Untitled - Aerobib - Universidad Politécnica de Madrid

140 CHAPTER 6. LAMINAR FLAMES
1
0.8
0.6
S b
0.4
A
B
0.2
0
0 0.2 0.4 0.6 0.8 1
T 0
/T f T /T i f
Figure 6.2: Schematic diagram showing the flame propagation velocity vs ignition temperature.
6.5 Propagation velocity of the flame
A simple check of the boundary conditions at the “reaction zone” shows that they
are superabundant. In fact, since we are dealing with a system of three equations of
first-order, the solution of the system is determined for each value of m by boundary
conditions (6.14), except for a translation, which is determined by the additional condition
that for x = 0 be T = T i as imposed by (6.13). In order to satisfy as well the
additional condition x = 0 : ε = 0, it is necessary that m takes a particular value: the
“eigenvalue” of the system which makes compatible boundary conditions (6.13) and
(6.14).
This eigenvalue will determine the propagation velocity of the flame, which by
virtue of (6.1) is given by
u 0 = m . (6.22)
ρ 0
6.6 Example
Even in the case of only two chemical species as considered in the preceding paragraphs,
the non-linear character of the flame equations and in particular the shape of
the reaction velocity add difficulties to the problem to such an extent that it becomes