Untitled - Aerobib - Universidad Politécnica de Madrid

10.1. INTRODUCTION 249
∆ P /∆ P t
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
f
Figure 10.5: Pressure drop as a function of the burnt fraction for incompressible flow and
λ = 6.
G.A. Ball, [3] and [4], has studied the same problem by applying the relaxation
method.
The assumption that density, for both unburnt and burnt gases, is constant,
is permissible only when the variations of the Mach number along the chamber are
small. Due to the acceleration produced by the pressure drop, the local Mach number
close to the walls and near the end of the chamber can be large, even for small entrance
Mach numbers. As a consequence a choking of the chamber can occur preventing the
complete combustion of the gas, as will be shown later. This choking phenomenon
cannot be predicted by Scurlock’s hydrodynamic theory.
H.S. Tsien in the U.S.A. [5], and J. Fabri, R. Siestrunck and C. Fouré in France,
[6], [7] and [8], have studied (simultaneous but independently) the influence of gas
compressibility on the problem.
By assuming at each section a linear velocity distribution for the burnt gases
(see Fig. 10.3), H. S. Tsien has demonstrated that an approximate analytical solution
of Scurlock’s problem can be easily obtained. The results are very close to those
obtained by Scurlock. The linear velocity a distribution is a good approximation to the
profile calculated by Scurlock. Moreover, Tsien has extended the analysis, taking into
account compressibility effects, by applying the same idea of a quasi one-dimensional
theory. He also assumes a linear velocity distribution for the burnt gases, showing that
in this case choking of the chamber can occur.