$($Problem $1-$ use MATLAB$)$A tubular reactor contains a UV$-$sensitive photocatalyst

and is used to carry out the conversion of a reactant $($R$)$

into a chemical product. When the catalyst is exposed to

ultraviolet light, the reaction proceeds over time $($ t $),$ and

continues as the reactant R moves down the reactor $($in

the x direction$).$ The concentration of species R can be

described with the partial differential equation:

$($delc$\_($R$))/($delt$)=$D$($del$^(2)$c$\_($R$))/($dx$^(2))-$v$($delc$\_($R$))/($delx$)-$kc$\_($R$)$

For a particular reactor $(30$m in length$),$ the value of the

axial dispersion coefficient is D$=0\mathrm{.}75($m$^(2))/(($s$)),$ the flow

velocity is v$=1\mathrm{.}8($m$)/($s$),$ and the rate constant is k$=0\mathrm{.}10$s$^(-1).$The entering concentration of species $R$ is $10\frac{\text{m}\text{o}\text{l}}{m^3},$ and

reaction stops at the reactor exit $($at $x=30m,\frac{del{c}_A}{\text{d}\text{e}\text{l}\text{x}}=0),$

where the photocatalyst bed ends. Initially $($before

ultraviolet light is turned on$),$ the reactor is full of a

depleted solution of species $R(5\frac{\text{m}\text{o}\text{l}}{m^3}).$

Perform a Numerical Method of Lines analysis to solve

for time$-$dependent $c_R$ values inside the reactor. For

consistency, divide the reactor into sections of $3$ meters

$($this will create $11$ total locations $-$ two on the edges,

nine in the middle$).$ Use only second$-$order finite

difference approximations where appropriate $($see the

"Finite Difference Formulas" document on Canvas$).$

$($a$)$ Create a properly formatted plot showing the

positional, time$-$dependent concentrations for the $9$

locations within the reactor over a time span of $25$

seconds.

$($b$)$ At what time $($seconds$)$ does the point $6m$ from the

reactor entrance reach $7\frac{\text{m}\text{o}\text{l}}{m^3}?$ Interpolate if needed.

$($c$)$ Display the entire program code used for your

solution in MATLAB. Include your name as a comment.