Question

A pump, submerged at the bottom of a well that is 35 m deep, is used to pump water uphill to a house that is 50 m the top of the well, as shown above. The density of water is 1,000 kg/m³. Neglect the effects of friction, turbulence, and viscosity.

 

(a) Residents of the house use 0.35m³ the day. of water per day. The day’s pumping is completed in 2 hours during 

i. Calculate the minimum work required to pump the water used per day 

ii. Calculate the minimum power rating of the pump. 

(b) In the well, the water flows at 0.50m/s and the pipe has a diameter of 3.0cm. At the house the diameter of the pipe is 1.25cm. 

i. Calculate the flow velocity at the house when a faucet in the house is open. 

ii. Calculate the pressure at the well when the faucet in the house is open.

Answer 1

(a) (i) The total mass of water moved can be found with the density and volume m = ρV = (1000)(0.35) = 350 kg of water. This water is moved a distance 85 m so the work done to move it is W = Fd = (350)(9.8)(85) = 291,500J.

(ii) The force needed to move the water = the weight of the water (mg). 

Using. P = Fd /t = (350)(9.8)(85) / (2hrs *3600 s/hr) = 40.5W

(b) i) Using fluid continuity. A₁v₁ = A₂v₂ π(.03/2)² (0.5) = π(0.0125/2)² v₂ 

v₂ = 2.88 m/s

(ii) Apply Bernoulli’s equation. 

P₁ + ρgy₁ + ½ ρv₁² = P₂ + ρgy₂ + ½ ρv₂² y₁ = 0 

P₂ = atmospheric (open faucet)

P₁ + 0 + ½ (1000) (0.5)² = 1.01 x 10⁵ + (1000)(9.8)(85) + ½ (1000)(2.88)² 

P₁ = 938000 Pa