Modeling of Nonlinear Multi Component Chromatography

8/3/2019 Modeling of Nonlinear Multi Component Chromatography

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Modeling of Nonlinear MulticomponentChromatographyT.Gul. G.-J. Tsai2. and G. T. Tsao3'Department of Chemical Engineering. Ohio University. Athens. OH 45701.USABuilding 130. Lederle Laboratories. Pearl River. NY 10965. USALaboratory of Renewable Resources Engineering. 1295 Potter Center. PurdueUniversity. West Lafayette. IN 47907.1295. USA

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .istofSymbols1 Review of Models for Chromatography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1 Equilibrium Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 Plate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3 RateModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.1 Rate Expressions and Particle Phase Governing Equations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . ..3.2 Adsorption Kinetics and Affinity Chromatography. . . . . . . . . . . . . . . . . . . . . . . . . . ..3.3 Governing Equations for Bulk-Fluid Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..4 General Multicomponent Rate Models1.5 Solution to the General Multicomponent Rate Models . . . . . . . . . . . . . . . . . . . . . .1.5.1 Discretization of Particle Phase Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . ..5.2 Discretization of Bulk-Fluid Phase Equations1.5.3 Solution to the ODE System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .A General Multicomponent Rate Model for Axial Flow Chromatography2.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. 2 ModelFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Numerical Solution to the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..1 Discretization3.2 Solution to the ODE System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..3 Isotherm Expressions . . . . . . . . . . . . . . . . . . . . . . . . . .Efficiency and Robustness of the Numerical Procedure

5 Extension of the Rate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1 Addition of Second Order Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5. 2 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3 Addition of Size Exclusion Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5. 4 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Other Extensions of the Rate Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Study of Stepwise Displacement7.1 Results and Discussion (70) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . ..2 Effect of Feed Concentration of Displacer (C,,) . . . . . . . . . . . . . . . . . ..3 Effect of Adsorption Equilibrium Constant of Displacer (b, )8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Biochemical Enginarin gBiotcchnology. Vol 49Managing Editor: A Ficchtcr0 pringer-VcrlagBerlin Hcidelberg 1993


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46 T. Gu, G.-J. Tsai and G. T. TsaoIn the age of rapid development of biotechnology, preparative and large scale chromatographybecomes more and m ore popular. Unlike analytical chromatography, dispersion and mass transfereffects are often significant in preparative and large scale chromatography. Conce ntration overloadoften leads to nonlinearity d t h e system. The study of nonlinear chromatograp hy becomes more andmore demanding. Much w ork has been done in the past two decades, but many topics of practicalimportance still need to be tackled. In this chapter, a review is given on different models forchromatograp hy. This chapter provides a brief review of different mathematical models for non-linear chromatography. A general multicomponent rate m odel, which accounts for various m asstransfer mechanisms and nonlinear isotherms is presented. This comprehensive model is a verypowerful tool for the study of the dynamics of nonlinear multicomponent chromatography. Thischapter also presents an efficient numer~calmethod for the solution of the model and its numerousextensions. As an example, the model is used for the study of some interesting effects of isothermcharacteristics of the displacer on the optimization of stepwise displacement.

List of SymbolsDescriptionconstant in Langmuir isotherm for component i, biCyadsorption equilibrium constant for component i, kai/kdiBiot number of mass transfer for component i, kiRp/(~p Dpi)rkiRp (&:iDpi)bulk phase concentration of component ifeed concentration profile of component i, a time dependent variableconcentration used for nondimensionalization, max{Cfi(t))concentration of component i in the stagnant fluid phase insideparticle macroporesconcentration of component i in the solid phase of particle (moleadsorbatelunit volume of particle skeleton)adsorption saturation capacity for component i (mole adsor-batelunit volume of particle skeleton)adsorption saturation capacity based on the unit volume of the bedconcentration of component i in the stationary phase based on theunit volume of the bedconcentration of component i in the fluid phase based on the unitvolume of the bed= c b i I C 0 i= Cp11c0i= Cii/COi= CiW/COi

axial or radial dispersion coefficient of component iDamkohler number for adsorption, L (h icoivDamkohler number for desorption, Lk&size exclusion factor for component i, E;~/E,,


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Modeling of Nonlinear Multicomponent Chromatography

film mass transfer coefficient of component iadsorption rate constant for component idesorption rate constant for component icolumn lengthnumber of interior collocation pointsnumber of quadratic elementsnumber of componentsPeclet number of axial dispersion for component i, vL/Dbiradial coo rdinate for particleparticle radius=R/Rptimeinterstitial velocityaxial coordinate= Z/L

Greek LettersEb bed void volume fraction&P particle porosity$i accessible particle porosity of component i

~p DpiL"l dimensionless constan t,--ivSi dimensionless constant for component i, 3Biiqi(l- b)/cbvtT dimensionless time, -LTimp dimensionless time duration for a rectangular pulse of the sample4 Lagrangian interpolation function


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48 T. Gu, G.-J. Tsai and G. T. Tsao1 Review of Models for ChromatographyA very comprehensive review on the dynamics and mathematical modeling ofadso rption and chro mato grap hy was given by Ruthven [I]. Mod els in this areaare generally classified in to three categories [I]: equilibrium theory, platemodels and rate models.

1 .I Equilibrium TheoryGlueckauf [2] is considered as being the first person t o develop the equilibriumtheory of multicomponent isothermal adsorption [I]. The theory further de-veloped in to the interference theory by Helfferich and K lein [3] is mainly aimedat stoichiometric ion-exchange systems with constant separation factors. Amathematically parallel treatise for systems with multicomponent Langmuirisotherms was developed by Rhcc and co-workers [4,5].Equilibrium theory assumes direct local equilibrium between the mobilephase and the stationary phase, neglecting axial dispersion and mass transferresistances. The theory gives good interpretation of experimental results forchrom atogra phic columns with fast mass transfer rates shown by many analyti-cal and some preparative columns. It can provide general location of theconcentration profiles of a chro mato grap hic system but fails to provide acc uratedetails if mass transfe r effects in the system a re significant [6]. Equ ilibriumtheory has been widely used for the study of multicomponent interference effects[3] an d ideal displacemen t development [5]. Ma ny cases of prac tical applica-tion have been reported [3, 7-12].

1.2 Plate ModelsGenerally speaking, there are two kinds of plate models, which may also becalled staged models o r staged theories [13]. The first kind is directly analog ousto the "tanks in series" model for nonideal flow systems [I]. In suc h a model, thecolumn is divided into a series of small artificial elements. Inside each elementthe content is assumed to be completely mixed. This gives a set of first orderordin ary differential equations (ODE 'S) that describe the adso rption a nd inter-facial mass transfer processes. Many researchers have c ontribu ted to this kind ofplate model [I, 14-16]. Howe ver, plate mode ls of this kind generally are notsuitable for multicomponent chromatography since the equilibrium stages maynot be assumed equal for different components.The other kind of platc m odel is formulated based on the distribution factorswhich determine the equilibrium of each component in each of the artificialstages, and the mode l solution involves recursive iterations, rather than solvingfor ODE systems. The most popular of this kind are the Craig distributionmodels. Considering the blockage effect, the Craig models are applicable to


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Modeling of Nonlinear M ulticomponent Chromatography 49multicomponent systems. Descriptions of Craig models were given by Eble et al.[17], Seshadri and Deming [18], and Solms et al. [19]. In recent years, Craigmodels have been extensively used for the study of column overload problems[17,20].

1.3 Rate ModelsThe word "rate" refers to the rate expression or rate equation for the masstransfer between the mobile phase and the stationary phase. A rate modelusually consists of two sets of differential mass balance equations, one for thebulk-fluid phase, the other for the particle phase. Different rate models havediffcrent complexities. A comprehensive review of rate models was given byRuthven [I].

1.3.1 Rate Expressions and Particle Phase Governing EquationsThe solid film resistance hypothesis was first proposed by Glueckauf and Coates[21]. It assumes a linear driving force between the equilibrium concentrations inthe stationary phase (determined from the isotherm) and the average fictitiousconcentrations in the stationary phase. Because of its simplicity, this rateexpression has been used by many researchers [I , 22-25] but this model cannotprovide the details of the mass transfer processes.

The fluid film resistance mechanism which also assumes a linear drivingforce is widely used [I]. It is often called external mass transfer resistance. If theconcentration gradient inside the particle phase is ignored, the model thenbecomes the lumped particle model, which has been used by some researchers[27-291. If the Biot number for mass transfer, which reflects the ratio of thecharacteristic rate of film mass transfer over that of intraparticle diffusion, ismuch larger than 1, the external film mass transfer resistance can be neglectedwith respect to pore diffusion.

In many cases both external mass transfer and intraparticle diffusion mustbe considered.A local equilibrium is often assumed between the concentrationin the stagnant fluid phase inside macropores and the solid phase of the particle.Such a rate mechanism is adequate to describe the adsorption and mass transferbetween the bulk-fluid phase and the particle phase, and inside the particlephase in most chromatographic processes. The local equilibrium assumptionhere is different from that made for the equilibrium model which assumesa direct equilibrium of concentrations in the solid and the liquid phase withoutany kind of mass transfer resistances.

1.3.2 Adsorption Kinetics and AfJinity ChromatographyIn some cases, the adsorption and desorption rates may not be high enough andthe assumption of the local equilibrium between the concentration in the


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50 T. Gu, G.-J. Tsai and G.T. Tsaostagn ant fluid phase inside macrop ores and the solid phase of the particle is nolonger valid. Kinetic models m ust be used. Some kinetic models were reviewedby Ruthven [I ] and Lee et al. [30]. The second order kinetics has been widelyused in affinity chromatography [31-391. If the saturation capacities for all thesolutes are the same, the second order kinetics reduces to the Langmuir iso-therm when equilibrium is assumed.

1.3.3 Governing Equations for Bulk-Fluid PhaseTh e partial differential equ ation for the bulk-fluid phase can be easily obtain edwith differential mass balances. They usually contain the following terms: axialdispersion, convection, transient, and the interfacial flux. Such eq uation s them-selves are generally linear if physical param eters are no t con centration depen-dent. They become nonlinear when coupled with nonlinear rate expressions.Analytical solutions may be obtain ed using Laplace transformation [39,40]for many isothe rmal, single compo nen t systems with linear isotherms. Th e linearoperator m ethod [41] can also be used to solve problem s in linear chrom atogr a-phy. Fo r mo re complex systems, especially those involving nonlinear isoth erms,analytical solutions generally cannot be derived [I]. With the rapid growth ofthe availability of fast and powerful computers and development of efficientnumerical method s, it is now possible to obtain num erical solutions to complexrate models that consider various forms of mass transfer mechanisms [42].Complex rate models are now becoming m ore and more p opular especially inthe study of preparative and large scale chromatography.

1.4 General Multicomponent Rate ModelsA rate mod el which considers axial dispersion, external mass transfer, intrapar-ticle diffusion and n onlinear iso therms is considered a general m ulticompo nentrate model. Such a general model is adequate in most cases to describe theadsorption and mass transfer processes in multicomponent chrom atography . Insome cases surface adsorp tion and size exclusion, adsorp tion kinetics, etc., mayhave to be included to give an adequate account for a particular system.Several groups of researchers have proposed and solved various generalmulticomponent rate models using different numerical approaches [42-451.

1.5 Solution to the General Multicomponent Rate ModelsA general multicomponent rate model consists of a coupled PDE system withtwo sets of mass balance equ ation s in the bu lk-fluid an d particle phases for eachcomponent, respectively. The transient PDE system becomes nonlinear if anynonlinear isotherms or nonlinear kinetics are involved in the system.


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Modeling of Nonlinear Multicomponent Chromatography 5 1Th e finite difference me thod is a very simple numerical proce dure a nd ca n be

directly applied for the solution t o the mode l [45,46], but this procedure oftenrequires a huge am oun t of memory space, and its efficiency an d accuracy are notcompetitive compared with other advanced numerical methods, such as ortho-gonal collocation (OC), finite element, and orthogonal collocation on finiteelement (OCFE ).The general strategy for solving a nonlinear transient PDE system numer-ically using the advance d nu merical methods is to discretize the spatial axes inthe model equations first, and then solve the resulting ODE system using anOD E solver.

1.5.1 Discretization of Particle Phase EquationsThe OC method is a very accurate, efficient and simple method for discreti-zation. It has been widely used for particle problems [47,48] and is obviouslythe best choice for the particle phase governing eq uation s of general multicom-ponent rate models [42,43,44].

1.5.2 Discretization of Bulk-Fluid Phase EquationsCon centra tion gradients in the bulk-fluid phase can be very stiff, thus, the O Cmethod is no long er suitable, since global splines using high o rder polynomialsare very expensive [48] and sometimes not stable. The OCFE method useslinear elements for global spline and collocation points inside each element. N onum erical integratio n for element matrices is needed because of the use of linearelements. This discretization method can be used for systems with stiff gradientsC481.The finite element method with a higher order of interpolation functions(typically quadratic, or occasionally cubic) is a very powerful method for stiffsystems. Its highly streamlined structure provides unsu rpassed convenience andversatility. This method is especially useful for systems with variable physicalparameters, such as radial flow chrom atography an d nonisothermal adsorptionwith or without chemical reaction. Chromatography of some biopolymers alsoinvolves variable axial dispersion coefficient [49].

1.5.3 Solution to the ODE SystemIf the finite element method is used for the discretization of bulk-fluid phaseequations and O C for the particle phase equations, an O D E system then results.The O D E system with initial values can be readily solved using an O D E solversuch as subroutine IVPAG of the IMSL [50], which uses the powerful Gear'sstiff meth od 1511.


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IT. Gu, G.-J. Tsai and G. T. Tsao

If the discretization of the bulk-fluid phas e equ ations is carried out using theO C FE m ethod, an O D E system coupled with some algebraic equations whichcome from the continuity of bou ndary fluxes results [44,48]. Th e system can besolved using a n available differential algebraic equation solver.Such a system can also be conveniently solved with an ODE solver if onemanipulates the user-supplied function su broutin e which evaluates the concen-tration derivatives for the O D E solver to eliminate those algebraic equa tions inan in situ fashion. This is possible since the trial concentrations are given asarguments for the subroutine. This approach helps reduce the total number ofequa tions in the final system. It was apparently adop ted by G ardini et al. [52] ,for a multi-phase reaction engineering problem.

2 A General Multicomponent Rate Modelfor Axial Flow Chromatography2.1 Model AssumptionsFigure 1 shows the anatom y of a chromato graph ic column w ith axial flow. Thefollowing basic assum ptions are needed for the form ulation of the general ratemodel.- Th e multicomponen t fixed-bed process is isothermal.- Th e bed is packed with porous a dsorb ents which are spherical and uniform in

size.-The concentration gradients in the radial direction of the bed are negligible.- Local equilibrium exists for each compon ent between the pore surface and thestagnant fluid phase in the macropores.- The diffusional and mass transfer coefficients are con stant a nd inde pendent ofthe mixing effects of the components.2.2 Model FormulationBased on these basic assumptions, the following governing equations can beformulated from the differential mass balances for each component in thebulk-fluid and the particle phases.


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Modeling of Nonlinear Multicomponent ChromatographyBulk-Fluid Phase

Effluent History(Chromatogram)

\-- cjiParticle PhaseFig. 1. Anatomy of a chromatographic column

with the initial and boundary conditions

Equations (1) and (2) are coupled via C p i ,R = Rp which is the concentration ofcomp onent i at the surface of a particle. In Eq. (2), C;i is the concentration ofcomp onent i in the solid phase of the adsorbents based o n the unit volume of thesolid, excluding pores. It is directly linked to the multicomponent isothermswhich couple the PD E system based on assumption (4). Concentrations CbiandCpiare b ased on th e unit volume of mob ile phase fluid. The effective diffusivities,D p i , n this work do not include the particle porosity.By introducing the following dimensionless terms


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54 T. Gu, G.-J. Tsai an d G. T. Tsaothe P D E system can be transformed into the following dimensionless forms.

For frontal adsorption Cfi(z)/Coi= 11 0II impFor elution Cfi(t),Coi= {o else

After the sample introduction (in the form of frontal adsorption):if component i is displaced, Cfi (t)/Coi = 0if component i is a displacer, Cfi(z)/Coi= 1

No te that all the dimensionless conc entration s are based on C oi which is equalto the maxihum of the feed profile C fi( t). Fo r example, in an elution, ifcom pone nt i is a sam ple solute in the sample which is injected as a rectangularpulse, the profile of C fi( r) s then of a rectangu lar shap e, and its upper bou ndaryvalue is the value of Coi.

3 Numerical Solution to the Model

An efficient and robust numerical procedure has been developed by Gu et al.[42] for the solution to the abo ve P D E system. It involves two parts. First, thespatial axes, z and r, are discretized. And then the resulting O D E system (withinitial values) is solved with an ODE solver (integrator). An overview of thegeneral strategy for the solution is shown in Figure 2.


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Coupled PDE System

Bulk Phase PDEs Particle Phase PDEs

Coupled ODE SystemNs(ZNe+ l) (N+l )

Gear's Stiff Method

Concentration Profile& Effluent HistoryI Inside Column Ic b i ( r ' z ) VS. Z

Fig. 2. Solution strategy for the general multicomponent rate model

3.1 DiscretizationEqu ations (9) and (10) can be d iscretized into a set of OD E'S by the finite elementand the O C methods respectively. Using the Galerkin approxim ation an d thefirst weak form [53], Eq. (9) becomes

where (DBi)&,,= f +,+,dz (18)

in which m, n E {1 , 2 , 3 } , and the superscript e indicates that the finite elementmatrices and vectors are evaluated over each individual element before globalassembly. Four point Gauss-Legendre quadratures [53] are used for integra-tions. The superscript p rime in this wo rk indicates a first order time derivative.The bold face variables indicate matrices or vectors. The natural boundary


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56 T. Gu, G.-J.Tsai and G. T. Tsaocondition (P B,) Z = = - hi + Cfi T ) / C ~ ,ill be applied to [AKB,] and [AFB,]at z = 0. (PB,) = 0 anywhere else.Using the same symme tric polynom ials as defined by Finlayson [48], Eq.(10) is transformed to the following equation by the O C method.

in which gi = (1 - p)cii+ cPcpi.Note that g i for each com ponent i contains thenonlinear m ulticompone nt isotherms. The value of (cP i),+, (i.e., cp i,,= ) can beobtained from the boundary condition at r = 1, which givesN + 1x AN+l, j(cP i) j B i i (cb i -~p i , r= l )j = l

(22)or

NB i i ~ b i- x AN+l,j(cpi)j

where the matrices A and B are the same as defined by Finlayson [48].

3.2 Solution to the ODE SystemIf Ne quadratic elements (i.e. (2Ne + 1) nodes) are used for the z-axis inbulk-fluid phase equations an d N interior O C points are used for the r-axis inparticle phase eq uations, the abov e discretization p rocedure gives Ns (2Ne + 1)(N + 1) OD E'S which are then solved simultaneously by G ear's stiff method[50]. A function subroutine must be supplied to the ODE solver to evaluateconcentration derivatives at each element node and OC point with given trialconcentration values. The conc entration derivatives at each element node (c&)are easily determined from Eq. (17). The concentration derivatives at each O Cpoint (c',,) are coupled because of the complexity of the isotherms which arerelated to gi via c",. At each interior O C point, E q. (21) can be rew ritten in thefollowing matrix form.

whereag dcpjG P i j= -, cb j = -, RH, = right hand side of Eq. (21)8% d~

Since the matrices [GP] and [RH] are known with given trial concentrations ateach interior O C point, the vector [cb] can be easily determined from Eq. (24).Using this approach, we can deal with complex nonlinear isotherms herewithout any iteration.


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Modeling of Nonlinear Multicomponent Chromatography3.3 Isotherm ExpressionsThe num erical procedure discussed above can accomm odate any type of nonlin-ear isotherms as long as they do not cause mathematical singularities. Thefollowing two common types of isotherms are used in this work.(1) Langmuir Isotherm

where ai = C m b i .Note that bjCojcan be treated as a dimensionless group foreach component. With this the entire model system can then be treated withdimensionless parameters alone. This helps reduce the total number of para-meters involved in discussions.(2) Stoichiometric Isotherm with Constant Separation Factors

where uij = l / a j i = aik k j , n d aii= 1. Ci is the concentration of ion compone nti in the stagnant fluid inside particles. C is the saturation capacity and isconsidered equal for all comp onents. Ci is the concen tration of ion com ponenti in the s olid of the particles. This type of isotherm is widely used in ion exchangeand all the concentrations are based on the unit volume of the column ratherthan on the respective phases as in the case of Langmuir isotherms [3].The stoichiometric isotherm can be converted into the isotherm shownbelow, which is the same algebraic expression as the La ngmuir isotherm exceptthat the "1 + " in the denominator of the Langmuir isotherm expression isdropped.

The following relationships are needed for the conversion.bi = ui ,Ns and ai = biCw = ai,N s c (i = 1 , 2 , . . . ,Ns )- ~ b ) ( l E p )

where ion compo nent Ns is assigned as the basis of the separation factors. Notethat the units of a i and bi in the Langmuir isotherm and the convertedstoichiometric isotherm are not the same. In the stoichiometric isotherm, theconcentrations of compone nts canno t all be zero a t the same time, which meansthat the column is never "empty."


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58 T. Gu, G.-J. Tsai and G. T. Tsao4 Efficiency and Robustness of the Numerical ProcedureThe solution to the rate model provides the effluent history and the movingconcen tration profiles inside the colum n for each com ponent. Th e concentrationprofile of each com ponent inside the stagnant fluid phase an d the solid phase ofthe particle can also be obtained, but they are rarely used for discussions.Generally speaking, one interior collocation point (N = 1) is sufficient insome cases, while more often N = 2 is needed, especially when D Pi values aresmall, which in turn give large Bii and small qi values. N = 3 is rarely needed.The number of elements Ne = 5-10 is usually sufficient for systems with non -stiff or slightly stiff concentration profiles. For very stiff cases, Ne = 20-30 isoften enough.Insufficient N tends to give diffused concentration profiles as shown in Figs.3 t o 5. Using N = 1 instea d of N = 2 in Figs. 3 to 5 (dashed lines) saves ab ou t60% C P U time on a S U N 41280 computer, but the conc entration profiles differto some extent from the converged ones (solid lines). In Fig. 5, the dotted linesare obtained by using three quad ratic finite elements (Ne = 3) and one interiorcollocation point (N = 1) with a CPU time of only 13.2 seconds. Thoughthe dotted lines show a certain degree of oscillation, they still providethe general shapes of the converged concentration profiles, which take6.9 minutes of CPU time. This means that one may use small Ne and Nvalues to get the rou gh conc entration profiles very quickly an d then d ecide whatto do next.The efficiency and robustness of the numerical pro cedurc are fu rther dem on-strated by more simulated effluent histories for the discussions in the followingparts of this chapter, including cases involving very stiff concentration profiles.

Dimensionless TimeFig. 3. Effect of the number of interior collocation points in the

C:0.-f! 1.5*IY1.0b

:.9a 0.5a

simulation

-- e d @ N=2; 12.1 min.----e=lO, N=l; 4.0 min-

-

frontal

Oo 1 2 3 I 5

adsorption


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1I -- N e ~ 7 , -2; 14.8mill

. - - - - - Ne.7, N=l; 5.4 min

2

10 20 30 40Dimensionless TimeFig. 4. Effect of the number of interior collocation points in the simulation of elution

I0 2 L 6Dimensionless Time

aS2 1.5*e86 1.08

0.5

IB 0Fig. 5. Convergence of the concentration profiles of a stepwise displacement system

- e.12, N=2; 6.9min- Ne. 12 N.1; 2.7 minN e d , N = l ; 1 3.2 s

1 2 (Dirplacet)

-

. . . .

The FO RT RA N code based on the numerical procedure discussed above iscapable of simulating many kinds of multicomponent chromatographic pro-cesses, including frontal analysis, displacement development, simple nongradi-ent elution, nonlinear gradient elution, an d some m ultistage operations. Eachmode of simulation is designated with a process index in the code, which isincluded in the dat a input.The input data for the FORTRAN code contains the number of compo-nents, elements and interior collocation points, process index, time con trol da ta,dimensionless parameters, isotherm type an d parameters. No te tha t the code isbased on the dimensionless PDE systems and CO i an be combined with bi to


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60 T. Gu , G.-J.Tsai and G. T. Tsaoform a dimensionless group biCoi.The initial conditions are reflected in theprocess index, or entered in the data file.

5 Extension of the Rate ModelThe assumption that a local equilibrium exists for each component between thepore surface and the stagnant fluid phase in the macropores (Sect. 2.1) may notbe satisfied if the adsorption and desorption rates are not high, or the masstransfer rates are relatively much faster. In such cases, isotherm expressionscannot be inserted into Eq. (2) to replace Cii. Instead, a kinetic expression isoften used. The so-called second order kinetics has been widely used to accountfor reaction kinetics in the study of affinity chromatography [31-33,3536,381.A general rate model with second order kinetics has been applied to affinitychromatography by Arve and Liapis [38].5.1 Addition of Second Order KineticsThe second order kinetics assumes the following reversible binding and dissocia-tion reaction.

where Pi is component i (macromolecule) and L represents the immobilizedligand. In this elementary reaction, the binding kinetics is of second order andthe disassociation first order, as shown by the rate expression below.

where kai and kdi are the adsorption and desorption rate constants for compo-nent i, respectively. The rate constant kai has a unit of concentration over timewhile the rate constant kdi has a unit of inverse time.If the reaction rates are relatively large compared to mass transfer rates, theninstant adsorption/desorption equilibrium can be assumed such that both sidesof Eq. (28)can be set to zero, which consequently gives the Langmuir isothermswith the equilibrium constant bi = kai/kdi or each component.Introducing dimensionless groups Daf = L (kaiCoi )/v nd Dap = Lkdi/vwhich are defined as the Damkohler numbers [54] for adsorption and desorp-tion, respectively, Eq. (28) can be nondimensionalized as follows.


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Modeling of Nonlinear Multicomponent Chromatography 61If the sa turation capacities are the same for all the comp onents, at equilibrium,Eq. (29) gives biCOi= Da;/Dap and ai = C W b i= cWDa;/Daf for the resultantmulticompo nent L angmu ir isotherm. The Dam kohle r numbers reflect the char-acteristic reaction times with that of the stoichiometric time. The bi = kai /kdivalues for affinity chr om ato gra phy are often very large [32], but it is erro neo usto jump to the conclusion depe nding on this alone that the desorption rate m ustbe much smaller than tha t of adsorp tion, since the two processes have differentreaction orders, and the concen tration C pi s often very small on the adso rptionside as expressed by the first term on th e right hand side of Eq. (28). It is obviou stha t the dimensionless Dam kohle r numbers provide a better comparison in thisregard.

5 .2 Solution StrategyAdding the second order kinetics to the general rate model does not com plicatethe numerical procedure for its solution since the discretization process isuntouched. O ne only has to add Eq. (29) in the final OD E system.The following equ atio n should be used to replace Eq. (21)

The final OD E system consists of Eqs. (17), (29) an d (30). With the tria l values ofchi, cpi and c", in the function su broutin e in the F OR TR AN code, their deriva-tives can be easily evaluated from the three O D E expressions.If Ne elements and N interior collocation points are used for the discreti-zation of the Eqs. (1) and (2), there will be Ns (2Ne + 1) (2N + 1) ODE's in thefinal OD E system, which a re Ns (2Ne + 1) N more than in the equilibrium case.These extra OD E's come from Eq. (29) at each element node an d each interiorcollocation point for each component.The relationship among the kinetic effects, reaction equilibrium and masstransfer rates were discussed by Gu [72].

5.3 Addition of Size Exclusion EflectsIn so me chro mato graph ic systems, large solute molecules have considerable sizeexclusion effects, which means tha t s uch large molecules either can not accesspar t of the small macropores in the particles or the entire particle a t all. This isespecially true in affinity chromatography in which large macromolecules areoften present, and sometimes even larger complexes can be formed between themacromolecules with the soluble ligands. Size exclusion effects reduces thesaturation capacity of a component with a large molecular size. A new isothermsystem was developed recently by Gu et al. [55], for the study of adsorptionsystems with uneven saturation capacities as a result of size exclusion.


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62 T. Gu, G.-J. Tsai and G. T. Tsao IIn recent years, there have been three ACS Symposium Series on size

exclusion chromatography [56,57,58]. Several mathematical models have beenproposed for size exclusion chromatography [59,60,61] among which themodel proposed by Kim and Johnson is particularly helpful for this work. Theirmodel is similar to the general rate model described in Sect. 1.2 of this work,except that their model considers size exclusion single component systems in-volving no adsorption. They introduced an "accessible pore volume fraction" toaccount for the size exclusion effect.In this work, the symbol E" is used to denote the accessible porosity (i.e.,accessible macropore volume fraction) for component i. It implies that for smallmolecules with no size exclusion effect, E B ~ ~ E,, and for large molecules that arecompletely excluded from the particles E" = 0. For any medium-sized molecules0 < ~i~ E,. It is convenient to define a size exclusion factor 0 I f"I suchthat &ti F:XcP.Ff" s a function of the distribution coefficient of component i.It is also a function of the particle size distribution if the particle sizes cannot beassumed to be equal C60J To include the size exclusion effect, Eq. (2) should bemodified as follows.

aciiwhere the first term (1 - E,)- should be dropped or set to zero, if a compo-atnent does not bind with the stationary phase. It should be pointed out again thatin the equation above Cii values are based on the unit volume of the solids of theparticles excluding the pores measured by the particle porosity E,. For a com-ponent which is completely excluded from the particles (i.e., E B ~ ~ 0),adsorbingonly on the outer surface of the particles, Eq. (31)degenerates into the followinginterfacial mass balance relationship.

This equation can be combined with the bulk-fluid phase governing equation(Eq. (1)) to give the following equation w,hich is similar to a lumped particlemodel.

where C", either follows the multicomponent isotherms or the expression forreversible binding, Eq. (28). If component i does not bind with the stationaryphase, CiiE 0 and the fourth term in Eq. (33) is dropped for that component. Asa reminder again, the solid phase concentration of component i, C",, is based onthe unit volume of the solid part of the particle excluding pores, i.e., the unit


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Modeling of Nonlinear Multicomponent Chromatography 63volume of the solid skeleton, The dimensional form of Eq. (33) is listed below.

5.4 Solut ion StrategyIf no component is totally excluded, the addition of the size exclusion effect inthe ra te models is very simple . One only has to use E ; ~ D ~ ~o replace cpD Pin theexpressions of Bi, and qi , and to use E;~c,, to replace e Pcpi n Eq. (10).Mathematically, a singularity occurs in the model equation system whena component (say, component i) is totally excluded from the particles (i.e.,E ; ~ = 0) if o ne does n ot use Eq. (34) to replace Eqs. (9)and (10). It turns out thatfor numerical calculation, there is no need to w orry a bou t this singularity, if E> isgiven a very small value below tha t of the tolerance of the ODE solver, which isset to lo -' throug hout this work. It is found that this treatm ent gives the resultswhich have the same values for the first five significant digits as those obtainedby using Eq. (34).One should be aware that the size exclusion of a component affects itssaturation capacity in the isotherm. It also affects the effective diffusivity of thecom pone nt since the tortuosity is related t o accessible porosity. It is clear thatusing size exclusion in a multicomponent model often leads to the use of unevensatura tion capacities for a componen t with significant size exclusion and a com -ponen t witho ut size exclusion. Th is may cause problems w hen the m ulticompo-nent Langmuir isotherm is used in terms of thermodynamic inconsistency [55].

6 Other Extensions of the Rate Model

The general rate model can also be modified to account for the interactionbetween adsorbates and soluble ligands as in affinity chromatography. Thisextension is considerably more complicated. Details were given by Gu 1721.

7 Study of Stepwise Displacement

Stepwise displacement chromatography has received considerable attentionrecently in microbiological processes for in situ removal of toxic product(s)[62,63 ,64]. Lee et al. [30], used a polyvinylpyridine (PV P) resin for the in situremoval of lactic acid during growth. This kind of in situ separation reduces


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64 T. Gu, G.-J. Tsai and G. T. Tsaoproduct inhibition, and thus enhances productivity. In such adsorption-com-bined processes, chromatographic columns are coupled with the bioreactor toremove the product(s) simultaneously via preferential adsorption and the adsor-b a t e ( ~ )s(are) then recovered throug h a displacement operation. This kind ofstepwise displacement is also widely used to recover biomolecules from a dilutesolution after they are adsorbed onto a column. In both cases, frontal adsorp-tion proceeds the displacement process which often concentrates the adsor-b a t e ( ~ ) y using a suitable displacer. This kind of stepwise displacement opera-tion is somewhat different from the classical displacement chromatography ordisplacem ent develop ment first classified by Tiselius [65] an d extensively re-viewed by Horvath and co-workers [ll,49,66,67].Classical displacement chromatograp hy was described by many researchersas a process in which a column packed with solid adsorben t is equilibrated withthe mobile phase that has no or weak affinity to the adsorbent. A sample ofmixtures is then introduced to the column. The sample usually takes upa fraction of the column volume in the inlet section. Subsequently, a develop-ment agent (called displacer) is pumped into the column. The displacer musthave a higher affinity to the statio nary phase tha n a ny of the com ponents in thesample, i.e., its adsorption isotherm overlies those of the feed components[l , 671. Provided tha t the column is sufficiently lon g, and iso therm curves are allfavorably shaped , sample componen ts will eventually migrate inside the columnwith the same speeds to form individual product zones. The series of such zonesis usually called a displacement train [49,66, 68,691. Figure 6 shows a displace-ment chromatogram with two sample solutes and one displacer. Parametervalues used in simulation are listed in Table 1. Compared with elutionchrom atography , the displacement development has two distinct advantages: (1)the displacement effect reduces tailing (Fig. 6); (2) sample loading can be higher

Fig. 6. Displacement chrom atogram

= 1.2.s??p 1.0P0 0.8v.8 0.6I-p 0.4E5 0.2

0-

- ith Displacer- ..........., Without Displaeer

3 (Uisplacer)- fI-

--

- 1, . - - - - -\ - - - _ _ 10 2 L 6 8 10Dimensionless Tim e


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Modeling of Nonlinear M ulticomponent Chromatography 65Table 1. Parameter values used for simulationFigures Species Physical Parameters NumericalParameters

In all cases, E, = 0.4 and E, = 0.5. For Figure 4, .rim, = 0.1 . The error tolerance of the O DE solver isto1 = Double precision is used in the Fortran code

[ll]. These features make the displacement development operation a veryattractive alternative to elution in preparative scale chromatography[67].

The main difference between the displacement development and the stepwisedisplacement studied in this chapter is often the operation purpose itself. Theformer desires the products to be separated into a displacement train containingindividual product zones in the effluent stream, while the latter requires theefficient displacement of the adsorbates. In other words, the desorption chroma-tography does not require a well-defined displacement train in the effluent;rather it requires the displacement of adsorbed component(s) with a minimumamount of displacer in a minimum length of time in order to obtain concen-trated product(s). The product(s) in the effluent after displacement may befurther purified if necessary after the stepwise displacement. A typical use of thestepwise displacement, as we have already mentioned, is the in situ separationduring product formation [30,62-641. Another important difference is that thedisplacement development takes a sample which usually occupies only a frac-tion of the column inlet section while the stepwise displacement has no such


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66 T. Gu, G.-J. Tsai and G. T. Tsaolimitation. The strong affinity of the displacer which is required in the displace-ment development should not be mistaken as a requirement for the stepwisedisplacement.

7.1 Results and Discussion [70]The multicomponent Langmuir isotherms (Eq. (25)) with uniform adsorptionsaturation capacities will be used in this study. For simplicity, only the twocompo nent displacement chromatog raphic processes will be discussed, in whichcomponent 1 is the component to be displaced and component 2 the displacer.

7 .2 Efect of Feed Concentration of Displacer (C02 )Figure 7 shows that the higher the displacer concentration in the mobile phase,the higher the roll-up peak on the concentration profile of component 1 (seeTable 1 for parameter values). This is due to the fact that a higher displacerconcentration i n the feed gives a faster migration rate for the concentration fron tof the displacer inside the column, a larger b2Cp2value and, hence, a betterdisplacement efficiency. Figure 8 shows a case in which the displacer does notgive much help in the displacement of compon ent 1 from the column because theconcentration of the displacer is too low. This kind of situation was mentionedby some researchers [5,71]. In Fig. 9, the affinity of the displacer is lower thanthat of the adsorbate. It shows that if the concentration of the displacer issufficiently high a desirable displacement of the adso rbate c an also be achieved.Th e study of the effect of eth ano l conc entra tion on the efficiency of the displace-

- Dimensionless Tim eFig. 7. Effect of displacer concentration on displacement


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Modeling of Nonlinear M ulticomponent Chromatography

2 I 6Dimensionless TimeFig. 8. Same conditions as Fig. 7, except that the concentration of the displacer is lower

Dimensionless TimeFig. 9. Effect of displacer concentration on displacem ent for a case in which b, < b,

ment of phenylalanine from a column packed with P-cyclodextrin-containingresin presented in Fig. 10[72] qualitatively proved this argum ent. Th e affinity ofethanol with fLcyclodextrin is much smaller that phenylalanine, but when theethanol concentration is sufficiently high, it still serves as an efficient displacerwhich gives good displacement results (Fig. 10).

7.3 Efect of Adsorption Equilibrium Constant of Displacer ( b z )The effect of the b 2 value on displacement performance is show n in Fig. 11. Itcan be seen that an increase in b2 delays the appe aranc e of the roll-up peak,


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100% (v ) EtOHiI

Breakthrough curve

50'10 EtOH 1

10'10 EtOH1I

Pure water 1

Fig. 10. Effect of ethanol concentration on displacement of phenylalanine

bz60.0- - - - - -- - -- 6.02.0

2

0 2 L 6Dimensionless T imeFig. 11. Effect of b, on displacement performance


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Modeling of Nonlinear M ulticomponent Chromatography 69gives a sharper displacer front, and reduces the tailing of the displaced compo-nent. The maximum roll-up peak occurs somewhere in the middle range of theb, value. If the primary goal of displacement is to obtain a large fraction of purecomponent 1, a larger b2 is obviously favorable. However, if the mixing ofdisplacer in the product is not a setback, such as in the case when the displacer isa volatile organic solvent and the product is readily recovered by evaporationafter the displacement, a larger b2 is not always favorable. As a matter of fact, ifthe displacement is terminated when the major portion of the product has beenrecovered, then a small b2 may be a better choice because the roll-up peakappears earlier.

Compared with the displacement development, the conclusion for stepwisedisplacement is somewhat different. The displacement development requiresa displacer which has an affinity higher than any other component in the sample.However, this is hardly true for the stepwise displacement as we have alreadydiscussed in the cases of Figs. 9 to 11.

8 SummaryAmong all kinds of models for nonlinear multicomponent chromatography, thegeneral multicomponent rate model is the most comprehensive one. It accountsfor various mass transfer mechanisms and nonlinear isotherms. It is a very usefultool for the study of the dynamics of nonlinear multicomponent chromatogra-phy. This chapter has presented an efficient numerical method for the solutionand the extensions of the model. The model was used for the study of someinteresting effects of isotherm characteristics of the displacer on the optimizationof stepwise displacement. It was concluded that a displacer with a high feedconcentration, and a suitable adsorption equilibrium constant is often a desir-able choice when the purpose of the displacement operation is to displace and toconcentrate the adsorbed species and to minimize the amount of displaceremployed.

9 References

1. Ruthven D (1984) Principles of adsorption and adsorption processes. Wiley, New York2. Glueckauf E (1949) Discuss Faraday Soc 7: 123. HelKerich F, Klein G (1970) Multicomponent chromatography theory of interference. MarcelDekker, New York4. Rhee H-K , Aris R, Amundson NR (1970) Philos Trans R Soc (London) Ser A 267: 4195. Rhee H-K, Amundson NR (1982) AIChE J 28: 4236. Lee CK, Yu Q, Kim SU, Wang N-HL (1989) J Chromatogr 484: 257. Glueckauf E (1947) J Chem Soc 1302


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70 T. Gu, G.-J. Tsai and G. T. Tsao8. Helfferich F, James DB (1970) J Chromatogr 46: 19. Bailly M, To ndeur D (1981) Chem Eng Sci 36: 45510. Frenz J, Horvath C (1985) AIChE J 31: 40011. Frenz J, Horvath C (1988) High Performance Liquid Chromatography 5: 21112. Yu Q, Yang J, Wang N-H L (1987) Reactive Polymers 6: 3313. Wankat PC (1986) Large-scale adsorption and chro matogr aphy, vol 1. CR C Press, Boca Raton,

FL14. Martin AJP, Synge RLM (1941) Biochem J 35: 135915. Yang CM (1980) Ph D Thesis, Purdue University, West Lafayette, IN16. Villermaux J (1981) In: Rodrigues AE, Tondeur D (eds) Percolation processes: Theory andapplications. Sijthoff and Noordhoff, Rockville, MD17. Eble JE, Grob RL, Antle PE, Snyder LR (1987) J Chromatogr 405: 118. Seshadri S, Deming SN (1984) Anal Chem 56: 156719. Solms DJ, Smuts TW, P retorius V (1971) J Chromatogr Sci 9: 60020. Eble JE, Grobe RL, Antle PE, Snyder LR (1987) J Chromatogr 384: 2521. Glueckauf E, Coates JI (1947) J Chem Soc 131522. Rhee H-K, Arnundson NR (1974) Chem Erlg Sci 29: 204923. Bradley WG, Sweed NH (1975) AIChE Symp Ser 71: 5924. Golshan-Shirazi S, Guiochon G (1988) J Chromatogr 461: 125. Golshan-Shirazi S, Guiochon G (1988) J Chromatogr 461: 1926. Farooq S, Ruthven DM (1990)27. Zwiebel I, Gariepy RL, Schnitzer JJ (1972) AIChE J 18: 113928. Santacesaria E, Morbidelli M, Servida A, Storti G, Ca rra S (1982) Ind Eng Chem Process DesDev 21: 44629. Santacesaria E, Morbidelli M, Servida A, Storti G, Car ra S (1982) Ind Eng Chem Process DesDev 21: 45130. Lee S, Tsai G-J, Seo JH, Tsao G T (1988) Third Chemical Congress Of North Am erica and 195thACS National Meeting. Toronto31. Chase HA (1984) Chem Eng Sci 39: 109932. Chase HA (1984) J Chromatogr 279: 17933. Arnold FH, Blanch HW, Wilke CR (1985) J Chromatogr 30: B934. Arnold FH, Blanch HW, Wilke CR (1985) J Chromatogr 30: B2535. Arnold FH, Schofield SA, Blanch HW (1986) J Chromatogr 355: 136. Arnold FH, Schofield SA, Blanch HW (1986) J Chromatogr 355: 1337. Arve BH, Liapis A1 (1987) AIChE J 33: 17938. Arve BH, Liapis A1 (1987) Biotechnol Bioeng 30: 63839. Arve BH, Liapis A1 (1988) Biotechnol Bioeng 31: 24040. Lee W-C (1989) PhD Thesis, Purdue University, West Lafayette, IN41. Ramkrishna D, Amundson NR (1985) Linear opera tor m ethods in chemical engineering withapplications to transport and chemical reaction systems. Prentice-Hall, Englewood Cliffs, NJ42. Gu T, Tsai G-J, Tsao GT (1990) AIChE J 36: 78443. Liapis AI, Rippin DWT (1978) Chem Eng Sci 33: 59344. Yu Q, Wang N-HL (1989) Computers Chem Eng 13: 91545. Mansour A (1989) Sep Sci Techno1 24: 104746. Mansour A, von Rosenberg DU, Sylvester N D (1982) AIChE J 28: 76547. Villadsen J, Michelsen ML (1978) Solutions of differential equation models by polynomialapproximation. Prentice Hall, Englewood Cliffs, NJ48. Finlayson BA (1980) Nonlinear analysis in chemical engineering. McGraw-Hill, New York49. Antia FD, Horvath C (1989) Ber Busenges Phys Chem 93: 96150. IMSL (1987) IMSL User's Manual, Version 1.0. IMSL, Inc. Houston 640-65251. Gear C (1972) Numerical initial-value problems in ordinary differential equations. Prentice-Hall, Englewood Cliffs, NJ52. Gardini L, Servila A, Morbidelli M , Carra S (1985) Computers Chem Eng 25: 49053. Reddy JN (1984) An introduction to the finite element method. McGraw Hill, New York54. Froment GF, Bischoff KB (1979) Chemical reactor analysis and design. Wiley, New York55. Gu T, Tsai G-J, Tsao GT (1991) AlChE J 37: 133356. Provder T (ed) (1980) ACS Sym p Series. No 13857. Provder T (ed) (1984) ACS Sym p Series. No 24558. Provder T (ed) (1984) ACS Sym p Series. No 352


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Modeling of Nonlinear Multicomponent Chromatography 7159. Yau W W, Kirkland JJ, Bly D D (1979) Modern size-exclusion liquid chrom atograp hy, Wiley,New York60. Kim DH, Johnson A F (1984) In: Provder T (ed) ACS Sym posium Series 245: 2561. Koo Y-M, Wa nkat PC (1988) Sep Sci Techno1 23: 41362. Yang X (1988) M.S. Thesis, Purdue University. West Lafayette, IN63. Yang X, Tsai G-J, Tsao G T (1988) Third Chemical Congress Of North America and 195th ACSNational Meeting, Toronto64. Yang X, Tsai G-J, Tsao G T (1989) AIChE Summer National Meeting, Philadelphia65. Tiselius A (1940) Ark K em Mineral G eol 14B: 166. Horvath C, Nahum A, Frenz JH (1981) J Ch roma togr 218: 36567. Horvath C (1985) In: B runer F (ed) The Sciedce of Chrom atograp hy. Elsevier, New York68. Phillips WM , Cramer SM (1988) J Chromatogr 454: 169. K atti AM , Guichon G A (1988) J C hromatogr 449: 2570. G u T, Tsai G-J, Tsao G T (1991) Biotechnol Bioeng 37: 6571. M orbidelli M, Storti G , Carra S, Niederjaufner G, Pontoglio A (1985) Chem Eng Sci 40: 115572. G u T (1990) Ph D Thesis, Purdue University, West L afayette, IN

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Modeling of Nonlinear Multi Component Chromatography

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