Untitled - Aerobib - Universidad Politécnica de Madrid

272 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
We have said in the preceding paragraph of this chapter, that Crocco has proven
that time lag τ ch is not constant but a function of the conditions in the chamber. In
his study he schematizes this by making τ ch depending on the pressure, as shown in
Eq. (11.36). In intrinsic instability the variation of τ ch with p plays the same part than
the consumption of fuel in Figure 11.1, since the element acting on the pressure of
the chamber is not the fuel rate injected into it, but the rate of fuel transformed into
products, which may vary from one instant to another, even if the former is constant,
when τ ch varies with pressure. Under such conditions, if a coupling of phases is
reached we will have a self-excited oscillation as in the preceding case.
Let us now see the rules obtained for the scaling of rockets, to maintain physical
similarity, so that both rockets be equaly stable with respect to the low-frequency
oscillations.
In such case, aside from the conditions derived in the preceding paragraph, it
is necessary that some additional ones be satisfied relative to feeding system, since its
influence is important.
In particular, we must keep the equality of the ratio of the pressure drop ∆p
through the injector to the pressure in the chamber, that is
the injector
∆p 1
p 1
= ∆p 2
p 2
. (11.49)
From here, one obtains the following scaling law for the pressure drop through
∆p 1
∆p 2
= p 1
p 2
. (11.50)
However, we must keep in mind that the pressure drop through the injector
determines velocity v i of the fuel when passing through the same, and that the ratio
of such velocities has been determined before by the condition of physical similarity
between velocity fields in Eq. (11.44). Consequently we cannot insure the simultaneous
satisfaction of conditions (11.50) and (11.44), except for certain particular cases
or by adopting special precautions. For instance, if the injectors have similar friction
characteristics, then the following ratio would exist between ∆p and v i
and we have
∆p ∼ v 2 i , (11.51)
( ) 2
∆p 1 vi1
= , (11.52)
∆p 2 v i2
which is not compatible with Eq. (11.50), except for n = 2, as it can readily be
verified.