Three Gorges on Yangtze River in China is the largest hydroelectric power plant

Question

Three Gorges on Yangtze River in China is the largest hydroelectric power plant in the world. Water enters a tunnel through an inlet whose cross sectional area is 530 m2 , then drops a vertical height of 46.0 m through a huge pipe connected to a generator. The cross sectional area of the pipe is 96.0 m2 , and the volume flow rate of water inside is 800 m3 /s. What is the pressure difference between the inlet and a point in the pipe just in front of the generator? Indicate which point is at a higher pressure.

Explanation / Answer

Using Continuity equation

Q = A*v

Q1 = Q2

Q1 = 800 m^3/sec = A1*v1

v1 = 800/530 = 1.51 m/sec

v2 = 800/96 = 8.33 m/sec

Using bernoulli's law:

P1 + 0.5*rho*v1^2 + rho*g*h1 = P2 + 0.5*rho*v2^2 + rho*g*h2

h1 - h2 = 46 m

rho = 1000 kg/m^3

P2 - P1 = 0.5*rho*(v1^2 - v2^2) + rho*g*(h1 - h2)

P2 - P1 = 0.5*1000*(1.51^2 - 8.33^2) + 1000*9.81*46

P2 - P1 = 4.177*10^5 Pa

Lower point will be at higher pressure.

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