Untitled - Aerobib - Universidad Politécnica de Madrid

162 CHAPTER 6. LAMINAR FLAMES
0.05
0.04
θ a
=8
(1−θ)e −θ a (1−θ)/θ
0.03
0.02
0.01
θ l
0.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
θ
Figure 6.15: Curve showing the typical values of the integrand of I as a function of θ.
The reason for the insensibility of the solutions to θ i appears clearly in Fig. 6.15
where it is seen that the values of θ i comprised between θ 0 and θ l do not influence in
the value of integral I, since in this zone the integrand is practically zero. To the
contrary, if θ i is comprised between θ l and 1, then one disregards in the solution the
contribution of the part corresponding to θ l < θ < θ i . Since Λ −1/2 is proportional to
√
I, this explains the fact that Λ −1/2 should decrease and tend to zero when is θ i → 1.
The fact that the combustion velocity tends to ∞ when θ i → θ 0 may be physically
explained since then all the mass of gas burns simultaneously giving infinite
wave velocity. In order to give a theoretical explanation it would be necessary to
analyze in detail the differential system.
The fact that Λ −1/2 is independent from θ i enables the elimination of this value
in all the equations of the preceding paragraphs, in special in the lower limit of integral
I (Eq. 6.74), by substituting it for zero without changing the result.
6.12 General equations for the combustion wave in the
case of more than two chemical species
Hirschfelder and his collaborators were the first to give the general equations for the
combustion wave in the case of more than two chemical species [7]. A derivation of
these equations may be found in a paper by Kármán and Penner [6].