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Hydrodynamic Simulation of Fluidization by Using a Modified Kinetic Theory Wei Wang* and Youchu Li Institute of Chemical Metallurgy, Chinese Academy of Sciences, Beijing 100080, China
For a pseudofluid consisting of a particle assembly, particle stress is transmitted through mutual contact between particles. When the particles are densely agglomerated, contacts are usually of long duration and frictional, and this part of the stress is the frictional stress. When the particles are sparsely spaced, on the other hand, contacts are temporary and collisional, and this part of the stress consists of kinetic and collisional stresses. In many cases the particle contact lies between these two extremes in a gas-solid fluidized bed, and all of these three parts of the stressskinetic, collisional, and frictional stressessplay important roles in particle-phase transport. However, the existing kinetic theory for granular flow (KTGF) only involves the kinetic and collisional parts of transport. In this paper, a frictional particle pressure was introduced for correction of KTGF in the case of highly dense flow, and the solid shear stress was corrected to be consistent with Einstein’s effective viscosity equation for dilute suspensions. This modified KTGF model may account for the stress over the entire range between two extremes of a densely packed state and a sparsely spaced state. As verification in the dense gas-solid flow, the timeaveraged total pressure drop and the particle pressure predicted by this modified KTGF model were found to be in agreement with the measurements in a cylindrical fluidized bed. The inflection point on the particle pressure curve, implying competition among the three transport mechanisms, was also predicted. Moreover, instantaneous formation of slugs starting from a homogeneous inflow condition was reproduced through simulation and the quantitative comparison of the slug velocity with empirical correlation was approving. For dilute gas-solid flow in a circulating fluidized-bed riser, the model predictions agree with the time-averaged solid viscosity in order of magnitude. Further modeling may require a better understanding of the drag force and turbulence. Introduction Quantitative understanding of the hydrodynamics of particle-fluid two-phase flow is of crucial importance for the design and operation of fluidized-bed reactors. A recent development of numerical methods provides us with a powerful tool to investigate the complex phenomena inherent in fluidization.1,2 Similar to the assumption of the original definition of fluidization, the pseudofluid models treat both gas and solid particles as continuous and fully interpenetrating fluids. Both phases are described in terms of conservation equations for mass and momentum respectively as follows:3
∂ (R F ) + ∇‚(RkFku b) k ) 0 ∂t k k
(1)
∂ (R F u b) + ∇‚(RkFku bu kb) k ) ∂t k k k g + ∇‚τk + β(u bl - u b) -Rk∇Pg + RkFkb k (2) where the overhead arrow stands for a vector and the double overbar for a tensor, subscript k ) g (gas phase) and s (solid phase), and l is the reversal of k. The solidphase stress τs can be written as T τs ) -Psδ + λs∇‚u bδ bs + (∇u b) s + µs[∇u s
bδ] (3) (2/3)∇‚u s * To whom correspondence should be addressed. Tel.: (86)10-62562448. Fax: (86)-10-62561822. E-mail: wangwei-cas@ 263.net.
where δ denotes a unit tensor. The above equation needs to be specified for particle pressure Ps, solid shear viscosity µs, and solid bulk viscosity λs. In some previous works,4 the solid viscosity was ignored or assumed constant, and the following empirical correlation was employed to account for the particle pressure.
∇Ps ) G(Rs) ∇Rs
(4a)
-G(Rg) ) 10-8.76Rg+5.43
(4b)
Here G(Rs) represents the solid-phase elastic modulus. However, its parameters were found to be variable in the literature.5 Because of a desire to develop a first-principle approach to calculate the solid-phase stresses, the kinetic theory for granular flow (KTGF), which is basically an extension of the Chapman-Enskog theory for dense gases,6 was widely used as a closure model for solidphase rheology in recent years.7-10 However, in the dense region of a fluidized bed, a high solid concentration may lead to multiple particle collisions or longduration contacts, while in the dilute region, the number density of particles may not be large enough to meet the requirement of the continuum assumption. Johnson and Jackson11 presented an expression for the particle phase stresses accounting for frictional and kineticcollisional effects in dense granular flow but did not include a correction factor for very dilute conditions. Therefore, the KTGF is still questionable in application and needs further improvement, especially in the simulation of the dense gas-solid flow in risers.12
10.1021/ie000989y CCC: $20.00 © 2001 American Chemical Society Published on Web 08/10/2001
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Figure 1. Three particle transport mechanisms: kinetic, collisional, and frictional transport.
The objective of this paper is to give a modified KTGF model to accommodate widely the extremely dilute and the highly dense flows. On the basis of Einstein’s equation for the effective viscosity of extremely dilute suspensions, the solids shear viscosity was corrected by a factor to avoid the physically implausible instability at the low solids fraction. Moreover, an empirical correlation for frictional pressure was added to the particle pressure. This modified KTGF was incorporated into the computer code to predict the nonuniform flow structure such as air slugs under realistic conditions. The computed results agree quantitatively well with experimentally measured particle pressure and the total bed pressure drop in a bubbling fluidized bed and qualitatively well with the experimentally determined solid viscosity in a fast fluidized-bed riser. Modified Kinetic Theory In the existing KTGF, the particle quantities (i.e., mass m, momentum of random motion mC, and kinetic energy of random motion 1/2mC 2) can be transported via kinetic motion and collisional interactions.13 However, the analogy between particles and molecules is not always valid because the particle concentration may be so high as to be close to the packed state, where the frictional contact may play a decisive role in the particlephase transport. Therefore, the frictional mechanism of particle-quantity transfer should be considered as well. As is known from Campbell and Wang’s experiment14 in a gas fluidized bed, the particle pressure decreases with increasing gas velocity under the densely packed condition as progressively more and more of the bed weight is supported by the drag force. It reaches a minimum at the incipient fluidization and rises when the bed is fluidized. This inflection point implies that different driving forces control the gas-solid flow at the two sides of incipient fluidization. At a packed state, the bed is crammed with particles, and hence the frictional force prevails over the other forces, while at a fluidized state, lasting contact gives way to free flight and brief collisions among particles. The competition and transformation of dominating forces lead to flow transition from the packed bed to fluidization. Subsequently, three mechanisms of the particle-phase transport result in two types of flow states, as shown in Figure 1. The frictional transport determines the behaviors at a close packed state, while the kineticcollisional transports cause a two-phase flow. So, the total stress may be approximated as the sum of frictional and kinetic-collisional contributions as if each of them acts alone.
Figure 2. Comparison of two moduli for particle pressure (Fs ) 2500 kg/m3 and Rsm ) 0.6436).
Frictional Particle Pressure The particle pressure is the normal part of the solid stress. For particles under the close-packed condition, such as a fixed bed or a fluidized bed with almost zero gas inflow, the solid momentum equation is reduced to the form
∇Psm ) RsmFsb g
(5)
where Rsm denotes the maximum solid volume fraction and Psm denotes the particle pressure at the corresponding solid concentration. Because no fluctuating motion occurs, the part of particle pressure due to kinetic motion and collisional interactions can be thought of as negligible. In this case, eq 5 is actually transformed to a differential definition of the frictional particle pressure. The experimental data,14 although somewhat scattered, could be correlated as the frictional particle pressure as follows.
Ps,frictional ) RsmFs exp[-100.6(Rsm - Rs) - 0.894] (6) Obviously, this pressure decreases quickly with increasing volume fraction of the solid. That means that the possibility of particle attrition decreases, as progressively the kinetic and collisional parts become important factors. Equation 4 is analogous to this pressure in form. However, their physical signification is different from each other as eq 4 is used to describe the whole particle pressure. Figure 2 gives a comparison of the elasticity modulus originated from the above two pressure expressions, revealing their remarkable difference in gradients. KTGF for Kinetic-Collisional Transport The KTGF developed for the kinetic-collisional transport follows a somewhat similar route.10,15,16 Assuming that the velocity distribution function of a large collection of particles satisfies the Boltzmann integraldifferential equation, multiplying the Boltzmann equation with a particle quantity φ, and taking the ensemble average yield the general Maxwell transport equation. Letting the quantity φ to be m, mC, and 1/2mC 2, the transport equation becomes the conservation equation of mass, momentum, and granular temperature, respectively. Supposing that the single-particle distribution function is subject to the Maxwellian velocity distribution (zero-order distribution) or its first-order perturbation and the Enskog assumption for the pair-distribution function is adopted, the shear viscosity from the
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Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 Table 1. Governing Equations for Gas-Solid Flow continuity equation (k ) g, s) ∂ (R F ) + ∇‚(RkFku b) k ) 0 ∂t k k momentum equation (k ) g, s; l ) s, g) ∂ b) + ∇‚(RkFku bu g + ∇‚τk + β(u bl - u b) (R F u kb) k ) -Rk∇pg + RkFkb k ∂t k k k gas-phase stress τg ) 2RgµgSg solid-phase stress
Figure 3. Reduced effective viscosity from different approaches (e ) 0.9).
zero-order and first-order solutions, the particle pressure, and the solid bulk viscosity could be written as13 (0)
µs
4 ) Rs2Fsdpg0(1 + e) 5
x
x
Θ π
(7a)
Θ 4 + µs(1) ) Rs2Fsdpg0(1 + e) 5 π 2µs,dilute 2 4 1 + (1 + e)Rsg0 (7b) 5 (1 + e)g0
[
]
Ps ) RsFs[1 + 2(1 + e)Rsg0]Θ
xΘπ
4 λs ) Rs2Fsdpg0(1 + e) 3
(7c) (7d)
where Θ ) 1/3〈C2〉 and µs,dilute ) 5/96FsdpxπΘ. These formulas of shear viscosity are consistent with each other under dense flow condition but yield various performances in dilute gas-particle flow.9,15 Accordingly, problem remains as to which distribution function is better. To choose an appropriate expression from various velocity distribution functions, we appealed to the effective viscosity for verification, and the effective viscosity in extremely dilute suspensions can be expressed as Einstein’s equation:
µeff ) µg(1 + 2.5Rs)
(8)
According to the definition of the effective viscosity for two-phase flow, it can be written as follows:17
Fg + Fsus/ug µeff ) µg Fg + Fsµgus/ugµs
τs ) [-ps + λs∇‚u bs]δ + 2µsSs deformation rate Sk ) (1/2)[∇u bk + (∇u bk)T] - (1/3)∇‚u bkδ solid-phase pressure Ps ) RsFsΘ + 2Rs2Fs(1 + e)g0Θ + RsmFs exp[-100.6(Rsm - Rs) 0.894] solid-phase shear viscosity µs )
2Rs0.96µs,dilute (1 + e)g0
[1 + 54(1 + e)g R ] + 54R 2
0 s
To examine the effective viscosity as a function of the solid concentration, experimental data18 of granular temperature measured in a riser were used to calculate the solid shear viscosity. The calculated relationship of the effective viscosity against the solid concentration is shown in Figure 3. According to Einstein’s equation, the effective viscosity should approach the gas viscosity in the extremely dilute regime. However, the zero-order solution underestimates while the first-order solution overestimates the effective viscosity. To be consistent with Einstein’s equation, a correction factor Rs0.96 for
Fsdpg0(1 + e)
xΘπ
µs,dilute ) (5/96)FsdpxπΘ solid-phase bulk viscosity λs ) (4/3)Rs2Fsdpg0(1 + e)(Θ/π)1/2 radial distribution function g0 ) [1 - (Rs/Rsm)1/3]-1 granular temperature equation 3 ∂ (R F Θ) + ∇‚(RsFsu bΘ) ) τs:∇u bs - ∇‚q - γ + βCg‚Cs - 3βΘ s 2 ∂t s s collisional energy dissipation γ ) 3(1 - e2)Rs2Fsg0Θ[(4/dp)(Θ/π)1/2 - ∇‚u bs] flux of fluctuating energy q ) -ζs∇Θ conductivity of the fluctuating energy ζs ) 2ζs,dilute[1 + (6/5)(1 + e)g0Rs]2/[(1 + e)g0] + ζs,collision ζs,dilute ) (75/384)dpFsxΘπ ζs,collision ) 2Rs2Fsdpg0(1 + e)xΘ/π drag coefficient β ) 150Rs2µg/(Rgdp2) + 1.75FgRs|u bg - u b|/d p for Rg e 0.80 s 2.65d ) for R > 0.80 β ) (3/4)CDsRgRsFg|u bg - u b|/(R g p g s where CDs ) 24[1 + 0.15Rep0.687]/Rep for Rep < 1000 CDs ) 0.44 for Rep g 1000 Rep ) RgFg|u bg - u b|d s p/µg
[
]
the first-order solution was introduced. Then the new shear viscosity becomes
x
Θ 4 + µs ) Rs2Fsdpg0(1 + e) 5 π 2Rs0.96µs,dilute (1 + e)g0
(9)
2 s
[1 + 54(1 + e)R g ] s 0
2
(10)
The modified shear viscosity gives a best fit to Einstein’s correlation as shown in Figure 3. As for the normal part of the solid stress, adding the frictional pressure to eq 7c gives a new total particle pressure as follows.
Ps ) RsFsΘ + 2Rs2Fs(1 + e)g0Θ + RsmFs exp[-100.6(Rsm - Rs) - 0.894] (11) Thus, the modified solid stress has been made to fit the experimental data over the whole range. The final model is listed in Table 1. Bagnold’s expression13 for the radial
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5069 Table 2. Parameter Settings for the Simulation of Slug Formation diameter dp material density Fs stuff width stuff height bed height
Hoomans’s
Stewart’s
4 mm aluminum 2700 kg/m3 150 mm 250 mm 500 mm
0.28 mm ballotini 2420 kg/m3 50.8 mm 500 mm 800 mm
Hoomans’s
Stewart’s
10 mm 20 mm 15 25 3m/s 0.9
2.54 mm 5.55 mm 20 144 20.4 cm/s 0.9
cell width ∆x cell height ∆y no. of x cells no. of y cells superficial gas velocity ug coefficient of restitution e
Table 3. Parameter Settings for the Simulation of Bed Pressure Drop and Particle Pressure particle diameter dp material density Fs stuff width stuff height bed height
0.848 mm, 1.063 mm glass beads 2500 kg/m3 100 mm 360 mm 720 mm
cell width ∆r cell height ∆z no. of r cells no. of z cells superficial gas velocity ug coefficient of restitution e
1.666 mm 10 mm 30 72 20-70 cm/s 0.9
Table 4. Parameter Settings for the Simulation of Solid Shear Viscosity in a Circulating Fluidized-Bed Riser particle diameter dp material density Fs riser diameter riser height superficial gas velocity ug solid mass flux Gs
0.14 mm sand 2500 kg/m3 0.15 m 8m 4.2-6.0 m/s 40 kg/m2 s
cell width cell height no. of x cells no. of z cells restitution coefficient e restitution coefficient at the wall ew specularity coefficient
5 mm 40 mm 30 200 0.99 0.95 0.5
distribution function g0, which is likely to describe the stress near the packed state, was adopted to account for the effects of particle crowding in a dense system. For bubbling fluidized beds, the two-phase covariance term βCg‚Cs in the granular temperature equation is negligible,15 while for a circulating fluidized-bed riser, however, the covariance term is assumed to be equal to the dissipation term 3βΘ due to turbulence.10 Simulation Test Cases. As a typical extremely heterogeneous flow structure, air slugs in fluidized beds attracted many works on simulation. The computation models are usually based on a discrete particle method.19-21 To examine the pseudofluid approach based on the modified KTGF, the same parameter settings as those used in Hoomans’s simulation21 and Stewart’s measurements22 were chosen as two typical test cases as shown in Table 2. As an important factor for the stability of fluidized beds, the particle pressure has been a focus of experimental measurements in recent years.5,14 Here Polashenski’s experimental settings5 were used in our simulations of the particle pressure as shown in Table 3. For simulation of solid stress in dilute gas-particle flow, the detailed parameter settings used in the simulation, which are similar to those of the experimental configuration,23 were provided in Table 4. Solution Procedure. The ICE scheme24 was used to solve the highly coupled equations. For bubbling fluidized bed, a uniform gas inflow was specified in the bottom row of cells. The particles were positioned at rest under gravity, and the voidage of the bed was set as 0.4 initially. The wall of the bed was modeled as a noslip rigid wall for the gas phase and a partial slip wall15 for the solid phase. For the simulation of particle pressure in a cylindrical rig, a free-slip wall was assumed to represent the symmetric line at the center. For a circulating fluidized-bed riser, the initial height of the slumped bed was 40 cm and its voidage was 0.4. The boundary condition applying for the two-phase velocities and granular temperature at the wall were the same as those in widely cited literature.8,10,11 For
Figure 4. Snapshots of slugging formation with homogeneous inflow conditions in a two-dimensional bed: (a) 1.0 s, (b) 1.2 s, (c) 1.6 s, (d) 2.0 s. e ) 0.9, ug ) 1.75umf, and 4 mm aluminum.
Figure 5. Pressure at 20 cm above the distribution plate (4 mm aluminum particle, ug ) 1.75umf).
all cases, zero-gradient assumptions were used at the outlet of the beds, and equally spaced grids were adopted. The time step was 10-5 s in all computations. Slugging Phenomena. The simulation for slugs on Hoomans’s settings lasted 7 s of real time. Some characteristic snapshots of slugging phenomena were drawn with the spectral mode in Figure 4. The formation of continuous slugs at given gas inflows can be clearly observed from these images. The pressure at the position of 20 cm above the distribution plate was plotted as a function of time as shown in Figure 5, where the sampling frequency was 10 Hz. The curve repro-
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slugs with regular shapes. Sometimes slugs with irregular motion will occur and the rising velocities are larger as expressed by
uA ) ug - umf + 0.35x2gD [cm/s]
Figure 6. Snapshots of slugging with 20.4 cm/s gas inflow (0.28 mm ballotini, ug ) 2.90umf).
Figure 7. Rising velocity of air slugs for 0.28 mm ballotini.
duced the characteristic pressure fluctuations corresponding to propagation of slugs. Their frequency (about 2 Hz) and amplitude (about 2000 Pa) were in accordance with the results of the discrete particle model. Compared with the discrete particle model, the pseudofluid approach is able to deal with the simulation for realistic systems with large quantities of particles similar to the settings of Stewart’s experiments,22 and, meanwhile, its demand for CPU capability is comparatively low. In Figure 6 snapshots of particle concentrations were given. The formation and rising of slugs can be clearly recognized. If the bottom line of slugs is used as a criterion for the position of the slugs, the rising velocity uA of slugs is obtainable. Figure 7 gave the relation of the slug rising velocity versus the superficial gas velocity. It was found that the simulated rising velocity of slugs was in rough agreement with the empirical correlation.25
uA ) ug - umf + 0.35xgD [cm/s]
(12)
where g is 980 cm/s2 and D is the bed width [cm]. The above correlation is based on cases of steadily rising
(13)
Our simulation result gave a regular shape and was hence closer to the bottom line. Total Pressure Drop and Particle Pressure. Measurement of the pressure drop with increasing gas inflow provides us insight into the flow structure of a fluidized bed. In this paper, Polashenski’s experimental data5 was used to compare with the simulation results. The effects of the superficial gas velocity on the total bed pressure drop and the particle pressure were investigated for particles of 0.848 and 1.063 mm in diameter. The time-averaged gas pressure was taken from the last 2 s of real time with a sampling frequency of 105 Hz. Figure 8 gives a comparison between the simulated and measured pressure drop. Considering that the model is based on a first-principle method, the agreement is approving. The inflection point on the particle pressure curves, which is caused by the transition of dominant driving forces, characterizes the transformation from a fixed bed to a fluidized bed. There is also an inflection point on the gas pressure curve between fast-fluidization and pneumatic transport.26 These two transitions mark two bifurcations27 inherent in fluidization. The difference between these two transitions lies in the fact that the former happens at a particle-dominating state, while the latter occurs at a gas-dominating state. So, the reversed quantities are the particle pressure and the gas pressure, respectively. Figures 9 and 10 show the simulated particle pressure at different positions and their comparison to the measurements. With introduction of a frictional particle pressure, the modified KTGF model predicts a decaying particle pressure with increasing gas input before a fully fluidized state. At the incipient point, the total bed is supported by interstitial gas, and the minimum particle pressure is reached. Further increasing gas inflow causes an increase of the particle pressure, because the kinetic motion and collisional interaction become important because of agitation of bubbles. However, probably because of the oversimplified model for the frictional particle pressure, the simulation underestimates the particle pressures at low gas inflow. Moreover, the discrepancy for the larger particles (1.063 mm) at the incipient fluidization point is possibly due to the irregular shape of particle materials.5 If the empirical correlation (eq 4) was used to predict the particle
Figure 8. Total bed pressure drop vs superficial gas velocity: (a) 0.848 mm glass beads; (b) 1.063 mm glass beads.
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5071
Figure 9. Particle pressure vs superficial gas velocity at 20 cm above the distributor for glass beads.
Figure 10. Simulated particle pressure vs superficial gas velocity at different heights for glass beads.
Figure 11. Radial profiles of computed (a and b) solid viscosity and (c and d) solid fraction for 0.14 mm sand where (a and c) ug ) 6.0 m/s, Gs ) 40 kg/m2 s and (b and d) ug ) 4.2 m/s, Gs ) 40 kg/m2 s.
pressure, a monotonically decreasing curve instead of a turning point , as the particle pressure in eq 4 decreases with increasing voidage. If the pure KTGF model was used, it would be hard to simulate the conditions with gas inflow lower than the incipient velocity, because it might cause a loss of bed weight.4 Solid-Phase Viscosity. Solid-phase viscosity plays an important role in circulating fluidized beds. It was believed that clusters formation and downflow along the riser walls is not possible to model without a solid
viscosity. Here a comparison of the computed results based on the modified KTGF model with the experimental results presented by Polashenski and Chen23 was provided for dilute gas-particle flow. The effect of the superficial gas velocity on the solid shear viscosity was investigated for 0.14 mm sand. The computation lasted 36 s, and the data were averaged from the last 8 s with a sampling frequency of 105 Hz. Figure 11 shows the time-averaged radial profiles of the solid viscosity and solid volume fraction as computed by the modified
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KTGF model and their comparisons with measurements. For the higher gas velocity and more dilute run, the measured solid viscosity decreases slightly with the radial position and does not vary much with the axial position except midway between the tube center and wall. Similarly, the computed solid viscosity varies little with the axial position because the solid fraction is almost the same along the axial direction. However, the model prediction does not capture the decreasing tendency in the region adjacent to the wall. For the lower gas velocity and more dense case, the computed solid viscosity shows slightly decaying trends along the radial direction similar to that of measurements. However, the predicted axial profile of the solid viscosity is more similar to that of the dilute run but different from that of the dense case. This contradiction probably comes from the fact that the computed solid fraction profile, almost uniformly dilute across the riser, deviates from the fast fluidization which is characterized by the coexistence of a dense-phase region in the lower section and a dilute-phase region in the upper section. In fact, we still need more knowledge on two-phase turbulence and drag correlation, etc., for more reliable predictions of high-velocity fluidization. It should also be noted that the current experimental data of the solid viscosity are not based on direct measurements and our computation results are strongly dependent on the particle elasticity and the boundary condition. Different boundary conditions may cause different trends of the solid viscosity along the radial direction,28,29 and a slightly adjusted coefficient of restitution may lead to a variation in order of magnitude.30 Although the simulation shows reasonable agreement in order of magnitude between measurement and computation with delicately specified parameters and boundary condition, the sensitivity of the simulation to these model parameters implies a long way to go for successful application. Conclusion The modified KTGF model in this paper was extended to the cases of extremely dilute and highly dense twophase flow. This modified model is consistent with Einstein’s equation for the effective viscosity of dilute suspension and coincides with the measured particle pressure in dense particle flow. A pseudofluid approach with this KTGF closure model was provided. For dense and moderately dense conditions, our model reproduced the slugging phenomena quantitatively and predicted in reasonable agreement with measurements on the time-averaged total pressure drop and particle pressure in a cylindrical fluidized bed. The measured inflection point on the particle pressure curve was also predicted. The existence of this inflection point is due to the transition of dominating forces among three mechanisms. For dilute gas-particle flow, comparison of the computed solid viscosity with measurements in a circulating fluidized-bed riser showed reasonable agreement in order of magnitude. Further modeling requires a deeper insight into the mechanism of turbulence in the presence of the solid phase, the drag correlation, and a more elaborate assignment on the coefficient of restitution and boundary condition. Nomenclature C ) fluctuating velocity of the particle dp ) particle diameter
e ) coefficient of restitution g ) gravity g0 ) radial distribution function u ) velocity P ) pressure R ) volume fraction β ) drag coefficient µ ) shear viscosity µeff ) effective viscosity λ ) bulk viscosity F ) density τ ) stress Θ ) granular temperature Subscripts g ) gas phase k ) phase k m ) maximum s ) solid phase
Literature Cited (1) Kuipers, J. A. M.; van Swaaij, W. P. M. Application of Computational Fluid Dynamics to Chemical Reaction Engineering. Rev. Chem. Eng. 1997, 13 (3), 1. (2) Li, Y. Hydrodynamics. In Advances in Chemical Engineering; Kwauk, M., Ed.; Academic Press: New York, 1994; Vol. 20, pp 85-146. (3) Anderson, T. B.; Jackson, R. A. Fluid Mechanical Description of Fluidized Beds. Ind. Eng. Chem. Fundam. 1967, 6, 527. (4) Gidaspow, D. Hydrodynamics of Fluidization and Heat Transfer: Supercomputer Modeling. Appl. Mech. Rev. 1986, 39, 1. (5) Polashenski, W., Jr.; Chen, J. C. Normal Solid Stress in Fluidized Beds. Powder Technol. 1997, 90, 13. (6) Chapman, S.; Cowling, T. G. The Mathematical Theory of Non-Uniform Gases; Cambridge University Press: Cambridge, U.K., 1970. (7) Lun, C. K. K.; Savage, S. B. B.; Jeffrey, D. J.; Chepurniy, N. Kinetic Theories for Granular Flow: Inelastic Particles in Couette Flow and Slightly Inelastic Particles in a General Flow Field. J. Fluid. Mech. 1984, 140, 223. (8) Sinclair, J. L.; Jackson, R. Gas-Particle Flow in a Vertical Pipe with Particle-Particle Interactions. AIChE J. 1989, 35, 1473. (9) Boemer, A.; Qi, H.; Renz, U. Eulerian Simulation of Bubble Formation at a Jet in a Two-Dimensional Fluidized Bed. Int. J. Multiphase Flow 1997, 23, 927. (10) Nieuwland, J. J.; van Sint Annaland, M.; Kuipers, J. A. M.; van Swaaij, W. P. M. Hydrodynamic Modeling of Gas/Particle Flows in Riser Reactors. AIChE J. 1996, 42, 1569. (11) Johnson, P. C.; Jackson, R. Frictional-Collisional Constitutive Relations for Granular Materials with Application to Plane Shearing. J. Fluid Mech. 1987, 176, 67. (12) Sun, B.; Gidaspow, D. Computation of Circulating Fluidized-Bed Riser Flow for the Fluidization VIII Benchmark Test. Ind. Eng. Chem. Res. 1999, 38, 787. (13) Gidaspow, D. Multiphase Flow and Fluidization: Continuum and Kinetic Theory Description; Academic Press: Boston, 1994. (14) Campbell, C. S.; Wang, D. G. Particle Pressures in GasFluidized Beds. J. Fluid Mech. 1991, 227, 495. (15) Ding, J.; Gidaspow, D. A Bubbling Fluidization Model Using Kinetic Theory of Granular Flow. AIChE J. 1990, 36, 523. (16) Cao, J.; Ahmadi, G. Gas-Particle Two-Phase Turbulent Flow in a Vertical Duct. Int. J. Multiphase Flow. 1995, 21, 1203. (17) Liu, D. Fluid Dynamics of Two-Phase Systems; High Education Press: Beijing, 1993. (18) Gidaspow, D.; Lu, H. Equation of State and Radial Distribution Functions of FCC Particles in a CFB. AIChE J. 1998, 44, 279. (19) Tsuji, Y.; Kawaguchi, T.; Tanaka, T. Discrete Particle Simulation of Two-Dimensional Fluidized Bed. Powder Technol. 1993, 77, 79. (20) Ouyang, J.; Li, J. Simulation of Gas-Solid Two-Phase Flow by a Particle-Motion-Resolved Model. Sci. China B 1999, 29, 29.
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5073 (21) Hoomans, B. P. B.; Kuipers, J. A. M.; Briels, W. J.; van Swaaij, W. P. M. Discrete Particle Simulation of Bubble and Slug Formation in a Two-Dimensional Gas-Fluidized Bed: a HardSphere Approach. Chem. Eng. Sci. 1996, 51, 99. (22) Stewart, P. S. B.; Davidson, J. F. Slug Flow in Fluidized Beds. Powder Technol. 1967, 1, 61. (23) Polashenski, W., Jr.; Chen, J. C. Measurement of Particle Phase Stresses in Fast Fluidized Beds. Ind. Eng. Chem. Res. 1999, 38, 705. (24) Harlow, F. H.; Amsden, A. A. Numerical Calculation of Multiphase Fluid Flow, J. Comput. Phys. 1975, 17, 19. (25) Davidson, J. F.; Harrison, D. Fluidization; Academic Press: London, 1971. (26) Reddy-Karri, S. B.; Knowlton, T. A Practical Definition of Fast Fluidized Bed. In Circulating Fluidized Bed Technology III; Basu, P., Hasatani, M., Horio, M., Eds.; Pergamon Press: Oxford, U.K., 1991; pp 67-72.
(27) Li, J.; Qian, G.; Wen, L. Gas-Solid Fluidization: a Typical Dissipative Structure. Chem. Eng. Sci. 1996, 51, 667. (28) Samuelsberg, A.; Hjertager, B. H. Computational Modeling of Gas/Particle Flow in a Riser. AIChE J. 1996, 42, 1536. (29) Neri, A.; Gidaspow, D. Riser Hydrodynamics: Simulation Using Kinetic Theory. AIChE J. 2000, 46, 52. (30) Samuelsberg, A.; Hjertager, B. H. An Experimental and Numerical Study of Flow Patterns in a Circulating Fluidized Bed Reactor. Int. J. Multiphase Flow 1996, 22, 575.
Received for review November 27, 2000 Revised manuscript received May 17, 2001 Accepted June 6, 2001 IE000989Y
