Studienarbeit (PDF format)

Prof. Dr. Volker Kasche

Simulation of Liquid Chromatography andSimulated Moving Bed (SMB) Systems

Studienarbeit

by

César LazoMatr. Nr: 12968

Tutor:

Dipl. Ing. Babbette Scheidat

Hamburg 19.07.99

TABLE OF CONTENTS

List of Figures------------------------------------------------------------------------------------------------ iii

List of Symbols and Abbreviations------------------------------------------------------------------------- v

1 Introduction --------------------------------------------------------------------------------------------- 1

2 Literature Review --------------------------------------------------------------------------------------- 2

2.1 Chromatographic Separation Operations ------------------------------------------------------------- 22.1.1 What is Chromatography?------------------------------------------------------------------------------------------- 22.1.2 Fixed-Bed Chromatography----------------------------------------------------------------------------------------- 32.1.3 Moving-Bed Chromatography-------------------------------------------------------------------------------------- 32.1.4 Simulated Moving-Bed Chromatography ------------------------------------------------------------------------ 4

2.2 Competitive Equilibrium Isotherms -------------------------------------------------------------------- 5

2.3 Theories for Nonlinear Multicomponent Liquid Chromatography------------------------------- 62.3.1 Equilibrium theory---------------------------------------------------------------------------------------------------- 62.3.2 Plate models ----------------------------------------------------------------------------------------------------------- 62.3.3 Rate models ------------------------------------------------------------------------------------------------------------ 6

2.4 Modeling Strategies for Simulated Moving Bed Chromatographic Processes ------------------ 9

3 General Multicomponent Rate Model for Column Liquid Chromatograpy-----------------11

3.1 Model Assumptions -------------------------------------------------------------------------------------- 11

3.2 Model Formulation--------------------------------------------------------------------------------------- 133.2.1 Continuity Equation in the Flowing Mobile Phase------------------------------------------------------------ 133.2.2 Continuity Equation inside the Macropores-------------------------------------------------------------------- 143.2.3 Boundary Conditions----------------------------------------------------------------------------------------------- 143.2.4 Dimensionless Equations ------------------------------------------------------------------------------------------ 15

3.3 Model Solution -------------------------------------------------------------------------------------------- 163.3.1 Finite Element Formulation for the Bulk-Fluid Phase Governing Equation------------------------------ 183.3.2 Orthogonal Collocation Formulation of the Particle Phase Governing Equation ------------------------ 193.3.3 Solution to the ODE System -------------------------------------------------------------------------------------- 20

3.4 Model Simulation ----------------------------------------------------------------------------------------- 20

3.5 MATLAB Code for the General Multicomponent Rate Model ---------------------------------- 253.5.1 Application Files Description------------------------------------------------------------------------------------- 253.5.2 The ODE solver ----------------------------------------------------------------------------------------------------- 29

4 General Rate Model Applied to Simulated Moving Bed Chromatography ------------------31

4.1 Modeling the SMB process------------------------------------------------------------------------------ 32

4.2 Model Parameters Estimation-------------------------------------------------------------------------- 33

4.3 Simulation Results---------------------------------------------------------------------------------------- 34

4.4 SMB MATLAB Program ------------------------------------------------------------------------------- 37

5 Conclusions and Further Work---------------------------------------------------------------------40

References-----------------------------------------------------------------------------------------------------43

Appendix ------------------------------------------------------------------------------------------------------47

A. Chroma Program ----------------------------------------------------------------------------------------- 47

B. Additional Files for the SMB Program-------------------------------------------------------------- 55

LIST OF FIGURES

Fig. 2.1 Chromatographic Column------------------------------------------------------------------------------------------- 2Fig. 2.2 Fixed-Bed Mode ------------------------------------------------------------------------------------------------------ 3Fig. 2.3 Moving-Bed Mode---------------------------------------------------------------------------------------------------- 3Fig. 2.4 Moving-Column Mode ----------------------------------------------------------------------------------------------- 4Fig. 2.5 Simulated moving-bed system--------------------------------------------------------------------------------------- 4Fig. 3.1 Anatomy of a chromatographic column --------------------------------------------------------------------------11Fig. 3.2 Solution strategy-----------------------------------------------------------------------------------------------------17Fig. 3.3 Single-component breakthrough concentration profiles inside the column----------------------------------22Fig. 3.4 Ternary frontal adsorption with two roll-up peaks--------------------------------------------------------------22Fig. 3.5 Ternary frontal adsorption, concentrations profiles of component 1 -----------------------------------------23Fig. 3.6 Ternary frontal adsorption, 3D representation of the concentration profiles of Component 2 -----------23Fig. 3.7 Ternary frontal adsorption, pseudocolor graph of concentration profiles component 1-------------------24Fig. 3.8 Calculated ternary adsorption breakthrough curves using one interior collocation point ----------------24Fig. 3.9 Jacobian sparsity pattern of three species multicomponent chromatography-------------------------------26Fig. 3.10 Mass matrix sparsity pattern for three species multicomponent chromatography ------------------------26Fig. 3.11 Graphic representation of the concentration column vector reordering for two species -----------------28Fig. 3.12 3D representation of the data structure for the particle phase-----------------------------------------------29Fig. 3.13 Flow chart of program Chroma.m-------------------------------------------------------------------------------30Fig. 4.1 Scheme of a Simulated Moving Bed Unit-------------------------------------------------------------------------31Fig. 4.2 Separation of EMD 53986 enantiomers. Steady state internal concentration profile. Variable selectivitymodified Langmuir isotherm -------------------------------------------------------------------------------------------------35Fig. 4.3 Separation of 1,1’-bi-2-naphtol enantiomers. Steady state internal concentration profile. Bi-Langmuirisotherms. -----------------------------------------------------------------------------------------------------------------------36Fig. 4.4 Axial profile of SMB unit for cyclic steady state at the end of a switching period. Linear isotherms ----36Fig. 4.5 Flow chart of SMB program ---------------------------------------------------------------------------------------38Fig. 4.6 Jacobian sparsity pattern for a SMB system. Nz = 21, Nr = 2, Ns = 2 --------------------------------------39

LIST OF SYMBOLS AND ABBREVIATIONS

A Matrix for the orthogonal collocation method, first derivative approximationA Column cross-section areaai Constant in Langmuir isotherm for component iAFBi Galerkin element matrix for component iAKBi Galerkin element matrix for component iB Matrix for the orthogonal collocation method, second derivative approximationbi Adsorption equilibrium constant for component iBii Biot number of mass transfer for component i, kiRp/(εpDpi)CAi Holdup capacity for component iC0i Concentration used for nondimensionalization, max{Cfi(t)}Cbi Bulk-fluid phase concentration of component iCfi Feed concentration profile of component i, a time dependent variableCpi Concentration of component i in the stagnant fluid phase inside the particle

macroporesC*

pi Concentration of component i in the solid phase of particle (based on unitvolume of particle skeleton)

C!i Adsorption saturation capacity for component i (based on unit volume of

particle skeleton)cbi = Cbi/C0I

cpi = Cpi/Coi

c*pi = C*

pi/Coi

DBi Galerkin element matrix for component iDbi Axial dispersion coefficient of component iDpi Effective diffusivity of component i, porosity not includedD Inner diameter of a columnGPij Jacobian matrix of the isothermki Film mass transfer coefficient of component iL Column length of section length for a SMBNr Number of interior collocation pointsNz Finite element nodesNs Number of componentsPBi Galerkin element matrix for component iPeLi Peclet number of axial dispersion for component i, νL/Dbi

Q Mobile phase volumetric flow rateqi Solid phase concentration (amount adsorbed) of species iR Radial coordinate for particleRHi Matrix notation for orthogonal collocation in particle phase for component iRp Particle radiusr = R/Rp

t Dimensional time (t = 0 is the moment a sample enters a column)tc Switching timeZ Axial coordinatez Dimensionless axial coordinate, Z/L

Simulation of Liquid Chromatography and Simulated Moving Bed (SMB) Systems

Technische Universität Hamburg-Harburg Arbeitsbereich Biotechnologie II

vi

Greek Letters

εb Bed void volume fractionεp Particle porosityε* Overall void fraction of the bed, defined as ε*=εb+(1-εb)εp

ηI Dimensionless constant, εpDpiL/(Rpν)ξI Dimensionless constant for component i, 3Biiηi(1-εb)/εb

ν Interstitial velocity, 4Q/(πd²εb)τ Dimensionless time, νt/Lτc Dimensionless switching timeτe Time value at which the breakthrough curve has already labeled offφ Lagrangian interpolation function

Subscripts

B bulk-fluid phaseC Operation cycle in an SMBD DesorbentF FeedI i-th componentJ j-th component or j-th section in an SMBL bulk-Fluid phaseR RaffinateP Particle phaseX ExtractI, II, III, IV Different sections in an SMB

Superscripts

* Particle phase concentration! Saturation capacity

11 IInnttrroodduuccttiioonn

Chromatography is a very important separation technique which has been applied tocontinuos operation either as true moving bed (TMB) or simulated moving bed (SMB)systems. The latter approach has excellent perspectives in the field of fine chemicalsseparations. The development of the SMB technology asks for close mathematicaldescriptions of its process dynamics. Under that guideline was elaborated the present work.

A literature review is done in chapter 2 about chromatographic separation operations ingeneral, and the different approaches to model liquid chromatography and SMB systems.Then a general multicomponent rate model based on differential mass balance equations ispresented in chapter 3 to simulate liquid chromatography, its numerical solution is outlined,and results of the simulation program in MATLAB are shown. The structure of theMATLAB simulation program is also discussed. The simulated moving bed is modeled inchapter 4 using the general multicomponent rate model approach. Simulation results areshown and the MATLAB SMB simulation program is described.

All the simulations were done taking as base data and results reported in the availableliterature about the subject. After comparison, it is concluded in chapter 5 that the presentmodel describes very well the dynamic behavior of the systems under study.

A number of simplifications have been assumed (chapter 3) in order to develop the models.However, the simulation programs can handle any type of equilibrium isotherms (as long as itis differentiable), multicomponent operations, different number of spatial discretizationpoints, different numbers of columns (SMB case), and different operating and systemparameters.

In particular the adopted numerical method of solution, (which is a combination of finiteelement method, orthogonal collocation method, and a common ODE solver) proved to bevery effective and robust.

The MATLAB simulation programs have been coded taking advantage of the excellentcapabilities of MATLAB to handle matrices. As a consequence the code is compact, clear,and easy to understand and modify. The codes are listed in the appendix and are explained inchapters 3 and 4.

Fig. 2.1 ChromatographicColumn

22 LLiitteerraattuurree RReevviieeww

2.1 Chromatographic Separation Operations

2.1.1 What is Chromatography?

The definition of Chromatography can be stated as:

Chromatography: "A method, often used in laboratories, which enables the easy and efficientseparation of mixtures of chemical compounds using the phenomenon of adsorption. Thetechnique's strengths are especially manifested in the separation of isomers and naturalmaterials.

In 1910, Tswett first used this method in the separation of leaf pigments, and the techniqueachieved rapid growth later when it was applied to the separation of carotinoid pigments,

among other uses.

The separation method called “column chromatography” uses a glass columnfilled with adsorbent though which passes a composite liquid mixture. Thetechnique operates such that each component will be separated in a differentsection of the column arranged by color according to the adsorption affinity ofeach material.

In performing column chromatography, a vertical glass column (Fig. 2.1) isfilled evenly with the proper amount of adsorbent. Next, a liquid mixture ispoured into the column and the liquid passes through the adsorbent. Eachcompound is absorbed into beds at different heights depending on thecomponent's individual adsorption affinity. At this stage, the sections into whicheach of the components has been absorbed are still not completely separated.However, if the appropriate desorbent is poured into the column, thecomponents which have been adsorbed on the adsorbent dissolve into thedesorbent and start moving downwards in the column. Each component moves

towards the lower part of the column, but the migration ratesagain differ according to each component's adsorptionaffinity. The components in the lower layers move faster, andin the end, each component will be clearly separated. This

stage of the column chromatography process is called “development”.

If the material being separated is a mixture of pigments, there will be colored zones atdifferent heights in the column filled with adsorbent. These colored zones are called achromatogram.

When the development stage is over and the pigments appear, the adsorbent is pushed out ofthe column, divided into each zone and the absorbates are extracted separately using adesorbent. This stage of the process is called “elution”. Alternately, without removing theadsorbent from the column, a desorbent may be successively poured in from above, and each

Chapter 2 Literature Review


3

Fig. 2.2 Fixed-Bed Mode

Fig. 2.3 Moving-Bed Mode

zone preferentially eluted by the desorbent will dissolve into it and trickle down the columnone at a time. The liquids can then be collected from the bottom of the column as they dripout.

2.1.2 Fixed-Bed Chromatography

This chromatographical technique is essentially thesame as the method used in laboratory analysis. Byhaving the targeted composite substance absorbed intothe adsorbent, the chromatographic process divides themixed substance into its parts. Now let us suppose thatthe adsorption affinity of substances A, B, C are as

follows, A<B<C. A column as shown in Fig. 2.2 ispacked with adsorbent, and then filled with liquiddesorbent. In this state, a fixed amount of the sampleliquid (a mixture of A, C) is fed in at one end of the column. Next, if we continually feed inliquid desorbent, each component of the sample (A and C) will move in the adsorbent layersat the migration rates Ua and Uc, each determined according to their individual adsorptionaffinity. Thus, if there is difference between the migration Ua and Uc, as component A andcomponent C move through the adsorbent layers they will separate from each other. In thisway, by collecting the liquids as they successively drip out of the column, the mixture can beseparated into fractions each abundant with one of the component materials.

Since the types and densities of the desorbents can easily be changed, fixed-bedchromatography can be applied to the separation of many types of useful substances.However, in cases where a more precise separation is required, the fact that the fixed-bedchromatographic process results in a material with a low concentration means that highercosts are entailed in once again raising the material's concentration. Thus, although it isdifficult to use this method in cases where the final product is not very expensive, the processis still one, which can easily be used to separate many components. However, there are someadditional deficiencies which surface when this method is used on a large scale:

1. The entire adsorbent bed is not efficiently utilized.2. A large quantity of desorbent is consumed, and the separated components are obtained in

a diluted state.3. In order to obtain a successful separation, a sufficiently large difference in the adsorption

affinities of the adsorbates is required4. The operation is not continuous. Due to these deficiencies, there have been many

innovative attempts to improve the fixed-bed mode so that it can be used as an industrialdevice.

2.1.3 Moving-Bed Chromatography

This section explains the moving-bed method, which isone of the ways by which the fixed-bed system can bemade continuous.

Let us say that in Fig. 2.3, the adsorbent is caused to movein the opposite direction of the desorbent (and component

A, C) at a migration rate between the velocities of



4

Fig. 2.4 Moving-ColumnMode

Fig. 2.5 Simulated moving-bedsystem

components A and C. This would be the situation in Fig. 2.3, where while the desorbent iscontinuously fed into one end of the column, the adsorbent is made to move in the directionopposite to that of the desorbent, while the feed mixture is also supplied at the middle of theadsorbent bed. At this point the adsorbent is caused to move at a migration rate, Us, which isbetween Ua and Uc, the velocities of components A and C in the fixed-bed mode, i.e.(Ua>Us>Uc).In such a situation, components A and C will move within the column at thespeeds, Ua-Us (>0) and Uc-Us (<0) respectively. Thus, starting from the point where the feedmixture is fed, components A and C will move in opposite directions from one another withinthe column. If the mixture is continuously fed into such a device, components A and C willcontinuously be separated to both ends of the column.

However, in a large industrial device, it is extremely difficult to uniformly move theadsorbent without disturbing the absorption band. Fromthis conception came the idea that rather than have theadsorbent continuously move, instead a column of fixedlength (packing bed) could move. This process is shown inFig. 2.4

With a device such as the one described in the precedingparagraph, a mixture can be separated in a continuousmanner using the principle of the moving-bed mode. Whileit is not impossible to move the column in smaller devices,in an industrial device, it becomes mechanically complex.The simulated moving-bed mode described in the

following section is a chromatographic separation methodinvented so as to gain the same level of separation as themoving-bed mode, without having to actually move theadsorbent.

2.1.4 Simulated Moving-Bed Chromatography

In Fig. 2.4, if we change our observation point of theadsorption column movement in the moving-bed mode,we will see that it is equal to continuously moving thepoints of the adsorbent, feed mixture and componentwithdrawal in the direction of the desorbent.

As shown in Fig. 2.5, approximately the same quality ofseparation as in the moving-bed mode can be obtained byseparating the adsorbent layers into several differentcolumns while moving the introduction point of the feedmixture successively along in the direction of the flow ofthe desorbent. This is the principle behind the simulated

moving-bed method.

Next, let us consider the actual simulated moving-bedchromatographic separation device. As shown in Fig.2 5,

the mouths of each column are connected to form a circular loop, with four openings for thefeeding and drawing of fluids set in each column.



5

While the fluids are circulating inside the column, assume that feed mixture F, desorbent D,component A, and component C are continuously allowed to enter or leave the column fromeach of the openings. Then, successively for the length of one full column at a time, thepositions of the openings for F, D, A, C are changed in the direction of the circular flow at aregularly fixed point in time. Here the migration rate at each opening (Uv = columnlength/switching time) is set so that it will be smaller than component A's migration rate, Ua,and larger than component C's migration rate, Uc, i.e. (Ua>Uv>Uc).

By operating the device under these conditions, it seems as if the adsorbent moved at amigration rate of Us ( = -Uv) in the opposite direction to the flow of the fluids. Therefore,components A and C move in the opposite direction from the feed mixture introduction point,F, and each component can be removed in a continuous manner from its respective withdrawal point A and C.

This is how the simulated moving-bed chromatographic separation system works. However,the weak point of the simulated moving-bed method is that it can only perform a separationresulting in two fractions. As can be seen in Fig. 2.5, with the simulated moving-bed device,there is always either components A or component C present in the area between the feedmixture introduction opening, F, and each opening for the withdrawal of components A and Crespectively. Therefore, even if there existed in the feed mixture a component B, having amigration rate in between that of components A and C, and it was collected in the vicinity ofthe feed mixture introduction point F, it would be impossible to extract this component B in ahighly purified state, since it would only exist in a mixed form with either component A andC.

Consequently, by a simulated moving-bed system, unless the position of elution of thecomponent targeted for extraction was either the fastest or the slowest among the separatedcomponents it was impossible to extract the component by a single separation operation. Incases where a targeted component is eluted in a narrow position in between two impurities orother components, one either had to carry out the separation operation twice, or link twodevices in series. Owing to this fundamental principle, the use of the simulated moving-bedsystem has been limited to 2 fractions. However, naturally there are many circumstanceswhen the object of separation exists in a multiple component form. A system which couldmaintain the high efficiency of the simulated moving-bed mode and perform separations intothree or more product divisions in a single step had long been awaited.

2.2 Competitive Equilibrium Isotherms

Since chromatography is a separation method [16], we consider mostly mixtures of dilutedspecies. In many cases, the mobile phase itself is not a pure solvent but a mixture and it maycontain additives which themselves are more or less strongly adsorbed. The concentration ofone of these components at equilibrium no longer depends only on its own concentration inthe mobile phase. It also depends on the concentration of all the other components involved.At high concentrations, the molecules of the various components of the mobile phase and ofthe feed compete for their involvement in the retention mechanism, which has a limitedcapacity. In liquid-solid chromatography, the competition comes from the finite capacity ofthe adsorbent surface.



6

Various modeling approaches can be considered for the correlation of equilibrium data. Thesimplest case is that of constant selectivities, where purely correlative, thermodynamicallyinconsistent models, such as the well-known multicomponent Langmuir equation withdifferent saturation loading for each component, can be used; even though it has beenchallenged because it does not agree with the Gibbs’adsorption isotherm unless all saturationcapacities are identical.

2.3 Theories for Nonlinear Multicomponent LiquidChromatography

Many researchers have contributed to the modeling of liquid chromatography. There exist adozen or more theories of different complexities. A comprehensive review on the dynamicsand mathematical modeling of isothermal adsorption and chromatography has been given byRuthven [43] who classified models into three general categories:

• Equilibrium theory• Plate models• Rate models

2.3.1 Equilibrium theory

Equilibrium theory [13] assumes a direct local equilibrium between the mobile phase and thestationary phase, neglecting axial dispersion and mass transfer resistance. It effectivelypredicts experimental retention times for chromatographic columns with fast mass transferrates. It provides general locations, or retention times of elution peaks, but it fails to describepeak shapes accurately if mass transfer effects are significant.

2.3.2 Plate models

Generally speaking [13], there are two kinds of plate models. One is directly analogous to thetanks in series model for nonideal flow systems. In such a model, the column is divided into aseries of small artificial cells, each with complete mixing. This gives a set of first order ODEsthat describe the adsorption and interfacial mass transfer. Many researchers have contributedto this kind of plate model. However, plate model of this kind are generally not suitable formulticomponent chromatography since the equilibrium stages may not be assumed equal fordifferent components.

The other kind of plate model is formulated based on the distribution factors that determinethe equilibrium of each component in each artificial stage. The model solution involvesrecursive iteration rather that solving ODE systems. The most popular are the Craigdistribution models.

2.3.3 Rate models

Rate models [13] refer to models containing a rate expression, or rate equation, whichdescribes the interfacial mass transfer between the mobile phase and the stationary phase. Arate model usually consists of two sets of differential mass balance equations, one for the



7

bulk-fluid phase, the other for the particle phase. Different rate models have varyingcomplexities.

2.3.3.1 Rate Expressions

The solid film resistance assumes a linear driving force between the equilibriumconcentrations in the stationary phase (determined from the isotherm) and the averagefictitious concentrations in the stationary phase. This simple rate expression has been used bymany researchers because of its simplicity, but this model cannot provide details of the masstransfer processes.

The fluid film mass transfer mechanism with a linear driving force is also widely used [13].The driving force is the concentration difference of a component between that on the surfaceof a particle and that in the surrounding bulk-fluid. It is assumed that there is a stagnant fluidfilm between the particle surface and the bulk-fluid. The fluid film exerts a mass transferresistance between the bulk-fluid phase and the particle phase, often called the external masstransfer resistance. If the concentration gradient inside the particle phase is ignored, thechromatography model then becomes a lumped particle model, which has been used by someresearchers. If the mass transfer Biot number, which reflects the ratio of the characteristic rateof film mass transfer to that of intraparticle diffusion, is much larger that one, the externalfilm mass transfer resistance can be neglected with respect to intraparticle diffusion.

In many cases, both the external mass transfer and the intraparticle diffusion must beconsidered. A local equilibrium is often assumed between the concentration in the stagnantfluid phase inside macropores and the solid phase of the particle. Such a rate mechanism isadequate to describe the adsorption and mass transfer between the bulk-fluid and particlephases, and inside the particle phase in most chromatographic processes. The localequilibrium assumption here is different from that made for the equilibrium of concentrationsin the solid and the liquid phases without any mass transfer resistance.

If the adsorption and desorption rates are not sufficiently high, the local equilibriumassumption is no longer valid. A kinetic model must be used. Second order kinetics has beenwidely used in kinetic models for process where the local equilibrium assumption isunrealistic, e.g. affinity chromatography.

2.3.3.2 Governing Equation for the Bulk-Fluid Phase

The governing partial differential equation [13] for the bulk-fluid phase can be easily obtainedfrom a differential mass balance of the bulk-fluid phase for each component. Axial dispersion,convection, transient, and interfacial flux terms are usually included. Such equationsthemselves are generally linear if physical parameters are not concentration dependent. Theybecome nonlinear when coupled with a rate expression involving nonlinear isotherms orsecond order kinetics.

For some rate models, such as models for isothermal, single component systems with linearisotherms, analytical solutions may be obtained using the Laplace transform. For morecomplex systems, especially those involving nonlinear isotherms, analytical solutionsgenerally cannot be derived. Numerical methods must be used to obtain solutions to complexrate models that consider various forms of mass transfer mechanism.



8

2.3.3.3 General Multicomponent Rate Models

A rate model that considers [13]:

• axial dispersion• external mass transfer• intraparticle diffusion• nonlinear isotherms

is called a general multicomponent rate model. Such a model is adequate in most cases todescribe the adsorption and mass transfer processes in multicomponent chromatography. Insome cases, surface adsorption, size exclusion and adsorption kinetics may have to beincluded to give an adequate description of a particular system.

2.3.3.4 Numerical Solutions

A general multicomponent rate model consists of a coupled PDE system with two sets ofmass balance equations, one for the bulk-fluid and one for the particle phases for eachcomponent, respectively. The finite difference method is a simple numerical procedure thatcan be directly applied to solution of the entire model. This often requires a huge amount ofcomputer memory during computation, and its efficiency and accuracy are not competitivecompared with other more advanced numerical methods, such as the orthogonal collocation(OC), finite element, or the orthogonal collocation on finite element (OCFE) methods.

For the particle phase governing equation, the OC method is the obvious choice. It is a veryaccurate, efficient and simple method for discretization. It has been widely used with successfor many particle problems [8].

Unfortunately, concentration gradients in the bulk-fluid phase can be very steep, thus the OCmethod is no longer a desirable choice, since global splines using high order polynomial areexpensive [8] and sometimes unstable. The method of OCFE uses linear finite elements forglobal splines and collocation points inside each element. No numerical integration forelement matrices is needed because of the use of linear elements. This discretization methodcan be used for system with stiff gradients [8].

The finite element method with higher order of interpolation functions (typically quadratic, oroccasionally cubic) is a very powerful method for stiff systems. Its highly streamlinedstructure provides unsurpassed convenience and versatility. This method is especially usefulfor systems with variable physical parameters, as in radial flow chromatography andnonisothermal adsorption with or without chemical reactions.

2.3.3.5 Solution to the ODE system

If the finite element method is used for the discretization of the bulk-fluid phase PDE and theOC method for the particle phase equations, and ODE system is produced [8]. The ODEsystem with initial values can be readily solved using and ODE solver such as the subroutine“ODE15S” of MATLAB which is a variable-order solver based on the numericaldifferentiation formulas (NDFs). Optionally it uses the backward differentiation formulas,BDFs, (also known as Gear’s method) that are usually less efficient [48].



9

2.4 Modeling Strategies for Simulated Moving BedChromatographic Processes

Several authors have developed models to predict the performance of an SMB separationprocess with reasonable agreement with experimental results [44]. There are two mainstrategies of modeling an SMB process [40] [4]:

1. The first method, known as the moving-bed approach, treats the simulated countercurrentprocess as equivalent to a true countercurrent system (Sec. 2.1.3)

2. The second technique, known as the fixed bed approach, considers the simulatedcountercurrent process as a series of fixed beds and incorporates the actual flow switchingof the process at fixed time intervals

The approaches can also be classified whether the bed elements are represented by acontinuous-flow model (plug flow or axial dispersed plug flow) or as a cascade or mixingcells.

Table 2.1 Classification of models for SMB counter-current processes

Representation ofthe system

Continuous Moving-BedModel

Intermittent Moving-Bed Model

Representation ofthe bed elements

Continuous Mixing cell Continuous Mixing cell

Each of these four models may be treated either according to equilibrium theory or byincluding an appropriate rate expression to account for mass transfer resistance, generally interms of a linear driving force model. Representative examples of the application of thesemodels are summarized in [44].

It has been shown that the TMB and SMB processes are equivalent [39]. However, an SMBdoes not exactly work in the steady state, predicted by a TMB, but in a periodic steady state:during a given period, the internal concentration profiles vary, but the internal profilesexamined at the same time for two successive periods are identical (except for a one-columntranslation). An SMB is actually operated under cyclic steady state. The real operation in theperiod between two shiftings of the input and output lines is a transient operation since thespecies concentrations in the output streams are not constants but functions of time.Therefore, for better understanding of the dynamics of the simulated moving bed, a transientmathematical model is essential [26]

The TMB approach is justified in the case of units with a large number of fixed beds and lowswitching interval times. If the bed were divided into very small elements with acorrespondingly short switch time, we would have a perfect analogue of countercurrent flow.In practice we have to deal with finite bed sized in order to retain mechanical simplicity of theprocess.

The motivation for modeling and equivalent unit (the countercurrent unit) instead of the realunit (the simulated moving-bed unit) is related to the different level of difficulty involved inthe solution of the tow models. In particular, the first refers to a steady-state operation, thesecond to a cyclic steady-state operation where simulation requires the solution of a transientmodel. Accordingly, models of the second type are usually characterized by one moreindependent variable, time.



10

It has been concluded from simulations results [26] that the continuous moving-bed model isacceptable for general separation performance studies of a simulated moving bed, but theintermittent moving-bed model is necessary for better understanding of the process dynamics.Also, with regards to case studies to optimize the process, it has been recommended to use therigorous dynamic SMB process model [47]. By taking into account the quasi-stationary,periodic process behavior, the potential for optimization is enhanced. The rigorous modelingapproach gives essential information for a better understanding of the chromatographicprocesses. Therefore it is a crucial aid to plan experiments, to develop control strategies and toanalyze the stability of SMB processes concerning to malfunctions.

As a consequence of this conclusions and recommendations, the rigorous dynamic SMBprocess model is the first choice for simulation.

33 GGeenneerraall MMuullttiiccoommppoonneenntt RRaattee MMooddeell ffoorr CCoolluummnnLLiiqquuiidd CChhrroommaattooggrraappyy

3.1 Model Assumptions

For the modeling of multicomponent liquid chromatography [13] [16], the column is dividedinto the bulk-fluid phase and the particle phase. The following equations are based on thehypothesis of an intraparticular mass transfer controlled by diffusion into macropores. Thisapproach considers three phases:

• the mobile phase flowing in the space between particles• the stagnant film of mobile phase immobilized in the macropores• the stationary phase where adsorption occurs

This approach models the interactions a the local level with a single porosity model. It is alsopossible to have systems with macro/micro porosity [10].

The anatomy of a fixed-bed axial flow chromatography [12] [52] column is given in Fig. 3.1.

Fig. 3.1 Anatomy of a chromatographic column

The following basic assumptions are made in order to formulate a general rate model.

1. The compressibility of the mobile phase is negligible.2. The partial molar volumes of the samples components are the same in both phases.

Bulk-Fluid Phase

L

Column

v DetectorCfi(t)

Cbi, Dbi, eeeeb

Particle Phase

Rp

ki

Cpi, Dpi, eeeep

R

Cpi*

Mobile Phase

Stagnant FilmStationary Phase

Film Mass Transfer Resistance

Z



12

3. The mobile phase (if pure) or the weak solvent (if the mobile phase is a mixture) is notadsorbed.

4. The chromatographic process is isothermal. There is no temperature change during a run.5. The porous particles in the column are spherical and uniform in diameter.6. The concentration gradients in the radial direction are negligible.7. The fluid inside particle macropores is stagnant, i.e., there is no convective flow inside

macropores8. An instantaneous local equilibrium exists between the macropore surface and the stagnant

fluid inside macropores of the particles.9. The film mass transfer mechanism can be used to describe the interfacial mass transfer

between the bulk-fluid and particle phases.10. The diffusional and mass transfer parameters are constant and independent of the mixing

effects of the components involved.11. All the mechanisms which contribute to axial mixing are lumped together into a single

axial dispersion coefficient (dispersion model).12. The “dead” volumes at both ends of the packed adsorbent are negligible

As a consequence of the first assumption, the mobile phase velocity can be consideredconstant along the column. Furthermore, as a consequence of the first two assumptions, themass balance equation of the mobile phase can be dropped. . Even in the case of a purecomponent band eluted by a pure mobile phase, two mass balances are in principle needed tocalculate the band profile. This is the situation in gas and in supercritical fluidchromatography where, because of the mobile phase compressibility and the differencebetween the partial molar volumes of the components in the two phases, the mobile phasevelocity and the carrier gas partial pressure vary significantly along the column, and acrossthe band profile itself.

Assumption two is legitimate in liquid chromatography, where the differences between thesevolumes do not exceed a few percent. Thus, we can consider that the mass transfer takingplace during the adsorption process are made at constant volume. This is a main differencefrom gas chromatography, where the partial molar volumes of the components in the mobilephase are at least two orders of magnitude larger than in the stationary phase. The sorptioneffect which is very important in gas chromatography, is negligible in liquid chromatography.

The isothermal condition is usually assumed for simple adsorption-desorption systems. If theheat of adsorption is high enough, the system can deviate significantly from isothermalbehavior. In such a case, it is necessary to complete the system of partial differential equationsof chromatography by two differential heat balance equations, for the mobile and stationaryphases. This could be also the case with reacting systems.

The influence of non-uniform particles has been studied [1]. Nevertheless, in order to obtainsimpler equations, an average radius is usually considered.

Concentration gradients in the radial direction are usually negligible, this is not the case inradial flow chromatography [50].

If the adsorption and desorption rates are not sufficiently high, the local equilibrium is nolonger valid. A kinetic model must be used. Second order kinetics has been widely used inkinetic models for affinity chromatography [13]. Also the kinetics of macromolecules showthat the well-known Langmuir kinetics equation, which does not take into account sterichindrance at high surface coverage, can not explain breakthrough curves of proteins at high

Chapter 3 General Multicomponent Rate Model for Column Liquid Chromatograpy


13

loading. Computer simulations that consider steric hindrance must be used to developmodified kinetic equations to fully explain the observed adsorption kinetics that are muchslower than predicted by the Langmuir kinetics [18].

The concentration independence of the mass transfer and diffusion parameters is true forchemicals of low or moderate molecular weight in the concentration range used in liquidchromatography, rarely exceeding 5% by weight. This may not be true when higher feedconcentrations are used, when highly viscous feeds are processed, or when proteins areseparated. The solutions of these compounds in the mobile phase are much more viscous thanthe pure mobile phase, hence the transfer parameters of this solutes are strongly dependent ontheir concentration.

The dispersion coefficient is not the molecular diffusion coefficient, but an axial dispersioncoefficient which includes the influence of the packing tortuosity (which slows axialmolecular diffusion by increasing the path length), as well as its anastomosis, which causesturbulent mixing by forcing the splitting and recombination of local streams flowing aroundthe packing particles. This latter effect is often referred to as eddy diffusion.

3.2 Model Formulation

Based on the assumptions above, the governing equations can be obtained from differentialmass balances of the bulk-fluid phase en the particle phase, respectively, for component i [1].The following equations can also be derived from equations of continuity provided by Bird etal. [2]:

3.2.1 Continuity Equation in the Flowing Mobile Phase

Eq. 3.1

Using Cpi, the concentration in the stagnant mobile phase, and writing the expression of theinterfacial flux leads to:

=−=

∂

∂

pRRpi,CbiCpRi3k

tiq

Eq. 3.2

Accumulation in thestationary phase

accumulation inthe mobile phase

transport by axial dispersionin the mobile phase

convective transport inthe mobile phase

( )0

1

Zbi

C!

2Z

biC2

biD =

∂

∂−+

∂

∂+

∂

∂+

∂

∂−

ti

q

b

btbi

C

ε

ε



14

Substitution of Eq. 3.1 in Eq. 3.2 give us:

( )0

pRRpi,

Cbi

C

pR

b"

b-1

i3k

Zbi

C!

2Z

biC2

biD =

=−+

∂

∂+

∂

∂+

∂

∂−

ε

tbi

C Eq. 3.3

3.2.2 Continuity Equation inside the Macropores

The particle phase continuity equation in spherical coordinates is:

Eq. 3.4

3.2.3 Boundary Conditions

Space and time boundary conditions [1] related to each chromatographic mode, i.e. elution,frontal and displacement mode, are in fact the only differences between the mathematicaldescription of these three modes. Whatever the mode, the column is initially equilibrated withthe carrier, either pure or nonpure. That initial state may be represented by the followingequations:

t=0, ( )Z0,bi

Cbi

C = Eq. 3.5

( )ZR,0,pi

Cpi

C = Eq. 3.6

Z=0, ( )( )tCCD

!Z

Cfi bi

bi

bi −=∂

∂Eq. 3.7

Z=L, 0Z

Cbi =∂

∂Eq. 3.8

R=0, 0R

Cpi =∂

∂Eq. 3.9

accumulation in themicroporus stationary phase

accumulation inthe macropores

Radial diffusion insidethe porous particle

( ) 022

1*piC

p"1 =∂

∂

∂∂

+∂

∂+

∂−

∂ R

piC

RRRpi

Dpt

piC

ptεε



15

R=Rp ( )pRRpi,bi

pip

ipiCC

D

k

R

C=−

ε=

∂

∂Eq. 3.10

Eq. 3.7 and 3.8 for the two column ends are the so called Danckwerts boundary conditions,which are mass balanced [13].

3.2.4 Dimensionless Equations

Defining the following dimensionless constants [13],

cbi=Cbi/Coi, cpi=Cpi/Coi, c*pi=C*

pi/Coi, τ=νt/L, r=R/Rp

z=Z/L, PeLi=νL/Dbi, Bii=kiRp/(εpDpi), ηi=εpDpiL/(Rp2ν)

ξi=3Biηi(1-εb)/εb

the model equations can be transformed into the following dimensionless equations:

( ) 0ccc

z

c

z

c

Pe

1- 1r,pibii

bibibi2

Li=−ξ+

τ∂∂

+∂

∂+

∂∂

= Eq. 3.11

( )[ ] 0r

cr

rr

1cc1

pi22ipip

*pip =

∂

∂

∂∂η−ε+ε−

τ∂∂ Eq. 3.12

In this equations, the Peclet number (PeLi) reflects the ratio of the convection rate to thedispersion rate while the Biot number (Bii) reflects the ratio of the external film mass transferrate to the intraparticle diffusion rate.

Initial conditions:

τ=0, ( )z,0cc bibi = Eq. 3.13

( )z,r,0cc pipi = Eq. 3.14

Boundary conditions:

z=0( )

τ−=

∂∂

oi

fibiLi

bi

C

CcPe

z

cEq. 3.15

where Coi is the concentration used to nondimensionalize other concentrations for componenti. It should be the highest concentration of component i ever fed to the column, i.e.,Coi=max{Cfi(t)} with -! < t < +!.

For frontal adsorption, Cfi(τ)/Coi=1

For elution, Cfi(τ)/Coi=1 0≤τ≤τ imp

Cfi(τ)/Coi=0 else

After the sample introduction (in the form of frontal adsorption):



16

If component i is displaced, Cfi(τ)/Coi=0

If component i is a displacer, Cfi(τ)/Coi=1.

At z=1, 0z

cbi =∂

∂Eq. 3.16

At r=0, 0r

cpi =∂

∂Eq. 3.17

At r=1, ( )1r,pibiipi

ccBir

c=−=

∂

∂Eq. 3.18

The concentration c*pi in Eq. 3.12 is the dimensionless concentration of component i in the

solid phase of the particles. It is directly linked to a multicomponent isotherm such as thefollowing commonly used multicomponent Langmuir isotherm:

∑=

+

=Ns

1jpjj

pii*pi

Cb1

CaC Eq. 3.19

In dimensionless form:

( )∑=

+

=Ns

1jpjj0j

pii*pi

cCb1

cac Eq. 3.20

3.3 Model Solution

If the general rate model is solved [13], it provides the effluent history (chromatogram),cbi z=1 vs. τ. In fact the model even provides transient concentration profiles anywhere insidethe column, either in the bulk-fluid [cbi(r, z)], or the stagnant fluid inside particle macropores[cpi(τ, r, z)], or in the solid skeleton of the particles [c*

pi(τ, r, z)]. Usually only the effluenthistory is used to study chromatographic phenomena.

The model itself is not necessarily nonlinear, but it becomes nonlinear whenever a nonlinearisotherm, such as the Langmuir isotherm, is used. A true multicomponent case is almostcertainly nonlinear, since no linear isotherm can be used to describe true multicomponentadsorption. For such a nonlinear multicomponent model, there is no analytical solution. Themodel equations must be solved numerically. Fig. 3.2 shows the strategy of the numericalmethod used to solve the PDE system in the model.



17

Fig. 3.2 Solution strategy

The numerical method of lines [45] is used in order to obtain, through space discretization ofthe PDE system, an ODE system which can be solved which a common ODE solver. Thebulk-fluid phase and the particle equations are first discretized using the FE (finite element)and the OC (orthogonal collocation) methods, respectively. The resulting ODE system issolved using an existing ODE solver provided by MATLAB® [48].

Coupled PDE SYSTEM

Particle Phase PDE'SNs

Bulk-Fluid Phase PDE'SNs

Orthogonal CollocationNsNzNr

Finite ElementNsNz

Coupled ODE SystemNsNz(Nr+1)

Numerical SolutionCp i (t,z,r), C bi (t,z)

Concentration ProfilesInside Column

C bi (t,z) vs. z

Effluent HistoryCbi( t,z) vs. t

Discretization

ODE Solver



18

3.3.1 Finite Element Formulation for the Bulk-Fluid Phase GoverningEquation

Applying the Galerkin weighted residual method [46] to Eq. 3.11 one obtains [13]

( )∫ =

−ξ+

τ∂∂

+∂

∂+

∂∂

φ =

B

A

z

z

1r,pibiibibibi

2

Lim 0dzcc

c

z

c

z

c

Pe

1- Eq. 3.21

where the integration limit {zA, zB} contains the two boundary points of a typical finiteelement. Rearrangement, using integration by parts on the second order partial differentialterm [46] in Eq. 3.21 gives,

∫ ∫ ∫∫ =φξ−

φξ+∂

∂φ+

τ∂∂

φ+∂

∂φ

−+

∂φ∂

∂∂

=

B

A

B

A

B

A

B

A

B

A

z

z

z

z

z

z

1r,pimibimibi

mbi

mzz

z

z

bim

Li

mbi

Li0dzcdzc

z

cdz

c

z

c

Pe

1dz

zz

c

Pe

1

Eq. 3.22

Inserting the interpolation form for quadratic elements ∑=

φ=3

1nn,binbi cc and ∑

==φ=

3

1nn,1r,pinpi cc

into Eq. 3.22 yields

∑ ∫ ∑ ∫ ∑ ∫= = =

=φφ+

φφξ+

∂φ∂

φ+∂φ∂

∂φ∂3

1n

z

z

3

1n

3

1n

z

z

nm'

n,binmin

mn,binm

Lin,bi

B

A

B

A

dzcdzz

cdzzzPe

1c

∑ ∫=

= φφξ+∂

∂φ

3

1n

z

z

nmin,1r,pizz

bim

Lidzc

z

c

Pe

1B

A

B

AEq. 3.23

Eq. 3.23 can be expressed in the matrix form as follows:

[DBi][c’bi] = [AKBi][cbi] + [PBi] + [AFBi][cpi,r=1] Eq. 3.24

where the bold face indicates a matrix or a vector for each quadratic element, and

[DBi]m,n = ∫ φφB

A

z

z

nm dz Eq. 3.25

[AKBi]m,n = ∫

φφξ+

∂φ∂

φ+∂φ∂

∂φ∂

−B

A

z

z

nmin

mnm

Lidz

zzzPe

1 Eq. 3.26

[PBi]m,n = B

A

zz

bim

Li z

c

Pe

1 ∂

∂φ Eq. 3.27

[AFBi]m,n = ∫ φφξB

A

z

z

nmi dz Eq. 3.28



19

in which m,n ∈ {1,2,3}. The finite element matrices and vectors are evaluated over eachindividual element before a global assembly. After the global assembly, the natural boundarycondition,

(PBi) z=0 = ( )i0

fibi C

Cc

τ+− Eq. 3.29

will be applied to [AKBi] and [AFBi] at z=0. (PBi)=0 anywhere else. The values of the finiteelement matrices for this system, can be derived from the Galerkin element matrices forquadratic shape functions, which can be found in [8].

The concentration derivatives at each element node [c’bi] are determined from Eq. 3.24.

3.3.2 Orthogonal Collocation Formulation of the Particle PhaseGoverning Equation

Using the same symmetric polynomials (Jacobi polynomials for problems with symmetry) asdefined by Finlayson [8], Eq.3.12 is transformed into the following equation by the OCmethod [13].

( )∑∑+

==η=

τ∂∂ 1Nr

1kkpik,li

l

Ns

1j

pj

pj

i ,cd

dc

c

gB l=1,2,...,Nr Eq. 3.30

in which

( ) pip*pipi cc1g ε+ε−= Eq. 3.31

Note that for component i, c*pi is related to cpj values for all the components involved via

multicomponent isotherms such as Eq. 3.20. The value of (cpi)N+1, i.e., cpi,r=1, can be obtainedfrom the boundary condition at r = 1, which gives

( )∑+

==+ −=

1Nr

1j1r.pibiijpij.1Nr ccBi)c(A Eq. 3.32

or

( )

i1Nr,1Nr

Nr

1jjpij,1Nrbii

1r,pi Bi

ccBi

c+

−

=++

=+

=

∑A

A

Eq. 3.33

In Eqs. 3.30, 3.31 and 3.32 the Matrices A and B are the same as defined by Finlayson [8],who also provided approximated values in his book. Exact values of this matrices can befound in [51].

The concentration derivatives at each OC point [c’pi] are coupled because of the complexity of

the isotherms that are related to gi via c*pi (Eq. 3.31) in multicomponent cases. At each interior

OC point, Eq. 3.30 can be rewritten in the following matrix form:



20

[GP][c’p]=[RH] Eq. 3.34

where GPij = "gi/"cpj, c’pj = dcpj/d#, and [RHi] = right hand side of Eq. 3.30. Since the matrix

[GP] and the vector [RH] are known with given trial concentrations values at each interiorOC point, the vector [c’p] can be easily calculated from Eq. 3.34. Using this approach, we candeal with complex nonlinear isotherms without iteration [14].

3.3.3 Solution to the ODE System

If Nz quadratic nodes are used for the z-axis in the bulk fluid phase equation and Nr interiorOC points are used for the r-axis in the particle phase equation, the above discretizationprocedure gives a total of Ns(Nz)(Nr+1) ODEs that are then solved simultaneously by any ofthe stiff ordinary differential equations solver provided by MATLAB [48].

3.4 Model Simulation

The general rate model presented in Chap. 3 with the multicomponent Langmuir isotherms isused in this section to do some simulations. The parameters used in the simulation are listedin Table 3.1. The results are presented in Fig. 3.3 to Fig. 3.8. and can be compared in [13].

Table 3.1 Parameters and CPU times for simulation in Chap. 3.

Physical Parameters Numerical ParametersFigure Species εεεεb εεεεp PeLi ηηηηi Bii ai bixC0i Nz Nr ODE’s

Time (s)1

Fig. 3.3 1 0.4 0.4 50 2 10 8 7x0.2 41 2 123 44.43Fig. 3.4Fig. 3.5Fig. 3.6Fig. 3.7

123

0.4 0.5 300 1 201

1020

2x0.120x0.140x0.1

17 2 153 63.27

Fig. 3.8123

0.4 0.5 300 1 201

1020

2x0.120x0.140x0.1

17 1 102 29.66

1 The CPU times are based in a Pentium II Intel microprocessor running at 350 Megahertz

Fig. 3.3 shows the concentration profiles corresponding to a single component for frontaladsorption. The figure shows the breakthrough curve, the concentration profiles in the fluid-bulk phase at different times, the concentration profiles in the center of the macropores atdifferent times, and the concentration profile in one characteristic particle, which has beenapproximated using Jacobi polynomials according to the orthogonal collocation method [51].

In the effluent history of a frontal analysis, each breakthrough curve can be integrated to seewhether it matches the dimensionless column holdup capacity for the correspondingcomponent, which is expressed by the following expression assuming that there is no sizeexclusion effect [13]



21

( )( ) ( )

b

bpb

j0

Ns

1jj

iipb

i

1

Cb1

Cb11

CAε

ε+εε−+

+

ε−ε−

=∑

=

∞

Eq. 3.35

In Eq. 3.35, CAi consists of three parts

• the amount of component i adsorbed onto the solid part of the particles• that in the stagnant fluid inside particles• and that in the bulk-fluid

Eq. 3.35 is actually equal to the first moment of a breakthrough curve. The hold-up capacityshould also be equal to the area integrated from

∫τ

= τ−τ=e

0

1zbiei dcCA Eq. 3.36

where τe is a time value at which the breakthrough curve has already leveled off. The abovetwo equations are very helpful in checking the mass balance on an effluent history in frontalanalysis and stepwise displacement.

The results of both calculations are given in Fig. 3.1. The area under the breakthrough curvewas calculated with the MATLAB routine TRAPZ which applies trapezoidal numericalintegration [3]. The calculations are in excellent agreement.

Fig. 3.4 to Fig. 3.8 are examples of the displacement effect in multicomponentchromatography involving competitive isotherms, in this case competitive Langmuirisotherms [15]. Fig. 3.8 in particular, is calculated with the same parameters as Fig. 3.4 butusing just one interior collocation points. The shape of the curves is essentially the same andthe calculation time needed in Fig. 3.8 is about one half of the time needed in Fig. 3.4 (Table3.1).



22

Fig. 3.3 Single-component breakthrough concentration profiles inside the column

Fig. 3.4 Ternary frontal adsorption with two roll-up peaks



23

Fig. 3.5 Ternary frontal adsorption, concentrations profiles of component 1

Fig. 3.6 Ternary frontal adsorption, 3D representation of the concentration profiles ofComponent 2



24

Fig. 3.7 Ternary frontal adsorption, pseudocolor graph of concentration profiles component 1

Fig. 3.8 Calculated ternary adsorption breakthrough curves using one interior collocationpoint



25

3.5 MATLAB Code for the General Multicomponent RateModel

The code for the solution of the general multicomponent rate model was written inMATLAB 5.2. It is not an stand-alone application, so that it needs the MATLAB

environment in order to run. The program is executed line by line. Programs which work inthat fashion are called “interpreters”.

Throughout the program, the matrix capabilities of MATLAB [31] are exploited. Thesolution algorithm has been vectorized as far as possible. Sparse matrices are usedeverywhere in order to make an efficient use of memory and speed up the operations. As aresult the discretized equations that describe the whole system are written in a rather compactway (File F.m in appendix A)

3.5.1 Application Files Description

Five files compose the solution program. Four of them deal with the parameters and solutionof the equation and one (Show_results.m) with the graphic output of the data. The files are thefollowing (appendix A):

Chroma.m: This is the main file. The parameters used in the simulation are declared (with adescription of their meaning) and other files are called in order to do specific tasks. It callsalso the ODE solver routine provided by MATLAB with the appropriate parameters.

Assign_datai.m: In this file, the parameters are assigned with specific values correspondingto a particular simulation. The Galerkin element and orthogonal collocation matrices are alsoconstructed. There is the possibility of having 1 or 2 interior collocation points, no more.Usually 2 interior collocation points for the particle phase are quite enough [13].

Mass_balancei.m: This routine makes the necessary setup so that the ODE solver works in aefficient way. It has two important outcomes:

1. Sparsity Pattern: Since the ODEs that results from the discretization process are stiff [5],the Jacobian [3] of the system needs to be calculated. The sparsity pattern is a sparsematrix with 1s where there are nonzero entries in the Jacobian. The MATLAB routine togenerate the Jacobian (NUMJAC) uses this pattern to generate a sparse Jacobian matrixnumerically. This accelerates execution greatly.The sparsity pattern for three species multicomponent chromatography can be seen in Fig.3.9. This pattern has been generated using trial concentrations with the mass balanceequations and the NUMJAC routine in order to identify the entries in the Jacobian matrixthat are nonzero. The matrix density, which shows the proportion of nonzero entries in thematrix, is very low (3.8 %). This information makes possible to calculate the Jacobian justin selected places of the Jacobian matrix and to ignore completely the zero entries (96.2%of the Jacobian matrix).



26

Fig. 3.9 Jacobian sparsity pattern of three species multicomponent chromatography

Fig. 3.10 Mass matrix sparsity pattern for three species multicomponent chromatography



27

2. Mass Matrix: The mass matrix of the system [48] is assembled from the mass matrices ofEq. 3.24 and Eq. 3.34 (left hand side matrices). The matrix coming from Eq. 3.24 is just aGalerkin element matrix already constructed in Assign_datai.m. The matrix coming fromEq. 3.34 is calculated according to the procedure described in Sec. 3.3.2 which is nothingbut the assembled Jacobian of the multicomponent equilibrium isotherms with respect tothe different species concentrations in every interior collocation point. The Jacobian iscalculated through the function “jacobianri” (appendix A file Mass_balancei.m) whichuses a numerical method, namely the Richardson extrapolation method [3]. Such anapproach makes possible to use any type of equilibrium isotherm as long as it isdifferentiable.The multicomponent equilibrium isotherm is defined in the function “isotherm” (appendixA file Mass_balancei.m). It should be written in a vectorized way.In principle the MATLAB routine NUMJAC could have been used to calculate thisJacobian but it would create conflicts with the calculation of the Jacobian ODE systemsince the NUMJAC routine uses a powerful algorithm based on working storage (on-linehelp, MATLAB ). The Jacobian matrix can also be calculated in an analytical way. Thiswould considerably decrease the CPU time indeed but generality would be lost because itwould be necessary to write a different analytical expression for every different type ofequilibrium isotherm.Because of the concentration dependence of the matrix coming from Eq. 3.34, the massmatrix of the system is calculated at every step time integration. Fig. 3.9 shows a typicalmass matrix sparsity pattern for three species multicomponent chromatography. Again thematrix density is very low (2.1 %). Recognizing this fact, saves a lot of CPU time whensolving the linear system of equations in order to calculate the derivatives defined byEq.3.24 and Eq. 3.34 [46] [48].

F.m: This file contains actually the definition in matrix notation of the right hand side of Eq.3.24 and Eq. 3.34. It also manipulates the order of the column vector of concentrations inorder to express the mass balance of the particle phase in a concise way.

The particle phase concentrations by default are ordered in an alternate way with regards tothe species (Fig. 3.11) because the left-hand side of Eq.3.34 implies interaction betweenspecies concentrations in a definite interior collocation point. This is not the case with theright hand side of Eq. 3.34 which implies interaction between the concentrations in thedifferent interior collocation points for a definite specie. In the latter case, a column vectorordering which keep adjacent all the concentrations for a definite specie is more convenient.The particle concentrations are first rearranged in a tri-dimensional array, whose structure canbe appreciated in Fig. 3.12, and then the calculation of the right hand side of Eq. 3.24 and Eq.3.34 is carried out. Afterwards the calculated values are rearranged to their original order.

The different arrangements of the matrices have been handled through standard MATLAB

routines for matrix manipulation [31].



28

Fig. 3.11 Graphic representation of the concentration column vector reordering for twospecies

Show_resultsi.m:

This routine displays the simulation results typically in 2D graphs. The breakthrough curves,concentration profiles in the bulk-fluid phase and stagnant film inside the macropores aredepicted. Also a characteristic concentration profile of the particle phase is calculated andplotted. This latter calculation approximates the concentration along the particle using Jacobipolynomials according to the orthogonal collocation method.

3D graphs showing the behavior of the concentration profile in the fluid-bulk phase vs. timeand distance are also plotted. In this case, since the data is non-uniformly sampled (due to theODE solver multi-step characteristics) a triangle based cubic interpolation method is used inorder to generate uniformly sampled data. Standard MATLAB routines are used [32].

Cb11Cb12

.

.

.Cb1NzCb21Cb22

.

.

.Cb2Nz

Cp111Cp211Cp112Cp212

.

.

.Cp121Cp221Cp122Cp222

Cp12NzCp12Nz

C p(Ns,Nr,Nz)

C b(Ns,Nz)

Reordering

Cb11Cb12

.

.

.Cb1Nz

Cp111Cp112

.

.

.Cp12Nz

Cb21Cb22

.

.

.Cb2Nz

Cp211Cp212

.

.

.Cp22Nz



29

Fig. 3.12 3D representation of the data structure for the particle phase

Show_resultsi.m:

This routine displays the simulation results typically in 2D graphs. The breakthrough curves,concentration profiles in the bulk-fluid phase and stagnant film inside the macropores aredepicted. Also a characteristic concentration profile of the particle phase is calculated andplotted. This latter calculation approximates the concentration along the particle using Jacobipolynomials according to the orthogonal collocation method.

3D graphs showing the behavior of the concentration profile in the fluid-bulk phase vs. timeand distance are also plotted. In this case, since the data is non-uniformly sampled (due to theODE solver multi-step characteristics) a triangle based cubic interpolation method is used inorder to generate uniformly sampled data. Standard MATLAB routines are used [32].

3.5.2 The ODE solver

Since the ODE system generated after the discretization of the PDE system is stiff, a stiffODE solver is necessary in order to solve the system. MATLAB provides the following stiffODE solvers:

• ODE15S: Stiff differential equations, variable order method• ODE23S: Stiff differential equations, low order method• ODE23T: Moderately stiff differential equations, trapezoidal rule• ODE23TB: Stiff differential equations, low order method

In principle all the solvers can solve our ODE system. An outstanding performance instability and time was observed in ODE15S so that this solver is our first selection. The solverODE15S is a powerful multi-step algorithm based on the numerical differentiation formulas

Cp(Nz,Nr,Ns)Axial Dimension

RadialDimension

Species



30

(NDFs). Optionally it uses the backward differentiation formulas, BDFs, (also known asGear’s method) that are usually less efficient. At every time integration, the solver reuses theformer calculated Jacobians to integrate de variables. Under this scheme very few newJacobians are formed. The solver parameters are tuned in such a way that the results areplotted as they are computed. Typically the breakthrough curves are plotted but other curvesare also possible [48].

A little modification was made on the original code of ODE15S. This was necessary becausethe code allowed passing information only about the time but not the variables to the routinein charge of computing the mass matrix. As a consequence, it is absolutely necessary to runthe model with this modified version of ODE15S, otherwise the whole program cannotrun!. Fig. 3.13 shows the flow chart of the program Chroma.m

Fig. 3.13 Flow chart of program Chroma.m

ODE15S

Chroma

Assing_datai

NUMJAC FJPattern

Mass_matrix

F

Jacobianri Isotherm

Integrator (ODE15S)

Cb, Cp

t >= t f

Show_results

No

Yes

44 GGeenneerraall RRaattee MMooddeell AApppplliieedd ttoo SSiimmuullaatteedd MMoovviinngg BBeedd CChhrroommaattooggrraapphhyy

In a simulated moving bed process [6], the countercurrent movement of the liquid and solid phases is achieved by sequentially switching the valves of the interconnected columns in the direction of the liquid flow to obtain a simulated moving bed (SMB). The process is shown in Fig. 4.1.

Fig. 4.1 Scheme of a Simulated Moving Bed Unit

The switching results in the feed and desorbent inlets, extract and raffinate outlets being located one columns downwards after each switching period. The process can be operated with varying numbers of columns. Fig. 4.1 shows a plant with eight columns which can be divided into four sections of two columns each. Units with two and three sections, or a



32

different number or columns per each section are also possible. Each section plays a specificrole in the operation. A section is defined by its position relative to the inlets and outletsrather than by a specific pair of columns.

The section numbers are determined as described in Fig. 4.1. Two components, A and B, anadsorptive and less adsorptive component, respectively, are separated by the simulatedmoving-bed adsorber. In the adsorber, each section has its inherent function: in section IV,component B remaining in the fluid stream from section III should be adsorbed. Thecomponent B adsorbed on the resin is carried to zone III. The components, A and B, fedbetween zones II and III migrate toward the withdrawal point of the raffinate stream (inlet ofsection IV). In section III, component A is adsorbed on the adsorbent and should not flow outin the raffinate stream. Section II functions to desorb the component B, which is partiallyadsorbed in section III, and to return it to section III. In section I, the strongly adsorptivecomponent A is desorbed to recover it in the extract stream. Thus sections II and III separatethe feed components, while section I regenerates the adsorbent and section IV purifies thedesorbent.

4.1 Modeling the SMB process

A dynamic model of the SMB process can be obtained by connecting the dynamic models ofthe single chromatographic columns while considering the cyclic port switching. In fact,simulated moving bed adsorbents coincide with fixed-bed adsorbers excepts at the moment ofrotating the column. On this base, the equations for a single chromatographic column (Sec.3.2), apply for the SMB system.

In general, the bulk-fluid velocity is different for each SMB section. As a consequence, inorder to have the same time basis for each section, a minor change is needed in the continuityequations of Sec. 3.2. In this case, we define the dimensionless time τ as:

ct

t=τ 0 ≤ τ ≤ 1 Eq. 4.1

where tc is the switching time. With this definition, the dimensionless equations that describethe chromatographic columns are

( ) 0ccc

z

c

z

c

Pe- 1r,pibiic

bibic

bi2

Li

c =−ξτ+τ∂

∂+

∂∂

τ+∂

∂τ= Eq. 4.2

( )[ ] 0r

cr

rr

1cc1

pi22icpip

*pip =

∂

∂

∂∂ητ−ε+ε−

τ∂∂ Eq. 4.3

where τc = νtc/L is a dimensionless cyclic time.

The fluid velocities and the inlet concentrations in the different sections can be calculated bymass balances around the inlet and outlet nodes [53].

Desorbent node (eluent):

Chapter 4 General Rate Model Applied to Simulated Moving Bed Chromatography


33

νIV+νD=νI Iin

I,iDD,iIVout

IV,i CCC ν=ν+ν Eq. 4.4

Extract draw-off node:

νI-νX=νII X,iin

II,iout

I,i CCC == Eq. 4.5

Feed node:

νII+νF=νIII IIIin

III,iFF,iIIout

II,i CCC ν=ν+ν Eq. 4.6

Raffinate draw-off node:

νIII-νR=νIV R,iin

IV,iout

III,i CCC == Eq. 4.7

The liquid flow rate (Qj) through column j is related to the liquid phase velocity by Qj=εbAνj,where A is the column cross-section area. This set of equations may be integrated, starting attime zero, with any specified initial profile. If the integration is started from an initially cleansystem, one obtains the transient response [Ci(t)] as well as the final steady-state profile andthe number of cycles required to approach steady state.

It is worth mentioning that because of the intrinsic mechanism of displacementchromatography, the heat production related to the adsorption of one component is balancedby the heat consumption due to the desorption of another one, thus leading in most cases topractically isothermal conditions [10].

4.2 Model Parameters Estimation

Several parameters appear in the model whose numerical values have to be properlyestimated. This can be done conveniently by dividing them into three classes [10]:

1. geometrical and physical characteristics of the column and of the adsorbent particles(column section, column length, particle size, void fraction, porosities)

2. parameters accounting for mass transport and axial mixing,3. adsorption equilibrium parameters

Since the numerical values of the parameters of class 1 are usually available, the remainingparameters have to be estimated through predictive semi-empirical relationships, whenavailable, or through direct experimental measurements.

Mass transfer parameters such as ki, Dbi, Dpi are often not available from literature, or not easilymeasured by experiments. However they can be estimated with certain accuracy. Fortunately,rate models are not very sensitive to mass transfer parameters. Errors up to a certain degree donot affect the outcome to any great extent [13]. Semi-empirical relationships can be found inthe literature with respect to the parameter of class 2 [10] [13] [43]. If a particular masstransfer parameter cannot be evaluated, a typical value may be used as a substitute. As amatter of fact, if one is not very interested in exact band widths of peaks, a set of values canbe artificially assigned to the three mass transfer related dimensionless group, PeLi, ηi and Bii,



34

to carry out an initial simulation [13]. This is the chosen approach in the following simulationresults.

More difficult is the evaluation of the adsorption equilibrium parameters (class 3); whichusually requires an accurate estimation because of their very important influence on theperformance of the adsorber. Isotherms, especially multicomponent isotherms are generallynot available for a particular system from existing publications in the literature. They mayhave to be measured experimentally. Isotherms data points are first measured experimentallyand then fitted or correlated with an isotherm model, such as the most commonly usedLangmuir isotherm model. Often, difficulties in characterizing the multicomponent adsorptionequilibria make it necessary to use rather empirical models based on Langmuirian isotherms,i.e., Langmuir, modified Langmuir or bi-Langmuir [10] [13] [36] [43] [49].

4.3 Simulation Results

The parameter and the references used in the simulations are listed in Table 4.1.

Table 4.1 system and operating parameters for the simulation examples

Fig. 4.2 Fig. 4.3 Fig. 4.4System Parameters

d 2.6 2.6 1.4εb 0.1987 0.2254 0.2584εp 0.1987 0.2254 0.2584L 8.5 10.5 47.5PeLi 500 2000 500Bii 10 10 10η i 1 1 1Type of Isotherm Modified Langmuir1 Bi-Langmuir type2 Linear3

Operating Parametersnumber of species 2 2 2Cf (both) [kg/m³] 3.5 2.9 50νI [m/s] 7.0917 x (10)-3 7.9146 x (10)-3 2.3407 x (10)-3

νII [m/s] 3.2379 x (10)-3 5.4106 x (10)-3 1.7549 x (10)-3

νIII [m/s] 4.0907 x (10)-3 5.9175 x (10)-3 2.1722 x (10)-3

νIV [m/s] 3.2221 x (10)-3 4.9273 x (10)-3 1.6719 x (10)-3

switching time t [min] 13.2 3 10.3Columns configuration 3-3-3-3 2-2-2-2 2-2-2-2

Numerical Algorithm ParametersNz 21 21 21Nr 1 1 1No. Eqs. 336 336 336steady state after (cycles) 25 35 30CPU time4 (min) ≈ 13 ≈ 25 ≈ 13steady state real time (h) 5.5 1.75 5.15

References[33] [39] [11] [40] [6] [7]

1

21

11

*1p C317.0C140.01

C07.7C96.3C

+++= and

21

22

*2p C317.0C140.01

C96.15C96.3C

+++=

2 21

2

21

1*1p C3C1

C10.0

C0466.0C0336.01

C69.2C

+++

++= and

21

2

21

2*2p C3C1

C3.0

C0466.0C0336.01

C73.3C

+++

++=

3 1*

1p C56.0C = and 2*

2p C23.0C =4 Based in a microprocessor Pentium II running at 350 Megahertz.



35

The selection of the operating conditions to achieve high separation performances is a majorproblem in running a SMB unit. Several approaches have been proposed [10] [11] [22] [27][33] [39] [44]. Without a systematical approach, the search of operational parameters thatensure separation for a given system is extremely difficult, even with the aid of amathematical model.

In this section, optimized data has been taken from the literature in order to carry outsimulations with the model and compare the results. The steady state of the simulations arepresented in Fig. 4.2, Fig. 4.3, and Fig. 4.4

In all the simulations a typical behavior was observed in the axial profile of the SMB units:

• first, the profiles of the components move closer to the corresponding product outlets• second, the concentration level in the columns increases from period to period.

The obtained results are in good agreement with the results reported in the literature (seereferences, Table 4.1).

Fig. 4.2 Separation of EMD 53986 enantiomers. Steady state internal concentration profile.Variable selectivity modified Langmuir isotherm



36

Fig. 4.3 Separation of 1,1’-bi-2-naphtol enantiomers. Steady state internal concentrationprofile. Bi-Langmuir isotherms.

Fig. 4.4 Axial profile of SMB unit for cyclic steady state at the end of a switching period.Linear isotherms



37

4.4 SMB MATLAB Program

The SMB system was simulated in MATLAB taking the program described in Sec. 3.5 asbasis to describe each chromatographic column in the system. The following programmingstrategies were taken in order to simulate the system:

1. Since the fluid velocity only differs from section to section, during one switching periodeach section can be modeled as one single column.

2. An axial coordinate system fixed to the functional sections of the SMB units rather than tothe chromatographic column is chosen.

3. The periodic switching is taken into account by using the axial profile of column j at theend of the previous switching period as initial condition for column j-1 for the newswitching period, i.e. the port switching mechanism is actually treated as columnmovement in the opposite direction.

The structure of the program is quite similar to that of the single chromatographic column(Sec. 3.5). Some additional treatment is necessary to consider the increased column numberand the periodical switching of the columns. In particular the concentration data structure isnow handled through a four-dimensional array: C(Nz, Nr, Ns, Nse), whose dimensions are:

1. Nz: axial dimension, finite element discretization points2. Nr: radial dimension, interior orthogonal collocation points3. Ns: number of species (usually 2)4. Nse: number of sections (4 sections with any number of columns per section. The number

of columns per section is the same for all sections)

Actually, the program is able to handle N-species, however the simulated SMB system is onlyable to separate binary mixtures.

In appendix B are listed the most important additions to the Chroma program (appendix A).The main file in this case is called SMB instead of CHROMA as the single chromatographiccolumn case. The listed files in appendix B are:

FSMB.M:

This file is similar to the file F.M (appendix A). It has just one more external loop in order toaccount for the four sections of the SMB system and carries out the mass balances given byEq. 4.4 - Eq. 4.7 in dimensionless form.

Initial_values.M:

This file accounts for the switching mechanism of the SMB system. It defines the initialconditions with which every cycle begins. The file accepts as input the final concentrations ofa cycle, reorders them according to programming strategy number 3 and gives them as outputfor the initial conditions of the following cycle. Since the Galerkin element matrices forquadratic elements have an odd structure, there is a difficulty in making the division(programming strategy number 3) when faced with even configuration of columns (e.g. 2-2-2-2 or 4-4-4-4). In such a case, the file has and additional routine that approximates theconcentrations with cubic splines and adds one more point to every section in order to have aneven number for the concentration displacement. Although this solution proved to work well,it is not the best one and it is subject to further modifications.

The Flow chart of the SMB program can be seen in Fig. 4.5.



38

Fig. 4.5 Flow chart of SMB program

ODE15S

SMB

Assing_data

NUMJAC FSMBJPattern

Mass_matrix

FSMB

Jacobianri Isotherm

Integrator (ODE15S)

Cb, Cp

t >= t c

Show_results

No

Yes

Initial_values

Show_results

Cycle>=Nc

Yes

No



39

As in the CHROMA program, in the SMB program sparse matrices are used everywhere inorder to speed up the calculations. One example is the Jacobian sparsity pattern for the SMBsystem shown in Fig. 4.6.

Fig. 4.6 Jacobian sparsity pattern for a SMB system. Nz = 21, Nr = 2, Ns = 2

55 CCoonncclluussiioonnss aanndd FFuurrtthheerr WWoorrkk

The solution of the general rate model for column liquid chromatography and simulatedmoving bed (SMB) systems developed in this work, accounts very well for the description ofthe simulated systems. Comparisons were made between the results obtained from the modelprograms and that reported in the literature. The agreement was very good. In particular, thechosen approach to solve the system, which discretize the partial differential equations using acombination of Galerkin finite element and orthogonal collocation methods, proved to be veryefficient and robust, superior by far to the finite differences procedure (comparisons notshown here [49]).

In order to describe appropriately particular chromatographic systems (e.g. affinitychromatography, macromolecules separation), surface adsorption, size exclusion, andadsorption kinetics can be readily included in the general rate model to give an adequatedescription of a particular system

The SMB model could be used as a tool to test or develop optimization strategies [11] [18][27] [33] [39], system identification parameters from measured data, and control strategiesassessment [7]. In the pursuit of that objectives, the appropriate MATLAB toolboxes couldbe used [23] [25] [28] [30].

With regard to the optimization strategies developed in [11] and [33], where and overall voidfraction (ε*) is introduced as a key parameter to determine optimal SMB operating conditions,the model here developed could be used to determine the influence of the different ratiosbetween the bed porosity (εb) and particle porosity (εp) (given a specific value of overall voidfraction of the bed ε*) in the performance of the SMB unit. Besides, analysis of the effects ofaxial dispersion and mass transfer resistances on SMB performances when high purities arerequired, is also possible.

It must be pointed out that different numbers of columns could be implemented in each zoneof a real system [9], but this possibility in not considered in the actual program. Minormodifications should be necessary in order to do so.

The programs could also be extended in order to support reacting systems [10] [18] [19] [22][49], different types of adsorption kinetics [10] [13] [18] [43] [49] [50], and gas andsupercritical fluid operations [10] [16] [22] [34] [42]. The extensions should be possible usingalways the general rate model approach.

As it is known, simulated moving-bed adsorbers are usually useful only for the separation oftwo components or the fractionation of multiple components into two groups. for amulticomponent system, a simulated moving bed can be used without technical modificationsif the product to be recovered is the first of the last eluted one in a multicomponent mixture.The main drawback of a simulated moving bed stems from its lack of versatility. Therefore,different approaches are being studied in order to adapt the SMB technology tomulticomponent separations e.g. columns packed with two kinds of resins (Hashimoto et al. in[10]), coupled loop chromatography [17], annular chromatography [20], a shut-off valve [21],non-isocratic separation [24], and additional zones [38]. This alternatives could be evaluatedthrough modifications in the programs.

Chapter 5 Conclusions and Further Work


41

The MATLAB programming environment based on matrices proved to be ideal forapplications of this type. Using MATLAB matrix notation and matrix processing routines itis possible to write a very compact code easy to understand, debug, and modify. The CPUtimes to run the SMB model are acceptable. However improvement could be done incalculating the mass matrices of the system (Jacobian matrices of the equilibrium isothermsSec. 3.3.2 and Sec. 3.5.1), either using a better numerical algorithm or using automatedanalytical expressions [35]. There is always the possibility of coding particular expression forparticular isotherms, but generality is lost with this approach.

Even though the advantages offered by MATLAB are great, it does have the disadvantage ofbeen an interpreter language. Unavoidably, the CPU times to run programs under such aplatform is longer than stand-alone external applications. If time is a factor, one can generateC or C++ code trough the MATLAB compiler [35] and then use a C or C++ compiler togenerate stand-alone external applications which wouldn’t need the MATLAB environmentto run.

REFERENCES

[1] Bellot JC , Condoret JS (1991) Liquid Chromatography Modelling: A Review. ProcessBiochemistry 26:363

[2] Bird RB, Steward WE, Lightfoot EN (1960) Transport Phenomena. John Wiley, NewYork

[3] Chapra SC, Canale RP (1998) Numerical Methods for Engineers. WCB/McGraw-Hill,Boston

[4] Chu KH, Hashim MA (1995) Simulated Countercurrent Adsorption Processes: AComparison of Modelling Strategies. The Chemical Engineering Journal 56:59

[5] Cooper JM (1998) Introduction to Partial Differential Equations with MATLAB.Birkhäuser, Boston

[6] Dünnebier G. Weirich I, Klatt KU (1998) Computationally Efficient Dynamic Modellingand Simulation of Simulated Moving Bed Chromatographic Processes with LinearIsotherms. Chemical Engineering Science 53:2537

[7] Dünnebier G, Engell S, Klatt KU, Schmidt-Traub H, Strube J, Weirich I (1998) Modelingof Simulated Moving Bed Chromatographic Processes with Regard to Process ControlDesign. Computers chem. Engng 22:S855

[8] Finlayson BA (1980) Nonlinear Analysis in Chemical Engineering. McGraw-Hill, NewYork

[9] Francotte E, Richert P, Mazotti M, Morbidelli M (1998) Simulated Moving BedChromatographic Resolution of a Chiral Antitussive. Journal of Chromatography A796:239

[10] Ganetsos G, Barker PE (editors) (1993) Preparative and Production ScaleChromatography. Marcel Dekker, New York

[11] Gentilini A, Migliorini C, Mazotti M, Morbidelli M (1998) Optimal Operation ofSimulated Moving-Bed Units for Nonlinear Chromatographic Separations II. Bi-Langmuir Isotherm. Journal of Chromatography A 805:37

[12] Gottschlich N (1997) Einsatz der Simulierten Gegenstromchromatographie zurReinigung hochmolekularer Bioprodukte. Shaker, Aachen.

[13] Gu T (1995) Mathematical Modeling and Scale-up of Liquid Chromatography. Springer,Berlin



44

[14] Gu T, Tsai GJ, Tsao GT (1990) New Approach to a General nonlinear MulticomponentChromatography Model. AIChE J 36:784

[15] Gu T, Tsai GJ, Tsao GT, Ladish MR (1990) Displacement Effect in MulticomponentChromatography. AIChE J 36:1156

[16] Guiochon G, Shirazi SG, Katti AM, (1994) Fundamentals of Preparative and NonlinearChromatography. Academic Press, Boston

[17] http://www.arifractal.com/ar00002.htm

[18] http://www.atom.ecn.purdue.edu/~biosep/research.html

[19] http://www.cheme.kyoto-u.ac.jp/3koza/smbr-E.html

[20] http://www.isopro.net/web6.htm

[21] http://www.organo.co.jp/technology/hisepa/en_hisepa/index.html

[22] http://www.rereth.ethz.ch/mavt/verfahrenstechnik/mazzotti/pj.04.html

[23] István Kollár (1998) Frequency Domain System Identification Toolbox. The Mathworks,Massachusetts

[24] Kasche V (?) Skript Chromatographie. Technische Universität Hamburg-Harburg

[25] Lennart Ljung (1998) System Identification Toolbox. The Mathworks, Massachusetts

[26] Lu ZP, Ching CB (1997) Dynamics of Simulated Moving-Bed Adsorption SeparationProcesses. Separation Science and Technology 32:1993

[27] Ma Z, Wang NHL (1997) Standing Wave Analysis of SMB Chromatography: LinearSystems. AIChE Journal 43:2488

[28] Mathworks (1998) Control System Toolbox. The Mathworks, Massachusetts

[29] Mathworks (1998) MATLAB Compiler Version 5.2. The Mathworks, Massachusetts

[30] Mathworks (1998) MATLAB Optimization Toolbox Version 5.2. The Mathworks,Massachusetts

[31] Mathworks (1998) Using MATLAB Version 5.2. The Mathworks, Massachusetts

[32] Mathworks (1998) Using MATLAB Graphics Version 5.2. The Mathworks,Massachusetts

[33] Mazzotti M, Storti G, Morbidelli M (1997) Optimal Operation of Simulated Moving BedUnits for Nonlinear Chromatographic Separations. Journal of Chromatography A 769:3

[34] Mazzotti M, Storti G, Morbidelli M (1997) Supercritical Moving Bed Chromatography.Journal of Chromatography A 786:309

http://www.arifractal.com/ar00002.htm

http://atom.ecn.purdue.edu/~biosep/research.html

http://www.cheme.kyoto-u.ac.jp/3koza/smbr-E.html

http://www.isopro.net/web6.htm

References


45

[35] Moler C, Costa PJ (1998) MATLAB Symbolic Math Toolbox. The Mathworks,Massachusetts

[36] Morgenstern AS (1995) Mathematische Modellierung der präparativen Flüssig-chromatographie. Deutscher Universitäts-Verlag, Wiesbaden

[37] Nakamura S (1996) Numerical Analysis and Graphic Visualization with MATLAB.Prentice Hall, New Jersey

[38] Navarro A, Caruel H, Rigal L, Phemius P (1997) Continuous ChromatographicSeparation Process: Simulated Moving Bed Allowing Simultaneous Withdrawal of threeFractions. Journal of Chromatography A 770:39

[39] Nicoud RM, Charton F (1995) Complete Design of a Simulated Moving Bed. Journal ofChromatography A 702:97

[40] Pais LS, Loureiro JM, Rodriguez AE (1998) Modeling Strategies for EnantiomersSeparation by SMB Chromatography. AIChE Journal 44:561

[41] Ramirez WF (1997) Computational Methods for Process Simulation. Butterworth-Heinemann, Oxford

[42] Reverchon E, Lamberti G. Subra P (1998) Modelling and Simulation of the SupercriticalAdsoption of Complex Terpene Mixtures. Chemical Engineering Science 53:3537

[43] Ruthven DM (1984) Principles of Adsorption and Adsorption Processes. Wiley, NewYork

[44] Ruthven (DM), Ching (CB) (1989) Counter-Current and Simulated Counter-CurrentAdsorption Separation Processes. Chemical Engineering Science 44:1011

[45] Schiesser WE (1991) The Numerical Method of Lines. Academic Press, San Diego

[46] Schiesser WE (1994) Computational Mathematics in Engineering and Applied Science.CRC Press, Boca Raton

[47] Schmidt-Traub H, Strube J. (1996) Dynamic Simulation of Simulated-Moving-BedChromatographic Processes. Computers chem. Engng 20:S641

[48] Shampine LF, Reichelt MW (1997) The MATLAB ODE Suite. SIAM Journal onScientific Computing 18:1

[49] Tien C (1994) Adsorption Calculations and Modeling. Butterworth-Heinemann, Boston.

[50] Tsao T (1993) Chromatography. Springer, Berlin

[51] Villadsen JV, Stewart WE (1967) Solution of Boundary-Value Problems by OrthogonalCollocation. Chemical Engineering Science 22:1483

[52] Wergin F (1997) Modellierung der Flüssigchromatographie und Überprüfung imExperiment. Studienarbeit in Arbeitsbereich Biotechnologie 2, Technische UniversitätHamburg-Harburg



46

[53] Zhong G, Guiochon G (1997) Simulated Moving Bed Chromatography. Comparisonbetween the Behaviors under Linear and Nonlinear Conditions. Chemical EngineeringScience 52:4403

APPENDIX

A. Chroma Program

FILE: CHROMA.M123

% Main script File45

% physical parameters6clear;7global Eb % bed void volume fraction [-]8global Ep % particle porosity [-]9

10% Langmuir isotherm model11global a % constant in Langmuir isotherm12global b % adsorption equilibrium constant [m³/Kmol]13

14% operational parameters15

16global Coi % maximum at the column entrance [kmol/m³]17global Cf % Concentration at the column entrance [kmol/m³]18

1920

%numeric algorithm parameters21global Nz % number of spatial grid points in the axial dimension [-]22global Nr % number of spatial grid points in the radial dimension [-]23global Nc % number of components [-]24global Ne % number of equation per component per column [-]25global dim % flag used to indicate a single or multicomponent case [-]26global dz % length between grid points in the axial dimension [m]27global dr % length between grid points in the radial dimension [m]28global c_i % init. concent. in the mob. phase and the statio. phase29[kmol/m³]30

31global Mz % mass matrix for the finite element method32global MS233global M1 % Galerkin element matrix34global M2 % Galerkin element matrix35global M3 % Galerkin element matrix36global M4 % Galerkin element matrix37global M5 % Galerkin element matrix38global A % matrix for orthogonal collocation with Jacobi polynomials39global B % matrix for orthogonal collocation with Jacobi polynomials40

414243

%dimensionless parameters44global Pe % Peclet number45global Bi % Biot number46global Ni % Dimensionless constant47global Ei % Dimensionless constant48



48

global z % Dimensionless axial coordinate49global r % Dimensionless radial coordinate50global ta % Final dimensionless time51

52assign_datai; % Parameters values and initial conditions53figure;54colordef black;55ind=[];56for i=1:Nc57 ind(i)=i*Nz;58end59options=odeset('OutputFcn','odeplot','OutputSel',[ind],JPattern','on','Mass60','on');61

62[t c S]= ode15s('Mass_balancei',[0 ta], c_i,options);63

64%show_resultsi (t,c);65

FILE: ASSIGN_DATAI.M123

function [] = assign_data4% Assigment of values to the parameters5

6% physical parameters7Eb=0.4;8Ep=0.5;9a(1)=1;10b(1)=2;11a(2)=10;12b(2)=20;13a(3)=20;14b(3)=40;15

16% numerical algorithm parameters17Nz=17;18Nr=2;19Ne=Nz*Nr+Nz;20Nc=2;21if Nc==122 dim=0;23else24 dim=1;25end26z=1;27r=1;28dz=2*(z-0)/(Nz-1);29

30% Dimensionless parameters31z=1;32r=1;33ta=8;34Pe(1:Nc)=300;35Ni(1:Nc)=1;36Bi(1:Nc)=20;37Ei=3*Bi.*Ni*(1-Eb)/Eb;38

39% adsorption case40Coi(1)=0.1;41

Appendix


49

Cf(1)=0.1;42Coi(2)=0.1;43Cf(2)=0.1;44Coi(3)=0.1;45Cf(3)=0.1;46

47c_i=zeros(size(1:Nc*Ne))';48

4950

% Stiffness Matrix5152

% bulk phase stiffness matrix5354

M1=[];M2=[];M3=[];M4=[];M5=[];d=[];v1=[];V2=[];MS1=[];MS2=[];5556

d(1)=2/15;57d(Nz)=2/15;58for i=2:Nz-159 if mod(i,2)==060 d(i)=8/15;61 v2(i-1)=-1/30;62 else63 d(i)=4/15;64 v2(i-1)=0;65 end66end67v1(1:Nz-1)=1/15;68M1=diag(d,0)+diag(v1,1)+diag(v1,-1)+diag(v2,2)+diag(v2,-2);69M1=sparse(M1);70

71d(1)=7/3;72d(Nz)=7/3;73for i=2:Nz-174 if mod(i,2)==075 d(i)=16/3;76 v2(i-1)=1/3;77 else78 d(i)=14/3;79 v2(i-1)=0;80 end81end82v1(1:Nz-1)=-8/3;83M2=(diag(d,0)+diag(v1,1)+diag(v1,-1)+diag(v2,2)+diag(v2,-2))/dz^2;84M2=sparse(M2);85

86d(1)=-1/2;87d(Nz)=1/2;88for i=2:Nz-189 if mod(i,2)==090 d(i)=0;91 v2(i-1)=1/6;92 else93 d(i)=0;94 v2(i-1)=0;95 end96end97v1(1:Nz-1)=2/3;98M3=(diag(d,0)+diag(v1,1)-diag(v1,-1)-diag(v2,2)+diag(v2,-2))/dz;99M3=sparse(M3);100

101M5=inv(M1);102



50

103Mz=M1;104for i=1:Nc-1105 ind=Nz*i+1:Nz*(i+1);106 Mz(ind,ind)=Mz(1:Nz,1:Nz);107end108Mz=sparse(Mz);109

110111

%macropores phase stiffness matrix112if Nr==2113 A=[-9/2+8/10*60^(1/2) -9/2-8/10*60^(1/2) 9];114 B=[-30+111/60*60^0.5 15+39/60*60^0.5 15-5/2*60^0.5; 15-39/60*60^0.5 -30-115111/60*60^0.5 15+5/2*60^0.5];116end117if Nr==1118 A=[-3.5 3.5];119 B=[-10.5 10.5];120end121

FILE: MASS_BALANCEI.M123

function varargout=Mass_balance (t, c, flag)4% set up of mass balances for the adsorption5

6switch flag7case '' % Return dy/dt = f(t,y).8 varargout{1}=f(t,c);9case 'jpattern' % Return sparsity pattern matrix S.10 varargout{1} = jpattern(t,c);11case 'mass' % Return mass matrix M(t) or M.12

varargout{1} = mass(t,c);13otherwise14error(['Unknown flag ''' flag '''.']);15end16% ------------------------------------------------------------17function S = jpattern(t,c)18

19global Ne20global Nc21

22T=0;23Y=(1:Ne*Nc)*7;24Y=Y';25FTY=f(T,Y);26THRESH(1:Ne*Nc,1)=1e-14;27FAC=[];28VECTORIZED=[];29[DFDY, FAC]=numjac('f',T,Y,FTY,THRESH,FAC,VECTORIZED);30S = spones(sparse(DFDY));31%---------------------------------------------------------------32function M = mass(t,c)33global Mz34global M135global Nr36global Nz37global Nc38global Ne39

Appendix


51

40%Mz=M1;41

42%for i=1:Nc-143% ind=Nz*i+1:Nz*(i+1);44% Mz(ind,ind)=Mz(1:Nz,1:Nz);45%end46

47cp=c(Nz*Nc+1:Nc*Ne);48for i=1:Nr49 for j=1:Nz50 ind=Nz*Nc*(i-1)+(j-1)*Nc+1:Nz*Nc*(i-1)+(j-1)*Nc+Nc;51 GP(ind,ind)=jacobianri(t,cp(ind));52 end53end54

55ind=Nz*Nc+1:Ne*Nc;56Mz(ind,ind)=GP;57

58M = sparse(Mz);59

6061

function Gi=jacobianri(t,cp)62% Jacobian calculation through Richarson Method63

64global Nc;65

66h(1)=0.01;67h(2)=h(1)/2;68for i=1:269 for j=1:Nc70 cp(j)=cp(j)+h(i);71 GG2(1:Nc,j)=isotherm (t,cp);72 cp(j)=cp(j)-2*h(i);73 GG1(1:Nc,j)=isotherm (t,cp);74 cp(j)=cp(j)+h(i);75 end76 G(:,:,i)=GG2-GG1;77end78Gi=1/h(1)*(4/3*G(:,:,2)-1/6*G(:,:,1));79

80function I=isotherm (t,cp)81% definition of equilibrium isotherms82

83global Ep;84global a;85global b;86global Coi;87global Nc;88

89%Langmuir isotherm90I=(1-Ep)*a(1:Nc)'.*cp/(1+sum(b(1:Nc)'.*Coi(1:Nc)'.*cp))+Ep*cp;91

FILE: F.M123

function dc=f(t,c)45

cbb=reshape(c(1:Nc*Nz),Nz,Nc);6



52

cpp=reshape(shiftdim(reshape(c(Nc*Nz+1:Ne*Nc),Nc,Nz,Nr),1),Nz,Nr,Nc);78

for q=1:Nc910

cb=cbb(1:Nz,q);1112

cp(1:Nr,1:Nz,q)=cpp(1:Nz,1:Nr,q)';1314

cp(Nr+1,1:Nz,q)=(Bi(q)*cb(1:Nz)'-A(1:Nr)*cp(1:Nr,1:Nz,q))/(A(Nr+1)+Bi(q));1516

%Mass balance in the bulk fluid phase along the z-axis17%boundary conditions in the bulk phase18

19S1(1)=-(cb(1)-Cf(q)/Coi(q))/dz;20S1(Nz)=0;21

22dcb(1:Nz,q)=-(M2/Pe(q)+M3+M1*Ei(q))*cb+M1*(Ei(q)*cp(Nr+1,:,q)')+S1'; %(1)23

24%Mass balance inside the macropores phase25%differential mass balance inside the particle macropores, in matrix26notation27

28dcp(:,:,q)=Ni(q).*(B*cp(:,:,q)); %(2)29

30end31

32dc=cat(1,reshape(dcb,Nz*Nc,1),reshape(permute(shiftdim(dcp,dim),[2,1,3]),Nr33*Nz*Nc,1));34

FILE: SHOW_RESULTSI.M123

function []= show_results (t,c)4%show in graphics the results of the calculations5

6global Nz7global Nr8global Nc9global Ne10global A11global B12global Bi13

1415

close;16figure;17 pig(1)='y';18 pig(2)='m';19 pig(3)='c';20 pig(4)='r';21 pig(5)='g';22 pig(6)='w';23 pig(7)='b';24 pig(8)='b';25

26colordef black;27points=size(t); % Points in the time interval28

29cbb=c(1:points(1),1:Nc*Nz)';30

Appendix


53

cbb=reshape(cbb,Nz,Nc,points(1));31cpp=c(1:points(1),Nc*Nz+1:Nc*Ne)';32cpp=reshape(cpp,Nc,Nz,Nr,points(1));33cpp=permute(cpp, [4 2 3 ,1]);34cpp=shiftdim(cpp,1);35cpp=reshape(cpp,Nr*Nz,Nc,points(1));36cc=cat(1,cbb,cpp);37cc=reshape(cc,Ne*Nc,points(1));38

3940

cc=reshape(cc,Nz,Nr+1,Nc,points(1));41box on;42hold on;43

44for q=1:Nc45 plot (t, squeeze(cc(Nz,1,q,:)),pig(q));46 tit(q,1)=num2str(q);47 legend (tit,0);48

title('Breakthrough Curve');49xlabel ('Dimensionless Time');50ylabel ('Dimensionless Concentration');51grid on;52

axis tight;53end54for q=1:Nc55 figure;56 xlin = linspace(0,1,Nz);57 ylin = linspace(0,t(points(1)),points(1));58 [X,Y] =meshgrid(ylin,xlin);59 Z = griddata(t,xlin,squeeze(cc(1:Nz,1,q,:)),X,Y,'cubic');60 surf(X,Y,Z);61 shading interp;62 light;63 lighting phong;64 axis tight;65 %colormap hsv;66 colormap jet;67 rotate3d;68 %camproj('perspective');69

title(strcat('Profile of Component No.',int2str(q)));70 xlabel('Dimesionless Time');71 ylabel ('Dimensionless z');72 zlabel ('Dimensionless Concentration');73 colorbar;74

7576

end7778

for q=1:Nc79figure;80subplot (2,2,1);81plot (t, squeeze(cc(Nz,1,q,:)),pig(q));82grid on;83title(strcat('Breakthrough Curve No. ',int2str(q)));84xlabel ('Dimensionless Time');85ylabel ('Dimensionless Concentration');86axis tight;87

88subplot (2,2,2);89hold on;90grid on;91



54

title('Profiles Inside the Column')92xlabel ('Dimensionless z, bulk phase');93ylabel ('Dimensionless Concentration');94

95No=7; %number of curves showing the axial concentration96del=(t(points(1))-t(1))/No;97

98j=1;99for i=1:No100 for j=1:points(1)101 if ((t(j))/t(points(1)))^(1/1.8)>=(del*i)/t(points(1))102 plot ((0:(Nz-1))/(Nz-1),cc(1:Nz,1,q,j),pig(i));103

break;104 end105 end106end107

108plot ((0:(Nz-1))/(Nz-1),cc(1:Nz,1,1,1));109axis tight;110hold off;111

112cpp(1:Nz,2:Nr+1,:)=cc(1:Nz,2:Nr+1,q,:);113

114if Nr==2115 X=[0 ((15-60^0.5)/33)^0.5 ((15+60^0.5)/33)^0.5 1]'; %coordinates of the116collocation points117 XX=inv([ones(Nr+1,1) X(2:Nr+2,1).^2 X(2:Nr+2,1).^4]);118else119 X=[0 (3/7)^0.5 1]'; %coordinates of the collocation points120 XX=inv([ones(Nr+1,1) X(2:Nr+2,1).^2]);121end122

123for i=1:points(1)124

cpp(1:Nz,Nr+2,i)=(Bi(q)*cc(1:Nz,1,q,i)'-125A(1:Nr)*cpp(1:Nz,2:Nr+1,i)')'/(A(Nr+1)+Bi(q));126 a=XX*cpp(1:Nz,2:Nr+2,i)';127 cpp(1:Nz,1,i)=a(1,1:Nz)';128end129

130subplot (2,2,4);131hold on;132grid on;133xlabel ('Dimensionless z, center of macropores');134ylabel ('Dimensionless Concentration');135

136points=size(t); % Points in the time interval137No=7;138del=(t(points(1))-t(1))/No;139T=[];140T='s';141j=1;142for i=1:No143 for j=1:points(1)144 if ((t(j))/t(points(1)))^(1/1.8)>=(del*i)/t(points(1))145

146 T(i,1:3)=num2str(t(j),'%1.1f');147 plot ((0:(Nz-1))/(Nz-1),cpp(1:Nz,1,j),pig(i));148

break;149 end150 end151end152

Appendix


55

153legend (T);154plot ((0:(Nz-1))/(Nz-1),c(1,1:Nz));155axis tight;156hold off;157

158subplot(2,2,3)159z=[];160for i=1:points(1)161 if t(i)>=(t(points(1))-t(1))/2162 row=i;163 break164 end165end166

167a=XX*cpp(round(Nz/2),2:Nr+2,row)';168Z = (0:0.05:1)';169

170if Nr==2171 Y=[ones(size(Z)) Z.^2 Z.^4]*a;172else173 Y=[ones(size(Z)) Z.^2]*a;174end175

176plot (Z,Y,'g',X,cpp(round(Nz/2),1:Nr+2,row),'mo');177

178grid on;179xlabel ('Dimensionless Particles Radio');180ylabel ('Dimensionless Concentration');181legend ('z=1/2, t=tmax/2',0);182axis tight;183

184end185

B. Additional Files for the SMB Program

FILE: FSMB.M123

function dc=fsmb (t, c)4% differential mass balances for the adsorption5

6c=reshape(c,Ne*Ns,Nse);7cbb=reshape(c(1:Nz*Ns,1:Nse),Nz,Ns,Nse);8cpp=permute(reshape(c(Nz*Ns+1:Ne*Ns,1:Nse),Ns,Ne-Nz,Nse),[2,1,3]);9c=cat(1,cbb,cpp);10

11Cf(1:Ns,1)=(c(Nz,1:Ns,4).*Coi*v(4)/v(1))';12Cf(1:Ns,2)=(c(Nz,1:Ns,1).*Coi)';13Cf(1:Ns,3)=((Coi*vf+c(Nz,1:Ns,2).*Coi*v(2))/v(3))';14Cf(1:Ns,4)=(c(Nz,1:Ns,3).*Coi)';15



56

16for s=1:Nse17

18sum1=zeros(Nr*Nz,1);19

20for q=1:Ns21

22 cb=c(1:Nz,q,s);23

24 cp(1:Nr,1:Nz,q)=reshape(c(Nz+1:Ne,q,s),Nz,Nr)';25

26 cp(Nr+1,1:Nz,q)=(Bi(q,s)*c(1:Nz,q,s)'-27A(1:Nr)*cp(1:Nr,1:Nz,q))/(A(Nr+1)+Bi(q,s));28

29 %Mass balance in the bulk fluid phase along the z-axis30 %boundary conditions in the bulk phase31

32 S1(1)=-(cb(1)-Cf(q,s)/Coi(q))/dz;33 S1(Nz)=0;34

35dcb(1:Nz,q)=ta(s)*(-36

(M2/Pe(q,s)+M3+M1*Ei(q,s))*cb+M1*Ei(q,s)*cp(Nr+1,:,q)'+S1'); %(1)3738

%Mass balance inside the macropores phase39 % differential mass balance inside the particle macropores,40

41dcp(:,:,q)=ta(s)*Ni(q,s).*(B*cp(:,:,q)); %(2)42

43end44

45dc(1:Ne,1:Ns,s)=cat(1,dcb,reshape(permute(dcp,[2 1 3]),Nr*Nz,Ns));46

47end48

49dcb=reshape(dc(1:Nz,1:Ns,1:Nse),Nz*Ns,Nse);50dcp=reshape(permute(dc(Nz+1:Ne,1:Ns,1:Nse), [2 1 3]),(Ne-Nz)*Ns, Nse);51dc=reshape(cat(1,dcb,dcp),Ne*Ns*Nse,1);52

FILE: INITIAL_VALUES.M123

function []=initial_values(cycle)4%initial values of the columns5global c_i;6global Ns;7global Nse8global Nz;9global Nr;10global Ne;11global Np;12global c;13%global t;14

15if cycle==116 c_i=zeros(size(1:Ns*Ne*Nse))';17else18 points=size(c);19 U=c(points(1),1:Ne*Ns*Nse);20

Appendix


57

U=U';21 U=reshape(U,Ne*Ns,Nse);22

23 cbb=reshape(U(1:Nz*Ns,1:Nse),Nz,Ns,Nse);24 cpp=permute(reshape(U(Nz*Ns+1:Ne*Ns,1:Nse),Ns,Ne-Nz,Nse),[2,1,3]);25 U=cat(1,cbb,cpp);26

27U=reshape(U,Nz,Nr+1,Ns,Nse);28

29 if mod(Np/Nse,2)==030 x1=(0:(Nz-1))/(Nz-1);31 x2=(0:((Nz+1)-1)-0)/((Nz+1)-1);32 for i=1:Nse33 for j=1:Ns34 for k=1:(Nr+1)35 U(1:Nz+1,k,j,i)=(spline(x1,U(1:Nz,k,j,i),x2))';36 end37 end38 end39 Nz=Nz+1;40 end41

Documents

Studienarbeit (PDF format)