Page 319 - Dust Explosions in the Process Industries

288 Dust Explosions in the Process Industries

The first two boundary conditions allow determination of the temperature profile, and
the third one specifies the burning velocity. By making certain assumptions, equations
(4.32)-(4.34) were solved to yield the temperature profile and the burning velocity:

(4.35)

(4.36)

This is the RCC model. To evaluate the relative importance of conduction, the thermal
conductivity can be set equal to 0, yielding the radiation and convection model (RC).
This results in the same temperature profile but a different expression for the burning
velocity:

(4.37)
It was found that the difference between burning velocities predicted by the RCC and
RC models was negligible. Hence, conduction was negligible compared to convection
and radiation. The predicted burning velocity was 0.27 m/s for a flame temperature of
1750 K and increased almost linearly with flame temperature to 0.37 m/s for the adia-
batic flame temperature 1950K. Predicted burning velocities in the range 0.27-0.37 ds
for flame temperatures in the range 1750-1950 K are in reasonable agreement with
experimental values.
Weber (1989) proposed a modificationof the approach by Ogle et al. He used the math-
ematical condition for an inflectionpoint (second derivative equal to 0) to obtain the burn-
ing velocity S, as an eigenvaluefrom the two-point boundary value problem for a linear,
second-order differential equation with arbitrary forcing. The flame was divided into a
preheating zone from To to Ti,where Tiwas the inflection point of the temperature-
versus-distanceprofile, and a reaction zone from Tjto TfThe applicationto dust flames,
with thermal radiation, was considered.

4.2.4.4
Theory by Nomura and Tanaka for Monosized Particles

In the theory for plane flames developed by Nomura and Tanaka (1978) for monosized
particles, it is assumed that the particles are initially arranged in a cubical pattern with
center-to-centerdistance L in all three main directions. The relationship between L and
the dust concentration C, is given by
l3 (4.38)

where Dpis the particle diameter and ppis the particle density. The flame propagation is
assumed to occur as a one-dimensional wave composed of identical parallel elements
of cross-sectional area L2,starting from a plane wall, as indicated in Figure 4.21.
Each particle is assumed to be located at the center of a cubical air element of volume
L3,indicated by the dotted lines in Figure 4.21. When particle number 1burns, the sur-
rounding gas element is heated adiabatically at constant pressure and expands in the x