$($a$)$ Write an expression for Darcy friction factor f for this pipe flow in terms of the given variables. Reduce your expression

as far as possible. Note: At this point, it is not known whether the flow in the pipe is laminar or turbulent.

$($b$)$ If the flow is turbulent, the liquid is water at $20\backslash$deg C flowing at a volume flow rate of $110\mathrm{.}8$ liters per minute, and L$=1\mathrm{.}07$

m$,$D$=2\mathrm{.}34$cm$,$HGL$\_(1)=1\mathrm{.}21$m$,$ and HGL$\_(2)=0\mathrm{.}135$m$,$ calculate Darcy friction factor f $($no units$)$ and pipe roughness $\backslash$epsi

$($in units of cm $).$ Hint: You may need to iterate using your equations and the Churchill equation.