Untitled - Aerobib - Universidad Politécnica de Madrid

9.2. CONDITIONS THAT MUST BE SATISFIED BY THE JUMP ACROSS A FLAME FRONT. 233
and v t are the normal and tangential components of this velocity. Moreover ρ and p
are, as usual, the density and pressure of the gases, and h is the total specific enthalpy,
that is, (as seen in chapter 1), the thermal enthalpy h T plus the formation enthalpy h f ,
h = h T + h f , (9.5)
where h T is expressed, in terms of the heat capacity c p at constant pressure and the
absolute temperature T , as follows
h T =
∫ T
0
which, for the special case c p = constant reduces to
c p dT, (9.6)
h T = c p T. (9.7)
At each side of the flame front, the pressure, density and temperature are related
by the state equation, which is assumed to be that of the perfect gases, that is
p 1
ρ 1
= R 1 T 1
and
p 2
ρ 2
= R 2 T 2 , (9.8)
where
R 1 = R M u
and R 2 = R M b
(9.9)
are the specific constants of the unburnt and burnt gases, respectively, and M u and M b
their respective average molecular masses.
Since the tangential component of the velocity is continuous throughout the
flame, there results that the incoming and outgoing velocities and the vector normal to
the combustion front are coplanar.
When the state of the unburnt gases and the incline α 1 of the incident flow
relative to the flame front, see Fig. 9.2, are known, the system of equations (9.1),
(9.2), (9.3), (9.4) and (9.8) determines the state of the burnt gases.
Let ϕ be the flame propagation velocity corresponding to the composition of
the unburnt gas mixture in the thermodynamic state defined by the values p 1 and ρ 1
of pressure and density. Obviously we have
ϕ = v n1 , (9.10)
and for the incline α 1 of the flame front relative to the incident flow
sin α 1 = v n1
v 1
= ϕ v 1
. (9.11)