Untitled - Aerobib - Universidad Politécnica de Madrid

10.2. TSIEN METHOD 251
As aforesaid, Tsien assumes that the velocity distribution of the burnt gases
between A and C is linear, that is, u (Fig. 10.6) is given by the expression
u = u 1 + (u e − u 1 )
(1 − y )
, (10.3)
y 1
where u e is the velocity on the axis of the chamber.
Furthermore, Tsien assumes that the ratio λ of the density of the unburnt gases
to that of the burnt gases at each point of the flame front is constant. 1 Then, it can be
easily shown that the density of the burnt gases is constant between A and C and equal
to ρ 1 /λ
In fact, in D we have
ρ = ρ 1
λ . (10.4)
ρ ′ 1
ρ ′ 2
= λ. (10.5)
But due to the fact that the expansions of both unburnt and burnt gases between
pressure p ′ in D and p in C are isentropic, there result
and
ρ ′ 1
ρ 1
=
( ) 1 p
′
γ
p
(10.6)
The combination of (10.5), (10.6) and (10.7) leads to (10.4).
ρ ′ ( ) 1
2 p
′
ρ = γ . (10.7)
p
Since pressure and density of the burnt gases are constant between A and C, the
temperature T of the burnt gases must also be constant between A and C, and equal to
T 2
T = T 2 = λT 1 = T e , (10.8)
where T e is the temperature at A.
The Bernoulli equation, when applied to the unburnt gas between u 0 and u 1 ,
gives
1
2 u2 0 + c p T 0 = 1 2 u2 1 + c p T 1 . (10.9)
If T 1 is eliminated by combining this equation with the equation
( ) 1
ρ 1 T1 γ − 1
=
ρ 0 T 0
(10.10)
1 Relation (9.36) in the preceding chapter proves that this ratio actually varies with the flame front incline.