Untitled - Aerobib - Universidad Politécnica de Madrid

110 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
Chapman-Jouguet detonations. As will presently be seen, the type of detonation that
occurs in each particular case depends on the boundary conditions on the burnt side of
the detonation wave. As for the deflagrations, Friedrichs obtained the conclusion that
strong deflagrations are impossible and that weak deflagrations are possible only for a
well defined value of the propagation velocity. This value depends on the state of the
unburnt gases, on the reaction rate and on the transport coefficients of the mixture.
The discussion of the differential system of combustion waves is considerably
simplified by analyzing, as done by von Kármán [2], the limiting system deduced from
the previous one by assuming that a characteristic time of the thermodynamic transformations,
for example the average time between two molecular collisions, is very small
compared to a characteristic time of the chemical transformations, for example the average
time between two collisions of molecules of different species under the required
circumstances for these molecules to react to one another. This assumption appears
justified if one considers that, in a mixture of reactants, of all the molecular collisions
only a few are accompanied by chemical transformations. In fact, for this to happen
a number of favourable circumstances must concur: for example, the molecules must
be of the adequate species, the orientation of the collision and the energy in certain
degrees of freedom must be the proper ones, etc. 3
In the following, the solutions of the aforementioned differential system will
be discussed mainly from this stand-point, following von Kármán’s reasoning.
5.3 Characteristic times
As a characteristic τ t of the thermodynamic transformations the following ratio is
adopted
τ t = µ p . (5.16)
This ratio is proportional to the average time between molecular collisions, which is
given by ratio l/¯v of the mean free path l of the molecules to the average velocity ¯v
of the molecular motion. In fact, the Kinetic Theory of Gases shows that µ is of the
form
µ ∼ ρ¯vl, (5.17)
and that ¯v is of the form
√ p
¯v ∼
ρ . (5.18)
3 See chapter 1.