Untitled - Aerobib - Universidad Politécnica de Madrid

120 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
The elimination of T , v, T 1 and v 1 between these equations and the state equations,
pτ = R m T and p 1 τ 1 = R m T 1 , (5.44)
gives the following Hugoniot relation for each value of the degree of advancement ε
of the combustion
γ + 1
2(γ − 1) (pτ − p 1τ 1 ) + 1 2 (p 1τ − pτ 1 ) = qε. (5.45)
This equation represents a family of hyperbolas. For each value of ε a hyperbola
is obtained, which represents the locus of the possible states of the gas when
burnt fraction is ε and the initial state is (p 1 , τ 1 ). The Hugoniot curve E’JE is obtained
for the particular case ε = 1. This curve corresponds to all the possible states of the
burnt gases, see Fig. 5.2. Similarly, for ε = 0 the Hugoniot curve PD is obtained,
which corresponds to all the shock waves that can form in the unburnt gases at the
initial state represented by point P . Between both curves lie those corresponding to
the intermediate states. Two of them are shown in Fig. 5.2, corresponding to ε = 0.3
and ε = 0.7.
Now, let us consider, for example, a Chapman-Jouguet detonation. Equation
(5.40) corresponding to this detonation is represented by the straight line PJC. The
transformations that occur throughout the shock wave, preceding the combustion,
carry the gas from the initial state P to the state C after the shock wave with no combustion.
However, the trajectory of the gases through the shock wave is not represented
by the straight line PC. In fact, we have seen that the thickness of the shock wave
is very small and, as aforesaid, the thermal conductivity and viscosity therein cannot
be neglected. In particular, Eq. (5.40) must be substituted by the following relation,
deduced from (5.3.a), which takes into account the viscosity, 12
p − p 1 = m 2 (τ 1 − τ) + 4 3 µ dv
dx . (5.46)
But since, through the shock wave the velocity decreases, dv/ dx is negative. Therefore,
the representative points of Eq. (5.46) lie below the straight line PC, as indicated
in Fig. 5.2. 13
The representative states of the combustion that follow the shock wave, that
is, those corresponding to section BE, Fig. 5.1, are obtained by traversal segment CJ,
starting from C. The points at which this segment intersects the successive Hugoniot
12 Assuming that the transformations through the shock wave can be described by variables corresponding
to a continuum. See chapter 3, §1.
13 Hirschfelder et al. ib. page 810.