Review of Numerical Methods for Multiphase Flow

Transcription

1 Title Review o Numerical Methods or Multiphase Flow M. Sommereld Mechanische Verahrenstechnik Zentrum ür Ingenieurwissenschaten Halle (Saale), Germany www-mvt.iw.uni-halle.de

2 Content o the Lecture Classiication o multiphase lows Classiication o numerical methods or multiphase lows. Particle-resolved direct numerical simulations Interace tracking method or bubbly lows Lattice-Boltzmann method (particle resolved) Direct numerical simulations with point-particles Approaches with turbulence modelling (point-particles) Euler/Euler (two luid) approach Euler/Lagrange approach Summary/Conclusions

3 Classiication o Multiphase Flows 1 Multiphase lows may be encountered in various orms: Transient twophase lows Separated twophase lows The two-luid concept is suitable, but requires methods or handling the interace (e.g.tracking, VOF,...) Dispersed twophase lows

4 Classiication o Multiphase Flows Examples o separated multiphase lows: slug low churn low Slug low in oil-water-gas pipe low

5 Classiication o Multiphase Flows 3 Dispersed two-phase low systems and important technical and industrial applications. Continuous/Dispersed Phase Gas-solid lows Industrial/Technical Application pneumatic conveying, particle separation in cyclones and ilters, luidised beds, coal combustion Liquid-solid lows hydraulic conveying, liquid-solid separation, particle dispersion in stirred vessels Gas-droplet lows spray drying, spray cooling, spray painting, spray scrubbers, spray combustion Liquid-droplet lows Liquid-gas lows mixing o immiscible liquids, liquid-liquid extraction bubble columns, aeration o swage water, lotation Moreover, numerous examples o multiphase lows may be ound in industry, such as: liquid-gas-solids reactors or spray scrubbers (droplets and particles dispersed in a gas low)

6 Classiication o Multiphase Flows 4 Examples o dispersed gas-solid lows: Dilute two-phase low aerodynamic transport Dense two-phase low particle-particle interaction Gas-Solid Flow; Euler/Lagrange Calculation (Helland et al. 000)

7 Classiication o Multiphase Flows 5 Estimation o the interaction between particles based on the inter-particle spacing or a regular cubic arrangement: 1/3 L π D P 6 αp Inter-particle distance or a cubic arrangement L Regimes o dispersed two-phase lows: Example: gas-solids low : η 1.0 α p p 500 kg/m 3 F 1.18 kg/m 3 L/D P One-Way Coupling inter-particle spacing L / D P Dilute Dispersed Two-Phase Flow Two-Way Coupling α P,max Dense Dispersed Two-Phase Flow Four-Way Coupling E-8 1E-7 1E-6 1E-5 1E-4 1E volume raction [-]

8 Classiication o Multiphase Flows 6 Dispersed multiphase lows are characterised by the ollowing properties o the particle phase. Integral values or characterising two-phase lows: volume raction o the dispersed phase mass raction, porosity Characteristics o the particles: size distribution particle shape, porosity surace and surace structure Particle motion in luids: velocity distribution o particles luctuating velocity o particles

9 Classiication o Multiphase Flows 7 Volume raction o the particle phase and porosity: Ni VPi i αp ε 1 α P V Eective density o both phases (or bulk density): b P cp αp Mixture density: b ( P ) F Number concentration (particles per unit volume): N n P P V P F 1 α ( αp ) F + αp P b b m F + P 1 Mass loading o particles (generally only or gas-solid lows): m P αp P UP η m 1 α U F ( P ) F F

10 Numerical Methods Multiphase Flow 1 Particle methods: Numerical calculation o the behaviour o bulk solids with and without low: Molecular dynamics (MD) Granular dynamics (GD) Discrete element methods (DEM) Discharge o bulk solids rom a silo Full direct numerical simulation (DNS): Resolution o particle contour and low around the particle: contour-adapted grid Interace tracking Volume o Fluid (VOF) Lattice Boltzmann (LBM) Pro. D.D. Joseph

11 Numerical Methods Multiphase Flow Direct numerical simulations (DNS) and large eddy simulations (LES): Point-particle assumption or turbulence studies Homogeneous isotropic turbulence Numerical methods or dispersed multiphase lows (RANS-type methods): Reynolds-averaged conservation equations with turbulence model, pointparticle assumption: Mixture models Euler/Euler (two-luid) approach Population balance method Euler/Lagrange approach

12 Numerical Methods Multiphase Flow 3 Application o dierent numerical approaches: Taylor-bubble with VOF Point-particle-DNS or turbulent low RANS-calculation or technical systems St 0.5 Increasing modelling requirements

13 Particle-Resolved Simulations 3 The volume-o-luid method (VOF): The interace is being reconstructed based on the solution o a transport equation or the volume raction. t + u 0 Real interace location VOF (Hirt & Nichols) Young`s VOF

14 Particle-Resolved Simulations 4 Inluence o shear low on bubble migration calculated by a VOF approach (Tomiyama et al. 1993) Eo 1, Mo 10-3 Eo 10, Mo 10-3 Eo g ( ) σ g D b Mo g µ 4 ( ) σ 3 g Determination o lit orces

15 Particle-Resolved Simulations 5 Flow around nearly spherical and large non-spherical bubbles simulated by VOF (Bothe et al. 007) Eo h g ( ) σ g D h C Tomiyama correlation A min ( 0.88 tan h ( 0.11 ReB )), ( Eoh ) ( Eo ) h ür : ür : Eo h < 4 4 Eo h 3 ( Eo ) Eo Eo Eo h h h h +

16 Particle-Resolved Simulations 6 Interace resolved direct numerical simulations allow a detailed analysis o the atomisation process Numerical grid: Mesh size:.36 µm Numerical grid: Mesh size: 1.17 µm BERLEMONT 008 Level Set/Ghost Fluid Method

17 Interace Tracking Approach 1 Direct numerical simulations o the bubble motion using a interace tracking approach (Pro. Tryggvason). Time-dependent solution o the three-dimensional, Navier-Stokes equations. u T + u u p + ( 0 ) g + µ ( u + u ) + Fs δ( x x ) da t The surace tension orce is calculated rom the curvature o the interace. Incompressible gas- und liquid-phase: Equation o state or the density and the viscosity: t u 0 + u 0 µ t + u µ 0 F Surace tension

18 Interace Tracking Approach Rectangular grid or the low domain. Interace grid to mark the position o the interace. The interace is resolved within -3 meshes o the grid resolving the low domain. Spatial Discretisation: nd-order central dierences Time Discretisation: nd-order explicit method Density ratio: b 0 Periodic boundary conditions at the cube aces

19 Interace Tracking Approach 3 Binary mixture o bubbles, instantaneous distribution, case 1, Simulation or 9 small and 9 large bubbles (α 6 %). volume ratio: diameter ratio: 1.6 Ga 600, 100 Eo 0.63, 1.0 Ga ( ) g µ g D 3 b Eo g ( ) σ g D b

20 Interace Tracking Approach 4 Binary mixture case (Göz and Sommereld 004): Volume Ratio: 8 Diameter Ratio: Ga 900, 700 Eo 1, 4 α 6 %: 40 small, 5 large α 1 %: 64 small, 8 large

21 Interace Tracking Approach 5 Direct numerical simulations o bubble motion in a cube by accounting or interace deormation. Behaviour o bi-disperse bubble systems (case ) α %: 8 small, 4 large α 6 %: 40 small, 5 large

22 Interace Tracking Approach 6 Turbulent kinetic energy or case 1 and : temporal development turbulent kinetic energy in dependence o gas hold-up

23 Interace Tracking Approach 7 Bubble Reynolds number in dependence o gas hold-up: Case 1 Re D µ b b d 1 U ( α ) b

24 Interace Tracking Approach 9 Trajectories o non-spherical bubbles obtained by interace tracking method (Hua et al. 007) zigzag trajectory helical trajectory

25 Lattice Boltzmann Method 1 Lattice-Boltzmann method is based on the simulation o discrete luid elements in order to predict the macroscopic low system. The LBM is very robust and suitable or complex geometries. The basic variable o the Boltzmann statistics is the distribution unction (x, v, t) which declares the number o luid elements having the velocity v at the location x and time t: Macroscopic properties are related to the moments o the probability unction: m ( x, t) 3 v ( x, t) u( x, t) 3 v 3 3 v x v d 3 v 3 3 ( x, v, t) d x d v d 3 v

26 Lattice Boltzmann Method Boltzmann equation (rate o change due to transport and collision) with single relaxation approach (Bhatnagar Gross Krook (BGK) equation): The Lattice Boltzmann equation (post collision distribution) arises rom the discretisation o the BGK equation in time, space and velocity: σi ( x + v σi + v t t, t ( 0 ( x, v, t) ( x, v, t) ) ( x, v, t) + t) Propagation term σi 1 τ ( x, t) ( ) t τ ( (0) ( x, t) ( x, t) ) σi σi Collision term τ Macroscopic properties, discretised: ( x, t) ( x, t) σ i σi ( x, t) u( x, t) v ( x, t) σ i σi σi v σi x t σi

27 Lattice Boltzmann Method 3 Discrete velocity vectors o the D3Q19 model: Three-dimensional Spatial discretisation by regular grid (voxels) 19 discrete velocity directions 1 1 Sequential solution: Propagation step Collision step Velocity vectors in the dierent directions: v σi (0,0,0), ( ± 1,0,0)c, (0, ± 1,0)c, (0,0, ± 1)c ( ± 1, ± 1,0)c, ( ± 1,0, ± 1)c, (, ± 1, ± 1)c, σ 0, σ 1, σ, v x t σi σi i 1 i 1 6 i 1 1 Lattice constant x c t

28 Lattice Boltzmann Method 4 Discrete equilibrium distribution unction (Maxwellian distribution or Kn << 1): 3v σi u(x, t) 9(vσ (x, t) ωσ 1+ + c u(x, t)) 4 c 3u (x, t) c (0) i σi Pressure (equation o state): p ( x, t) (x, t) c s c s c 3 ω σ 1, 3 1, 18 1, 36 σ 0, σ 1, σ, i 1 i 1 6 i 1 1 From a series expansion around the equilibrium distribution (Chapman- Enskog-Expansion) the dependence o the viscosity on the relaxation parameter (e.g. τ ollows: ν 1 c 6 ( τ t)

29 Lattice Boltzmann Method 5 Standard wall boundary condition: Curved wall boundary condition: Local grid reinement: Forces over a particle are obtained rom a momentum balance (relection o the luid elements) 6 Cells per Agent Particle

30 C LS [ - ] I N L E T 10 1 Validation: Particle Sitting on a Wall under Shear Flow Moving Wall Non-Slip O U T L E T Resolution: 40 grid cells o the inest mesh Re S [ - ] Saman Leighton & Acrivos sim. Derksen & Larsen sim. Zeng et al. correlation Zeng et al. present LBM C D [ - ] x/d 18, y/d 1 Re s z / D [ - ] Re D P Domain Size x/d 18, y/d 18, z/d 18 Three reinement regions S µ G

31 Microscale Simulations by LBM Lattice-Boltzmann simulations are perormed or a carrier particle (100 µm) randomly covered with the drug particles (3, 5 or 10 µm); Cui et al. 014: The particle cluster is centrally ixed in a cubic domain I N L E T Symmetric Symmetric O U T L E T Re < 100: X 7.8 D carrier ; Y Z 6.5 D carrier Re > 100: X 10.4 D carrier ; Y Z 9.1 D carrier Determination o luid orces on the drug particles or low conditions obtained by RANS calculations

32 Flow Structure at Dierent Re Present simulations Measurement o Taneda Re 16 Re 17.9 Re 3 Re 37.7 Re 100 Re 104 Re 00 Re 0

33 Force on Drug Particles Depending on Position Angle Total orce on all the ine particles in dependence o position angle or our dierent random distributions (colored dots) and resulting polynomial itting curve (Re 100, coverage degree 50 %, D ine /D carrier 5/100). 4.0x10-9 Data points Polynomial it Re x10-9 F total [N].0x x Position Angle [degree] Dierent drug particle distribution

34 Dierent Reynolds Number Fitting curves or the normal orce on ine particles as a unction o position angle or dierent Reynolds numbers (coverage degree 50 %, D ine /D carrier 5/100) F vdw 35 nn.0x x10-9 Fitting curve: Re 70 Re 140 Re 00 F 0.0 n [N] Position Angle [degree] -1.0x x10-9 Direct lit-o is not likely to occur, hence detachment occurs by sliding or rolling

35 Flow Resistance o Agglomerates Resistance coeicients or dierent agglomerates (Dietzel & So 013): O1 O Aerodynamic coeicient [-] drag lit torque O3 O4 0 L_30 C_30 VES Agglomerate type agglomerates with identical volume equivalent diameter 30 primary particles 50 grids per volume equivalent size porosity o convex hull 0 (dendrite: 0.78, compact: 0.57) 0 Reynolds number o agglomerate: 0.3 Drag and lit coeicient c_d, c_l [-] c_d Sphere Agg_O1 Agg_O Agg_O3 Agg_O4 c_l

36 Direct Numerical Simulations 1 Direct numerical simulations (DNS) or dispersed turbulent two-phase lows by considering the particles as point-particles and using a Lagrangian approach to simulate the dispersed phase (all real particles). The grid needs to resolve all turbulence structures (i.e. Kolmogorov scale). The calculations are limited to smaller low Reynolds numbers. The particles need to be smaller than the grid size and smaller than the Kolmogorov scale. point-particles!!! The equation o motion needs to be solved by accounting or all relevant particle orces (generally Stokes low). DNS has been applied mainly to basic turbulence research, in order to analyse the particle behaviour in turbulent lows and to derive closure relations or modelling approaches.

37 Direct Numerical Simulations Direct Numerical Simulation (DNS): time-dependent solution o the threedimensional conservation equations by resolving the smallest turbulent scales. Continuity equation: Momentum equations: t ( u ) + x i i 0 ( u ) ( u j ui ) t i + x j p x i τ x ij j + S P,u i + g i Stress tensor: τ ij µ u x i j + u x j i + 3 µ u x j j δ ij Kronecker symbol: δ ij Particle source terms are obtained rom all luid dynamic orces acting on all particles in a control volume

38 Direct Numerical Simulations 3 Direct numerical simulations (96 3 ) on turbulence modiication by particles in isotropic turbulence (Bovin et al. 1998): Re L T E 0.0 m / τ 6 10 F ( τ / τ ) 1.6, 4.49, λ K K φ 0 Turbulent kinetic energy Particle dissipation Dissipation rate St St

39 Direct Numerical Simulations 4 Analysis o particle preerential accumulation in homogeneous isotropic turbulence (Scott and Shrimpton 007): Domain size: 64 3 Enstropy Velocity ield St 4.0 St 0.5

40 RANS-Approaches 1 The numerical calculation o industrial low processes is generally based on the Reynolds-averaged Navier-Stokes (RANS) equations. Turbulence modelling (e.g. k-ε turbulence model, Reynolds-stress model) Approaches or dispersed multiphase lows: Euler/Lagrange Approach Euler/Euler Approach

41 Modelling Strategy With reduced complexity o the numerical method used, the modelling (closure) requirements are increasing: Particle resolved simulations (ull DNS) RANS methods with point particles Increased modelling Modelling approach: Modelling o microphysical phenomena: non-spherical particles particle-wall collisions droplet coalescence bubble break-up Experimental studies Direct numerical simulations The resulting analytical and/or empirical models have to be implemented and validated based on detailed experiments.

42 Two-Fluid Approach 1 Euler/Euler approach (two-luid approach) Both (multiple) phases are treated as interpenetrating continua. The properties o the dispersed phase have to be averaged or the control volumes (D P << x). Similar sets o conservation equations are obtained or both phases, allowing or identical solution algorithms. Requires considerable modelling work to describe the relevant microphysics (closure o the conservation equations): Interaction between the phases Turbulent dispersion o particles (luid-particle correlation) Wall collisions o particles (wall roughness eect) The consideration o size distributions requires the solution o several sets o conservation equations (or each phase). Numerical diusion at particle phase boundaries may result in errors. This approach is especially suitable or high volume ractions o the dispersed phase. The two-luid approach might be also coupled to a population balance.

43 Two-Fluid Approach Classiication o multi-luid models: Mixture Models Drit lux model: The slip between the phases is calculated by analytical correlations Homogeneous model: All phases share the same velocity ield (no slip) Complete Multi- Fluid Model Reduced turbulence model: Turbulence model only or the continuous phase The luctuation energy o the dispersed phase is related to the luid turbulence by appropriate correlations Multiphase turbulence model: For each phase conservation equations are solved or the turbulence properties including coupling

44 Two-Fluid Approach 3 The averaging o the conservation equations or a multiphase low results in the ollowing set o equations (Simonin 000): Continuity equations: Momentum equation continuous phase: ( ) ( ) ( ) ( ) ( ) n n,i n n i n n i i C m U x t 0 U x t α + α α + α ( ) α α + µ + α α N n Pi n n j,,i j j j,i i j,i j,,i m F u u x x x U x P x U U t U Mass transer due to collision/coalescence

45 Two-Fluid Approach 4 Momentum equation dispersed phase: α n n U t n,i + α n n, j Momentum transer due to drag orce: n U U x ( u ) Turbulent dispersion described by drit velocity: n,i j 18 µ n + C m P x n i x n,i j + α ( α u u ) n n n g n i n, j + α {( U U ) V } D F n,i m P,i n,i + P DP n,i n n,i n F m n,i n V n,i D n,ij 1 α n α x n j 1 α α x j

46 Diusion coeicient: D τ u u n,ij n Two-Fluid Approach 5,i n, j Integral time scale o turbulence seen by the particles (Csanady): τ τ ll n n T L 1+ C T β L 1+ 4 C ξ β r ξ r ξ r u 3 k C 0.45 β Requires inormation on turbulent integral time scale

47 Two-Fluid Approach 6 Simple correlations or stationary homogeneous isotropic turbulence: Fluid-Particle velocity correlation: u,i u n, j u,i u, j ηr 1+ η Kinetic energy o particle luctuating motion: r η r τ τ n n u n,i u n, j u,i u, j ηr 1+ η r In the case o more complex two-phase lows additional transport equations or both properties have to be solved. Simonin, 1991, 000

48 Fluidised Bed Numerical computation o a bubbling luidised bed with the two-luid approach (see Sommereld 013). d p ~ 140µm p / g ~ U g ~.5m/s Δx 30cm Δx 14cm Δx 10cm Δx 3cm

49 Euler/Lagrange Approach 1 Euler/Lagrange Approach The luid low is calculated by solving the Reynolds-averaged Navier- Stokes equations (or LES) with an appropriate turbulence model. The dispersed phase is simulated by tracking a large number o particles through the low ield (representative particles). A statistically reliable determination o the particle phase properties and source terms requires a large number o particles to be tracked. This method is a hybrid approach and requires coupling iterations between Eulerian and Lagrangian part. The particles are point-particles which have to be considerably smaller then the size o the grids (D P << x). The particle size distribution may be considered with good resolution. The relevant micro-processes may be : Wall collisions o particles Inter-particle collisions and agglomeration Droplet/bubble coalescence and break-up For high particle concentration the standard approaches may cause considerable convergence problems.

50 Euler/Lagrange Approach Fluid low; continuous phase (Sommereld 1996; Lain & Sommereld 01): General orm o the conservation equations in connection with the k-ε turbulence model: t σ x ( σ φ) + ( σ U j φ) Γ + S + φ Sφ P ( α ) 1 j P x j φ x j eective density φ S φ S φ,p Γ φ U i k U p j U x Γ + i j x i x i ε k C G G k ε ε ( C ε) G U 1 k U U i j i k µ t +, x j x i x j g i SU i,p µ Cµ t S µ k,p S µ ε,p k ε µ + µ t + µ σ µ t k + t σ ε constants C µ 0.09 C C 1.9 σ k 1.0 σ ε 1.0

51 Euler/Lagrange Approach 3 Dispersed phase: ( ) ( ) ( ) ( ) i P i P P F P F LR p F F p F P LS p F P P,i i D P P P P,i P F 1 g m u u u u C D 4 u u D C D 4 u u u u C m D 4 3 d t d u m + + Ω Ω π + ω π + The Lagrangian approach is based on tracking a large number o representative particles through the low ield considering all relevant orces. drag orce gravity/ buoyancy slip-shear lit other orces, e.g. electrostatic slip-rotation lit Ω Ω ω R 5 p F p p C D dt d I Rotation: p p u dt x d Models elementary processes: turbulent dispersion particle-rough wall collision inter-particle collisions

52 Euler/Lagrange Approach 6 Grid generation, Boundary conditions, Inlet conditions Eulerian Part Calculation o the luid low without particle phase source terms Lagrangian Part Tracking o parcels without interparticle collisions, Sampling o particle phase properties and source terms Two-way coupling approach or stationary lows Under-relaxation o source terms: S i+ 1 φp i i+ 1 ( 1 γ) Sφ P + γ Sφ P( calculated) Under-relaxation o the source terms improves convergence behaviour!!! (Kohnen et al. 1994) Coupling Iterations no Eulerian Part Calculation o the luid low with particle phase source terms: Converged Solution Solution with a ixed number o iterations Lagrangian Part Tracking o parcels with inter-particle collisions, Sampling o particle phase properties and source terms Convergence two-way coupling yes Output: Flow ield, Particle-phase statistics

53 Conclusion Numerical methods or multiphase lows were subdivided into three classes: Full DNS methods resolving the particles; or or interacial systems. Methods or dispersed multiphase lows with point particle approximation (DNS, LES and RANS). Discrete particle methods (e.g. DPM and DEM) or dense particle systems (with and without low). Particle resolved numerical simulations (ull DNS) become increasingly important or analysing micro-physical phenomena and providing results or modelling. Point-particle DNS is mainly used or turbulence studies. For analysing or optimising processes o technical relevance, RANS approaches are still very important (i.e. Euler/Euler and Euler/Lagrange). However, the importance o LES is continuously increasing. The Lagrangian approach has considerable advantages in modelling elementary processes and is preerred or systems with particle size distributions.