Page 443 - Dust Explosions in the Process Industries

4 7 0 Dust Explosions in the Process Industries

The spark discharge time is very short, less than 0.1 ps for a 1J spark and less than
0.01 ,us for a 1 mJ spark. This means that the spark discharge is completed before any
significant expansion of the hot gas has taken place.
The maximum temperature (i.e., the temperature immediately after completion of
the very short heating period and prior to the onset of the subsequent expansion of
the hot gas) is estimated at 60,000 K, based on the peak temperature of 50,000 K
in a 2 ps, 1J spark found experimentally by Krauss and Krempl(l963).
The initial spark is spherical;the rapid expansion of the hot gas sphere to ambientpres-
sure, following the discharge, is adiabatic; and a rectangular radial temperature dis-
tribution in the hot gas is maintained throughout this process. The equation of state
for ideal gases and the expressions C, = 5l2R and Cp/Cv= 1.5 apply.
After completion of the rapid expansion, the hot gas is cooled to ambient tempera-
ture by heat conduction into the surrounding gas. This process, involving diffusion
of both heat and mass, is described by the equation
2 au
-au = .3{$+;%---} (5.23)
ae

where u is a dimensionless function of the spark temperature, x is a dimensionless
expression of the distance from the spark center, and 8is a dimensionless expression
of the time.
The upward movement of the hot gas due to buoyancy is neglected.
The radial distribution of gas pressure is assumed rectangular throughout the super-
sonic expansion of the hot gas to ambient pressure.
The particles are first accelerated by the extremely rapid passage of the shock front
through the particle and by the rapid outward flow of expanding gas following the
shock front. At a certain point, the particle velocity, because of the inertia, overtakes
the gas velocity; and from this stage on, the particle velocity gradually decreases.
Depending on the Reynolds number, Re, either the laminar drag

24 P
K,-*-V 2 A
Re 2 (5.24)
or the turbulent drag

P2
K, =-V Ap (5.25)
2
acts on the particles during the acceleration as well as during the subsequent retarda-
tion process.
The theoretical treatment by Enstad confirmed that a dust-free zone, separating the
dust cloud from the hot gas core, may in fact be established. As an example, the theo-
retical results for a “short” 1.5J spark discharge in a cloud of lycopodium in air are sum-
marized in Figure 5.16. The distance of a dust particle from the spark center is given
as a function of the time after spark discharge and the initial position of the particle.
Beyond a given instant, depending on the initial particle position, the particle to spark
center distance decreases with time. This is because, beyond this point, the settling