Entrainment rate of coarse particles at different temperatures in gas fluidized beds.

Choi, Jeong-Hoo ; Kim, Sang Done ; Grace, John R. 等

INTRODUCTION

Particles with terminal velocities greater than the superficial gas velocity commonly appear among the particles captured by cyclones downstream of gas-fluidized bed reactors. There is evidence that the entrainment of these coarse particles is caused by the fine particles. Geldart et al. (1979) and Geldart and Pope (1983) reported that the presence of fine particles greatly enhances entrainment of coarse particles from a fluidized bed containing binary mixtures of fine and coarse particles. Geldart et al. (1979) proposed a correlation considering the effect of the fines. Their results were correlated in terms of an effective gas density based only on the particles actually carried out of the column. Geldart and Pope (1983) found that the correlation of Geldart et al. (1979) was oversimplified. Kwauk (1992) reported an analytical study of polydisperse systems, showing that the upward motion of fine particles caused coarse particles to be entrained. Choi et al. (2001) investigated the effect of fine particles on the carryover of coarse sand particles from a gas-fluidized bed of sand at room temperature as bed material. The entrainment rate of the coarse particles increased with the superficial gas velocity and the proportion of fines in the bed. It was nearly proportional to the upward momentum of the fine particles based on the composition of bed particles at a specified gas velocity. The effect of the fines on the entrainment of the coarse particles decreased as the gas velocity increased. The influence of the particle size distribution in the bed on the elutriation rate of fine particles was negligible. An improved correlation to determine the entrainment rate of the coarse particles was proposed for sand particles at room temperature.

The effect of gas properties is important, affecting the hydrodynamics and performance of gas-fluidized bed reactors. However, few studies have been carried out with gases other than air at atmospheric temperature and pressure, and these have generally covered limited experimental ranges, as discussed by Choi et al. (1997a, 1998, 1999). Romanova et al. (1980) and Milne et al. (1993) reported that particle entrainment increased with temperature. However, they did not isolate the effect of temperature since their results included the effect of gas velocity, which increased with temperature. For the temperature range between 27 and 172[degrees]C, George and Grace (1981) found that the effect of temperature on the entrainment rate was small. In fluidized-bed combustors (Merrick and Highley, 1974; Choi et al., 1989; Lee et al., 1990; Park et al., 1990; Lee et al., 1992), the particle entrainment rate decreased with increasing temperature. According to Wouters and Geldart (1998), the entrainment rate of FCC particles decreased with an increase in temperature up to 400[degrees]C. However, in their case the particles were so small (7 to 48 [micro]m) that interparticle forces were important.

Choi et al. (1997a, 1998) investigated the qualitative effect of temperature on particle entrainment at the exit of a gas-fluidized bed. Their results indicated an increase in particle entrainment rate, after an initial decrease, as temperature increased. Hence, a minimum entrainment rate occurred at a certain temperature. At constant gas velocity, nearly the same trend with temperature appeared for all particle sizes considered. However, they focused on the elutriation of fine particles. They explained the trend as resulting from decreasing gas density and increasing gas viscosity, with interparticle effects neglected. Over the experimental range investigated, the variation of entrainment rate with temperature was very similar to the variation of the particle size at which the terminal velocity was equal to the superficial gas velocity.

Chan and Knowlton (1984) found that the entrainment rate was linearly proportional to gas density in a pressurized fluidized bed. Knowlton (1992) and Knowlton et al. (1990) reported that entrainment was augmented by increasing gas viscosity or gas density, corresponding to a decreased particle terminal settling velocity. Chan and Knowlton (1984), Knowlton (1992) and Knowlton et al. (1990) all considered the effect of gas properties.

The Wen and Chen (1980) correlation obtained from data on fluidized beds at room temperature predicts that particle entrainment from the bed surface should decrease with increasing temperature. However, the validity of their correlation at different temperatures and pressures has not been tested. Choi et al. (1997a, 1998) showed that existing correlations were unable to represent the effect of temperature on the particle entrainment rate because they were based on experiments conducted at room temperature, or over very narrow temperature ranges. Choi et al. (1999) developed an empirical equation based on comprehensive experimental data, including variations of column size, gas velocity, temperature, particle size, and particle density. Their correlation mainly focused on the elutriation of fine particles and does not include the effect of fine particles on the entrainment of coarse particles.

The goal of this paper is to present a comprehensive correlation to predict the entrainment of relatively coarse particles from gas-fluidized beds, accounting for the effects of the fine particles, temperature, particle size and density, gas velocity, and column size.

DERIVATION OF CORRELATION

Particle entrainment is so complicated that no generalized model is yet able to give reliable predictions over wide ranges of particle and gas properties and operating conditions, even though extensive experimental and theoretical studies have been carried out. As a result, empirical or semi-empirical correlations are usually applied over limited ranges to predict the particle entrainment rate.

In the freeboard region solid particles either rise or fall depending on their size, density, gas properties, gas velocity and interaction with other particles. Particles can be divided into two groups on the basis of the reference particle diameter [d.sub.crit], defined as the particle diameter at which the terminal settling velocity is equal to the superficial gas velocity. For particles larger than [d.sub.crit] ([U.sub.t]>U), Geldart (1985) called the particles coarse, whereas particles whose terminal settling velocities are less than U (i.e., [d.sub.p]<[d.sub.crit]) are referred to as fines. This study investigates the entrainment of coarse particles, defined in this manner.

Experimental entrainment rates of coarse particles measured in relatively cold fluidized beds by George and Grace (1981) and Choi et al. (1989, 1997b, 2001) and at elevated temperature by Choi et al. (1997a, 1998) are used to develop an empirical relationship. These experiments were carried out in bubbling and turbulent fluidized beds for ranges of column size, gas velocity, temperature, particle size, particle density, and particle size distribution. The experimental conditions are summarized in Table 1. In order to provide consistency, data are included only for cases where there was an abrupt exit from the bed and a steady-state recirculation system, as described by Kunii and Levenspiel (1991). As mentioned by Choi et al. (1997a, b, 1998, 2001), changes of bed particle size distribution were minimized by controlling the sampling time and by including changes of particle size distribution in the bed in determining the entrainment rate constant.

The correlation of Choi et al. (2001) was first applied to the data in Table 1 excluding their own data. The correlation coefficient ([r.sup.2]) was 0.735 and the correlation, did not express the effect of gas velocity properly, because the exponent on the gas velocity is nearly fixed in the correlation, whereas the exponent on gas velocity decreases with increasing gas velocity in reality. When the same parameters as in the correlation of Choi et al. (2001) are applied to the complete range of data, including the data obtained in that study, a correlation that fits the [K.sub.i.sup.*] of coarse particles ([d.sub.p] > [d.sub.crit]), was obtained:

[K.sub.i.sup.*]/[C.sub.d][[rho].sub.g]U = 1.25 x [10.sup.-2]P.sup.0.575][F.sub.d.sup.5.03]/[F.sub.g.sup.0.892] (1)

where

[F.sub.g] = 2g[d.sub.p]([[rho].sub.p]-[[rho].sub.g])/3, [F.sub.d] = [C.sub.d][[rho].sub.g][U.sup.2]/2, P = [SIGMA](U-[U.sub.t]) for all [d.sub.p] < [d.sub.crit] (SI units throughout) (2)

[C.sub.d] = 24/[Re.sub.p] for [Re.sub.p] [less than or equal to] 5.8,

[C.sub.d] = 10/[Re.sub.p.sup.0.5] for 5.8 < [Re.sub.p] [less than or equal to] 540, [C.sub.d] = 0.43 for 540 < [Re.sub.p] (3)

with

[Re.sub.p] = [d.sub.p]U[[rho].sub.g]/[micro] (4)

Here [F.sub.g] and [F.sub.d] are the gravity force minus buoyancy force per projected area of the particles, and the drag force acting on the particle divided by the projected area, respectively. The regression coefficient for the above correlation ([r.sup.2]) is 0.792.

As shown in Equation (1), [K.sub.i.sup.*] of the coarse particles increases with increasing upward momentum of fine particles (P) calculated on the basis of bed composition for different temperatures, particle densities and column sizes. In Equation (1), the exponent of the gravitational force term is less than half of that in the correlation of Choi et al. (2001). Since Choi et al. (2001) used only sand particles Equation (1) seems to be more realistic. However, Equation (1) also cannot express the effect of gas velocity on entrainment rate properly because the exponent of gas velocity in the experimental data decreased with increasing gas velocity.

As a result of trial and error, we found that the best method of predicting entrainment of the coarse particles is to add terms to an existing correlation to account for the effect of the fines. Improved prediction of the entrainment rate of coarse particles was achieved by beginning with the correlation of Choi et al. (1999), which well describes effects of other variables, including temperature, in spite of its complicated form. In this relationship, the entrainment flux, [K.sub.i.sup.*], of particles of size [d.sub.p] at the exit considers a cluster flux ([K.sub.ih.sup.*]) and a dispersed non-cluster flux ([K.sub.i[infinity].sup.*]), as proposed by Hazlett and Bergougnou (1992). The cluster flux is assumed to decrease exponentially with height in the freeboard. On the other hand, the dispersed non-cluster flux remains constant over the freeboard height, corresponding to the elutriation flux above the transport disengaging height. The decay constant of the cluster flux is assumed to be equal to that of the axial solid holdup profile. The influence of the exit configuration is assumed to be minor. The correlation of Choi et al. (1999) was able to predict the particle entrainment rates at the exit for particle diameters from 0.021 to 0.710 mm, particle densities from 2400 to 6200 kg/[m.sup.3], gas velocities from 0.15 to 2.8 m/s, temperatures from 12 to 600[degrees]C, pressures from 101 to 3200 kPa, column diameters or hydraulic diameters from 0.1 to 0.91 m and column heights from 2.0 to 9.1 m. However, this correlation did not account for the influence of fine particles on the entrainment of coarse particles.

The following correlation was found to provide the best fit to the experimental data of Choi et al. (1989, 1997a, b, 1998, 2001) and George and Grace (1981) for [d.sub.p] > [d.sub.crit], with a better correlation coefficient ([r.sup.2]) of 0.86.

[K.sub.i.sup.*] = 15.9 ([K.sub.ih.sup.*] + [K.sub.i[infinity].sup.*]) [P.sup.0.724][F.sub.d.sup.-1.19][F.sub.g.sup.-0.234] (5a)

where

Kih*dp/ = CdRep exp(-9.12-0.0153a(Ht-Hb)) (5b)

[K.sb.i[infinity].sup.*] [d.sub.p]/[micro] = [Ar.sup.0.5] exp(6.92-2.39[F.sub.g.sup.0.303]-13.1/[F.sub.d.sup.0.902] (5c)

with

Ar = g[d.sub.p.sup.,3][[rho].sub.g]([[rho].sub.p]-[[rho].sub.g])/[[[micro].sup.2] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

As shown in Equation (5c), gas velocity has a positive effect in the final -13.1[F.sub.d.sup.-0.902] term. In the present correlation, the effect of gas velocity is also included in the [P.sup.0.724] term. The net effect of gas velocity is positive in spite of the [F.sub.d.sup.-1.19] term. However, because of the -13.1[F.sub.d.sup.-0.902] term, the net exponent on gas velocity decreases with increasing gas velocity.

Equation (7) represents the bed expansion of fluidized beds (Choi et al., 1999). [H.sub.t], [H.sub.b] and [H.sub.mf] are the column height from the distributor plate to the gas exit in the freeboard, the expanded height of the dense bed, and the bed height at minimum fluidization, respectively. Equation (7) covers a range of particle diameters from 0.096 to 1.147 mm, particle densities from 1400 to 6200 kg/[m.sup.3], minimum fluidizing velocity of 0.0043-0.50 m/s, gas velocity from 0.015 to 1.4 m/s, and bed voidage from 0.45 to 0.78, including both the bubbling and turbulent fluidization flow regimes.

The decay constant, a, is obtained from Choi et al. (1997c):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where [U.sub.mf] is the minimum fluidizing velocity of the bed based on the specific surface mean diameter of bed particles, and dp is the particle diameter for which [K.sub.i.sup.* is determined, not the mean diameter of bed particles (Choi et al., 1999). Choi et al. (1997c) determined particle diameter by sieving. Equation (8) is based on column diameters from 0.05 to 0.4 m, particle diameters from 46 to 720 [micro]m, particle densities from 930 to 3050 kg/[m.sup.3], superficial gas velocities from 0.3 to 6.2 m/s and temperatures from 24 to 600[degrees]C. The above equations do not consider effects of interparticle adhesion forces and therefore are not intended for use when these forces are significant compared with gravitational and hydrodynamic forces.

RESULTS AND DISCUSSION

Predictions from the above new correlation are compared with experimental entrainment rates (Choi et al., 2001) of coarse particles having different mass fractions of these particles in the bed ([x.sub.i]), gas velocities, and particle diameters in Figure 1. These results are for room temperature with sand as the bed material. [U.sub.t] is the terminal settling velocity of particles of diameter [d.sub.p] in the figure. All superficial gas velocities were lower than the terminal velocity of the bottom size (0.3 mm) for the 0.363 mm size fraction. [K.sub.i.sup.*] decreases for [d.sub.p] = 0.363 mm with increasing [x.sub.i] in the range of gas velocity covered, as shown in Figure 1(a). At this condition, the proportion of fines decreased with increasing proportion of 0.363 mm size fraction in their bed particles (Choi et al., 2001). This trend means that [K.sub.i.sup.*] for the coarse particles ([d.sub.p]>[d.sub.crit]) increases with increasing mass fraction of fine particles ([d.sub.p]< [d.sub.crit]) in the bed, as noted by Choi et al. (2001). This indicates that the entrainment of coarse particles is promoted by transfer of momentum from the fine particles (Geldart et al., 1979; Geldart and Pope, 1983; Kwauk, 1992; Choi et al., 2001). It is also seen that the effect of the fines on the entrainment of the coarse particles decreased as the gas velocity increased. This occurred because the proportion of momentum gained from the fine particles decreased relative to that from the gas phase as U increased (Choi et al., 2001).

[FIGURE 1 OMITTED]

For the conditions of Figure 1b, the proportion of fines decreased with increasing the proportion of 0.256 mm size fraction in their bed particles. However, for the conditions of Figure 1c, the proportion of fines increased with increasing proportion of the 0.181 mm size fraction in their bed particles (Choi et al., 2001). Therefore, for the 0.256 mm and 0.181 mm size fractions, the trend of [K.sub.i.sup.*] shown in Figures 1b and 1c was practically the same as the trend showing [K.sub.i.sup.*] of coarse particles ([d.sub.p]> [d.sub.crit]) increasing with the proportion of fines ([d.sub.p]< [d.sub.crit]), as noted by Choi et al. (2001). The present correlation fits the measured entrainment rate of coarse particles well, increasing with more fines in the bed, while also giving a good representation of the influence of superficial gas velocity.

The present correlation predicts a lower entrainment rate of coarse sand particles than that measured by Geldart et al. (1979) (Figure 2) who employed sand-alumina mixtures of different relative proportions as bed materials at room temperature (solid density: 2650 kg/[m.sup.3] for sand, 1830 kg/[m.sup.3] for alumina; 98.7% by mass in the size range 125-355 m for sand, all less than 125 [micro]m for alumina). The deviation may result from the density difference between the alumina and the sand. Segregation tests with binary particle mixtures of different densities (Rowe and Nienow, 1976) show that lower-density particles tend to be enriched at the bed surface relative to the overall mean composition. The present correlation is not completely able to account for segregated beds of particles of different densities.

Figures 2 and 3 compare the present correlation and the entrainment rates measured by George and Grace (1981) and Choi et al. (1989) for columns of 0.25 m x 0.43 m x 3.0 m tall and 0.38 m i. d. x 9.1 m tall, respectively at room temperature, with sand as the bed material. Figure 3 covers a wide range of particle sizes (0.090-1.06 mm). The present correlation is seen to be in reasonably good agreement with the entrainment rate of coarse particles for different column sizes and a wide range of particle sizes. However, the effect of column diameter and height could not be shown in this study because the size distribution of bed particles, column diameter and column height all differed in the studies considered.

[FIGURE 2 OMITTED]

Figure 4 shows the effect of superficial gas velocity on the entrainment of coarse particles for different particle densities and diameters at three temperatures, as measured by Choi et al. (1997a, 1998). The experimental data are compared with predictions from the correlation. As the gas velocity increases, the entrainment rate increases due to increased drag on the particles, but the slope decreases. The present correlation is seen to describe the gas velocity effect reasonably well in both the cold and higher temperature fluidized beds for various particle sizes and densities.

Figure 5 depicts the effect of temperature on the particle entrainment rate for different particle sizes and densities at different superficial gas velocities as reported by Choi et al. (1997a, 1998). In previous studies (Choi et al., 1997a, 1998), the measured entrainment rate increased after an initial decrease as temperature increased. This is believed to be caused by decreasing gas density and increasing gas viscosity with increasing temperature. As shown in Figure 5, the present correlation is able to describe the temperature effect on the entrainment rate of coarse particles reasonably well. However, it shows some deviation from the measured value at 500 and 600[degrees]C for 0.256 and 0.363 mm particles because the minimum entrainment rate is predicted at a higher temperature than measured. Considering the limitation of previous studies, more experimental data are needed to enable a better correlation at high temperatures.

In summary, for negligible interparticle forces, the present correlation give good predictions of the entrainment rate of coarse particles at the exit of gas-fluidized beds over a wide experimental range: particle diameter 0.090-1.1 mm, particle density 2500-6200 kg/[m.sup.3], superficial gas velocity 0.38-2.44 m/s, temperature 12-600[degrees]C, column diameter 0.1-0.38 m and column height 1.97-9.1 m. Given the considerable scatter in entrainment rates measured by different groups and the wide deviations between commonly used correlations and experimental data, our new equation appears to be promising as a mean of predicting the entrainment of coarse particles.

[FIGURE 3 OMITTED]

CONCLUSIONS

A comprehensive correlation to predict the entrainment rate of coarse particles at the exit of dense fluidized beds is developed based on extensive available experimental data. The correlation successfully includes the effects of gas velocity, temperature, particle density, particle size distribution, column diameter and height. The entrainment rate of coarse particles increases with increasing gas velocity and proportion of fine particles in the bed. The effect of the fine particles is expressed in terms of the upward momentum of fine particles per unit mass at a specified gas velocity. Fines play a lesser role on the entrainment of coarse particles as the gas velocity increases. The proposed correlation correctly predicts that the entrainment rate of coarse particles increases after initially decreasing with increasing temperature.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

ACKNOWLEDGEMENT

This work was supported by Konkuk University, Korea.

NOMENCLATURE

a decay constant ([m.sup.-1])
Ar Archimedes number, [gd.sub.p.sup.3][[rho].sub.g]
 ([[rho].sub.p]-[[rho].sub.g])/[[micro].sup.2] (-)
[C.sub.d] drag coefficient based on superficial gas velocity (-)
[d.sub.crit] [d.sub.p] of particle with terminal velocity equal to
 U (m)
[d.sub.p] particle diameter (m)
[D.sub.t] column diameter (m)
[F.sub.d] drag force on particle per projected area (Pa)
[F.sub.g] (gravity minus buoyancy force) per projected area of
 particle (Pa)
g gravitational acceleration (9.8 m/[s.sup.2])
[H.sub.b] bed height (m)
[H.sub.mf] bed height at minimum fluidization (m)
[H.sub.t] column height (m)
[K.sub.i] entrainment rate of particles of size dp
 (kg/[m.sup.2]s)
[K.sub.ih] * cluster flux of entrained particles of size
 [d.sub.p] (kg/[m.sup.2]s)
[K.sub.ih]* dispersed non-cluster flux of entrained particles of
 size [d.sub.p] or elutriation rate constant of
 particles of size [d.sub.p] above TDH (kg/[m.sup.2]s)
P upward momentum of fine particles per unit mass of
 entire bed material (m/s)
[Re.sub.p] particle Reynolds number, [d.sub.p]U [[rho].sub.g]/
 [micro](-)
T temperature ([degrees]C)
TDH transport disengaging height, distance above bed
 surface beyond which entrainment rate is virtually
 constant (m)
U superficial gas velocity (m/s)
[U.sub.mf] minimum fluidization velocity of bed particles (m/s)
[U.sub.t] particle terminal velocity (m/s)
x mass fraction of a size-cut of fine particles
 ([d.sub.p] < [d.sub.crit]) in the bed (-)
[x.sub.i] mass fraction of a size-cut of coarse particles
 ([d.sub.p]>[d.sub.crit]) in the bed (-)

Greek Symbols

[micro] gas viscosity (Pa s)
[[rho].sub.g] gas density (kg/[m.sup.3])
[[rho].sub.p] particle density (kg/[m.sup.3])

Manuscript received June 5, 2006; revised manuscript received September 4, 2006; accepted for publication January 5, 2007.

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* Author to whom correspondence may be addressed. E-mail address: choijhoo@konkuk.ac.kr

Jeong-Hoo Choi [1] *, Sang Done Kim [2] and John R. Grace [3]

[1.] Department of Chemical Engineering, Konkuk University, Seoul 143-701, Korea

[2.] Department of Chemical and Biomolecular Engineering, KAIST, Daejeon 305-701, Korea

[3.] Department of Chemical and Biological Engineering, The University of British Columbia, Vancouver, BC, Canada V6T 1Z3

Table 1. Summary of previous studies used for present correlation for
[K.sub.i] *

Authors Particles [rho].sub.p] [d.p[rho] [mm]
 [kg/[m.sup.3]]
George and Sand 2630 0.030-
Grace (1981) 0.272 x0.43

Choi et al. Sand 2630 0.053-
(1989) 1.06

Choi et al. Sand 2598 0.075-
(1997b) 0.425

Choi et al. Sand 2509 0.075-
(1997a, 1998) 0.425

 Emery 3981 0.075-
 0.425

 Cast 6158 0.075-
 iron 0.425

Choi et al. Sand 2465-2562 0.075-
2001 0.6

Authors Column size [H.sub.t] [U.sub.mf] U [ms/s]
 [m] [m] [m/s]
George and 0.25 3.0 0.015 0.2-1.2
Grace (1981)

Choi et al. 0.38 i.d. 9.1 0.11 0.38-2.01
(1989) 0.086 0.95-2.44

Choi et al. 0.1 i.d. 2.25 0.09 0.81-2.8
(1997b)

Choi et al. 0.1 i.d. 1.97 0.009- 0.65-2.3
(1997a, 1998) 0.093

 0.1 i.d. 1.97 0.032- 0.8-2.2
 0.043

 0.1 i.d. 1.97 0.039- 0.8-2.2
 0.060

Choi et al. 0.1 i.d. 1.97 0.0182- 0.8-2.2
2001 0.0745

Authors T [[degrees]C]

George and 27, 162, 172
Grace (1981)

Choi et al. 30, 45
(1989)

Choi et al. 29
(1997b)

Choi et al. 20-600
(1997a, 1998)

 12-600

 12-600

Choi et al. 20-26
2001