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Chapter 8: Problem 102

Water is pumped between two large open reservoirs through \(1.5 \mathrm{km}\) of smooth pipe. The water surfaces in the two reservoirs are at the same elevation. When the pump adds \(20 \mathrm{kW}\) to the water, the flowrate is \(1 \mathrm{m}^{3}\) is. If minor losses are negligible, determine the pipe diameter.

Short Answer

Expert verified
The diameter of the pipe is given by the formula \(D = \sqrt{\frac{4ρQ^{2}g}{πP}}\).

Step by step solution

01

Determine velocity of water

First, calculate the velocity of water. The power (\(P\)) supplied by the pump is equal to the product of the force exerted on water, its velocity and the efficiency of the pump. The velocity (\(v\)) is given by \(v = \frac{P}{{mg}}\), where \(m\) is the mass flow rate of water which is the product of the water density (\(ρ\)) and the volume flow rate (\(Q\)), and \(g\) is the acceleration due to gravity. Substituting for \(m\) and rearranging gives: \(v = \frac{P}{{ρQg}}\).
02

Calculate flow area of water

Next, calculate the flow area (\(A\)) of water. This is given by \(A = \frac{Q}{v}\). Substituting the expression for \(v\) from the previous step gives: \(A = \frac{ρQ^{2}g}{P}\).
03

Calculate pipe diameter

Finally, the pipe diameter (\(D\)) can be obtained by equating the flow area to the area of a circle: \(A = \frac{π D^{2}}{4}\). Rearranging for \(D\) gives: \(D = \sqrt{\frac{4A}{π}}\). Substituting the expression for \(A\) from the previous step gives: \(D = \sqrt{\frac{4ρQ^{2}g}{πP}}\).

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Most popular questions from this chapter

A viscous fluid flows in a 0.10 -m-diameter pipe such that its velocity measured \(0.012 \mathrm{m}\) away from the pipe wall is \(0.8 \mathrm{m} / \mathrm{s}\) If the flow is laminar, determine the centerline velocity and the flowrate.

For fully developed laminar pipe flow in a circular pipe, the velocity profile is given by \(u(r)=2\left(1-r^{2} / R^{2}\right)\) in \(\mathrm{m} / \mathrm{s},\) where \(R\) is the inner radius of the pipe. Assuming that the pipe diameter is \(4 \mathrm{cm},\) find the maximum and average velocities in the pipe as well as the volume flow rate.

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