$4)$ Branching flows $(25$ points$).$ In order to cool a solution, it is pumped through a direct heat exchanger $($a coolant tank with a manifold of $\backslash ($ n $\backslash )$ tubes$),$ with a total volumetric flow rate of $\backslash ($ Q $\backslash ).$ However, the temperature of the solution is dropping too much and you need to reduce the overall contact of the solution tubes $($each with length $\backslash ($ L $\backslash )$ and diameter $\backslash ($ D $\backslash ))$ with the coolant$-$you can either reduce $\backslash ($ D $\backslash ),$ reduce $\backslash ($ L $\backslash ),$ or reduce $\backslash ($ n $\backslash ).$

Assuming the flow is laminar, with negligible minor losses, how does each of these three approaches affect the pump head $($that is$,$ the pressure difference between the pump and the exit of the manifold$)?$ Show all work.

a$)$ Reduce each tube diameter by $\backslash (20\backslash \%\backslash )$

b$)$ Reduce the number of tubes by $\backslash (20\backslash \%\backslash )$

c$)$ Reduce the tube length by $\backslash (20\backslash \%\backslash )$