Please show the full steps!!! Thanks !!!! A hydropower scheme is shown in Figure

Question

Please show the full steps!!! Thanks !!!!

A hydropower scheme is shown in Figure 4. A 0.5-meter pipe of length 200m discharges into a tailrace after passing through a turbine. Determine the electrical power that is required to transfer water from the lower reservoir to the upper reservoir at a flow rate is 4m3/s. Assume that the pump operates at 70% efficiency and that the head loss in the pipe between A and B equals to 5V2 / 2g Also assume alpha = 1.0 at all locations. V = average velocity in the pipe.

Explanation / Answer

Applying Bernoulis Equation b/w A and B (in terms of head)

Pa/(rho*g) + Va^2/(2*g) + Za + hloss = Pb/(rho*g) + Vb^2/(2*g) + Zb + Hpump

Here

Pa = Pb = Patm

Va = Vb = 0

hloss = 5*V^2/(2*g) (given in the problem)

Also

Q = A*V = 4 m^3/s

Pi/4*0.5^2*V = 4

V = 20.38 m/s

hloss = 105.87 m

Substitute in above expression we get

65 + 105.87 = 10 + Hpump

Hpump = 160.87

Power, P = rho*g*H*Q = 998*9.81*160.87*4 W

P = 6.23 MW (This is for 100% efficient pump)

For 70% efficieny

Pactual = P/0.7 = 9 MW

Power of the pump is 9 Mega-Watts

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