Title Strategy in modelling irregular shaped particle behaviour in confined turbulent flows M. Sommerfeld F L Mechanische Verfahrenstechnik Zentrum Ingenieurwissenschaften 699 Halle (Saale), Germany www-mvt.iw.uni-halle.de T F D
Content of the Lecture Introduction to non-spherical-particles treatment in a Lagrangian approach Statistical modelling of irregular shaped particles Evaluation of flow resistance coefficients for irregular-shaped particles by the Lattice-Boltzmann method Derivation of PDF`s for the resistance coefficients Experimental analysis of irregular particle wall collisions using high-speed cameras Derivation of PDF`s for the restitution ratios Validation of model calculations for a horizontal channel flow Conclusions and outlook
Introduction 1 Numerical predictions of dispersed two-phase flows by the Euler/Euler- or Euler/Lagrange approach quite often rely on the assumption of spherical particles (fluid-dynamic forces and wall collision process). However, in most practical situations and industrial processes the particles are irregular in shape or have certain geometrical shapes, (e.g. fibres, cylinders, discs, ) or are even agglomerates. granulates irregular particles
Introduction 2 Lagrangian treatment of non-spherical particles Regularly shaped nonspherical particles Irregular non-spherical particles Deterministic (forces) tracking of particles, new location, linear and translational velocities Requires additionally tracking of the particle orientation using Euler angles/parameters or quaternions Particle orientation-dependent resistance coefficients are required (Hölzer and Sommerfeld 29) Wall collisions: Solving the impulse equations for non-spherical particles considering particle orientation (Sommerfeld 22) Stochastic tracking of particles, new location, linear and translational velocities Determine the instantaneous values of the resistance coefficients from a- priory determined distribution functions (random process), e.g. by Lattice-Boltzmann simulations Apply these random resistance coefficients in the equation of motion each time step Calculate the particle velocity change during a wall collision process from measured distribution functions of the restitution ratios
Lattice-Boltzmann Method 1 Lattice-Boltzmann equation: Behaviour of fluids on mesoscopic level f σi ( x + ξ t, t + t) f ( x, t) = f ( x, t) f ( x, t) σi Key variable: Discrete distribution function fσi Discretization of space by a regular grid Discretization of the velocity space: D3Q19 Macroscopic parameters (density, momentum): Derived as moments of f σi Iteration loop: Relaxation (t+) & Propagation (t+ t) σi t τ ( σi σi ) + t Fσ i Point of time: t Point of time: t+ Point of time: t+ t
Lattice-Boltzmann Method 2 Standard wall boundary condition: Curved wall boundary condition: Forces over a particle are obtained from a momentum balance (reflection of the fluid elements) Local grid refinement
Simulations for Non-Spherical Particles 1 Numerical calculation of a plug flow about an irregular shaped particle fixed in a cubic domain using the boundary conditions specified below. Determination of the resistance coefficients (drag, lift and torque) in dependence of particle Reynolds number. symmetry boundary condition dv dy = I N L E T Symmetric O U T L E T stress-free boundary condition dv dx = Symmetric
Simulations for Non-Spherical Particles 2 The simulations were conducted for four different shaped particles with about the same sphericity (D VES,hull 1 µm, ψ.87) Sample 1 Sample2 Sample3 Sample4 For each Re-number the particle was randomly rotated to 71 positions with respect to the inflow in order to derive distributions of the resistance coefficients. ρ U D Particle Reynolds number: Drag coefficient: Lift coefficient: Torque coefficient: C C D L = = ρ 2 ρ U U F 2 F 2 VES VES 2 D A L A C T Re = ρ 2 = VES U 2 D T µ VES A VES
Simulations for Non-Spherical Particles 3 Study on the grid-independence of the results: Normalized C D [ - ] 3. 2.5 2. 1.5 1. 2 cells per D VES, hull Re = 1 Re = 1 Re = 5.5 5 1 15 2 25 Domain size / D VES [ - ] Normalized C D [ -] 1.1 1.5 1..95 Re = 1 Re = 1 Re = 5 domain size: 15 D VES, hull.9 1 2 3 4 Cells per D VES [ - ] Reynolds number Computational domain Refinement regions Cells per D VES, hull Re = 1, 1, 5 8 x 8 x 8 2 2 Re = 1 16 x 16 x 16 1 2 Re = 2 16 x 16 x 16 2
Simulations for Non-Spherical Particles 4 Velocity about an irregular particle at different Re: Re p = 1. Re p = 2
Resistance Coefficients 1 Typical PDF`s for the drag coefficient resulting from the different orientations and the 4 similar particles (D VES,hull 1 µm, ψ.87) Counts [ - ] 8 7 6 5 4 3 2 1 Re = 1 3 32 34 36 38 4 42 44 46 C D [ - ] The PDF`s can be approximated by a normal distribution Counts [ - ] 8 7 6 5 4 3 2 1 Re = 1.8 1. 1.2 1.4 1.6 1.8 C D [ - ]
Resistance Coefficients 2 Typical PDF`s for the lift coefficient resulting from the different orientations and the 4 similar particles (D VES,hull 1 µm, ψ.87) Counts [ - ] 9 8 7 6 5 4 3 2 1 Re = 1-1 1 2 3 4 5 C L [ - ] Counts [ - ] 8 7 6 5 4 3 2 1 Re = 1 -.2..2.4.6.8 C L [ - ]
Resistance Coefficients 3 Typical PDF`s for the moment coefficient resulting from the different orientations and the 4 similar particles (D VES,hull 1 µm, ψ.87) 1 8 Re = 1 Counts [ - ] 6 4 2 -.5..5 1. 1.5 C T [ - ] Counts [ - ] 8 7 6 5 4 3 2 1 Re = 1 -.1..1.2.3.4 C T [ - ]
Resistance Coefficients 4 Resistance coefficients as a function of particle Reynolds number with standard deviation (vertical bars), ψ.87 resistance coefficients [ - ] 1 1.1 C D C L C M 1 1 1 Re [ - ]
Resistance Coefficients 5 Resistance coefficients as a function of particle Reynolds number and standard deviation, ψ.87 mean resistance coefficients [ - ] 1 1.1 1 1 1 Re [ - ] C D C L C M rms resistance coefficients [ - ] 1.1 C D C L C M 1 1 1 Re [ - ]
Resistance Coefficients 6 Drag coefficient as a function of particle Reynolds number, comparison with different correlations, ψ.87 C D, mean, all [ - ] 1 1 C D Simulations (ψ =.87) C D Haider & Levenspiel (1983) C D Sphere 1 1 1 1 Re [ - ]
Wall-Collision Experiments 1 Experimental studies on wall collisions of non-spherical particles Impact angles: 5 to 85 Impact velocities: 8 and 21 m/s Photron High Speed Video Camera, FASTCAM SA4 RV operated at 6, fps LED illumination back-lightning Telecentric lens of T8 series Magnification ratio of 1:1
Wall-Collision Experiments 2 Properties of particles considered in the experiments Material Density (kg/m 3 ) Size Distribution Diameters (μm) Mean size (μm) Sphericity Granulates 112 5 x 5 48.778 MC4 Duroplast 148 25-35 3.773 Quartz Sand 265 2-3 25.851 Cylinders 11 5 x 15 825.778
Wall-Collision Experiments 3 Wall collision of an irregular particle (Duroplast particles MC-4, projection diameter 3 µm, ψ =.773, velocity: 16.3 m/s) horizontal camera before and after collision vertical camera Before and after collision
Wall-Collision Experiments 4 Videos wall collision of Duroplast particles (MC-4) horizontal camera view vertical camera view
Wall-Collision Experiments 5 The restitution ratios are determined from the ratio of the velocities obtained by the particle tracking routine: e = V 2 V 1 e n = V yy V yy e t = V xx V xx From the impulse equations for spherical particles the friction coefficient is obtained in the following way: μ = V xx V xx 1 + e V yy
Absolute Impact Velocity [m/s] Normal Restitution Ratio [ - ] 3 25 2 15 1 5 quartz sand Wall-Collision Results 1 1 2 3 4 5 6 7 8 9 Impact Angle [degree] 5 4 3 2 1 quartz sand 1 2 3 4 5 6 7 8 9 Impact Angle [degree] tangential restitution Ratio [ - ] Correlations between wall collision properties and impact angle quartz sand 4.5 4. 3.5 quartz sand 3. 2.5 2. 1.5 1..5. 1 2 3 4 5 6 7 8 9 Impact Angle [degree]
number [ - ] Wall-Collision Results 2 Normal restitution quartz sand 2 18 16 α 14 = P 5o 12 1 8 6 4 2..5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 normal restitution [ - ] 2 number [ - ] 14 12 1 8 6 4 2 α P = 45o..2.4.6.8 1. normal restitution [ - ] 15 α P = 85o number [ - ] 1 5..2.4.6.8 normal restitution [ - ]
Wall-Collision Results 3 Tangential restitution quartz sand number [ - ] 3 25 2 15 1 5 α P = 5o.5.6.7.8.9 1. 1.1 tangential restitution [ - ] 3 number [ - ] 18 16 14 12 1 8 6 4 2 α P = 45o.4.5.6.7.8.9 1. 1.1 tangential restitution [ - ] number [ - ] 25 2 15 1 5 α P = 85o 1 2 3 4 5 tangential restitution [ - ]
Wall-Collision Results 4 Dependence of normal and tangential restitution ratios on mean impact angle quartz sand 3. Restitution Ratios [ - ] 2.5 2. 1.5 1..5 normal tangential. 1 2 3 4 5 6 7 8 9 Impact Angle [degree]
COR (e) COR (e) Wall-Collision Results 5 Dependence of total restitution coefficient on impact velocity: 1 15 2 25 3 1.1.9 1.5 1..95.9.85.8 1..95.9.85.8.75.7.65.6.55.5 5 Quartz 35 7 9 11 13 15 17 19 Impact Velocity (V imp ).85.7 COR (e).8.75.5.85.55.5.8.35.75.3.7 1 15 2 25 3.8.75.7.4 3 5 7 9 11 13 7 9 11 13 15 17 19 Impact Velocity (V imp ) Impact Velocity (V imp ) 1 15 2 25 3 1 15 2 25 3.9.65 COR (e).75.65.6.55.45.6.45.4.25.2 55 85 9 11 13 15 17 19 Impact Velocity (V imp )
Impact Velocity [m/s] 3 25 2 15 1 5 Duroplast Wall-Collision Results 6 Correlations between wall collision properties and impact angle Duroplast Normal Restitution Ratio [ - ] 1 2 3 4 5 6 7 8 9 Impact Angle [degree] 2. 1.5 1..5 Duroplast. 1 2 3 4 5 6 7 8 9 Impact Angle [degree] Tangential Restitution Ratio [ - ] 2. 1.5 1..5 Duroplast. 1 2 3 4 5 6 7 8 9 Impact Angle [degree]
Wall-Collision Results 7 Dependence of normal and tangential restitution ratios on mean impact angle 1.4 Normal Restitution Ratio [ - ] 1.2 1..8.6.4.2 Duroplast Particles. 1 2 3 4 5 6 7 8 9 Impact Angle [degree] Tangential Restitution Ratio [ - ] 1.6 1.4 1.2 1..8.6.4.2 Duroplast Duroplast Particles. 1 2 3 4 5 6 7 8 9 Impact Angle [degree]
Comparison 1 Rebound angle versus impact angle for quartz and Duroplast Rebound Angle [degree] 1 9 8 quartz sand 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 Impact Angle [degree] Rebound Angle [ - ] 9 8 7 6 5 4 3 2 1 Duroplast 1 2 3 4 5 6 7 8 9 Impact Angle [degree]
Comparison 2 Comparison of the results for the irregular non-spherical particles Normal Restitution Ratio [ - ] 1.4 1.2 1..8.6.4 Quartz Duroplast.2 1 2 3 4 5 6 7 8 9 Impact Angle [degree] SD Normal Restitution [ - ] 1..8.6.4.2 Quartz Duroplast. 1 2 3 4 5 6 7 8 9 Impact Angle [degree]
Comparison 3 Comparison of the results for the irregular non-spherical particles Tangential Restitution Ratio [ - ] 1.8 1.6 1.4 1.2 1..8.6.4 Quartz Duroplast.2 1 2 3 4 5 6 7 8 9 Impact Angle [degree] SD Tangential Restitution [ - ] 1.4 1.2 1..8.6.4.2 Quartz Duroplast. 1 2 3 4 5 6 7 8 9 Impact Angle [degree]
Friction Coefficient [ - ] Comparison 4 Comparison of the results for the irregular non-spherical particles.3.25.2.15.1.5 Quartz Duroplast. 1 2 3 4 5 6 7 8 9 Impact Angle [degree] SD Friction Coefficient [ - ].2.15.1.5 μ = V xx V xx 1 + e V yy Quartz Duroplast. 1 2 3 4 5 6 7 8 9 Impact Angle [degree]
Measurements in the Horizontal Channel For providing experimental data for validation, measurements in a horizontal channel (35 mm x 35 mm) or a pipe may be performed by using laser-doppler anemometry and imaging techniques (Kussin 24). Tracer: Glass beads 2 µm Particle phase (without tracer): Reflection: 12 Tracer particles in two-phase flow: Refraction: 33.5 Conveying velocity: 19 2 m/s
Validation of Measurement Procedure For the validation of the measurement procedure the velocity distribution (stream-wise component) of tracer and quartz particles (185 µm) in the centre of the channel was analysed. Single phase flow Two phase flow (η =.3) 3 7 3 25 6 25 samples [ - ] 2 15 1 samples [ - ] 5 4 3 2 2 15 1 5 1 5 12 14 16 18 2 22 24 26 28 U [m/s] 12 14 16 18 2 22 24 26 28 U [m/s]
Particles 1 Properties of the considered particles: T L = 3.66 ms Particles Glass beads Duroplast I Glass beads Quartz sand Duroplast II Particle size 13 µm 18 µm 2 µm 185 µm 24 µm Density 245 kg/m 3 148 kg/m 3 245 kg/m 3 265 kg/m 3 148 kg/m 3 Relaxation time 81.1 ms 85.3 ms 143.3 ms 138. ms 123.1 ms Stokes number 22.17 23.31 39.15 37.69 33.64
Particles 2 Properties of the considered particles: Particle Cylinder Polystyrene Glass beads Particle size 48 µm 66 µm 625 µm Density 112 kg/m 3 15 kg/m 3 245 kg/m 3 Relaxation time 216.8 ms 282.5 ms 461.1 ms Stokes number 59.24 77.19 126. T L = 3.66 ms
Euler/Lagrange Approach 1 The fluid flow is calculated by solving the Reynolds-averaged conservation equations (steady or unsteady) by accounting for the influence of the particles (full two-way coupling). Two-way coupling iterations (PSIC) Turbulence models with coupling: Reynolds-stress turbulence model The Lagrangian approach relies on tracking a large number of representative particles (point-mass) through the flow field accounting for particle rotation and all relevant forces like: Models for elementary processes: turbulent dispersion particle wall collisions (roughness) inter-particle collisions (spherical) drag force gravity/buoyancy slip/shear lift (not) slip/rotation lift (not) torque on the particle Particle properties and source terms result from ensemble averaging for each control volume
Euler/Lagrange Approach 2 Particle Phase: simultaneous, time-dependent tracking of a large number of particles dx dt pi = u pi du pi m p m p g ρ = i 1 + FDi + F dt ρ p dω p i I p = T dt i Li F D 3 ρ = mpcd ( u u p ) u u 4 D p π 2 FL = ρcl Dp u u 2 4 u u p tˆ = tˆ nˆ = u u T P 1 2 p 1 3 2 T p u u p = ρc 2 F tˆ' = F p res res π D 8 F res tˆ ' nˆ' = = F D p nˆ nˆ' + F L
Euler/Lagrange Approach 3 C x Stochastic Coefficients Determination First approach: Gaussian random process Generate at each time step the instantaneous flow coefficients: C D, C L, C T ( Re,t) = C ( Re ) +σ ( Re ) N(,1) P x P Instantaneous normal restitution coefficient: e n e n Instantaneous tangential restitution coefficient: e t N (, 1): standard Gaussian random process C ( α,t ) = e ( α ) + σ n x e n P ( α )N(,1) e ( α,t ) = e ( α ) + σ ( α )N(,1) t t e t Counts [ - ] Number [ - ] 14 12 1 8 7 6 5 4 3 2 1 Re = 1.8 1. 1.2 1.4 1.6 1.8 C D [ - ] 8 6 4 2 Quartz α P =17.3O..2.4.6.8 1. 1.2 1.4 1.6 1.8 2. Normal Restitution Ratio [ - ]
Normalised height [ - ] 1..8.6.4.2 Validation Horizontal Channel 1 Exp. particles Num. non-spherical no coll. Num. non-spherical coll. Num. spherical coll. Horizontal Channel: Length: 6 m Width: 35 mm Height: 35 mm U av = 2 m/s Quartz sand: 185 µm Roughness: 2.3 µm (1.5 o )..2.4.6.8 1. 1.2 1.4 1.6 1.8 Normalised Particle Mass Flux [ - ] Euler/Lagrange computations (Lain and Sommerfeld 28): Reynolds-stress turbulence model Two-way coupling Wall collisions with roughness Inter-particle collisions Normalised height [ - ] 1..8.6.4.2. Exp. Fluid Exp. Particles Num. Fluid Num. non-spherical no coll. Num. non-spherical coll. Num. spherical coll..5.6.7.8.9 1. 1.1 Normalised Mean Velocity [ - ]
Validation Horizontal Channel 2 Velocity rms values Normalised Height [ - ] 1..8.6.4.2 Normalised Height [ - ] 1..8.6.4.2 Exp. Fluid Exp. Particles Num. Fluid Num. non-spherical no coll. Num. non-spherical coll. Num. spherical coll...2.4.6.8.1.12.14.16 Normalised Streamwise rms [ - ]..1.2.3.4.5 Normalised Vertical rms [ - ]
Properties of Different Particles 1 Comparison for different particles with almost identical Stokes number (St ~ 36.8): Normalised number flux 1..8 glass beads 195 µm Duroplast irregular 24 µm quartz particle 185 µm y / H [ - ] 1. glass beads 195 µm Duroplast irregular 24 µm.8 quartz particle 185 µm.6.4 η =.6 y / H [ - ].6.4.2 η =.3...5 1. 1.5 2. 2.5 N P / N av [ - ].2...5 1. 1.5 2. 2.5 3. 3.5 N P / N av [ - ] U av = 19.7 m/s Low roughness R = 2.2 µm
Properties of Different Particles 2 Comparison for different particles with almost identical Stokes single phase number (St ~ 36.8): 1. Stream-wise mean velocity glass beads 195 µm Duroplast irregular 24 µm quartz particle 185 µm.8 y / H [ - ] single phase glass beads 195 µm Duroplast irregular 24 µm quartz particle 185 µm 1..8.6.4 η =.6 y / H [ - ].6.4.2 η =.3..5.6.7.8.9 1. 1.1 1.2 U / U av [ - ].2..5.6.7.8.9 1. 1.1 1.2 U / U av [ - ] U av = 19.7 m/s Low roughness R = 2.2 µm
Fluctuating velocities: Properties of Different Particles 3 1..8 y / H [ - ] 1..8.6.4.2 horizontal St ~ 36.8 y / H [ - ]...5.1.15.2 u rms / U av [ - ].6.4.2 single phase glass beads 195 µm Duroplast irregular 24 µm quartz particle 185 µm...2.4.6.8.1 v rms / U av [ - ] U av = 19.7 m/s, η =.6 single phase glass beads 195 µm Duroplast irregular 24 µm quartz particle 185 µm vertical Low roughness R = 2.2 µm
Conclusions A statistical Lagrangian model was developed in order to allow the numerical calculation of wall bounded particle-laden flows with irregular shaped particles was developed. o Statistical generation of resistance coefficients along particle trajectories based on DNS by LBM o Statistical treatment of wall collisions based on experimental studies In further studies slip-shear lift and slip rotation lift needs to be evaluated for irregular shaped particles by DNS Change of particle angular velocity through a wall collision??? An estimate of angular velocity change may be possible through a momentum balance The data base for the resistance coefficients and wall collision parameters will be further extended for other shapes of irregular particles
Workshop 215