a review of mathematical modeling of fixed-bed columns for ... · (ruthven, 1984). the gas–solid...

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chemical engineering research and design 9 2 ( 2 0 1 4 ) 961–988

Contents lists available at ScienceDirect

Chemical Engineering Research and Design

j ourna l h omepage: www.elsev ier .com/ locate /cherd

review of mathematical modeling of fixed-bed columnsor carbon dioxide adsorption

ohammad Saleh Shafeeyan, Wan Mohd Ashri Wan Daud ∗, Ahmad Shamiriepartment of Chemical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia

a b s t r a c t

Carbon dioxide emissions must be stabilized to mitigate the unfettered release of greenhouse gases into the atmo-

sphere. The removal of carbon dioxide from flue gases, an important first step in addressing the problem of CO2emissions, can be achieved through adsorption separation technologies. In most adsorption processes, the adsor-

bent is in contact with fluid in a fixed bed. Fixed-bed column mathematical models are required to predict the

performance of the adsorptive separation of carbon dioxide for optimizing design and operating conditions. A com-

prehensive mathematical model consists of coupled partial differential equations distributed over time and space

that describe material, energy, and the momentum balances together with transport rates and equilibrium equa-

tions. Due to the complexities associated with the solution of a coupled stiff partial differential equation system, the

use of accurate and efficient simplified models is desirable to decrease the required computational time. The simpli-

fied model is primarily established based on the description of mass transfer within adsorption systems. This paper

presents a review of efforts over the last three decades toward mathematical modeling of the fixed-bed adsorption

of carbon dioxide. The nature of various gas–solid equilibrium relationships as well as different descriptions of the

mass transfer mechanisms within the adsorbent particle are reviewed. In addition to mass transfer, other aspects of

adsorption in a fixed bed, such as heat and momentum transfer, are also studied. Both single- and multi-component

CO2 adsorption systems are discussed in the review.

© 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

ling; Mass transfer; Linear driving force approximation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962

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Keywords: Adsorption; Carbon dioxide; Fixed bed; Mode

ontents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Overview of the prediction of adsorption column dynam3. Development and analysis of a mathematical model . . .

3.1. Fluid phase material balance . . . . . . . . . . . . . . . . . . . . . . 3.2. Complexity of kinetic models . . . . . . . . . . . . . . . . . . . . .

3.2.1. Local equilibrium model . . . . . . . . . . . . . . . . . . 3.2.2. Mass transfer resistance models . . . . . . . . .

3.3. Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Gas phase energy balance . . . . . . . . . . . . . . . . 3.3.2. Solid-phase energy balance . . . . . . . . . . . . . . .3.3.3. Wall energy balance. . . . . . . . . . . . . . . . . . . . . . .

3.4. Momentum balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Corresponding author. Tel.: +60 3 79675297; fax: +60 3 79675319.E-mail addresses: [email protected] (M.S. Shafeeyan), ashriReceived 16 March 2013; Received in revised form 22 July 2013; Accept

263-8762/$ – see front matter © 2013 The Institution of Chemical Engittp://dx.doi.org/10.1016/j.cherd.2013.08.018

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@um.edu.my (W.M.A. Wan Daud).ed 19 August 2013neers. Published by Elsevier B.V. All rights reserved.

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962 chemical engineering research and design 9 2 ( 2 0 1 4 ) 961–988

1. Introduction

Concerns over the gradual increase in the atmospheric con-centration of CO2 and its impact on climate change haveprompted a global research effort to capture CO2 from pointsource emissions and stabilize its concentration in the atmo-sphere (Gomes and Yee, 2002; Grande and Rodrigues, 2008;Plaza et al., 2007; Shafeeyan et al., 2010). The most impor-tant sources of CO2 emissions are power plants that generateelectricity from fossil fuels (coal, oil, and natural gas) (Dantaset al., 2011a; Grande et al., 2008; Grande and Rodrigues, 2008;Kikkinides et al., 1993; Mulgundmath et al., 2012; Park et al.,2002; Shafeeyan et al., 2012). Therefore, it is critical to sepa-rate and recover carbon dioxide from the flue gases emittedby power plants to avoid excess CO2 emissions (Chou andChen, 2004; Ko et al., 2005; Mulgundmath et al., 2012). Vari-ous separation techniques, such as liquid solvent absorption,membrane separation, cryogenic techniques, and adsorptionover solid sorbents, are increasingly used to reduce CO2 emis-sions (Gomes and Yee, 2002; Takamura et al., 2001). At present,the most widely used technology for the removal of CO2 fromgaseous mixtures is amine absorption (Delgado et al., 2006b;Leci, 1996). However, this process is energy-intensive duringthe regeneration of solvent and is also plagued by exten-sive corrosion of the process equipment (Chue et al., 1995;Gray et al., 2004, 2005; Ko et al., 2005; Shafeeyan et al., 2011).It is therefore important to explore economical and energy-efficient alternative approaches for CO2 separation (Grandeet al., 2008; Xu et al., 2005).

Recently, it was reported that the cost associated withCO2 capture can be reduced below the cost of conventionalabsorption with liquid solvents by using adsorption separa-tion technologies (Ho et al., 2008; Radosz et al., 2008). Severaltechnological advances in the field of CO2 capture by adsorp-tion have been developed around the world, demonstratingthe attractiveness of this technique for post-combustion treat-ment of flue gas (Dantas et al., 2011a,b; Grande et al., 2008). Twomain adsorption technologies are viewed as feasible for CO2separation and purification on a large scale: pressure/vacuumswing adsorption (PSA/VSA) and temperature swing adsorp-tion (TSA) (Chue et al., 1995; Clausse et al., 2004; Plaza et al.,2009, 2011). Recent developments have demonstrated that PSAis a promising option for separating CO2 due to its ease ofapplicability over a relatively wide range of temperature andpressure conditions, its low energy requirements, and its lowcapital investment costs (Agarwal et al., 2010b; Cen and Yang,1985; Delgado et al., 2006b; Gomes and Yee, 2002). Many stud-ies concerning CO2 removal from various flue gas mixturesby means of PSA processes have been addressed in the lit-erature (Agarwal et al., 2010b; Chaffee et al., 2007; Chou andChen, 2004; Chue et al., 1995; Grande et al., 2008; Ho et al., 2008;Kikkinides et al., 1993; Ko et al., 2003; Mulgundmath et al., 2012;Na et al., 2001; Reynolds et al., 2005; Sircar and Kratz, 1988;Xiao et al., 2008). Prior to the design of an adsorption process,selecting an appropriate adsorbent with high selectivity andworking capacity, as well as a strong desorption capability, iskey to separating CO2. As a result, a wide variety of adsorbents,such as activated carbons, synthetic zeolites, carbon molecu-lar sieves, silicas, and metal oxides, have been investigated inrecent years for this purpose (Chue et al., 1995; Dantas et al.,2011a,b; Moreira et al., 2006; Plaza et al., 2011; Xu et al., 2005).

The design of an appropriate adsorption process requires

the development of a model that can describe the dynam-ics of adsorption on a fixed bed with the selected adsorbent

(Dantas et al., 2011a,b; Delgado et al., 2006a; Lua and Yang,2009). The absence of an accurate and efficient adsorptioncycle simulator necessitates the use of data from experimen-tal units to develop new processes. This empirical design ofan adsorption column through extensive experimentation onprocess development units tends to be expensive and timeconsuming (Siahpoosh et al., 2009). A predictive model usingindependently established equilibrium and kinetic parame-ters may provide, in principle, a method of estimating thecolumn dynamic capacity without extensive experimentation.A fixed-bed column mathematical simulation that consid-ers all relevant transport phenomena is therefore requiredto obtain a better understanding of the behavior of newadsorbents during the adsorption/desorption cycles and foroptimization purposes. Moreover, these models are capableof estimating the breakthrough curve and temperature pro-file for a certain constituent in the bulk gas at all locationswithin the packed column. This experimentally verified modelis then used to conduct an extensive study to understand theeffects of various process parameters on the performance ofthe PSA cycle. These are the main reasons why the mathemati-cal modeling of adsorption processes has attracted a great dealof attention among researchers.

In general, prediction of column dynamics behaviorrequires the simultaneous solution of a set of coupled partialdifferential equations (PDEs) representing material, energy,and momentum balances over a fixed bed with the appro-priate boundary conditions (Hwang et al., 1995). Because thesimultaneous solution of a system of PDEs is tedious andtime consuming, the use of simplified models capable of sat-isfactorily predicting fixed-bed behavior is desirable. Manyattempts have been made to evaluate and develop simplify-ing assumptions to decrease computational time and facilitateoptimization studies. A review of the literature reveals thedevelopment of simplifying assumptions mainly on the repre-sentation of mass transfer phenomena within the adsorbentparticles as an alternative pathway to simplify fixed-bedadsorption calculations. Modeling and optimization of thefixed-bed adsorption of CO2 has developed over the past threedecades and is still of great interest to investigators. Thisreview presents a fairly extensive survey of previous stud-ies on the mathematical modeling of the CO2 adsorptionprocess in a packed column. Various models for gas–solidadsorption equilibria as well as different descriptions of themass transfer mechanisms within the adsorbent particle arereviewed. In addition to concentration variation, other aspectsof adsorption in a fixed bed, such as temperature and pressurevariations, are also studied. The purpose of this study was toinvestigate the mathematical models capable of simulatingthe dynamic behavior of the fixed-bed adsorption of carbondioxide.

2. Overview of the prediction of adsorptioncolumn dynamics

In most adsorption processes, the adsorbent is in contact witha fluid in a packed bed. An understanding of the dynamicsbehavior of such systems is therefore required for rationalprocess design and optimization (Rutherford and Do, 2000a).The dynamics behavior of an adsorption column system canbe classified based on the nature of the gas–solid equilibriumrelationship of fluid constituents and the complexity of the

mathematical model required for describing the mechanismby which the mass transfer from the fluid to the solid phase

chemical engineering research and design 9 2 ( 2 0 1 4 ) 961–988 963

oigttetleshcTotppadafGnh

tfdlpaaegsomtmtttramgtmdatai

3m

Tptprte

ccurs (Ruthven, 1984). The gas–solid adsorption equilibriumndicates the limiting capacity for solute separation from theas phase into the solid phase. It is the most important processhat controls the dynamics behavior of a packed column sohat the general nature of a mass transfer zone is determinedntirely by the equilibrium isotherm. Therefore, due to varia-ions in the composition/temperature with respect to time andocation within the adsorption column and the consequentffects on the adsorption equilibrium relation, a comprehen-ive gas–solid equilibrium model is needed. Several authorsave reported experimental evidence of these effects in aolumn packed with microporous adsorbents (Carta, 2003).he complexity of the mathematical model, in turn, dependsn the concentration level, the choice of rate equation, andhe choice of flow model (Ruthven, 1984). In addition, tem-erature changes may also affect the concentration profiles,articularly for high-concentration feeds in which the heat ofdsorption generates thermal waves in both axial and radialirections. Therefore, apart from the mass transfer effects ondsorption rate, the effects of heat generation and heat trans-er in the adsorbent bed must also be considered (Rezaei andrahn, 2012). Moreover, the axial pressure along the bed mayot be constant. As a consequence, a momentum balance alsoas to be included in the model.

Table 1 provides a comprehensive classification scheme ofhe summary of the fixed-bed column mathematical modelsor carbon dioxide adsorption developed over the last threeecades. All of the models assume that the gas phase fol-

ows the ideal gas law. The flow pattern is described by thelug flow or axially dispersed plug-flow model. It is furtherssumed that the radial gradients of concentration and, wherepplicable, temperature and pressure are negligible (with thexception of models 4 and 20). The assumption that the radialradient is negligible has been widely accepted in many othertudies (Jee et al., 2002; Kim et al., 2006, 2004). The majorityf the models reviewed here include the effects of the finiteass transfer rate, resulting in a theoretical representation

hat more closely approaches a real process. Most of the afore-entioned models use a linear driving force approximation

o describe the gas–solid mass transfer mechanism. Some ofhese models consider the effects of heat generation and heatransfer in the adsorbent bed, which may affect the adsorptionates. Moreover, in modeling the non-isothermal operation ofdsorption processes occurring in packed beds, it is also com-only assumed that the heat transfer resistance between the

as and the solid phases is negligible and that they reachhermal equilibrium instantaneously. With the exception of

odels 15–17, 21, 24, 26–27, 29–30, 31, and 33, the pressurerop across the adsorbent bed is neglected, and the column isssumed to operate at constant pressure. Most of the adsorp-ion equilibrium is described using non-linear isotherms suchs the Langmuir isotherm or a hybrid Langmuir–Freundlichsotherm; only rarely have linear isotherms been used.

. Development and analysis of aathematical model

he fixed-bed column mathematical models are used toredict the transient behavior of the concentration andemperature profiles for any defined changes in the initialarameters such as feed concentration, temperature, and flowate. A complete mathematical model capable of describing

he dynamics behavior of a fixed-bed adsorption system isstablished based on a set of fairly complex partial differential

and algebraic equations (PDAEs) constructed from conser-vation of mass, energy, and momentum and augmentedby appropriate transport rate equations and equilibriumisotherms (Hwang et al., 1995). The models used to represent aPSA process differ mainly in the form of the mass transfer rate,the form of the equilibrium isotherm, thermal effects, andthe pressure drop along the bed. General descriptions of theabove-mentioned items are presented in the following subsec-tions. Many mathematical models for gas–solid adsorption inan adsorption column have been published over the past fewdecades, and there is still interest in developing a descriptionof the dynamic evolution of such systems (Afzal et al., 2010;Leinekugel-le-Cocq et al., 2007).

3.1. Fluid phase material balance

The transient gas phase component mass balance, whichincludes the axial dispersion term, convection flow term,accumulation in the fluid phase, and source term caused bythe adsorption process on the adsorbent particles, can be rep-resented by the following equation for a differential controlvolume of the adsorption column (Ruthven, 1984; Yang, 1987):

−Dzi∂2ci∂z2

+ ∂∂z

(uci) +∂ci∂t

+(

1 − εbεb

)�p

∂qi∂t

= 0 (1)

where ci represents the adsorbate concentration in the fluidphase; z is the distance along the bed length; u is the fluidvelocity; t denotes time; εb is the bed void fraction; �p is the par-ticle density; qi denotes average concentration of componenti in adsorbent particle, which forms a link between the fluidand solid-phase mass balance equations; and the effects of allmechanisms that contribute to axial mixing are lumped into asingle effective axial dispersion coefficient, Dzi, which can beestimated using the following correlation (Da Silva et al., 1999;Ruthven, 1984; Wakao and Funazkri, 1978; Welty et al., 2000;Yang, 1987):

εbDziDmi

= 20 + 0.5ScRe (2)

where Dmi is the molecular diffusivity of component i and Scand Re are the Schmidt and Reynolds numbers, respectively.

The above equation, Eq. (1), is used to find the distributionof gas composition along the bed. Assuming no radial depend-ence of concentration and solid loading, ci and qi, representcross-sectional average values (these variables are functionsof t and z).

The well-known Danckwert’s boundary conditions for adispersed plug flow system can be assumed as follows(Khalighi et al., 2012; Wehner and Wilhelm, 1956):

Dzi∂ci∂z

∣∣∣z=0

= −u|z=0 (ci∣∣z=0− − ci|z=0) (3)

∂ci∂z

∣∣∣z=L

= 0 (4)

where ci|z=0− represents the feed composition for componenti and L is the bed length.

In the above model, Eq. (1), if the flow pattern is representedas plug flow, axial dispersion can be neglected, and thereforethe term −Dzi

(∂2ci/∂z

2)

can be dropped, reducing Eq. (1) to a

first-order hyperbolic equation. This is a reasonable approx-imation, particularly for large industrial units, for which the

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Table 1 – Summary of the dynamics models for fixed-bed adsorption of carbon dioxide.

Model assumptions

No.Equilibriumrelationship

Flow patternMass transferrate model

Heat effects Others Application Solution method Results and comments Ref.

1 Linearequilibriumisotherm

Plug flow Local equilibriummodel

Isothermal No radial variation inconcentrationNegligible pressure dropTrace system*

PSA separationof carbon dioxidefrom a He–CO2mixture usingsilica gel

Analytical resultsfrom a linearmathematicalmodel obtainedby the method ofcharacteristics

The model provided aqualitative or semiquantitative processdescription. Due toneglecting the effects ofmass transfer resistancesome of the detailedbehavior differed fromexperimental results

Shendalman andMitchell (1972)

2 A hybridLangmuir–Freundlichisotherm

Plug flow Local equilib-rium/lineardriving force(LDF)approximationmodel

Non-isothermal No radial variations inconcentration andtemperatureThermal equilibriumbetween the fluid andparticles

Separation ofcoal gasificationproductscontaining H2,CO, CH4, H2S,and CO2 by PSAusing activatedcarbon

The model wassolved using animplicit finitedifferencemethod whichwas stable andconvergent

Poor comparison withexperimental data forthe predictiveequilibrium model. Themajor discrepancy wasin the CO2concentration. Theresults of the LDF modelwere in fair agreementwith the experimentaldata. Mass transfercoefficient for CO2 wasdetermined empirically

Cen and Yang(1985)


Axial dispersedplug flow

LDFapproximationwithnon-constantcoefficient

Isothermal Negligible radial gradientof concentrationNegligible pressure dropTrace system

PSA separationof carbon dioxidefrom a He–CO2mixture usingsilica gel

The solution tothe modelequations wasobtained byorthogonalcollection andusing finitedifferencemethods withconsistentresults

The theoretical curvesbased on theassumption of inversedependence of the masstransfer coefficient withthe pressure provided agood representation ofthe experimental results

Raghavan et al.(1985)



Pore diffusionmodel

Non-isothermal Negligible radialconcentration gradientRadial temperature profilein the column/uniformtemperature over thecolumn cross-sectionNegligible axial pressuregradientConstant temperature ofthe column wall

Theoretical andexperimentalstudies on theCO2 capture in acolumn packedwith activatedcarbon particles

Analyticalsolution wasperformed in theLaplace domainunder thecondition of asemi-infinitecolumn

Thecentral-axis-thermalwaves measured atvarious axial locationsin the column were ingood agreement withthose predicted

Kaguei et al.(1989, 1985)

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5 A hybridLangmuir–Freundlichisotherm

Plug flow Local equilibriummodel.Pore/surfacediffusion models

Non-isothermal Negligible radial gradientsin temperature andconcentrations. Thermalequilibrium between thefluid and particlesNegligible pressure dropsin the bed

Separation of gasmixturescontaining CO2,CH4, and H2(one-third eachby volume) byPSA usingactivated carbon

The models weresolvednumerically byemploying finitedifferencemethod

The Knudsen plussurface diffusion modelprovided the best fitwhen compared to theexperimental data. Dueto the assumption ofinfinite pore diffusionrate, the ILE modelpredicted a laterbreakthrough plus alower concentration forCO2

Doong and Yang(1986)

6 Langmuirisotherm

Plug flow LDF approxima-tionmodel with acycletime-dependentcoefficient

Isothermal Negligible radialconcentration gradientNegligible pressure drop

PSA separation ofa CO2 (50%)–CH4(50%) mixtureusing a carbonmolecular sieve

The model wassolved using animplicitbackward finitedifferencescheme, whichwas both stableand convergent

The model predictionswere reasonable and theaverage differencebetween the modelprediction andexperimental result waswithin 3.0%

Kapoor and Yang(1989)

7 Langmuirisotherm

Plug flow Local equilibriummodel

Non-isothermal(adiabatic)

No radial variations inconcentration andtemperatureThermal equilibriumbetween the fluid andparticlesNegligible pressure drop

Separation ofcarbon dioxidefrom binary gasmixtures(CO2/N2,CO2/CH4, andCO2/H2) usingBPL carbon and5A zeolite

A set of PDEs wasreduced to ODEsand solved byusing thenumericaltechnique offinite differences

The adiabaticsimulation of theblowdown step showedthat an isothermalityassumption isinadequate for processdesign. However, itcould be an excellenttool for predicting thecolumn behavior andtrends in a semiquantitative manner

Kumar (1989)

8 Langmuirisotherm

Plug flow LDFapproximationModel

Non-isothermal Negligible radialtemperature andconcentration gradientsThermal equilibriumbetween the gas and solidphasesNegligible pressure dropthrough the bed

CO2 capture froma mixture of N2(90%)–CO2 (10%)by PSA using 5Amolecular sieve

The non-linearrate equationswere solvedusingRunge–Kutta–Merson method.Adsorbateconcentrationand temperatureprofiles werepredicted usingan implicitbackwarddifferenceapproximation

A comparison ofexperimentalbreakthrough andtemperature profileswith model predictionsrevealed that the modelreproduced theexperimental datasatisfactorily, whichindicates that theassumptions the modelis based on are valid forthis system

Mutasim andBowen (1991)

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Table 1 (Continued)

Model assumptions




9 Langmuirisotherm


LDFapproximationmodel

Isothermal No radial variations inconcentrationNegligible pressuregradient across the bed

Investigation ofadsorption anddesorptionbreakthroughbehaviors of COand CO2 onactivated carbon

A set of PDEs wassolved by themethod oforthogonalcollection. Theresulting set ofODEs was solvednumerically inthe time domainby using DGEARof theInternationalMathematicaland StatisticalLibrary (IMSL)which employsGear’s stiffmethod withvariable orderand step size

The experimentaladsorption anddesorption curves werepredicted fairly well bythe LDF model and thepressure dependentmass transfercoefficients calculatedfrom a singlecomponent systemprovided a reasonablygood representation ofadsorption anddesorption data for amulti-componentsystem

Hwang and Lee(1994)

10 Langmuirisotherm/Idealadsorbedsolution theory(IAST)

Plug flow LDFapproximationmodel

Non-isothermal No radial concentrationand temperaturegradientsNegligible axial pressuregradient

Fixed-bedadsorption of aN2 (85%)–CO2(15%) mixtureusing a of X-typezeolite

A set ofdifferentialequation withthe initial andboundaryconditions wassolved by usingthe solver LSODA

A comparison betweenconcentration andtemperature historycurves with theoreticalresults revealed that thepresented model couldpredict the dynamicbehavior of theadsorption bed, eventhough a slightdeviation was observedafter the maximumpoint

Kim et al. (1994)

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11 Langmuirisotherm

Plug flow LDFapproximationmodel withlumped masstransfercoefficient

Non-adiabatic,adiabatic, andisothermal

Negligible radial velocity,temperature, andconcentration gradientsNegligible pressuregradient across the bed

Fixed-bedadsorption ofcarbon dioxide(with helium asthe carrier gas)on activatedcarbon

A set of PDEs wassolved by thenumericalmethod of lines.The resulting setof ODEs wassolved by usingthe subroutineDIVPAG of theIMSL library,while thenon-linearalgebraicequation wassolved by usingthe subroutineDNEQNF of thesame library

The model provided agood representation ofthe experimentalbreakthrough andtemperature curves.Since the mass transfercoefficients weredetermined by fittingthe experimental data,the disadvantage of thismodel is thedetermination of a newvalue for the effectivemass transfercoefficient for each run

Hwang et al.(1995)

12 ExtendedLangmuir–Freundlichisotherm

Plug flow LDFapproximationmodel with asingle lumpedmass transfercoefficient

Non-isothermal Negligible radial gradientsin temperature andconcentrationsThermal equilibriumbetween the fluid andparticlesNegligible pressure dropsin the bed

Separation of abinary mixtureH2 (70%)–CO2(30%) by PSAusing zeolite 5A

A set of PDAEsrepresenting thepacked columnwere solved by aflux correctedthird-orderupwind method.Numericaloscillation,which oftenappears when aconvectionequation issolved, iseliminated by theflux correctedscheme

The predicted valuesmatched significantlywith the experimentalresults at shorteradsorption time. Theerrors at longeradsorption time wereattributed to a partialbreakthrough of masstransfer zone duringcocurrentdepressurization and/orblowdown/purge steps

Yang et al. (1995)

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Table 1 (Continued)

Model assumptions




13 Langmuirisotherm


Isothermal No radial variation inconcentrationNegligible pressure dropTrace system

Removal andconcentration ofCO2 dilute gasfrom air by PSAusing three typesof commercialmolecular-sievezeolites (13X, 5A,and 4A)

A set ofequationsdescribing thesystem wassolved by Euler’smethod

Good agreementbetween the model andthe experimentalresults was obtainedparticularly for values ofbetween the ratio offeed/enriched productflow rates = 3 and 6.Also, at the point theratio of feed/leanproduct flow rates

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LDFapproximationmodel with asingle lumpedmass transfercoefficient

Non-isothermal Negligible radial gradientsof concentration, pressureand temperatureThermal equilibriumbetween the fluid andparticlesPressure drop along thebed was calculated by theErgun equation

Layered-bed PSAseparation of acoke oven gascontaining H2,CH4, CO, N2, andCO2 usingactivated carbonand zeolite 5A

A set of coupledPDEs was solvedusing a finitedifferencemethod. Thespatialdimension wasdiscretized byusing asecond-ordercentraldifference and asecond-orderbackwarddifference for thesecond-orderand thefirst-order spacederivatives,respectively

In spite of the frozensolid-phase model, theutilized LDF modelcould predict a transientvariation of the effluentstream duringpressurization anddepressurization stepsand simulated results ofthe dynamic modelagreed well with thePSA experimentalresults. Theexperimental dataresulted in slightlyhigher recovery thanpredicted (4% error)

Lee et al. (1999)

16 Langmuir–Freundlichisotherm



Non-isothermal No radial variations intemperature, pressure,and concentrationThe Ergun equation wasused to estimate thepressure drop

Packed bedadsorption ofcarbon dioxide,nitrogen, andwater onmolecular sieve5A

A set of partialdifferentialequations wassolved usingfinite differencesand Newmans’smethod

The model provided areasonable fit toexperimentaladsorption data.However, comparing theexperimental data withthe model predictionsuggested that a 2Dmodel is required foraccurate simulation ofthe average columnbreakthroughconcentration

Mohamadinejadet al. (2000)

17 Langmuirisotherm


Local equilibriummodel/LDFmodel based onpore diffusion

Non-isothermal No radial concentration,pressure and temperaturegradientsPressure distribution wasdescribed by the ErgunequationThermal equilibriumbetween the gas andparticles

Hightemperaturecarbon dioxideadsorption onhydrotalciteadsorbent

The equationswere solved inthe gPROMSmodelingenvironment.The spatialdiscretizationmethod oforthogonalcollocation onfinite elementswas employed

The LDF model wasfound to give a gooddescription of theadsorption anddesorption dataespecially for high feedCO2 concentrations. TheILE model failed to givean adequate descriptionof the desorptionkinetics

Ding and Alpay(2000)

970

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ical

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g resea

rch a

nd

desig

n

9 2

(

2 0

1 4

)

961–988

Table 1 (Continued)

Model assumptions




18 Langmuirisotherm


Isothermal Negligible radial gradientof concentrationNegligible pressure drop

CO2 recoveryfrom boilerexhaust gas(containing CO2and N2) by PSAusing Na-X andNa-A typezeolites

A set ofequationsdescribing thesystem wasdiscretized inspace and theresulting set ofordinarydifferentialequations with avariable timestep was solved

Both simulation andexperimental resultsshowed the same trendof the recoveryefficiency and the CO2concentration of therecovery gas withrespect to the variationof the feed gas flow rate

Takamura et al.(2001)

19 ExtendedLangmuirisotherm



Negligible gradients inradial concentration andtemperatureNegligible pressure drop

CO2 recoveryfrom a flue gas(containing 83%N2, 13% CO2, and4% O2) by PSAusing zeolite 13X

A MATLABfunction basedon sequentialquadraticprogramming(SQP) methodwas used to solvethe constrainednon-linearprogrammingoptimizationproblem

The analysis ofbreakthrough curvesshowed good agreementwith simulation data.However, analysis oftemperature changes inthe adsorption bedsrevealed somediscrepancy betweensimulations andexperiments

Choi et al. (2003)

20 Ideal adsorptionsolution theory(IAST)

Non-Darcianflow model (2Dflow)


Non-isothermal Variations in temperature,concentration, andvelocity along the radialdirection of column

Carbon dioxideadsorption froma mixture of(CO2, N2, andH2O) in a columnpacked withzeolite 5A

A set of coupledPDEs wasdiscretized byfirst- orsecond-orderdifferences intime and spatialdimensions. Theset of discretizedfinite differenceequations wassolvedsimultaneouslyby the implicitmethod ofNewman

The model prediction ofbreakthrough curvesdefinitely matched theobtained experimentaldata. The temperatureprofile results of 2Dmodel also estimatedthe experimental datafairly well. The fewdegree discrepancybetween the model andexperimental data wasattributed to predictionof heat transfercoefficients

Mohamadinejadet al. (2003)

chem

ical

eng

ineerin

g resea

rch a

nd

desig

n

9 2

(

2 0

1 4

)

961–988

971

21 O’Brien–Myersisotherm


LDFapproximationmodel with anadjustable masstransfercoefficient

Adiabatic,near-adiabatic,and isothermal

Negligible radialand angulargradients inconcentration,temperature andvelocityThermalequilibriumbetween the gasand the adsorbentThe momentumequationrepresented byErgun’s equation

Adsorption of a30% CO2–10%C2H6 mixture innitrogen (inertcarrier gas) byTSA usingAmbersorb 600adsorbent

To solve a set ofdifferentialequations, thenumerical method oflines was retained.For the spatialdiscretization, afinite volumesscheme with twostaggered grids waschosen: one for thevelocity and one forthe temperature,pressure andconcentrations (gasand adsorbedphases). The set ofPDAEs obtained wasintegrated byemploying anintegrator (DASPK2.0)

For low mass transfercoefficients (

972

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nd

desig

n

9 2

(

2 0

1 4

)

961–988

Table 1 (Continued)

Model assumptions




23 Langmuirisotherm



Isothermal No radial variations inconcentration

Prediction of thedynamics of CO2breakthrough ina carbonmonolith column

The system ofpartialdifferential-algebraicequations(PDAEs) wascoded in gPROMSsoftware toobtain anumerical model

The model that includedthe detailed structure ofthe monolith providedan excellent match toexperimental resultswhereas the modelbased on the equivalentsingle channel approachincorrectly predictedhigher separationefficiencies at differentconcentrations

Ahn andBrandani (2005)

24 MultisiteLangmuirisotherm


A double LDFapproximationmodel to expressmacropore andmicroporediffusionequations


Negligible heat, mass, andmomentum transport inthe radial direction of thecolumnPressure drop wasdescribed using Ergunequation

Separation of amixture of CH4(55%)–CO2 (45%)by VSA–PSAtechnology usinga Takeda carbonmolecular sieve3 K

The fixed-bedmodel wassolved ingPROMS (PSEEnterprise,London, U.K.)using orthogonalcollocationmethod on finiteelements with 25finite elementsand 2 interiorcollocationpoints perelement

The proposed modelwas able to predict wellthe behavior of thebinary mixture in afixed bed. Darken’s lawprovided a successfulcorrection of themicropore diffusioncoefficients in thenon-linear regions ofthe isotherms

Cavenati et al.(2005)

25 Langmuirisotherm



Isothermal Negligible radialconcentration gradientNegligible pressure drop

Adsorption ofcarbon dioxidefrom mixtures ofCO2 diluted inhelium onto ahydrotalcite-likeAl–Mgcompounds in afixed bed

A set of PDAEswas solved usingthe PDECOLpackage in theFORTRANlanguage, whichis based on themethod oforthogonalcollocation offinite elementsfor partialdifferentialequations indouble precision

The dispersion andmass transfercoefficients werecalculated by theoreticalcorrelations and themodel described quitevery well the dynamicsof CO2 adsorption in afixed bed

Moreira et al.(2006)

chem

ical

eng

ineerin

g resea

rch a

nd

desig

n

9 2

(

2 0

1 4

)

961–988

973



General LDFmodel withconstant diffu-sivity/modifiedLDF model withconcentration-dependentdiffusivity

Non-isothermal Negligible radialconcentration andtemperature gradientsThermal equilibriumbetween fluid andparticlesThe pressure drop alongthe bed was calculated bythe Ergun equation

PSA separationof a mixture ofCH4/CO2(50/50 vol%)using Takeda 3Acarbonmolecular sieve

The gPROMSmodeling tooldeveloped byProcess SystemsEnterprise Ltd.was used toobtain thesolution of thedynamicsimulation of themodel

Compared with the LDFmodel with constantdiffusivity theexperimentalbreakthrough curvesand adsorptiondynamics, waswell-predicted by usingthe proposednon-isothermal andnon-adiabatic modifiedLDF model withconcentration-dependentdiffusivity

Kim et al. (2006)

27 ExtendedLangmuirisotherm



Non-isothermal Negligible gradients inradial concentration andtemperatureThermal equilibriumbetween the gas andparticlesThe momentum balancerepresented by Ergun’sequation

Fixed-bedadsorption ofbinary gasmixtures(CO2/He, CO2/N2,and CO2/CH4)onto silicalitepellets, sepiolite,and a basic resin

The completemodel wassolvednumericallyusing thePDECOL programthat usesorthogonalcollocation onfinite elementstechnique

The model describedadequately thebreakthrough curves forthe experiments withlow CO2 concentration,whereas the error washigher for the runs withhigher CO2concentration. Thequality of the predictionwas improvedintroducing interactionfactors in this model,because of the stronginteractions betweenthe adsorbed CO2molecules

Delgado et al.(2006a,b, 2007a,b)

28 Langmuirisotherm


A double LDFapproximationmodel

Non-isothermal Negligible radialvariations inconcentration andtemperatureThermal equilibriumbetween the gas and solidphasesNegligible pressure drop

Adsorption of amixture of CH4(70%)–CO2 (30%)in a columnpacked withbidisperseadsorbent (5Azeolite)

Orthogonalcollocations wereused as a spatialdiscretizationmethod. Theresultingordinarydifferential-algebraic systemof equations wassolved by theDDASPGintegrationsubroutine (IMSLlibrary), based onthe Petzold–GearBDF method

The presented modelfitted well withexperimental data, forboth outlet compositionand bed temperature.This indicated that theapproximationsproposed in this studygive a goodrepresentation of theintraparticle masstransfer

Leinekugel-le-Cocq et al.(2007)

974

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9 2

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2 0

1 4

)

961–988

Table 1 (Continued)

Model assumptions




29 MultisiteLangmuirisotherm


A rigorousdescription formacroporediffusion modelas well as an LDFapproximationfor microprediffusions

Non-isothermal No radial variations intemperature, pressure,and concentrationThe pressure drop wasdescribed by Darcy’s law

Low-concentrationCO2 removalfrom flue gasstreams byelectric swingadsorption usingmonolith

The mathematicalmodel was solvedusing gPROMS 3.01(PSE Enterprise,United Kingdom)

The results showed thatboth adsorption anddesorption curves werenot symmetrical, whichcannot be accuratelydescribed by theemployed mathematicalmodel. This wasattributed to thechannels with differentsizes in the boundariesof the honeycomb

Grande andRodrigues (2008)

30 Virial isothermmodel


A double LDFapproximationmodel to expressmacropore andmicroporediffusionequations

Non-isothermal No mass, heat or velocitygradients in the radialdirectionThermal equilibriumbetween the gas and solidphasesThe momentum balancerepresented by Ergun’sequation

Separation of gasmixturescontaining CO2,CH4, CO, N2 andH2 by PSA usingactivated carbon

The numericalsolutions wereperformed withgPROMS (PSEEnterprise, UK) usingthe orthogonalcollocation on finiteelements as thenumerical method

A good agreement wasobserved between theexperimental and thepredicted concentrationhistory at the end of thecolumn and also thetemperature evolutionwithin the column. Adeviation between theadsorbed amountobtainedexperimentally and thepredicted by the modelwas lower than 10%

Grande et al.(2008)

31 Toth isotherm Axial dispersedplug flow


Non-isothermal(adiabatic/non-adiabatic)

Negligible radialconcentration andtemperature gradientsThe momentum balancerepresented by Ergun’sequation

Fixed-bedadsorption ofbinary gasmixtures(CO2/He andCO2/N2) usingactivated carbonand zeolite 13X

The mathematicalmodel was solvedusing thecommercialsoftware gPROMS(Process SystemEnterprise Limited,UK). The orthogonalcollocation methodon finite elementswas used with sixfinite elements andthree collocationpoints in eachelement of theadsorption bed

The model acceptablyreproduced theexperimental data forthe different feedconcentrations andtemperatures. By usingthe Toth equation forpure components, thesimulated curve fittedwell the experimentaldata, whereas adeviation was observedfor multicomponent

Dantas et al.(2009, 2011a,b)

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)

961–988

975

32 Virial isothermmodel/MultisiteLangmuirisotherm


A rigorousdescription ofmacropore andmicroporediffusion models

Isothermal Negligible radialconcentration gradientConstant velocity withinthe column

Adsorption ofCO2 onpitch-basedactivated carbon

Simulations ofthe presentedmathematicalmodel wereperformed ingPROMS (PSEEnterprise, UK)using theorthogonalcollocation onfinite elements

An exponentialdependence of themicropore diffusivitywith temperature wasshown to correctlydescribe theexperimental datawithin the temperaturerange studied

Shen et al. (2010)

33 Dual-siteLangmuirisotherm

Plug flow LDFapproximationmodel withlumped masstransfercoefficient

Non-isothermal No radial variations intemperature, pressure,and concentrationThe gas and the solidphases are in thermalequilibriumThe pressure drop alongthe bed was calculated bythe Ergun equation

CO2 capture froman 85% to 15%N2–CO2 feedmixture usingPSA cycles/CO2capture from asynthesis gasfeed mixture(55% H2 and 45%CO2) using PSAcycles

A completediscretizationapproach thatuses the finitevolume methodwas applied inboth spatial andtime domains,and the resultinglarge-scalenon-linearprogrammingproblem (NLP)was solved usingan interior pointNLP solver

The results indicatedthe potential of thesuperstructureapproach to predict PSAcycles with up to 98%purity and recovery ofCO2. Verifications of theaccuracy of thediscretization schemeshowed this approach isreasonably accurate incapturing the dynamicsof PSA systemsgoverned by hyperbolicPDAEs and steepadsorption fronts, andcan be used for PSAsystems with efficientNLP solvers

Agarwal et al.(2010a,b)

34 Langmuirisotherm


LDFapproximationmodel forexternal fluidfilm masstransfer/arigorousdescription ofpore diffusionmodel forintraparticlemass transfer

Non-isothermal Negligible radialtemperature andconcentration gradientsNegligible pressure drop

Fixed-bedadsorption ofcarbon dioxidefrom a CO2–N2gas mixture (10%CO2 in 90% N2)using zeolite 13X

– The curvature of theconcentrationbreakthrough curveincluding the notedtailing was predictedwith good accuracy.Energy profile waspredicted with loweraccuracy. However, thepoint at which thetemperaturebreakthrough occurswas estimated withgood accuracy which isthe most importantfactor for industrialapplications

Mulgundmathet al. (2012)

∗ The adsorbable component is present at only low concentration in an inert carrier.


term representing the axial dispersion is very small comparedto the convection term (Simo et al., 2008).

As a result of pressure and temperature variations, gas den-sity and hence gas velocity vary along the bed. The followingequation expressing the overall material balance for the bulkphase in the adsorption column is used to find the velocitydistribution through the bed (Ko et al., 2005):

−Dzi∂2C

∂z2+ ∂ (uC)

∂z+ ∂C

∂t+

(1 − εb

εb

)�p

n∑i=1

∂qi∂t

= 0 (5)

where C is the total concentration in the bulk phase and n isthe number of components.

Applying the ideal gas law (ci = yiP/RTg), the overall massbalance equation can be expressed as follows (Ahn et al., 2001;Lee et al., 1999):

−Dzi∂2P

∂z2+ ∂P

∂t+ P ∂u

∂z+ u ∂P

∂z+ PTg

[−Dzi

∂2(1/Tg)

∂z2+ ∂(1/Tg)

∂t+ u ∂ (1/Tg)

∂z

]

−2DziTg∂(1/Tg)

∂z

∂P

∂z+

(1 − εb

εb

)�pRTg

n∑i=1

∂qi∂t

= 0(6)

where yi is the mole fraction of component i in the gas phase,P is the total pressure, Tg is the gas temperature and R is theuniversal gas constant.

3.2. Complexity of kinetic models

The term ∂qi/∂t in Eq. (1) represents the overall rate of masstransfer for component i (at time t and distance z) averagedover a particle. The mass balance for an adsorbent particleyields the adsorption rate expression, which may be writtenas

∂qi∂t

= f (qi, ci) (7)

For an isothermal system, the expressions for the concen-tration profiles in both phases,

[ci(z, t), qi(z, t)

], is given by the

simultaneous solution of Eqs. (1) and (7), subject to the initialand boundary conditions imposed on the column. For non-isothermal systems, an energy balance must also be takeninto account. In this case, all equations are coupled because,in general, both the equilibrium concentration and the ratecoefficients are temperature dependent.

Although the mass transfer rate expression, Eq. (7), waswritten here as a single equation, it commonly consists of a setof equations comprising one or more diffusion equations withtheir associated boundary conditions. It is worth noting that akinetic model is basically a mass balance that involves differ-ent variables describing mass transfer mechanisms within theadsorbent particle (Chahbani and Tondeur, 2000). A variety ofmass transfer kinetic models with different degrees of com-plexity can be found in the literature. Mass transfer kineticmodels can be classified into two main categories based onthe assumption of local equilibrium or the existence of masstransfer resistance between the adsorbent particle and thefluid phase. They are introduced in the following subsections.

3.2.1. Local equilibrium modelThis model is expressed by the existence of an instantaneouslocal equilibrium (ILE) between the solid and fluid concentra-

tions. If the mass transfer rate is relatively rapid, one mayassume that local equilibrium is always maintained between

the gas phase and the adsorbed phase within the particle atall points in the column. In other words, the local equilibriumassumption relates to the negligible effect of mass transferresistance through the particles. As a result, it is assumedthat, in this model, the adsorptive quantity is equal to theequilibrium adsorptive quantity:

∂qi∂t

= ∂q∗i

∂t(8)

In the above equation q∗i

is the adsorbed-phase con-centration of species i in equilibrium with the fluid phaseconcentration.

Equilibrium theory is aimed at identifying the generalfeatures of the dynamic response of the column withoutdetailed calculations, as the overall pattern of the responseis governed by the form of the equilibrium relationshiprather than by kinetics. However, in practice, because axialmixing and mass transport resistances are neglected, break-through curves predicted by equilibrium models fail to givequantitatively satisfactory results and give only approximaterepresentations of the behavior observed (Hwang et al., 1995).Although such systems are not common in practice, theiranalytical solution can provide useful information about theprocess dynamics and system behavior, which is quite valu-able for preliminary design and analysis, leading to a greaterunderstanding of the behavior of more complex systems.

Based on the classification presented in Table 1, the sim-plest case to consider is an isothermal system with no axialdispersion in which a trace-level component is adsorbed froma non-adsorbing carrier gas with the assumption of negligiblemass transfer resistance (model 1). Because the adsorbablecomponent is present at a low concentration (trace levelassumption), variation in the fluid velocity across the masstransfer zone is considered to be negligible, and the superfi-cial velocity calculated based on the flow at the inlet can betreated as a constant. For these systems, the differential gasphase mass balance, Eq. (1), reduces to

u∂ci∂z

+ ∂ci∂t

+(

1 − εbεb

)�p

∂q∗i

∂t= 0 (9)

Analytical determination of the concentration front in apacked adsorption column is limited to a few simple cases.Using a linear equilibrium isotherm, it is possible to obtain ananalytical solution for isothermal or adiabatic systems withnon-disperse behavior. Assuming a constant pattern profile,an improved analytical solution was obtained for the systemthat used a non-linear equilibrium isotherm such as the Lang-muir isotherm, but there are some assumptions that restrictits application (Yang, 1987). In the case of the PSA processes,the first analytical solution of the equilibrium model wasobtained by Shendalman and Mitchell (1972) for the sepa-ration of CO2 from a He–CO2 mixture using silica gel as anadsorbent (model 1). They implemented the linear equilibriumrelation for isothermal adsorption of a trace-level componentin a one-dimensional system with no axial dispersion. Theassumption of linear equilibrium for one adsorbable compo-nent permitted them to obtain a solution to the equationsby the characteristic method. However, the experimental datarevealed rather large deviations from the equilibrium theorypredictions, suggesting that effects of mass transfer resistance

are likely important. Chan et al. (1981) extended the localequilibrium theory to the separation of two-component


gtTrmaeHesi

3Tsfttftsbtpptwacoa

pbatoomedbapmptf(tdstcwibp

3taas

aseous mixtures via the PSA process in an isothermal sys-em in which both the carrier and the impurity are adsorbed.hey analytically studied the assumption of instant equilib-ium between the adsorbate and the adsorbent when the

ore strongly adsorbed component is present at a trace levelnd the linear isotherms for both components were consid-red. Later, Fernandez and Kenney (1983) and Knaebel andill (1985) solved the model for binary mixtures with lin-ar adsorption isotherms, and Serbezov (1997) extended theolution to multi-component mixtures with linear adsorptionsotherms.

.2.2. Mass transfer resistance modelshe modeling of transport equations in a packed columntrongly depends on the mechanism by which the mass trans-er from the fluid to the solid phase occurs. In fact, equilibriumheory is confined to systems in which the adsorptive selec-ivity depends on a difference in equilibrium and is not usefulor systems in which separation is based on kinetic selec-ivity (Hassan et al., 1986). An example of kinetic adsorptiveeparation is the separation of a CO2/CH4 mixture using a car-on molecular sieve, in which the separation is achieved byhe large difference in diffusion rates between the two com-onents (Diagne et al., 1996). Therefore, in modeling a realractical non-equilibrium packed column, the effects of massransfer resistance between the fluid and the particle andithin the particle must be considered (Hwang et al., 1995). In

n attempt to construct a theoretical representation that morelosely approximates a real process, researchers have devel-ped dynamic models that consider effects due to dispersionnd a finite mass transfer rate.

The mass transfer of solute from bulk gas into the solidhase is driven by equilibrium isotherms, whereas the massalance equation inside the adsorbent particle depends on thedsorbent structure. At the microscopic level, the diffusion ofhe adsorbate into the adsorbent particles before adsorptionnto the micropore surface (or adsorption onto the macrop-re surface, if no micropores exist) involves different transferechanisms. The adsorbate molecules initially must cross the

xternal film surrounding each adsorbent particle and theniffuse through and along the porous structure of the adsor-ent, as illustrated in Fig. 1. Depending on the specific systemnd the conditions, any one of the three different types ofotential resistance to mass transfer may be dominant, andore than one resistance may be significant. These three

otential resistances are the external fluid film resistance andhe intraparticle diffusional resistances, the macropore dif-usional resistance and the micropore diffusional resistanceLeVan et al., 1999). In general, the mass transfer processhrough such a heterogeneous system can be expressed byetailed models identifying the film resistance around theolid particles and macropore/micropore resistances insidehe particles. The most general case in adsorption pro-ess modeling is the case of macropore/micropore diffusionith external film resistance. Consequently, the discussion

n the following subsections will focus on the case of theidisperse pore diffusion model with clearly distinct macro-ore/micropore diffusion.

.2.2.1. External fluid film resistance. External fluid film massransfer is defined based on the concentration differencecross the boundary layer surrounding each adsorbent particle

nd is strongly affected by the hydrodynamic conditions out-ide the particles (as characterized by the system’s Sherwood,

Reynolds, and Schmidt numbers) (LeVan et al., 1999). Indeed,it is supposed that the mass transfer resistance between thebulk phase and the macro-porous gas phase is localized toan external film around the adsorbent particles. By assumingsteady-state conditions at the fluid–solid interface, the masstransfer rate across the external film is supposed to be equalto the diffusive flux at the particle surface (Farooq et al., 2001).In fact, because no accumulation of adsorbates is allowed,the film transfer and macropore diffusion can be treated assequential steps, and mass conservation assumption is appli-cable. It can be expressed as the following equation (Jin et al.,2006; LeVan et al., 1999):

∂qi∂t

= 3kfiRp

(ci − cpi

∣∣(t,Rp)

)= 3

RpεpDpi

∂cpi

∂R

∣∣∣(t,Rp)

(10)

where kfi is the external film mass transfer coefficient, Rp isthe macroparticle radius, cpi is the adsorbate concentrationin the macropore, which is a function of radial position inthe particle, εp is the adsorbent porosity, Dpi is the macrop-ore diffusivity, and R is the distance along the macroparticleradius.

The external film mass transfer coefficient, kfi, around theparticles can be estimated from the following correlation,which is applicable over a wide range of conditions (Wakaoand Funazkri, 1978):

Sh = 2kfiRpDmi

= 2 + 1.1 Sc1/3Re0.6 (11)

In most gas adsorption studies, the intraparticle diffusionalresistance is normally much greater than the external fluidfilm resistance (intraparticle transport of the adsorbate is theslower step). Therefore, it is reasonable to assume negligiblegas-side resistance and simulate adsorption systems based ona diffusion model (Carta and Cincotti, 1998; Farooq et al., 2001;Raghavan et al., 1985). An accurate kinetic model that accountsfor the intraparticle diffusional resistances can provide reli-able simulations of kinetically controlled PSA processes.Indeed, neglecting intraparticle mass transfer kinetics leads tosignificant deviations from the exact solution (Chahbani andTondeur, 2000).

3.2.2.2. Macropore diffusional resistance. Diffusion in suffi-ciently large pores (macro- and mesopores) such that thediffusing molecules escape from the force field of the adsor-bent surface is often referred to as macropore diffusion (orpore diffusion). Depending on the relative magnitude of thepore diameter and the mean free path of the adsorbatemolecules, transport in a macropore can occur by differentmechanisms (Karger and Ruthven, 1992). For gas phase dif-fusion in small pores at low pressure, when the molecularmean free path is much greater than the pore diameter, Knud-sen diffusion dominates the transport mechanism. In thiscase, the resistance to mass transfer mainly arises from colli-sions between the diffusing molecules and the pore wall. TheKnudsen diffusivity (Dki) is independent of pressure and variesonly weakly with temperature as follows (Karger and Ruthven,1992; Ruthven, 1984; Suzuki, 1990; Yang, 1987):

√

Dki = 9700 rp

T

M(12)


Fig. 1 – Schematic diagram showing various resistances to the transport of adsorbate as well as concentration profilesthrough an idealized bidisperse adsorbent particle demonstrating some of the possible regimes: (1) + (a) rapid mass transfer,equilibrium through particle; (1) + (b) micropore diffusion control with no significant macropore or external resistance;(1) + (c) transport controlled by the resistance at the micropore interior; (1) + (d) controlling resistance at the surface of themicroparticles; (2) + (a) macropore diffusion control with some external resistance and no resistance within themicroparticle; (2) + (b) all three resistances (micropore, macropore, and film) are significant; (2) + (c) diffusional resistancewithin the macroparticle with some external film resistance together with a restriction at the micropore interior (2) + (d)diffusional resistance within the macroparticle in addition to a restriction at the micropore mouth with some external filmresistance.

where rp is the mean macropore radius in cm, T isthe temperature, and M is the molecular weight of theadsorbate.

By contrast, when the molecular mean free path is smallrelative to the pore diameter, the bulk molecular diffusionwill be the dominant transport mechanism and can be esti-mated from the Chapman–Enskog equation (Bird et al., 2002;Ribeiro et al., 2008b; Ruthven, 1984; Sherwood et al., 1975)for binary systems or the Stefan–Maxwell equation for multi-component systems (Suzuki, 1990). In the case of moleculardiffusion, the collisions between diffusing molecules are themain diffusional resistance. For the intermediate case, both

mechanisms are of comparable significance, and thus thecombined effects of the Knudsen and the molecular diffusion

constitute the rate-controlling mechanism. The effectivemacropore diffusivity (Dp) is obtained from the Bosanquetequation (Grande et al., 2008; Yang, 1987):

1Dpi

= �(

1Dki

+ 1Dmi

)(13)

where � is the pore tortuosity factor.As discussed above, in macropore diffusion, transport

occurs within the fluid-filled pores inside the particle (LeVanet al., 1999; Ruthven, 1984). In this situation, a differential

mass balance equation for species i over a spherical adsor-bent particle may be written as follows (Do, 1998b; Gholami


a2

gtta(mcGQ

b

ε

o

3wutSvosiisdtttte

e1

o

nd Talaie, 2009; Jin et al., 2006; LeVan et al., 1999; Qinglin et al.,003a):

∂cpi

∂t+

(1 − εp

εp

)∂qi∂t

= 1R2

∂

∂R

(R2Dpi

∂cpi

∂R

)(14)

This equation is used to determine the composition of theas penetrating macropore volume at each radial position. Inhe above equation, qi is the average adsorbed-phase concen-ration of component i in the micropore, which is related to thedsorbate flux at the micropore mouth by either Eq. (18) or Eq.19), depending on the expression of the dominant transport

echanism in the micropore. The corresponding boundaryonditions for macropore balance are as follows (Do, 1998b;holami and Talaie, 2009; Jin et al., 2006; LeVan et al., 1999;inglin et al., 2003a,b):

∂cpi

∂R

∣∣∣(t,0)

= 0 (15)

The external fluid film resistance can be reflected in theoundary condition as follow:

p Dpi∂cpi

∂R

∣∣∣(t,Rp)

= kfi(

ci − cpi∣∣(t,Rp)

)(16)

r cpi(t, Rp) = ci for no external film resistace (when pureadsorbate is fed to the column) (17)

.2.2.3. Micropore diffusional resistance. In very small pores inhich the pore diameter is not much greater than the molec-lar diameter, the adsorbing molecules can never escape fromhe force field of the pore wall, even at the center of the pore.uch a mechanism, in which transport may occur by an acti-ated process involving jumps between adsorption sites, isften called micropore diffusion (also known as solid diffu-ion) (LeVan et al., 1999; Ruthven, 1984). In this situation, thentraparticle gas phase is neglected, and diffusion through its supposed to be null (Chahbani and Tondeur, 2000). Con-equently, the material balance equation in the microporesoes not contain any gas phase accumulation term. As illus-rated in Fig. 1, transport in the micropores may occur byhree different mechanisms: barrier resistance (confined athe micropore mouth), distributed micropore interior resis-ance, and the combined effects of both resistances (Cavenatit al., 2005; Farooq et al., 2001; Srinivasan et al., 1995).

The mass transfer rate across the micropore mouth can bexpressed by the following equations (Buzanowski and Yang,989; Jin et al., 2006; LeVan et al., 1999; Qinglin et al., 2003b):

∂qi∂t

= kbi(q∗i − qi ) when the gas diffusion is controlled by the

barrier resistance (18)

r = 3 D�i∂qi

∣∣∣ when the distributed micropore interior

Rc ∂r (t,Rc)

resistance is dominant (19)

where kbi is the barrier transport coefficient, Rc is the micropar-ticle radius, qi is the distributed adsorbate concentration in themicropore, D�i is the micropore diffusivity of component i, andr is the distance along the microparticle radius.

The strong dependence of the micropore diffusivity onconcentration can be expressed using Darken’s equation(Cavenati et al., 2005; Chihara et al., 1978; Do, 1998a; Kawazoeet al., 1974; Khalighi et al., 2012; Ruthven et al., 1994):

D�i = D∞�id ln(pi)d ln(qi)

∣∣∣∣T

(20)

where D∞�i

is the micropore diffusivity of component i at infi-nite dilution and pi is the partial pressure of component i,which is in equilibrium with the adsorbed concentration inthe micropore.

The temperature dependence of the corrected diffusivityand the surface barrier mass transfer coefficients follows anArrhenius-type form, as described by the following (Cavenatiet al., 2005; Gholami and Talaie, 2009; Grande and Rodrigues,2004, 2005; Khalighi et al., 2012; Qinglin et al., 2003b)

D∞�i = D0�iexp(

− EaiRgTs

)(21)

kbi = k0biexp(

− EbiRgTs

)(22)

where D0�i

and k0bi

are the temperature-independent pre-exponential constants, Rg is the universal gas constant, Ts isthe solid temperature, and Eai and Ebi are the activation energyof micropore diffusion and the activation energy of surfacebarrier resistance for component i, respectively.

When the resistance distributed in the micropore interiordominates the transport of species i, the mass balance equa-tion for micropore diffusion is the following (Jin et al., 2006;LeVan et al., 1999; Qinglin et al., 2003b):

∂qi∂t

= 1r2

∂

∂r

(r2D�i

∂qi∂r

)(23)

The corresponding boundary conditions for the micropar-ticle balance are as follows (Jin et al., 2006; LeVan et al., 1999;Qinglin et al., 2003b):

∂qi∂r

∣∣∣(t,0)

= 0 (24)

When a combination of barrier and distributed microporeinterior resistaces is dominant, the barrier resistance can bereflected in the boundary condition as follows:

3Rc

D�i∂qi∂r

∣∣∣(t,Rc)

= kbi(q∗i − qi∣∣(t,Rc)

) (25)

or qi(t, Rc) = q∗i for no barrier resistace (26)

The adsorbed amount at a certain time for component ibased on particle volume can be calculated by volume inte-gration of the concentration profiles in the macropores andmicropores (Jin et al., 2006; Khalighi et al., 2012; Qinglin et al.,2003a,b, 2004):

∫ Rp ∫ Rp

qi = εp

3

R3p 0cpiR

2dR + (1 − εp) 3R3p 0

qiR2dR (27)


where:

qi =3

R3c

∫ Rc0

qir2dr (28)

In most kinetically selective processes, the controllingresistance for the uptake of sorbates is typically diffusion inthe micropores (Cavenati et al., 2005; Farooq et al., 2001; Lamiaet al., 2008). Micropore diffusion can contribute significantlyto the overall intraparticle mass transport, primarily due tothe higher concentration of the adsorbed phase, althoughthe mobility of molecules in the adsorbed phase is gener-ally much smaller than in the gas phase (Kapoor and Yang,1990). Doong and Yang (1986) reported that micropore dif-fusion contributed as much as 50% to the total flux in theactivated carbon pores during the PSA separation of CO2, H2,and CH4 (model 5). Liu and Ruthven (1996) gravimetricallymeasured the diffusion of CO2 in a carbon molecular sievesample and concluded that the data were consistent with thebarrier resistance model at lower temperatures, while the dis-tributed micropore interior resistance model adequately fittedthe data at higher temperatures. They found that the resultssuggested a dual resistance model with varying importanceof the two components depending on pressure and temper-ature. In another study, Rutherford and Do (2000b) fitted theuptake of CO2 in a sample of a carbon molecular sieve (Takeda5A) using a model based on distributed diffusional resistancein the micropore interior. The model simulation results werein fair agreement with the experimental data. Qinglin et al.(2003a,b) investigated the diffusion of carbon dioxide in threesamples of carbon molecular sieve adsorbent. They indicatedthat transport of gases in the micropores of these samplesis controlled by a combination of barrier resistance at themicropore mouth followed by a distributed pore interior resis-tance acting in series. The proposed dual resistance modelwas shown to be able to fit the experimental results overthe entire range covered in that study. Cavenati et al. (2005)studied diffusion of CO2 on the carbon molecular sieve 3 Kand reported that the initial difficulty associated with diffu-sion due to the surface barrier resistance was not observedin the uptake of CO2. A successful description of diffusionin micropores was achieved using the distributed microporeinterior resistance model without the need for the surface bar-rier resistance model at the mouth of the micropore (model24). They attributed the absence of surface barrier resistanceto performing the activation protocol at a higher temperature.Shen et al. (2010) studied diffusion of CO2 on pitch-based acti-vated carbon beads using diluted breakthrough experimentsperformed at different temperatures. To simulate the break-through curves, they developed a mathematical model basedon a rigorous description of macropore and micropore dif-fusion with a non-linear adsorption isotherm and assumedthat the process was isothermal (model 32). The experimen-tal results demonstrated that micropore resistances controlthe diffusion mechanism within the adsorbent. More recently,Mulgundmath et al. (2012) investigated concentration andtemperature profiles of CO2 adsorption from a CO2–N2 gasmixture in a dynamic adsorption pilot plant unit to betterunderstand the adsorbent behavior. A dynamic model basedon an exact description of pore diffusion was developed forthe simulation of non-isothermal adsorption in a fixed bed(model 34). The proposed model was able to adequately pre-

dict the experimental data at all three ports for the durationof the experiment.

3.2.2.4. Linear driving force model. Although the diffusionalmodels are closer to reality, due to the mathematical complex-ities associated with such equations for the exact descriptionof intraparticle diffusion in adsorbent particles, simplerrate expressions are often desirable (Carta and Cincotti,1998; Zhang and Ritter, 1997). Simplified models are gener-ally adopted by using an expression of the particle uptakerate, which does not involve the spatial coordinates. Theapproximations express the mass exchange rate between theadsorbent and its surroundings in terms of the mean con-centration in the particle, regardless of the actual nature ofthe resistance to mass transfer (Lee and Kim, 1998). Simpli-fying assumptions should increase the practical applicabilityof the model without reduction of accuracy. The most fre-quently applied approximate rate law is the so-called lineardriving force (LDF) approximation, which was first proposedby Glueckauf and Coates (1947). They originally suggested thatthe uptake rate of a species into adsorbent particles is propor-tional to the linear difference between the concentration ofthat species at the outer surface of the particle (equilibriumadsorption amount) and its average concentration within theparticle (volume-averaged adsorption amount):

∂qi∂t

= ki(q∗i − qi) (29)

As can be seen, the overall resistance to mass transfer islumped into a single effective linear driving force rate coef-ficient, ki. Glueckauf demonstrated that the LDF overall masstransfer coefficient for spherical particles was equal to 15De/R2p(Glueckauf, 1955). The above equation has been shown to bevalid for dimensionless times (Det/R2p) > 0.1, where De is theeffective diffusivity (accounts for all mass transfer resistances)and t is the time of adsorption or desorption (Yang, 1987).Although the LDF model deals with the average concentra-tions of the adsorbate within the adsorbent particle, Liaw et al.(1979) demonstrated that the same value for ki could be sim-ply obtained by assuming a parabolic concentration profilewithin the particle. This assumption was later shown to beacceptable, as the exact solution to the concentration profilehas almost always been found to be a parabolic function (Doand Rice, 1986; Patton et al., 2004; Tsai et al., 1983, 1985; Yangand Doong, 1985). Sircar and Hufton (2000a) demonstratedthat the LDF model approximation is in accordance with anycontinuous intraparticle concentration profile within a spher-ical particle when a numerical constant other than 15 is usedin the expression of the LDF rate coefficient. The literatureincludes many attempts to develop new correlations for theaccurate prediction of the overall LDF rate constant (Gholamiand Talaie, 2009). When both the macropore and the microporediffusions are dominant, the overall LDF mass transfer coeffi-cient can be expressed by defining a single effective diffusivityrelated to both macropore and micropore diffusivities. Thefollowing correlation was proposed by (Farooq and Ruthven,1990), in which more than one mass transfer resistance (i.e.,film, macropore, and micropore resistances) is considered sig-nificant:

1ki

= Rpq03kfic0

+R2pq0

15εpDpic0+ R

2c

15D�i(30)

where q0 is the value of q at equilibrium with c0 at feed tem-perature.


ain

Let(tfiliaeLwa1

tdmaeosCmsbtiCmtsiatsdgmofifo

tefcfiatHfd(wrii

The above equation is actually an extension of the Glueck-uf approximation, which, apart from validity for a linearsothermal system, is also known to work reasonably well foron-linear systems.

Recently, the Stefan–Maxwell approach (Do and Do, 1998;iow and Kenney, 1990) or the dusty gas model (Mendest al., 1995; Serbezov and Sotirchos, 1998) has been proposedo describe adsorption kinetics. However, Sircar and Hufton2000b) indicated that the LDF model is adequate to cap-ure gas adsorption kinetics because in the estimation of thenal process performance, the detailed characteristics of a

ocal adsorption kinetic model are lumped during repeatedntegrations (Agarwal et al., 2010a,b). Indeed, although thisdsorption rate model is rather simple, it can predict thexperimental data with satisfactory accuracy (Yang andee, 1998). Consequently, this approximation has foundidespread application in modeling fixed-bed and cyclic CO2

dsorption processes (Hwang and Lee, 1994; Raghavan et al.,985).

A dynamic model that included finite mass transfer resis-ance based on a linear driving force assumption was firsteveloped by Mitchell and Shendalman (1973) for the isother-al removal of CO2 (a strongly adsorbed component in a trace

mount) from He (an inert product) using silica gel. How-ver, the model was found to provide a poor representationf the experimental data. Cen and Yang (1985) performedeparation of a five-component gas mixture containing H2,O, CH4, H2S, and CO2 by PSA. Both equilibrium and LDFodels were employed to develop a mathematical model for

imulating the PSA process (model 2). The results predictedy the equilibrium model, particularly for CO2 concentra-ion, were in poor agreement with the experimental data,ndicating the significant role of mass transfer resistance inO2 adsorption/desorption. The simulation results of the LDFodel were in generally good agreement with the experimen-

al data. Raghavan et al. (1985) simulated an isothermal PSAeparation of a trace amount of an adsorbable species from annert carrier using a linear equilibrium isotherm and with thessumption of a linear driving force for mass transfer resis-ance (model 3). The theoretically predicted behavior of theystem was shown to provide a good fit with the experimentalata of Mitchell and Shendalman (1973) for the CO2–He–silicael system. The major difference between this model and theodel of Mitchell and Shendalman (1973) is the assumption

f an inverse dependence of the effective mass transfer coef-cient on the total pressure. Such behavior is to be expectedor a system in which the uptake is controlled by external filmr pore diffusional resistance (Raghavan et al., 1985).

Kapoor and Yang (1989) also studied the kinetic separa-ion of a CO2/CH4 mixture on a carbon molecular sieve. Thexperimental results were simulated using a linear drivingorce model approach with a cycle time-dependent LDF rateoefficient (model 6). The cycle time-dependent LDF coef-cient included all mass transfer resistances such as filmnd intraparticle diffusion and was determined by matchinghe model simulation results with the experimental results.owever, the experimental estimates of this parameter dif-

ered considerably from the predictions of a priori correlationseveloped by Nakao and Suzuki (1983) and Raghavan et al.

1986). Diagne et al. (1996) developed a new PSA processith the intermediate feed inlet position operated with dual

efluxes for separation of CO2 dilute gas from air. They stud-

ed the influence of different CO2 feed concentrations and feednlet positions on CO2 product concentration. An isothermal

model based on LDF approximation was developed (model 13)to explore the effects of various combinations of the operat-ing variables and to analyze semi-quantitatively the effects ofthe main characteristic parameters such as the dimensionlessfeed inlet position and the stripping-reflux ratio. Good agree-ment between the model prediction and the experimentalresults was obtained.

In another study, low-concentration CO2 separation fromflue gas was performed by PSA using zeolite 13X as theadsorbent (Choi et al., 2003). To further assess the effectsof adsorption time and reflux ratio on product purity andthe recovery, dynamic modeling of the PSA process based onan LDF approximation was developed (model 19). The com-parison of the numerical simulation-based and experimentalresults demonstrated that the model adequately describes theexperimental breakthrough curves and temperature changesin the bed. Delgado et al. (2006a,b, 2007a,b) investigated thefixed-bed adsorption of binary gas mixtures (CO2/He, CO2/N2,and CO2/CH4) onto silicalite pellets, sepiolite, and a basicresin. The experimental breakthrough curves were simulatedby a model based on the LDF approximation for the masstransfer that considered the energy and momentum balancesand used the extended Langmuir equation to describe theadsorption equilibrium isotherm (model 27). They proposeda lumped mass transfer coefficient instead of consideringtwo mass transfer resistances in a bidisperse adsorbent. Acomparison between the experimental and theoretical curvesdemonstrated that the model reproduces the experimentaldata satisfactorily for the different feed concentrations, flowrates, and temperatures used. More recently, Dantas et al.(2011a) studied the fixed-bed adsorption of carbon dioxidefrom CO2/N2 mixtures on a commercial activated carbon. Amodel based on the LDF approximation for the mass trans-fer that considered the energy and momentum balances wasused to simulate the adsorption kinetics of carbon dioxide(model 31). They considered an overall LDF mass transfercoefficient in which the effects of film, macropore, and micro-pore resistances were assumed to be significant. The proposedLDF model acceptably reproduced the experimental data forthe different feed concentrations/temperatures and was suit-able for describing the dynamics of CO2 adsorption from themixtures. The importance of the external and internal masstransfer resistances was determined by performing a sensitiv-ity analysis, which concluded that micropore resistances arenot very important in the studied system. Moreover, it wasdeduced that, in the case of macropore resistances only, themolecular diffusivity is predominant.

If one neglects diffusion through macropores, the masstransfer rate through micropore volumes can be simplified byapplying the LDF model approximation, which is mathemati-cally equivalent to the modeling of transport through a barrierresistance confined at the micropore mouth (Cavenati et al.,2005; Grande and Rodrigues, 2007; Srinivasan et al., 1995):

∂qi∂t

= K�i(q∗i − qi) (31)

K�i =1

1/kbi + R2c /15D�i(32)

where K�i is the LDF constant for mass transfer in the micro-pores for component i (Grande and Rodrigues, 2007).

When there is no surface barrier resistance in the mouthof the micropores, the first term in the denominator of Eq.


(32) vanishes (Cavenati et al., 2005). This model, which hasbeen referred to as the LDFS model, is simply obtained fromEq. (23) if the intraparticle concentration profile of the adsor-bate is assumed to be parabolic (Carta and Cincotti, 1998;Chahbani and Tondeur, 2000; Do and Rice, 1986; Liaw et al.,1979; Siahpoosh et al., 2009). The mathematically simpleLDF approximation permits the direct use of the averagedadsorbed concentration in the interior of the adsorbent par-ticle and thus eliminates the need for the integration stepat the particle level, in contrast to the solid diffusion model(Chahbani and Tondeur, 2000; Sircar and Hufton, 2000b).

If the adsorbed-phase diffusion is neglected, a similar lin-ear driving force model based on the gaseous phase can beused to approximate the diffusive process in macropore resis-tance as follows (Khalighi et al., 2012):

εp∂cpi

∂t+ �p ∂qi

∂t= Kpi(ci − cpi) (33)

Kpi = εp15Dpi

R2p

BiiBii + 1

(34)

where Kpi is the LDF constant for mass transfer in the macro-pores for component i, cpi is the mean intraparticle gas phaseconcentration of species i, and Bii = Rpkfi/(5εpDpi) is the massBiot number, which represents the ratio of internal macroporeto external film resistances.

As can be seen, the proposed effective LDF rate coefficient,Kpi, is a combination of external fluid film transport, molecular,and Knudsen diffusions in the macropores. This model, whichhas been referred to as the LDFG model, can be derived fromthe pore diffusion model, Eq. (14), based on the assumptionof a parabolic gas phase concentration profile in the particle(Chahbani and Tondeur, 2000; Leinekugel-le-Cocq et al., 2007;Serbezov and Sotirchos, 2001; Yang and Doong, 1985). Sucha space-independent expression for the adsorption rate cantransform the PDE expressing mass conservation for gas pen-etrating pores into an ODE, and therefore the solutions aremathematically simpler and faster than the solution of thediffusion models.

Lai and Tan (1991) developed approximate models for porediffusion inside the particle with a non-linear adsorptionisotherm based on a parabolic concentration profile assump-tion for the summation of the gas and adsorbed phases.They developed a rate expression model that depends on theslope of the adsorption isotherm at the external surface ofthe sorbent. Ding and Alpay (2000) studied high-temperatureCO2 adsorption and desorption on hydrotalcite adsorbentat a semi-technical scale of operation. They presented adynamic model based on a linear driving force approxima-tion to describe intraparticle mass transfer processes (model17). To address the importance of intraparticle mass trans-fer resistances during different steps of operation, they alsodeveloped an adsorption model based on ILE assumptionbetween the gas and adsorbed phases. Overall, they con-cluded that although the ILE model failed to give an adequatedescription of the desorption kinetics, the LDF model basedon pore diffusion and accounting for the non-linearity ofthe isotherm provides an adequate approximation of theadsorption and desorption processes. Grande and Rodrigues(2008) studied the operation of an electric swing adsorp-tion process for low-concentration CO2 removal from flue

gas streams using an activated carbon honeycomb monolithas an adsorbent. To explore the dynamics behavior of the

system, the authors developed a mathematical model thatincluded bidisperse resistances within the porous structureof the monolith (model 29). A rigorous description and a lin-ear driving force approximation were employed for macroporeand micropore diffusion, respectively. Adsorption/desorptionbreakthrough experiments were performed to determine thevalidity of the proposed mathematical model. A comparisonof simulated breakthroughs and experimental data showedthat the dynamic model incorporating mass, energy, andmomentum balances agreed well with the experimentalresults.

If macropore or the adsorbed-phase diffusion cannot beignored, the mass transfer rate expression can be expressedusing a double LDF model, through which the macroporeand the micropore diffusion are both represented by LDFapproximations taken in series (Da Silva et al., 1999; Doongand Yang, 1987; Kim, 1990; Leinekugel-le-Cocq et al., 2007;Mendes et al., 1996). Cavenati et al. (2005) studied the sep-aration of a methane–carbon dioxide mixture in a columnpacked with bidisperse adsorbent (carbon molecular sieve3 K). To reduce the computational time required for the sim-ulations, macropore and micropore diffusion equations weredescribed using a bi-LDF simplification instead of the massbalances in macropores and in micropores (model 24). Theyassumed that the macropore dif

a review of mathematical modeling of fixed-bed columns for ... · (ruthven, 1984). the gas–solid...

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