Chemical Engineering Science 66 (2011) 1953–1961

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m

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

1D and 2D simulations of partially wetted catalyst particles: A focuson heat transfer limitations

Frederic Bazer-Bachi, Frederic Augier n, Bruno Santos

IFP Energies nouvelles, Rond-point de l’echangeur de Solaize, BP3, 69360 Solaize, France

a r t i c l e i n f o

Article history:

Received 30 September 2010

Received in revised form

19 January 2011

Accepted 22 January 2011Available online 2 February 2011

Keywords:

Trickle bed

Partial wetting

Heat/mass transfer

Catalyst shape

Effectiveness factor

Evaporation

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ces.2011.01.052

esponding author. Tel.: +33 4 37 70 21 42; fa

ail address: frederic.augier@ifpenergiesnouve

a b s t r a c t

Hydrodynamics and transport phenomena inside Trickle Bed Reactors are strongly modified when

superficial liquid velocity is not sufficient to insure a perfect wetting of catalyst particles. For this

reason, the impact of partial wetting on catalyst effectiveness has been widely studied in past, in case of

isothermal reactions. Heat transfer limitations, inside or outside catalyst particles, have also been

investigated, but only in case of total wetting.

In the present study, the effect of partial wetting is quantified numerically for various kinetics, mass

and heat transfer limitations. When possible, an analogy is proposed between 1D and 2D models of

particles. Robust rules are proposed to take into account for partial wetting in 1D-radial models of

particles.

The case of partial evaporation inside catalyst pores is also studied. Evaporation phenomena is

approximated following a simple approach. Performed calculations show that the effect of partial

wetting can affect very strongly catalyst effectiveness factors, especially in case of evaporation in the

pores. The use of 1D approximations gives a good prediction of the global catalyst efficiency, as far as

boundary conditions are symmetric between heat and mass transfer. Otherwise, 1D models are not

relevant anymore.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Trickle Bed Reactors (TBRs) are fixed-bed gas–liquid–solidcatalytic reactors operated in co-current downward direction.They are extensively used in refining industry thanks to theirsimplicity and low operating costs. Partial wetting phenomenonoccurs at surface of catalyst particles when liquid superficialvelocity (vsl) is below 1 cm s�1 approximately. Major part ofindustrial TBRs are operated at vsl41 cm s�1, and are not con-cerned by poor wetting efficiency. This is the case of middledistillate desulphurisation and various hydrogenation reactors.But partial wetting is of major importance in case of pilotreactors, as they are often operated at very low liquid flow rates,due to their small height and the similitude of mean residencetime with industrial columns. Moreover, with the increasinginterest on valorisation of heavy oil cuts or residues, industrialTBRs operated at low vsl tend to be more common.

Many studies have been done in the past 4 decades in order topredict both the wetted surface of catalyst (f) and the catalysteffectiveness factor (Z) of a partially wetted particle. Concerningthe prediction of f, some of the most known correlations are those

ll rights reserved.

x: +33 4 37 70 20 08.

lles.fr (F. Augier).

given by Al-Dahhan and Dudukovic (1995), Burghardt et al.(1995), Mills and Dudukovic (1981), and Larachi et al. (2001).Liquid flowrate is always reported as the most important para-meter on the wetting efficiency f. Recent works have also beendone in order to include effects of various geometrical, flow orphysical properties (Palmisano et al., 2003; Baussaron et al., 2007;Lappalainen et al., 2008; Julcour-Lebigue et al., 2009). The otherfield of research deals with the prediction of catalyst effectivenessfactors Z under partial wetting conditions. As reactions operatedin TBRs often involve hydrocarbons and hydrogen, diffusionlimitations, if any, generally rather concern the liquid reactant.For this reason a large number of publications deals withapparent kinetics of only one liquid reactant. In these conditions,partial wetting reduces catalyst effectiveness when diffusion isthe limitative parameter. As a result, Thiele modulus F can bemodified to take into account for partial wetting f as follows:

Fapp ¼Ff

ð1Þ

Modification of the Thiele modulus by f can be seen as amodification of the diffusivity of the reactant inside the particleby the partial wetting, as proved by Julcour-Lebigue et al. (2007)by a theoretical approach

De,app ¼ f 2De ð2Þ

F. Bazer-Bachi et al. / Chemical Engineering Science 66 (2011) 1953–19611954

or as a modification of the characteristic length L used in theThiele Modulus, for a first-order reaction (Dudukovic, 1977):

Fapp ¼V

Sw

ffiffiffiffiffiffik

De

sð3Þ

In this case, L is the ratio between the particle volume V and thewetted surface Sw. Classical analytical rules to calculate Z as afunction of F for different particle shapes can then be used. Forinstance, for a spherical particle

Z¼ 1

Fappcothð3FappÞ�

1

3Fapp

� �ð4Þ

Modification of Thiele modulus by f is also usable for higher ordersof reactions (Goto et al., 1981). Tan and Smith (1980), Sakornwimonand Sylvester (1982) and Zhenglu et al. (1984) proposed similarapproximate analytical solutions to include in Eq. (4) effects ofpartial wetting and external mass transfer limitation:

Z¼ f

Fcothð3ðF=f ÞÞ�ðf=3FÞ

1þð3F=BimÞðcothð3ðF=f ÞÞ�ðf=3FÞÞð5Þ

where Bim is the mass Biot number.For spherical particles, Yentekakis and Vayenas (1987) and Lai

and Tan (1991) studied the effect of the shape of the wettedsurface and showed that for a given wetting efficiency f, effec-tiveness factor depends also on the number of wetted zones andtheir location: fractioning the wetted surface enhances catalystperformance. Same tendencies were obtained by Tsamatsoulisand Papayannakos (1996) considering cylindrical catalyticparticles. Augier et al. (2010) showed that the modification ofthe Thiele modulus by f is usable for various shapes of catalystparticles, as cylinders, trilobes or quadrilobes.

Van Houwelingen et al. (2007) reported the importance ofthe wetting efficiency distribution inside TBRs to estimate Z,rather than the mean wetted surface fraction per particle, asdone usually. This observation leads to a channelling modelincluding wetting efficiency distribution (Van Houwelingenet al., 2007).

Yentekakis and Vayenas (1987) and Zhu and Hofmann (1997)studied the case of kinetics involving gas and liquid reactants. Thecase differs from single reactant kinetics only when apparentThiele modulus of each reactant, as defined by Bischoff (1965), ishigh (i.e. 40.5). In this case, the effect of partial wetting can bevery complex. When transfer of gas phase reactant is limitant,either because of a high Thiele modulus or low external liquid/solid or gas/liquid mass transfer coefficients, catalyst efficiencycan be increased by partial wetting. Indeed, gaseous reactant candirectly transfer into the particle through its dry surface fraction.

Heat transfer potentially has a very strong impact on catalystefficiency. For totally wetted particles, Weisz and Hicks (1962)showed that a low thermal conductivity or very high heat ofreactions can lead to important temperature gradients insideparticles. In such cases, reaction rates are increased comparedto isothermal conditions. In case of partial wetting, there is animportant lack of knowledge to estimate the impact of heattransfer on catalyst effectiveness. Moreover, when exothermicityis important inside grains, possible depletion or partial evapora-tion of the liquid phase occurs inside a fraction of the pores,depending on local capillary pressure and partial vapour pressureof the different species (Jaguste and Bhatia, 1995; Mikhailovaet al., 2003). Reactivity in the gas phase is often very differentfrom the liquid one, and consequently catalyst effectiveness isstrongly affected by evaporation phenomena.

The object of this study is to investigate reactions whentemperature is not homogeneous inside particles, and to quantifythe impact of partial wetting in various cases. Evaporation phenom-ena induced by heat of reaction are also discussed. Another purpose

of the study is to supply simple rules to include partial wettingeffects into classical fixed bed reactor models. Reactor modelsdeveloped in oil refining processes are often based on simpleplug-dispersion 1D-axial description concerning hydrodynamicsand transport of species, plus a 1D-radial description of the catalystparticles. This kind of model is called 2�1D models as onedimension describes axial evolution inside the reactor and onedimension describes concentration and temperature gradientsinside catalyst particles. Different cases are investigated, includingexternal mass/heat transfer and exothermic reactions, in presence ofpartial wetting, in order to provide modifications of apparentphysical properties well suited for 2�1D models.

2. Modelling

Alternatively to dual series solutions (Mills and Dudukovic,1981) and fundamental solutions (Palmisano et al., 2003), a finite-element method is used to perform 1D and 2D effectiveness factorcalculations. Porous particles are supposed to be filled with liquid,even in case of partial external wetting (evaporation inside thepores is discussed separately in the last paragraph). Calculationsare performed with the commercial solver Comsol 3.5. Transportequation of chemical species is

�Der2cþRx ¼ 0 ð6Þ

Heat transport equation is

�lr2TþRxDH¼ 0 ð7Þ

Boundary conditions at the surface of particles are

ce�cs ¼1

Bimrc9s ð8Þ

Te�Ts ¼1

BiHrT9s ð9Þ

With different values of Bim or BiH following the surface type(wetted or dry).

When 1D-radial simulations are performed, the boundarycondition at the centre of the particle is a symmetry.

Catalyst effectiveness factor Z is defined as the ratio betweenthe real particle productivity and its value in case of homoge-neous concentration and temperature profile (c¼cs, T¼Ts),i.e. when reaction is not limited by diffusion for isothermalconditions:

Z¼RRR

V Rxðr,yÞdV

Rx,maxVð10Þ

Studied kinetics are of nth order (0ono3): Rx ¼ kcn, or ofLangmuir type: Rx ¼ kc=ð1þkLcÞ. k follows an Arrhenius law:k¼ k0 expð�EA=RTÞ. Generalised Thiele modulus is then calculatedas established by Bischoff (1965)

F¼LRxðcsÞffiffiffi

2p

Z cs

0DeRxðcÞdc

� ��1=2

ð11Þ

When non-isothermal reactions are studied, Prater (1958)and Weisz and Hicks (1962) showed that pertinent parametersfor the determination of Z are

b¼csð�DHÞDe

lTsð12Þ

and

g¼ �EA

RgasTsð13Þ

b has been referred as thermicity or heat generation function(Satterfield, 1970). It represents the ratio between heat

F. Bazer-Bachi et al. / Chemical Engineering Science 66 (2011) 1953–1961 1955

generation and heat conduction in the grain and characterises themaximal difference of temperature in the particle divided bythe temperature at its surface (DT=Ts). g is the Arrhenius group(Satterfield, 1970) and represents non-dimensional activationenergy for the chemical reaction.

Different shapes of wetting surfaces are studied. Polar ‘‘cap’’shapes are generally considered concerning spherical particles,but bipolar or equatorial shapes are also simulated. Concerningextruded particles as cylinders or trilobes, axial or radial wettedsurface are studied.

2.1. Evaporation phenomenon

Evaporation appears in pores when local pressure, equal tobulk pressure plus capillary pressure, is below the partial pressureof the liquid present in the catalyst particles. For the developmentof the physical model, general assumptions were made, based onthe works developed earlier on this subject (Harold, 1988; Kirillovet al., 2002; Mikhailova et al., 2003). Compared to cited previousworks, the present study involves a simplistic transport modellinginside the pores, as classic diffusion model is used instead of acapillary penetration model. But the model is applied to a morerealistic 2D geometry and focuses on the specific effect of partialwetting on catalyst effectiveness. The grain is a spherical porouscatalyst particle. A pore density distribution function is used todescribe the porous structure of the grain

f ðrÞ ¼r expð�r=rÞ

r2ð14Þ

Volume fractions of liquid and gas phases in the catalyst arecalculated by integrating the pore density distribution function,assuming that the pores with a radius smaller than a critical poreradius (rn) are filled with liquid

al ¼

R r�0 f ðrÞdrR10 f ðrÞdr

¼ 1�exp �r�rð Þ r�rþ1ð Þ ð15Þ

Capillary pressure is assumed to be given by the followingexpression (Defay and Prigogine, 1966):

Pcap ¼2scosj

rð16Þ

in pores of radius rn, P¼Pcap+Pe¼Pvap then

r� ¼2scosjPvap�Pe

ð17Þ

This expression is only usable when PvapZPe. Vapor–liquidequilibrium in the pores is modelled by the generalised Kelvinequation (from Jaguste and Bhatia, 1995)

He¼Pe

Rgas T

rl

Mw

Pvap

Peexp �

2scosjRgas T r�

Mw

rl

� �ð18Þ

where He is the constant of Henry defined by

He¼ CgCl ð19Þ

The existence of two phases with different properties in the sameporous media, and the dynamic modelling and simulation of theevaporation phenomena is a great challenge due to the difficultyto take into account the thermodynamic equilibrium at gas–liquidinterface and to insure the continuity of mass flux between gasand liquid. A work developed by Haroun et al. (2010) presents asolution based on a numerical model that solves for the equationsof mass transport and volume fraction assuming the resolution ofa single equivalent fluid. The mathematical model developedintroduces an extra flux c in the mass diffusion equation (23)that arises from the discontinuity of concentration at the gas–liquid interface, resulting from the Henry’s law.

Equivalent concentration is defined as an average of gas andliquid concentration pounded by their respective volume fraction

C ¼ agCgþalCl ð20Þ

Following authors, equivalent diffusivity D is calculated as

D¼DlDg

agDlþalDgð21Þ

Equivalent thermal conductivity is calculated following anequivalent expression:

l¼lllg

agllþallgð22Þ

The transport equation of c is then

rðDrCþcÞþalRxlþagRxg ¼ 0 ð23Þ

where c is extra flux that arises from concentration discontinuityat interfaces

c¼�Dð1�HeÞClral ð24Þ

Liquid and gas concentrations are calculated from concentra-tion fields of C by the following expression:

Cl ¼Cg

He¼

C

alþð1�alÞHeð25Þ

3. Results and discussion

A common presentation of results is to report Z versus themodified Thiele modulus Fapp (¼F/f). Thus, comparison betweeneffectiveness factors at different values of f, including f¼1, showshow 2D cases (when fo1) can be reduced to equivalent 1D caseswith different property modifications.

3.1. Isothermal reactions

First-order isothermal reactions are studied with no externaltransfer limitations. Reactant is considered non-volatile and thereis no external mass transfer limitation on the wetted surface:Bim,d¼0 and Bim,w¼N. Concentration profiles inside half-wettedspherical particles are shown in Fig. 1, for two different Thielemodulus: F¼1 and 5.

Comparison between effectiveness factors at different valuesof F and f are given in Fig. 2a. At equal F/f, catalyst efficiencies aresimilar for 0.4r fr1, but small discrepancies are observed atf¼0.2 in the intermediate range 0.5rF/fr5. The case of verylow wetting efficiencies is not discussed here, as the validity ofsome assumptions is questionable. For instance, total internalwetting of the particles is not evident at fo0.2, and should beexperimentally verified before any theoretical analysis. Fig. 2band c reports, respectively, the effect of the particle shape andkinetics (Langmuir type (k¼1, kL¼10) or second-order reaction).Results show that approximately when f40.3, Eqs. (1) and (2) areusable for any shape of catalyst and any single reactant kinetics,as Z only depends on F/f.

The effect of the shape of the wetted surface is also investi-gated, in two different cases. First case reported in Fig. 3a,corresponds to a spherical particle wetted by different shapes ofsurfaces: single polar surface, bipolar, bipolar+equatorial belt and‘‘uniform’’ wetting, i.e. wetted surface is uniformly spread on theparticle, as described by Lai and Tan (1991). Z is sensitive to thewetted surface shape in the intermediate range of F [0.3–10]. Inthis range, Z increases with the spreading of the wetted surface.This result is coherent with results from Lai and Tan (1991)and Yentekakis and Vayenas (1987). Results also complete thoseof Palmisano et al. (2003). They studied the effect of the shape on

Fig 2. Z¼ f(F/f) under different conditions: (a) first-order kinetics at various f; (b) v

fo1 (plots).

Fig. 1. Snapshot of concentration profiles inside spherical particles. Case of

isothermal reactions, first-order kinetics, non-volatile liquid reactant: f¼0.5,

F¼1 and 5 and C¼1 (black), C¼0 (white).

Fig. 3. Effect of the shape of the wetted surface (a) on sph

F. Bazer-Bachi et al. / Chemical Engineering Science 66 (2011) 1953–19611956

the wetting efficiency, and showed that the numerical methodbased on fundamental solution is as accurate as finite-elementsolutions. Concerning a slab geometry, they showed that thespreading of the wetted surface is positive on the effectivenessfactor.

The second case deals with the wetting of a cylindrical particle,for which ‘‘axial’’ and ‘‘radial’’ wetting are studied. Tsamatsoulisand Papayannakos (1996) reported an effect of the shape of thewetted surface, confirmed by the results obtained. For example,Z decreases significantly for very thin cylinders with a radialwetting, as observed in Fig. 3b, L/rp¼10 cylinder.

3.2. Effects of mass transfer limitations

Liquid reactant is considered as non-volatile. Thus, the bound-ary condition applied on the dry surface is Bim,d¼0. Simpleconsideration on external mass transfer coefficient K should leadto its multiplication by f to take into account for the loss of wettedsurface. Without other consideration, mass Biot number Bimshould be multiplied by f also. But the very good agreement ofEq. (5) with numerical simulations performed by Sakornwimonand Sylvester (1982) proves that the apparent mass Biot number,is rather modified by f as follows:

Bim,app ¼Bim

fð26Þ

arious catalyst shapes; (c) second-order and Langmuir kinetics: f¼1 (lines) and

eres and (b) on cylinders. First-order kinetics: f¼0.4.

F. Bazer-Bachi et al. / Chemical Engineering Science 66 (2011) 1953–1961 1957

This is due to the modification of the apparent diffusivity De,multiplied by f2 as in Eq. (2), in addition to the modification ofK(Kapp¼Kf). Fig. 4 illustrates this point by comparing differentresults of 1D and 2D simulations with Eq. (5).

3.3. Heat transfer effects

Heat transfer phenomenon affects catalyst effectiveness factorin different possible cases. First, when conductivity inside parti-cles is not sufficient to insure internal homogeneous temperatureprofiles. When values of b deviate sufficiently from 0, a fraction ofthe particle volume is heated or cooled by the reaction. Followingthe value of g, temperature deviations can lead to very importantmodifications of effectiveness factor.

The other thermal effect corresponds to an external heattransfer limitation. When Biot number relative to heat transferBiH is too low compared to the heat generated or consumed by thereaction, temperature inside particles also deviates from the bulktemperature. This can lead to thermal runaway or ‘‘hot-spots’’inside fixed bed reactors. This phenomenon can occur even forsmall values of b and g, for example in case of hydrocarbonhydrogenation or hydrodesulphurisation. Complex effects are alsopossible when external heat and mass transfer are limitative.

Fig. 4. Catalyst effectiveness factor versus F/f for different values of f and

Bim¼0.1. Case of a spherical particle. Comparison between 2D simulations (dots)

and 1D model of Sakornwimon (lines, Eq. (5)).

Fig 5. Z versus F/f for at g¼20, b¼0.1: (a) wetted surface Bim¼BiH¼N; dry surf

In regard with the difference of thermal conductivity betweengas (ll) and liquid (lg), BiH is often much higher at the wettedsurface than at the dry surface. As a first approximation, BiH isconsidered null at the dry surface of the particles, and infinite at thewetted surface. Thus there is a symmetry between heat and massboundary conditions. In these asymptotic conditions, comparisonsbetween 1D simulations, at f¼1, and 2D simulations of partiallywetted particles tends to prove that the b parameter can beconsidered constant in regards with f. This is illustrated in Fig. 5awhere calculations are operated at constant b value and equivalentheat and mass transfer boundary conditions. As b involves the ratioDe/l, it suggests that l is modified by f in similar manner than De, i.e.

lapp ¼ f 2l ð27Þ

Similar calculations are reported in Fig. 5b considering also noheat transfer limitation on the dry side of the particle (BiH,d¼N).In this case, partial wetting decreases slightly Z in the range ofintermediates F/f values, as an additional external heat exchangeon the non wetted side occurs. In this case, forms of mass andheat transport boundary conditions are no longer symmetric.Some discrepancies appear between 1D (f¼1) and 2D (fo1)models, in the range of Thiele modulus 0.4oFo2. However,constant b value is still the most acceptable approximation.

Concerning the hydrogenation of petroleum cuts, partial wettingefficiencies occur only with heavy molecules as gasoil or heavier. Asreaction rates are rather slow, reactors are operated at low super-ficial liquid velocities. Heavy molecules are generally characterisedby low volatility. Thus external mass transfer is usually negligible onthe dry surface. As discussed earlier, heat transfer is generally smallon the dry surface compared to the wetted one, but exceptions arepossible in case of very high superficial gas velocity. Consequently,the assumption of symmetric heat and mass boundary conditions isgenerally relevant in this field of application.

Limitation by external heat transfer appears when heatexchange at the surface of the particles is not sufficient to transferthe heat produced by the reaction. Energy balance at the surfaceof a particle is written as

VZkcsDH¼�ShðTs�TeÞ ð28Þ

Heat transfer limitation leads to an increase of the temperature at thesurface of the particles Ts. For a given value of BiH, it exists amaximum value of F above which the temperature in the particlestarts to increase, leading quickly to safety problems or undesirablereactions. Weisz and Hicks (1962) proposed a conservative estimationof the critical Thiele modulus in case of total wetting condition. Incase of partial wetting, different phenomena occur with oppositeeffects on Z. The partial wetting decreases the reaction rate, especiallyat high F, because of the loss of inlet surface for liquid reactants. But

ace Bim¼BiH¼0 and (b) BiH¼N; wetted surface Bim¼N; dry surface Bim¼0.

F. Bazer-Bachi et al. / Chemical Engineering Science 66 (2011) 1953–19611958

in the same time, if heat transfer is lower on the dry surface, for agiven heat flux produced in the particle, partial wetting decreases alsothe amount of heat exchange. Apparent heat transfer coefficient ismultiplied by f, as mass transfer coefficient. And as b is unchanged byf, apparent heat Biot number is

BiH,app ¼BiHf

ð29Þ

As an example, Fig. 6 presents different simulations performedat fixed BiH/f. Results show also that Z depends only on b, F/f andBiH/f. In this case, when F/f47, and for BiH/f¼0.1 and g¼20, anincrease of Z is observed for any value of f. It corresponds to thebeginning of heat transfer limitation and a temperature increaseinside the particle. Calculations are not possible above F/f410because of numerical divergence due to catalyst thermal runaway.

Another treated example cumulates mass and heat transferlimitations, in case of more realistic values of b concerninghydrogenation reactions. Following conditions are used: b¼0.01or 0.001 at f¼0.5, Bim¼40, BiH¼0.5. Results are compared tosimulations operated under different conditions (see Fig. 7):

–

Figg¼

f¼1 and no external limitations

– f¼0.5 and external mass transfer limitation only (Bim¼40,

BiH¼1000)

– f¼1 but Bim/f and BiH/f are kept constant (Bim/f¼80, BiH/f¼1)

. 6. Comparison of simulations performed at different f and equal BiH/f¼0.1:

20.

Fig. 7. Comparison between simulations at f¼1 or 0.5, at fixed b, F

At b¼0.001 reported in Fig. 7a, heat transfer is not limitativeand BiH¼0.5 or 1000 give equal results. When F410, a limitationby mass transfer appears and Z differs from the case without anylimitations. Results at f¼0.5 are well predicted by the 1D modelat f¼1, when F/f and BiH/f are kept constant between simulations(case f¼1, BiH¼1, Bim¼80 in Fig. 7a).

At b¼0.01 (Fig. 7b), in addition to external mass transfer, heattransfer is also sensitive because Z is higher than without anythermal effects in the range 1oF/fo30. This is due to the poorcooling of the particle, and temperature inside the particle ishigher than in the bulk. But at higher values of F/f, mass transferlimitation strongly reduces Z and heat transfer limitation tends todisappear: results tend to those obtained with external masstransfer limitation only. This result is explained by the fact thatthe heat flux JH at the surface of the particle is drived by themaximum flux in regard with mass transfer, given by

JH ¼ SKfCeDH ð30Þ

Simulations at f¼0.5 are also well predicted by 1D simulationswith f¼1 when F/f and BiH/f are kept constant (case f¼1, BiH¼1,Bim¼80 in Fig. 7b).

3.4. Evaporation

Calculation parameters are chosen to be representative of thea-methylstyrene to cumene hydrogenation. An average pore size(r) of 10�6 m is assumed. Liquid–solid properties are charac-terised by s¼ 0:025Jm�2; f¼ p=3. The vapour pressure is calcu-lated using Antoine equation, with general formula (31) and theparameters presented in Table 1

Pvap ¼ aexp b�c

T½K��d

� �ð31Þ

As a-methylstyrene and cumene have close physical proper-ties, It was assumed in this simple model that only the propertiesof a-methylstyrene at T¼435 K, P¼105 Pa, reported in Table 2,are significant to describe evaporation. The exothermic reaction isdescribed for a first-order reaction and it proceeds inside thecatalyst particle in wetted and dry pores:

Rxi¼ k0,i exp �

EA

RgasT

� �Ci ð32Þ

/f and BiH/f. Spherical particles: (a) b¼0.001 and (b) b¼0.01.

Table 1Antoine equation parameters for a-methylstyrene (Kirillov et al., 2002).

Parameter a (Pa) b c (K) d (K)

Value 131.57 16.45 3644.3 64.55

Fig. 8. Gas phase volume fraction inside particles for cases (a) and (b): f¼0.4

(gas¼white, liquid¼black), F¼0.57.

Fig. 9. Comparison between 1D and 2D simulations in case of partial evaporation

in the pores, at f¼0.4 (cases (a) and (b)).

Table 2

a-methylstyrene physical properties in gas and liquid phase (Aspen PlusTM), at

T¼435 K and P¼105 Pa.

Properties CPi

(J kg�1 K�1)

Di (m2 s�1) ki

(W m�1 K�1)

qi

(kg m�3]li (Pa s)

Gas 1741 0.32�10�4 0.02 3.37 8.8�10�6

Liquid 2157 0.9�10�9 0.1 777 2.4�10�4

Table 3Modifications of physical properties and non-dimensional parameters to use in 1D

models of catalyst particles in case of partial wetting.

Properties 2D model 1D model

Wetted fraction f 1

Diffusion coefficient D Df2

Thiele modulus F F/f

External mass transfer coefficient K Kf

Mass Biot number Bim Bim/f

Thermal conductivity coefficient l lf2

Prater number b bExternal heat transfer coefficient hext hextf

Heat Biot number BiH BiH/f

F. Bazer-Bachi et al. / Chemical Engineering Science 66 (2011) 1953–1961 1959

where k0,g and k0,l are assumed to be equal in the study and aremodified to fit the value of the Thiele modulus between simula-tions. Following Mikhailova et al. (2003), activation energy EA isfixed at 72 kJ mol�1. Flux of evaporation per unit of volume isextracted from the mass balance performed on the gas phase

�agDgrðrCgÞ ¼ agRxg þFvap ð33Þ

Evaporation heat (Hvap¼3.43�105 J kg�1) is introduced in Eq. (7)by addition of the term Hvap � Fvap.

Different external and heat transfer limitations are studied.First case (a) assumes BiH¼BiM¼N on the wetted surface andBiH¼BiM¼0 on the dry surface. Second case (b) is similar to case(a) but BiH¼N on the dry surface (non-symmetric boundaryconditions). Operating conditions are P¼105 Pa and T¼435 K.They are chosen to facilitate evaporation as temperature is closeto the bubble point of a-methylstyrene: Tev¼436 K at standardpressure. The constant of Henry is calculated following Eq. (18)and is found equal to 0.02.

Evaporated gas phase volume fractions corresponding to f¼0.4and F¼0.57 are reported in Fig. 8 for cases (a) and (b). In bothcases evaporation occurs in an important fraction of pores. Butwhen heat transfer is present on the whole surface, evaporation issmaller close to the surface, as the temperature is also lower. Theimportant differences of gas volume fraction in the lower part ofthe particles, i.e. close to the dry surface, show the strong effect ofboundary conditions concerning heat transfer.

Catalyst effectiveness factors are also computed and compared to1D simulations. Results are presented in Fig. 9. Case (a) is wellpredicted by 1D model for any value of the apparent Thiele modulus.

In presence of partial evaporation in the pores, 1D approximation stillleads to good approximations. Case (b) is well predicted by 1D modelonly when F43. Otherwise, when external heat transfer is notnegligible at the dry surface, current 1D approximation is notappropriate. As without evaporation, this could result from thecomplex interaction between non-symmetric boundary conditionsbetween external heat and mass transfer, and reactivity inside theparticles. But evaporation strongly amplifies discrepancies between1D and 2D models due to non symmetrical heat transfer boundaryconditions.

4. Conclusions

The effect of partial wetting on catalyst effectiveness has beenstudied in the case of an exothermic reaction, in presence of externalheat or mass transfer limitations. Simple rules have been confirmedor proposed to include partial wetting effects when 1D models ofcatalyst particles are used. These rules are based on the use ofapparent physical properties, i.e. real physical properties modified bythe wetting efficiency, as synthesised in Table 3. Deduced modifica-tions of non-dimensional relevant parameters are also reportedin Table 3.

The case of partial evaporation inside the pores has been alsomodelled by a simple approach. It has been shown that, in case ofpartial evaporation, the use of 1D models that include partialwetting effects is possible when heat transfer on dry surface ofparticles is negligible. This preliminary study provides

F. Bazer-Bachi et al. / Chemical Engineering Science 66 (2011) 1953–19611960

complementary results to previous works on evaporation model-ling inside catalyst particles (Kirillov et al., 2002; Mikhailova et al.,2003).

Perspectives of this study are first to identify relevant 1Dapproximation rules in case of non-symmetric boundary conditionsbetween heat and mass transfer, i.e. when heat transfer at the drysurface is not negligible. Another axis of research will be to investigategas reactant limitations in addition of the liquid reactant one. Indeed,transport of gas reactant can partially limit catalyst efficiency indifferent industrial fixed bed reactors, including diesel fuel hydro-desulphurisation or hydrogenation of vegetable oils. Finally, evapora-tion model have to be improved by including a more appropriatephase transition model, based on binary equilibrium.

Nomenclature

BiH Biot number relative to heat transfer (¼ hR=l)Bim Biot number relative to mass transfer (¼ KR=De)c reactant concentration (mol m�3)D, De diffusivity (m2 s�1)EA activation energy (J mol�1))f wetted surface fraction (dimensionless)Fvap flux of evaporation (mol m�3)L characteristic length (m) (¼V/S)h external heat transfer coefficient (W m�2 K�1)Hvap enthalpy of vaporisation (J mol�1)He constant of Henry (dimensionless)JH energy flux (W m�2)k kinetics constant (s�1) (¼k¼ k0expð�EA=RTÞ)k0 pre-exponential constant (s�1)kL Langmuir constant (m3 mol�1)K external mass transfer coefficient (m s�1)Mw molar mass (kg mol�1)r radial coordinate (m) or pore radius (m)rn critical pore radius (m)r mean pore radius (m)P pressure (Pa)Rgas universal gas constant (¼8.314 J mol�1 K�1)Rx reaction rate (mol m�3 s�1)rp radius of cylindrical particles (m)S surface of particle (m2)T temperature (K)V volume of catalyst particle (m3)vsl superficial liquid velocity (m s�1)

Greek letters

a phase fraction (dimensionless)b Prater number (¼DHrDeCe=lTe)DH heat of reaction (J mol�1)Z effectiveness factor ¼

RRRV Rxðr,yÞdV=Rx,maxV

� �l thermal conductivity (W m�1 K�1)g Arrhenius number (g¼�EA=RT)j contact angle (dimensionless)F Thiele modulus (¼L

ffiffiffiffiffiffiffiffiffiffiffik=De

p)

r mass density (kg m�3)c flux defined by Eq. (23) (mol m�2 s�1)y angular coordinate (dimensionless)s surface tension (N m�1)

Indices

app apparentcap capillary

d drye external (bulk)g gasi phase index (¼g or l)l liquids surfacevap vaporisationw wetted

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