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Chapter 8: Problem 73

At a ski resort, water at \(40^{\circ} \mathrm{F}\) is pumped through a \(3-\mathrm{in}\). diameter, 2000 -ft-long steel pipe from a pond at an elevation of \(4286 \mathrm{ft}\) to a snow-making machine at an elevation of \(4623 \mathrm{ft}\) at a rate of \(0.26 \mathrm{ft}^{3} / \mathrm{s}\). If it is necessary to maintain a pressure of \(180 \mathrm{psi}\) at the snow-making machine, determine the horsepower added to the water by the pump. Neglect minor losses.

Short Answer

Expert verified
The horsepower added to the water by the pump is approximately 440.3 hp.

Step by step solution

01

Calculate Force

The force exerted on the water by the pump is equal to the pressure difference between the two ends of the pipe times the cross-sectional area of the pipe. First, convert the pressure from psi to lb/ft^2 by multiplying by \(144 \, \mathrm{lb/ft^2/psi}\). Then, calculate the area of the pipe in square feet: \(\pi(1.5/12)^2 \, \mathrm{ft^2}\). The force is then \(180 \times 144 \times \pi(1.5/12)^2 \, \mathrm{lb}\).
02

Calculate Distance

The water is pumped a distance of \(4623 - 4286 = 337 \, \mathrm{ft}\), so this is the distance used in the work calculation.
03

Calculate Work

The work done on the water by the pump is the force times the distance, or \((180 \times 144 \times \pi(1.5/12)^2) \times 337 \, \mathrm{ft \cdot lb}\).
04

Calculate Time

The rate at which the water is pumped is \(0.26 \, \mathrm{ft^3/s}\). Since the volume of the pipe is \(2000 \, \mathrm{ft} \times \pi(1.5/12)^2 \, \mathrm{ft^3}\), the time to pump the water is \(2000 \times \pi(1.5/12)^2 / 0.26 \, \mathrm{s}\).
05

Calculate Horsepower

Finally, divide the work by the time and convert from ft·lb/s to horsepower by dividing by 550: \(((180 \times 144 \times \pi(1.5/12)^2) \times 337) / (2000 \times \pi(1.5/12)^2 / 0.26) / 550 \, \mathrm{hp}\).

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