Surta, Osaka 565 (Japan)

Powder Technology 77 (1993) 79-87 79 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 (PA Cundall and ODL 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 ntroduction 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 t 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 f 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 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 n their simulation, particle assemblies were treated three-dimensionally but the fluid motion was treated one-dimensionally n 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 0032-5910/93/$6 00 0 1993 - Elsevrer Sequora All rights reserved

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 Equations Particle motion Equations of translational and rotational particle motion are given by 3, =ijrn +g (1) t=?j (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 n 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 At (5) 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 dash-pot (a) @ Fig 1 Models of contact forces: (a) normal force; (b) tangential force jti= -k&v?,, (7) 3, = (i$ $5 (7) jet= -k& 7$, (8) i$=i;,--3 @> 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 f, however, the following relation is satisfied lfctl > El&ll (9) where pt is the coefficient of friction, then sliding is taken to occur, and the tangential force is given by fct = - cl& li (10) t is the unit vector defined by i= &/(?,( n dense phase flows, a particle usually touches several others at any time n 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 f 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 Shape: spherical Diameter d,: 4 mm Density p,: 2700 kg me3 Stifiess k: 800 N m- Coefficient of restitution e: 09 Coefficient of friction h: 03 Number: 2400 Conditions of gas Density p: 1205 kg mm3 Vlxosity CL: 180~ lo- N mm2 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 n fluidized beds or pneumatic conveying, the most dominant force causing particle motion is that of fluid n 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 f we assume a stiffness too much smaller than the actual value, the predicted particle motion changes n 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 Fluid motion t 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+ ; (a,)=0 The equation of fluid motion is taken to be $ (ezlj4,)= - $ g +f, J (11) (12) 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 fs,=p(u,-up (13) The above coefficient /3 depends on the void fraction When the void fraction p is less than 08, the coefficient /? is deduced from the well-known Ergun equation for the packed bed When the void fraction is larger than 08, 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 (~908) 150 ( - )cl + 175p&-ii1 d, P= (14) c,= (~>08) 24( 1 + 015Re, )/Re (Re < 1000) 043 (Re > 1000) (15) Re=]iis-+ffd,/~ (16) where d, is the particle diameter and p is the dynamic fluid viscosity, and! 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 SMPLE method (Semi-mplicit 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

: : 0 ) t j, 0 : :- :< i i / j,, : :- i< t ( _ 012345678 150 mm ) ~_~~~ Fig 2 Flow field and cells * _ 177 m s- and the terminal velocity is 165 m s-l The superficial gas velocity ranged from 20 to 26 m s-l of these particles are shown in Table 1 Concerning the wall properties such as stiffness and friction, the same values as the particle were given Fluid motion was neglected Using the DEM, we calculated the motion of these particles falling from the initial still condition under the effect of gravity and neglecting mteractton with the fluid n this calculation, cases based on various time steps were examined The oscillation period of the spring-mass system used to model contacting particles is given by r= 2& (17) One half of the above period were divided by a factor of Z to obtain the time step n this calculation, six cases were considered for, ie, = 3,4, 5, 10, 20 and 100 Table 2 shows the time step deduced in that way As a criterion for the stability of calculation, we gave attention to the total kinetic energy of particles from the time they began to fall to the time they were settled Figure 3 shows the kinetic energy plotted against time The results of different time steps are compared in the figure t was found that the results were the same for n > 5 n the case of it =4, the energy dissipated, but more slowly than for the shorter time steps When n <3, we were not able to obtain the result, because the calculation was unstable We accordingly concluded that the time step can be obtained by dividing one half of the natural oscillation period by a factor of 5 TABLE 2 Time step n 3 4 5 10 20 100 ht (x10-4 s) 352 264 211 106 053 011 Time step The time step should be set smaller than a certain critical value to make the calculation stable However, if the time step is made too small, an unnecessarily long computation time is needed Therefore, a proper time step should be chosen Two kmds of time step must be considered: that required for calculation of particle motion and that corresponding to fluid motion From the viewpoint of the stability of calculation, the critical time step for particle motion is much smaller than that of fluid motion Cundall and Strack [3] proposed a method for determining the time step for calculating particle motion, which is based on the characteristic natural frequency of a spring-mass oscillation system Referring to their method, we determined the time step by the following procedure First, we randomly distributed 150 particles in a horizontal pipe with an internal diameter 30 mm and a length 260 mm as the initial condition according to a uniform probability distribution The properties 0 01 02 Time s Fig 3 Effect of time step on energy dlsslpation

83 n the simulation of the fluidized bed, 20X 10e4 s was used as the time step The coefficient of viscous dissipation 77 is given by n=2y&% (18) Flle open, fntlallzatlon _of_ all data y can be determined e as r=a/jit2 from the coefficient of restitution a+= -(l/r) n e (20) See the Appendix for a derivation of the above equation The calculation process is shown as a flow chart in Fig 4 Figure 4(a) shows the main flow of calculation and Fig 4(b) shows the flow chart of particle motion 1 Calculation of void fraction 1 t-vt+at alculatlon of fluld motion d End (4 Repeat N times 1 Calculation of particle motion 1 1 Data save 1 Calculation Calculation,+ Calculation with the wall of fluid motion of nter-particle yes of contact force, Experiment Figure 5 shows the experimental apparatus Air supplied by a blower was regulated through a sandwiched packed bed and issued into the container from a nozzle of 10 mm width at the center of the bottom wall The particles were aluminium spheres with a mean diameter of d, = 40 mm ( f 005 mm) and a density of 2700 kg mm3, which are the same conditions as given in Table 1 To obtain a similar condition to two-dimensional calculation, the flow was confined between two parallel plates, 22 mm apart The breadth of the container was 150 mm The initial bed height was about 220 mm Here, the difference between the simulation and physical experiment should be noted n the simulation the thickness of the bed corresponds to the particle size; that is, the bed contains one row of particles The experimental fluidized bed has a thickness of about five particle diameters Patterns of fluidization were recorded using a VTR to compare the results with calculation Also, pressure fluctuations were measured 200 mm downstream from the bottom wall Calculation of fluid force acting on the particle L Calculation of partlcle velocity and acceleration 1 Calculation of particle posltlon \1 Data save (b) Fig 4 Flow chart of calculation motion (a) mam flow, (b) particle Fig 5 Experimental apparatus air

Result and discussion Calculated results are shown in Fig 6(a)-(d), where cases of four different air velocities are presented The conditions of calculation are listed in Table 1 The initial bed height was assumed to be equal to that of the experiment, ie, about 220 mm The dimensions of the container were almost the same as the experimental apparatus, the only difference being that the thickness of the fluidized bed was set equal to the (4 Fig 6 Behawor of particles m fluldrzed bed- (a) u0=20 m s-l, (b) u0=2 2 m s-l, (c) u,=24 m s-l; (d) u,=26 m S-

85 particle diameter (4 mm) in the simulation, while it was 22 mm in the experiment t was, however, assumed in the calculation that the contact forces between the particles and the wall (front and back) do not act on particles The flow field was divided into 10X 20 mm rectangular cells as shown in Fig 2 and was calculated up to a distance of 100 mm above the free boundary of the bed The motion of particles can be observed by following the colored particles At a low air velocity below 20 m s-, the bed expands but no circulation is observed At higher air velocities, particles circulate and are mixed in a region whose size increases with the velocity at the nozzle At r+, = 20 m s-l, (a), p a rt ic 1 es near the free surface are not mixed as shown in Fig 6(a) At u0 = 22 m s-l, (b), the particles near the surface come into the mixed motion only gradually Bubbles (t =04 s) or slugs (t= 13 s) are generated at u,=24 m s-l, (c), and as a result all particles are mixed At u,=26 m s-l, (d), periodic generation of bubbles is clearly observed The above results of circulation are roughly the same as the experimental ones For example, particles were mixed only in the lower region at u0 < 24 m s-l, and bubbles began to appear at u0 = 24-25 m s- in the experiment Figure 7 shows the velocity vectors of particles corresponding to the above four air velocities The scale of particle mixing becomes larger as the air velocity increases This expansion of the region of particle mixing when the air velocity was increased was also observed in the experiment We did not measure particle velocities in the physical experiment and therefore most comparisons should be qualitative As long as we saw actual phenomena with our eyes, the observed flow field showed qualitatively good agreement with calculated results When comparison between the simulation and experiment is made in detail, some quantitative differences are noted The scale of circulations in the experiment was smaller than that in the simulation These differences are probably caused by simplifications in the present model which neglects rolling friction and the frictional forces with the front and rear walls The inviscid flow assumption may also cause differences between simulation and experiment We were interested in the large pressure fluctuations in the fluidized bed because such fluctuations are considered to be caused by bubbles and slugs To confirm this experimentally, we measured the pressure fluctuations 200 mm downstream from the bottom wall with a pressure transducer and at the same time measured the frequency of bubble formation with the VTR t was found from comparison between pressure wave forms and videotaped pictures that the frequency of the pressure fluctuations agrees with that of bubble formation n the simulation the instantaneous pressure in each cell was calculated, allowing pressure fluctuations equivalent to the experiment to be obtained To see how well the simulation predicts bubble or slug formation, pressure wave forms were examined Rx- u0=20 m/s ua=2 2 m/s ua=24 m/s uc=26 m/s Fig 7 Velocity vectors of pactdes

List of symbols (4 Time s 0 20 40 60 @ Time s Fig 8 Pressure fluctuation (uo=26 m s-l): (a) expenment; (b) calculation perimental and calculated wave forms are shown in Fig 8 (a) and (b) Agreement is good for the frequency but not so good for the amplitude of fluctuation drag coefficient particle diameter coefficient of restitution sum of forces acting on particle contact force fluid drag force gravity acceleration vector inertial moment of particle stiffness particle mass unit vector in the normal direction pressure particle Reynolds number defined by eqn (16) position vector of particle gravity center torque time unit vector in the tangential direction time step bulk gas velocity gas velocity of the zth direction in the coordinate particle velocity coordinate Greek letters E void fraction (porosity) 77 damping coefficient 4 friction coefficient P kinetic viscosity of gas P air density (3 angular velocity vector Conclusions Discrete particle simulation was attempted for a twodimensional fluidized bed Motion of individual particles was numerically Calculated using Cundall and Strack s DEM model, and taking into account the interaction with fluid motion An experiment was made for comparison with the simulation The results are summarized as follows Qualitatively, the results of the simulation are satisfactory in many respects, such as particle circulating motion and mixing Not only qualitative but quantitative agreement was obtained for the following quantities: (1) the air velocity corresponding to the onset of bubbling and (2) the frequency of pressure fluctuations There are some differences between the experiment and simulation To resolve these differences, further studies are required to extend the work to three-dimensional calculation Subscripts 0 initial value n normal component t tangential component References JW Prichett, TR Blake and SK Garg, AChE Symp Ser, 176 (1978) 134 JX Bouillard, RW Lyczkowski and D Gldaspow, AZChE J, 35 (1989) 908 PA Cundall and 0D L Strack, Geotechnlque, 29 (1979) 47 Y Tsuji, T Tanaka and T shlda, Powder Technol, 71 (1992) 239 T B Anderson and R Jackson, &EC Fundamentals, 6 (1967) 527 S V Patankar, Numerical Heat Transfer and Flurd Flow, Hem- sphere, New York, 1980

87 Appendix The equation of motion for the oscillation system consisting of a mass, a spring and a dash-pot is ti+ryl+kx=o (Al) The solution of this equation under the initial condition that x=0 and i=v, at t=o is given by x = 64/q) sir@) exp( - ~4 (A3 i = (02) exp( - 7W){q co@) - ~ti sin(qt)) (A% where U0-G (Ad) -- 17 y- am W q=c&q 646) The oscillation period of this system is 2dq A particle colliding with another particle at the time t = 0 detaches at the time t= T/q The velocity at the time = r/q is given by vu0 =& + = - 21, exp(y~04q) Therefore, the coefficient of restitution becomes e = - v,/v, = exp( - yawdq) (A3 (A8) f the coefficient of restitution is regarded as a constant empirical parameter, the damping coefficient may be determined from the above equations