Consider the alternative design for the combustion chamber of Problem $12.$ The carbon surface is now porous, with porosity $=0\mathrm{.}4$ and tortuosity $=5.$ The molar fraction of $O_2$ at the carbon surface is $0\mathrm{.}21.$ For simplicity, assume that the combustion of $O_2$ is complete,

$C+O_{2}C{O}_2$

and follows a first order reaction rate,

$r_{O2}=-k{C}_{O2}$

where $C{o}_2$ is the concentration of $O_2.$ Derive an expression to calculate the depth $()$ in the carbon bed where the concentration of $O_2$ is $1\mathrm{.}0\%$ of the surface concentration.

What would this depth be if $\frac{k}{D_{AB}}=0\mathrm{.}1c{m}^{-2}?$ Assume steady$-$state.

$A=0\mathrm{.}21$@$z=0$

$A=0\mathrm{.}0021$@$z=$

Carbon surface $($porous$)$

The total concentration of reactants and products, and $D_{AB}$ may be considered constant.