Measurement of Two Kinds of Granular ... - ACS Publications

Ind. Eng. Chem. Res. 2005, 44, 1329-1341

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Measurement of Two Kinds of Granular Temperatures, Stresses, and Dispersion in Bubbling Beds Jonghwun Jung and Dimitri Gidaspow* Department of Chemical and Environmental Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

Isaac K. Gamwo National Energy Technology Laboratory, U.S. Department of Energy, Pittsburgh, Pennsylvania 15236

A CCD camera technique was developed to measure instantaneous particle velocities in a thin bubbling bed for fluidization of 530 µm glass beads. The hydrodynamic velocities were computed by averaging the instantaneous velocities over the velocity space using the concepts of kinetic theory. Laminar-type kinetic stresses and granular temperatures were computed from the measurement of instantaneous velocities. Bubblelike granular temperatures were computed from the hydrodynamic velocities. The measured Reynolds normal stresses per unit bulk density in the vertical direction were 8 times larger than the measured Reynolds normal stresses per unit bulk density in the lateral direction because of higher velocity fluctuations for particles in the bubble-flow region. The sum of the measured shear stresses was equal to the pressure drop minus the weight of the bed of solids within experimental error. The restitution coefficients for 530 µm glass beads, estimated from the ratio of shear to normal stresses, are in the range of 0.99. The mixing in the bubbling and turbulent fluidized beds is due to laminarlike particle oscillations measured by the conventional granular temperature and due to bubblelike granular temperatures produced by the motion of bubbles. The bubblelike granular temperature is much larger than the particle granular temperature. In the center of the riser, the particle granular temperature was about 3 times larger than the Reynolds-like granular temperature. These observations are consistent with the literature of particle dispersion in bubbling beds, such as the early Ruckenstein analysis of homogeneous and bubbling beds. Introduction A quantitative understanding of the hydrodynamics of fluidization is needed for the design and scale-up of efficient new reactors in the petroleum, chemical, and electric power industries.1-4 It is well-known that the flow regimes vary from a bubbling, through turbulent, fast or circulating fluidized bed (CFB) to pneumatic transport with an increase in the gas velocity.2,5 The main advantage of fluidization is its continuous powder handling ability and its good heat- and mass-transfer characteristics. A measure of the quality of mass transfer is the particle diffusivity or dispersion coefficient. A review of the literature6-8 shows that they vary by 5 orders of magnitude. We believe these properties in the fluidized-bed reactors are due to random oscillations of nearly elastic particles suspended in fluids. Hence, the kinetic theory is ideally suited to describe such flows. Savage and others9,10 showed that the dense-phase kinetic theory, as presented by Chapman and Cowling,11 can be applied to granular flow of the particles. Gidaspow5,12 has reviewed this theory. In granular flow, the particles dissipate energy because of inelastic collisions and because of drag between the fluid and particles. In the kinetic theory of granular flow, the basic concept is the granular temperature, which is like the thermal temperature in the kinetic theory of gases. It measures the random oscillations of * To whom correspondence shoul be addressed. Tel.: 1-312567-3045. Fax: 1-312-567-8874. E-mail: [email protected].

particles, which is the average of the three variances of the instantaneous velocities of the particles. The CCD camera method shown in this paper describes the measurement of this fundamental quantity. The first method of measuring the granular temperature in a bubbling bed was done by Cody’s group at Exxon.13 They measured the acoustic shot noise with a vibration meter. A very significant finding was a maximum in the particle fluctuation velocity for Geldart group A particles.14 Wildman and Huntley15 used a video camera to measure two granular temperatures in a binary vibrofluidized bed.16 Gidaspow and Huilin17 have shown that the viscosity obtained from the kinetic theory formulas agrees with macroscopic measurements in a vertical pipe. They18 have also shown that there exists an analogue of the ideal equation of state for particles. Campbell and Wang19 and Polashenski and Chen20,21 measured particle pressures using the particle-pressure transducer. Polashenski and Chen estimated a reasonable particle viscosity from their measurements. The existence of particulate and Reynolds stresses was recognized early in the development of the equations of motion for fluidization, as reviewed in the 1994 Flour-Daniel award lecture by Jackson.22 Buyevich23 also discussed the existence of such particulate and Reynolds-like stresses. Koch and Sangani24 derived expressions of particulate granular temperature for homogeneous, nonbubbling, fluidization. In their approximate expressions for the granular temperature, the production is due to shear and the dissipation is due to fluid-particle drag and/or inelastic collisions. Conduc-

10.1021/ie0496838 CCC: $30.25 © 2005 American Chemical Society Published on Web 01/04/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005

Figure 2. Typical particle streak images captured by the CCD camera. Particle diameter: 530 µm glass beads. Particle density: 2500 kg/m3. Superficial inlet velocity: 0.587 m/s. Exposure time (∆t): 0.004 s.

Figure 1. Experimental schematic diagram for a two-dimensional rectangular fluidized bed [0.154 m (width) × 0.022 m (depth) × 0.58 m (height)].

tion is not included. Their earlier theories predicted the existence of anisotropy observed in this study. A more complete review of theories for riser flow has been recently published by Gidaspow12 and Tartan and Gidaspow,25 who obtained granular temperatures from stress measurements in the IIT riser. Two types of granular temperatures were identified. In this study, we have extended the CCD camera technique for the granular temperature in the IIT riser to the bubbling and turbulent fluidization regimes. This particle image velocimetry method is similar to the stress measurements in a gas-liquid bubble column reported by Mudde et al.26 The observed anisotropy in riser flow and now in bubbling beds demands the development of anisotropic kinetic theory. Such a development27 has begun based on the theory of Strumendo and Canu.28 The large contribution of bubbles to the stresses necessitates the development of a computational fluid dynamics (CFD) code with a good bubble resolution. In a companion paper,29 we used a modified CFD code to compute the bubblelike granular temperatures measured here. The two granular temperatures reported in the simulation study29 approximately agree with those of the CFD simulations. The mixing of solids was estimated from the early random walk theory of Ruckenstein.30,31 The diffusivity and dispersion coefficients roughly scaled with the two granular temperatures. Experimental Setup and Procedure The experimental setup used for fluidization of Geldart group B particles is shown schematically in Figure 1. A thin rectangular bed was constructed from glass sheets to prevent particles from sticking to the walls of the bed and to facilitate visual observation and video recording of bed operation, such as bubbling, bed expansion, and mixing and segregation of particles. The bed height was 0.58 m, and the cross section was 0.154 × 0.022 m. To achieve a uniform fluidization, a rectangular gas distributor with a height of 0.18 m was used. It was covered with 165 × 1400 mesh 304L stainless steel wire (Newark Wire Cloth Co.). Air was passed through a bed of activated silica gel. The air flow was

measured by means of rotameters. A three-dimensional rectangular bed [0.133 m (width) × 0.094 m (depth) × 0.62 m (height)] was constructed with a central rectangular jet [0.018 m (width) × 0.094 m (width)]. A cylindrical bed with a diameter of 0.076 m and a height of 0.3 m was also constructed for viscosity measurements. The fluidization solids were glass spheres (Potters Industries Inc.) with an average particle diameter of 530 and 42 µm and a density of 2500 kg/m3. The minimum fluidization velocity was estimated by means of a pressure drop measurement by varying the superficial air velocity. The solids were charged into the fluidized bed to give a static bed height of 0.14 m and then fluidized with a uniform inlet gas velocity. Particle Velocity. The instantaneous particle velocity (c) was measured by means of the CCD color video camera shown in Figure 1. In this technique, the particle velocity was measured by means of a length of a streak divided by the elapsed time. It was described earlier.17,18 To get a good visualization of the microscopic movement of particles, a fiber-optic light at the front or backside was reflected on the field of view in the experiments. A rotating colored transparent disk was placed in front of the light source to decide the direction of flow. As the particles were fluidized inside the bed, the camera with a zoom lens, 0.018-0.108 m, and closeup focus transferred its field of view to the monitor with streak lines. These streak lines represented the space traveled by the particles in a given time interval specified on the camera. The images were then captured and digitized by a microimaging board and analyzed using Image-Pro Plus software. Vertical and lateral velocity measurements were conducted at different locations inside the bed. The velocity vector of a single particle shown in Figure 2 was calculated as

cx(r,t) )

∆L cos R ∆t

cy(r,t) )

∆L sin R ∆t

(1)

where ∆L is the distance traveled, R is the angle from vertical, ∆t is the inverse of the shutter speed, and cx and cy are the vertical and lateral instantaneous particle velocity components, respectively. Bubble Size and Velocity. Bubbles were recorded by a digital video camcorder (Canon) and then captured and analyzed using Image-Pro Plus software. The bubble size was averaged at a bed height of 0.1 m for

Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1331

over 600 pictures, captured at 30 frames/s. Only bubbles that were fully enclosed were used in the count, where the center of the bubbles was at a bed height of 0.1 m. The bubble diameter was an equivalent diameter. The bubble velocity was measured by the travel length of a bubble between two frames at a bed height of 0.1 m. Solids Volume Fraction. The time-averaged solids volume fraction was obtained by means of the γ-ray adsorption techniques at a designated location shown in Figure 1. The γ-ray densitometer has been used previously to measure porosities of fluidized beds and solids concentrations in nonaqueous suspensions.5,32 This technique is based on the fact that the reading of the transmitted γ-rays can be described as a linear function of the porosity of the system. The γ-ray densitometer was calibrated using the same methods as those described earlier by Seo and Gidaspow.32 The γ-ray densitometer assembly and the method are that described by Gidaspow.5 Particle Viscosity. The effective particle viscosities were measured using the Brookfield digital viscometer (model LVDV-II+) with a spring torque of 6.737 × 10-5 N‚m. The viscometer was placed at the top of the fluidized bed and secured over the centerline of the bed. The cylindrical spindle of LV2 or LV3 was attached to the bottom of the viscometer without the guard and was lowered inside the fluidized bed until it was completely immersed in the initial bed height of 0.17 m during measurements. The viscosities were measured with a rotational speed of 100 rpm and then averaged for 2 min. The calibration of the viscometer-spindle apparatus was done using a Newtonian liquid, namely, water. Results and Discussion Particle Velocity. The nonhomogeneous fluidized bed can be described as a two-phase mixture consisting of dense gas-solid particles, in which the solid particles are uniformly distributed, and a gaseous phase traversing the bed as bubbles.31 To get a good microscopic movement of the particles, the image of the streak lines was divided into a dense flow region and a dilute bubbleflow region. Figures 3 and 4 show typical vertical and lateral instantaneous particle velocity (c) histograms for a frame captured by the CCD camera technique in the nonhomogeneous bubbling bed with a particle diameter of 530 µm in the center region at a bed height of 0.14 m. The view area in a dense flow region was 0.003 44 × 0.002 58 m with an exposure time of 1/250 s. The view area in a bubble-flow region was 0.006 88 × 0.005 16 m with the same exposure time, to prevent the particles from moving out of the field of view. The number of streak lines for each frame was between 20 and 60. Hence, the small volume for averaging of the point variable is large enough to contain a great number of particles while still possessing small dimensions compared with the system dimensions. The standard deviation of the vertical and lateral instantaneous particle velocity for a dense flow region is about 0.0347 m/s. The standard deviation of the vertical instantaneous particle velocity for a bubble-flow region is 4 times higher than that of the lateral instantaneous particle velocity of 0.0359 m/s. They show the isotropy in the dense flow region and the anisotropy in the bubble-flow region. The particle velocity fluctuation is given by the measured instantaneous particle velocity [c(r,t)] minus the hydrodynamic velocity for particles [v(r,t)]. The

Figure 3. Histograms of vertical (A) and lateral (B) instantaneous particle velocities (c) in the dense flow region at a bed height of 0.14 m.

hydrodynamic velocity is defined as5

vi(r,t) )

1

n

∑ cik(r,t)

(2)

nk)1

where i represents x, y, and z directions and n is the number of particles per unit volume at each frame. The mean particle velocity fluctuation [〈C〉(r,t)] is zero relative to the hydrodynamic velocity, as shown in the following.

〈C〉 ) 〈c〉 - v ) 0

(3)

Figure 5 shows typical time series of vertical and lateral hydrodynamic velocities for particles. A series of 240 consecutive pictures captured by the camera at 30 frames/s were constructed. The hydrodynamic velocities for each frame are calculated from the average of the instantaneous velocities, as given by eq 2. Their frequency distributions are shown in Figure 6. The main frequency for the vertical direction is 2.28 Hz, with a spectral magnitude of 13.6, and the main frequency for the lateral direction is 2.63 Hz, with a spectral magnitude of about 4.0. Here the mean free path is much larger than the diameter of the glass beads used. The dominant frequency (f) of porosity oscillations in the bubbling bed is given as33

f)

( )[

1 g 2π H0

1/2

]

(3s/g + 2)s s0

1/2

(4)

where s0 and H0 are some initial solids volume fraction

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Figure 4. Histograms of vertical (A) and lateral (B) instantaneous particle velocities (c) in the dilute bubble-flow region at a bed height of 0.14 m.

and height of a bed filled with particles. s0 ) 0.6 and H0 ) 0.14 m for this study. The solids volume fraction measured by the γ-ray densitometer is 0.40. The main frequency for porosity oscillations is 2.17 Hz. This shows a reasonably good agreement with the main frequency for the hydrodynamic velocity shown in Figure 6. Particle Stresses and Granular Temperature. The particle stresses (〈CC〉) per unit bulk density over a given frame are calculated from the definition of kinetic stresses.5

〈CiCj〉(r,t) )

1

Figure 5. Typical time series of vertical (A) and lateral (B) hydrodynamic velocities (v) for particles in the center region at a bed height of 0.14 m.

n

∑ [cik(r,t) - vi(r,t)][cjk(r,t) - vj(r,t)] nk)1

(5)

The particle granular temperature (θparticle) over a single frame is defined as the mean of the particle normal stresses per unit bulk density, shown in eq 5.

1 θparticle(r,t) ) 〈CiCi〉(r,t) 3

(6)

The particle granular temperature, which is 2/3 of the random particle kinetic energy, varies with time and position. Such a granular temperature is due to the oscillation of particles in small regions for a small time period. Figure 7 shows the typical time series of particle normal stresses per unit bulk density calculated from the measured peculiar particle velocity at each frame. The value for the vertical direction is an order of magnitude larger than that for the lateral direction in a dilute bubble-flow regime. A typical time series of the measured particle granular temperature is shown in Figure 8. By assuming that the particle fluctuations are

Figure 6. Frequency and power spectral magnitude of hydrodynamic velocities shown in Figure 5. Main frequency: vertical ) 2.28 Hz; lateral ) 2.63 Hz.

equal in the lateral and depth directions, the definition of eq 6 can further be simplified to

1 2 θparticle(r,t) = 〈CxCx〉 + 〈CyCy〉 3 3

(7)

Such an assumption is reasonable. It was approximately true for the riser flow, with a high gas velocity.25 The particle granular temperatures oscillate with time. They have larger values because of much higher particle oscillations at times of bubble formation in a nonhomogeneous bubbling bed. Figure 9 shows the time-averaged particle normal stresses per unit bulk density measured by a CCD camera technique in the center region at a


Figure 9. Time-averaged particle normal stresses per unit bulk density measured by a CCD camera technique (h ) 0.14 m).

Figure 7. Typical time series of particle normal stresses per unit bulk density obtained from the measured peculiar particle velocity (C ) c - v) at each frame: (A) vertical direction; (B) lateral direction.

Figure 10. Typical time series of Reynolds normal stresses per unit bulk density calculated from variances of hydrodynamic velocities for particles shown in Figure 5: (A) vertical direction; (B) lateral direction. Figure 8. Typical time series of particle granular temperatures calculated from Figure 7.

bed height of 0.14 m. The particle normal stresses in the vertical direction are about 4 times larger than the particle normal stresses in the lateral direction. This anisotropy is due to the larger particle fluctuation velocity of the vertical direction in the dilute bubbleflow regime, as shown in Figure 7. The time-averaged particle normal stresses give the granular temperatures.

1 1 2 θparticle(r) ) 〈CiCi〉(r) = 〈CxCx〉 + 〈CyCy〉 3 3 3

(8)

They are shown in Figure 13 for a comparison with the bubblelike granular temperature.

Reynolds Stresses and Bubblelike Granular Temperature. Variations of the hydrodynamic velocity for the particles, as shown in Figure 5, permit the determination of Reynolds (θbubblelike) stresses per unit bulk density. Equations given in the following are similar but not identical to those given for the particle stresses.

Vi′Vj′(r) )

1

m

∑ [vik(r,t) - vji(r)][vjk(r,t) - vjj(r)] mk)1 vj i(r) )

1

(9)

m

∑ vik(r,t)

mk)1

(10)

The mean velocity (vj ) is calculated by averaging the hydrodynamic velocity for particles shown in Figure 5,

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Figure 11. Time-averaged Reynolds normal stresses per unit bulk density (h ) 0.14 m).

Figure 14. Mean particle velocity (vj x) for the vertical direction in Figure 13.

Reynolds normal stresses, with an assumption that the fluctuations of hydrodynamic velocities are equal in the lateral and depth directions.

1 1 2 θbubblelike(r) ) Vi′Vi′ = Vx′Vx′ + Vy′Vy′ 3 3 3

Figure 12. Time-averaged particle and Reynolds shear stresses per unit bulk density (h ) 0.14 m).

The bubblelike granular temperatures shown in Figure 13 have much higher values than those of the particle granular temperatures. These high values are due to the motion of bubbles in the bubbling fluidized beds. Shear Stresses for Particles. Figure 12 shows the time-averaged shear stresses for particles obtained from the time series of shear stresses per unit bulk density. The particle shear stresses are very low, close to zero. The Reynolds shear stresses have the low value of about 2 × 10-3 m2/s2. The sum of particle plus Reynolds shear stresses should be equal to the pressure drop minus the weight of the bed of solids, similar to the computation of the shear Reynolds stresses for a single phase flow34

τyx ) sFs〈CxCy〉 + sFsVx′Vy′ = -

Figure 13. Comparison of particle (eq 14) and bubblelike (eq 13)13 granular temperatures calculated from time-averaged particle and Reynolds normal stresses for 530 µm glass beads. (U0/Umf ) 2.5, s ) 0.4, h ) 0.14 m, µs/κs ) 4/15 for particles.5) θparticle ) (1/3)〈CiCi〉; θbubblelike ) (1/3)Vi′Vi′.

where m is the total frame number, 240 over a given time period. Figure 10 shows the typical time series of Reynolds normal stresses per unit bulk density. They have much higher values than the particle normal stresses shown in Figure 7. Figure 11 shows the timeaveraged Reynolds normal stresses per unit bulk density at a bed height of 0.14 m. The ratio of Reynolds normal stresses in the vertical direction to that in the lateral direction is about 8. This is larger than that of particle normal stresses. The bubblelike granular temperature (θbubblelike) is calculated from time-averaged

(11)

∆p L∆x

∫0ysFsg dy

(12)

where ∆p/∆x is the pressure drop per unit length and L is the lateral distance. The pressure drop is 2060 ( 15 N/m2. This value almost equals the weight of the bed of solids. The total shear stress (τyx) calculated from Figure 12 is very small, about 2 N/m2. Hence, the relation of eq 12 has a reasonable agreement with our experiments. Comparison of Particle and Bubblelike Granular Temperatures. Figures 9 and 11 show the timeaveraged normal stresses per unit bulk density. They show anisotropic flow in the bubbling bed. Figure 13 shows a comparison of measured particle and bubblelike granular temperatures for 530 µm glass beads. Both granular temperatures are flat in the center region of the bed. They are approximately 2.8 × 10-3 and 1.85 × 10-2 m2/s2, respectively. The measured mean particle velocity for the vertical direction is shown in Figure 14. The turbulence intensity, defined as the average granular temperature scaled with the mean particle velocity (xθave/vj x), is 1.08 for the bubblelike granular temperature and 0.42 for the particle granular temperature. The turbulence intensity drops with a decrease of the normal stresses. This analysis is similar to that of the time series for turbulence in a gas-liquid bubble column


reported by Mudde et al.,26 in which the removal of the contribution of the vortices gave relatively flat normal stresses. Their stress data showed an isotropic flow for liquids in bubble columns. The mean of the squares of high-frequency velocity fluctuations was about 3.5 times lower than that of total velocity fluctuations of 1.8 × 10-2 m2/s2 with large-scale vortical structures. According to our interpretation, the former corresponds to the particle granular temperature and the latter is the bubblelike granular temperature. Degaleesan et al.35 found that Reynolds normal stresses for liquid in bubble columns are flat across the column, except near the wall region from computer-automated radioactive-particle tracking experiments. Their stress measurements showed that turbulence is anisotropic for higher superficial gas velocities in bubble columns. The axial Reynolds normal stresses of about 0.13 m2/s2 were 3 times higher than the radial and tangential Reynolds normal stresses. The Reynolds shear stresses were zero in the center and at the walls. They were higher in the region between the center and the wall. Matonis et al.36 also showed a constant granular temperature of 1.5 × 10-2 m2/s2 for 800 µm leads glass beads in a slurry bubble column, which was obtained from the sum of all frames over time and velocity space using our CCD camera techniques. Such a granular temperature is similar to the bubblelike granular temperature, according to our interpretation. The small error is primarily due to insufficient frames in the velocity space and due to the use of an equal weight of the streak lines for all frames. Cody et al.13 were the first to measure the granular temperature using an acoustic shot noise technique in gas-solids bubbling beds. The vibrational signal at the wall due to the random particle impact was converted to give the average granular temperature over the surface area of the bed. They correlated their data Geldart B glass spheres, for gas superficial velocities greater than twice the minimum fluidization velocity, by the expression

granular temperature ≈ (110 µm/dp)2U02 (13) Equation 13 gives a granular temperature of 1.47 × 10-2 m2/s2 for the 530 µm glass beads in our experimental conditions. As pointed out by Cody et al., bubbles may contribute indirectly to their averaged granular temperatures. This granular temperature is our bubblelike granular temperature. Our measured value in the two-dimensional bed is about 25% higher than Cody’s value. This small difference may be due to the difference between the cylindrical bed used by Cody and the twodimensional bed used in this study. We also measured granular temperatures for 530 µm glass beads in the central jet region of a threedimensional rectangular bed. Figure 15 shows the measured bubblelike granular temperatures at three different vertical positions in the central jet flow region. The jet velocity was 0.56 m/s, and the two grid velocities were the minimum fluidization velocities of particles. Granular temperatures increase from 5 × 10-3 to 1.2 × 10-2 m2/s2 with a decrease of the solids volume fraction. This is due to the increase of the mean free path of particles in the collisional regime.5 The granular temperature at the solids volume fraction of 0.4 is 1.2 × 10-2 m2/s2. This is 20% lower than Cody’s value in the cylindrical bed. This shows that the granular temperatures depend on flow patterns due to the fluidized-bed

Figure 15. Measured bubblelike granular temperatures for 530 µm glass beads in a central jet [0.018 m (width) × 0.094 m (depth)] region of a three-dimensional rectangular bed [0.133 m (width) × 0.094 m (depth) × 0.62 m (height)]. Central jet gas velocity Ujet ) 0.56 m/s.

geometry. Our measurements in the two-dimensional bed represent the flow of particles in the center region, while our measurements in the three-dimensional bed represent the flow of particles near the wall, in which the effect due to bubble formation is reduced. Hence, our value for granular temperatures is roughly the same as Cody’s measurements. Equation 13 is a measure of the bubblelike granular temperature, in the Geldart B fluidization regime, as shown in Figure 13. Gidaspow and Mostofi37 derived the relation between the granular temperature and the mean particle velocity from a balance between conduction and generation of the granular temperature for the flow of elastic particles in a pipe. For flow between parallel flat plates, this relation can be shown to be as follows:

θmax - θw )

()

3 µs 2 vj 4 κs x

(14)

where the ratio of viscosity (µs) to conductivity (κs) of particles is 4:15 from the kinetic theory of gases. With neglect of the granular temperature (θw) at the wall, the maximum granular temperature (θmax) is 3.2 × 10-3 m2/ s2. This is shown in Figure 13. Equation 14 is mathematically identical with the relation between the thermal temperature and the mean velocity for laminar flow between parallel plates derived in many books on fluid mechanics or transport. Bubble Size. Figure 16 shows the equivalent bubble diameter for 530 µm glass beads captured by a video camcorder at a bed height of 0.1 m. Darton et al.38 predicted the growth of the bubble size as a function of the distance above the distributor in a bubbling fluidized bed. Davidson’s bubble-growth model38 for Gedart group B particles is

DB ) 0.54(U0 - Umf)0.4(h + 4 xA0)0.8/g0.2

(15)

where DB is the equivalent bubble diameter, U0 is the superficial gas velocity, Umf is the minimum fluidization velocity, h is the height above the distributor, g is the gravity, and A0 is the catchment area. 4xA0 is 0.03 m for a porous-plate gas distributor. The solid line in Figure 16 represents the empirical correlation obtained from eq 15 at a bed height of 0.1 m. They are consistent with equivalent bubble diameters obtained from our experimental conditions. From

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Figure 16. Bubble size for 530 µm glass beads captured by a video camcorder at a bed height of 0.1 m. The line represents an empirical correlation obtained from the Davidson model.38

Figure 18. Ratio of shear to normal stress obtained from the measured particle granular temperature for 530 µm glass beads.

pression of particles. g0 is the radial distribution function for particles41

g0 ) [1 - (s/s,max)1/3]-1

Figure 17. Particle viscosities measured by means of a Brookfield viscometer in the bubbling bed. The lines represent empirical correlations.

the snapshots made by a video camcorder, we see that bubbles form at the bottom of the bed, grow, coalesce near a bed height of 0.1 m, and then burst at the top of the bed. For a superficial gas velocity of 0.578 m/s, there are large bubbles above a bed height of 0.1 m due to the bubble burst. Above this velocity, bubbles are larger than those predicted by Davidson’s model with an oval shape. This may be due to the effect of a twodimensional bed, with a depth of 0.022 m. Viscosity. Figure 17 shows the particle viscosities measured by means of a Brookfield viscometer in the cylindrical bubbling bed. The measured particle viscosity shows an infinite value with no gas flows and a low value near the minimum fluidization velocity. It increases to a high value in the viscous region and then decreases with a decrease of the solids volume fraction in the grain-inertia region. In the viscous region, the frictional stresses from particle-particle and wallparticle play an important role in determining the fluidized-bed behavior.39,40 A high peak may be due to the compressive yield stresses in a low gas velocity. This may be the cause of a maximum granular temperature measured by Cody et al.13 In the grain-inertia region, the dominant effects arise from particle collisions.41 The frictional stresses can be neglected. The empirical correlation for the particle viscosity is µs ) 0.033s1/3 for 530 mm glass beads and µs ) 0.007s1/3g0 for 42 µm glass beads. Gidaspow and Huilin42 showed a similar empirical correlation for Fluid Catalytic Cracking (FCC) particles. Here, the 1/3 variation of the volume fraction of particles, s, is obtained from “isotropic”-type com-

(16)

where s,max is the maximum solids packing fraction of 0.6413. Restitution Coefficients. The restitution coefficients (e) for particle-particle collisions are an empirical input for CFD simulations based on the kinetic theory of granular flow. From the theoretical analysis,43,44 the restitution coefficient is not a material physical property but depends strongly on the particle impact velocity and hardness of the material. At low velocities, the restitution coefficient is very nearly equal to unity; that is, the particles are elastic. Such a restitution coefficient was approximately estimated from the kinetic theory of dense granular flow in the simple shear flow.5,45 The ratio of shear stress to normal stress is

|Shear•stress| 6 ) (3π)-1/2R* Normal•stress 5

(17)

where dimensionless group R* introduced by Lun and Savage46 is

R* )

dp |∂vx/∂y|

(18)

〈3θ〉1/2

Figure 18 shows the ratio of shear to normal stress obtained from the measured particle granular temperatures for 530 µm glass beads. The shear rates are obtained from the velocity gradient for particles, between a bubble and a dense flow region, over a given space by considering shear due to the motion of the bubbles. The ratio of shear to normal stress increases with the shear rate, as shown in Figure 18. The restitution coefficient is calculated from the productiondissipation balance5 for simple shear flow.

(1 - e)θ )

( )

1 ∂vx 15 ∂y

2

dp2

(19)

The average restitution coefficient is a constant value of 0.99 from the measured particle granular temperature and shear rate. It has a good agreement with the theory of Lun and Savage46 over a given experimental range. From their analysis,46 the restitution coefficient was assumed to decay exponentially with an


Figure 19. Variation of the restitution coefficient as a function of the impact velocity for glass spheres.

increase in the impact velocity

e ) exp[-δ(Fp/E*)1/2|kc12|]

(20)

Figure 20. Solids volume fraction profiles for 42 µm glass beads measured with a γ densitometer in a three-dimensional rectangular bed with a central jet. Ujet ) 1.36 m/s. Initial static bed height: 0.08 m. x: distance from the center. Symbols: ([) x ) 0.0 cm; (9) x ) 0.7 cm; (2) x ) 1.5 cm; (b) x ) 2.5 cm; (0) x ) 3.5 cm; (O) x ) 4.5 cm; (4) x ) 5.5 cm.

where δ is a nondimensional coefficient and E* is the modulus of elasticity. c12 ) c1 - c2 is the relative velocity, and k is the unit vector between the centers of the two colliding particles. Figure 19 shows the restitution coefficient as a function of δ(Fp/E*)1/2 and the mean particle impact velocity (V h s) for glass spheres. The results from the particle impact experiments43,46 showed that the average restitution coefficient is 0.95 in the range of an impact velocity between 1 and 2.5 m/s. The shear rate was less than 300 s-1 for this range. In our study, the impact velocity46 is estimated from the particle granular temperature.

[( ) ]

(21)

Figure 21. Comparison of measured particle and bubblelike granular temperatures for 42 µm glass beads in a central jet region of a three-dimensional rectangular bed. Ujet ) 1.36 m/s.

The restitution coefficient is 0.97 in the range of an estimated impact velocity between 0.6 and 0.9 m/s. The restitution coefficient estimated from the productiondissipation balance for simple shear flow is 0.99, as shown in Figure 18. The small difference between the two methods may be due to air, which acts as a buffer film, preventing direct contact between the particle surfaces. This effect is known as the lubrication force. The measurement of radial distribution functions of statistical mechanics18 showed that particles fluidized in water fly apart well before impact, at a radius of about 50% larger than the particle radius, while particles fluidized in air come close to contacting each other. Nevertheless, a thin air film must exist at contact. In our CFD simulation, the restitution coefficient, as an empirical input, will be assumed to be a constant value of 0.99 for the bubbling fluidized beds. Granular Temperature for Geldart A 42 µm Glass Beads. Figure 20 shows solids volume fraction profiles for 42 µm glass beads. They were measured with a γ-ray densitometer in a three-dimensional rectangular bed with a central jet. The particles were loaded to give a static bed height of 0.08 m. The velocity of the central jet was 1.36 m/s, and the superficial velocity of both sides was 0.01 m/s. Particles entrained by the gas were returned into the fluidized bed through a cyclone. The solids volume fraction profiles in the central jet region increase and then decrease with height, as shown in Figure 20. The maximum solids volume fraction in the

central jet region is 0.4. The main frequencies away from the central jet are 1.6 Hz, with a power spectral magnitude of under 0.6. However, the main frequency in the central jet region is approximately 3.1 Hz. Figure 21 shows the two types of granular temperatures for 42 µm glass beads, measured with the CCD camera technique in the central jet region between 0.11 and 0.18 m. The particle velocities were about 1 m/s over the range of measurements. The bubblelike granular temperature increases to about 0.2 m2/s2 for a solids volume fraction of about 0.08, reaches a maximum value, and then decreases with an increase of the solids volume fraction. The particle granular temperature has the same trend as the bubblelike granular temperature. The maximum particle granular temperature is about 8.5 × 10-2 m2/s2. The appearance of maximum values can be explained using the kinetic theory of granular flow.5,18 The granular temperature rises as the 2/3 power of the solids volume fraction because of a compression effect in the dilute kinetic-flow regime. As the mean free path of the particles decreases with a higher solids volume fraction of dense suspensions, the granular temperature decreases as the square of the mean free path in the collisional flow regime. Such a maximum granular temperature can also be obtained from a random fluctuations theory in a homogeneous system.47 Two Kinds of Granular Temperatures. Figure 22 shows a comparison of the two kinds of granular temperatures. Two kinds of granular temperatures can

V hs )

s,max 24 g 1/2 0 s π

1/3

- 1 θ1/2

1338


Figure 22. Comparison of two kinds of granular temperatures for (1) 530 µm glass beads in the bubbling beds, (2) 42 µm glass beads in the central jet of a three-dimensional rectangular bed, and (3) 530 µm glass beads in the IIT riser.25

be identified for gas-solids flows. Granular temperatures increase with the gas velocity. The bubblelike granular temperature due to the motion of the bubbles is much higher than the particle granular temperature due to the particle oscillations. In the riser flow, without the formation of bubbles, a Reynolds-like granular temperature due to particle clusters was much lower than the particle granular temperature.25 The motion of the bubbles in the bubbling beds is analogous to the motion of the particle clusters in the risers. The dimensionless group [θ/(µs/κs)vj x2] scaled with the particle granular temperature is close to 1 in the bubbling bed with a low gas velocity and in the riser. In a turbulent flow, like that of 42 µm glass beads, this value becomes smaller because of large dissipations. The granular temperature relation given by Cody et al., eq 13, does not work because of the breakup of bubbles in this kind of flow. The value predicted from eq 13 for 42 µm glass beads is too large. The bubblelike granular temperature varies with the intensity of the bubbles. The two kinds of granular temperatures relate to the particle diffusivity and dispersion coefficients for mixing in the fluidized beds. Diffusivity or Dispersion Coefficients. Fluidization has been widely used industrially because of its continuous powder handling ability and its good heatand mass-transfer characteristics. A measure of the quality of mass transfer is the particle diffusivity (D) or dispersion coefficient (E). A review of the literature6-8 shows that they vary by 5 orders of magnitude and that there exists no reliable predictive theory for estimating these diffusivities. In agreement with the early random walk theory of Ruckenstein,30,31 one can express the particle diffusivity,5 using the concepts of the kinetic theory, as follows:

diffusivity (D) ) mean free path × oscillating velocity (22) where the oscillating velocity is the square root of the granular temperature. This gives eq 235 as approximately

1 xθdp D) 3xπ s

(23)

This particle diffusivity is analogous to the molecular diffusivity. It is due to the oscillation of particles and

Figure 23. Diffusivities for risers and dispersion coefficients for bubbling and turbulent fluidized beds for Geldart A and B particles.1,49-57 The bed diameter is between 7.5 and 15 cm.

particle granular temperatures, as described previously. The dispersion coefficient depends on the hydrodynamics of the system. For a bubbling fluidized bed, using eq 22, it becomes essentially the product of the bubble diameter and the bubble velocity.31,48 A relation was established for nonhomogeneous fluidization31

3 E ) φDBUB π

(24)

where φ is the volume fraction of the fluidized bed occupied by the bubble, DB is the bubble diameter, and UB is the bubble rise velocity. Shi and Fan48 derived a similar relation. It is due to the formation of bubbles in fluidized beds. For a single-size particle mixture, it can be, in principle, computed using available theory and CFD codes because bubble sizes and their velocities have been computed.5 For mixtures of particles of various sizes, where there may be segregation, and for the turbulent fluidization regime, there is no theory at the present time. Figure 23 gives a summary of dispersion and diffusion coefficients from the literature and from our work for Geldart A and B particles. The dispersion and diffusion coefficients for A particles are substantially larger than those for the B particles. They appear to correlate roughly with the granular temperature shown in Figure 24. They are very high in the poorly understood but industrially important turbulent region (expanded top fluid bed). They decrease by an order of magnitude for the riser flow because in the riser they are particle diffusivities. For 530 µm glass beads in this study, the dispersion coefficient calculated from eq 24 is about 5.8 × 10-3 m2/s. This agrees with the literature data, as shown in Figure 23. The dispersion coefficient can be estimated approximately from the bubblelike granular temperature.

E=

1 - s θ f

(25)

For Geldart B particles in the bubbling and turbulent fluidized beds, the diffusion coefficients show a good agreement with eq 25. For Geldart A particles of 42 µm glass beads, the diffusion coefficient is 4.5 × 10-2 m2/s


Figure 24. Granular temperature of Geldart type A and B particles.13,17,19-21,25 Data from Figure 13 for Geldart B 530 µm glass beads.

Figure 25. Effect of the bed diameter on dispersion coefficients for bubbling and turbulent fluidized beds for Geldart A and B particles.1,8,49-51,54,56,58,59 The ratio of gas velocity to minimum fluidization velocity for Geldart B particles is 10 times lower than that for Geldart A particles.

from eq 25, where the main frequency is 3.1 Hz, the bubblelike granular temperature is 0.2 m2/s2, and the void fraction is 0.7. This agrees with the experimental value from the literature, as shown in Figure 23. However, there are more requirements for experimental data of Geldart A particles. Figure 25 shows that the dispersion coefficients increase with the bed diameter, as suggested by eq 24, where the ultimate bubble diameter increases with the bed diameter. This effect is making the scale-up of fluidized beds difficult and often leads to the choice of more reliable fixed-bed reactors.60 The long-term objective of this study is to make the scale-up of fluidized beds reliable. Conclusions Granular temperatures were measured in a twodimensional bed using the IIT CCD camera technique similar to that developed for flow in a riser.25 Similar to the riser study, two granular temperatures can be identified. The first is due to particle oscillations obtained by averaging over the velocity space per a given frame. The second is due to the bubble motions

and is like the average Reynolds normal stress. Both granular temperatures were flat in the center region of the fluidized beds. The granular temperature due to bubble motion was much higher than that due to particle oscillations in the bubbling and turbulent fluidized beds. In the riser flow with no bubbles, the former was lower than the latter.25 For the new data in the central jet region of a rectangular bed, both granular particle and bubblelike temperatures have a maximum near 9% of solids. Such a maximum was first obtained by Gidaspow and Huilin18 for flow of FCC particles in a riser. The represented granular temperature was probably the sum of the temperatures. A random fluctuations theory proposed by Buyevich and Kapbasov47 has a similar maximum in the granular temperature. The measured normal stresses per unit bulk density in the vertical direction were larger than the measured normal stresses per unit bulk density in the lateral direction because of higher velocity fluctuations for particles in the bubble-flow region. Similar anisotropy was observed in the riser.25 The sum of the measured shear stresses equals the pressure drop minus the weight of the bed of solids within experimental error, which is large because of subtraction of two large numbers. The frequency obtained from the hydrodynamic velocity for particles agreed with the frequency measured for the void fraction. The viscosity measured by a Brookfield viscometer has a high peak because of the yield stresses in the viscous region. An empirical correlation for the particle viscosity was obtained from the measurements. The restitution coefficients for 530 µm glass beads were estimated from the ratio of shear to normal stress to be in the range of 0.99. They are a function of the particle impact velocity. For low particle velocities, in the range of 0.1 m/s, the restitution coefficients are closer to unity than those for a riser at a velocity of 1 m/s. Previous shear cell studies produced less elastic coefficients. A review of the literature shows that the dispersion coefficients for A particles are substantially larger than those for the B particles for both the bubbling and turbulent fluidized beds. They are very high in the turbulent region. This means that mixing in the bubbling and turbulent fluidized beds is due to bubble motion, as well as due to particle oscillations, similar to molecular diffusion in fluids. The dispersion coefficients increase with the granular temperature. Acknowledgment This study was supported by the University of Pittsburgh NETL student Partnership Program, partially by the National Science Foundation grant to professor Gidaspow and by the ORISE program. Nomenclature ci ) instantaneous particle velocity in the ith direction Ci ) peculiar particle velocity in the ith direction, ci - vi D ) particle diffusivity DB ) bubble diameter dp ) particle diameter e ) restitution coefficient E ) dispersion coefficients for particles f ) frequency g ) gravity go ) radial distribution function for particles

1340


h ) height above the distributor m ) number of total frames n ) number of particles per unit volume at each frame r ) position vector t ) time UB ) bubble rise velocity Umf ) minimum fluidization velocity U0 ) superficial gas velocity vi ) hydrodynamic velocity for particles in the ith direction vj i ) mean particle velocity in the ith direction Vi′ ) hydrodynamic velocity relative to the mean velocity, vi - vj i Vs ) mean particle impact velocity Greek Letters k ) volume fraction of phase k κs ) granular conductivity µk ) shear viscosity of phase k θ ) granular temperature Fk ) density of phase k τ ) stress Subscripts g ) gas phase i or j ) x, y, and z directions max ) maximum s ) particle (solids) phase x ) vertical direction y ) lateral direction

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Received for review April 17, 2004 Revised manuscript received October 31, 2004 Accepted November 2, 2004 IE0496838

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