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6.4. MODIFICATION OF THE CONDITIONS AT THE “COLD BOUNDARY” 137
6.4 Modification of the conditions at the “cold
boundary”
There are several solutions available in order to elude the difficulty of the cold boundary,
which has passed unnoticed until recently. The history of the evolution of thought
on this subject may be followed by consulting references [17] through [19]. Herein
we will only study the two solutions currently used, which consist in introducing an
ignition temperature T i and a flame holder.
Ignition temperature
By adopting the same assumption used in the classical theories of Combustion, that
is, by assuming that there is an ignition temperature T i , such that reaction velocity of
the mixture is zero for T < T i , this temperature divides the combustion wave into
two zones: a “heating and diffusion zone”, corresponding to temperatures under T i , at
which no chemical reaction takes place, and a “reaction zone”, which corresponds to
temperatures over T i .
The differential equations for the reaction zone are the same given in the preceding
paragraph, which are summarized in the following, as well as the boundary
conditions, assuming that the origin of coordinates x = 0 is located at ignition point
T = T i .
Reaction zone, x > 0.
m dε = w(Y, ρ, T ), (6.2)
dx
ρD dY
dx
= m(Y − ε), (6.3)
λ dT
dx = mc p
The boundary conditions for this system will be
(
(T − Tf ) + (T f − T 0 )(1 − ε) ) . (6.5.a)
x = 0 : ε = 0, T = T i , (6.13)
x → +∞ : ε = 1, T = T f , Y = 1. (6.14)