Untitled - Aerobib - Universidad Politécnica de Madrid

6.3. BOUNDARY CONDITIONS 135
c) Diffusion equation.
d) Momentum equation.
e) Energy equation.
f) State equation.
ρD dY
dx
= m(Y − ε) (6.3)
p = const. (6.4)
c p T − qε − λ m
dT
dx = e (6.5)
p
ρ = R gT (6.6)
In this system, m and e are two constants, whose values will result from the
boundary conditions. The elimination of ρ through (6.6) reduces the system to three
first order differential equations (6.2), (6.3) and (6.5) with three unknown ε, Y and T .
6.3 Boundary conditions
In order for the solution of this system to represent a stationary wave, it is necessary
that it satisfies the following boundary conditions, which insures the transition from an
uniform state of the unburnt gases before the wave to an uniform state of the products
in chemical equilibrium after it,
Unburnt gases, x → −∞ : Y → 0, ε → 0, T → T 0 . (6.7)
Products, x → +∞ : Y → 1, ε → 1, T → T f . (6.8)
These conditions imply, as well, the following
x → ±∞ :
dY
dx → 0,
dε
dx → 0,
dT
dx
→ 0. (6.9)
The first of conditions (6.9) is satisfied by virtue of (6.3), since ε and Y take
the same values both in x = +∞ and in x = −∞, by virtue of (6.7) and (6.8).
The third condition (6.9) is also satisfied by virtue of (6.5) when the following
values are assigned to q and e
q = c p (T f − T 0 ),
e = c p T 0 ,
(6.10)