Carbon monoxide oxidation,

$CO+\frac{1}{2}{O}_{2}C{O}_2,$

occurs in a catalytic reactor at temperature $T_p=398\mathrm{.}15($unit: $K)$ and the pressure is $P=1($unit: atm$).$ The

reaction occurs inside a spherical pellet of radius $R_p=0\mathrm{.}175($unit: $cm).$ The carbon monoxide diffusivity is

$D_{CO}=0\mathrm{.}0487($unit: $c\frac{m^2}{s}).$ The $CO$ and $O_2$ fractions in the bulk fluid are $f_{CO,b}=0\mathrm{.}01$ and $f_{O_{2,b}}=0\mathrm{.}005,$

respectively. Treating the oxygen concentration as a constant, the $($positively defined$)$ reaction rate can be

approximated as

$R_1=\frac{k_1*c_{CO}}{(1+K_{CO}*c_{CO}{)}^2},$

$($Langmuir$-$Hinshelwood$-$Hougen$-$Watson linetics$)$ where $k_1=6\mathrm{.}962*{10}^{11}$exp$(-3,\frac{210}{T_p})*c_{O_2,b}($unit: $\{$:$s^{-1})$

and $K_{CO}=8\mathrm{.}1*{10}^5$exp$(\frac{409}{T_p})($unit: $c\frac{m^3}{m}ol).$ The cxternal mass transfer cofficient is $h_{CO}=3\mathrm{.}90($unit:

$\{$:$cm*s^{-1}).$

Question $1.$

Files to upload on CANVAS: One $m$ file called HW$\_21m$ with the solutions of all points below and one

function $m$ file called laplacian$\_$mixed.m which returns the discretised Laplacian. You can reuse the function

$m$ file given in the workshop $3.$

$($a$)$

Use MATLAB ${?}^{}$ and the Finite Difference Method to find the carbon monoxide radial concentration,

$c_{CO}(r),$ across the pellet. Use spherical coordinates $($soe Appendix A of the lecture notes$)$ with

reflective boundary condition at the centre of the sphere and mixed boundary condition $($i$.$e$.,$ external

mass transfex 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. $[$Hint: Use dimensionless units and start from the workshop $3$ example. Sct the

dimensionless reaction rate equal to ${}^2*\frac{x_{CO}}{(1+*\frac{x_{CO}}{(+{10}^{-9})}{)}^2},$ where $=\sqrt[\phantom{2}]{R_p^2\frac{k_1}{D_{CO}}}$ is the

Thicle modulus, $\frac{x_{cO}}{}=\frac{\text{c}\text{c}\text{O}}{c_{CO},(b)}$ is the dimensionless concentration, $=\frac{r}{R_p}$ is the dimensionless radial

coordinate, and ${}^{**}=K_{CO}*c_{CO,b}.$ The small factor ${10}^{-9}$ at denominator must be added to aroid

indeterminate expressions that MATLAB ${?}^{}$ cannot resolve.$]$

$(20$ marks$)$

$($b$)$

Use MATLAB ${?}^{}$ to plot and compare the pellet stationary radial concentration, $c_{CO}(r),$ obtained

in point $(1$a$)$ against the exact radial concentration of the stationary reaction$-$diffusion problem with

$R_1=k_1*c_{CO}($first order linetics, see Addendum at the end of this document$)$ and the same value

of $.[$Hint: Plot the two concentrations on the same figure.$]$

$(15$ marks$)$

$($c$)$ Use MATLAB ${?}^{}$ to compute the effectiveness factor defined as $($dimensionless integral$)$

$=3(1+{)}^2{\displaystyle \underset{0}{\overset{1}{}}}[\frac{x_{CO}}{(1+*\frac{x_{CO}}{(+{10}^{-9})}{)}^2}]d,$

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$)$