Untitled - Aerobib - Universidad Politécnica de Madrid

138 CHAPTER 6. LAMINAR FLAMES
The equations for the heating and diffusion zone, x < 0, may be obtained
from those given §2 by introducing in them the conditions expressing the absence of
chemical reaction, that is: w = ε = 0. There resulting
Heating and diffusion zone, x < 0.
ρD dY
dx
The corresponding boundary conditions are
= mY, (6.15)
λ dT
dx = mc p(T − T 0 ). (6.16)
x = −∞ : T → T 0 , Y → 0, (6.17)
x = 0 : T = T i . (6.18)
Since for x → −∞ both dY / dx and dT / dx tend to zero, as it can be verified
by taking (6.17) into (6.15) and (6.16), the introduction of an ignition temperature
eliminates the difficulty at the cold boundary. Furthermore by comparing the equations
for the heating and diffusion zone with those for the reaction zone, given in the
preceding paragraph, and the boundary conditions (6.13) and (6.18), it may be readily
verified that the transition from one zone to another at point x = 0 is continuous for
all variables when adopting the additional condition that the value for Y at x = 0,
which is undetermined, be the same for both solutions
x = 0 : Y (0 − ) = Y (0 + ). (6.19)
The objection to this way of dealing with the difficulty at the cold boundary
states that the solution obtained, and in special the propagation velocity for the flame,
will depend on the value adopted for T i , which does not actually exist. Nevertheless, it
will be proved further on, when studying the solutions for the combustion wave, that as
long as the velocity of the chemical reaction depends substantially on the temperature
of the mixture, it will happen that the solution obtained will be independent from the
values of T i , except for values of this temperature very close to T 0 or to T f . Under
such conditions the influence of the ignition temperature will vanish completely from
the result, when calculating the propagation velocity of the flame.
The presence of a flame holder
Another solution proposed to the problem of the cold boundary consists in imagining
the presence of a flame holder placed in front of the wave which has the double mission