. Effects of these three constraints on the optimal design are also considered. In addition to the effects of these constraints, the effects of the splitting strategy on the total cost are studied. Distribution of ZnCl2 in ring A is allowed in case II, but not in case I (Figure 3). As listed in Table 3, eight combinations are possible. Additionally, the effect of having identical columns in rings A and B is discussed. Overall, 10 cases are studied (Figure 3). The column length is fixed at 15 cm in cases a, b, and c, but it is included as one of the optimization variables in cases d and e. No constraint is placed on the column diameter in cases a and d. However, in cases b, c, and e, column diameter is constrained to be 45 cm or less. Furthermore, identical columns are used in case c. 2.2. Optimization Algorithm for a Single SMB. Figure 4 shows the flow sheet of the algorithm used to search for optimal parameters for a four-zone single SMB. The algorithm consists of one process box (SWD) and three loops (innermost, inner, and outer loops). The process box finds the zone velocities and the switching time that satisfy the standing wave equations. The SWD for ternary separation in a tandem SMB was developed

previously.5 For a fixed feed linear velocity, zone configuration, number of columns, and column length in a given SMB unit, the SWD determines the four zone interstitial velocities and the switching time by solving the following five equations for systems with linear isotherms

uI0 ) (1 + Pδk)ν + ∆Ik

(1)

II uII 0 ) (1 + Pδ1)ν + ∆1

(2)

III uIII 0 ) (1 + Pδ2)ν - ∆2

(3)

IV uIV 0 ) (1 + Pδ1)ν - ∆1

(4)

II uF ) uIII 0 - u0

(5)

where the mass-transfer correction terms of component i in zone j are defined as

∆ji

)

βji

[

NjcolLc

j Eb,i

+

]

P(δiν)2 j Kf,i

(6)

The subscripts 1, 2, and k stand for the three components in the tandem SMB; the superscripts I, II, III, and IV stand for the four zones; ν is the average port velocity () column length/switching time); u0 is the interstitial velocity; uF is the feed linear velocity; P is the phase ratio, (1 - b)/b, where b is the interparticle voidage; δ is defined as Kep + (1 - Kep)a, where p is the particle porosity and a is the partition coefficient; Lc is the single-column length; Eb is the axial dispersion coefficient; and Kf is a lumped mass-transfer parameter as defined by Ma et al.4 β is the logarithm of the ratio of the highest concentration to the lowest concentration of a standing wave in a particular zone. β is an index of product purity and yield; the larger the β value, the higher the product purity and yield.4,5,21 To consider any extra-column dead-volume effects, the definition of δ in this work is modified as described in a previous study.3 In ring A, the subscript k is 2 if distribution of the highest-affinity component is allowed and 3 if not.3,5 In

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ring B, the subscript k is always 2 because ring B corresponds to an SMB for binary separation. In the innermost loop, the feed linear velocity is varied while zone configuration, number of columns, and column length are fixed. For each feed linear velocity, the SWD determines a set of operating parameters. The feed linear velocity is limited by one of the following two constraints: (1) maximum pressure drop or linear velocity and (2) mass-transfer rate. The specific sizeexclusion gel used in our previous studies, and consequently used here also, is compressible and has a maximum allowable flow rate of 9.5 mL/min for a column that is 5.1 cm in diameter and 15 cm in length.3 The bed void fraction is 0.35. The maximum linear velocity for the 15-cm column is 1.3286 cm/min, which is obtained by dividing the maximum flow rate by the available cross-sectional area for flow. The limit on the zone linear velocity for an arbitrary column length is

maximum allowable zone linear velocity ) 15 cm 1.3286 cm/min × (7a) Lc (cm) If mass transfer is limiting, the maximum allowable feed linear velocity (uF,max ) is determined from the SWD equations as follows5

uF,max ) 4

(

P(δ2 - δ1)2 2 βIII 2 δ2

III KIII f,2 LcNcol

+

2 βII 1 δ1 II KII f,1LcNcol

(

)

-

III βIII 2 Eb,2

LcNIII col

+

)

II βII 1 Eb,1

LcNII col

(7b)

The optimal feed linear velocity, subject to the two constraints (eqs 7a and 7b), is found as the total cost reaches its minimum. To maintain a high accuracy and reduce the computation time for the search, a variable step size is used in the search for the optimal feed linear velocity. The step size is set at 0.002 cm/min initially. Total cost is evaluated at each step. A preliminary optimal region where the minimum cost occurs is found first. In the resulting region, the optimal feed linear velocity is sought with a smaller step size, 0.0005 cm/ min, followed by the smallest step size, 0.000 05 cm/ min. The entire procedure is repeated three times using three different starting values such as 0.003, 0.01, and 0.02 cm/min. The optimal feed linear velocity is finally determined by comparing the resulting costs from the three independent searches. The aforementioned step sizes are based on a column length of 15 cm. If the column length is greater than 15 cm, the search region for feed linear velocity is reduced as a result of a lower limit on the zone linear velocity. In this case, the step size is reduced proportionally to maintain high accuracy. The inner loop in the algorithm is a variation of the zone configuration and number of columns. In this study, the number of columns is varied from 6 to 20, and all zone configurations are searched. The number of possible configurations (assuming at least one column per zone) in a single four-zone SMB is given by

number of possible configurations ) 20

(Ncol - 1)!

∑ N )6(N col

col

- 4)!3!

(8)

Figure 4. Flow sheet of a proposed algorithm for the optimization j represents the of a four-zone single SMB for minimum cost. u0,max maximum allowable zone interstitial velocities due to the mechanical strength of the stationary phase. uF,max is the maximum feed linear velocity, which is limited by mass-transfer effects. No constraints are placed on the column diameter in this example. Rectangle, calculation; parallelogram, comparison; rounded rectangle, start/end.

The computations above are repeated for various column lengths in the outer loop. In this study, the column length is varied from 5 to 100 cm. The search for the optimal column length is performed using variable step sizes, as in the search for the optimal feed linear velocity. First, a crude search is performed with a step size of 9.5 cm over the entire range, followed by searches with smaller step sizes, 0.95 and 0.12 cm, in the reduced regions. 2.3. Cost Analysis Functions of Single and Tandem SMBs. In this work, the total purification cost (TC) consists of the resin cost (RC), the solvent cost (SC), and the equipment cost (EC). Costs for labor, utilities, facilities, waste disposal, and other factors are excluded in the evaluation of total purification cost. Below is a definition for each of the costs considered.

RC ($/kg of product) ) resin price (9a) resin lifetime × productivity SC ($/kg of product) ) solvent usage × solvent price × (1 - solvent recycle) (9b) EC ($/kg of product) ) [equipment price]/[depreciation time × maximum production rate × utilization factor] (9c) TC ) RC + SC + EC (10) where solvent recycle in eq 9b is the fraction of solvent that can be reused after the removal of impurities. Each piece of SMB equipment is usually designed to meet a

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certain range of production rates. The ratio of the desired production rate to the maximum production rate is defined as the equipment utilization factor. Therefore, multiplication of the maximum production rate by the utilization factor in eq 9c gives the desired production rate. The aforementioned cost function is specific for insulin purification. It should be modified for non-pharmaceutical products. For example, for a commodity biochemical such as a sugar product, the cost of evaporation following the SMB step is a major cost. In this case, product dilution carries a major penalty. Therefore, the cost function needs to be modified to include the evaporation cost. In contrast, for insulin purification, the output from the SMB process can be directly used in the subsequent crystallization process. No evaporation is required prior to crystallization. For this reason, no cost of evaporation is considered in the cost function in this study. For less valuable products such as non-pharmaceuticals, unnecessarily high yields might result in a high solvent cost, high product dilution, and consequently a high evaporation cost. For such products, yield should be included as an optimization variable. The total cost function should be modified to take into account the cost of lost product for non-pharmaceutical products as follows

TC ) RC + SC + EC + FC FC ($/kg product) )

(11)

feed price product yield

FfeedNSMBCF,PYP uFbCF,PYP ) (13) BVFs LcNcolFs

solvent usage

) (kgL ofof solvent product)

NSMB(Ffeed + FD) uF + uD ) (14) NSMBFfeedCF,PYP uFCFPYP

equipment price ($) ) (fEP,f + fEP,NNcol)NSMB NSMB ) int

[

4 × PR uFπbCF,PYP(1 - tdown)dc,limit2

]

(15a) (15b)

where Ffeed is the feed flow rate; BV is the bed volume; Fs is the resin packing density; NSMB is the number of SMB units; CF,P and YP are the feed concentration and product yield, respectively; FD is the desorbent flow rate; uD is the desorbent linear velocity; fEP,f is the fixed part of SMB equipment price; fEP,N relates the number of columns to the SMB equipment price; PR is the production rate requirement; dc,limit is the maximum allowable

(BVA + BVB)Fs

[

LAc

solvent usage )

)

uAF bCF,PYP NAcol

+

LBc

NBcol

() ] dBc 2NBSMB

dAc NASMB

NASMB(FAfeed + FAD) + NBSMB FBD NASMB FAfeedCF,PYP uAF + uAD +

(16)

Fs

)

()

dBc 2NBSMB B uD dAc NASMB

uAF CF,PYP

(17)

equipment price ) (fEP,f + fEP,NNAcol)NASMB + (fEP,f + fEP,NNBcol)NBSMB (18a) NASMB ) int

product ) (kgkgof ofresin × year)

FAfeed NASMBCF,PYP

productivity )

(12)

where FC is the feed cost. In this study, however, the product yield is kept at the highest possible value of 99.99% because insulin is a highly valuable product. For this case, the feed cost remains unchanged and has no effect on the optimization results. For this reason, the feed cost term is excluded in the evaluation of the total purification cost below. Evaluation of the total purification cost requires information on productivity, solvent usage, and equipment price. For a single SMB, these functions are given by

productivity

column diameter; and tdown is the fraction of down time for the maintenance of SMB equipment. The operator “int” is used to round the value inside brackets up to the next highest integer. For a tandem SMB, the functions of productivity, solvent usage, and equipment price are expressed as follows

NBSMB

(

[

4 × PR 2 - tdown)dc,limit

uAF πbCF,PYP(1 A

]

1 uext 4 × PR ) int B A uF uF πbCF,PYP(1 - tdown)dc,limit2

(18b)

)

(18c)

where the superscripts A and B stand for ring A and ring B, respectively; dc is the inner column diameter; and uext is the extract linear velocity. The number of SMB units (NSMB) is an integer, which can be more than unity if the column diameter required to meet the desired production rate exceeds the limit. Notice in eqs 13 and 14 that both the productivity and the solvent usage for a single SMB are determined by the linear velocities of the feed and eluent. They are independent of the column diameter. Therefore, the resin cost (RC) and the solvent cost (SC) for a single SMB are independent of the column diameter. However, the productivity (eq 16) and the solvent usage (eq 17) for a tandem SMB are affected by the diameter ratio (dBc /dAc ). For this reason, the cost function of a tandem SM contains the diameter raito. This means that the diameter ratio is also one of the variables to be optimized in a tandem SMB. 2.4. Optimization of a Tandem SMB. Two different optimization methods are studied here. In the first method, the system and operating parameters of ring A and ring B are optimized individually for the lowest total cost for each ring. Then, the individually optimized ring A is connected to the individually optimized ring B. Therefore, the optimization method for a single SMB is applied twice (once for ring A and once for ring B). The optimal column diameter of the ring A SMB (dA* c ) is obtained from the production rate requirement (PR) and the optimal feed linear velocity for the ring A SMB (uA* F ) as follows

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dA* c (cm) )

x

4 × PR

A uA* F πbCF,PYPNSMB(1

(19)

- tdown)

The optimal column diameter of the ring B SMB (dB* c ) is determined by matching the output of the individually optimized ring A SMB with the input of the individually optimized ring B SMB A* dB* c (cm) ) dc

x

A uA* ext NSMB

(20)

B uB* F NSMB

A* A Because dA* c , uext, and NSMB are determined from the individual optimization of ring A SMB, they are independent of dB* c in eq 20. In the second method, the overall optimization method, the system and operating parameters of rings A and B are optimized simultaneously for the lowest total cost of the tandem SMB. Any interaction between rings A and B can be taken into account in the overall optimization method. The optimal uAF value depends not only on ring A but also on ring B. In addition, the optimal uBF value should satisfy eq 20. Once the optimal values of uAF and uBF are determined simultaneously, the optimal values of dAc and dBc can be calculated from eqs 19 and 20, respectively. 2.5. Analysis of RC, SC, and EC in Single and Tandem SMBs. In the overall optimization method, the total cost (TC) of the tandem SMB can be expressed as a function of the total costs of rings A and B as follows

(TC of a tandem SMB) ) f1[(TC of ring A),(TC of ring B)] (21a) where

(TC of ring A) ) (RC of ring A) + (SC of ring A) + (EC of ring A) (21b) (TC of ring B) ) (RC of ring B) + (SC of ring B) + (EC of ring B) (21c)

FAext - FAfeed ) [Pν(δk - δ1) + ∆Ik + ∆III 2 ]bS (24)

The above equations can be rearranged as follows

(TC of a tandem SMB) ) (RC + SC) of a tandem SMB + EC of a tandem SMB ) f2{[(RC + SC) of ring A],[(RC + SC) of ring B]} (22) + f3[(EC of ring A),(EC of ring B)] An examination of f2 and f3 can help elucidate the nature of the additional term that is considered in the overall optimization but not in the individual optimizations. Below is the expression for f2

[(RC + SC) of a tandem SMB] ) 1 [(RC + SC) of ring A] + YBP FAext FAfeed

{

}

SP(1 - SR) 1 [(RC + SC) of ring B] A YP CF,PYAP YBP

respectively. The input parameters SP and SR are the solvent price and solvent recycle fraction, respectively. The derivation of eq 23 is presented in Appendix A. Equation 23 shows that the cost functions of the two single SMBs (ring A and ring B) are linked by FAext/FAfeed in the cost function of a tandem SMB. This implies that, in the overall optimization, FAext/FAfeed should also be minimized in addition to the individual optimizations of ring A and ring B. The physical meaning of a decrease in FAext/FAfeed is a reduction in product dilution in ring A, which, in turn, decreases the bed volume of ring B. Furthermore, it can also result in a reduction of the number of SMB units in ring B if there is a limit on the column diameter. Unlike resin and solvent costs, equipment costs are largely affected by constraints on the column diameter. If the column diameter calculated from the PR and uF values exceeds a set limit, more than two SMB units are needed. For insulin purification, the SMB equipment cost is much higher than the combined costs of resin and solvent. Therefore, the main factor in both optimization methods is minimization of the number of SMB units. Equation 18c suggests that the number of B ) can be further reduced by SMB units in ring B (NSMB A A A decreasing uext/uF , or Fext/FAfeed, in the overall optimization. However, in the individual optimizations, the values of uAext and uAF are determined from the optimization of ring A, and they are fixed in ring B. This can cause a large difference in the equipment cost between the two methods, as shown in the Results and Discussion section. If there is no limit on the column diameter, only one SMB unit is needed in each ring, and thus, the two optimization methods result in the same number of SMB units. For this reason, the difference between the overall optimization method and the individual optimization method is mainly due to resin and solvent costs. It is worth mentioning that low values of FAext/FAfeed (or low dilution in ring A) reduces the purification cost in a tandem SMB. Equations 1-5 can be used to express the value of (FAext - FAfeed) as

(23)

where YAP and YBP are the yields of rings A and B,

where S is the cross-sectional area and ∆ji is the masstransfer correction term of component i in zone j (eq 6). To decrease FAext/FAfeed or product dilution in ring A, the right-hand side of eq 24 should be reduced. Therefore, the following methods can decrease the total purification cost of a tandem SMB: (1) increase the length of zone I or zone III in ring A by increasing either Njcol or Lc, or both; (2) distribute the highest-affinity component in ring A because this allows a component with a smaller retention factor to be standing in zone I;5 or (3) if the column length, zone configuration, and splitting strategy are fixed, reduce FAext/FAfeed by decreasing the feed linear velocity of ring A. The third option is possible because a decrease in the feed linear velocity reduces the amount of desorbent required to overcome masstransfer resistance, resulting in less dilution of the extract. 2.6. Optimization Algorithm for a Tandem SMB. Because there are many optimization variables, simultaneous optimization of rings A and B results in an enormous search space. An efficient methodology is needed to reduce the computation time. Notice in Table

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depends on the limit on the column diameter (eqs 18c and 23). In this case, additional loops must be included in the algorithm in Figure 5. First, the optimal feed linear velocity of ring B for each number of columns is determined for the case without the diameter limit using the algorithm in Figure 5. If this linear velocity is equal B ), to the maximum feed linear velocity of ring B (uF,max then the linear velocity is the optimal value for ring B for that given number of columns. If the linear velocity B B , this implies that uF,optimal lies (uBF ) is lower than uF,max B B between uF and uF,max. The total cost of the tandem SMB in this region is evaluated to find the value of B . uF,optimal 2.7. Matching Ratio in the Optimized Tandem SMB. The matching ratio is defined as the ratio of the output flow rate of ring A to the maximum allowable input flow rate of ring B, i.e.

Matching ratio )

Figure 5. Flow sheet of a proposed algorithm for the optimization of a four-zone tandem SMB for minimum cost. The yield of each ring is fixed. No constraints are placed on the column diameter in this example.

2 that the equipment cost remains constant if the number of columns is fixed. Equation 23 indicates that, if the total numbers of columns in rings A and B are fixed and if there is no limit on the column diameter, then the minimum cost of ring B always leads to the minimum cost of the tandem SMB. Therefore, one can optimize the linear velocities and switching time of ring B independently of ring A to reduce the search space, as shown in Figure 5. The first step is to optimize ring B individually under a fixed number of columns. The resulting optimal zone configuration and operating parameters of ring B for each number of columns are coupled to ring A at a given number of columns and zone configuration. Then, the innermost loop searches for the optimal feed linear velocity in ring A for the minimum cost of the tandem SMB according to eq 23. This procedure is repeated for various zone configurations and total numbers of columns in ring A (inner loop). Finally, the total number of columns in ring B is varied in the outer loop. This methodology can save computation time significantly by reducing the search regions. If the column lengths in rings A and B are to be optimized, two more outer loops can be added to Figure 5. As a result, a two-dimensional search plane is constructed, and a simultaneous search is needed. In this case, a crude search in the entire plane followed by a fine search in the reduced plane has a possibility of missing the optimum. To overcome this problem, the first step is to search in the entire plane three times, gradually increasing the number of elements each time, e.g., 10 × 10, 20 × 20, and 30 × 30. If the optimum values found from the three different searches fall in a small area, then a search using a finer mesh is carried out in the reduced area. This method can reduce the computation time while keeping the number of meshes sufficiently high. If there is a limit on the column diameter, the optimization of ring B is coupled to the optimization of ring A. This is because the number of SMB units

FAext B Ffeed,max

(e1.0)

(25)

B is the maximum allowable feed flow where Ffeed,max rate of ring B at the optimal column length, number of columns, column diameter, and zone configuration for the lowest total cost of the tandem SMB. Keep in mind B is limited by either the zone linear velocthat Ffeed,max ity limit (eq 7a) or the mass-transfer effects (eq 7b). As the matching ratio approaches unity, the feed linear velocity of ring B becomes closer to the maximum feed linear velocity. Complete matching of the flow rates (i.e., a matching ratio value of 1.0) indicates that ring B is fully utilized, resulting in a low resin cost. In addition, complete matching usually, but not always, reduces the number of SMB units and results in a low equipment cost, as shown in eq 18c. However, to maintain a matching ratio of unity, a large amount of desorbent is required to overcome the mass-transfer resistances in ring B. This results in an increase in solvent cost. For this reason, a smaller matching ratio (