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FTFS Chap13 P059

# FTFS Chap13 P059 - Chapter 13 Momentum Analysis of Flow...

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Chapter 13 Momentum Analysis of Flow Systems Review Problems 13-59 Water is flowing into and discharging from a pipe U-section with a secondary discharge section normal to return flow. Net x- and y- forces at the two flanges that connect the pipes are to be determined. Assumptions 1 The flow is steady, one-dimensional, and incompressible. 2 The weight of the U-turn and the water in it is negligible. Properties We take the density of water to be 1000 kg/m 3 . Analysis The flow velocities of the 3 streams are m/s 51 . 0 ] 4 / m) 05 . 0 ( )[ kg/m (1000 kg/s 30 ) 4 / ( 2 3 2 1 1 1 1 1 D m A m V m/s 80 . 2 ] 4 / m) 10 . 0 ( )[ kg/m (1000 kg/s 22 ) 4 / ( 2 3 2 2 2 2 2 2 D m A m V m/s 3 . 11 ] 4 / m) 03 . 0 ( )[ kg/m (1000 kg/s 8 ) 4 / ( 2 3 2 3 3 3 3 3 D m A m V We take the entire U-section as the control volume. We designate the horizontal coordinate by x with the direction of incoming flow as being the positive direction and the vertical coordinate by y . The momentum equation for steady one-dimensional flow is i i e e m m F V V . We let the x- and y- components of the anchoring force of the cone be F Rx and F Ry , and assume them to be in the positive directions. Then the momentum equations along the x and y axes become 0 - 0 ) ( 3 3 3 3 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 V V V V V V m F m F m m A P A P F m m A P A P F Ry Ry Rx Rx Substituting the given values, N 666 N k F Rx 666 . 0 m/s kg 1000 kN 1 m/s) 1 kg/s)(0.5 30 ( m/s kg 1000 kN 1 m/s) 0 kg/s)(2.8 22 ( 4 m) (0.10 ] kN/m ) 100 150 [( 4 m) (0.05 ] kN/m ) 100 200 [( 2 2 2 2 2 2 N 90.4 2 m/s kg 1 N 1 m/s) 3 kg/s)(11. 8 ( Ry F The negative value for F Rx indicates the assumed direction is wrong, and should be reversed. Therefore, a force of 666 N acts on the flanges in the opposite direction. A vertical force of 90.4 N acts on the flange in the vertical direction. Discussion To assess the significance of gravity forces, we estimate the weight of the weight of water in the U-turn and compare it to the vertical force. Assuming the length of the U-turn to be 0.5 m and the average diameter to be 7.5 cm, the mass of the water becomes kg 2 . 2 m) (0.5 4 m) (0.075 ) kg/m 1000 ( 4 2 3 2 L D AL V m whose weight is 2.2 9.81 = 22 N, which is much less than 90.4 N, but still significant. Therefore, disregarding the gravitational effects is a reasonable assumption if great accuracy is not required. 13-60 A fireman was hit by a nozzle held by a tripod with a rated holding force. The accident is to be investigated by calculating the water velocity, the flow rate, and the nozzle velocity.

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FTFS Chap13 P059 - Chapter 13 Momentum Analysis of Flow...

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