Question $6$:

$($a$)$ In controliing the pH of a drilling mud, sodium carbonate in form of powder is constantly fed into a drilling mud tank at the rate of $0\mathrm{.}5k\frac{g}{s},$ and the mixture is kept uniform by stirring.

The tank initially contains $5m^3$ of drilling mud with no sodium cabonate in it$.$ If during the addidition of sodium carbonate, nothing is being withdrawn from the tank after $300$ seconds.

Hints: The mass balance over the tank is; $m_{in}-m_{\text{o}\text{u}\text{t}}=\frac{dm}{dt}$

$($b$)$ The rate of cooling a body according to Newton's Law of cooling can be expressed as; $\frac{dT}{dt}=-k(T-T_s),$ At time $t=0,T=T_0$

Where $T=$ Temperature of the cooling object at time $t,$

$T_5=$ Temperature of the surrounding $($assumed constant$)$

$k=$ Thermal constant related to the object

If a metal ball heated to $90C$ is drop into water that is held at a constant temperature of $T_5=20C,$ calculate how long st takes the ball $10$ cool $1030C$ if $k=0\mathrm{.}1mi{n}^{-1}.$

Hints; The energy balance equation is given as$,\frac{dT}{dt}=-k(T-Ts)$