Characteristics of flow in wet conical spouted beds of unequal-sized spherical Particles

FLUID DYNAMICS; HEAT AND MASS TRANSFER; AND OTHER TOPICS

Characteristics of flow in wet conical spouted beds of unequal-sized spherical Particles

M. S. Bacelos; M. L. Passos and J. T. Freire^*

Departamento de Engenharia Química, Centro de Secagem de Pastas, Suspensões e Sementes, Universidade Federal de São Carlos, Fax: +(55) (16) 3351-8266, Rod. Washington Luiz km 235, Cx.P 676, CEP: 13565-905, São Carlos, SP, Brazil E-mail: [e-mail]

ABSTRACT

Interparticle forces, developed in wet spouted beds composed of a mixture of spherical particles with different size distributions, intensify particle segregation mechanisms interfering in gas distribution inside the bed and, consequently, in the spouting flow characteristics. Therefore, this paper is aimed at describing the effect of interparticle forces on the air-solid flow distribution in conical spouted beds of unequal-sized particles coated by a thin glycerol film. Experimental results show that both the minimum spouting airflow rate and the minimum spouting pressure drop decrease as the amount of glycerol added to the bed increases. In addition, simulated results of the annular air velocity along the bed height showed that, at the base of the column, the radial component of the inertial force is high enough to break liquid bridges between particles and carry these particles out along the spout. Moreover, as the glycerol concentration increases, the spout diameter increases along the bed height. Such changes in the air-solid flow can maintain the spouting regime for higher glycerol concentrations as shown by experimental data.

Keywords: Wet conical spouted bed; Minimum spouting conditions; Cohesive forces; Unequal-sized spherical particles.

INTRODUCTION

Over the fifty years since its discovery, researchers have tried to apply the spouted bed technique in many industrial processes, such as pneumatic conveyance (Ferreira and Freire, 1992; Ferreira and Narimatsu, 2001; Costa et al., 2004); catalytic polymerization (Bilbao et al., 1987); pyrolysis of sawdust (Aguado et al., 2000); drying of grains (Mathur and Gishler, 1955; Mathur and Epstein, 1974; Devahastin et al., 1998), pastes or suspensions (Pham, 1983; Barret and Fane, 1990; Passos et al., 1997; Correa et al., 2000, 2004a,b; Spitzner Neto et al., 2001; Souza and Oliveira, 2005; Cordeiro and Oliveira, 2005); coating seeds (Duarte et al., 2004) and tablets (Publio and Oliveira, 2004); producing fuel powders (Passos et al., 2004), pharmaceutical powders (Jono et al., 2000) and fertilizers (Ayub et al., 2001).

In small-scale production, the conical spouted bed dryer is pointed to as a more promising technique for processing viscous and thermal sensitive suspensions (Spitzner Neto et al., 2002; Trindade et al., 2004; Bacelos et al., 2005, 2007) than other techniques, such as fluidization (Daleffe and Freire, 2004) or spray-drying (Birchal et. al., 2005, 2006).

In conical spouted beds, high rates of heat and mass transfer coupled with the intensive attrition between particles induce high shear rates in the thin suspension layer that coats particles, reducing its apparent viscosity and making feasible the drying operation for powder production (Passos et al., 2004). Moreover, in the drying of pastes or suspensions some problems that reduce powder production, such as particle agglomeration and particle adhesion to dryer walls, must be overcome. Thus, in order to produce optimum values of either paste or moisture material content in the bed, Spitzner Neto et al. (2001, 2002) verified that monitoring the bed suspension feed can control these conditions. Passos et al. (2004) proposed and proved that the problem of particle agglomeration can be well overcome by feeding the suspension into the conical spouted bed of inert particles intermittently under a rigid control of outlet air temperature. For a pseudoplastic suspension, such as black liquor, this rigid control maintains the liquor viscosity low enough to make feasible the continuous production of powdery fuel from eucalyptus and bamboo black liquor.

In spite of the many potential applications of the conical spouted bed technique, as pointed out in the literature (Mathur and Epstein, 1974; Passos et al., 1997; Devahastin et al., 1998) the thermal performance of air is still low for industrial applications. Recently, Fernandes and Correa (2005) showed how to recycle the exhaust gas and recover its sensitive heat in order to increase the energy efficiency of the operation of drying of pastes in conical spouted beds.

Experimental results together with validated models to predict momentum, heat and mass transfer in porous media allow researchers to simulate optimum routes for drying and powdering suspensions. Therefore, efforts have been made to describe fluid flow characteristics in conical spouted beds (Charbel et al., 1999; Spitzner Neto et al., 2001; Costa Jr. et al., 2004; Duarte et al., 2005; Bacelos et al., 2005) as well as drying mechanisms (Passos et al., 2004; Trindade et al., 2004; Costa Jr. et al., 2006).

Initially, Costa Jr. et al. (2001) proposed a simulator predicting the drying of diluted Newtonian suspensions in conical spout-fluid beds, ignoring the effect of interparticle forces. This simulator consists of three interactive modules: gas flow (module 1), solid flow (module 2) and drying (module 3). According to these authors, the main feature of modules 1 and 2 is to predict gas and solids flow behavior as a function of the spout formation mechanism, which is specified by range of parameter (A) as shown in Table 1. This parameter (A) relates the minimum inlet energy required to form the spout to the minimum frictional energy lost to maintain the spout throughout its area (Littman and Morgan, 1988):

where r_g and r_s are respectively the gas and solid density, U_mf is the minimum fluidization gas superficial velocity, U_T is the terminal particle velocity, g is gravity force and d_i is the inlet nozzle diameter.

Output variables from module 1 and module 2 enter into module 3, where mass and energy balance equations are developed for solids, gas and suspension. These equations are solved along the bed height and together with the drying kinetics correlation for the suspension. These three modules supply all data needed for describing the operation of drying diluted suspension.

Charbel et al. (1999) showed that the changes in gas-flow characteristics are related to the failure mechanism of the bed and depend on geometric parameters alone. Moreover, these authors verified that for each specified range of parameter A presented in Table 1, there is a relation between parameter A, type of spout formation mechanism, spout shape and gas flow behavior. Such a relation was considered in the simulator proposed by Costa Jr. et al. (2001), which estimates either spout diameter or gas superficial velocity as a function of axial direction for different ranges of parameter A.

Trindade et al. (2004) proposed a methodology to introduce into this simulator the effect of interaction forces between inert particles coated by suspension. This technique is based on correlations developed by Passos and Mujumdar (2000), which consider that the effect of interparticle forces can be quantified by two specific curves: (Qmj/Qmj0) vs. p1 = e(1-e_ag)/ e_ag and (DPmj/ DPmj0) vs. p2 = p₁f /d_p. Then, by changing the type and amount of suspension injected into the bed of inert particles, the minimum spouting gas flow rate (Qmj) changes in relation to the one obtained in the bed of dry particles (Qmj0). Consequently, this induces changes in the minimum spouting pressure drop (D Pmj) in relation to that obtained in the bed of dry particles (D Pmj0). The type of interaction forces is directly related to the suspension properties, but the strength of these forces depends on the thickness of the suspension layer that covers the particle surface as well as the bed failure mechanism during spout formation (Charbel et al., 1999).

As reported in the literature, changes in the behavior of fluid dynamic variables, such as D Pmj and Qmj, are associated with gas flow rate crossing the spout-annulus interface. Moreover, any change in the inlet gas flow rate produces modifications in the shape of spout diameter along the bed height. Therefore, Trindade et al. (2004) used two corrective functions with adjustable parameters to modify the simulator. The first one, f1, depends on p1 and z/H and is applied to the spout-diameter model equation. The second corrective function, f2, depends on p2 and z/H (in the same way that f1 depends on p1 and z/H) and is applied to the gas flow rate crossing the spout-annulus interface. These authors tested this methodology to include the effect of interparticle forces between polypropylene particles coated by Eucalyptus black liquor in a conical spouted bed with A > 0.02. Based on photographic results, they showed that the spout closes as the amount of suspension injected into the bed is increased. A simulator using optimization routes to define either the type of these corrective functions (f₁, f₂) or their adjustable parameters can effectively predict this behavior.

Recently, Costa Jr. et al. (2006) have applied this methodology to describe flow characteristics of a conical spouted bed of spherical glass beads at A = 0.015 (transitional bed failure mechanism 0.014 < A < 0.02) during the drying of egg emulsion. Their simulated results show good agreement with the characteristics of bed behavior reported during experiments.

However, for more general purpose, this modified simulator should be applied for other column dimensions (at least A < 0.014), unequal-sized particles and different suspensions. Note that wet beds of unequal-sized particles may simulate quite well the effect of interparticle binding forces on the drying of suspension since, in practice, the agglomeration phenomena should cause nonuniformity in the equal-sized particle packing interfering with gas-solid flow characteristics. Using unequal-sized particles as inert particles, such interference is intensified.

Therefore, as a first step, this paper is aimed at applying this methodology to describe the effect of interparticle forces on airflow distribution in a wet conical spouted bed of spherical unequal-sized particles. To simulate the effect of a highly viscous suspension, glycerol, a nonvolatile liquid, is used. As shown in Table 2, this Newtonian liquid has physical properties similar to those of pseudoplastic suspensions without the effect of the moisture concentration on these properties.

INTERPARTICLE FORCES IN CONICAL SPOUTED BEDS OF UNEQUAL-SIZED PARTICLES

The spouting regime in conical spouted beds is achieved as the inertial force from the inlet gas jet is high enough to break the packed bed structure, usually in the central column axis, and to carry out particles, forming a fountain above the bed. This means that this inertial force on particles must overcome gravity and any other interparticle force. Therefore, due to the action of the inertial inlet gas force, a continuous and cyclic particle movement is established inside the column, characterizing the spouting regime. This regime consists of a central core dilute phase (called spout) moving upward; a fountain, formed of particles that leave the spout and rain back onto the annulus surface; and an annular dense phase, characterized by the sliding motion of particles in countercurrent to the gas flow.

For conical spouted beds, San José et al. (1995) presented the velocity vector map as it is schematically shown in Figure 1. At a given axial bed position (z), at the spout, the air velocity vectors (i.e., the interstitial ones) present a quasi-flat profile with r-direction and become an approximately parabolic profile with r-direction at the annulus. In addition, at a given radial bed position (r), the velocity vectors decrease with an increase in z-direction. Thus, the highest radial and axial velocity components are seen at the bottom of the bed, as shown in Figure 1.

According to Charbel et al. (1999), especially for the conical spouted bed used in this research with A = 0.015 (i.e., in a range of 0.014<A<0.02), there is a characteristic flow behavior, which was previously shown in Table 1. Such behavior can be explained by the air inertial force attained in the spouting regime. The air injected vertically through an inlet nozzle diameter flares out into the annulus as it travels upward (i.e., at air velocity v =1.1v_mj, about 40-54% of the air injected into the bed crosses into the annulus region [see Mathur and Epstein, 1974; Charbel et al., 1999]). At the bottom of the bed, the airflow behavior is due to the contribution of both radial and axial air velocity components in the inertial force. Thus, the axial inertial force component shears the particle layer at the spout-annulus interface and the radial one presses the assembly firmly, enlarging the spout region. On the other hand, as the air travels upward in the bed carrying particles, it loses energy by friction with solids, decreasing its inertial force components. This can be proved by the decrease in magnitude of air velocity vectors as the z-direction is increased (see Figure 1).

Moreover, as a conical spouted bed characteristic, the annulus area increases with the increase in bed height, inducing higher resistance to air crossing this interface in the direction of the annular region. Therefore, at a given bed axial position (z), enlargement of the spout starts to decrease, so that, it narrows back. Thereafter the spout remains practically constant close to the top of the bed as shown in Table 1. In addition, note that, as liquid is added to the bed, this spout enlargement with the axial direction becomes more pronounced (see point b in Figure 2). Such behavior is strongly dependent on interparticle forces developed between wet particles under the minimum spouting condition. These forces interfere with the magnitude of inertial force at the base and top of the column (Bacelos et al., 2007).

In conical spouted beds of unequal-sized particles, the particle size segregation can occur as schematized in Figure 2. This phenomenon is predominantly governed by the action of the particle inertia and the gravitational force. Note that, in the fountain region, particles move approximately in parabolic trajectories before falling onto the top of the annular region. As shown in Figure 2, the smaller particles reach the top of the annulus at regions near the column wall as they travel longer radial distances than those traveled by the coarser ones. This solids flow behavior in the bed explains the formation of liquid bridges between particles of different sizes. At the spout-annulus interface, liquid bridges exist between coarser particles; however, as liquid moves radially outward in the bed, liquid bridges can also occur between unequal-sized particles and in regions close to the column wall, liquid bridges between smaller particles can be found.

As reported in the literature (see Seville et al., 2000; Hotta et al., 1974), forces on bridges such as capillary and boundary forces (F_cb), increase with the increase in the particle diameter. For two given unequal particles on the bridge, F_cb is stronger the lower the difference between particle diameters (Bacelos et al., 2007). The viscous force on particle (F_v) depends on particle size as well as the magnitude of axial air velocity, as this can induce a relative movement between particles on bridges. Thus, due to the particle size and air-flow behavior in this conical spouted bed of unequal-sized particles shown in Figure 2, the total interparticle force (F = F_v+F_cb) increases radially from the column wall to the spout-annulus interface and axially from the top to bottom of the bed.

Finally, as shown in Figure 2, concentrations of coarser particles (i.e., those of highly cohesive liquid bridges) are higher at the spout-annulus interface, and then a greater inertial force is required from the air jet to break off these liquid bridges. Consequently, an increase in minimum spouting velocity is required to maintain the spouting regime. As noted, higher interparticle forces are achieved at the spout-annulus interface, increasing bed resistance to air crossing this interface into the annulus region. Therefore, the airflow into the spout is eased (i.e., less particle cross flow from annulus to spout), decreasing the minimum spouting pressure drop significantly for distributions containing a high weight fraction of coarse particles.

EXPERIMENTAL METHODOLOGY

Material and Apparatus

Table 3 shows the type and dimensions of the conical spouted bed column and inert particles used in this work in comparison to those used by Trindade et al. (2004) and analyzed by Costa Jr. et al. (2006).

As noted in Table 3, the inert particles are a mixture of spheres with three different diameters (small – d_pS = 2.18 mm, medium – d_pM = 2.58 mm and large – d_pB = 3.67 mm) in the following mass proportions: 22 % small, 36 % medium and 42 % large. This results in a flat size distribution (i.e., a linear relationship between particle mass fraction and diameter) with a Sauter diameter (average diameter) equal to that of the medium-sized particles (d_S = 2.58 mm), as reported by Bacelos and Freire (2005, 2006).

Figure 3 shows the detailed experimental apparatus used in this work. This comprises a conical spouted bed with the dimension specified in Table 3, an air compressor to blow air into the bed, a flow rate Venturi meter to measure the air inlet flow rate and manometers to measure the pressure drop along the bed. In addition, pressure transducers are used to capture analogical pressure signals of both bed and flow rate Venturi meter. Data are logged and processed by a supervisory system, which consists of a conditioning module, an acquisition board and software developed in the graphic program language, using Labview 7.0 by National Instruments.

Experimental Procedure

Glycerol, used to simulate the presence of paste or suspensions in the bed, creates the cohesion between inert particles. A gradual buret is employed to add a known volume of glycerol to the bed, which is already in stable spouting regime. The addition of liquid under this condition assures a good homogeneity for liquid distribution on inert particles. For each given experimental condition, a measuring tape (in millimeters) fixed to the column wall is used to measure bed height. Thus, at a given glycerol concentration, the overall bed porosity, e_a, in the annular region is determined. The dry-based bed porosity in the annulus region, e_ag, is obtained by subtracting (v_l/v_bed) from the e_a value.

Moreover, by keeping constant the mass of particles (M = 7 kg) and changing the glycerol concentration (0 < v_l/v_p< 0.008), the DP (pressure drop) vs. Q (airflow rate) curve is determined by decreasing Q for both dry and wet spouted beds. Based on these curves, the dimensionless ratios Q_mj/Q_mj0 and DP_mj/DP_mj0 can be evaluated as a function of p₁ and p₂.

Data Simulation

As experimental data on Q_mj/Q_mj0 vs. p₁ and DP_mj/DP_mj0 vs. p₂ have been evaluated, the two corrective functions, f1and f2 , are primarily chosen to be the same as those reported by Trindade et al. (2004):

As mentioned earlier, f1 is applied to the spout diameter model equation and depends on p1 and z/H and f2 corrects the rate of airflow crossing the spout-annulus interface and depends on p2 and z/H. Note that these two functions reach unit values in beds without suspension (p₁ = p₂ = 0). The procedure used to determine the six adjustable parameters, a₁, a₂, b₁, b₂ and c₁, c₂, is the one proposed by Trindade et al. (2004) by minimizing the objective function defined as

As suggested by these authors, the heuristic particle swarm optimization method, developed by Kennedy and Eberhart (1995), is chosen to optimize f_obj, as it is robust and efficient enough to guarantee that the adjustable parameter values obtained are those representative of the global minimum of the objective function.

RESULTS AND DISCUSSION

Table 4 shows the optimum values for the six parameters of the corrective functions in Equation 2. Based on these corrective functions of this work, the spouted bed operational variables, Q_mj, DP_mj, r_mj, U_a and P_j, can be simulated.

Figures 4a and 4b contain the experimental and simulated curves of operational spouted bed variables: the dimensionless minimum spouting flow rate ratio (Q_mj/Q_mj0) as a function of p₁ and the dimensionless minimum spouting pressure drop ratio (DP_mj/DP_mj0) as a function of p₂. Based on the statistical analysis, the correlation coefficient (r) between these experimental and simulated data is 0.95 for the Q_mj/Q_mj0 vs. p₁ curve and 0.85 for the (DP_mj/DP_mj0) vs. p₂ curve. This indicates that the model with the corrective functions in Table 4 is suitable for predicting quite well the Q_mj/Q_mj0 vs. p₁ curve (mean error = 0.075 %) and reasonably well the (DP_mj/DP_mj0) vs. p₂ curve (mean error = -4.87 %). In addition, simulated errors are lower than experimental ones and acceptable statistically within the 95% of confidential interval. Therefore, one can conclude that this model with the corrective functions in Table 4 can be used to estimate the other flow variables as a function of the amount of glycerol in the bed.

Furthermore, as noted in Figures 4a and 4b, Q_mj/Q_mj0 and DP_mj/DP_mj0 decrease with the increase in glycerol concentration (e.g. as p₁ and p₂ rise). The behavior of the Q_mj/Q_mj0 curve is due to the formation of a liquid bridge and it increases as liquid binding forces between particles increase (Bacelos et al., 2007). These contribute to reduce the airflow rate crossing the spout-annulus interface and the particle motion in the annulus region. Thus, with the increase in glycerol concentration, a lower flow rate is needed to maintain the stable spouting regime as shown in Figure 4a. The trends of the DP_mj/DP_mj0 vs. p₂ curve can be explained by the increase in bed resistance to the passage of air through the spout-annulus interface in the presence of glycerol. This prevents part of the inlet air from crossing the interface, causing a reduction in bed pressure drop, as shown in Figure 4b. Therefore, the simulated P_j(z) curves, shown in Figure 5, corroborate that the increase in glycerol concentration in the bed results in a decrease in gauge pressure at the spout-annulus interface. This indicates that less air crosses this boundary, in agreement with the DP_mj/DP_mj0 experimental trend behavior.

Figures 6a and 6b show the simulated profiles of the superficial air velocity in the annulus region, U_a(z), and of the spout radius, r_mj(z), as a function of glycerol concentration. In the range of 0 < v_l/v_p< 0.003, these profiles are similar (not shown in Figs. 6a and 6b). Remarkable changes in the shape of the U_a(z) curve are reported for v_l/v_p> 0.004. Note that, for a given glycerol concentration, U_a(z) increases until reaching a maximum point close to z/H < 0.125 and thereafter begins decreasing at z/H > 0.125. Such behavior can be explained due to the enlargement of the spout radius (r_mj), in spite of the decrease in airflow rate from spout to annulus region. As shown in Figure 6b, the enlargement of spout radius (r_mj) is more pronounced from z/H = 0 to z/H = 0.125 than for the radius of the annular region. Moreover, this is a characteristic shape for conical spouted beds in a specific range of values for parameter A (0.014 < A < 0.02) presented in Table 1.

In spite of the fact that the U_a(z) and r_mj(z) profiles are only simulated, these are in agreement with experimental trends reported for the values for the magnitude of liquid binding forces on bridges in conical spouted beds, as shown by the theory of cohesive forces. According to previous work (see Bacelos et al., 2007), the highest values for magnitude of these interparticle forces are found at the spout-annulus interface, as high values of viscous force are attained. Thus, only at the base of the column, the radial air velocity component has enough radial inertial force components to break off the liquid bridges between inert particles and carry them along the spout until reaching the bed surface. This behavior not only maintains the stable spouting regime, but probably enlarges the spout at the column base, as shown in Figure 6b. Moreover, in practical applications such as in coating of pellets and in drying of pastes, these expectations imply that particle agglomeration will start at z/H > 0.125, rather than at the column base, as this latter region has high values of U_a (see Figure 6a)

By comparing this set of data to one reported for egg emulsion (see Costa Jr. et al., 2006), it can be noted that both sets of data are in agreement, as their spout radius increases with the increase in liquid concentration (glycerol or egg emulsion). This implies an increase in U_a at the column base and its reduction at the top (see Figure 6a), favoring particle agglomeration on the bed surface when the suspension is fed in the column top alone, as verified by Spitzner Neto et al. (2002) and by Bacelos et al. (2005).

On the other hand, data obtained here differ from those reported for eucalyptus black liquor (see Costa Jr. et al., 2006), as in this latter the spout radius narrows with the increase in concentration of eucalyptus black liquor added to the bed. As well, U_a(z) becomes higher at top of the column than at the base. This indicates that particle agglomeration starts near the base of the column due to the reduction in U_a(z) in this region, as pointed out by Trindade et al. (2004).

From these comparisons, one can see that the data presented are consistent with those for egg emulsion, which are obtained using the same equipment and inert particles. Consequently, the same transitional bed failure mechanism (0.014 < A < 0.02) is attained in both set of experiments, in spite of the differences between these pastes. Thus, these sets of data differ from those on eucalyptus black liquor because, in the conical spouted bed of propylene inert particles, a different type of bed failure mechanism (A > 0.02) is obtained (see Table 1).

CONCLUSIONS

Based on these experimental data, one can note there is evidence showing that the behavior of conical spouted bed fluid dynamics changes significantly with mixture of particles, pastes or suspensions. As has been reported in the literature, such changes can be predicted well taking into account the effect of cohesive forces on the fluid dynamics behavior of the conical spouted bed. Therefore, the previous model has been demonstrated to be effective in predicting spouted bed operational variables. As shown in the Discussion section model prediction plays an important role in processes of drying of pastes. It allows determination of spout shape and annular superficial velocity profiles as a function of the suspension added to the spouted bed of inert particles, and shows where in the bed particle agglomeration should start. This permits researchers to better understand both fluid dynamics and drying phenomena in wet spouted beds. Moreover, by comparing the model verification with the same bed and with different suspensions, it can be noted that the type of bed failure mechanism dictates the spouting airflow behavior in the wet bed, whereas the paste characterizes the magnitude of change in the operational spouted bed variables.

ACKNOWLEDGEMENTS

The authors express their gratitude to Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for its financial support in carrying out this research, and especially to Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for the D.Sc. and Ph.D. scholarship respectively awarded M. S. Bacelos.

NOMENCLATURE

ai, bi, ci

adjustable parameters of fi

(-)

A

parameter defined in Equation 1

(-)

C_ss

solid concentration (% in mass)

(-)

d_b

diameter of the base of the column

m

d_c

diameter of the top of the column

m

d_I

inlet nozzle diameter

m

dj

spout diameter

m

d_mj

spout diameter under minimum condition

m

d_p

particle diameter

m

d_pI

particle diameter of I-type particles(I = S – small; I = M – medium; I = B – large; see Tab.2)

m

d_S

Sauter particle diameter

m

e

suspension layer thickness = (dp/2)[(1+V)^1/3 -1]

µm

f_i, f_obj

corrective and objective functions respectively

(-)

g

gravity acceleration

m/s²

H

bed height

m

p₁

= e (1- e_ag)/ e_ag

mm

p₂

= e (j /dp) (1- e_ag)/ e_ag

(-)

Q_mj

gas flow rate at minimum spouting

m³/s

Q_mj0

gas flow rate at minimum spouting for dry particles

m³/s

r

correlation coefficient

(-)

r_mj

spout radius at minimum spouting (= dmj/2)

m

R_c

column radius

m

U_a

gas superficial velocity in the annulus region

m/s

Uj

gas superficial velocity in the spout region

m/s

Umf

gas superficial velocity at minimum fluidization

m/s

UT

terminal particle velocity

m/s

v_bed

volume of the bed

m³

v_l

volume of suspension

m³

_Vp

volume of particle

m³

X_pI

mass fraction of I-type particles I= S – small, I= M – medium, I= B – large)

(-)

z

axial coordinate of the column

m

Greek Symbols

a

included column angle

º

DPmj

minimum spouting pressure drop

Pa

DPmj0

minimum spouting pressure drop for dry particles

Pa

e_a

overall bed porosity

(-)

e_a0

bed porosity in the annulus for dry bed

(-)

e_ag

dry-based porosity in the annulus

(-)

(Received: January 16, 2006 ; Accepted : September 26, 2007)

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*

To whow correspondence should be addressed

Publication Dates

• Publication in this collection
  28 Apr 2008
• Date of issue
  Mar 2008

History

• Received
  16 Jan 2006
• Accepted
  26 Sept 2007