Untitled - Aerobib - Universidad Politécnica de Madrid

264 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
e) Diffusion equation.
( )
D0 D
′
v 0 l 0 v ′
dY
dx
− Y + ε = 0. (11.11)
The set of dimensionless coefficients P i characteristic of the system is
P 1 = ρ 0v 0
, P 2 = p 0
l 0 w 0 ρ 0 v0
2
P 5 =
q
c p0 T 0
, P 6 =
, P 3 = µ 0
,
ρ 0 v 0 l 0
P 4 = v2 0
,
c p0 T 0
λ 0
, P 7 = µ 0v 0
,
l 0 ρ 0 v 0 c p0 l 0 ρ 0 c p0 T 0
P 8 = D 0
.
v 0 l 0
(11.12)
The physical similarity requires that the values of all these coefficients be equal for
the two processes compared, when using as characteristic values of the variables for
their calculations those at corresponding points and instants.
Such equality guarentees the identity of the left-hand sides of Eqs. (11.7) to
(11.11) in both phenomena. If, furthermore, we guarentee as well the equality of the
right-hand sides, which correspond to the boundary conditions of the problem, we will
have insured the identity of the solution, thus reaching the physical similarity of the
phenomena. Let us proceed to discuss the set of parameter of Eq. (11.12).
We readily observe that P 7 is equal to the product of P 3 by P 4
P 7 = P 3 · P 4 , (11.13)
while the remaining parameters are independent from one another. Consequently, the
number of independent dimensionless parameters is seven. Let us see which are these
seven.
by it
It is clear that P 3 is reciprocal to the Reynolds number and may be substituted
1) Re = ρ 0v 0 l 0
µ 0
. (11.14)
The combination of P 2 and P 4 gives the following parameter P ′ 4 which may
substitute P 4
P ′ 4 =
p 0
ρ 0 c p0 T 0
. (11.15)
Yet, the state equation p 0
ρ 0
= R g T 0 allows P ′ 4 to be written
P ′ 4 = R g
c p0
= c p0 − c v0
c p0
= 1 − 1 γ . (11.16)
Consequently, the constancy of P ′ 4 is equivalent to the constantcy of relation (11.16)
between specific heats, which supplies the second characteristic dimensionless parameter