Untitled - Aerobib - Universidad Politécnica de Madrid

154 CHAPTER 6. LAMINAR FLAMES
whose solution is
θ ≃ 1 : 1 − Y ≃ L
1 − θ 0
(1 − θ), (6.86)
this approximation must replace Eq. (6.76) when L is different from unity and furthermore
the value taken for L must be that corresponding to the temperature and
composition of the hot boundary conditions of the flame, T = T f .
If the activation energy has a small value and an approximation of Y vs θ that
will be valid within a wider range of temperatures is desired, then the linear approximation
(6.86) may be substituted by the following parabolic
Y ≃
( θ − θ0
1 − θ 0
) L
, (6.87)
which for θ = 1 coincides with (6.86) and for θ ≃ θ 0 it behaves like the solution
corresponding to the heating zone (ε ≡ 0). This can readily be verified. Fig. (6.10)
shows some of the curves (6.86) and (6.87).
1.0
0.8
L=0.5
L=0.75
0.6
L=1.5
L=2.0
Y
0.4
L=1.0
0.2
0.0
0.125 0.2 0.4 0.6 0.8 1.0
θ
Figure 6.10: Linear and parabolic approximations of Y vs θ given by Eqs. 6.86 and 6.87
for θ 0 = 0.125.
The preceding considerations prove that it is generally sufficient, when calculating
the integral to take for λ/λ f the value unity which corresponds to the hot
boundary, or some other approximation at the neighborhood of this point. Frequently
the following approximations are used
λ
λ f
≃ θ,
λ
λ f
≃ √ θ. (6.88)