Untitled - Aerobib - Universidad Politécnica de Madrid

92 CHAPTER 4. COMBUSTION WAVES
This curve is obtained from the elimination of v 1 and v 2 between Eqs. (4.1), (4.2) and
(4.3). There results for H, the following equation
h 2 − h 1 − 1 2 (τ 1 + τ 2 ) (p 2 − p 1 ) = 0. (4.10)
This curve has a detonation branch and a deflagration branch.
Let p 2 be the final pressure corresponding to an adiabatic combustion at constant
volume τ 1 . Since the reaction is exothermic p 2 > p 1 and the representative point
of the state (p 2 , τ 1 ), will be, for instance, point A. The detonation branch starts at this
point. The propagation velocity corresponding to this detonation at constant volume
is, according to Eq. (4.8), infinite. Hence, at this point the combustion propagates
instantaneously throughout the mass.
Similarly, let τ 2 be the specific volume corresponding to an adiabatic combustion
at constant pressure p 1 . Here τ 2 > τ 1 , and point B, corresponding to the state p 1 ,
τ 2 , is the starting point of the deflagration branch. The propagation velocity of this
limit constant pressure deflagration is, according to (4.8), zero.
Hereinafter, it is assumed that the Hugoniot curve satisfies the following conditions
( ) ( ∂p
∂ 2 p
< 0 ,
∂τ
H
∂τ
)H
2 > 0, (4.11)
where subscript H indicates differentiation along the Hugoniot curve. These conditions
mean that H is monotonically decreasing and turns its concavity to the axis
p > 0. Both conditions correspond to the curves generally observed in practice. In
general, curve H has an asymptote, parallel to the pressure axis, for τ = τ 0 > 0, and
cuts the specific volume axis at point τ 3 . Therefore, the general form of H is the one
shown in Fig. 4.2.
Let us consider a straight line that starts from point P , corresponding to the
initial state, and enters in region III of detonations. Let α, Fig. 4.3(a), be the angle
between this line and the negative direction of the axis. Depending on the values of α,
the three following cases are possible:
1) If α is smaller than a certain limit value α min which corresponds to the tangent
from P to H, the straight line corresponding to α cannot cut the Hugoniot curve.
2) If α = α min , the corresponding line is, as aforesaid, tangent to H at point J.
3) If α > α min , the corresponding line cuts the Hugoniot curve at two points E
and E ′ at different sides of point J.