Ind. Eng. Chem. Res. 2003, 42, 2627-2633
2627
Numerical Simulation of Gas and Particle Flow in a Rotating Fluidized Bed A. Ahmadzadeh, H. Arastoopour,* and F. Teymour Department of Chemical and Environmental Engineering, Illinois Institute of Technology, Chicago, Illinois 60616
Fluidized-bed reactors are widely used in chemical and biochemical processes, including polymerization (Kunii, D. G.; Levenspiel, O. Fluidization Engineering; Wiley: New York, 1969). To enhance the efficiency of the fluidized-bed reactors, more flexibility in controlling the residence time of particles of different sizes, particularly very fine particles, is needed. To achieve this goal, we studied gas and particle flow patterns in a vertical rotating fluidized bed (RFB). We limited our study to the quasi-three-dimensional simulation of an isothermal RFB with constant particle size and no chemical reaction to examine the predictability of a computational fluid dynamics approach to the analysis of RFB systems. We used 200-µm particles (with properties similar to polyethylene) as the particulate phase and air as the gas phase in our simulation. The instantaneous velocity, volume fraction, and pressure drop profiles for the gas and particulate phases were calculated. Simulation results showed that the particle residence time distribution could be controlled by the manipulation of rotational speed and inlet gas velocity. Also our simulation results detected the beginning of fluidization at the inner surface of the bed. The pressure drop curve of Geldart group A particles showed that under higher “g” they fluidized like group B particles. 1. Introduction Fluidized-bed reactors are widely used in many chemical and biochemical processes including gas-solid catalytic reactions and Ziegler-Natta-catalyzed polymerization of olefins.1,2 However, these systems are still limited to moderate throughputs because of the relatively low fluidization velocity required. We expect rotating fluidized beds (RFBs) have the potential of being an alternative technology, mostly because they permit higher flow rates and controlled residence time, because the particles are fluidized in a controlled centrifugal field, as opposed to the constant gravitational force encountered in a conventional fluidized bed. Qualitative and applied research work has been done on rotating beds as early as the 1960s. Kroger et al.3 studied the effect of different parameters on the pressure drop in packed and fluidized rotating beds. They concluded that a more complete computational fluid dynamics (CFD) model is needed to predict the particle distribution and local fluidization throughout the bed. Later Levy et al.4 used a simplified two-dimensional model to study the elutriation of particles from RFBs. They found that the air flow rate and rotational speed have strong effects on particle loss and that the bed thickness has a significant influence on the initiation of elutriation and the elutriation rate. Chen5 observed a plateau profile for pressure drop beyond the critical fluidizing velocity, contrary to the results of Takahashi et al.6 and Fan et al.,7 whose observed pressure drop profiles show a maximum. Later, in a series of papers, Qian et al.8-10 performed a more extensive experimental study on RFBs. They measured the pressure drop across beds of alumina particles and glass beads in a horizontal RFB and investigated the effect of different types of gas distributors on the pressure drop. They observed different fluidization characteristics for two types of gas
distributors.8 Later they performed another set of experiments for several types of particles and studied their mixing characteristics.9 Finally, they discussed the behavior of various Geldart-type particles in a centrifugal field. They reported that the particle fluidization behavior changes by varying the centrifugal force. These authors concluded that applying a higher acceleration force causes a shift in the transition lines defining the various Geldart particle groups.10 Kao et al.,11 Chen,5 and Fan et al.7 have used onedimensional models based on the conservation equations to describe the steady-state behavior of a RFB. Chen5 considered the variation of the voidage in his model and predicted the layer-by-layer fluidization in RFBs. The goal of this study was to obtain a fundamental understanding of gas-solid flow patterns in a RFB using CFD. The fast rate of development of CFD techniques in the past decade has made it possible to conduct detailed simulations of multiphase systems. As reviewed by Arastoopour,12 CFD has been used extensively to study the flow behavior of several multiphase systems. For example, Benyahia et al.13 and Sun and Gidaspow14 successfully used CFD codes to predict the flow patterns of the PSRI riser. There are several approaches to numerical modeling of multiphase flows, including Eulerian and Lagrangian approaches, which have been reviewed by Sinclair.15 In the Eulerian or two-fluid model approach, which is used in this study, constitutive relations (closures) are expressed for each phase based on different theories.16 In this work we used a quasi-three-dimensional unsteady-state model to study the detailed flow patterns and particle concentrations, together with the overall behavior of the RFBs such as pressure drop, bubble formation, and particle elutriation. Furthermore, for the
10.1021/ie020740b CCC: $25.00 © 2003 American Chemical Society Published on Web 04/24/2003
2628
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003
The conservation of momentum equations for the gas and solid phases respectively can be written as
∂ B ) + ∇‚(gFgU B gU B g) ) ( F U ∂t g g g -∇‚P + ∇‚Tg - β(U Bg - U B s) + Fggb g (2) ∂ B ) + ∇‚(sFsU B sU B s) ) ( F U ∂t s s s -∇‚Ps + ∇‚Ts - β(U Bs - U B g) + Fssb g (3) To complete the governing equations (1-3), closure equations are needed and were used as follows. The solid pressure, Ps, is assumed to consist of two separate contributions: a kinetic term, which dominates in the dilute flow regions, and a collision term, which is significant in the dense flow regions. Thus, it is expressed as Figure 1. Geometry of the RFB used in this simulation.
Ps ) Fssθs + 2Fs(1 + e)s2g0θs
first time the expected layer-by-layer fluidization phenomenon was confirmed by numerical simulation. Fluent 4.5 software with modified user subroutines was used to simulate the transient gas-particle flow in a RFB. The computer code uses the finite-volume technique17 and is capable of handling a complex geometry.
where e is the restitution coefficient determined experimentally16 and g0 is the radial distribution function given by
g0 )
[ ( )]
s 3 15 s,max
1/3 -1
(5)
The gas-solid drag coefficient was used as suggested by Arastoopour et al.:18
2. System Description The overall objective of this study is to simulate the gas-solid flow patterns expected to emerge in a representative polymerization-separation process. At this stage, we are only concerned with simulating the fluidization behavior of typical polymer particles in the absence of the reaction and thermal effects. The understanding gained from this study is expected to help guide further more complex analysis of the full reaction system. The RFB considered in this simulation consists of a cylindrical tank with a radius and height of 0.15 m and a draft tube of 0.075 m radius and 0.05 m height, located at the top of the cylindrical tank (see Figure 1). Initially, in a typical simulation, the solid particles are at rest at the bottom of the bed. The gas and solid velocities are set to zero. The reactor starts rotating and, under the action of the centrifugal force, the solid particles move radially outward, forming compact vertical layers of solids near the wall. At this point, the gas is introduced into the system radially from the side walls.
β)
Fg|U Bg - U B s| (1 - g)g-2.8 dp
+ 0.336) (17.3 Re
(6)
where the Reynolds number expression is given by
Re )
Bg - U B s| Fgdp|U µg
(7)
Finally, the stress tensors for both phases, Tg and Ts, are expressed by
B g + ∇U B gT) Tg ) gµg(∇U
(8)
and
2 Ts ) sµs(∇U B s + ∇U B sT) + s λs - µs ∇‚U B sI 3
(
)
(9)
In eq 9, the solid viscosity, µs, was chosen to be expressed as
µs ) 0.5s
3. Mathematical Modeling Because of the low-pressure drop in the bed, both the particulate and gas phases were assumed incompressible. The continuity equation for phase i (i ) g for gas and s for solid) is
∂ ( F ) + ∇‚(iFiU B i) ) 0 ∂t i i
(4)
(1)
which assumes zero mass transfer between the phases in the absence of the reaction. The volume fractions must also satisfy the constraint ∑i ) 1.0.
Inclusion of the appropriate boundary conditions is necessary for the development of a realistic simulation program. In this study, Neumann boundary conditions are applied at the outflow boundary for all of variables except pressure, which is specified as atmospheric pressure. In addition, axisymmetric conditions are assumed around the vertical axis of symmetry. The “no-slip” boundary condition has been used for the gas phase at the wall, with only the tangential velocity having a prescribed value proportional to the rotational speed. For the solid phase, the “no-slip” condition was used for the tangential velocity but a nonzero or “partial-slip” velocity was defined for each
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2629
of the radial and axial velocities. The partial-slip velocity is based on the Jackson-Johnson19 equation
τs,w )
πFsb us,wφxθs
(
)
s,max2/3 s,max x 2 3 s 2/3 s
(10)
where τs,w is the solid shear stress at the wall, b us,w is the solid velocity at the wall, φ is the specularity coefficient, and θs is the granular temperature. The specularity coefficient is a measure of the friction between a particle and the wall. A value of zero indicates that the particle will slip freely at the wall, a value of 1 indicates that the particle will adhere to the wall, and a value between 0 and 1 allows for partial slip. A value of 0.01 was used for this variable.20 Finally, for the inlet boundary, the Dirichlet conditions are used for the tangential velocity and radial inflow velocity. Initially, the system is at rest and all velocities are zero. A rotating frame of reference21,22 was used in this simulation. The rotational speed of the reference frame was defined such that no angular velocity is defined for either phase at the boundaries, contrary to the case of a stationary-frame approach. The mass and momentum equations for the gas and particulate phases were solved (Eulerian approach) using the Fluent 4.5 software. The drag force constituted the only link between the two phases. The standard k- turbulence model was used in the code to describe the turbulence behavior of the gas phase in the RFB.23 All of the equations were solved in the cylindrical coordinates using the rotating frame of reference. The kinetic theory approach was not used in this study, but the solid pressure was calculated using an equation that has been obtained from the kinetic theory. The granular temperature was considered to be constant at a value prescribed initially. We used unsteady-state quasi-three-dimensional equations to obtain numerical results. Even though this approach is more numerically complex, it is essential in obtaining a more reliable understanding of the hydrodynamics of a RFB by using CFD. 4. Numerical Simulation Because of the axisymmetric condition, only half of the system was simulated, and a total of 5084 (82 × 62 × 1) grid cells and a time step of 10-3 s were used in the simulations. In the Eulerian approach, used in Fluent 4.5, each phase is considered separately as a continuum and the mass and momentum conservation equations are developed for each phase. With no mass transfer between phases, the phases only exchange momentum through the drag force. The equations were first discritized on a nonstaggered grid using the finite-volume method17 and then solved in their algebraic form in the Fluent computer program. A fully coupled implicit algorithm, called the full elimination algorithm (FEA), is used in Fluent 4.5.24 Also the default discritization scheme in the program is the power law interpolation, which is more accurate than a first-order scheme but less involved than a second-order scheme. Finally, the SIMPLE (semi-
Figure 2. Gas pressure (Pa) and velocity vectors for the singlephase RFB simulation with no gas inflow (rotation only, t ) 40 s, Ω ) 30 rad/s, 14g).
implicit method for a pressure-linked equation) algorithm17 was used for the pressure correction calculations. 5. Results and Discussion 5.1. Single-Phase (Gas) Rotating Bed. To obtain a better understanding of the dynamic behavior of gassolid flow, a simple case of a single-phase (gas) rotating bed was simulated first. Air (F ) 1.293 kg/m3 and µ ) 1.72 × 10-5 kg/ms) was used as the gas phase in this simulation. The bed rotation was started with no gas entering from the side-wall distributor with an initial angular velocity of Ω ) 3 rad/s (0.14g), which was linearly increased until it reached 50 rad/s (38g) in 40 s. The velocity vectors in Figure 2 show that a large vortex, starting and ending at the top opening, was created and propagated to the side wall. Increasing the angular velocity resulted in a larger vortex. At some point, the bottom wall begins to act like a rotating disk. At Ω ) 50 rad/s (38g), the vorticity at the bottom boundary spread further upward until it almost covered the entire container. The system then became very unstable and resulted in a high gas pressure oscillation; possibly numerical instabilities are at the root of this behavior. Therefore, the simulation was continued with the stable angular velocity of 30 rad/s (14g). After the reactor reached steady state, the gas was introduced gradually from the side-wall distributor. The gas inlet velocity was increased linearly for 30 s until the inlet velocity reached 1 m/s. After introduction of the gas, high pressure began to build up at the inlet as the gas inlet velocity increased. The gas velocity vectors in Figure 3 show that the gas radial velocity profile is uniform along the vertical direction and that there is no flow reversal and recirculation outside the core region. Levy et al.4 obtained experimentally similar results for the condition where the inlet gas velocity is high and no overall circulation is formed. This situation is close to a single-pass (plug-flow) reactor. Circulation of the gas was limited to the central core region, which
2630
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003
Figure 3. Change of gas pressure and flow field after introduction of the inlet gas from the side wall (t ) 70 s, Vin ) 1 m/s, Ω ) 30 rad/s, 14g).
is due to the radial pressure drop occurring from the inlet toward the center. The gas inward axial velocity is decreased because of the gas input through the sidewall distributor. The gas pressure at the wall distributor and throughout the rotating container, except at the exit zone, increased with increasing gas velocity. 5.2. Gas-Solid RFB. Initially, 0.223 kg of polyethylene particles (F ) 900 kg/m3 and dp ) 200 µm) was loaded at the bottom of the reactor, with a solid volume fraction of 0.4. The reactor started rotating, and the rotational speed increased linearly up to 30 rad/s (14g) after 5 s of startup. At this time, when the solids formed a vertical layer along the reactor wall, air was introduced radially inward. Simulation results showed that the solids void fraction increased and reached packed-bed conditions instantly after startup under the action of the gravitational force. The increase in the rotational speed caused the solids movement toward the wall gas distributor (see Figure 4) and formation of a packed-bed layer at the wall (see Figure 5). At the lower angular velocity, the gravitational force is the dominant force acting on the particles. As the rotational velocity increases, the centrifugal acceleration becomes more dominant and solids form a layer by the side wall of the cylindrical reactor. Our simulation shows that the bed thickness at 30 rad/s (14g) angular velocity is not uniform because of the downward effect of the gravitational force and possibly lower solid viscosity that was considered in the constitutive equations for the particulate phase. As a result, the bed thickness was observed to decrease gradually with the height of the cylinder. When the particles formed a packed layer and reached the quasi-steady-state condition, the air was introduced radially inward. Figure 6 shows the gas inlet velocity profile (case 1). As the gas flows inward, its velocity
Figure 4. Solid-phase volume fraction and velocity arrow plot after 2 s of startup, showing the radial movement toward the wall (Vin ) 0 m/s, Ω ) 30 rad/s, 14g).
Figure 5. Volume fraction and velocity arrow plot of the solid phase after 5 s, indicating the formation of a packed-bed layer by the wall (Vin ) 0 m/s, Ω ) 30 rad/s, 14g).
increases, because of a decrease in the cross-sectional flow area; this, in turn, will result in a higher drag force exerted by the gas on the particles. Therefore, by moving from the gas distributor to the center of the cylinder, the drag force increases and the centrifugal force decreases. Thus, there is a location where the magnitudes of these antagonist forces are equal. This location is always at the inner surface of the bed, contrary to the fluidized-bed systems where fluidization begins at the distributor. In other words, the minimum fluidization velocity is first reached at the interface between the gas and particles inside the bed. The velocity at
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2631
Figure 6. Gas inlet velocity profile for case 1 (Ω ) 30 rad/s, 14g).
which the first layer of the bed becomes fluidized is called the surface minimum fluidization velocity. Chen,5 Kao et al.,11 and Kroger et al.3 obtained the surface fluidization phenomena theoretically and verified their results experimentally. Figure 7a shows that the fluidization of the interface zone occurred at a gas inlet velocity of 0.14 m/s, at r ) 0.125 m with the local superficial velocity of approximately 0.16 m/s. At a gas inlet velocity of 0.30 m/s, the whole bed was fluidized (see Figure 7b). The velocity, at which the whole bed is fluidized, is called the critical minimum fluidization velocity.5 The difference between the surface and critical minimum fluidization velocities depends on the bed thickness.5 Our simulation agrees qualitatively with Chen,5 Kao et al.,11 and Kroger et al.3 results. As the gas inlet velocity was increased, the bed expanded more and its void fraction increased. A further increase of the gas inlet velocity to about 0.4 m/s enhanced the bed expansion as bubbles started forming (see Figure 7c).
Figure 8 shows the gas pressure drop profile for the case where the gas inlet velocity was increased slowly. It shows that, by increasing the gas inlet velocity up to 0.14 m/s, the pressure drop will increase up to the point where the surface minimum fluidization is achieved and a smooth change in the slope of the curve is observed. With a further increase in the gas inlet velocity up to 0.20 m/s, the pressure drop reaches a maximum and the whole bed is fluidized at critical minimum fluidization velocity. The pressure drop remains nearly constant after reaching this point. The value of these velocities obtained from the pressure drop curve is consistent with the literature data. The particles used in our simulation had a diameter of 200 µm and a density of 900 kg/m3, which was classified as Geldart group A. By comparison of the calculated pressure drop curve (see Figure 8) and group A fluidization curves, it can be seen that these particles do not exhibit group A particle behavior; instead, they behave more like group B particles. As Qian et al.10 reported, by applying high g (using a rotational fluidized bed), the transition line between different groups of particles in the Geldart classification will move toward smaller particle size. Our simulation confirms this experimental behavior reported by Qian et al.10 To obtain a better understanding of the effect of the inlet gas flow on the fluidization behavior of the RFB system, we ran a simulation with a more rapid increase of the gas inlet velocity. Figure 9 shows the gas inlet velocity profile (case 2). Figure 10 a-c shows the gas-solid flow patterns as a function of time. At time 20 s (see Figure 10a), when the gas superficial velocity is 1 m/s, which is higher than the critical minimum fluidization velocity, the whole bed was fluidized and bubbles were formed. The bed expan-
Figure 7. Solid volume fraction and velocity arrows at different gas inlet velocities, showing different stages of fluidization in RFB (Ω ) 30 rad/s, 14g).
2632
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003
toward the distributor because of the centrifugal force, and as soon as they reached the dense phase zone, they fluidized. 6. Conclusions
Figure 8. Gas pressure drop (Pa/m) profile of a two-phase RFB (case 1), indicating Geldart type B behavior (Ω ) 30 rad/s, 14g).
Figure 9. Gas inlet velocity profile for case 2 (14g).
sion can be seen after 40 s (see Figure 10b), when the gas inlet velocity is 2.5 m/s. Some solid particles began leaving the system because the gas local superficial velocity at the inner layer was almost 4 m/s, which is higher than the terminal velocity of particles at the computed local centrifugal force. Figure 10c shows gassolid flow patterns, after 65 s, when the gas inlet velocity is 4.5 m/s. In this high gas inlet velocity most of the solids have left the system. We studied the effect of mixing of the feeding particles in a continuous RFB by injecting the solid particles from the bottom of the reactor. Our calculation showed that, for the single-size particle case, the solid particles moved
The CFD model presented in this paper was successfully used to predict the experimentally observed gassolid flow behavior in a RFB. The single-phase simulation showed that an increase in the rotational speed increases the gas entrainment from the top opening in the case of no gas inflow from the side wall. The flow behavior was changed by introducing the gas into the cylinder. No vortex was observed in this condition, showing a uniform gas radial velocity profile with height. In the two-phase flow system, the centrifugal force caused the particles to move from the bottom to the side wall. When there was no gas inlet, the particles formed a dense layer by the gas distributor. The experimentally observed layer-by-layer fluidization of the particles was predicted by introducing the gas radially from the sidewall distributor. By an increase in the gas velocity to values higher than the terminal velocity of the particles, elutriation of the particles was predicted. The particle elutriation is dependent on the rate of gas inflow and angular velocity. Thus, when the optimum residence time of the particles and the desirable production rate corresponding to a certain gas flow rate are known, the proper rotational speed could be obtained. It is known that Geldart group A and B particles behave differently at normal gravity conditions in fluidized beds. The pressure drop vs inlet gas velocity plot obtained from this numerical simulation showed that the Geldart group A particles used in this study behave like Geldart group B under high acceleration force. It shows that the transition line between these two groups is shifted to the left side of the powder classification curve toward smaller particle size. It may be speculated that Geldart group C particles, which are
Figure 10. Solid volume fraction contour of gas-solid RFB at high gas inlet velocity, showing the dependency of particle residence time on the gas inlet velocity (14g).
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2633
difficult to fluidize because of their cohesive forces, could also be fluidized like group A particles in a RFB under proper centrifugal force. Nomenclature dp ) particle diameter e ) restitution coefficient b g ) gravity vector g0 ) radial distribution function k ) turbulent kinetic energy t ) time b us,w ) solid velocity at the wall I ) unit tensor P ) pressure Ps ) solid pressure Re ) particle Reynolds number Ti ) stress tensor of phase i U B i ) velocity vector of phase i Greek Letters β ) drag coefficient ) turbulent dissipation energy i ) volume fraction of phase i θ ) granular temperature µi ) viscosity of phase i λ ) bulk viscosity Fi ) density of phase i Ω ) angular velocity b τs,w ) solid shear stress φ ) specularity coefficient
Literature Cited (1) Kunii, D. G.; Levenspiel, O. Fluidization Engineering; Wiley: New York, 1969. (2) Choi, K. Y.; Ray, W. H. The Dynamic behavior of Fluidized Bed Reactors for Solid Catalyzed Gas-Phase Olefin Polymerization. Chem. Eng. Sci. 1985, 40, 2261-2279. (3) Kroger, D. G.; Levy, E. K.; Chen, J. C. Flow Characteristics in Packed and Fluidized Rotating Beds. Powder Technol. 1979, 24, 9-18. (4) Levy, E. K.; Shakespeare, W. J.; Tabatabaie-Raissi, A.; Chen, J. C. Particle Elutriation from Centrifugal Fluidized Beds. AIChE Symp. Ser. 1981, 205 (77), 86-95. (5) Chen, Y. M. Fundamentals of a Centrifugal Fluidized Bed. AIChE J. 1987, 33, 722-728. (6) Takahashi, T.; Tanaka, Z.; Itoshima, A.; Fan, L. T. Performance of a Rotating Fluidized Bed. J. Chem. Eng. Jpn. 1984, 17 (3), 333-336. (7) Fan, L. T.; Chang, C. C.; Yu, Y. S.; Takahashi, T.; Tanaka, Z. Incipient Fluidization Condition for a Centrifugal Fluidized Bed. AIChE J. 1985, 31 (6), 999-1009.
(8) Qian, G. H.; Bagyi, I.; Pfeffer, R.; Shaw, H.; Stevens, J. G. A Parametric Study of a Horizontal Rotating Fluidized Bed using Slotted and Sintered Metal Cylindrical Gas Distributors. Powder Technol. 1998, 100, 190-199. (9) Qian, G. H.; Bagyi, I.; Pfeffer, R.; Shaw, H. Particle Mixing in a Rotating Fluidized Beds: Inferences about the Fluidized State. AIChE J. 1999, 45 (7), 1401-1410. (10) Qian, G. H.; Bagyi, I.; Burdick, I. W.; Pfeffer, R.; Shaw, H. Gas-Solid Fluidization in a Centrifugal Field. AIChE J. 2001, 47 (5), 1022-1034. (11) Kao, J.; Pfeffer, R.; Tardos, G. I. On Partial Fluidization in Rotating Fluidized Beds. AIChE J. 1987, 33 (5), 858-861. (12) Arastoopour, H. Numerical Simulation and Experimental Analysis of Gas/Solid Flow Systems: 1999 Fluor-Daniel Plenary Lecture. Powder Technol. 2001, 119, 59-67. (13) Benyahia, S.; Arastoopour, H.; Knowlton, T.; Massah, H. Simulation of Particles and Gas Flow Behavior in the Riser Section of a Circulating Fluidized Bed using the Kinetic Theory Approach for the Particle Phase. Powder Technol. 2000, 112, 24-33. (14) Sun, B.; Gidaspow, D. Computation of Circulating Fluidized Bed Riser Flow for the Fluidization VIII Benchmark Test. Ind. Eng. Chem. Res. 1999, 38, 787-792. (15) Sinclair, J. Hydrodynamic Modeling. Circulating Fluidized Bed; Grace, J. R., Avidan, A. A., Knowlton, T. M., Eds.; Blackie Academic and Professional: London, 1997; pp 149-180. (16) Gidaspow, D. Multiphase Flow and Fluidization Continuum and Kinetic Theory Description; Academic Press: New York, 1994. (17) Patankar, S. V. Numerical Heat Transfer and Fluid Flow; Hemisphere: Bristol, PA, 1983. (18) Arastoopour, H.; Pakdel, P.; Adewumi, M. Hydrodynamic Analysis of Dilute Gas-Solids Flow in a Vertical Pipe. Powder Technol. 1990, 62, 163-170. (19) Johnson, P. C.; Jackson, R. Frictional-Collisional Constitutive Relations for Granular Materials, with Application to Plane Shearing. J. Fluid Mech. 1987, 176, 67-93. (20) Sinclair, J. L.; Jackson, R. Gas-Particle Flow in a Vertical Pipe with Particle-Particle Interactions. AIChE J. 1989, 35 (9), 1473-1486. (21) Kundu, P. Fluid Mechanics; Academic Press: San Diego, 1990. (22) Batchelor, G. K. An Introduction to Fluid Mechanics; Cambridge University Press: Cambridge, U.K., 1999. (23) Launder, B. E.; Spalding, D. B. Lectures in Mathematical Models of Turbulence; Academic Press: London 1972. (24) Fluent 4.4, User’s Guide; Fluent: Lebanon, 1997.
Received for review September 20, 2002 Revised manuscript received March 10, 2003 Accepted March 18, 2003 IE020740B
