Question $4$

A spherical tank, with diameter of $4m,$ is used mix products from two streams with different

concentration of $A.$ The first stream has a volumetric flow rate $(F_1)$ of $2\frac{m^3}{s}$ and concentration of

A $(C_{A1})$ of $1k\frac{g}{m^3}.$ The second stream has a volumetric flow rate $(F_2)$ of $3\frac{m^3}{s}$ and concentration

of $A(C_{A2})$ of identifier $k\frac{g}{m^3}.$ The effluent has a volumetric flow rate $(F_3)$ of $4\frac{m^3}{s}.$ The identifier is

dependent on the last digit of your URN.

As an example, students with last digit of URN of $0$ have $C_{A2}$ of $5k\frac{g}{m^3}.$

a$.$ Generate an ordinary differential equation to represent the liquid level $(d\frac{h}{d}t)$ inside the

tank$,$ evaluate $d\frac{h}{d}t$ at $h=3m,$ and provide generic solution that represents $h(t).[10]$

b$.$ If initially, the initial liquid volume and liquid concentration inside the tank is $5m^3$ and $5$

$k\frac{g}{m^3},$ respectively, describe the dynamics of concentration of A inside the tank $(C_{A3})$ i$.$e$.$

$C_{A3}=f(t)$ and determine the concentration under steady state condition.