Consider a fluidized coal reactor operating at $1150K$ where oxygen reacts to form carbon

monoxide:

$2C+O_{2}2CO$

At the conditions operated, the process is controlled by the diffusion of $O_2$ to the surface of

particles $($with counterflow of produced CO$).$ Some approximations that can be applied:

The coal particles are pure carbon with a density ${}_C($~~$1\mathrm{.}28\frac{kg}{m^3})$ and the particles are

spherical with known initial radius $(R_i(t=0)).$ The radius will be reduced with time as

the reaction proceeds.

The reaction at the surface is very rapid and the mole fraction of the oxygen at the surface

$x_{O_2|R}$ is known and practically very small $($~~$0).$

Air, far away from the particles has mole fractions of $21\%O_2$ and $79\%N_2.$

At the conditions of the combustion process $($constant pressure or total mole concentration

$C$ surrounding the particle and high temperature$)$ the diffusivity of $O_2($pseudobinary$)$ is

known $($ ~~$1\mathrm{.}3{10}^{-4}\frac{m^2}{s}).$

No reaction takes place between the diffusing $CO$ and $O_2.$

Use the pseudosteady state approximation and determine the overall oxygen rate of transfer

$W_{O_2}=4r^2{N}_{O_2,r}$ at a distance $r$ from a single spherical particle $($do not perform actual

arithmetic calculations $-$ just use all known variables$).$ Describe when should the pseudosteady

state approximation be reasonable.