Untitled - Aerobib - Universidad Politécnica de Madrid

10.3. METHOD OF FABRI-SIESTRUNCK-FOURÉ 257
Moreover, it is immediately obtained that
λ ′ = n − U ′ 2
1 − U ′ 2 , (10.28)
from which results
′′
ρ 1 T n − U ′ 2
=
ρ
′′
T 1 1 − U ′ 2 . (10.29)
Bernoulli’s equation when applied to E, gives
U ′′ 2 +
T ′′
T s,1
= n. (10.30)
Likewise, Bernoulli’s equation when applied to C for the unburnt gases, gives
U 2 + T 1
T s,1
= 1. (10.31)
By combining (10.29), (10.30) and (10.31), the following is obtained for U ′′
√
U ′′ nU
=
2 − U ′2 (n − 1 + U 2 )
1 − U ′ 2
. (10.32)
Substituting (10.29) and (10.32) into (10.26), and making use of relation
ρ 1
ρ 0
=
the following equation is finally obtained
( 1 − U
2
1 − U 2 0
) 1
γ − 1 , (10.33)
( ) 1
U 1 − U
2
γ − 1 − 1 + ψ
U 0 1 − U0
2 ∫ ψ
= √ U (n − U ′2 )
√ n
0
(1 − U ′2 )
(U 2 − U ′ 2 n − 1 + U 2 ) dψ′ .
n
(10.34)
This expression is an integral equation which determines ψ as a function of U. Once
the solution is known, the value of ψ gives the burnt fraction as a function of the
velocity U of the gases at the point where the streamline ψ crosses the flame front.
The flame width η = y/h is given by the expression
η = 1 − ρ 0U 0
(1 − ψ). (10.35)
ρ 1 U
To determine the shape of the flame front, the abscissa ξ = x/h corresponding to ψ
must also be known. This abscissa is given by the following expression
ξ =
∫ ψ
0
ρ 0 U 0
ρ 1 ψ ′ dψ′ , (10.36)