Lattice Boltzmann Study of the Drag Correlation in Dilute and Moderately Dense Fluid-Particle Systems

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1 Lattice Boltzmann Study of the Drag Correlation in Dilute and Moderately Dense Fluid-Particle Systems Simon Bogner, Swati Mohanty1 and Ulrich Rüde Chair for System Simulation (LSS), University of Erlangen-Nürnberg, Germany 1 CSIR-Institute of Minerals and Materials, Bhubaneswar, India 1

2 Outline Motivation Lattice Boltzmann Method: Two-relaxation-time (TRT) collision operator Drag force computations Results Importance of drag correlations Drag law: Reynolds Re from 0 to 300 and bed densities ϕ from 0.01 to 0.35 Comparison to other literature Conclusion 2

3 Part I I.) Motivation 3

4 Drag Correlations Fluid-solid systems Important in chemical engineering (fluidized beds, hydrocyclone, thickener, flotation columns) Relate the drag force per particle to the local particle Reynolds number (relative velocity) and solid volume fraction. Examples: Wen & Yu (1966), Ergun (1952) 4

5 Fluid-Solid Flow Simulation DNS can be used to obtain drag correlations! Here: Lattice Boltzmann method (LBM) to simulate flow through fully resolved particle beds Emphasis on dilute case: Regime: Paper draft: 5

6 Part II II.) Lattice Boltzmann Method wikipedia.org 6

7 Lattice Boltzmann Method Lattice Boltzmann Equation (LBE) Non-local part, Linear propagation to neighbors (stream step) Local part, Non-linear operator, (collide step) wikipedia.org Each node stores the vector The number N of Particle Distribution Functions (PDFs) depends on the lattice model ( Qian et al. 1992). 7

8 TRT Collision Operator Two-Relaxation-Time (TRT) model: Different relaxation times for even and odd parts of the distribution function Other collision models may NOT yield viscosity-independent results... 8

9 Viscosity Independence (cont.) Optimal choice for second relaxation time Numerical artifact: Permeability obtained from simulation depends on relaxation times (viscosity) BGK (single relaxation time) Permeability (normalized) (Pan et al. 2005) (taken from Pan et al. 2005) TRT (*) BGK TRT (*): optimal choice for second parameter ( Ginzburg, Adler, 1994) Using the optimal choice, the TRT-approach yields viscosity-independent results Use TRT instead of SRT/BGK!!! 9

10 Drag Computation (1) Random particle beds (homogeneous): Number of particles N = 27 Spherical particles of volume Solid volume fraction Cubic domain length L with Periodic boundaries along x, y and z direction! Simulation of flow: Accelerating gravity Reynolds number where is the average flow rate. 10

11 Drag Computation (2) Average drag:.. is estimated for each simulation (steady state). The (total) hydrodynamic force is The dimensionless drag is defined by (normalized using the Stokes drag). 11

12 Flow field in bed of density

13 Vorticity in bed of density 0.01 Domain size: Re = 300, 13

14 Part III III.) Results 14

15 New Drag Correlation 5 runs (beds) for each pair Ansatz function: Undetermined: Optimization (conjugate gradient) to minimize where is the averaged dimensionless drag obtained from simulations. 15

16 New Drag Correlation Finally, the drag correlation reads Average absolute percentage error: 9.7 % 16

17 Comparison to previous studies Our main references for comparison are: Tenneti et al., 2011 (pseudo-spectral) Pseudo-spectral Navier-Stokes solver Reynolds up to 300 Bed density Beetstra et al., 2007 Single-relaxation time (SRT/BGK) LBM (not viscosity-independent!) Reynolds up to 1000 Bed density However: The correlations obtained by Tenneti et al. and Beetstra deviate with a relative error of up to 35%!!! Beetstra may be under-resolved (Tenneti et al, 2011) Compared to our study: SRT/BGK may be inaccurate! 17

18 Comparison to previous studies 18

19 Comparison to previous studies Comparison to previous studies Maximum average deviation +12% at compared to Tenneti et al. (2011) Maximum average deviation -20% at compared to Beetstra et al. (2007) Note: Of the two studies, the later work by Tenneti should be regarded as more reliable. 19

20 Part IV IV.) Conclusion 20

21 Conclusion Lattice Boltzmann study of drag law Drag correlation for Emphasizing the dilute case (good agreement with Wen & Yu) Comparison to previous studies still indicates uncertainty (which correlation is most reliable?) Outlook: More accurate simulations by using LBM with higher order boundary conditions Future: Inclusion of higher Reynolds numbers and bed densities 21

22 End Thank you very much for listening! Get paper draft at: 22