Fines Deposition Dynamics in Packed-Bed Bubble Reactors

Ind. Eng. Chem. Res. 2003, 42, 2441-2449

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Fines Deposition Dynamics in Packed-Bed Bubble Reactors Ion Iliuta† and Faı1c¸ al Larachi* Chemical Engineering Department & CERPIC, Laval University, Que´ bec G1K 7P4, Canada

When liquid suspensions containing fine solids are treated in packed-bed bubble reactors, bed plugging develops and increases the resistance to two-phase flow. Accumulation of fines in the catalyst bed increases the reactor pressure gradient until eventually the unit must be shut down and the physically deactivated catalyst replaced. Currently, physical models linking the twophase flow to the space-time evolution of fines buildup are virtually nonexistent. An attempt has been made with this contribution to fill in this gap by developing a unidirectional dynamic multiphase flow model based on the volume-average equations of mass and momentum balance for the gas and suspension and the species balance for the fines. Coherent with experimental observations, the model hypothesizes that plugging develops through deep-bed filtration mechanisms. The model incorporates physical effects of porosity and effective specific surface area changes due to the fines capture by the collecting catalyst particles, inertial effects in the gas and suspension, and coupling effects between the filtration parameters and the interfacial momentum exchange force terms. For the rationalization of deep-bed filtration phenomena in packed-bed bubble reactors, parametric studies of the effects of liquid velocity and viscosity, gas density and velocity, fines concentration in the influent suspension, and fines diameter on the plugging dynamics are discussed. 1. Introduction Packed-bed bubble reactors (gas-liquid-solid fixedbed reactors operating in the bubble-flow regime) experiencing cocurrent two-phase downflow or upflow are frequently used in petroleum, petrochemical, and chemical industries, in waste treatment, and in biochemical and electrochemical processing.1-7 For the bubble-flow regime to take place, high-liquid-holdup conditions are required. In trickle beds (i.e., cocurrent downflow), it is achieved at quite high liquid-to-gas throughput ratios (L/G). In packed bubble columns (i.e., cocurrent upflow), such a flow pattern is more spontaneous and is observed even at low L/G. In this regime, the solid particles are totally wetted by the continuous liquid phase while the gas phase is dispersed in the form of bubbles that clear their path upwardly through the packing channels. When liquid suspensions containing a low concentration of nonfilterable fine solid impurities are treated in packed-bed bubble reactors, plugging develops and leads to the progressive obstruction of the bed that is accompanied with a rise in pressure drop. In the petroleum refining industry, the fines, typically in the fewmicron-size range, are often troublesome and can be of various origins, e.g., the iron sulfides coming from the corrosion of upstream equipment,8 the naturally occurring fine clays in the oil sand bitumen,9 the in situ formed coke fines,10,11 or organic precipitates.12 Circumstantially, the projected capacity increase for upgrading/ hydrotreating the bitumen-derived crude from Canadian oil sands, which represents one-third of the world’s useful petroleum resources,13 unavoidably turns the problem of fines buildup in hydrotreaters into an acute one. Current industry response is to leave the bed collecting these fines until the pressure drop climbs to * Author to whom correspondence should be addressed. E-mail: [email protected]. Tel.: 1-418-656-3566. Fax: 1-418 656-5993. † E-mail: [email protected].

a critical level, forcing reactor shutdown.12 Hence, the reactor is emptied and reloaded afresh with pristine catalyst. When this replacement occurs before the catalyst activity has been fully exploited, the process operating costs may seriously become a major obstacle. Numerous investigators in the past have attempted to theoretically describe the transient behavior of deepbed filtration in the case of single-phase flow through porous media.14-26 However, despite the critical operational problem of fines in the petroleum refining industry, there is a lack of fundamental and theoretical work for the elucidation of the process of fines accumulation and for the planning of strategies against plugging in gas-liquid packed-bed bubble reactors. It is therefore vital to gain new fundamental knowledge by tackling the complex hydrodynamics and surface phenomena involved in the plugging with fines of packed-bed bubble reactors especially under the conditions of interest to the oil refining industry. This contribution is offered as a step forward in this direction. The present work’s aim is the development of a physical model for the description of two-phase flow and space-time evolution of fines deposition in packed-bed bubble reactors based on the volume-average mass, momentum, and species balance equations. Coupling between the liquid suspension and solids is monitored via the fines filtration rate equation and the interaction drag or momentum exchange force terms. Monolayer and multiple-layer deposition mechanisms are accounted for by including the appropriate filter coefficient expressions.27 The incidence of fines buildup is evaluated in terms of pressure drop rise as a function of time, as well as in terms of plugging patterns for the local porosity and the specific deposit versus bed depth. The repercussions of liquid superficial velocity and viscosity, gas superficial velocity and density, fines concentration in the feeding suspension, and fines diameter on pressure buildup are theoretically analyzed. This contribution is forcibly illustrative to describe two-phase flow and space-time evolution of fines deposi-

10.1021/ie0205039 CCC: $25.00 © 2003 American Chemical Society Published on Web 11/12/2002

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Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003

tion in packed-bed bubble reactors. It is indeed only illustrative because there is no single work in the literature dealing with experiments for two-phase flow and which could have been used for validation purposes. This contribution has to be viewed as an incitement to academia and industry to explore and further our understanding of this category of processes. 2. Governing Equations for the Fines Dynamics in Packed-Bed Bubble Reactors A cocurrent two-phase bubble flow through a porous medium of uniform initial porosity ° and single-sized catalytic particles is considered. For each flowing phase, we shall assume unidirectional, isothermal, incompressible, viscous Newtonian, and passive (no chemical reaction) conditions. The gas/liquid/fine/porous medium multiphase system is viewed as a system of three interpenetrated continua: (i) a flowing gas phase; (ii) a dilute pseudohomogeneous suspension phase consisting of the liquid and seeding fines, in short henceforth referred to as fluid; (iii) a stationary pseudocontinuous solid phase made up of the packing particlessalso referred to as collectors in this worksconstituting the clean porous medium as well as of the fines that get captured onto their surface. Further assumptions inherent to the formulation developed here are as follows: (a) The fluid properties, e.g., density, viscosity, holdup, are equal to those of the embracing liquid (inlet fines volume fraction usually down to 0.1‰). (b) The fluid phase behaves as a continuum so that the macroscopic differential balance equations can be applied. (c) The packing surface is totally covered by a continuous liquid film, and the gas is dispersed in the form of spherical bubbles (with the same size) flowing through the packing channels. (d) The momentum balance equation for the dispersed gas is set from the motion equation for a single bubble integrated over all of the bubbles in a unit control volume of the reactor.28,29 (e) The solid stress balance is not considered. (f) The coupling between the fluid and solid phases is monitored via the fines filter rate equation and the interfacial drag forces. (g) Only the inlet liquid was considered as a source for fines without considering fines generated in the catalyst bed (because of attrition, byproduct buildup, etc.). (h) The fines seeded in the liquid are considered single-sized with density Ff and diameter df. (i) Reentrainment of the deposited fines due to hydrodynamic drag forces is precluded. (j) Bed plugging by the blocking and the sieving modes does not take place.20,30 (k) The filtration mechanism to occur is of the deepbed filtration type and not of the cake filtration type;10 in other words, fine-fine attractive interactions within the fluid phase are assumed absent or at least small enough to preclude flocculation. (l) The gas-liquid interface is impervious to the fines. (m) The net sink in the fluid momentum balance due to the mass transfer of fines from the fluid to the collector is negligibly small. The model is based on the volume-average form of the transport equation developed for multiphase systems.29,31 The model equations consist of the conserva-

tion of volume, the continuity and the Navier-Stokes equations for the gas and fluid phases, the continuity equation for the solid stationary phase, and the species balance equation for the fines undergoing migration from the fluid phase to the solid phase.

Conservation of volume G + L )

(1)

Continuity equations for the gas, fluid, and solid phases ∂ ∂ F + F u )0 ∂t G G ∂z G G G

(2)

∂ ∂ F + F u ) -FfN ∂t L L ∂z L L L

(3)

∂ [(1 - °)Fs + (1 - d)(° - )Ff] ) FfN ∂t

(4)

Species balance equation for the fines ∂ ∂ ∂2 Lc + uL Lc ) -N + DL 2 Lc ∂t ∂z ∂z

(5)

Momentum balance equations for the dispersed gas and continuous fluid phases ∂ ∂ F u + uG GFGuG ) ∂t G G G ∂z -G

∂ P + GFGgz + FG (6) ∂z G

∂ ∂ ∂2 FLLuL + uL LFLuL ) LµeL 2 uL ∂t ∂z ∂z ∂ L PL + LFLgz + FL (7) ∂z where

µeL ≈ FLuLdp FG ) -FG-L -

and

(8)

FL ) -FL-S + FG-L (9)

∂PL ∂PG ∂P ))dz dz dz

(10)

gz ) -g (for two-phase upflow) and gz ) +g (for two-phase downflow) (11) In the model above, P stands for pressure, R, FR, and uR represent respectively the holdup, the density, and the longitudinal (interstitial) velocity of phase R (gas or fluid phases), while the subscripts s and f refer to the solid phase (or collectors) and fines, respectively. FR-β represents the interfacial drag force per unit reactor volume exerted at the interface between mutually interacting R and β phases (gas, fluid, and solid phases). The fluid-phase effective viscosity, µeL, which arises from the combination of the viscous and the pseudoturbulence stress tensors is formulated as proposed by Dankworth et al. (eq 8).32 In addition, g is the acceleration due to gravity, is the local porosity at time t, ° is the initial clean bed porosity, d is the fines deposit porosity, c is the local fines volumetric concentration, and N is the local filtration rate. Note that in

Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2443

the formulation of the momentum balance equations, the capillary pressure between phase L and phase G is neglected (eq 10). The fines are imposed in the influent stream as a fluid step-increase function after suddenly switching from steady-state flow of clean or fines-free liquid flow through the immaculate porous medium. Solution of this initial state is obtained by solving for c ) 0 and N ) 0 the conservation of volume, the continuity, and the momentum balance equations for the gas and unloaded liquid:

G° + L° ) °

(12)

∂ ∂ G°FG° + G°FG°uG° ) 0 ∂t ∂z

(13)

∂ ∂ °F + °F u ° ) 0 ∂t L L ∂z L L L

(14)

as it travels a unit distance through the bed.27 The form of the filter coefficient is dictated by the nature of the capture phenomena in play and by the amount of capture as bed plugging proceeds. (a) For σ e σcr (initial stage: fines adhere individually as a monolayer on the collector surface)

3 λ ) λ° ) (1 - °)1/3(η°/dp°) 2 (sphere-in-cell model18) (19) where

3 η° ) As°(1 - °)2/3NR2 2 2 1/8 -1/8 9 N NR N 6/5NR-12/5 + 4As1/3Pe-2/3 (20) + 3 L 4000 G

[

]

2(1 - p5) , p ) (1 - °)1/3, w w ) 2 - 3p + 3p5 - 2p6 (21)

A s° )

∂ ∂ F ° °u ° + uG° G°FG°uG° ) ∂t G G G ∂z ∂ -G° P + G°FG°gz + FG° (15) ∂z ∂ ∂ F °u ° + uL° L°FLuL° ) ∂t L L L ∂z ∂2 ∂ j eL° 2 uL° - L° P + L°FLgz + FL° (16) L° µ ∂z ∂z where the superscript ° stands for the clean bed flow and state. The problem depicted by the above seven conservation equations (eqs 1-7) is 1 - D + t. It contains seven unknowns, i.e., L, G, , uL, uG, P, and c. It further contains three dependent unknowns, the filtration rate, N, and the interfacial gas-liquid and liquid-solid drag forces, FG-L and FL-S, which require additional closure formulations.

(b) For σ > σcr (multilayer deposition)34

[

]

17/24 As λ σ 1+ + ) B1 λ° A s° (1 - °)(1 - d) 4.4/3 As σ B2 1+ + As° (1 - °)(1 - d) As 1/3 σ B3 1+ As° (1 - °)(1 - d)

[

( )[

]

]

4/9

(22)

where

B1 )

(1 - °)2/3 As°NL1/8NR15/8 η°

B2 ) 3.376 × 10-3

(23)

(1 - °)2/3 As°NG1.2NR-0.4 (24) η°

3. Closure Models

3

3.1. Filtration Rate Model. The filtration (or deposition) rate determines the degree of collection of fines. The filtration rate is related to the specific deposit, σ, which represents the volume of fines deposited per unit reactor volume.27

N(σ,c,t,z) ) ∂σ/∂t

(17)

Iwasaki33

The logarithmic law of was used to express the dependence of the deposition rate on the local fines concentration and the interstitial fluid velocity:

N ) λcuL

(18)

Equation 18 is among the simplest forms to express the dependence among the local filtration rate, the local fines concentration, and the local interstitial fluid velocity. It is assumed to hold in the particular gasliquid flow context of the bubble-flow regime being studied in this work. This assumes that the fluid flow field is affected only hydrodynamically by the presence of the flowing gas phase. In other words, the presence of gas is felt via the interstitial velocity, uL, which in turn affects the fluid holdup, L, and the fines concentration, c. In eq 18, λ is the filter coefficient, which can be thought of as the probability for a fine to be captured

B3 )

[

]

4xAs°Pe-2/3 η°

(25)

As 1 - (1 - )5/3 ) A s° 1 - (1 - °)5/3 2 - 3(1 - °)1/3 + 3(1 - °)5/3 - 2(1 - °)2

[

2 - 3(1 - )1/3 + 3(1 - )5/3 - 2(1 - )2

]

(26)

The critical specific deposit, σcr, corresponds to the amount of fines required for completing a monolayer having a coating porosity d. σcr is calculated assuming the sphere-in-cell model configuration:25

σcr )

[(

) ]

dp° + 2df dp°

3

- 1 (1 - d)(1 - °)

(27)

3.2. Interfacial Drag Forces. In the momentum balance equations (6) and (7), constitutive equations are required for the interfacial drag forces. The resultant of the forces exerted on the liquid phase (FL) involves two components: one for the liquid-solid interactions (FL-S) and another for the gas-liquid interactions (FG-L). With the packing being completely wet, the gasphase drag will have contributions only due to contact

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with the liquid phase (FG-L). Moreover, the liquid phase behaves as a continuous medium for which the KozenyCarman theory can be applied for predicting the liquidsolid drag forces:28

FL-S )

{

| |}

E2 (1 - ) E1 2(1 - )2 a j j µL + a FL vSL vSLL 3 36 6 L L3 (28)

The gas-liquid interaction force per unit reactor volume, FG-L, is obtained from nbfG-L, where fG-L is the drag force exerted on a single bubble and nb is the number of bubbles per unit reactor volume in the bubble-flow regime.28 fG-L is obtained by superimposing the drag force exerted on a bubble in the viscous and turbulent regimes. Finally, by introduction of the gas drift flux in the resulting FG-L expression, a form similar to that of the Kozeny-Carman equation was established by Attou and Ferschneider:28

FG-L )

{(

}

G 127 µL + Fm jG jGG 2 80 d 2 G b

||

12 G 2 G 1 db

)

(29)

where

(

jG ) G 1 -

)

G (uG - uL)

β)1-

(31)

db = 1.15(σL/FL)3/5eL-2/5

(32)

where the rate of dissipation is obtained from the mechanical energy balance applied to the fluid phase:

)

(33)

3.3. Effective Specific Surface Area for LiquidSolid Momentum Transfer. With the progress of fines deposition, the solid-liquid surface area gets altered by two opposing phenomena:35 an increase in the surface area through the addition of the area of the captured fine exposed to the streamline flow and a loss of area A∆ due to the shadow effect20 (the shadow effect refers to the fact that once a particle is captured by a collector certain parts of the collector surface near the deposited particle are not accessible to approaching particles). The effective surface area can be expressed as the summation of the surface area of the porous medium and the surface area of the fines:

Ncγπdp2(t) + Nc∂Nf[γπdf2 - A∆(t)] a j) π π Nc (dp°)3 + Nf df3 6 6

(34)

The first term in eq 34 reflects the area of the porous medium, i.e., the collectors, and the second, the area of the fines that becomes available for particle intercep-

[ ][

1 dp° 2 dp(t)

2

1-

dp(t) dp(t) + 2df

(35)

]

(36)

The collector area loss, A∆, occasioned by the shadow effect per attached fine is estimated from the shadow left by an equilateral triangle in which the fine is inscribed:35

(30)

The bubble diameter is determined by considering its proportionality with the scale of liquid turbulence:28

(

∂Nf ) 4β(1 - d)[dp(t)/df]2

A∆ )

G L Fm ) FG + FL

vSL ∆P eL ) + FLgz FL H

tion. Not all of the original media collector or the deposited fine is available for momentum transfer because of the packing structure, fracturing, or cul-desac formation. The geometrical parameter γ, which represents the percentage of a single-particle surface area truly available for momentum transfer, has been included to account for this loss of area. According to the simulation results, approximately 55% of the surface area is available for momentum transfer and thus for particle capture. This value is in close agreement with previously recommended values proposed by Liu et al.,36 Stephan and Chase,35 and Ortiz-Arroyo et al.,26 which state that particle collisions mostly exploit the upper frontal hemisphere of the collectors and the fines alike. ∂Nf represents how many fines, per collector, culminate at the periphery of the collecting assemblage at time t24 (the peripheral fines populate the area corresponding to the cross-sectional fraction β26):

(

[ ][

1 dp(t) 2 2

2

df df 2x3 - sin 2x3 dp(t) dp(t)

)]

(37)

The increase in the collector diameter was calculated as a function of the specific deposit assuming spherein-cell model configuration:25 3

dp(t) ) dp°

x

1+

σ (1 - d)(1 - °)

(38)

In the absence of fines in the liquid phase, the effective specific surface area is calculated using the classical relationship

a j ) 6/dp°

(39)

3.4. Liquid Axial Dispersion Coefficient. The extent of backmixing in the gas-liquid upflow packedbed bubble reactors is quantified in terms of an axial dispersion coefficient which is evaluated using a recent comprehensive Bodenstein number correlation.37 For the cocurrent downflow case, previous experimental and theoretical investigations indicated marginal importance of axial mixing in the bubble-flow regime so that plug flow is assumed for the fluid phase (DL ) 0 in eq 5). 4. Method of Solution To make the model (eqs 1-7) solvable, boundary and initial conditions need to be specified for the system in the bubble-flow regime. It is assumed that there is an inlet of gas and fluid at the top (or bottom) of the reactor and an outlet at its bottom (or top). The pressure, the fines concentration, the liquid and gas holdups, and the gas and liquid interstitial velocities are specified at the inlet. The spatial discretization is performed using the standard cell-centered finite difference scheme. The


Figure 1. Variation of the bed porosity with the dimensionless axial distance at different superficial liquid velocities: (a) twophase downflow; (b) two-phase upflow.

Figure 2. Variation of the specific deposit with the dimensionless axial distance at different superficial liquid velocities: (a) twophase downflow; (b) two-phase upflow.

Table 1. Model Parameters Properties of Materials liquid kerosene viscosity: 2 mPa‚s density: 801 kg/m3 gas air density: 1.3-10.3 kg/m3 fines Kaolinite average diameter: 10 and 26 µm density: 2000 kg/m3 porosity of the deposit layer: 0.827 influent concentration: 33-50 mg/L packing spherical catalyst particles diameter: 0.004 m bed porosity: 0.385 Geometry of the Fixed-Bed Reactor diameter: 0.038 m height: 0.9 m

GEAR integration method for stiff differential equations was employed to integrate the time derivatives. Transient flow simulations in a clean bed are first performed until the (pressure, velocity, and holdup) flow

fields reach steady state. Under these circumstances, the conservation equations (eqs 12-16) are solved in the absence of fines in the liquid along with the effective specific surface area calculated using eq 39. Starting from these solutions, transient simulations with finescontaining liquid are then resumed by solving eqs 1-7. Simulations are carried out on a dual-processor Pentium III running at 1000 MHz each. 5. Simulation Filtration in two-phase flow porous media systems is complex so that flow-field imaging of the transient phenomena accompanying fluid flow and fines deposition is not trivial and has not yet been experimentally attempted. It is proposed here to test the model potentiality through simulations of different experimental configurations by solving the transport equations for packed-bed bubble reactors experiencing deep-bed filtration. Plugging tests (room temperature) are run using kerosene as the liquid phase (seeded with kaolinite fines) and air as the gas phase. Spherical (γ-alumina) catalytic particles are used as collectors. The simulated conditions are listed in Table 1 and coincide with typical hydrotreating conditions as discussed by Gray and co-

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Figure 3. Impact of the liquid superficial velocity and fines concentration on the transient behavior of the two-phase pressure drop: (a) two-phase downflow; (b) two-phase upflow.

workers.10-12 To illustrate the quantitative and qualitative features of the plugging in the bubble-flow regime, the simulated results to follow are presented (i) in terms of longitudinal profile snapshots of the local porosity and specific deposit as well as (ii) in the form of transient pressure drop buildup, at different liquid superficial velocities and viscosities, influent fines concentrations, gas superficial velocities and densities, and fines diameters. Figures 1 and 2 show a snapshot at t ) 280 min of the longitudinal profiles of the local porosity (normalized with respect to the initial uniform porosity, °) and the corresponding specific deposit for two different liquid superficial velocities in both downflow and upflow configurations. Figures 1a and 2a show that the extent of fines deposition depends on the axial distance in the case of two-phase downflow. There is a noticeable decrease in the local porosity in the entrance section of the catalyst bed that is coherent and also mirrored by the increased specific deposit. The high interaction regime and low axial dispersion of the fluid phase in the case of two-phase downflow may be an explanation of this behavior. In the case of two-phase upflow, fines

Figure 4. Two-phase pressure drop ratio versus time at different liquid viscosities: (a) two-phase downflow; (b) two-phase upflow.

deposition is relatively uniformly distributed along the bed (Figures 1b and 2b) and may presumably be ascribed to the high axial dispersion coefficient in the fluid phase. When a liquid suspension passes through a bed, some of the fines are retained by the bed (the deposition being caused by diffusion, interception, gravitational collection, and collection due to surface forces) and two-phase pressure drop (∆P) increases compared to that for the clean bed without fines (∆P°), irrespective of whether downflow or upflow takes place; see Figure 3a,b. The major change resulting from fines deposition involves bed porosity, and therefore the increase of two-phase pressure drop is the result of a decrease in the bed porosity. Figure 3 shows an evident increase of the twophase pressure drop ratio and bed-average specific deposit with an increase of the superficial liquid velocity. Higher superficial liquid velocity means shorter residence time for each element in the suspension resulting in a lower specific deposit. Simultaneously, higher liquid holdups (at higher liquid velocities) mean higher fluxes of fines to impinge on the collectors surface, and thus conversely higher specific deposits are to be expected. Because the superficial liquid velocity and liquid holdup appear to exert opposite effects on the overall specific


Figure 5. Two-phase pressure drop ratio versus time at different gas superficial velocities: (a) two-phase downflow; (b) two-phase upflow.

Figure 6. Impact of the gas density on the transient behavior of the two-phase pressure drop ratio: (a) two-phase downflow; (b) two-phase upflow.

deposit, the net outcome may be case-dependent for the filtration performance. In the simulation of the 26 µm fines shown in Figure 3a,b, the higher bed-average specific deposit at vSL ) 30 mm/s explains the higher pressure drop ratio as opposed to the run at vSL ) 20 mm/s where the influent concentration is kept constant at co ) 0.05 kg/m3. Holding constant the quantity covSL, which measures the mass flux of fines entering the reactor, yields higher specific deposit and higher pressure drop ratios at lower liquid velocity (respectively higher influent concentration); see Figure 3a,b. The contrast is more pronounced for two-phase downflow than for upflow. Also, Figure 3 shows the effect of the inlet fines concentration on the ∆P/∆P° ratio. At higher fines concentration, the filtration rate is higher, giving rise to higher overall specific deposits. As a result, the bed porosity decreases and the two-phase pressure drop ratio increases. As the liquid viscosity decreases, the two-phase pressure drop ratio increases (Figure 4) with more pronounced sensitivity to viscosity in downflow. When the migration of fines to the collector surface is impeded,

more viscous liquids seem to yield lower values for the overall specific deposit, yielding thus less porosity reduction in the bed and as a consequence lower twophase pressure drop ratios. The effect of the gas phase on the pressure drop can be split into an effect of the superficial gas velocity and another of the reactor pressure (or gas density). A comparison of the plots of the total pressure drop ratio versus time (Figure 5) indicates that the fines deposition process is influenced by the gas flow rate. In the case of two-phase downflow, the pressure drop ratio decreases with an increase of the superficial gas velocity (Figure 5a) even if at high gas velocity the overall specific deposit is higher. Note, however, that the difference (∆P - ∆P°) increases with an increase of the superficial gas velocity. On the other side, in the case of two-phase upflow, the pressure drop ratio, overall specific deposit, and ∆P - ∆P° increase with an increase of the superficial gas velocity (Figure 5b). A comparison of the predicted pressure drop histories at different values of the gas density (Figures 3 and 6) indicates that the influence of the gas density is not very important in the deposition process. This is because the bubbleflow regime is intrinsically high liquid holdup, so that

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and volume-average species balance equation for the fines was proposed for the description of two-phase flow and space-time evolution of the deposition of fines in packed-bed bubble reactors. The coupling between the fluid and solid phases was monitored via the fines filter equation and the interaction drag or momentum exchange force terms. Both monolayer and multiple-layer deposition mechanisms were accounted for by including the appropriate filter coefficient formulation. The impact of fines was analyzed in terms of the pressure drop rise as a function of time, as well as in terms of plugging patterns. The effects of liquid superficial velocity and viscosity, gas superficial velocity and density, influent fines concentration, and fines diameter on pressure buildup were analyzed. The following conclusions can be drawn from the simulation results: (a) the two-phase pressure drop ratio and volume-average specific deposit increase with an increase of the liquid flow rate; (b) the two-phase pressure drop ratio increases as the liquid viscosity decreases; (c) the influence of the gas density is not a determining factor in the fines deposition process; (d) the two-phase pressure drop ratio decreases as the inlet fines concentration decreases; (e) the twophase pressure drop ratio increases as the fines diameter increases. This contribution is to be viewed as an incitement to explore experimentally in more detail the hydrodynamic effects for this important category of multiphase flows. Experimental work on deep-bed filtration phenomena in packed-bed reactors is recommended to understand their dependence on the flow and bed variables, as well as on the physicochemical phenomena taking place at the fines-collector interfaces. The redistribution of flow, the continued buildup of fines at a new location, and the impact of bed-scale maldistribution should be taken into account to improve the model. Notation

Figure 7. Two-phase pressure drop ratio versus time at different values of fines diameters: (a) two-phase downflow; (b) two-phase upflow.

both the liquid holdup and two-phase pressure drop are only marginally influenced by the gas density at least in the 1.3-10 kg/m3 range explored in the simulation. Figure 7 shows the effect of the fines diameter on the ∆P/∆P° ratio. At higher fines diameter the interception dimensionless group is higher and the filtration rate is higher, giving higher overall specific deposits. As a result, the bed porosity decreases and the two-phase pressure drop increases. Figure 7 also shows that, for the same operation conditions, the largest fines diameter yields an overall specific deposit higher in upflow than in downflow. Conversely, the smallest fines diameter yields an overall specific deposit higher in downflow than in upflow. In general, there is an agreement between the behavior unveiled by the simulations in the dispersed bubble flow discussed above and those obtained for the case of the trickle-flow regime38 where experimentally measured pressure drop buildup has been successfully described by a similar deep-bed filtration model. 6. Conclusion A one-dimensional dynamic model based on the volume-average mass and momentum balance equations

a j ) effective specific surface area, m-1 A∆ ) collector area loss, m2 c ) fine volumetric concentration (liquid volume basis) db ) bubble diameter, m dc ) column diameter, m dp ) effective particle diameter, m DBM ) Brownian diffusion coefficient ) 2kBT/6πµLdf DL ) axial dispersion coefficient in the liquid phase, m2/s eL ) rate of mechanical energy dissipation per unit mass of the liquid phase E1, E2 ) Ergun constants FG-L ) gas-liquid drag force, N/m3 FL-S ) liquid-solid drag force, N/m3 g ) gravity acceleration, m/s2 G ) gas mass flux, kg/m2‚s H ) Hamaker constant, J jG ) gas drift flux, m/s kB ) Boltzmann’s constant, J/K nb ) number of bubbles per unit volume of the reactor ) 6G/πdb3 N ) filtration rate (reactor volume basis), s-1 Nc ) number of collectors in grid cell volume v ) (6v/πdp°3) × (1 - °) Nf ) number of trapped fines in grid cell volume v ) (6v/ πdf3)(° - )(1 - d) ∂Nf ) number of peripheral fines per collector NG ) gravitational dimensionless group ) (Ff - FL)df2g/ 18πµLuL NL ) London-van der Waals dimensionless group ) 4H/ 9πµLdf2uL

Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2449 NR ) interception dimensionless group ) df/dp° P ) pressure, Pa Pe ) Brownian diffusion group ) dp(t) uL/LDBM R ) constant of ideal gas, J/kg‚K ReR ) R-phase Reynolds number ) vSRdpFR/µR T ) temperature, K uR ) average interstitial velocity of the R phase, m/s v ) grid cell volume, m3 vSR ) R-phase superficial velocity, m/s Greek Letters β ) collector cross-sectional fraction ) bed porosity 〈〉 ) bed volume-average porosity d ) porosity of the deposits G ) gas holdup L ) liquid holdup γ ) surface area parameter η ) collector efficiency λ ) filter coefficient, m-1 λo ) clean filter coefficient, m-1 µR ) R-phase dynamic viscosity, kg/m‚s µeL ) liquid-phase effective viscosity (combination of bulk and shear terms), kg/m‚s FR ) density of the R phase, kg/m3 Fm ) density of the medium around a bubble, kg/m3 σ ) specific deposit (reactor volume basis) ) (° - )(1 d) 〈σ〉 ) bed volume-average specific deposit ) ∫10σ(t) dξ σL ) surface tension, N/m ξ ) dimensionless axial distance ) z/H Subscripts f ) fine G ) gas phase L ) liquid phase s ) solid phase Superscripts ° ) clean bed state

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Received for review July 3, 2002 Revised manuscript received September 13, 2002 Accepted September 25, 2002 IE0205039

Fines Deposition Dynamics in Packed-Bed Bubble Reactors

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