Untitled - Aerobib - Universidad Politécnica de Madrid

126 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
5.7 Transition from deflagration to detonation
The preceding study presents deflagrations and detonations as independent phenomena.
Both, deflagrations and detonations can be initiated by means of various procedures.
For example, deflagrations can be initiated by means of an electric spark within
a combustible mixture, and detonations by making a shock wave cross the detonant
mixture. Such procedure was applied, for example, by Fay [14] using a shock tube.
However, a detonation can also be produced starting from a deflagration by compression
of the unburnt gases and acceleration of the combustion wave. This can occur, for
example, when a detonant mixture filling a tube is ignited at the closed end. In such
a case a deflagration wave initiates at the ignition point. This deflagration accelerates
as it propagates along the tube, producing a succession of intermediate states, nonstationary,
which end with the establishment of a stable Chapman-Jouguet detonation,
Hereinafter, we shall restrict ourselves to a qualitative description of the process, following
the lines of Zeldovich’s work [8], who carefully studied this phenomenon. For
this purpose the flame front will be considered as a discontinuity that travels along the
tube, as done in the preceding chapter.
Let ϕ be the propagation velocity of the flame through the unburnt gases, and
S the area of the cross-section of the tube. If the flame front is plane and normal to the
tube axis, the mass of the gases burnt per unit time is ρ 1 ϕS, where ρ 1 is the density
of the unburnt gases before the flame. Before burning, this mass occupies a volume
ϕS. Since the gases expand as they burn, the volume that must occupy the said mass
at the same pressure 15 is nϕS. Here n is the ratio of the temperature of the burnt gases
to temperature of the unburnt gases. Consequently, the increase in volume due to the
combustion is (n − 1)ϕS. Such an increase in volume sets the unburnt gases into
motion in front of the combustion wave to empty the necessary space. The total mass
of unburnt gases is not set into motion simultaneously, but progressively by a pressure
wave that travels through the mass with a velocity close to the velocity of sound. The
situation is shown schematically in Fig. 5.7. The gases are at rest within the region
between O and the flame front A. Between the front A and the pressure wave B, lie the
unburnt gases, compressed and moving towards the right-hand side. Starting from the
pressure wave B and towards tho right-hand side, lie the unburnt gases in the initial
state and at rest. The intensity of the pressure jump across the pressure wave is such
that the burnt gases are at rest. Let v p1 and v p2 be the velocities of the unburnt gases
before and after the pressure wave, measured with respect to this wave. The velocity
at which the pressure wave travels along the tube is v p1 . The velocity of the unburnt
15 The pressure drop across the flame is negligible.