radius $R_1.$ The biomass is surround by air. Assume a spherical shell of gas film around the

biomass ball. The oxygen in the air has to diffuse into the spherical gas film before reaching

the biomass surface where the following reaction takes place.

$C_5{H}_{12}(s)+6O_2(g)4CO(g)+C{O}_2(g)+6H_{2}O(g)$

The reaction products then diffuse from the biomass surface to the bulk gas phase. Assume

that the bulk gas consists of air supplied at a high circulation rate so that the mole fraction

of oxygen is $x_{O_2}$ and that the reaction proceeds to completion $($instantaneous reaction$)$ at the

biomass surface so that $x_{O_2}$~~$0$ at $r=R_1.$

$($a$)$ Write an expression for the flux of oxygen by reducing the Fick's Law

$N_{O_2,r}=-c{D}_{O_2,\text{m}\text{i}\text{x}}\frac{d{x}_{O_2}}{dr}+x_{O_2}(N_{O_2,r}+N_{C{O}_2,r}+N_{CO,r}+N_{H_{2}O,r})$

$($b$)$ Write a shell balance in the gas film layer using the flux of oxygen derived in part $($a$).$

Derive an expression for the concentration profile of $O_2$ in the gas film.

Note:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{dx}{1+-Kx}=+-\frac{1}{K}ln(1+-Kx)$

$($c$)$ Compute the molar rate of oxygen through the film at $r=R_1.$ Can you write a mass

circuit?

$($d$)$ Derive an expression for time $t$ it takes for the biomass to disappear completely.