Untitled - Aerobib - Universidad Politécnica de Madrid

100 CHAPTER 4. COMBUSTION WAVES
4.5 Applications
As an application of the previous study we shall consider the propagation of a combustion
wave, assuming that both the unburnt and burnt gases are perfect gases and
that the chemical composition of the burnt gases is independent from their state. In
this case, the enthalpy of the burnt gases can be expressed in the form
∫ T2
h 2 = h 21 + c p2 dT, (4.33)
T 1
where h 21 is their enthalpy at the temperature T 1 . Moreover, it is assumed that the
heat capacity c p2 is independent from the temperature in the interval (T 1 , T 2 ). Then
Eq. (4.33) reduces to
h 2 = h 21 + c p2 (T 2 − T 1 ) . (4.34)
Now, by taking this expression into the Hugoniot equation (4.10), we have
c p2 T 2 = (h 11 − h 21 ) + c p2 T 1 + 1 2 (τ 1 + τ 2 ) (p 2 − p 1 ) , (4.35)
where h 11 − h 21 is the difference between the formation enthalpies of the burnt and
unburnt gases at the temperature T 1 of the unburnt gases. Let Q be this value. Then
Eq. (4.35) can be written in the form
c p2 T 2 = Q + c p2 T 1 + 1 2 (τ 1 + τ 2 ) (p 2 − p 1 ) . (4.36)
Let
p 2 τ 2 = R 2 T 2 (4.37)
be the state equation of the burn gases, and γ 2 = c p2 /c v2 the ratio of their capacities.
Furthermore, let x = τ 2 /τ 1 and y = p 2 /p 1 the specific volume and pressure of the
burnt gases, referred to the corresponding values for the unburnt gases, and
q = 2
( ) ( )
γ2 − 1 Q
+ 2γ 2ν
γ 2 + 1 p 1 τ 1 γ 2 + 1 − γ 2 − 1
γ 2 + 1 , (4.38)
where ν = τ ′ 1/τ 1 and τ ′ 1 is the specific volume that would correspond to the burnt
gases at the pressure p 1 and temperature T 1 of the unburnt gases, that is,
ν = R 2
R 1
= M u
M b
, (4.39)
where M u and M b are the average mole masses for the unburnt gases and for the
combustion products respectively.