The Vaporization-Condensation Phenomenon in a Linear Heat Wave
Authors: Chu Chieh 
Published: December 2007

Abstract
A theoretical investigation has been made of the forward combustion process using a one-dimensional linear mathematical model, taking into consideration the effect of the vaporization-condensation which occurs on the leading edge of the heat wave. This work involves the solution of five coupled partial differential equations. Besides the vaporization-condensation phenomenon, these equations account for conduction, convection, combustion, heat loss, diffusion and bulk fluid flow. For the one-dimensional linear model studied, the vaporization-condensation phenomenon does not induce appreciable change in the temperature at the combustion front; and its primary effect is to create a steam plateau and to increase the length of the heated zone ahead of the combustion front. This effect becomes more pronounced at lower pressures, higher porosities or reduced gas saturations. The peak temperature and the temperature profile on the leading edge of the heat wave stabilize after a certain period. The length of the steam bank remains practically constant, although the length of the water bank increases as the heat wave advances.

Introduction
The existence of the vaporization-condensation phenomenon in the heat-wave process and the important role played by the phenomenon have been recognized by several investigators. Kuhn and Koch stated that steam plateaus were frequently observed on the temperature records of their experiments. The steam plateaus were attributed primarily to the vaporization and subsequent condensation of the interstitial water existing in the oil sand. Szasz suggested that both lighter hydrocarbons and water are vaporized on the leading edge of the heat wave, carried forward in the gas stream, and then condensed to create banks of oil and water. Martin et al. suggested that the vaporization- condensation phenomenon is one of the main mechanisms of the heat-wave process, along with thermal expansion and viscosity reduction. Wilson et al. reported the existence of a steam plateau several inches in length in their small-scale tube-run experiments. However, this important phenomenon has never been taken into consideration in the numerous theoretical analyses by various authors. The purpose of this work was to study the thermal aspects of a linear heat wave, taking into consideration the vaporization- condensation on the leading edge of the wave, to determine the effect of this phenomenon on the temperature profile of the reservoir, and to investigate how this effect varies when other process variables are changed.

THEORY
We consider a reservoir of porous medium of cross-sectional area A, extending from x=0 to x=L. This reservoir contains, aside from the solid matrix itself, a gas phase and a "combined liquid phase" which is a combination of two immiscible liquid phases - namely, an oil phase and a water phase. The oil present in the reservoir is assumed to consist of three fractions, a noncondensable gas, a nondistillable residuum, and a vaporizable oil fraction which may be present in both the gas phase and liquid phase. Before the heat-wave process begins, preheating has taken place and has imparted an initial temperature distribution To(x) to the reservoir, At the start of the process, a stream of oxygen-bearing gas is introduced through the face at x=0. This gas supports the combustion of the residual fuel and supplies the heat throughout the process.
SPEJ
P. 85ˆ