Untitled - Aerobib - Universidad Politécnica de Madrid

252 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
of the reversible adiabatics, it results for ρ 1 /ρ 0
(
ρ 1
= 1 − γ − 1 M0 2 (U 2 − 1)
ρ 0 2
) 1
γ − 1 = λ
ρ
ρ 0
, (10.11)
where M 0 = u 0 / √ (γ − 1)c p T 0 is the Mach number at the inlet section of the combustion
chamber.
On the other hand, Bernoulli’s equation applied to the burnt gases between O
and A, taking (10.8) into account, gives
1
2 u2 0 + λc p T 0 = 1 2 u2 e + λc p T 1 . (10.12)
Finally, the elimination of T 0 and T 1 between (10.9) and (10.12) gives for u e
u e
u 0
= √ 1 + λ(U 2 − 1). (10.13)
When (10.3), (10.11) and (10.13) are substituted into (10.2.a) and the integration
performed, the following expression for η as a function of U is obtained
η =
(
U −
1
2λ
1 − γ − 1 M0 2 (U 2 − 1)
2
) 1 −
γ − 1
(
(2λ − 1)U − √ 1 + λ(U 2 − 1)
The fraction f of the burnt gases is given by the expression
f = ρ 0u 0 h − ρ 1 u 1 (h 1 − y 1 )
ρ 0 u 0 h
=1 − ρ 1u 1
ρ 0 u 0
(1 − η)
) . (10.14)
(
=1 − U(1 − η) 1 − γ − 1
) 1
M0 2 (U 2 γ − 1
− 1) . (10.15)
2
System (10.14) and (10.15) allow computation of the fraction f of burnt gas as a
function of the flame width η, through parameter U. Fig. 10.7 shows several of the
results given by Tsien in his work for the case λ = 6.
The main result of Tsien’s work is the following: for each value of λ there is a
critical Mach number M cr for the flow at the inlet section, such that when M 0 < M cr
the combustion is complete and the flame reaches the chamber walls. On the other
hand, if M 0 > M cr the maximum burnt fraction f max is smaller than unity and the
flame cannot reach the walls. This is known as the choking phenomenon. It shows