Question

In rural areas, water is often extracted from underground by pumps. Consider an underground water source whose free surface is 60 m below ground level. The water is to be raised 5 m above the ground by a pump. The diameter of the pipe is 10 cm at the inlet and 15 cm at the exit. Neglecting any heat interaction with the surroundings and frictional heating effects, determine the power input to the pump required for a steady flow of water at a rate of 15 L/s (=0.015 m3/s)

Answer 1

Answer:

W = 9533.09 Watt

Explanation:

given,

diameter of pipe inlet, d₁ = 10 cm

                                      r₁ = 5 cm

diameter of pipe outlet, d₂ = 15 cm

                                      r₂= 7.5 cm

head upto water level is to rise = 60 + 5

                                          = 65 m

flow rate = 0.015 m³/s

we know

A₁ v₁ = A₂ v₂ = Q  

 π r₁² v₁ = π r₂² v₂  = 0.015

 $v_1= \dfrac{r_2^2}{r_1^2} v_2$

 $v_1= \dfrac{7.5^2}{5^2} v_2$

 $v_1= 2.25 v_2$

 $v_2 = \dfrac{0.015}{\pi r_2^2}$

 $v_2 = \dfrac{0.015}{\pi 0.075^2}$

    v₂ = 0.848 m/s

    v₁ = 1.908 m/s

Applying Bernoulli's equation

 $P_p = \dfrac{1}{2}\rho (v_2^2-v_1^2)+ \rho g h$

 $P_p= \dfrac{1}{2}\times 1000\times (0.848^2-1.908^2)+ 1000\times 9.8\times 65$

 $P_p= 635539.32 Pa$

 P_p is the pump pressure

Power of the pump

W = P_p x Q

W = 635539.32 x 0.015

W = 9533.09 Watt