Chapter 3 - Wave Equations of Heat and Mass Transfer

Abstract

This chapter describes the main thermodynamic aspects of the solution to the so-called paradox of infinite speed of propagation of infinite speed of propagation of thermal and concentration disturbances in the processes of heat and mass diffusion. Under highly nonstationary conditions, when relaxation of diffusive fluxes of energy and matter is essential, an approach based on irreversible thermodynamics leads to the conclusion about the existence of extra terms of inertial type in the related phenomenological equations. Consequently, unsteady transport phenomena in a typical fluid can be described by an extended model of a viscoelastic body, which takes into account the relaxation of diffusional fluxes of mass, heat, and momentum. Therefore, the equations of change assume a hyperbolic form, thus describing a transport model which is non-Fourier and non-Fick in character. Various hyperbolic equations of change are presented. Properties of solutions of parabolic and hyperbolic partial differential equation of heat are compared. It is described how the attempts of getting rid the paradox of infinite propagation influence the construction and development of the important thermodynamic theory now called extended irreversible thermodynamics (EIT). A plausible concept of relaxation time is introduced, and a simple evaluation of relaxation times for heat, mass, and momentum in ideal gases with a single propagation speed is described. The simplest theory of coupled wave heat and mass transfer is constructed based on a disequilibrium entropy. The analysis of the entropy source yields a matrix formula for the relaxation matrix, τ. This matrix holds for the coupled transfer processes, and corresponds to the well-known values of relaxation times of pure heat transfer and isothermal diffusion. Various useful forms of wave equations for coupled heat and mass transfer are discussed. Extensions of the second differential of the entropy and the excess entropy production are given, which allow to prove the stability of coupled heat and mass transfer equations by the second method of Lapunov. Dissipative variational formulations are presented, leading to approximate solutions by direct variational methods. Applications of the hyperbolic equations in the description of short-time effects and high-frequency behavior are outlined. Qualitative role of the relaxation effects and transfer of harmonic thermal disturbances in solids are characterized.