10.1.1.112.617

Accepted for publication in Powder Technology

1

Simulation of Particles and Gas Flow Behavior in the Riser Sectionof a Circulating Fluidized Bed Using the Kinetic Theory

Approach for the Particulate Phase

S. Benyahia a, H. Arastoopour a,* , T.M. Knowlton b, and H. Massah ca Illinois Institute of Technology, Chicago, IL 60616b Particulate Solid Research Inc., Chicago, IL. 60632

c Fluent, Inc., Lebanon, NH 03766

Abstract

Gas/particle flow behavior in the riser section of a circulating fluidized bed (CFB)

was simulated using a computational fluid dynamics (CFD) package by Fluent Inc. Fluid

Catalytic Cracking (FCC) particles and air were used as the solid and gas phases,

respectively.

A two-dimensional, transient and isothermal flow was simulated for the

continuous phase (air) and the dispersed phase (solid particles). Conservation equations

of mass and momentum for each phase were solved using the finite volume numerical

technique. This approach treats each phase separately, and the link between the gas and

particle phases is through drag, turbulence, or energy dissipation due to particle

fluctuation.

Gas and particle flow profiles were obtained for velocity, volume fraction,

pressure, and turbulence parameters for each phase. The computational values agreed

reasonably well with the available experimental results. Our computational results

showed that the inlet and outlet design have significant effects on the overall gas and

solid flow patterns and cluster formations in the riser. However, the effect of the initial

condition tended to disappear after some time. The main frequencies of oscillations of the

* Corresponding author. Tel.: +1-312-567-3038; Fax: +1-312-567-8874; E-mail:[email protected]


2

system were obtained in different regions of the riser. These frequencies are important in

comparing the computational results with the available time-averaged experimental data.

Keywords: Kinetic theory; Numerical simulation; Circulating beds; Fluidization; FCC particles

Introduction

Gas/solid flow systems are an essential part of many chemical processes, and

contributions to the understanding of the behavior of such flow systems can significantly

enhances the design and, in turn, the productivity of such processes.

The first step in the fundamental understanding of fluidization is usually

attributed to Davidson [1] for his analysis of a single bubble motion in an infinite fluid

bed. The development of Davidson's model was carried by Jackson [2], Murray [3],

Pigford and Baron [4], and Soo [5,6]. Solid viscosity and pressure are closures that can

either be determined experimentally or theoretically by using more fundamental

approaches. In any case, solid viscosity is an important parameter to include in

simulations of the riser section of a circulating fluidized bed (CFB) since wall effects can

be significant.

In recent years, the dynamics of inter-particle collisions based on the concept of

the work of Chapman and Cowling [7] have been considered by Soo [5], Jenkins and

Savage [8], Lun et al. [9], Johnson and Jackson [10], Sinclair and Jackson [11], and Ding

and Gidaspow [12]. The kinetic theory approach uses a one equation model to determine

the turbulent kinetic energy of the particles (granular temperature) and assumes either a

Maxwellian distribution for the particles, or a non-Maxwellian distribution [13], which

considers both dilute and dense cases. Moreover, Kim and Arastoopour [14] considered


3

the cohesiveness of the particles and extended the kinetic theory model for cohesive

particles. The kinetic theory approach for granular flow allows the determination of the

pressure and viscosity of the solid in place of empirical relations.

The interaction between gas and particles has been considered in the form of drag

force in most of the publications in the literature. Louge et al. [15] were probably the first

to include both gas turbulence and particle-particle interaction. Recently, most of the

investigations have focused on very dilute systems with gas turbulence and, in some

cases, large-scale particulate phase fluctuations [16, 17, 18]. This could be due to the fact

that most gas phase turbulence models [19] have been developed for single-phase flow,

and their application could most probably be limited to a very dilute flow of particles. For

the case of dense particle flows, the interaction between gas turbulence and particle

fluctuation is significant and, in some cases, the flow tends to be similar to laminar flow.

The kinetic theory formulation includes an interaction term between gas turbulence and

particle fluctuations in the granular temperature equation [13]. However, the validity of

this expression to describe the interactions between gas turbulence and particle

fluctuation is yet to be investigated.

FCC units usually operate at high solid throughput of 400-1200 kg/m2s. The riser

section of a CFB can be characterized by a core-annular flow regime with particles

forming structures at the walls in the form of clusters and sheets. Lateral segregation in

high density CFB's was probably first found experimentally by Bader et al. [20] who

simultaneously measured the solid density and mass flux across the diameter of the riser.

Simulation of lateral segregation in the riser is possible only by using at least a two-

dimensional model. Tsuo and Gidaspow [21] were among the first to predict the core-


4

annular flow regime using multiphase flow equations. They successfully used an

empirical relation for the solid viscosity that was later used by Bing [22] and Benyahia et

al. [23]. However, theoretically derived solid viscosity and pressure are the preferred

relations to fully predict the gas/solid in the riser.

Recently, Knowlton et al. [24] presented a challenge problem at the fluidization

VIII conference to compare different CFB hydrodynamic models. The experimental data

were produced mainly at the Particulate Solid Research Inc. (PSRI). These experimental

data included the solid concentration and flux at a certain location of the riser, as well as

the axial gas pressure drop. Benyahia et al. [23] conducted a simulation using an

empirical relation for solid viscosity and compared their results with the pilot-scale

experimental data. The analysis in this paper will include the use of the kinetic theory for

solid particles; the main goal is to investigate the effect of initial conditions, inlet and

outlet design, and increased riser diameter on the overall gas/solid mixing in the riser.

Furthermore, frequency analyses of the oscillatory gas/solid flow were performed. This

enables an accurate comparison of the time-averaged computational results with the

experimental data.

Gas/Solid Multiphase Flow Model Description

The complexity of the hydrodynamic equations makes obtaining an analytical

solution very unlikely. Therefore, numerical solutions should be considered. In the Fluent

computer program that was used in this study, the governing equations were discretized

using the finite volume technique [25]. The discretized equations, along with the initial

and boundary conditions, were solved to obtain a numerical solution.


5

Conservation equations of mass and momentum were developed using the

Eulerian approach, and were solved simultaneously. This approach treats each phase

separately, and the only link between the two phases is through the drag force in the

momentum equations.

The discretization scheme for the convection terms in the momentum equations

uses the power law interpolation scheme which provides a formal accuracy between first

and second order. This scheme is more robust and less computationally intensive than

higher order schemes. The SIMPLE iterative algorithm [25] is used by Fluent to relate

the velocity and pressure corrections to recast the continuity equation in terms of a

pressure correction calculation. The Inter-Phase Slip Algorithm (IPSA) method that uses

the Partial Elimination Algorithm (PEA) developed by Spalding [26] is used by Fluent-

4.4. The Full Elimination Algorithm (FEA), which is a fully coupled implicit approach

used in the new version of Fluent-4.5, significantly enhances convergence of the

numerical scheme.

Governing Equations

The following are the equations of conservation of mass and momentum for the

gas and particulate phases as well as the particulate phase fluctuating energy [13, 27]:

The conservation equation of mass of phase i (i = gas, solid)

( ) ( ) 0=∇+ iiiii .t

Uερερ∂∂ (1)

with the constraint

1=∑ iε (2)


6

The conservation of momentum of phase I (i = gas, solid, k≠ i)

( ) ( ) ( ) gUUTUUU iikiiiiiiiii .P.t

ερβεερερ∂∂ +−−∇+∇−=∇+ i (3)

The fluctuations that occur in the solid phase were modeled from the kinetic theory of

gases modified to account for inelastic collisions between particles. The equation for the

turbulent fluctuating energy of solid, called granular temperature, sΘ ( is U31=Θ ), may

be written as:

( ) ( ) ( ) sgssssssssssss kt

CCU:TU ⋅+−−∇∇+∇=∇+ βΘβγΘΘερΘερ∂∂ 3..

23 (4)

The last two terms in Equation (4) represent the interaction between gas turbulence and

particle fluctuation, and their derivation may be found in Gidaspow [13]. Numerically,

these terms can be important. For example, the term before the last term may be about

three times larger than the dissipation term. However, these terms were not included in

the present model since no turbulence model in the gas phase was used.

Constitutive Equations

Constitutive relations are needed to close any governing relations. The following

represent the constitutive relations used in the current model.

Arastoopour et al. [28] used a gas/solid drag force that gives continuous values over all

ranges of solid volume fractions:

( )

g

gsgpe

ggsgp

g

e

dR

dR

µ

ρ

εερ

β

UU

UU

−=

−−

+= − 8.21336.03.17

(5)


7

Solid phase stress:

( ) ssssbsss P SIUT µεµε 2+⋅∇+−= (6)

Deformation rate:

( )[ ] ( )IUUUS sT

sss ⋅∇−∇+∇=31

21 (7)

Gas phase stress:

gggg ST µε2= (8)

The solid pressure ( sP ), viscosity ( sµ ), and conductivity ( sλ ) are composed of two parts:

a kinetic term that dominates in the dilute flow regions and a collision contribution that is

significant in the dense flow regions.

( ) sossssss geP ΘερΘρε 212 ++= (9)

Where e is the restitution coefficient, and go is the radial distribution function which

becomes very large as the solid volume fraction of solid approaches the maximum

packing ( max,sε = 0.6 in this paper). The following expression was used by Ding and

Gidaspow [12]:

1

31

153

−

−=

max,s

sog

εε (10)

Solid phase shear viscosity:

( ) ( ) ( )2/12

, 1541

541

12

++

++

+=

πΘρεε

µµ s

opsssoo

dilss gedeg

ge (11)

Solid phase dilute viscosity:

965

,πΘρ

µ spsdils

d= (12)


8

The solid bulk viscosity accounts for the resistance of the granular particles to

compression and expansion, and is given by:

( )2/1

134

+=πΘρεµ s

opssb egd (13)

The diffusion coefficient for the solid phase energy fluctuation is:

( ) ( )2/1

22

121561

)1(384150

++

++

+=

πΘρεε

πΘρ sopssso

o

spss egdeg

ged

k (14)

The collisional dissipation of energy fluctuation is:

( )

∇−

−= s

s

psosss d

eg U.4132/1

22

πΘ

Θρεγ (15)

The dissipation of energy fluctuation due to interaction of particles with the gas phase

was not considered. The term ( s.U∇ ) in Equation (15) can be positive or negative

depending on the direction of the flow in the riser. Since the term ( s.U∇ ) was about two

orders of magnitude less than the positive term on its left, Equation (15) will always be a

dissipative term in the granular temperature Equation (4). It is important to note that the

strong dependency of the dissipation of the granular temperature on the restitution

coefficient (e) appears in term ( 21 e− ). An increase in the restitution coefficient from 0.9

to 0.99 would lead to an order of magnitude decrease in the dissipation ( sγ ). This

decrease in the collisional dissipation yields to a significant increase in the granular

temperature. Pita and Sundaresan [29,30] pointed out the high sensitivity of their

computed solids density and mass flux to the restitution coefficient. They showed that

only a value of e = 1 was appropriate to accurately compute the experimental data of

Bader et al. [20]. However, this value of restitution coefficient is not realistic since the


9

particle-particle collisions in the FCC riser are not elastic. For particles like FCC

catalyst, the restitution coefficient may be deduced from indirect measurement. Gidaspow

et al. [31] estimated the restitution coefficient to be close to unity from measurements of

the Reynolds stress and granular temperature. In this study, a particle-particle restitution

coefficient (e) of 0.95, and a particle-wall restitution coefficient (ew) of 0.9 were used. In

the literature, the reported measured restitution coefficients were for larger particles than

FCC particles [32].

Boundary Conditions

At the inlet, all velocities and volume fractions of both phases were specified. The

pressure was not specified at the inlet because of the incompressible gas phase

assumption (relatively low pressure drop system).

At the outlet, only the pressure was specified (atmospheric). The other variables

were subject to the Newmann boundary condition.

At the wall, the gas tangential and normal velocities were set at zero (non-slip

condition). The solid normal velocity was also set at zero. The following boundary

equations apply for the solid tangential velocity and granular temperature at the wall:

nv

gv w,s

sss

max,ssw,s ∂

∂−=

Θερφπεµ

03

6 (16)

w,smax,s

/sslip,sssw,s

w,s

ssw,s

gv

nk

γε

ΘερφπΘγΘΘ

6

3 230

2

+∂

∂−= (17)

These equations were developed by Hui et al. [33], and Johnson and Jackson [10].

According to their work, the slip velocity between particles and the wall can be obtained

by equating the tangential force exerted on the boundary and the particle shear stress

close to the wall. Similarly, the granular temperature at the wall was obtained by equating


10

the granular temperature flux at the wall to the inelastic dissipation of energy, and to

generation of granular energy due to slip at the wall region. Later, these boundary

conditions were successfully applied to dense riser flow [34].

Simulation and Results of Gas/Solid Flow in a 2-D Riser

System Description

Figure 1 shows the riser section of a CFB used in the present numerical

simulation of gas/solid flow. The geometry of the 20-cm riser is similar to the

experimental set-up used in the challenge problem by Knowlton et al. [24] for the case of

489 kg/m2s solid mass flux inside the riser. Solid particles were fed from both sides of the

riser near minimum fluidization conditions. At these conditions, an expected low value of

granular temperature was assigned to the particulate phase.

It is important to note that the real geometry of the riser is cylindrical with the

solid entering from one side only. To obtain mixing at the entrance zone similar to the 3-

D experiment, a two-inlet geometry was selected. A one-inlet design for the 2-D riser

could lead the inlet gas to flow to the opposite side of the solid inlet, therefore, creating

low mixing throughout the height of the riser. The effects of inlet design will be

demonstrated in this paper.

Initially, the velocity of both the gas and solid was set at zero. Gas and solid

exited the system through two symmetric side outlets set 0.3 m below the closed top of

the riser with a width of 0.1 m (similar to the side-inlets). The total height of the riser was

14.2 m. The solid particles in this simulation consisted of FCC material of 76 mµ in


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diameter and 1712 kg/m3 in density. The simulated system was isothermal at 300 K, and

the initial pressure was set at 1 bar.

In this simulation, the computational domain consisted of 18 grids radially and

210 non-uniform grids along the axis of the riser. A total of 3780 fluid cells resulted from

this grid distribution. A constant time step of 5.10-4 sec was used, and the simulation was

conducted on an SGI Origin 200 computer for 40 sec of real fluidization time

corresponding to 3-4 weeks of computational time.

Results and Discussion

The flow patterns in the 20-cm riser section of a CFB consisted of a core-annular

flow regime typical of a high density CFB as shown in Figure 11. The solid density is

higher near the wall region and more dilute at the center of the riser. The solid velocity is

high at the center of the riser and solid down-flow at the wall regions is shown in Figure

11 by the arrow plot of the solid velocity vectors. Moreover, the solid down-flow was

mostly in the form of clusters descending at low velocities at the wall region.

Figure 2 shows the comparison between the calculated time-averaged (from 15 to

40 sec) solid density distribution ( ss ερ ) at 3.9 m height inside the riser with the

experimental data taken at the same height. This figure shows a dilute region in the center

of the pipe and a high concentration of particles at the walls. This distribution reflects the

establishment of a core-annular regime shown by the experiments and the computational

results. However, contrary to the experiments, the computational results showed a smaller

core width in the core-annular system. By means of mass conservation, the solid

concentration at the walls was predicted to be less than the experimental values.

Moreover, it was experimentally proven that bigger particles in a mixture (as in an FCC


12

particle size distribution) have a greater tendency than the smaller particles to

accumulate at the wall regions [35, 36]. This could result in a higher solid concentration

at the wall than that predicted in this study using mono-sized particles. In addition, a

more accurate boundary condition that accounts for the possible electrostatic forces at the

walls could increase the solid density to a value closer to the experimental data.

Figure 3 shows the comparison between the calculated time-averaged solid mass

flux distribution in the riser at a height of 3.9 m with the experimental data taken at the

same height. The computational results agreed reasonably well with the experimental

data. The solid flux was at its maximum value at the center of the riser, although the solid

density was at its lowest value. This is due to the very high solid (and gas) axial velocity

at the center of the riser. Near the wall region of the riser, there was down-flow of solid

mainly in the form of clusters (see Figure 11). The solid down-flow was due to the

weight of the solid in the annular region that exceeded the axial pressure drop. The

velocity boundary condition used in this case successfully captured the velocity

magnitude of solid down-flow at the walls.

Figure 4 shows the time-averaged axial pressure drop in the riser compared with

experimental data. The high pressure drop at the bottom of the riser was due to the effect

of solid feeding in that region. The pressure drop then decreased along the height of the

riser due to the decrease in the solid concentration. The pressure drop was in reasonable

agreement with the experimental data, although it was under-predicted. This could be

mainly due to the prediction of a bigger dilute core in the 2-D riser (see Figure 11), which

yielded lower flow resistance and, in turn, a lower calculated pressure drop.


13

Figure 5 shows the instantaneous variation of the granular temperature with the

solid volume fraction after 10 sec of simulation. Gidaspow and Huilin [37] showed that,

in the dilute regions, the granular temperature is proportional to the solid volume fraction

raised to the power of 2/3. This is similar to the increase of an ideal gas temperature upon

compression. In the dense regions, the decrease in the granular temperature is due to the

decrease of the mean free path of the particles. The trend of the computed results shown

in Figure 5 agreed well with the experimental results of Gidaspow and Huilin [37].

However, the granular temperature was an order of magnitude lower than the

experimental data. This could be due to the neglect of inlet gas and, in turn, large-scale

fluctuations and the relatively low restitution coefficient used in the simulation. An

increase in the restitution coefficient will definitely increase the granular temperature

since the dissipation of fluctuating energy will decrease considerably. Figure 5 also

shows that the granular temperature was very low in the dense flow regions. The most

probable reason could be that the large-scale oscillations that were measured

experimentally were not included in the computational results [38]. The time-averaged

experimental measurements always include small- and large-scale oscillations [39], and

the computational results, in this case, include only the small-scale oscillations.

Therefore, a direct comparison between the computation and experimental data would be

accurate only if the large-scale oscillations are subtracted from the experimental data.

Unfortunately, Knowlton et al. (1995) did not obtain a time series analysis of their

experimental data. Therefore, calculations of the large-scale oscillations for this

experiment were not possible. On the other hand, large-scale oscillations may be


14

calculated by computing the time-averaged quantity: ss UU using the results of our

simulation.

Figure 6 shows the variations of the viscosity of the solid particles with the solid

volume fraction in the riser after 10 sec. The solid viscosity measured by Gidaspow and

Huilin [37] was also found to be an order of magnitude higher than the computed results

in Figure 6. This was mainly due to the under-prediction of the granular temperature (see

Figure 5). In fact, a correct granular temperature will lead to a more realistic prediction of

viscosity using Equation 11, and a linear increase of the solid viscosity with the solid

volume fraction would be obtained [37].

Figure 7 shows the power spectrum density at different frequencies of oscillations

in the riser. The Fourier transform method has been used to calculate the power spectrum

of the solid volume fraction. This analysis was done in the dilute core region of the riser

at a height of about 12 m with a sampling time of 0.1 sec. The calculated average solid

volume fraction was about 5%. The main frequency of the solid particles was 0.14-0.2

Hz, which corresponds to 5-7 sec time-period. A similar frequency has been reported

experimentally in a different riser by Gidaspow et al. [40] and computed successfully by

Neri [34] using the kinetic theory approach for the solid phase.

Figure 8 shows the power spectrum of different frequencies of oscillations in the

dense region at about 3 m above the bottom of the riser. The main frequency of

oscillations of particles in the dense region was about 0.14 Hz, which corresponded to 7-8

sec time-period. The dense region had an average solid volume fraction of about 13%.

The main reason for calculating the frequency of oscillations of the gas/solid flow was to

know the minimum time required to conduct proper time averaging of the computational


15

data. In the specific simulation that was run, there was not a significant difference

between the frequencies of oscillations in the dilute and dense regions. This could be due

to the small difference between what we called the dense and dilute solid densities. In the

case of larger differences in solid density, Gidaspow et al. [40] showed that large

differences could exist between the main frequencies of the dense and dilute regions. It is

important to mention that the computational values for the frequency analysis were taken

at least 3 m away from the solids inlet zone. In the inlet region, a high concentration of

solids along with solids inlet configuration led to a very small oscillation. Therefore, it is

advised to perform time series analysis away from the inlet region in order to capture the

right oscillatory flow patterns inside the riser. Experimentally, the measurements of

Knowlton et al. [24] did not include a frequency analysis of their system, and comparison

with the experimental data was, therefore, not possible.

Effects of Initial Conditions

A case where initially the solid particles filled the bottom of the riser to a height of 3

m at minimum fluidization conditions ( mf,sε = 0.4) was considered. All the other initial

and boundary conditions are the same as the previous case study.

Figure 9 shows the effect of initial conditions on the solid density distribution in

the riser at 3.9 m height compared with experimental data taken at the same height. At the

beginning of the simulation, the solid initially present in the riser moved toward the wall

region. This enhanced the solid density at the walls in the beginning of the simulation.

However, after a longer period of time, the solid initially present at the bottom moved

toward the top and mixed with the particles that were initially in the bed. The longer

averaging time (10-40 sec) showed no significant difference between this simulation and


16

the previous one starting with an empty riser (see Figure 2). Therefore, the effect of the

initial condition used in this simulation tends to disappear after several seconds of

operation.

Effects of Inlet and Outlet Design

To investigate the inlet and outlet effects, only one inlet and outlet that are

diametrically symmetric are considered. Geometrically, this simulation seems to be in

many features closer to the experiment since the 3-D experiment consists of only one

inlet and outlet. The inlet solid velocities were U = V = 0.505 m/s with a feed inclination

of 45o, and a minimum fluidization solid volume fraction of 0.4. This assured a 489

kg/m2s solid mass flux in the riser similar to the previous cases.

Figure 10 shows the time-averaged solid density at 3.9 m height in the riser

compared with the experimental data at the same height. The solid density was not

symmetric due to inlet and outlet effects. Most of the solid particles are concentrated at

the inlet side of the riser, and the opposite side of the riser was maintained at a relatively

dilute concentration. The typical core-annular flow regime was not clearly observed in

this simulation. It is important to mention that the experimental data were taken from one

side of the riser to its center only. Therefore, it is not certain that the experimental data

were symmetric. In the previous figures, radial symmetric behavior of the data was

assumed for the sake of comparison with the computational values. The exact

experimental inlet and outlet configuration and conditions cannot be implemented unless

a 3-D simulation is conducted in a geometry that is more complex. This requires very

high computational time with today's workstations capabilities.


17

Effects of Doubling the Riser Diameter

To investigate the effect of increasing the riser diameter on the flow profile and

pressure drop along the riser, a 40-cm riser operating at the same solid flux of 489 kg/m2s

as the 20-cm riser was considered. The diameter of the side inlets and outlets was 20 cm.

All other initial, inlet, and outlet conditions are described in Figure 1.

Figure 11 shows a comparison of the flow behavior in a 20- and 40-cm riser after

40 sec. The core-annular flow regime can be seen clearly in this figure since the core of

both risers is more dilute than the near wall regions. The velocity of the particulate phase

is higher at the center, and down-flow of solids occurred at the wall of the two risers

mainly in the form of clusters. It is noticeable that the length of the core of the 40-cm

riser was relatively smaller than that of the 20-cm riser. This was due primarily to solid

side inlet design. A change in the inlet design could significantly change the solid mixing

and flow patterns in the lower region of the riser.

Figure 12 shows a comparison between the time-averaged axial pressure drop

inside the 20- and 40-cm risers. From single-phase flow calculations, it could be shown

[41] that a higher diameter would lead to a lower pressure drop inside the riser. However,

the effects of solid inlet conditions significantly affected the mixing in the inlet region of

the riser and, in turn, resulted in higher pressure drop in the riser with a larger diameter.

The higher pressure drop was also due to the higher solid hold up in the riser as shown in

Figure 11. Therefore, special care should be taken when designing the solid inlets and

outlets.


18

Conclusions

A two-dimensional transient model incorporating the kinetic theory for the solid

particles used in the Fluent code was capable of predicting reasonably well the complex

gas/solid flow behavior in the riser section of a CFB.

The core-annulus flow observed in the dense riser flow was predicted by this

model. The flow regime was significantly affected by the solid down-flow in the form of

clusters near the walls of the riser.

The calculated solid flux and the axial pressure drop inside the riser compared

reasonably well with the available large-scale experimental data. However, the calculated

solid density deviated from the experimental data at the wall region. We believe that

experimentally verified boundary conditions that account for particle structures due to

electrostatic effects, particles cohesiveness, and multi-sized particles are needed to

accurately predict the solid density at the wall region. The kinetic theory should be

extended for multi-sized particles with the proper interaction terms between particles of

different sizes.

The kinetic theory for solid particles predicts well the trends and behavior that are

experimentally observed for the granular temperature and solid viscosity. The differences

are due to the restitution coefficient, large-scale fluctuations, and use of proper gas phase

turbulence interaction with the particulate phase.

The fluctuations in gas/solid flow predicted by this model showed that the main

frequency of the oscillations was about 0.15 Hz in both the dilute and dense regions (e.g.,

5% and 13% of solid volume fraction). Therefore, an adequate averaging time is

necessary for comparison with the time-averaged experimental results.


19

The initial presence of solids inside the riser did not have a significant effect on

the solid density distribution. The initial condition used in this simulation tended to

disappear after several seconds of simulation.

The effects of inlet and outlet design are significant in the overall solid flow

patterns and pressure drop along the riser. Real inlet and outlet conditions can be

implemented only when using a 3-D simulation with complex geometry. However,

computational time is still the major limiting factor to simulate gas/solid flow in complex

3-D geometry.

Nomenclature

2-D, 3-D - Two-dimensional, three-dimensionaliC - Instantaneous velocity vector of phase ipd - Particle diameter

e - Particle-particle restitution coefficientew - Wall-particle restitution coefficient

og - Radial distribution functionsk - Diffusion coefficient of granular temperature

P - PressuresP - Solids pressure

Re - Particle Reynolds number

iT - Stress tensor of phase iU i - Velocity vector of phase iβ - Inter-Phase drag coefficient

n∂∂ - Normal gradient to the wall

iε - Volume fraction of phase iφ - Specularity coefficient (φ = 0.01)

sγ - Collisional dissipation of granular temperatureiµ - Viscosity of phase iiρ - Density of phase IsΘ - Granular temperature


Figure 1 Schema with Inlet and

Figure 3 Solid M the Riser at 3.9 with Experime

m

Gas + SolidOutlet

1

0m

Uniform Gas InlDistribution, Vg =

g =ε1s =Θ

350ExperimentalComputation

R-0.10 -0.05

Solid

Mas

s Flu

x (k

g/m

2 s)

-200

0

200

400

600

800

1000

0.2

kg/m

3 )

250

300

4.2 m
.1
2205 s/me −

Gas + Solid Inlet

Solid

Den

sity

(

100

150

200

.1 m
50 0.3
tic Drawi

Initial Con

ass Flux D m Heightntal Data.

et 5.2 m/s

Radial Position (m)-0.10 -0.05 0.00 0.05 0.10

adial Position 0.00

ExperimCompu

Us = Ug = 0.476 m/ssε = 0.4,

22

At Time = 0Vs = Vg = 0

20

Figure 2 Solid Density Distribution in the Riser at 3.9 m Comparedng of a 2-D Riser with Experimental Data.ditions.

istribution in Figure 4 Time-Averaged Axial Pressure Compared Drop in the Riser.

051 s/mes −=Θ

(m)0.05 0.10

entation

Height (m)0 2 4 6 8 10 12 14

Pres

sure

Dro

p (P

a/m

)

0

1000

2000

3000

4000

5000

6000

7000

8000

ExperimentalComputation


21

Figure 5 Granular Temperature Variation Figure 6 Solid Viscosity Variation withwith Solid Volume Fraction in the Riser Solid Volume Fraction in the RiserAfter 10 sec. After 10 sec.

Figure 7 Power Spectrum Density Figure 8 Power Spectrum DensityAnalysis of the Solids Density Analysis of Solid Density FluctuationFluctuation in the Dilute Region in the Dense Region of the Riserof the Riser (12 m Above the Inlet). (3 m Above the Inlet).

Frequency (Hz)0 1 2 3 4 5

Pow

er S

pect

rum

Den

sity

0

1

2

3

Frequency (Hz)0 1 2 3 4

Pow

er S

pect

rum

Den

sity

0

5

10

15

20

25

Solid Volume Fraction0.00 0.05 0.10 0.15 0.20

Gra

nula

r Tem

pera

ture

(m2 /s

2 )

0.0

0.2

0.4

0.6

Solid Volume Fraction0.0 0.1 0.2 0.3 0.4 0.5 0.6

Solid

Vis

cosi

ty (P

a.s)

0.00

0.01

0.02


22

Figure 9 Effect of Initial Conditions Figure 10 Solid Density Distribution aton the Solid Density Distribution in 3.9 m Height in the Riser with One Inletthe Riser at 3.9 m Height. Compared with Experimental Data.

Figure 12 Comparison of the Time AveragedAxial Pressure in a 20 and 40-cm Riser.


Solid

Den

sity

(kg/

m3 )

50

100

150

200

250

300

350

400

450ExperimentalComputation (10-15 sec)Computation (10-40 sec)


Solid

Den

sity

(kg/

m3 )

50

100

150

200

250

300

350

400

450

ExperimentalComputation

Height (m)

0 2 4 6 8 10 12 14

Pres

sure

Dro

p (P

a/m

)

0

1000

2000

3000

4000

5000

6000

7000

8000

20-cm Riser40-cm Riser


23

20-cm Riser 40-cm Riser

Figure 11 Comparison of the Solid Volume Fraction and Velocity Profiles in a 20 and 40-cm Diameter Riser After 40 sec.


24

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