Combustion theory

Combustion refers to processes where fuel is converted into non-reacting endproducts by an exothermic chemical reaction. Several kinds of combustion can be distinguished on the basis of physical and chemical characteristics, such as, e.g., solid/gaseous fuel; flames with injection of premixed fuel, diffusion flows; detonations, explosions (high Mach numbers, cf. Mach number); various types of reaction kinetics; etc.

Mathematical models in combustion theory consist of a number of coupled partial differential equations (cf. Differential equation, partial). In a gaseous mixture these partial differential equations describe the conservation of species, energy, momentum, mass, and the equation of state. Variables that play a role are concentrations of the reactions, temperature, velocity components of the gas, pressure, and density. Due to thermal expansion the convection/diffusion equations describing the reaction dynamics are intrinsically coupled with the fluid dynamics (see also Diffusion equation; Diffusion process).

From a mathematical point of view combustion theory is a rather recent field of interest. One of the main problems is the non-linearity due to the reaction kinetics. The systems of partial differential equations describing combustion can be studied using numerical or analytical techniques. The numerical approach has led to nice computer graphics simulating flames.

Analytically one constructs approximations based on asymptotics with respect to a small or large parameter, e.g. the high activation energy of the reaction (cf. Small parameter, method of the). Thus one obtains simplified models where an unknown free surface, the flame front, separates the region of the unburnt mixture from the region field with endproducts. In the unburnt mixture, temperature rises until it reaches the critical value at the flame front where in a small region the burning takes place at a high burning rate. Such simplified models are useful to study theoretically the influence of various parameters, such as heat loss, Lewis number, etc. Flames can be constructed in several geometrically different situations (planar, cylindrical, spherical) and their properties such as stability or bifurcation to more complex shapes (cell, etc.) or oscillatory states can be analyzed.

References

[a1]	M.M.R. Williams, "Mathematical methods in particle transport theory" , Butterworths (1971)
[a2]	J.W. Buckwaster, G.S.S. Ludford, "Lectures on mathematical combustion" , SIAM (1983)