(199e) Solution of the Boltzmann Equation by the FCMOM

The fluid dynamics of the particulate phase in gas-solid flows (except in the regions where frictional forces are dominating) is described by the Enskog-Boltzmann equation for inelastic particles; 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.

The first application of the method of moments to the Boltzmann equation was developed by Grad [1], who solved the Boltzmann equation for simple (elastic) gases using a third order approximation of the particle velocity distribution. Strumendo and Canu [2] generalized the Grad approach and were able to compute the anisotropy of the granular flows, however their technique is not successful when extended to higher order moments.

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; more specifically, the Boltzmann for bi-dimensional systems (hard disks) is a bi-variate population balance equation, while the Boltzmann for tri-dimensional systems (hard spheres) is a tri-variate population balance equation. 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.

Recently, several variations of the method of moments (QMOM, DQMOM, FCMOM) were developed and were capable to provide numerically efficient and accurate solutions of population balance equations. It is interesting to investigate whether and to which extent such techniques can be applied for solving the Boltzmann equation.

Fox [3] explored the possibility of applying quadrature-based closures in the solution of the kinetic equation for dilute gas-particle flows, thus enlarging the spectrum of applications of some of the core ideas of the QMOM/DQMOM techniques.

In this work, the FCMOM technique is extended for solving the Boltzmann equation. The FCMOM (Finite size domain Complete set of trial functions Method Of Moments) was developed by Strumendo and Arastoopour ([4], [5]) 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).

The proposed method is applied to the Boltzmann equation for bi-dimensional systems (hard disks) and is illustrated through homogeneous and in-homogeneous applications. In the homogeneous case, the relaxation of elastic particles to the Maxwellian state, and the asymptotic behaviour of inelastic particles (Homogeneous Cooling State solution) are considered. In the in-homogeneous case, the impulsive start-up problem is considered.

[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).