Untitled - Aerobib - Universidad Politécnica de Madrid

198 CHAPTER 6. LAMINAR FLAMES
Hence, dissociation of bromine reduces the flame propagation velocity basically because
such dissociation reduces the molar fractions of Br 2 at temperatures close to
T f . Dissociation also influences the value for J, as shown by Eq. (6.283). However,
this influence is small and it acts opposite to the influence of thermal conductivity; the
value for J is very small and its influence is negligible. This last conclusion will be
verified when numerical calculations are performed later on.
Introducing Eq. (6.298) into Eq. (6.295) we find, for θ ≃ 1, the following
behavior of X 4
X 4 ≃ 1.676 e −θ r
√
RT f
p (α 1 − α 3 ), (6.300)
thus X 4 is a linear function of (α 1 − α 3 ) and, for θ ≃ 1, X 4 ≫ X 2 .
Evaluation of X 1 and X 3
The evaluation of X 1 and X 3 can be easily performed from the diffusion relations
Eqs. (6.264) and (6.265).
Equations (6.229) and (6.254) permit us to express ε 2 and ε 3 as functions of ε 1
and ε 4 as follows
and
ε 2 = M 4
M 1
(ε 1f − ε 1 ) − ε 4 , (6.301)
ε 3 = ε 3f + M 1 − M 4
M 1
(ε 1f − ε 1 ). (6.302)
By introducing these expressions, together with Eqs. (6.285) and (6.286) , into
Eq. (6.264) and (6.265), and if ε 4 , X 2 and X 4 are expressed as functions of (α 1 − α 3 )
by use of the relations given in Eqs. (6.251), (6.298) and (6.300), two equations are
obtained which depend only on θ, (ε 1f − ε 1 ), α 1 and α 3 . These expressions can be
developed in series of these variables near θ = 1. The results are
dX 1
dθ
dX 3
dθ
[( =λ f RT f M4 X 1f
+ M 1 − M 4
pc p q 1 M 1 M 2 D 12 M 1 M 3
(
α1 α 3
+O , ,
ε 1f − ε 1 ε 1f − ε 1
[( =λ f RT f M4 X 3f
− M 1 − M 4
pc p q 1 M 1 M 2 D 23 M 1 M 3
(
α1 α 3
+O , ,
ε 1f − ε 1 ε 1f − ε 1
X 1f
D 13
+
1 )
X 3f
M 1 D 13
)
1 − θ
+ higher order terms
ε 1f − ε 1
X 1f
D 13
−
1 )
X 3f
M 1 D 13
)
1 − θ
+ higher order terms
ε 1f − ε 1
]
, (6.303)
]
. (6.304)