22PvPv1A2B ??z1???z2??hf?水g2g?水g2g
又?vA?vB?v
?g?hPPP?P ?hf?1?z1?2?z2?12?(z1?z2)?汞??h?水g?水g?水g?水g
lv2v2
又hf?h??h????? d2g2g 2glv22g?汞?hl2?9.8?0.02136000.8???(h??)?(??h)????(?1)?0.0285f
v2d2gv2?水d2.12210000.05
?0.64
15. Pipe AB is of uniform diameter and h=10.5 m The pressure at A is 170 kPa and at B is 275 kPa. Find the direction of the flow, and what is the pipe friction head loss in meters of the fluid if the liquid has a specific gravity of (a) 0.85; (b) 1.45? (10分)
Solution: (a)
2 v2v2pAvA170000? ?10.5??30.89?(m) H A ? ? ? z A ? 2 g 0 .85?98102g2g 2
v2v2pv275000EE H ? ? z ? ? ?0??32.98?(m)BE2g2g ? 2 g 0 .85 ? 9810 ∵ HA?HB ∴ the flow is from B to A 22vv ? H h f ? H?(30.89?)?2.09(m)BA?32.98?2g2g
2 (b) v2v2pAvA170000? z H A ? ? A ?2g?1.45?9810?10.5?2g?22.45?2g(m) 222vvpv275000 H ? E ?E?? z?0??19.33?(m)BE?2g1.45?98102g2g
HA?HB ∵
∴ the flow is from A to B 22vv h ? H ? H?22.45??(19.33?)?3.12(m)fAB2g2g
16. A curved pipe is attached to a tank, and liquid flows out into atmosphere through the curved pipe. Determine the resultant force on the curved pipe. All the significant data are given in the figure. Assume an ideal liquid with γ=55 lb/ft3. Neglect the weight of the liquid. (10分)
Solution: 22pvpv3311 ?z1???z3??2g?2g
2v3 30?144?35?20? 552?32.2 ?v3?77.6(fps)
v?A377.6?9 v2?3??43.6(fps)A162
22 p3v3p2v2?z2???z3? ?2g?2g 22v3?v277.62?43.62p2??(z3?z2)??55?(20?10)?
2g2?32.2
?4075.2(lb/ft2)?28.3(psi)
Fx?F2x?F3x?Fbx??Q(v3x?v2x)
Fy?F2y?F3y?Fby??Q(v3y?v2y)
?D32?D22?Fbx??v3(v3cos20?v2)?p2?165(lb)
44
Fx??Fbx??165(lb) 2?D3 Fby??v3(v3sin20??0)??173(lb)4
Fy??Fby?173(lb)
22
F?Fx?Fy?239
??17. An 180-mm-diameter pipeline (f=0.032) 150 m long discharges a 60-mm-diameter water jet into the atmosphere at a point that is 80 m below the water surface at intake. The entrance to the pipe is reentrant, with ke=0.9, and the nozzle loss coefficient k2=kn=0.055. Find the flow rate and the pressure at B. (15分)
Solution: (a)
22 p0v0p2v2?z0???z2??hL ?2g?2g 22v12v2v2lv12z0?z2?f?ke?k2?
d2g2g2g2g
2v1D26021
?2??v2D1 18029 v2?9v1 2v150113.02221 80?(0.032?0.9?92?0.055?92)?v10.182?9.812?9.81
v1?3.73(m/s)
??0.182
Q?v1.A1?3.73??0.0948(m3/s) 4 (b) v2?9v1?33.54(m/s)22 pBvBp2v2?zB???z2??hL
?2g?2g 222v?vv2B2 pB??(?k2)2g2g
22v2?v12v2
??(?k2)?9810?59.78?5.86?105(Pa)?586(kPa) 2g2g
18. What minimum value of b is necessary to keep the rectangular stone from sliding if it weighs 160 lb/ft3, a=14ft, c=16ft, and the coefficient of friction is 0.45? With this minimum b value, will it also be safe against overturning? Assume that water dose not get underneath the stone block. (10分)
Solution1: 2a14 Fw??whcA?62.4??a?B?62.4??B?6115.2B(lb)22
Ff?f?sbcB?0.45?160?b?16?B?1152bB(lb)
to keep the rectangular stone from sliding, then
Ff?Fw
1152bB?6115.2B
b?5.31(ft)
Since a14Mw?Fw??6115.2B??28537.6B(lb.ft) 33 b5.31Ms??sbcB??160?5.31?16B??36091(lb.ft)?Mw 22 ∴ it is also safe against overturning.
19. A flow is defined by u=2(1+t), v=3(1+t), w=4(1+t). What is the velocity of flow at the point (3,2,4) at t=2? What is the acceleration at that point at t=2? (10分)
Solution: At t=2
u?2(1?t)?2?(1?2)?6
v?3(1?t)?3?(1?2)?9
w?4(1?t)?4?(1?2)?12
?u?u?u?uax??u?v?w?2
?t?x?y?z
?v?v?v?v ay??u?v?w?3?t?x?y?z
?w?w?w?w
az??u?v?w?4 ?t?x?y?z
222222?V?u?v?w?6?9?12?16.16
222 a?ax?ay?az?22?32?42?5.39
20. Water flows over the spillway of constant section. Given that y1 = 4.2m and y2 = 0.7m, determine the resultant horizontal force on the spillway per meter of spillway width (perpendicular to the spillway section). Assume ideal flow. (10分)
Solution: v1y1?v2y222 p1v1p2v2?y1???y2?
?2g?2g
将y1 = 4.2m, y2 = 0.7m, p1 = p2 , γ=9810 N/m2 , 代入得
4.2v1?0.7v2
22vv 124.2??0.7?2?9.812?9.81
解得：v1 =1.4 , v2 =8.4
Fx??Q(v2x?v1x)
F1?F2?Fb??v1y1B(v2?v1)
yy
Fb???1y1B??2y2B??v1y1B(v2?v1) 22 Fb4.220.72??9810??9810??1000?1.4?4.2?(8.4?1.4)??42960(N/m) B22 F ? ? F To the right
b?42.96(kN)
21. Water and oil in an open storage tank. (a) Find the total forces exerted by the fluids on a tank wall, and (b) the location of the center of pressure. (10分) Solution: (a) F?F1?F2'?F2''
F1??oilhc1A1
1.5 ?0.8?9800??(1.5?2)2
?17640(N)
F2'??oilh1A2 ?0.8?9800?1.5?(1?2)
?23520(N)
F2''??waterhc2A2?9800?1(N)2?(1?2)?9800
F?F1?F2'?F2''?17640?23520?9800?50960(N)1.5(b)
yp1?1??1.5(m) 31yp2'??0.5(m)
2
1yp2''??0.333(m)
?3