Quadrature-Based Moment Model for Moderately Dense Polydisperse Gas−Particle Flows

Abstract

A quadrature-based moment model is derived for moderately dense polydisperse gas-particle flows starting from the inelastic Boltzmann-Enskog kinetic equation including terms for particle acceleration (e.g., gravity and fluid drag). The derivation is carried out for the joint number density function, f(t,x,m,u), of particle mass and velocity, and thus, the model can describe the transport of polydisperse particles with size and density differences. The transport equations for the integer moments of the velocity distribution function are derived in exact form for all values of the coefficient of restitution for particle-particle collisions. For particular limiting cases, the moment model is shown to be consistent with hydrodynamic models for gas-particle flows. However, the moment model is more general than the hydrodynamic models because its derivation does not require that the particle Knudsen number (and Mach number) be small. A quadrature-based moment model is derived for moderately dense polydisperse gas-particle flows starting from the inelastic Boltzmann-Enskog kinetic equation including terms for particle acceleration (e.g., gravity and fluid drag). The derivation is carried out for the joint number density function, f (t,x,m,u), of particle mass and velocity, and thus, the model can describe the transport of polydisperse particles with size and density differences. The transport equations for the integer moments of the velocity distribution function are derived in exact form for all values of the coefficient of restitution for particle-particle collisions. For particular limiting cases, the moment model is shown to be consistent with hydrodynamic models for gas-particle flows. However, the moment model is more general than the hydrodynamic models because its derivation does not require that the particle Knudsen number (and Mach number) be small.

A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

“…However, it is possible to approximate the collision integral using the weights and abscissas of the quadrature approximation. The procedure to derive the closure with hard-sphere collisions is fairly complicated, and is explained in [22]. Only the resulting moment source terms are reported here.…”

“…For the system under consideration the off-diagonal components should be null for all times, and this is confirmed by the simulations. For finite-size particles (e.g., the MD simulations), the stress tensor contains an additional term due to the collisional fluxes [22,23], not present in the QMOM calculations, which scales like Tα 2 s . Thus, we can anticipate that Eq.…”

“…The heat fluxes in the directions parallel to the walls should be null for all times, and this is confirmed by the simulations. As with the stress tensor, the heat flux in the MD data contains an additional contribution due to the collisional fluxes [22,23], not present in the QMOM calculations. We can again anticipate that (5.4) will underestimate the MD results.…”

“…The stress-tensor components σ 22 and σ 33 are equal, as a consequence only the first one is reported in the figures. Likewise, as noted earlier, for this system with no mean velocity gradients the off-diagonal components of the stress tensor are null.…”

“…6. It is worth noting that similar considerations cannot be applied for σ 22 , because M 2 020 does not produce a flux.…”