Discrete particle simulation of two-dimensional fluidized bed

79

Powder Technology 77 (1993) 79-87

Discrete particle simulation of two-dimensional

fluidized bed

Y. Tsuji, T. Kawaguchi and T. Tanaka Department of Mechanrcal Engineering, Osaka Unrvemity

Surta, Osaka 565 (Japan)

(Received December 1, 1992; m revrsed form April 19, 1993)

Abstract Numerical simulation, in which the motion of individual particles was calculated, was performed of a twodimensional gas-fluidized bed. Contact forces between particles are modeled by Cundall’s Distinct Element Method (P.A. Cundall and O.D.L. Strack, Geotechntque, 29 (1979) 47), which expresses the forces with the use of a spring, dash-pot and friction slider. The gas was assumed to be invrscid and its flow was solved simultaneously with the motion of particles, taking into account the interactton between parttcles and gas. The simulation gives realistic pictures of particle motion. Formatron of bubbles and slugs and the process of particle mixing were observed to occur in the same way as in experiments. The calculated pressure fluctuations compared well wtth measurements.

Introduction From a macroscopic viewpoint, the solid phase in a fluidized bed behaves like a kind of fluid. Thus, most numerical simulations of fluidized beds are based on theories assuming that the solid phase is a continuum. Many such simulations have used a two-fluid model, which regards a solid-fluid mixture as consisting of two kmds of fluids. For example, Pritchett et al. [l] simulated a two-dimensional fluidized bed using the two-fluid model and showed the formation of bubbles. Bouillard et al. [2] used the two-fluid model to investigate numerically a fluidized bed with an inserted body. It is necessary in the two-fluid model to assume constitutive equations for the solid phase. The problem of the method is that parameters included in the constitutive equations lack generality. If good agreement with experiment is required, some parameters in the constitutive equations should be determined empirically; sometimes even from experiments similar to the simulation to be done. Discrete particle simulation has been used mainly for dilute solid-fluid flows. Computers with large memories and high calculation speed make it possible to treat a large number of particles. Cundall and Strack’s [3] DEM (Distinct Element Method) opened up new possibilities for using discrete particle simulation to calculate the dense phase flows such as fluidized beds. Particles in granular flows interact with each other through contact forces. These forces were calculated with simple mechanical models such as a spring, a dashpot and a friction slider. Though this method was

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originally proposed to predict the behavior of soil, it has been found to be applicable to many other phenomena concerned with granular materials. The DEM has several advantages over the two-fluid model. For instance, the particle size and density distribution can be directly taken into account m the simulation if necessary, because characteristic properties of individual particles such as size and density can be specified. Another advantage is that parameters, affecting interparticle contacts, can be determined from the properties of materials such as the Young’s modulus, Poisson ratio and coefficient of restitution. This means that the discrete particle approach needs fewer assumptions than the two-fluid model. The principal limitation of the method is that when the number of particles is of the same order as in real flows of fine materials, the computation time becomes extremely long. Therefore, it is expensive or impossible to apply the discrete particle simulation to the case of fine powder. Tsuji et al. [41] used the DEM to simulate densephase pneumatic conveying (plug flow), taking into account the fluid forces. In their simulation, particle assemblies were treated three-dimensionally but the fluid motion was treated one-dimensionally. In spite of the simple assumption concerning fluid motion, the results were quite satisfactory. However, one-dimensional treatment of fluid motion is not adequate for the present simulation, because the flow field in the fluidized bed is a recirculating one. Physically the fluid moves in the narrow space between particles in the fluidized bed, and the flow field therefore has a complicated micro-structure. But in the present calculation

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80

fluid motion around each particle was not considered, which is extremely difficult, and an approximate method was used for fluid motion. Calculation of fluid motion was made two-dimensionally but based on calculation cells whose size was substantially larger than individual particles, modelling the particles as a continuous porous medium to calculate pressure gradient. The calculated flow velocities might therefore be thought of as spatial averages over these cells. The present simulation dealt with a spouting bed where gas issued from a small nozzle on the bottom of the container. Moreover, an experiment of the twodimensional fluidixed bed with the same size as the simulation was made. Particle motion was observed using a VTR, and fluctuating pressure was measured.

dash-pot

(a)

@I

Fig 1. Models of contact forces: (a) normal force; (b) tangential force.

jti=

-k&v?,,

(7)

3, = (i$ $5

(7)’

l

Equations

jet=

Particle motion

i$=i;,--3 ”

Equations of translational motion are given by

and rotational

particle

3, =iJrn +g’

(1)

t= ?JI

(2)

where Y; is the particle velocity vector, m is the particle mass, F is the sum of forces acting on the particle, g” is the gravity acceleration vector, 3 is the angular velocity vector, f is the net torque caused by the contact force, Z is the moment of inertia of the particle, and t) denotes a time derivative. In general, the force F consists of contact forces and fluid forces. The new velocities and position after the time step At are calculated explicitly: 3,=3,+i;,,

b

(3)

G=&+v’,, At

(4)

&=;o+;o

(5)

At

where subscript 0 denotes the initial value. Modehg of cqntact forces The force F can be divided into the contact

force

and fluid force as 5=_& +&

(6)

The fluid force will be explained in the next section. The contact force based on the DEM was explained in detail in our previous paper [4] but it will be briefly described again for convenience. The contact force is further divided into the normal force&, and tangential forcei,,. Figure 1 shows Cundall and Strack’s model for the forces, which they modelled by

-k&

7$,

(8) @>’

where i,, and & are the particle displacements in the normal and tangential directions, respectively, 3, is the relative velocity, k is the stiffness of the spring and 17 is the coefficient of viscous dissipation. If, however, the following relation is satisfied

lfctl> El&Ill

(9)

where pt is the coefficient of friction, then sliding is taken to occur, and the tangential force is given by fct = -

t’is

cL&“li

(10)

the unit vector defined by i= &/(?,(. In dense phase flows, a particle usually touches several others at any time. In such circumstances, the total contact force is found by summing over all the contacts. As seen in the above equations, the stiffness, coefficient of viscous dissipation and friction coefficient must be specified to use the model shown in Fig. 1. These parameters can be determined from the physical properties of the particles. The Hertzian contact theory is useful for determining the stiffness as was described in our previous paper [4]. The coefficient of viscous dissipation can be determined from the coefficient of restitution e. The relation between the coefficient of viscous dissipation and the coefficient of restitution was described in our previous paper [4]. The time step depends on the stiffness. The larger the stiffness, the smaller the time step should be. If we determine the time step from an actual stiffness, we get a very small value, which requires very long computation time. To save computation time, we assumed a value of the stiffness smaller than that estimated from the physical properties of the particle, as shown in Table 1 of the next section. The justification for using such a small value is as follows.

TABLE 1 Conditions of particles and gas Conditions of particle

Conditions of gas

Shape: spherical Diameter d,:4 mm Density p,: 2700 kg me3 Stifiess k: 800 N m-’ Coefficient of restitution e: 0.9 Coefficient of friction h: 0.3 Number: 2400

Density p: 1.205 kg mm3 Vlxosity CL:1.80~ lo-’ N mm2

fs,=P(u.,-UP

Note: The coeffiaent of restitution e was not used for calculating particlevelocity before and after colhsion but used for determining the coefficient of viscous dissipation.

In fluidized beds or pneumatic conveying, the most dominant force causing particle motion is that of fluid. In such cases, the effect of stiffness on particle motion is secondary, which was confirmed in the preliminary calculation. Calculation based on a smaller stiffness than an actual one does not show a large difference in particle motion from the calculation based on the actual stiffness. Of course, this is the matter of degree. If we assume a stiffness too much smaller than the actual value, the predicted particle motion changes. In the present simulation, we assumed the stiffness which is convenient for obtaining realistic particle motion within bearable computation time. The time step will be discussed again later. Fluidmotion It is almost impossible even for modem super-computers to solve the instantaneous flow field on both a small scale as relevant to distances between moving particles in the fluidized bed and on a large scale which is of interest in phenomena such as bubbles. Therefore it is reasonable to make a calculation based on locally averaged quantities following Anderson and Jackson [5]. As is usual in many numerical calculations of flow fields, the finite difference method was used in the present simulation. The flow domain was divided into cells, the size of which is smaller than the macroscopic motion of bubbles in the fluidized bed but larger than the particle size. All quantities such as pressurep and velocity u are averaged in the cell using a weight function. The void fraction of each cell can be defined by the number of particles existing in the cell. The equation of continuity is given by %

e+ ;

(11)

(a,)=0 I

The equation of fluid motion is taken to be

$ (ezlJ.4,)= - $g +f., J

I

where p is the fluid density. The flow is assumed to have inviscid behavior. Only when considering the fluid drag on particles is viscosity taken into account. The last term f., in eqn. (12) denotes the effect of particles on fluid motion through the fluid drag force. Following Prichett et al. [l], fs, is given by

(12)

(13)

The above coefficient /3 depends on the void fraction. When the void fraction p is less than 0.8, the coefficient /? is deduced from the well-known Ergun equation for the packed bed. When the void fraction is larger than 0.8, the particle’s motion is taken to be only weakly affected by other particles, and a modified equation of the fluid resistance for a single particle is used for the dilute region. The expressions of /3 are as follows 150 (I - l )cL + 1.75p&-ii1 d, (~90.8)

P=

(14)

I (~>0.8) 24( 1 +0.15Re” ,‘)/Re

(Re < 1000)

I 0.43

(Re > 1000)

c,=

(15)

Re=]iis-+ffd,/~

(16)

where d, is the particle diameter and p is the dynamic fluid viscosity, and I!. is the particle velocity vector averaged in a cell. The fluid motion was solved simultaneously with the motion of particles. Once both motions have been solved, the drag force on each single particle is calculated from local gas and particle velocities obtained. The drag and contact forces acting on individual particles are put into eqns. (1) and (6) which determine the position and velocity of particles at the next step. As a numerical method, the SIMPLE method (Semi-Implicit Method for Pressure-Linked Equation) developed by Patankar [6] was used. Conditions of simulation The conditions of particles and gas are given in Table 1. The container (150 mm breadth) and cells (10 mm x 20 mm) are shown in Fig. 2. Gas is issued from the center nozzle, whose width is 10 mm, with uniform velocity. Motion in the direction perpendicular to the paper was not considered. The height of the particle layer at rest is 220 mm. The fluidizing velocity of the particles is

I

. . . .:. . . .‘_._ . .:_._ .

.

.:.

0 ) t

:_ j,. :_ 0

.

:’ ::