Untitled - Aerobib - Universidad Politécnica de Madrid

3.10. STATIONARY, ONE-DIMENSIONAL MOTION OF IDEAL GASES WITH HEAT ADDITION 83
corresponds to subsonic velocities. This case has great importance in technical applications
as it determines, for example, the maximum velocity permissible at the inlet
of a constant cross-section combustion chamber as a function of the heat released in
the same. 23 The value of the corresponding Mach number M 0,max is given by
√
2M0,max
∗2
M 0,max =
(γ + 1) − (γ − 1) M0,max
∗2 . (3.106)
The values of M 0,max and M ∗ 0,max as a function of n are represented in Fig. 3.4.
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1
14
subsonic
supersonic
n (supersonic)
12
M * 0,max , M 0,max (subsonic)
1.0
0.8
0.6
0.4
M 0,max
M * 0,max
10
8
6
4
M * 0,max , M 0,max (supersonic)
0.2
2
M 0,max
0.0
0
1 3 5 7 9 11 13 15 17 19 21 23
n (subsonic)
Figure 3.4: Values of M 0,max and M ∗ 0,max as a function of n.
It is easily seen that this choking effect imposes an important limitation to the Mach
number at the inlet of the combustion chamber.
With respect to the law of variation of temperature as a function of Q it is
interesting to remark that the maximum temperature has the value
θ max =
(γ + 1)2
, (3.107)
4γ
and it is reached for a value of n slightly smaller that its maximum value as well as
for a subsonic velocity. This is due to the fact that for velocities slightly subsonic the
acceleration produced by the added heat is so large that the heat received is unable to
balance the cooling produced by the expansion of the accelerating gas. In chapter 5,
where these results are applied, it is seen that this effect produces in detonation.
23 See chapter 10.