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Modelling and simulation of nonlinear chromatographic separation processes: a comparison of different modelling approaches
Chemical Engineering Science 55 (2000) 373}380
Modelling and simulation of nonlinear chromatographic separation processes: a comparison of di!erent modelling approaches G. DuK nnebier, K.-U. Klatt* Process Control Laboratory, Department of Chemical Engineering,University of Dortmund, 44221 Dortmund, Germany Received 31 March 1999; accepted 1 April 1999
Abstract Chromatographic processes provide a powerful tool for the separation of multicomponent mixtures in which the components have di!erent adsorption a$nities. As an alternative to conventional batch chromatography as a discontinuous process, the simulated moving bed (SMB) process as a continuous chromatographic separation process gained more and more impact recently. Mathematical modelling and simulation of batch chromatographic processes is widely documented in the literature. For the modelling of SMB chromatographic processes two di!erent strategies exist. The "rst one considers the corresponding true moving bed process neglecting the cyclic switching and thus the dominating dynmics of the process. The second alternative which is considered within this framework is to connect dynamic models of single chromatographic columns while considering the cyclic port switching in order to represent the dynamics of the real process correctly. We here describe three di!erent approaches for the modelling and simulation of chromatographic separation processes with nonlinear adsorption thermodynamics described by the Langmuir isotherm. The "rst one is a rigorous modelling and numerical simulation approach, the second one is based on the equilibrium theory and proceeds from the analytical solution for binary separation processes based on nonlinear wave propagation. The third approach only includes nonidealities caused by the limited mass transfer rates. All approaches are analyzed and compared with respect to complexity, accuracy and computational time. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Chromatography; Simulated moving bed; Process modelling; Dynamic simulation; Nonlinear wave theory
1. Introduction Chromatographic processes provide a powerful tool for the separation of multicomponent mixtures in which the components have di!erent adsorption a$nities, especially when the components show separation factors near one and high resolutions, yields and purities are required. Typical applications are found in the pharmaceutical industry and in the food industry where standard thermal unit operations like distillation are not suitable. Conventional batch chromatography is a discontinuous process resulting in highly diluted products. To increase the separating power of the system, a continuous countercurrent operation is desirable but the real countercurrent of the solid phase leads to serious operating problems. In this context, the simulated moving bed (SMB) technology is
* Corresponding author. Tel.: #00-49-231-755-5124. E-mail address: [email protected] (K.-U. Klatt)
becoming an important technique for large-scale continuous chromatographic separation processes. The SMB process is realized by connecting several single chromatographic columns in series. The countercurrent movement is approximately realized by a cyclic switching of the feed stream and the inlet and outlet ports in the direction of the #uid stream. The main application in the food industry is the sugar separation, especially the separation of fructose and glucose, but also sucrose from molasses, where the SMB technology has growing economic impact (Rearick, Kearney & Costesso, 1997). In the pharmaceutical industry, the separation of enantiomers of chiral drugs on the preparative scale has recently been discovered as an e!ective application for the SMB technology Francotte & Richert, 1997; Schulte, Kinkel, Nicoud & Charton 1996; Strube, AltenhoK ner, Meurer & Schulte, 1997; Pais, Loureiro & Rodrigues, 1997). From the mathematical modelling point of view, it is useful to distinguish chromatographic processes by the type of adsorption isotherms. The type of isotherm
0009-2509/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 3 3 2 - 2
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describing the thermodynamic behavior of the modelled system in#uences the structure of the resulting mathematical problem substantially. Processes with linear adsorption isotherms (DuK nnebier, Weirich & Klatt, 1998; DuK nnebier, Engell, Klatt, Schmidt-Traub, Strube & Weirich, 1998a), as most sugar separation processes, can be modelled and simulated much easier than those with nonlinear isotherms, e.g. competitive Langmuir or Bi-Langmuir isotherms. The separation of enantiomers is a process of the last type, and this paper focusses on di!erent modelling and simulation approaches for nonlinear chromatographic processes. The rigorous dynamic modelling and simulation of chromatographic processes using "nite di!erence, "nite element or orthogonal collocation methods is widely documented in the literature and leads in general to very complex models with an enormous need for computational power (Ching & Lu, 1998; Strube & SchmidtTraub, 1996; Kaczmarski & Antos, 1996; Lu & Ching, 1997; Ma & Guiochon, 1991). The successful design and operation of SMB separation units depend on the correct choice of the operating conditions, particularly of the #ow rates in each section. Because of the complex dynamics of the process, the choice of the operating parameters is not straightforward. The known design approaches are all based on a steady-state model of the corresponding true moving bed process, e.g. (Ma & Wang, 1997; Mazzotti, Storti & Morbidelli, 1997). Due to the simplicity of the modelling approaches used here, these design strategies only give initial guesses for the optimal operating conditions. For this task, a detailed and reliable dynamic model of the process is necessary which includes the continuous dynamics of the single columns as well as the discrete events resulting from the cyclic operation. Most dynamic modelling and simulation approaches from the literature have only been applied to batch chromatography before, and their application to the SMB process with its switching pattern and the complex initial and boundary conditions is the main and novel scope of this paper. In the sequel, this paper describes the SMB process and the dynamic modelling approach of the process chosen in Sections 2 and 3, respectively. The dynamic modelling approach of the process is based on dynamic models of single chromatographic columns. Those models can be classi"ed by the way of modelling the kinetic and dispersive e!ects which is directly linked to the model complexity. The modelling approaches are presented here in the order of descending complexity. In Section 4, a computationally e$cient solution strategy for a rigorous modelling approach is presented. In Section 5, we show the limitations of a simpli"ed approach lumping all kinetic e!ects in a single parameter. Finally, in Section 6, a solution approach for an ideal model, neglecting all kinetic e!ects, is presented.
2. The simulated moving bed process In most areas of chemical engineering, countercurrent unit operations are preferable for thermodynamic reasons. The standard concept of a countercurrent unit operation in chromatography implies a moving solid phase. However, a moving solid phase is not realizable e!ectively in practice due to technical limitations. An alternative is a simulated moving bed unit which consists of several chromatographic columns connected in series (Ruthven & Ching, 1989). The inlet and outlet ports are located between the columns. The counter current movement of the liquid and solid phases is achieved by sequentially switching the valves of the interconnected columns in the direction of the liquid #ow. In the limit of an in"nite number of columns and in"nitely short switching periods, we obtain the true moving bed process. The process is shown schematically in Fig. 1. The feed and desorbent inlets and extract and ra$nate outlets are switched one column downstream after each switching period. For economic reasons, a small number of columns is desirable. Here we consider a plant with eight columns which can be divided into four sections of two columns each. Each section has a speci"c role within the operation. A section is de"ned by its position relative to the inlets and outlets rather than by a speci"c pair of columns. The stationary regime of this process is a cyclic steady state, in which in each section an identical transient during each period between two valve switches takes place. The cyclic steady state is practically reached after a certain number of valve switches, but the system states are still varying over time because of the periodic movement of the inlet and outlet ports along the columns. In general, this dynamic behavior of the process cannot be approximated by a model of the equivalent countercurrent process. This is especially true for an economically viable SMB process with a rather small
Fig. 1. Scheme of a simulated moving bed unit.
G. Du( nnebier, K.-U. Klatt / Chemical Engineering Science 55 (2000) 373}380
number of columns and SMB pocesses with nonlinear adsorption isotherms. We therefore choose a dynamic modelling approach for the purpose of system analysis and process optimization. Our model is assembled of a number of connected dynamic models of single chromatographic columns. The di!erent modelling and simulation approaches investigated are presented below.
3. Modelling the SMB process A dynamic model of the SMB process can be obtained by connecting the dynamic models of the single chromatographic columns while considering the cyclic port switching. The #uid velocities and the inlet concentrations in the di!erent sections can be calculated by mass balances around the inlet and outlet nodes (node model, Ruthven & Ching, 1989). In these equations, Q ,Q ,Q and Q are the #ow rates through the corre' '' ''' '4 sponding process sections, Q is the desorbent #owrate, " Q the feed #ow rate, Q the extract #ow rate and Q the $ # 0 ra$nate #ow rate: Desorbent node: Q #Q "Q , (1) '4 " ' c Q "c Q , i"A,B, (2) G '4 '4 G ' ' Extract node: Q !Q "Q , (3) ' # '' Feednode: Q #Q "Q , (4) '' $ ''' c Q #c Q "c Q , i"A,B, (5) G '' '' G $ $ G ''' ''' Ra$nate node: Q !Q "Q . (6) ''' 0 '4 Dynamic modelling approaches for single chromatographic columns are presented in the following sections.
4. The general rate model Most rigorous modelling approaches found in the literature can be seen as di!erent versions of a general rate model. It takes into account all the important e!ects of nonidealities in the column, which is axial dispersion, pore di!usion and the mass transfer between liquid and solid phase. Only di!erent ways of describing these effects distinguish many modelling approaches. We consider a single chromatographic column with void fraction e and length ¸ through which a #uid mixture containing two di!erent solutes #ows with the velocity u. We assume that the solid phase consists of porous, uniform and spherical particles with void fraction e and N that local equilibrium is established within the pores. Let c represent the concentration of the solute i in the #uid G phase and q its concentration in the solid phase, respecG tively. D is the axial dispersion coe$cient, c the equi G
375
librium concentration, R the particle radius and r the N axial coordinate of the particle, k the "lm mass transfer J G resistance. The di!usion coe$cient within the particle pores is denoted with D , the concentration in the pores N G of the particle with c . Assuming no radial distribution N G of u and c , the material balance for each solute componG ent between planes at distances x and x#dx from the entrance of the bed over a time period t to t#dt in the limit yields to the following set of coupled partial di!erential equations: *c (1!e)3k *c *c G# J G(c !c (r"R ))!D G#u G"0, G N G N *t eR *x *x N (7)
e*q *c 1 * *c G#e N G!e D (1!e ) r N G N e*t N *t N N G r *r *r
"0.
(8)
The most common nonlinear adsorption equilibrium relation is the competitive Langmuir isotherm with N , k G G being equilibrium constants: N k c G G G . q" G 1# LA k c I I I
(9)
This isotherm takes into account both e!ects of saturation and interaction between di!erent components. A numerical solution strategy for this model is given by Gu (1995). To be solved is a coupled PDE system with partial di!erential equations for both #uid and solid phase. In the original work, a "nite element formulation is used to discretize the #uid phase and orthogonal collocation for the solid phase (Finlayson, 1980). The resulting ODEs can be solved using a conventional solver. The use of this model formulation and solution approach for the dynamic simulation of SMB chromatographic processes is a novelty and has not yet been reported. (Ching & Lu, 1998) compared the prediction of a linear driving force (LDF) model with a general rate model for the SMB case and concluded that both approaches are valid for SMB processes with nonlinear adsorption isotherms. The LDF model is based on the assumption of a parabolic concentration pro"le in the particle. Using a single interior point collocation in combination with the symmetry properties of a spheric particle, we can exactly model a parabolic pro"le. We did therefore only use one interior collocation point in our simulation study, resulting in the same number of ODEs to be solved for the more complex general rate model than for the LDF approach. Due to the complexity of the modelling approach, there is a need for simpli"ed modelling and simulation approaches. Furthermore, especially for on-line optimization and process control purposes, the complex models do not allow deep structural insight in the process dynamics.
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5. Kinetic approach It has been shown by Van Deemter, Zuiderweg and Klinkenberg (1956), that the nonideal e!ects of axial dispersion and mass transfer resistance and particle di!usion are additive in the case of a linear isotherm under a few assumptions. By adjusting the corresponding parameters, one can therefore simplify the model by considering only one of the e!ects. This relation does not strictly hold in case of a nonlinear isotherm. In case of a Langmuir isotherm it has been shown by Golshan-Shirazi and Guiochon (1992) that, while adjusting the model parameters of several kinetic models to experimental data, the quality of the prediction of simpler models only considering mass transfer resistance does not di!er significantly from those of models with both axial dispersion and mass transfer resistance. The conclusion can be drawn that the assumption of Van Deemter is at least a useful approximation of the reality even in case of a Langmuir Isotherm. Using this simpli"cation, the general rate model reduces to the following set of partial di!erential equations; *c 1!e *q *c G# G"!u G, *t e *t *x
(10)
*q G"k (c !c). ?N G G G *t
(11)
Tan (1997) proposed a solution approach for this model equations for the case of the multicomponent competitive Langmuir Isotherm (see Eq. (9)) based on a nonlinear transformation combined with an explicit Euler method. The numerical solution of an implicit correlation for just a single reference species is required, which makes the approach very promising with respect to computational cost especially for multicomponent batch chromatography. The main drawback of this approach is the dependency of the extention of the dimensionless domain for which the system has to be solved on the numerical values of the parameters k . The values of these para?N G meters describing the nonideality of the system tend towards in"nity if the system approaches the properties of an ideal system. In this case, the extension of the dimensionless domain tends to in"nity as well. Though the solution of the system for a single gridpoint can be obtained very fast, the number of gridpoints has to be increased in order to ensure stability and desired accuracy. Technical desirable chromatography systems should have a possibly high e$ciency, which is re#ected in relatively large numerical values of the parameter k . The advantages of this solution approach ?N G reported, for example systems by Tan (1997), do not hold in this case. This basically re#ects the di$culties commonly encountered when applying classical numerical
schemes to hyperbolic partial di!erential equations. To be able to approximate speci"c features of the solutions (e.g. dicontinuous solutions, shock waves) using a conventional discretization scheme a very "ne or automatically adapted grid is needed. This basic limitation can not be circumvented by the approach presented by Tan. A solution approach especially developed for hyperbolic PDEs is presented in the following section.
6. The ideal model Neglecting all nonideal e!ects and assuming overall equilibrium between liquid and solid phase, the ideal model of chromatography is obtained. It is usually formulated after de"ning a set of dimensionless variables with ¸ being a characteristic length of the column: x"z/¸, q"eut/¸.
(12)
A di!erential material balance for all components leads in the limit to the following set of quasilinear partial di!erential and algebraic equations: *c *f G# G"0, *x *q
(13)
ec #(1!e)q "f . (14) G G G With the Langmuir isotherm as in Eq. (9), this gives a system of coupled hyperbolic "rst-order partial di!erential equations which can either be solved numerically or the system might be solved in closed form for some cases of initial and boundary conditions using the theory of nonlinear wave propagation. The details of this solution procedure have already been presented by DuK nnebier and Klatt (1998) and are brie#y summarized in the sequel. 6.1. Summary of solution procedure The solution procedure can be summarized as follows: In the xrst step, the system is transformed with a nonlinear transformation (h-transformation) to decouple the equations. The solution for a Riemann problem (step input and constant initial conditions) can then be found analytically using the method of characteristics (Hel!erich & Whitley, 1996; Hel!erich, 1997; Rhee, Aris & Amundsen, 1989,1970). This methodology can, in its standard form, only be applied to systems with competitive Langmuir adsorption isotherms (see Eq. (9)), but an extension to more complex isotherms can be found in (ZenhaK usern & Rippin, 1998). The solutions of these problems are travelling waves, and with the theory of wave interaction the solution of several Riemann problems can be superposed. The straightforward application of this approach is limited, since more complex
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interaction patterns might require the solution of a partial di!erential equation themselves. Since the usual initial and boundary conditions in most applications as the SMB processes are more complex than one single step, in the second step, the initial and boundary conditions have to be discretized as a series of steps using front tracking (Wendro!, 1992). Assuming a su$cient discretization, the whole solution can now be calculated analytically in the transformed space using the rules of wave propagation and wave interaction of shock waves. In the third step, the solution has to be transformed back into the original concentration space. The approach has previously been only applied to batch chromatography by (Wendro! (1993). The implementation to study a simulated moving bed process as a novel application is a challenging task due to the far more complex initial and boundary conditions in this case.
377
streams are chosen as follows: Q "0.0475 cm/s with $ a concentration of c "0.23 g/cm, Q "0.828 cm/s, $ G # Q "0.8773 cm/s and Q "0.0577 cm/s. " '4 7.1. General rate model For the discretization of the #uid phase, 12 cubic "nite elements are used for each column of Length ¸"25 cm. This results in computation times of approx. 30 cpu seconds on a PC PII266 for one switching period. 7.2. Ideal model The sampling parameter for the front tracking is chosen as 0.01 (see DuK nnebier & Klatt, 1998). This results in computation times of approx. 12 cpu seconds on a PC PII266 for one switching period. 7.3. Comparison of simulation results
7. Simulation example for SMB processes In this simulation example we consider a SMB unit with one column each in the "rst and fourth section and three columns in the second and third section, which re#ects an industrial separation of cyclic hydrocarbons. The adsorption equilibrium of the system can be described with a competitive Langmuir Isotherm as given in Eq. (9). The purpose of this example is to illustrate the capabilities of the modelling and simulation approaches described above. The simulation results of the general rate model are compared with the predictions of the ideal model, simulated using the method of characteristics and front tracking. For the reasons presented in Section 5, the kinetic approach has not been implemented for simulation. The isotherm parameters N have to be adjusted G for the two modelling approaches. In the rigorous model, the adsorbed solid concentration q1 is expressed G per volume of solid. In the ideal model, the adsorbed solid concentration q. is expressed per particle volume. G With q."q1(1!e )#c e G G N N G N
(15)
the following relation holds:
e LA N."N1(1!e )# N k c . G G N I I k G I
(16)
The system parameters of the example system chosen are N."0.8763, N."0.6812, k "1.85 cm/g, k "5.94 c m/g, all columns have a void fraction of e"0.4, a diameter of d"5 cm, a length of ¸"25 cm. The parameters N1 have been estimated for the interesting G concentration range using Eq. (16). The switching is performed after each t "3500 s, and the feed and outlet
In Fig. 2, the axial pro"les of the central columns of the SMB unit at di!erent instances during the startup period are displayed. Starting with at the top of the "gure, a comparison of the axial pro"les at the end of the "rst switching period leads to the following conclusions: E The solution of the ideal model forms a kind of skeleton for the prediction of the general rate model. E If the equilibrium is reached despite the kinetic e!ects (as in the "rst column after the feed inlet), both pro"les are nearly identical. This is only the case if the extension of the concentration plateaus is su$ciently large. E If the equilibrium is not reached due to the kinetic e!ects (as for component B at the frontal adsorption wave at the beginning of the second column after the feed inlet), the plateau level predicted by the two models is di!erent. Since the concentration level a!ects the speed of propagation in case of a nonlinear isotherm, the speed of propagation of the concentration waves is not predicted correctly by the ideal model in this case. This is a speci"c feature of the nonlinear adsorption isotherm, a very good agreement between the ideal and the general rate model can be obtained in case of a linear adsorption isotherm (DuK nnebier et al., 1998a,b). E This deviation is particulary large in Section III (between feed inlet and ra$nate outlet) and grows during the simulation due to the integration, as can be seen in the two lower parts of the "gure showing the axial pro"les at the end of the second and "fth switching periods, respectively. The axial pro"le of the column in cyclic steady state, which is approximately reached after the 70th switching period, as predicted by both modelling approaches is
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Fig. 2. Development of axial pro"les during startup procedure.
displayed in Fig. 3. One can consider the solution of the ideal model as a skeleton for the solution of the general rate model, and the predicted pro"les do agree closely in most parts of the process.The small error in the calculated speed of propagation of the concentration waves leads to an error in the prediction of the pro"les in steady state due to the integrating e!ect of the long time horizon. The concentration levels of the constant states are slightly di!erent, and some e!ects, like the kinks predicted by the ideal model in sections two and three, are predicted more smoothly by the general rate model due to the e!ects of mass transfer resistance and axial dispersion. The predicted location of the desorption front of component A at the extract outlet and the adsorption front of component B at the ra$nate outlet of both models show good agreement and the chosen operating conditions lead to the desired separation in both cases. As a "rst initial guess, a design of a SMB unit using the ideal model might therefore be feasible. But the di!erent prediction of the product #ows
of the impurities leads to signi"cantly di!erent product purities in both cases:
Purity [%]
Ideal model General rate model
Extract
Ra$nate
100 99.45
78 87.64
Due to the unrealistic assumptions of ideality, the accuracy of prediction of this modelling approach is not su$cient for design, control or optimization purposes. Those deviations between the two models are not due to the front tracking procedure, since the same e!ect can also be found in the case of batch chromatography, where analytical solutions for the ideal model are available. Great care has to be taken when using design approaches based on ideal models, as the one by
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379
Fig. 3. Comparison of axial pro"les of SMB unit for cyclic steady state at the end of a switching period.
Mazzotti et al. (1997). Besides the questionable assumption of ideality, these approaches use a (steady-state) model of the corresponding true moving bed process introducing an additional error. The physical plane portrait (showing the propagation directions of the di!erent waves in a q vs. x graph) as one of the results of the ideal model solution might be used to obtain an understanding of the complex and interacting process dynamics. By following the shock paths through space and time, one can examine the complex interaction patterns. An example is given in DuK nnebier and Klatt (1998). Therefore, the ideal model can be useful for the analysis of the qualitative behavior of the process and the in#uence of di!erent process parameters. Especially the possible insight in the dynamics which can be gained by examination of the shock paths helps to understand and analyse the process dynamics and the startup procedure. The ideal model should not be chosen in any application where precise quantitative predictions are needed since those can not be given due to the unrealistic modelling assumptions.
8. Conclusions Three di!erent modelling and simulation approaches for chromatographic separation process have been presented and analyzed, starting with the rigorous general rate model considering various kinetics e!ects. Two simpli"ed modelling approaches have been presented in the sequel, one lumping the kinetic e!ects into a single parameter, the second (being the ideal model) neglecting all kinetic nonidealities. The suitable modelling and simulation approach has to be chosen depending on the speci"c application. This paper presented a case study extending and applying two modelling and simulation approaches to the complex simulated moving bed (SMB) process as a novel applica-
tion. The case study was used to illustrate the capabilities of the proposed approaches. The ideal model and the solution approach based on the method of characteristics and front tracking allows speci"c qualitative insight into the complex interacting dynamics of this process especially during the startup procedure but lacks of a su$cient accuracy for quantitative predictions due to the unrealistic modelling assumptions. The approach proposed based on a kinetic model seems to be very elegant and e!ective at "rst glance but is limited to columns with low e$ciency which are very rarely used in practical applications today. The general rate model and the numerical solution approach presented here allow for a quantitative analysis of the process. As to be expected, those calculations tend to be more computational expensive than the solution of the ideal model, though far less as reported otherwise in the literature.
Acknowledgements The authors are indebted to S. Engell for his support and valuable input to this work. The "nancial support of the Bundesministerium fuK r Bildung und Forschung under the grant number 03D0062B0 is very gratefully acknowledged. We also kindly acknowledge the help and support of T. Gu.
Notation Variables c concentration of component i in bulk phase, G g/cm cCO equilibrium concentration of component i, G g/cm c concentration of component i in particle pores, N G g/cm D axial dispersion coe$cient, cm/s ?V
G. Du( nnebier, K.-U. Klatt / Chemical Engineering Science 55 (2000) 373}380
380
D N G
Q R N q t u x Z
particle di!usivity coe$cient of component i, cm/s void fraction of the bed, dimensionless void fraction of the particle, dimensionless "lm mass transfer resistance of component i, cm/s lumped mass transfer resistance of component i, 1/s Langmuir Isotherm coe$cient of component i, cm/g column length, cm Langmuir isotherm constant, g/cm concentration of component i in the solid phase, g/cm volumetric #ow rate, cm/s particle radius, cm dimensionless time, dimensionless valve switching interval for SMB process, s #uid velocity, cm/s axial coordinate, cm dimensionless axial coordinate, dimensionless
Indices i in j out R E F D
component i"A,B column inlet section j"I,II,III,IV column outlet ra f f inate extract feed desorbate
e e N k J G k ?N G k G ¸ N G q G
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Modelling and simulation of nonlinear chromatographic separation processes: a comparison of di!erent modelling approaches G. DuK nnebier, K.-U. Klatt* Process Control Laboratory, Department of Chemical Engineering,University of Dortmund, 44221 Dortmund, Germany Received 31 March 1999; accepted 1 April 1999
Abstract Chromatographic processes provide a powerful tool for the separation of multicomponent mixtures in which the components have di!erent adsorption a$nities. As an alternative to conventional batch chromatography as a discontinuous process, the simulated moving bed (SMB) process as a continuous chromatographic separation process gained more and more impact recently. Mathematical modelling and simulation of batch chromatographic processes is widely documented in the literature. For the modelling of SMB chromatographic processes two di!erent strategies exist. The "rst one considers the corresponding true moving bed process neglecting the cyclic switching and thus the dominating dynmics of the process. The second alternative which is considered within this framework is to connect dynamic models of single chromatographic columns while considering the cyclic port switching in order to represent the dynamics of the real process correctly. We here describe three di!erent approaches for the modelling and simulation of chromatographic separation processes with nonlinear adsorption thermodynamics described by the Langmuir isotherm. The "rst one is a rigorous modelling and numerical simulation approach, the second one is based on the equilibrium theory and proceeds from the analytical solution for binary separation processes based on nonlinear wave propagation. The third approach only includes nonidealities caused by the limited mass transfer rates. All approaches are analyzed and compared with respect to complexity, accuracy and computational time. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Chromatography; Simulated moving bed; Process modelling; Dynamic simulation; Nonlinear wave theory
1. Introduction Chromatographic processes provide a powerful tool for the separation of multicomponent mixtures in which the components have di!erent adsorption a$nities, especially when the components show separation factors near one and high resolutions, yields and purities are required. Typical applications are found in the pharmaceutical industry and in the food industry where standard thermal unit operations like distillation are not suitable. Conventional batch chromatography is a discontinuous process resulting in highly diluted products. To increase the separating power of the system, a continuous countercurrent operation is desirable but the real countercurrent of the solid phase leads to serious operating problems. In this context, the simulated moving bed (SMB) technology is
* Corresponding author. Tel.: #00-49-231-755-5124. E-mail address: [email protected] (K.-U. Klatt)
becoming an important technique for large-scale continuous chromatographic separation processes. The SMB process is realized by connecting several single chromatographic columns in series. The countercurrent movement is approximately realized by a cyclic switching of the feed stream and the inlet and outlet ports in the direction of the #uid stream. The main application in the food industry is the sugar separation, especially the separation of fructose and glucose, but also sucrose from molasses, where the SMB technology has growing economic impact (Rearick, Kearney & Costesso, 1997). In the pharmaceutical industry, the separation of enantiomers of chiral drugs on the preparative scale has recently been discovered as an e!ective application for the SMB technology Francotte & Richert, 1997; Schulte, Kinkel, Nicoud & Charton 1996; Strube, AltenhoK ner, Meurer & Schulte, 1997; Pais, Loureiro & Rodrigues, 1997). From the mathematical modelling point of view, it is useful to distinguish chromatographic processes by the type of adsorption isotherms. The type of isotherm
0009-2509/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 3 3 2 - 2
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describing the thermodynamic behavior of the modelled system in#uences the structure of the resulting mathematical problem substantially. Processes with linear adsorption isotherms (DuK nnebier, Weirich & Klatt, 1998; DuK nnebier, Engell, Klatt, Schmidt-Traub, Strube & Weirich, 1998a), as most sugar separation processes, can be modelled and simulated much easier than those with nonlinear isotherms, e.g. competitive Langmuir or Bi-Langmuir isotherms. The separation of enantiomers is a process of the last type, and this paper focusses on di!erent modelling and simulation approaches for nonlinear chromatographic processes. The rigorous dynamic modelling and simulation of chromatographic processes using "nite di!erence, "nite element or orthogonal collocation methods is widely documented in the literature and leads in general to very complex models with an enormous need for computational power (Ching & Lu, 1998; Strube & SchmidtTraub, 1996; Kaczmarski & Antos, 1996; Lu & Ching, 1997; Ma & Guiochon, 1991). The successful design and operation of SMB separation units depend on the correct choice of the operating conditions, particularly of the #ow rates in each section. Because of the complex dynamics of the process, the choice of the operating parameters is not straightforward. The known design approaches are all based on a steady-state model of the corresponding true moving bed process, e.g. (Ma & Wang, 1997; Mazzotti, Storti & Morbidelli, 1997). Due to the simplicity of the modelling approaches used here, these design strategies only give initial guesses for the optimal operating conditions. For this task, a detailed and reliable dynamic model of the process is necessary which includes the continuous dynamics of the single columns as well as the discrete events resulting from the cyclic operation. Most dynamic modelling and simulation approaches from the literature have only been applied to batch chromatography before, and their application to the SMB process with its switching pattern and the complex initial and boundary conditions is the main and novel scope of this paper. In the sequel, this paper describes the SMB process and the dynamic modelling approach of the process chosen in Sections 2 and 3, respectively. The dynamic modelling approach of the process is based on dynamic models of single chromatographic columns. Those models can be classi"ed by the way of modelling the kinetic and dispersive e!ects which is directly linked to the model complexity. The modelling approaches are presented here in the order of descending complexity. In Section 4, a computationally e$cient solution strategy for a rigorous modelling approach is presented. In Section 5, we show the limitations of a simpli"ed approach lumping all kinetic e!ects in a single parameter. Finally, in Section 6, a solution approach for an ideal model, neglecting all kinetic e!ects, is presented.
2. The simulated moving bed process In most areas of chemical engineering, countercurrent unit operations are preferable for thermodynamic reasons. The standard concept of a countercurrent unit operation in chromatography implies a moving solid phase. However, a moving solid phase is not realizable e!ectively in practice due to technical limitations. An alternative is a simulated moving bed unit which consists of several chromatographic columns connected in series (Ruthven & Ching, 1989). The inlet and outlet ports are located between the columns. The counter current movement of the liquid and solid phases is achieved by sequentially switching the valves of the interconnected columns in the direction of the liquid #ow. In the limit of an in"nite number of columns and in"nitely short switching periods, we obtain the true moving bed process. The process is shown schematically in Fig. 1. The feed and desorbent inlets and extract and ra$nate outlets are switched one column downstream after each switching period. For economic reasons, a small number of columns is desirable. Here we consider a plant with eight columns which can be divided into four sections of two columns each. Each section has a speci"c role within the operation. A section is de"ned by its position relative to the inlets and outlets rather than by a speci"c pair of columns. The stationary regime of this process is a cyclic steady state, in which in each section an identical transient during each period between two valve switches takes place. The cyclic steady state is practically reached after a certain number of valve switches, but the system states are still varying over time because of the periodic movement of the inlet and outlet ports along the columns. In general, this dynamic behavior of the process cannot be approximated by a model of the equivalent countercurrent process. This is especially true for an economically viable SMB process with a rather small
Fig. 1. Scheme of a simulated moving bed unit.
G. Du( nnebier, K.-U. Klatt / Chemical Engineering Science 55 (2000) 373}380
number of columns and SMB pocesses with nonlinear adsorption isotherms. We therefore choose a dynamic modelling approach for the purpose of system analysis and process optimization. Our model is assembled of a number of connected dynamic models of single chromatographic columns. The di!erent modelling and simulation approaches investigated are presented below.
3. Modelling the SMB process A dynamic model of the SMB process can be obtained by connecting the dynamic models of the single chromatographic columns while considering the cyclic port switching. The #uid velocities and the inlet concentrations in the di!erent sections can be calculated by mass balances around the inlet and outlet nodes (node model, Ruthven & Ching, 1989). In these equations, Q ,Q ,Q and Q are the #ow rates through the corre' '' ''' '4 sponding process sections, Q is the desorbent #owrate, " Q the feed #ow rate, Q the extract #ow rate and Q the $ # 0 ra$nate #ow rate: Desorbent node: Q #Q "Q , (1) '4 " ' c Q "c Q , i"A,B, (2) G '4 '4 G ' ' Extract node: Q !Q "Q , (3) ' # '' Feednode: Q #Q "Q , (4) '' $ ''' c Q #c Q "c Q , i"A,B, (5) G '' '' G $ $ G ''' ''' Ra$nate node: Q !Q "Q . (6) ''' 0 '4 Dynamic modelling approaches for single chromatographic columns are presented in the following sections.
4. The general rate model Most rigorous modelling approaches found in the literature can be seen as di!erent versions of a general rate model. It takes into account all the important e!ects of nonidealities in the column, which is axial dispersion, pore di!usion and the mass transfer between liquid and solid phase. Only di!erent ways of describing these effects distinguish many modelling approaches. We consider a single chromatographic column with void fraction e and length ¸ through which a #uid mixture containing two di!erent solutes #ows with the velocity u. We assume that the solid phase consists of porous, uniform and spherical particles with void fraction e and N that local equilibrium is established within the pores. Let c represent the concentration of the solute i in the #uid G phase and q its concentration in the solid phase, respecG tively. D is the axial dispersion coe$cient, c the equi G
375
librium concentration, R the particle radius and r the N axial coordinate of the particle, k the "lm mass transfer J G resistance. The di!usion coe$cient within the particle pores is denoted with D , the concentration in the pores N G of the particle with c . Assuming no radial distribution N G of u and c , the material balance for each solute componG ent between planes at distances x and x#dx from the entrance of the bed over a time period t to t#dt in the limit yields to the following set of coupled partial di!erential equations: *c (1!e)3k *c *c G# J G(c !c (r"R ))!D G#u G"0, G N G N *t eR *x *x N (7)
e*q *c 1 * *c G#e N G!e D (1!e ) r N G N e*t N *t N N G r *r *r
"0.
(8)
The most common nonlinear adsorption equilibrium relation is the competitive Langmuir isotherm with N , k G G being equilibrium constants: N k c G G G . q" G 1# LA k c I I I
(9)
This isotherm takes into account both e!ects of saturation and interaction between di!erent components. A numerical solution strategy for this model is given by Gu (1995). To be solved is a coupled PDE system with partial di!erential equations for both #uid and solid phase. In the original work, a "nite element formulation is used to discretize the #uid phase and orthogonal collocation for the solid phase (Finlayson, 1980). The resulting ODEs can be solved using a conventional solver. The use of this model formulation and solution approach for the dynamic simulation of SMB chromatographic processes is a novelty and has not yet been reported. (Ching & Lu, 1998) compared the prediction of a linear driving force (LDF) model with a general rate model for the SMB case and concluded that both approaches are valid for SMB processes with nonlinear adsorption isotherms. The LDF model is based on the assumption of a parabolic concentration pro"le in the particle. Using a single interior point collocation in combination with the symmetry properties of a spheric particle, we can exactly model a parabolic pro"le. We did therefore only use one interior collocation point in our simulation study, resulting in the same number of ODEs to be solved for the more complex general rate model than for the LDF approach. Due to the complexity of the modelling approach, there is a need for simpli"ed modelling and simulation approaches. Furthermore, especially for on-line optimization and process control purposes, the complex models do not allow deep structural insight in the process dynamics.
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5. Kinetic approach It has been shown by Van Deemter, Zuiderweg and Klinkenberg (1956), that the nonideal e!ects of axial dispersion and mass transfer resistance and particle di!usion are additive in the case of a linear isotherm under a few assumptions. By adjusting the corresponding parameters, one can therefore simplify the model by considering only one of the e!ects. This relation does not strictly hold in case of a nonlinear isotherm. In case of a Langmuir isotherm it has been shown by Golshan-Shirazi and Guiochon (1992) that, while adjusting the model parameters of several kinetic models to experimental data, the quality of the prediction of simpler models only considering mass transfer resistance does not di!er significantly from those of models with both axial dispersion and mass transfer resistance. The conclusion can be drawn that the assumption of Van Deemter is at least a useful approximation of the reality even in case of a Langmuir Isotherm. Using this simpli"cation, the general rate model reduces to the following set of partial di!erential equations; *c 1!e *q *c G# G"!u G, *t e *t *x
(10)
*q G"k (c !c). ?N G G G *t
(11)
Tan (1997) proposed a solution approach for this model equations for the case of the multicomponent competitive Langmuir Isotherm (see Eq. (9)) based on a nonlinear transformation combined with an explicit Euler method. The numerical solution of an implicit correlation for just a single reference species is required, which makes the approach very promising with respect to computational cost especially for multicomponent batch chromatography. The main drawback of this approach is the dependency of the extention of the dimensionless domain for which the system has to be solved on the numerical values of the parameters k . The values of these para?N G meters describing the nonideality of the system tend towards in"nity if the system approaches the properties of an ideal system. In this case, the extension of the dimensionless domain tends to in"nity as well. Though the solution of the system for a single gridpoint can be obtained very fast, the number of gridpoints has to be increased in order to ensure stability and desired accuracy. Technical desirable chromatography systems should have a possibly high e$ciency, which is re#ected in relatively large numerical values of the parameter k . The advantages of this solution approach ?N G reported, for example systems by Tan (1997), do not hold in this case. This basically re#ects the di$culties commonly encountered when applying classical numerical
schemes to hyperbolic partial di!erential equations. To be able to approximate speci"c features of the solutions (e.g. dicontinuous solutions, shock waves) using a conventional discretization scheme a very "ne or automatically adapted grid is needed. This basic limitation can not be circumvented by the approach presented by Tan. A solution approach especially developed for hyperbolic PDEs is presented in the following section.
6. The ideal model Neglecting all nonideal e!ects and assuming overall equilibrium between liquid and solid phase, the ideal model of chromatography is obtained. It is usually formulated after de"ning a set of dimensionless variables with ¸ being a characteristic length of the column: x"z/¸, q"eut/¸.
(12)
A di!erential material balance for all components leads in the limit to the following set of quasilinear partial di!erential and algebraic equations: *c *f G# G"0, *x *q
(13)
ec #(1!e)q "f . (14) G G G With the Langmuir isotherm as in Eq. (9), this gives a system of coupled hyperbolic "rst-order partial di!erential equations which can either be solved numerically or the system might be solved in closed form for some cases of initial and boundary conditions using the theory of nonlinear wave propagation. The details of this solution procedure have already been presented by DuK nnebier and Klatt (1998) and are brie#y summarized in the sequel. 6.1. Summary of solution procedure The solution procedure can be summarized as follows: In the xrst step, the system is transformed with a nonlinear transformation (h-transformation) to decouple the equations. The solution for a Riemann problem (step input and constant initial conditions) can then be found analytically using the method of characteristics (Hel!erich & Whitley, 1996; Hel!erich, 1997; Rhee, Aris & Amundsen, 1989,1970). This methodology can, in its standard form, only be applied to systems with competitive Langmuir adsorption isotherms (see Eq. (9)), but an extension to more complex isotherms can be found in (ZenhaK usern & Rippin, 1998). The solutions of these problems are travelling waves, and with the theory of wave interaction the solution of several Riemann problems can be superposed. The straightforward application of this approach is limited, since more complex
G. Du( nnebier, K.-U. Klatt / Chemical Engineering Science 55 (2000) 373}380
interaction patterns might require the solution of a partial di!erential equation themselves. Since the usual initial and boundary conditions in most applications as the SMB processes are more complex than one single step, in the second step, the initial and boundary conditions have to be discretized as a series of steps using front tracking (Wendro!, 1992). Assuming a su$cient discretization, the whole solution can now be calculated analytically in the transformed space using the rules of wave propagation and wave interaction of shock waves. In the third step, the solution has to be transformed back into the original concentration space. The approach has previously been only applied to batch chromatography by (Wendro! (1993). The implementation to study a simulated moving bed process as a novel application is a challenging task due to the far more complex initial and boundary conditions in this case.
377
streams are chosen as follows: Q "0.0475 cm/s with $ a concentration of c "0.23 g/cm, Q "0.828 cm/s, $ G # Q "0.8773 cm/s and Q "0.0577 cm/s. " '4 7.1. General rate model For the discretization of the #uid phase, 12 cubic "nite elements are used for each column of Length ¸"25 cm. This results in computation times of approx. 30 cpu seconds on a PC PII266 for one switching period. 7.2. Ideal model The sampling parameter for the front tracking is chosen as 0.01 (see DuK nnebier & Klatt, 1998). This results in computation times of approx. 12 cpu seconds on a PC PII266 for one switching period. 7.3. Comparison of simulation results
7. Simulation example for SMB processes In this simulation example we consider a SMB unit with one column each in the "rst and fourth section and three columns in the second and third section, which re#ects an industrial separation of cyclic hydrocarbons. The adsorption equilibrium of the system can be described with a competitive Langmuir Isotherm as given in Eq. (9). The purpose of this example is to illustrate the capabilities of the modelling and simulation approaches described above. The simulation results of the general rate model are compared with the predictions of the ideal model, simulated using the method of characteristics and front tracking. For the reasons presented in Section 5, the kinetic approach has not been implemented for simulation. The isotherm parameters N have to be adjusted G for the two modelling approaches. In the rigorous model, the adsorbed solid concentration q1 is expressed G per volume of solid. In the ideal model, the adsorbed solid concentration q. is expressed per particle volume. G With q."q1(1!e )#c e G G N N G N
(15)
the following relation holds:
e LA N."N1(1!e )# N k c . G G N I I k G I
(16)
The system parameters of the example system chosen are N."0.8763, N."0.6812, k "1.85 cm/g, k "5.94 c m/g, all columns have a void fraction of e"0.4, a diameter of d"5 cm, a length of ¸"25 cm. The parameters N1 have been estimated for the interesting G concentration range using Eq. (16). The switching is performed after each t "3500 s, and the feed and outlet
In Fig. 2, the axial pro"les of the central columns of the SMB unit at di!erent instances during the startup period are displayed. Starting with at the top of the "gure, a comparison of the axial pro"les at the end of the "rst switching period leads to the following conclusions: E The solution of the ideal model forms a kind of skeleton for the prediction of the general rate model. E If the equilibrium is reached despite the kinetic e!ects (as in the "rst column after the feed inlet), both pro"les are nearly identical. This is only the case if the extension of the concentration plateaus is su$ciently large. E If the equilibrium is not reached due to the kinetic e!ects (as for component B at the frontal adsorption wave at the beginning of the second column after the feed inlet), the plateau level predicted by the two models is di!erent. Since the concentration level a!ects the speed of propagation in case of a nonlinear isotherm, the speed of propagation of the concentration waves is not predicted correctly by the ideal model in this case. This is a speci"c feature of the nonlinear adsorption isotherm, a very good agreement between the ideal and the general rate model can be obtained in case of a linear adsorption isotherm (DuK nnebier et al., 1998a,b). E This deviation is particulary large in Section III (between feed inlet and ra$nate outlet) and grows during the simulation due to the integration, as can be seen in the two lower parts of the "gure showing the axial pro"les at the end of the second and "fth switching periods, respectively. The axial pro"le of the column in cyclic steady state, which is approximately reached after the 70th switching period, as predicted by both modelling approaches is
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Fig. 2. Development of axial pro"les during startup procedure.
displayed in Fig. 3. One can consider the solution of the ideal model as a skeleton for the solution of the general rate model, and the predicted pro"les do agree closely in most parts of the process.The small error in the calculated speed of propagation of the concentration waves leads to an error in the prediction of the pro"les in steady state due to the integrating e!ect of the long time horizon. The concentration levels of the constant states are slightly di!erent, and some e!ects, like the kinks predicted by the ideal model in sections two and three, are predicted more smoothly by the general rate model due to the e!ects of mass transfer resistance and axial dispersion. The predicted location of the desorption front of component A at the extract outlet and the adsorption front of component B at the ra$nate outlet of both models show good agreement and the chosen operating conditions lead to the desired separation in both cases. As a "rst initial guess, a design of a SMB unit using the ideal model might therefore be feasible. But the di!erent prediction of the product #ows
of the impurities leads to signi"cantly di!erent product purities in both cases:
Purity [%]
Ideal model General rate model
Extract
Ra$nate
100 99.45
78 87.64
Due to the unrealistic assumptions of ideality, the accuracy of prediction of this modelling approach is not su$cient for design, control or optimization purposes. Those deviations between the two models are not due to the front tracking procedure, since the same e!ect can also be found in the case of batch chromatography, where analytical solutions for the ideal model are available. Great care has to be taken when using design approaches based on ideal models, as the one by
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379
Fig. 3. Comparison of axial pro"les of SMB unit for cyclic steady state at the end of a switching period.
Mazzotti et al. (1997). Besides the questionable assumption of ideality, these approaches use a (steady-state) model of the corresponding true moving bed process introducing an additional error. The physical plane portrait (showing the propagation directions of the di!erent waves in a q vs. x graph) as one of the results of the ideal model solution might be used to obtain an understanding of the complex and interacting process dynamics. By following the shock paths through space and time, one can examine the complex interaction patterns. An example is given in DuK nnebier and Klatt (1998). Therefore, the ideal model can be useful for the analysis of the qualitative behavior of the process and the in#uence of di!erent process parameters. Especially the possible insight in the dynamics which can be gained by examination of the shock paths helps to understand and analyse the process dynamics and the startup procedure. The ideal model should not be chosen in any application where precise quantitative predictions are needed since those can not be given due to the unrealistic modelling assumptions.
8. Conclusions Three di!erent modelling and simulation approaches for chromatographic separation process have been presented and analyzed, starting with the rigorous general rate model considering various kinetics e!ects. Two simpli"ed modelling approaches have been presented in the sequel, one lumping the kinetic e!ects into a single parameter, the second (being the ideal model) neglecting all kinetic nonidealities. The suitable modelling and simulation approach has to be chosen depending on the speci"c application. This paper presented a case study extending and applying two modelling and simulation approaches to the complex simulated moving bed (SMB) process as a novel applica-
tion. The case study was used to illustrate the capabilities of the proposed approaches. The ideal model and the solution approach based on the method of characteristics and front tracking allows speci"c qualitative insight into the complex interacting dynamics of this process especially during the startup procedure but lacks of a su$cient accuracy for quantitative predictions due to the unrealistic modelling assumptions. The approach proposed based on a kinetic model seems to be very elegant and e!ective at "rst glance but is limited to columns with low e$ciency which are very rarely used in practical applications today. The general rate model and the numerical solution approach presented here allow for a quantitative analysis of the process. As to be expected, those calculations tend to be more computational expensive than the solution of the ideal model, though far less as reported otherwise in the literature.
Acknowledgements The authors are indebted to S. Engell for his support and valuable input to this work. The "nancial support of the Bundesministerium fuK r Bildung und Forschung under the grant number 03D0062B0 is very gratefully acknowledged. We also kindly acknowledge the help and support of T. Gu.
Notation Variables c concentration of component i in bulk phase, G g/cm cCO equilibrium concentration of component i, G g/cm c concentration of component i in particle pores, N G g/cm D axial dispersion coe$cient, cm/s ?V
G. Du( nnebier, K.-U. Klatt / Chemical Engineering Science 55 (2000) 373}380
380
D N G
Q R N q t u x Z
particle di!usivity coe$cient of component i, cm/s void fraction of the bed, dimensionless void fraction of the particle, dimensionless "lm mass transfer resistance of component i, cm/s lumped mass transfer resistance of component i, 1/s Langmuir Isotherm coe$cient of component i, cm/g column length, cm Langmuir isotherm constant, g/cm concentration of component i in the solid phase, g/cm volumetric #ow rate, cm/s particle radius, cm dimensionless time, dimensionless valve switching interval for SMB process, s #uid velocity, cm/s axial coordinate, cm dimensionless axial coordinate, dimensionless
Indices i in j out R E F D
component i"A,B column inlet section j"I,II,III,IV column outlet ra f f inate extract feed desorbate
e e N k J G k ?N G k G ¸ N G q G
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