Untitled - Aerobib - Universidad Politécnica de Madrid

106 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
For simplicity, the study herein will be restricted to the case of a perfect gas
with heat capacity and number of moles constant throughout the wave. The limiting
solutions, obtained by assuming that a characteristic time of the thermodynamic
transformations is very small compared to a characteristic time of the chemical transformations
(as explained in §3), will be carefully analyzed. This plausible assumption
enables an essential simplification of the equations, favouring the study of the properties
of the corresponding solutions. This analysis will follow, mainly, the line of von
Kármán’s reasoning [2]. More detailed studies, for example those of Friedrichs [3] and
Hirschfelder [4], which take into account the influence of the terms neglected herein,
show that the discrepancy between both cases is very small. Therefore, the method
followed in this study is well justified. Furthermore, these studies allow one to derive
conclusions for the cases in which, due to the existence of ”abnormal” reaction rates,
the simplified treatment developed herein is not applicable.
In the following analysis special attention is given to the study of detonations
(see §5), since deflagrations are the subject of a detailed analysis in the following
chapter.
5.2 Wave equations
Let us consider an indefinite plane wave which propagates in undisturbed uniform
combustible mixture. We shall adopt a reference system fixed to the wave were the x
axis is parallel to the propagation direction and positive towards the burnt gases. With
respect to this reference system, the process is stationary and x is the only independent
variable. The values of x vary from x = −∞ for the unburnt gases to x = +∞ for
the burnt gases.
For simplification it will be assumed that the following considerations are satisfied
throughout the wave:
1) The mixture behaves as a perfect gas.
2) The heat capacity at constant pressure c p is independent from the temperature
and composition of the mixture.
3) The ratio γ of the heat capacities is independent from the composition of the
mixture.
4) The chemical composition of the mixture at each point of the wave is determined
by only one chemical parameter. We shall adopt as chemical parameter the degree
of advancement of the combustion, measured by the mass fraction ε(x) of
the gas that has burnt when the point x of the wave is reached. Due to the action