Untitled - Aerobib - Universidad Politécnica de Madrid

6.9. SOLUTION OF THE FLAME EQUATIONS 155
1.4
1.3
Λ −1/2 / Λ −1/2
λ=λ λ=λ θ
f f
1.2
1.1
θ 0
=0.250
θ 0
=0.125
1.0
2 4 6 8 10 12 14 16
θ a
Figure 6.11: Ratio of flame velocities corresponding to λ = λ f and λ = λ f θ as a function
of the activation temperature θ a for two different values of the initial temperature
θ 0.
Figure 6.11, taken from Ref. [22], sets forth the scarce influence of the variation
of λ with temperature on the flame velocity, specially for high activation temperature.
Two different solution are compared in this figure, one assuming that λ is
constant through the flame λ = λ f ; and another where λ varies linearly with temperature
λ = λ f θ, in accordance with (6.88). The solution was computed for two different
values of the temperature of unburnt gases The deviations are under 10 per cent.
Once established that the preceding considerations solve the problem of the
calculation of integral I we shall proceed with the study of J.
Approximation of integral J
In order to approximate integral J, we must first consider that in the normal cases, this
is to say when θ a ≫ 1, this integral is much smaller than the other term of the left
hand side of Eq. (6.73), that is
J ≪ 1 − θ 0
. (6.89)
2
Such a property is illustrated in Fig. 6.12 which corresponds to a typical case
where the value of J is shown by a dot-area while the value for (1 − θ 0 )/2 is represented
by the area of triangle ABC.