A Continuous Stirred$-$tank reactor $($CSTR$)$ is one of the classic conceptual models for a process reactor, that can be easily derived from applying a material balance, but keeping the unsteady state portion of the equation. In the simplest embodiment, you have an isothermal system with a single stream in and a single stream out. We assume that the mixing within the tank is so effective, fast and efficient, that the concentration of the reactant leaving the tank equals the concentration of the reactant in the tank. If there is no reaction in the tank and there is a change in concentration of compound A $($Ca$,$in$)$ in the tank feed, there will be a lag in time before the concentration out of the tank will equal the concentration fed into the tank.$(5$ points$)$ Using the appropriate Material balance, write an equation that describes a CSTR in unsteady state

conditions.

$(5$ points$)$ using this equation, write an appropriate ODE function file based upon this equation. Assuming $V$

$=1000L,$ and Qin $=25\frac{L}{m}in.$ The tank is at steady state with Ca$=0.$ At t$=0,$ Ca$,$o $=5\frac{g}{l}.$

$(5$ points$)$ Using ODE$45,$ numerically solve the ODE.

$(5$points$)$ Analytically solve the equation in step $1,$ and create an anonymous function using the same

constants.

$(5$ points$)$ In a single plot, compare the trend from the analytical solution with the numerical solution.

Determine at what time it takes for the concentration out to equal $99\%$ of the concentration in$.$

$(10$points$)$ In the plant, the flow meters ceased functioning. As such, all we have available to determine the

flow rate is the concentration profile in the tank as a result of a concentration step change. Assuming the

data attached $($ CME$220$ Project$2$Data.xIsx darr, and V $=1000$L$,$ find the flow rate and the Concentration in

$($Cin$)$ that best fits this data. To do this, use Matlab and the nonlinear regression approach coupled with the

ODE solver. $($Hint: to do this, it will be easier to create an anonymous ODE function within your SSE

function$)$

$(10$points$)$ For comparison, solve the same problem using excel, but fitting the analytical solution and not

the ODE.

$(5$points$)$ write a brief reflection on the comparison of these two approaches and when one may be superior

to the other.