Macropore diffusion dusty-gas coefficient for pelletised zeolites from breakthrough experiments in...

Chemical Engineering Science 60 (2005) 4593–4608

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Macropore diffusion dusty-gas coefficient for pelletised zeolites frombreakthrough experiments in theO2/N2 system

Richard S. Todd, Paul A.Webley∗

Department of Chemical Engineering, Monash University, Clayton, Victoria, Australia

Received 8 December 2004; received in revised form 2 March 2005; accepted 10 March 2005Available online 13 May 2005

Abstract

Simulation of air separation in zeolites requires an accurate mass transfer model particularly under rapid cycling conditions. An earlierstudy [Todd et al., 2005. Adsorption 11, 427–432] examined the application of the dusty gas model to the simulation of the adsorptionof O2/N2 system in lithium-exchanged low silica-to-alumina ratio x-type zeolite (LiLSX) pellets. In that study we characterised Knudsendiffusion (CK ) and viscous flow (Cv) structural parameters, two of three parameters that constitute the viscous flow plus dusty gas model(VF + DGM) intrapellet flux equation for a macroporous pellet under adsorbing conditions. This paper quantifies the third structuralparameter (Cm), related to molecular diffusion that arises when a multicomponent gas mixture is introduced into the pore network of asorbent pellet. With these three parameters defined, the mass transfer behaviour for the O2/N2 system in our adsorbent is completelycharacterised (under the assumption of no surface diffusion). To experimentally characteriseCm, a series of breakthrough experiments wereperformed with a packed bed of LiLSX pellets. Six independent breakthrough runs over a range of conditions were performed with one ofthese runs used to determineCm. The remaining five runs were used for validation. Our adsorption simulator previously described [Toddet al., 2003. Computers and Chemical Engineering 27, 883–899] was enhanced with the addition of a rigorous wall model and was used toperform all breakthrough simulations. A value ofCm = 0.122 was obtained giving a molecular diffusion tortuosity of 2.5. This is typicalof tortuosities for porous adsorbent pellets. Experiments and simulations revealed operating conditions were very close to adiabatic duringeach experimental breakthrough run while a sensitivity analysis on the intrapellet structural coefficients revealed the sensitivity coefficientsof Cm andCK are similar in magnitude suggesting both mechanisms of diffusion are important. Viscous flow was several orders ofmagnitude smaller and hence is of minor importance in breakthrough simulations. Simulations also suggest the level of radial discretisationwithin a sorbent pellet is an important aspect of a discretised pellet model to consider when simulating a non-isothermal and non-isobaricbreakthrough experiment. Relatively high pellet discretisations are needed to obtain physically meaningful structural coefficients.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Zeolites; Adsorbent pellets; Dusty-gas model

1. Introduction

The diffusion of gases into and out of adsorbent pelletsis an important topic of fundamental and engineering inter-est. Adsorptive separation processes all rely on this featureto either effect a separation (such as kinetic separation ofO2 from N2 in N2PSA systems) or to improve the sepa-ration in an equilibrium-based process. Modeling the dif-fusive transport of gases in adsorbents is often the most

∗ Corresponding author. Tel.: +61399051874; fax: +61399055686.E-mail address:[email protected](P.A. Webley).

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.03.024

difficult step in a mathematical simulation of adsorption pro-cesses and short-cut models are frequently employed. How-ever, accurate modeling is needed especially for modelinghigh throughput PSA systems. In this context, the dusty gasmodel (DGM) has frequently been used to describe the fun-damental fluxes within adsorbents. Inclusion of viscous flowalong with the mechanisms of molecular and Knudsen dif-fusion is important for cases where total pressure changesoccur rapidly across the adsorbent dimension.Several researchers have used the chromatographic

technique for the extraction of mass transfer informa-tion. Hashimoto and Smith (1974)obtained macro- and

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4594 R.S. Todd, P.A. Webley / Chemical Engineering Science 60 (2005) 4593–4608

micropore diffusion coefficients using moments analysis forthe adsorption of 0.5mol%n-butane from a helium car-rier stream over bidisperse alumina pellets.Ruthven et al.(1975)assumed isobaric, non-isothermal and plug flow op-erating conditions using the LDF model to describe microp-ore diffusion control of propylene, 1-butene andcis-2-buteneflowing in an inert carrier stream over 5A zeolite pellets.Andrieu and Smith (1980)obtained micropore diffusion co-efficients of carbon dioxide in a packed bed of activated car-bon using a moments analysis.Cui et al. (1990)compare thetortuosity coefficient predicted from the chromatographicand Wicke–Kallenbach methods for microporous alumina.They highlighted the inconsistent tortuosity coefficient pre-dicted from a Wicke–Kallenbach cell when processes otherthan diffusion, such as adsorption or chemical reaction oc-cur within the pellet.Boniface and Ruthven (1985), Kim(1990), Farooq (1995), Nijhuis et al. (1997), Noordhoek etal. (1998)andChun and Lee (2000)have also used the chro-matographic method to determine intrapellet diffusion coef-ficients over a range of sorbent materials.In an earlier study (Todd et al., 2005) the Knudsen (CK )

and viscous flow (CV ) structural coefficients were obtainedfor the viscous flow plus dusty gas model (VF+DGM) frompure gas experiments with an uptake apparatus leaving themolecular diffusion parameterCm to be determined. In thisstudy we show how performing a chromatographic experi-ment on a binary gas mixture allows this coefficient to beindependently determined. Literature related to the extrac-tion of VF+ DGM parameters for macropore-based diffu-sion control in an adsorbent pellet has received very little at-tention (Hartmann and Mersmann, 1994). For this reason noinformation is readily available that identifies for which sys-tem(s) the benefits of a VF+DGM approach are important.This is mainly due to the computational overheads associ-ated with a simulator that discretises the model equationsover an adsorbent pellet. This study includes non-isothermaland non-isobaric behaviour for a system where the adsorp-tion of a bulk component occurs. The important characteris-tics that must be considered with any numerical scheme forsuch a system is the need to maintain closure in mass andenergy balances while maintaining a fast yet efficient nu-merical algorithm for spatial and temporal integration. Sev-eral techniques have been recently identified in the literaturethat offers new prospects for adsorption simulation, such asa wavelet-based adaptive grid method (Cruz et al., 2002)and multi-resolution method (Cruz et al., 2003). However,these techniques are yet to be adopted within a simulatorthat considers the mechanisms addressed in this paper. Thesolution-adaptive finite volume method in conjunction withtwo novel CPU optimisation techniques (Todd et al., 2003),have been successfully used to simulate the N2–O2–zeolitesystem. From this perspective, future research efforts shouldbe directed towards exploring and validating these new solu-tion techniques for adsorption simulation with the solution-adaptive finite volume technique as one benchmark. Withthis said, we have obtained independent experimental data

from a rapid pressure swing adsorption (RPSA) pilot plantthat shows a VF+DGMapproach to intrapellet mass transferdoes correctly predict trends that a simpler model, the LDF,does not using the solution-adaptive finite volume method.This will be the subject of a future paper (Todd and Webley,in press).

2. Simulation and intrapellet flux equations

The VF+DGM is considered here to represent intrapel-let mass transfer during a breakthrough experiment. The ad-sorption process simulator that has been developed and ver-ified in our research group uses conservation of mass andenergy for the interpellet gas phase with discretised pelletmodels (DPMs) to represent the intrapellet region (Webleyand He, 2000; Todd et al., 2003).Conservation of mass, interpellet region:

1

RT

�pBi

�t− pB

i

RT 2�T�t

= − ��z

(pBi vB

RT

)+(1− �B

�B

)6

dPNi,surface

for i = 1, . . . , Nc. (1)

Conservation of energy, combined inter- and intrapelletregions(TB = TP = T ):

�(∑Nc

i=1�Bg,iU

Bg,i

)�t

+ 24(1− �B)�B

×∫ dP /2

0

�P

�(∑Nc

i=1�Pg,iU

Pg,i

)�t

+�P

�(∑Nc

i=1UPa,i

)�t

+ �P

�UPS

�t

r2 dr

d3P

= − ��z

[∑Nc

i=1

(pBi vB

RTHB

g,i

)]− 4hB,W

dB�B(T − TW).

(2)

Conservation of mass, intrapellet region:(�PRT

+ �P

�nPi,eq�pP

i

)�pP

i

�t+

Nc∑j=1,j �=i

(�P

�nPi,eq�pP

j

)�pP

j

�t

+(

�P

�nPi,eq�T

− �PpPi

RT 2

)�T�t

= − 1r2

��r

(r2Ni)

for i = 1, . . . , Nc (3)

whereR is the ideal gas constant in J gmol−1 K−1, �Bg is the

gas phase density in gmolm−3, �P is the density of the pel-lets in kgm−3, TW is the wall temperature in K,hB,W is thebed-to-wall heat transfer coefficient in Wm−2 K−1 andnPeqrepresents the equilibrium amount adsorbed in gmol kg−1.

R.S. Todd, P.A. Webley / Chemical Engineering Science 60 (2005) 4593–4608 4595

UBg andUP

g are the internal energy of the inter- and in-

trapellet gas phases, respectively, in J gmol−1 while UPa

and UPS are the internal energy of the adsorbed and solid

phases, respectively, in J kg−1. Nc represents the number ofcomponents in the system, which is 2 in this study.Ni inEq. (1) is evaluated at the surface of the pellet,r = dP /2.The radial integral of Eq. (2) becomes a summation overNr

spherical shells when the finite volume method is applied.It is assumed in this study that interpellet gas temperatureTB changes directly with pellet temperatureTP such thatTB = TP = T . Serbezov and Sotirchos (1998)have shownthe pellet-to-gas heat transfer resistance is negligible underthe relatively high gas velocity conditions typical of a PSAcycle. Simulations with varying magnitudes on the pellet-to-gas heat transfer coefficient have also confirmed this re-sult (Todd, 2003). Time integration was performed with theintegrator DVODE (Brown et al., 1989).Todd et al. (2005)used a pure gas to reduce the intrapel-

let flux equation for the VF+ DGM to a particularly sim-ple form equivalent to Ficks first law of diffusion. In thepresent study we are using binary gas mixtures in break-through experiments. An analytical expression for the fluxof componenti, Ni using the VF+DGM can be derived bysolving the governing constitutive equations simultaneouslyfor components 1 and 2 (Todd, 2003) to give:

Ni =− 1

RT

�pPi

�r− 1

RT

yPi

Dem,ij

(1

DeK,j

+ yPi

Dem,ij

)−1

× �pPj

�r− 1

RT

(BepP

�P

)

× yPi

DeK,i

+ yPi

Dem,ij

(1

DeK,j

+ yPi

Dem,ij

)−1yPj

DeK,j

× �pP

�r

/[

1

DeK,i

+ yPj

Dem,ij

− yPi yPj

(Dem,ij )

2

×(1

DeK,j

+ yPi

Dem,ij

)−1for i = 1,2 and i �= j . (4)

The Knudsen, viscous and molecular diffusion coefficientsfor porous media are related to their non-porous analogsthrough structural coefficients that may also be expressed interms of tortuosity

DeK,i = CKDK,i = �p

�KDK,i ,

Be = CvB = �p�v

B,

Dem,i = CmDm,ij = �p

�mDm,ij . (5)

The voidage correction factor�P indicates the volume avail-able for diffusion is only a small fraction of the total intrapel-

let volume. The tortuosity correction factor�, usually greaterthan 1, reduces the magnitude of the true or unhindered dif-fusion coefficient by an amount that represents the intercon-nectedness of randomly oriented pore channels surroundingthe structural matrix for each transport mechanism. The in-troduction ofCK ,Cv andCm into their unhindered diffusioncounterparts gives rise to an effective diffusion coefficient,represented by a superscript “e”. Although tortuosities canbe calculated these require specification of appropriate pel-let voidages,�p, which can only at best represent an aver-age over the pore size distribution. Eq. (4) is a complicatednon-linear function that does not provide an analytical so-lution when incorporated into the intrapellet conservationequations. The structural parametersCK andCv withinDe

K,i

andBe, respectively, have been found previously (Todd etal., 2005) while T, pP and yPi are found by discretisingthe model equations over the radial co-ordinate and solvingthe resulting conservation equations within each sphericalcontrol volume. From this, a DPM with the VF+DGM in-trapellet flux equation forNc = 2 within a chromatographicsystem reduces Eq. (4) to a non-linear parameter fit forCm

throughDem,ij .

Although the intrapellet domain can be reduced to findingjust the one parameter, flowing a gas mixture through apacked bed of sorbent pellets introduces three additionaldispersive phenomena, pressure drop, external mass transferand axial dispersion of the interpellet gas phase. Interpelletpressure drop is described by the Ergun equation (Ergun,1952):

�pB

�z= −

[�viscous�B

d2P

(1− �B

�B

)2]vB

−[

�kinetic �Bg

dP

(1− �B

�B

)]vB |vB |, (6)

wherepB is the pressure of interpellet gas in Pa,�B is theviscosity of interpellet gas in Pa s,dP is the equivalent di-ameter of the pellet in m,�B is interpellet voidage,vB isinterstitial gas velocity through the packed bed in m s−1 and�Bg is the density of interpellet gas in kgm

−3. The viscousand kinetic constants of Eq. (6) have been found experimen-tally to be�viscous= 154± 14 and�kinetic= 1.47± 0.13 forlithium-exchanged low silica-to-alumina ratio x-type zeolite(LiLSX) sorbent pellets (Todd, 2003). External mass trans-fer at the pellet surface and axial dispersion are discussed inmore detail in Section 7 of this paper.The experimental apparatus described in the next section

comprises mainly a column that operates at conditions closeto adiabatic. The conservation of energy equation used inour simulator was enhanced to include a wall heat transferterm and rigorous wall energy model to establish the extentof adiabatic behaviour actually attained. In addition to thesteady-state momentum equation, the interpellet conserva-tion of energy has introduced another parameter, the bed-to-wall heat transfer coefficienthB,W , that needs to be resolved


Table 1Summary of experimental parameters for LiLSX pellets

LiLSX properties

dP 1.7± 0.2 ×103m�P 926± 77 kgm−3�P 0.62± 0.04 —�P,macro 0.31± 0.04 —�B 0.35± 0.06 —� 2.5± 0.4 ×107mCK 0.083± 0.018 —Cv 0.061± 0.026 —�viscous 154± 14 —�kinetic 1.47± 0.13 —

The LiLSX adsorbent was provided in spherical pellet form by AirProducts and Chemicals Inc.

Fig. 1. Equilibrium isotherm for pure component nitrogen and oxygenobtained at three different temperatures using a pure component equilib-rium measuring device, along with the corresponding analytical fit usingregressed dual-site Langmuir (DSL) isotherm parameters.

beforeCm can be found.Ruthven (1984, p. 218)found sat-isfactory agreement between experimental heat transfer co-efficients ofRuthven et al. (1975)and those from Leva’scorrelation for the adsorption of propylene,cis-2-butene and1-butene from an inert carrier stream in a packed bed of 5Amolecular sieve sorbent. For this study it is assumed Leva’scorrelation is appropriate for the estimation of the bed-to-wall heat transfer coefficient.Table 1summarises physicalproperties for the system whileFig. 1shows the equilibriumisotherm for LiLSX pellets.

3. LUB apparatus for binary chromatographicexperiments

To acquire experimental mass transfer information underadsorbing conditions a length of unused bed (LUB) appara-tus was constructed (Todd, 2003). This apparatus comprisesa single cylindrical column of internal diameter 0.050m(dB)

and packed bed length of 2m(LB) with associated valvingand pipe work required to pass gas to and from the column.The adsorbent was LiLSX beads and the gas mixtures con-sisted of O2/N2/Ar with compositions indicated later in thisstudy.A breakthrough run involves the measurement of the mass

transfer zone (MTZ) that traverses the packed bed at the on-set of a change in feed gas composition. A basic schematicarrangement of a breakthrough run is depicted inFig. 2.Data acquisition and control was performed from a local PCvia a custom written QuickBasic (QB) program. This QBprogram was specifically developed for rapid data collectionwith one scan time through the entire code lasting approxi-mately 0.1 s. All analog signals from pressure, temperature,composition and flow metering devices were converted to12 bit digital integers through two Advantech PCL818HGcards and then logged to file. Valve switching and data log-ging was activated via user-entered keystrokes through theQB program.Before solenoid valve (SV) SV2 was opened to administer

feed gas into the adsorption column the packed bed was firstpurgedwith oxygen for approximately 10min to remove pre-adsorbed nitrogen and provide a uniform initial conditionfor simulation (Fig. 2a). Once the bed was purged and thetemperature was uniform to within 0.5K, feed gas flow wasintroduced. To ensure a bumpless transfer in pressure andflow occurred during the valve switch, feed gas was initiallydiverted through a bypass line and adjusted using a pressuremodulating valve until the desired conditions were achieved.Once pressure, temperature and composition were at

steady state, the inlet valve manifold was switched andSV2 opened while SV1 and SV3 simultaneously closed(Fig. 2b). As the nitrogen/oxygen feed gas mixture entersthe bed, a MTZ forms that progressively moves with speedvMTZ. As the MTZ passes a thermocouple and nitrogenuptake from the inter- to intrapellet region occurs, an in-crease in temperature is observed that plateaus to a steadyvalue once the MTZ has passed. Given temperature andcomposition waves move with the same speed for a typeI system (Pan and Basmadjian, 1967) temperature profilesindicate the location of the composition wave. The oxygenanalyser was placed at a relatively high position within theadsorbent bed to ensure any small fluctuations in pressureand flow that occurred around the valve switch had suffi-ciently decayed by the time the MTZ reached the analyser(Fig. 2c). The time for the oxygen analyser reading to fallto feed gas composition is defined as the breakthroughtime, t100% break. Once breakthrough has occurred and dataacquisition ceased, the bed was regenerated with oxygen inreadiness for the next breakthrough run.To validate measured process signals from each equip-

ment item, a series of calibration tests were performed.Temperature was measured using junction exposed T-typethermocouples that were deemed accurate to±0.5K as ob-tained from several water bath measurements and KTU ex-periments (Todd, 2003). Response time for these junction


Fig. 2. Schematic diagram outlining the procedure initiated for one breakthrough experiment performed on the LUB apparatus: (a) the bed is initiallypurged with oxygen until all measured pressure, composition and temperature profiles are at steady state; (b) at time zero, solenoid valve SV2 is openedwhile solenoid valves SV1 and SV3 are simultaneously closed to invoke a “bumpless” transfer from purge to feed gas; and (c) the MTZ has progressedto the oxygen analyser and the corresponding breakthrough curve is measured.

exposed thermocouples was found to be less than 1 s fromvarious step tests performed between ambient air and con-trolled water bath conditions.Oxygen compositionmeasured at axial positionz=1.55m

in the column was obtained from a Servomex ZR733 fastresponse zirconia oxygen analyser with a rated accuracy of±0.5mol% O2. The oxygen analyser is connected to the col-umn through a116 inch stainless steel tube of length 0.07mthat is located at the centre of the packed section within thecolumn. Step tests performed on this arrangement revealeda total response time of 0.4 s was required when detectinga transient change from 100 to 22mol% O2. Absolute pres-sure readings are accurate to±1 kPa. The DP cell has a ratedresponse time of 0.5 s due to signal damping while abso-lute pressure transducers (PTs) have a rated response timeless than 0.1 s (i.e., less than one scan time of the QB dataacquisition program).Each of the above-mentioned equipment items are deemed

“primary measured variables” given the experimental signalobtained from each run is used directly for data analysis.Feed flow rate, on the other hand, is classified a “secondarymeasured variable” as this parameter requires experimen-tal measurements of temperature, absolute pressure and

differential pressure (DP) drop across the annubar to inferflow rate. The rated accuracy of experimentally measuredflow rates is therefore a function of the combined errorsfor each individual element present within the equationprovided from the manufacturer, Fisher–Rosemount, for an-nubar flow. An approximate estimate for the combined erroron feed flow, reported in this dissertation as a velocity, is±0.20m s−1. The DP cell mounted directly on the annubarprovides the major limitation towards response time, whichis 0.5 s due to signal damping on the output signal.Although the adsorption column is relatively small in

diameter (dB = 0.050m), the ratio of column-to-pelletdiameters is approximately 30 and the assumption ofnegligible channeling, at least numerically, is satisfied(Mohamadinejad et al., 2000). The advantage of usinga small column diameter that just satisfies thedB : dPratio is that relatively high velocities can be maintainedfor relatively low feed volume flow rates. The adsorp-tion column itself is a section of PVC pipe with plas-tic flanges welded at both ends. This arrangement hasgood mechanical and insulation properties over the du-ration of a single breakthrough run for pressures upto 200 kPa. Wrapped around the outside of the column


wall is a 0.020m thick layer of fibreglass insulation tominimise heat exchange with the environment. This meansenergy accumulation in the wall, heat conduction along thewall and heat transfer to/from the environment is minimal.Conditions close to adiabatic are maintained within the col-umn to replicate operating conditions typically encounteredwithin an industrial scale unit. This also implies radial tem-perature gradients will be minimal over the duration of asingle breakthrough run (Pentchev et al., 2002).To assess the extent of adiabatic behaviour, our simula-

tor is run in two different modes. The first involved runswith a rigorous wall model and the wall-to-ambient heattransfer coefficient set to zero representative of conditionsthat prevail during a single breakthrough run. Under theseconditions the rigorous wall model takes on the followingform:

�W cW�TW�t

= �W�2TW�z2

+ 4dBhB,W(d2col − d2B)

(TB − TW). (7)

Within Eq. (7)dcol=dB +2�zW where�zW is the thicknessof the PVC pipe (�zW =0.005m).�W represents the densityof the wall material in kgm−3, cW the specific heat capacityof the wall material in J kg−1 K−1 and�W the thermal con-ductivity of the wall material in Wm−1 K−1. The secondmode was a true adiabatic case with the internal heat transfercoefficient from the bed-to-wall set to zero(hB,W = 0) andthe rigorous wall model deactivated, the ideal case closelyapproached within an industrial scale unit.The experimental arrangement also contains a finite void

space between the inlet valve manifold and packed bedinterface at the bottom of the column. A CSTR modelwas therefore used in our simulator to account for thisregion (lower void volume= 5.70× 10−4m3). Althoughpressures of 200 kPa could be achieved within the col-umn, instabilities in pressure and flow readings at the onsetof breakthrough could not be avoided at pressures above130 kPa so bed pressure was limited to 125 kPa for thisstudy. Despite a relatively small pressure window beingused, a wide variation in feed gas composition was con-sidered. Nitrogen–oxygen feed gas mixtures were providedfrom three different Linde gas cylinders with the followingcompositions:

(i) 10.2mol% N2 and 89.8mol% O2, denotedmixtureA,(ii) 20.0mol% N2 and 80.0mol% O2, denotedmixtureB,

and(iii) Industrial air, denotedmixture C. This is essentially

dry air at a dew point of−60◦C so the compositionis assumed to be 78.0mol% N2 and 22.0mol% O2 +Ar given argon behaves like oxygen (Chou and Chen,1994).

While mixture C lumps oxygen and argon together as theone component, mixtures A and B do not require this as-sumption as both gas bottles were purchased from the ven-dor with the above mentioned certified composition. Given

Table 2Summary of experimental conditions investigated across each break-through run

Run descriptor Feed gas Feed gas Bed pressuremixture velocity(ms−1) at z = 1.86m (kPa)

Breakthrough run 1 A 2.95± 0.26 112± 1Breakthrough run 2 A 4.15± 0.21 125± 1Breakthrough run 3 B 3.12± 0.29 112± 1Breakthrough run 4 B 4.44± 0.23 126± 1Breakthrough run 5 C 2.43± 0.35 111± 1Breakthrough run 6 C 2.89± 0.28 125± 1Note that breakthrough run 3 was used to independently adjustCm.The remaining five breakthrough runs provide validation data usingCm

obtained from run 3.

absolute pressure arises withinDm,ij it is generally as-sumed that different absolute pressure windows are requiredto observe any variation in the magnitude of moleculardiffusion. Although absolute pressure does appear withinDm,ij Eq. (4) reveals molecular diffusion arises within theintrapellet molar flux as a ratio with mole fraction. Thefirst approximation to the Chapman–Enskog procedure atlow gas densities (Hirschfelder et al., 1954, p. 539) allowsDe

m,ij /yPk to be written as follows:

Dem,ij

yPk= Cm

(Dm,ij

yPk

)

=Cm

[1.883× 10−7

√T 3(Mi + Mj)/MiMj

pPk �2ij

]

for i = 1, . . . , Nc, i �= j

and k = 1, . . . , Nc, (8)

where �ij is the Lennard–Jones collision diameter inAngstrom and is the dimensionless collision integral.Eq. (8) shows partial rather than absolute pressure is foundin terms that represent intrapellet molar flux for moleculardiffusion. Keeping total pressure relatively constant whilevarying composition and hence partial pressure will havethe same effect on Eq. (8) as holding composition steadywhile varying total pressure. Two different bed pressures ateach feed gas composition were performed experimentallywith bed temperature initially at ambient conditions. Ex-perimental boundary conditions measured during each runis summarised inTable 2. Absolute pressure atz = 1.86mrepresents the boundary condition invoked at the top ofthe sorbent bed (PT3 onFig. 2 where PT stands for pres-sure transducer). Feed gas velocity measured from theinlet flow meter was used as the lower boundary condi-tion at the valve seat of SV2 (FM onFig. 2 where FMstands for flow meter). One last issue regarding numer-ical simulation of breakthrough runs will be addressedbefore simulated results are compared with experimentaldata.


4. Axial discretisation and breakthrough simulationtimes

Webley and He (2000)highlight the importance in theaxial discretisation node distribution when modelling break-through curves within a packed bed that is initially cleanwith respect to the strongly adsorbed component. They indi-cate 10 nodes should be maintained across the MTZ to min-imise the effect of numerical dispersion. Given a structuralparameter is being calibrated against experimental data, nu-merical simulations must ensure the MTZ is adequately re-solved and that numerical dispersion is not limiting the pa-rameter fit. For this reason control volume discretisation wasperformed in a manner that ensured a minimum of 10 axialnodes were present across the MTZ atz=1.55m for break-through run 3. An appropriate a priori estimate of intrapelletradial discretisation, on the other hand, is not readily knownand for this reason a range of discretisation levels will beconsidered to assess the impact ofNr on the optimal molec-ular structural parameter. With an appropriate estimate ofCm, the remaining breakthrough runs were simulated usingNr = 5 within each axial control volume.AlthoughNr >5 was considered when fittingCm, RPSA

pilot plant simulations presented inTodd and Webley, inpress, appliedNr =5 to reduce computational overheads. Atrade-off between reasonable simulation time and model ac-curacy had to be made. Errors of only a few percent in work-ing capacity arise when comparing discretisation levels ofNr =5 and 30 over an individual sorbent pellet operated un-der isothermal (Todd and Webley, 2002) or non-isothermal(Todd, 2003) RPSA conditions. This provides some justifi-cation thatNr = 5 will be a suitable discretisation level fora first approach simulation of RPSA pilot plant data withthe DPM. This issue will be addressed further inTodd andWebley (in press).For gas mixtures A and B that involve relatively low feed

concentrations of nitrogen, axial control volume spacing of0.01m adequately resolved the MTZ. Although the LUBpacked section is 2m in length, the simulator models thepacked bed up to a height of 1.86m as the boundary con-dition invoked by PT3 fromFig. 2 renders the remaining0.14m section of sorbent irrelevant for modelling purposes.An optional input within the adsorption simulator allows theuser to manually set each axial node coordinate within thepacked bed. For breakthrough runs 1 through 4 this optionwas activated and node spacing at 0.01m increments fromthe inlet end was specified along with additional points ofexperimental interest that do not lie exactly on this uniformnode spacing. In total, 187 axial nodes were used in thebed withLB = 1.86m numerically. Activating the rigorouswall model, the DPM withNr = 5 and lower void volumeresulted in a total of 2632 simultaneous ordinary differen-tial equations (ODEs) being passed to the variable coeffi-cient ordinary differential equation solver (VODE) (Brownet al., 1989). To reproduce a breakthrough run of 160 s thatincludes 60 s of oxygen purge before the valve switch re-

quired approximately 4 h of simulation time on a 650MHzCompaq XP1000 Dec-Alpha workstation.Although mixtures A and B result in a relatively broad

MTZ, gasmixture C involved bulk adsorption of the stronglyadsorbed component and consequently axial node distribu-tion had to be refined to 0.005m. In this case 373 axialnodes were used through the sorbent bed, resulting in a totalof 5236 simultaneous ODEs with the rigorous wall modelactivated. Simulating 140 s of a LUB breakthrough run in-cluding 60 s of oxygen purge in this case exceeded three anda half days on the same Dec-Alpha workstation. Even withthis level of discretisation and simulation time, the num-ber of axial nodes observed across the MTZ was still foundto be slightly under 10. While a more refined level of ax-ial discretisation would have been preferred, breakthroughruns 5 and 6 were close to the run-time memory limit ofthe Dec-Alpha workstation when 100% CPU time was al-located.

5. Comparing experimental and numericalbreakthrough curves

To directly compare experimental and numerical resultspoint-by-point, the absolute oxygen reading is normalisedwith respect to the upper and lower limits from the break-through curve:

yBi = yBi −yBi,99% break

yBi,1% break−yBi,99% break

for i=2 at z=1.55m. (9)

Using the difference in time betweenyBi,1% break and

yBi,99% breakand knowing the speed of the MTZ, the lengthof the MTZ can be found through the following:

LMTZ= vMTZ (t99% break− t1% break). (10)

Webley and Todd (2003)highlight the most reliable exper-imental estimate ofvMTZ is obtained by dividing the mid-point time from each temperature transient with the heightof this thermocouple in relation to the sorbent bed interface.There are five thermocouples located within the bed so eachprovides an independent estimate ofvMTZ. Experimentalerror bounds onvMTZ reflect 95% confidence limits ob-tained from this technique. To remove any dependence onthe point in time where incipient breakthrough occurs, timeis reassigned such that zero now represents the midpoint ofthe breakthrough curve (i.e., time is given bytvalve switch−t50% break where t50% break occurs whenyBi = 0.50 fromEq. (9)). Reassigning time in this manner allows the shapeof experimental and numerical breakthrough curves to be di-rectly overlaid. The predicted value oft50% breakwas within2–8% of the same value measured experimentally on theLUB apparatus.


Fig. 3. GSSE as a function of the one independent fitting parameterCm

for breakthrough run 3 withNr =3 and the rigorous wall model activated.

5.1. FittingCm to breakthrough run 3

Breakthrough run 3 fromTable 2was arbitrarily desig-nated the fitting run on the basis that runs with higher andlower nitrogen–oxygen partial pressures could be used asindependent verification checks on the resulting magnitudeof Cm. Since only one parameter is being adjusted, a customwritten optimisation routine that employed a quadratic in-terpolation scheme (Chapra and Canale, 1998, pp. 349–351)was incorporated around the simulator that systematicallyvariedCm until a minimum in the global sum squared error(GSSE) along the breakthrough front was obtained:

GSSE=1∑

m=1

(yBi,LUB,m − yBi,NDGNAS,m

)2for i = 2, (11)

where1 indicates the number of experimental data pointsalong the breakthrough curve.In total, 15–20 individual numerical simulations were re-

quired at each level of radial discretisation to converge onan optimalCm value that was accurate to three significantfigures. To demonstrate the impact ofCm on the obtained fitto the breakthrough curve,Fig. 3 plots the GSSE at smallincrements around the optimalCm when Nr = 3. Fig. 3demonstrates a distinct minimum exists in the GSSE at oneparticularCm value. A monotonically increasing trend inthe GSSE is observed both above and below the optimalvalue, a result of the numerical MTZ becoming to sharp andbroad, respectively, in relation to the experimental profile.Identical trends in GSSE versusCm are observed for higherdiscretisation levels with the minimum in GSSE located ata slightly differentCm value. FromFig. 3 the global min-imum in GSSE forNr = 3 occurs atCm = 0.218. With anincreasing number of radial nodes within the sorbent pellet,a decrease in the optimalCm value was observed (Table 3).

Radial discretisation above 14 was close to the limit inrun-time memory allocation on the Dec-Alpha workstationand hence is as close toNr → ∞ as could be achieved withthe computing resources available. Numerical simulationsconsumed two and a half weeks of CPU time to converge onthe optimisedCm parameter withNr =14. WhenNr =5 wehave approached 82% of the value forCm whenNr = 14.If the value ofCm is to be interpreted physically however,a value free of discretisation error must be obtained. Thediscretisation error inCm follows 1/Nr proportionality ascan be seen inTable 3. From this knowledge, extrapolationto Nr → ∞ givesCm = 0.122.The trend ofCm versusNr fromTable 3suggests intrapel-

let profiles are sharp across the MTZ and approximately10 nodes are required to maintain a relatively consistentestimate onCm, at least to the level of radial discretisa-tion that could be achieved in this study. It is importantto estimate whether experimental uncertainty or radial dis-cretisation limits the determination ofCm. A propagation oferror analysis (not included here for brevity) suggests thaterrors in experimentally obtained parameters for the pellet(Table 1) are less than radial discretisation error across thepellet. To keep computation time within reasonable bounds,we appliedCm = 0.166 forNr = 5 to all of the remainingbreakthrough simulations to ensure adequate resolution ofthe MTZ is maintained with respect to breakthrough run 3.In essence, a higher value ofCm compensates for additionalnumerical dispersion giving the same breakthrough curvewhich would be obtained withCm = 0.122 and an infinitenumber of radial nodes. It is important to note that deter-mination ofCK was limited by uncertainty in experimentalparameters for the pellet and not byNr (Todd et al., 2005).This allowed the value ofCK asNr → ∞ (or Nr = 50 nu-merically) to be applied as a universal value onCK for anyNr . The viscous flow coefficient,Cv, was also relativelyinsensitive toNr so Cv assumes the value obtained whenNr → ∞ (Todd et al., 2005).

5.2. Discussion on the magnitude of each intrapelletstructural parameter

One way to examine the magnitudes ofCK , Cv andCm isto transform these into tortuosity coefficients. Using Eq. (5)assuming we are dealing with macropore-based intrapelletdiffusion control gives the following (Todd, 2003):

�m = �P,macro/Cm = 0.31/0.122= 2.5, (12)

�K = �P,macro/CK = 0.31/0.083= 3.7, (13)

�v = �P,macro/Cv = 0.31/0.061= 5.1. (14)

Tortuosity coefficients have been extensively discussed inthe literature and nominal values have been suggested.Ruthven (1984, p. 134)indicates experimental tortuosityfactors should lie somewhere between 2 and 6 for most


Table 3Dependence on the level of radial discretisation within the pellet to the optimised molecular diffusion structural parameter obtained from breakthrough run 3

Parameter Radial discretisation within the pellet,Nr

3 4 5 6 8 10 14

Molecular parameter,Cm 0.218 0.183 0.166 0.156 0.146 0.141 0.136Number of ODEs 1884 2258 2632 3006 3754 4502 5998

porous materials of practical interest, whileSatterfield andSherwood (1963, p. 21)indicate tortuosity coefficients varyover a 20-fold range from a review of the published litera-ture on catalytic materials.Feng and Stewart (1973)foundthe ratio�/� varied between 0.009 and 0.364 for a range ofporous materials using published data. Mathematical esti-mates on these tortuosity coefficients include:

• 1.5 (� = 1 − 12 ln �P,macro) for a porous medium that

contains randomly packed spheres of either uniform ornon-uniform size that do not overlap (Weissberg, 1963).

• 2.9 (� = 1/�P,macro) for macro-, micro and seriesmacro–micropore diffusion in a bidisperse medium(Wakao and Smith, 1962).

• 3.0 for a network of randomly interconnected uniform di-ameter cylindrical pores with no dead ends (Johnson andStewart, 1965).

• 5.0 (� = √3/�P,macro) for an array of cells that are vi-

sualised as cubes or spheres with a fraction�P,macro oftheir surface open to the neighbouring cell (Weisz andSchwartz,1962).

Assessing tortuosity coefficients obtained in Eqs. (12)–(14)with these estimates suggests the results for Knudsen andmolecular tortuosity are in good agreement with typical mag-nitudes obtained in the literature while the value obtainedfor viscous flow is slightly high. It should be pointed out,however, that a majority of these correlations assume oneoverall tortuosity coefficient for the intrapellet pore networkand rarely attribute tortuosity coefficients to each individualmechanism, as was the case in this study.

5.3. Comparing the adiabatic versus rigorous wall modelsimulation

Using the calibrated molecular structural parameter atNr = 5, the simulator was run under the adiabatic as-sumption and numerical results compared to the equiva-lent results obtained with the rigorous wall model.Fig. 4compares both numerical breakthrough curves against thecorresponding experimental profile. Both numerical profilesshow negligible variations along the entire breakthroughcurve, providing some justification that the breakthroughcurve is being measured at conditions close to adiabatic. Inaddition, both numerical profiles match LUB data to withinexperimental error. The time when 50% of the breakthrough

Fig. 4. Numerical breakthrough curves obtained from the rigorous wallmodel and adiabatic simulations of breakthrough run 3 in comparison toLUB data.

curve has passed the oxygen analyser is predicted to within2 s numerically, a favourable result given:

i. errors in predicted equilibrium isotherm capacity andselectivity using ideal adsorbed solution theory;

ii. small fluctuations in velocity and pressure at the onsetof a valve switch;

iii. error in the flow reading obtained at the inlet to thecolumn from the annubar-DP transducer arrangement;and

iv. real operating conditions that reside somewhere betweenwell mixed and plug flow are modelled as well mixedonly within the bottom void volume

all manifest in the prediction oft50% break.To further examine the extent of adiabatic behaviour

achieved within the column,Fig. 5compares predicted tem-perature profiles from both numerical simulations againstLUB data.Fig. 5demonstrates the final temperature plateauachieved after the MTZ has traversed the bed is predicted towithin experimental error with both simulations. Note errorbars of±0.5K are not shown over LUB data inFig. 5 toprevent the graph becoming too cluttered. Temperature doesnot show any obvious decrease experimentally or numeri-cally with the rigorous wall model activated for the entireduration of the breakthrough experiment, indicating heatloss to the wall and conduction along the wall is negligible


Fig. 5. Numerical temperature profiles obtained from the rigorous wallmodel and adiabatic simulations of breakthrough run 3 in comparison toLUB temperature profiles.

and near-adiabatic behaviour is experimentally attained.The time at which each temperature profile begins to rise

from the initial condition is also well predicted along thebed, indicating the speed of the thermal zone and henceMTZis well captured for the duration of the breakthrough run.The simulator imposes a bumpless transfer numerically bymaintaining the same boundary condition across the purgeand feed steps, so matching experimental temperature pro-files in absolute time indicates the transition from oxygen tofeed gas was achieved with minimal impact on experimentalpressure and velocity profiles.One mismatch observed between the LUB and corre-

sponding numerical profile for both the adiabatic and rigor-ous wall model simulations, however, is the rate at whicheach temperature profile increases as the MTZ traverses thebed. Simulated temperature transients rapidly increase as theMTZ passes each point in the bed, mimicking the early shapeof the breakthrough curve. However, experimental profilesshow a slow asymptotic approach to final bed temperatureonce local equilibrium is attained behind the MTZ. We be-lieve this discrepancy is due to a feature of the thermocouple-to-gas heat transfer resistance that is currently the subject offurther investigation.

5.4. Sensitivity analysis of breakthrough run 3

Before the remaining five breakthrough runs are dis-cussed, it is useful to investigate the importance of each in-trapellet structural parameter on the predicted breakthroughcurve. To do this sensitivity coefficients as a function ofthe three structural parametersCK , Cv andCm are derived.Lagrange interpolating polynomials are used to curve fit theresponse of each breakthrough curve at each increment intime with one parameter systematically varying around itsoptimal value (eitherCK , Cv orCm) while the other two pa-

Fig. 6. First-order sensitivity coefficients ofyBias a function ofCK , Cv

andCm obtained from breakthrough run 3. The inlay plot magnifies they-axis for viscous flow only.

rameters remain constant. Analytically, these breakthroughcurves as a function of time and structural parameter canbe represented in the following form:

yBi (tq , x) ≈NL∑m=0

yBi,m(tq)Lm(x) for i = 2

where

Lm(x) =[∏NL

n=0,n�=m(x − xn)]

[∏NL

n=0,n�=m(xm − xn)] . (15)

Within Eq. (15) x represents eitherCK , Cv or Cm andtq = (0.02q ± t50% break) for q = 1–350.NL is the order ofthe Lagrange interpolating polynomial. To find first-ordersensitivity coefficients from Eq. (15) the first derivative withrespect tox is taken

�yBi (tq , x)

�x≈

NL∑m=0

yBi,m(tq)dLm(x)

dxfor i = 2

where

dLm(x)

dx={∑NL

n=0,n�=m

[∏NL

p=0,p �=m,p �=n(x − xp)]}

[∏NL

n=0,n�=m(xm − xn)] . (16)

For this analysis two values above and below the optimalstructural parameter were considered soNL=3. A cubic La-grange interpolating polynomial for breakthrough run 3 wasadequate to resolve each sensitivity coefficient. Each struc-tural parameter was varied in fractional increments aroundits optimal valuewith the Knudsen structural parameter vary-ing between 0.073�CK �0.095, viscous flow parameter be-tween 0.055�Cv�0.067 and molecular parameter between0.131�Cm�0.226. Sensitivity coefficients obtained from


Table 4Tabular summary of LUB versus numerical data from each breakthrough run obtained withNr = 5 andCm = 0.166Parameter (mixture A runs) Breakthrough run 1 Breakthrough run 2

LUB Numerical LUB Numerical

t50% break(s) 100.5± 0.1 108.3 71.1± 0.1 73.6t4% break− t50% break(s) −9.9± 0.5 −9.9 −9.9± 1.1 −9.0t80% break− t50% break(s) 4.6± 0.1 4.9 3.8± 0.1 4.6t96% break− t50% break(s) 11.5± 0.9 10.4 7.9± 0.3 9.8vMTZ (m s

−1) 0.0145± 0.0009 0.0146 0.0234± 0.0015 0.0217LMTZ (m) 0.311± 0.040 0.296 0.417± 0.060 0.407Axial nodes in MTZ N/A ≈ 29 N/A ≈ 40

Parameter (mixture B runs) Breakthrough run 3a Breakthrough run 4LUB Numerical LUB Numerical


−1) 0.0183± 0.0002 0.0183 0.0273± 0.0028 0.0275LMTZ (m) 0.187± 0.006 0.187 0.267± 0.039 0.263Axial nodes in MTZ N/A ≈ 18 N/A ≈ 26

Parameter (mixture C runs) Breakthrough run 5 Breakthrough run 6LUB Numerical LUB Numerical


−1) 0.0237± 0.0050 0.0222 0.0288± 0.0010 0.0274LMTZ (m) 0.042± 0.012 0.028 0.048± 0.007 0.039Axial nodes in MTZ N/A ≈ 5 N/A ≈ 7aThis run was used to experimentally calibrateCm.

this analysis are presented inFig. 6. Fig. 6reveals molecularand Knudsen diffusion is of a similar order of magnitude forthe entire passage of the breakthrough curve. While molec-ular and Knudsen diffusion both contribute significantlyto the shape of the breakthrough curve, the inlay plot onFig. 6 indicates viscous flow is more than two orders ofmagnitude smaller. Sensitivity analysis of single componentuptake rates indicated a similar difference between Knudsenand viscous flow sensitivity coefficients (Todd, 2003), sug-gesting the importance of Knudsen diffusion over viscousflow for a pure component system carries similar weightwhen a binary gas mixture in introduced into the intrapelletpore network.

6. LUB and numerical results from remainingbreakthrough runs

With a calibrated set of experimental parameters deter-mined for LiLSX pellets, the simulator was used to pre-dict the remaining breakthrough runs usingNr = 5 andCm =0.166.Table 4summarises all six LUB and numericalruns in terms of some prescribed points along the break-

through curve and derived quantities. This table comple-ments the graphical presentation of each breakthrough curveobtained numerically and experimentally inFig. 7. All re-sults presented in this subsection, includingTable 4, relateto rigorous wall model simulations. Experimental versus nu-merical results for the speed and width of the MTZ gener-ally agree to within experimental error. Numerical simula-tions also capture the shape of each breakthrough curve well,from a disperse wave that takes approximately 30 s to passthe analyser to a sharp wave that is complete in less than 2 s.Although it may seem breakthrough runs 1–4 provide a bet-ter prediction of the breakthrough curve, the time scale formixture C runs are very short and in general the deviationbetween numerical and LUB data is less than 0.2 s, which isclose to the scan rate of the LUB data acquisition program.Runs that involve a low nitrogen feed concentration,

breakthrough runs 1–4, indicate the MTZ is adequatelyresolved in accordance with theWebley and He (2000)criterion discussed in Section 4. Resolving the packed bedinto 0.005m increments for mixture C runs, on the otherhand, meant that only 5 and 7 axial nodes for breakthroughruns 5 and 6, respectively, were achieved across the MTZ.Breakthrough runs 5 and 6 were close to the upper limit of


Fig. 7. Comparing LUB and numerical breakthrough curves obtained with the rigorous wall model for the five remaining breakthrough runs usingCm = 0.166: (a) breakthrough run 1; (b) breakthrough run 2; (c) breakthrough run 4; (d) breakthrough run 5; and (e) breakthrough run 6.

run-time memory available on the Dec-Alpha workstationas discussed in Section 4. Although the number of nodesacross the MTZ is low, results fromTable 4andFigs. 7dand e show simulation results are still in good agreement

with experimental data. This discussion does highlight theefficacy in developing a discretisation scheme that can dy-namically adjust node spacing to follow a steep movingfront such as those experienced in the passage of a MTZ


during a breakthrough run (Sereno et al., 1991; Cruz etal., 2002, 2003). These techniques maintain the axial nodedistribution density very high across the MTZ but allow theremaining regions of the packed bed to be relatively sparsein node distribution density. ProvidedLMTZ/LB � 1, theentire axial domain no longer requires node distribution tobe dense, just the region where the MTZ is passing through,significantly reducing the number of ODEs to solve andhence CPU time.Breakthrough curves obtained experimentally and numer-

ically for one particular feed gas mixture at two differentvelocities and pressure did not show any significant varia-tion. This suggests large changes in pressure and velocityare required to effect any change on the breakthrough curvefor one particular feed gas mixture. Given pressures above130 kPa could not be used, the alternative method of vary-ing feed gas composition was found to significantly alter theshape of the breakthrough curve. This result justifies the useof different feed gas mixtures when examining moleculardiffusion through Eq. (8).

7. Verifying intrapellet mass transfer as the limitingresistance

To minimise mass transfer resistance at the external sur-face of the pellet, relatively high interpellet gas velocitiesare required to ensure the Biot mass transfer numberBimassis greater than 1 (Yang, 1997, p. 124). Interpellet gas veloc-ities must also ensure the Peclet numberPeB,massis greaterthan 100 to eliminate axial dispersion of the gas phase (Liaoand Shiau, 2000). With each intrapellet structural parameternow ascertained the remaining discussion shall quantify theomission of axial dispersion and external film mass trans-fer resistance. Second-order mass dispersion is characterisedthrough the axial Peclet number as follows:

PeB,mass= vBz

Deax. (17)

To determinePeB,massa correlation forDeax is required that

accounts for intrapellet mass transfer. The model ofGunn(1987)is a suitable candidate, which takes on the followingform:

Deax=

v2Bd2P (1− Pax)

2

23.14Dm,ij

(�B

1− �B

)

+ v3Bd3PPax(1− Pax)

3

535.3D2m,ij

(�B

1− �B

)2

×{exp

[ −23.14Dm,ij

Pax(1− Pax)vBdP

(1− �B

�B

)]− 1

}

+ Dm,ij

�ax. (18)

Within Eq. (18) Pax represents a probability factorfor axial dispersion, given by the expression 0.17 +0.33 exp(−24�B/vB�BdP �B

g ) for spherical geometry.�ax

is a tortuosity coefficient that defines the distance a packetof gas travels within the interpellet void space in relationto the unhindered distance (=1.4 for spherical geometry).Eq. (18) is applicable to bulk conditions away from the re-taining column walls across all ranges ofReP investigated inthis paper (Gunn, 1987). To confirm the plug flow assump-tion across each breakthrough run, output profiles from thesimulator were used to calculateDe

ax from Eq. (18) and thenPeB,mass. Results from this calculation atz = 1.55m, theaxial location where the oxygen analyser was located, areshown inFig. 8 for each feed gas mixture considered. Themagnitude ofPeB,massnever drops below 700 at any pointduring each simulation. According toLiao and Shiau (2000)PeB,massgreater than 100 indicates convective gas motiondominates interpellet mass transfer, justifying the plug flowassumption during the experimental determination ofCm.In addition toPeB,massfor breakthrough runs 1, 3 and 6,

Fig. 8shows the corresponding trends forBimass. Themagni-tude ofBimasswas calculated using output profiles obtainedfrom the simulator through Eq. (19) with the Bosanquet for-mula used to estimate the effective intrapellet diffusion co-efficientDe

i (Yang, 1997, p. 123):

Bimass= kP,BdP

2Dei

= 0.179�B

Dm,ij

Dei

Re0.641P Sc1/3, (19)

wherekP,B is the external film mass transfer coefficient inm s−1,ReP is the dimensionless pellet Reynolds number andSc the dimensionless Schmidt number. For each feed gasmixtureBimassis of order 30 or greater for the entire dura-tion of each breakthrough run atz=1.55m, indicating exter-nal film mass transfer resistance was of minimal importanceover the period of experimental interest (Bimass is at leastone order of magnitude greater than 1). This analysis, how-ever, has neglected the impact of radial dispersion across theinterpellet domain within the column. This parameter is notas simple to quantify for an adsorption system and as suchis assumed to have minimal impact on experimental profileswhen near-adiabatic conditions arise (Pentchev et al., 2002).In the absence of multiple oxygen analysers placed at oneaxial location but different radial coordinates, this mecha-nism of dispersion is very hard to quantify. Similarly, a ra-dial plus axial discretisation scheme across the interpelletdomain, in addition to intrapellet discretisation would sig-nificantly increase computational time beyond those alreadyobserved for breakthrough simulations of this study.

8. Conclusions

Molecular diffusion in beaded LiLSX pellets in theO2/N2system was studied theoretically and experimentally usingthe DGM incorporating viscous flow. Unlike lumped mod-els such as the linear driving force model, which requiresindependent parameters for each gas species, the dusty gasmodel requires only structural coefficients related to theadsorbent. The single structural parameterCm has been


Fig. 8. Peclet mass transfer number at the bed scale and Biot mass transfer number for each gas mixture atz = 1.55m, the location of the oxygenanalyser, for breakthrough runs 1, 3 and 6.

quantified using breakthrough curves obtained from a suit-ably instrumented LUB apparatus. The physically realisticvalue of Cm = 0.122 was found (free of numerical dis-persion error) which resulted in a tortuosity of 2.5, in therange expected. A sensitivity analysis revealed that the twostructural parameters,CK andCm are of similar importancewhen examining the passage of the MTZ within a packedbed of LiLSX pellets. Viscous flow was found to be oflesser importance.The molecular diffusion structural parameter was cali-

brated using one breakthrough curve, breakthrough run 3. Inconjunction with Knudsen diffusion and viscous flow struc-tural parameters obtained from an independent set of ex-periments (Todd et al., 2005), good agreement was foundacross all of the remaining five breakthrough runs. Exper-imental composition response across each LUB run showsgood agreement with numerical predictions.

Notation

B permeability coefficient of the intrapellet porenetwork, m2

Bimass Biot mass transfer number, dimensionlesscW heat capacity of the wall material per unit kg of

sorbent, J kg−1 K−1CK , Cv,Cm

structural parameters within effective diffusioncoefficients related to Knudsen diffusion, vis-cous flow and molecular diffusion, respectively,dimensionless

dcol, dB ,dP

external diameter of the adsorption column, in-ternal diameter of the adsorption column andequivalent diameter of the adsorbent pellets, m

Dax axial dispersion coefficient, m2 s−1DK , Dm Knudsen and molecular diffusion coefficients,

m2 s−1hB,W heat transfer coefficient between interpellet gas

and column wall, Wm−2 K−1HB

g enthalpy of interpellet gas phase, J gmole−1kP,B external film mass transfer coefficient, m s−1Li the ith term of anNLth order Lagrange interpo-

lating polynomial, dimensionlessLB ,LMTZ

length of the adsorption column and length ofthe mass transfer zone, respectively, m

M molecular weight, g gmole−1Nc number of components present within sys-

tem,dimensionlessNi intrapellet molar flux per unit area for compo-

nent i, gmolem−2 s−1Nr number of radial control volumes within pellet,

dimensionlessNL order of Lagrange interpolating polynomial, di-

mensionlesspB , pP pressure on absolute scale in inter- and intrapel-

let regions, respectively, PaPax probability factor for axial dispersionPe Peclet number, dimensionlessr intrapellet radial coordinate for spherical geom-

etry, mR universal gas constant,(8.314 J gmole−1 K−1)Re Reynolds number, dimensionlessSc Schmidt number, dimensionless


t time, sT , TW combined temperature of inter- and intrapellet

regions, wall temperature, KUBg , U

Pg internal energy of the inter- and intrapellet gas

phases, respectively, J gmole−1UPa , U

PS internal energy of the adsorbed and solid phases,

respectively, J kg−1vB ,vMTZ

interstitial gas velocity and velocity of masstransfer zone, m s−1

y, y mole fraction in gas phase and ratio of molefractions for analysing breakthrough curves, di-mensionless

z length coordinate variable, m

Greek letters

�B , �P void volume of the inter- and intrapellet regions,respectively, dimensionless

�viscous,�kinetic

viscous and kinetic constants of pressure dropequation, dimensionless

�W thermal conductivity of the wall material,Wm−1 K−1

�B , �P inter- and intrapellet gas phase viscosity, respec-tively, Pa s

�B , �g,�P

density of the interpellet gas, intrapellet gas andadsorbent pellet, kgm−3

� Lennard–Jones collision diameter,A�ax tortuosity coefficient related to axial dispersion

through the interpellet domain, dimensionless�K , �m,�v

tortuosity coefficient for Knudsen diffusion,molecular diffusion and viscous flow, dimen-sionless

dimensionless transport collision integral, di-mensionless

Acknowledgements

The authors gratefully acknowledge the financial supportof Air Products and Chemicals, Inc. and the Department ofChemical Engineering, Monash University.

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