Untitled - Aerobib - Universidad Politécnica de Madrid

290 CHAPTER 12. DIFFUSION FLAME
Moreover, as done in the theory of free jets, here only some terms of the viscosity
forces need to be preserved. 4
2) The influence of mass forces is important. Hydrostatic equilibrium in the fluid
undisturbed by the flame determines the pressure field. If ρ 0 is the density of the
undisturbed fluid
−∇p + ρ 0 ¯F = ¯0, (12.5)
which gives for Eq. (12.3)
ρ(¯v · ∇)¯v = ∇ · τ ev + (ρ − ρ 0 ) ¯F . (12.6)
Furthermore, in each case only the significant terms of ∇ · τ ev will be retained. So
far a solution that includes the influence of the last term of Eq. (12.6) has not been
obtained.
Energy equation
Kinetic energy of motion, energy dissipated by viscosity and work done by mass
forces are negligible. Hence, equation (3.38) reduces to
(
ρ(¯v · ∇)h − ∇ · (λ∇T ) + ∇ · ρ ∑ )
Y i h i¯v di = 0. (12.7)
i
Diffusion equations
Such equations are given by system (3.8) which takes the form
∑
(
Y j ¯vdi − ¯v dj
+ ∇Y i
− ∇Y )
j
= ¯0, (i = 1, 2, . . . , l)
M
j j D ij Y i Y j
∑
Y j ¯v dj = ¯0,
j
(12.8)
where pressure and thermal diffusion are not included since they are negligible.
State equation
This is equation (3.39)
p
ρ = R mT. (12.9)
4 See Fay, Ref. [15].