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Computers and Chemical Engineering 31 (2007) 1663–1670

An agent-based solution of Lagrange equations for adsorption processes

Edgar Salgado ∗, Juan ArandaDepartamento de Bioingenierıa, Unidad Profesional Interdisciplinaria de Biotecnologıa, Instituto Politecnico Nacional,

Mexico. Av. Acueducto SN, La Laguna Ticoman, Mexico 07340, D.F., Mexico

Received 20 April 2006; received in revised form 24 January 2007; accepted 30 January 2007Available online 4 February 2007

bstract

A simple numerical method for solving the rate equation of adsorption processes is presented. The method starts with the mass balance forhe fluid phase in its lagrangian form and the corresponding equation for the solid phase; these equations are then used to specify the governingnteraction rules of discrete elements, dubbed agents. In the calculation code, ALEAP, the calculation is carried out as a series of cycles in which the

gents, representing the adsorption process, interact according to these rules. In this paper we present the results obtained for linear isotherms fromo transfer to high transfer rate. The method is surprisingly efficient for finding the right solution for the problem of dispersion with no adsorptionnd superior, in terms of computer processing time, to other methods for the simulation of the adsorption process with linear or non-linear isotherms.

2007 Elsevier Ltd. All rights reserved.

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(m

eywords: Fixed bed adsorption; Chromatography; Numerical simulation; Mo

. Introduction

The adsorption of substances onto solid surfaces is very use-ul for the recuperation and purification of products, in particulars chromatographic processes, in which a mixture is fed to a fixeded of adsorbent where, as the fluid pass through, solutes arexchanged at different rates between the fluid and the particles ofdsorbent (Fig. 1). It is the differences in adsorption and desorp-ion rates that are exploited to separate the mixture. In spite of theigh power of resolution of these separation processes, their useas been rather limited for they are not easily optimized becausehis requires that different conditions be evaluated and this, inurn, requires the availability of fast and accurate calculation

ethods. In general, there are two modeling approaches of chro-atographic processes: the plate models and the rate models.In plate models the problem is represented as a tank in series

ituation where equilibrium is usually assumed to be reached

n each tank. This was the first approach that gave acceptableesults but it is only adapted to linear processes (Gu, Tsai, &sao, 1993). Dispersion and mass transfer effects can be taken

nto account by using HETP theory following the results of vaneemter, which are in general valid for linear isotherms and pro-

∗ Corresponding author. Tel.: +52 5729 6000x56335;ax: +52 5729 6000x56338.

E-mail address: esalgado@ipn.mx (E. Salgado).

a2rocsait

098-1354/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2007.01.011

lements; Lagrange; Linear driving force

esses where mass transfer is not very slow (Guiochon, Shirazi,Katti, 1994; Velayudhan & Ladisch, 1993) and give an effec-

ive number of equilibrium plates. A relevant example of thiselations is Eq. (20) of this paper, other expressions can be foundlsewhere (Guiochon et al., 1994).

Rate models are developed from mass balances includingpecific expressions for the mass exchange between the solidadsorbent and solutes) and liquid (solvent and solutes) phases.ome exact solutions exist for the simpler models, but unfortu-ately, their applicability is limited to linear isotherms. For theost complete models, several numerical methods have been

roposed. Perhaps the most popular ones are the finite differenceethods (FDM) and finite element methods (FEM), in particularith orthogonal collocation of mesh nodes.In FDM such as Crank–Nicolson and the method of lines

MOL), derivatives in the governing equations are approxi-ated by using finite differences. Details about the stability

nd suitability of FDM can be found elsewhere (Alhumaizi,004; Botte, Ritter, & White, 2000). In general, they requireelatively large computation times for an accurate trackingf the derivatives, in particular when dealing with steeponcentration gradients. This hard-to-follow changes require

pecial schemes for controlling the numerical oscillations thatrise from the high-order interpolating polynomials involvedn the calculations (Alhumaizi, 2004). Some of the methodshat have been developed are the flux-corrected transport

1664 E. Salgado, J. Aranda / Computers and Chem

Nomenclature

a bed specific interfacial area (m−1)c solute’s concentration in liquid phase (kg/m3)c0 input solute’s concentration in liquid phase

(kg/m3)Dz axial dispersion coefficient (m2/s)I0 zero-order Bessel function of the first kindK equilibrium constantKA affinity constant (kg/m3)KS global mass transfer coefficient relative to solid

phase (m/s)n number of agents or axial position nodesN solute’s interfacial mass flux (kg/m2 s)Pe Peclet numberq solute’s concentration in the solid phase (kg/m3)qmax saturation capacity of the solid for the solute

(kg/m3)q* solute’s solid phase concentration in equilibrium

with c, (kg/m3)R dimensionless contact timet time (s)u dimensionless liquid phase concentrationv liquid interstitial velocity (m/s)z axial dimension (m)ZB bed length (m)

Greek lettersΓ equilibrium solid loadε bed porosityζ dimensionless positionθ dimensionless timeθH fluid retention timeξ dimensionless parameter (Eq. (19))

((L(d(ca

s

Kctfiic1ad

iscmiapcHw

ifidi

tstiotegcpsca

1

F

τ dimensionless parameter (Eq. (19))ω dimensionless solid phase concentration

FCT), the monotone upstream scheme for conservation lawsMUSCL), weighted essentially non-oscillatory (WENO) (Lim,e Lann, & Joulia, 2001) and the space–time CE/SE method

Lim, Chang, & Jørgensen, 2004). Additionally, when axialispersion is low the numerical dispersion is very importantRenou, Perrier, Dochain, & Gendron, 2003) and the method

an fail, unless a judicious choice of time increments is done tovoid the problem (Kaczmarski & Antos, 1996).

Orthogonal finite element methods are best suited in general,ee for instance Kaczmarski (1995) or Piatkowski, Antos and

Fig. 1. A chromatographic separation.

g

a

wsfiea

ical Engineering 31 (2007) 1663–1670

aczmarski (2003), although they are somewhat more compli-ated and they can also fail when dealing with steep concentra-ion fronts because they require then low-order polynomials orne meshes for keeping the accuracy. These solutions, however,

ncrease the calculation time substantially. Moving elementsan also be used (Kaczmarski, Mazzotti, Storti, & Morbidelli,997, Coimbra, Sereno, & Rodrigues, 2004) but this require andjustable parameter that makes the method become applicationependent (Liu, Cameron, & Bhatia, 2001).

As expected, other methods have been proposed. Fornstance, Liu et al. (2001) proposed a wavelet technique forolving adsorption problems but still based on orthogonal collo-ation. More recently, Renou et al. (2003) suggested a solutionethod for the related tubular reactor problem that solves the

nvolved phenomena (convection, dispersion and reaction) sep-rately at each time step. This simple sequencing-method (SM)erformed better than a FDM when dealing with low dispersionases (Pe = 108), obtaining the exact solution with 500 nodes.owever, the SM method failed to reproduce the inlet profileshen dispersion was relatively high (Pe = 104).Asplund and Edvinsson (1996) proposed a method consisting

n a set of point masses and compared it to the Craig solution,nding it superior. However, it still requires the estimation ofispersion as HETP, based on van Deemter results, and to datet seem it has not been applied to non-linear isotherms.

To our knowledge, no method has yet succeeded in all cases,hen a compromise must be done between the accuracy of theolution and the computation time. In general, it is acceptedhat the methods have difficulties with the numerical dispersionntroduced by the discretization and the mathematical solutionf the resulting set of equations, in particular when the concen-ration front is steep. Also, it is acknowledged that the morefficient methods are those involving the movement of the meshrid or nodes. These considerations lead us to think that a dis-rete, moving agents based method could be well suited to theroblem. In this work we compared the solutions obtained byuch a method when adsorption isotherms are linear since in thisase, there are several solutions with which to compare and sossess the accuracy of the numerical solution.

.1. Theory

Considering an adsorption process like the one shown inig. 1, mass balances for the solutes in the fluid and solid phasesive:

∂c

∂t= Dz

∂2c

∂z2 − v∂c

∂z− Na

ε(1)

nd

∂q

∂t= Na

1 − ε(2)

here c is the solute’s concentration in the fluid phase, q the

olute’s concentration in the solid phase, Dz the dispersion coef-cient, v the interstitial fluid velocity, N the mass flux of solutexchanged between the phases, a the bed specific interfacialrea, t the time and z is the axial position.

Chem

trpi

N

Hl

s

ω

wfltΓ

td

d

u

a

ω

tc

2

mciesflu

Fa

ofif

IspN

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cw

sfl

Nte

u

Tricam

H

E. Salgado, J. Aranda / Computers and

The flux between the phases depends on the external massransfer rate, the diffusion rate inside the solid and the adsorptionate within the solid. However, it is often possible to model theserocesses with the so called “linear driving force model” whichn its “solid” version reads:

= KS(q∗ − q) (3)

ere q* is the solid phase concentration in equilibrium with theiquid concentration.

The model specified by Eqs. (1)–(3) is more convenientlyolved when written in the following dimensionless form:

∂u

∂θ= 1

Pe

∂2u

∂ζ2 − ∂u

∂ζ− Γ

∂ω

∂θ(4)

∂ω

∂θ= R(ω∗ − ω) (5)

∗ = q(c)

q(c0)(6)

here u = c/c0 is the liquid phase concentration, θH = ZB/v theuid retention time, Pe = vZB/Dz the Peclet number, θ = t/θH

he dimensionless time, ζ = z/ZB the dimensionless position,= ((1 − ε)/ε)(q(c0)/c0) the equilibrium solid load, ω = q/q(c0)

he solid phase concentration and R = (KSa/(1 − ε))θH is theimensionless contact time.

The equations are solved with the following boundary con-itions:

(θ, 0) = 1,∂u

∂θ= 0 at ζ = 1

nd the following initial conditions:

(0, ζ) = 0, u(0, ζ) = 0

hat corresponds to a step input at time zero to a solute-freeolumn.

. Agent-based solution

One can imagine that in the adsorption bed fluid elements areoving and, while doing this they are exchanging solute with

ontiguous fluid elements and with solid elements. This views somewhat similar to the abstraction made in plate models,

xcept that in this case, the fluid “plates” are moving while theolid ones remain motionless, whereas in plate models staticuid plates interact. As the elements are moving and interactingnder some specific rules, we named them “agents” (Fig. 2).

ig. 2. Abstraction of adsorption processes as interacting agents. Fluid agentsre moving, solid agents are not.

oos

u

ω

ω

w

ical Engineering 31 (2007) 1663–1670 1665

The interaction rules for the defined moving agents arebtained from the dimensionless governing equations by writingrst the fluid mass balance in terms of the substantial derivativeollowing Lagrange mechanics:

Du

Dθ= 1

Pe

∂2u

∂ζ2 − Γ∂ω

∂θ(7)

n this form, the convection term is implicitly included in theubstantial time derivative and so only two terms remain: dis-ersion in the fluid phase and exchange between the phases.ext, Eq. (7) is written in its discrete version:

uk+1i − uk

i

θ=(

1

Pe

uki+1 − uk

i

ζ2 − Γωk

i

θ

)(8)

here k and k + 1 indicate time changes, while i and i + 1 indicateosition changes.

In Eq. (8) the second derivative has been approximated byentered difference of concentration that can be considered a for-ard difference of the first derivatives. As the spatial incrementζ defines agent size, the characteristic time, or time increment

hould be that available for the interaction, this is simply theuid retention time in each element so, by definition of ζ and θ:

ζ = θ = 1

n(9)

otice that this implies that fixing space increments also fixesime increments. Including Eq. (9) in (8) leads to the followingxpression of the interaction rules:

k+1i = uk

i + 1

Pe

uki+1 − uk

i

ζ− Γ ωk

i (10)

he second right-hand term of Eq. (10) specifies the interactionule for the fluid elements, whereas the third one specifies thenteraction with the solid agent. It is interesting to note thatonvection is not considered in the interactions because it islready considered in the displacement of fluid agents. This is aajor change of what is normally done.For ω we used first the following expression:

ωki = θR(ω∗

i − ωki ) (11)

owever, the use of this equation resulted in an overestimationf the mass transfer between the phases, even when the numberf agents was high, so we devised a simple predictor–correctorcheme for the solid–fluid interaction:

ωki = θR(ω∗(uk

i ) − ωki ) (12)

ˆ k+1i = uk

i − Γ ωki (13)

ˆ k+1i = ωk

i + ωki (14)

ωk+1i = θR(ω∗(uk+1

i ) − ωk+1i ) (15)

k+1i = ωk

i + ωki + ωk+1

i

2(16)

ith these equations the results were accurate.

1 Chemical Engineering 31 (2007) 1663–1670

RepnIeiw

2

am

1

2

345

asaeflfl

tanE

fluaauttdtMu

3

3

ciri

Fa

TfaatstnAsup

ldb

sstion (Levenspiel, 1962):

σ2 = θ2H

[2

Pe− 2

Pe2 (1 − e−Pe)

](17)

666 E. Salgado, J. Aranda / Computers and

The proposed method is somewhat similar to the method ofenou et al. (2003) for the tubular reactor problem, in whichach phenomenon is solved successively at each time step. Inarticular, their mesh choice is not far from ours, however, we doot separate the convection phenomenon but solve it implicitly.ndeed, we based the solution in the lagrangian form of the ratequation instead of the eulerian form that, apparently is the start-ng point of other methods, even when the equation is frequentlyritten in its lagrangian form.

.1. ALEAP encoding

Based on the notion that agents interact in time cycles char-cterized by the number of agents (according to Eq. (9)), theethod consist of the following steps:

. Calculate fluid–fluid interaction from concentration differ-ences (Peclet term in Eq. (10)).

. Calculate solid–fluid interaction with the simple predictor–corrector scheme (Eqs. (12)–(16)).

. Apply interactions results to each agent (Eq. (10)).

. Advance fluid agents to next position.

. Calculate next interaction cycle (step 1).

Profiles are directly obtained from each agent concentrationnd the breakthrough curve is obtained from the agent leaving theystem in each cycle. Because of the boundary conditions, whenn agent is leaving the system, it interacts only with the fluidlement that follows it and with the last solid agent, whereas auid agent entering the system interacts only with the precedinguid agent and the first solid agent.

The method was programmed in Delphi®, for it is an easyo use object oriented, high-level programming language thatllows a simple manipulation of dynamic arrays of objects. Weamed the code ALEAP, for Agent-based solution of Lagrangequations for Adsorption Processes.

For comparison, the rate equations (Eqs. (4)–(6)) with theuid mass balance written in its Euler form (Eq. (4)) were solvedsing a FDM in MatLab®. For this, the spatial derivatives werepproximated by centered formulae, except in the boundaries,nd the results were obtained by the Method of Lines (MOL)sing Gear’s method for solving the ODEs system (ode15s withhe BDF option in MatLab®). As when the isotherm is linearhe model has analytical solution in the time and the frequencyomains, these results were also used in the comparisons. Inhe case of non-linear isotherms, the model was solved with the

OL-GEAR for the Langmuir isotherm and the results weresed for comparison.

. Results

.1. Non-adsorptive solid: dispersion and steep fronts

When there is no adsorption in the packed bed (Γ = 0), the

oncentration front is only modified by dispersion. When thiss low, i.e. the Peclet number is high, the concentration profileemains virtually unaffected as it moves along the bed: If a stepnput is introduced, it will breakthrough virtually unchanged.

Ft

ig. 3. Breakthrough curve for a step input and negligible dispersion (Pe = 109)nd no transfer to solid phase.

his profile is not easily followed by numerical methods, inact, the MOL we programmed was unable to give an accept-ble output. In Fig. 3 the results obtained with the agents methodre shown and compared with the MOL. It can be appreciatedhat ALEAP succeeds in reproducing the curve when it is veryteep while MOL oscillates at the same discretization. In fact,he latter method did not give accurate results even when theumber of elements was increased to 800. Interestingly, whenLEAP results are inspected it is seen that virtually no disper-

ion occurred, for instance at θ = 0.99, u = 10−11 and at θ = 1.1,= 0.99999992, so the method seems to follow this difficultroblem surprisingly well.

In Fig. 4 ALEAP results are shown for systems from neg-igible dispersion to high dispersion. As can be seen, whenispersion is high the step spreads consequently and, as itecomes lower, the step input is almost maintained.

To verify that the method gave the right solution when disper-ion was perceptible, its results were compared with the normalolution by using the dispersion provided by the following equa-

ig. 4. Breakthrough curve for a step input for different Peclet numbers and noransfer to solid phase.

E. Salgado, J. Aranda / Computers and Chem

Fig. 5. Breakthrough curve for a step input with high dispersion (Pe = 500)without mass transfer to solid phase. ALEAP method, time is corrected.

Table 1Second moment of numeric solutions for Pe = 500 (σ2 = 0.004)

n ALEAP MOL

s2 s2/σ2 s2 s2/σ2

100 0.00397 0.992 0.00395 0.98850 0.00393 0.982 0.00421 1.052

TncFena

sqtih

Fw

3

q

wmC

u

w

ξ

aafa

σ

F

t

tc

FmMHiIo

25 0.00385 0.962 −0.00231 −0.57910 0.00360 0.900 0.02990 7.476

he comparison is shown in Fig. 5, in function of the agentumber for Pe = 500. It must be noted that, in this plot, theoncentrations have been modified considering agents “size”.rom the second moment of the results (Table 1), it becomesvident that the approximation is quite good, even when theumber of agents is very small: about 10% error with only 10gents.

In Fig. 6 the results for the same conditions and the MOL arehown. It is obvious that MOL does not give accurate results,uite probably because of numeric dispersion. Accuracy of

he results improve as the number of elements increases buts not as good as that of ALEAP and the calculation time isigher.

ig. 6. Breakthrough curve for a step input with high dispersion (Pe = 500)ithout mass transfer to solid phase. MOL method, time is corrected.

c(tA

Fm

ical Engineering 31 (2007) 1663–1670 1667

.2. Adsorptive solid, linear isotherm

If the equilibrium isotherm is linear:∗ = Kc (18)

here K is the equilibrium constant. In this case, the approxi-ate solution of Hiester and Vermeulen can be applied (Douglas,arta, & Yon, 1997):

= 1 −(∫ ξ

0exp(−τ − ξ)I0

(2√

τξ)

dξ

)(19)

here

= KSaθH1 − ε

ε, τ = KSa

K(t − θH)

nd I0 is the zero-order Bessel function of the first kind. Also,n exact solution can be obtained in the frequency domain, androm this the second moment in the time domain can be obtaineds (Guiochon et al., 1994):

2 = θ2H

[2

Pe− 2

RΓ

(Γ

1 + Γ

)2]

(20)

rom this variance and the solute’s retention time:

R =(

1 + 1 − ε

εK

)θH (21)

he breakthrough curve can be obtained as a cumulative normalurve.

Comparison between the different solutions are shown inig. 7. As can be seen, they are very similar although someinor differences are perceived. Consequently, both methods,OL and ALEAP, converge to the proper solution in this case.owever, it should be noted that, as the MOL requires a good

ntegration scheme it takes substantially more time than ALEAP.t could be thought that the difference is due to the fact thatne code was being interpreted (MOL), whereas the other was

ompiled (ALEAP). However, the difference is so importantseconds and minutes versus second fractions) that this explana-ion can be ruled out. We believe that the superior performance ofLEAP comes from the absence of polynomial approximations

ig. 7. Comparison among ALEAP (n = 200), MOL (n = 200), Hiester and Ver-eulen and Normal solution (σ2 = 0.0102). Pe = 2500, Γ = 1 and R = 50.

1668 E. Salgado, J. Aranda / Computers and Chemical Engineering 31 (2007) 1663–1670

F(

asstted

rtb

pbct

3

q

Faa

FRa

wabn

alCGts

3

ig. 8. Breakthrough curve for a step input with increasing transfer ratesPe = 2500, Γ = 1 and R variable).

nd matrix manipulations. Additionally, when the equations areolved by the MOL, numerical dispersion causes a small, butpurious breakthrough before the fluid can actually flow throughhe bed. This is not observed with the agents method. Evenhought it is arguably that the false breakthrough has someffect on the results, it is certainly a clear evidence of numericalispersion that, eventually, might be important.

In Fig. 8, ALEAP results are shown for different exchangeates. The curves behave as expected. It is however clear thathe number of agents must be increased as the exchange rateecomes more important.

Finally, in Fig. 9 ALEAP profiles at different times are com-ared to those predicted by Eq. (19) when behavior is controlledy the transfer between the phases, according to Eq. (20). Asan be seen, they are almost undistinguishable what supportshe fidelity of the ALEAP solution.

.3. Adsorptive solid, Langmuir isotherm

A commonly used non-linear isotherm is that of Langmuir:

= qmaxc

KA + c(22)

ig. 9. Concentration profiles at different times as given by ALEAP (continuous)nd Hiester and Vermeulen (dotted) when dispersion is low (Pe = 2500, R = 50nd Γ = 1) for 200 agents.

m

u

wt

1

t

n

w

Γ

wi

ig. 10. Breakthrough curves for non-linear Langmuir isotherm. Pe = 2500,= 50, a value of 1 for the parameter KA/(KA + c0) indicates linear conditions,

s the value increases so does non-linearity (n = 200).

here qmax is the saturation capacity of the solid for the solutend KA is the affinity constant. Even if the equation is an hyper-ole, when the liquid concentration is very small the behavior isearly linear. We have taken the parameter:

KA

KA + c0(23)

s a relative measure of the non-linearity: a value of 1 indicatesinearity while higher values indicate non-linear conditions.onsidering this, a comparison was made between the MOL-EAR and the ALEAP, both with 200 nodes. As a reference,

he normal solution for the linear case was also plotted. Results,hown in Fig. 10, are nearly indistinguishable in all cases.

.4. Stability issues

Combining Eqs. (6), (9), (10), (12) and (18), the base of theethod for the linear case, we obtain:

k+1i =

[ n

Pe

]uk

i+1 +[

1 − 2n

Pe− ΓR

n

]uk

i

+[ n

Pe

]uk

i−1 + ΓR

nωk

i (24)

hen von Newman stability analysis is applied on this equationhe following conditional stability is defined:

− 2n

Pe− ΓR

n≥ 0 (25)

his inequality cannot be solved for n but can be arranged as:

≤ Pe

2

(1 − ΓR

n

)(26)

hich requires that:

R ≤ n (27)

hat gives an approximate lower limit for n. From the former,t can be observed that when Γ , R or its product is small in

E. Salgado, J. Aranda / Computers and Chem

Fw5

c

n

wutuiFemlatwaflbsm

Fif

cbs

4

esflbeDhta

1

2

cbol

A

fi

R

ig. 11. Concentration profiles in function of agent number. Linear isothermith Γ = 1, Pe = 2500 and R = 50. Stability limits, from Eqs. (27) and (28) are0 < n < 125.

omparison with Pe:

≤ Pe

2(28)

hat gives an approximate upper value for n. The exact val-es are that given by Eq. (26). It is somewhat counterintuitivehat increasing the number of elements makes the methodnstable, as for most of the other methods reducing step sizesncreases precision and stability. For illustrating this point, inig. 11 ALEAP concentration profiles are shown near the liquidntrance when integrating close to stability limits and at an inter-ediate, stable value. Instability is clearly observed in the upper

evel and less obviously in the lower case, manifesting itself asn impossible solution with u > 1. So, Eqs. (27) and (28) predicthe limits reasonably well even when there is no adsorption inhich case only Eq. (28) apply. It is interesting to note that, from

n agent point of view, when the exchanged solute between the

uid agents travels faster than the agents themselves the solutionecomes unstable and, when the exchange of solute between theolid and the fluid is faster than the time of the interaction theethod also becomes unstable.

ig. 12. Concentration profiles in function of agent number. Non-linearsotherm, KA/(KA + c0) = 10, with Γ = 0.1, Pe = 2500 and R = 50. Stability limits,rom Eqs. (27) and (28) are 50 < n < 125.

A

A

B

C

D

G

G

K

K

ical Engineering 31 (2007) 1663–1670 1669

Although von Newman analysis is not valid for the non-linearase, the limit seem to hold for the upper case as shown in Fig. 12ut not as good for the lower case as the method still gives a stableolution.

. Conclusions

A new numerical scheme is presented for solving rate mod-ls of adsorption and chromatography. The approach is verytraightforward considering the convection of fluid as movinguid agents and the mass transfer between agent fluids andetween fluid and solid agents as interaction rules, which arexplicitly derived from model equations in its lagrangian form.epending on agent’s size and fluid velocity, the fluid agentsave a well-defined time for interacting between them and withhe immobile solid agents. The two main differences with otherpproaches are:

. Time and spatial increments are the same in dimensionlessunits.

. Convection is not included explicitly in the equations but itis in the algorithm.

The proposed method follows unexpectedly well the con-entration front when there is no adsorption, a case that cannote solved efficiently by a MOL. It is also faster than MOL forbtaining an accurate solution when there is adsorption withinear or non-linear (Langmuir) isotherms.

cknowledgements

The authors thank COFAA and SIP of the IPN for theirnancial support of this work.

eferences

lhumaizi, K. (2004). Comparison of finite difference methods for the numeri-cal simulation of reacting flow. Computers and Chemical Engineering, 28,1759–1769.

splund, S. E., & Edvinsson, R. K. (1996). A method for the rapid simulation ofpreparative liquid chromatography. Computers and Chemical Engineering,20, 507–516.

otte, G. G., Ritter, J. A., & White, R. E. (2000). Comparison of finite differenceand control volume methods solving differential equations. Computers andChemical Engineering, 20, 2633–2654.

oimbra, M. C., Sereno, C., & Rodrigues, A. (2004). Moving finite elementmethod: Applications to science and engineering problems. Computers andChemical Engineering, 28, 597–603.

ouglas, M., Carta, G., & Yon, C. (1997). Adsorption and ion exchange. InR. Perry & D. W. Green (Eds.), Perry’s chemical engineers’ handbook (pp.16-1–16-66). New York: McGraw-Hill.

u, T., Tsai, G. J., & Tsao, G. T. (1993). Modeling of nonlinear multicomponentchromatography. Advances in Biochemical Engineering, 49, 45–71.

uiochon, G., Shirazi, S. G., & Katti, A. M. (1994). Fundamentals of preparativeand nonlinear chromatography. Boston: Academic Press.

aczmarski, K. (1995). Use of orthogonal collocation on finite elements withmoving boundaries in the simulation of non-linear multicomponent chro-

matography. Influence of fluid velocity variation on retention time in LC anHPLC. Computers and Chemical Engineering, 20, 49–64.

aczmarski, K., & Antos, D. (1996). Fast finite difference method for solv-ing multicomponent adsorption-chromatography models. Computers andChemical Engineering, 20, 1271–1276.

1 Chem

K

L

L

L

L

P

R

670 E. Salgado, J. Aranda / Computers and

aczmarski, K., Mazzotti, M., Storti, G., & Morbidelli, M. (1997). Modelingfixed-bed adsorption columns through orthogonal collocations on movingfinite elements. Computers and Chemical Engineering, 21, 641–660.

evenspiel, O. (1962). Chemical reaction engineering. New York: John Wileyand Sons.

im, Y. I., Le Lann, J. M., & Joulia, X. (2001). Moving mesh generation for track-ing a shock or steep moving front. Computers and Chemical Engineering,

25, 653–663.

im, Y.-I., Chang, S.-C., & Jørgensen, S. B. (2004). A novel partial differ-ential algebraic equation (PDAE) solver: Iterative space–time conservationelement/solution element (CE/SE) method. Computers and Chemical Engi-neering, 28, 1309–1324.

V

ical Engineering 31 (2007) 1663–1670

iu, Y., Cameron, I. T., & Bhatia, S. K. (2001). A wavelet-based adaptive tech-nique for adsorption problems involving steep gradients. Computers andChemical Engineering, 25, 1611–1619.

iatkowski, W., Antos, D., & Kaczmarski, K. (2003). Modeling of preparativechromatography processes with slow intraparticle mass transport kinetics.Journal of Chromatography A, 988, 219–231.

enou, S., Perrier, M., Dochain, D., & Gendron, S. (2003). Solution of the

convection–dispersion–reaction equation by a sequencing method. Comput-ers and Chemical Engineering, 27, 615–629.

elayudhan, A., & Ladisch, M. R. (1993). Plate models in chromatography:Analysis and implications for scale-up. Advances in Biochemical Engineer-ing, 49, 123–144.