Untitled - Aerobib - Universidad Politécnica de Madrid

148 CHAPTER 6. LAMINAR FLAMES
Once Λ is known then the velocity u 0 of the flame is given by
√
Bλ f e
u 0 =
−θa
ρ 2 0 c p(1 − θ 0 ) Λ−1/2 . (6.72)
6.9 Solution of the flame equations
As before said, the preceding system cannot by integrated analytically. Hence, it
becomes necessary to resort either to numerical solutions or else to semi-analytical
approximation methods.
The numerical methods are very cumbersome since we are dealing with an
eigenvalue problem with given boundary conditions at both extremes of the interval,
and, therefore, we would have to make numerous tentatives before reaching the exact
solution. Generally, the use of electronic computers is required and in particular they
were widely utilized by Hirschfelder and his collaborators.
In recent years, several approximate methods have been proposed generally
based in the unusual behavior of the solutions due to the presence of factor e −θa/θ in
the reaction velocity. In fact, when the reduced activation temperature θ a is large, as
should be expected in combustion reactions, due to this factor the result is determined,
essentially, by the form of the solution close to the hot boundary and this enables
the development of methods, as those proposed by Zeldovich-Frank-Kamenetskii-
Semenov [4] and the one by Boys-Corner [5], Adams [20], Wilde [21] and von Kármán
[6], with different variations and approximations. Only very recently a comparative
study of these methods has been performed. Information on the subject will be found
in the work by von Kármán [10] and, specially, in the more complete study carried
out by Millán, Sendagorta and Da Riva [22]. Figures 6.5 and 6.6, taken from this
study show a comparison between the approximate values for Λ −1/2 ( to which the
flame velocity is proportional by virtue of (6.72)), obtained through several of those
methods, with the exact value, given by a numerical integration. This comparison was
performed for several values of the temperature of the unburnt gases and of the activation
energy of the reaction. All the results shown in these figures correspond to a
Lewis-Semenov number equal to unity and to a constant mean molar mass of the mixture.
However, the unpublished results of the calculations performed for more general
cases indicate that the same conclusions are still valid.
Figures 6.5 and 6.6 show that the Boys-Corner method in its first iteration
gives defect values with an important deviation. The same happens with the method
by Zeldovich et al., but giving excess values. The approximation improves with the