Untitled - Aerobib - Universidad Politécnica de Madrid

6.12. GENERAL EQUATIONS FOR THE COMBUSTION WAVE 165
On the other hand, considering that 8
and
Y j = M j
M m
X j (6.117)
ρ = pM m
RT , (6.118)
the elimination of v di , v dj , Y j and ρ between these three last equations and (6.115),
finally gives the following system of diffusion equations in the form which will be
applied to the solution of the flame equations
dX i
dx = mRT
p
l∑
j=1
(
)
1 ε j ε i
X i − X j , (i = 1, 2, . . . , l). (6.119)
D ij M j M i
Energy equation
The energy equation is the one give in Eq. (3.74), when we disregard in it the terms
due to kinetic energy and viscosity, thus reducing it to the following
m
l∑
j=1
ε j h j − λ dT
dx
where e is a constant defined by the initial or final conditions.
= e, (6.120)
Before transforming this equations into the form in which it will be applied we
shall first consider the boundary conditions.
Boundary conditions
These conditions are analogous to those applied in the case of two chemical species,
namely,
Cold boundary, x → −∞:
T → T 0 , ε i → ε i0 , X i → X i0 ,
dT
dx → 0, dε i
dx → 0, dX i
→ 0. (6.121)
dx
Hot boundary, x → +∞:
T → T f , ε i → ε if , X i → X if ,
dT
dx → 0, dε i
dx → 0, dX i
→ 0. (6.122)
dx
8 See Chap. 1, Eq.1.34