Untitled - Aerobib - Universidad Politécnica de Madrid

9.5. ENTROPY JUMP ACROSS THE FLAME FRONT 241
that gives the incline of the flame front, and the relation
M 1 ≃ v t
a , (9.50)
which, as seen in the preceding paragraph, is valid for the very inclined flame fronts,
Bernoulli equation for unburnt gases can be written
1 + γ 1 − 1
M1 2 =
2
(
a01
ϕ
) 2
M 2 1 tan 2 α 1 , (9.51)
where a 01 is the sound speed at the stagnation point of the unburnt gases.
By eliminating M 1 between this equation and equation (9.39), λ can be expressed
as a function of the flame incline, as follows
λ = n +
2
γ 1 − 1
n − c p1
(
a01
ϕ
c p2
) 2
tan 2 α 1 − 1. (9.52)
When the flame incline is increased, α 1 decreases, and relation (9.52) show that λ
increases. Therefore, the entropy jump ∆S increases when the flame incline is increased.
Such behavior is the opposite to that of a shock wave where the entropy jump
decreases when the wave incline is increased. See Fig. 9.7 [5] . The influence of
this variation of the entropy jump on the flow will be studied in the following paragraph.
The simplified expression (9.48) for the entropy jump will be used, and since
the constant S 02 − S 01 has no influence on the flow, we shall express in short
∆S = c p ln λ. (9.53)
∆ S
ω 2 ∆ S
ω 2
(a) Flame
(b) Shock wave
Figure 9.7: Entropy jump across a flame front and a shock wave.