Let\'s say that for a given power plant the top of the water carrying tube is co

Question

Let's say that for a given power plant the top of the water carrying tube is connected to the bottom of a dam of a large lake. This tube has a radius of 40.0 cm and the tube is located 60.0 m below the surface of the lake. The bottom end of the tube, where the hydroelectric plant is located, is another 250.0 m below the top of the tube (and 310.0 m below the surface of the lake). The outlet nozzle of the tube, which feeds the turbine, has a radius of 20.0 cm.


If water wasn't flowing, you could easily calculate the pressure at the inlet to the pipe (60.0 m below the lake surface). How much greater or less than this pressure is the pressure at the inlet when water is flowing? (if the pressure is less when water is flowing, your answer should be negative)

Explanation / Answer

Given Radius of tube r1 = 40 cm                           = ( 40 cm ) ( 0.01 m / 1 cm )                           = 0.4 m Radius of turbine r2 = 20 cm                               = ( 20 cm ) ( 0.01 m / 1 cm )                               = 0.2 m Area of the tube A1 = r12                                = ( 0.4 m ) 2                                = 0.5026 m2 Area of turbine A2 = r22                                 = ( 0.2 m ) 2                               = 0.1256 m 2 From equation of continuity                     A1v1 = A2v2                          v1 = A2v2 / A 1                                 = 0.125 v2 / 0.5026                              =   0.2499 v 2                                                                                                  ...... (1) According to bernoullis equation                     p2 + ( 1 /2 ) v22 + gh2 = p1 +( 1 /2 ) v12 + gh1                                         p2 - p1        = (1/2) ( v12 - v 22 ) + g ( h 1 - h2 ) From equation (1)                                         p2 - p1        = (1/2) ( 0.623v22 - v 22 ) + g ( h 1 - h2 )                    ...... (2)      Here water was not flowing so v 2 = 0    Therefore                         p2 - p1        = ( 1000 kg /m3 ) ( 9.8 m/s2 ) ( 60 m )                                                           = 5.88 * 105 Pa ________________________________________________________________________ If water is flowing we required the velocity of water at the inlet or outlet. So plug the valkues in equation (2) and get the answer ________________________________________________________________________ If water is flowing we required the velocity of water at the inlet or outlet. So plug the valkues in equation (2) and get the answer

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