At a ski resort, water at 40 F is pumped through a 3-in diameter, 2000ft long, steel pipe from a pond at an elevation of 4286ft to a snow-making machine at an elevation of 4623ft at a rate of 0.26ft^3/s. It is necessary to maintain a pressure of 180 psi at the snow making machine. Determine the horsepower added to the water by the pump. Neglect minor loses.

Total power supplied to the water by the pump shaft in the absence of losses = 22.2 hp

Explanation:

Neglecting the losses, it means the pump operates at a 100% efficiency and total power supplied by shaft = power gained by the fluid

Power gained by the fluid is converted to the power to overcome pressure difference, Power in kinetic energy form and Power in potential energy form

Power to overcome pressure difference = Q(ΔP)

Q = volumetric flow rate = 0.26 ft³/s = 0.00736 m³/s

ΔP = 180 psi = 1.241 × 10⁶ Pa

Power to overcome pressure difference = 0.00736 × 1.241 × 10⁶ = 9134.2 W

Power in kinetic energy form = ṁ(v₂² - v₁²)/2

ṁ = mass flow rate = Density × volumetric flow rate = 1000 × 0.00736 = 7.36 kg/s (Density of water = 1000 kg/m³

since the water was initially at rest, v₁ = 0m/s

v₂ = volumetric flowrate/cross sectional area

Cross sectional Area of pipe = 2πrl

r = 3/2 in = 0.0381 m, l = 2000 ft = 609.6m

v₂ = 0.00736/(2π× 0.0381 × 609.6) = 0.000504 m/s

Power in kinetic energy form = ṁ(v₂² - v₁²)/2 = 7.36(0.0000504²)/2 = (9.4 × 10⁻⁸) W

Power in potential energy = ṁg(z₂ - z₁)

z₂ - z₁ = change in elevation = (4623 - 4286) = 337 ft = 102.72 m

Power in potential energy = ṁg(z₂ - z₁) = 7.36 × 9.8 × 102.72 = 7408.99 = 7409 W

Total power gained by fluid = 9134.2 + (9.4 × 10⁻⁸) + 7409 = 16543.2 W

Total power supplied by shaft = Total power gained by fluid = 16543.2 W

In horsepower,

746 W = 1 Horsepower

16543.2 W = 22.185 hp = 22.2 hp

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