Theoretical Analysis of the Particle Collision Model for Multi-Phase-Particle-in-Cell (MP-PIC) Calculation of Dense Particle Flows with Application to Simple Shear Flow and Bi-Disperse Particle Settling

The purpose of this communication is to present a theoretical analysis of collision model for multi-phase-particle-in-cell (MP-PIC) method by comparison of the first order particle velocity moment equations (number density, mass, momentum and second-order velocity correlation) derived from the corresponding PDF equation with the ones derived in the frame of kinetic theory of granular media.

The solid phase prediction in the frame of the MP-PIC method (Andrew and O’Rourke, 1996) is based on the Lagrangian computation of stochastic numerical particles (parcels) in order to solve a kinetic (or Liouville) equation of the particle distribution function , if particles have a distribution of velocity ( ) and mass ( ). One of the main advantage of such an approach, compared to more standard Eulerian particle moment approach derived in the frame of kinetic theory (Gidaspow, 1986) is that stochastic Lagrangian approaches are meshless and lead to very few numerical diffusion effect. In particular, they allow to predict (for an affordable cost) the small particle flow structures (clusters) which are known to play a crucial role in the macroscopic hydrodynamic behavior of fluidized beds (Agrawal et al., 2001).

In dilute flows, for solid volume fraction less than 1 %, particle-particle collision effect can be accounted for by using Monte Carlo (DSMC) method (see for example, Fede et al. ; 2015). For larger solid volume fraction, the DSMC method is no longer valid and, according to Andrew and O’Rourke (1996), particle interaction effect may be accounted for using an additional contribution in the particle acceleration term of the kinetic equation which represents the averaged effect of the surrounding particles. This term was originally written to account only for the collisional stress in the momentum equation (Andrew and O’Rourke, 1996) neglecting the dissipation and redistribution effect due to inter-particle collision which plays a crucial role, in non-homogeneous flows, on the particle fluctuating velocity variances and on the effective kinetic viscosity predictions (Boelle et al., 1995) and on the particle velocity in polydisperse flows (Gourdel et al., 1999). More recently, O’Rourke and Snider (2010) proposed to add a collision damping term which tends to equilibrate the numerical particle velocities to the local mass-averaged particle velocity with a collision damping time scale determined in order to give the correct decay rate of the fluctuating kinetic energy. Indeed, the expression of the collision damping term is written by comparison of the fluctuating kinetic energy equations derived from the corresponding MP-PICS kinetic equation with the ones derived in the frame of kinetic theory of granular media.

However, it is interesting to notice that the collision damping term added in the PDF equation leads to an averaged momentum and fluctuating kinetic energy transfer term between particle species in polydisperse solid mixture as shown by kinetic theory and Discrete Particle Simulation (DPS) results (Gourdel et al., 1999). Unfortunately, according to the proposed determination of the collision damping term, these momentum and energy transfer effects between particle species are equal to zero when the particle to particle collision are elastic while they should reach maximum values. Similarly, it can be shown that the redistribution effect between the separate components of the particle kinetic stress (or second-order velocity correlations) is not correctly accounted for by the proposed approach and is equal to zero for elastic particles, in contrast with kinetic theory of granular media and DPS results (Boelle et al., 1995 ; Parmentier and Simonin, 2012).

Detailed analysis of the collision terms appearing in the mean velocity and fluctuating kinetic energy, equation derived separately for each particle species in a bi-disperse mixture, and in the particle kinetic stress transport equations, in a mono-disperse granular flow, shows that the collision damping term cannot account for both dissipation and redistribution effect. Therefore, an additional collision term corresponding to a diffusion effect in the PDF equation, and to a Wiener process in the stochastic Lagrangian treatment of the numerical particles, is proposed. The explicit form is derived in order to satisfy both the dissipation and the redistribution effect for the particle kinetic stress tensor components according to kinetic theory of granular media. Then the corresponding collision transfer terms (averaged momentum and fluctuating kinetic energy) between particle species for a bi-solid mixture are derived and compared with the kinetic theory closure model (Gourdel et al., 1999).

References

Agrawal, K., Loezos, P.N., Syamlal, M., Sundaresan, S. 2001, “The role of meso-scale structures in rapid gas–solid flows”, J. Fluid Mech, Vol. 445, pp 151–185.

Andrews, M.J., O’Rourke, P.J., 1996, "The multiphase particle-in-cell (MP-PIC) method for dense particulate flows", Int. J. Multiphase Flow, Vol. 22, pp 379-402.

Boelle, A., Balzer, G., Simonin, O., 1995, "Second-Order Prediction of the Particle-Phase Stress Tensor of Inelastic Spheres in Simple Shear Dense Suspensions", in Gas-Particle Flows, ASME FED, Vol. 228, pp 9-18.

Gidaspow, D., 1994, Multiphase Flow and Fluidization Continuum and Kinetic Theory Descriptions, Academic Press, Boston, USA.

Gourdel, C., Simonin, O., Brunier, E., 1999, "Two-Maxwellian Equilibrium Distribution Function for the Modelling of a Binary Mixture of Particles", Circulating Fluidized Bed Technology VI, Proc. of the 6th Int. Conference on Circulating Fluidized Beds, J. Werther (Editor), DECHEMA, Frankfurt am Main (Germany), pp. 205-210.

O’Rourke, P.J., Snider, D.M., 2010, "An improved collision damping time for MP-PIC calculations of dense particle flow with applications to polydisperse sedimenting beds and colliding particle jets", Chem. Eng. Sci. 65, 6014–6028.

Parmentier, J.F., Simonin, O., 2012, “Transition models from the quenched to ignited states for flows of inertial particles suspended in a simple sheared viscous fluid”, J. Fluid Mech., Vol. 711, pp 147-160.

Snider, D.M., Clark, S.M., O’Rourke, P.J., 2011, "Eulerian-Lagrangian method for three-dimensional thermal reacting flow with application to coal gasifiers", Chem. Eng. Sci., 66, 1285-1295.