Untitled - Aerobib - Universidad Politécnica de Madrid

5.7. TRANSITION FROM DEFLAGRATION TO DETONATION 127
O
A
B
P P
n ϕ
2
1
P v p − v p = (n−1)ϕ v p1
3 1 2
P 3
P 2
P, v
(n−1)ϕ
P 1
Figure 5.7: Schematic diagram of the flame propagation in a closed tube.
gases after the wave, with respect to the tube, is v p1 −v p2 . The condition that the burnt
gases must be at rest is
v p1 − v p2 = (n − 1)ϕ, (5.53)
as it can easily be verified. The velocity v a at which the flame travels along the tube is
v a = nϕ, (5.54)
where ϕ is the propagation velocity of the flame throughout the gases in their state
after the pressure wave. Such propagation velocity differs from that corresponding
to the state of the gases before the flame. This state of affairs can be maintained
unchanged along the tube. For the detonation to occur it is necessary to activate the
combustion, increasing the burnt mass per second. Such an increase is necessary in
order to increase the intensity of the pressure jump across the pressure wave, up to the
point of self-ignition of the mixture after the wave, needed to maintain the detonation.
The slight compression of the mixture, produced by the compression wave, is not
sufficient to increase, appreciably, the propagation velocity of the flame. Therefore,
the increase of burnt mass per second must be due to other effects. The explanation
can be found in the influence of the tube walls. In fact, due to friction, the distribution
of velocities in the cross-section of a tube is not uniform but maximum at the axis
and zero on the walls. Due to this non-uniformity of the velocity distribution, the
flame front curves increasing its surface. Due to this surface increase, the burnt mass
per second increases and therewith the velocity of the gases and the intensity of the
pressure jump, producing an interaction of both effects such that when the tube is
sufficiently long it leads to the establishment of the detonation wave. This standpoint
is corroborated by the fact that for tubes with rough walls or small diameter the