A New Algorithm to Solve Efficiently the Boltzmann Equation for Inelastic Particles

Granular flows are described by the Enskog-Boltzmann equation for inelastic particles (except in the regions where frictional forces are dominating); in dilute systems, the Enskog-Boltzmann equation reduces to the Boltzmann equation. Several methods are available to solve the Boltzmann equation, i.e. generalizations of the Chapman-Enskog expansion, method of moments, Monte Carlo simulations, discretization of the Boltzmann equation in the velocity space, Lagrangian methods; in this work, the focus is on the solution of the Boltzmann equation by the method of moments [1,2,3].

The Boltzmann equation can be interpreted as a multi-variate population balance equation, in which the internal variables are the components of the particle velocities. In this perspective, concepts and techniques developed in the solution of population balance equations can be used for the solution of the Boltzmann equation, and vice versa.

The FCMOM (Finite size domain Complete set of trial functions Method Of Moments) was developed by Strumendo and Arastoopour ([4]-[6]) and by Strumendo ([7]) to solve mono-variate and bi-variate population balance equations. The fundamental idea is to construct a method of moments on a finite domain of the internal variables (typically the particle size in mono-variate population balance equations, or the particle velocities in the Boltzmann equation). An advantage of the FCMOM is that it provides both the moments and the reconstructed distribution of the particle internal variables. Further, the domain of the internal variables is always well defined (a property which is relevant in multi-variate applications). In the applications developed, the numerical solutions provided by the FCMOM converged rapidly to the exact solutions.

In principle, the FCMOM technique could be extended for solving the Boltzmann equation for inelastic particles. However, some difficulties arise when applying the FCMOM algorithm to the Boltzmann equation. In fact, when inelastic particles are considered the numerical solution converges slowly and the algorithm performance is poor. This appears clearly in the numerical solution of the Homogeneous Cooling State problem.

Additionally, several different regimes need to be considered when testing if a numerical technique can solve efficiently the Boltzmann equation for granular systems, namely low/high Knudsen number flows and low/high Mach number flows (low/high Stokes number flows need to be considered as well if the fluid-particle interaction is included in the kinetic equation). Therefore, it is crucial to demonstrate the capability of solving simultaneously the Boltzmann equation in different regimes, and thus the transition between different regimes.

In this work, a new technique, which modifies the FCMOM algorithm, is proposed for solving efficiently the Boltzmann equation for inelastic particles. The proposed method is illustrated through homogeneous and in-homogeneous applications (relaxation of elastic particles to the Maxwellian state, Homogeneous Cooling State solution for inelastic particles, impulsive start-up problem).

[1] Grad, H., “On the Kinetic Theory of Rarified Gases”, Communications on Pure and Applied Mathematics, 2, 331-407, (1949).

[2] Strumendo, M., Canu P., “Method of Moments for the Dilute Granular flow of Inelastic Spheres”, Physical Review E 66, 041304/1-041304/20, (2002).

[3] Fox, R.O., “A quadrature-based third-order moment method for dilute gas-particle flows”,
Journal of Computational Physics, Volume 227, Issue 12, 6313-6350, (2008).
[4] Strumendo, M., Arastoopour, H., “Solution of PBE by MOM in Finite Size Domains”, Chemical Engineering Science 63, 2624-2640, (2008).

[5] Strumendo, M., Arastoopour, H., “Solution of Bivariate Population Balance Equations Using the FCMOM”, Industrial and Engineering Chemistry Research 48(1), 262-273, (2009).

[6] Strumendo, M., Arastoopour, H., “Solution of population balance equations by the FCMOM for in-homogeneous systems”, Industrial and Engineering Chemistry Research, 49(11), 5222-5230, 2010.

[7] Strumendo, M., “Solution of population balance equations using the FCMOM”, PBM 2010. Proceedings of the Fourth International Conference on Population Balance Modelling, Berlin, Germany, September 15-17, 2010, 293-300.