Scenario $2$:

Water flows through a $2$ cm diameter pipe at $1\mathrm{.}6$m$/$s with kinematic viscosity of $1\mathrm{.}31$ x$10^(-6)$m$^(2)/$s and

density of $1000$kg$/$m$^(3).$ Using the Buckingham Pi Theorem:

f$)$ Calculate the Reynolds number.

g$)$ Find the velocity required to give the same Reynolds number when the same pipe is transporting

air.

h$)$ Obtain the ratio of pressure drops in the same length of pipe for both cases.

i$)$ Reflect on the case studies provided$-$Food Processing Industry, Chemical Manufacturing, and

Water Treatment Facilities. Evaluate how the dimensional analysis using the Buckingham

Pi Theorem applies to these scenarios and discuss the specific challenges encountered,

such as variable fluid properties and complex fluid dynamics which are not always captured

by the theorem. Offer practical recommendations for engineers to tailor dimensional

analysis techniques to the unique requirements of each case study scenario, optimizing design

and operational efficiency.