and Three-Phase Flows in Bubble Columns - American Chemical

detailed local information. The present study uses such a CFD model to predict flow in two-phase and three- phase slurry bubble columns. A large body ...
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Ind. Eng. Chem. Res. 1998, 37, 2284-2292

Two- and Three-Phase Flows in Bubble Columns: Numerical Predictions and Measurements D. Mitra-Majumdar† and B. Farouk† Department of Mechanical Engineering, Drexel University, Philadelphia, Pennsylvania 19104

Y. T. Shah* Research Office, 300 Brackett Hall, Clemson University, Clemson, South Carolina 29634-5701

N. Macken Department of Engineering, 500 College Avenue, Swarthmore College, Swarthmore, Pennsylvania 19081

Y. K. Oh Department of Mechanical Engineering, Cho Sun University, Kwang-ju City, Korea

Multiphase flows are widely encountered in many chemical reactor processes. Multiphase flow reactors in the form of bubble columns are used in many industrial applications such as hydrogenation of heavy oils, fermentation processes, Fischer-Tropsch reactors, etc. Past work in the modeling of these flows was based on empirical correlations. Even though these correlations fit the data well, they were applicable for narrow ranges of experimental parameters. Moreover, detailed information regarding the local flow parameters was not available through these correlations. The present study attempts to overcome some of these deficiencies. It must also be pointed out that detailed experimental data for three-phase flows through bubble columns are scarce in the literature. These flows are very complex, and understanding them would greatly help in the design and scale-up of the chemical process reactors. Introduction A conventional gas-sparged slurry bubble column (SBC) is a vertical, tubular column through which a three-phase (gas-solid-liquid) mixture flows in a cocurrent or countercurrent manner (Figure 1). In a typical slurry bubble column reactor, the solid phase can be either a reactant or a catalyst. The gas and liquid phases are generally reactants. The present study involved both experimental and numerical investigations. Traditionally, the design and scale-up of slurry reactors were largely based on multiphase and multidimensional dispersion models (Shah, 1979; Smith et al., 1986; Fan, 1989; Joshi et al., 1990; Deckwer, 1992). In these models, the design parameters were largely evaluated by empirical correlations which are applicable to narrow operating ranges. Due to the advent of more powerful computers and a better understanding of the multiphase flows in recent years, computational fluid dynamic (CFD) models based on the fundamental governing equations to predict the transport and mixing in multiphase flows are gaining popularity (Rizk and Elgobashi, 1989; Ding and Gidaspow, 1990; Turkoglu and Farouk, 1991; Jakobssen et al., 1993; Alajbegovich et al., 1996). These models are general in nature and give detailed local information. The present study uses such a CFD model to predict flow in two-phase and threephase slurry bubble columns. * Author to whom correspondence should be addressed. Phone: 864-656-7701. Fax: 864-656-7700. E-mail: ytshah@ clemson.edu. † Phone: 215-895-2287. ‡ Phone: 610-328-8073.

Figure 1. Schematic of the flow through a slurry bubble column.

A large body of experimental work has been done on two- and three-phase bubble columns in the past (Hills, 1974; Begovich and Watson, 1978; Badgujar et al., 1986; Torvik and Svendsen, 1990; Kim and Kim, 1991; Tzeng et al., 1993). However, a detailed knowledge of the phase holdups in three-phase flows, both radial and axial variations, for the whole flow domain for flows through bubble columns is lacking. There is, in general, a paucity of data for the validation of the three-phase flow models. The present experimental program focused on setting up an experimental facility, along with the diagnostic tools, for the measurement of phase holdups in a SBC (for both air-water and air-water-glass beads flows). In the experiments, both the radial

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Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2285

variation of the gas holdups and the axial variations of the solid and gas holdup were obtained for different flow conditions. These data can be directly compared to the model predictions, and some of them have been used for model validation. A schematic of the problem of interest is shown in Figure 1. In the present study, the height of the cylindrical column (H) was 2.0 m and the diameter of the column (D) was 0.15 m. Gas, liquid, and solid particles are introduced at the bottom of the column. Air, water, and glass beads (diameter 450 µm) were used in this study. The objectives of the study were the following: (i) Setup an experimental facility and diagnostic tools to obtain the radial and axial variations of the gas volume fractions and the axial variation of the solid volume fractions in the SBC. (ii) Validate the experimental data by comparisons to the predictions of the CFD model and to existing correlations in the literature. Mathematical Model The mathematical formulation used in the present study was based on the multifluid model approach (Spalding, 1980). A k- turbulence model was used to include the effects of turbulence for the multiphase flows considered. Mass conservation equations, momentum equations, and equations for the turbulence parameters, together with the equations for the interphase momentum transfer, were solved numerically to determine the flow structure in the bubble columns. The Reynolds averaged field variables for each phase were weighted by the volume fraction of that phase. The following assumptions were made in the derivations of the governing equations: (a) The pressures in all phases are the same within a computational cell. (b) The flow is axisymmetric. (c) The effect of the gas-phase turbulence on the flow field is negligible. (d) The solid particles are much smaller in size than the computational cells considered. The governing equations are presented below (Rosner, 1986). Mass Conservation Equation. In the absence of phase change, the mass balance for phase i with volume fraction Ri (where RL + RG + RS ) 1.0) yields the following:

∂ (R F ) + div(RiFiUi) ) 0 ∂t i i

(1)

Momentum Equations. The momentum equation for each phase can be written as

∂ (R F U ) + div(RiFiUiUi) ) div(π) + Ri(B′′) + Fint i ∂t i i i (2) is the where B′′ is the body force term and Fint i interphase momentum transfer term. The π term in eq 2 is the contact stress operator and consists of the pressure gradient and the viscous stress terms (Rosner, 1986). Equations for the Turbulence Parameters. Reynolds averaged forms of eqs 1 and 2 are used in the present study. The solid-phase effective kinematic viscosity was derived following the work of Rizk and

Elgobashi (1989) and Choi and Chung (1983), where the solid-phase effective viscosity is formulated as a function of the continuous-phase viscosity. The concept on which this is based was used primarily for the evaluation of the laminar viscosity of a dispersed phase based in a two-phase flow. However, various workers, specifically Choi and Chung (1983), extended this to obtain the effective eddy viscosity of the dispersed phase as a function of that of the continuous phase. In the present study, the following correlation was used.

µeff,S )

[RSFS/RLFL]1.5

µeff,L (1.0 + FS/FL)0.5

(3)

The bubble motion induces turbulence (or recirculation) in the liquid phase. A k- model has been employed to illustrate the effects of the turbulence in the liquid-phase only. The transport equation for the turbulent kinetic energy for a phase can be given by

((

) )

µt,i ∂ (RiFiki) + ∇(RiFiUiki) - ∇ µi + ∇ki ) Ri(Pi + ∂t σk Gbi - Fii) (4) where Pi is the shear production of the turbulent kinetic energy and Gbi is the production rate of the turbulence energy by the body forces. The transport equation for the dissipation rate of the turbulent kinetic energy () is given by

((

) )

µt,i ∂ (F  ) + ∇(FiUii) - ∇ µi + ∇i ) ∂t i i σ i2 (5) (Pi + C3 max(Gbi,0)) - C2Fi ki In the turbulence model described above, standard values of the turbulence model constants have been used (Spalding, 1980). The effect of the presence of the bubbles on the turbulence structure of the liquid phase has also been included in the present study (MitraMajumdar et al., 1995). The turbulence will also have an impact on the solid holdup. For example, at higher gas velocity, bubble motion would create larger backmixing and thereby more uniform solids distribution in the axial direction. Interphase Momentum Exchange. The major component of the forces due to interactions between the phases is due to the drag, caused by the slip between the phases. Various other interfacial forces are present in multiphase flows. These include the Magnus effect, virtual mass effect, pressure gradient force, wall and indirect bubble-bubble interaction, etc. The effect of these forces on the prediction of gas-liquid flows has been discussed in detail by Mitra-Majumdar et al. (1995). The Magnus effect and the virtual mass force were also included in the present simulations. However, it was found that the drag force formulation for the phases has to be changed to consider the effect of the presence of the third phase on the flow field. In the present study, the interphase momentum transfer has been modeled by including the interactions between the gas and liquid phase and the solid and liquid phase. For the purpose of calculating the interfacial area and the drag coefficient, the gas and solid phases have been considered as the dispersed phases and the liquid phase

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has been considered as the continuous phase. Since the gas phase will carry the solid particles with it, this effect has been modeled indirectly as a reduction of the drag force on the solid particles (as compared to that calculated from the solid-liquid interactions only). The other effect, i.e., the increase in the resistance to the motion of the gas due to the presence of the solid particles, has been modeled as an increase in the effective drag force acting on the gas bubbles (as compared to that calculated from the gas-liquid interactions only). The methodology followed in incorporating these into the present model is outlined below. The interphase drag force, Fdr, per unit volume between two phases can be written as

Fdr ) CfVr

(6)

where Cf is the interphase friction coefficient and Vr is the slip between the phases (in the present model this includes the slip between gas-liquid and solid-liquid). This formulation was modified in the present model to consider the effect of the presence of the third phase in the flow field. Cf can be written as (Ishii and Zuber, 1979)

Cf ) (0.75Cd|Vr|RdFc)/d

(7)

where d is the bubble/particle diameter and Rd is the dispersed-phase volume fraction. In the present study, the drag force formulation of the bubble-liquid interaction is a function of the local Reynolds number, ReG, based on the slip velocity between the gas and liquid phase (Clift et al., 1978), and the drag force formulation of the particles-liquid interaction is a function of the local Reynolds number, ReS, based on the slip velocity between the solid and liquid phase. In three-phase flows, not only will the gas phase face a drag force due to the presence of the liquid phase but its motion will also be hindered by the presence of solid particles in the flow field. The slip between the gas bubbles and the solid particles is large, with the bubbles moving at a much higher speed than the particles. This is due to the large difference in the density of the two phases and the effect of the buoyancy. Thus, a large hindrance to the motion of the gas bubbles will be present in the flow due to the presence of the solid particles. On the other hand, the particles will be lifted by both the liquid and gas phase. These effects have been included in the present model. Details regarding the formulation has been described in MitraMajumdar et al. (1997). It should be noted that the model also accounts for the solid-solid interactions following the work of Yasuna et al. (1995). The present model does not implicitly consider bubble coalescence/breakage phenomenon. While this phenomenon is not very important in the homogeneous bubble flow regime (for superficial gas velocity below about 7 cm/s), it is very important in transition and particularly in the churn-turbulent flow regime (for superficial gas velocity higher than approximately 10 cm/s). The present model will have to be further refined for these latter two regimes to include the effects of bubble coalescence/breakage phenomenon on the column dynamics. Boundary Conditions. The boundary conditions for the dependent variables used in the calculations are described as follows:

Figure 2. Schematic of the experimental setup for three-phase flow experiments.

(1) At the inlet of the slurry bubble column, the mass in-flow rates of the phases are specified. Both the inlet velocities and the volume fractions have to be specified for the calculations. Inlet values of k and  were also specified (Spalding, 1980). (2) Along the walls, the velocities satisfy the no-slip boundary conditions. (3) Along the axis, the symmetry boundary conditions apply.

∂uG ∂uL ∂uS ∂k ∂ ) ) ) ) )0 ∂r ∂r ∂r ∂r ∂r

(8)

∂uG ∂RL ∂RS ) ) )0 ∂r ∂r ∂r

(9)

vJ ) vL ) vS ) 0

(10)

(4) At the top surface of the computational domain, i.e., at the outlet of the column, fixed-pressure boundary conditions apply. The outlet pressure is set to atmospheric. The gradients of the dependent variables are set to zero at the outlet. Experimental Setup The experimental facility used for the study of airwater and air-water-glass beads flow through a bubble column is described in Figure 2. It was used to measure the local gas volume fraction in two- and three-phase flows and the axial variation of the radially averaged solid volume fraction in three-phase flows through the bubble column. In the experiments, the gas was introduced at the bottom of the column from an existing air line in the

Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2287 Table 1. Parameters for Gas-Liquid-Solid Flow Experiments size of particles (µm) superficial gas velocity (m/s) superficial liquid velocity (m/s) operating pressure operating temperature

450 0-0.11 0-0.4 ambient ambient

laboratory. The gas passed through a gas distributor plate with 100-µm holes. The gas was allowed to leave the column in the gas-slurry disengaging section at the top of the column. Most of the liquid was injected through the liquid line at the side of the column, using a pump which was fed from a water tank. A flowmeter was placed on this line to measure the main liquid flow rate. An additional liquid flow line was introduced to the solids tank to keep the solids fluidized and facilitate the flow of the slurry into the main SBC. An additional flowmeter was introduced on this line to measure the secondary liquid flow rate. A one-way valve was placed on the secondary liquid line which prevented the backflow of the slurry to the pump. The solid particles were fed from the solid storage tank through the solid line on the side of the column. The slurry and the air flowed cocurrently upward through the column. The gas-slurry disengaging section was a rectangular tank and had separate outlets for the removal of the solid particles and water. In this section, the solids settled down and the slurry was removed through an opening at the bottom of the tank. The slurry flowed into the solids tank, which fed the slurry bubble column through the solid line. The flow from the slurry line from the bottom of the top tank could be directed to two different paths using the threeway valve as shown in Figure 2. During the operation of the experiment, the three-way valve was opened to allow the slurry to flow continuously into the solid tank following path 1 (see Figure 2). To measure the solid flow rate during the experiment, the three-way valve directed the flow of slurry into a graduated cylinder through path 2. The slurry was collected over a short time period (10 s). The solid particles settled down into the bottom of the graduated cylinder and the solid circulation rate could then be measured. Multiple measurements were done during the experiments. In the two-phase experimental setup, the solids tank and the slurry return line were absent. The water in the top tank was removed using the water recovery line. To ensure that the solid particles did not reach the water tank and subsequently the pump, an in-line filter was used to remove the solid particles. An additional filter was placed on the inlet line of the pump to ensure that no solid particles enter the pump. Moreover, the beginning of the liquid removal line had a mesh attached to it. This mesh could keep out particles larger than 100 µm. The parameters for the experiments are shown in Table 1. The volume fraction measurements in the three-phase flows take on additional complexity due to the fact that the values of three volume fractions are to be quantified. The radially averaged volume fractions of the three phases are related by

R hS + R hL + R h G ) 1.0

(11)

where the quantities on the left-hand side of eq 11 are radially averaged volume fractions of the phases. In the present method the gas volume fraction at any location in the bubble column was measured indepen-

Figure 3. Radial variation of the local gas volume fraction in airwater flow in a bubble column, for UG ) 5.05 cm/s. Table 2. Flow Conditions Considered for the Air-Water Flow through the Bubble Column case

superficial gas velocity (UG), cm/s

superficial liquid velocity (UL), cm/s

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

5.05 7.64 10.08 5.05 7.64 10.08 5.05 7.64 10.08

0.0 0.0 0.0 1.88 1.88 1.88 3.77 3.77 3.77

dently using an electroresistivity probe. The details of the construction of the probe and the measurement technique have been described in Mitra-Majumdar (1997). The pressure drop method was used to obtain the axial variation of the volume fractions of the phases. In this method, pressure taps are placed along the length of the column and the pressure drop between successive taps is measured (Mitra-Majumdar, 1997). The pressure drop between successive taps is related to the volume fractions of the phases by the following correlation (Liang et al., 1995):

dp/dh ) -g(FGRG + FLRL + FSRS)

(12)

In eq 12, the volume fractions are the radially averaged volume fractions. The radially averaged gas volume fractions were calculated from the measured local gas volume fractions using the following expression:

RG )

∫0R(rRG) dr

8 D2

(13)

In eq 13, D refers to the column diameter and r refers to the local radius where the local gas volume fraction (RG) was measured. Thus, eqs 11-13 can be used to obtain the axial variations of the radially averaged gas, liquid, and solid volume fractions in the slurry bubble column. Results The flow conditions for the study of air-water flows through the bubble column are summarized in Table 2. Superficial velocity (for gas and liquid) and circulation velocity for the particles are defined as the volumetric flow rate of the phases per unit cross-sectional area of the bubble column. The local gas volume fraction profile at the superficial gas velocity of 5.05 cm/s is shown in Figure 3. The void fraction peaks at the center of the column and decreases toward the wall. This profile of the radial void fraction

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Figure 4. Comparison of the present experimental data (airwater flows) with correlations from the literature.

distribution was seen in all the cases studied. This is due to the effect of the buoyancy on the gas phase. The gas phase accelerates, aided by buoyancy, and, in order to move the quickest, concentrates near the axis of the column. It can be seen that the liquid flow rate had little effect on the gas volume fraction in the column. However, an increase in the gas flow rate increased the gas volume fraction in the column. The experimental data obtained in the study were compared to existing experimental data/correlations in the literature. The experimental data of Oels et al. (1978) and Hikita et al. (1980) were chosen for the comparison. The results are shown in the Figure 4. The correlation given by Hikita et al. (1980) is:

( ) ( ) () ()

RG ) 0.672

UGµL σ

0.578

UL4g -0.131 FLσ3

FG FL

0.062

µG µL

0.107

(14)

where RG is the domain-averaged gas volume fraction. The present experimental data are close to the values predicted by the correlation of Oels et al. (1978). The discrepancy between the two is larger at higher gas flow rate. There is considerable difference between the present date and that given by the correlation of Hikita et al. (1980). This is due to the difference in the gas sparger platessHikita et al. used a single nozzle for gas injection in the column, while a distributor plate was used in the present experiments. The comparison between the CFD model prediction and experimental data of the local gas volume fraction at an axial location of 1.23 m from the inlet, for case 1.9 (Table 2), is shown in Figure 5. The predictions and the experimental data are in close agreement. The model performance in predicting the local gas volume fraction in two-phase flows is adequate. The flow conditions, for the investigations of the three-phase flows through the bubble column, are summarized in Table 3. The radial variations of the local void fraction for case 1.1 are shown in Figure 6. The radial variations of the gas volume fractions are similar to those observed in the air-water flow through the bubble column. The void fractions show a peak at the axis of the column and decreases toward the wall of the column. Along the axial height, the gas volume fractions are seen to be highest near the inlet of the column and decrease with increasing axial height. This is because the gas phase accelerates due to the effect of the buoyancy as it moves up in the column. This increase in velocity causes a decrease in the gas volume

Figure 5. Comparison of the predicted radial gas volume fraction (in air-water flows) with experimental data.

Figure 6. Radial variation of the gas volume fraction for threephase flow in the slurry bubble column, case 1.1. Table 3. Experimental Parameters in Air-Water-Glass Beads Flow through the Slurry Bubble Column

case

superficial gas velocity (UG), cm/s

superficial liquid velocity (UL), cm/s

solid circulation velocity (US), cm/s

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

5.05 7.64 10.08 5.05 7.64 10.08 5.05 7.64 10.08

2.8 2.8 2.8 3.3 3.3 3.3 3.9 3.9 3.9

0.04 0.06 0.07 0.06 0.07 0.09 0.09 0.10 0.11

fraction. The gas volume fraction also increased with increasing gas superficial velocity. This indicates that, as in air-water flows through the bubble column, the void fraction in the column depends strongly on the gas flow rate. The axial variations of the radially averaged gas volume fraction for cases 1.1-1.3 are shown in Figure 7. As mentioned earlier, the gas volume fraction shows a decrease as the axial height increases. Also, it can be seen that the gradient of the variation decreases with the axial location. The decrease in the gas volume fraction is more pronounced in the lower section of the column than in the upper section. This is due to the fact that the gas phase accelerates rapidly in the lower section. The effects of the drag force and other mechanisms of interphase momentum transfer hinder the motion of the gas phase, and the velocity of the bubbles tends toward a terminal velocity. These interfacial momentum transfer mechanisms, which act against the buoyancy-driven acceleration of the bubbles, cause the change in the gradient of the slope of the axial variation of the gas volume fraction. Also, it can be seen clearly

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Figure 7. Axial variation of the radially averaged gas volume fraction for three-phase flow in the slurry bubble column, cases 1.1-1.3.

Figure 8. Axial variation of the solid volume fraction for threephase flow in the slurry bubble column, cases 1.1-1.3.

that the higher the gas flow rate, the higher the values of the gas volume fraction in the column. The axial variations of the solid volume fraction in the column, for cases 1.1-1.3, are shown in Figure 8. The solids volume fraction is highest at the bottom of the column and shows a decrease with the axial height in the column, for all three cases shown in the figure. The highest solid volume fraction is seen for the highest gas flow rate. As the gas flow rate is increased, i.e., with an increase in the momentum of the gas phase through the inlet, the flow in the column is able to sustain a higher solids flow rate. This translates into a larger amount of solids being transported and, hence, a higher solids volume fraction in the column. The axial variations of the gas volume fractions, for cases 1.7-1.9, are shown in Figure 9. The gas volume fractions again decrease with an increase in the axial height and increase with the gas flow rate. The rate of decrease in the values of the gas volume fractions slows down at higher elevations. Moreover, the axial volume fraction profiles indicate that the bubbles are attaining terminal velocity quicker in this set of experiments than in the earlier ones. The profiles are flatter than the earlier sets. At the highest gas flow rate, the axial variation of the gas volume fraction in the bubble columns is not substantial. It was also seen that, for a particular gas flow rate, the gas volume fraction in the column increased with an increase in the liquid flow rate, unlike in the two-phase flows. The reason for this

Figure 9. Axial variation of the radially averaged gas volume fraction and solid volume fraction for three-phase flow in the slurry bubble column, cases 1.7-1.9.

observation is that, with higher gas and liquid flow rates, the solid volume fraction increases in the slurry bubble column. The presence of larger amounts of solid particles in the flow domain creates a larger hindrance to the flow of the bubbles, resulting in their slower movement. In a three-phase column, the gas volume fraction increases as the liquid flow rate is increased. This observation is opposite to that seen in gas-liquid flow, where the liquid flow rate has no effect on the gas volume fraction. The axial variations of the solids volume fraction for these three cases (cases 1.7-1.9) are also shown in Figure 9. The solid volume fraction again shows a high value at the lower part of the column and decreases with the axial height. However, by comparison of the results with those obtained with the earlier cases, it can be seen that an increase in the liquid flow rate increases the solid transport, resulting in large solid volume fractions. In these three cases also, an increase in the gas flow rate increases the solid flow rate. Thus, both the gas and liquid flow rates play a major role in the determination of the solid transport. It can also be seen that the solid volume fraction profile shows lesser and lesser variation of the values of the solid volume fraction between the lower portion of the column as compared to that seen at higher elevations in the slurry bubble column. The combination of the highest gas flow rate and the largest liquid flow rate produces an almost completely fluidized and homogeneous slurry bubble column (case 1.9). Variations of the average solid volume fraction in the slurry bubble column as a function of the gas flow rate, at three different liquid flow rates, are shown in Figure 10. As was mentioned earlier, the solid flow rate is increased with an increase an the gas flow rate. The reason for such an increase is 2-fold: (a) A larger gas flow rate indicates an increased capacity of the gas phase to carry the solid particles with it. (b) The gas bubbles cause a larger acceleration of the liquid phase, at higher gas flow rates. This, in turn, increases the liquid-phase momentum and thus allows more solid particles movement. Variations of the average solid volume fraction in the slurry bubble column as a function of the liquid flow rate, at three different gas flow rates, are shown in Figure 11. As was mentioned earlier, the solids flow rate is increased with an increase in the liquid flow rate.

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Figure 10. Effect of superficial gas velocity on the average solid volume fraction in the slurry bubble column.

Figure 13. Comparison of the numerically predicted axial variation of the solids volume fraction for three-phase flows with the experimental data, case 1.9.

Figure 14. Comparison of the predicted radial variation of the gas volume fraction with the present experimental data, case 1.9. Figure 11. Variation of the average solid volume fraction in the slurry bubble column with the liquid superficial velocity.

Begovich and Watson (1978). Their correlation is given as:

RG ) 0.058UG0.748db0.229D-0.05

Figure 12. Comparison of the average gas volume fraction from the present experimental data with the correlation of Begovich and Watson (1978).

This is due to the increased momentum of the liquid phase. A comparison of Figures 10 and 11 shows that the liquid phase has a more dominant effect on the transport of the solid particles in the slurry bubble column. This is probably due to the larger density of the liquid phase as compared to the gas phase. The average gas volume fractions for air-water-glass beads flow through a slurry bubble column obtained in the present experimental study are compared in Figure 12 with those generated from the correlation given by

(15)

Air and water were used as the gas and liquid phases and alumina beads (6.2-mm diameter) and glass beads (4.6- and 6-mm diameters) were used as the solid phase in the experiments. The experiments of Begovich and Watson (1978) were carried out in two different columns of diameters 0.076 and 0.152 m. The present experimental data for the average gas volume fraction in the column and those predicted by the correlation are in close agreement. A maximum of 12% difference was seen between the two sets. The comparison of the numerically predicted axial variation of the solid volume fraction with experimental data, case 1.9, is shown in Figure 13. The model predictions and the experimental data are in good agreement. The predicted data show a decrease in the initial portion of the column and a flat profile over the rest of the axial height. The numerically predicted radial variation of the local gas volume fraction is compared in Figure 14 with the experimental data. The predicted values of the local gas volume fraction are higher than that seen in the experiments. However, the profile of the radial variation of the gas volume fraction is similar to that seen in the experimental data. The differences in the predictions and the experimental data could be attributed to the fact that two-phase flow interphase transfer correlations were modified to predict the interactions between the phases in the three-phase flows. Even though the modifications proved to be

Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2291

adequate for the predictions of the average solid volume fractions in the column, they failed to capture the detailed local interactions between the phases. At this time no correlation exists for the calculations of drag between the phases in three-phase flows. It is the belief of the authors that the use of such correlations, when they become available, would greatly enhance the capabilities of these CFD models to predict the local flow structure in three-phase flows.

eff ) effective G ) gas phase i ) phase index in ) inlet L ) liquid phase S ) solid phase t ) turbulent

Summary

Greek Symbols

A study to obtain the phase volume fraction distribution in both two- and three-phase flows through a SBC has been carried out. It involved both numerical and experimental investigations. In the experimental studies it was found that the gas volume fractions peaked near the axis of the column and decreased toward the wall in both two- and three-phase flows. In the threephase flow experiments, both the solid and gas volume fractions decreased with axial height. However, the bubbles attained terminal velocity within a short distance from the entrance, and the volume fraction of the gas phase became constant. The solids volume fraction in the column increased with an increase in both the liquid and gas flow rates. The experimental data from the present study compared favorably with existing correlations. The CFD model predicted the local gas volume fraction in the two-phase flows and the axial variation of the average solid volume fraction. However, the model did not adequately predict the local gas volume fraction in the three-phase flows. Further improvements to the model are necessary and are being pursued.

 ) dissipation rate of turbulent kinetic energy, m2/s3 µ ) dynamic viscosity, Pa‚s F ) density, kg/m3

Nomenclature B′′ ) body forces, N/m4 Cd ) drag coefficient Cf ) interphase friction coefficient for unit volume and unit relative velocity, kg/m4 d ) solid particle/bubble diameter, m D ) diameter of the column, m F ) interfacial force, N/m4 Gb ) production of turbulent kinetic energy by body forces, kg/s3 Gs ) shear production of the turbulent kinetic energy, kg/ s3 H ) height of the column, m k ) turbulent kinetic energy, m2/s2 p ) pressure, Pa r ) local radius, m R ) volume fraction ReS ) local Reynolds number of flow (based on the slip velocity between the solid and liquid phases and the solid particle diameter) ReG ) local Reynolds number of flow (based on the slip velocity between the gas and liquid phases and the bubble diameter) v ) radial velocity component, m/s u ) axial velocity component, m/s Vr ) relative velocity vector, m/s U ) superficial velocity (volumetric flow rate per unit crosssectional area of the column), m/s U ) velocity vector, m/s Subscripts b ) bubbles d ) dispersed e ) bubble/particle

Superscript int ) interfacial

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Received for review January 12, 1998 Revised manuscript received March 23, 1998 Accepted March 25, 1998 IE980022I