Introduction and Kinetics of Particles

Abstract

There are two main approaches to solving the transport equations (heat, mass, and momentum) computationally: continuous and discrete. In the continuous approach, ordinary or partial differential equations can be obtained by applying conservation of energy, mass, and momentum with an infinitesimal control volume. Since it is difficult to solve the governing differential equations for many reasons (nonlinearity, complex boundary conditions, complex geometry, etc.), one uses finite difference, finite volume, and finite element methods, among others, to convert the governing differential equations with a given boundary and initial conditions to a system of algebraic equations. Those equations can be solved iteratively until convergence is ensured. Let us discuss the procedure in more detail for a given problem in which the governing equations need to be identified (mainly partial differential equations). This step is called mathematical modeling, which depends on the physics of the problem (and perhaps on the chemistry as well). The next step is to discretize the domain into finite volumes, grids, or elements, depending on the method of the solution. We can consider this step as assigning to each of the finite volumes or nodes or elements a collection of particles (a large number, on the order of $10 ^{16}$). The scale is macroscopic. The velocity, pressure, and temperature of all the particles are represented by a nodal value, or averaged over a finite volume, or simply assumed to vary linearly or bilinearly from one node to another. The phenomenological properties such as viscosity, thermal conductivity, and heat capacity are in general known parameters (input parameters, except for inverse problems). For inverse problems, one or more thermophysical properties may be unknown.