Untitled - Aerobib - Universidad Politécnica de Madrid

250 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
Fabri, Siestrunck and Fouré have developed a similar theory. They also include
the effect of density variations, but do not postulate a linear velocity profile for burnt
gases. Their stating of the problem leads then to an integral equation for the stream
function. For some typical cases they have integrated this equation numerically. Furthermore,
they extended the analysis to other types of chambers. For instance, the
cylindrical chamber with circular cross section and different arrangements of the flame
stabilizer.
As a result of their studies Fabri, Siestrunck and Fouré also predict the occurrence
of choking when the velocity at the inlet section is larger than a critical velocity.
In the following sections the approximate method of Tsien will be described
first and then the method developed by Fabri-Siestrunck-Fouré.
10.2 Tsien method
h
p’ p
B
ρ’
1
D
ρ’
2
C
y
u 0
ρ
1
u e
ρ
1
y
u
1
u
0
A
Figure 10.6: Notation for the Tsien method.
Figure 10.6 contains the necessary elements for Tsien’s approximate calculation.
obtained
By applying the continuity equation to section AB, the following equation is
∫ y1
ρ 1 u 1 (h − y 1 ) + ρu dy = ρ 0 u 0 h. (10.2)
0
By introducing η = y 1 /h and U = u 1 /u 0 it can be written in the following dimensionless
form
∫
ρ η
1
ρ u
( y
)
U(1 − η) + d = 1. (10.2.a)
ρ 0 0 ρ 0 u 0 h