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5.5. DETONATIONS 121
curves, represent the successive states of the mixture for the corresponding values of
the degree of advancement of the combustion.
Now, let us consider a strong detonation, which in Fig. 5.2 would be represented
by a straight line such as PED. The initial shock wave produces a jump from
state P to state D. From D to E the reaction takes place and is accompanied by an
expansion and acceleration of the gases. The intermediate states are represented by
the points of segment DE.
Figure 5.2 also shows that weak detonations are not possible. In fact, once
point E is reached, point E’ corresponding to a weak detonation can only be reached
by means of an expansion wave between E and E’, which is impossible, or else by
means of an endothermic reaction, corresponding to segment EE’.
The temperature variation throughout the wave can be analyzed in the same
manner as that used for the pressure variations. For this purpose it is sufficient to
eliminate p and p 1 from Eq. (5.40), by making use of (5.44). Thus, we obtain
T
= (1 + γM1 2 ) τ ( ) 2 τ
− γM1
2 , (5.47)
T 1 τ 1 τ 1
in which the value of the Mach number M 1 of the unburnt gases has been made explicit.
To each value of the Mach number M 1 corresponds a different parabola, on
which lie the representative points of the successive states of the mixture within the
combustion zone of the detonation wave. In Fig. 5.3 two parabolas are shown. One
corresponds to the Chapman-Jouguet detonation and the other to a strong detonation.
The same letters are used to designate homologous points in Figs. 5.1 and 5.2.
The Hugoniot curves are obtained from Eq. (5.45) eliminating p and p 1 by
means of Eq. (5.44), as has been done to obtain Eq. (5.47). There results
( ) 2 τ
+ γ + 1 T τ
− T ( 2γ qε
−
+ γ + 1 ) τ
= 0. (5.48)
τ 1 γ − 1 T 1 τ 1 T 1 γ − 1 c p T 1 γ − 1 τ 1
These curves are a family of hyperbolas. A hyperbola is obtained for each value of
ε. In particular, for ε = 1 one obtains the Hugoniot curve of the burnt gases and for
ε = 0 the curve of the shock waves of the unburnt gases. Both have been taken into
Fig. 5.3. Their intersection with curve of Eq. (5.47), representative of the intermediate
states, determines section CJ or DE, corresponding to the said intermediate states. The
state corresponding to a given fraction of burnt gases is given by the intersection of
curve of Eq. (5.48), corresponding to this fraction, with section CJ, or DE, as shown
in Fig. 5.3 for the values ε = 0.3 and ε = 0.7.
The preceding considerations give an idea of the structure of a detonation wave.
The initial shock wave compresses, decelerates and heats suddenly the gasses, within a