Untitled - Aerobib - Universidad Politécnica de Madrid

5.6. DEFLAGRATIONS 125
8
7
v/v 1
=T/T 1
6
5
4
3
−∆ P/(1/2)ρ 1
v 1
2
T 2
/T 1
=8
ρ D c p
/λ=1.5
2
1
Y
0
0 0.2 0.4 0.6 0.8 1
ε
Figure 5.6: Typical values of T , v, Y and ∆p in a deflagration as a function of the fraction
of burnt gases .
In this system the reaction rate w is a known function of Y and T . The pressure
p is constant in virtue of Eq. (5.31). One must look for a solution of this system, in
the interval 0 ≤ ε ≤ 1, satisfying the following boundary conditions
ε = 0 : Y = 0, T = T 0 , (5.51)
ε = 1 : Y = 1, T = T f . (5.52)
Since this is a system of two first order equations, the four conditions (5.51)
and (5.52) cannot be satisfied unless parameters e and m take well defined values.
Parameter e is given by the expression: e = q − c p T f , which results from expressing
the condition w = 0 for ε = 1, that is, at the end of the combustion. As for m it must
taken well defined value, so that the previous differential system to be compatible.
This means that weak deflagrations can only propagate with a well defined velocity
which depends on the state of the mixture, its reaction rate and its coefficients of thermal
conductivity and diffusion. The problem of determining the propagation velocity
of the flame through a combustible mixture appears, thus, as an “eigenvalue” problem.
The solution to this problems will be studied in the following chapter.
The same conclusions are reached when taking into account the influence of all
the terms in the differential equations of the combustion wave, instead of considering
a limiting solution as done herein.