Untitled - Aerobib - Universidad Politécnica de Madrid

142 CHAPTER 6. LAMINAR FLAMES
In order to find the solution, it is convenient to eliminate x from the preceding
system. This is done by dividing (6.24) and (6.25) by (6.26), thus resulting
In this system
dY
dθ = L Y − ε
θ − 1 + (1 − θ 0 )(1 − ε) , (6.27)
dε
dθ = Λ 1 − Y
θ − 1 + (1 − θ 0 )(1 − ε) . (6.28)
L =
λ
ρDc p
(6.29)
is the Lewis-Semenov number of the mixture, which is constant.
Λ is a dimensionless parameter, also constant, defined by
Λ = λρ 0k
m 2 c p
. (6.30)
The problem lies, precisely, in determining the eigenvalue of this parameter. Once it is
known, the velocity u 0 of the flame propagation may be derived from it and Eq. (6.22),
thus obtaining
√
λk
u 0 = Λ −1/2 . (6.31)
ρ 0 c p
The system of equations (6.27) and (6.26) holds only within the reaction zone, where
boundary conditions are
where we have written θ i = T i /T f .
θ = θ i : ε = 0, (6.32)
θ = 1 : ε = θ = 1, (6.33)
Precisely because these three conditions exist for a system of second-order, it
is necessary to look for the value of Λ which makes them compatible.
The system is readily integrated by testing lineal solutions of the form
1 − ε = α(1 − θ), (6.34)
1 − Y = β(1 − θ), (6.35)
which satisfy boundary conditions (6.33) at the hot boundary.
Condition (6.32) at the cold boundary gives the following relation
1 = α(1 − θ i ). (6.36)