Untitled - Aerobib - Universidad Politécnica de Madrid

5.5. DETONATIONS 119
This relation shows that the pressure at each point of the wave is determined
only by the corresponding value of the density. This relation plays here the same part
as, for example, in Gas Dynamics the relation p = Cρ γ for the isentropic motions of
non-reacting gases.
In order to represent the result in the diagram (p, τ), as done in chapter 3, the
specific volume τ = 1/ρ must be introduced in place of the density in Eq. (5.39),
which then takes the form
p − p 1 = m 2 (τ 1 − τ). (5.40)
This equation is represented in diagram (p, τ), see Fig. 5.2, by a straight line joining
the representative points of the initial and final states. On this line also lie all the
representative points of the intermediate states corresponding to the reaction zone. 11
70
60
D
p/p 1
E
50
C
Trayectory through shock wave
40
30
ε=1.0
J
20
E‘
ε=0.3 ε=0.7
10
ε=0
1
P
0.0 0.2 0.4 0.6 0.8 1.0
τ /τ 1
Figure 5.2: Hugoniot curves from different values of ε.
Bringing forth in Eqs. (5.1.a), (5.28.a) and (5.29.a) the specific volume and the
initial conditions corresponding to the unburnt gases, there results
v
τ = v 1
τ 1
, (5.41)
p + v2
τ = p 1 + v2 1
τ 1
, (5.42)
c p T + 1 2 v2 − qε = c p T 1 + 1 2 v2 1. (5.43)
11 W. Michelson was the first to postulate that within the reaction zone of the detonation wave the linear
relation (40) is verified.