6 th International Conerence on Multiphase Flow, Numerical Simulation o Particle Mixing in Dispersed Gas-Liquid-Solid Flows using a Combined Volume o Fluid and Discrete Particle Approach Niels G. Deen, Martin van Sint Annaland and J.A.M. Kuipers University o Twente, Faculty o Science and Technology, Institute o Mechanics Processes and Control Twente PO Box 17, NL-7500 AE Enschede, The Netherlands E-mail: N.G.Deen@utwente.nl Keywords: volume o luid model, discrete particle model, particle mixing Abstract In this paper a hybrid model is presented or the numerical simulation o gas-liquid-solid lows using a combined Volume O Fluid (VOF) and Discrete Particle (DP) approach applied or respectively dispersed gas bubbles and solid particles present in the continuous liquid phase. The hard sphere DP model, originally developed by Hoomans et al. (1996) or dense gas-solid systems, has been extended to account or all additional orces acting on particles suspended in a viscous liquid and has been combined with the VOF model presented recently by van Sint Annaland et al. (005) or complex ree surace lows. In this paper the physical oundation o the combined VOF-DP model will be presented together with illustrative computational results highlighting the capabilities o this hybrid model. The eect o bubble-induced particle mixing has been studied ocusing on the eect o the volumetric particle concentration. In addition particle mixing was studied in systems with two co-laterally and three oblique rising bubbles. Introduction Multiphase systems are requently encountered in a variety o industrial processes involving a.o. coating, granulation, drying and synthesis o uels (Fischer Tropsch) and base chemicals. The hydrodynamics o multiphase systems is dictated by the motion o the individual phases and the complex mutual interactions and as a direct consequence thereo CFD-based modeling o these systems has proven notoriously diicult. In literature both the Eulerian approach (Torvik and Svendsen, 1990; Mitra-Majumdar et al., 1997) and Lagrangian approach (Bourloutski and Sommereld, 00) have been adopted to study gas-liquid-solid three-phase lows. Although these computational methods are in principle well-suited to simulate the large-scale low behavior o slurry bubble columns and three-phase luidized beds (i.e. the time-averaged circulation pattern and spatial gas holdup distribution), problems arise related to the representation o the interactions between the individual phases leading to considerable closure problems. To overcome this problem detailed microscopic models can be used to generate closure laws which are needed in coarse-grained simulation models which are used to describe the macroscopic behavior and as a practical consequence thereo do not resolve all the relevant length and time scales. This multi-scale modeling approach has been used by the authors previously or dense gas-solid lows (van der Hoe et al., 004) and or dispersed gas-liquid two-phase lows (van Sint Annaland et al., 003 and Deen et al., 004a, 004b) and can in principle be used or other types o multiphase low as well. Our model is based on a combined Volume O Fluid (VOF) and Discrete Particle (DP) approach applied or respectively dispersed gas bubbles and solid particles present in the continuous liquid phase. Volume O Fluid (VOF) methods (Hirt and Nichols, 1981; Youngs, 198; Rudman, 1997, 1998; Rider and Kothe, 1998; Scardovelli and Zaleski, 1999; Popinet and Zaleski, 1999; Bussman et al., 1999) employ a color unction F(x,y,z,t) that indicates the ractional amount o luid present at a certain position (x,y,z) at time t. The evolution equation or F is usually solved using special advection schemes (such as geometrical advection, a pseudo-lagrangian technique), in order to minimize numerical diusion. In addition to the value o the color unction the interace orientation needs to be determined, which ollows rom the gradient o the color unction. Roughly two important classes o VOF methods can be distinguished with respect to the representation o the interace, namely Simple Line Interace Calculation (SLIC) and Piecewise Linear Interace Calculation (PLIC). Earlier work is generally typiied by the SLIC algorithm due to Noh and Woodward (1976) and the Donor-Acceptor algorithm published by Hirt and Nichols (1981). Modern VOF techniques include the PLIC method due to Youngs (198). The accuracy and capabilities o the modern PLIC VOF algorithms greatly exceeds that o the older VOF algorithms such as the Hirt and Nichols VOF method (Rudman, 1997). A drawback o VOF methods is the so-called artiicial (or numerical) coalescence o gas bubbles which occurs when their mutual distances is less than the size o the computational cell. Discrete Particle (DP) methods employ particle tracking taking into account i) all relevant external orces (gravity, pressure, drag, lit and virtual mass orces) acting on the particles and ii) collisions between particles and conining walls. Finally, two-way coupling, which becomes important at high solids volume raction, is also taken into account. In
this study the hard sphere DP model, originally developed by Hoomans et al. (1996) or gas-solid systems (also see Hoomans, 000), has been extended to account or all additional orces acting on particles suspended in a viscous liquid and has been combined with the VOF model presented recently by van Sint Annaland et al. (005). The direct numerical simulation has been pioneered by Fan and co-workers (Li et al., 1999, 000, 001; Zhang et al., 000a, 000b) who used a two-dimensional model based on the VOF method to track the bubble interace. The organization o this paper is as ollows: irst the description o the model is given ollowed by a description o the numerical solution method. Subsequently the results are presented and discussed and inally the conclusions are presented. Model description Our model consists o two major sub-models, the model dealing with the motion o the gas bubbles in the continuous phase (consisting o liquid and suspended solid particles) and the DP model dealing with the motion o the suspended particles taking into account the action o external orces and non-ideal particle-particle and particle-wall collisions. First the VOF model is briely described ollowed by a brie description o the DP model. Volume o luid model The governing conservation equations or unsteady, Newtonian, multi-luid lows are given by the ollowing expressions: ε + ( ε u ) = 0 (1) t ρ[ ( ε u) + ( ε uu)] = ε p + ρg F s t () T + ( ε μ[( u) + ( u) ]) + F σ The source terms F σ and F s account respectively or the orces due to surace tension and two-way coupling due to the presence o the suspended solid particles. Furthermore ε represents the volume raction o the continuous liquid phase. The mixture density ρ and viscosity μ are evaluated rom the local distribution o the phase indicator or color unction unction F which is governed or by: DF F = + ( u F) = 0 (3) Dt t expressing that the interace property is advected with the local luid velocity. For the local average density ρ linear weighing o the densities o the liquid (l) and gas phase (g) is used: ρ = Fρ + (1 F) ρ (4) l g Similarly, the local average dynamic viscosity can also be obtained via linear averaging o the dynamic viscosities o the liquid and gas phase. As an alternative, more undamental approach recently proposed by Prosperetti (001), the local 6 th International Conerence on Multiphase Flow, average viscosity can be calculated via harmonic averaging o the kinematic viscosities o the involved phases according to the ollowing expression: ρ l g = F ρ + (1 F) ρ (5) μ μ μ l g In all computations reported in this paper Eq. (5) was used to compute the local average viscosity. The volumetric surace tension orce appearing in the momentum Eq. () acts only in the vicinity o the interace Discrete particle model The motion o the suspended solid particles is given by the Newton s second law given by the ollowing expression which accounts or the action o gravity, ar-ield pressure, drag, lit and added mass orces. d ( m ) p v = F + g F + p F + d F + L F VM dt V β p p = mg V p+ ( u v) p p (1 ε ) CVρ ( v u) ( u) L p l D ρ V C [ ( v u) + ( v u) ( u) ] l p VM Dt This equation is similar to the one used by Delnoij et al. (1997) in their discrete bubble model or dispersed gas-liquid two-phase low. Since the size o the particles typically is (much) smaller than the size o the computational grid required or capturing the bubble dynamics, a closure equation is required or the eective drag (riction) acting on the particles. Following Hoomans et al. (1996) the riction between the continuous liquid phase and the suspended solid particles is given by the Ergun equation in the dense regime (ε < 0.8) and the Wen and Yu equation in the dilute regime (ε > 0.8) respectively given by (ε < 0.8): (1 ε ) μ ρ l l β = 150 + 1.75(1 ε ) u v p ε ( d ) d p p (7a) and (ε > 0.8): β 3 ε (1 ε ) (6).65 = C ρ u v ( ε ) (7b) p d l 4 d p where the drag coeicient C d ollows well-known equations or drag coeicients or spheres whereas C VM = 0.5 and C L = 0.5 was assumed. Alternatively drag closure equations obtained rom LB models can be used or the eective drag closure (van der Hoe et al., 004, 005). For the (possible) non-ideal particle-particle and particle-wall collisions we use a three parameter model accounting or normal and tangential restitution and riction. The associated collision parameters will respectively be denoted by e n, e t and μ t.
type 1 type type 3 Figure 1: Five generic types o interace conigurations considered in the computation o the luxes through the cell aces. Numerical solution method type 4 type 5 Computation o the low ield The Navier-Stokes equations can be solved with a standard inite volume technique on a staggered rectangular three-dimensional grid using a two-step projection-correction method with an implicit treatment o the pressure gradient and explicit treatment o the convection and diusion terms. A second order lux delimited Barton-scheme (Centrella and Wilson, 1984) is used or the discretization o the convection terms and standard second order central inite dierences or the diusion terms. We use a robust and very eicient Incomplete Cholesky Conjugate Gradient (ICCG) algorithm to solve the Pressure Poisson Equation (PPE). Computation o the surace orce In the CSF model (Brackbill et al., 199) the surace tension orce acts via a source term F σ in the momentum equation which only acts in the vicinity o the interace. The expression or F σ is given by 6 th International Conerence on Multiphase Flow, cell-centered components are computed by averaging over the eight surrounding vertices. To minimize the number o possible interace conditions which need to be considered, irst a number o transormations at the level o computational cells are carried out to achieve that the components n 1, n and n 3 o the normal to the interace are always positive and satisy the ollowing inequality: n1 < n < n 3 (11) where the subscript reers to the co-ordinate direction. These transormations involve i) change o co-ordinate directions ii) mutual interchange o co-ordinate directions and iii) interchange o the dispersed and continuous phase. Through the use o these transormations it is possible to reduce the total number o possible interace conigurations rom sixty our to ive generic ones which are schematically shown in Fig. 1. From these ive generic interace types the particular type prevailing in a certain Eulerian cell needs to be determined on basis o the known interace orientation (i.e. the normal vector to the interace) and the F-value o the interace cell. For the computation o the luxes through the cell aces the equation or the planar interace segment cutting through the Eulerian cell needs to be considered. This equation is given by: nξ + n ξ + nξ = d (1) 1 1 3 3 where ξ i (i = 1..3) represents the dimensionless co-ordinate in direction i given by: F = Fσκ m (8) σ x i ξ = i Δ x i (13) where the expression or the curvature is obtained rom the divergence o the unit normal vector to the interace: κ = 1 ( m n) = m ( ) m m m The normal to the interace is computed rom the gradient o the smoothed color unction. The smoothing technique used in this paper will be discussed later. Solution o the F-advection equation The integration o the hyperbolic F-advection equation is the most critical part o the VOF model and is based on geometrical advection which can be viewed as a pseudo-lagrangian advection step. The advantage o the geometrical advection is given by the act that a very sharp interace is maintained during the simulations. First or each Eulerian cell containing an interace the unit normal vector to the interace is estimated rom the gradient o the color unction F: n F F (9) = (10) In our model irst the components o the normal vector are computed at the vertices o the Eulerian cell and then the where Δx i represents the grid-spacing in co-ordinate direction x i (i = 1..3). The value o the plane constant d can be determined by equating the expression or the dimensionless liquid volume (volume below the planar interace segments shown in Fig. 1) to the known ractional amount o liquid or the F-value in the interace cell. The value o d can be obtained readily rom the root o these non-linear equations using the Newton-Raphson method which needs however to be done with care in order to ind the correct root o the cubic equations. As an alternative the Regula Falsi method can be used, which requires however an interval in which the root can be ound. This interval can be obtained on basis o the known interace orientation (i.e. components o the normal to the interace) and the ractional amount o liquid in the interace cell (i.e. the F-value) using simple geometrical considerations. One should keep in mind here that the solution o the non-linear equation needs to be carried out only or the interace cells. Once the aorementioned steps have been taken, inally the amount o liquid luxed through each o the aces o the Eulerian cells during a time step Δt can be computed. The F-advection equation is discretized with an explicit treatment o the convections terms, where a straightorward generalization o the D geometrical advection method given by Delnoij (1999) is used (also see Scardovelli and Zaleski, 1999). In our implementation o this method we have adopted the split advection scheme. Because the expressions or the luxes through the cell aces are quite lengthy they are
not given here. Finally the computed new F-values are corrected or (small) non-zero divergence o the velocity ield due to the iterative solution o the Pressure Poisson Equation (PPE). Smoothening o the color unction F As indicated beore the interace orientation (i.e. the normal to the interace) is computed rom the gradient o the color unction F according to Eq. (8). Basically this involves numerical dierentiation o a discontinuous unction leading in practice to (small) inaccuracies. This problem can be overcome however by making use o a smoothed color unction F or the computation o the unit normal to the interace using Eq. (8) with F replaced by F obtained rom: (14) F ( x) = D( x x ) D( y y ) D( z z ) F( xm ) m m m m where the smoothening unction D is given by the unction proposed by Peskin (1977): Dx ( ) = 1 x (1+ cos( π )) (15) h h or as an alternative by a suitable polynomial expression as the one proposed by Deen et al. (004b): 15 1 x x Dx = 16 h h h 4 ( ) ( ) ( ) 1 + (16) where h represents the width o the computational stencil used or the smoothening. The summation in Eq. (14) only involves the grid points with distance (in each separate co-ordinate direction) equal or less then the smoothening or ilter width h. We typically use h = Δ where Δ represents the Eulerian grid size and, unless otherwise stated. The width o the computational stencil or the smoothening should be selected careully. When the width is too small numerical instabilities may arise, especially in case the coeicient o surace tension is high. On the other hand when the width o the computational stencil is chosen too large, excessive smoothening ( thickening o the interace) is obtained which is undesirable. For the simulations reported in this paper we used Eq. (16) and additionally we used the smoothed color unction F instead o F in Eq. (8). It should be stressed here that this smoothed color unction is only used in conjunction with the estimation o the unit normal to the interace and not in the computation o the material luxes through the aces o the computational cells or which the unsmoothed color unction was used. Computation o the particle motion and particle-luid coupling The equation o motion or the particles is solved with a simple irst order integration scheme with linear implicit treatment o the eective drag and added mass orces. Where required the Eulerian quantities such as the liquid phase volume raction, pressure and velocity components as well as the spatial derivatives o these quantities are interpolated 6 th International Conerence on Multiphase Flow, rom the staggered Eulerian grid using volume-weighing. The non-ideal particle-particle and particle-wall collisions are taken into account with an event-driven computational scheme using eicient Molecular Dynamics (MD) type o schemes (see Hoomans et al., 1996 and Hoomans, 000). In the present version o the model we have not accounted or changes in the collision dynamics in case the particles are very close to each other (see Li et al., 1999). The particle-luid coupling requires Euler-Lagrange and Lagrange-Euler mapping or which we use a volume-weighing technique. Results Prior to the presentation o the main results the veriication o our computational model will be briely discussed which involves tests or the low solver and the two-way coupling o the dispersed phases. For a detailed test o the encounter model we reer to Hoomans et al. (1996) and Hoomans (000). Model veriication In our irst test the correctness o the implementation o the low solver was tested or ully developed laminar low in a square channel with diameter a. For this particular case the expression or the ully developed velocity proile can be derived analytically and is given by: a Δp x v = 1 ( ) z μl a n x y 1 1 ( 1) cos[( n+ ) π ] cosh[( n+ ) π ] n = 4 a a 1 3 1 n= 0 (( n+ ) π) cosh[( n+ ) π] (17) where the z-coordinate corresponds with the main low direction. For the simulation the density and viscosity o the luid were set to respectively 1000 kg/m3 and 1.0 kg/(m.s) whereas the z-velocity at the inlet was set to 1.0 m/s. In Fig. the dimensionless velocity proiles are shown or the central plane cutting through a square duct with diameter 0.0 m obtained rom the analytical solution and the solution obtained rom the numerical model using 40 computational cells in both the x- and y-direction and 100 cells in the low direction (duct length 1.0 m). In this igure the average velocity, obtained by integrating Eq. 17 over the cross-sectional area o the column, was used to obtain the dimensionless velocity. As evident rom Fig. the agreement between the two solutions is very good with a maximum relative error near the wall o 1.6%. No-slip boundary conditions were used or the walls o the duct, whereas or the inlet and outlet respectively prescribed (uniorm) inlow and prescribed pressure boundary conditions were taken.
6 th International Conerence on Multiphase Flow,.5.0 1.5 φ [ ] 1.0 0.5 0.0 0.000 0.005 0.010 0.015 0.00 x [m] Figure : Dimensionless velocity proiles or the central plane cutting through a square duct with diameter 0.0 m obtained rom the analytical solution (+) and the numerical solution (x) using 40 computational cells in both the x- and y-direction and 100 cells in the low direction (duct length 1.0 m). In our second test the correctness o the implementation o two-way coupling or the particles was tested by comparing the computed pressure gradient or liquid low through a ixed bed o particles with the pressure gradient according to the Ergun equation: dp (1 ε ) (1 ε ) = 150 + 1.75 dz ε d d μ U l 0 ρu l 0 3 3 ε p p (18) In this test the liquid phase low ield and associated pressure drop was computed or a stationary array o 5x5x15 particles with a diameter o 3.95 mm packed in a square column with a diameter o 0.1 m and a length o 1.0 m. Free-slip boundary conditions were used or the walls o the duct whereas or the inlet and outlet respectively prescribed (uniorm) inlow and prescribed pressure boundary conditions were taken. In this case the bed porosity equals 0.49579. The density and viscosity o the luid were set to respectively 1000 kg/m3 and 0.01 kg/(m.s) whereas the supericial velocity at the inlet was set to 0.1 m/s. For these conditions the pressure gradient according to the Ergun equation equals 0,335 Pa/m. From the numerical model, using 10 computational cells in both the x- and y-direction and 50 cells in the low direction, an average pressure gradient o 19,73 Pa/m was obtained leading to a relative error o 5.%. This error can be reduced urther by using more computational cells. Finally van Sint Annaland et al. (005) perormed extensive calculations using their VOF-model or gas bubbles rising in quiescent viscous liquids and demonstrated that the computed terminal rise velocities and shapes o the bubbles agreed very well with those obtained rom the Grace diagram (see next section) over a very wide range o Eötvös and Morton numbers, while using a high density and viscosity ratio characteristic or gas-liquid systems. They also showed that i) the size o the computational domain in the lateral directions should equal 3 to 4 times the diameter o the bubble to reduce the wall eect to an acceptable level ii) or Figure 3: Bubble diagram o Grace (1973) or the shape and terminal rise velocity o gas bubbles in quiescent viscous liquids. the gas bubble typically 16 to 0 computational cells (in each direction) are required to correctly predict the bubble characteristics (rise velocity and shape). Van Sint Annaland et al. (005) also successully computed the co-axial and oblique coalescence o two bubbles in a viscous liquid, a process which has been studied experimentally by Brereton and Korotney (1991). Clearly the agreement between the computed results and the results obtained rom the analytical expressions is very good and thereore our model can be applied with conidence to more complex problems such as gas-liquid-solid systems. Single gas bubbles Grace (1973) has analyzed a large body o experimental data on shapes and rise velocities o bubbles in quiescent viscous liquids and has shown that this data can be condensed into one diagram, provided that an appropriate set o dimensionless numbers is used. Although this diagram is strictly not valid or gas-liquid-solid systems it is nevertheless useul or reerence purposes. A copy o this diagram, taken rom Clit et al. (1978) is reproduced in Fig. 3 where the dimensionless Morton (M), Eötvös (Eo) and Reynolds (Re b ) are given by: g Δ M = μ ρ ρ σ 4 l 3 l * (19) gδρd Eo = e σ (0) ρ vd l e Re = b μ (1) l
6 th International Conerence on Multiphase Flow, Table 1: Data used or the simulations shown in Fig. 4. Computational grid 40x40x100 (-) Grid size 0.001 m Number o particles 0 (a) 0,000 (b) 40,000 (c) 60,000 (d) 80,000 (e) 100,000 () Time step 10-4 s Initial bubble radius 0.01 m Initial bubble position, (0.0, 0.0, 0.0) m (x 0,y 0,z 0 ) Particle density 3000 kg/m 3 Particle diameter 400 μm Collision parameters e n, e t, μ t 0.9, 0.33, 0.10 (-) Liquid density 1000 kg/m 3 Liquid viscosity 0.1 kg/(m.s) Gas density 10 kg/m 3 Gas viscosity 1.0 10-3 kg/(m.s) Surace tension 0.1 N/m Morton number 9.71 10-4 (-) Eötvös number 38.8 (-) t = 0.05 s t = 0.10 s t = 0.15 s t = 0.0 s Figure 4: Rise o a single gas bubble (initial diameter 0.0 m, Eo = 38.8, M = 9.71 10-4 ) through a suspension o N p solid particles (See Table 1 or additional data). From top to bottom N p increases rom 0 to 100,000. where the eective diameter d e is deined as the diameter o a spherical bubble with the same volume as the bubble under consideration. In our simulations a ixed density and viscosity ratio o one hundred was used (viscosity and density o the continuous liquid phase equal one hundred times the viscosity and density o the dispersed gas phase). First the eect o the presence o solid particles on the rise velocity o isolated gas bubbles is assessed. A typical result is presented in Fig. 4, showing the rise o a single gas bubble through a suspension o N p solid particles. The particles are initially at rest and are randomly distributed over the bottom section o the domain, excluding the volume occupied by the gas bubble. Initially the bubble is spherical and the suspension is quiescent. For all simulations reported in this paper ree-slip boundary conditions were applied at the conining walls. The data used or these simulations are given in Table 1. For this particular case the Eötvös and Morton number, based on the physical properties o the pure (i.e. without solid particles) liquid, are respectively equal to 38.8 and 9.71 10-4, which according to the Grace diagram should correspond to a terminal bubble Reynolds number Re b o 60. For the pure liquid (i.e. without particles) the computed Re b equals 46 or all cases and is lower than the value according to the Grace diagram. This discrepancy can be attributed to the relatively small lateral size o the container. For the investigated range o solids volume raction the retarding eect o the particles on the rise velocity o the bubble is quite small, although small changes in the bubble shape can be discerned. From the series o snapshots given in Fig. 4, the particle drit caused by the rising gas bubble can be clearly seen. While the bubble ascends through the suspension the particles are pushed aside and accumulate at the bubble base.
6 th International Conerence on Multiphase Flow,.0 3.0 1.5.5 X d [-] 1.0 X d [-] 0.5.0 0.0 0.00 0.05 0.10 0.15 0.0 0.5 0.30 t [s] Figure 5: Dimensionless volume X d o the drit zone behind the rising bubble as a unction o time. Total number o particles: 100,000 (case given in Table 1). In Fig. 5 the dimensionless volume o the drit zone X d is plotted as a unction o time or a solids volume raction ε s = 0.055 (case in Table 1). All particles which are present above the initial (horizontal) interace o the suspension were assigned to the drit zone and rom this number o particles N d the equivalent volume o the initially uniorm suspension was calculated. This volume was inally divided by the bubble volume V b to obtain the dimensionless volume o the drit zone X d : X N V d s = () d N V p b where V s is the initial volume occupied by the suspension. From this igure it can be seen that this dimensionless volume increases steadily and reaches quite high values (corresponding to twice the bubble volume) indicating appreciable transport o the particles due to the passage o the bubble. In Fig. 6 the dimensionless volume o the drit zone X d is plotted as a unction o the solids volume raction ε s (cases b to in Table 1) 0.30 s ater bubble release. Apparently the bubble becomes less eective with respect to the vertical particle transport with increasing solids volume raction. Multiple gas bubbles The irst example involves the co-lateral rise o two gas bubbles in an initially quiescent suspension where the bubbles are initially spherical with their centers separated by 3. bubble radii (see Table or additional inormation). In Fig. 7 the computed evolution o the bubble shapes is shown as the bubbles rise through the suspension. The Morton and Eötvös number or this case are respectively equal to 10-4 and 16 (based on the individual bubbles) which, according to the Grace diagram, would correspond with a terminal Reynolds number o 55 (case indicated with an asterisk in Fig. 3) which agrees reasonably well with the computed value o 40. The behavior o the bubbles is rather similar to that o that o a single bubble as can be seen clearly rom Fig. 7. The bubbles do neither attract nor repel each other. 1.5 0.00 0.01 0.0 0.03 0.04 0.05 0.06 ε s [-] Figure 6: Dimensionless volume X d o the drit zone as a unction o the solids volume raction ε s at t = 0.30 s ater bubble release. The second example involves the oblique coalescence o three gas bubbles in an initially quiescent suspension where the bubbles are initially spherical and positioned as one leading bubble, ollowed by two trailing bubbles. Compared to the irst case the two trailing bubbles are at the same position, while the leading bubble is newly added to the simulation domain. In this case, all parameters are kept unchanged, i.e. the Morton and Eötvös number are the same as or the irst case. In Fig. 8 the computed evolution o the bubble shapes is shown beore and ater the coalescence process. Similar to the previous case the leading bubble behaves as an isolated bubble although it is lattened, as the two trailing bubble enter its wake region. The trailing bubbles approach each other and experience considerable shape deormation when they enter the wake region o the leading bubble. Subsequently they coalesce and eventually they catch up with the leading bubble. Table : Data used or the multiple bubble simulations. Computational grid 80x80x160 (-) Grid size 5 10-4 m Number o particles 100,000 Time step 5 10-5 s Initial bubble radius 0.005 m Initial bubble positions, (x 0,y 0,z 0 ) (0.01, 0.0, 0.01) m, (0.08, 0.0, 0.01) m, and (0.0, 0.0, 0.05) m Particle density 3000 kg/m 3 Particle diameter 00 μm Collision parameters e n, e t, μ t 0.9, 0.33, 0.10 (-) Liquid density 1000 kg/m 3 Liquid viscosity 4.667 10 - kg/(m.s) Gas density 10 kg/m 3 Gas viscosity 4.667 10-4 kg/(m.s) Surace tension 6.131 10 - N/m Morton number.00 10-4 (-) Eötvös number 15.8 (-)
6th International Conerence on Multiphase Flow, Paper No 71 t = 0.000 s t = 0.05 s t = 0.050 s t = 0.075 s t = 0.100 s t = 0.15 s t = 0.150 s t = 0.175 s t = 0.00 s Figure 7: Snapshots at dierent times o the co-lateral rise o two initially spherical bubbles o 0.010 m diameter released rom positions (0.01 m, 0.00 m, 0.010 m) and (0.08 m, 0.00 m, 0.010 m) in an initially quiescent liquid in a square column o 0.04 m x 0.04 m x 0.08 m, using a 80x80x160 grid and a time step o 5 10-5 s.
6th International Conerence on Multiphase Flow, Paper No 71 t = 0.000 s t = 0.05 s t = 0.050 s t = 0.075 s t = 0.100 s t = 0.15 s t = 0.150 s t = 0.175 s t = 0.00 s Figure 8: Snapshots at dierent times o the oblique rise o three initially spherical bubbles o 0.010 m diameter released rom positions (0.01 m, 0.00 m, 0.010 m), (0.08 m, 0.00 m, 0.010 m) and (0.00 m, 0.00 m, 0.05 m) in an initially quiescent liquid in a square column o 0.04 m x 0.04 m x 0.08 m, using a 80x80x160 grid and a time step o 5 10-5 s.
X d [-] 10 8 6 4 0 0.00 0.05 0.10 0.15 0.0 0.5 0.30 Figure 9: Dimensionless volume X d o the drit zone behind the rising bubble as a unction o time or co-laterally rising bubbles (closed symbols), and the oblique rising bubbles (open symbols). In Fig. 9 the results o these two calculations are compared in terms o X d computed as a unction o time. It is seen that the volume o the drit zone gradually increases and shows a ew local maxima, which are indicated in the igure by the arrows. At those moments bubbles break through the top o the suspension. As the bubbles themselves do not contain any particles, the breakthrough leads to small dips in the X d curve. The case with the three oblique bubbles gives rise to the largest drit volume, which is our to ive times larger than the drit volume o a single bubble. Conclusions In this paper a hybrid model has been presented enabling the direct numerical simulation o gas-liquid-solid lows using a combined Volume o Fluid (VOF) and Discrete Particle (DP) approach applied or respectively dispersed gas bubbles and solid particles present in the continuous liquid phase. The hard sphere DP model, originally developed by Hoomans et al. (1996) or dense gas-solid systems, has been extended to account or all additional orces acting on particles suspended in a viscous liquid and has been combined with the VOF model presented recently by van Sint Annaland et al. (005) or complex ree surace lows. Our model generates very detailed inormation on bubble induced particle mixing and can in principle be used to study and quantiy the eect o all important physical properties o the gas-liquid-solid system and other variables (such as bubble size and shape) as well. The model has been successully applied to compute the particle drit behind rising gas bubbles and revealed that in case multiple bubbles rise, the bubbles compete with each other or the particles driting behind them. Clearly computational resources constitute a limiting actor at present. Our model is quite general and it can in principle also be used to study the behavior o small bubbles (replacing the particles) in the presence o large bubbles provided that appropriate closure equations are used or the (eective) orces acting on the small (unresolved) bubbles. t [s] 6 th International Conerence on Multiphase Flow, Nomenclature a Computational domain in x-direction (m) C d Drag coe. or an isolated spherical particle (-) C L Lit coeicient (-) C VM Virtual mass coeicient (-) d Plane constant or interace segment cutting through Eulerian cell (-) d e Equivalent bubble diameter (m) d p Particle diameter (-) D Distribution or smoothening unction (-) e n Normal coeicient o restitution (-) e t Tangential coeicient o restitution (-) Eo Eötvös number (-) F Fractional amount o liquid (-) F Smoothed color unction (-) h Width o the computational stencil or the smoothening unction (m) Measure or Eulerian grid size (m) L Length o square duct (m) M Morton number (-) m p Particle mass (kg) N p Number o suspended solid particles (-) n i i th component o the unit normal vector (-) p Pressure (N/m ) Re b Reynolds number (-) t Time (s) U 0 Supericial velocity (m/s) V b Bubble volume (m 3 ) V p Particle volume (m 3 ) V s Initial suspension volume (m 3 ) v Terminal bubble rise velocity (m/s) X d Dimensionless volume o drit zone (-) x i i th co-ordinate direction (m) x x co-ordinate (m) y y-co-ordinate (m) z z-co-ordinate (m) Greek letters β p Eective luid-particle riction coeicient (kg/(m 3.s)) φ Dimensionless liquid velocity (-) ε Liquid phase volume raction (-) ε s Solid phase volume raction (-) κ Curvature (m -1 ) μ Dynamic viscosity (kg/(m.s)) μ t Coeicient o riction (-) ρ Density (kg/m 3 ) Δρ Density dierence (kg/m 3 ) Δx i Grid spacing in i th co-ordinate direction (m) σ Surace tension (N/m) Dimensionless i th co-ordinate direction ξ i Vectors Gravity orce (N) F g F p Pressure orce (N) F d Drag orce (N) F L Lit orce (N) F VM Virtual mass orce (N) F σ Volumetric surace tension orce (N/m 3 ) F Volumetric orce due to luid-solid coupling (N/m 3 ) s
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