Untitled - Aerobib - Universidad Politécnica de Madrid

254 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
U η=1
10
8
6
4
2
M η=1
M cr
U η=1
1.0
0.8
0.6
0.4
0.2
M cr
, M η=1
0
0.0
0 2 4 6 8 10 12
λ
Figure 10.8: Critical conditions for the complete combustion as a function of the density
ratio λ.
Figure 10.8, in which the values of M cr thus calculated have been taken from
[5], shows that the choking effects impose an important limitation to the velocity at
which gases enter the combustion chamber. Choking effects can be avoided by using
an expanding combustion chamber. It is interesting to point out that when the maximum
Mach number is calculated at critical conditions, that is, at the initiation of the
choking phenomenon, this number is slightly larger than one. This value corresponds
to the unburnt gases at the point where the flame reaches the chamber wall. Therefore,
in the neighbourhood of this point, there exists a small area where the flow is slightly
supersonic. Details of the calculation can be found in Tsien’s work. Furthermore,
Tsien’s analysis shows that Scurlock’s hydrodynamic theory gives satisfactory results
when the entry Mach number is not too close to the critical.
Recently, G. Ernst [9] has applied Tsien’s method to the same problem as well
as to the study of other types of combustion chambers and other arrangements of the
stabilizer. Ernst substitutes Tsien’s linear velocity profile for a parabolic one with
an exponent 1.33, which is the one that best fits the profiles obtained by Scurlock.
Moreover, instead of assuming a constant density for the burnt gases at each cross
section of the chamber, he includes the effect of the slight density variation of the burnt
gases. However, Ernst’s results only differ slightly from those obtained by Tsien.