simulation of dense two phase particle flow

8/6/2019 Simulation of Dense Two Phase Particle Flow

1/12

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

Direct Numerical Simulation of DenseGas-Solid Two-PhaseFlowsY.ZhulinThermal Energy Engineering Research Institute, Southeast University,Nanjing 210018, P.R. CHINA

Based on Newton's law and the classical physical laws, Eulerian and Lagrangianmethods are respectively used to deal with gas-field and discrete particles. The three-dimensional viscid air-field and three-dimensionaldiscrete particle ield are solved ineach time step At. Collision and friction between individual panicles are taken intoaccount when establishing the mathematical models, including individual particlediameter, density, st if iess and fiction coeflcient. Particles mixing in ball mills,particles droppingfrom hoppers, and particles fluidizing influidized be& are used asexamplesof the simulations. Selected simulated results are compared to experimentalresults.

IntroductionIn industry there are many dense gas-solid two-phase flows, such as gas-solid flows influidized beds. The comm on characteristics are that bubbles and particle clustersexist. The dismbution of particles is extrem ely inhomogeneous and there are intensecollisions and friction between particles as well as between particles and the vesselwalls. T he 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. Inthese cases either the solid phase is considered as a con tinuous medium and treated bya Eulerian method, or it is based on the gas kinetic theory and treated by a L agrangian(Correspondenceby email: [email protected] or fa.086-25-7714489)

207


2/12

Y.Zhulin

method causing large errors in practice. With rapid computer developments, it hasbeen possible to follow an individual particle in a discrete particle field [l-31. In thispaper, for each time step At, a three-dimensional viscous air-field is solved by theEulerian method and the three-dimensional discrete particle field is calculatedsimultaneously. Collision and friction are directly taken into account whenestablishing the mathematical models, thus reducing the num,ber of inherentassumptions.

Mathematical Model of Particle PhaseI TheArurlyses of Partick MotionParticles 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 solidborder; (iii) the entrainment force between particle and its surrounding air due torelative velocity; (iv) gravitational force. These are the main forces. Also theparticles are subjected to other forces, such as the Magnus force acting upon a movingrotational particle in air field, the electric-field force generated by the friction betweenparticles, 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 andthus they are neglected.

Based on the classical physical laws, when two spherical particles move in a Iinein opposing directions and impact with each other, elastic deformation at the contactpoint will occur. The extent of deformation will depend upon the relative velocity ofthe particles and the stiffness of particle material. Particles are subjected to elasticresistance after collision in their direction of motion. The resistance force is directlyproportional to the displacement of deformation(6) nd the stiffness (k)of the particlematerials. When the displacement reaches its maximum value the particles will ceaseforward motion. Affected by this force, the collision particles will rebound along theoriginal direction. An energy transformation will occur during the collision, part ofthe kinetic energy transforms to heat. The loss of kinetic energy relates to theproperties of the particle material and the relative velocity of collision between the

208


3/12

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

particles. The loss of kinetic energy can be considered as a resistance force exertedduring collision, the magnitude of the force is equal to the product of relative velocityand a parameter which is usually called the damping coefficient [4,5].

When non-central collision occurs, the contact face at the collision point can beresolved in the normal direction and in tangential direction. The componentsfcn, fe,can be calculated by normal displacement 6 , and tangential displacement 61respectively. The exerting force as a result of the normal component forcef, is thesame as for central collision, and the tangential component force will produce atorque to the particle center. The torque makes the particle rotate and the accelerationof angular velocity can be calculated from the torque and the inertial moment of theparticle. The maximum value of tangential force is limited by the product of thefriction coefficient at the particle surface and the normal component force en . Whenthe 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 pointneeds to be calculated since the particIe is rotating. The existence of the extra slipvelocity will increase the damping loss.

For dense phase flows, particlei usually collides with several other particles at thesame time, the resultant force and resultant torque can then be obtained by summationof all component vectors. The resultant force on each particle produces the movingacceleration, while the resultant torque produces rotational acceleration.

N The Determinationof Partick OriginalPositionsThe height from which the particles are dropped into the fluidized-bed beforeexperiments is noted for each individual particle. As soon as the particles aredropped, they are followed in each time step. After collisions with the bottom of bedthen rebounding and colliding with each other multiple times, they find their staticequilibrium positions at the bottom of the bed. The equilibrium positions are used asthe original positions for the simulation.

209


4/12

Y.Zhulin

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

(3)-VSii =TSj- CnJ * ii)E + r ( q +a j ) iiwhere fc is the contact force, k is the stiffness of the particle material, n' is thevector of elastic deformation, q is the coefficient of damping dissipation, s' is theunit vector in the normal direction, v', is the relative veIocity of particle i to particle jand Cs s the slip velocity at the contact point, r is the particle's radius, dj is theangular velocity vector of particles. Subscripts n and t represent the normalcomponent and tangential component respectively, i and j denote the two collisionparticles. If the component of contact force in the tangential direction is larger thanthe maximum friction force, slip occurs and the contact force in the tangentialdirection takes on the maximum value of frictionas shown below:

(6)- - -tii= vsu1Vsvlwhere 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'smass, g' is the gravitational acceleration, T is the resultant torque, I is the particledmoment of inertia.

210


5/12

Direct Numerical Simulation of Dense Gas-S olid Two-Phase Flows

v'= Po + G A tJ = i ' , + v ' t8=f.Ijo+8At

where v' is the velocity vector of particle, A t is the time step, 7 is the position vectorof particle's center, ?and are the moving acceleration and rotation accelerationrespectively. Subscripto denotes the old value of the formerAt.

The physical parameters involved in the mathematical models can be obtainedfrom appropriate data handbooks. The damping coefficientq can be calculated by thefollowing equations[6]:

q,,= 2 J m l k,,q, = 2 J G

Mathematical Model of GasPhaseFor dense gas-solid two-phase flows, the particles exert a large influence on the airfield. Presently it is difficult to solve the instantaneous flow field accurately behindor between moving particles due to the large number of calculations. A feasiblemethod is to divide the flow domain into cells which are a size larger than individualparticles but smaller than the bubble occuning in the fluidized bed. All quantitiessuch as velocity and void fraction are considered uniform in one cell, and the finitedifference method can be used to solve the air field [7]. Since each particle isfollowed when calculating the discrete particle field, the void fraction of each cell inevery A t can be determined accurately by the particle data in each cell. Amathematical model of three-dimensional viscid air-field is compiled and solved bytheSIMPLEmethod.

211


6/12

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:

i23z =-- divvS, + 2 p ,

where$ is the force of particles acting on fluids, z is the viscous force, g is thegravitational acceleration, ~1 is the coefficient of viscosity, c p and Fr are particlevelocity and fluid velocity, j3 is the drag coefficient. Subscript i denotes particles. nis the total number of particles in the cell and 6, and E~ are two mathematicaloperators.

The turbulence equation:

V(cp@)=v[[.4 + ?i7(&)] +G-&pE- E p (19)atwhere k is the turbulent kinetic energy, pt is the viscosity of turbulent flow, T ~s aconstant,E is the dissipated energy of turbulent flow.

G and E,, are the generating terms of turbulent kinetic energy and the disturbingenergy coming from particles respectively, they are expressed as:

G = p,(vij)*

212


7/12

Direct Numerical Simulation of Dense Gas-Solid T wo-Phase Flows

1

where Vpiand Vpre the velocity modules of particle and air.

The dissipation equation of turbulent energy is:

&Ek k k (22)E &E2+ CiG- C2G- C3E p-where o,,CI,,,C3are constants.

Drag coefficients are:E


8/12

Y. Zhulin

Resultsfrom the Direct SimulationMethod(i) Numerical Sirnubtion of Ball MiuS and Particles Droppingfiom HoppersThe fluid force can be neglected for particle movements in a ball mill and for particlesdropping in a hopper, although the force is very important in fluidized beds andpneumatic transportation. The simula tion results of the particle moving in the tube-shaped ball mill with a semicircular jacket on the inner wall are shown n Figure 1.The simulation results of the particles dropping from a hopper are shown in Figure 2.

Figure 1. Simulation results o r partic les mixing in a rotating ball mill.

Figure 2. Simulation results fo r particles dropping from a hoppel:

214


9/12

Direct Numerical Simulation ofDense Gas-Solid Two-Phase Flows

(i i) Numerical Simulrrtion of the Spouted Bed Start-up ProcessFigure 3 shows the animations editing of the spout bed start-up process simulated in a3D particle and air field. In this figure, bubbles form from the bed bo ttom in thebeginning, gradually they expand and burst. Over time, the bed layer rises graduallyand reaches a stable height in the end.

(iii) Comprving the Results of the Numerical Simulation with the Results of theExperimentsTo check the validity of simulation, experiments have been carried out in a spou t bedexperiment device similar to the conditions used in the simulation. The bed for theexperiment was made by transparent polymethyl methacrylate. Experiment particleswere made from polystyrene resin with diameter d, = 9mm and density p = 1042kg/m3. The contrasts between results of simulation and experiment are shown inFigure 4. As shown, the fluidized statuses of the two are accordant.

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

215


10/12

Y.Zhulin

Figure 4. Comparison between simulation and experimental results (the number ofparticles used was800).

216


11/12

Direct Numerical Simulation ofDense Gas-Solid Two-PhaseFlows

ConclusionsSince fewer assum ptions were adopted, the results of the direct numerical simulationfit the experimental results perfectly. This method has enormous potential with thedevelopment of computer hardware and the improvement of simulation precision.Comparing with other multiphase flows sirnufation methods, the direct simulationmethod takes more computation time especially while there are a large numbers ofparticles in the flow field. At present, the number of particles in the flow fieldsimulated by microcom puter is restricted to 100,OOO. So, for dense gas-solid flowswith larger particle diameter and more intense collision and friction between particles,the direct simulation method is recommended. However for dilute flows, othersimulation methods are more applicable.

References1.2.3.4.5 .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 wo-phase flow in a verticalpipe: on the effect of inter-particle collision,FED, as-Solid Flows, 1991, ~01.121, p.123-128.Kawaguchi, T. Discrete particle simulation of two-dimensional luidized bed, Powder Technol., 1993,Mindlin, R.D. Compliance of elastic bodies in contact, Appl. Mech.(Trans.ASME), 1949, Vo1.16,

Mindlin.R.D.,nd 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 bubblingfluidizedmodel 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.

217


12/12

218

simulation of dense two phase particle flow

Documents