Untitled - Aerobib - Universidad Politécnica de Madrid

224 CHAPTER 8. IGNITION, FLAMMABILITY AND QUENCHING
A correct stating of the problem would require first an adequate formulation of
the same, in an analogous way to the one used in Chap. 6 for the stationary wave. In
the second place it would be necessary to perform an analysis of the type of solutions
for the system of equations obtained under the set of initial and boundary conditions
corresponding to the model previously described.
If we consider the more simple case of an indefinite plane wave corresponding
to a one-dimensional problem, in which case the initial energy must be stored between
two parallel layers, it may be easily verified that the system of equations of Chap. 6,
corresponding to a stationary wave, keeping the same notation, must be substituted by
the following:
a) Continuity equation.
b) Energy equation.
c) Diffusion equation.
∂ρ
∂t + ∂(ρv) = 0. (8.7)
∂x
( ∂T
ρc p
∂t + v ∂T )
= ∂ (
λ ∂T )
+ qw. (8.8)
∂x ∂x ∂x
ρ
( ∂Y
∂t + v ∂Y )
= ∂ (
ρD ∂Y )
+ w. (8.9)
∂x ∂x ∂x
Thus we obtain a system of three equations for the three unknowns T , v and Y .
Since the process takes place at constant pressure, ρ is already determined, as function
of T .
With reference to initial and boundary conditions corresponding to the model
under consideration, if 2d is the width of the heated slab and the origin of coordinates
is fixed at the central point of the slab, we will have:
a) Initial conditions (t = 0).
0 < x < d : T = T f ,
d < x < ∞ : T = T 0 ,
(8.10)
b) Boundary conditions (by symmetry).
0 < x < ∞ : v = Y = 0.
x = 0 :
∂T
∂x = v = ∂Y
∂x = 0.
Furthermore an additional condition must be introduced expressing that w must
be zero for T = T 0 , as it was done for the case of a stationary wave.