Untitled - Aerobib - Universidad Politécnica de Madrid

194 CHAPTER 6. LAMINAR FLAMES
near θ = 1, where the exponent n takes the following values
X 3,0 < 0.5 : n = 1,
X 3,0 = 0.5 : n = 7 4 ,
(6.275)
X 3,0 > 0.5 : n = 5 4 .
The approximation given in Eq. (6.274) holds only for values of θ very close
to unity. In fact, the values for ε 1f − ε 1 given by such an approximation decrease very
rapidly with 1 − θ, reaching the value zero for values of θ very close to unity. A better
approximation for ε 1f − ε 1 can be obtained as follows. In Eq. (6.270), (1 − θ) as well
as q 4 (ε 4f − ε 4 ) are much smaller than q 1 (ε 1f − ε 1 ) for the interesting range of values
of θ. 13 Therefore, a good approximation for Eq. (6.270) can be obtained if these terms
are neglected, thus yielding
(ε 1f − ε 1 ) dε 1
dθ = Λ X 3 X 3/2
2 e −θ 1 − θ
a
θ
q 1 θ(X 2 + 0.119X 1 ) . (6.276)
Equation (6.276) may be integrated from the hot boundary forward. The result is
√ ∫ 1
ε 1f − ε 1 = √ 2Λ q 1
θ
f(θ ′ ) dθ ′ , (6.277)
where
f(θ) = X 3X 3/2
2 e −θ 1 − θ
a
θ
θ(X 2 + 0.119X 1 ) . (6.278)
It can be seen that Eq. (6.277) is an approximation for (ε 1f −ε 1 ) which is valid
for a range of temperatures far more extensive than that covered by Eq. (6.274).
The improved approximation is necessary when the velocity of the flame is
computed, as it is done here, without neglecting the influence of ε 4 and X 4 . In fact, if
an approximation similar to Eq. (6.274) is used in this case, when computing the integral
on the left-hand side of Eq. (6.273), the contribution of this integral would be considerably
overestimated, as can be verified easily. The approximation of Eq. (6.277)
gives
√
dε 1
dθ = Λ
2q 1
13 However a very small interval near θ = 1 must be excluded.
f(θ)
√ ∫ . (6.279)
1
θ f(θ′ ) dθ ′