HW3_solution - Minxue(Kevin He Homework#3 Solution Set...

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Minxue (Kevin) He CEE151 W08 Homework #3 Solution Set Page - 1 - of 11 1. Solution: Assume that three pipes shown in Figure P4.5.2 are pipe A, pipe B, and pipe C from the left to right, respectively. Head loss in these pipes ( li h ) can be expressed as: 2 2 2 2 2 2 A A lA A A B B lB B B C C lC C C L V h f D g L V h f D g L V h f D g = = = ……………………………………(1-1) where , , i A B C = ; li h is the head loss in pipe i ; i f is the friction coefficient in pipe i ; i L is the length of pipe i ; i D is the diameter of pipe i ; i V is the flow velocity in pipe i . It can be observed that there is a sudden expansion from pipe A to pipe B. According to Equation (4.3.20) in the textbook, this head loss ( lAB h ) can be expressed as: 2 ( ) 2 A B lAB V V h g - = ……………………………….…..(1-2) It can also be observed that there is a sudden contraction from pipe B to pipe C. From Table 4.3.2 in the textbook, the head loss here ( lBC h ) can be expressed as: 2 2 C lBC C V h K g = ……………………………………..(1-3) Using the center line of the pipes as the base line (elevation datum), then the elevation head at both points 1 and 2 are zero. Applying energy equation between point 1 and point 2 with Equations (1-1) to (1-3) considered: 2 2 2 2 2 2 2 1 2 ( ) 0 0 2 2 2 2 2 2 2 C C C C A A A B B A B A B C C A B C V L V V P V P L V L V V V f f f K g g D g D g D g g g γ γ - + + = + + + + + + + ….…(1-4) where 1 P and 2 P represent the pressure at points 1 and 2, respectively. From continuity equation: 3 2 2 2 3 2 2 3 2 2 4 4 0.05 / 2.831 / (0.15) 4 0.05 / 0.874 / (0.27) 4 0.05 / 1.316 / (0.22) A A A B C Q Q m s V m s A D m m s V m s m m s V m s m π π π π × = = = = × = = × = = ……….…..(1-5) Assume 3 9810 / N m γ = , substituting the value of 1 P and Equation (1-5) above into Equation (1-4) gives:
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Minxue (Kevin) He CEE151 W08 Homework #3 Solution Set Page - 2 - of 11 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 260000 2.831 / 9810 / 2(9.81) / 1.316 / 300 2.831 / 250 0.874 / 9810 / 2(9.81) / 0.15 2(9.81) / 0.27 2(9.81) / 400 1.316 / (2.831 0.874) / 0.22 2(9.81) / 2(9.8 A B C Pa m s N m m s P m s m m s m m s f f N m m s m m s m m s m m s m s f m m s + = + + + - + + 2 2 2 2 2 1.316 / 0.05 1) / 2(9.81) / m s m s m s + ……(1-6) Equation (1-6) above can be simplified as: 2 [ ] 26.624 817.000 36.050 160.490 [ ] 9810 A B C P m f f f m = - - -
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