Untitled - Aerobib - Universidad Politécnica de Madrid

258 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
which is deduced from (10.19) by making v ≃ −ψ. When computing this expression
the following relation must be used
ρ ′ 1
ρ 0
=
(
) 1
1 − U ′ 2 γ − 1
1 − U0
2
(10.37)
as well as the values ψ ′ = ψ(U ′ ) given by the solution of (10.34).
Equation (10.34) can be integrated numerically by substituting the integral for
a summation with a finite number of terms. Fabri, Siestrunck and Fouré have, thus,
calculated the solutions corresponding to several typical cases. Some of their results
can be found in the references included herein.
When calculating the solution it is found that for each value of n there is a
corresponding value U 0,cr of U 0 such that if U 0 > U 0,cr the curve of ψ = ψ(U)
presents an horizontal tangent for a value of ψ smaller than one. This means that in
such a case the burnt fraction must be smaller than unity. Therefore, the existence
of a critical Mach number for the inlet flow is also obtained here, thereupon choking
occurs. Furthermore, it can be proved that the value of U 0,cr is equal to the value
U 0,cr = (√ n − √ n − 1 ) √ γ − 1
γ + 1 , (10.38)
even by the one-dimensional theory that results from the assumption that at each crosssection
of the chamber the distribution of velocities is uniform. For such a case the
value of the final Mach number of the burnt gases is unity. 3
Fig. 10.10, taken from Ref. [8], shows a solution of (10.34) that corresponds
to the case n = 6 for the three following values of U 0 : subcritical value U 0 = 0.05,
critical value U 0 = U 0,cr = 0.087, and supercritical value U 0 = 0.100. In the latter,
the burnt fraction would be only ψ max = 0.8.
The experimental evidence available is not enough to judge the good approximation
of these methods.
10.4 Cylindrical chambers
As Fabri, Siestrunck and Fouré have demonstrated [6] the case of a cylindrical chamber
of circular cross-section reduces to the two-dimensional problem studied in the
preceding paragraph. In fact, in such case equation (10.18), which defines the stream
3 See chapter 3, §9.