On excel and matlab Take a look at the stirred tank time dependent concentration problem. There is a short video attached as well. Al you really need here is the final equation, but you most definitely should be able to derive it$!$ Say we have a $10$ liter tank which is stirred to provide uniform concentration. It is initially full of pure water... so the concentration of salt $C_i=0.$ Water flows into and out of the tank at $1$ liter$/$minute$.$ The concentration of salt coming in is $0\mathrm{.}2\frac{g}{?}$ liter.

a$.$ Using the analytical solution to this ODE, what is

i$.$ The steady state concentration in the tank $($you can actually just state this without running any numbers $..$ by logical deduction$)$

ii$.$ Make a table with time steps of $\mathrm{.}2$ minutes and run it out to $1$ hour.

iii. Calculate the analytical solution and plot it

iv$.$ Use Euler to do the same numerically.

v$.$ Use RK$-4$ to do this. Note that for the first couple of steps you need to use the modified Euler method until you have enough point of history to switch to RK$4.($NOTE: Your modified Euler output can be copy and pasted.

vi$.$ Optional try to use Adams Moulton method. Again you will need modified Euler and $($optional$)$ RK$2$ to get started. This is $95\%$ the same as the RK$4.$ Essentially only one line changes.

vii. Plot all of these to compare.

viii. Now change $C_0$ to $0$ and set $C_i$ to $\mathrm{.}5$ and replot. This is a dilution problem. You may note that in theory, the tank never gets to pure $..$ but as a practical matter it does. Report how many tank volumes it takes to remove $99\mathrm{.}95\%$ of the salt form the tank. $($VERY significant practical problem... like how many times do you need to do a full rinse on a glass before its "clean". Suppose it contained HCN $..$ how many times would you wash it before you used it knowing that a spec of HCN the size of a grain of salt is lethal?$)$