Journal of Porous Media, 14 (6): 555–563 (2011)

THERMAL NON-EQUILIBRIUM MODELING OFCOUPLED HEAT AND MASS TRANSFER IN BULKADSORPTION SYSTEM OF POROUS MEDIA

Xuewei Zhang & Wei Liu∗

School of Energy and Power Engineering, Huazhong University of Science and Technology,Wuhan 430074, Peoples Republic of China

∗Address all correspondence to Wei Liu E-mail: w liu@hust.edu.cn

Original Manuscript Submitted: 10/13/2007; Final Draft Received: 10/26/2009

According to the volume-averaging method, a two-dimensional thermal non-equilibrium mathematical model was de-veloped to simulate the coupled heat and mass transfer in porous media with strong adsorption, in which local fluidand solid temperatures were dealt with separately. The temperature-dependent Langmuir isotherm was applied to de-scribe the equilibrium characteristics of binary gas CO2 adsorbed by zeolite 13X. The adsorption-generated heat wasconsidered in the energy equation of the solid matrix. The coupling of two energy equations was made by consideringthe interfacial heat transfer term. The adsorption feature of a packed bed filled with porous adsorbent and the effects ofbulk flow velocity, particle diameter, effective thermal conductivity of solids, and specific heat of adsorbent on productconcentration were investigated numerically. The numerical simulation shows that the heat transfer characteristic hasnotable effects on the quantity of adsorption and the purity of the product. High adsorption efficiency can be achievedby improving the adsorbent physical properties and reducing the temperature span of the pressure swing adsorptionprocess. Experimental data reported by other researchers validate the accuracy of the present model.

KEY WORDS: heat and mass transfer, non-equilibrium, porous media, pressure swing adsorption, carbondioxide

1. INTRODUCTION

Pressureswing adsorption(PSA) technologybasedontheadsorptioncharacteristicsof thecomponentshasbeenwidely applied to separategas mixtures.Recently, thePSAprocesswith strongadsorptionhasattractedspecialattentionin separatingcarbondioxide, one of the mostpredominantgreenhousegases,from flue gas.This pro-cessis characterizedby several uniquefeatures.First ofall, masstransferplaysan importantrole in the periodicadsorptionprocessunderarapidcyclecondition.Second,the operatingtemperaturespancausedby absorptionorreleaseof adsorption-generatedheatis commonlymorethan10◦C whichleadsto asignificantchangein adsorbedquantity. Finally, heatandmasstransferarecoupledwitheachother. Therefore,it is necessaryto considerbothheat

andmasstransferprocessesin analysisof the adsorber,especiallyin somelarge-scalePSAequipment(Steinbery,1992;Kikkinidesetal., 1993).

Heatandmasstransferin porousmediahave beenin-vestigatedfor several decades(Nield and Bejan, 1999).The local thermal equilibrium model (Cheng, 1978;Whitaker, 1986; Kim and Vafai, 1989; Reesand Vafai,1999) and the local thermal non-equilibrium model(Quintardand Whitaker, 1993; Amiri and Vafai, 1998;Kim et al. 2000) have beenusedin dealing with heattransportin porousmedia.Theformeris relatively simpleHowever it is valid only whenthetemperaturedifferencebetweensolid andfluid phasesis very smallThelatter isrelatively complex, asheatexchangebetweensolid andfluid phaseshasto beconsidered.Accordingto thecrite-rion proposedby Kim andJang(2002),thethermalnon-

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556 Zhang & Lui

NOMENCLATURE

Bi Langmuir parameter (1/Pa) Greek SymbolsRa Rayleigh number ϕ porosity of bedNu Nusselt number ρ density of fluid (kg/m3)T temperature (K) ρb density of bed (kg/m3)t time (s) µf viscosity of fluid [kg/(m/s)]Ci mass fraction of speciesi in the gas phase λf thermal conductivity of coefficientCf specific heat of fluid [J/(kg/K)] [kg/(m2/K)] fluid [W/(m/K)]Cs specific heat of solid [J/(kg/K)] λs thermal conductivity of heatv velocity (m/s) transfer coefficient [W/(m3/s)]u uniform velocity solid [W/(m/K)]Db mass diffusivity (m2/s)dp particle diameter (m) SubscriptsF geometric function (m−3/2) eff effectiveH adsorption heat (J/kg) i speciesihsf solid-to-fluid heat transfer coefficient k species number

[W/(m2/K)] qmax,i maximal adsorbed amount ofhm,sf solid-to-fluid mass transfer speciesi (kg/kg)hv volumetric solid-to-fluid qi adsorption quantity of speciesihm,v volumetric solid-to-fluid mass (kg/kg)K permeability of porous media (m2)L length of adsorber (m) SuperscriptP pressure (Pa) ∗ equilibrium state

equilibriummodelshouldbe usedto describethestrongadsorptionprocessesin theporouspackedbed.

Thesimplestandmostcommonlyappliedmasstrans-fer modelis thelineardriving force(LDF) model,whichwasfirst proposedby GlueckaufandCoates(1947).TheLDF model was derived by assumingthat the drivingforce of masstransferwithin a chromatographicprocesstakeson a simple linear form betweenthe compositionsin equilibrium with the bulk phase(Carta,1993;Hsuen,2000)Veldsinket al. (1995)adoptedtheFick modelandthe dusty-gasmodel to the studymasstransportprocessaccompaniedwith thechemicalreactionin aporouscata-lyst.

However, attentionhasnot beenfully paid to the ef-fect of heat transferon masstransfer for studying thecouplingof heatandmasstransferin porousmediasub-jected to chemicalreactionsor physical adsorptionSo,a mathematicalmodel describingthe coupled processof bulk adsorptionhas beenestablishedin this paper.Both the Forchheimerand Brinkman effects have been

taken into accountin the momentumequationThe ther-malnon-equilibriummodelhasbeenadoptedandtwo en-ergy equationshavebeencoupledwith theinterfacialheattransferterm,in which theadsorption-generatedheathasbeenconsideredin the energy equationof the solid ma-trix. The temperature-dependentLangmuir isothermhasbeenusedto describethe equilibrium characteristicsofCO2 andN2, andthe masstransfercoefficient hasbeenapproximatedastheexternalmasstransfercoefficient ofthe particle. The adsorptionfeaturesof the packed bedalsohavebeeninvestigatednumerically.

2. MATHEMATICAL MODELING

2.1 Problem Description

Theschematicdiagramof thephysicalmodelis shown inFig. 1. The cylindrical bedwith a 257 mm diameteranda lengthof 1000mm wasfilled with a homogenousandisotropicsphericaladsorbentsubjectedto aconstanttem-

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Thermal Non-Equilibrium Modeling of Coupled Heat and Mass Transfer 557

FIG. 1: Schematicdiagramof a packed bedfilled withzeolite13X

peratureboundarycondition.The fluid wasinjectedat aconstantratewith uniform velocity u0. Due to the sym-metryof thepackedbed,only half of it wasconsideredinthenumericalsimulation

Thefollowing assumptionsaremadein themodeling:

1. Idealgaslaw is applicable.

2. Thetemperatureof aparticleis uniform.

3. Naturalconvectionandradiationheattransferarene-glected.

4. The physical propertiesof the gasphase,adsorbentandcolumnwall areconstant.

5. Theheatcapacityof theadsorbedgasis neglected.

Under the above assumptions,the governing equationsareobtainedasfollows:

1. For overallmassbalance:

∂ϕρ

∂t+

∂ρu

∂x+

1r∂ (rρv)

∂r+ ρb

n∑

k=1

∂qk

∂t= 0 (1)

2. For massbalancefor gasi (CO2 or N2):

∂ϕρCi

∂t+

∂

∂x(ρuCi) +

1r

∂

∂r(rρvCi)− ∂

∂x

×(Db

∂Ci

∂x

)− 1

r

∂

∂r

(rDb

∂Ci

∂r

)+ρb

∂qi

∂t= 0 (2)

3. For the Darcy–Forchheimer–Brinkman momentumequation (Sozen and Kuzay, 1996):

ρf

(∂~V

∂t+~V · ∇~V

)= −∇p+µeff∇2~V − µf

K~V

− ρfFϕ√

K

∣∣∣~V∣∣∣ ~V (3)

Thesecond,third andfourth termson therighthandsideof Eq.(3) representtheviscouseffect,frictional resistanceandinertial forcerespectively.

ThepPermeabilityof theporousbed(Ergun,1952)andgeometricfactorF in themomentumequationaregivenas(QuintardandWhitaker, 1993)

K =ϕ3d2

p

150 (1−ϕ)2(4)

F =1.75

150ϕ3/2(5)

Theenergy conservationequationsfor thefluid andsolidmatrixaregivenas

∂ [ϕCfρTf ]∂t

+∂ (CfρuTf )

∂x+

1r

∂ (rCfρvTf )∂r

=∂

∂x

(λf,eff

∂Tf

∂x

)+

1r

∂

∂r

(rλf,eff

∂Tf

∂r

)

+ hv (Ts − Tf ) (6)

∂ [CsρbTs]∂t

=∂

∂x

(λs,eff

∂Ts

∂x

)+

1r

∂

∂r

(rλs,eff

∂Ts

∂r

)

− hv (Ts − Tf ) +n∑

k=1

Hkρb∂qk

∂t(7)

Themasstransferis describedby theLDF model,

ρb∂qi

∂t= hm,v (Ci − C∗i ) (8)

Thefluid andsolideffective thermalconductivitiesare

λf,eff = ϕλf (9)

λs,eff = (1−ϕ) λs (10)

Thevolumetricsolid-to-fluidheatandmasstransfercoef-ficientshv andhm,v are

hv = 6hsf (1−ϕ)/dp (11)

hm,v = 6hm,sf (1−ϕ)/dp (12)

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558 Zhang& Lui

whereheattransfercoefficient hsf is givenby (Wakaoetal., 1979)

Nu =hsfdp

λf= 2.0 + 1.1Pr1/3Re0.6

p

Pr =Cpµ

k, Rep =

ρudp

µ

(13)

whereReynoldsnumberRep coverstherangeupto 8500.Masstransfercoefficienthm,sf is givenby

Sh =Dihm,sf

Dm= 2.0 + 0.6Sc1/3Re1/2

Sc =µ

ρfDm, Re=

ρfV ϕdf

µ

(14)

The temperature-dependentLangmuir isotherm(HarlickandTezel,2001)is usedto representthegas–solidequi-librium isotherm:

qi =qmax,ibipC∗i

1 +n∑

k=1

bkpC∗k

(15)

where qmax,i = B1,i exp (B2,iT ) and bi = B3,i exp(B4,iT ).

Theparametervaluesusedin thesimulationarelistedin Table 1, in which the Langmuir parametersfor CO2

andN2 weremeasuredby theconstantvolumemethodatdifferentpressures(Suzuki,1990).

The initial andboundaryconditionsaregiven as fol-lows:

1. Initial conditions(t = 0):

U = U0, V = 0Tf = Ts = Tw = 330K

CCO2 = 0, CN2 = 1.0qCO2 = 0, qN2 = q∗N2

(CN2 = 1.0)

TABLE 1: Parameter values in simulation of tempera-ture-dependent Langmuir isotherm

Langmuir parameters CO2 N2

B1 (kg/kg) 2.2 0.28B2 (1/T) −0.01 −0.01

B3 (1/atm) 500 60B4 (1/T) −0.01 −0.02

2. Boundaryconditions(t > 0):

x = 0 : U = U0, V = 0, CCO2 = 0.1,

CN2 = 0.9, Tf = Ts = Tw = 330K

r = 0 :∂U

∂r= 0,

∂V

∂r= 0,

∂CCO2

∂r= 0,

∂CN2

∂r= 0,

∂Ts

∂r= 0,

∂Tf

∂r= 0

r = R : U = 0, V = 0,∂CCO2

∂r= 0,

∂CN2

∂r= 0.0, Tf = Ts = Tw = 330K

3. RESULTS AND DISCUSSIONS

The numerical solution is obtainedby employing thefinite-volume method.The convective flux term is dis-cretizedby thesecond-ordercentraldifferencingschemethat is blendedwith the upwind differencingschemeus-ing the deferredcorrection method. The SIMPLE al-gorithm for pressure–velocity coupling is applied.Thesourceterm of the equationis linearizedin sucha waythatensuresdiagonaldominancein thecorrespondingdis-cretizedequation(Ferzigerand Peric, 1999). The dis-creteequationscannotbe solved independentlybecauseof the cross-couplingbetweenmassandenergy balanceequations.Therefore,theiterativemethodwasadoptedtosolve the temperatureand concentrationfields. Conver-genceis checkedin termsof themaximumchange(10−4)for every variableduringthe iterationat eachtime incre-ment.All computationshave beencarriedout for a half-cylindrical reactorwith non-uniformgrid arrangements.Differentgrid sizes,suchas50×25 100×50, 200×100,400 × 200, and800 × 400 have beenusedin thedomainof L × R. As thecomputationalaccuracy by takinggridsizesof 200 × 100, 400 × 200, and 800 × 400 are al-mostthesame,wechosethegrid sizeof 200× 100 in thecomputations.

In orderto verify the presentmodel,numericalsimu-lation wasperformedunderthesameconditionsasthoseof theexperimentconductedby Hwanget al. (1995).Asshown in Fig. 2, a comparisonof outlet CO2 concentra-tion wasmadebetweenthe numericalsolutionsandex-perimentaldata, in which agreementbetweenthe theo-reticalcalculationandexperimentalmeasurementcanbeobserved,which validatesthepresentmodelto someex-tent.

The four main parametersimpacting an adsorptionprocess,suchasbulk flow velocity, particlediameter, ef-

Journal of Porous Media

Thermal Non-Equilibrium Modeling of Coupled Heat and Mass Transfer 559

FIG. 2: ComparisonbetweenoutletCO2 concentrationsobtainedby the presentmodel and the experimentcon-ductedby Hwangetal. (1995)

fectivesolid thermalconductivity, andspecificheatof ad-sorbent,areanalyzedin thefollowing discussions.

3.1 Effect of Bulk Flow Velocity

The influences of the bulk flow velocity on the transportand adsorption characteristics are shown in Figs. 3–6. InFig. 3 it can be seen that the pressure drop in the packedbed filled with adsorbent increases with the increase of

FIG. 3: Effect of bulk flow velocity on the pressure dropbetween the inlet and outlet

FIG. 4: Effect of bulk flow velocity on CO2 concentra-tion at the outlet

FIG. 5: Effect of bulk flow velocity on outlet fluid tem-perature

FIG. 6: Effectof bulk flow velocityonaverageadsorbenttemperature

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560 Zhang& Lui

bulk flow velocity, which indicatesthat more energy isconsumed.Figure4 shows that the larger the bulk flowvelocity, theshortertheadsorptiontime will be.Thusthetransitionpointof thebreakthroughcurveappearsearlier.Therefore,increasingthe bulk flow velocity canchangethe capacityof per-volume adsorbent.Figures5 and 6demonstratethat the outlet fluid temperatureand aver-ageadsorbenttemperatureincreasemorerapidlywith theincreaseof bulk flow velocity in the upgoing stageofcurves.However, the maximumoutlet fluid temperaturedecreaseswith anincreaseof thebulk flow velocity, whilethis tendency is inversedfor themaximumaverageadsor-bent temperature.The reasonsfor thosephenomenacanbeexplainedastheincreaseof theaverageadsorbenttem-peraturebeingdependentonthereleaserateof adsorptionheat,theheattransferratebetweensolidandfluid andtheheatcapacityof thesolid adsorbentwhile theincreaseofbulk fluid temperaturedependson the heattransferratebetweensolid andfluid andthe heatcapacityof fluid. Ifthe bulk flow velocity is higher, the adsorptionratewillincrease,andadsorption-generatedheatwill releasemorequickly, which leadsto aquicker increasein thesolid ad-sorbenttemperatureMeanwhile,ahigherbulk flow veloc-ity resultsin a relatively shortertime for fluid residence.Thus heatexchangebetweenfluid and adsorbentis notsufficient,which leadsto a loweroutletfluid temperature.

3.2 Effect of Partic le Diameter

Figures 7–10 display the effect of the particle diameter ofadsorbent on the transport and adsorption characteristicsof the packed bed. From Fig. 7 it can be seen that pressure

FIG. 7: Effect of particle diameter on the pressure dropbetween the inlet and outlet

FIG. 8: Effect of particle diameter on outlet CO2 concen-tration

FIG. 9: Effect of particle diameter on outlet fluid temper-ature

FIG. 10: Effectof particlediameteronaverageadsorbenttemperature

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ThermalNon-EquilibriumModelingof CoupledHeatandMassTransfer 561

dropdecreasesgreatlywith anincreaseof theparticledi-ameter. It is expectedthat thepermeabilityof thepackedbedincreasessignificantlywhentheparticlediameterbe-comeslarger. This indicatesthat adoptingbiggeradsor-bentparticlescansave moreenergy. Figure8 shows thatthe larger the particlediameter, the higherthe CO2 con-centrationin the outflow will be. It is notedfrom Fig. 9that the outlet fluid temperatureincreasesmore slowly,asthe particlediameterincreases,while this tendency isinversedfor theaverageadsorbenttemperatureshown inFig.10.It isexpectedthatwhenotherconditionsarefixed,the heatandmasstransfercoefficientsdecreasewith theincreaseof theparticlediameter. In thecaseof largerpar-ticles,theincreaseof averageadsorbenttemperatureleadsto a lower equilibriumadsorptionquantity. Consequentlytheadsorptionperformancewill beslightly worse.

3.3 Effect of Effective Solid ThermalConductivity

As shown in Fig. 11, the time it takesuntil the appear-anceof CO2 in theoutflow is almostthesamefor differentcurves.After that,however, theoutletCO2 concentrationincreases,andits increaserateis relatively slower whenλs,eff is larger. As shown in Figs.12 and13, for boththeoutlet fluid temperatureand the averageadsorbenttem-peraturetheincreaseratesarealsorelatively slowerwhenλs,eff is larger. It can be explained as follows: when ef-fective solid thermal conductivityλs,eff is raised, moreadsorption heat is transferred to the surroundings, whichmakes the average adsorbent temperature increase moreslowly and the equilibrium adsorption quantity increase

FIG. 11: Effect ofλs,eff on outlet CO2 concentration

FIG. 12: Effect ofλs,eff on outlet fluid temperature

FIG. 13: Effect of λs,eff on averageadsorbenttempera-ture

As aresult,theadsorptionperformanceof thepackedbedwill beimproved.

3.4 Effect of Solid Specific Heat

Figures14–16demonstratethat when the solid specificheatis larger it takesmore time until the appearanceofCO2 in theoutflow, andtheoutletCO2 concentration,out-let fluid temperatureandaverageadsorbenttemperatureincreasemoreslowly. Thesephenomenacanbeexplained

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562 Zhang & Lui

FIG. 14: Effect of solid specific heat on outlet CO2 con-centration

FIG. 15: Effect of solid specific heat on outlet fluid tem-perature

FIG. 16: Effect of solid specificheaton averageadsor-benttemperature

as follows: when the sameamountof heatis absorbed,highersolidspecificheatmeanslargerheatcapacityof theadsorbentwhich resultsin a smallertemperatureswingThus,accordingto the temperature-dependentLangmuirisotherm,the equilibrium adsorptionquantity increases.Therefore,more gas mixturescan be separatedand thepurity of the productioncanbe raisedby increasingthesolidspecificheat

4. CONCLUSIONS

A local thermalnon-equilibriummathematicalmodelde-scribing the coupledheatandmasstransferin a porousadsorbentbedwasestablished.Themodelwassolvedbythe finite-volumemethodandverified throughthe com-parisonbetweensimulationresultsandexperimentaldatareportedby otherresearchers.Theeffectsof thebulk flowvelocity, particle diameter, effective solid thermal con-ductivity andspecificheatof theadsorbentwereinvesti-gatednumerically. Thecomputationresultsshow that(1)increasingthebulk flow velocity canenhancethecapac-ity of per-volume adsorbent,but worsenthe heattrans-fer performanceandconsumemoreenergy; (2) increas-ing the particle diametercan slightly worsenheat andmasstransferperformance,but flow resistancedecreasessignificantly; (3) adsorptionperformancecanbe slightlyimproved when the effective solid thermalconductivitybecomeslarger; and (4) adsorptionperformancecanbesignificantlyimprovedwhenthesolid specificheatis in-creased,while energy consumptionremainsalmost thesame.Thoseresultsmayprovidesomeusefulinformationfor operatingaPSAprocess.

ACKNOWLEDGEMENTS

This work was funded by the National Key Basic Re-searchDevelopmentProgramof China(2007CB206903)andtheNaturalScienceFoundationof China(51036003).

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