Small hydroelectric plants in the mountains sometimes bring the water from a res

Question

Small hydroelectric plants in the mountains sometimes bring the water from a reservoir down to the power plant through enclosed tubes. In one such plant, the intake tube in the base of the dam, with diameter of 1.00×10^2cm, is 50.0m below the reservoir surface. The water drops 2.00×10^2m through the tube before flowing into the turbine through a nozzle with a 50.0cm diameter.
a.) What is the water velocity into the turbine?
b.) By how much does the inlet pressure difference from the hydrostatic pressure at that depth?

Explanation / Answer

a)Using Bernoulli's equation:

Patm + rho g y = Patm + 1/2 rho v^2

v = sqrt (2 g y)

v = sqrt (2 x 9.8 x 2 x 10^2) = 62.61 m/s

Hence, v = 62.61 m/s

b)from the continuity eqn

v1 A1 = A2 V2

v1 = r2^2/r1^2 sqrt (2 g y)

From Bernoulli's eqn

Patm + rho g y = 1/2 rho v2^2 + rho g y1

y - y1 =d

P = Ps - rho g y1 (r2/r1)^4

or P = rho g y1 (r2/r1)^4

P = 1000 x 9.8 x 200 (50/100)^4 = 122500 Pa

Hence, P = 122500 Pa = 1.225 atm

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