Untitled - Aerobib - Universidad Politécnica de Madrid

6.12. GENERAL EQUATIONS FOR THE COMBUSTION WAVE 167
In this equation, expression ∑ c pj ε j depends on temperature T and on the values for
j
mass fluxes ε j of the species. In order to simplify this expression we adopt a mean
value c p independent from temperature and the composition of the mixture so that
Eq. (6.126) may be written in the form
λ dT (∑
)
dx = m h 0 j(ε j − ε jf ) + c p (T − T f ) . (6.127)
j
The value for c p is derived from (6.127) when expressing the condition that
(6.127) vanishes at the cold boundary in agreement with (6.121) obtaining
∑
j
c p =
h0 j (ε j0 − ε jf )
, (6.128)
T f − T 0
or else since, as before said, ε jf = Y jf and ε j0 = Y j0 due to the fact that diffusion is
absent in both limits
c p =
∑
j h0 j (Y j0 − Y jf )
T f − T 0
. (6.129)
If, as before done, we introduce dimensionless temperature θ = T/T f into Eq. (6.127),
this may be written
λ dθ (
dx = mc p θ − 1 +
l∑
j=1
h 0 )
j
(ε j − ε jf ) . (6.130)
c p T f
It is also convenient here to eliminate variable x, by dividing the reaction equation
(6.109.a) and diffusion Eq. (6.119) by (6.130), as it was done for the case of two
chemical species. Thus the system of the flame equations reduces to the following
Reaction equations
dε i
dθ = λ λ f
r∑
j=1
(ν ′′
ij − ν ′ ij)Λ ij θ δ j−n j
e −θ aj/θ
θ − 1 +
l∑
q j (ε j − ε jf )
j=1
l ∏
s=1
X s ν ′ sj
, (i = 1, 2, . . . , l), (6.131)
where, by analogy with the case of two chemical species, we have taken
Λ ij = M iλ f B j T δ j−n j
f
m 2 , (6.132)
c p
θ aj =
E j
RT f
, (6.133)