A fluidized coal reactor has been proposed for a new power plant. If operated at $1000K,$ the process

will be limited by the diffusion of oxygen countercurrent to the carbon dioxide, $C{O}_2,$ formed at the

particle surface. Assume that the coal is pure solid carbon with a density of $1\mathrm{.}28{10}^{3}k\frac{g}{m^3}$ that the

particle is spherical with an initial diameter of $2\mathrm{.}5{10}^{-4}m(250m).$ Air and $\{$:$79\%N_2)$ exists

several diameters away from the sphere. Under the conditions of the combustion process, the diffusivity

of oxygen in the gas mixture at $1000K$ is $2\mathrm{.}5{10}^{-4}c\frac{m^2}{s}.$ If a steady$-$state process is assumed, calculate

the time necessary to reduce the diameter of the carbon particle to $50m.$ The surrounding air serves as

an infinite source for $O_2$ transfer, whereas the oxidation of the carbon at the surface of the particle is

the sink for $O_2$ mass transfer.

The reaction at the surface is: $C(s)+O_2(g)C{O}_2(g).$ At the surface of the coal particle, the reaction is

so rapid.