direct numerical simulation of dense gas-solid two-phase flows

Dev. Chem. Eng. Mineral Process., 8(3/4), pp.207-217,2000.

Direct Numerical Simulation of Dense

Gas-Solid Two-Phase Flows

Y. Zhulin Thermal Energy Engineering Research Institute, Southeast University, Nanjing 210018, P. R. CHINA

Based on Newton's law and the classical physical laws, Eulerian and Lagrangian

methods are respectively used to deal with gas-field and discrete particles. The three-

dimensional viscid air-field and three-dimensional discrete particle field are solved in

each time step At. Collision and friction between individual panicles are taken into

account when establishing the mathematical models, including individual particle

diameter, density, stifiess and fiction coeflcient. Particles mixing in ball mills,

particles dropping from hoppers, and particles fluidizing in fluidized be& are used as

examples of the simulations. Selected simulated results are compared to experimental

results.

Introduction In industry there are many dense gas-solid two-phase flows, such as gas-solid flows in

fluidized beds. The common characteristics are that bubbles and particle clusters

exist. The dismbution of particles is extremely inhomogeneous and there are intense

collisions and friction between particles as well as between particles and the vessel

walls. The diameters of the particles cover a wide range and they are relatively large.

All these factors exert a significant influence on the behavior of gas-solid flows. In

these cases either the solid phase is considered as a continuous medium and treated by

a Eulerian method, or it is based on the gas kinetic theory and treated by a Lagrangian

(Correspondence by email: [email protected] or fa. 0086-25-7714489)

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Y. Zhulin

method causing large errors in practice. With rapid computer developments, it has

been possible to follow an individual particle in a discrete particle field [l-31. In this

paper, for each time step At, a three-dimensional viscous air-field is solved by the

Eulerian method and the three-dimensional discrete particle field is calculated

simultaneously. Collision and friction are directly taken into account when

establishing the mathematical models, thus reducing the num,ber of inherent

assumptions.

Mathematical Model of Particle Phase

I The Arurlyses of Partick Motion Particles in dense gas-solid two-phase flow can be acted by the following forces: (i)

the contact force between particles; (ii) the contact force between particle and solid

border; (iii) the entrainment force between particle and its surrounding air due to

relative velocity; (iv) gravitational force. These are the main forces. Also the

particles are subjected to other forces, such as the Magnus force acting upon a moving

rotational particle in air field, the electric-field force generated by the friction between

particles, etc. These forces can be taken into account in direct numerical simulation,

but they are relatively small for dense gas-solid two-phase flows of large particles and

thus they are neglected.

Based on the classical physical laws, when two spherical particles move in a Iine

in opposing directions and impact with each other, elastic deformation at the contact

point will occur. The extent of deformation will depend upon the relative velocity of

the particles and the stiffness of particle material. Particles are subjected to elastic

resistance after collision in their direction of motion. The resistance force is directly

proportional to the displacement of deformation (6) and the stiffness (k) of the particle

materials. When the displacement reaches its maximum value the particles will cease

forward motion. Affected by this force, the collision particles will rebound along the

original direction. An energy transformation will occur during the collision, part of

the kinetic energy transforms to heat. The loss of kinetic energy relates to the

properties of the particle material and the relative velocity of collision between the

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Direct Numerical Simulation of Dense Gas-Solid Two-Phase Flows

particles. The loss of kinetic energy can be considered as a resistance force exerted

during collision, the magnitude of the force is equal to the product of relative velocity

and a parameter which is usually called the damping coefficient [4,5].

When non-central collision occurs, the contact face at the collision point can be

resolved in the normal direction and in tangential direction. The components fcn, fe, can be calculated by normal displacement 6, and tangential displacement 61 respectively. The exerting force as a result of the normal component forcef, is the

same as for central collision, and the tangential component force will produce a

torque to the particle center. The torque makes the particle rotate and the acceleration

of angular velocity can be calculated from the torque and the inertial moment of the

particle. The maximum value of tangential force is limited by the product of the

friction coefficient at the particle surface and the normal component force fen. When

the tangential component& is larger than the product, slip occurs at the contact point.

The most general case is the non-central collision of two rotating particles.

Despite 6, and 61, the extra slip velocity in the tangential direction at the contact point

needs to be calculated since the particIe is rotating. The existence of the extra slip

velocity will increase the damping loss.

For dense phase flows, particle i usually collides with several other particles at the

same time, the resultant force and resultant torque can then be obtained by summation

of all component vectors. The resultant force on each particle produces the moving

acceleration, while the resultant torque produces rotational acceleration.

N The Determination of Partick Original Positions

The height from which the particles are dropped into the fluidized-bed before

experiments is noted for each individual particle. As soon as the particles are

dropped, they are followed in each time step. After collisions with the bottom of bed

then rebounding and colliding with each other multiple times, they find their static

equilibrium positions at the bottom of the bed. The equilibrium positions are used as

the original positions for the simulation.

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Y. Zhulin

This physical process can be described by the following mathematical model:

(3) - VSii = TSj - (CnJ * ii)E + r ( q + aj) x ii

where fc is the contact force, k is the stiffness of the particle material, n' is the

vector of elastic deformation, q is the coefficient of damping dissipation, s' is the

unit vector in the normal direction, v', is the relative veIocity of particle i to particle j

and Cs is the slip velocity at the contact point, r is the particle's radius, dj is the

angular velocity vector of particles. Subscripts n and t represent the normal

component and tangential component respectively, i and j denote the two collision

particles. If the component of contact force in the tangential direction is larger than

the maximum friction force, slip occurs and the contact force in the tangential

direction takes on the maximum value of friction as shown below:

(6) - - - tii = vsu /1Vsvl

where p, is coefficient of friction, is the unit vector in the tangential direction.

F = I c + & (7)

i j = P / r n + g (8)

cii=TIZ (9) where is the resultant force, fF is the fluid entrainment force, rn is the particle's

mass, g' is the gravitational acceleration, T is the resultant torque, I is the particle d

moment of inertia.

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v'= Po +GAt J = i ' , + v ' t

8=f.Ijo+8At

where v' is the velocity vector of particle, At is the time step, 7 is the position vector

of particle's center, ?and are the moving acceleration and rotation acceleration

respectively. Subscript o denotes the old value of the former At.

The physical parameters involved in the mathematical models can be obtained

from appropriate data handbooks. The damping coefficient q can be calculated by the

following equations [6]:

q,, = 2Jml k,, q, = 2 J G

Mathematical Model of Gas Phase For dense gas-solid two-phase flows, the particles exert a large influence on the air

field. Presently it is difficult to solve the instantaneous flow field accurately behind

or between moving particles due to the large number of calculations. A feasible

method is to divide the flow domain into cells which are a size larger than individual

particles but smaller than the bubble occuning in the fluidized bed. All quantities

such as velocity and void fraction are considered uniform in one cell, and the finite

difference method can be used to solve the air field [7]. Since each particle is

followed when calculating the discrete particle field, the void fraction of each cell in

every At can be determined accurately by the particle data in each cell. A

mathematical model of three-dimensional viscid air-field is compiled and solved by

the SIMPLE method.

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Y. Zhulin

The continuity equation:

where E is void fraction, p is density of air, v’ is air velocity and t is time.

The momentum equation:

i

2 3

z = - - pdivv’S, + 2 p ,

where$ is the force of particles acting on fluids, z is the viscous force, g is the

gravitational acceleration, ~1 is the coefficient of viscosity, cp and Fr are particle

velocity and fluid velocity, j3 is the drag coefficient. Subscript i denotes particles. n

is the total number of particles in the cell and 6, and E~ are two mathematical

operators.

The turbulence equation:

V(cp@) = v[ [ j.4 + ?i7(&)] + G - &pE - Ep (19) at

where k is the turbulent kinetic energy, pt is the viscosity of turbulent flow, C T ~ is a

constant, E is the dissipated energy of turbulent flow. G and E,, are the generating terms of turbulent kinetic energy and the disturbing

energy coming from particles respectively, they are expressed as:

G = p,(vi’j)*

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1

where Vpi and Vpre the velocity modules of particle and air.

The dissipation equation of turbulent energy is:

&E k k k (22) €E &E2 + CiG - - C2G - - C3 E p -

where o,, CI, C,, C3 are constants.

Drag coefficients are:

E < 0.8

E 2 0.8

(25) 24 Re

Re<lOOO Cd =-[1+0.15Reo.687]

Re 2 1000 c, =0.44 (26)

where d is the diameter of particle, c d is the coefficient of resistance,

C,, =0.09, C, ~ 1 . 4 4 , C2 =1.92, C3 ~ 1 . 2 ,

CT, = 1, CT, = 1.33 are the calculated constants.

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Y. Zhulin

Results from the Direct Simulation Method

(i) Numerical Sirnubtion of Ball MiuS and Particles Droppingfiom Hoppers

The fluid force can be neglected for particle movements in a ball mill and for particles

dropping in a hopper, although the force is very important in fluidized beds and

pneumatic transportation. The simulation results of the particle moving in the tube-

shaped ball mill with a semicircular jacket on the inner wall are shown in Figure 1. The simulation results of the particles dropping from a hopper are shown in Figure 2.

Figure 1. Simulation results for particles mixing in a rotating ball mill.

Figure 2. Simulation results for particles dropping from a hoppel:

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(ii) Numerical Simulrrtion of the Spouted Bed Start-up Process

Figure 3 shows the animations editing of the spout bed start-up process simulated in a

3D particle and air field. In this figure, bubbles form from the bed bottom in the

beginning, gradually they expand and burst. Over time, the bed layer rises gradually

and reaches a stable height in the end.

(iii) Comprving the Results of the Numerical Simulation with the Results of the Experiments To check the validity of simulation, experiments have been carried out in a spout bed

experiment device similar to the conditions used in the simulation. The bed for the

experiment was made by transparent polymethyl methacrylate. Experiment particles

were made from polystyrene resin with diameter d, = 9mm and density p = 1042

kg/m3. The contrasts between results of simulation and experiment are shown in

Figure 4. As shown, the fluidized statuses of the two are accordant.

Figure 3. Simulation results for the start-up process of a spouted bed.

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Y. Zhulin

Figure 4. Comparison between simulation and experimental results (the number of

particles used was 800).

216


Conclusions Since fewer assumptions were adopted, the results of the direct numerical simulation

fit the experimental results perfectly. This method has enormous potential with the

development of computer hardware and the improvement of simulation precision.

Comparing with other multiphase flows sirnufation methods, the direct simulation

method takes more computation time especially while there are a large numbers of

particles in the flow field. At present, the number of particles in the flow field

simulated by microcomputer is restricted to 100,OOO. So, for dense gas-solid flows

with larger particle diameter and more intense collision and friction between particles,

the direct simulation method is recommended. However for dilute flows, other

simulation methods are more applicable.

References 1. 2.

3.

4.

5.

6.

7.

Tsuji, Y Discrete particle simulation of gas-solid flows, KONA, 1993, No.11. pp.57-68. Tan& T. Numerical simulation of gas-solid two-phase flow in a vertical pipe: on the effect of inter- particle collision, FED, Gas-Solid Flows, 1991, ~01.121, pp.123-128. Kawaguchi, T. Discrete particle simulation of two-dimensional fluidized bed, Powder Technol., 1993,

Mindlin, R.D. Compliance of elastic bodies in contact, Appl. Mech. (Trans. ASME), 1949, Vo1.16,

Mindlin. R.D., and Deresiewicz H. Elastic spheres in contact under varying oblique forces, J. Appl.

Cundall, P.A., and Strack, O.D. A discrete numerical model for granular assemblies, Geotechnique,

Ding, J., and Gidaspow, D. A bubbling fluidized model using kinetic theory of granular flow, AICHE

N0.77, pp.79-87.

pp.259-276.

MWh. (T-. ASME), 1953, V01.20, pp.327-341.

1979,29(4), pp.523-538.

J., 1990,36(4), pp.523-538.

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direct numerical simulation of dense gas-solid two-phase flows

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