Untitled - Aerobib - Universidad Politécnica de Madrid

240 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
where S 01 depends only on the composition of the mixture. Likewise, the entropy of
the burnt gases is
S 2 = S 02 + c p2 ln p1/γ2 2
, (9.42)
ρ 2
Therefore, the entropy jump ∆S across the flame, is
∆S = S 2 − S 1 = (S 02 − S 01 ) + c p2 ln p1/γ 2
2
− c p1 ln p1/γ 1
1
. (9.43)
ρ 2 ρ 1
As previously seen, the pressure drop across the flame is very small. Therefore, in
(9.43) we can take
p 2 ≃ p 1 = p. (9.44)
Making use of this simplification and of the relation (9.28) between p 1 and p 2 , the
following expression is obtained for ∆S
being
∆S = S 2 − S 1 = (S 02 − S 01 ) + c p2 ln λ + (c p2 − c p1 ) ln p1/γ 12
ρ 1
, (9.45)
γ 12 = c p2 − c p1
c v2 − c v1
, (9.46)
where c v1 and c v2 are the heat capacities at constant volume of the unburnt and burnt
gases respectively.
In the particular case
c p2 = c p1 = c p , (9.47)
the entropy jump across the flame reduces to
∆S = (S 02 − S 01 ) + c p ln λ. (9.48)
Equation (9.39) shows that λ can vary along the flame front when M 1 varies. Therefore,
if the local conditions of the flow vary, the entropy jump across the flame front
can vary considerably from one point to another. Due to this variation of the entropy
jump along the flame front, the motion after the front can be rotational, even
if the motion before the front is potential. This is similar to what occurs in the case
of supersonic motions with shock waves, when their incline varies from one point to
another. 4
If the motion before the flame is isentropic, the value λ depends only on the
incline α 1 of the flame front. In fact, by using the relation,
57.
tan α 1 = v n1
v t
= ϕ v t
, (9.49)
4 See A. Ferri: Elements of Aerodynamic of Supersonic Flows. Mac Millan Comp., New York, 1949, p.