Untitled - Aerobib - Universidad Politécnica de Madrid

262 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
In the following we will first deduce the dimensionless parameters of Aerothermochemistry
starting from the fundamental equations governing motion. Thereafter
two practical application will be performed, one to the rule of the scaling of rockets
and another to the problem of flame stabilization.
11.2 Dimensionless parameters of Aerothermochemistry
The physical similarity between two analogous processes consists in the proportionality
between corresponding magnitudes (velocities, temperatures, etc.) through space
and time. In the first place, it implies a geometrical similarity and, moreover, proportionality
of times.
The origin of this similarity and the required conditions for it to occur may be
conceived as follows. Let us consider the equations of the process and write them in
dimensionless form, referring each one of the variable quantities comprised in them
(length, pressure, velocity, temperature, etc.) to a characteristic value of the same.
Thus, we shall obtain a system of equations between dimensionless variables, with
coefficients that will also be dimensionless combinations of such characteristic values.
All the combinations of these values for which the coefficients are invariable will
correspond to processes represented by identical systems of equations. If, furthermore,
we keep invariable the dimensionless expressions of the initial and boundary
conditions, then the solution of the system thus obtained will correspond to a physically
similar set of processes. We will pass from one another of these processes
by introducing changes into the characteristic values which keep invariable the said
coefficients of the equations and their initial and boundary conditions. Hence, these
coefficients are the dimensionless parameters of the physical similarity. Therefore, the
problem reduces to finding, among them, those independent from one another, and to
select the most simple expressions possible for the same.
Let us see now in what way this can be attained, utilizing, for simplicity and
clearness, the equations corresponding to an one-dimensional stationary flow with
only two chemical species, since the generalization to other cases is straightforward
and it only requires multiplying the number of parameters.
The system of equations that we must apply to this study was deduced in §2 of
Chap. 5, but it is reproduced here in order to assist the reader:
a) Equation of mass conservation.
ρv = m. (11.1)