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as a preparative separation step for life science products. The simulated moving bed ... A new multicolumn continuous chromatographic process was rece...
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Ind. Eng. Chem. Res. 2002, 41, 4328-4337

Optimal Operation of Continuous Chromatographic Processes: Mathematical Optimization of the VARICOL Process Abdelaziz Toumi,* Felix Hanisch, and Sebastian Engell Department of Chemical Engineering, University of Dortmund, 44221 Dortmund, Germany

Continuous chromatographic processes have gained a lot of attention due to the increasing use as a preparative separation step for life science products. The simulated moving bed (SMB) process is a well-known example. A new multicolumn continuous chromatographic process was recently introduced as the VARICOL process [Ludemann-Hombourger, O. et al. The “VARICOL” Process: A New Multicolumn Continuous Chromatographic Process. Sep. Sci. Technol. 2000, 35(12), 1829]. Although in SMB chromatography a continuous operation is realized by periodic and simultaneous shifting of all inlet and outlet ports, VARICOL uses individual movements of the ports resulting in a more efficient use of the expensive adsorbent. This paper presents a more general definition of the VARICOL process than given previously. For this flexible process description, an optimization strategy for determining optimal operating parameters is presented. Based on a detailed mathematical process model, optimal operating conditions are computed for two separation tasks: the separation of two amino acids, tryptophan and phenylalanine [Wu, D.-J. et al. Design of Simulated Moving Bed Chromatography for Amino Acid Separations. Ind. Eng. Chem. Res. 1998, 37, 4023], with minimal desorbent consumption, and the separation of highly concentrated sugars, glucose and fructose [Jupke, A. et al. Experimental Verification of a Process Model, Simulation and Optimisation of SMB Chromatography. SPICA, 2000, 200010-11, Zurich], with maximum throughput. It is shown that a significant increase in profitability can be achieved by exploiting the flexibility of the VARICOL process by mathematical optimization of its operating parameters. 1. Introduction In the development of Life Science products, an important step is the determination and the design of cost-efficient unit operations for purification. Pharmaceuticals often have to be nearly 100% pure because of regulatory demands. Sometimes their physicochemical properties differ little from those of byproducts, and they may be thermally unstable. In these cases, standard separations such as distillation are not applicable. Therefore, in recent years, chromatographic separation processes gained a lot of attention not only for analytical applications (HPLC and GC) but also for preparative separations of products in the food and pharmaceutical industries. In this area, liquid chromatography is used where the substances to be separated are solved in a desorbent. The chromatographic separation is based on the different adsorptivities of the components to a specific adsorbent that is fixed in a chromatographic column. The most simple process, batch chromatography, involves a single column which is charged with impulses of the feed solution. These feed injections are carried through the column by pure desorbent. While traveling through the column, the more adsorptive species is retained longer by the adsorbent thus leaving the column after the less adsorptive species. The separated peaks can be withdrawn as different fractions at the end of the column with the desired purity. Batch chromatography is a highly flexible1 yet not very efficient mode of operation. * To whom all correspondence should be addressed. Fax: +49 231 755-5129. Phone: ++49 231 755-5173. E-mail: [email protected].

In SMB chromatography, the adsorbent is distributed over a number of columns, 6-24 in standard applications.2 Desorbent and feed streams enter the process continuously. A counter-current solid stream is approximated by shifting the inlet ports periodically in the direction of the internal recycle flow (compare Figure 1). Thus, the solid stream is “simulated” by a discrete movement of columns rather than using a continuous stream. This leads to the desired separation of the feed components which can be withdrawn at the extract and raffinate ports which are shifted in the same manner as the inlets. Inlet and outlet ports divide the SMB process into four zones with characteristic functionalities: zone I, desorption of component A (purification of adsorbent); zone II, desorption of component B; zone III, adsorption of component A; zone IV, adsorption of component B (purification of desorbent). Designing SMB processes involves, among other tasks, the selection of the number of columns per zone. A satisfactory operation is achieved by proper selection of five flow rates (feed, desorbent, raffinate, extract, and recycle) and of the switching period. The asynchronous SMB or VARICOL process3 is characterized by an asynchronous shifting of the inlet and the outlet ports. This results in a varying number of columns per zone over time. The same regime of switching operations, however, is repeated from period to period. Thus an average number of columns per zone can be calculated over one cycle of the VARICOL process. The main innovation of the VARICOL process is a more flexible allocation of the adsorbent to each of the four zones of continuous chromatography according to the needs of a specific separation task, as the average number of columns can assume rational values.

10.1021/ie0103815 CCC: $22.00 © 2002 American Chemical Society Published on Web 07/24/2002

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Figure 1. SMB process and axial concentration profile in cyclic steady state.

In this paper, the definition of the VARICOL process presented in ref 3 is extended to a more general process. Based on this description, a physically realizable subset of VARICOL processes is selected for closer analysis. A rigorous mathematical model of VARICOL processes is presented. The main innovation is the systematic optimization of the VARICOL operating parameters, including the flow rates as well as the individual switching times for each of the inlet and outlet ports. In the design of SMB processes, the right distribution of the columns over the zones is an important task and leads to a mixed integer nonlinear program. This problem can be solved directly by using special mathematical methods4 or by calculating the optimal operating point of each possible configuration, which is in both cases computationally demanding. In contrast, in the VARICOL case, the numbers of columns per zone are rational and can be determined together with the flow rates and the switching times by solving only one nonlinear program. The power of this approach is demonstrated on two examples, the separation of tryptophan and phenylalanine5 with minimal desorbent consumption and the separation of glucose and fructose6 with maximum feed throughput. 2. Continuous Chromatographic Processes: SMB vs VARICOL For a formal description of SMB and VARICOL processes, the nomenclature introduced by LudemannHombourger et al.3 is adopted here. SMB operation is characterized by a synchronous shifting of all external ports. (In reality, one port per period is shifted asynchronously to compensate the dead volume of the recycle pump; this is a technical detail that does not affect the comparison of SMB and VARICOL and will therefore be neglected for the remainder of the paper.) The position of an inlet or outlet line shall be denoted as Line(n), e.g., Ex(2) means the extract is withdrawn at the inlet of column 2. For a six-column SMB process, an initial configuration could be given by

t0: De(1)/Ex(2)/Fe(4)/Ra(6) S p(t0) ) [d0 ) 1, x0 ) 2, f0 ) 4, r0 ) 6] (1) This corresponds to 1/2/2/1 number of columns in zones I-IV. After the switching time ∆T has passed, the lines move one column downstream following the internal flow rate and thus moving the columns up one position against the internal flow:

t0 + ∆T: De(2)/Ex(3)/Fe(5)/Ra(1) S p(t0 + ∆T) ) [d ) 2, x ) 3, f ) 5, r ) 1] (2)

Now the raffinate line has passed the recycle pump, and the raffinate is withdrawn before column 1. It is obvious that in this case after six periods one SMB cycle is completed and the initial configuration of lines is reached again. Note that the number of columns per zone will be constant in the SMB mode implying constant zone lengths as well. In a more general notation, let Nci be the number of columns in zone i, i ) I-IV, and Nc ) ∑(Nci) the number of columns in the process. If d0, x0, f0, and r0 define the initial positions of the lines at t0, then (d0 + n) modulo Nc or in a more compact form [d0 + n]Nc gives the current line position in period t0 + n.3 Similar formulations apply for x, f, and r. The resulting process exhibits mixed discrete-continuous dynamics because of the interaction of continuous fluid flows and discrete switching events. Consequently, the process never reaches a steady state, as other purely continuous operations would (e.g., continuous distillation columns) but converges toward a cyclic steady state (CSS). In the CSS, the axial concentration profile of the components, plotted over all columns as in Figure 1, remains the same from period to period at a certain point of time in the period, e.g., at the end of a period. Although for SMB an equivalent steady-state TMB configuration exists, this is no longer the case for the new VARICOL process described in the following paragraphs.3 This is due to the fact that the solid velocity is not constant with respect to the inlet/outlet ports. Let us again consider a six-column continuous chromatographic process but this time operated in the asynchronous VARICOL mode. Starting from the same initial configuration, now the lines are switched one at a time. If the period ∆T is divided into four intervals of equal length and one line is moved downstream one column after each interval, as shown in Figure 2, the process can be described formally corresponding to the nomenclature defined above in Table 1. In this example, after 1/4∆T the feed line switches, followed by the extract line at 1/2∆T and the desorbent line at 3/4∆T. When the raffinate line switches after one full period, the original relative positions of the lines are resumed, however, shifted one column downstream. Because the number of columns per zone varies over the subperiods (right column of the table), an average number of columns per zone can be calculated over one period, e.g., (1 + 1 + 2 + 1)/4 ) 1.25 for zone I. The choice of the sequence of line switches (in the example above: Fe w Ex w De w Ra), and the lengths of the subperiods determines the average number of columns per zone. Extending the description given in refs 3, 7, and 8, a general formulation of the VARICOL process can be expressed as follows. We define d, x, f, and r line position for desorbent, extract, feed, and raffinate ∆T length of the main period in [s] Nsub number of subperiods within the main period δti switching time of at least one line characterizing the sub-period i ) 1...Nsub, normalized to the length of the period 0 e δti e 1 δniL number of columns that line L ∈ {d,x,f,r} is moved at instant δti, i ) 1...Nsub δni vector of the number of columns that the lines are moving at instant δti, δni) [δnid,δnix,δnif,δnir]

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Figure 2. Example of port switching for six-column VARICOL process. Table 1. Example of a Six-Column VARICOL Process line position t0: t0 + 1/4∆T: t0 + 1/2∆T: t0 + 3/4∆T:

columns per zone

p(t0) ) [d0, x0, f0, r0] p(t0 + 1/4∆T) ) [d0, x0,[f0 + 1]Nc, r0] p(t0 + 1/2∆T) ) [d0,[x0 + 1]Nc,[f0 + 1]Nc, r0] p(t0 + 3/4∆T) ) [[d0 + 1]Nc,[x0 + 1]Nc,[f0 + 1]Nc, r0] average number of columns per zone:

t0 + ∆T:

p(t0 + ∆T) ) [[d0 + 1]Nc,[x0 + 1]Nc,[f0 + 1]Nc,[r0 + 1]Nc]

1/2/2/1 1/3/1/1 2/2/1/1 1/2/1/2 (5/4)/(9/4)/(5/4)/(5/4)/ 1/2/2/1

Table 2. General Definition of the VARICOL Process t0: tj ) t0 + δtj∆T:

line position

columns per zone

p(t0) ) [d0, x0, f0, r0] p(tj) ) p(t0) + δnj

Z(t0) ) [Nc,I,0, Nc,II,0, Nc,III,0, Nc,IV,0] Z(tj) ) [Nc,I,j, Nc,II,j, Nc,III,j, Nc,IV,j]

average number of columns per zone: t0 + ∆T:

p(t0 + ∆T) ) p(t0) + ∆N

∆NL number of columns that each line is moved Nsub (δnjL), L ∈ {d,x,f,r} during the whole period ∆NL ) ∑j)1 ∆N vector summarizing the number of columns all lines are moving during the whole period ∆N ) [∆Nd,∆Nx,∆Nf,∆Nr] Nc,i,j number of columns in zone i ) I...IV and subperiod j ) 1...Nsub N h c,i average number of columns per period in zone i Nsub (δtjNc,i,j) ) I, ..., IV N h c,i ) ∑j)1 Nc total number of columns in the process Nc ) IV /N h c,i. ∑i)I Then one period of the VARICOL process can be described at initialization, during one period and at the end of one period as shown in Table 2. The following remarks illustrate some features of this definition; reasonable restrictions for practical implementations will be given below. (1) As in the SMB mode, it is possible to move one line more than one column downstream at one switching instant δtj, i.e., δnjL > 1, L ∈ {d,x,f,r} is a feasible option. (2) One could include the case δnjL < 0, i.e., a specific line would be moved upstream temporarily before being switched back again. This could be of interest for

Z h )N h c,I,N h c,II,N h c,III,N h c,IV] Z(t0 + ∆T) ) Z(t0)

example for the feed line in order to increase the loading factor of a column that has been partly eluted before. However, lines must not cross each other when allowing δnjL < 0. (3) Consequently, it could make sense to increase the number of subperiods to more than one per line, i.e., Nsub > 4. (4) The above formulation includes the SMB case, Nsub ) 1, δt1 ) 1.0 and δN1L ) ∆N1L ) 1. (5) Although lines must not cross each other, it is a feasible situation that certain zones cease to exist temporarily, e.g., raffinate is withdrawn at the same node where desorbent enters the system. In this case, it needs to be ensured in practice that the desorbent enters right after the port where the raffinate is withdrawn to avoid short cut streams. Practical restrictions of the general process description based on physical considerations and process understanding are as follows: (1) δnjL ) 1 or 0. At a specific switching instant δtj, line L ∈ {d, x, f, r} is moved either one column downstream or not moved at all. (2) However, more than one line can be moved at a time, e.g., δn ) [0, 1, 0, 1].

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Figure 3. Reachable column partitions for zones II and III with initial configuration Z(t0) ) [2, 2, 2, 2]. Table 3. Comparison of the Different Optimization Problems Discussed in This Paper design problem

DOF

opt. variables

design parameters (fixed)

SMB

5

VARICOL1

5

VARICOL2

9

QDe, QIV, QEx, QFe ∆T QDe, QIV, QEx, QFe ∆T QDe, QIV, QEx, QFe ∆T,δtd ) δtx ) δtf ) δtr

N h c,i ) Nc,i,0 ∈ N, i ) 1, ..., IV δtd ) δtx ) δtf ) δtr ) 1.0 Nc,i,0 ∈ N, i ) 1, ..., IV δtd ) δtx ) δtf ) δtr ∈ R w N h c,i ∈ R Nc,i, 0 ∈ N, i ) 1, ..., IV

(3) Each line can be moved only once during one period, i.e., Nsub e 4. In this case, each switching time δtj corresponds to the movement of only one port. They therefore are indexed in the sequel within the port notation as δtL, L ∈ {d,x,f,r}. (4) Two cases have to be distinguished in the calculation of the average zone length N h c,i of zone i depending on the chronological order of the port-switching δtL(i) and δtL(i+1). L maps hereby the zone index i ) I, ..., IV to the port index L(i) ∈ {d,x,f,r} according to: {L(I) ) L(V) ) d, L(II) ) x, L(III) ) f, L(IV) ) r}.

if δL(i) g δL(i+1) N h c,i ) δL(i+1)Nc,i,0 + (δtL(i) - δtL(i+1)) (Nc,i,0 + 1) + (1 - δtL(i))Nc,i,0 else δL(i) < δL(i+1) N h c,i ) δNc,i,0 + (δtL(i+1) - δtL(i)) (Nc,i,0 - 1) + (1 - δtL(i+1))Nc,i,0 In both cases N h c,i results as

N h c,i ) Nc,i,0 + δtL(i) - δtL(i+1), i ) I, ..., IV

operating parameters which are optimized together with the flow rates and ∆T (VARICOL2). The column partition is no longer a fixed design parameter, but now it becomes an operating parameter. The degrees of freedom and the design parameters of the three design problems (SMB,VARICOL1,VARICOL2) are listed in Table 3. 3. Modeling A lot of work has been published on modeling of chromatographic processes.9,14 Only the relevant aspects will be briefly discussed here. High fidelity dynamic models of multicolumn continuous chromatographic processes consist of dynamic models of each column, under consideration of the periodic port switching. From mass balances around the inlet and outlet nodes, the internal fluid velocities of the columns and the product concentrations are calculated (node model,12):

desorbent node (3)

It is noteworthy that this expands the region of reachable column partitions to the areas depicted in Figure 3. For an eight-column process with initial column h c,2 and N h c,3 can be chosen partition Z(t0) ) [2, 2, 2, 2], N anywhere in the shaded areas for VARICOL (depending on the choice of N h c,1), whereas SMB is limited to the integer values, marked as dots. In the context of operating parameter optimization of the asynchronous VARICOL process, two different approaches can be taken: 1. In the “classic” approach, as described in ref 1, the numbers of columns per zone are selected a priori, similar to the design of SMB but rational column partitions N h c,i are allowed. A fixed switching sequence and fixed lengths of subperiods are determined to map the N h c,i to a physically meaningful switching regime (VARICOL1). The remaining operating parameters are the same as in SMB separations, i.e., the flow rates Qi and the length of the period ∆T. 2. The approach discussed below considers the switching times δtj within one period as additional variable

QIV + QDe ) QI out in QIV ) ci,I QI ci,IV

i ) A, B

QI - QEx ) QII

extract node feed node

(4)

QII + QFe ) QIII out in QII + ci,FeQFe ) ci,III QIII ci,II

raffinate node

QIII - QRa ) QIV

(5) (6) (7)

i ) A, B (8) (9)

Here QI...IV denote the flow rates in the corresponding zones, QDe, QEx, QFe, and QRa are the external flow rates, and cin and cout i i denote the concentrations of component i in the stream leaving or entering the respective zone. A single chromatographic column is here described by the general rate model10,11,13,14 which accounts for all important nonidealities of the column, axial dispersion, pore diffusion, and mass transfer between liquid and solid phase. We assume that the solid phase consists of porous, uniform and spherical particles (radius Rp,

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radial coordinate r) with void fraction p and that a local equilibrium is established within the pores. The concentration of solute i is denoted by ci in the fluid phase and by qi in the solid phase. b is the void fraction of the bulk phase, Dax is the axial dispersion coefficient, is the equilibrium concentration, kl,i is the film ceq i mass transfer resistance, and Dp,i is the diffusion coefficient within the particle pores. The concentration within the pore is denoted by cp,i. Then, the following set of partial differential equations can be derived from a mass balance around an infinitely small cross-section of the column, if constant radial distribution of u and ci is assumed:

∂ci (1 - b) 3 kl,i ∂2ci + (ci - cp,i (r ) Rp)) - Dax 2 + ∂t 6Rp ∂x ∂ci u ) 0 (10) ∂x

[ ( )]

∂qi ∂cp,i 1 ∂ 2∂cp,i r + p - pDp,i 2 (1 - p) ∂t ∂t ∂r r ∂r

)0

Hikici , i ) 1, 2 1 + k1c1 + k2c2

(14)

which can be checked numerically as

||Φ(cax,k) - cax,k|| e epssteady

(15)

epssteady is a small value between 10-7 and 10-6. Then, given a SMB or VARICOL1 process with a certain number of columns and a fixed column partition, the optimization problem can be stated as follows:

min Costspec(QDe, QFe, QEx, QIV, ∆T) s.t. ||Φ(cax,k) - cax,k||e epssteady ∆T

∑0 cA,Ex(t) dt

(12)

and a parabolic isotherm for the sugars glucose and fructose3

qi ) Hici + kici2 + kijcicj, i, j ) 1, 2 and i * j

cax,k+1 ) Φ(cax,k)

(11)

The adsorption equilibrium has to be determined experimentally for each mixture and adsorbent and is described by the adsorption isotherm. The isotherms of the mixtures that serve as example systems in this publication are described by a Langmuir isotherm for the amino acids tryptophan and phenylalanine5 with Hi and ki being equilibrium constants

qi )

operating parameter optimization become closely related issues. The problem of finding optimal operating parameters for SMB or VARICOL can be formulated in the same way. In this section we will differentiate between VARICOL1 and VARICOL2 (see Table 3), stating the optimization problem for SMB and VARICOL1 first. The goal is to minimize specific separation costs for a given plant meeting the required product purities after the process has reached the cyclic steady state (CSS). For the specification of the CSS, the operator Φ is introduced, which represents the process dynamics and the switching operations between two switching intervals. In the CSS, the axial concentration profile cax,k at the end of period k does not change from period to period, i.e.:

∑0 [cA,Ex(t) + cB,Ex(t)] dt ∆T

(13)

These isotherms represent different types of nonlinearities and couplings between components due to different adsorption mechanisms. The resulting system of coupled partial differential equations can be solved efficiently using the approach introduced by Gu,14 where a finite element discretization of the bulk phase is combined with an orthogonal collocation of the solid phase. This method was applied to SMB processes by Du¨nnebier and Klatt.10 This very accurate process model can be solved about 2 orders of magnitude faster than real-time and is the basis of the systematic optimization of operating parameters for both the SMB and the VARICOL process. 4. Optimization Strategy

g PurEx,min

∆T

∑0 cA,Ra(t) dt

g PurRa,min

∆T

∑0 [cA,Ra(t) + cB,Ra(t)] dt Q1 e Qmax

(16)

Inequality constraints are imposed on the product purities and on the flow rate in zone I. Because zone I exhibits the highest flow rate and because of the pressure drop in the plant, the flow rate must not exceed a certain limit. It can be calculated according to Darcy’s law:

Qmax ) k0

∆pmax L

(17)

al.,15

model-based optimization of operatIn Klatt et ing parameters of continuous chromatographic processes was introduced in the framework of online-control of chromatographic processes. The goal is to find optimal operating parameters for a given separation task on a given plant. Design optimization is a different issue for SMB chromatography that has been dealt with in other publications (see, e.g., Biressi et al.16). An interesting feature of the VARICOL process, which will be illustrated in more detail below, is that design and

∆pmax is the maximal allowable pressure in the plant and k0 is an experimental proportional factor. The objective function must be specified based on the available data: If cost data exist for adsorbent, desorbent, investment, depreciation, etc., a detailed cost function can be minimized. In other cases, only dominant factors are considered, e.g., the desorbent consumption for a given throughput, which often has a strong impact on the specific separation costs because of costly solvent

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recycling, or the objective can be to maximize feed throughput (S min(-QFe)), if desorbent costs are negligible, which is the case in our second example, the separation of glucose and fructose, where water is used as the solvent. In both cases, the natural degrees of freedom are the flow rates of desorbent QDe, extract QEx, feed QFe, and recycle QIV and the switching period ∆T. However, this results in an ill-conditioned optimization problem. The numerical tractability can be improved by introducing the so-called β factors via a nonlinear transformation of the natural degrees of freedom:12

QS ) β1 )

( (

(1 - b)AL ∆T

) )

( (

) )

Q1 1 ∂gA Q2 1 ∂gB - / |cF,i, β2 ) - / | QS F ∂cA QS F ∂cB cF,i

Q3 1 ∂gA Q4 1 ∂gB 1 1 ) - / |cF,i, ) - / | β3 QS F ∂cA β4 QS F ∂cB cF,i gi ) pci + (1 - p)qi(cA, cB), F )

1 - b b

(18)

min Costspec(βi, QFe, δtj) s.t. ||Φ(cax,k) - cax,k|| e epssteady ∆T

∆T

g PurEx,min

∑0 [cA,Ex(t) + cB,Ex(t)] dt ∆T

∑0 cA,Ra(t) dt ∆T

g PurRa,min

∑0 [cA,Ra(t) + cB,Ra(t)] dt Q1 e Qmax 0 e δtj e 1, j ∈ {d, x, f, r}

run 1 exp. (Wu et al.) sim. (Wu et al.) sim. general rate

run 2

PurEx

PurRa

PurEx

PurRa

85.1% 86.8% 86.1%

91.4% 96.7% 95.3%

99.7% 99.7% 99.5%

96.7% 99.7% 98.3%

SQP algorithm. The routine E04UCF from the NAG library was used, where some of the constraint gradients with respect to the optimization variables could be specified explicitly, whereas other constraint gradients as well as the gradients of the objective function are evaluated by perturbation methods (see ref 11 for details of the implementation). The potential of the flexible formulation and the optimization of the VARICOL process is illustrated on two examples. The starting points for all optimizations are taken from literature data, previous work, or the triangle theory by Morbidelli et al.17,18 5. Case Study 1: Separation of Amino Acids

where QS is the apparent solid flow rate and qi ) f (cA, cB) describes the adsorption equilibrium isotherm. In the case of VARICOL2, the degrees of freedom, i.e., the number of optimization variables, increase by the number of the sub-periods δtj:

∑0 cA,Ex(t) dt

Table 4. Model Validation for Separation of Amino Acids

(19)

Inequality constraints have to be added for the lengths of the subperiods, which have to lie within one regular period ∆T and are normalized to 1. For the solution of the optimization problem (16), a sequential algorithm was proposed in ref 11. The degrees of freedom βi and QFe, to which the switching times δtj are added in the VARICOL case, are chosen by the optimization algorithm in an outer loop and then transformed back into flow rates and switching times. In the inner loop, the CSS is calculated by direct dynamic simulation. The purity constraints are evaluated by integration of the elution profiles. The nonlinear program in the outer loop can be solved by a standard

The separation of tryptophan and phenylalanine on a poly-4-vinylpyridine resin was studied by Wu et al.,5 and detailed experimental data and physical properties are given in that publication. For a 10-column SMB process with column partition Z ) [2, 3, 3, 2], operating points were determined by standing wave design based on a true moving bed model. The operating parameters were tested experimentally and adjusted to achieve the desired performance. Two operating points were given in ref 5 and used for the validation of our general rate SMB model (see Table 4). The general rate model fits the experimental data slightly better than the TMB model used by Wu et al. For the operating point of interest, run 2, the purities predicted by the general rate model are higher than required. For the following studies, these higher purities will be used as constraints for the optimization runs. The objective of the operating parameter optimization is to minimize the desorbent consumption QDe for a given fixed feed flow rate QFe ) 0.25 [cm3/s] which is held constant for all runs. Extract and raffinate purities must be at least 99.5% and 98.3%, respectively. The optimization runs were initialized at a feasible starting point, with the values computed by Wu et al. Table 5 summarizes the results. It can be seen that all systematic optimization approaches exploit significant potential for desorbent savings. In all cases, the purity requirements are exactly met because the optimal solution is found at the intersection of the nonlinear constraints. In the case of Nc ) 10 columns, an attempt was made to reduce the desorbent consumption further by increasing the average number of columns in the central separation zones II and III (runs 4 and 5). Then VARICOL2 was optimized (run 6, where the optimization algorithm determined the optimal switching instants δtL), and this configuration leads to the best result with respect to the desorbent consumption. Similar observations can be made in the cases of Nc ) 8 and 6 columns. Again a systematic optimization of the average number of columns per zone (VARICOL2) leads to better results. The case of fewer columns, Nc ) 6, can only be treated using a dynamic model that includes the hybrid discrete/continuous dynamics. Op-

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Table 5. Optimization Results for a Separation of Amino Acidsa run

process

Nc [-]

β1 [-]

β2 [-]

β3 [-]

β4 [-]

δtd,δtx,δtf,δtr [-]

2 3 4 5 6

SMB Wu et al. SMB VARICOL1 VARICOL1 VARICOL2

10 10 10 10 10

1.10 1.11 1.07 1.04 1.01

0.97 0.94 0.94 0.96 0.99

1.43 1.20 1.19 1.18 1.18

1.53 1.10 1.23 1.16 1.17

1.0/1.0/1.0/1.0 1.0/1.0/1.0/1.0 1.0/0.75/0.5/0.25 1.0/0.5/1.0/0.5 1.0/0.44/1.0/0.46

a

7 8 9 10 11

SMB VARICOL1 VARICOL1 VARICOL1 VARICOL2

8 8 8 8 8

1.21 1.39 1.10 1.13 1.10

1.00 1.05 1.07 0.98 1.04

1.29 1.36 1.22 1.25 1.21

1.13 1.09 1.22 1.28 1.31

1.0/1.0/1.0/1.0 0.25/0.5/0.75/1.0 1.0/0.5/1.0/0.5 1.0/0.75/0.5/0.25 0.95/0.67/0.43/0.17

12 13

SMB VARICOL2

6 6

2.26 1.21

1.01 1.05

1.33 1.22

1.67 1.60

1.0/1.0/1.0/1.0 0.8/0.3/0.9/0.8

QDe,rel [%] 100.0 44.1 42.8 40.7 39.5 56.0 72.6 50.3 48.5 47.5 124.3 84.1

Nc,i,0 is selected in the 10-, 8-, and 6-column cases respectively as [2 3 3 2]/[2 2 2 2]/[1 2 2 1], and QF is fixed to 0.25 cm3/s.

Table 6. Comparison of Calculation Times for Four Optimizations

run 3 (SMB) 6 (VARICOL) 7 (SMB) 11 (VARICOL) a

error calc. in calc. time total mass time for function CPU balance per periods CSS calls to time (%) period to CSS (min) optimum (h) 0.36 0.35 0.32 0.24

8s 7s 5s 4s

130 132 129 122

17 19 10 11

92 130 78 112

26a 41 13 20

All calculation times on a PC PII266.

erating parameters can be found for a reduced number of columns while maintaining the same throughput and product purities. Considering the high cost of adsorbent, this may lead to big reductions in investment cost as shown by runs 12 and 13. Finally, it should be pointed out that VARICOL2 usually showed much better convergence properties in the optimization runs compared to SMB or VARICOL1. A good starting point is essential for any of the runs. VARICOL2 could always be optimized starting from the corresponding SMB operating point (δtL ) 1, L ∈ {d,x,f,r}). The optimizer E04UCF from the NAG library used in this work does not perform a global optimization. Depending on the initial point, the optimizer may converge toward a subminima. To exclude this case, the optimizations are restarted from different initial points. All optimizations could be performed numerically stably and reliably. As shown in Table 6, the computation times in the VARICOL case are significantly higher than in the SMB case. This is due to a larger number of degrees of freedom, and consequently, more CPU-time is needed for the evaluation of the gradients. When introducing new optimization variables, it is of interest how sensitive the optimization result is against variations of these variables. Figure 4 shows the extract purity of the VARICOL process from run 11 in case of deviations from the nominal values of the switching times. Here, the switching times of the extract (δte) and the feed line (δtf) were disturbed in a range of (2% of the nominal value, while keeping δtd,r ) const. The processes were simulated until the cyclic steady state was reached. It can be seen that the purity varies from -0.5% to +0.2% and that negative deviations in δte cause a decrease and positive deviations an increase in extract purity. The opposite is true for δtf: negative deviations cause the extract purity to increase,

Figure 4. VARICOL(var): extract purities in case of deviations in δte and δtf.

and positive deviations cause the extract purity to decrease. This can be explained by the corresponding changes in the average number of columns per zone: smaller h c,I to increase and N h c,II to decrease; values of δte cause N N h c,II is further decreased by larger values of δtf, which also leads to larger N h c,III. The effect on the axial concentration profile is shown in Figure 5. For δte ) 0.98 δt0e and δtf ) 1.02 δt0f , the desorption front in zone I moves to the right because zone I becomes bigger. It gets closer to the next desorption front because of a smaller zone II, resulting in a decrease of extract purity. The adsorption fronts are separated better in zone III that is larger than in the nominal case. To summarize the sensitivity analysis, it can be said that the VARICOL process is sensitive to the switching times. Therefore, they can be used as compensation variables in advanced control strategies for continuous chromatographic separation processes. 6. Case Study 2: Separation of Sugars The separation of concentrated glucose and fructose on an ion-exchange resin was studied by Jupke et al.6 It is a typical example of a separation task in the food industry. The process can be described by a parabolic adsorption isotherm (eq 13) in the concentration range up to 300 mg/cm3. In the group of Schmidt-Traub, the process was analyzed experimentally on a laboratoryscale SMB process (NOVASEP Licosep 12-26). Because water is used as desorbent in this process, desorbent

Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4335

Figure 5. VARICOL(var): axial concentration profiles for deviations in δte and δtf. Table 7. Optimization Results for a Separation of Sugarsa run

process VARICOL2

Nc [-]

β1 [-]

β2 [-]

β3 [-]

β4 [-]

δtd,δtx,δtf,δtr [-]

QF [cm3/s]

Nc [-]

1 2 3 4 5 6

SMB [1 3 3 1] VARICOL1 VARICOL1 SMB [2 2 2 2] VARICOL1b

8 8 8 8 8 8

2.18 2.25 2.00 1.73 1.49 1.30

1.22 1.20 1.21 1.22 1.21 1.27

0.98 1.00 1.05 1.04 1.06 1.02

2.01 1.68 1.29 1.28 1.14 1.27

0.63/0.56/1.0/0.48 1.0/1.0/1.0/1.0 0.5/1.0/0.5/0.0 0.75/1.0/0.75/0.25 1.0/1.0/1.0/1.0 1.0/0.5/1.0/0.5

0.045 0.044 0.038 0.036 0.028 0.027

100 96 83 79 61 59

7 8 9 10 11

VARICOL2 VARICOL1 SMB [1 2 2 1] VARICOL1 VARICOL1

6 6 6 6 6

3.29 2.63 2.19 1.71 1.50

1.23 1.21 1.21 1.21 1.22

1.00 1.00 1.05 1.06 1.07

2.44 4.43 1.55 1.27 1.15

0.5/0.95/0.95/0.31 0.6/1.0/0.6/0.2 1.0/1.0/1.0/1.0 0.75/0.5/0.75/1.0 0.5/0.0/0.5/1.0

0.032 0.030 0.027 0.022 0.017

93 88 78 64 49

12 13 14 15

VARICOL2 VARICOL1 VARICOL1 SMB [1 1 2 1]

5 5 5 5

2.42 1.68 1.50 1.33

1.22 1.21 1.23 1.38

1.02 1.06 1.13 0.97

2.00 1.35 1.15 1.66

0.50/0.97/0.29/0.30 0.5/0.5/1.0/0.5 0.5/0.25/1.0/0.75 1.0/1.0/1.0/1.0

0.022 0.016 0.010 0.009

79 56 35 32

a N b c,i,0 was chosen in the 8-, 6-, and 5-column cases respectively as [1 3 3 1]/[1 2 2 1]/[1 1 2 1]. initial number of columns per zone Z ) [2 2 2 2].

minimization is not as critical as maximization of throughput. Thus, the optimization problem in this case is formulated as

min (-QFe) s.t. ||Φ(cax,k) - cax,k|| e epssteady PurEx g 99.5% PurRa g 99.5% Q1 e 0.5 cm3/s

(20)

For SMB and VARICOL1, δtj are fixed, whereas for VARICOL2, they are optimized as part of the nonlinear program. The starting point of this study was the original configuration as an eight-column SMB plant with Z ) [2, 2, 2, 2] columns in zones I-IV. It was compared to a number of SMB and VARICOL configurations, also with fewer columns. As a performance

criterion the productivity per amount of adsorbent

Pr )

QFe NcVcol

(21)

where Vcol is the fixed volume of one chromatographic column, is used. The productivity here is calculated under the assumption that both fractions are value products. Consequently, the feed flow rate determines the product throughput if the product purity is satisfied in both fractions. This is guaranteed because of the constraints enforced in the optimization algorithm. The productivity was finally normalized to the maximum value of all runs. The results are presented in Table 7. Comparing VARICOL2 and VARICOL1 for different numbers of columns Nc, it becomes apparent that the advantage of VARICOL2 increases when less columns are available. This is obvious because the optimal allocation of adsorbent in each zone is more critical if there are fewer columns. Another expected result can be seen when comparing runs 5, 11, and 14 (see Table 8): In these runs, the amount of adsorbens varies (Nc

4336

Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 Table 9. Amino Acids parameter L dp keff,A KHB kA D b

value 68.6 cm 3.9 × 10-4 cm 139 × 10-6 0.1050 61.0300 2.54 cm 0.4

parameter keff,B cF,A kB p Dp KHA cF,B

value 139 × 10-6 9.3 × 10-4 g/cm3 15.3400 0.55 1.0917 × 10-6 cm2/s 0.0666 2.0 × 10-3 g/cm3

Table 10. High Concentrated Sugar parameter

Figure 6. VARICOL, reachable column partitions for zones II and III with initial configuration Z(t0) ) [1, 3, 3, 1]. Table 8. β Factors for Evenly Distributed Columns over All Zones run

N h c,I

N h c,II N h c,III N h c,IV

5 11 14

2 2 2 1.5 1.5 1.5 1.25 1.25 1.25

β1

β2

β3

β4

2 1.49 1.21 1.06 1.14 1.5 1.50 1.22 1.07 1.15 1.25 1.50 1.23 1.13 1.15

QFe[cm3/s] 0.0278 0.0166 0.0101

) 8, 6, and 5), but the adsorbent is always distributed evenly over all zones. The resulting β factors are about the same. Note that, because the feed rate varies for the compared processes (due to different adsorbent volume), the resulting flow rates, which can be calculated from the β factors using eq 4, vary as well. This demonstrates the advantage of using β factors as operating parameters, which are a measure of the relative fluid and solid streams velocities rather than absolute values depending on a specific configuration. In the optimization of the VARICOL2 process, the initial distribution of the columns over the zones Z(t0) must be chosen carefully. Although run 1 was initialized as Z(t0) ) [1, 3, 3, 1], a different result is obtained starting at Z(t0) ) [2, 2, 2, 2]. Under the restrictions introduced in section 2 for this class of VARICOL processes, any configuration starting at Z(t0) ) [2, 2, 2, 2] cannot exceed N h c,I ) 3 (see Figure 3), whereas for Z(t0) ) [1, 3, 3, 1], the reachable range of column partitions is depicted in Figure 6. Thus, for the initial configuration Z(t0), values have to be chosen which yield the largest reachable region as shown in Figure 6. Apart from the above-mentioned considerations regarding the initial configuration, the optimal number of columns per zone is determined by the optimization algorithm. This simplifies the design procedure for continuous chromatographic processes: Finding an optimal SMB configuration is a mixed-integer nonlinear program and requires an iterative solution, where the length of the columns, the number of columns, and the columns per zone have to be determined including the optimization of operating parameters for a variety of configurations. For VARICOL(var), only the total amount of adsorbent needs to be prespecified. Then, the adsorbent is distributed over a convenient number of columns, e.g., 5 or 6, and the size of the zones is determined by the optimization scheme. 7. Conclusion In the framework of continuous chromatography, two processes were compared regarding their optimal operating parameters: SMB chromatography and VARI-

L dp keff,A KHB kA D b

value 56 cm 3.25 × 10-4 cm 5.0 × 10-5 0.26 0.12 2.6 cm 0.37

parameter keff,B cF, A p Dp KHA cF,B k21

value 7.0 × 10-5 9.3 × 10-4 g/cm3 0.01 1.0 × 10-3 cm2/s 0.46 2.0 × 10-3 g/cm3 0.1

COL. For the VARICOL process, two alternative design descriptions were introduced, VARICOL1 and VARICOL2, depending on whether the switching times are fixed or are part of the set of the optimization variables. The VARICOL process allows rational average numbers of columns per zone. The rigorous optimization strategy for the operating parameters of continuous chromatography as presented by Du¨nnebier et al.11 was extended to the VARICOL case with variable switching times for each of the input and output ports. The power of this approach was demonstrated by two case studies: the separation of tryptophan and phenylalanine with minimum desorbent consumption and the separation of glucose and fructose with maximum throughput. In all cases, VARICOL proved to be the best alternative for the desired objectives resulting in considerable savings for desorbent consumption or required adsorbent. Finally, it was pointed out how VARICOL2 combines the tasks of finding the optimal design as well as optimal operating parameters. By systematic optimization, flow rates and switching times for each input and output port are determined that achieve an optimal allocation of adsorbent to each of the four separation zones. Future work will deal with the sensitivity of the resulting continuous chromatographic processes to variations of the operating parameters. This must be extended to the influence of uncertainties in the parameters of the first-principles model because it is important to quantify the robust region of operation near the mathematical optimum that is attainable for practical realizations of the processes. This involves the incorporation of a parameter estimation tool as described by Zimmer et al.19 for systems with linear adsorption isotherms and its extension to systems with nonlinear isotherms. Acknowledgment The authors are indebted to H. Schmidt-Traub and his research group for their support and valuable input to this work. We thank the anonymous reviewers for helpful input and suggestions. The financial support of the Deutsche Forschungsgemeinschaft, in the context of the research cluster “Integrated Reaction and Separation Processes” at the University of Dortmund (SCHM 808/5-1) under Grant (DFG En 152/26), is very gratefully acknowledged.

Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4337

Nomenclature {d, x, f, r} ) line position for desorbent, extract, feed and raffinat βi ) dimensionless parameter ci ) concentration in the bulk phase [g/cm3] cp,i ) concentration in the particle pores [g/cm3] D ) column diameter [cm] δtj ) scaled switching time of a line within a period δniL ) number of columns that line L is moved in the subperiod i dp ) particle diameter [cm] Dax ) axial dispersion coefficient [cm2/s] ∆T ) switching time [s] ∆NL ) number of columns each line L is moved during one period Dp,i ) particle diffusion coefficient [cm2/s] b ) void fraction bulk p ) void fraction particle F ) ratio of the liquid phase to the solid phase Hi ) Henry coefficient k0 ) coefficient of the Darcy-law for the pressure drop among the columns [bar s/cm2] kl,i ) mass transfer resistance in liquid film [cm/s] ki ) Langmuir isotherm parameter [cm3/g] L ) column length [cm] Nc ) Total number of columns Nc,i,j ) Number of columns in zone i during the sub-period j N h c,i ) Average number of columns per period in zone i Nsub ) Number of sub-periods within one period p ) position of the lines {d, x, f, r} Puri ) product purity [%] qi ) concentration in the solid phase [g/cm3] Qi ) flow rate in the respective zone of the SMB process [cm3/s] QDe ) flow rate of the Desorbent stream [cm3/s] QEx ) flow rate of the Extract stream [cm3/s] QFe ) flow rate of the Feed stream [cm3/s] Qmax ) Maximum flow rate which is allowed in the plant [cm3/s] QS ) Apparent flow rate of the solid [cm3/s] QIV ) flow rate of the Recycle stream [cm3/s] r ) radial coordinate [cm] rp ) particle radius [cm] Reci ) product recovery [%] ti, t ) time variables [s] u ) interstitial velocity [cm/s] Vcol ) column volume [cm3] x ) axial coordinate [cm] Z ) distribution of the columns over the zones

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Received for review April 30, 2001 Revised manuscript received April 24, 2002 Accepted May 1, 2002 IE0103815