Untitled - Aerobib - Universidad Politécnica de Madrid

8.3. FLAMMABILITY LIMITS 225
The preceding system of equations allows an easy simplification, by using the
stream function ψ, frequently applied to the problem of Fluid Mechanics.
The said function is defined by the following conditions
ρ = ∂ψ
∂x ,
ρv = −∂ψ
∂t , (8.11)
through which Eq. (8.7) is identically satisfied. When taking Eq. (8.11) into Eqs. (8.8)
and (8.9), this system reduces to the following
∂T
∂t = ∂ ( ρλ ∂T
∂ψ c p ∂ψ
∂Y
∂t = ∂ ( ρ 2 D
∂ψ
Thus the variable v has been eliminated.
c p
∂Y
∂ψ
)
+ q c p
w
ρ , (8.12)
)
+ w ρ . (8.13)
The integration of the above system has not yet been performed. However,
Spalding [7] has obtained graphical solutions of Eq. (8.12) disregarding diffusion.
These solutions show the existence of a critical value d cr for d under which the wave
extinguishes, while for d > d cr it propagates indefinitely.
There are some experimental observations on the evolution followed by an
ignited mass before the flame establishes [8].
8.3 Flammability limits
The theoretical studies performed so far have shown that the combustible mixture is
always capable of maintaining a flame. Experimentally, however, it has been verified
that it is not so, since there are numerous mixtures which, in practise, are not able to
propagate a flame, even when theory predicts otherwise.
In fact, experiments disclose that when analyzing the behavior of a combustible
mixture under pressure and temperature constant, but changing its composition, for instance
a mixture of fuel and air, the flame velocity is maximum for a definite value
of the composition which is normally very close to the stoichiometric one, and it decreases
for mixtures either rich or loan, up to values known as inflammability limits of
the mixture, beyond which it cannot propagate a flame. Moreover the flame velocity
does not vanish at the inflammability limits but it generally reaches values of a few
centimeters per second. Inflammability limits depend also on pressure and they are
difficult to determine experimentally because it is necessary to operate under marginal
conditions, on which the experimental techniques might have an influence. Tables