Water flows through a $30-cm-$diameter pipe that is $60m$ long and made from commercial steel. At the upstream point, the elevation is $5m$ and the pressure is $240$kPa. The pipe slopes downward such that the downstream point has an elevation of $2m$ and the pressure is $135$kPa.

a$)$ Using an iterative scheme, what is the flowrate through the pipe? Show at least an initial guess and one iteration by hand calculations. See the in$-$class Type III problem for an example of what these hand calculations should look like.

b$)$ Estimate the flowrate using the Darcy$-$Weisbach $+$ Colebrook equation. How does this compare to your answer in part a$)?$