$2\mathrm{.}39.$ Water at $20C$ is pumped from a lower reservoir to an upper reservoir as shown in Figure $2\mathrm{.}39.$ The riveted steel pipeline has a length of $1km,$ a diameter of $500mm,$ and the difference between the watersurface elevations in the upper and lower reservoirs is $20m.$ An antiquated gate valve controls the flow in the pipeline, and when it is fully open it has a head$-$loss coefficient of $45.$ The currently installed pump delivers $250kW$ of power to the water when it is turned on$,$ and produces a flow of $500\frac{L}{s}$ when the gate valve is fully open. $($a$)$ Estimate the equivalent sand roughness of the pipeline. $($b$)$ If the gate valve is replaced by a new one such that the head$-$loss coefficient is reduced by $70\%,$ what pumping power is needed to generate the same flow rate?

Figure $2\mathrm{.}39$: Pumping from a lower to an upper reservoir