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Radial profile of the solid fraction in a liquid-solid circulating fluidized bed
CHINA PARTICUOLOGY Vol. 1, No. 2, 84-87, 2003
z Shorter Communication
RADIAL PROFILE OF THE SOLID FRACTION IN A LIQUID-SOLID CIRCULATING FLUIDIZED BED Tiefeng Wang, Jinfu Wang*, Jing Lin and Yong Jin Department of Chemical Engineering, Tsinghua University, Beijing 100084, P. R. China *Author to whom correspondence should be addressed. Tel: 0086-10-62772051, E-mail: [email protected]
bounded shear flow with particle Reynolds number Rep much less than unity. They found that the lateral force acting on a stationary sphere in a linear shear flow always acts toward the higher-fluid-velocity region. However, their results cannot be applied to large particle Reynolds number of Rep >>1. Jordan & Fromm (1972) numerically estimated the lateral force acting on a cylinder with a higher particle Reynolds number of Rep=400, and showed a different direction for the lateral force. Kurose et al. (1999) studied the effects of fluid shear on the lateral force acting on a rotating sphere in linear shear flow by three-dimension numerical simulations for 1≤Rep≤500. They defined t h e l a t e r a l f o r c e c o e ff i c i e n t f o r a pa r t i c l e a s : FL,p . (2) C L' = 0.5ρ l u l2 Ap Their simulation showed that in the low particle Reynolds number range (Rep<1), the lateral force acts from the low-fluid-velocity side to the high-fluid-velocity side. However, the lateral force changed direction with an increase in Rep.To study the mechanism of generation of the negative 0.8
0.4
0.2
0.0
Model Development Identifying lateral and turbulent dispersion forces Auton (1987) and Drew & Lahey (1987) proposed an expression for the shear-induced lateral force per unit volume of suspension as: du (1) FL = −CLα s ρ lup,slip l , dr CL is the lateral force coefficient, which may change its value and sign with the flow condition. Saffman (1965) studied the lateral force acting on a sphere in linear un-
C'L,p C'L,f C'L
0.6
C'L
Liquid-solid circulating fluidized beds have a number of attractive features suitable for processes where liquid-solid contact is important (Liang et al., 1996; Zhang et al., 2002). Liang et al. (1996) and Zheng et al. (2002) studied the radial profile of the solid fraction in the liquid-solid circulating fluidization regime and found that it is not uniform, unlike the conventional liquid-solid fluidized bed. This non-uniformity can affect reactant concentration distribution, mass transfer and ultimately reactant conversion. Therefore, information on the radial flow structure is crucial to reactor design and process optimization. Most studies on the radial non-uniformity in a liquid-solid circulating fluidized bed have been experimental except the work by Liang et al. (1996) in which a core-annulus model was proposed. The core-annulus model assumes that there are two homogeneous flow regions (core and annulus) in the radial direction of the bed and uses the differences between them to describe the radial non-uniform flow structure. However, this model cannot provide a detailed radial profile of the solid fraction. Lucas et al. (2001) proposed a method for the prediction of the radial gas profile for a given bubble size distribution in gas-liquid systems. It is based on the equilibrium of the forces acting on a bubble perpendicular to the flow direction. Likewise, in a liquid-solid circulating fluidized bed, lateral forces acting on the particles determine the movement of the particle in the radial direction, and this creates a radial profile for the solid fraction. This work proposes a model to calculate the radial profile of the solid fraction in a liquid-solid circulating fluidized bed based on the balance of the lateral force and the turbulent dispersion force.
1
10
100
1000
Rep Fig. 1
Contribution of the pressure force coefficient C’L,p and viscous force coefficient C’L,f to the total lateral coefficient C’L for α*=0.2 as simulated by Kurose et al. (1999).
lateral force, they investigated the contributions of the pressure force and the viscous force to the total lateral force and found that both the coefficients for the pressure force and for the viscous force changed sign from positive to negative in the range 1
Wang, Wang, Lin & Jin: Radial Profile of the Solid Fraction in a Liquid-Solid Circulating Fluidized Bed
85
The particle Reynolds number in the liquid-solid circulating fluidized bed employed in the present study is in the range where the lateral force coefficient has a negative value. Similar to the diffusion of a component from a high concentration region to a low concentration region, solid particles in a high-αs region will diffuse to a low-αs region in a manner analogous to Fick’s law (Drew & Passman, 1998): dα v dα s (3) J s = −Γt s = − t σΓ dr , dr where Γt is the turbulent diffusivity which is assumed to be proportional to the turbulent viscosity vt. This assumption is based on the similarity between turbulent momentum transfer and turbulent mass transfer. Under turbulent dispersion a particle will move with a velocity up,TD and a turbulent dispersion force per unit volume of suspension of: 3 2 (4) / dp , FTD = α s ρ lCDup,TD 4 where the drag coefficient CD is related to Rep,TD, in the viscous region as 24 24 μ l (5) = CD = Rep,TD ρ l d pup,TD . Combining Eqs. (4) and (5) yields: 18μ FTD = 2 l α sup,TD , dp Also, Js is defined as J s = α sup,TD .
(6)
(7)
The turbulent dispersion force acting on the particles per unit volume of suspension can be obtained from Eqs. (3) to (7) as: 18μ v dα s dα (8) FTD = − 2 l t = −CTD vt s d p σΓ dr dr , where CTD is the coefficient for the turbulent dispersion force.
Balancing lateral and turbulent dispersion forces In the fully developed flow region in a liquid-solid circulating fluidized bed, the balance between the lateral force and the turbulent dispersion force per unit volume of suspension can be written, as shown in Fig. 2, as FL + FTD = 0 . (9) Substituting Eqs. (1) and (8) into Eq. (9) yields: dα s du (10) C TD v t + C L u p,slip ρ l l α s = 0 . dr dr Equation (10) is a first-order differential equation for the solid fraction αs. This equation can be solved when the coefficient of turbulent dispersion force CTD, the turbulent viscosity vt, the particle slip velocity up,slip, and the lateral force coefficient CL are known.
Fig. 2
Lateral and turbulent dispersion forces acting on the solid particles.
The slip velocity up,slip can be determined from the balance between drag and buoyancy: u p,slip =
4d p ( ρ p − ρ l ) g 3ρ l C D
,
(11)
where CD is the drag coefficient of a particle in a liquid-solid suspension, which can be obtained by modifying the drag coefficient of a single particle in a stagnant liquid as (Mostoufi & Chaouki, 1999): C D = βC D0 , (12) where CD0 is the drag coefficient for a single sphere in a stagnant liquid: 24 0.413 CD0 = , (13) (1 + 0.173Rep0.657 ) + Rep 1 + 16300 Rep−1.09 and the modifying factor β can be correlated with the solid fraction by: β = (1 − α s ) − m , (14) where
m = 3.02 Ar 0.22 Re P−0.33 .
(15)
The turbulent viscosity as defined for a given liquid velocity profile, du (16) τ = ρ ls ( vl + v t ) l , dr can be expressed in terms of the linear variation of the shear stress τ from the wall stress τw with radial position r/R: τ /τ w = r / R , (17) where τw is calculated by: 1 (18) τ w = fρ ls ul2 , 2 in which the friction factor f is determined from the Blasius equation:
f = 0.0791Rels−0.25 .
(19)
Results Eq. (10) shows that the radial profile of the solid fraction
αs is only dependent on the ratio of CL to CTD. By using the ratio CL/CTD as a model parameter, an iteration procedure was used to solve Eq. (10). The radial profiles of the liquid velocity and the solid
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CHINA PARTICUOLOGY Vol. 1, No. 2, 2003
fraction measured by Zhu et al. (2000) and Zheng et al. (2002) were used to validate the proposed model. The liquid velocity profile was correlated by a cubic polynomial as shown in Fig. 3, which in turn was used to calculate the turbulent viscosity from Eqs. (16) to (19). The best fit value -6 for CL/CTD was 1.0×10 from minimizing the deviations between calculated and experimental data. Fig. 4 also shows that there is a good agreement between the meas-
ured and model predicted profiles of the solid fraction. It can be seen from Fig. 3 that the radial non-uniformity of the solid fraction increases with increasing radial non-uniformity of the liquid velocity. This phenomenon is because the lateral force, which acts on particles near the wall and makes the solid fraction higher in the wall region, increasing with an increase in the gradient of the liquid velocity.
Fig. 3 Liquid velocity ul and solid fraction αs profiles for different superficial liquid velocities: Symbols: experimental data from Zhu et al. (2000); lines: fit by cubic polynomial.
(a) dp=0.50 mm Fig. 4
(b) ρp=2490 kg.m-3
Influence of particle density ρp and particle diameter dp on the radial profile of the solid fraction αs with the given liquid velocity profile indicated by ▲ in Fig. 3 and CL/CTD being 1.0×10-6.
Increases in the particle density and diameter cause an increase in the particle slip velocity, which in turn increases the lateral force. Therefore, with a given liquid velocity profile, the radial non-uniformity of the solid fraction will increase with increases in the particle density and diameter. The model predicted results are shown in Fig. 4. Zheng et al. (2002) reported that the radial profile of the solid fraction
is more uniform in a liquid-solid circulating fluidized bed with plastic beads than with glass beads of the same diameter. A quantitative comparison between the model prediction and the measured data is not possible because the liquid velocity was not reported by Zheng et al. (2002); nevertheless the model is in a qualitative agreement with measured data.
Wang, Wang, Lin & Jin: Radial Profile of the Solid Fraction in a Liquid-Solid Circulating Fluidized Bed
Nomenclature Ar AP CD CD0 CL C′L C′L,f C′L,p CTD dp FL FL,P FTD F Gs H Js m R r Re ul Ul up,slip up,TD
αs α* β ρl ρls Γt σΓ μl
Archimedes number, dp3ρl(ρp-ρl)g/μl2 2 cross area of the particle, m drag coefficient for a particle in a liquid-solid suspension drag coefficient for a particle in a stagnant liquid lateral force coefficient defined by Eq. (1) lateral force coefficient defined by Eq. (2) viscous force coefficient pressure force coefficient coefficient of the turbulent dispersion force particle diameter, m -3 lateral force per unit volume of suspension, N.m lateral force acting on a particle, N turbulent dispersion force per unit volume of -3 suspension, N.m friction factor -2 -1 particle circulating rate, kg.m .s distance form the inlet, m particle flux due to the turbulent dispersion force, -1 m .s model parameter in Eq. (14) radius of the riser, m radial coordinate, m Reynolds number -1 local liquid velocity, m.s -1 superficial liquid velocity, m.s slip velocity of the particle in a liquid-solid sus-1 pension, m.s particle velocity due to the turbulent dispersion -1 force, m.s solid volume fraction dimensionless shear rate of the fluid, (dp/2ul)·(dul/dr) modifying factor for the drag coefficient -3 liquid density, kg.m -3 density of the liquid-solid suspension, kg.m 2 -1 turbulent diffusivity, m .s turbulent Prandtl number
liquid molecular viscosity, Pa.s
vl vt
τ τw
87
2 -1 liquid kinetic molecular viscosity, m .s 2 -1 turbulent viscosity, m .s -2 shear stress at r, N.m -2 wall shear stress, N.m
References Auton, T. R. (1987). The lift force on a spherical body in a rotational flow. J. Fluid Mech., 183, 199-218. Drew, D. A. & Lahey, R. T., Jr. (1987). The virtual mass and lift force on a sphere in a rotating and straining flow. Int. J. Multiphase Flow, 13, 113-121. Drew, D. A. & Passman, S. L. (1998). Theory of Multicomponent Fluids, Appl. Math. Sci., Vol. 135. New York: Springer. Jordan, S. K. & Fromm, J. E. (1972). Laminar flow past a circle in a shear flow. Phys. Fluids A, 16, 972-976. Kurose, R., Misumi, R. & Komori, S. (1999). Drag and lift forces on a rotating sphere in a linear shear flow. J. Fluid Mech., 384, 183-206. Liang, W. G., Zhu, J. X., Jin, Y., Yu, Z. Q., Wang, Z. W. & Zhou, J. (1996). Radial non-uniformity of flow structure in a liquid–solid circulating fluidized bed. Chem. Eng. Sci., 51, 2001-2010. Lucas, D., Krepper, E. & Prasser, H. M. (2001). Prediction of radial gas profiles in vertical pipe flow on the basis of bubble size distribution. Int. J. Therm. Sci., 40, 217-225. Mostoufi, N. & Chaouki, J. (1999). Prediction of effective drag coefficient in fluidized beds. Chem. Eng. Sci., 54, 851-858. Saffman, P. G. (1965). The lift on a small sphere in a slow shear flow. J. Fluid Mech., 22, 385-400. Zhang, H., Wang, T. F., Wang, J. F. & Jin, Y. (2002). Study on the axial velocity of particles in liquid-solid circulating fluidized bed using the ultrasound-Doppler technique. J. Chem. Eng. Chin. Univ., 16(4), 408-414. (In Chinese) Zheng, Y., Zhu, J. X., Marwaha, N. S. & Bassi, A. S. (2002). Radial solids flow structure in a liquid-solids circulating fluidized bed. Chem. Eng. J., 88, 141-150. Zhu, J. X., Zheng, Y., Karamanev, D.G. & Bassi, A.S. (2000). (Gas)-liquid-solid circulating fluidized beds and their potential applications to bioreactor engineering. Can. J. Chem. Eng., 78, 82-94. Manuscript received March 10, 2003 and accepted April 15, 2003.
z Shorter Communication
RADIAL PROFILE OF THE SOLID FRACTION IN A LIQUID-SOLID CIRCULATING FLUIDIZED BED Tiefeng Wang, Jinfu Wang*, Jing Lin and Yong Jin Department of Chemical Engineering, Tsinghua University, Beijing 100084, P. R. China *Author to whom correspondence should be addressed. Tel: 0086-10-62772051, E-mail: [email protected]
bounded shear flow with particle Reynolds number Rep much less than unity. They found that the lateral force acting on a stationary sphere in a linear shear flow always acts toward the higher-fluid-velocity region. However, their results cannot be applied to large particle Reynolds number of Rep >>1. Jordan & Fromm (1972) numerically estimated the lateral force acting on a cylinder with a higher particle Reynolds number of Rep=400, and showed a different direction for the lateral force. Kurose et al. (1999) studied the effects of fluid shear on the lateral force acting on a rotating sphere in linear shear flow by three-dimension numerical simulations for 1≤Rep≤500. They defined t h e l a t e r a l f o r c e c o e ff i c i e n t f o r a pa r t i c l e a s : FL,p . (2) C L' = 0.5ρ l u l2 Ap Their simulation showed that in the low particle Reynolds number range (Rep<1), the lateral force acts from the low-fluid-velocity side to the high-fluid-velocity side. However, the lateral force changed direction with an increase in Rep.To study the mechanism of generation of the negative 0.8
0.4
0.2
0.0
Model Development Identifying lateral and turbulent dispersion forces Auton (1987) and Drew & Lahey (1987) proposed an expression for the shear-induced lateral force per unit volume of suspension as: du (1) FL = −CLα s ρ lup,slip l , dr CL is the lateral force coefficient, which may change its value and sign with the flow condition. Saffman (1965) studied the lateral force acting on a sphere in linear un-
C'L,p C'L,f C'L
0.6
C'L
Liquid-solid circulating fluidized beds have a number of attractive features suitable for processes where liquid-solid contact is important (Liang et al., 1996; Zhang et al., 2002). Liang et al. (1996) and Zheng et al. (2002) studied the radial profile of the solid fraction in the liquid-solid circulating fluidization regime and found that it is not uniform, unlike the conventional liquid-solid fluidized bed. This non-uniformity can affect reactant concentration distribution, mass transfer and ultimately reactant conversion. Therefore, information on the radial flow structure is crucial to reactor design and process optimization. Most studies on the radial non-uniformity in a liquid-solid circulating fluidized bed have been experimental except the work by Liang et al. (1996) in which a core-annulus model was proposed. The core-annulus model assumes that there are two homogeneous flow regions (core and annulus) in the radial direction of the bed and uses the differences between them to describe the radial non-uniform flow structure. However, this model cannot provide a detailed radial profile of the solid fraction. Lucas et al. (2001) proposed a method for the prediction of the radial gas profile for a given bubble size distribution in gas-liquid systems. It is based on the equilibrium of the forces acting on a bubble perpendicular to the flow direction. Likewise, in a liquid-solid circulating fluidized bed, lateral forces acting on the particles determine the movement of the particle in the radial direction, and this creates a radial profile for the solid fraction. This work proposes a model to calculate the radial profile of the solid fraction in a liquid-solid circulating fluidized bed based on the balance of the lateral force and the turbulent dispersion force.
1
10
100
1000
Rep Fig. 1
Contribution of the pressure force coefficient C’L,p and viscous force coefficient C’L,f to the total lateral coefficient C’L for α*=0.2 as simulated by Kurose et al. (1999).
lateral force, they investigated the contributions of the pressure force and the viscous force to the total lateral force and found that both the coefficients for the pressure force and for the viscous force changed sign from positive to negative in the range 1
Wang, Wang, Lin & Jin: Radial Profile of the Solid Fraction in a Liquid-Solid Circulating Fluidized Bed
85
The particle Reynolds number in the liquid-solid circulating fluidized bed employed in the present study is in the range where the lateral force coefficient has a negative value. Similar to the diffusion of a component from a high concentration region to a low concentration region, solid particles in a high-αs region will diffuse to a low-αs region in a manner analogous to Fick’s law (Drew & Passman, 1998): dα v dα s (3) J s = −Γt s = − t σΓ dr , dr where Γt is the turbulent diffusivity which is assumed to be proportional to the turbulent viscosity vt. This assumption is based on the similarity between turbulent momentum transfer and turbulent mass transfer. Under turbulent dispersion a particle will move with a velocity up,TD and a turbulent dispersion force per unit volume of suspension of: 3 2 (4) / dp , FTD = α s ρ lCDup,TD 4 where the drag coefficient CD is related to Rep,TD, in the viscous region as 24 24 μ l (5) = CD = Rep,TD ρ l d pup,TD . Combining Eqs. (4) and (5) yields: 18μ FTD = 2 l α sup,TD , dp Also, Js is defined as J s = α sup,TD .
(6)
(7)
The turbulent dispersion force acting on the particles per unit volume of suspension can be obtained from Eqs. (3) to (7) as: 18μ v dα s dα (8) FTD = − 2 l t = −CTD vt s d p σΓ dr dr , where CTD is the coefficient for the turbulent dispersion force.
Balancing lateral and turbulent dispersion forces In the fully developed flow region in a liquid-solid circulating fluidized bed, the balance between the lateral force and the turbulent dispersion force per unit volume of suspension can be written, as shown in Fig. 2, as FL + FTD = 0 . (9) Substituting Eqs. (1) and (8) into Eq. (9) yields: dα s du (10) C TD v t + C L u p,slip ρ l l α s = 0 . dr dr Equation (10) is a first-order differential equation for the solid fraction αs. This equation can be solved when the coefficient of turbulent dispersion force CTD, the turbulent viscosity vt, the particle slip velocity up,slip, and the lateral force coefficient CL are known.
Fig. 2
Lateral and turbulent dispersion forces acting on the solid particles.
The slip velocity up,slip can be determined from the balance between drag and buoyancy: u p,slip =
4d p ( ρ p − ρ l ) g 3ρ l C D
,
(11)
where CD is the drag coefficient of a particle in a liquid-solid suspension, which can be obtained by modifying the drag coefficient of a single particle in a stagnant liquid as (Mostoufi & Chaouki, 1999): C D = βC D0 , (12) where CD0 is the drag coefficient for a single sphere in a stagnant liquid: 24 0.413 CD0 = , (13) (1 + 0.173Rep0.657 ) + Rep 1 + 16300 Rep−1.09 and the modifying factor β can be correlated with the solid fraction by: β = (1 − α s ) − m , (14) where
m = 3.02 Ar 0.22 Re P−0.33 .
(15)
The turbulent viscosity as defined for a given liquid velocity profile, du (16) τ = ρ ls ( vl + v t ) l , dr can be expressed in terms of the linear variation of the shear stress τ from the wall stress τw with radial position r/R: τ /τ w = r / R , (17) where τw is calculated by: 1 (18) τ w = fρ ls ul2 , 2 in which the friction factor f is determined from the Blasius equation:
f = 0.0791Rels−0.25 .
(19)
Results Eq. (10) shows that the radial profile of the solid fraction
αs is only dependent on the ratio of CL to CTD. By using the ratio CL/CTD as a model parameter, an iteration procedure was used to solve Eq. (10). The radial profiles of the liquid velocity and the solid
86
CHINA PARTICUOLOGY Vol. 1, No. 2, 2003
fraction measured by Zhu et al. (2000) and Zheng et al. (2002) were used to validate the proposed model. The liquid velocity profile was correlated by a cubic polynomial as shown in Fig. 3, which in turn was used to calculate the turbulent viscosity from Eqs. (16) to (19). The best fit value -6 for CL/CTD was 1.0×10 from minimizing the deviations between calculated and experimental data. Fig. 4 also shows that there is a good agreement between the meas-
ured and model predicted profiles of the solid fraction. It can be seen from Fig. 3 that the radial non-uniformity of the solid fraction increases with increasing radial non-uniformity of the liquid velocity. This phenomenon is because the lateral force, which acts on particles near the wall and makes the solid fraction higher in the wall region, increasing with an increase in the gradient of the liquid velocity.
Fig. 3 Liquid velocity ul and solid fraction αs profiles for different superficial liquid velocities: Symbols: experimental data from Zhu et al. (2000); lines: fit by cubic polynomial.
(a) dp=0.50 mm Fig. 4
(b) ρp=2490 kg.m-3
Influence of particle density ρp and particle diameter dp on the radial profile of the solid fraction αs with the given liquid velocity profile indicated by ▲ in Fig. 3 and CL/CTD being 1.0×10-6.
Increases in the particle density and diameter cause an increase in the particle slip velocity, which in turn increases the lateral force. Therefore, with a given liquid velocity profile, the radial non-uniformity of the solid fraction will increase with increases in the particle density and diameter. The model predicted results are shown in Fig. 4. Zheng et al. (2002) reported that the radial profile of the solid fraction
is more uniform in a liquid-solid circulating fluidized bed with plastic beads than with glass beads of the same diameter. A quantitative comparison between the model prediction and the measured data is not possible because the liquid velocity was not reported by Zheng et al. (2002); nevertheless the model is in a qualitative agreement with measured data.
Wang, Wang, Lin & Jin: Radial Profile of the Solid Fraction in a Liquid-Solid Circulating Fluidized Bed
Nomenclature Ar AP CD CD0 CL C′L C′L,f C′L,p CTD dp FL FL,P FTD F Gs H Js m R r Re ul Ul up,slip up,TD
αs α* β ρl ρls Γt σΓ μl
Archimedes number, dp3ρl(ρp-ρl)g/μl2 2 cross area of the particle, m drag coefficient for a particle in a liquid-solid suspension drag coefficient for a particle in a stagnant liquid lateral force coefficient defined by Eq. (1) lateral force coefficient defined by Eq. (2) viscous force coefficient pressure force coefficient coefficient of the turbulent dispersion force particle diameter, m -3 lateral force per unit volume of suspension, N.m lateral force acting on a particle, N turbulent dispersion force per unit volume of -3 suspension, N.m friction factor -2 -1 particle circulating rate, kg.m .s distance form the inlet, m particle flux due to the turbulent dispersion force, -1 m .s model parameter in Eq. (14) radius of the riser, m radial coordinate, m Reynolds number -1 local liquid velocity, m.s -1 superficial liquid velocity, m.s slip velocity of the particle in a liquid-solid sus-1 pension, m.s particle velocity due to the turbulent dispersion -1 force, m.s solid volume fraction dimensionless shear rate of the fluid, (dp/2ul)·(dul/dr) modifying factor for the drag coefficient -3 liquid density, kg.m -3 density of the liquid-solid suspension, kg.m 2 -1 turbulent diffusivity, m .s turbulent Prandtl number
liquid molecular viscosity, Pa.s
vl vt
τ τw
87
2 -1 liquid kinetic molecular viscosity, m .s 2 -1 turbulent viscosity, m .s -2 shear stress at r, N.m -2 wall shear stress, N.m
References Auton, T. R. (1987). The lift force on a spherical body in a rotational flow. J. Fluid Mech., 183, 199-218. Drew, D. A. & Lahey, R. T., Jr. (1987). The virtual mass and lift force on a sphere in a rotating and straining flow. Int. J. Multiphase Flow, 13, 113-121. Drew, D. A. & Passman, S. L. (1998). Theory of Multicomponent Fluids, Appl. Math. Sci., Vol. 135. New York: Springer. Jordan, S. K. & Fromm, J. E. (1972). Laminar flow past a circle in a shear flow. Phys. Fluids A, 16, 972-976. Kurose, R., Misumi, R. & Komori, S. (1999). Drag and lift forces on a rotating sphere in a linear shear flow. J. Fluid Mech., 384, 183-206. Liang, W. G., Zhu, J. X., Jin, Y., Yu, Z. Q., Wang, Z. W. & Zhou, J. (1996). Radial non-uniformity of flow structure in a liquid–solid circulating fluidized bed. Chem. Eng. Sci., 51, 2001-2010. Lucas, D., Krepper, E. & Prasser, H. M. (2001). Prediction of radial gas profiles in vertical pipe flow on the basis of bubble size distribution. Int. J. Therm. Sci., 40, 217-225. Mostoufi, N. & Chaouki, J. (1999). Prediction of effective drag coefficient in fluidized beds. Chem. Eng. Sci., 54, 851-858. Saffman, P. G. (1965). The lift on a small sphere in a slow shear flow. J. Fluid Mech., 22, 385-400. Zhang, H., Wang, T. F., Wang, J. F. & Jin, Y. (2002). Study on the axial velocity of particles in liquid-solid circulating fluidized bed using the ultrasound-Doppler technique. J. Chem. Eng. Chin. Univ., 16(4), 408-414. (In Chinese) Zheng, Y., Zhu, J. X., Marwaha, N. S. & Bassi, A. S. (2002). Radial solids flow structure in a liquid-solids circulating fluidized bed. Chem. Eng. J., 88, 141-150. Zhu, J. X., Zheng, Y., Karamanev, D.G. & Bassi, A.S. (2000). (Gas)-liquid-solid circulating fluidized beds and their potential applications to bioreactor engineering. Can. J. Chem. Eng., 78, 82-94. Manuscript received March 10, 2003 and accepted April 15, 2003.