Untitled - Aerobib - Universidad Politécnica de Madrid

256 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
Let us consider the streamline ψ that crosses the flame front at section AB
(Fig. 10.9) at point C. At this section, the streamline separates the unburnt from the
burnt gases. To the first correspond values ψ ′ of the stream function between ψ and 1.
To the latter correspond values ψ ′ between 0 and ψ. By separating in (10.21) both
intervals, the following is obtained;
∫ ψ
0
∫
ρ 0 u 1
0 ρ 0 u 0
ρ ′′ u ′′ dψ′ +
ψ ρu dψ′ = 1, (10.22)
where ρ ′′ and u ′′ are the corresponding values of ρ and u, on the streamline ψ ′ at point
E on section AB.
As previously said, both density and velocity of the unburnt gas are constant
at each cross section of the chamber. If ρ 1 and u are their values at section BC, the
second integral in (10.22) can be integrated, obtaining
∫ 1
This value, when carried into (10.21), gives
∫ ψ
This equation can be written
0
ψ
ρ 0 u 0
ρ 1 u dψ′ = ρ 0u 0
(1 − ψ). (10.23)
ρ 1 u
ρ 0 u 0
ρ ′′ u ′′ dψ′ = 1 − ρ 0u 0
(1 − ψ). (10.24)
ρ 1 u
∫
ρ 1 u
ψ
ρ 1 u
− 1 + ψ =
ρ 0 u 0 0 ρ ′′ u ′′ dψ′ . (10.25)
Hereinafter all velocities will be referred to the maximum velocity u max =
√
2cp T s,1 of the unburnt gases, and the ratio named U. Equation (10.25) is then
written as follows
∫
ρ 1 U
ψ
ρ 1 U
− 1 + ψ =
ρ 0 U 0 0 ρ ′′ U ′′ dψ′ . (10.26)
The problem lies now in expressing ρ 1 U
ρ ′′ U ′′ as a function of U and of the dimensionless
velocity U ′ at the point D where the streamline ψ ′ crosses the flame,
Fig. 10.9. Let us see how it can be done.
Proceeding in the same manner as done for the deduction of equations (10.4)
and (10.8) in Tsien’s method, that is, by comparing the expansions along ψ and ψ ′ ,
the following is obtained
λ ′ = ρ ′′
1 T
= . (10.27)
ρ
′′
T 1