On the mathematical model simplification using constant Lewis number – Impact assessment on heterogeneous char conversion process

Reactive systems in a thermochemical conversion domain are modelled considering N-specie, 1-energy and 2-mass conservation equations assuming negligible pressure gradient resulting in N+3 non-linear coupled PDE system with dependency on thermodynamic and transport properties. Typically, simplistic temperature-dependent polynomials are chosen for estimating thermal conductivity and specific heat, however, the estimation of mass diffusion coefficient (Di;mix) follows a complicated procedure involving kinetic theory culminating in Chapman-Enskog equation. This renders the solution computationally intensive. The complexity is simplified by assuming a constant Lewis (Le) number, a standard practice in the analytical solution for conventional reactive systems. In fixing Le, (Di;mix) is equated to thermal diffusivity (a ratio of thermodynamic properties) resulting in the specie and energy equation yielding a similar solution and collapse of N+3 system of simultaneous equations to 3 equations. The current article explores the validity and limitation of assuming constant Le in the simulation of char conversion process in air and steam. Results of char conversion are compared for fixed Le and D estimated with Chapman{Enskog expresion. The analysis suggests that Le remains invariant only under a severely restricted set of conditions. Fixing Le influences, the conversion process either over-/under-predicting the conversion time scales and the product gas composition.

The relativistic Enskog equation near the vacuum in the Robertson–Walker space-time

In this paper, we consider the Cauchy problem for the relativistic Enskog equation with near vacuum data for a hard sphere gas in the Robertson–Walker space-time. We prove an existence and uniqueness result of the global (in time) mild solution in a suitable weighted space. We also study the asymptotic behavior of the solution as well as the {L^{\infty}} -stability.

The Random Gas of Hard Spheres

The inconsistency between the time-reversible Liouville equation and time-irreversible Boltzmann equation has been pointed out by Loschmidt. To avoid Loschmidt’s objection, here we propose a new dynamical system to model the motion of atoms of gas, with their interactions triggered by a random point process. Despite being random, this model can approximate the collision dynamics of rigid spheres via adjustable parameters. We compute the exact statistical steady state of the system, and determine the form of its marginal distributions for a large number of spheres. We find that the Kullback–Leibler entropy (a generalization of the conventional Boltzmann entropy) of the full system of random gas spheres is a non-increasing function of time. Unlike the conventional hard sphere model, the proposed random gas system results in a variant of the Enskog equation, which is known to be a more accurate model of dense gas than the Boltzmann equation. We examine the hydrodynamic limit of the derived Enskog equation for spheres of constant mass density, and find that the corresponding Enskog–Euler and Enskog–Navier–Stokes equations acquire additional effects in both the advective and viscous terms.

Unified theory for a sheared gas–solid suspension: from rapid granular suspension to its small-Stokes-number limit

A non-perturbative nonlinear theory for moderately dense gas–solid suspensions is outlined within the framework of the Boltzmann–Enskog equation by extending the work of Saha & Alam (J. Fluid Mech., vol. 833, 2017, pp. 206–246). A linear Stokes’ drag law is adopted for gas–particle interactions, and the viscous dissipation due to hydrodynamic interactions is incorporated in the second-moment equation via a density-corrected Stokes number. For the homogeneous shear flow, the present theory provides a unified treatment of dilute to dense suspensions of highly inelastic particles, encompassing the high-Stokes-number rapid granular regime ($St\rightarrow \infty$) and its small-Stokes-number counterpart, with quantitative agreement for all transport coefficients. It is shown that the predictions of the shear viscosity and normal-stress differences based on existing theories deteriorate markedly with increasing density as well as with decreasing Stokes number and restitution coefficient.

Invariant Forms of Conservation Equations for Reactive Fields and Hydro-Thermo-Diffusive Theory of Laminar Flames

A scale-invariant model of statistical mechanics is described leading to invariant Enskog equation of change that is applied to derive invariant forms of conservation equations for mass, thermal energy, linear momentum, and angular momentum in chemically reactive fields. Modified hydro-thermo-diffusive theories of laminar premixed flames for (1) rigid-body and (2) Brownian-motion flame propagation models are presented and are shown to be mathematically equivalent. The predicted temperature profile, thermal thickness, and propagation speed of laminar methane–air premixed flame are found to be in good agreement with existing experimental observations.