Untitled - Aerobib - Universidad Politécnica de Madrid

6.9. SOLUTION OF THE FLAME EQUATIONS 153
is small, and therefore the influence of L is more obvious than it would be for larger
values of θ a .
On the other hand, due to the presence of factor e −θ 1 − θ
a
θ
in the integral of
the right hand side of Eq. (6.73) it occurs that for large values of θ a , as being those
normally appearing in combustion, the value of the integral is influenced only by the
value taken at the neighborhood of θ = 1 by the quantity under the integral.
1.0
0.8
θ a
=4
L=0.5
ε
0.6
θ
Y
0.4
0.2
θ 0
=0.125
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ε, Y
Figure 6.9: Solutions of the flame equations 6.67 and 6.68 for L = 0.5, θ a = 4, θ 0 =
0.125, θ i = 0.4, n = 1, a = 0 and δ = 1.
In order to obtain the desired approximation Y vs θ it is sufficient to observe
Fig. 6.3 corresponding to the example treated in §7, since this figure shows that even
when the activation energy θ a is zero it happens that near θ = 1 is 1 − ε ≫ 1 − Y ,
and also 1 − ε ≫ 1 − θ. This fact appears more evident for reactions with a high
activation temperature as shown in Fig. 6.9 which was calculated for a more realistic
case. Eq. (6.68) may be written
dY
dθ = L (1 − ε) − (1 − Y )
1 − θ 0
(1 − ε) − 1 − θ , (6.84)
1 − θ 0
and, in accordance with the preceding considerations, both 1 − Y and 1 − θ
1 − θ 0
are negligible
close to θ = 1 and the following approximation may be obtained for Eq. (6.84)
θ ≃ 1 :
dY
dθ ≃ L , (6.85)
1 − θ 0