$($a$)$ Use COMSOL ${?}^{}$ to find the carbon monoxide concentration across the pellet $(3$D model$).$ Use

spherical coordinates $($see Appendix A of the lecture notes$)$ with reflective boundary condition at the

centre of the sphere and mixed boundary condition $($i$.$e$.,$ external mass transfer limitation$)$

$-D_{CO}\frac{del{c}_{CO}}{\text{d}\text{e}\text{l}\text{r}}{|}_r=R_p=-h_{CO}(c_{CO,b}-c_{CO}(R_p)),$

at the surface. Add two plots. The first plot will include the radial concentration, $c_{CO}(r),$ and

the exact radial concentration of the stationary reaction$-$diffusion problem with $R_1=k_1*c_{CO}($first

order kinetics, see Addendum at the end of this document$)$ and the same value of $.$ The second

plot will include the radial reaction rate, $R_1(r),$ and the exact radial reaction rate of the stationary

reaction$-$diffusion problem with $R_1=k_1*c_{CO}($first order kinetics$)$ and the same value of $.[$Hint:

Use a cut line to plot the radial concentrations and rates. Each plot will contain two lines.$]$

$(20$ marks$)$

$($b$)$ Use COMSOL ${?}^{}$ to evaluate the effectiveness factor $($volume integral$)$

$=\frac{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{R}_1(r)dV}{\frac{4}{3}{R}_p^3*R_{1,b}},$

and compare it against the exact effectiveness factor of the stationary reaction$-$diffusion problem with

$R_1=k_1*c_{CO}($first order kinetics$)$ and the same value of $.$

$(15$ marks$)$

$($c$)$ Discuss briefly $($up to $100$ words of text without counting tables, equations, and figures$)$ the differ$-$

ences between the results obtained for the stationary reaction$-$diffusion problem with: $1)$ Langmuir$-$

Hinshelwood$-$Hougen$-$Watson kinetics and $2)$ exactly solvable first order kinetics.

$(15$ marks$)$