Untitled - Aerobib - Universidad Politécnica de Madrid

114 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
The second case is represented by the system of equations (5.1a), (5.2a), (5.31)
and (5.32a), that is, by
Case B: α ≪ 1, α/M 2 ∼ 1 .
ρD
m
dY
dx
c p T − qε − λ m
ρv = m,
m dε
dx = w,
(5.1.a)
(5.2.a)
− Y + ε = 0, (5.15)
p = i, (5.31)
dT
dx = e,
(5.32.a)
in which the influence of the kinetic energy and viscosity has disappeared. This system
represents a combustion at constant pressure, as limiting case of the weak deflagrations
that are physically observed.
In the following paragraphs both cases will be studied separately.
5.5 Detonations
Let us first discuss the solutions represented by system A).
The three equations (5.1.a), (5.28.a) and (5.29.a), together with the state equation
(5.6), allows one to express the variation laws of p, T and v as function of the
degree of advancement ε of the combustion, or of the mass fraction Y = ε of the
burnt gases in the mixture. This is the problem of heat addition in the ideal gas in
one-dimensional and stationary motion. 6 In particular, the following equation for v is
obtained
v 2 γ i − 1)
− 2 v + 2(γ (e + qε) = 0, (5.33)
(γ + 1) m γ + 1
which shows that to each value of ε satisfying the condition
(
0 ≤ ε ≤ 1 γ 2 ( ) 2 i
q 2(γ 2 − e)
, (5.34)
− 1) m
corresponds two different real values of v, given by the expression
( √
)
γ i
v =
1 ± 1 − 2 (γ2 − 1)
( m
) 2
(e + qε)
(γ + 1) m
γ 2
. (5.35)
i
6 See chapter 3, paragraph 9.