Untitled - Aerobib - Universidad Politécnica de Madrid

268 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
Let
K = l 1
l 2
(11.33)
be the linear relation of sizes. Then from (11.28), (11.29), (11.30) and (11.33) one
derives
p 1
= 1 p 2 K , (11.34)
τ ch1
= K.
τ ch2
(11.35)
The first condition imposses that the variation of pressure be inversely proportional
to size. The second one cannot be insured since in a rocket, as we have seen,
τ ch is the resultant of a complicated physico-chemical process. Crocco, by means of
a detailed analysis [11], has reached the conclusion that τ ch may be represented, in
many cases, through a decreasing function of pressure in the form
where n is an exponent differing slightly from unity.
τ ch ∼ p −n , (11.36)
If n = 1, by taking (11.36) into (11.34) and comparing with (11.35), we see
that the later is satisfied, thus obtaining a complete physical similarity.
If n is different from unity the complete similarity is not possible. In such case,
if we disregard the condition of equality of the Mach numbers, which is actually not
too important under steady-state conditions, since velocities at the combustion chamber
are very small and compressibility effects can be neglected, the following conditions
are obtained, instead of (11.29). By eliminating v 1 and v 2 between Eqs. (11.28)
and (11.26), which remain valid, and taking into account Eq. (11.33), it results
p 1
p 2
= τ ch1
τ ch2
K −2 . (11.37)
If expression (11.36) is assumed for the time lag, the following condition is obtained
for the ratio of pressures
p 2
1
= K − 1 + n , (11.38)
p 2
whereas, from here and Eqs. (11.28) and (11.33), it results for the ratio at velocities
1 − n
v 1
= K 1 + n . (11.39)
v 2
Penner and Tsien [6] adopt a different view point. They also neglect the equality
of Mach numbers, but assume that both rockets operate at the same pressure
p 1 = p 2 . (11.40)