Chemical Engineering Science - Transfer · chromatographic models and, hence, an efﬁcient numerical method is required to obtain accurate and physically realistic solutions. Discontinuous

Chemical Engineering Science 90 (2013) 17–31

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Science

0009-25

http://d

n Corr

Technic

Tel.: þ4

E-m

qamar@

journal homepage: www.elsevier.com/locate/ces

Analysis and numerical investigation of two dynamic models forliquid chromatography

Shumaila Javeed a,b,c, Shamsul Qamar a,b,n, Waqas Ashraf b, Gerald Warnecke c,Andreas Seidel-Morgenstern a

a Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, 39106 Magdeburg, Germanyb Department of Mathematics, COMSATS Institute of Information Technology, Park Road Chak Shahzad Islamabad, Pakistanc Institute for Analysis and Numeric, Otto-von-Guericke University, 39106 Magdeburg, Germany

H I G H L I G H T S

c The equilibrium dispersive and lumped kinetic chromatographic models are studied.c The first three moments for both models are analytically and numerically calculated.c A close connection between the models is analyzed for linear adsorption isotherms.c The discontinuous Galerkin (DG) method is applied to solve lumped kinetic model.c Results of DG-method are compared with analytical and finite volume schemes results.

a r t i c l e i n f o

Article history:

Received 27 August 2012

Received in revised form

23 November 2012

Accepted 6 December 2012Available online 19 December 2012

Keywords:

Chromatographic models

Moments analysis

Discontinuous Galerkin method

Dynamic simulation

Adsorption

Mass transfer

09/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ces.2012.12.014

esponding author at: Max Planck Institute

al Systems, Sandtorstrasse 1, 39106 Magdebu

9 391 6110454; fax: þ49 391 6110500.

ail addresses: [email protected]

mpi-magdeburg.mpg.de, shamsul.qamar@com

a b s t r a c t

This paper presents the analytical and numerical investigations of two established models for simulating

liquid chromatographic processes namely the equilibrium dispersive and lumped kinetic models. The

models are analyzed using Dirichlet and Robin boundary conditions. The Laplace transformation is applied

to solve these models analytically for single component adsorption under linear conditions. Statistical

moments of step responses are calculated and compared with the numerical predictions for both types of

boundary conditions. The discontinuous Galerkin finite element method is proposed to numerically

approximate the more general lumped kinetic model. The scheme achieves high order accuracy on coarse

grids, resolves sharp discontinuities, and avoids numerical diffusion and dispersion. For validation, the

results of the suggested method are compared with some flux-limiting finite volume schemes available in

the literature. A good agreement of the numerical and analytical solutions for simplified cases verifies the

robustness and accuracy of the proposed method. The method is also capable to solve chromatographic

models also for non-linear and competitive adsorption equilibrium isotherms.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Chromatography is a highly selective separation and purificationprocess used in a broad range of industries, including pharmaceuticaland biotechnical applications. In recent years, this technologyemerged as a useful tool to isolate and purify chiral molecules,amino-acids, enzymes, and sugars. It has capability to provide highpurity and yield at reasonable production rates even for difficultseparations, for instance to isolate enantiomers or to purify proteins.During the migration of the mixture components through tubularcolumns filled with suitable particles forming the stationary phase,

ll rights reserved.

for Dynamics of Complex

rg, Germany.

(S. Javeed),

sats.edu.pk (S. Qamar).

composition fronts develop and propagate governed by the adsorp-tion isotherms providing a characteristic retention behavior of thespecies involved. Separated peaks of desired purity can be collectedperiodically at the outlet of the columns.

Different models were introduced in the literature for describ-ing chromatographic processes, such as the general rate model,various kinetic models, and the equilibrium dispersive model(EDM), see e.g. Ruthven (1984), Guiochon and Lin (2003), Felingeret al. (2004), and Guiochon et al. (2006).

In this paper, the EDM and a non-equilibrium adsorption lumpedkinetic model (LKM), suggested initially by Lapidus and Amundson(1952), are solved analytically and numerically. For this purpose, theLaplace transformation is utilized as a basic tool to transform thepartial differential equations (PDEs) of the models for linearisotherms to ordinary differential equations (ODEs). The corre-sponding analytical solutions of EDM and LKM are obtained alongwith Dirichlet and Robin boundary conditions. If no analytical

www.elsevier.com/locate/ces

www.elsevier.com/locate/ces

http://dx.doi.org/10.1016/j.ces.2012.12.014



http://crossmark.crossref.org/dialog/?doi=10.1016/j.ces.2012.12.014&domain=pdf



mailto:[email protected]




S. Javeed et al. / Chemical Engineering Science 90 (2013) 17–3118

inversion could be performed, the numerical inversion is used togenerate the time domain solution for both types of boundaryconditions. To analyze the considered models, the momentmethod is employed to get expressions for retention times, bandbroadenings, and front asymmetries. Such moment analysisapproach has been found instructive in the literature, see forexample Guiochon et al. (2006), Kubib (1965a,b), Kucera (1965),Miyabe and Guiochon (2000, 2003), Miyabe (2007, 2009), Ruthven(1984), Schneider and Smith (1968) as well as Suzuki (1973).

For non-linear adsorption isotherms, analytical solutions of themodel equations cannot be derived. For that reason, numericalsimulations are needed to accurately predict the dynamic behaviorof chromatographic columns. Steep concentration fronts and shocklayers may occur due to the convection dominated PDEs ofchromatographic models and, hence, an efficient numerical methodis required to obtain accurate and physically realistic solutions.

Discontinuous Galerkin (DG) methods have been widely usedin the computational fluid dynamics and could be a good choice tosolve non-linear convection-dominated problems, see e.g. Bassiand Rebay (1997), Bahhar et al. (1998), Aizinger et al. (2000), Holik(2009). This method was introduced by Reed and Hill (1973) forhyperbolic equations. Afterwards, various DG methods weredeveloped and formulated for non-linear hyperbolic systems, seefor example Cockburn and Shu (1989, 1998, 2001), Cockburn et al.(1990). The DG methods are robust, high order accurate andstable. They use discontinuous approximations which incorporatethe ideas of numerical fluxes and slope limiters in a very naturalway to avoid oscillations in the region of sharp variations. Theirhighly parallelizable nature make them easily applicable tocomplicated geometries and boundary conditions. In this work,the Runge–Kutta discontinuous Galerkin (RKDG) method is for thefirst time applied to solve the more general lumped kineticchromatographic model. The scheme uses the DG scheme in spacecoordinates that converts the given PDEs of the model to a systemof ODEs. An explicit and non-linearly stable high order Runge–Kutta method is used to solve the resulting ODEs system. Thescheme satisfies the total variation bounded (TVB) property thatguarantees the positivity of the scheme, for example the non-negativity of the mixture concentrations in the current study.

This paper is arranged as follows. In Section 2, the non-equilibriumlumped kinetic model and as its special case the equilibriumdispersive model are introduced. In Section 3, analytical solutions ofthe two models are discussed. In Section 4, the RKDG method isderived. In Section 5, the results of the moment analysis are presentedfor the considered models. Numerical test problems are discussed inSection 6 including the case of binary mixtures for non-linearconditions. Finally, Section 7 gives the conclusions.

2. The lumped kinetic model (LKM)

The following assumptions are used in the derivation of LKM:

1.
The column is packed homogeneously and is isothermal. 2. The radial gradients of concentrations in the column are
neglected.
3. The volumetric flow rate remains constant. 4. The model lumps contribution of internal and external mass
transport resistances with a mass transfer coefficient k.
5. Additionally, the axial dispersion coefficient D is considered
constant for all components.

In the light of above assumptions, the mass balance laws of amulti-component LKM are expressed as

@cn

@tþu

@cn

@z¼D

@2cn

@z2�

k

E ðqn

n�qnÞ, ð1Þ

@qn

@t¼

k

1�Eðqn

n�qnÞ, ð2Þ

qn

n ¼ f ðcnÞ, n¼ 1,2, . . . ,Nc: ð3Þ

In the above equations, Nc represents the number of mixturecomponents in the sample, cn denotes the n-th liquid concentra-tion, qn is the n-th solid concentration, u is the interstitial velocity,E is porosity, t is time, and z stands for the axial-coordinate. Thethree characteristic times in the model Eq. (1) are defined as

tC ¼L

u, tD ¼

D

u2, tMT ¼

1

k: ð4Þ

The ratios of these characteristic times provide dimensionlessquantities as

~t1 ¼tC

tD¼

Lu

D, ~t2 ¼

tC

tMT¼

Lk

u: ð5aÞ

Here, ~t1 typically is frequently called the Peclet number Pe

Pe¼ ~t1 ¼Lu

D, ð5bÞ

where L denotes the length of the column.In Eq. (3), the isotherm qn

n describes an equilibrium relation-ship between the concentrations of n-th components in thestationary and mobile phases. Isotherms provide thermodynamicinformation for designing a chromatographic separation process,while a non-linear isotherm describes a non-linear relationshipbetween the liquid and solid phases. The frequently appliedconvex non-linear Langmuir isotherm is defined as

qn

n ¼ancn

1þPNc

~n ¼ 1 b ~n c ~n, n¼ 1,2, . . . ,Nc , ð6Þ

where the an represent Henry’s coefficients and the b ~n quantifythe non-linearity of the single component isotherms. For dilutedsystems or small concentrations, Eq. (6) reduces to linear iso-therms

qn

n ¼ ancn, n¼ 1,2,3, . . . ,Nc: ð7Þ

In above equations, an denotes the n-th Henry’s coefficient. Theinitial conditions for fully regenerated columns are given as

cnð0,zÞ ¼ 0, qnð0,zÞ ¼ 0: ð8Þ

To solve Eqs. (1)–(3), inflow and outflow boundary conditions(BCs) are required. Two types of boundary conditions are appliedin this paper to solve the above model.

2.1. Boundary conditions of type I: Dirichlet boundary conditions

For sufficiently small dispersion coefficient, for exampleD¼ 10�5 m2=s, the Dirichlet boundary conditions can be used atthe column inlet

cn9z ¼ 0 ¼ cn,0: ð9aÞ

A useful and realistic outlet boundary conditions are

cnð1,tÞ ¼ 0: ð9bÞ

2.2. Boundary conditions of type II: Robin type boundary conditions

The following accurate inlet boundary conditions are typicallyused, see e.g. Danckwerts (1953) and Seidel-Morgenstern (1991).This Robin type of boundary conditions is known in chemicalengineering as Danckwerts conditions

cn9z ¼ 0 ¼ cn,0þD

u

@cn

@z, n¼ 1,2,3, . . . ,Nc: ð10aÞ

S. Javeed et al. / Chemical Engineering Science 90 (2013) 17–31 19

These inlet conditions are usually applied together with thefollowing outlet conditions:

@cnðL,tÞ

@z¼ 0: ð10bÞ

Apart from the two sets used in this work, other boundaryconditions can also be applied to solve the model Eqs. (1)–(3).

2.3. The equilibrium dispersive model (EDM)—a limiting case of

LKM

The basic assumption of EDM is that the kinetics of masstransfer in the chromatographic column and the kinetics ofadsorption–desorption are fast. This means, the equilibriumbetween stationary and mobile phases at all positions ofthe column is achieved instantaneously. Further, the valueof the mass transport coefficient k used in the LKM, Eqs. (1)and (2), becomes larger, i.e. k-1, the model Eqs. (1)–(3) changeto the equilibrium dispersive model ð@q=@t¼ @qn=@tÞ as givenbelow

@cn

@tþF

@qnn

@tþu

@cn

@z¼Dapp

@2cn

@z2, n¼ 1,2,3, . . . ,Nc : ð11Þ

Here, F is the phase ratio related to porosity, i.e. F ¼ ð1�EÞ=E, Dapp

is the apparent dispersion coefficient related to the Peclet numberby Pe¼ Lu=Dapp.

3. Analytical solutions of EDM and LKM for linear isotherms

In this section, single component (Nc¼1) linear chromato-graphic models are considered. Analytical solutions are derived ina Laplace domain for linear isotherms (Eq. (7)) with Dirichlet (Eq.(9a)) and Danckwerts (Eq. (10a)) inlet boundary conditions. Tosimplify the notations, we consider cðx,tÞ ¼ c1ðx,tÞ.

3.1. Analytical solution of EDM

The Laplace transformation is defined as

Cðx,sÞ ¼

Z 10

e�stcðx,tÞ dt, s40: ð12Þ

After normalizing Eq. (11) by defining

x¼ z=L, Pe¼ Lu=Dapp ð13Þ

and by applying the Laplace transformation (12) to Eq. (11) withNc¼1 and cinitðt¼ 0,zÞ ¼ 0, we obtain

@2C

@x2�Pe

@C

@x�s Pe

L

uð1þaFÞC ¼ 0: ð14Þ

The solution of this equation is given as

Cðx,sÞ ¼ A expðl1xÞþB expðl2xÞ, ð15Þ

l1,2 ¼Pe

28

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2þ4 Pe

L

uð1þaFÞs

r: ð16Þ

After applying the Dirichlet boundary conditions in Eqs. (9a) and(9b), the values of A and B are given as

A¼c0

s, B¼ 0: ð17Þ

Then, Eq. (15) takes the following simple form:

Cðx,sÞ ¼c0

sexp

Pe

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2þ4 Pe

L

uð1þaFÞs

r !x: ð18Þ

The solution in the time domain cðx,tÞ, can be obtained by usingthe exact formula for the back transformation as

cðx,tÞ ¼1

2pi

Z gþ i1

g�i1etsCðx,sÞ ds, ð19Þ

where g is a real constant that exceeds the real part of all thesingularities of Cðx,sÞ. On applying Eq. (19) to Eq. (18), we obtain

cðx,tÞ ¼c0

2erfc

ffiffiffiffiffiPe

2

rL=uð1þaFÞx�tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L

utð1þaFÞ

r0BB@

1CCA

þc0

2expðx PeÞ erfc

ffiffiffiffiffiPe

2

rL=uð1þaFÞxþtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L

utð1þaFÞ

r0BB@

1CCA, ð20Þ

where erfc denotes the complementary error function.If we consider the second set of boundary conditions given by

Eqs. (10a) and (10b), the values of A and B take the followingforms:

A¼c0

s

l2 expðl2Þ

1�l1

Pe

� �l2 expðl2Þ� 1�

l2

Pe

� �l1 expðl1Þ

, ð21Þ

B¼�c0

s

l1 expðl1Þ

1�l1

Pe

� �l2 expðl2Þ� 1�

l2

Pe

� �l1 expðl1Þ

: ð22Þ

With these values A and B, the back transformation of Eq. (15) isnot doable analytically. However, well established numericalinverse Laplace transformation could be used to get cðx,tÞ. In thispaper, the Fourier series approximation of Eq. (19) is used, see e.g.Rice and Do (1995).

3.2. Analytical solution of LKM

After applying the Laplace transformation to the single com-ponent LKM with Dirichlet boundary conditions, Eqs. (1) and (2)take the forms

@2C

@x2�Pe

@C

@x�

Pe L

uCs�

L2

EDakCþ

L2

EDkQ ¼ 0, ð23Þ

sQ ¼k

1�EðaC�Q Þ ) Q ¼

ka

ð1�EÞ

sþk

ð1�EÞ

C: ð24Þ

On putting the value of Q in Eq. (23), we obtain

@2C

@x2�Pe

@C

@x�Pe

L

uCs�

L2

ED akCþL2

ED

k2a

ð1�EÞ

sþk

ð1�EÞ

C ¼ 0 ð25Þ

or

@2C

@x2�Pe

@C

@x� Pe

L

us�

L2

ED

k2a

ð1�EÞ

sþk

ð1�EÞ

þL2

EDak

0BBB@

1CCCAC ¼ 0: ð26Þ

Thus, the Laplace domain solution is given as

Cðx,sÞ ¼ A expðl1xÞþB expðl2xÞ, ð27Þ


where

l1,2 ¼Pe

28

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe

2

� �2

þ PeL

us�

L2

ED

k2a

ð1�EÞ

sþk

ð1�EÞ

þL2

ED ak

0BBB@

1CCCA

vuuuuuut : ð28Þ

For the simplified boundary conditions (9a) and (9b), we haveagain

A¼co

s, B¼ 0: ð29Þ

Using these values of A and B in Eq. (27), we obtain

Cðx,sÞ ¼co

sexp

Pe

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe

2

� �2

þ PeL

us�

L2

ED

k2a

ð1�EÞ

sþk

ð1�EÞ

þL2

EDak

0BBB@

1CCCA

vuuuuuut

0BBBB@

1CCCCAx

ð30Þ

or

Cðx,sÞ ¼co

sexp

Pe

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2þ4 Pe

L

us 1þ

aF

1þsð1�EÞ

k

0B@

1CA

vuuuut0BB@

1CCAx: ð31Þ

For sufficiently large values of k in Eq. (31), i.e. when k-1, thetransformed solution Cðx,sÞ for LKM becomes the solution of EDMgiven by Eq. (18). Once again, the numerical inverse Laplacetransformation is employed to find the original solution cðx,tÞ ofEq. (31). For Danckwerts boundary conditions in Eqs. (10a) and(10b), the solution in the Laplace domain takes the same form asgiven by Eq. (27). The values of A and B are provided by Eqs. (21)and (22), whereas l1,2 can be found in Eq. (28). The numericalinverse Laplace transformation was employed to find the originalsolution cðx,tÞ.

4. Reduced EDM and LKM: moment models

Moment analysis is an effective strategy for deducing impor-tant information about the retention equilibrium and masstransfer kinetics in the column, see e.g. Guiochon et al. (2006),Kucera (1965), Miyabe (2007, 2009), Ruthven (1984), Schneiderand Smith (1968) as well as Suzuki and Smith (1971). The Laplacetransformation can be used as a basic tool to obtain moments. Thenumerical inverse Laplace transformation of the equations pro-vides the optimum solution, but this solution is not helpful tostudy the behavior of chromatographic band in the column. Theretention equilibrium-constant and parameters of the masstransfer kinetics in the column are related to the moments inthe Laplace domain. In this section, a method for describing

Table 1Analytically determined moments for EDM and LKM for x¼1 and c0 ¼ 1, m0 ¼ 1 and m1

Models and BC’s m02

EDM (Dirichlet) 2LDappð1þaFÞ2

u3

EDM (Danckwerts) 2LDapp

u3ð1þaFÞ2 1þ

Dapp

Luðe�Lu=Dapp�1Þ

� �

LKM (Dirichlet) 2LDð1þaFÞ2

u3þ

1

k

2LaFð1�EÞu

� �

LKM (Danckwerts) 2LD

u3ð1þaFÞ2 1þ

D

Luðe�Lu=D�1Þ

� �þ

1

k

2LaFEu

� �

chromatographic peaks by means of statistical moments is usedand the central moments up to third order are calculated for twosets of boundary conditions. The moment of the band profile atthe exit of chromatographic bed of length x¼L is

mi ¼

Z 10

Cðx¼ L,tÞti dt: ð32Þ

The i-th initial normalized moment is

mi ¼

R10 Cðx¼ L,tÞti dtR10 Cðx¼ L,tÞ dt

: ð33Þ

The i-th central moment is

m0i ¼R1

0 Cðx¼ L,tÞðt�m1Þi dtR1

0 Cðx¼ L,tÞ dt: ð34Þ

In this study, the first three moments are calculated for theequilibrium dispersive and lumped kinetic models. The formulasfor the finite moments m0, m1, m02, m03 corresponding to the EDMand LKM with both sets of boundary conditions are given inTable 1. Complete derivations of these moments for EDM andLKM are presented in Appendix A. It is well known that the firstmoment m1 corresponds to the retention time tR. The value of theequilibrium constant a can be estimated from the slopes of astraight lines, m1 ¼ tR over 1=u for constant column length andporosity. It is shown (cf. Table 1) that effects of longitudinaldiffusion are not significant with respect to retention time or firstmoment. The second central moment m2 or the variance of theelution breakthrough curves or peaks provides significant infor-mation related to mass transfer processes in the column. Thequantitative value of m2 or variance indicates the band broad-ening or width of breakthrough curves or peaks and helps tocalculate the HETP (height equivalent to theoretical plates). Azeroth value of third statistical moment m03 designates symmetriccurves. Finally, the third central moment m03 was analyzed whichevaluates front asymmetries. The second and the third centralmoments for the more general Danckwerts BCs reduce to themoments for the Dirichlet BCs in case Dapp approaches to zero. Acomparison of analytical moments and numerical momentsobtained by the proposed numerical scheme is given in the nextsection.

5. Numerical scheme for solving LKM

In this section, the Runge Kutta discontinuous Galerkinmethod is applied to lumped kinetic model, e.g. Cockburn andShu (1989, 2001). The scheme is second order accurate in theaxial-coordinate. The resulting ODE-system is solved by using athird-order Runge–Kutta ODE-solver. For simplicity, a single

¼ L=uð1þaFÞ.

m03

12LD2app

u5ð1þaFÞ3

12LD2appð1þaFÞ3

u51þ

2Dapp

Lu

� �e�Lu=Dapp þ 1�

2Dapp

Lu

� �� 12LD2

u5ð1þaFÞ3þ

1

k

12LDð1þaFÞaFð1�EÞu3

� �þ

1

k2

6LaFð1�EÞ2

u

!

12LD2ð1þaFÞ3

u51þ

2D

Lu

� �e�Lu=Dþ 1�

2D

Lu

� ��

þ1

k

12LDaFEð1þaFÞ

u3

D

Lue�Lu=DþF�

D

Lu

� �� þ

1

k2

6LaF3E2

u

!


component model is considered to derive the numerical scheme

@c

@tþ@

@zuc�D

@c

@z

� �¼�

k

E ðqn�qÞ, ð35Þ

@q

@t¼

k

1�E ðqn�qÞ: ð36Þ

Let us define

gðcÞ :¼ffiffiffiffiDp @c

@z, f ðc,gÞ :¼ uc�

ffiffiffiffiDp

gðcÞ: ð37Þ

Then, Eq. (35) changes to the following system of PDEs:

@c

@t¼�

@f

@z�

k

E ðqn�qÞ, ð38Þ

g ¼ffiffiffiffiDp @c

@z, ð39Þ

@q

@t¼

k

1�E ðqn�qÞ: ð40Þ

The axial-length variable z is discretized as follows. Forj¼ 1,2,3, . . . ,N, let zjþ 1

2be the cell partitions, Ij ¼ �zj�1

2,zjþ 1

2½ be the

domain of cell j, Dzj ¼ zjþ 12�zj�1

2be the width of cell j (cf. Fig. 1).

Moreover, I¼UIj be the partition of the whole domain. We seekan approximate solution chðt,zÞ to cðt,zÞ such that for each timetA ½0,T�, chðt,zÞ belongs to the finite dimensional space

Vh ¼ fvAL1ðIÞ : v9Ij

APmðIjÞ, j¼ 1,2,3, . . . ,Ng, ð41Þ

where PmðIjÞ denotes the set of polynomials of degree up to m

defined on the cell Ij. Note that in Vh, the functions are allowed tohave jumps at the cell interface zjþ 1

2. In order to determine the

approximate solution chðt,zÞ, a weak formulation is needed. Toobtain weak formulation, Eqs. (38)–(40) are multiplied by anarbitrary smooth function v(z) followed by integration by partsover the interval Ij, we getZ

Ij

@cðt,zÞ

@tvðzÞ dz¼�ðf ðcjþ 1

2,gjþ 1

2Þvðzjþ 1

2Þ�f ðcj�1

2,gj�1

2Þvðzj�1

2ÞÞ

þ

ZIj

f ðc,gÞ@vðzÞ

@z

� �dz�

k

E

ZIj

ðqn�qÞvðzÞ dz, ð42Þ

ZIj

gðcÞvðzÞ dz¼ffiffiffiffiDpðcjþ 1

2vðzjþ 1

2Þ�cj�1

2vðzj�1

2ÞÞ�

ffiffiffiffiDpZ

Ij

cðzÞ@vðzÞ

@zdz,

ð43Þ

ZIj

@q

@tvðzÞ dz¼

k

1�E

ZIj

ðqn�qÞvðzÞ dz: ð44Þ

One way to implement Eq. (41) is to choose Legendre polyno-mials, Pl(z), of order l as local basis functions. In this case, the L2-orthogonality property of Legendre polynomials can be exploited,namelyZ 1

�1PlðsÞPl0 ðsÞ ¼

2

2lþ1

� �dll0 : ð45Þ

For each zA Ij, the solutions ch and gh can be expressed as

chðt,zÞ ¼Xml ¼ 0

cðlÞj jlðzÞ, ghðchðt,zÞÞ ¼Xml ¼ 0

gðlÞj jlðzÞ,

zj−1 zj zj+1

zj− 12

z

zj+ 12

Fig. 1. Discretization of the computational domain.

qhðt,zÞ ¼Xm

l ¼ 0

qðlÞj jlðzÞ, ð46Þ

where

jlðzÞ ¼ Plð2ðz�zjÞ=DzjÞÞ, l¼ 0,1, . . . ,m: ð47Þ

If m¼0 the approximate solution ch uses the piecewise constantbasis functions, if m¼1 the linear basis functions are used, and soon. In this paper the linear basis functions are taken into account,therefore l¼0, 1. By using Eqs. (45)–(47), it is easy to verify that

wðlÞj ðtÞ ¼2lþ1

Dzj

ZIj

whðt,zÞjlðzÞ dz, wAfc,g,q,qng: ð48Þ

Then, the smooth function v(z) can be replaced by the testfunction jlAVh and the exact solutions c and g by the approx-imate solutions ch and gh. Moreover, the function f ðcjþ 1

2,gjþ 1

2Þ ¼

f ðcðt,zjþ 12Þ,gðcjþ 1

2ÞÞ is not defined at the cell interface zjþ 1

2. There-

fore, it has to be replaced by a numerical flux that depends on twovalues of chðt,zÞ at the discontinuity, i.e.

f ðcjþ 12,gjþ 1

2Þ � hjþ 1

2¼ hðc�

jþ 12,cþ

j�12

Þ: ð49Þ

As g :¼ gðcÞ, it can be dropped from the arguments of h forsimplicity. Here

c�jþ 1

2:¼ chðt,z�

jþ 12Þ ¼

Xm

l ¼ 0

cðlÞj jlðzjþ 12Þ,

cþj�1

2

:¼ chðt,zþj�1

2

Þ ¼Xm

l ¼ 0

cðlÞj jlðzj�12Þ: ð50Þ

Using the above definitions, the weak formulations in Eqs. (42)and (43) simplify to

dcðlÞj ðtÞ

dt¼�

2lþ1

Dzjðhjþ 1

2jlðzjþ 1

2Þ�hj�1

2jlðzj�1

2ÞÞ

þ2lþ1

Dzj

ZIj

f ðch,ghÞdjlðzÞ

dz

� ��

k

E ðqnðlÞj �qðlÞj Þ, ð51Þ

gðlÞj ðtÞ ¼2lþ1

Dzj

ffiffiffiffiDp

� cjþ 12jlðzjþ 1

2Þ�cj�1

2jlðzj�1

2Þ�

ZIj

chðt,zÞdjlðzÞ

dzdz

!, ð52Þ

dqðlÞj ðtÞ

dt¼

k

1�E ðqnðlÞj �qðlÞj Þ: ð53Þ

The initial data for the above system are given as, cf. Eq. (48),

cðlÞj ð0Þ ¼2lþ1

Dzj

ZIj

cð0,zÞjlðzÞ dz, gðlÞj ð0Þ ¼ gðcðlÞj ð0ÞÞ,

qðlÞj ð0Þ ¼ qðcðlÞj ð0ÞÞ: ð54Þ

It remains to choose the appropriate numerical flux function h.The above equation defines a monotone scheme if the numericalflux function hða,bÞ is consistent, hðc,cÞ ¼ f ðc,gðcÞÞ, and satisfies theLipschitz continuity condition, i.e. hða,bÞ is a non-decreasingfunction of its first argument and non-increasing function of itssecond argument. In other words, a scheme is called monotone ifit preserves the monotonicity of the numerical one-dimensionalsolution when passing from one time step to another. In thispaper, the local Lax–Friedrichs flux was used which satisfies theabove mentioned properties, see e.g. Kurganov and Tadmor(2000), LeVeque (2003), Zhang and Liu (2005)

hLLFða,bÞ ¼

1

2½f ða,gðaÞÞþ f ðb,gðbÞÞ�Cðb�aÞ�, ð55Þ

C ¼ maxmin ða,bÞr srmax ða,bÞ

9f 0ðs,gðsÞÞ9: ð56Þ

Table 2Parameters for Section 6.1 (linear isotherm).

Parameters Values

Column length L¼ 1:0 m

Porosity E¼ 0:4

Interstitial velocity u¼ 0:1 m=s

Characteristic time tC ¼ 10 s

Dispersion coefficient for EDM Dapp ¼ 10�4 m2=s

Peclet no for EDM ~t1 ¼ Pe¼ 103

Characteristic time for EDM tD ¼ 0:01 s

Dispersion coefficient for LKM D¼ 10�5 m2=s

Peclet no for LKM ~t1 ¼ Pe¼ 104

Characteristic time for LKM tD ¼ 0:001 s

Mass transfer coefficient k¼ 100 1=s

Characteristic time tMT ¼ 0:01 s

Dimensionless number ~t2 ¼ 103

Concentration at inlet c1,0 ¼ 1:0 g=l

Henry coefficient a¼0.85


The Gauss–Lobatto quadrature rule of order 10 was used toapproximate the integral terms appearing on the right hand sideof Eqs. (51) and (52).

In order to achieve the total variation stability, some limitingprocedure has to be introduced. For that purpose, it is needed to

modify c7jþ 1

2

in Eq. (49) by some local projection. For more details,

see e.g. Cockburn and Shu (1989). To this end, we write (50) as

c�jþ 1

2¼ cð0Þj þ

~cj, cþj�1

2

¼ cð0Þj �c j, ð57Þ

where

~cj ¼Xml ¼ 1

cðlÞj jlðzjþ 12Þ, c j ¼�

Xm

l ¼ 1

cðlÞj jlðzj�12Þ: ð58Þ

In this study, we consider the linear basis functions, therefore

l¼0, 1. In above equation, when m¼0, ~cj ¼ c j ¼ 0 and when m¼1,

~cj ¼ c j ¼ 6cð1Þj , etc. Next, ~cj and c j can be modified as

~c ðmodÞj ¼mmð~cj,Dþ cð0Þj ,D�cð0Þj Þ,

cðmodÞj ¼mmðc j,Dþ cð0Þj ,D�cð0Þj Þ, ð59Þ

where D7 :¼ 7 ðcj71�cjÞ and mm is the usual minmod function

defined as

mmða1,a2,a3Þ

¼s � min

1r ir39ai9 if signða1Þ ¼ signða2Þ ¼ signða3Þ ¼ s,

0 otherwise:

(ð60Þ

Then, Eq. (57) modifies to

c�ðmodÞ

jþ 12

¼ cð0Þj þ~c ðmodÞ

j , cþðmodÞ

j�12

¼ cð0Þj �cðmodÞj ð61Þ

and replaces (49) by

hjþ 12¼ hðc�ðmodÞ

jþ 12

,cþðmodÞj�1

2

Þ: ð62Þ

This local projection limiter does not affect the accuracy in thesmooth regions and convergence can be achieved without oscilla-tions near shocks, e.g. Cockburn and Shu (1989). Finally, a Runge–Kutta method that maintains the TVB property of the scheme isneeded to solve the resulting ODE-system. Let us rewrite Eqs. (51)and (53) in a concise form as

dch

dt¼ Lhðch,tÞ: ð63Þ

Then, the m-order TVB Runge–Kutta method can be used toapproximate Eq. (63)

ðchÞm¼Xm�1

l ¼ 0

½amlðchÞðlÞþbmlDtLhððchÞ

ðlÞ,tsþdlDtÞ�,

m¼ 1,2, . . . ,r, ð64Þ

where based on Eq. (54)

ðchÞð0Þ¼ ðchÞ

m, ðchÞðrÞ¼ ðchÞ

mþ1: ð65Þ

Here, s denotes the s-th time step. For second order TVB Runge–Kutta method the coefficients are given as

a10 ¼ b10 ¼ 1, a20 ¼ a21 ¼ b21 ¼12,

b20 ¼ 0, d0 ¼ 0, d1 ¼ 1: ð66Þ

While, for the third order TVB Runge–Kutta method the coeffi-cients are given as

a10 ¼ b10 ¼ 1, a20 ¼34 , b20 ¼ 0, a21 ¼ b21 ¼

14 , a30 ¼

13,

b30 ¼ a31 ¼ b31 ¼ 0, a32 ¼ b32 ¼23 ; d0 ¼ 0, d1 ¼ 1, d2 ¼

12:

ð67Þ

The CFL condition is given as

Dtr1

2mþ1

� �Dzj

9u9, ð68Þ

where m¼1, 2 for second- and third-order schemes, respectively.Boundary conditions: Let us put the boundary at z�1

2¼ 0. The

left boundary condition given by Eqs. (10a) and (10b) can beimplemented as

c��1

2ðtÞ ¼ cð0Þ0 þ

D

u

cð0Þ1 �cð0Þ0

Dz, ð69Þ

~c ðmodÞ0 ¼mmð~c0,Dþ cð0Þ0 ,2ðcð0Þ0 �cinjÞÞ,

cðmodÞ0 ¼mmðc0,Dþ cð0Þ0 Þ: ð70Þ

Outflow boundary condition is used on the right end of thecolumn, cðlÞNþ1 ¼ cðlÞN .

6. Numerical test problems

In order to validate the results, several numerical test pro-blems are solved. Here, linear basis functions are considered ineach cell, giving a second order accurate DG-scheme in axial-coordinate. The ODE-system is solved by a third-order Runge–Kutta method given in Eq. (66). The program is written in theC-language under a Linux operating system and was compiled ona computer with an Intel(R) Core 2 Duo processor of speed 2 GHzand memory (RAM) 3.83 GB.

6.1. Single component breakthrough curves for linear isotherms

6.1.1. Error analysis for LKM

Here, a comparison of different numerical schemes is pre-sented for lumped kinetic model. The parameters of the problemare given in Table 2. The Koren (1993) scheme was alreadypresented by Javeed et al. (2011) for the EDM and the remainingflux limiters are given in Table 3. The numerical results at thecolumn outlet are shown in Fig. 2. In that figure, the concentrationprofiles generated by using different numerical schemes on 100grid points are compared with the analytical solution obtainedfrom the Laplace transformation. It can be observed that DG andKoren methods have better accuracy as compared to other fluxlimiting finite volume schemes. Further, the zoomed plot of Fig. 2shows that the solution of DG-scheme is closest to the analyticalsolution. The L1-error in time at the column outlet was calculated

Table 4Errors and CPU times at 50 grid points for linear isotherm.

Limiter L1-error Relative error CPU (s)

DG-scheme 0.4011 0.0100 5.86

Koren 0.5155 0.0137 4.90

Van Leer 0.9324 0.0248 8.26

Superbee 1.0732 0.0286 9.34

MC 0.9762 0.0260 8.82

Table 5Errors and CPU times at 100 grid points for linear isotherm.

Limiter L1-error Relative error CPU (s)

DG-scheme 0.0139 3:71� 10�4 8.46

Koren 0.0153 4:08� 10�4 7.82

Van Leer 0.2255 0.0060 14.90

Superbee 0.2966 0.0079 17.73

MC 0.2477 0.0066 14.96

Table 6Parameters for Section 6.2 (non-linear isotherm).

Parameters Values

Column length L¼ 1:0 m

Porosity E¼ 0:4

Interstitial velocity u¼ 0:1 m=s

Dispersion coefficient D¼ 10�4 m2=s

Adsorption rate k¼ 103 1=s

Characteristic time tC ¼ 10 s

Characteristic time tD ¼ 0:01 s


using the formula

L1-error¼XNT

n ¼ 1

9cnR�cn

N9Dt: ð71Þ

The relative error can be defined as

relative error¼

PNT

n ¼ 1 9cnR�cn

N9PNT

n ¼ 1 9cnR9

Dt, ð72Þ

where cRn denotes the Laplace solution at the column outlet for

time tn and cNn represents the corresponding numerical solution.

Moreover, NT denotes the total number of time steps and Dt

represents the time step size. Note that, numerical accuracy canbe influenced by the choice of time step size Dt. Comparisons ofL1-errors, relative errors, and computational times of schemes aregiven in Tables 4 and 5 for 50 and 100 grid points, respectively. Itcan be observed that the DG-scheme produces small errorscompared to the other schemes for both 50 and 100 grid cells,but efficiency (or CPU time) of the Koren scheme is better thanthe other schemes for both numbers of grid points. It can benoticed that relative errors of the DG and Koren schemes are verylow for 100 grid points. For that reason, to achieve reliable resultswith better accuracy, 100 mesh points are chosen for furthernumerical simulations discussed in this section. On the basis ofthese results, one can clearly conclude that the DG method can bean optimal choice to approximate chromatographic models.Therefore, we are relying on the results of the DG scheme forthe remaining problems of this paper.

6.1.2. Dispersion and mass transfer effects

In this next problem, both the single component EDM and LKMmodels are considered and compared. The parameters used to

Table 3Different flux limiters.

Flux limiter Formula

Van Leer (Leer, 1977)jðrÞ ¼

9r9þr

1þ9r9Superbee (Roe, 1986) jðrÞ ¼maxð0,minð2r,1Þ,minðr,2ÞÞ

Minmod (Roe, 1986) jðrÞ ¼maxð0,minð1,rÞÞ

MC (Leer, 1977) jðrÞ ¼maxð0,minð2r, 12 ð1þrÞ,2ÞÞ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

c [g

/l]

Laplace solutionDG−schemeKoren methodVan LeerSuperbeeMC method0.9 1 1.1

0

0.2

0.4

0.6

0.8

1

t /µ1

Fig. 2. Breakthrough curves (BTC) at x¼1. Comparison of different numerical

schemes for LKM with tC ¼ 10 s, tD ¼ 0:01 s (D¼ 10�5 m2=s or ~t1 ¼ Pe¼ 1000Þ,

and tMT ¼ 0:01 s ðk¼ 100 1=s or ~t2 ¼ 103Þ.

Characteristic time tMT ¼ 0:001 s

Peclet no ~t1 ¼ Pe¼ 103

Dimensionless number ~t2 ¼ 104

Concentrations c1,0 ¼ c2,0 ¼ 10 mol=l

Henry coefficients a1¼0.5, a2 ¼ 1

Constants used in Eq. (6) b1 ¼ 0:05 l=mol, b2 ¼ 0:1 l=mol

Injection time tin ¼ 12 s

solve the model equations are taken from Lim and Jorgen (2009)and are given in Table 2. Fig. 3 (top: left) depicts the dispersioneffects of EDM by considering different values of Dapp or char-acteristic times tD. It demonstrates that smaller values of Dapp

produce steeper fronts. Fig. 3 (top: right) shows similar dispersioneffects by varying D as described by the LKM keeping tC ¼ 10 sand tMT ¼ 0:01 s ðk¼ 100 1=s, ~t2 ¼ 103

Þ fixed. The similarity withthe corresponding figure for the EDM is due to the relative largevalue for k. In Fig. 3 (bottom: left), different values of the masstransfer coefficients k for fixed D¼ 10�5 m2=s ðPe¼ 104

Þ andtC ¼ 10 s are used for the LKM. This figure shows that increasingvalues of k has a similar effects as produced by decreasing D. InFig. 3 (bottom: right), dispersion coefficient D¼ 10�3 m2=sðPe¼ 102

Þ is taken into consideration for different values of k. Itis evident that even for a large magnitude of the mass transfercoefficient k, sharp fronts are not possible due to significantdispersion effects. All trends generated numerically are well-known and realistic for the adsorption community.

6.1.3. Comparison of analytical and numerical solutions

This part focuses on the comparison of analytical and numer-ical results from the EDM and the LKM for both Dirichlet andDanckwerts boundary conditions. In Fig. 4 (left), the exact solu-tion obtained for the EDM with Dirichlet boundary conditions iscompared with the numerical Laplace inversion and DG-scheme

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.2

0

0.2

0.4

0.6

0.8

1

1.2

c [g

/l]

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

c [g

/l]

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

c [g

/l]

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

c [g

/l]

Dapp = 0Dapp = 10−5

Dapp = 10−4

Dapp = 10−3

EDM

k = 100

D = 0D= 10−5

D = 10−4

D = 10−3

D = 10−5 D = 10−3

k = 1k = 10k = 100k = 1000

t /µ1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6t /µ1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6t /µ1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6t /µ1

k = 1k = 10k = 100k = 1000

LKM

LKMLKM

Fig. 3. BTC at x¼1. Top (left): Dispersion effects as described by EDM keeping tC ¼ 10 s constant and varying tD (or Dapp). Top (right): Dispersion effects for LKM keeping

tC ¼ 10 s and tMT ¼ 0:01 s ðk¼ 100 1=s or ~t2 ¼ 103Þ constant, and varying tD (or D). Bottom (left): Mass transfer effects for LKM taking tC ¼ 10 s,

tD ¼ 0:001 s ðD¼ 10�5 m2=s or ~t1 ¼ Pe¼ 104Þ, and varying tMT (or k). Bottom (right): Mass transfer effects taking tC ¼ 10 s, tD ¼ 0:1 s ðD¼ 10�3 m2=s or Pe¼ 102

Þ, and

varying tMT (or k).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.2

0

0.2

0.4

0.6

0.8

1

1.2

c [g

/l]

EDM with Dirichlet BCs

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.2

0

0.2

0.4

0.6

0.8

1

1.2

c [g

/l]

LKM with Dirichlet BCs

Laplace numerical inversionDG−scheme

t /µ1 t /µ1

Analytical solutionLaplace numerical inversionDG−scheme

Fig. 4. BTC at x¼1. EDM predictions using Dirichlet boundary conditions (9a), left: comparison of analytical solution (cf. Eq. (19)), Laplace numerical inversion and

DG-method solutions with tC ¼ 10 s, tD ¼ 0:02 s ðDapp ¼ 2� 10�4 m2=s or Pe¼ 500Þ. LKM predictions using Dirichlet boundary conditions (9a), right: comparison of

Laplace numerical inversion and DG-scheme solutions with tC ¼ 10 s, tD ¼ 0:01 s ðD¼ 10�4 m2=s or Pe¼ 1000Þ, and tMT ¼ 0:0667 s ðk¼ 15 1=s or ~t2 ¼ 150Þ.


results. Good agreement of these profiles verifies the accuracy ofnumerical Laplace inversion and the proposed numerical scheme.Moreover, the numerical Laplace inversion technique is found tobe a reliable method to solve such model problems and will beused below in the subsequent case studies. In Fig. 4 (right), theresults of the DG method for the LKM and Dirichlet boundaryconditions are compared with the numerical Laplace inversionsolution. No analytical back transform solution was available forthe LKM using Dirichlet boundary conditions. Fig. 5 (left) and(right) validates the results of numerical Laplace inversion andthe DG-scheme for the EDM and LKM with Danckwerts boundary

conditions, respectively. These profiles show the high precision ofthe numerical Laplace inversion technique and the suggestednumerical scheme. Thus, it can also be concluded that theconsidered numerical Laplace inversion technique is an effectivetool for solving these linear models.

6.1.4. Effect of boundary conditions

For the sake of generality, the Peclet number, cf. Eq. (13), istaken as parameter, while ~t2 ¼ 103

ðtMT ¼ 0:01 sÞ and tC ¼ 10 s areassumed to be constant for this problem. The results shown in

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.2

0

0.2

0.4

0.6

0.8

1

1.2EDM with Danckwerts BCs

c [g

/l]

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

c [g

/l]


LKM with Danckwerts BCs

t /µ1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6t /µ1


Fig. 5. BTC at x¼1. EDM predictions using Danckwerts boundary conditions (10a), left: comparison of Laplace numerical inversion and DG-method solutions with

tC ¼ 10 s, tD ¼ 0:02 s ðDapp ¼ 2� 10�4 m2=s or Pe¼ 500Þ. LKM predictions using Danckwerts boundary conditions (10a), right: comparison of Laplace numerical inversion

and DG-scheme solutions with tC ¼ 10 s, tD ¼ 0:01 s ðD¼ 10�4 m2=s or Pe¼ 1000Þ, and tMT ¼ 0:0667 s ðk¼ 15 1=s or ~t2 ¼ 150Þ.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.10.20.30.40.50.60.70.80.9

1

c [g

/l]

00.10.20.30.40.50.60.70.80.9

1c

[g/l]

Dirichlet BCs with Pe = 1Danckwerts BCs with Pe = 1Dirichlet BCs with Pe = 2Danckwerts BCs with Pe = 2Dirichlet BCs with Pe = 10Danckwerts BCs with Pe = 10

EDM LKM

t /µ1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6t /µ1

Dirichlet BCs with Pe = 1Danckwerts BCs with Pe = 1Dirichlet BCs with Pe = 2Danckwerts BCs with Pe = 2Dirichlet BCs with Pe = 10Danckwerts BCs with Pe = 10

Fig. 6. BTC at x¼1. Effect of boundary conditions for different values of Peclet number. Symbols are used for Dirichlet BCs (9a) and solid lines for Danckwerts BCs (10a),

left: EDM with tC ¼ 10 s, and Pe or tD varies, right: LKM with tC ¼ 10 s, tMT ¼ 0:01 s ðk¼ 100 1=s or ~t2 ¼ 103Þ, and Pe or tD varies.


Fig. 6 illustrate the importance of using more accurate Danck-werts boundary conditions for chromatographic model equations,in the case of relatively small Peclet numbers, e.g. Peo10. Forsuch values, there are visible differences between the resultsobtained by using Dirichlet and Danckwerts boundary conditions.On the basis of these results, one can conclude that the imple-mentation of Dirichlet boundary conditions is not adequate forlarge dispersion coefficients. For large values of Peclet numberðPeb10Þ or small axial dispersion coefficients as typicallyencountered in chromatographic columns well packed with smallparticles, there is no difference between Dirichlet and Danckwertsboundary conditions. The described behavior was observed in thesolutions of both EDM and LKM.

6.1.5. Discussion on analytically and numerically determined

moments

In this work, only step inputs are taken into account. Theanalytical moments are calculated by the formulas presented inTable 1 and in Appendix Appendix A. The formulas given belowuse derivatives to approximate the moments and transform thestep response to a pulse response which is the requirement for

finite results of numerical integration. The simulated momentsare obtained from our proposed numerical methods by using thefollowing formulas for the first normalized, second central andthird central moments, respectively:

m1 ¼

R10

dCðx¼ 1,tÞ

dtt dt

R10

dCðx¼ 1,tÞ

dtdt

, m02 ¼

R10

dCðx¼ 1,tÞ

dtðt�m1Þ

2 dt

R10

dCðx¼ 1,tÞ

dtdt

,

m03 ¼

R10

dCðx¼ 1,tÞ

dtðt�m1Þ

3 dt

R10

dCðx¼ 1,tÞ

dtdt

: ð73Þ

The trapezoidal rule is applied to approximate the integrals in Eq.(73). The quantitative comparison of first moment or (retentiontime) over the flow rate u for the EDM, LKM, and the analyticalformula (cf. Table 1) can be seen in Fig. 7 (left). The results are ingood agreement with each other, verify the high precision of ournumerical results and reveal the expected linear trends.

For calculations of second moments m02 and third moments m03,the analytical formulas of Table 1 are used.


Matching m02EDM and m02LKM for the Dirichlet boundary condi-tions generates the following relationship between Dapp, D and k:.

Dapp ¼Dþ1

k

að1�EÞ2

E 1það1�EÞE

� �2

0BBB@

1CCCAu2: ð74Þ

For given a, E, and u, the above equation can be used to findpossible connections between the kinetic parameters of twomodels which should provide very similar elution profiles. Forgiven reference values of a and E (cf. Table 2), for a velocityun ¼ 0:35 m=s and a given Dapp ¼ 10�4 m2=s for EDM, there is aninfinite number of combinations for k and D in the LKM. Forexample D¼ 5� 10�5 m2=s, we obtain k¼ 362 1=s from the for-mula (74). The crossing point for the second moments of the twomodels marked in Fig. 8 (left) indicates that for this particularflow rate un, both models will produce almost the same concen-tration profiles, as verified in Fig. 8 (right). For other than thisspecific value of flow rate, keeping Dapp, D, and k fixed, the resultsof equilibrium dispersive and lumped kinetic models should

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

first

mom

ent,

µ 1 [s

]

Analytical moments (both EDM and LKM)EDM numerical momentsLKM numerical moments

1/u [s/m]

Fig. 7. First moments m1 of EDM and LKM versus different flow rates u. For EDM,

Eq. (A.5) for analytical & Eq. (73) for numerical moments are used, considering

Dapp ¼ 10�4 m2=s. For LKM, Eq. (A.25) for analytical & Eq. (73) for numerical

moments are used, considering D¼ 5� 10�5 m2=s and k¼ 30 1=s.

0 10 20 30 40 50 60 700

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

seco

nd m

omen

ts, µ

’ 2 [s

2 ]

Analytical EDM momentsnumerical EDM momentsAnalytical LKM momentsnumerical LKM moments

reference: u* = 0.35

1/u3 [s3/m3]

Fig. 8. Left: second moments m02 for both models, analytical versus numerical with D

un ¼ 0:35 m=s. Right: two predicted breakthrough curves for un ¼ 0:35 m=s and u¼ 0:1

deviate from each other. This is clearly depicted in Fig. 8 (right)for u¼ 0:1 m=s.

To study the third moments in more detail, we varied u andincreased the values of Dapp and D by an order of magnitude, i.e.Dapp ¼ 10�3 m2=s and D¼ 5� 10�4 m2=s, and specified corre-sponding k values exploiting Eq. (74). Fig. 9 (left) shows the thirdmoments m03 for different flow rates and reveals the difference inthe third moments for the perfect match of the second moments.A good agreement between analytical and numerical moments ofour proposed numerical scheme guarantees the high precision ofsimulation results one more time. For velocity u¼ 0:1 m=s, wherethe large difference between two models is seen in Fig. 9 (left),time derivatives of two breakthrough curves for both models areplotted in Fig. 9 (right). It can be observed that EDM gives largevalues of m03 as compared to LKM. Thus, EDM produces moreasymmetry in the concentration profiles which is visible in Fig. 9(right). The fronting edge is steeper and the tailing edge is moredisperse for the EDM predictions as a clear indication of largerasymmetry compared to the LKM predictions (cf. Fig. 9 (right)).However, it is evident that the difference in the profiles of bothmodels is still very small. This justifies the use of the simpler EDMfor linear isotherm involving just one parameter Dapp as comparedto the more complicated LKM which involves two parameters D

and k.

6.2. Two component elutions: extension to competitive non-linear

adsorption isotherm and finite feed volumes (numerical simulations)

After validating the proposed numerical scheme for single com-ponent linear adsorption, this section is intended to extend our studyto a non-linear problem. On the basis of accurate results for linearmodels, we concluded that the suggested numerical technique couldproduce reliable solutions for non-linear models as well. In this testproblem, the two components lumped kinetic model along with anon-linear Langmuir isotherm (cf. Eq. (6)) is considered for finite feedvolumes. A rectangular pulse of a liquid mixture of width tinA ½0,12�is injected to the column. The boundary conditions are given by Eqs.(10a) and (10b) for two components (Nc¼2). The parameters of thisproblem are taken from Shipilova et al. (2008) and are provided inTable 3. The numerical results are shown in Fig. 10 (left) by using 150spatial grid points. The figure elucidates that the DG-scheme givesbetter resolution of the rectangular profiles as compared to theKoren scheme. These results also agree with those obtained byShipilova et al. (2008), even for coarse mesh cells. The simulationresults validate the importance of suggested method for approximat-ing non-linear chromatographic models. Fig. 10 (right) describesnon-equimolar injection concentrations with c1,0 ¼ 4 mol=l and

0.85 0.9 0.95 1 1.05 1.10

0.2

0.4

0.6

0.8

1

1.2

c [g

/l]

EDM with u* = 0.35, m/sLKM with u* = 0.35, m/sEDM with u = 0.1, m/sLKM with u = 0.1, m/s

t /µ1

app ¼ 10�4 m2=s for EDM, D¼ 5� 10�5 m2=s and k¼ 362 1=s for LKM. Crossing at

m=s, respectively.

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

third

mom

ents

, µ’ 3

[s3 ] Analytical EDM moments

numerical EDM momentsAnalytical LKM momentsnumerical LKM moments

correspondsto u = 0.1, m/s

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

dc (x

= 1

,t)/d

t

0.6 0.65 0.70

0.005

0.01

EDMLKM

1.35 1.4 1.45 1.50

0.005

0.01

1/u5 [s5/m5] x 104 t /µ1

Fig. 9. Left: third moments m03 versus 1=u5. For EDM Dapp ¼ 10�3 m2=s, and for LKM, D¼ 5� 10�4 m2=s are used. Moreover, k is adjusted for each u via Eq. (74) to match m02.

Danckwerts BCs are taken into account. Right: Time derivatives of breakthrough curves. Illustration of effects of differences in third moments generated by EDM and LKM

for u¼ 0:1 m=s (corresponding to 1=u5 ¼ 105).

1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

14

16

18

t u / L

1.4 1.45 1.5 1.5502468

1012141618

Solid lines: reference solutiondash lines: DG−schemedash−dot lines: Koren method

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t u / L

solid lines: c1,0 = 4 mol/l, c2,0 = 2 mol/l, dashed lines: c1,0 = 2 mol/l,c2,0 = 1 mol/ldashed−dotted lines:c1,0 = 1 mol/l,c2,0 = 0.5 mol/l

c i [g

/l]

c i [−

]

Fig. 10. Left: Two component non-linear elutions profile at the column outlet. Parameters are given in Table 6. Right: different injection volumes are used, first injection:

c1,0 ¼ 4 mol=l and c2,0 ¼ 2 mol=l, second injection: c1,0 ¼ 2 mol=l and c2,0 ¼ 1 mol=l, third injection: c1,0 ¼ 1 mol=l, c2,0 ¼ 0:5 mol=l.


c2,0 ¼ 2 mol=l, c1,0 ¼ 2 mol=l and c2,0 ¼ 1 mol=l as well as c1,0 ¼

1 mol=l and c2,0 ¼ 0:5 mol=l, respectively. The results in Fig. 10(right) illustrate the well-known fact that strong non-linearitiesproduce overshoots in the profiles. The efficiency and accuracy ofthe schemes can be graphically seen in Fig. 11. These plots highlightthat errors of the DG-scheme are lower than the other schemes. It isprobably worthwhile to conclude that an increase in the number ofgrid points produces smaller errors, but the computational time of thenumerical schemes increases. The computational times of the DG andthe Koren scheme are comparable, while the time taken by the otherschemes is significantly higher. From the above observations, weconclude that the DG-scheme could be a better option for solvingsuch models.

7. Conclusion

This paper described analytical and numerical investigationsof equilibrium dispersive and lumped kinetic models by consider-ing Dirichlet and Danckwerts boundary conditions. The Laplacetransformation was used as a basic tool to transform the single-component linear sub model to a linear ordinary differential

equation which is then solved analytically in the Laplace domain.The inverse numerical Laplace formula was employed to get backthe time domain solution due to the unavailability of exactsolution. A moments analysis of both models was carried outanalytically and numerically for linear isotherms. Good agree-ment up to third moments assure the better accuracy of numer-ical solutions. The close connection between EDM and LKM wasanalyzed for linear isotherms, a concordance formula was derivedand the strength of the simpler EDM was illustrated. For thenumerical solution of considered models, the DG-scheme wasapplied. The presented scheme satisfies the TVB property andgives second-order accuracy. The method incorporates the ideasof numerical fluxes and slope limiters in a very natural way tocapture the physically relevant discontinuities without producingspurious oscillations in their vicinity. The accuracy of proposedscheme was validated against flux-limiting finite volume schemesand analytical solutions for linear isotherms. The DG-scheme wasfound to be a suitable method for the simulation of linear andnon-linear chromatographic processes in terms of accuracy andefficiency.

The present contribution was focused on the numericalapproximation of chromatographic process in a single column.

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

number of grid points

L1 −er

ror

DG−schemeKoren methodVan LeerSuperbeeMC method

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5


L1 −er

ror


0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

1600


Cpu

tim

e (s

econ

ds)


Fig. 11. Error analysis for two component non-linear elutions: top: L1-Error for

component 1, middle: L1-Error for component 2, bottom: CPU time of different

schemes. Parameters are given in Table 6.


However, the current scheme can also be used to deal with thenon-linearity of more complicated processes, such as multi-column, moving-bed, and periodic operations. Especially, to

develop an efficient and accurate numerical scheme for simulatedmoving bed will be the main focus of our future research work.

Nomenclature

an
Henry constants of component n, (–) cn liquid phase concentration of component n, (mol/l) d column diameter, (m) Dapp apparent dispersion coefficient in EDM, ðm2=sÞ D dispersion coefficient in LKM, ðm2=sÞ h numerical flux function Ij jth mesh interval, (–) k mass transfer coefficient, (1/s) L column length, (m) mm minmod limter function Nc number of components, (–) N number of grid cells, (–) Pe Peclet number, (–) Pl Legendre polynomial of order l (–)
qnn
solid phase concentration of component n, (mol/l)
t
time, (s) u interstitial velocity, (m/s) x dimensionless distance, x¼ z=L
z
spatial coordinate, (m)
Greek symbols

Dt
time step
Dxj
width of mesh interval Ij
Dzj
width of mesh interval Ij
E
external porosity tC characteristic time ðL=uÞ, (1/s) tD characteristic time ðD=u2Þ, (1/s) tMT characteristic time ð1=kÞ, (1/s)
fl
local basis function of order l
mn
n-th initial normalized moment
m0n
n-th central moment
Subscripts

in
injection j number of discretized cells n number of Nc components
Abbreviations

BCs
boundary conditions DG discontinuous Galerkin EDM equilibrium dispersive model LKM lumped kinetic model ODEs ordinary differential equations PDEs partial differential equations RKDG Runge–Kutta discontinuous Galerkin TVB total variation boundedness
Acknowledgment

The authors gratefully acknowledge the International MaxPlanck Research School Magdeburg and the Higher EducationCommission (HEC) of Pakistan for financial support.

Appendix A

Here, the complete derivations of moments are presented forboth equilibrium dispersive and lumped kinetic models withDirichlet and Danckwerts boundary conditions.


A.1. Equilibrium dispersive model with Dirichlet boundary

conditions

In this part, the moments of equilibrium dispersive model withDirichlet boundary conditions are derived. By taking x¼1 andc0 ¼ 1, the Eq. (18) can be written as

Cðx¼ 1,sÞ ¼1

sexp

Pe

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2þ4 Pe

L

uð1þaFÞs

r !: ðA:1Þ

Let us define

Pe¼Lu

Dapp, b1 ¼ Peð1þaFÞ

L

u: ðA:2Þ

The moment generating property of the Laplace transform is usedexploiting (e.g. Van der Laan, 1958)

mi ¼ ð�1Þi lims-0

diðsCÞ

dsi, i¼ 0,1,2,3, . . . : ðA:3Þ

Thus, the zeroth moment is given as

m0 ¼ lims-0

sCðx¼ 1,sÞ ¼ lims-0

expPe

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2þ4b1s

q� �¼ 1: ðA:4Þ

The first initial moment can be obtained from Eq. (A.3) as

m1 ¼ ð�1Þlims-0

dðsCÞ

ds¼ lim

s-0

expPe

2�

1

2


q� �b1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pe2þ4b1s

q , ðA:5Þ

Thus,

m1 ¼b1

Pe¼

L

uð1þaFÞ: ðA:6Þ

The second initial moment can be derived from the relation givenin Eq. (A.3) as

m2 ¼ ð�1Þ2 lims-0

d2ðsCÞ

ds2, ðA:7Þ

where

d2ðsCÞ

ds2¼

2b21 exp

Pe

2�

1

2


q� �ðPe2þ4a1sÞ3=2

þ

b21 exp

Pe

2�

1

2


q� �Pe2þ4a1s

:

ðA:8Þ

Thus, the second initial moment is given as

m2 ¼b2

1ð2þPeÞ

Pe3¼

2L

u3Dappð1þaFÞ2þ

L2

u2ð1þaFÞ2: ðA:9Þ

The second central moment or the variance is given by thefollowing expression:

m02 ¼ m2�m21 ¼

b21

Pe3¼

2L

u3Dappð1þaFÞ2: ðA:10Þ

Finally, the third initial moment is again obtained using Eq. (A.3)

m3 ¼ ð�1Þ3 lims-0

d3ðsCÞ

ds3¼

b31

Pe5ð6Peþ12þPe2

Þ ðA:11Þ

or

m3 ¼L3

u3ð1þaFÞ3

6Dapp

Luþ

12D2app

L2u2þ1

!: ðA:12Þ

The third central moment can be calculated from the momentsgiven above using the subsequent formula

m03 ¼ m3�3m1m2þ2m31: ðA:13Þ

Thus

m03 ¼12LD2

app

u5ð1þaFÞ3: ðA:14Þ

A.2. Equilibrium dispersive model with Danckwerts boundary

conditions

Here, the moments of the equilibrium dispersive model withDanckwerts boundary conditions are presented.

The zeroth moment is given as

m0 ¼ lims-0

A expðl1xÞþB expðl2xÞ ¼ 1: ðA:15Þ

The first initial moment is obtained as

m1 ¼b1

Pe¼

L

uð1þaFÞ: ðA:16Þ

The second initial moment is calculated as

m2 ¼2b2

1

Pe4�1þPeþ

Pe2

2þe�Pe

!ðA:17Þ

or

m2 ¼2D2

appð1þFaÞ2

u4�1þ

Lu

Dappþ

L2u2

2D2app

þe�Lu=Dapp

!: ðA:18Þ

The second central moment is given as

m02 ¼2b2

1e�Pe

Pe4ð�ePeþePePeþ1Þ

¼2L

u3Dappð1þaFÞ2 1þ

Dapp

Luðe�Lu=Dapp�1Þ

� �: ðA:19Þ

Lastly, the third initial moment is provided as

m3 ¼b3

1

Pe6ð�24þ6Peþ6Pe2

þPe3þ24e�Peþ18e�PePeÞ: ðA:20Þ

The third central moment formula based on (A.13) is given below

m03 ¼12LD2

appð1þaFÞ3

u51þ

2Dapp

Lu

� �e�Lu=Dappþ 1�

2Dapp

Lu

� �� :

ðA:21Þ

A.3. Lumped kinetic model with Dirichlet boundary conditions

This part presents the moments for the lumped kinetic modelwith Dirichlet boundary conditions. For x¼1 and c0 ¼ 1, Eq. (30)can be rewritten as

Cðx¼ 1,sÞ ¼1

sexp

b

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þ4a1s�

4a2

sþa3þ4a4

r !: ðA:22Þ

With

b¼ Pe¼Lu

D, a1 ¼ Pe

L

u, a2 ¼

L2

EDk2a

ð1�EÞ ,

a3 ¼k

ð1�EÞ , a4 ¼L2ak

ED : ðA:23Þ

The zeroth moment is given as

m0 ¼ lims-0

expb

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þ4a1s�

4a2

sþa3þ4a4

r !¼ 1: ðA:24Þ


The first initial moment is calculated as

m1 ¼ lims-0

4 a1þa2

ðsþa3Þ2

!exp

b

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þ4a1s�

4a2

sþa3þ4a4

r� �

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þ4a1s�

4a2

sþa3þ4a4

rðA:25Þ

or

m1 ¼a1a2

3þa2

a23b

¼L


The second initial moment is again derived using Eq. (A.7) with

d2ðsCÞ

ds2¼

4 a1þa2

ðsþa3Þ2

!exp

b

2�

1

2


4a2

sþa3þ4a4

r� �

8 b2þ4a1s�

4a2

sþa3þ4a4

� �3=2

þ

2a2 expb

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þ4a1s�4

a2

sþa3þ4a4

r� �

b2þ4a1s�

4a2

sþa3þ4a4

� �1=2

ðsþa3Þ3

þ

4 a1þa2

ðsþa3Þ2

!exp

b

2�

1

2


4a2

sþa3þ4a4

r� �

16 b2þ4a1s�

4a2

sþa3þ4a4

� � :

ðA:27Þ

Then

m2 ¼1

a43b3ð2a2

1a43þ4a1a2

3a2þ2a22þ2a2a3b2

þba21a4

3þ2ba1a23a2þba2

2Þ

ðA:28Þ

or

m2 ¼2LDð1þaFÞ2

u3þ

1

k

2LaFð1�EÞu

þL2

u2ð1þaFÞ2

!: ðA:29Þ

Exploiting Eq. (A.10), the second central moment is defined as

m02 ¼2

a43b3ða2

1a43þ2a1a2

3a2þa22þa2a3b2

Þ

¼

2LD 1þa1�EE

� �2

u3þ

1

k

2Lað1�EÞ2

Eu

0BB@

1CCA ðA:30Þ

or

m02 ¼2LDð1þaFÞ2

u3þ

1

k

2LaFð1�EÞu

� �: ðA:31Þ

The third initial moment is obtained using again Eq. (A.11) with

�d3ðsCÞ

ds3¼

3 4a1þ4a2

ðsþa3Þ2

!3

expb

2�

1

2


4a2

sþa3þ4a4

r� �

16 b2þ4a1s�

4a2

sþa3þ4a4

� �5=2

þ

3a2 4a1þ4a2

ðsþa3Þ2

!exp

b

2�

1

2


4a2

sþa3þ4a4

r� �

b2þ4a1s�

4a2

sþa3þ4a4

� �3=2

ðsþa3Þ3

þ

3 4a1þ4a2

ðsþa3Þ2

!3

expb

2�

1

2


4a2

sþa3þ4a4

r� �

32 b2þ4a1s�

4a2

sþa3þ4a4

� �2

þ

6a2 expb

2�

1

2


4a2

sþa3þ4a4

r� �

b2þ4a1s�

4a2

sþa3þ4a4

� �1=2

ðsþa3Þ4

þ

3a2 4a1þ4a2

ðsþa3Þ2

!exp

b

2�

1

2


4a2

sþa3þ4a4

r� �

2 b2þ4a1s�

4a2

sþa3þ4a4

� �3=2

ðsþa3Þ3

þ

4a1þ4a2

ðsþa3Þ2

!3

expb

2�

1

2


4a2

sþa3þ4a4

r� �

64 b2þ4a1s�

4a2

sþa3þ4a4

� �3=2:

ðA:32Þ

Then

m3 ¼1

a63b5ð12a3

1a63þ36a2

1a43a2þ36a1a2

3a22þ12a3

2þ12a2a33b2a1

þ12a22a3b2

þ6ba31a6

3þ18ba21a4

3a2þ18ba1a23a2

2þ6ba32þ6a2a2

3b4

þ6a2a33b3a1þ6a2

2a3b3þb2a3

1a63þ3b2a2

1a43a2þ3b2a1a2

3a22þb2a3

2Þ

ðA:33Þ

or

m3 ¼L3

u3ð1þaFÞ3

6D

Luþ

12D2

L2u2þ1

!

þ6L2ð1þaFÞað1�EÞ

u3k

2D

Lþð1�EÞu2

Lkð1þaFÞþu

� �: ðA:34Þ

Using (A.13), the third central moment is given as

m03 ¼12LD2

u5ð1þaFÞ3þ

6L2ð1þaFÞaFð1�EÞ

ku3

2D

Lþð1�EÞu2

Lkð1þaFÞ

� �ðA:35Þ

or

m03 ¼12LD2

u5ð1þaFÞ3þ

1

k

12LDð1þaFÞaFð1�EÞu3

� �þ

1

k2

6LaFð1�EÞ2

u

!:

ðA:36Þ

A.4. Lumped kinetic model with Danckwerts boundary conditions

This part discusses the derivation of moments for lumpedkinetic model with Danckwerts boundary conditions.

The zeroth moment is again given as

m0 ¼ 1: ðA:37Þ

The first initial moment corresponding again to Eq. (A.6)

m1 ¼a1a2

3þa2

a23b

¼L


The second initial moment is given as

m2 ¼e�b

a43b4ð4a1a2

3a2�2eba21a4

3þ2ebba22þebb2a2

2�4eba1a23a2

þ2ebba21a4

3þebb2a21a4

3þ2a2eba3b3þ4ebba1a2

3a2

þ2ebb2a21a2

3a2þ2a21a4

3þ2a22�2beba2

2Þ: ðA:39Þ

The second central moment is defined as, cf. (A.10)


m02 ¼2L

u3Dð1þaFÞ2 1þ

D

Luðe�Lu=D�1Þ

� �

þ1

k

2LaFEu

� �: ðA:40Þ

The third initial moment is given as

m3 ¼e�b

a63b6ð24a3

1a63þ24a3

2þ72a21a4

3a2þ72a1a23a2

2

þ12a22a3b2

þ18ba31a6

3

þ54ba21a4

3a2þ54ba21a2

3a22þ6a2eba2

3b5þ6a3

1a63ebb2

�12a22eba3b2

þ12a22eba3b3

þ6ebba31a6

3þ6a22a3b4

þebb3a31a6

3þ18ba32þ6ebba3

2

þebb3a32�24eba3

1a63

þ6ebb2a32�24eba3

2þ18ebba21a4

3a2�12a2eba33b2a1

þ12a2eba33b3a1þ18ebba1a2

3a22

þ6eba2a33b4a1þ3ebb3a2

1a43a2þ3ebb3a1a2

3a22þ18a2

1a43ebb2a2

þ18a1a23ebb2a2

2

�72eba21a4

3a2�72eba1a23a2

2þ12a2a33b2a1Þ: ðA:41Þ

The third central moment is using again (A.13) as

m03 ¼12LD2

ð1þaFÞ3

u51þ

2D

Lu

� �e�Lu=Dþ 1�

2D

Lu

� ��

þ1

k

12DLaFEð1þaFÞ

u3

D

Lue�Lu=DþF�

D

Lu

� �� þ

1

k2

6LaF3E2

u

!:

ðA:42Þ

References

Aizinger, V., Dawson, C., Cockburn, B., Castillo, P., 2000. Local discontinuousGalerkin method for contaminant transport. Adv. Water Res. 24, 73–87.

Bahhar, A., Baranger, J., Sandri, D., 1998. Galerkin discontinuous approximation ofthe transport equation and viscoelastic fluid flow on quadrilaterals. Numer.Meth. Partial Diff. Equations 14, 97–114.

Bassi, F., Rebay, S., 1997. A high-order accurate discontinuous finite elementmethod for the numerical solution of the compressible Navier-Stokes equa-tions. J. Comput. Phys. 131, 267–279.

Cockburn, B., Shu, C.-W., 1989. TVB Runge–Kutta local projection discontinuousGalerkin finite element method for conservation laws II. General framework.Math. Comput. 52, 411–435.

Cockburn, B., Hou, S., Shu, C.-W., 1990. TVB Runge–Kutta local projectiondiscontinuous Galerkin finite element method for conservation laws IV: themultidimensional case. Math. Comput. 54, 545–581.

Cockburn, B., Cockburn, C.-W., 1998. The Runge–Kutta discontinuous Galerkinfinite element method for conservation laws V: multidimensional systems.J. Comput. Phys. 141, 199–224.

Cockburn, B., Shu, C.-W., 2001. Runge–Kutta discontinuous Galerkin methods forconvection-dominated problems. J. Sci. Comput. 16, 173–261.

Danckwerts, P.V., 1953. Continuous flow systems. Chem. Eng. Sci. 2, 1–9.Felinger, A., Cavazzini, A., Dondi, F., 2004. Equivalence of the microscopic and

macroscopic models of chromatography: stochastic-dispersive versus lumpedkinetic model. J. Chromatogr. A 1043, 149–157.

Guiochon, G., Lin, B., 2003. Modeling for Preparative Chromatography. AcademicPress.

Guiochon, G., Felinger, A., Shirazi, D.G., Katti, A.M., 2006. Fundamentals ofPreparative and Nonlinear Chromatography, 2nd ed. ELsevier Academic Press,New York.

Holik, M., 2009. Application of discontinuous Galerkin method for the simulationof 3D inviscid compressible flows. In: Proceedings of the ALGORITMY, pp. 304–312.

Javeed, S., Qamar, S., Seidel-Morgenstern, A., Warnecke, G., 2011. Efficient andaccurate numerical simulation of nonlinear chromatographic processes. Com-put. Chem. Eng. 35, 2294–2305.

Koren, B., 1993. A robust upwind discretization method for advection, diffusionand source terms. In: Vreugdenhil, C.B., Koren, B. (Eds.), Numerical Methodsfor Advection–Diffusion Problems, Notes on Numerical Fluid Mechanics, vol.45. Vieweg Verlag, Braunschweig, pp. 117–138. (Chapter 5).

Kurganov, A., Tadmor, E., 2000. New high-resolution central schemes for nonlinearconservation laws and convection–diffusion equations. J. Comput. Phys. 160,241–282.

Kubin, M., 1965a. Beitrag zur Theorie der Chromatographie. Collect. Czech. Chem.Commun. 30, 1104–1118.

Kubin, M., 1965b. Beitrag zur Theorie der Chromatographie. 11. Einfluss derDiffusion Ausserhalb und der Adsorption Innerhalb des Sorbens- Korns.Collect. Czech. Chem. Commun. 30, 2900–2907.

Kucera, E., 1965. Contribution to the theory of chromatography: linear non-equilibrium elution chromatography. J. Chromatogr. A 19, 237–248.

Lapidus, L., Amundson, N.R., 1952. Mathematics of adsorption in beds, VI. Theeffect of longitudinal diffision in ion exchange and chromatographic columns.J. Phys. Chem. 56, 984–988.

Leer, B.V., 1977. Towards ultimate conservative finite difference scheme 4. A newapproach to numerical convection scheme. J. Comput. Phys. 12, 276–299.

LeVeque, R.J., 2003. Finite Volume Methods for Hyperbolic Systems. CambridgeUniversity Press, Cambridge.

Lim, Y.I., Jorgen, S.B., 2009. Performance evaluation of the conservation elementand solution element method in the SMB process simulation. Chem. Eng.Process.: Process Intensification 59, 1931–1947.

Miyabe, K., Guiochon, G., 2000. Influence of the modification conditions of alkylbonded ligands on the characteristics of reversed-phase liquid chromatogra-phy. J. Chromatogr. A 903, 1–12.

Miyabe, K., Guiochon, G., 2003. Measurement of the parameters of the masstransfer kinetics in high performance liquid chromatography. J. Sep. Sci. 26,155–173.

Miyabe, K., 2007. Surface diffusion in reversed-phase liquid chromatography usingsilica gel stationary phases of different C1 and C18 ligand densities.J. Chromatogr. A 1167, 161–170.

Miyabe, K., 2009. Moment analysis of chromatographic behavior in reversed-phase liquid chromatography. J. Sep. Sci. 32, 757–770.

Rice, R.G., Do, D.D., 1995. Applied Mathematics and Modeling for ChemicalEngineers. Wiley-Interscience, New York.

Roe, P.L., 1986. Characteristic-based schemes for the Euler equations. Ann. Rev.Fluid Mech. 18, 337–365.

Ruthven, D.M., 1984. Principles of Adsorption and Adsorption Processes. Wiley-Interscience, New York.

Reed, W.H., Hill, T.R., 1973. Triangular mesh methods for the neutron transportequation. Los Alamos Scientific Laboratory Report LA-UR, pp. 73–79.

Schneider, P., Smith, J.M., 1968. Adsorption rate constants from chromatography.A.I.Ch.E. J. 14, 762–771.

Seidel-Morgenstern, A., 1991. Analysis of boundary conditions in the axialdispersion model by application of numerical Laplace inversion. Chem. Eng.Sci. 46, 2567–2571.

Shipilova, O., Sainio, T., Haario, H., 2008. Particle transport method for simulationof multicomponent chromatography problems. J. Chromatogr. A 1204, 62–71.

Suzuki, M., Smith, J.M., 1971. Kinetic Studies by Chromatography. Chem. Eng. Sci.26, 221–235.

Suzuki, M., 1973. Notes on determining the moments of the impulse response ofthe basic transformed equations. J. Chem. Eng. Jpn. 6, 540–543.

Van der Laan, Th., 1958. Letter to the editors on notes on the diffusion type modelfor the longitudinal mixing in flow. Chem. Eng. Sc. 7, 187–191.

Zhang, P., Liu, R.-X., 2005. Hyperbolic conservation laws with space-dependentfluxes: II. General study of numerical fluxes. J. Comput. Appl. Math. 176,105–129.

Documents

Chemical Engineering Science - Transfer · chromatographic models and, hence, an efﬁcient numerical method is required to obtain accurate and physically realistic solutions. Discontinuous