Untitled - Aerobib - Universidad Politécnica de Madrid

150 CHAPTER 6. LAMINAR FLAMES
The other methods are considerably better mainly when applied to normal activation
temperatures. This is specially true about Wilde’s method, the second and third
alternatives of von Kármán’s one and that proposed by Sendagorta [23], which is the
best and does not require more work than the others.
1.0
0.8
BOYS−CORNER
0.6
EXACT
θ
0.4
0.2
KARMAN 3
0.0
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
ε
Figure 6.7: Comparison of the distributions of ɛ vs θ obtained by two approximate methods
with the exact result.
Figure 6.7 shows, for one of the case calculated, the law of the variation of ε
with θ of the Boys-Corner approximation, Kármán’s third alternative and the exact
solutions. It is evident that the approximation reached with Kármán’s method is very
satisfactory. The following studies the nature of this method after referring the reader
to the above references for a more detailed analysis of the other methods mentioned.
If in Eq. (6.67) we bring the denominator of the right hand side into the left one,
and this equation is integrated between the cold boundary (6.71) and the hot boundary
(6.70), we obtain
1 − θ 0
2
−
∫ 1
0
∫ 1
( ) n
λ 1 − Y
(1 − θ) dε = Λ(1 − θ 0 ) θ δ−n e −θ 1 − θ
a
θ dθ.
θ i
λ f 1 + aY
(6.73)
Consequently, in order to determine Λ, the problem reduces now to obtaining
an approximation of θ vs ε on the integral of the left side of Eq. (6.73) and an approximation
of Y vs θ on the integral of the right side of the same equation. These integrals