Question $3$:

You are asked to size the pump for a large scale pumped hydro$-$electric storage system. The system is to pump water from a lake to a man$-$made reservoir on a mountain, $350$ m above the lake. It should be capable of delivering volumetric flow rates up to $100\frac{m^3}{s}$ and you can assume there is no frictional head loss in the pipeline $($i$.$e$.$ no major or minor losses$).$ A dimensional analysis yields the following PI groups where $P$ is pressure; $Q^{}$ is volumetric flow rate; $$ is fluid density; $$ is rotational speed; and $D$ is diamter:

$\frac{O}{\frac{P}{{}^2{D}^2},\frac{(Q^{})}{D^3}}$

A novel impellor design $($i$.$e$.$ the rotating part of the pump$)$ is being proposed but the relationship between PI groups is unknown. It was decided to conduct some laboratory experiments on a small scale model of the impellor which is $0\mathrm{.}3$ m in diameter. These experiments were conducted at various rotational speeds and with three different working fluids while pressure rise across the pump and volumetric flow rate were measured. The results are indicated in the table below.

$\backslash$table$[[\backslash$table$[[$Working$],[$Fluid$]],\backslash$table$[[$Fluid$],[$Density$],[(k\frac{g}{m^3})$