Automatic Control of Simulated Moving Beds II: Nonlinear Isotherm

Ind. Eng. Chem. Res. 2004, 43, 3895-3907

3895

Automatic Control of Simulated Moving Beds II: Nonlinear Isotherm Gu 1 ltekin Erdem,† Stefanie Abel,‡ Manfred Morari,*,† Marco Mazzotti,‡ and Massimo Morbidelli§ Swiss Federal Institute of Technology, ETH-Zu¨ rich, Physikstrasse 3, CH-8092 Zu¨ rich, Switzerland

Simulated moving bed (SMB) chromatography has become a widely used separation technology in the pharmaceutical and fine chemical industries. However, operation of SMBs at their economic optimum is still an open issue because of the absence of proper process control schemes. This work proposes an on-line optimization-based control concept to avoid the tradeoff between the robustness and productivity of the process and to accelerate the development of SMB separations. The control concept requires minimal information on the system, i.e., Henry’s constants and the average porosity of the columns. This paper illustrates how a controller that is based only on the linear adsorption isotherm parameters can find the correct operating conditions for the SMB applied to a system characterized by a nonlinear competitive isotherm in order first to fulfill the product specifications and then to optimize the economics of the operation. The performance of the controller is assessed for several scenarios addressing the main challenges in SMB operation. 1. Introduction The underlying principle of simulated moving bed (SMB) technology is countercurrent separation involving a liquid phase and a solid phase. The mixture to be separated enters at the feed port, and in the case of a binary system, its two components move in opposite directions, i.e., one propagates in the direction of the solid and the other in the direction of the liquid stream, according to their affinities to either of the phases. The two purified components can be collected at the two outlet ports left and right of the feed port, i.e., in the extract and raffinate streams. A process as described above involving real countercurrent contact is called a true moving bed (TMB). Because it is very difficult in practice to move the solid phase without damaging it or changing its properties, TMBs have been regarded as infeasible. The practical implementation of this concept is the SMB, which comprises a loop of several fixed-bed chromatographic columns (see Figure 1). The most widely adopted SMB configuration has a total of four inlet and outlet streams that divide the unit into four sections. Each of the sections consists of one or more identical fixed-bed columns. The SMB overcomes the difficulties connected with the movement of the solid phase by “simulating” such movement: all inlet (feed, desorbent) and outlet (raffinate, extract) ports of the unit are switched synchronously and periodically by one column position in the same direction as the fluid flow. The solid-phase velocity is defined as the ratio of the column length to the switching time, i.e., the period of time between two successive port switches. * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: +41 1 632 76 26. Fax: +41 1 632 12 11. † A ¨ utomatic Control Laboratory, ETH-Zu¨rich, Physikstrasse 3, 8092 Zu¨rich, Switzerland. ‡ Institute of Process Engineering, ETH-Zu ¨ rich, Sonneggstrasse 3, 8092 Zu¨rich, Switzerland. § Institute for Chemical and Bio-Engineering, ETH-Zu ¨ rich, 8093 Zu¨rich, Switzerland.

Figure 1. Scheme of a simulated moving bed (SMB) unit. Dash arrow indicates the inlet-outlet positions after the first switch.

Each section of the SMB plays a specific role in the process. The functionalities of the sections are explained briefly below, whereas a more detailed description of the SMB process can be found elsewhere.1 Section I between the desorbent inlet and the extract outlet port has the task of regenerating the solid phase. Before a column leaves this section and is switched to section IV, both components have to be completely desorbed by the entering fresh solvent. Section II between the extract outlet and the feed inlet has the task of desorbing the less-retained species completely to avoid polluting the extract with it. Section III between the feed inlet and raffinate outlet has the task of adsorbing the more-retained species completely to avoid polluting the raffinate with it. Section IV between the raffinate outlet port and the desorbent inlet has the task of regenerating the liquid

10.1021/ie0342154 CCC: $27.50 © 2004 American Chemical Society Published on Web 04/22/2004

3896 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004

phase by adsorbing all components before the liquid is recycled as fresh solvent to section I. SMB technology is a cost-efficient separation technique with a characteristically high productivity per unit mass of solid phase and a low solvent consumption compared to other separation techniques such as batch chromatography. Both the popularity of SMB technology and the significance of the chiral drugs on the global drug market have increased, e.g., single-enantiomer drugs comprised 40% of global drug sales in 2000.2 This is not a coincidence, because many chiral separations are characterized by a low selectivity and their cost is dominated by solvent consumption and expensive chiral stationary phases. SMB technology can satisfy the needs for both the development and further production stages of chiral drugs1. A thorough understanding of the SMB process has ledtooptimizationtoolswithdifferentlevelsofcomplexity3-5 and to new SMB schemes, e.g., VARICOL,6 PowerFeed,7,8 which allow for further improvements in the efficiency of SMB units. On the other hand, robust operation of SMB units at their optimal operating conditions is still an open issue.9 SMB separations close to their optimal operating conditions in terms of productivity and solvent consumption are highly sensitive to disturbances, e.g., aging of the stationary phase; feed composition or temperature changes; and uncertainties in the system, such as column characteristics or adsorption isotherms that are hard to measure precisely. This can lead to suboptimal operating conditions in the bestcase scenario and off-specification production in general.10 Therefore, it is a common practice to operate SMB units far from the optimal operating conditions to guarantee a certain level of robustness. Because SMB chromatography is a slow process, it can, in principle, be controlled manually when it is operated at robust operating conditions, i.e., operating conditions far from the optimal ones. On the other hand, the need for feedback control of SMB units becomes crucial when these units are operated close to their optimal operating conditions, where any type of uncertainty can lead to significant consequences as mentioned above. Automatic control of SMB units has the potential to deliver the full economic potential of the technology, and several different SMB control approaches have been proposed recently.11-14 The main drawback of these approaches is the necessity of obtaining accurate physical data on the separation system under consideration, which is already the bottleneck for the optimal operation of SMB units. In this work, a recently introduced model predictive control (MPC) method, i.e., the so-called repetitive model predictive control (RMPC) that was particularly formulated for periodic process control problems,15 is applied. The feasibility of the on-line optimization and control concept in simulation has been shown previously for SMB separations applied to systems characterized by linear isotherms.16-19 This paper addresses the on-line optimization and control of SMBs applied to systems with nonlinear adsorption isotherms. This is a key improvement, as it is well-known that SMBs are mostly operated under overloaded chromatographic conditions, corresponding to nonlinear competitive adsorption behavior. The most important difference from other control concepts is that minimal information on the separation system is sufficient. The paper is organized as follows: In the second section, the SMB unit and the binary system under

consideration are described. In the third section, the implemented automatic control concept is introduced. The fourth section addresses the performance of the developed controller for several simulated scenarios, and finally, conclusions are drawn in the last section. 2. Process Description The SMB unit under consideration is a closed-loop four-section eight-column plant arranged in a 2-2-2-2 configuration (see Figure 1). The columns constituting the unit have a length and diameter of 10 and 1 cm, respectively. The dynamic model of the SMB process is obtained by interconnecting the simulation models of single chromatographic columns for which an equilibrium dispersive model (EDM) is employed.20

h

/ ∂qi,h ∂ci,h ∂ci,h Qh ∂ci,h + (1 - h) + ) hDap,i 2 ∂t ∂t Acr ∂z ∂z

(1)

In the equation above, Dap,i is the apparent axial dispersion coefficient lumping the mass-transfer resistance and axial dispersion. Qh and h are the volumetric flow rate and the total packing porosity in the hth column, respectively. The relation between the concentrations of component i in the solid phase (q/i ) and in the liquid phase (ci) is given by the adsorption isotherm model. / ) f(ci,h) qi,h

i ) A, B

(2)

A number of isotherm models for liquid-solid equilibrium are suggested and available in the literature.20,21 Any type of isotherm model that suits best to describe the system under consideration can be used in the EDM. It is assumed that the columns constituting the SMB unit are initially saturated with mobile phase in equilibrium with the stationary phase and that the concentrations of both components are zero along all the columns.

ci,h ) 0

∀ z, t ) 0 for h ) 1, ..., 8

(3)

The assumed boundary conditions are as follows in ) ci,h|z)0+ ci,h

∂ci,h | )0 ∂z z)L

∀ t for h ) 1, ..., 8

(4)

∀ t for h ) 1, ..., 8

(5)

in is the concentration of component i fed into where ci,h the column h. The node balances given below complete the mathematical modeling of the SMB process

Q1 ) Q8 + QI/O 1

(6)

in out I/O ) Q8ci,8 + QI/O Q1ci,1 1 ci,1

(7)

I/O Qh+1 ) Qh + Qh+1

for h ) 1, ..., 7

(8)

in out I/O I/O ) Qhci,h + Qh+1 ci,h+1 for h ) 1, ..., 7 (9) Qh+1ci,h+1 out where ci,h is the concentration of component i at the outlet of the column h. QI/O h identifies the flow rate of the inlet or outlet stream entering or leaving the SMB I/O is the concentration loop just before column h, and ci,h

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3897 Table 1. Flow Rates of the Inlet/Outlet Streams Entering/Leaving the SMB Loop and the Concentration of Component i in the Corresponding Stream at Successive Switch Periods 0-t* h 1 2 3 4 5 6 7 8

t*-2t*

QI/O h

chi,I/O

QI/O h

chi,I/O

QD 0 -QE 0 QE 0 -QR 0

cD i

0 QD 0 -QE 0 QE 0 -QR

0 cD i 0 out ci,3 0 cFi 0 out ci,7

0 out ci,2 0 cFi 0 out ci,6 0

of component i in the corresponding stream. Note that volumetric flow rates in columns belonging to the same section are identical, i.e., Q1 ) Q2 ) QI, Q3 ) Q4 ) QII, Q5 ) Q6 ) QIII, Q7 ) Q8 ) QIV for the SMB configuration given in Figure 1 and Q2 ) Q3 ) QI, Q4 ) Q5 ) QII, Q6 ) Q7 ) QIII, Q8 ) Q1 ) QIV after the first switch. The hybrid nature of the process is explicitly considered by enforcing the inlet/outlet switching mechanism on the node balances. For instance, the node balances applicable to the SMB configuration given in Figure 1 during the first and second switch intervals are defined in Table 1. The system of partial differential equations (PDEs) of the SMB model can be reduced to a system of ordinary differential equations (ODEs) by a finite-difference approximation of the first-order spatial derivatives. The boundary conditions and the node balances can be substituted in the system of ODEs of the SMB model. It is a common practice to replace the apparent dispersion term by numerical dispersion to obtain a computationally efficient solution of the EDM.22 The system of ODEs is integrated in time by a commercial stiff integrator. Note that a different set of ODEs is applicable for each port configuration of the SMB unit because the input/output switching mechanism is explicitly imposed on the node balances. For the sake of clarity, the simulated SMB plant will be referred to as the “SMB plant” throughout this work. The binary system under consideration consists of the enantiomers of the antitussive agent guaifenesin, and it is described by the binary Langmuir adsorption equilibrium isotherm / qi,h )

Hici,h 1 + KAcA,h + KBcB,h

i ) A, B

(10)

where Ki and Hi are the equilibrium and Henry’s constants, respectively, of the ith species. A and B indicate the more-retained enantiomer, i.e., (S)-(+)guaifenesin, and the less-retained enantiomer, i.e., (R)(-)-guaifenesin, respectively. The isotherm parameters are taken from available literature data, i.e., HA ) 3.5, HB ) 1.4, KA ) 0.0550, and KB ) 0.0135.23 The variations in the column properties due to velocity differences are neglected, and the apparent axial dispersion is defined such that each column has 100 theoretical stages with respect to each solute. A final remark is that the differences due to packing heterogeneity are considered by assigning different porosity values to each column constituting the SMB unit. The average porosity of the columns constituting the SMB unit is taken as ave ) 0.7, whereas the single-column

Figure 2. Scheme of the on-line optimization-based automatic control concept.

porosities are given in section 4 together with each scenario under consideration. 3. Control Concept It has been shown that the possible gain from optimization of SMB processes is significant.3,4,10 On the other hand, regardless of their complexity, the performance of optimization algorithms is limited by the quality of the available physical data. Unfortunately, it is also a fact that the precise measurement of multicomponent adsorption isotherms, i.e., the key physicochemical properties characterizing the separation, is a rather difficult task. It is common experience, especially at the development stage of new chiral drugs, that the pure enantiomer is not available or the amount is very limited; therefore, measurement of the isotherm of the target drug might not be possible at all. Another issue is the significant dependency of the optimal conditions on the operating parameters, i.e., feed concentration/composition, and the physical parameters of the system, i.e., the adsorption isotherm and column properties, which are both subject to change. Therefore, it is hard to discuss a set of a priori fixed optimal operating conditions for SMB units. Recharacterization of the system and reoptimization of the process are required to account for the changes. Thus, the automatic control of SMB units presents itself as a dynamic optimization problem rather than a simple regulation or tracking problem, and integration of feedback control and on-line optimization is desirable to exploit the full economic potential of SMB technology. The challenges in the automatic control of an SMB unit arise from its underlying characteristics, such as its nonsteady-state, nonlinear, and hybrid nature (inlet and outlet port switches), as well as the long delays in exhibiting the effect of disturbances. Fortunately, firstprinciple SMB models with different complexities are available, providing a thorough understanding of the process,20 and such models can be utilized for control purposes. In this work, an explicit model of the process is used to predict the future evolution of the plant on the basis of its current state (see Figure 2). The internal flow rates in four sections, i.e., Qj, where j ) I, ..., IV, and the concentration levels in the two outlet streams, i.e., extract and raffinate, are chosen as the manipulated and measured variables, respectively. Detailed information on state-of-the-art on-line concentration measurement techniques can be found elsewhere.24 The RMPC formulation requires a defined period; therefore, the global period of the process, i.e., a complete cycle,


comprising eight switch times, is fixed a priori, and the switching time (t*) is not employed as a manipulated variable (t* ) 400 s in this case). In practice, internal flow rates can be manipulated by acting on QI and on the external flow rates QE, QE, and QR. A periodic timevarying Kalman filter provides a recursive correction of the model errors by combining the model estimation and the available measurements in an optimal sense.25 The performance of the plant is optimized over a defined future time window, i.e., the prediction horizon, and the optimal changes in the manipulated variables, i.e., the changes resulting in the optimal performance, are determined for a chosen control horizon. Here, the control and prediction horizons are chosen as 1 and 2 cycles, respectively. 3.1. Simplified SMB Model. The model of the process is a critical element of the control scheme. A model that represents the key characteristics of the process, i.e., its hybrid and periodic nature, explicitly is desirable for automatic control purposes. We make use of the first-principle modeling concept described in section 2 and obtain a linearized time-varying SMB model for on-line optimization and control. Another critical issue is the amount of knowledge required for controller design. It is desirable to have a control strategy that is based on minimal information about the system. Note that the information about the competitive adsorption isotherm is not easily accessible for the reasons mentioned above, whereas the linear adsorption isotherm parameters, i.e., Henry’s constants, can be determined easily and precisely by measuring the retention time of a pulse of each species at very low fluid-phase concentrations corresponding to linear chromatographic conditions, i.e., cFT ) cFA + cFB approaches zero. Moreover, their determination does not require pure enantiomers but only the racemate, i.e., 1:1 mixture of the enantiomers, because there is no competition between the species under linear chromatographic conditions. It is worth mentioning that the dynamics of SMB units operating under nonlinear and linear conditions differ in many ways, but on the other hand, the underlying dynamics and operating principles are very similar, e.g., the functionalities of the sectional flow rates and their influences on the outputs. Therefore, the proposed control concept is based only on the knowledge of the linear adsorption isotherm, i.e., the isotherm model given in eq 2 is described with the linear isotherm equation / qi,h ) Hici,h

i ) A, B

(11)

where HA ) 3.5 and HB ) 1.4 and the overall average packing characteristic of the SMB unit is described by ave ) 0.7 in our case. The modeling concept is briefly described here; a detailed description can be found elsewhere.17,19 With reference to the modeling concept given in section 2, each set of ODEs belonging to different input/output configurations can be recast in the following form

∂c ) f p(c,Q) ∂t cout ) g p(c)

for p ) 1, ..., 8

(12)

where f p is a vector-valued function and p is an index indicating the different input/output configurations of

the SMB unit. Each ODE set comprises neq ) ns × ng × ncol equations ,where ns, ng, and ncol are the number of species, number of grid points defined along each column to discretize the first-order space derivatives, and number of columns constituting the SMB unit, respectively (ns ) 2, ng ) 40, and ncol ) 8 in our case). Q and c are vectors of the volumetric flow rates in the four sections, i.e., QI, ..., QIV, and the concentration values along the unit, i.e., ci,h,g(t) for i ) A, B; h ) 1, ..., 8; and g ) 1, ..., ng, respectively. g p is another vectorvalued function relating the process outputs, cout, to the state variables, c. cout comprises the process outputs, which are the concentration levels of the two components in the raffinate and the extract outlets, i.e., cRA(t), cRB(t), cEA(t), and cEB(t). The nonlinear terms in the ODEs, namely, the convective terms containing the product of concentrations and flow rates, can be linearized at their steadystate values, but because the cyclic steady-state profile of the SMB is time-varying, the duration of a cycle is divided into N sample points (N ) 64 in our case), and the nonlinear model of the system (eq 12) is linearized at each sample point using the corresponding cyclic steady-state values, i.e., reference values. This procedure results in N different continuous-time linear statespace model equations describing the dynamics of a complete cycle. The discrete-time state-space equations are obtained by using a zero-order hold on the inputs and integrating the continuous-time linear equations with a time step equal to the duration between two successive linearization sample points, i.e., ncolt*/N.

xk(n + 1) ) A(n) xk(n) + B(n) uk(n) yk(n) ) C(n) xk(n) for n ) 0, ..., N - 1

(13)

The transition from one cycle to the next can be written as

xk+1(0) ) xk(N)

(14)

where k is the cycle index and n is the time index running within the cycle index. x, u, and y are the state, input, and output vectors, respectively, defined in terms of deviation variables. For instance, the state vector comprising the internal concentration values along the unit is defined as x(n) ) c(n) - cref(n), and similarly, the manipulated variable vector is defined as u(n) ) Q(n) - Qref(n). Equation 14 implies that the space composition profiles at the end of the previous cycle, i.e., k, are used as initial conditions for the next cycle, i.e., k + 1. There is a critical issue concerning the sampling of the plant outlets. If one inspects the concentration changes at the extract and raffinate outlets between two successive switches, one can see that they attain their highest values at the beginning and end of the switching time at the extract and raffinate outlets, respectively, i.e., they show opposite behavior such that, when the concentrations of both components are decreasing in the extract, they are increasing in the raffinate and vice versa (see the concentration profiles of extract and raffinate outlets given in Figure 3). Therefore, we defined the output of the model in such a way that the extract outlet is sampled just after the switch, whereas the raffinate outlet is sampled just before the switch (see the sampling points indicated by O and 0 in Figure

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3899

Figure 3. Sampling of the plant outlets: O, sampling points for the extract outlet; 0, sampling points for the raffinate outlet.

3 for the extract and raffinate outlets, respectively). This method is straightforward to implement because the necessary information is part of the states, i.e., internal concentration profile, and can be extracted easily. From a modeling point of view, this leads to a model in which the dynamics of the most critical period of time for both the extract and raffinate outputs are well taken into account. It also allows for a more accurate calculation of the outlet purities. On the other hand, in practice, the discontinuity due to the port switching cannot result in two different outlet concentration values at the same time. In other words, in reality, one cannot see a difference in concentration levels at the plant outlets just before the switch and just after the switch. Note that the order of the model is equal to the number equations constituting the ODE system, i.e., neq ) ns × ng × ncol, which is 640 in this case, and it is too large for on-line optimization applications. The order of the model is reduced to 40 by the balanced model reduction technique.26 Because available model reduction techniques are applicable to time-invariant systems, the model is first lifted for a complete cycle as a cycle-to-cycle time-invariant model. Then, it is unlifted again into the time-to-time formulation after the model reduction. Details of the model reduction procedure are given elsewhere.19 Repetitive model predictive control (RMPC) was formulated particularly for periodic processes. Its underlying assumption is that the possible model prediction errors and effect of disturbances on the plant outputs tend to repeat from period to period. Therefore, it is possible to estimate these errors and effects by utilizing the available output measurements and to correct for them by storing the information for the current cycle and passing it to the future cycles as part of the states. The obtained time-varying linear model (eqs 13 and 14) constitutes the basis for the formulation of the control problem along the lines of RMPC. The special formulation of RMPC is not given here; we refer the interested reader to the available literature for its description and implementation details.15,19 3.2. Optimization. For a given plant layout, i.e., total number of columns constituting the SMB unit, their dimensions and distribution in the sections, and a predefined switching time t*, the optimization of the SMB process reduces to the maximization of the productivity, i.e., throughput, and minimization of the

solvent consumption given that the process specifications arising from the hardware limitations, such as maximum allowable pressure drop in the columns or for the pumps, and the product specifications, i.e., the purities of the raffinate and extract outlets, are fulfilled. These specifications can be considered explicitly as constraints in the optimization problem. The switching time is not included as a decision variable in the optimization problem, because the SMB model used in the control algorithm is developed on the basis of a defined process period. Note that the raffinate and extract purities can be calculated directly from the model outputs (eqs 13 and 14) and the corresponding outlet flow rates, i.e., QE ) QI - QII and QR ) QIII - QIV. On the other hand, because of the cyclic steady-state nature of SMB units, the concentrations of the species at the plant outlets and so the outlet purities show a periodic time-varying behavior within a cycle, so that defining the product specifications on the instantaneous outlet purities would be too conservative. Of main interest is the delivery of products with the given specifications averaged over cycles. Thus, the constraints on the product specifications are defined on the average purities of the extract and raffinate outlets over the prediction horizon, i.e., np ) 2 cycles. The average purity expressions are nonlinear in the outlet flow rates and concentrations; therefore, they are linearized recursively and introduced to the optimization problem as linear inequality constraints.

Pave g Pmin E E

(15)

min Pave R g PR

(16)

The output concentration values must be nonnegative so that the predicted product purities are calculated correctly. This can be enforced by the following constraints on the predicted outputs p cnout g0

(17)

The pressure drop limitations for the columns in use can be fulfilled by defining upper bounds on the internal flow rates.

Qj e Qmax for j ) I, III

(18)

0 e Qexternal

(19)

Also

0 e Qj

for j ) I, ..., IV

(20)

Excessive sudden pressure changes in the columns constituting the SMB unit can be avoided by introducing constraints on the maximum allowable internal flow rate changes.

|∆Qj| e ∆Qmax j

for j ) I, ..., IV

(21)

For on-line applications, it is common practice to transform the hard constraints on states, such as those given by eqs 15-17, into soft ones by introducing slack variables in order to avoid infeasibility problems. Therefore, the constraints on the product specifications (eqs 15 and 16) and the plant outputs (eq 17) are defined as


soft constraints by introducing the slack variables s1, s2, and s3. min Pave - s1 E g PE

(22)

min - s2 Pave R g PR

(23)

p g 0 - s3 cnout

(24)

si > 0

(25)

Excessive usage of slack variables can be avoided by penalizing them in the cost function of the optimization problem

min (λDQnDc - λFQnFc + λ1s1 + λ2s2 +λ3s3) Qnc,s

(26)

where nc stands for the control horizon (one cycle in our case) and QnDc and QnFc are the cumulative solvent consumption and throughput, respectively, over the control horizon. Qnc comprises the manipulated variables, i.e., internal flow rates, for the whole control horizon. λD and λF are the weights of the cumulative desorbent and throughput terms. λ1, λ2, and λ3 are the weights of the corresponding slack variables in the cost function, which are kept relatively large in order to punish the excessive use of the slack variables, i.e., s1, s2, and s3. In this work, the weights λD and λF were chosen as 4 and 20, respectively, whereas all of the weights for the slack variables, i.e., λ1, λ2, and λ3, were chosen as 100. Note that the weights should be chosen by considering the order of magnitude of the corresponding terms in the cost function. We refer to the available literature for the formulation and implementation details of the optimization problem.19 The linear constraints (eqs 18-25) and the linear cost function (eq 26) constitute a linear program (LP), for which well-established algorithms and commercial solvers are available. On-line solution of the constructed optimization problem provides the optimal flow rate sequence, which is implemented according to a receding horizon strategy, i.e., the element corresponding to the current time n is implemented on the plant and the remaining elements are discarded. As new information becomes available with the new measurements collected from the plant, a new optimization problem is solved at time n + 1 using the updated state of the plant. It is important to note that the constraints (eqs 1825) and the cost function (eq 26) comprising the optimization problem should be considered as a formulation that can be adjusted for specific applications. On the other hand, because the control problem is formulated as a general optimization problem rather than a tracking or regulation problem and because the cost function is in terms of manipulated variables only, it does not make sense to add penalty terms on the manipulated variables to avoid their excessive use, which is a common practice in MPC formulations to achieve a smooth operation. Here, we enforce identical flow rates over the whole control horizon for the internal flows of sections II and III. Also, the controller is allowed to change the flow rates in sections I and IV only once at the beginning of each cycle. These flow rates are then kept constant for the rest of the cycle. This was done to demonstrate the possibility of enforcing smoother flow rate changes if desired. The operation of the SMB unit

with varying flow rates, i.e., similar to the so-called PowerFeed7,8 operating mode, was reported previously.19 Clearly, different alternatives can be proposed to suit specific applications. ILOG CPLEX 7.0 was used as the commercial LP solver, and the maximum calculation time to solve the LP was found to be 1.2 s on a PC with a 3-GHz processor, which is far below the sampling time. The available CPLEX license was server-based, and one can expect even shorter calculation times with a local license. 4. Controller Performance “Triangle theory” provides explicit criteria for the choice of the operating conditions of SMB units to achieve the desired separation performance. These criteria are very practical because they allow for the determination of optimal and robust operating conditions of SMB units using simple algebraic equations.10 According to triangle theory, the key operating parameters are the ratio of the net fluid- and solid-phase flow rates in each section of the SMB unit

Qjt*-V

mj )

V(1-)

j ) I, II, III, IV

(27)

For the sake of clarity, we provided a short summary of the triangle theory results; a detailed description of the theory and its experimental verification can be found elsewhere.10 In the framework of triangle theory, the separation conditions are derived on the basis of the equilibrium theory model, i.e., on the assumptions that axial mixing and mass-transfer resistance are negligible and adsorption equilibrium is established everywhere in the columns instantaneously. The necessary and sufficient conditions for the complete separation of a system characterized by a linear adsorption isotherm (eq 11) are given by the following inequalities

HA < mI < ∞

(28)

HB < mII < mIII < HA

(29)

0 < mIV < HB

(30)

under the assumption that nonporous particles constitute the solid phase. The first and last constraints given by eqs 28 and eq 30 guarantee the complete regeneration of the liquid and solid phases, respectively. On the other hand, the constraints on operating parameters mII and mIII guarantee the complete separation of the species in the two central sections. mII < mIII implies a positive feed flow rate. Given that the solid and liquid phases are regenerated completely, i.e., inequalities 28 and 30 are fulfilled, the position of the operating point in the (mII, mIII) plane allows one to make a prediction of the separation performance (see Figure 4). Triangle Theory provides the necessary and sufficient conditions on the flow rate ratios mj also for the systems characterized by nonlinear Langmuir adsorption equilibrium isotherms (eq 10), i.e.

HA ) mI,min < mI < ∞

(31)

mII,min < mII < mIII < mIII,max

(32)

0 < mIV < mIV,max

(33)


Figure 4. mII-mIII operating spaces for linear and nonlinear isotherms.

where

mII,min ) f(mIII,Hi,Ki,cFi )

i ) A, B

(34)

f(mII,Hi,Ki,cFi )

i ) A, B

(35)

mIII,max )

1 mIV,max ) {HB + mIII + KBcFB(mIII - mII) 2

x[HB + mIII + KBcFB(mIII - mII)]2 - 4HBmIII}

(36)

Operating parameters mI and mIV have explicit lower and upper bounds, and the constraints on mI does not depend on the other flow rate ratios (see eqs 31 and 33). On the other hand, the upper limit for mIV is an explicit function of flow rate ratios mII and mIII (eq 36). The constraints on mII and mIII (eq 32) do not dependent on flow rate ratios mI and mIV; hence, they define an operating region in the (mII, mIII) plane as in the case of the linear isotherm (see Figure 4). The boundaries of the complete separation region are functions of the adsorption isotherm parameters and the feed composition (eqs 34 and 35).10 It is important to note that the nonlinear Langmuir isotherm (eq 10) approaches the linear isotherm (eq 11) at very low fluid-phase concentrations, i.e., where cFT ) cFA + cFB approaches zero. As a consequence, the operating constraints on the flow rate ratios mj for the Langmuir isotherm (eqs 31-33) approach those for the linear adsorption isotherm (eqs 2830) at low concentrations. In this section, several scenarios illustrating the performance of the controller under rather difficult conditions are considered. The first example shows how

the controller based on only linear isotherm information can find the correct operating conditions for the separation of a system characterized by a competitive nonlinear Langmuir adsorption isotherm and, moreover, can optimize the economics of the operation. It also shows how the controller responds to a step change in the total feed concentration to recover the required purities and to drive the operation of the plant to the new optimal operating conditions, which were altered significantly by the disturbance. The second example addresses the controller’s performance for an SMB unit comprising columns with low efficiency and different packing characteristics. Finally, the last example illustrates how the controller handles a technical difficulty arising from the on-line measurement system, which is typical in practice. In all cases, the minimum purities for both outlets ) Pmin ) 99%. Zero-mean are chosen as 99%, i.e., Pmin E R white noise with a standard deviation of 2% of the measured concentration value is added to all of the outlet measurements. The control algorithm is initiated by open-loop, i.e., without the controller, operation of the SMB unit for a complete cycle. Because it is assumed that the nonlinear adsorption isotherms are unknown to the SMB operator, it is also assumed that an initial guess of the necessary startup operating conditions based on the isotherm information, e.g., by using triangle theory, is not possible. Therefore, the plant is simply started-up with the same operating parameters ref as used for linearization, i.e., mref I ) 3.57, mII ) 1.3, ref ref mIII ) 3.59, and mIV ) 1.37. Because the control concept aims to optimize the economics of the operation, the production cost is given for each scenarios in the following. The production cost F is defined similarly to the cost function of the optimization problem (eq 26), but the contribution of the slack variables is excluded. ave F ) λDQave D - λFQF

(37)

In this equation, λD and λF are the constants reflecting the prices of the solvent and product, respectively, and they are the same as used in the optimization problem, ave i.e., λD ) 4 and λF ) 20. The quantities Qave D and QE are the average values of the solvent and feed flow rates implemented throughout a cycle, respectively. 4.1. Operation under Overloaded Chromatographic Conditions. This example illustrates the ability of the controller based on only the linear isotherm parameters, i.e., the Henry constants, to find the correct operating conditions for the SMB unit operated under nonlinear competitive adsorption conditions, first to fulfill the required product specifications and then to optimize the separation performance. In this case, the feed mixture has a total concentration of 9 g/L, i.e., cFA ) 4.5 g/L and cFB ) 4.5 g/L. As discussed in detail in section 4.3, column-to-column variations are hard to avoid in practice. Therefore, it is assumed that the single-column porosities of the columns constituting the SMB unit are different, i.e., the average porosity is ave ) 0.7 and the standard deviation is 2.7%. The assigned porosity values are given in Table 2. The plant is started-up and operated in open-loop mode for a complete cycle with the reference operating conditions, i.e., mref j , before the controller is activated. The startup operating conditions are within the pureextract region in the (mII, mIII) plane for the nonlinear separation and far from the corresponding complete

3902 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 Table 2. Porosities of the Columns Constituting the SMB Unit column, h

h

column, h

h

1 2 3 4

0.735 0.714 0.658 0.728

5 6 7 8

0.721 0.686 0.693 0.665

separation area. The parameter mref I barely fulfills the operating constraint on section I, whereas mref IV is not low enough to achieve complete regeneration of the liquid phase in section IV, thus leading to pollution of the extract outlet. Figure 5 shows the evolution of the product purities at the controlled plant outlets. The first 100 cycles of the operation comprising the startup phase and the steady-state operation of the SMB unit correspond to the scenario given above; the rest of the operation, i.e., after the disturbance, is discussed in section 4.2. It can be seen that the controller can maintain the specified extract purity at all times. On the other hand, it takes 11 cycles to fulfill the required raffinate purity, which is fairly satisfactory considering that the startup operating conditions (indicated as the point p1 in Figure 6) are quite far from the conditions necessary for pure raffinate. One can also see that the production cost (F) is minimized and reaches a steady-state value, as do the outlet purities, after 30 cycles. Figure 6 illustrates the trajectory of the operating point in the (mII, mIII) plane under the action of the controller. The corresponding mj values are calculated with the implemented internal flow rates averaged over one cycle and using the average porosity of the columns. Note that the operating conditions are driven into the complete separation region within five cycles and the steady-state operating point indicated by p2 in Figure 6 is very close to the vertex of the triangle, which is the optimal operating point in terms of productivity and solvent consumption per unit mass of solid phase according to the triangle theory. The instantaneous external flow rates of desorbent, extract, raffinate, and feed calculated from the implemented internal flow rates are given in Figure 7. 4.2. Step Change in the Feed Concentration. The feed concentration is one of the most important variables characterizing an SMB separation in the case of nonlinear systems. It has a significant impact on the operating conditions to achieve complete separation and regeneration. Triangle theory gives explicit bounds on

Figure 5. Average output purities for the controlled plant over cycles and production cost (F).

Figure 6. Controller action represented in mII-mIII operating parameter space. mj values are calculated with implemented internal flow rates averaged over a cycle and the average porosity of the columns constituting the SMB unit, i.e., ave ) 0.7. p1, startup operating conditions; p2, steady-state operating conditions before the disturbance; p3, steady-state operating conditions after the change in total feed concentration takes place.

Figure 7. Instantaneous external flow rates calculated from the internal flow rates implemented by the controller.

the operating parameters as a function of feed composition (see eqs 31-36) that provide a thorough insight into the effects of the feed composition not only on the necessary operating conditions but also on the optimal operating conditions. In practice, SMB separations are designed and optimized with a priori fixed feed concentrations and with the assumption that this feed composition is not subject to change. On the other hand, differences between different feed batches are a part of practical experience. It is common practice to adapt the operating conditions with the help of available shortcut design tools, e.g., based on triangle theory, to account for the changes in the feed concentration/composition. The effort and time needed to find the necessary operating conditions that fulfill the product specifications depend on the amount and accuracy of the available information on the separation system, i.e., primarily the nonlinear adsorption isotherm, as well as the experience of the SMB operator. If SMB chromatography is used as part of a continuous process, this approach might not be possible.


This example assesses the performance of the controller when the total feed concentration changes significantly during steady-state operation, i.e., at cycle 100 in Figure 5. It is increased by more than 30% from 9 to 12 g/L, i.e., cFA ) 6 g/L and cFB ) 6 g/L. The change in the feed concentration necessitates different operating conditions in sections II-IV. In particular, the complete separation region in the (mII, mIII) plane becomes smaller and sharper, and accordingly, the unit operation becomes less robust (see the complete separation region with dotted boundaries in Figure 6). The vertex of the triangle, i.e., the optimal operating point, shifts downward and to the left, which implies that operating parameters mII and mIII have to be decreased in order to maintain the optimal performance. On the other hand, the vertex moves closer to the diagonal, i.e., the difference (mIII - mII) decreases, which means that the feed flow rate has to be decreased. The complete separation region reflects the increasing nonlinearity of the SMB dynamics caused by the higher feed concentration. Figure 5 shows that the controller can adapt the operating conditions so that the obtained extract purity fulfills the specifications at all times and the required raffinate purity is achieved again within five cycles after the cFT change. The process reaches the new steady-state operating conditions after about 10 cycles. The higher feed concentration necessitates lower feed flow rates, i.e., lower (mIII - mII) values, and higher solvent consumption to maintain the proper regeneration of the liquid phase in section IV, i.e., lower mIV values according to eq 36. As a result, the production cost in terms of throughput and solvent consumption increases after the change in the total feed concentration (see Figure 5). Actually, the overall productivity, which is proportional to the product of the total feed concentration and the feed flow rate, increases more than 20%. The decreased feed and increased solvent flow rates implemented by the controller are illustrated in Figure 7. It is worth noting that the new steady-state operating conditions indicated as point p3 in Figure 6, are very close to the vertex of the new complete separation region, i.e., the optimal operating point for the system with the new total feed concentration 12 g/L. As a final comment about this example, it is worth noting that the disturbance considered here should be regarded as an extreme case. In real plant operation, the changes in feed composition are going to be smaller than the 30% change considered here. It is also possible that the feed composition changes gradually and not suddenly. In all cases, the task of the controller would be easier than that considered here.19 This implies that one can expect better performance for the more realistic situations. 4.3. Column Variations and Efficiency. During the design of an SMB separation, it is common practice to assume that the set of chromatographic columns comprising the SMB unit have identical packing characteristics, i.e., porosities, although this is hardly probable because of the practical impossibility of packing columns in exactly the same way.27 Therefore, in reality, the columns constituting an SMB unit have different column packing characteristics that would lead to different retention times of the feed components in each column. One can see from eq 27 that the flow rate ratio in each section, i.e., the mj value, is a function of the column porosity. As a result, one might conclude that, because the columns have different porosities, each

Table 3. Porosities of the Columns Constituting the SMB Unit column, h

h

column, h

h

1 2 3 4

0.595 0.700 0.600 0.805

5 6 7 8

0.800 0.650 0.750 0.700

column in section j will experience a different mj value. According to the results of triangle theory (eqs 31-33), complete separation requires that the operating conditions of all columns in section j fulfill the constraints for that section, e.g., all of the columns in sections II and III should operate with mj values that lie within the triangle given in Figure 4. It has been shown previously that column-to-column variations can deteriorate the separation performance, especially for SMB units with a small number of columns, and their influence becomes crucial if the columns have low efficiencies. On the other hand, SMB units with more than one column per section are more robust with respect to changes in the column characteristics.28,29 The example presented here analyzes the controller performance for an SMB unit consisting of columns with significantly varying porosities. The porosity values of the single columns have a standard deviation of 7.67%, whereas the overall porosity of the unit, ave, is 0.7 (see Table 3). Moreover, the efficiency of the chromatographic columns is reduced by defining the apparent axial dispersion so that each column has 50 theoretical stages for each solute. The system under consideration has a total feed concentration of 8 g/L, i.e., cFA ) 4 g/L and cFB ) 4 g/L. Because the SMB under consideration is an eightcolumn unit with a 2-2-2-2 configuration, each column constitutes a part of each section twice in every cycle. The columns constituting each section for eight different port configurations (indicated by index p), together with the corresponding average porosities of the sections, are given in Table 4. Figure 8 illustrates the evolution of the product purities for the controlled SMB unit. The controller is activated after open-loop operation of the plant for one complete cycle, as in the previous example. It can be seen that the controller can maintain the purity specification for the extract outlet at all times, whereas the product specification for the raffinate outlet is fulfilled after cycle 17. It is clear that the better the startup operating conditions, the shorter time required for the controller to fulfill the purity requirements. It can also be observed that the economics of the operation is optimized after the product specifications are fulfilled (see the production cost in the same figure) and the steady-state is reached after 40 cycles. The internal flow rates implemented by the controller are illustrated in Figure 9. Note that the steady-state flow rates are constant, as in conventional SMB operation. On the other hand, the flow rate ratios mj vary from switch to switch within a cycle, because the pairs of columns constituting the sections as well as their porosities are different for each configuration. The mj values of section j for each configuration calculated with the steady-state internal flow rates are given in Table 4. Figure 10 shows the steady-state operating conditions in the (mII, mIII) plane for each port configuration, i.e., p ) 1, ..., 8. One can see that the average operating point, i.e., the point indicated by 0 in the preceding figure, is very close to the vertex of the triangle, which

3904 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 Table 4. Column Configuration for Each Port Configuration, Corresponding Average Sectional Porosities, and Operating Parameters of Each Section p 1 2 3 4 5 6 7 8

section I

section II

section III

section IV

column

I

mI

column

II

mII

column

III

mIII

column

IV

mIV

1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-1

0.648 0.650 0.703 0.803 0.725 0.700 0.725 0.648

3.334 3.350 3.765 5.165 3.992 3.742 3.992 3.334

3-4 4-5 5-6 6-7 7-8 8-1 1-2 2-3

0.703 0.803 0.725 0.700 0.725 0.648 0.648 0.650

1.180 1.272 1.195 1.179 1.195 1.152 1.152 1.153

5-6 6-7 7-8 8-1 1-2 2-3 3-4 4-5

0.725 0.700 0.725 0.648 0.648 0.650 0.703 0.803

2.931 2.770 2.931 2.506 2.506 2.517 2.785 3.689

7-8 8-1 1-2 2-3 3-4 4-5 5-6 6-7

0.725 0.648 0.648 0.650 0.703 0.803 0.725 0.700

1.308 1.241 1.241 1.242 1.285 1.430 1.308 1.283


Figure 9. Instantaneous internal flow rates implemented by the controller.

represents the optimal operating conditions according to triangle theory. It is clear that the purities of both outlets averaged over one switch period vary because of the varying mj values (see Figure 10), whereas the average purity over the global period, i.e., one cycle consisting of eight switches, is constant (see Figure 8). In principle, the process is periodic with period equal to t*, and it would be possible to model the process on the basis of periodic dynamics over successive switches. This would lead to a smaller model, i.e., the size of the model would be ncol times smaller, and to a smaller optimization problem to be solved on-line than with the approach adopted here. On the other hand, in practice, the process is periodic only over a complete cycle, because the effects of extra-column dead-volume differences among the sections and of column-to-column

variations repeat themselves over cycles. This example shows the advantage of formulating the process model on the basis of the global period, which allows one to correct for periodwise-persistent effects such as those discussed here. It is also worth noting that, because the purity constraints are defined as the average purities over cycles, the controller is not affected by variations of the average purities over switches, as long as their average values fulfill the specifications over the whole cycle. 4.4. Limitations Imposed by the On-line Monitoring System. State-of-the-art measurement systems for the on-line monitoring of absolute enantiomer concentrations are based on the coupling of a UV detector with a polarimeter at each outlet of the SMB unit. The UV detector provides the absorbance of the mixture, which is proportional to the sum of the outlet concentration, i.e., cRA + cRB and cEA + cEB, whereas the polarimeter provides a signal that is proportional to the concentration difference, i.e., cRA - cRB and cEA - cEB. The two detectors, i.e., UV and polarimeter, can be calibrated via standard calibration techniques, and their signals can be combined to calculate the absolute concentration of each species at the outlets.24 Recently, detectors for high-pressure applications have been introduced, i.e., UV detectors and polarimeters with high-pressure cells. On the other hand, even though they can be regarded as robust in a constantpressure environment, available polarimeters show significant sensitivity toward pressure variations, which can occur easily as a result of port switching, columnto-column differences, or flow rate changes during SMB operation. Therefore, it is common practice to locate the detectors at the outlets of the unit, i.e., after the raffinate and extract pumps where the pressure is atmospheric.24 Actually, locating the detectors at the low-pressure side after the pumps instead of directly at the column outlets has a significant drawback for control applications, because the unavoidable dead volume in the line from the internal loop of the SMB to the detectors, i.e., the volume due to the multiposition valves, the piping, and the pumps, introduces mixing effects that have an impact on the outlet concentration profiles. This implies that the measured concentration profiles are smoother than those that would be observed at the withdrawal position in the internal SMB loop and that would correspond to the concentration profiles normally calculated through simulations. One can model the effect of the dead volume on the output profiles, e.g., by adding a well-stirred tank connected to each outlet of the SMB unit. Figure 11 illustrates the concentration profile of the extract outlet for two different cases. The solid lines are the rather sharp profiles calculated with zero dead volume. The dashed line shows the significantly broader profiles


Figure 10. Steady-state operating parameters represented in mII-mIII operating parameter space: p, index for port configuration; O, operating conditions corresponding to each port configuration; 0, average operating condition.

Figure 11. Concentration profile of component A in the extract outlet: solid line, output profile without dead volume; dashed line, output profile with dead volume. The mixing effect of dead volume is simulated by a well-mixed tank with a volume of 1.5 mL connected to the extract outlet.

obtained by connecting a well-mixed tank with a volume of 1.5 mL to the extract outlet. It is worth mentioning that the areas under the peaks for the two different profiles are the same, as they must be to fulfill the material balances. This implies that the mixing effect of the dead volume does not change the average purities of the product streams, which are the quantities that matter in evaluating the separation performance. The example presented here addresses the performance of the controller when the output concentration profiles are affected by the effect of dead volume in the measurement line. The mixing effect caused by the dead volume is simulated by two well-mixed tanks (with a volume of 1.5 mL) connected to the extract and raffinate outlets. The system under consideration has a total feed concentration of 15 g/L, i.e., cFA ) 7.5 g/L and cFB ) 7.5 g/L. The same porosity value, i.e., ) 0.7, is assigned to all SMB columns so that the outlet concentration profiles are not affected by column-to-column variations. The startup operating conditions are the same as in the


previous examples, i.e., mref j . It can be seen from Figure 12 that the controller fulfills the purity requirements for extract and raffinate within 10 and 20 cycles, respectively. Fulfillment of the product specifications is followed by the optimization of the operation economics. The production cost decreases continuously and reaches its minimum at cycle 80. It is observed that the optimization of the economics is relatively slow compared to the cases presented in sections 4.1 and 4.3. This is due to the combined effect of increased nonlinearity because of the higher total feed concentration and the damped dynamics of the plant outputs. The implemented internal flow rates are given in Figure 13, and Figure 14 shows the evolution of the corresponding operating points in the (mII, mIII) plane. One can see that the triangle area became smaller and closer to the diagonal than in Figure 6 because of the higher feed concentration. The controller drives the operating point into the correct separation region within three cycles, but it takes more time to tune the conditions. The controller moves the operating conditions to the vertex of the triangle within 80 cycles. The steady-state operating point is very close to the vertex of the complete


5. Conclusions

Figure 13. Instantaneous internal flow rates implemented by the controller.

Figure 14. Controller action represented in mII-mIII operating parameter space. mj values are calculated with implemented internal flow rates averaged over a cycle.

separation area, i.e., the optimal operating point for the system under consideration. As a final remark, it is worth pointing out the slight differences between the specified purities and the attained purities that can be observed in Figures 5 and 8. It is important to note that the output profiles are sampled only N times (N ) 64 in our case) within a cycle, and therefore, the product purities are calculated on the basis of limited information of the plant outputs. On the other hand, the product purities given in these figures are calculated by continuous integration of the outputs. Therefore, there can be slight differences between the specified purities and the physical purities. The magnitude and also possibly the sign of the difference depends on the shape of the output profiles, e.g., the differences are different in the cases given in Figures 5 and 8. One can use estimation methods or simply increase the number of sampling points to obtain a more accurate approximation of the output profiles, if necessary. On the other hand, these profiles are generally smoother in real applications, and therefore, one can expect that the difference will be negligible in real applications. This can also be observed for the case given above. The profiles become smoother because of the mixing effects of the dead volume in the measurement line, and the difference between the set values and the values of the plant output vanishes (see Figure 12).

In this paper, we have presented the first automatic control concept for SMBs that is based on the linear adsorption isotherm only and can be applied to SMBs operated under overloaded chromatographic conditions characterized by nonlinear adsorption isotherms. We have illustrated that the controller can find the necessary conditions to fulfill the required product specifications for the separation of a system characterized by nonlinear competitive Langmuir adsorption isotherm and also optimize the economics of the operation. The performance and the robustness of the controller have been assessed through simulations by considering different scenarios. Market competition forces pharmaceutical companies to reduce the development time of new drugs; the proposed control strategy has the potential to address the crucial needs of these companies by reducing the development time of new SMB separations. In fact, this approach requires minimal information on the behavior of the mixture to be separated in order to fulfill the required separation quality and to optimize process performance. In principle, it is applicable to different operating modes of SMBs, e.g., PowerFeed, ModiCon,5 and VARICOL, for predefined switching intervals. The only limitation of the concept is that the switching time is not a decision variable for the optimization algorithm, and therefore, the controller can only deliver the optimal operating conditions for the assigned value of the switching time t*. This is an important issue that will be addressed in our future work. Moreover, the implementation and performance of the controller for SMBs with small numbers of columns needs to be investigated, because the smaller the number of columns constituting the SMB unit, the more sensitive the unit becomes toward uncertainties and disturbances. Such units are preferable for industry because of their reduced investment costs. Finally, it is worth mentioning that laboratory work has been undertaken to achieve experimental validation of the control concept. Acknowledgment The authors are grateful to Marc Lawrence for his software support. The support of ETH Zurich through Grant TH-23′/00-1 is gratefully acknowledged. We are pleased to dedicate this paper to Art Westerberg in recognition of his role as a founder of the field of process systems engineering and in gratitude for his inspirational leadership and personal friendship over the past three decades. Notation A, B, C ) discrete time state-space matrices Acr ) column cross-sectional area (cm2) c ) concentration (g/L) c ) vector consisting of the concentration values along the unit cout ) vector consisting of the process outputs Dap ) apparent axial dispersion coefficient (cm2/s) f ) vector-valued function g ) vector-valued function H ) Henry’s constant k ) cycle index L ) length of the column (cm) N ) number of time steps within a cycle n ) time index within a cycle

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3907 nc ) control horizon ncol ) number of columns constituting the SMB unit neq ) number of equations in the SMB ODE system ng ) number of grid points defined along each column np ) prediction control ns ) number of species p ) index for different input/output configuration, p ) 1, ..., 8 PE ) purity of the extract outlet PR ) purity of the raffinate outlet Q ) volumetric flow rate (mL/s) Q ) vector consisting of volumetric flow rates in four sections, Qj (mL/s) q* ) adsorbed-phase concentration (g/L) s1, s2 ) slack variables for purity constraints s3 ) vector consisting of the slack variables for output concentrations t) time (s) t* ) period of time between two successive switches, i.e., switching time (s) u ) input vector x ) state vector y ) output vector z ) axial coordinate (cm) Greek Letters ) bed void fraction λ ) weighting factors in the cost function λ ) vector consisting of the weighting factors Subscripts and Superscripts ave ) average D ) desorbent E ) extract F ) feed g ) space index (grid point number) h ) column position index i ) component index (i ) A, B) I/O ) inlet/outlet stream in ) column inlet j ) section index (j ) I, ..., IV) max ) maximum min ) minimum out ) column outlet R ) raffinate ref ) reference values used for linearization

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Received for review October 28, 2003 Revised manuscript received March 1, 2004 Accepted March 2, 2004 IE0342154

Automatic Control of Simulated Moving Beds II: Nonlinear Isotherm

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