Untitled - Aerobib - Universidad Politécnica de Madrid

function of p 2 and ρ 2
h 2 = h 2 (p 2 , ρ 2 ) . (4.5)
90 CHAPTER 4. COMBUSTION WAVES
Unburned gases
Burned gases
V
1
V 2
p ρ 1 T p ρ T
1 1
2 2 2
Combustion wave
Figure 4.1: Schematic diagram of a combustion wave to obtain the relations between initial
and final states.
Similarly, if the burnt gases are in thermodynamic equilibrium, h 2 is only a
When equations (4.4) and (4.5) are substituted into (4.3), this equation together
with Eq. (4.1) and (4.2) form a system of three equations with six unknowns: p 1 , ρ 1 ,
v 1 , p 2 , ρ 2 and v 2 . If three of these values are known, the system determines the values
for the other three. As said before, the state of the unburnt gases is defined by the
values p 1 and ρ 1 . Therefore, in order to determine the propagation, another of these
values must be known, for instance that of the propagation velocity v 1 . The analysis
of the propagation velocity shows the existence of two types of essentially different
waves. In fact, the elimination of v 2 between Eq. (4.1) and (4.2), gives for v 1 ,
v 1 = 1 ρ 1
√ √√√ p 2 − p 1
1
ρ 1
− 1 ρ 2
. (4.6)
The study is simplified by introducing in the above system the specific volume
τ, which is related to the density ρ by
τ = 1 ρ , (4.7)
thus obtaining for v 1
v 1 = τ 1
√
p2 − p 1
τ 1 − τ 2
. (4.8)
satisfied
Since the propagation velocity must be real, the following condition must be
p 2 − p 1
τ 1 − τ 2
> 0. (4.9)