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Coexistence of Solidlike and Fluidlike States in a Deep Gas-Fluidized Bed Junwu Wang,* M. A. van der Hoef, and J. A. M. Kuipers Faculty of Science and Technology, UniVersity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
Characterizing regime transition in gas-fluidized beds is of fundamental importance for the successful applications of fluidization technology. In this study, we show that a state-of-the-art two-fluid model has the ability to correctly predict the transition from packed bed to fully bubbling fluidized beds. To this end, we have studied a deep gas fluidized bed, in which the process of transition is shown much more clearly (see Figure 1) compared to relatively shallow fluidized beds. Particularly, we show that in the process of regime transition, the solidlike and fluidlike states can coexist. We conclude this on the basis of visual observation, global bed pressure drop signals, local solid volume fraction signals, cross-sectional averaged axial granular temperature profiles, and radial particle velocity profiles. The presence of the coexisting phases can be easily understood from the fact that the gas phase is compressible. 1. Introduction A typical gas-fluidized bed consists of a vertical vessel, a bed of particles, and fluidizing air, which is fed from the bottom of the bed and exerts drag force on the particles. When the drag force is less than the particle weight, the bed is operated in the fixed bed regime. By increasing the superficial gas velocity (Ug) to the minimum fluidization velocity (Umf), the drag force becomes sufficient to balance the particle weight, and the bed is said to be fluidized. Further increasing Ug will give rise to a variety of fluidization regimes.1 It is well-known that fluidization regime transition is directly dependent on the particle properties.2 For Geldart B particles, once Umf is exceeded, there is no interval of homogeneous fluidization and gas bubbles appear directly.2 For Geldart A particles, with increasing Ug, the bed transforms from a fixed bed to a homogeneous fluidized bed, and finally to a bubbling fluidized bed. A recent study3 has revealed that homogeneous fluidization actually consists of solidlike and fluidlike states, where the bed transforms from a solidlike state to a fluidlike state when Ug is increased from Umf to the minimum bubbling velocity (Umb). Note that the fluidlike state is actually not a true homogeneous state as in ordinary fluids, since there exist mesoscale pseudoturbulent structures and short-lived voids.4,5 In this study, we study a deep gas-fluidized bed, in which the solidlike and fluidlike states can actually coexist at specific operating conditions. We use Geldart B particles as bed material, because it has been demonstrated extensively in previous studies that state-of-the-art Eulerian models can correctly predict many important features of Geldart B and D particles.6-8 2. Numerical Model and Simulation Layout The numerical model used here is a state-of-the-art Eulerian model. Unfortunately, a reliable and comprehensive physical model for particulate phase stresses of the simulated system, including a friction-dominated solidlike state, a collisiondominated fluidlike state and the transition state (the solidlike and fluidlike coexistence state), is not yet available.9,10 For the present study we assume that a linear combination of the kinetic theory of granular flows proposed by Nieuwland et al.11 and the frictional model derived by Srivastava and Sundaresan12 is * To whom correspondence should be addressed. E-mail: J.Wang@ tnw.utwente.nl. Phone: +31-53-4892370. Fax: +31-53-4892882.
sufficient to represent the particulate phase stresses of the present system. The governing equations and its constitutive relations are given in Table 1, more details of the model can be found in the work of Patil et al.7 The physical properties of gas and solid and parameters used in the simulations are summarized in Table 2. Note that in all simulations, the angle of internal friction φ is 30° A 2D gas-fluidized bed, 0.267 m i.d. and 2.5 m height, was simulated. At the wall, the no-slip condition is applied for both phases. At the bottom inlet, a uniform gas velocity is specified, at the top outlet, atmospheric pressure (101325 Pa) is prescribed. At the initial state, the particles are packed in the bottom section of the bed with specified height (H0 ) 1.6 m) and voidage (ε0 ) 0.44). The physical model is numerically solved by a finitedifference method with a second-order upwind scheme for the convective term, details of which can be found in the work of van der Hoef et al.13 It is well-known that the constitutive laws used in two-fluid models can have an influence on the simulation results. To this end, a parametric study is carried out to investigate the effect of the particulate phase stress and interphase drag force correlation that we use; furthermore, additional simulations are performed to show the effect of particle size, details of which are given in the Appendix. From the results, it is clear that the qualitative observations of this paper (to be discussed in the next section) do not change with the used constitutive laws and the particle size. Another point we should address is the grid size we used, which is about 15 particle diameters (when the particle diameter is 3.5 × 10-4 m). One may argue that the grid size is not fine enough; therefore, the bubble prediction is possibly suppressed; we believe that the grid size in our study is less critical, since the main purpose is to qualitatiVely show the coexistence of solidlike and fluidlike states (the quantitative results depend on the constitutive laws as shown in the Appendix). Furthermore, the results using coarser particles (7.0 × 10-4 m), shown in the Appendix, indicate that the coexistence of solidlike and fluidlike states also exist, while in that case the grid size is about 7.5 particle diameters, which is assumed to be sufficiently fine to obtain grid-size-independent results according to many previous studies, for example, refs 14 and 15. In the remainder of the paper, only the simulation results for a particle diameter of 3.5 × 10-4 m using the drag force correlation of Gidaspow16 with the effect of the frictional stress model12 are analyzed in detail.
10.1021/ie901555p 2010 American Chemical Society Published on Web 02/05/2010
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Table 1. Governing Equations for Gas-Solid Flow and Constitutive Law continuity equations for gas and solid phases ∂ (ε F ) + ∇ · (εgFgb u g) ) 0 ∂t g g ∂ (ε F ) + ∇ · (εsFsb u s) ) 0 ∂t s s momentum equations for gas and solid phases ∂ (ε F b u ) + ∇ · (εgFgb u gb u g) ) -εg∇p + ∇ · (εgcτg) + εgFgb g + β(u bs - b u g) ∂t g g g ∂ (ε F b u ) + ∇ · (εsFsb u sb u s) ) -εs∇p - ∇ps + ∇ · (εsτcs) + εsFsb g + β(u bg - b u s) ∂t s s s energy conservation equation for granular temperature 3 ∂(εsFsΘs) + ∇ · (εsFsb u sΘs) ) (-pscI + εsτcs):∇u bs - ∇ · (εsqs) - γ - 3βΘs 2 ∂t
[
]
gas phase density Fg )
Mg p RTg
stress-strain tensor for gas and solid phases 2 τcg ) µg(∇u bg + ∇u bTg ) - µg(∇ · b u g)Ic 3 2 bs + ∇u bsT) + λs - µs (∇ · b u s)Ic τcs ) µs(∇u 3
(
)
{
solid pressure
εs e 0.5
0
ps ) εsFsΘs + 2Fs(1 +
e)εs2g0Θs
+ pf(εs),
pf(εs) )
0.05
(εs - 0.5)2 (εsmax - εs)5
εs > 0.5
solid bulk viscosity 4 λs ) εsFsdsg0(1 + e) 3
Θs π
solid shear viscosity 4 µs ) εsFsdsg0(1 + e) 5
5Fsdp√πΘs Θs 4 8 + 1.016 1 + (1 + e)εsg0 1 + g0εs + π 96εsg0 5 5
(
where
)(
)
pf(εs)√2sin φ c ij:D c ij + Θs /dp2 2εs√D
1 c ij ) 1 (∇u bs + (∇u u scI D bs)T) - ∇ · b 2 3
radial distribution function g0 ) 1 + 4εs
1 + 2.5000εs + 4.5904εs2 + 4.515439εs3 εs 3 0.67802 1εs,max
[ ( )]
3. Visual Observations In Figure 1, we plot grayscale snapshots of the solids volume fraction distribution at various superficial gas velocities at t ) 20 s (dp ) 3.5 × 10-4 m). It can be seen that with increasing Ug, in the fixed bed regime (Ug < 0.12 m/s), the bed is slightly expanded, which seems inconsistent with the previous conclusion that, in the fixed bed regime, the bed height is a constant. It should be noted however that recent experiments17,18 and
discrete particle simulations19 also showed that a slight bed expansion can be detected in the fixed bed regime. When Ug ) 0.13 m/s, the top of the bed becomes unstable, and very small gas bubbles can be visually observed near the confining walls. At the same time, the bottom section of the bed is homogeneous and actually still solidlike. It is important to point outsand we will discuss later in more detailsthat this solidlike state is not a fixed bed state, but rather a homogeneous fluidized state. Figure 1 thus shows
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Table 1. Continued pseudo-Fourier flux of kinetic fluctuation energy
qs ) -ks∇Θs
ks ) 2εsFsdp(1 + e)g0
75Fsdp√Θsπ Θs 6 12 + 1.02513 1 + (1 + e)εsg0 1 + εsg0 π 384εsg0 5 5
(
collisional dissipation of energy γ ) 3(1 - e2)εs2Fsg0Θs
interphase drag coefficient β)
CD )
{
{
-4
3.5 × 10 m, 7.0 × 10 1780 kg/m3 1.85 × 10-5 kg/(m · s) 0.98 0.64356
dp Fs µg e εs,max
m
150
εs2µg εgdp2
+ 1.75
(
)
2
)
]
Θs - (∇ · b u s) π
εs e 0.2
ug - b u s| Fgεs | b εs > 0.2 dp
εgFgdp | b ug - b u s| (24/Re)(1 + 0.15Re0.687), Re < 1000 , Re ) µg 0.44, Re g 1000
∆t grid number Mg Tg R
-5
5.0 × 10 s 50 × 400 28.8 × 10-3 kg/mol 293.0 K 8.314 J/(mol · K)
that there is a coexistence of solidlike and fluidlike states around Ug ) 0.14 m/s. Note that if we calculate Umf using the value εmf ≈ 0.46 observed in the simulations and the well-known correlation by Kunii and Levenspiel,20 1.75 dpUmfFg µg εmf3
4 dp
ug - b u s | -2.65 3 Fgεgεs | b C εg 4 D dp
Table 2. Summary of Parameters Used in Numerical Simulations -4
[
)(
150(1 - εmf) dpUmfFg dp3Fg(Fp - Fg)g + ) µg εmf3 µg2 (1)
we indeed obtain the value Umf ) 0.13 m/s. However, if in the simulation, we “measure” Umf as one would do in
experimental studies, namely, the gas velocity at which pressure drop reaches the value of particle weight, Umf is about 0.12 m/s, which is slightly lower than the value calculated from eq 1. From Figure 1, we can see that for the particles we studied (Group B type according to Geldart’s classification2), a short but clearly recognizable homogeneous fluidization can be found, which seems inconsistent with the previous conclusion that, for this type of particle, there is no interval of homogeneous fluidization or the minimum bubbling velocity should equal the minimum fluidization velocity. However, it is interesting to note that carefully conducted fluidization experiments did observe a short but clearly visible interval of homogeneous fluidization of Geldart B particles.21,22 Upon further increasing Ug, a fluidization front is seen to extend downward through the bed, which means that the solidlike state is gradually changed into fluidlike state. When Ug ) 0.15 m/s, the bed is fully operated
Figure 1. Snapshots of solids volume fraction distribution at various superficial gas velocities at t ) 20 s, dp ) 3.5 × 10-4 m.
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at the bubbling fluidization regime, that is, the bed is completely fluidlike. Further increasing the gas velocity, the bed will possibly transit to a slugging fluidization regime as shown in Figure 1; see Ug ) 0.2 m/s. The qualitative characteristics obtained from numerical simulation are in agreement with previous experimental studies under low operating pressures,23-25 which report that, at some specific superficial gas velocities, the coexistence of solidlike and fluidlike states (or a partially bubbling bed) are present. In particular, Wraith and Harris25 have studied the details of this phenomenon over a range of absolute pressure between 67 and 400 Pa and observed that, with the increase of superficial gas velocity, a fluidization front is seen to extend downward through the bed, as what have observed in our numerical simulations. Note that, it is impossible to perform corresponding numerical simulations and then compare directly with their experimental results, since at the given pressure (from 67 to 400 Pa), the continuum assumption of gas phase at the level of the solid particles is not valid anymore.25 Figure 2 shows snapshots of the solids volume fraction distribution of the partially bubbling fluidized bed at 5, 10, 15, and 20 s at a constant fluidization velocity (Ug
Figure 4. Normalized pressure drop (squares) and average bed height (triangles) as a function of superficial gas velocity. The error bars indicate the standard deviation of the pressure drop fluctuation.
) 0.14 m/s). It is evident that the simultaneous presence of solidlike (homogeneous) and fluidlike (heterogeneous) states is not a transient state, but a stable phenomenon under specific operating conditions; that is, we can speak of a true coexistence of the solidlike and fluidlike states. 4. Detailed Analysis of the Fluidization Characteristics
Figure 2. Snapshots of solids volume fraction distribution at different times, demonstrating the stability of the coexistence of solidlike (homogeneous) and fluidlike (heterogeneous) states (Ug ) 0.14 m/s, dp ) 3.5 × 10-4 m).
Figure 3. Transient bed pressure drop signals for the fixed bed (Ug ) 0.06 m/s), the partially bubbling fluidized bed (Ug ) 0.14 m/s), and the fully bubbling fluidized bed (Ug ) 0.2 m/s).
In Figure 3, we plot the transient bed pressure drop signals for the fixed bed, partially bubbling fluidized bed, and fully bubbling fluidized bed regimes. It can be seen that the fluctuation of the pressure drop of the fixed bed is visually nonexistent, which indicates that the bed is very stable and solidlike. The pressure drop is less than the particle weight per unit cross-sectional area (about 1.55 × 104 N/m2), indicating that part of the particle weight is supported by frictional stress. When the bed is operating as a partially bubbling fluidized bed, the pressure drop fluctuates around a mean value with a small standard deviation, which is caused by the appearance of small bubbles or voidlike structures in the upper part of the bed. The fluctuation of pressure drop becomes more significant when the bed is in the fully bubbling fluidization regime. In Figure 4, we show the pressure drop (normalized by the bed weight) and the average bed height as a function of Ug. The average bed height is defined as the height where the solid volume fraction is equal to half the average solid volume fraction in the dense section. It can be seen that in the fixed bed regime the pressure drop increases with increasing Ug. When Ug reaches a certain critical value, the pressure drop reaches a maximum, while upon further increasing Ug, the pressure drop first decreases very slightly, and finally attains at a constant value. However, a different trend is observed for the average bed height, which increases monotonously when Ug passes through all of these regimes. The slight increase of the average bed height in the fixed bed regime indicates the existence of bed expansion in this regime as seen in Figure 1. Note that in the fully bubbling fluidization regime, the normalized pressure drops are slightly less than one, and the presence of such an offset in pressure drop is also found in previous experimental studies,21,22,26 implying that the particles are not fully fluidized by the gas and that a small part of its weight is being supported by the distributor and the tube wall.22,26 In Figure 5, we show the transient local solids volume fraction signals measured at the center of the bed (H ) 0.6 m) for the different regimes. It can be seen that no fluctuation is present
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in the fixed bed (Ug ) 0.08 m/s); when the bed is in the partially bubbling fluidized state, fluctuation of the solids volume fraction can be detected (Ug ) 0.145 m/s), which corresponds to the formation of small bubbles or short-lived voids and, therefore, is fluidlike, as is also clearly seen in Figure 2. Upon further increasing Ug to 0.24 m/s, the bed is in the fully bubbling fluidization regime and the bubble frequency is obviously far larger compared to a partially bubbling fluidized bed. Figure 6 shows the cross-sectional averaged axial granular temperature profiles at different regimes. The granular temperature of the fixed bed (Ug ) 0.1 m/s) is 10-12 (m/s)2, except at the surface of the bed, where small fluctuations can be detected. Note that the value (10-12 (m/s)2) is the minimum granular temperature (Θmin) set in numerical simulations. For Ug ) 0.20 m/s (it is operated in the fully bubbling fluidization regime), a granular temperature which is 8 orders of magnitude larger than Θmin appears in the entire suspension, where the maximal granular temperature again prevails at the surface of the bed, which may be caused by the breakup of large bubbles in the splash zone. In the case of Ug ) 0.14 m/s (partially bubbling fluidized bed), we find that when the height is in between 0 and about 0.5 m, the granular temperature is 10-12 (m/s)2, which indicates this part of the bed is actually solidlike. From the snapshots in Figure 1 and the pressure drop characteristic in Figure 4, it is clear that the solidlike state in the partially bubbling fluidization state should not be interpreted as fixed bed but as homogeneous fluidization, as in previous experimental studies.21,22,26 For this reason, the detected homogeneous fluidization can not be explained by the particulate phase pressure arising from particle velocity fluctuation as in the studies of Sergeev et al.27 and Cody et al.28 but can be easily explained in terms of the frictional stresses or yield stresses associated with enduring particle contact as in previous experiments.21,22,29,30 Figure 7 shows the simulated axial distribution of the time-averaged solid volume fraction at the bottom portion of a partially bubbling fluidized bed. It can be seen that in the solidlike state (H < 0.5 m; see the analysis of Figure 6) the mean solid volume fraction is larger than 0.5, which means the frictional stress does exist according to the numerical model we used (see the equation for the solid pressure in Table 1), although the value is very small compared to the particle weight. This is the reason why the simulated pressure drop is very close to but slightly less than the particle weight (about 99% of the particle weight), as shown in Figure 4. Also, this is the reason why solidlike homogeneous fluidization can be achieved according to the mechanics proposed by Loezos et al.,22 that is, bubbling occurs when the yield stress or frictional stress in the particle assembly is small enough so that it can be overcome by spatial and/or temporal fluctuations in the velocities. At heights in between 0.5 and about 1.8 m, significant particle velocity fluctuation is observed, which indicates that this part of the bed is fluidlike; again, this is obviously due to the bubble motion, see Figure 1. At heights larger than 1.8 m, the granular temperature is again 10-12 (m/s)2, but now this is obviously due to the absence of particles in this region. Figure 8 shows the time-averaged radial profiles of particle velocities at four different heights for different regimes, which further indicate the coexistence of solidlike and fluidlike states in the gas-fluidized bed. When Ug ) 0.1 m/s, the particle velocities at all of the four bed heights are zero, indicating there is no particle motion, which is consistent with the basic nature of a fixed bed. When Ug ) 0.135 m/s, the particle velocities at
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Figure 5. Transient local solids volume fraction at the center of the bed (H ) 0.6 m) under the fixed bed (Ug ) 0.08 m/s), the partially bubbling fluidized bed (Ug ) 0.145 m/s), and the fully bubbling fluidized bed (Ug ) 0.24 m/s).
Figure 6. Axial granular temperature profile of the fixed bed (Ug ) 0.10 m/s), the partially bubbling fluidized bed (Ug ) 0.14 m/s), and the fully bubbling fluidized bed (Ug ) 0.2 m/s).
Figure 7. Axial distribution of the time-averaged solid volume fraction at Ug ) 0.14 m/s.
H ) 0.4 and 0.6 m are zero, but not for H ) 0.8 and 1.1 m, which indicates that the upper part of the bed is fluidlike and the lower part of the bed is solidlike. Upon further increasing of Ug to 0.14 m/s, the fluidization front is extended downward to H ) 0.6 m, but the particle velocities at H ) 0.4 m are still zero. At Ug ) 0.2 m/s, particle motion can be detected at all bed heights, the time-averaged radial profiles of particle velocities presented here indicate the solid circulation pattern
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Figure 8. Radial profiles of particle velocities at four different heights for the fixed bed (Ug ) 0.10 m/s), the partially bubbling fluidized bed (Ug ) 0.135 and 0.14 m/s), and the fully bubbling fluidized bed (Ug ) 0.2 m/s).
erogeneous) states in a deep gas-fluidized bed. Can we qualitatively explain the underlying mechanics of such phenomenon? First, we note that the particle pressure in the frictional stress model we used increases with increasing solid volume fraction and tends to infinity when the solid volume fraction is approaching to the solid volume fraction at packed conditiona (εsmax). Consequently, the mean solid volume fraction should decrease toward the top of the bed in the solidlike state (it can be either fixed bed or homogeneous fluidization), since the frictional stress within the particle assembly must increase toward the bottom of the bed, in order to counterbalance the increasing particle weight. Second, in the solidlike state, the particle velocity is essentially zero; therefore, the interphase drag force can be calculated from the Ergun equation, F ) 150
εs2µg εgdp
u + 1.75 2 g
[
) K1 Figure 9. Snapshots of solids volume fraction distribution at various superficial gas velocities with a constant gas density of 1.2 kg/m3. All other parameters are exactly the same as those used in the figures in the whole manuscript.
of a deep bed, that is solids ascending at the center and descending at the wall. 5. Explanation of the Coexistence State The foregoing study has shown that there can be a coexistence of solidlike (homogeneous) and fluidlike (het-
(
)
Fgεs 2 K2εs εs2 ug ) K1 + u dp 1 - εs 1 - εs g εs2
(1 - εs)
2
+
K2εs (1 - εs)2
]
Ug(2)
where K1 and K2 are constants, since for a bed of uniform cross-section as in the present study, the throughout of gas in a unit area Qg/A ) Fgug(1-εs) ) FgUg is a constant. From eq 2, it is clear that if the effect of gas compressibility is negligible (so that Ug is constant), with the increase of gas velocity, the interphase drag force will balance the particle weight first at the part of the bed where the solid volume fraction is highest. This means that if the gas density is a constant, the fluidization will be triggered at the bottom
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portion of the bed and then proceeds upward, as in the experimental and theoretical study of Tsinontides and Jackson.21 However, as shown in Figure 1, in our study the bed tends to fluidize from the free surface and then proceeds downward. This is because in the simulated deep fluidized bed, the effect of gas compressibility plays a key role. Now let us get back to eq 2. If we assume that the effect of gas compressibility overwhelms the effect of the axial variation of solid volume fraction, we then can assume that the axial solid volume fraction is a constant (this is approximately validated in our cases, as can be seen in Figure 7), that is, K1[εs2/(1 - εs)2] + K2[εs/(1 εs)2] is a constant. Therefore, we have
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Ftop Ug,top Fg,bottom poutlet + FsgH0(1 - ε0) ) ) ≈ ) Fbottom Ug,bottom Fg,top poutlet 1.154(3) This means that there is a drag force gradient across the bed: the drag force at the upper part is larger than that of the lower part. This is why the solidlike and fluidlike can coexist in the bed, since under some specific operating conditions, the drag force at the upper part is sufficient to balance the particle weight; therefore, the fluidization will be triggered, whereas the drag force at the lower part is insufficient to support the particle weight and frictional stress is still necessary to keep force balance. That is possibly the reason why this phenomenon has thus far only been observed for gas fluidization with very low absolute pressures, since it is much easier to achieve a
Figure 10. Snapshots of solids volume fraction distribution at various superficial gas velocities, dp ) 7.0 × 10-4 m.
Figure 11. Snapshots of solids volume fraction distribution at various superficial gas velocities. All parameters are exactly the same as those used in Figure 1, but now without the frictional stress model proposed by Srivastava and Sundaresan.12
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Figure 12. Snapshots of solids volume fraction distribution at various superficial gas velocities. All parameters are exactly the same as those used in Figure 1, but now using a drag correlation derived from extensive lattice Boltzmann simulations.31
sufficiently large pressure gradient or gas density gradient across the bed under vacuum conditions. Note that in a previous experimental study,25 the pressure drop was several times that of the absolute operating pressure; therefore, they can even clearly observe this phenomenon in a shallow bed (initial bed height ) 0.27 m) because of the very large pressure gradient achieved. In order to verify our assumption that the gas compressibility is indeed the key to the coexistence state, simulations are carried out with a constant gas density of 1.2 kg/m3. The results are shown in Figure 9. It can be seen that, in case of constant gas density, the bed is still solidlike when the gas velocities are 0.13 and 0.14 m/s, which are already in the partially bubbling bed if the gas is compressible as shown in Figure 1. When the superficial gas velocity is 0.15 m/s, the bed is operated at the fully bubbling bed, which means that the regime transition happens suddenly from Ug ) 0.14 to 0.15 m/s without the existence of a partially bubbling bed. The conclusion is thus that the compressibility of gas is indeed the source of the observed partially bubbling bed. 6. Conclusion A state-of-the-art Eulerian model is used to study phase coexistence in a gas-solid fluidized bed. It is shown that the solidlike and fluidlike states can coexist in a deep bed, which is due to the compressibility of gas phase. Obviously, the quantitative characteristics depend on the constitutive laws we used, especially the bed expansion characteristics at fixed bed and partially bubbling fluidized bed regimes. For this reason, further development of a reliable particulate phase stress model is needed. Once such physical model is available, one can readily use it to refine the results presented in this study. Acknowledgment We would like to thank the anonymous reviewers for invaluable suggestions and comments, which significantly improved the quality of this article. This research was financially supported by NWO top grant “Towards a reliable model for industrial gas-fluidized bed reactors with polydisperse particles”.
Appendix: The Effect of Particle Size and Constitutive Laws on the Simulation Results Figure 10 shows the regime transition from a fixed bed to a fully bubbling bed for a particle diameter of dp ) 7.0 × 10-4 m, which shows the same qualitative characteristics of the coexistence of solidlike and fluidlike states as for dp ) 3.5 × 10-4 m systems, studied in the main paper. Figures 11 and 12 show the effect of particulate phase stress model and interphase drag force correlation, respectively. It can be seen that quantitative results depend on the constitutive laws used, but again the qualitative observations do not change. Note that without inclusion of frictional stress model, no convergence in the flow solver can be achieved for superficial gas velocities, which are less than 0.06 m/s. The reason is that when there is no frictional stress to support the particle weight, the drag force is not sufficient to balance the particle weight, so that, the bed starts compressing and the solid volume fraction reaches a value very close to εsmax. This causes serious numerical problems. Notation A ) cross-section area, m2 dp ) particle diameter, m g ) gravitational acceleration, m/s2 g0 ) radial distribution function H ) bed height, m H0 ) initial bed height, m Qg ) throughout of gas, kg/s Mg ) molar mass of gas, kg/mol p ) gas pressure, Pa ps ) particle pressure, Pa R ) gas constant, J/(mol · K) Tg ) gas temperature, K b ug, b us ) gas and solid velocity vectors, m/s Ug ) superficial gas velocity, m/s Umb ) minimum bubbling velocity, m/s Umf ) minimum fluidization velocity, m/s Greek Symbols β ) drag coefficient for a control volume, kg/m3 · s εg ) voidage
Ind. Eng. Chem. Res., Vol. 49, No. 11, 2010 ε0 ) voidage at initial state εmf ) voidage at minimum fluidization εs ) solid volume fraction εsmax ) solid volume fraction at packed condition φ ) angle of internal friction, deg µg, µs ) fluid and solid viscosity, Pa · s Θs ) granular temperature, m2/s2 Fg, Fs ) fluid density and solid density, kg/m3
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ReceiVed for reView October 5, 2009 ReVised manuscript receiVed January 15, 2010 Accepted January 25, 2010 IE901555P
