Question $1$

$($a$)$ Briefly explain what dimensional analysis is and why it is commonly used over analytical or numerical methods when solving engineering problems involving fluids.

$[5]$

$($b$)$ Conduct a dimensional analysis to obtain the pi groups required when analysing pressure drop in pipe flow systems and explain how engineers use these dimensionless groups to analyse pipe flow systems.

$[6]$

$($c$)$ Oil with a density of $950k\frac{g}{m^3}$ and viscosity of $1{10}^{-2}$ Pa$.$s flows through a $0\mathrm{.}2$ m diameter, $500$ m long pipe on a downware incline of $1$ m in $50$ m in the flow direction. The roughness ratio of the pipe is $\frac{}{D}=0\mathrm{.}0002$ and the following flanged fittings are incorporated: one fully open swing check valve and five $90$ long radius elbows.

$($i$)$ Determine the pressure differential required to pump fluid through the pipeline if the desired volumetric flow rate is $150\frac{L}{s}$

$[6]$

$($ii$)$ Determine the volumentric flow rate that would naturally result from the head difference across the pipeline alone, i$.$e$.$ without any pump.

$[8]$

$[$Total $25]$