Measurement of Particle Phase Stresses in Fast Fluidized Beds

The particle distribution in FFB risers is often quite nonuniform, with higher concentrations occurring near the riser walls and base. Depending on th...
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Ind. Eng. Chem. Res. 1999, 38, 705-713

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Measurement of Particle Phase Stresses in Fast Fluidized Beds William Polashenski, Jr.,* and John C. Chen Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

Despite concentrated research efforts devoted to the study of the two-phase fluid mechanics of fast fluidized beds (FFBs), the particle phase stresses in FFBs are not well-characterized. To address this problem, measurements were made of the local time-averaged solid pressure in a FFB riser for a wide range of flow conditions and bed locations. The solid-pressure values, which were several orders of magnitude smaller than the gas pressure, were found to vary primarily with the local time-averaged solid concentration. Using constitutive relations from the kinetic theory of particulate flow, values of local time-averaged solid-phase shear viscosity were calculated from the measured solid pressure and solid fraction. The calculated viscosity values were found to be intermediate within the rather wide range of estimated values previously used in various two-fluid models for dilute vertical gas-particle flow. Introduction Recently much attention has been focused on understanding the gas-particle fluid mechanical behavior in fast fluidized bed (FFB) risers. The particle distribution in FFB risers is often quite nonuniform, with higher concentrations occurring near the riser walls and base. Depending on the operating conditions, a high degree of backmixing can occur, with net solids downflow near the walls. As the experimental database on the particle distributions in FFB risers has become more extensive, increasingly sophisticated efforts have been made to model the gas-particle hydrodynamics. Because of computing limitations and the lack of analytical solutions to the transport equations for gas-particle mixtures, early modeling efforts were limited to solutions for the axial development of a cross-sectionally averaged solid fraction. However, the presence and significance of radial nonuniformities in the particle concentration require a more detailed representation of the flow physics. In his recent assessment of the field, Jackson1 has noted that, given current computational abilities, posing the appropriate form for the equation set is now the key challenge involved in gas-particle flow modeling. The need for detailed knowledge of the particle distributions in FFB risers has led to the development of complex models for gas-particle flow. Because of the fluidlike behavior of fluidized particles, many of these models are based on a two-fluid approach, which treats the gas and particles as interpenetrating continuum fluid phases. A derivation of the basic equation set for gas-particle flow based on local volume averaging was first proposed in the classic work of Anderson and Jackson;2 equation sets derived using other averaging techniques were subsequently developed.3,4 The transport equations for the gas and particle phases comprise a set of strongly coupled, highly nonlinear partial differential equations. The key difficulty in solving this equation set, apart from having sufficient computing strength, is posing appropriate constitutive relations required for closure. Among these relations, arguably * Corresponding author. Current address: Lomic, Inc., 200 Innovation Blvd., Suite 241, State College, PA 16803. E-mail: [email protected]. Fax: 814-238-5315.

the most important are those for the interfacial drag coefficient and particle phase stresses. For the interfacial drag coefficient, the usual form is that for singleparticle drag, modified by a correction factor proposed by Richardson and Zaki.5 For the particle phase stresses, a Newtonian form is typically assumed, although in fact the stress-strain relationship in particulate assemblies can be quite nonlinear. If the solid stresses are represented by a Newtonian form, appropriate relations must then be proposed for the particle “pressure” and “viscosity”. Mutsers and Rietema6 proposed an empirical expression for the particle pressure, based on data from powder compaction experiments, which has been often used in gasparticle flow models.

∇ps ) G(s) ∇s

(1)

In the above, G(s) is the solid-phase elastic modulus, which has the following general form:

G(s) ) 10as+b

(2)

No similar empirical expression has been reported for the solid viscosity; in some two-fluid models for dilute gas-solid flows, solid viscosity was simply assumed to be constant7,8 or some simple function of the solid fraction.9 Recently, a first-principles method was developed to determine the solid-phase stresses in particulate assemblies. The dense-gas kinetic theory development of Chapman and Cowling10 was modified to apply to particulate flows by allowing for inelastic interparticle collisions. A number of kinetic theory based derivations have been developed for particulate flows.11-15 Included in the kinetic theory derivations is an equation for the calculation of the fluctuating kinetic energy of the particles, as characterized by their granular (pseudothermal) temperature, which is analogous to the thermal temperature of standard gas kinetic theory. Furthermore, constitutive relations can be derived for the particle normal and shear stresses in terms of the particle properties, local solid fraction, and local granu-

10.1021/ie980354n CCC: $18.00 © 1999 American Chemical Society Published on Web 01/20/1999

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Figure 2. Mechanism of solid-pressure measurement. Note that the face of the diaphragm is exposed to both gas and particle pressures, while a screened vent allows the back of the diaphragm to experience only the gas pressure. Figure 1. Values of solid viscosity reported in the literature for dilute gas-solid flow. Values from experimental setups are taken at a radial location r/R ) 0.9 at an axial location far from the solids inlet.

lar temperature. As an example, the following are the solid stress constitutive relations of Lun et al.:12

ps ) sFsTs(1 + 4ηsg0) µs )

5πFsdp

(3)

(1 + 58η g )(1 + 58η(3η - 2) g ) +

96g0η(2 - η)

s 0

s 0

8 F d g η 2 5 s p 0 s

x

Ts (4) π

In the above, ps and µs represent the particle phase pressure and shear viscosity, respectively, s denotes the local solid fraction, Ts represents the granular temperature, go denotes the radial distribution function, and η ) 0.5(1 + e), where e is the coefficient of restitution for particle-particle collisions. Although kinetic theory based equation sets were initially developed to describe dense particulate flows, they were later applied to more dilute gas-particle flows. In their pioneering work, Sinclair and Jackson16 presented the first two-fluid model using kinetic theory to describe the particle phase in dilute vertical fully developed gas-solid flows. A number of similar models were proposed shortly thereafter.17-20 Later, predictions were reported by various authors for developing gas-particle flows.21-26 For the most part, however, the authors presented only limited comparisons with experimental data and little information on the calculated solid stresses. Figure 1 compares some of the values of the solid shear viscosity reported in the literature, covering experimental efforts, empirical models, and kinetic theory based models. Notice that the values vary by several orders of magnitude for flows of rather similar overall solid concentration. The wide variation among these viscosity values raises the question of whether the proposed two-fluid models are, in fact, capturing correctly the essential flow physics. Hence, experimental measurements of the solid stresses are needed to verify the relative importance of the solid stresses to the overall fluid dynamic behavior in fast fluidized beds. Few attempts have been made to determine experimentally the solid-phase stresses in dilute-phase fluidized beds. Adewumi and Arastoopour7 deduced values of solid viscosity from the measurements of Cutchin27 of the falling velocity of a large sphere in a mixture of gas and fine particles. However, Stokes’ and Kay’s laws

were applied to arrive at these viscosity values, and it is unclear as to whether these laws are, in fact, valid for the conditions of the measurements. Tsuo and Gidaspow8 and Miller and Gidaspow28 backcalculated values of solid viscosity by first measuring the local pressure gradient, solid fraction, and solid mass flux, then using these data in the mixture momentum equation to deduce values of the solid stress term, and then determining the shear rate from the solid velocity profile in order to obtain the solid viscosity values. However, sizable uncertainties were propagated through the calculations, thus limiting the precision of the resulting viscosity values. In fact, the two studies yielded very different values of solid viscosity for rather similar operating conditions. More recently, Gidaspow and Huilin29 used a video-digital camera technique to measure particle velocities in gas-solid flows of various overall solid concentrations. From these data, they calculated granular temperature and solid viscosity values, the latter of which were found to agree well with the values reported by Miller and Gidaspow.28 Clearly, some direct measurement of the solid-phase stresses in a FFB riser would be desirable in order to establish unequivocally their order of magnitude. This paper describes a probe used to isolate and measure the particle pressure in a fluidized bed and then gives experimental results for measurements of time-averaged solid pressure made in a pilot-scale FFB riser. Relations from the kinetic theory of particulate flows are used to calculate values of local solid shear viscosity, which can be compared with results from several twofluid models proposed for dilute gas-particle flow. Solid-Pressure Probe The concept of the solid-pressure measurement is illustrated in Figure 2. A sensitive transducer with a piezoresistive sensing element was the key component of the probe. The diaphragm was mounted flush with the face of the transducer. A screened vent tube proceeding to the rear of the diaphragm permitted access for the gas, but not for the particles. Hence, the net response of the diaphragm should correspond to the impact of the particles in the direction normal to the probe face. This method of isolating the particle pressure was pioneered by Campbell and co-workers,30-32 who made measurements at the wall of a bubbling fluidized bed. Polashenski and Chen33 used the probe of Figure 2 in a bubbling bed to measure solid pressure for a wide range of particle sizes. They also presented preliminary results related to the present work at a recent AIChE

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Figure 5. Needle capacitance probe used for solid-fraction measurement.

Figure 3. Probe used for solid-pressure measurements at the riser wall.

Figure 6. Schematic of a pilot-scale FFB (10.8 m height and 0.15 m inside diameter). Figure 4. Probe used for solid-pressure measurements in the interior of the riser.

meeting;34 the present work will present extended data and more detailed analysis for solid-pressure measurements both at the wall and in the interior of a FFB. After the AIChE presentation, a similar technique for solid-pressure measurement in a FFB was reported,35 but these measurements were limited to data at the wall. Two different holding devices were fabricated to allow for the measurement of solid-phase normal stress. Figure 3 illustrates a device used for measurements at the bed wall. The wall holder, constructed of brass, was threaded so that the probe could be held snugly in place without its electronics being affected. A screened access tube allowed the gas to pass to the vent tube of the transducer. Also, an arc profile was machined into the face of the holder so that it could be mounted flush with the riser wall. Figure 4 shows a simple holder used for internal measurements. The transducer was simply inserted into the holder and secured in place, with its face flush with the end of the holder. Although this setup was expected to cause some flow interference, most of the interference would be expected to be around the body of the holder, not at the sensing surface of the transducer. Along with the solid-pressure measurements, solidfraction measurements were simultaneously made.

Figure 5 depicts a needle capacitance probe used to determine the local solid concentration. Capacitance probes have been used by a number of researchers36,37 to measure the local solid concentration in the interior of FFB risers. The operating principle of these probes is the relation between the dielectric constant and particle concentration in a two-phase mixture. A number of relations linking the dielectric constant and particle concentration of fluid-particle mixtures have been proposed;38 the results presented here were calculated via the classical relation of Maxwell.39 Measurements of solid pressure and solid fraction were made at several radial locations at each of several elevations in the riser of a pilot-scale FFB (Figure 6) which had an inside diameter of 0.15 m and a height of 10.8 m. To avoid interference between the probes, measurements were made at different circumferential locations; for a number of runs, the positions of the probes were switched to ascertain that no circumferential effects existed. The outputs from the probes were sent to signal conditioners for rectification and amplification and were subsequently recorded on a microcomputer by a software program designed for highfrequency multichannel data recording. The data acquisition software was set to a recording frequency of 200 Hz, and measurements were made for a duration of 160 s at each location.

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Figure 7. Radial and axial variation in the local time-averaged solid pressure (a) and solid fraction (b) for 140 µm sand for the case where vsup ) 6.0 m/s and Gs,nom ) 40 kg/m2‚s.

Solid-Pressure Measurements Measurements of solid pressure were made in the riser of the pilot-scale FFB for two particles, a Geldart40 Group B sand (dp ) 140 µm, Fs ) 2500 kg/m3) and a Group A FCC (dp ) 94 µm, Fs ) 1500 kg/m3). The ranges of superficial gas velocity and overall solid mass flux were chosen to cover a wide range of flow conditions spanning the fast fluidization operating regime. Results of the axial and radial variation of local timeaveraged solid pressure and solid fraction for a sample case of dilute flow of the 140 µm sand (vsup ) 6.0 m/s, Gs,nom ) 40 kg/m2‚s) are depicted in Figure 7. As shown in Figure 7a, the solid pressure was found to be rather low in magnitude for this case, ranging from 10 to 15 Pa both near the base of the riser and at a position far downstream from the solids inlet. Figure 7b shows that the solid fraction was not very high for this case, ranging from about 0.002 to 0.01 throughout most of the riser, except near the wall at the base of the riser, where it was found to be about 0.04. Note that, although a noticeable radial variation in solid fraction occurred even well downstream from the solids inlet, the solid pressure varied little with radial position. It is worth noting that the solid-phase momentum equations for time-independent, fully developed flow indicate that no radial variation in solid pressure should exist for fully developed conditions, even though a radial variation in solid fraction may exist. Figure 8 displays the solid-pressure and solid-fraction data for a sample case of dense flow of the 140 µm sand (vsup ) 4.2 m/s, Gs,nom ) 40 kg/m2‚s). For these flow

Figure 8. Radial and axial variation in the local time-averaged solid pressure (a) and solid fraction (b) for 140 µm sand for the case where vsup ) 4.2 m/s and Gs,nom ) 40 kg/m2‚s.

conditions, Figure 8a indicates that solid pressure was relatively large at the base of the riser, reaching nearly 100 Pa near the wall, and had a pronounced radial nonuniformity. However, as the flow developed, the solid pressure continually decreased in magnitude and became more radially uniform. Figure 8b shows that the solid fraction was quite high at the base of the riser but dropped off quickly at higher elevations. In addition to examination of sample cases for different overall particle concentrations, trends with the key operating parameters are also worth observing. Figures 9 and 10 show solid-pressure and solid-fraction results for three cases for the 94 µm FCC at two different elevations in the riser, one near its base and one a considerable distance downstream, in which the superficial gas velocity was held constant at 3.0 m/s and the overall solid mass flux was varied. Parts a and b of Figure 9 show that at an elevation 1.5 m above the riser inlet, for the two more dilute cases, solid pressure was nearly constant with radial location at a value near 10 Pa. However, for the run where Gs ) 35 kg/m2‚s, the solid pressure increased from about 10 Pa in the core region to over 100 Pa near the wall. Parts a and b of Figure 10 show that at an elevation 5.5 m above the inlet, where the particle concentrations were lower, solid pressure varied little with radial position and had a magnitude of about 10 Pa. For these cases the solid fraction was quite low, below about 0.05, except near the wall at the base of the riser for the case with the highest solid mass flux. In Figures 11 and 12, the results are shown for the 94 µm FCC where the overall solid mass flux was held constant at 25 kg/m2‚s and the superficial gas velocity

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Figure 9. Radial variation in the local time-averaged (a) solid pressure and (b) solid fraction at an elevation 1.5 m above the solids inlet for 94 µm FCC for various values of the overall solid mass flux at a constant vsup ) 3.0 m/s.

Figure 10. Radial variation in the local time-averaged (a) solid pressure and (b) solid fraction at an elevation 5.5 m above the solids inlet for 94 µm FCC for various values of the overall solid mass flux at a constant vsup ) 3.0 m/s.

was varied. Again, for the more dilute cases (higher values of vsup), solid pressure is seen to vary little with axial and radial location in the riser. However, for the case where vsup ) 2.4 m/s, marked variations with position in the riser are evident. Near the riser base, where the solid concentrations approached those normally associated with dense-phase fluidized beds, the solid pressure reached several hundred pascals in magnitude. At the higher elevation, where the particle phase became more dilute, solid pressure decreased both axially and radially. Hence, it appears that solid pressure does not vary in a linear fashion with superficial gas velocity or overall solid mass flux but rather remains approximately constant until the flow becomes moderately dense and then reaches larger values in areas of the riser with relatively high local solid concentrations. The plots examined for the variation in time-averaged solid pressure seem to indicate that the strongest correlating factor was the local solid fraction. Figure 13 shows a plot of all of the solid pressure data taken for both particles as a function of the local solid fraction. A clear trend is observed, with the solid pressure remaining nearly constant for low values of the solid fraction and steadily increasing once the solid fraction exceeds about 0.05. The local time-averaged solid pressure was found to correlate with the local time-averaged solid fraction according to the following relation:

correlated for two different particles for a wide variety of operating conditions; however, it is not yet known if riser geometry has any effect on the local solid pressure. Nonetheless, the above correlation represents a relation which can be used for the prediction of solid pressure within the fast fluidization operating regime.

ps [Pa] ) [(16.5s0.1)5 + (518s1.1)5]0.2

(5)

The line representing this correlation is also shown in Figure 13. As mentioned earlier, the results were

Kinetic Theory Calculations Because the solid pressure was found to be several orders of magnitude lower than the gas pressure even for regions of high solid concentration in the riser, it can be concluded that the solid pressure would not have a noticeable impact on the numerical results from a twofluid model. However, the same cannot be said a priori for the solid shear stress. As mentioned earlier, the solid viscosity is rather difficult to measure accurately in fluidized beds. However, the mathematical development for the kinetic theory of particulate flows generated a set of constitutive relations for solid pressure and viscosity in terms of the local solid fraction, local granular temperature, and particle properties. Assuming that the particle stresses in FFB risers are generated primarily by particle-particle interactions, which induce a fluctuating component of particle velocity, the solid pressure data can be used in conjunction with the local solid fraction values and particle properties to calculate values of granular temperature and solid viscosity from the appropriate kinetic theory relations. Figures 14-17 depict solid viscosity values calculated from the solid-pressure and solid-fraction data presented in Figures 7-12 via the solid stress constitutive

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Figure 11. Radial variation in the local time-averaged (a) solid pressure and (b) solid fraction at an elevation 1.5 m above the solids inlet for 140 µm sand for various values of the superficial gas velocity for a constant Gs,nom ) 25 kg/m2‚s.

relations of Lun et al.12 given by eqs 3 and 4. Figure 14 shows that, for the more dilute run for the 140 µm sand, the solid viscosity decreases with radial position and does not vary much with axial position except perhaps midway between the tube center and wall. Figure 15 shows that, for the more dense case for the 140 µm sand, solid viscosity varies little with radial position in the dense region near the base of the riser but increases in magnitude in the downstream region and shows a radial profile similar to that of the dilute case. At first it may seem unusual that solid viscosity should decrease with increasing solid fraction. However, it should be noted that the granular temperature of the particles, which at low concentrations has a stronger effect on the solid viscosity than does the solid fraction (as can be deduced from eq 4), was found to increase as the concentration decreased. The trends in solid viscosity with operating parameters are shown for the 94 µm FCC in Figures 16 and 17. In Figure 16a, it is seen that, for a constant superficial gas velocity of 3.0 m/s, no clear trend exists with solid mass flux near the base of the riser. The solid viscosity was found to be nearly constant with radial position for the most dense case and to decrease with radial position in general for the more dilute cases. Figure 16b shows that further downstream from the inlet solid viscosity decreased with increasing mass flux and had its maximum values roughly halfway between the riser center and wall. Figure 17a shows the varia-

Figure 12. Radial variation in the local time-averaged (a) solid pressure and (b) solid fraction at an elevation 5.5 m above the solids inlet for 140 µm sand for various values of the superficial gas velocity for a constant Gs,nom ) 25 kg/m2‚s.

Figure 13. Variation in the local time-averaged solid pressure with the local time-averaged solid fraction for all CFB data. The solid line represents the correlation given in eq 5.

tion of solid viscosity with gas velocity for a constant net mass flux of 25 kg/m2‚s. The viscosity increased with increasing gas velocity in the core region and showed the opposite trend near the wall. The viscosity was again seen to vary little with radial position for the most dense case and to decrease in general with radial position for the more dilute cases. At the 5.5 m elevation (Figure 17b), solid viscosity was highest near the riser centerline and did not vary significantly with solid mass flux. At this point, the calculated solid viscosity values can be compared with the previously reported values. Figure 18 compares representative values of solid viscosity for both the sand and FCC particles, as measured in the

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Figure 14. Radial and axial variation in calculated local solid viscosity for 140 µm sand for the case where vsup ) 6.0 m/s and Gs,nom ) 40 kg/m2‚s.

Figure 15. Radial and axial variation in calculated local solid viscosity for 140 µm sand for the case where vsup ) 4.2 m/s and Gs,nom ) 40 kg/m2‚s.

present study (solid bars), with the viscosity values shown in Figure 1 (shaded bars). Note that the calculated values fall intermediate within the reported range of values. Also note that the overall magnitude of solid viscosity, which was on the order of 0.01 Pa‚s, is generally much lower than the solid viscosity values reported in bubbling fluidized beds (which tend to be in the range 0.1-1.0 Pa‚s). Hence, the extrapolation of bubbling bed viscosities to fast fluidized beds would seem to be inappropriate. Figure 19 displays the solid viscosity data for all bed locations and operating conditions as a function of the local solid fraction. Note that solid viscosity has an overall downward trend with solid fraction until the latter quantity exceeds 0.10, at which point the trend reverses. Also plotted in Figure 19 are lines representing the correlations of Arastoopour et al.9 (thin line) and Miller and Gidaspow28 (thick line) for solid viscosity as a function of solid fraction. Although the present data and correlated solid viscosity values are of the same order of magnitude, the trends of solid viscosity with solid fraction are different. For a given particle, the present method of calculating viscosity depends on the local solid fraction and granular temperature, whereas the correlations are functions of solid fraction only and are unable to account for the intensity of the interparticle interactions. It should be noted that the data on which the correlation of Miller and Gidaspow28 was based are limited to the region adjacent to the riser wall, where the intensity of particle-particle interactions is

Figure 16. Radial variation in the calculated local solid viscosity at elevations (a) 1.5 m and (b) 5.5 m above the solids inlet for 94 µm FCC for various values of the overall solid mass flux for a constant vsup ) 3.0 m/s.

much lower than that throughout the rest of the bed. Furthermore, Arastoopour et al.9 acknowledged that their expression was hypothetical and was meant to give only the correct order of magnitude of the solid viscosity. It should also be noted that the current data are limited to the fast fluidization regime, where the particles can still be represented as a continuum fluid phase. As the flow regime changes from the fast fluidization regime to the pneumatic transport regime, the intensity of particle-particle interactions will quickly diminish. The solid viscosity will then decrease, approaching zero as the solid concentration approaches zero. On the basis of the above analysis, it is apparent that expressions for the solid viscosity used in numerical models for FFB flow should take into account both the local solid concentration and the intensity of particle-particle interactions. In closing, it should be noted that the experiments in this work were conducted in a pilot-scale FFB with a cylindrical riser of fixed diameter (0.15 m). A parametric study considering the effects of riser geometry and diameter was beyond the scope of this effort. However, industrial-scale risers, which could be as large as several meters in diameter, have been shown18,23 to exhibit steady-state multiplicity within certain ranges of operating conditions. As such, additional measurements should be performed on risers of larger diameter to verify that the trends observed and correlations presented in this work are indeed valid for industrial-scale FFBs as well as to study the solid stress behavior for flow conditions where multiplicity occurs.

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Figure 19. Variation of the solid viscosity with the local solid fraction for various riser locations and operating conditions. Also shown are the correlations of Miller and Gidaspow28 (thick line) and Arastoopour et al.9 (thin line).

Figure 17. Radial variation in the calculated local solid viscosity at elevations (a) 1.5 m and (b) 5.5 m above the solids inlet for 140 µm sand for various values of the superficial gas velocity for a constant Gs,nom ) 25 kg/m2‚s.

averaged local solid fraction. A correlation was developed for the solid pressure as a function of the solid fraction. This correlation, along with constitutive relations from the kinetic theory of particulate flows, was used to develop a method to calculate the local solid viscosity in a fast fluidized bed. The resulting solid viscosity values were found to be intermediate within the range of previously reported values for FFB risers. Having established the magnitude of the solid-phase stresses in fast fluidized beds, we hope that these results can be used toward improving the modeling of the fluid dynamics of FFB risers. Notation dp ) particle diameter, µm e ) coefficient of restitution g0 ) radial distribution function G(s) ) solid-phase elastic modulus, Pa Gs,nom ) overall solid mass flux, kg/m2‚s ps ) solid pressure, Pa Ts ) granular temperature, m2/s2 vsup ) superficial gas velocity, m/s Greek Letters s ) solid fraction η ) 0.5(1 + e) µs ) solid viscosity, Pa‚s Fs ) particle density, kg/m3

Figure 18. Comparison of representative solid viscosity values for sand and FCC particles calculated in this work (solid bars) at an elevation of 5.5 m at a radial location r/R ) 0.9 with the solid viscosity values presented in Figure 1 (shaded bars).

Summary A probe was developed to isolate and measure the particle-phase normal stress in gas-particle flows. This probe was used to obtain the first known measurements of solid pressure at the wall and in the interior of the riser of a fast fluidized bed. Solid pressure was found to be a rather small component of the total pressure, ranging from about 10 Pa for dilute conditions to 100300 Pa in particularly dense regions for a pilot-scale FFB operated at near-atmospheric pressure. Time-averaged local solid pressure was found to vary in a monotonically increasing fashion with the time-

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Received for review May 29, 1998 Revised manuscript received November 11, 1998 Accepted November 19, 1998 IE980354N