Can you provide a solution for this example in the image attached, the expected answer for a$)$ Ca$($t$)=0\mathrm{.}02[1-$exp$(-$t$/$tau$),$ b$)$ t$=0\mathrm{.}138$s

Question $1$: A mixing tank with no reaction is shown in Figure$-1,$ where $F_0$ and $F_1$ are

inlet and outlet mass flow rates, and $V$ is the tank volume. Initially, the concentration

of component A$,C_A,$ is zero in the tank $(C_A(t=0)=0).$ At $t=0,$ the amount of

component $A,C_{A0}(t),$ is added to the tank, which results in changes in the

concentration in the tank and at outlet, $C_A(t).$

Figure$-1$: Mixing tank of the concentration of component A$,C_A.$

The mass balance for the concentration of component $A$ in the mixing tank is:

$\frac{d{C}_A(t)}{dt}=C_{A0}(t)-C_A(t)$

with $=\frac{V}{F}$ and $F=F_0=F_1.$

Answer the following questions:

a$)$ At $t=0,$ there is a unit$-$step change of the concentration of species $A$ in the

inlet stream, $C_{A0}(t)=C_{A0}u(t),$ with $C_{A0}=0\mathrm{.}02\frac{\text{m}\text{o}\text{l}}{m^2}.$ Use Laplace Transform

to find the time evolution of concentration of component $A,C_A(t).$

$[20$ marks$]$

b$)$ How long does the concentration of component $A,C_A(t),$ take to reach

$0\mathrm{.}015\frac{\text{m}\text{o}\text{l}}{m^2}$ with an inlet flow rate of$,F=5\frac{m^3}{s},$ and a tank of volume of$,V=$

$2m^3?$

You can use the following table for the Laplace transforms:

$[5$ marks$]$