Consider the reduction of a cylindrical FeO pellet by a $CO-C{O}_2$ gas mixture $($see figure below$).$ The length of the pellet $(L)$ is much larger than its initial radius $(r^0)($i$.$e$.,L{r}_{\text{e}\text{e}\text{o}}^0)$ and the reduction is controlled by $1-$dimensional gas$-$film diffusion in the $r$ direction. Let the gas$-$film thickness be $(($:$r_{\text{F}\text{e}\text{o}}^0\}$

a$)$ Perform a shell mass balance for $CO$ in the $-r$ direction by considering a cylindrical shell of length $L,$ radius $r$ and thickness $r($as shown in the figure$)$ and show that in the limit $r0,$ the ordinary differential equation for the variation of molar flux of $CO(N_{CO(-r)})$ with $r$ under steady state can be written as

$\frac{1}{r}*\frac{d}{dr}(r*N_{CO(-r)})=0$

b$)$ Obtain the concentration profile of $CO$ i$.$e$.,$ express $C_{CO}$ as a function $r$ in the range here is measured from the centre of the cylinder $($that is $r=0$ at the centre of the cylinder$)$

c$)$ Obtain an expression for the fraction of FeO transformed $(x)$ as function of time $(t).$

d$)$ Using the result in $($c$),$ now calculate the time required to completely reduce a cylindrical FeO pellet having an initial diameter of pellet is $0\mathrm{.}2m,$ by a $CO-C{O}_2$ gas mixture which has a bulk composition of $90\%CO$ and $10\%C{O}_2,$ at $1000K.$ Assume that the reduction is controlled by gas$-$film diffusion and the spherical gas$-$film around the pellet is $1mm$ thick. Also, assume a total pressure of $1$atm.

Given:

i$.,D_{CO-C{O}_2}$ at $)=(72$gra$\frac{m}{m}$ole

ii$.$ For the reaction: FeO$(s)+CO(g)=Fe(s)+C{O}_2(g)$;$G^0=-18700+22\mathrm{.}46T\frac{J}{m}$ole $(T$ in Kelvin$)$