$3$

$($a$)$ Explain, with the aid of suitable diagrams, how a rotameter measures flowrate.

$($b$)$ Starting from Bernoulli's equation:

$\frac{P_1}{(g)}+\frac{u_1^2}{2(g)}+z_1=\frac{P_2}{(g)}+\frac{u_2^2}{2(g)}+z_2$

show that for a Pitot tube, the volumetric flowrate can be given by the expression

$Q=C_{d}A\sqrt[\phantom{2}]{2\frac{P}{p}}$

Explain why $C_d$ is introduced.

$($c$)$ Crude oil of density $900k\frac{g}{m^3}$ flows in a $100$ mm diameter pipe. Upstream of a Pitot tube, the pressure is $150000\frac{N}{m^2}$ whereas at the Pitot tube itself the pressure is $200000\frac{N}{m^2}.$ Assuming $C_d$ to be $0\mathrm{.}65,$ what is the flowrate of the oil in $\frac{m^3}{s}?$

$(6)$