Untitled - Aerobib - Universidad Politécnica de Madrid

102 CHAPTER 4. COMBUSTION WAVES
By subtracting from the left hand side of Eq. (4.42) the right hand side multiplied
by two, we obtain
y
γ 2
x = y − 1
1 − x , (4.48)
which, together with Eq. (4.47), show that at the Chapman-Jouguet points, the velocity
of the burnt gases with respect to the combustion point is equal to the sound velocity.
For example, by taking the typical values p 1 = 1 atm, τ 1 = 1000 cm 3 /gr,
T 1 = 300 K, γ 1 = γ 2 = 1.4, Q = 1000 cal gr −1 , c p2 = 0.46 cal gr −1 K −1 and ν = 1
the following is obtained for the Hugoniot curve
xy = 15.23 + 1 (y − x) . (4.49)
6
This curve has been represented in Fig. 4.7. The branch of detonations (Fig. 4.7(a))
corresponds to the values of x in the interval
1
6 ≤ x ≤ 1 . (4.50)
The branch of deflagrations (Fig. 4.7(b)) corresponds to the interval
13.2 ≤ x ≤ 91.4 . (4.51)
The Chapman-Jouguet points for detonations and deflagrations are, respectively,
Detonations: x 1 = 0.58, y 1 = 35.7 .
Deflagrations: x 2 = 24.82, y 2 = 0.45 .
The corresponding propagation velocities are
Chapman-Jouguet detonation:
Chapman-Jouguet deflagration:
v 1
= 7.68 .
a 1
v 1
= 0.126 .
a 1
4.6 Remarks
If the variation of the chemical composition along the Hugoniot curve is taken into
account, that is the influence of the dissociation of the combustion products on the
characteristics of the wave, the problem is far more complicated. To attain a solution
one must resort to the equations of thermodynamic equilibrium, which must be combined
with the state equations and the invariants across the wave. Then, the Hugoniot
curve must be drawn point by point. For additional information see Ref. [4].