Onset of pulsing in gas–liquid trickle bed filtration

Chemical Engineering Science 59 (2004) 1199–1211www.elsevier.com/locate/ces

Onset of pulsing in gas–liquid trickle bed &ltration

I. Iliuta, F. Larachi∗

Department of Chemical Engineering, Laval University, Pouliot Building 15120, Cite Universitaire Ste-Foy, Qu$ebec, Canada G1K 7P4

Received 17 September 2003; received in revised form 17 December 2003; accepted 22 December 2003

Abstract

When liquid suspensions containing low concentration of &ne solids are treated in catalytic packed bed gas–liquid–solid reactors,which are operated in trickle 2ow or near the transition between trickle and pulse 2ow, plugging develops and increases the resistanceto two-phase 2ow. Also due to obstruction, such accumulation of &nes in the catalyst bed shifts progressively the 2ow pattern fromtrickling to pulsing 2ow. The progressive onset of pulsing 2ow along the packed bed was estimated using a sequential approach based oncombining a “large time-scale” unsteady-state &ltration solution of two-phase 2ow with a “short time-scale” solution of a linear stabilityanalysis of two-phase 2ow. Space–time evolution and two-phase 2ow of the deposition of &nes in trickle bed reactors under trickle 2owregime was described using a one-dimensional two-2uid model based on the volume-average mass and momentum balance equationsand volume-average species balance equation for the &nes. The model hypothesized that plugging occurred via deep-bed &ltration andincorporated physical e9ects of porosity and e9ective speci&c surface area changes due to the capture of &nes, inertial e9ects of phases,and coupling e9ects between the &nes &lter rate equation and the interfacial momentum exchange force terms. The transition betweentrickle 2ow and pulse 2ow regimes was described from a stability analysis of the solution of the transient two-2uid model around anequilibrium state of trickle 2ow under pseudo steady state conditions. The impact of liquid super&cial velocity, viscosity and surfacetension, gas super&cial velocity and density, feed &nes concentration, and &nes diameter on the transition between trickle and pulse 2owsin the presence of &nes deposition was analyzed.? 2004 Elsevier Ltd. All rights reserved.

Keywords: Trickle bed; Flow regime transition; Filtration; Two-phase 2ow; Plugging; Hydrotreating; Colloidal and non-colloidal &nes

1. Introduction

Oil sand bitumen is a promising alternative to counterdeclines in conventional crude oil sources. For instance,Western Canada tar oil sands are estimated to contain ca.1.7 trillion barrels of oil, of which, 300 billion barrelsare recoverable. Bitumen upgrading yields the so-calledbitumen-derived crude (BDC). Primary upgrading, viaC-rejection (i.e., delayed and 2uid coking) or H-addition(LC-Fining), produces synthetic crude high in sulfur andnitrogen, whereas secondary upgrading (hydrotreating) isnecessary to comply with the ultra-low sulfur speci&cationsin light crude oil. Heavy gas oil (HGO), which amounts to40% of BDC liquid yield (Ring, 2002), is the worst to re&neinto the fuels boiling range. This state of a9airs arises notonly because of HGO high aromaticity but also because of

∗ Corresponding author. Tel.: +1-418-656-3566;fax: +1-418-656-5993.

E-mail address: [email protected] (F. Larachi).

its content in non-&lterable &nes, typically ¡ 20 �m. These&nes escape the coker and entail economic and processingsetbacks in the downstream HGO trickle-bed hydrotreaters.Of various origins, these &nes: (i) occur naturally in theform of reservoir-mud mineral solids (Narayan et al., 1997),(ii) or represent recalcitrant clay intruders that worm intothe mineral processing upstream units, (iii) or originateas coke &nes in the cokers through thermally-triggeredasphaltene/resin condensation/polymerization of aromaticrings (Wang et al., 1999, 2001; Tanabe and Gray, 1997),(iv) or &nally may be corrosion-induced in upstreamequipment such as iron sul&de scales (Wang et al., 2001,Brossard, 1996).Trickle bed hydrotreaters use &xed beds of catalyst that

operate at high pressure (10 MPa) and temperatures (350–400◦C) in a H2-rich atmosphere to process such HGO/&nessuspensions. H2 and HGO 2ow co-currently downwards in atrickle bed where the hydrotreating reactions remove the ob-jectionable S and N impurities, and reduce HGO aromatic-ity. These reactions are severely hampered by the interfering

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1200 I. Iliuta, F. Larachi / Chemical Engineering Science 59 (2004) 1199–1211

&nes conveyed in HGO as dilute suspension (as a matter offact &ltration prior to hydrotreaters can e9ectively interceptthe ¿ 20 �m particles but not all the ¡ 20 �m &nes).WhenHGO suspensions are hydrotreated, plugging devel-

ops yielding progressive bed obstruction. Although the &nesconcentration is in the 100-range weight ppm, the cumula-tive e9ect of thousands of barrels of feed each day divertsthe catalyst bed from its primary function to that of a gi-gantic &lter collecting hundreds of kilograms of &nes. Finesaccumulation causes the pressure to rise by restricting the2ow. Eventually, the pressure drop becomes so high that thehydrotreaters are shutdown and the chemically-active cata-lyst replaced. Hence, &nes deposition prematurely shortenshydrotreater cycle life, increases operational problems andmaintenance work leading to poor energy eIciency.Inspection by Narayan et al. (1997) of the morphology of

the deposits indicates complex capture mechanisms. Drivenby &ne–&ne interceptions, scattered 2ocs form at low liq-uid velocities. The &nes preferentially deposit adjacent tothe contact points between collectors, presumably becausethese are sites of static hold-up with poor liquid renewal. Athigh rates, collector–&ne interactions occur via 2uid drag,gravity-buoyancy, van der Waals and Brownian forces, toyield scattered thin patches of deposits over collectors. Col-lector coordination (number of contact points with neighborsper collector) varies typically in the 4–12 range in randompackings (Ortiz-Arroyo et al., 2003). Therefore, the &nescollection is believed to be strongly associated with catalystcontact points where the hydrodynamic forces funnel &nestrajectories to coincide with static hold-up pockets (Cushingand Lawler, 1998).Over the past years, numerous investigators have at-

tempted to describe the transient behavior of deep-bed &l-tration in single-phase 2ow through porous media (Herziget al., 1970; O’Melia and Ali, 1978; Vigneswaran andTulachan, 1988; Tien, 1989, Narayan et al., 1997; Stephanand Chase, 2000; Ortiz-Arroyo et al., 2002). In contrast,the literature still remains short about the hydrodynamicsinvolved in the plugging with &nes of gas–liquid packedbed reactors despite the critical operational problem of &nesin the bitumen re&ning industry. Thus, a limited number ofexperimental works on the process of &nes accumulation(Chan et al., 1994; Wang et al., 1999, 2001, Gray et al.,2002) and only some attempts concerning the modelingand conceptualization of &nes deposition dynamics havebeen reported (Iliuta et al., 2003, Iliuta and Larachi, 2003).Short-duration experiments by Chan et al. (1994) showedthat even small quantities of &ne solids can increase pres-sure drop in two-phase 2ow and the &nes accumulationoccurs even when the reactors operate in the pulse 2owregime. The works of Wang et al. (1999, 2001) suggestedthat the rate of &ltration of &nes may be a9ected by the hy-drogenation reaction environment due to changes in surfacechemistry. Gray et al. (2002) showed that the pressure dropincreases monotonically with the concentration of &nes andthis increase correlates well with the reduction in e9ective

porosity due to the accumulation of &nes. All the experi-mental studies showed that the major change resulting from&nes deposition involves the e9ective size and geometry ofcatalyst particles, the surface characteristics of the catalystparticles, and the bed porosity.Iliuta et al. (2003) and Iliuta and Larachi (2003) devel-

oped a one-dimensional two-2uid model for the descriptionof two-phase 2ow and space–time evolution of the deposi-tion of &nes in &xed bed reactors. The impact of &nes wasevaluated in terms of pressure drop rise as a function of thespeci&c deposit (or time), as well as in terms of pluggingpatterns, e.g., local porosity, and &ne concentration in thesuspension versus bed depth. Coherent with experimentalobservations, the model hypothesizes that plugging devel-ops through deep-bed &ltration mechanisms. The model in-corporates physical e9ects of porosity and e9ective speci&csurface area changes due to the &nes capture by the collect-ing catalyst particles, inertial e9ects in the gas and the sus-pension, and coupling e9ects between the &ltration param-eters and the interfacial momentum exchange force terms.Both mono- and multiple-layer deposition mechanisms dur-ing the ripening stages were accounted for by including theappropriate &lter coeIcient formulation.One of the important aspects that have not deserved atten-

tion in the literature is in what extent plugging with &nes isresponsible for the shift from the trickle 2ow regime to thepulse 2ow regime as a result of &nes deposition. It is indeedcrucial to develop phenomenological frameworks that en-able to assess how initially clean beds experiencing trickle2ow regime evolve to pulse 2ow regime strictly as a result of2ow obstruction when &nes get deposited and reshape per-manently the porous medium geometry. Hence, dependingon how fast &nes deposition occurs in the bed, the porousmedium backbone is patterned through deposition in termsof local properties such as porosity, collector diameter, spe-ci&c surface area, and speci&c deposit. For &nes in the col-loidal range (¡ 1 �m), the deposits have been reported todistribute relatively evenly along the bed (Gray et al., 2002).Therefore, it is expected that when instabilities develop, theshift from trickle 2ow to pulse 2ow during deposition oc-curs as a one shot-event a9ecting the entirety of the bed. Fornon-colloidal &nes, on the contrary, there is likelihood ofnon-uniformly distributed deposits along the bed with moredeposits in the upper section that in the lower section aspredicted theoretically by Iliuta and Larachi (2003). In thiscase, the bed upper section is prone to 2ow instabilities ear-lier than the lower section. The eventual consequence is thatthe bed becomes the siege of pulse 2ow in the upper sectionwhereas trickle 2ow may be the dominating 2ow pattern inthe lower section of the bed. Ultimately, with the progressof &ltration, pulse 2ow &lls up the whole column as 2owinstabilities are susceptible to form everywhere in the bed.Knowledge of the 2ow regime is thus very important for thedesign of trickle bed hydrotreaters because variables such asliquid holdup, pressure drop and mass transfer coeIcientsare known to change signi&cantly when regime changeover,

I. Iliuta, F. Larachi / Chemical Engineering Science 59 (2004) 1199–1211 1201

such as the transition from trickle 2ow to pulse 2ow regime,takes place in the reactor (Gianetto et al., 1973; Specchiaet al., 1974; Charpentier and Favier, 1975).

In this work, the progressive onset of pulse 2owalong the trickle bed reactor is estimated using a se-quential strategy consisting of combining the solution oftwo-phase 2ow deep-bed &ltration with a linear stabilityanalysis of two-phase 2ow. Two-phase 2ow and space–time evolution of the deposition of &nes are describedusing a one-dimensional two-2uid model based on thevolume-average mass and momentum balance equations andthe volume-average species balance equation for the &nes.The transition between trickle 2ow and pulse 2ow regimesis described from the stability analysis of the solution ofthe transient two-2uid model around an equilibrium state oftrickle 2ow. It will be shown later that the progressive de-position of &nes in the bed causes gradual increase in liquidholdup which sets up competition between the inertial andcapillary forces so that it would no be possible to sustainsteady-state trickle 2ow solutions under certain conditions.The loss of stability of the steady-state solution implies theonset of time-dependent solution, i.e., trickling-to-pulsingtransition as studied by Grosser et al. (1988) a decade agoin the case of clean trickle bed two-phase 2ows.This work is forcibly illustrative of the potential of the

proposed approach because not a single experimental workin the literature has dealt with this peculiarity of trickle bedsthat could have been used for validation. It is hoped thatit will trigger renewed interest in trickle bed fundamentalstudies dealing with an industrially important category ofcoupled catalytic-&ltration multiphase reactors.

2. Modeling transition to pulse �ow in �ltration conditions

In the conditions of spontaneous inception of pulsing,pulse propagation velocities up to 1:2 m=s have been re-ported for a variety of packings experiencing broad rangesof gas and liquid super&cial velocities (Boelhouwer et al.,2002). This leads to pulse characteristic times, namely, pulsedurations and frequencies, neighboring a few tenths of a sec-ond and a few Hertz, respectively. Plugging, on the otherhand, is a slow process which develops throughout sev-eral hours. Therefore, it is reasonable to assume that thetime scales of plugging and of pulse 2ow inception are toofar apart from one another. This enables to treat separatelythe large time-scale unsteady-state deposition phenomenain trickle bed &ltration from the short time-scale phenom-ena triggered by the instabilities that drive the reactor fromtrickle 2ow regime to pulse 2ow regime as a consequenceof bed geometry changes. For the short time-scale analysis,the trickle bed can be viewed as in a pseudo-steady state andbarely feeling the accumulation of &nes. Hence, the gradualonset of pulse 2ow along the packed bed can be estimatedfrom the superimposition of the unsteady-state solution ofthe two-phase 2ow deep-bed &ltration model (Eqs. (1)–(7))

with a linear stability analysis of the model in which thecontribution of deposition is neglected over brief periods oftime.The solution of the overall problem proceeds in &ve steps,

where � = 1 at the start:

• The macroscopic model for two-phase 2ow coupled withdeep-bed &ltration (Eqs. (1)–(7)) is solved from t =(� − 1)Ot to t = �Ot.

• At t=�Ot, two-phase 2ow is considered in pseudo-steadystate and the longitudinal pro&les of the e9ective surfacearea, the modi&ed particle diameter and the bed porositydue to the &nes deposition process are obtained.

• The pseudo steady-state model of two-phase 2ow without&nes deposition (Eq. (26)) is solved for every discretizedlongitudinal location in the bed (every di9erential elementis viewed as a virtual packed bed with a modi&ed particlediameter, e9ective surface area and bed porosity due tothe &nes deposition process).

• The stability criterion (Eq. (38)) for steady-state solu-tions is estimated (when the stability criterion is negative,then steady state is stable and trickle 2ow regime pre-vails; when the stability criterion is positive, steady stateis unstable meaning that pulse 2ow is likely to occur).

• � is incremented and the process is repeated for the nextperiod Ot.

2.1. Macroscopic model for two-phase 4ow coupled with5nes deposition (large time scale)

The non-steady-state model describing the deep-bed &l-tration of &nes in trickle 2ow regime is based on the adapta-tion of the macroscopic volume-average form of the trans-port equations for multiphase systems (Whitaker, 1973).The two-phase 2ow is assumed unidirectional, isothermal,non-reactive (i.e., no &nes generation in the catalyst bed andconstant gas and liquid throughputs) and both the 2owingphases are viscous Newtonian. Each 2uid phase is viewedas a continuum and the packing surface is totally coveredby a liquid &lm and the gas 2ows in the remaining intersti-tial void. The properties of the liquid/&ne suspension (den-sity, viscosity, holdup) are equal to those of the embracingliquid (inlet &nes volume fraction is ¡ 0:1%). Only the in-let liquid was considered as a source for (the single-sized)&nes. No bed plugging by the blocking or the sieving modes(Tien and Payatakes, 1979; Choo and Tien, 1995a) and noreentrainment of the deposited &nes due to hydrodynamicdrag forces are allowed. The gas–liquid interface is imper-vious to the &nes. The net sink in the 2uid momentum bal-ance due to the mass transfer of &nes from the 2uid to thecollector is neglected.The model consists of the conservation of volume, conser-

vation of mass or continuity, conservation of momentum forthe gas and liquid, continuity for the solid stationary phase(i.e., the &xed bed), and species balance for the &nes under-going displacement from the suspension to the solid phase.


Conservation of volume:

�‘ + �g = �: (1)

Continuity for the gas and 4uid phases:@@t

�g�g +@@z

�g�gug = 0; (2)

@@t

�‘�‘ +@@z

�‘�‘u‘ + �fN = 0: (3)

Continuity for the solid phase:@@t

[(1− �0)�s + (1− �d)(�0 − �)�f

]= �fN: (4)

Species balance for the 5nes:@@t

�‘c +@@z

�‘cu‘ + N = 0: (5)

Momentum balance equations for the gas and 4uid phases:

@@t

�g�gug + ug@@z

�g�gug = �g�eg@2

@z2ug

−�g@@z

Pg + �g�gg− Fg‘; (6)

@@t

�‘�‘u‘ + u‘@@z

�‘�‘u‘ = �‘�e‘@2

@z2u‘

−�‘@@z

P‘ + �‘�‘g+ Fg‘ − F‘s: (7)

Here P�, ��, ��, and u� represent, respectively, the pressure,the holdup, the density, and the longitudinal (interstitial)velocity of phase �, while the subscripts s and f refer tothe solid phase and &nes, respectively. F�� represents theinterfacial drag force per unit reactor volume exerted at theinterface between mutually interacting � and � phases (gas,liquid and solid phases). The �-phase e9ective viscosity, �e

�,which arises from the combination of the viscous and thepseudo-turbulence stress tensors is formulated as proposedby Dankworth et al. (1990). In addition, g is the accelerationdue to gravity, � is the local porosity at time t, �0 is theinitial clean bed porosity, �d is the &nes deposit porosity,c is the local &nes volumetric concentration, and N is thelocal &ltration rate.Filtration rate model. The &ltration rate was related to

the speci&c deposit, �, which represents the volume of &nesdeposited per unit reactor volume (Tien, 1989)

N (�; c; t; z) =@�@t

: (8)

The logarithmic law of Iwasaki (1937) is used to expressthe dependence of the deposition rate on the local &nes con-centration and the interstitial 2uid velocity

N = �cu‘: (9)

Eq. (9) expresses the dependence between the local &ltrationrate, the local &nes concentration and the local interstitialliquid velocity. As discussed elsewhere (Iliuta et al., 2003;Iliuta and Larachi, 2003), its extension to two-phase

2ow packed beds is assumed to be valid. In Eq. (9), � rep-resents the &lter coeIcient, or the probability for a &ne tobe captured as it travels a unit distance through the bed(Tien, 1989). The form of the &lter coeIcient is determinedby the nature of the capture phenomena, i.e., (monolayer)&ne–collector (for �6 �cr) (Rajagopalan and Tien, 1976)or (multilayer) &ne–&ne interactions (for �¿ �cr) (Tienet al., 1979). The correlations for representing the changeof the &lter coeIcient as a function of the speci&c depositare presented in our previous work (Iliuta et al., 2003) andthe critical speci&c deposit, �cr, is estimated according toChoo and Tien (1995b) as

�cr =

(1 + 2

df

d0p

)3− 1

(1− �d)(1− �0): (10)

Interfacial drag forces. In the momentum balanceEqs. (6) and (7), a set of constitutive equations is requiredfor the interfacial drag forces. The assumption of bed fullwetting indicates that the gas-phase drag will only havecontributions due to e9ects located at the gas–liquid inter-face. The resultant of these forces denoted,−Fg‘, is the dragforce exerted on the gas phase as a result of the relative mo-tion between the 2owing phases to oppose slip. Similarly,the resultant of the forces exerted on the liquid involves: (i)the drag force, −F‘s, experienced by the liquid due to theshear stress nearby the liquid–solid boundary, (ii) and thegas–liquid interfacial drag, Fg‘. Under the circumstancesof trickle 2ow and at a given depth z in the bed, the dragforces take the following forms (Iliuta et al., 2002):

F‘s ={E1

36C2

wa2cf

(1− �)2

�3‘�‘

+E2

6Cwi(1 + g‘)acf

(1− �)�3‘

�‘|v‘|}

v‘�‘; (11)

Fg‘ ={E1

36C2

wa2cf

(1− �)2

�3g�g

+E2

6Cwi(1 + g‘)acf

(1− �)�3g

�g|vg − �gu∗|}

×(vg − �gu∗)�g: (12)

Here v‘ and vg are the super&cial velocities of the 2owingphases based on the total cross-sectional area of the reac-tor. E1 and E2 are the bed Ergun viscous and inertial con-stants. Cw and Cwi are the viscous and inertial wall correc-tion functions (Liu et al., 1994). g‘ is a complex functionthat depends on the bed hydrodynamics and is incorporatedto account for the enhancement of pressure drop due to thegas–liquid interactions in trickle 2ow (Iliuta et al., 2002).Each one of the drag force expressions above involves a

viscous contribution proportional to velocity and an inertialterm expressed as quadratic velocity dependence. To adaptto the deep-bed &ltration context, these drag equations arerecast as functions of the local instantaneous values of thebed porosity and of the e9ective speci&c surface area of the


solid (collector + &nes assemblage) acf. This latter is usedinstead of the collector diameter because the e9ective spe-ci&c surface area increases with time as &nes are capturedonto the surface of the collectors (Stephan and Chase, 2000;Ortiz-Arroyo et al., 2002). In a previous work (Iliuta et al.,2003), a constitutive relation accounting for the changes un-dergone by acf due to accumulation of &nes was developed.To represent the gas–2uid relative motion intervening in

Fg‘, the e9ect of slip between the gas and the liquid isaccounted for by means of the velocity u∗ at the gas–liquidinterface derived in Iliuta et al. (2002) and estimated fromthe slit model as

u∗ =72E1

�‘(1− �)2a2cf�‘

×[12

(−OP

H+ �‘g

)�‘ +

(−OP

H+ �gg

)�g

]:

(13)

2.2. Simpli5ed macroscopic model of two-phase 4owaround �Ot (short time scale)

Equations of motion. Over time intervals relatively shortto be compatible with the short time scales associated withthe inception of 2ow instabilities causing termination of thetrickle 2ow, the deposition of &nes can be barely felt. Thisallows to re-express the conservation equations (Eqs. (1)–(7)) around time �Ot as if &nes deposition is absent andthe two-phase 2ow is totally indi9erent to &ltration:

�‘ + �g = ��Ot (14)

@��@t

+@@z

(��u�) = 0; �= g; ‘; (15)

�g�g

(@ug

@t+ ug

@ug

@z

)= �g�e

g@2

@z2ug

−�g@@z

Pg + �g�gg− Fg‘; (16)

�‘�‘

(@u‘

@t+ u‘

@u‘

@z

)= �‘�e

‘@2

@z2u‘

−�‘@@z

P‘ + �‘�‘g+ Fg‘ − F‘s: (17)

The drag forces are estimated using Eqs. (11) and (12), andthe e9ective speci&c surface area is evaluated as a summa-tion of the number of particles multiplied by the true sur-face area available for deposition on each particle (Iliutaet al., 2003). The modi&ed particle diameter is calculated asa function of the speci&c deposit, assuming the sphere-in-cellmodel such as that of Choo and Tien (1995b):

d�Otp = d0

p3

√1 +

��Ot

(1− �d)(1− �0): (18)

To complete the model, an additional relation between thegas and liquids pressures, Pg and P‘, has to be introduced.The gradient of the capillary pressure Pc =Pg −P‘ dependson the gradient of liquid holdup, therefore, for a fully estab-lished gas–liquid 2ow, it can be reasonably neglected com-pared with the gradient of pressure in each 2uid. While thisassumption is acceptable for predicting pressure drop andliquid holdup in steady state, it cannot be retained for pre-dicting the transient phenomenon of the transition betweentrickle 2ow and pulse 2ow. Indeed, close to the occurrenceof the transition phenomenon, the gradient of liquid holdupbecomes of a great importance due to the unstable natureof the heterogeneous 2ow. Therefore, the capillarity e9ectsmust be taken into account in these conditions. In this work,the capillary pressure was calculated using the correlationdeveloped by Attou and Ferschneider (2000) based on themomentum balance analysis applied at the vicinity of thegas–liquid interface in the situation close to the loss of sta-bility of the liquid &lm at the pore scale:

Pg − P‘ = 2�‘

(1− �1− �g

)0:33( 1d�Otp

+1

d�Otmin + (�‘=as)(1− (d�Ot

min =d�Otp ))

): (19)

where dp is the particle diameter and dmin is the minimumequivalent diameter of the area between three spheres incontact:

d�Otmin =

(√3

%− 1

2

)1=2d�Otp : (20)

Steady-state model: The steady-state di9erential-macroscopicbalance equations for the gas and liquid phases can bewritten as follows:

�‘(z) + �g(z) = ��Ot ; (21)

ddz

(��u�) = 0; �= g; ‘; (22)

�g�gug@ug

@z= �g�e

g@2

@z2ug − �g

@Pg

@z+ �g�gg− Fg‘; (23)

�‘�‘u‘@u‘

@z= �‘�e

‘@2

@z2u‘ − �‘

@P‘

@z

+ �‘�‘g+ Fg‘ − F‘s: (24)

It is straightforward to combine Eqs. (21)–(24) and obtain[G2

�g�2g+

L2

�‘�2‘− �e

‘u‘

�‘− �e

gug

�g− 0:66�‘

×(

1dp

+1

dmin + (�‘=as)(1− (dmin=dp))

)


×(

1− �1− �g

)−0:66 1− �(1− �g)2

]@�‘@z

−(�e‘u‘

�‘+ �e

gug

�g

)(@�‘@z

)2

+(�e‘u‘ − �e

gug)@2�‘@z2

=−F�Ot

g‘ + F�Ot‘s

�‘− F�Ot

g‘

�g

+(�g − �‘)g; (25)

where G=�g�gug and L=�‘�‘u‘ are the mass 2uxes of thegas and liquid throughout the column.As the liquid hold-up does not change signi&cantly with z

for a steady-state fully established 2ow, the one-dimensionalmodel for the uniform state is

F�Otg‘

��Otg

+−F�Ot

‘s + F�Otg‘

��Ot‘

+ (�‘ − �g)g= 0; (26)

where F�Otg‘ and F�Ot

‘s denote the drag forces evaluated at theconditions corresponding to the pseudo-steady state (�Ot).

2.3. Linear stability analysis of the uniform steady state

To analyze the conditions at which the trickling-to-pulsingtransition occurs according to the model considered here,in&nitesimal perturbations around the steady state describedby Eq. (26) are superimposed. In the analysis, the dependentvariables are expressed in terms of perturbations from theuniform state (Grosser et al., 1988):

��(z; t) = ��Ot� + ��1(z; t); (27)

u�(z; t) = u�Ot� + u�1(z; t); (28)

P�(z; t) = P�Ot� (z) + P�1(z; t); �= g; ‘: (29)

The drag forces terms are linearized by expanding in a Tay-lor series and keeping only the terms in the perturbations asfollows:

Fg‘ = F�Otg‘ +

(@Fg‘

@ug

)�Ot

ug +(@Fg‘

@�g

)�Ot

�g = F�Otg‘ + �gug + �g�g; (30)

F‘s = F�Ot‘s +

(@F‘s

@u‘

)�Ot

u‘ +(@Fg‘

@�‘

)�Ot

�‘ = F�Ot‘s + �‘u‘ + �‘�‘: (31)

Introducing these de&nitions in Eqs. (15)–(17) and lineariz-ing, one obtains@��1@t

+ ��Ot�

@u�1

@z+ u�Ot

�@��1@z

= 0; �= g; ‘; (32)

�g��Otg

@ug1

@t+ �g��Ot

g u�Otg

@ug1

@z

=��Otg �e

g@2ug1

@z2− �g1

@P�Otg

@z− ��Ot

g@Pg1

@z

+ �g1�gg− �gug1 − �g�g1; (33)

�‘��Ot‘

@u‘1

@t+ �‘��Ot

‘ u�Ot‘

@u‘1

@z

=��Ot‘ �e

‘@2u‘1

@z2− �‘1

@P�Ot‘

@z− ��Ot

‘@P‘1

@z

+ �‘1�‘g+ �gug1 + �g�g1 − �‘u‘1 − �‘�‘1: (34)

After linearization of the full model equations (Eqs. (15)–(17)), a general solution can be postulated using a harmonicperturbation

y =˙y exp(st + j!z); (35)

where ! is the wave number of the perturbation and s isthe complex wave velocity. Substitution of Eq. (35) into thelinearized equations gives a polynomial in s and !. Thispolynomial (Eq. (36)) can be solved for the real and imag-inary parts of s in terms of the wave number and severalcoeIcients (+i), which depend only on the 2uid velocitiesas well as the 2uid and bed properties:

s2+1 + s(+2 + !2+6 + 2!j+3)

−!2+4 − j(!+5 − !3+7) = 0; (36)

where the coeIcients +i are given by

+1 =�g

��Otg

+�‘

��Ot‘

;

+2 =�g

��Ot‘ ��Ot

g+

�g

(��Otg )2

+�‘

(��Ot‘ )2

;

+3 =�gu�Ot

g

��Otg

+�‘u�Ot

‘

��Ot‘

;

+4 =�g(u�Ot

g )2

��Otg

+�‘(u�Ot

‘ )2

��Ot‘

− 0:66�‘

×(

1d�Otp

+1

d�Otmin +(��Ot

‘ =a�Ots )(1−(d�Ot

min =d�Otp ))

)

× 1− ��Ot

(1− ��Otg )2

(1− ��Ot

1− ��Otg

)−0:66

;

+5 =F�Otg‘

(��Ot‘ )2

− F�Otg‘

(��Otg )2

− F�Ot‘s

(��Ot‘ )2

− �gu�Otg

��Ot‘ ��Ot

g

− �gu�Otg

(��Otg )2

+�g

��Ot‘

+�g

��Otg

− �‘u�Ot‘

(��Ot‘ )2

+�‘

��Ot‘

;

+6 =�eg

��Otg

+�e‘

��Ot‘

+7 =�egu

�Otg

��Otg

+�e‘u

�Ot‘

��Ot‘

:

By splitting the complex wave velocity into its real andimaginary parts, the real part indicates the disturbance


growth rate, while the imaginary part is related to the speedat which a disturbance would propagate in the bed. Thenecessary condition for stability is obtained by requiring thereal part of the attenuation wave velocity to be negative:

,1 − 2!2,2 + !4,3 ¡ 0; (37)

where,

,1 =+1+25 + 2+2+3+5 ++2

2+4;

,2 =+7(+1+5 ++2+3)−+6(+3+5 ++2+4);

,3 =+1+27 − 2+3+6+7 ++2

6+24 :

A condition similar to Eq. (37) has been obtained byDankworth et al. (1990) but the coeIcients +i are di9er-ent and inherent to the used drag force model formulation.Assuming that the e9ective viscosities are zero, the abovestability criterion Eq. (37) becomes formally similar to thatderived by Grosser et al. (1988) and Attou and Ferschneider(2000) but with di9erent coeIcients +i:

+1+25 + 2+2+3+5 ++2

2+4 ¡ 0: (38)

This inequality is valid irrespective of the wave number.

2.4. Method of solution

To make the two-phase 2ow coupled with deep-bed &l-tration model (Eqs. (1)–(7)) solvable, boundary and initialconditions need to be speci&ed for the system in trickle 2owregime. It is assumed that there is an inlet of gas and 2uidat the top of the reactor and an outlet at its bottom. Thepressure, the &nes concentration, the liquid and gas holdups,the gas and liquid interstitial velocities are speci&ed at theinlet. The spatial discretization is performed using the stan-dard cell-centered &nite di9erence scheme. The GEAR in-tegration method for sti9 di9erential equations is employedto integrate the time derivatives. Transient 2ow simulationsin a clean bed are &rst performed until the (pressure, veloc-ity and holdup) 2ow &elds reach steady state. Under thesecircumstances, the conservation equations are solved in theabsence of &nes in the liquid. Starting from these solutions,transient simulations with &nes-containing liquid are thenresumed by solving Eqs. (1)–(7). The two-phase 2ow with-out &nes deposition steady state model (Eq. (26)) is solvedby means of an iterative Newton–Raphson algorithm.

3. Results and discussion

3.1. Experimental veri5cation

There is no single work in the literature about trickle-to-pulse 2ow transition in the presence of &nes deposition thatcould be used for validation of the proposed approach. In-stead, we validated separately the model for two-phase 2owcoupled with &nes deposition process and the model fortransition to pulse 2ow in the absence of &nes deposition.

Validation of the model for two-phase 2ow coupled with&nes deposition process is based on the experimental workof Gray et al. (2002) obtained using kerosene as the liq-uid phase, seeded with kaolin &nes of 0:7 �m, air as thegas phase and spherical (--alumina) catalytic particles ascollectors. The experimental conditions are listed in Table 1.Fig. 1a shows the comparison between the experimental re-sults, expressed as two-phase pressure drop ratios OP=OP0

versus time, and the results obtained with the present model.OP0 represents the steady-state pressure drop of the cleanbed before &ltration is resumed. It can be seen that the pre-dicted pressure build-up behavior is found to adherer quitewell with physical reality.The transition between trickle and pulse 2ow regimes

has been studied experimentally by a great number of re-searchers and it has become a practice to present the ex-perimental data in the form of a Charpentier-like diagramof L� =G as a function of G=�� (Gianetto et al., 1978). Fig.1b shows a substantial scatter in the literature data. How-ever, the trend predicted by our model for air/water andair/kerosene systems (the operational conditions are listedin Table 1) is in qualitative agreement with the bulk of theexperimental data.

3.2. Simulations

It is proposed here to test the model potentiality throughsimulations of di9erent experimental con&gurations by solv-ing the transport equations for the trickle 2ow deep bed &l-tration. The gradual onset of the pulse 2ow along the packedbed is estimated using the sequential approach explained inSection 2. The simulated conditions are listed in Table 2 andcoincide with typical hydrotreating conditions (Gray et al.,2002). The simulated results are presented in the form of the(large time scale) instants versus bed depth plots highlight-ing the space–time progress of when Eq. (38) is no longerful&lled, i.e., become positive, thus favoring inception ofpulsing 2ow.Fig. 2 shows the e9ect of the &nes diameter on the tran-

sition between trickle and pulse 2ow. Initially, in the ab-sence of &nes deposition, the reactor is operated in trickle2ow regime. Irrespective of the &nes size, the transition be-tween trickle and pulse 2ow is progressive and takes placeat the same value of the local slice-averaged speci&c de-posit, 〈�(z)〉, between z and z + dz. At a given instant,larger diameter &nes induce higher &ltration rates and spe-ci&c deposits than smaller ones. As a result, the transitionbetween trickle and pulse 2ow occurs earlier for the larger&nes. Larger diameter &nes promote more con&ned pluggingin the entrance region (Iliuta et al., 2003). This results in amore pronounced reduction in bed porosity in this sectionof the catalyst bed. As a result, the transition to pulse 2owestablishes quite rapidly in the entrance region of the bed.The instability point for inception of pulsing then evolvesdownstream along the bed quite progressively. With smaller


Table 1Model parameters used in validation

Experimental validation of two-phase 2ow Experimental validation of the model formodel coupled with deep-bed &ltration transition between trickle and pulse 2owprocess (experimental conditions—Gray et al., 2002)

Properties of materials Properties of materialsLiquid Liquid

Kerosene KeroseneViscosity: 2 mPa s Viscosity: 2 mPa sDensity: 801 kg=m3 Density: 801 kg=m3

Gas WaterAir Viscosity: 1 mPa sDensity: 1:3 kg=m3 Density: 1000 kg=m3

Fines GasKaolinite AirAverage diameter: 0:7 �m Density: 1:3 kg=m3

Density: 2000 kg=m3 PackingPorosity of deposit layer: 0.80 Spherical catalyst particles

Packing Diameter: 0:004 mSpherical catalyst particles Bed porosity: 0.385

Diameter: 0:004 m Size of the &xed bed reactorBed porosity: 0.385 Diameter: 0:038 m

Size of the &xed bed reactor Height: 0:9 mDiameter: 0:038 mHeight: 0:9 m

0

1

2

3

4

5

0 100 200 300

Time, min

∆P/∆

P0

experimental

calculated

df=0.7 µmdp=0.004 mv l =0.0011 m/svg=0.223 m/s

ρg=1.3 kg/m 3

1

10

100

1000

0.01 0.1 1 10

G/λε, kg/m2s

L/G

λψ

d

dc

a

b

e

a - Gianettoet al. (1978)b - Sato et al. (1973)c - Charpentieret al. (1975)d - Chouet al. (1977)e - Specchia and Baldi (1977)

- present model (air-water system)

- present model (air-kerosene system)

Pulsing flowTrickling flow

(a) (b)

Fig. 1. (a) Representative plot of pressure drop buildup with time in trickle-2ow reactors—spherical catalyst (data from Gray et al., 2002); (b) trickle-topulse 2ow transition diagram.

diameter &nes (6 1 �m), plugging spreads out more uni-formly along the bed (Gray et al., 2002) turning thus any lo-cation in the bed to be propitious to 2ow regime changeover.The simulation suggests that the bed turns to pulsing all ofa sudden throughout its whole length for micron-size &nes.Whereas with &nes in the dozen micron size, it may hap-pen that the upper part of the bed, especially in the very tallones, operates in pulse 2ow while the lower part is still intrickle 2ow regime. Since the &nes deposition is more no-ticeable at high &nes diameters, the following simulationsare undertaken for these conditions.Fig. 3 shows the impact of &nes feed concentration on the

transition between trickle and pulse 2ow. According to Eq.

(9), higher &nes concentrations give rise to higher &ltrationrates and consequently higher speci&c deposits. As a result,the transition between trickle and pulse 2ow occurs earlierthe higher the suspension feed concentration. The transitionbetween trickle and pulse 2ow takes place when the samevalue of the local volume-averaged speci&c deposit 〈�(z)〉is reached.A decrease in the value of porosity �d of the &nes de-

posit means more compact deposited layer and a higher bedporosity. This expectedly translates into delayed transitionbetween trickle and pulse 2ow (Fig. 4). However, the tran-sition between trickle and pulse 2ow takes place at the samevalue of the local volume-averaged speci&c deposit.


Table 2Model parameters used in simulations

Properties of materialsLiquid

KeroseneViscosity: 2 mPa sDensity: 801 kg=m3

Interfacial tension: 0:025 N=mGas

AirDensity: 1.3–13 kg=m3

PackingSpherical catalyst particles

Diameter: 0:004 mBed porosity: 0.385

Size of the &xed bed reactorDiameter: 0:038 mHeight: 0:9 m

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

z/H

Tim

e, m

in

df=1 microndf=10 micronsdf=26 microns

<σ>=0.660 kg fines/m3 bed

<σ>=0.657 kg f ines/m3 bed

Fig. 2. Impact of &nes diameter on the transition between trickle and pulse2ow (c0=0:142 kg=m3, v‘=0:003125 m=s, vg=0:41 m=s, �g=1:3 kg=m3).

Fig. 5 shows the e9ect of the liquid super&cial velocity onthe transition between trickle and pulse 2ow in the presenceof the &nes deposition process at a constant gas velocity. Atlower liquid velocity, larger values of local volume-averagedspeci&c deposits are required to bring about the transitionfrom trickle to pulse 2ow (Fig. 5a). A decrease in v‘ meanslower liquid holdup (Fig. 5c) which translates into delayedinception of pulsing as well as larger amounts of &nes be-ing captured. Furthermore, the simulations suggest that theliquid super&cial velocity at the transition decreases linearlywith increasing the local volume-averaged speci&c deposit(Fig. 5b).Holding constant the liquid velocity, but decreasing gas

velocity, delays the trickle-to-pulse 2ow transition betweentrickle and pulse 2ow regimes in the presence of &nes de-position process (Fig. 6a). Additionally, at lower gas ve-locities the transition to pulse 2ow occurs at higher localvolume-averaged speci&c deposit. Similarly to liquid veloc-

0

50

100

150

200

250

300

350

0 0.2 0.4 0.6 0.8 1z/H

Tim

e, m

in

c0=0.071 kg/m3

c0=0.1 kg/m3

c0=0.142 kg/m3




Fig. 3. Impact of &nes concentration on the transition between trickleand pulse 2ow (df = 10 �m, v‘ = 0:003125 m=s, vg = 0:41 m=s,�g = 1:3 kg=m3).

0

40

80

120

160

200

0 0.2 0.4 0.6 0.8 1z/H

Tim

e, m

in Porosity deposit=0.75

Deposit porosity=0.8

<σ >=0.661 kg f ines/m3 bed

<σ > =0.656 kg fines/m3 bed

Fig. 4. Impact of deposit porosity on the transition between trickleand pulse 2ow (c0 = 0:142 kg=m3, df = 10 �m, v‘ = 0:003125 m=s,vg = 0:41 m=s, �g = 1:3 kg=m3).

ity, the gas transition velocity decreases linearly with thelocal volume-averaged speci&c deposit (Fig. 6b).As suggested from Eq. (19), the liquid surface tension is

expected to a9ect the regime changeover via the capillarypressure Pc. The model predicts that an increase in surfacetension tends to delay the inception of pulsing to later in-stants while allowing simultaneously more &nes to get cap-tured (due to increasingly local volume-averaged deposits),see Fig. 7. Retardation in the inception of pulsing is anindication that capillary forces increasingly overcome thedestabilizing e9ect of the inertial forces. The inertial forceshave the tendency to destabilize the &lm 2ow structure char-acterizing trickle 2ow, whereas the capillary forces coun-teract to oppose this destabilizing e9ect. On the one hand,the inertial forces exerted on the bulk 2uids contribute to


0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1z/H

Tim

e, m

in

<σ >=0.657 kg f ines/m3 bed


0

0.001

0.002

0.003

0.004

0 2 4 6 8

Local volume-averaged specific deposit, kg/m3

Liqu

id p

hase

tran

sitio

n ve

loci

ty ,

m/s <ε> = 0.385

<ε>=0.374 <ε>=0.37

0.995

1

1.005

1.01

1.015

1.02

0 50 100 150 200 250

εl/ ε

l0

Superficial liquid velocity=0.00281 m/s (εl0=0.0908)

Superficial liquid velocity=0.002667 m/s (εl0=0.0892)

Time, min

(a) (b)

(c)

1200 Superficial liquid velocity=0.003125 m/sSuperficial liquid velocity=0.00281 m/sSuperficial liquid velocity=0.002667 m/s

Fig. 5. (a) Transition between trickle and pulse 2ow at di9erent super&cial liquid velocities; (b) liquid phase transition velocity as a function oflocal volume-averaged speci&c deposit; (c) variation of liquid holdup in time in trickle 2ow regime (c0 = 0:142 kg=m3, df = 26 �m, vg = 0:41 m=s,�g = 1:3 kg=m3).

0

100

200

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8 1

z/H

Tim

e, m

in

Superficial gas velocity=0.41 m/sSuperficial gas velocity=0.3 m/s



0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20

Local volume-averaged specific deposit, kg/m3

Gas

pha

se tr

ansi

tion

velo

city

,m/s

<ε> = 0.344

<ε> = 0.385

<ε> = 0.363

(a) (b)

Fig. 6. (a) Transition between trickle and pulse 2ow at di9erent super&cial gas velocities; (b) gas phase transition velocity as a function of localvolume-averaged speci&c deposit (c0 = 0:142 kg=m3, df = 26 �m, v‘ = 0:003125 m=s, �g = 1:3 kg=m3).

the growth of interfacial waves, and on the other hand thecapillary force manifesting at the gas–liquid interface con-tribute to mitigate these interfacial waves. Hence, a surface

tension increase translates into higher capillary force thusdelaying transition from trickle to pulse 2ow regimes at ahigher local volume-averaged speci&c deposit (Fig. 7). This


0

50

100

150

200

250

300

350

400

450

0 0.2 0.4 0.6 0.8 1

z/H

Tim

e, m

in

Surface tension=0.035 N/mSurface tension=0.03 N/mSurface tension=0.025 N/m



Fig. 7. Impact of surface tension on the transition between trickle andpulse 2ow (c0 =0:142 kg=m3, df =26 �m, v‘=0:005 m=s, vg=0:3 m=s,�g = 1:3 kg=m3).

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

z/H

Tim

e, m

in

Liquid viscosity=0.002 kg/m sLiquid viscosity=0.0015 kg/m sLiquid viscosity=0.001 kg/m s



Fig. 8. Impact of liquid viscosity on the transition between trickle andpulse 2ow (c0=0:142 kg=m3, df=26 �m, v‘=0:01 m=s, vg=0:065 m=s,�g = 1:3 kg=m3).

behavior is coherent with the surface tension e9ects pre-dicted theoretically from Attou and Ferschneider (2000) andfrom Grosser et al. (1988) linear stability analysis of trickleto pulse 2ow transitions in clean trickle beds. The narrow-ing of the trickle 2ow region with decreasingly surface ten-sions aligns also with the systematic experimental study ofChou et al. (1977) on regime changeover.The model predicts that an increase in liquid viscosity

tends to shift the trickle-to-pulse boundary line to lower val-ues of local volume-averaged deposit (Fig. 8). Increasingliquid viscosity induces high liquid holdup translating intoan early transition between trickle and pulse 2ow regimes atlower local volume-averaged speci&c deposit (Fig. 8). Forhigh-viscosity liquids, the shear stress at the liquid–solidinterface is of great importance compared with the gravita-

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

z/H

Tim

e, m

in

Gas density=1.3 kg/m3

Gas density=13 kg/m3



Fig. 9. E9ect of gas density on the transition between trickle and pulse2ow (c0 = 0:142 kg=m3, df = 26 �m, v‘ = 0:009 m=s, vg = 0:1 m=s).

tional force, and accordingly the liquid mean residence timebecomes more signi&cant. As a consequence of the increaseof liquid holdup, smaller quantities of &nes are suIcient totrigger transition.Finally, the model predicts that the trickle-to-pulse

2ow transition is delayed and shifted to higher localvolume-averaged deposit with increasing gas density atconstant gas and liquid super&cial velocities (Fig. 9). Thee9ect of increased gas density at constant super&cial gasvelocity is attributed to the reduction of liquid holdup(Wammes et al., 1991; Al-Dahhan et al., 1997) and thusto the availability of more interstitial space to be pluggedbefore the instabilities trigger the shift to pulse 2ow. Inthis work, for the conditions of high pressure, the capillarypressure was corrected by means of a factor F dependingon the 2uid density ratio (Attou and Ferschneider, 2000):

F(�g

�‘

)= 1 + 88:1

�g

�‘: (39)

Experimental work on trickle-to-pulse transition in thepresence of &nes deposition in trickle bed reactors is rec-ommended to elucidate the dependence between depositionand transition with respect to the 2ow and bed variables, andto the 2uids’ physical properties (especially liquid surfacetension).

4. Conclusion

When liquid suspensions containing a low concentrationof &ne solids are treated in trickle bed reactors, which areoperated in the trickle 2ow or near the transition betweentrickle and pulse 2ow, plugging develops and leads to theprogressive transition to pulse 2ow. The progressive on-set of pulse 2ow along the packed bed was estimated us-ing a sequential strategy based on combining the solutionof two-phase 2ow coupled with &nes deposition model and


a linear stability analysis of two-phase 2ow. The e9ectsof liquid super&cial velocity, viscosity and surface tension,gas super&cial velocity and density, in2uent &nes concen-tration, and &nes diameter on the transition between trickleand pulse 2ows in the presence of &nes deposition were an-alyzed. For larger &nes diameter, the transition to pulse 2owoccurs rapidly in the entrance region of the bed with a rel-atively slow progression along the bed. For &nes diameterin the micron-range, plugging is more uniformly distributedalong the bed and the transition between trickle and pulse2ow regimes rapidly &lls up the whole packed bed. Liquidand gas super&cial velocities at the transition decrease lin-early with the local volume-averaged speci&c deposit. Anincrease in the values of surface tension (or viscosity) trans-lates into a higher value of capillary force (or liquid holdup)which enables the widening (or shrinking) of the trickle2ow region with respect to the pulse 2ow region. Similarly,the trickle-to-pulse 2ow transition is delayed and shifted tohigher local volume-averaged deposit with an increase ingas density.To closely approach physical reality of industrial hy-

drotreating trickle beds, future re&nements of the model canbe implemented to account for temperature gradients alongthe bed, gas and liquid 2ow rate changes due to the catalyticreaction, and ultimately a sink (or source) point in the con-servation equations to account for the formation of coke andeventually asphaltenes precipitation.

Notation

acf e9ective speci&c surface area, m−1

c &ne volumetric concentration (liquid vol-ume basis), dimensionless number

dp particle diameter, mD column diameter, mE1; E2 Ergun constants, dimensionless numberFg‘ gas–liquid drag force, N=m3

F‘s liquid–solid drag force, N=m3

g gravitational acceleration, m=s2

G gas mass 2ux, kg=m2sH bed height, mL liquid mass 2ux, kg=m2sN &ltration rate (reactor volume basis), s−1

P� pressure of �-2uid, PaPc capillary pressure, Pat time, sT temperature, Ku� average interstitial velocity of �-2uid, m/su∗ interfacial velocity, m/sv� � phase super&cial velocity, m/sz axial coordinate, m

Greek letters

- &ne cross-section fraction, dimensionlessnumber

� bed porosity, dimensionless number〈�〉 bed volume-averaged porosity,

dimensionless number�d deposit porosity, dimensionless number�g gas holdup, dimensionless number�‘ liquid holdup, dimensionless number� &lter coeIcient, m−1

�� phase dynamic viscosity, kg=m s�e� � phase e9ective viscosity (combination

of bulk and shear terms), kg=m s�� density of � phase, kg=m3

� speci&c deposit (reactor volume basis),�=(�0−�)(1−�d), dimensionless number

�‘ surface tension, N/m g‘ gas–liquid interaction factor, dimension-

less number

Subscripts/superscripts

cr criticald depositf &neg gas‘ liquid0 clean bed states solid

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