Page 332 - Dust Explosions in the Process Industries

Propagation of Flames in Dust Clouds 301

Equation (4.63) differs somewhat from equation (4.22) derived by Cassel et al. (1949)
but rests on a similar basic philosophy.
The third model element was the equation for the rate of pressure rise:

(4.64)

where yis the specific heat ratio, r is the radius of the spherical flame, and p, and pbare
the densities of the unburned gas and the combustiongases. Equation (4.64) is based on
the approximation drldt = (p,/p,)S,.
Estimates for S, for aluminum dust clouds, using the theory by Ogle et al., gave con-
siderablyhigher burning velocities,by a factor of 4, than experimentalvalues from lam-
inar burners.

4.2.5.5
Computer Model by Continillo

The laminar flame propagation through a coal dust/air suspension in a spherical enclo-
sure was studied by Continillo (1988a) by means of a one-dimensional, spherically
symmetricmathematicalmodel. An Eulerian formulationwas adopted for the gas phase
mass continuity, species, and energy balance equations,while a Lagrangianformulation
was employed for the mass, energy, and momentum balance equations for the particles.
For the “gas phase,” the following assumptions were made: The flow is laminar and
sphericallysymmetric.The viscous dissipationrate is negligible and the pressure is uni-
form in space (low Mach number) but varies in time. The gas mixture is thermally per-
fect. Binary diffusion coefficients for each pair of species are taken to be equal, thermal
mass diffusion is neglected. Mass diffusion and heat conduction are governed by Fick’s
and Fourier’s laws, respectively.The diffusion coefficient varies with temperature and
pressure. The Lewis number is unity. Radiative heat transfer is neglected. The combus-
tion chemistry is described by means of a single-step,irreversible reaction of the volatiles
with the oxygen, and Arrhenius-type kinetics with nonunity exponents for fuel and
oxygen concentrations apply. The equations also include coupling terms accountingfor
mass, momentum, and energy exchanges between the gas phase and particle phase.
In the simplified treatment of the “particle phase,” a coal particle was represented by
a spherecontainingash, fixed carbon, and volatiles in specified initial fractions.The par-
ticle was considered to remain spherical and conserve its volume. The temperature was
considered uniform in the particle, including its surface. The transport processes in the
gas film next to the particle were assumed to be quasi-steady, and the thermophysical
properties of the aidfuel vapor mixture were assumed uniform and evaluated at a con-
veniently averaged value of the temperature in the gas film. The fuel vapor production
rate was assumed to depend on the particle temperature and global composition only.
During the particle heat-up, the volatiles were assumed to be released according to a
simple one-step Arrhenius pyrolysis reaction. Due to the highly transient character of
the particle history in this kind of phenomena, surface oxidation reactions were not
considered. This eliminated the need to consider the mass transfer processes in the film.
All the volatiles released by the particle were immediately available in the gas phase.
The model accounted for the effects of the convective transport caused by the gas/particle