Question $3$

a$.$ A cylindrical holding tank is used to store a liquid product from two streams. The first and

second incoming stream has a volumetric flow rate of $([3])\frac{m^3}{s}$ and $(3+t)\frac{m^3}{s}.$

The effluent flow rate is set at $(0\mathrm{.}1h+2)\frac{m^3}{s}.$ The tank has the surface area of $2m^2.$

where $t$ is the time $($s$),h$ is the liquid level $($m$),$

i$.$ Set up an ordinary differential equation to represent the dynamics of liquid level

$($h$).$ State your assumptions for establishing the equation.

ii$.$ Noting that, the initial liquid level is $10m,$ determine the liquid level when $t=10s.$

Comment whether the equation that represents liquid level is physically realistic. If

not, provide your reasons.

b$.$ Based on your answers in part $($a$),$

i$.$ Propose a suitable equation to describe the effluent volumetric flowrate noting that

the piping has cross$-$sectional surface area of $a{m}^2.$ You may refer to your

knowledge on physics for this.

ii$.$ Based on your proposed equation, generate a revised ordinary differential

equation, and solve it to represent the dynamics of liquid level. If it is not

technically possible to solve it$,$ provide the reasons.