ARTICLE pubs.acs.org/IECR
Effect of Change in Fluidizing Gas on Riser Hydrodynamics and Evaluation of Scaling Laws Naoko Ellis,†,* Min Xu,† C. Jim Lim,† Schalk Cloete,‡ and Shahriar Amini‡ † ‡
Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, BC, Canada V6T 1Z3 Flow Technology research group, SINTEF Materials and Chemistry, Richard Birkelands Vei 3, 7034 Trondheim, Norway ABSTRACT: Riser hydrodynamics in a circulating fluidized bed are investigated experimentally and numerically by changing the composition of the fluidizing gas. Helium gas is added to air in the range of 0 to 96 vol % for fluidizing FCC particles 78 μm in size and density of 1560 kg/m3. Increasing He concentration decreases the fluidizing gas density and viscosity. The effect of change in gas composition is measured through the change in voidage along the riser from the pressure drop and the solids circulation rate, while analyzed by changes in slip velocity and particle Reynolds number. Numerical simulations using Computational Fluid Dynamics (CFD) have successfully predicted the experimental measurements over the entire range of fluidizing gas densities investigated. Simulations have also revealed that interphase momentum exchange in the bottom, accelerated region of the riser is dominated by cluster formation, while individual particle drag was dominant in the upper, more dilute regions. Given that CFD simulations have successfully reproduced these results, a scaling scheme is investigated whereby a hot model unit is simulated keeping either the Archimedes number or the density ratio of particle to gas constant. The results indicated better agreement between experimental and numerical voidage profiles for the density ratio scaling. Using the full set of scaling laws produced excellent prediction of the upper, fully developed region of the riser, but failed in the bottom regions. The scaling error in the bottom region was attributed to the momentum interaction in this region being dominated by cluster formation and not by the drag force on individual particles for which the scaling laws were derived. CFD has shown to be an effective tool in evaluation of scaling laws in risers.
1. INTRODUCTION Hydrodynamics in a circulating fluidized bed (CFB) riser continue to be a subject of investigation owing to its significance in many industrial processes including coal combustion, catalytic cracking, and gasification. Solids hold-up, solids mass flux, riser geometry and particle characteristics have been used to characterize the hydrodynamics through various models.1 More recently, particle motion has been tracked in the accelerating (bottom) zone of the riser, and characterized by the slip factor.2 The main mechanism for momentum transfer in gassolid fluidized beds is the drag force resulting from the velocity differences between the phases, and mathematically expressed as 1 ð1Þ FD ¼ CD Ap Fg ðυg - υp Þjυg - υp j 2 Various correlations for the drag coefficients have been developed to predict the drag forces in fluidized bed systems.3 Drag coefficient expressions useful for predicting the hydrodynamics of fluidized beds are those which take into account the effect of the surrounding particles. Thus, voidage (or solids holdup) and drag forces are interdependent variables not easily resolved. On the other hand, various scaling laws have been analyzed using the full set, simplified set, viscous-limiting set, and others containing two to five dimensionless groups to model the fluidization behavior of fast fluidization flow regime.4,5 Conducting cold flow model hydrodynamics studies for scaling, of often large hot units, require clever manipulation of dimensionless r 2011 American Chemical Society
numbers. In order to achieve the hydrodynamic similarity between the hot unit and cold-flow circulating fluidized bed system, a full set of the scaling law requires6 3 U 2 Fs Fs Fg d p g G s d p , , , , Ψ, bed geometry, , gD Fg μ2 Fs U D
particle size distribution
ð2Þ
Through the nondemensionalization of drag forces, the simplified scaling law considers the viscous-limit scaling set. However, omitting the ratio of the particle diameter to tube diameter may result in hydrodynamic similitude not being achieved.6 Patience et al., on the other hand, have incorporated the dependence of gas velocity, riser diameter and particle properties and proposed a slip factor based on Froude number as the scaling law component.7 Changing the fluidizing gas density in a CFB alters the drag force exerted on the particles without changing the particleparticle and particle-wall interactions. We report on the experimental and numerical investigations of the effect of fluidizing gas density on the hydrodynamics in a riser. The change in fluidizing gas density results in a change in Archimedes numbers ranging from 4.31 to 19.4 without changing the particle characteristics. This allows us to examine the difference in drag forces while keeping the interparticle forces similar. Experimental study is Received: May 22, 2010 Accepted: February 17, 2011 Revised: January 26, 2011 Published: March 08, 2011 4697
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Table 1. Experimental Conditions for Cold-Model case 1
conducted in a 0.102 m diameter Plexiglas of 5.64 m height with FCC particles. Fluidizing gas consists of air with Helium concentrations of 0, 20, 40, 60, 80, and 96 mol %. The numerical studies have been performed using computational fluid dynamics (CFD). The validity of CFD models have been assessed by experimental results. CFD holds the advantage that gas density, viscosity and particle characteristics can be simulated without any physical restrictions. This characteristic makes CFD an effective tool for investigating of scaling, since full control over all the important dimensionless quantities is possible. However, there are some limitations on CFD simulations because of computational intensity. Therefore, the numerical study has been performed on a 2D plane to maintain a spatial and temporal resolution fine enough to resolve the dynamic cluster formation.
2. EXPERIMENTAL SECTION The basic dimensions of the reactor system are: a riser 5.6 m in height with 0.1 m ID, and a secondary reactor (bubbling fluidized bed) with 1.5 m in height with ID 0.29 m, as shown schematically in Figure 1a. The length of the horizontal sections of loopseal1 and loopseal2 are 0.4 and 0.17 m, respectively. The vertical distances between J and H in loopseal1 and between L and K in loopseal2 are 0.7 and 0.8 m, respectively. The distances between B and L in loopseal1 and between A and I in loopseal2 are 1.1 and 0.9 m, respectively. Both loopseals are connected to the
3
4
5
6
He %
0
20
40
60
80
96
U (m/s)
5.0
5.0
5.0
5.0
5.0
6.0
Fg (kg/m3)
1.127
0.9328 0.7385 0.5443 0.3500 0.1946
Gs (kg/s/m2)
26.64
32.93
gas flow rate (kg/s)
0.0457 0.0378 0.0299 0.0221 0.0142 0.0079
solids flow rate (kg/s) 0.216
Figure 1. (a) Schematic diagram of a cold-flow reactor at UBC. Legend: 1, riser; 2, primary cyclone; 3, secondary cyclone; 4, dipleg; 5, downcomer 1; 6, downcomer 2; 7, internal cyclone; 8, downcomer 3; 9, secondary reactor; 10, loop-seal 1; 11, loop-seal 2; 12, ports for aeration gas of loop-seal 1; 13, ports for aeration gas of loop-seal 2; (b) locations of the inlet and outlet boundary conditions in the 2D plane geometry for simulation.
2
0.267
34.61 0.281
42.55 0.345
42.56 0.345
47.31 0.384
secondary reactor at an angle of 30°, with further details given elsewhere.8 The dual fluidized bed looping system was operated with spent FCC particles with density of 1560 kg/m3, mean particle diameter of 78 μm, and total solid inventory of 92 kg. The helium concentration in the fluidizing gas was maintained at 0, 20, 40, 60, 80, and 96 vol %, and the corresponding Fs/Ff varied at 8210, 4457, 2889, 2108, 1677, 1381, respectively. The superficial gas velocity in the riser, U, varied from 2.5 to 8 m/s; and the corresponding superficial gas velocity in the secondary reactor was adjusted from 0.028 to 0.090 m/s in order to achieve a constant input volumetric flow rate ratio of gases to both the riser and the secondary reactor. The required maximum flow rate of helium in the experiments is around 136.4 N m3/h and one helium cylinder can last only for a few minutes at the above flow rate for an open experimental system. Thus, the fluidizing gas had to be recycled in a closed experimental system through a blower. The temperature in the reactors was higher than the room temperature because of the heat generated by the blower. However, there was still some gas leakage between the gas blower and atmosphere. In order to compensate for the gas loss from the blower, fresh helium was added to the recycle stream. A Micro-GC (Varian Inc.) was used to constantly monitor the helium concentration at the gas inlets to the reactor. The aeration gas mixture of He and air for the loop-seals set at 4Umf and 1.5Umf for loopseals 1 and 2 (ports 12 and 13 in Figure 1a), respectively, was used to control the solids flow rate. As shown in Table 1, He concentration in the fluidizing gas was altered which resulted in the change in Ar and density ratios. Time average pressure measurements using differential pressure transducers (OMEGA PX142) were logged at 15 positions along the riser wall. Data were collected at a sampling frequency of 10 Hz for 105 s each to obtain the time average pressure measurements. Given the range of Gs and U, the flow regimes under this study will cover dilute riser flow to core-annulus flow.2
3. SIMULATIONS 3.1. Model Equations. The numerical simulation was carried out using a standard Eulerian two-fluid model approach. Conservation equations were solved both for the gas and solids phases. The continuity equations for the gas and solids phases phase are given below
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D ðRg Fg Þ þ rðRg Fg ! υ gÞ ¼ 0 Dt
ð3Þ
D ðRs Fs Þ þ rðRs Fs ! υ sÞ ¼ 0 Dt
ð4Þ
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Momentum conservation for the two phases was written as follows D ðRg Fg ! υ gÞ Dt ! ! þ rðRg Fg υ g υ g Þ ¼ - Rg rp þ rτ g þ Rg Fg gB þ Ksg ð! υs-! υ gÞ
ð5Þ
D ðRs Fs ! υ sÞ Dt þ r 3 ðRs Fs ! υ s! υ sÞ
¼ - Rs rp - rps þ r 3 τ s þ Rs Fs gB þ K ð! υ -! υ Þ gs
g
s
ð6Þ
Interphase momentum exchange was modeled following Wen and Yu9 for a relatively dilute system. Solids phase stresses were determined according to the Kinetic Theory of Granular Flow analogy where the random particle motion is linked to the thermal motion of molecules in a gas. This analogy allows the particulate phase to be simulated as a continuum and treated as a fluid with a modeled viscosity. The kinetic energy contained in these random particle motions is quantified in terms of a granular temperature and conserved as follows D ! ðRs Fs Θs Þ þ r 3 ðRs Fs υ s Θs Þ Dt ¼ ð-p I þ τ Þ : r! υ s
s
s
þ r 3 ðkΘs rΘs Þ - γΘs þ φgs
ð7Þ
The diffusion of granular temperature (kΘsrΘs) was modeled according to Gidaspow et al.10 The two final terms in eq 7 are the collisional dissipation of energy11 and the interphase exchange between the particle fluctuations and the gas phase.10 Solids stresses are calculated in the standard form by expressing the stress-strain tensor according to calculated shear and bulk viscosities.11 The modeled viscosities are based on the dispersive influence caused by particle collisions and translation. The shear viscosity consists of three parts: collisional,10,12 kinetic,10 and frictional.13 Granular temperature as well as the radial distribution function14 (measure of the average distance between particles) is used to determine the solids pressure11 (ps in eq 6). The particle-particle restitution coefficient, used to describe the elasticity of interparticle collisions, was set to 0.9. 3.2. Boundary Conditions. The wall boundary condition was set to a simple nonslip for the gas phase, while the Johnson and Jackson model15 was used to characterize the behavior of the solids. Two constants have to be set in this model: the specularity coefficient which describes the ease of slippage between the particle and the wall; as well as the particlewall restitution coefficient describing the elasticity of collisions between the particle and the wall. The specularity coefficient was set to 0.01 and the particle-wall restitution coefficient as 0.9. The gas and solids inlets were designated as mass flow inlets with the flow rate adjusted to the 2D geometry and varied pertaining to the simulation run in question. The 2D adjustment was made by multiplying the experimental mass flow rate with
Figure 2. Effect of He concentration in fluidizing gas on experimental pressure profile: U = 0.5 m/s, T = 40 °C, P = 1 atm, FCC particles.
the ratio in cross sectional area that would be presented by the 2D and 3D domains D/(πD2/4) = 4/(πD). Solids were injected uniformly across the inlets at a volume fraction of 0.5 and at a 60° downward angle according to the experimental setup. 3.3. Flow Solver and Solver Settings. The commercial software package, ANSYS FLUENT 12.116 was used as the flow solver. The phase-coupled SIMPLE algorithm17 was selected for pressure-velocity coupling, while the QUICK scheme18 was employed for discretization of all remaining equations. Second order implicit time stepping was used. 3.4. Geometry. A 2D plane section of the domain, 5.61 m in height and 0.1 m in width, was modeled as illustrated in Figure 1b. In the experiments, the riser consisted of a 3D pipe geometry with the two solids inlets located at an angle of 60° apart from each other. Thus, not on complete opposite sides of the riser as shown in Figure 1b. This arrangement was made in the model to allow an even distribution of solids to rise along the riser. For example, if both inlets were located on one side of the 2D riser, a very asymmetrical radial solids distribution would arise since solids will have to pass through the fluidizing gas stream to get to the opposite side of the riser. In 3D, the solids can migrate around the central gas stream implying that the time averaged radial solids distribution will revert to symmetry more easily. The domain was meshed with cells that are 20 particle diameters in width and 40 particle diameters in height (i.e., 1.56 by 3.12 mm). According to a detailed grid independence study conducted in a much denser system,19 a grid width of at most 10 particle diameters is required to accurately resolve the clusters. Cluster formation in the dilute system was much less pronounced; however, a grid independence study revealed that virtually identical results could be attained with the grid width of 20 particle diameters. 4699
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Figure 3. Effect of He concentration on time-average axial voidage profile: U = 0.5 m/s, T = 40 °C, P = 1 atm, FCC particles.
Table 2. Calculated Values for Experimental Conditions case 1
a
2
3
4
5
6
He %
0
20
40
60
80
96
Ar
22.3
18.5
14.7
10.9
7.0
3.9
Ut (m/s)a
0.234
0.237
0.241
0.245
0.250
0.256
Rg
0.9936
0.9916
0.9892
0.9826
0.9813
0.9837
νp (m/s)
2.69
2.51
2.05
1.57
1.46
1.86
νg [m/s]
5.03
5.04
5.06
5.09
5.10
6.10
νslip (m/s)
2.34
2.53
3.01
3.52
3.64
4.24
Rep
10.70
9.563
8.990
7.703
5.123
3.330
Calculated on the basis of Haider and Levenspiel model.20
and the gas phase where Figure 4. Solids circulation flux versus helium concentration in gas mixture for different superficial gas velocities in the riser.
U ¼ υg R g the slip velocity can be defined as υslip ¼ υg - υs
4. RESULTS AND DISCUSSION 4.1. Experimental Results. As shown in Figure 2, the experi-
mental pressure profile decreases with increasing He concentration of the fluidizing gas. This decrease in the absolute pressure with increasing He results from the decreased fluid friction on the wall from the reduced fluid density. Calculating the time-average voidage from differential pressure measurements assuming a fully developed flow and ignoring wall shear stress -
dp g½Fg Rg þ Fs Rs dz
ð8Þ
As shown in Figure 3, the solids fraction at the bottom of the riser only slightly increases with increasing He. Compared to the external solids circulation rate from Gs ¼ υs Rs Fs
ð9Þ
ð10Þ ð11Þ
The overall increase in Gs is a result of increasing solids fraction and decreasing slip velocity with increasing He concentration, as the particle terminal velocity increases with increasing He concentration. Furthermore, the achieved solids circulation rate is determined by the combination of the maximum carrying capacity of the gas-solid suspension and the pressure head of the return leg feeding solids back to the riser. The maximum carrying capacity decreases as the fluid density decrease with increasing He concentration. At the same time, the lower absolute pressure at the bottom of the riser creates higher pressure head from the loop, injecting more solids. The overall balance of the phenomena results in an increase in Gs with increasing He concentration, which is also experimentally shown in Figure 4. In all cases, voidage above 1.83 m from the bottom of the riser indicate a flat profile, signifying the “fully developed” section beyond the acceleration region. As summarized in Table 2, singleparticle free-falling velocity increases slightly with increasing He 4700
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Figure 5. Voidage comparison between experimental profiles calculated from differential pressure measurements and simulation. (Case 1) He = 0%; (Case 2) He = 20%; (Case 3) He = 40%; (Case 4) He = 60%; (Case 5) He = 80%; (Case 6) He = 96%.
concentration in the fluidizing gas. However, this change is insignificant relative to the change shown in the decrease of local particle velocity, υs, calculated from eq 10, based on experimentally determined Gs. As a result, the slip velocity increases considerably with increasing He. 4.2. Numerical Simulation Results and Validation. The axial solids concentration profiles from the simulations are compared to experimental data as presented in Figure 5.
The trend of increasing gas volume fraction from the lower to the upper regions of the riser is well predicted. The gas and solids flow rate input provided from experimental values are shown to predict the trend of increasing solids concentration with increasing He concentration near the bottom of the riser generally well. The Eulerian two-fluid model approach limits the particle size to a single average value which will result in inherent difference between the experimental values. Volume fractions extracted 4701
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Figure 6. Time and area averaged slip velocities along the height of the riser for the six different cases run as defined in Table 1.
from the 2D simulations were also adjusted to generate a pseudo3D value. This adjustment was done by weighing the radial volume fraction distribution when calculating the average so that the weight would increase with radial distance as it would in an actual cylindrical geometry. This adjustment improved the results especially in the lower region of the domain where a strong core annular flow regime was present. The largest discrepancies occurred in the lower regions of the riser, close to the solids inlets. The 2D simulations were not expected to do well these regions for two reasons: the inlet specification; and the distinct core annulus flow regime observed in the simulations. As described earlier, the solids inlets were placed on opposite sides of the riser in order to avoid a highly asymmetrical solids distribution. Naturally, the difference between the experimental and numerical setup will contribute to certain degrees of discrepancy. Arguably the largest source of error, however, will result from the inherent difference between space in a planar 2D and a cylindrical 3D geometry. The regions close to the walls, where the majority of solids reside will comprise a much greater volume in 3D cylindrical than in 2D planar. When appropriately averaged, the 2D planar geometry will thus exhibit a lower volume fraction than the 3D cylindrical geometry for the same radial solids distribution. The magnitude of this error will therefore contribute to the core annular flow pattern more profoundly. Furthermore, the bottom zone of a riser may be more dominated by acceleration of particles, attributing to the disagreement with the experimental results. The effect of interparticle forces resulting in cluster formation and further influencing the bed hydrodynamics is also not well predicted through CFD, resulting in further discrepancies. Much better agreement between the simulation and the experiments is observed toward the upper, more dilute regions of the riser. In these regions, the flow is more dilute and the simulated core annular profile is less distinct (radial distribution of solids is more uniform). The error resulting from 2D simulation is therefore not as great as can also be seen from the similarity between the 2D and pseudo 3D simulation data sets shown in Figure 5. 4.3. Numerical Studies on the Effect of Helium Concentration. An increase in helium concentration reduces the density of the fluidizing gas, proportionately reducing the drag force according to eq 1. Yet, to fluidize a fixed size particle falling at terminal velocity, the drag force required will be constant if
Figure 7. Comparison of simulated time average radial voidage profiles between Cases 2 and 5 at: (a) z = 1.0 m; and (b) z = 5.6 m.
buoyancy force is negligible. Thus, the reduction in density has to be compensated by an increase in slip velocity. As shown in Figure 6, a substantial increase in slip velocity is shown from simulation results as the He concentration is increased from Case 1 to Case 6, especially in the lower regions of the riser. It is also shown that the slip velocity decreases from the bottom to the top of the riser since the injected solids have to be accelerated upward by the fluidizing air. The lower regions of the riser also displayed a much higher volume fraction which allowed for the formation of clusters. Cluster formation substantially increases the slippage between solids and gas, since these large masses of particles can fall downward much more freely than single dispersed particles. As a result, higher drag force is exerted on the clusters. From the slip velocity profiles in Figure 6, the riser can be roughly divided into three parts: the dense bottom zone (z < 1 m), a transition zone (1 m < z < 2 m) and a dilute, fully developed region (z > 2 m). As shown in Figure 7, the simulated time average radial voidage profiles between Cases 2 and 5 at two axial locations, i.e., z = 5.6 and 1.0 m, are compared. The increase in experimental Gs from 32.9 to 42.6 kg/m3.s is reflected in the increase in solids fraction for both cases. More profound core-annulus flow structure is shown in the lower part of the column. Next, the difference in drag interaction between the bottom dense zone and the dilute upper zone are analyzed separately. The dilute upper region falls within the validity of the chosen drag formulation.9 Cluster formation in this region is almost 4702
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Table 3. Values Used for Establishing the Proportionality in the Top Region of the Riser Based on Simulation Res
case
Fg (kg/m3)
νslip (m/s)
μg (Pa.s) -5
Rg
CD
1.559
1
1.225
0.292
1.79 10
0.996
18.60
1.383 1.152
2 3
1.005 0.790
0.318 0.339
1.80 10-5 1.81 10-5
0.994 0.994
20.71 24.41
0.841
4
0.577
0.340
1.82 10-5
0.993
32.54
0.576
5
0.368
0.373
1.86 10-5
0.991
46.31
0.291
6
0.203
0.355
1.94 10-5
0.993
88.46
Figure 9. Proportionality between Fg and [CD(υBs-υBg)|υBs-υBg|]-1 in the dense region.
Figure 8. Proportionality between Fg and [CD(υBs-υBg)|υBs-υBg|]-1 in the dilute region.
negligible and a pure particle drag law should capture the physics correctly. Thus, according to eq 1, the gas density should be inversely proportional to the product of the drag coefficient and the square of the slip velocity. This is true because a constant force will be needed to fluidize the constan-sized particles in this region. The drag coefficient employed is formulated as follows9 CD ¼
24 ½1 þ 0:15ðRg Res Þ0:687 Rg Reg
ð12Þ
where Res ¼
Fg ds j! υs-! υ gj μg
ð13Þ
The relation of whether the drag formulation holds true by ensuring an inversely proportional relationship between Fg and (CD(υBs-υBg)|υBs-υBg|) is assessed by averaging the slip velocities and volume fractions for measurements at z > 1.82 m, as shown in Table 3. It is interesting to see that the slip velocity does not increase as much as expected with the decrease in gas density. This is because the selected model for the drag coefficient increases the particle drag with decreasing gas density. The linear fit between Fg and [CD(υBs-υBg)|υBs-υBg|]-1 is shown in Figure 8. When the same procedure is repeated for the bottom, dense region, however, the relationship does not hold well as indicated in Figure 9. For the dense cluster region, the selection of a model for the drag coefficient became insignificant as the drag force being directly resolved on the surface on the dense cluster and not modeled on a particle level any longer. However, a better fit is
Figure 10. Proportionality between Fg and [(υBs-υBg)|υBs-υBg|]-1 in the dense region (without CD).
obtained when the particle drag coefficient is omitted in comparing the proportionality between Fg and [(υBs-υBg)|υBs-υBg|]-1 as plotted in Figure 10. By neglecting the drag coefficient from the proportionality, it can be shown that the resolved stagnation pressure below the cluster is solely responsible for the interphase momentum transfer. A cluster will require the same pressure below it to be fluidized by a fluid of a given density. This pressure below a cluster originates due to the fluid stagnating below the obstruction created by the dense cluster. Bernoulli’s equation (K1 = 1/ 2Fυ2 þ Fgz þ P) can now be considered at a point on the bottom surface of the cluster and another point in the free stream in a lower point relative to the previous one. These two points are close to each other in the length of cluster formation, and then the term (Fgz) can be seen as constant at both points. Furthermore, if the velocity is taken relative to the cluster velocity, the velocity on the cluster surface will be zero and the velocity in the free stream will be the slip velocity. The pressure exerted on the cluster will be the difference between the pressure on the cluster face and the pressure in the free stream. In the relative framework used here, this can be called the stagnation pressure. When the Bernoulli equation is solved at the two points under these conditions, it is found that pstagnation = K2Fυ2slip. Because the 4703
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Table 4. Parameters for Simulation Comparison case 1
H1a
H1b
6
H6a
H6b
He %
0
0
0
96
0
0
T (K)
313
1073
1073
313
1183
1183
P (bar)
1.0
1.0
1.0
1.69
1.69
1.69
U (m/s)
5.0
5.0
5.0
6.0
6.0
6.0
Fg (kg/m3)
1.127
0.329
0.329
0.1946 0.504
Fs/Fg
1384
12,170 12,170 8016
8016
8016
Ar
22.3
22.2
28.0
28.0
22.2
3.9
0.504
gas flow rate (kg/s) 0.0457 0.0133 0.0133 0.0079 0.0245 0.0245 solids flow rate (kg/s) 0.216 0.216 0.559 0.384 0.384 0.995 total flow rate (kg/s)
0.262
0.229
0.572
0.392
0.401
1.92
Figure 12. Case 1 simulation results when the full set of scaling laws is implemented.
Figure 11. (a) Case H1b simulation results compared to experimental and simulation results from Case 1 (Archimedes scaling); (b) Case H6b simulation results compared to experimental and simulation results from Case 6 (density ratio scaling).
stagnation pressure below the cluster should be constant for all densities considered, i.e., K3 = Fυ2slipand F µ 1/υ2slip, which is plotted in Figure 10. With regards to grid resolution, these findings suggest that a fine grid, such as the one used in this study, will only be necessary
in the bottom regions where clusters are formed. In these regions, the drag interaction is resolved directly on the clusters and the selected formulation for drag coefficient does not play a role. In the upper, dilute parts, however, the selected drag formulation is very important because the flow is dilute enough so that individual particle drag is the dominant interphase momentum exchange mechanism. This finding also suggests that much coarser grids can be used in these regions because no clusters can be resolved in such dilute flows. 4.4. Evaluation of Scaling with Numerical Simulations. On the basis of the acceptable agreement between cold model experiments and simulations, the next step is to use CFD to predict the hydrodynamics of a hot model riser based on the scaling laws. To confirm the validity of the scaling laws, a set of simulations were conducted for simulating the hot model. The first three cases shown in Table 4, i.e., cases 1, H1a, and H1b, compare the scenario where Ar is kept constant while the temperature and particles are changed, whereas for the last three cases, i.e., cases 6, H6a, and H6b, the density ratio of solids to gas is kept constant. Figure 11 depicts the scaled simulation results in comparison with the corresponding experimental and numerical results. It is shown that the scaling only according to the Archimedes number produces a very poor fit throughout the length of the riser, whereas that according to the density ratio produces results that are poor at the bottom region of the riser, but better toward the top. These results indicate that the density ratio is probably the more important parameter to be kept constant, but might not be appropriate as a scaling rule on its own. Accordingly, a final simulation was run where Case 1 was scaled according to the full set of scaling laws proposed by Glicksman,5 as shown in Figure 12. It is clear that there is a large discrepancy in the bottom region of the riser, but the match in the top half is much better. The scaling laws are valid for fully developed flow and were derived from the basis of a particle drag law. Flows in the bottom of the present riser are not fully developed and drag interaction is dominated by denser cluster formation. It is therefore expected that the scaling should fail in the bottom regions of the riser. In the top half of the riser, however, the flow can be considered as fully developed and momentum exchange is controlled by individual particle drag. 4704
dx.doi.org/10.1021/ie101141f |Ind. Eng. Chem. Res. 2011, 50, 4697–4706
Industrial & Engineering Chemistry Research Thus, the scaling laws hold true in this region and produce an acceptable prediction.
5. CONCLUSIONS Experimental results showed that the solids holdup increases as the concentration of helium in the fluidizing gas is increased. This trend was successfully reproduced by means of numerical modeling. Simulation results were subsequently analyzed to identify two regions with distinctly different drag interactions within the riser. The top, dilute phase region was found to occur at a height of 2 m and above. Within this region, cluster formation is insignificant and the slip velocity is quite small for all cases. Interphase momentum exchange is dominated by the drag between the gas and individual particles as modeled by a particle drag coefficient. The correct selection of a particle drag law is therefore very important in this region. However, dense cluster formation is significant in the lower, dense region (H < 1 m). In this region, large differences in the particle slip velocity were observed with a change in the fluidizing gas density. The particle drag formulation was found to be insignificant in this region because the momentum exchange occurred predominantly between the gas phase and the resolved clusters. A sufficiently fine grid resolution is therefore required to resolve the clusters or an appropriate filtered drag law should be used. Different approaches to scaling have been tested using the simulation results. Scaling according to matching the Archimedes number did not result in satisfactory comparison. When the gas/solids density ratio is kept constant, somewhat better agreement is achieved. By using the full set of scaling laws, excellent agreement is achieved in the upper, fully developed region of the riser where the particle drag is controlling. However, predictions in the bottom region of the riser where cluster formation dominates the momentum interaction have been poor. ’ AUTHOR INFORMATION Corresponding Author
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[email protected]. Tel: 1-604-822-1243. Fax: 1-604822-6003.
’ ACKNOWLEDGMENT We acknowledge the financial support through the Natural Science and Engineering Research Council of Canada, and SINTEF Materials and Chemistry, Norway, under the CLC SEP 2009 grant. ’ NOMENCLATURE Ap Area of particle tangential to the fluid flow, m2 Ar Archimedes Number Ar = (Fg(Fs-Fg)d3s g/(μ2g)) Drag coefficient, CD d Diameter, m D Fluidized bed diameter, m Particle diameter, m dp Drag force FD Fr Froude number g Gravitational acceleration, m/s2 Gs Solids circulation rate, kg/m2.s I Identity tensork Diffusion coefficient, kg/(m s) K Momentum exchange coefficient, kg/(m3 s) P Pressure, Pa
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Pstagnation Stagnation pressure, Pa Particle Reynolds number Rep Solids Reynolds number Res = ((Fgds|υBs-υBg|)/(μg)) Res t Time, s U Superficial gas velocity, m/s Minimum fluidization velocity, m/s Umf Terminal velocity, m/s Ut z Axial location, m R Volumetric fraction, φ Energy transfer rate, W/m3 γ Energy dissipation rate, W/m3 μ Viscosity, Pa s Granular temperature, m2/s2 Θs Fg Gas density, kg/m3 Fs Particle density, kg/m3 Shear stress at the wall, N/m2 τs B τ Stress tensor, kg/(m s2) υB Velocity vector, m/s Local gas velocity, m/s υg Local solids velocity, m/s υs Slip velocity, m/s υslip Ψ Sphericity, -
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