Untitled - Aerobib - Universidad Politécnica de Madrid

11.3. SCALING OF ROCKETS 269
From here and Eq. (11.37), one obtains
τ ch1
τ ch2
= K 2 . (11.41)
The authors propose that these condition be satisfied by proper control of the
time lag, which may be reached, for instance, if the mean size of the droplets is conveniently
varied, since the evaporation time of a droplet depends on its size as will be
shown in chapter 13.
Now let us see the scaling rules for the thrust and injector.
Physical similarity implies that the number of injectors be the same in both
rockets and identically distributed.
Be e the thrust, g the flow rate of fuel, v 1 its velocity through the injector and
d its diameter. It is promptly verified that the following system of relations is valid
provided the temperature is preserved.
e ∼ g ∼ v i d 2 ∼ ρvl 2 ∼ pvl 2 , (11.42)
From here, the following system of scaling conditions is derived
e 1
= g 1
= v i1 d 2 1
e 2 g 2 v i2 d 2 = p 1v 1 l1
2
2 p 2 v 2 l2
2 . (11.43)
Moreover, since the velocity fields must be similar, it results
v i1
= v 1
. (11.44)
v i2 v 2
When Eqs. (11.28) and (11.33), which in all cases remain valid, are taken into
Eq. (11.43), one obtains for the ratio of thrusts
e 1
e 2
= K. (11.45)
The combination of this relation with Eqs. (11.43) and (11.44) gives for the ratio of
injector diameters
√
d 1
= K v 2
. (11.46)
d 2 v 1
Hence, the ratio of diameters depends on the scaling law for the velocities at
the chamber, which, as we have seen, depends on the rule applied. For example, for
the Penner-Tsien rule it results, by virtue of Eq. (11.28),
d 1
d 2
= K, (11.47)
whereas for Crocco’s rule, from Eq. (11.39) it is obtained
n
d 1
= K 1 + n . (11.48)
d 2