Untitled - Aerobib - Universidad Politécnica de Madrid

98 CHAPTER 4. COMBUSTION WAVES
p
C
E’
S 2
S 2
J
E
A
p
1
P
τ 0 1
Figure 4.5: Position of the isentropic curves in the detonation branch of the Hugoniot curve.
τ
τ
Hence the velocity of the burnt gases is supersonic in the weak detonations and subsonic
in the strong detonations.
Similarly, it can be demonstrated that the entropy of the burn gases in the deflagration
branch is maximum at J ′ . It can also be demonstrated that the velocity of
the unburnt gases is subsonic in the weak deflagrations, sonic in the Chapman-Jouguet
deflagration and supersonic in the strong deflagrations.
4.4 Propagation velocity
In order to complete the study, we shall analyze the character of the propagation velocity
v 1 , by comparing its values with the value of the velocity a 1 of the sound propagation
in the state (p 1 , τ 1 ). The study can be made in an identical manner to that
previously used for v 2 , that is by considering all the states (p 1 , τ 1 ) of the unburnt gases
that are compatible with a given state (p 2 , τ 2 ) of the burnt gases. All these states lie
on a Hugoniot curve H ′ which is obtained by fixing in Eq. (4.10) the values of p 2 and
τ 2 and varying p 1 and τ 1 . This curve, like curve H, has two branches (Fig. 4.6): a
lower detonation branch and an upper deflagration branch. It can easily be proved that
in the former, the entropy of the unburnt gases decreases starting from A, and in the
latter increases starting from B. Furthermore by studying the variation of function F ,
which is defined by (4.24) fixing the values of p 2 and τ 2 and varying p 1 and τ 1 , we
obtain that H ′ divides the plane (p, τ) in two regions. The upper region containing