Untitled - Aerobib - Universidad Politécnica de Madrid

6.9. SOLUTION OF THE FLAME EQUATIONS 151
will be written for shortness as
∫ 1
( ) n
λ 1 − Y
I =(1 − θ 0 ) θ δ−n e −θ 1 − θ
a
θ dθ, (6.74)
λ f 1 + aY
J =
∫ 1
0
θ i
(1 − θ) dε. (6.75)
With this notation, the fundamental Eq. (6.73) may be written
1 − θ 0
2
− J = ΛI.
The following is a separate study of both approximations.
(6.73.a)
Approximation of integral I
Heat transfer λ is a function of temperature θ of the mixture and its composition
defined by the value of Y . 6 Consequently, in order to calculate I the problem reduces
to finding an approximation for Y as a function of θ. The solution depends on the
value of the Lewis-Semenov number.
Lewis-Semenov number equal to unity
If L = 1, then Y is a lineal function of θ in the following form
In fact, when condition
Y = θ − θ 0
1 − θ 0
. (6.76)
L ≡
is satisfied, the diffusion Eq. (6.65) may be written in the form
λ
ρDc p
= 1 (6.77)
λ dY
dx = mc p(Y − ε). (6.78)
By adding to it the energy Eq. (6.66), multiplied by an arbitrary constant C,
λ d
(
dx (Y + Cθ) = mc p Y + C(θ − θ 0 ) − ( C(1 − θ 0 ) + 1 ) )
ε . (6.79)
This equation is satisfied identically with the following two conditions
6 See Chap. 2, §4.
C(1 − θ 0 ) + 1 = 0, (6.80)
Y + C(θ − θ 0 ) = 0, (6.81)