Untitled - Aerobib - Universidad Politécnica de Madrid

190 CHAPTER 6. LAMINAR FLAMES
Equations (6.228) through (6.232), (6.241), (6.242) and the four expressions
contained in Eq. (6.243) form a system of eleven equations for the computation of
the eleven unknowns of the flame. In the following section we simplify the problem
by introducing the steady-state approximation for the concentration of bromine and
hydrogen atoms.
The steady-state approximation
The following is the expression of the steady-state assumption for bromine and hydrogen
atoms, respectively
w 4 = 0, (6.244)
w 5 = 0. (6.245)
If Eqs. (6.239) and (6.240) are combined with Eqs. (6.244) and (6.245), we obtain
for the mole fractions of Br and H the following relations in terms of the principal
components and of the temperature
X 4 =
√
k1
k 5
√
RT f
√
θX2 , (6.246)
p
and
X 5 = k √ √
2 k1 RT f
k 3 k 5 p
X 3
√ θX2
X 2 + k 4
k 3
X 1
. (6.247)
From Eqs. (6.238), (6.242) and (6.246) the following expression is deduced for w 1
( ) 2 p
w 1 = 2M 1 θ −2 k 3 X 2 X 5 . (6.248)
RT f
Using in Eq. (6.248) the value for X 5 given in Eq. (6.247), we obtain
( ) 3/2
√
p
k1
w 1 = 2M 1 k 2 θ −3/2 X 3 X 3/2
2
RT f k 5
X 2 + k . (6.249)
4
X 1
k 3
Equation (6.249) expresses the rate of formation of HBr as a function of the mole
fractions of the principal components and of temperature. Introduction of Eq. (6.249)
into Eq. (6.230) leads to the result
m dε ( ) 3/2
√
1
p
dx = 2M k1
1
k 2
RT f
θ −3/2 X 3 X 3/2
2
k 5
X 2 + k 4
k 3
X 1
. (6.250)