Untitled - Aerobib - Universidad Politécnica de Madrid

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 199
dX 1
dθ
Hence, for θ ≃ 1, we find
≃ λ (
f RT f M4 X 1f
+ M 1 − M 4 X 1f
+
pc p q 1 M 1 M 2 D 12 M 1 M 3 D 13
1 )
X 3f
≡ β 1 , (6.305)
M 1 D 13
dX 3
dθ
≃ λ (
f RT f M4 X 3f
− M 1 − M 4 X 1f
−
pc p q 1 M 1 M 2 D 23 M 1 M 3 D 13
1 )
X 3f
≡ −β 3 , (6.306)
M 1 D 13
which leads to the following approximations for X 1 and X 3
X 1 ≃ X 1f − β 1 (1 − θ), (6.307)
X 3 ≃ X 3f + β 3 (1 − θ). (6.308)
Thus
where
α 1 − α 3 = (β 1 − β 3 )(1 − θ) = β 2 (1 − θ), (6.309)
β 2 = β 1 − β 3 . (6.310)
near θ = 1
In short, the following approximations for X 1 , X 2 , X 3 and X 4 are obtained
X 1 = X 1f − β 1 (1 − θ), (6.311)
X 2 =
(√
g(θ)2 + β 2 (1 − θ) − g(θ)) 2
, (6.312)
X 3 = X 3f − β 3 (1 − θ), (6.313)
(√ )
X 4 = 2g(θ) g(θ)2 + β 2 (1 − θ) − g(θ) . (6.314)
Evaluation of f(θ), I and J
When computing f(θ), the only values of X 1 , X 2 and X 3 needed are those for θ near
1, since the reduced activation temperature θ a is much larger than unity. The relevant
expressions are given in Eqs. (6.311) through (6.313). Once f(θ) is known, the
integral I can be computed by numerical of graphical integration. Likewise, knowing
f(θ) and X 4 , the integral J can be obtained either numerically or graphically.
Computation of the flame velocity
The preceding results may be summarized by the following method for the computation
of the velocity of the flame.