Page 330 - Dust Explosions in the Process Industries

Propagation of Flames in Dust Clouds 299

density pp,the particle diameter Dp,and the dust concentration C, was defined as

L= [;j- DP (4.60)

which differs from equation (4.38) by the factor (~/6)l’~.
Equation (4.41) was used in a simplified form by neglecting all thermal radiation
except that from the flame front to the next particle shell. The resulting equation for the
maximum rate of pressure rise in a spherical vessel with central point ignition was

(4.61)

which conforms with the “cube root law” as long as all constants at the right-hand side
are independent of the vessel radius R. It is implicitly assumed, during the derivation of
this equation, that the thickness of the flame zone is negligible compared to the vessel
radius R. The constant a in equation (4.61) has the dimensions of mass per unit volume
and equals the effective dust concentration that can burn completely consuming the
oxygen available. For dust concentrations C, up to stoichiometric the parameter a = Cd,
whereas for higher concentrations, it maintains the stoichiometric value.
The At is the time required for the flame to propagate from the (n - 1)th to the nth
particle shell. For starch dusts of Dp < 50 pm, At was found to be independent of n for
n > 30. Therefore, the burning velocity equals S, =L/At,, as defined by equation (4.43).
Nomiura and Tanaka derived At, as a complex function of particle and combustion
properties.
Nomura and Tanaka (1980) also extended their theoretical treatment to nonspherical
vessel shapes. This was done by maintaining spherical flame propagation for any part
of the flame that had not reached the vessel wall. As soon as a part of the flame reached
the wall, flame propagation stopped for that part. Heat loss to the vessel wall was not
considered. Under these conditions the theoretical analysis showed that the “cube root”
relationship was valid even for elongated, cylindrical vessels, as long as they were geo-
metrically similar.
Figure 4.23 illustrates the theoretical development of pressure with time in an elon-
gated cylinder. At time tl, the spherical flame reached the cylinder wall, and at time to,
the entire dust cloud has burned.
Nomura and Tanaka tried to correlate their theoretical results for laminar flame prop-
agation with experimental data from dust explosions in closed vessels. However,
inevitable and unknown turbulence in the experimental dust clouds could not be accounted
for, and the value of the correlation therefore seems limited.

4.2.5.4
Simplified Theory by Ogle, Beddow, and Vetter

Ogle, Beddow, and Vetter (1983) proposed a simplified three-element theory for the
development of a dust explosion in a closed vessel. The first element was a model for