Untitled - Aerobib - Universidad Politécnica de Madrid

184 CHAPTER 6. LAMINAR FLAMES
where
Taking (6.195) into (6.202)
( ) 3/2 1 − Y
e −θ 1 − θ
a
θ
dε
dθ = Λ 1 + Y
(1 − θ 0 )(1 − ε) − (1 − θ) , (6.206)
Λ = 128 × 10 11 e −θ a
If a linear variation of λ with temperature is assumed
we obtain from Eq. (6.207).
Λ = 128 × 10 11 e −θ a
( p
) λ
RT m 2 . (6.207)
c p
λ = λ f θ, (6.208)
( ) p λf
RT f m 2 . (6.209)
c p
Then the problem reduces to the integration of the system of Eqs. (6.203) and (6.206),
with the following boundary conditions
θ = θ 0 : ε = 0, (6.210)
θ = 1 : ε = 1, Y = 1. (6.211)
The value of Λ which makes compatible both boundary condition will be the
eigenvalue for the system, and it determines the flame propagation velocity.
Flame velocity
The flame velocity is given by
u 0 = m ρ 0
= 1 ρ 0
(
128 × 10 11 e −θ a
p
RT f
where Λ is the eigenvalue referred to.
When Eq. (6.206) is written
1 − θ 0
2
−
∫ 1
0
λ f
c p
) 1/2
Λ −1/2 , (6.212)
∫ 1
( ) 3/2 1 − Y
(1 − θ) dε = Λ
e −θ 1 − θ
a
θ dθ, (6.213)
θ 0
1 + Y
the problem reduces (following the idea of Karman’s method) to finding an approximation
of θ vs ε for the calculation of integral
∫ 1
Y vs θ for the computation of the integral of the right hand side.
0
(1 − θ) dε, and an approximation of