Untitled - Aerobib - Universidad Politécnica de Madrid

11.3. SCALING OF ROCKETS 267
Furthermore, both the scaling under steady operation as well as under low and
high frequency oscillations are important in practice. This problem was recently studied
by Penner-Tsien [6], Rose [7], Crocco [8], Barrère [9] and Penner-Fuhs [10].
We have seen in §2 that physical similarity imposses that the seven characteristic
parameters be equal, when passing from the model to the rocket under steady state
conditions and some additional ones under non-steady conditions.
We shall now study the scaling rules imposed by these equalities. If we assume
that we are working with the same mixture of gases and at the same temperature, the
equality of γ, as well as that of the Prandtl and Schmidt numbers and of Da 2 , is insured.
Consequently we only have to insure the equality of the other three parameters.
Disregarding subscript zero and using subscripts 1 for the model and 2 for the rocket,
we will have:
a) Equality of the Reynolds Number. The equality of temperatures makes µ 1 = µ 2 .
Furthermore, ρ may be substituted by p, to which it is proportional. Therefore,
this condition reduces to the following
p 1 v 1 l 1 = p 2 v 2 l 2 . (11.28)
b) Equality of the Mach Number. The equality of temperature insures the equality
of the velocity of sound a 1 = a 2 . Hence, this condition reduces to the following
v 1 = v 2 , (11.29)
this is to say that the velocity must be the same in both the model and rocket.
c) Equality of Da 1 . This condition gives
l 1
v 1 τ ch1
= l 2
v 2 τ ch2
. (11.30)
Since l is the length of the combustion chamber and v the gas velocity through it,
relation
τ r = l v
(11.31)
is the residence time of the gases in the chamber. τ ch is a physico-chemical time, characteristic
of the process, the so called time lag or delay from the instant at which the
fuel enters the chamber to the instant at which it transforms into products. Therefore,
it accounts for the time needed by the fuel to form droplets, to evaporate, to mix with
the gases and burn. Hence, Da 1 can be written in the form
Da 1 = τ r
τ ch
. (11.32)