Optimization of SMB Systems with Linear ... - ACS Publications

At the core of this model are the standing wave equations, which are algebraic relationships that offer a significant modeling advantage in optimizing...
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Optimization of SMB Systems with Linear Adsorption Isotherms by the Standing Wave Annealing Technique Fattaneh G. Cauley Seaver College, Pepperdine University, Malibu, California 90263

Yi Xie† and Nien-Hwa Linda Wang* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283

In this paper a nonlinear mathematical programming model is formulated and is solved by an innovative stochastic optimization algorithm, Simulated Annealing. The model is designed for simultaneous optimization of all variables that determine the design and construction as well as the operating costs of an SMB system. At the core of this model are the standing wave equations, which are algebraic relationships that offer a significant modeling advantage in optimizing SMB systems. Specifically, by solving these equations the desired purity and yield for nonideal systems is guaranteed. They also provide computational efficiency and tractability by eliminating numerical solutions of ordinary or partial differential equations. The method introduced in this paper, the Standing Wave Annealing Technique (SWAT), will extend the applicability of the standing wave equations to solve efficiently any single or multiobjective optimization problem for binary SMB systems with linear adsorption isotherms. Two examples from the literature are used in this study to illustrate the flexibility and computational efficiency of SWAT: (1) the separation of glucose from sulfuric acid and (2) the separation of insulin from an impurity. The optimization results obtained are similar or superior to those previously reported. The computational time for SWAT, compared to a grid search method, is at least an order of magnitude shorter. This study shows that SWAT has the capability to efficiently solve wide range of SMB optimization problems. 1. Introduction Chromatography processes have been widely used for the recovery and purification of biochemicals and pharmaceuticals. A variety of adsorbent-solute interactions in chromatography make it a flexible and powerful separation tool. Its mild operating conditions provide a suitable environment for bioseparation. However, chromatography, when operated in batch mode, is often inefficient for large-scale purification. In low-pressure chromatography mass-transfer limitations cause bands of similar compounds to overlap, making it difficult to achieve both high purity and high yield. This difficulty can be overcome in a simulated moving bed (SMB) system.1 A typical SMB system for binary separation consists of four or more chromatographic columns that are connected to form a circuit. The circuit is divided into four regions (zones) of different flow rates, by four ports. Two of these ports are for inlet streams, a feed solution and an eluent (or desorbent). The other two are for outlet streams, an extract, which is enriched with the compound with high affinity for the stationary phase, and a raffinate, which is enriched with the compound with low affinity. Figure 1 shows a schematic diagram of a four-zone SMB system. The inlet and outlet ports move periodically in the direction of the fluid flow to follow the migrating bands. * To whom correspondence should be addressed. Tel.: (765) 494-4081. Fax: (765) 494-0805. E-mail: [email protected]. † Current address: Eli Lilly and Company, Lilly Corporate Center, Indianapolis, IN 46285.

In a four-zone SMB system, when the average port movement velocity 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 shift toward the extract port and the fast solute toward the raffinate port.2 This difference in rate of migration provides the mechanism to separate the two and to collect pure products from the outlet ports. This strategy is based on the idea that zones II and III can be used to achieve partial separation, while zones I and IV are used to achieve adsorbent regeneration and product enrichment, respectively. In 1997 Ma and Wang2 proposed the concept of standing waves. That is, the flow rate in each zone can be chosen so that the key concentration wave (boundary of a solute band) in each zone migrates at the same speed as the ports. As a result, all band boundaries (or waves) remain “standing” with respect to the ports. As such, the concentration profiles are stable, and the product composition remains constant. In an ideal system, where there is no mass-transfer effect, general equilibrium analysis, Triangle Theory,3,4 or standing wave analysis can provide the values of the operating parameters for a system and ensure high purity and yield. Furthermore, it has been shown that for an ideal system numerical results obtained from the standing wave equations correspond to the vertex of the triangle region in the Triangle Theory.5 In a high-pressure SMB system, the columns are packed with fine particles with a diameter of a few microns, thus mass-transfer effects are minor. Such a system can be treated as an ideal system without significant loss of product purity and yield. However,

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Figure 1. Schematic diagram of a four-zone SMB system: (a) Step N and (b) Step N+1.

to reduce capital and operation costs, low-pressure SMB systems with large particles are often used in large-scale production. In these systems (nonideal systems) masstransfer effects are significant and must be considered in order to achieve high purity and yield.2 For nonideal systems the operating parameters determined by the Triangle Theory, which is applicable only to ideal systems, do not guarantee high purity and yield. For this reason, simulations of SMB processes, that require solutions of ordinary or partial differential equations, are needed to ensure that the purity and yield requirements are satisfied. In ref 2 Standing Wave (SW) equations for nonideal systems were proposed. This system of algebraic equations link product purity and yield to the zone lengths, zone interstitial velocities, port switching time, isotherms, and mass-transfer parameters (axial dispersion and lumped mass-transfer coefficients). The SW equations provided a simple and systematic procedure for finding the optimal port movement and zone interstitial velocities for nonideal systems. These algebraic relationships offer a significant modeling advantage. Specifically, the SW equations include mass-transfer effects as a simple function of the decision variables, and by solving these equations the desired purity and yield for nonideal systems is guaranteed. In contrast employing Triangle Theory requires numerical solutions of ordinary or partial differential equations to ensure desired product purity and yield. The simplicity of SW equations has facilitated the development of new SMB processes. The method was

employed in research on SMB processes for the separation of phenylalanine and tryptophan in ref 6 and for the purification of paclitaxel in ref 7. Xie et al.8 have shown that the SW equations can be applied to nonideal systems, even when detailed mass-transfer parameters are unavailable. Hritzko et al.9 extended the use of SW equations to multicomponent systems and applied this method to design a process to separate two sugars from two acids. Mallmann et al.10 applied the method to a binary nonlinear, ideal system (without mass-transfer effects). Xie et al.5 have further extended that method to nonlinear nonideal systems. Computer simulations based on a detailed rate model (VERSE) have verified that SW equations guarantee the desired purity and yield for nonideal systems. The SW equations are the key component of the model proposed in this study. Over the past decade SMB technologies has become more widespread in industry. Thus, research on the design issues that affect the long run costs of production has become increasingly important. Biressi et al.11 and Jupke et al.12 considered mass-transfer effects in finding optimal operating parameters of an SMB system and employed simulations and trial-and-error. Dunnebier et al.13 presented a model-based optimization strategy to find the optimal operating parameters and feed concentration in their examples. All of the above studies assumed the total number of columns and zone configuration as given for an SMB system. However, Azevedo and Rodrigues14 and Pynnonen15 pointed out that zone configuration is an important issue in the design of SMB systems. More recently, Zhang et al.16 presented a comparison of SMB and Varicol processes for a nonlinear system and optimized each system using a Genetic Algorithm. However, in the above study several variables were considered as fixed, only configurations with a total of 5 or 6 columns were considered, and more importantly the procedure lacked computational efficiency because the solutions of ordinary differential equations were needed to ensure that the product purity and yield are satisfied. Mun et al.17 proposed the first systematic method for simultaneous optimization of system and operating variables of an SMB system. Their approach, which is based on the SW equations, consisted of a grid search to systematically hunt for the optimal values for all decision variables. The grid search was originally designed for two SMB systems used in tandem for multicomponent separation but could be used for optimizing a single SMB system as well. The procedure, which was written and compiled in FORTRAN 77, although accurate, is time-consuming, and the choice of grid size is often problem dependent. Finally, it lacks flexibility to be easily changed or enhanced in order to meet new constraints and other user requirements. In this paper a nonlinear mathematical programming model is formulated and is solved by an innovative stochastic optimization algorithm, Simulated Annealing. The model is in effect a reformulation of the SW equations augmented with additional variables and constraints to extend its applicability to solve efficiently any single or multiobjective optimization problem for binary SMB systems with linear adsorption isotherms. The Simulated Annealing algorithm provides a powerful tool for overcoming the computational challenges resulting from the nonconvexity in the model. The close alliance of the mathematical programming model and the computational tractability obtained due to the

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Simulated Annealing algorithm has resulted in the first method that simultaneously and efficiently optimizes all variables that determine the design and construction as well as the operating costs of an SMB system. We will refer to this method as the Standing Wave Annealing Technique (SWAT). This paper will illustrate, by use of a number of examples from the literature, the broad applicability and computational efficiency of SWAT. Based upon this analysis it can be concluded that SWAT has the ability to efficiently solve a wide range of constrained and unconstrained optimization problems in the design of SMB systems. 2. A Mathematical Programming Model SWAT contains three nested tiers of decision variables. At each tier the investigator can either fix the value of the variables or allow the optimization algorithm to determine their values. There are four decision variables in Tier1, which are referred to as the system variables. These variables determine the fixed cost and physical capacity of an SMB system. Once purchased and placed in use they cannot be changed without incurring substantial additional costs. Tier2 consists of the configuration variables of an SMB system. These can be altered and optimized as needed (e.g., in response to variations in the desired level of output). Tier3 consists of the five operating variables of an SMB system, which together with the configuration variables determine the incremental costs of production. The clustering of the decision variables into three nested tiers offers many advantages. It is shown in section 4 that several previous papers2,6,8,18 on SMB systems are special cases of the model presented here where the decision variables in the first two tiers are regarded as fixed. In addition, short-term and long-term cost benefit analysis for an SMB system can be performed by fixing the variables in one or more of the tiers. 2.1. Variables. Tier1 variables are column length (Lc), column cross sectional area (S), number of columns (Ncol), and particle diameter (dp). Tier2 variables are the number of columns (N jcol) in zone j, j ) I, II, III. It should be noted that in this model (N IV col) is not a variable but is uniquely determined by the other variables (Ncol and the Tier2 variables). Tier3 consists of the variables: interstitial velocity (u j0) in the zone j, j ) I, II, III, IV, and v is the average velocity of port movement. 2.2. Constraints. The model contains five distinct groups of constraints. As was previously mentioned the SW equations given below constitute the core constraints:

∆ji )

β ji N jcolLc

(

j E b,i +

)

Pδ2i v2 K ji

, for i ) 1, 2 and j ) 1, 2, 3, 4 (2)

The superscript j designates the zone number; the subscript i (equal to 1 in zones II and IV and equal to 2 in zones I and III) designates the component. P is the phase ratio, (1 - b)/b, where b is the interparticle voidage. For each component δi is defined as Ke,ip + (1 - Ke,ip)ai, where Ke,i is the size-exclusion factor, p is the particle porosity, and ai is the partition coefficient; Eb is the axial dispersion coefficient; and K is the over all mass-transfer coefficient as defined by Ma and Wang.2 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; it is in effect an index of product purity and yield. The greater the value of β, the higher the product purity and yield. Appendix A contains a detailed description of the above terms as functions of the decision variables. It should be noted that the intrinsic adsorption in the SW equations, that are independent of the scale or the operating conditions of the system, are determined through a series of batch chromatography tests. The requirements for mass balance at the feed port and a physically feasible solution of eq 1 result in an equation for the maximum feed velocity uF,max for a given configuration.2

uF,max ) 4

(

P 2(δ2 - δ1)2 2 PβIII 2 δ2

III K III 2 N colLc

+

2 PβII 1 δ1 II K II 1 N colLc

(

)

-

III βIII 2 E b2

N III colLc

+

II βII 1 E b1

N II colLc

)

. (3)

The next group of constraints, primarily dictated by SW equations, are all linear except for the first where the nonlinearity is due to the uF,max term: II uIII 0 - u0 e uF,max IV uIII 0 (Y1 - 1) + u0 > 0

III uII 0 < u0

I uIV 0 < u0

uI0 ) (1 + Pδ2)v + ∆I2 II uII 0 ) (1 + Pδ1)v + ∆1

I uII 0 < u0

III uIII 0 ) (1 + Pδ2)v - ∆2 IV uIV 0 ) (1 + Pδ1)v - ∆1

Lc - vts,min g 0,

III uIV 0 < u0

(4)

(1)

where the mass-transfer correction terms are defined as

The pressure drop at each zone is bounded by a maximum pressure drop, ∆pmax. The Ergun equation19 can be used to relate ∆pmax to particle size, zone length,

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and other parameters. The resulting four constraints are

∆pj )

(

( )

150µu j0N jcolLc 1 - b b d2

p j 2 j 1.75F(u 0) N colLc

dp

2

106 + 6

)

1 - b 1 14.7 e ∆pmax b 3.6 1.013 × 105 (5)

j ) I, II, III, IV It should be noted that the pressure drop constraints can vary with the locations of the pumps (pump configuration). The annual production of the product Q is not a variable of the model but is a nonlinear function of a subset of the decision variables. This function is entered as a constraint with upper and lower bounds for Q. This provides flexibility in the model for analyzing the economic viability (e.g., determining long run costs) of an SMB system: II Qmin e Q e Qmax for Q ) S(uIII 0 - u0 )ΦP

(6)

In the above, ΦP is a function of interparticle voidage, feed concentration of the product, product yield, and duration of production. In addition to the constraints described above the model contains upper and lower bounds for all Tier1 variables. Also each zone (Tier2 variables) has a lower bound of 2 columns to ensure that the assumptions of the SW equations are valid.2 2.3. Objective Function. The objective function for SWAT is determined by the user and the problem under consideration. In section 4 the SWAT method will be employed for single objectives as well as multiobjective optimization of SMB systems. Data for two problems: (1) separation of insulin from an impurity and (2) separation of glucose from sulfuric acid are used for these illustrations. 3. Optimization Techniques The models presented and solved in section 4 contain nonlinear and nonconvex objective function and constraints. This section discusses the computational challenge of searching for the global optimum of a nonconvex model and some options that are available for their solution.20 3.1. Historical Development. The choice of an algorithm and the rate of its convergence to a solution, or obtaining a solution at all, are greatly influenced by the model formulation.21 As the complexity of the optimization problems being considered has increased, exploiting the structure of a problem has become increasingly important. Classical methods, such as local search, hill climbing, and gradient-based numerical optimization, were designed for solving models that satisfied convexity requirements (which this problem does not). Their efficiency stems from their inherent greediness to move in the direction of largest improvement. This trait will become a potential pitfall in the absence of convexity, and it may result in stopping at a local rather than a global optima. Enumeration, when possible, seems an attractive alternative, but this approach is not practical for large problems or problems with continuous variables. A

natural progression from complete enumeration is taking a large sample of the feasible set. Originally sampling was deterministic, such as grid search.17 However, given the computational requirements of grid search typically increases exponentially in the number of decision variables, this technique is relatively slow and has limited usefulness. During the last two decades stochastic optimization techniques, where sampling for the global optimum follows a probability distribution, have been developed. 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 (e.g., acceptable moves do not always have to be associated with better values of the objective function). With these algorithms, trials starting at the same point would likely follow different paths, take different quantities of computer time, and may end at different places. 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 a physical system. GA, developed by Holland,22 provides a method, based on survival of the fittest, to intelligently search the feasible set for global optimum; although there is no guarantee that at termination the algorithm has located the global optimum. In practice, GA has a strong record of finding a good solution for unconstrained optimization problems.23,24 GA typically handles constraints in optimization by adding the constraint to the objective function with a large coefficient as a penalty for violating the constraint. This tends to increase the difficulty of the search for feasible points. Also, GA inherently follows a greedy algorithm approach. This property, as indicated earlier, will increase the likelihood of getting stuck in local optima. In contrast SA algorithms not only are well suited for problems with a large number of constraints, but they do not behave as a greedy algorithm.25 SA algorithms have been successfully used in other areas of chemical engineering,26-31 and as it will be concluded in this study SA as implemented in the SWAT method provide a powerful tool to efficiently solve optimization problems in the design of SMB systems. 3.2. An Introduction to Simulated Annealing Algorithms. SA is based on an analogy taken from thermodynamics. The algorithm mimics the behavior of a physical system that is heated and then cooled slowly, such as growing crystals or annealing metals. Nature aids the physical system to reach its minimum energy if the cooling process proceeds slowly. Rapid cooling results in the energy of the physical system being trapped in a higher level, and irregularities in the product will be present. This is analogous to stopping a minimization problem at local optima. The process of annealing has been called nature’s minimization algorithm.32 During the mid 1980s, scientists at IBM,33 and Cerny34 independently, suggested the analogy between annealing and the solution of an optimization problem. SA has shown promise in solving combinatorial or multivariate constrained optimization problems.35-38 Often in the SA literature, the physical system terminology is used to describe the behavior of the algorithm. The analogy between a physical system and a standard optimization problem is described in Table 1. For example, in the SA literature the state of the physical system is equivalent to a feasible solution of

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Table 1. Terminology Equivalence simulated annealing

optimization

state energy move class temperature ground state

feasible solution objective function neighborhood relation control parameter optimal solution

the optimization problem, and the energy of the physical system at each instant is equivalent to the lowest value of the objective function achieved so far. Throughout this paper we will use standard optimization terminology. A formal description of SA and its implementation in SWAT follows. 3.2.1. The SA Algorithm. Let S represent the feasible set of a multivariate optimization problem, and elements x ∈ S are vectors representing a complete specification of the variables. The objective function C is a real valued function defined on the feasible set, C:SfR. SA works by searching the set S looking for low values of C. The search is not exhaustive; rather it is based on the principles of statistical mechanics. A system in thermal equilibrium at temperature T follows the Boltzmann probability distribution for reaching state C

PT(C) ∝ KT(C)e(-C/T)

(7)

where KT(C) is Boltzmann’s constant for temperature T. In the physical system context, this probability represents the fraction of time that the system has value C while it has equilibrated at temperature T. The analogy with optimization is that by employing SA we in effect defined the Boltzmann distribution on S. Thus, providing probabilities for visiting elements of S. This is equivalent to the algorithm providing opportunities for jumps from the current location to other parts of the feasible set to continue the search; this in turn helps the algorithm to escape entrapment in local optima and eventually reach the global optimum.25,39 The algorithm can be thought of as being composed of three steps. Formally, when solving an optimization problem by SA, at each instant, there is a current feasible point, x. Step1. Define a neighborhood that is a subset of the feasible set and choose a point x ′ from this neighborhood at random. A procedure that generates neighboring points to x is called a Move Class. Step2. Compare C(x ′) to C(x) and consider whether to move to x ′ and proceed from there or to stay at x and try again. The decision to accept or reject a move is controlled by a criterion called the acceptance rule. Step3. Continue the above steps as the Annealing Schedule dictates until a prescribed condition is arrived at. The choices of the Move Class, Acceptance Rule, and Annealing Schedule determine how the algorithm proceeds and conducts the search for an optimum. 3.2.2. Move Class. Choice of the Move Class can make a large difference in the rate of convergence to the minimum, perhaps more so than any of the other known “improvements” to the algorithm.25 SA was originally developed and tested on combinatorial problems, and the Move Classes discussed in the literature were designed for problems with discrete variables. SWAT is a mixed problem; it contains both integer and continuous variables. That is, the variables that relate to the number of columns of the system are integer

valued, but the other eight variables are continuous.40 A sophisticated scheme called Basin Hopping was chosen as our move class. This procedure has been successfully applied to problems that contain continuous variables.41 As the name indicates this procedure consists of moves among local minima. Basin Hopping is implemented in Step1 of the algorithm as follow. Assuming x is a local minima, the procedure starts by generating a neighboring point x ′, using known bounds and Gaussian perturbation: xn′ ) xn + N(0,σn), where σn ) (Un - L n) /6. The upper and lower bounds, Un and Ln, for uj0 and v are generated from the ideal case, and the bounds for the other variables are given as inputs in the model. Next, from the new point x ′, a gradientbased optimizer locates a local minima and updates x ′ to that point. It is of course important that the perturbation generates a point sufficiently far from the starting point so we do not end up in the same local minima repeatedly. 3.2.3. Acceptance Rule. In 1953 Metropolis et al.42 proposed the following widely used acceptance rule: Accept the move to x ′ if

(

min 1, exp

-∆C g U(0,1) T

)

(8)

where ∆C ) C(x ′) - C(x), and U(0,1) is a randomly generated number from the uniform (0,1) distribution. When ∆C > 0, the move is uphill, and ∆C e 0 indicates a downhill move. The Metropolis acceptance rule always accepts downhill moves (i.e., lower values of the objective function in a minimization problem), while it sometimes accepts uphill moves. The ability of accepting uphill moves allows the algorithm to escape entrapment in local minima and the possibility of finding a better value later on. The control parameter T in eq 8, called the temperature of the system in the SA literature, is the driving force behind accepting an uphill move. For T infinite, all moves will be accepted, while for T of zero only downhill moves are accepted. For intermediate values of T, the probability of acceptance decreases exponentially in ∆C. That is, large uphill moves are nearly always rejected, but small uphill moves will be considered for moderate values of T. Use of Metropolis acceptance rule results in a Boltzmann probability distribution on S.25 3.2.4. Annealing Schedule. The Annealing Schedule of the algorithm refers to how T is successively decreased; this in turn determines how the iterations of the algorithm proceed. Because it is important to be able to explore the whole feasible set at the start of the algorithm, T is initially assigned a relatively large value, thus the search is purely random. As T is decreased slowly, the mean of the Boltzmann distribution decreases (this is referred to as cooling the system), and toward termination of the algorithm only downhill moves no matter how small will be accepted. The implementation of SA algorithm in SWAT uses an exponential annealing schedule, which was described in ref 33.

T(φ) ) T0 r φ, φ ) 0,....,Γ

(9)

The parameter 0 < r < 1 represents the decay rate for T. The algorithm starts with φ ) 0, Tstart ) T(0) ) T0, and ends with Tstop ) T(Γ) ) T0 r Γ. For each value of φ, the algorithm repeats Step1 and Step2 for a prescribed number M. Choices of φ, T0, and M are typically problem

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dependent, and some preliminary sensitivity analysis of the model will provide values that ensure rapid convergence of the algorithm. See ref 43 for a comparison of Annealing Schedules. A Branch and Bound (B&B) procedure augments the SA algorithm in SWAT.44 B&B is the most widely used addition to algorithms for optimizing mathematical programming models that contain integer variables. Loosely described, the procedure starts by relaxing the integer requirement for all integer variables and optimizing the associated problem (by Simulated Annealing in SWAT). The relaxation step typically results in solutions for integer variables that are not integer values. For example, for the integer variable Ncol with lower and upper bounds 8 and 20, respectively, we may see the solution Ncol ) 10.3. The separation step then gives rise to two new descendent problems. Each is made of the previous problem in its entirety but has new bounds for Ncol; 8 to 10 and 11 to 20, respectively (Figure 2(a)). Each new problem is also relaxed and solved. It is possible for a descendent problem to be infeasible. However, if the problem is feasible, the solution could give rise to additional descendants, or it could result in the problem being fathomed. A relaxed problem is fathomed, which implies that it will not be considered any further, if it has an integer solution, or its value of objective function is inferior to the current best value. The word “Branch” refers to the separation step where the feasible set is divided into subsets along the boundary of integer variables, and the word “Bound” refers to the fact that certain regions need not be considered further. From the above discussion it should be clear that at the start of the algorithm it is difficult to predict how many optimizations by the end would be performed. Figure 2(a) illustrates a B&B procedure in general. Figure 2(b) details a flowchart of the SWAT method. For the examples illustrated in this paper, the B&B procedure solved anywhere from one to fifteen problems. Most optimizations considered less than six problems before the optimal solution was determined. 4. Results and Discussion In this section results obtained by using SWAT for the design of an SMB system are presented. The versatility and efficiency of SWAT are compared to other procedures used in the literature for single and multiobjective optimization of SMB systems. For all examples presented in this section SWAT was run on a 2.2 GHz Pentium. 4.1. Numerical Comparison of Solving SWAT for Tier3 Variables to Previous Solutions of SW Equations. In 1997 Ma and Wang2 introduced the SW equations for linear systems with mass-transfer resistances. The SW equations were employed to find the operating variables (Tier3 in SWAT) for fructose and raffinose separation in an SMB system. A conclusion of their analysis was that use of SW equations leads to large improvements in purities for both extract and raffinate compare with other procedures available at that time. Over the past seven years the procedure for solving the SW equations has been fine-tuned to a point where it takes under a second on a personal computer to obtain a solution. To show the versatility and benchmark SWAT, it was used to solve the problem of the separation of glucose and sulfuric acid presented in ref 9. This problem was simplified from the separation of glucose from sulfuric

Figure 2. Flowcharts of the optimization program. (a) Branch and Bound algorithm. (b) SWAT algorithm.

acid and acetic acid.9 The solution reported by the authors is given in the first column of Table 2. SWAT was reduced to the same problem by using the inputs and fixing the Tier1 (system) and Tier2 (configuration) variables at values used in ref 9. Thus, by optimizing the Tier3 (operating) variables SWAT is in effect solving the SW equations. The results of this run which took 1.7 s were identical to those obtained by Hritzko et al.9 Next, SWAT optimized all variables using the objective of minimizing the solvent usage. Not surprisingly the optimal result shows an almost 5% reduction in solvent usage compared to the solution obtained earlier.

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Table 2. Comparison of SWAT Solutions with Standing Wave Solutions standing wave method solution to the SW equations Tier1

Tier2

Tier3

SU TBV

S(cm2) (dc(cm)) Lc(cm) dp(µm) Ncol N Icol N II col N III col u I0(cm/min) u II 0 (cm/min) u III 0 (cm/min) u IV 0 (cm/min) v(cm/min) L/kg kg/L/day

8.92 (3.37) 100.00 320.00 20 3 4 9 18.08 7.60 17.12 7.57 6.17 3.67 8.93

Table 3. Comparison of SWAT Solutions with Grid Search Solutionsa

optimization by SWAT (minimize solvent usage) Tier1 & Tier2 no tiers fixed fixeda 8.92 (3.37) 100.00 320.00 20 3 4 9 18.08 7.60 17.12 7.57 6.17 3.67 9.93

417.19 (23.05) 200.00 50.00 20 7 4 6 0.36 0.16 0.36 0.16 0.13 3.50 .009

a The following bounds for Tier1 variables were used in this optimization: S(cm2) 0.79-125663.71, Lc(cm) 10-200, dp(µm) 50200, Ncol 8-20.

However, SWAT achieved the requested objective by reducing the throughput to only 1% of the previous case, a highly undesirable outcome. A closer examination reveals that by reducing the linear velocities u0 j, by 2 orders of magnitude, the mass-transfer correction terms in eqs 1 and 2 became negligible. Thus, the optimized system approaches an ideal system, which has the minimal solvent usage. Such a system of course is not practical, but the analysis clearly shows the importance of multiobjective optimization of SMB systems. The above results are summarized in Table 2. 4.2. Comparison of Optimization of an SMB System by SWAT to Grid Search. In this section optimization results obtained from SWAT are compared to results obtained from an implementation of a grid search method,17 another technique for simultaneous optimization of all variables contained in SWAT. For this comparison the problem of separating insulin from impurities presented in ref 17 were used. The authors analyzed a tandem SMB (two SMB systems in a series) for the separation of insulin from two impurities, highmolecular-weight protein (HMWPs) and zinc chloride (ZnCl2) using size exclusion chromatography. The comparison with SWAT is based on the first step of that process with some differences in assumptions and the values of the inputs. Specifically, extra column dead volume is assumed to be zero, ∆pmax is set at 1 psi for each zone, and the axial dispersion coefficient is estimated from the Chung and Wen correlation. Xie et al.18 reported the intrinsic parameter values, and the cost inputs used are set at values given in ref 17. Bounds for the Tier1 variables used in all optimizations here are given in Table 3. In ref 17 this example is analyzed fully including extensive sensitivity analysis on Tier1 variables. Two distinct cases were solved by both SWAT and grid search. Both cases are for the annual production of 10 metric tons of insulin. The optimization results obtained using SWAT are similar or superior to those previously reported. The computation time for SWAT compared to the grid search is at least an order of magnitude shorter, 1-4 min compared to 2545 min.

minimum purification cost production of 10 metric tons

cost

TC ($/ton) EC ($/ton) RC ($/ton) SC ($/ton) Tier1 S(cm2) (dc(cm)) Lc(cm) dp(µm) Ncol Tier2 N Icol N II col N III col Tier3 u I0(cm/min) u II 0 (cm/min) u III 0 (cm/min) u IV 0 (cm/min) v(cm/min) TBV kg/L/day

SA result

grid search result

6959.64 3428.57 340.56 3190.51 1135.18 (38.02) 10.00 85.51 8 2 2 2 1.91 1.11 1.83 1.09 0.84 0.30

6959.73 3428.57 337.23 3193.93 1,124.11 (37.83) 10.00 86.00 8 2 2 2 2.00 1.16 1.91 1.14 0.85 0.30

max TBV production of 10 metric tons SA result

grid search result

8148.05 8160.88 3428.57 3428.57 177.06 177.09 4542.42 4555.22 590.18 590.29 (27.41) (27.42) 10.00 10.00 147.74 148.00 8 8 2 2 2 2 2 2 5.69 5.70 3.15 3.16 4.52 4.53 2.88 2.88 2.29 2.63 0.58 0.58

a The following bounds for Tier1 variables were used in all optimizations: S(cm2) 0.79-1963.49, Lc(cm) 10-100, dp(µm) 50200, Ncol 8-20. ∆pmax ) 1 (psi) per zone.

4.2.1. Purification Cost Minimization for Design and Operation of an SMB System. The objective function for the cost of purification can be separated into the fixed costs (FC) and variable costs (VC) of production where fixed cost ) annual equipment cost (AEC) + annual adsorbent cost (AAC), and

AEC )

CE + CcolNcol + CSS rE

(10)

CRSLcFsNcol rR

(11)

and

AAC )

The numerator in eq 10 is the construction cost of an SMB system, where CE is the required fixed payment, Ccol is the fixed cost per column, and CS is cost per cm2 of the cross sectional area of a column. The denominator is the depreciation rate of the equipment. In eq 11, the numerator represents the initial cost of adsorbent for an SMB system, where CR is the cost per gram of adsorbent, and FS (g/mL), the adsorbent packing density. The denominator is the adsorbent depreciation rate. Equivalently, variable cost ) annual solvent cost (ASC) + annual feed cost (AFC), where III II ASC ) CsolSb[(uI0 - uIV 0 ) + (u0 - u0 )](1 - SR)t (12)

and

AFC ) CF

Q YP

(13)

In eq 12, Csol represents the cost per mL of solvent, SR is the solvent recycle ratio, and t, measured in minutes,

Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7595

is the time available annually for production. In the next equation, CF is the price per gram of feed, Q is the annual production quantity, and YP is the product yield. Note that all three terms in eq 13 remain fixed throughout this analysis and have no effect on the optimization result. For this reason, AFC is excluded from the analysis. The objective function is then written as

Min(AEC + AAC + ASC)

(14)

SWAT and grid search produced similar values for the objective function (14). The configuration that produces the lowest annual purification cost is (2,2,2,2). The annual purification cost/metric ton of production is $6960, where 49% is spent on equipment, 5% for adsorbent, and 46% for solvent. The detailed results including values for all variables are shown in Table 3. 4.2.2. Maximization of Throughput per Bed Volume of an SMB System. Throughput per bed volume (TBV) of an SMB system is defined as

TBV )

Q V(365)

(15)

In the above equation Q is the annual production as given in eq 6, and V is the total bed volume of the system. TBV, as defined above, has the units of kg/L/ day and is proportional to the customary definition of TBV which is the ratio of feed flow rate to total bed volume.7 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. Upon simplification eq 15 results in the following objective function:

Max

(uIII 0

uII 0)

LcNcol

(16)

Again, SWAT and grid search produced similar optimal configuration of (2,2,2,2), but slightly different values for the other variables with the computation time associated with SWAT being at least an order of magnitude shorter. The TBV given by SWAT is slightly larger, and the cost is islightly smaller when compared to the result of grid search. Thus, SWAT result is superior. It should be noted that SWAT solution coincides with the lower bounds of both Lc and Ncol. This result was caused by the pressure drop constraint that is bounded by the very low value of ∆pmax ) 1 psi. To maximize TBV, one should maximize the numerator ( II uIII 0 - u0 ) or minimize the denominator of eq 16. As shown in eq 5, one can increase dp to increase (uIII 0 ) while satisfying ∆p ) 1. However, as d inuII max p 0 creases, the overall mass-transfer coefficient K and uF,max both decrease (see the derivation of overall masstransfer coefficient in Appendix A and eq 3). Since (uIII 0 - uII 0 ) is bounded also by uF,max as shown in eq 4, one cannot increase the dp beyond 147.7 µm to further increase TBV. The solution presented by SWAT is the optimal system to yield the maximum TBV. Detailed results are presented in Table 3. 4.3. Multiobjective Optimization of an SMB System by SWAT. For many real world problems it is difficult to decide on a single objective. Often there are several objectives that are relevant and expected to be included in the decision process.23 As an illustration in this section the Pareto-efficient curve is developed for

the two objectives: maximization of TBV and minimization of purification cost. These objectives were used to illustrate single objective optimization in section 4.2. A Pareto-efficient or a tradeoff curve, named after the Italian sociologist who introduced the concept, consists of the set of efficient points. That is, these points represent the solutions (values) to the two objectives where any attempt to make one objective better will make the other objective worse.45 Thus, the Paretoefficient set is the only set of solutions that the decision makers need to consider, and the decision maker’s preferences will dictate which element of this set will be adopted. 4.3.1. Maximization of TBV and Minimization of Purification Cost. A closer look at the result of section 4.2.1 shows that when optimization was performed for the single objective of minimizing purification cost, a TBV of 0.30 (kg/L/day) was realized. However, this is only about 51% of the maximum TBV of 0.58 (kg/L/day) that is possible according to the results of section 4.2.2. It is clear that a decision maker may be willing to incur higher purification cost if it would result in higher TBV. To decide what the right trade off is, the Pareto-efficient curve for the two objectives was developed. The mathematical presentation of the two objective functions for this problem is written as II (uIII 0 - u0 ) Max LcNcol

(17)

Min(AEC + AAC + ASC)

(18)

and

The actual problem solved was the total cost minimization model of section 4.2.2, where the constraint set given in section 3 was augmented with an additional constraint requiring a lower bound on the value of TBV: II uIII 0 - u0 gb LcNcol

(19)

SWAT was run multiple times, and for each run a new value of b, from the range 0.30-0.58, was used in eq 19. The results are shown in Table 4, and the values of the two objectives are graphed in Figure 3. From this figure we see that the minimum purification cost for achieving a desired level of TBV increases exponentially. It is interesting to observe that the SWAT solution for the single-objective optimization for maximum TBV (TBV ) 0.58 kg/L/day, Table 3) also gives the minimum purification cost for providing a TBV of 0.58 kg/L/day (case G, Table 4). Because the solution in Table 3 satisfies the SW equations, it automatically provides the lowest solvent consumption, thus the lowest solvent cost. Furthermore, the resin cost and equipment cost are also automatically minimized when TBV is maximized. For these reasons, the purification cost, which is the sum of the solvent, resin, and equipment costs, is the minimum cost of providing a TBV of 0.58 kg/L/ day. A close examination of the cases A-G in Figure 3 and Table 4 indicates that SWAT finds increasing the particle size is the best way of increasing TBV, while keeping the overall purification cost at a minimum. The column length and the number of columns in each zone

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Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004

Table 4. SWAT Solutions Selected from the Pareto-Efficient Curve (Figure 3) for Multiobjective Optimization of Minimization of Purification Cost and Maximization of TBV cost EC RC SC Tier1

Tier2

Tier3

TBV

$/ton $/ton $/ton $/ton S(cm2) (dc(cm)) Lc(cm) dp(µm) Ncol N Icol N II col N III col uI0(cm/min) uII 0 (cm/min) uIII 0 (cm/min) uIV 0 (cm/min) v(cm/min) kg/L/day

A

B

C

D

E

F

G

6959.64 3428.57 340.56 3190.51 1135.18 (38.01) 10.00 85.51 8 2 2 2 1.91 1.11 1.83 1.09 0.84 0.30

6979.19 3428.57 286.91 3263.70 956.39 (34.89) 10.00 94.54 8 2 2 2 2.33 1.32 2.20 1.32 1.02 0.36

7022.13 3428.57 255.34 3338.21 851.15 (32.92) 10.00 101.68 8 2 2 2 2.69 1.56 2.51 1.52 1.17 0.40

7117.57 3428.57 224.47 3464.53 748.23 (30.86) 10.00 111.06 8 2 2 2 3.21 1.85 2.93 1.78 1.38 0.46

7245.64 3428.57 204.67 3612.39 682.24 (29.47) 10.00 119.42 8 2 2 2 3.71 2.12 3.31 2.03 1.58 0.50

7570.97 3428.57 184.34 3958.04 614.49 (27.97) 10.00 133.19 8 2 2 2 4.62 2.60 3.92 2.44 1.91 0.56

8148.05 3428.57 177.06 4542.42 590.18 (27.41) 10.00 147.74 8 2 2 2 5.69 3.15 4.52 2.88 2.29 0.58

particularly attractive alternative, because the computation time of using SWAT will increase linearly in the number of decision variables considered, whereas that of grid search increases exponentially. The flexibility and efficiency of SWAT was also illustrated with an example of a multiobjective optimization for insulin purification. The results strongly suggest that SWAT has the capability to efficiently solve a wide range of constrained and unconstrained SMB optimization problems. Acknowledgment Figure 3. Maximization of TBV and minimization of purification cost.

remained the same for all seven cases. As particle size increases in cases A to G, the zone linear velocities and II thus the feed linear velocity (uIII 0 -u0 ) can increase according to the Ergun equation, as shown in eq 5. SWAT finds the smallest diameter columns that can meet the production requirement in order to achieve the max TBV. However, as the zone linear velocities increase, to maintain the desired purity and yield, the mass transfer corrections terms, eq 2, also increase, resulting in a higher solvent cost (Table 4). For this reason, the overall purification cost increases from case A to case G. 5. Summary and Conclusions In this paper a nonlinear mathematical programming model of an SMB system for linear binary separation is formulated and solved. The process called SWAT optimizes simultaneously all variables that determine the design and construction as well as the operating cost for binary SMB systems with linear adsorption isotherms. At the core of SWAT are the SW equations, 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. Comparison of SWAT to a grid search method showed either similar or superior optimization results. However, the computational time required by SWAT was at least an order of magnitude shorter than the grid search method. SWAT may be a

Support from the Dean’s Research Fund at Pepperdine University, NSF 0215146 SBIR grant, and Indiana 21st Century Research and Technology Fund is gratefully acknowledged. Portions of this research were completed when the author was a Visiting Scholar at The Bioseparation Group, School of Chemical Engineering, Purdue University. The authors are thankful to Mr. Stephen F. Cauley for computational implementation and to Dr. Sungyon Mun for providing comparison numbers from the grid search. Appendix A: Mathematical Programming Model This model represents the objective of minimizing the total cost of annual production of Q grams. Objective Function. Minimize Total Cost ) Fixed Costs + Variable Costs, where

Fixed Costs ) AEC + AAC; AEC ) CRSLcFsNcol CE + CcolNcol + CSS , AAC ) rE rR Variable Costs ) ASC + AFC; ASC ) CsolSb × III II [(uI0 - uIV 0 ) + (u0 - u0 )](1 - SR)t, AFC ) CF

Q YP

Variables. Tier1 variables: S, dp, Lc, Ncol; Tier2 variables: N jcol, j ) I, II, III; Tier3 variables: u j0, j ) I, II, III, IV, v. Derived Variables Included in the Model. The axial dispersion coefficients Eb are calculated from Chung and Wen correlation:

Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7597

uI0b(dp × 10-4)

I Eb,2 )

II Eb,1 )

III Eb,2 )

IV Eb,1

)

0.2 + 0.011(FuI0bdp × 10-4/µ)0.48 u0IIb(dp × 10-4)

(

I II uIII 0 (u0 - u0 )(1 - Y2) I II uIV 0 (u0 - u0 )Y1

IV uI0(uIII 0 - u0 )(1 - Y1)

-4 uIII 0 b(dp × 10 ) -4 0.48 0.2 + 0.011(FuIII 0 bdp × 10 /µ) -4 uIV 0 b(dp × 10 ) -4 0.48 0.011(FuIV 0 bdp × 10 /µ)

0.2 +

( ( ( (

) ) ) )

bdp × 10-4 D∞,2

-2/3

-2/3

II 1/3 kII f,1 ) 1.09(u0 )

bdp × 10-4 D∞,1

bdp × 10-4 D∞,2

-2/3

1/3 1.09(uIII 0 )

bdp × 10-4 D∞,1

-2/3

kIf,2

)

)

1.09(uI0)1/3

IV 1/3 kIV f,1 ) 1.09(u0 )

(

βII 1 ) ln

NIII colLc

NII colLc

Subject to constraints

Ncol,min e Ncol e Ncol,max IV uIII 0 (Y1 - 1) + u0 > 0 III uII 0 < u0 I uIV 0 < u0 I uII 0 < u0 III uIV 0 < u0 II uIII 0 - u0 e uF,max

( ( ( (

) ) ) )

uI0 βI2 Pν2δ22 I ν+ Eb2 + )0 1 + Pδ2 (1 + Pδ )NI L KI2 2 col c

(dp × 10-4)2 dp × 10-4 1 ) + 60Ke,1pDp1 KIV 6kIV 1 f,1

βII Pν2δ21 uII 1 0 II + Eb1 + )0 ν1 + Pδ1 (1 + Pδ )NII L KII 1

follows:9

I II uIV 0 (u0 - u0 )(1 - Y2)

II KII 1 NcolLc

( )

II βII 1 Eb1

I II III NIV col ) Ncol - (Ncol + Ncol + Ncol)

(dp × 10 ) dp × 10 1 ) + 60Ke,2pDp2 KIII 6kIII 2 f,2

(

2 PβII 1 δ1

+

Njcol g 2, j ) I, II, III

-4

IV uI0(uIII 0 - u0 )Y2

III KIII 2 NcolLc

+

III βIII 2 Eb2

Lc,min e Lc e Lc,max

(dp × 10-4)2 dp × 10-4 1 ) + 60Ke,1pDp1 KII 6kII 1 f,1

βI2 ) ln

(

2 PβIII 2 δ2

-

Smin e S e Smax

(dp × 10-4)2 dp × 10-4 1 ) + KI2 60Ke,2pDp2 6kIf,2

The β values are calculated as

4

P2(δ2 - δ1)2

)

dp,min e dp e dp,max

The overall mass-transfer coefficients 1/Kji are calculated as follows:

-4 2

uF,max )

)

IV II I II (uIII 0 - u0 )(u0 Y2 + (u0 - u0 ))

βIV 1 ) ln

0.2 + 0.011(Fu0IIbdp × 10-4/µ)0.48

The film mass-transfer coefficients kf are calculated from Wilson and Geankoplis correlation:

kIII f,2

(

βIII 2 ) ln

1

βIII Pν2δ22 uIII 2 0 III Eb2 + III ) 0 ν1 + Pδ2 (1 + Pδ )NIIIL K

)

2

)

III III IV (uI0 - uII 0 )(u0 Y1 - (u0 - u0 )) III IV uII 0 (u0 - u0 )(1 - Y1)

col c

col c

2

βIV Pν2δ21 uIV 0 1 IV Eb1 + IV ) 0 ν1 + Pδ1 (1 + Pδ )NIV L K 1

col c

Lc - vts,min g 0

1

)

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Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004

∆pI )

(

p 1.75F(uI0)2NIcolLc

∆pII )

dp

(

2

(

dp

106 + 6

)

1 - b 1 14.7 e ∆pmax b 3.6 1.013 × 105

( )

II 150µuII 0 NcolLc 1 - b b d2

p 2 II ) 1.75F(uII 0 NcolLc

∆pIII )

( )

150µuI0NIcolLc 1 - b b d2

2

106 + 6

( )

III 150µuIII 0 NcolLc 1 - b b d2 p

)

1 - b 1 14.7 e ∆pmax b 3.6 1.013 × 105 2

)

106 + 6

2 III 1.75F(uIII 0 ) NcolLc 1 - b 1 14.7 e ∆pmax dp b 3.6 1.013 × 105

∆pIV )

(

( )

IV 150µuIV 0 NcolLc 1 - b b d2 p

2

106 + 6

)

2 IV 1.75F(uIV 0 ) NcolLc 1 - b 1 14.7 e ∆pmax dp b 3.6 1.013 × 105 II Q ) S(uIII 0 - u0 )ΦP

Qmin e Q e Qmax Nomenclature ai ) linear equilibrium distribution coefficient of component i, cm3/cm3 SV AAC ) annual adsorbent cost, $ AEC ) annual equipment cost, $ AFC ) annual feed cost, $ ASC ) annual solvent cost, $ CCol ) price per column of SMB equipment, $/col CE ) fixed price of SMB equipment, $ CS ) price of column cross section of SMB equipment, $/cm2 CF ) feed price, $/g CR ) adsorbent price, $/g CSol ) solvent price, $/mL CF,P ) feed concentration of the product, g/mL dc ) inner column diameter, cm dp ) particle diameter, µm dp,max ) maximum particle diameter, µm dp,min ) minimum particle diameter, µm Dp,i ) intraparticle diffusivity of component i, cm2/min D∞,i ) brownian diffusivity of component i, cm2/min j ) axial dispersion coefficient of component i in zone j E b,i j k f,i ) lumped mass-transfer coefficient of component i in zone j, min-1 Ke,i ) size exclusion factor for component i K ji ) overall mass-transfer coefficient of component i in zone j Lc ) single column length, cm Lc,max ) maximum column length, cm Lc,min ) minimum column length, cm Ncol ) total number of columns in SMB Ncol,max ) maximum number of columns Ncol,min ) minimum number of columns N jcol ) number of columns in zone j P ) 1 - b/b ) phase ratio Q ) annual production, kg Qmax ) maximum production Qmin ) minimum production rE ) depreciation rate of equipment, year

rR ) depreciation rate of adsorbent, year S ) column cross sectional area, cm2 Smax ) maximum inner column cross sectional area, cm2 Smin ) minimum inner column cross sectional area, cm2 SR ) solvent recycle ratio t ) duration of production, min ts,min ) minimum step time, min uF,max ) maximum feed interstitial velocity, cm/min u j0 ) interstitial velocity in zone j, cm/min v ) average port moving velocity, cm/min V ) total bed volume, cm3 Yi ) yield of component i YP ) yield of product Greek Letters β ji ) decay factor of standing wave i in zone j δi ≡ Ke,ip + (1 - Ke,ip)ai ) retention factor of component i ∆p j ) pressure drop for zone j, psi ∆pmax ) maximum pressure drop, psi b ) interparticle voidage p ) intraparticle voidage (or particle porosity) ΦP ) defined as bCF,PYPt F ) mobile phase density, g/mL Fs ) resin packing density, g/mL µ ) viscosity, g/cm/s Subscripts and Superscripts i ) component index j ) zone number index

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Received for review February 26, 2004 Revised manuscript received July 16, 2004 Accepted July 19, 2004 IE049842N