Untitled - Aerobib - Universidad Politécnica de Madrid

6.13. OZONE DECOMPOSITION FLAME 175
But this zone is unimportant when studying the flame structure and in particular its
propagation velocity.
The applicability of the steady state assumption is an important contribution
to the theory of flames which essentially simplifies. The success reached with ozone
flames has encouraged the idea of applying this assumption to the study of similar
practical cases, as will be seen later on.
Recently, however, the applicability of such an assumption has been discussed,
among others, by Campbell [26], Giddings and Hirschfelder [27] and Spalding [28].
They based their objections first on the fact that the crossing of the flame is so rapid
that there is not sufficient time for the formation of radicals to reach the level required
by the condition of equilibrium established by the said assumption. On the other hand,
even if such level were reached the diffusion of radicals would change substantially
the distribution. Although other studies by Gilbert and Altman [29] and by Millán
and Sanz [30] tend to confirm the applicability of the steady state assumption for the
cases studied in the foregoing paragraphs, however, the problem cannot be considered
as solved, until a general study enables the establishment of the conditions that must
be satisfied in a flame so that this assumption be valid.
Let us proceed with the study of the ozone flame. The application of the steady
state assumption eliminates Eq. (6.151) and reduces (6.152) to the following
where, to simplify notation, we have written
dε 3
dθ = −2Λ λ λ f
θ −3/2 e −θa1/θ X 3
θ − 1 + q 3 ε 3
, (6.155)
Λ 13 = Λ = M 1λ f B 3
. (6.156)
m 2 c p T 3/2
f
Taking into account the preceding considerations, the diffusion equation reduces to
where also, for shortness, we write
dX 3
dθ = L 3
2 X 3 − ε 3 − 1 2 X 3ε 3
θ − 1 + q 3 ε 3
, (6.157)
L 13 = L =
λRT
M 1 c p pD 13
. (6.158)
The problem reduces now to the integration of these two equations with the following
boundary conditions
θ = θ i : ε 3 = 0, (6.159)
θ = 1 : ε 3 = X 3 = 0. (6.160)