Ring main for fire hydrants—D. Prove for flow of water in an inclined pipe that: 32.2Δz + 74.3Δp + 4.00fFQ2 L D5 = 0. Here, the symbols have their usual meanings, but the following units have been...

Ring main for fire hydrants—D. Prove for flow of water in an inclined pipe that: 32.2Δz + 74.3Δp + 4.00fFQ2 L D5 = 0. Here, the symbols have their usual meanings, but the following units have been used: Δz (ft), Δp (psi), Q (gpm), L (ft), and D (in.).

Fig. P3.14 shows a pump that takes its suction from a pond and discharges water into a ring main that services the fire hydrants for a chemical plant. All pipe is nominal 6-in. I.D. At the pump exit, the pipe immediately divides into two branches. Points 1 and 2 are essentially at the same elevation, and losses before the pump may be neglected. If the pump exit pressure—which is also the pressure p2 at the dividing point 2—is 85 psig, determine the total flow rate coming from both branches through a fire hydrant at point 3, whose elevation is 100 ft above the pump exit, if the delivery pressure is p3 = 20 psig. The effective distances (including all fittings) between points 2 and 3 are 1,000 ft and 2,000 ft for the shorter and longer paths, respectively. Take only a single estimate of the Fanning friction factor—don’t spend time refining it by iteration. Also, if the pump and its motor have a combined efficiency of 75%, what power (HP) is needed to drive the pump?