Untitled - Aerobib - Universidad Politécnica de Madrid

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 193
is the reduced activation temperature for the formation of HBr. These authors assume
also that the coefficient of thermal conductivity λ is independent of the composition
of the mixture and that it varies proportionally with the square root of the temperature,
viz.,
λ = λ f
√
θ. (6.269)
If Eqs. (6.266), (6.267), and (6.269) are introduced into Eq. (6.263), we obtain
where
dε 1
dθ = X 3 X 3/2
Λθ−1 2 (X 2 + 0.119X 1 ) −1 e −θ 1 − θ
a
θ
, (6.270)
θ − 1 + q 1 (ε 1f − ε 1 ) − q 4 (ε 4f − ε 4 )
Λ = 1.6 × 10 14 λ ( ) 3/2
f p
m 2 M 1 e −θ a (6.271)
c p RT f
is the eigenvalue which determines the propagation velocity u 0 of the flame through
the relation
u 0 = 1.265 × 10 7 1 ρ 0
√
λ f M 1
c p
( p
RT f
) 3/4
e −θ a/2 Λ −1/2 . (6.272)
The eigenvalue Λ can be obtained by applying the method of von Kármán
and Penner [6] which involves integration of Eq. (6.270) between the cold and hot
boundaries of the flame in the following form
ε 2 ∫ ε1f (
) ∫ 1
1f
q 1
2 − X 3 X 3/2
2 e −θ 1 − θ
a
θ
1 − θ + q 4 (ε 4f − ε 4 ) dε 1 = Λ
0
θ 0
θ(X 2 + 0.119X 1 )
The problem lies now in obtaining:
dθ. (6.273)
1) Approximate expressions for (1 − θ) and (ε 4f − ε 4 ) as function of (ε 1f − ε 1 )
for the computation on the integral on the left hand of Eq. (6.273).
2) Approximate expressions for X 1 , X 2 , and X 3 as functions of θ, for the computation
of the integral appearing on the right hand side.
Von Kármán and Penner [40] performed this computation by assuming that
X 4 as well as ε 4 may be neglected (compared with unity). In this case, the diffusion
equations (for Lewis number close to unity) show that X 1 , X 2 , and X 3 are linear
functions of 1 − θ near θ = 1; a study of Eq. (6.270) shows that ε 1f − ε 1 is a function
of θ of the form
ε 1f − ε 1 = (1 − θ) n (6.274)