Instability in a combined heat and mass transfer problem in porous media

Abstract

Exothermic chemical reactions can influence natural convection effects in a porous medium. Such phenomena may occur in tubular reactors, oxidation of solid material in large containers, chemical vapor deposition systems, liquid explosives, and others. Experimental evidence indicates that the influence of natural convection in many chemically reacting systems cannot be neglected. In the present work, transient effects of a two-dimensional convection generated and sustained by an endothermic chemical reaction and a constant heat flux are studied. The Darcy-Boussinesq equations are used to describe fluid flow through porous medium with different orders of chemical reaction ( n). The complete mass, energy, and momentum balance equations are solved with the Arakawa-Dufort Frankel numerical scheme, which is particularly suitable for stationary solutions over a range of various parameters. Flow behavior is governed by several parameters, including thermal ( ra 1) and concentration ( Ra 2) Rayleigh numbers, Lewis number ( Le), Peclet number ( Pe), aspect ratio (α), and angle of inclination, ξ. The solution structure, i.e. the interconnections between the various branches appear quite complicated. Determination of linear stability on these solution branches reveal that all two-dimensional, stationary solutions develop some form of instability at different ranges of thermal or concentration Rayleigh numbers. The nature and frequency of these periodic solutions depend on the route followed in the parameter space of Ra 1, Ra 2, α, and ξ. The various routes to chaos are identified in this parameter space. A detailed study of frequency of oscillations and power spectrum reveals that instability in this system is reached through several routes. However, in most cases, determination of the dimension of chaotic attractors led to the assessment of chaotic flow, whereas frequency analysis detected clearly discernible spectral peaks with little frequency interlocking. This possible contradiction between Reulle-Takens instability with other more conventional approaches is highlighted in this paper. It is pointed out that instabilities due to simultaneous thermal and concentration effects may be unique. A complete stability map for different parameter spaces, e.g. Ra 1and Ra 2, ξ, α, and n is presented. Also presented in this map is the location of different types of instabilities, namely, periodic, quasi-periodic, and chaotic solutions.