Untitled - Aerobib - Universidad Politécnica de Madrid

232 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
of the flame. In fact, if the radius of curvature and the flame thickness are of the
same order of magnitude the flame speed depends not only on the thermodynamic
conditions of the unburnt gases, but also on the shape of the flame in the neighborhood
of the point under consideration. Such is the case in the flame tip of a Bunsen burner.
Here, the flame speed is increased due to the favorable effect of the curvature on heat
transfer and diffusion of the active particles towards the unburnt gases.
V n1
V t1
p , T ,
ρ
1 1 1
V t2
V
n2
p , T ,
2 2
ρ
2
Figure 9.1: Schematic diagram of a surface element of a flame.
Thus, let it be a flame front of small thickness and small curvature as indicated
in Fig. 9.1, and let us consider a surface element of this front. By applying the continuity,
momentum, and energy equations to the gases at both sides of this element, as
it is done in the study of the invariants across a shock wave 2 , the following conditions
are obtained which relate the state of the unburnt and burnt gases at both sides of the
elements:
1) Continuity equation.
ρ 1 v n1 = ρ 2 v n2 . (9.1)
2) Momentum equation.
a) Normal component.
ρ 1 v 2 n1 + p 1 = ρ 2 v 2 n2 + p 2 . (9.2)
b) Tangential component.
v t1 = v t2 . (9.3)
3) Energy equation.
1
2 v2 1 + h 1 = 1 2 v2 2 + h 2 . (9.4)
Subscripts 1 and 2 refer to conditions before and after the flame, respectively,
v is the velocity of the gases relative to the flame front, assumed stationary, and v n
2 See chapter 6.