Untitled - Aerobib - Universidad Politécnica de Madrid

136 CHAPTER 6. LAMINAR FLAMES
which, furthermore, enables (6.5) to be written as
λ dT
dx = mc (
p (T − Tf ) + (T f − T 0 )(1 − ε) ) , (6.5.a)
in which form it will be used further on.
As for the second condition (6.9) it is also satisfied for x = +∞, since the
condition of chemical equilibrium for the products may be expressed as
w(1, ρ f , T f ) = 0. (6.11)
To the contrary, condition dε → 0 for x → −∞, will not be satisfied unless
dx
w(0, ρ 0 , T 0 ) = 0, (6.12)
which, generally, will be in contradiction with the laws of Chemical Kinetics.
This circumstance implies a fundamental difficulty, whose origin lies upon the
fact that it is impossible to maintain invariable the composition of the unburnt gases
located before the wave, as imposed by boundary conditions (6.7) and (6.9), when the
reaction velocity of such gases is not zero.
The existence of stationary combustion waves observed in practice may be
explained by the fact that such waves do not correspond exactly to the one-dimensional
model proposed in this work, since the mixing of reactant species forms, in general,
shortly before they reach the wave, which always looses a certain amount of heat
through the walls of the flame holder, for example, through the walls of the burner.
Furthermore, these walls act as chain breakers for the chemical reactions of the wave.
There is a possibility of eluding the above mentioned difficulty by incorporating
to the one-dimensional model the effect of all these complex circumstances, which
prevent the reaction of the unburnt gases in a real flame, through a slight modification
of the boundary conditions at the neighborhood of the “cold limit” of the flame, which
may be attained in several different ways. The practical interest of these solutions
is based on the fact that for reactions with an appreciable activation energy, as those
expected in combustion, the structure and propagation velocity of the flame are insensible
to a modification of the said boundary conditions, which makes it unnecessary to
define them exactly, within a wide range of values of the defining parameters. Moreover,
the proposed solutions are equivalent since they lead to the same fundamental
results. The following paragraph is devoted to a discussion of these solutions.