Untitled - Aerobib - Universidad Politécnica de Madrid

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 197
where
θ √
g(θ) = 0.838 e − r
θ θ RT f
p . (6.294)
Likewise, when Eq. (6.291) is substituted into Eq. (6.246), one finds
√
X 4 = 1.676 e − θr
θ θ RT √
f X2 . (6.295)
p
Equations (6.293) and (6.295) permit the computation of X 2 and X 4 as soon as α 1
and α 3 are known. Nevertheless, before initiating the calculation of these quantities
let us study the behavior of X 2 as a function of θ, since this relation influences more
than any other the value of the flame velocity, as shown by Eq. (6.282).
For θ → 1, the following conditions are satisfied
√
g(θ) → 0.838 e −θ RT
r f
≠ 0,
p
α 1 − α 3 → 0.
(6.296)
Therefore, for θ near 1
α 1 − α 3
g(θ) 2 ≪ 1, (6.297)
it is now simple to show from Eq. (6.293) that, near θ = 1, X 2 behaves as follows
X 2 ≃ 1
2θ r
p
θ −1 e θ (α 1 − α 3 ) 2 . (6.298)
2.808 RT f
√
On the other hand, when the difference 1−θ increases, the term 0.838 e −θr/θ θ RT f
p
decreases very rapidly since θ r ≫ 1, whilst the term α 1 − α 3 increases. Therefore,
when the difference 1 − θ is not very small compared to unity, X 2 behaves as follows
X 2 ≃ α 1 − α 3 . (6.299)
In the different behaviors of Eqs. (6.298) and (6.299) lie essentially the influence
of the dissociation of bromine on the propagation velocity of the flame. In fact,
if the term X 4 is neglected in Eq. (6.255) when computing X 2 , as was done in [40],
then the approximation given in Eq. (6.299) holds for all temperatures. On the contrary,
when the influence of X 4 is taken into account, the values for X 2 near θ = 1
are far smaller for than those given by Eq. (6.299), since then X 2 varies as the square
of (α 1 − α 3 ), as shown by Eq. (6.298). Since the values for X 2 are smaller, the value
for the integral I is also smaller, as shown by Eq. (6.28). Finally, if I decreases, √ Λ
increases as shown by Eq. (6.284). Therefore, according to Eq. (6.272), u 0 decreases.