50 Example 1 A jet of water from a 50 mm diameter nozzle impinges normally with a velocity of 30 meters per second on a stationary flat plate. Find the force exerted by the jet on the plate. Solution: Diameter of jet, d = 50 mm = 0.05 m Velocity of the jet, v = 30 meters per second Area of the jet, a = 4 (0.05) 2 = 0.00196 metre2 Force exerted by the jet on the plate, P = 2 = 1000 x 0.00196 x 302 = 1764 N Example 2 A jet of water 250 mm in diameter impinges normally on a flat plate moving at 2 meters per second in the same direction as that of the jet. If the discharge is 0.490 m3 /sec, find the force exerted by the jet on the plate. Find also the work done on the plate per second. Solution: d = 250 mm, v = 2 m/sec, Q = 0.490 m3 /sec Area of the jet, a = 4 (0.25) 2 = 0.049 metre2 Velocity of the jet = v = = 0.490 0.049 = 10 / Force exerted by the jet on the plate, P = ( − ) 2 = 1000 x 0.049 x (10 – 2)2 = 3136 N
51 Work done per second = 3136 x 2 = 6272 Nm/s Example 3 A jet 150 mm in diameter moving at 25 meters per second strikes a plate which remains at rest. Find the force exerted on the plate normal to it when (i) the plate is held normal to the jet, and (ii) when the plate makes an angle of 30o to the jet. Solution: Area of jet, a = 4 (0.15) 2 = 0.0177 metre2 Velocity of jet = V = 25 / Case (i). When the plate is normal to the jet Force exerted by the jet, P = 2 = 1000 x 0.0177 x 252 = 11062.5 N Case (ii). When the plate makes an angle of 30o to the jet Force exerted normal to the plate, P = 2 = 11062.5 sin 30 = 5531.25 N Example 4 A 75 mm diameter jet having a velocity of 30 meters per second strikes a flat plate. Find the normal pressure on the plate (i) when the plate is stationary, and (ii) when the plate is moving with a velocity of 15 m/sec and away from the jet. Also determine the power and efficiency of the jet when the plate is moving. Solution: Area of jet, a = 4 (0.075) 2 = 0.004418 metre2
52 Velocity of jet = V = 30 / Case (i). When the plate is stationary Force acting normal to the plate, P = 2 = 1000 x 0.004418 x 302 = 3976.2 N Case (ii).When the plate moves at 15 m/sec away from the jet. Force acting normal to the plate, P = ( − ) 2 = 1000 x 0.004418 (30 - 15)2 = 994.05 N Work done/sec, Pv =994.05 x 15 = 14910.75 Nm/s Power developed in this arrangement, 14910.75 Watt Kinetic energy of the jet = 3 2 = 1000×0.004418×303 2 = 59643 Hydraulic efficiency = 14910.75 59643 × 100 = 25% 4.3 Exercises 1. A jet of water from a 70 mm diameter nozzle impinges normally with a velocity of 50 meters per second on a stationary flat plate. Find the force exerted by the jet on the plate. [Answer: P = 9620 N] 2. A jet of water 300 mm in diameter impinges normally on a flat plate moving at 3 meters per second in the same direction as that of the jet. If the discharge is 0.590 m3/sec, find the force exerted by the jet on the plate. Find also the work done on the plate per second. [Answer: P = 2020 N, Work done per second = 6060 Nms-1 ]
53 3. A jet 200 mm in diameter moving at 30 meters per second strikes a plate which remains at rest. Find the force exerted on the plate normal to it when (i) the plate is held normal to the jet, and (ii) when the plate makes an angle of 40o to the jet. [Answer: 28278 N, 18177 N] 4. A 80 mm diameter jet having a velocity of 35 meters per second strikes a flat plate. Find the normal pressure on the plate (i) when the plate is stationary, and (ii) when the plate is moving with a velocity of 20 m/sec and away from the jet. Also determine the power and efficiency of the jet when the plate is moving. [Answer: 6158 N, 1131 N, 22620 N, 21 %] 5. A water jet of 50 mm diameter has a velocity of 26 m/s. Determine the pressure on the plate if the water jet hit normally onto (i) a fixed plate (ii) a plate that moves with a velocity of 10 m/s in jet direction upon impact (iii) a plate that moves with a velocity of 7 m/s in the opposite direction of the water jet (iv) power of jet and efficiency of the system for case (iii). [Answer: 1327 N, 502.7 N, 2138 N, 17251 W, 86.8 %] 6. The force exerted by a 25 mm diameter water jet against a flat plate held normally is 700 N. Calculate the velocity of jet in m/s. [Answer: 37.76 m/s] 7. A 80 mm diameter jet having a velocity of 35 meters per second strikes a flat plate. Find the normal pressure on the plate (i) when the plate is stationary, and (ii) when the plate is moving with a velocity of 20 m/sec and away from the jet. Also determine the power and efficiency of the jet when the plate is moving. [Answer: 6158 N, 1131 N, 22620 Nm/s, 20.99 %]
54 8. A jet 160 mm in diameter moving at 26 meters per second strikes a plate which remains at rest. Find the force exerted on the plate normal to it when (i) the plate is held normal to the jet, and (ii) when the plate makes an angle of 35o to the jet. [Answer: 13588 N, 7794 N] 9. A 75 mm diameter jet having a velocity of 30 meters per second impinges normally on a flat plate. Find the pressure on the plate when the plate is moving with a velocity of 15 m/sec and away from the jet. [Answer: 994.05 N] 10. A jet of water 500 mm in diameter impinges normally on a flat plate moving at 4 meters per second in the same direction as that of the jet. If the discharge is 0.98 m3 /sec, find the force exerted by the jet on the plate. [Answer: 193.2 N] 11. A jet of water from a 75 mm diameter nozzle impinges normally with a velocity of 40 meters per second on a flat plate which remains at rest. Find the force exerted by the jet on the plate. [Answer: 7068.8 N] 12. The force exerted by a 30 mm diameter jet of water against a fixed flat plate held normal to the jet is 800 N. What is the flow? [Answer: 0.0238 m/s] 13. A 100 mm diameter jet moving at 15 m/s impinges normally on a flat plate moving at 3 m/s in the same direction as that of the jet. Find the force exerted by the jet on the plate. Find also the work done on the plate per second. [Answer: 1130.976 N, 3.393 kW] 14. A nozzle discharges a 60 mm diameter jet at a velocity of 18 m/s at right angles to a plate moving away from nozzle at 6 m/s. find the force on the plate, the work done on the plate per second and the efficiency. [Answer: 407.15 N, 2442.9 Nm/s, 29.6 %]
55 15. A jet 50 mm in diameter has a direct impact on a fixed plate and exerts a force of 1225 N over it. Find the rate of discharge. [Answer: 0.049 m3 /sec] 16. Find the ratio of forces exerted by a jet when it is made to have an impact on (a) a flat stationary flat plate held normal to the jet (b) a flat plate moving in the direction of the jet at one half the velocity of the jet. [Answer: 4:1] 17. A 50 mm diameter jet impinges on a flat plate with a velocity of 30 m/s. The jet is inclined at 45⁰ to the normal to the plate. Find the force exerted on the plate normal to the plate, when (a) the plate is stationary (b) the plate is moving away with a velocity of 15 m/s in the direction of motion of the jet (c) find also work done per second. [Answer: 1249.6 N, 312.39 N, 3313.35 Watt]
56 4.4 Force Exerted on Curved Vane Example 5 A 60 mm diameter jet of water moving at velocity 45 m/sec impinges on a fixed curved symmetrical vane at the center. Find the force exerted by the jet in its direction if the vane deflects the jet on either side by 120o .
57 Solution: Jet diameter, d = 60 mm Area of jet, A = 4 (0.06) 2 = 0.0028274 2 Discharge, Q = AV = 0.0028274 x 45 m3 /s = 0.1272 m3 /s Jet velocity at inlet = V = 45 m/s Velocity of the jet at outlet in the direction of the jet at inlet = V cos 60o = 45 cos 60o = 22.5 m/s (<--) Force exerted by the jet in the direction of the jet at, = (change in velocity in the direction of the jet) = 1000 x 0.1272 (45 + 22.5) = 8586 N Example 6 A bent plate deflects a 75 mm diameter jet of water through an angle of 45o . The jet has a velocity of 45 m/s. If the plate remains stationary, find the X and Y components of the force exerted by the jet on the plate.
58 Solution: Area of jet, a = ×0.0752 4 = 4.418 × 10−32 Discharge, Q = 4.418 x 10-3 x 45 = 0.1988 m3 /s Force exerted along x-axis = ρQ (Vx1 – Vx2) = 1000 x 0.1988 (45 – 45 cos 45o ) = 2620 N (-->) Force exerted along y-axis = ρQ (Vy1 – Vy2) = 1000 x 0.1988 (0 – 45 sin 45o ) = -6326 N (↓) Example 7 A jet of 15 cm diameter strikes at the center of a smooth semi-spherical vane. The velocity of the jet is 15 m/sec. Find the thrust if the vane moves at a velocity of 5 m/sec. Solution: = ( − ) 2 (1 + cos ) = 1000 × 4 × 0.15 × 0.15 × (15 − 5) 2 (1 + cos 0°) = 3534 4.5 Exercises 18. A 50 mm diameter jet of water moving at velocity 35 m/sec impinges on a fixed curved symmetrical vane at the center. Find the force exerted by the jet in its direction if the vane deflects the jet on either side by 125o .
59 [Answer: 3785 N] 19. A bent plate deflects a 70 mm diameter jet of water through an angle of 44o . The jet has a velocity of 40 m/s. If the plate remains stationary, find the X and Y components of the force exerted by the jet on the plate. [Answer: 1728 N, -4276 N]
60 20. A jet of 20 cm diameter strikes at the center of a smooth semi-spherical vane. The velocity of the jet is 17 m/sec. Find the thrust if the vane moves at a velocity of 7 m/sec. [Answer: 6283 N] 21. A 30 mm diameter jet of water moving at velocity 22.5 m/sec impinges on a fixed curved symmetrical vane at the center. Find the force exerted by the jet in its direction if the vane deflects the jet on either side by 120o . [Answer: 537 N] 22. A bent plate deflects a 70 mm diameter jet of water through an angle of 40o . The jet has a velocity of 40 m/s. If the plate remains stationary, find the X and Y components of the force exerted by the jet on the plate. [Answer: 1440 N, -3957 N] 4.6 Magnitude and Direction of Resultant Force on Pipe Bend Example 8 A 300 mm diameter pipe carries water under a head of 20 meters with a velocity of 3.5 m/s. If the axis of the pipe turns through 45o , find the magnitude and the direction of the resultant force at the bend.
61 Solution: Diameters of the pipe at extremities of bend = d1 = d2 = 0.30 m Area, a1 = a2 = ×0.3 2 4 = 0.0707 2 Velocity, v1, v2 = 3.50 m/s Discharge, Q = 0.0707 x 3.5 = 0.2475 m3 /s Pressure intensity, p1 = p2 = 9810 x 20 = 196.2 kPa Force along X-axis = Fx = Dynamic force + Static Force = ρQ (v1 – v2 cos 45o ) + p1a1 – p2a2 cos 45o = 1000 x 0.2475 (3.5 – 3.5 cos 45o ) + 196200 x 0.0707 - 196200 x 0.0707 cos 45o = 4.3165 kN (→) Force along Y-axis = Fy = Dynamic force + Static Force = ρQ (0 – v2 sin 45o ) – p2a2 sin 45o = 1000 x 0.2475 (0 – 3.5 sin 45o ) - 196200 x 0.0707 sin 45o = - 10.421 kN (↓) Resultant Force, F = √4.31652 + 10.42102 = 11.28 tan = 10.421 4.3165 = 67°30′
62 Example 9 A 200 mm diameter pipe conveys water at a pressure of 147.15 kPa at a velocity of 2 m/s. If the axis of the pipe has a 45o bend, find the magnitude and the direction of the resultant force on the bend. Take specific weight of water equal to 9.81 kN/cum. Solution: d1 = d2 = 0.20 m Area, a1 = a2 = ×0.2 2 4 = 0.03142 2 v1 = v2 = 2 m/s Discharge, Q = 0.03142 x 2 = 0.06284 m3 /sec p1 = p2 = 147.15 kPa
63 Force along X-axis (in the direction of the pipe axis before the bend) = Fx = Dynamic force + Static Force = ρQ (v1 – v2 cos 45o) + p1a1 – p2a2 cos 45o = 1000 x 0.06284 (2 – 2 cos 45o ) + 147150 x 0.03142 - 147150 x 0.03142 cos 45o = 1.39047 kN (→) Force along Y-axis = Fy = Dynamic force + Static Force = ρQ (0 – v2 sin 45o ) – p2a2 sin 45o = 1000 x 0.06284 (0 – 2 sin 45o ) - 147150 x 0.03142 sin 45o = - 3.35815 kN (↓) Resultant Force on the bend, F = √1.390472 + 3.358152 = 3.635 Let the resultant force be at α with the X-axis, tan = 3.35815 1.39047 = 2.41512 = 67.5° Example 10 A curved pipe was deflected to reduce the pipe diameter from 600 mm to 300 mm. The deflection of fluid is 60o . The pressure at the bend = 172 kN/m2 . The flow rate is 0.85 m3 /s. Based in figure below, calculate magnitude of resultant force at the bend and direction of resultant force.
64 Solution: d1 = 0.6 m, d2 = 0.3 m Area, a1= ×0.6 2 4 = 0.2827 2 , a2 = ×0.3 2 4 = 0.07069 2 v1= 0.85 0.2827 = 3.007 −1 , v2 = 0.85 0.0769 = 11.05 −1 1 + 1 2 2 = 2 + 2 2 2 172000 1000×9.81 + 3.0072 2×9.81 = 2 1000×9.81 + 11.052 2×9.81 p2 = 115464 Nm-2 Force along X-axis (in the direction of the pipe axis before the bend), Fx = Dynamic force + Static Force = ρQ (v1 – v2 cos 60o ) + p1a1 – p2a2 cos 60o = 1000 x 0.85 (3.007 – 11.05 cos 60o ) + 172000 x 0.2827 - 115464 x 0.07069 cos 60o = 42403 N (→)
65 Force along Y-axis, Fy = Dynamic force + Static Force = ρQ (0 – v2 sin 60o ) – p2a2 sin 60o = 1000 x 0.85 (0 – 11.05 sin 60o ) - 115464 x 0.07069 sin 60o = - 15203N (↓) Resultant Force on the bend, F = √424032 + 152032 = 45046 Direction, = −1 ( 15203 42403) = 19.72° 4.7 Exercises 23. A 400 mm diameter pipe carries water under a head of 30 meters with a velocity of 4.5 m/s. If the axis of the pipe turns through 43o , find the magnitude and the direction of the resultant force at the bend. [Answer: 28983 N, 68.5⁰] 24. A 300 mm diameter pipe conveys water at a pressure of 150 kPa at a velocity of 3 m/s. If the axis of the pipe has a 55o bend, find the magnitude and the direction of the resultant force on the bend. Take specific weight of water equal to 10 kN/cum. [Answer: 10380 N, 62.5⁰] 25. A curved pipe was deflected to reduce the pipe diameter from 500 mm to 250 mm. The deflection of fluid is 50o . The pressure at the bend = 150 kN/m2 . The flow rate is 0.75 m3 /s. Based in figure below, calculate magnitude of resultant force at the bend and direction of resultant force.
66 [Answer: 25811 N, 23.5⁰] 26. A 300 mm diameter pipe carries water under a head of 20 meters with a velocity of 3.5 m/s. If the axis of the pipe turns through 40o , find the magnitude and the direction of the resultant force at the bend. [Answer: 10081 N, 70⁰] 27. A 300 mm diameter pipe conveys water at a pressure of 150 kPa at a velocity of 3 m/s. If the axis of the pipe has a 46o bend, find the magnitude and the direction of the resultant force on the bend. Take specific weight of water equal to 9.81 kN/cum. [Answer: 8783 N, 70⁰] 28. A 200 mm diameter pipe carries water under a head of 18 meters with a velocity of 2.9 m/s. If the axis of the pipe turns through 47o , find the magnitude and the direction of the resultant force at the bend. [Answer: 4635 N, 67⁰]
67 29. A 450mm diameter pipeline conveying 1.0 m3 /s of water contains a 22.5° bend in the horizontal plane. If the pressure in the bend is 250 KN/m2 , calculate the magnitude and direction of the force on the bend. [Answer: 17.96 kN, 78.75⁰] 30. A 300mm diameter pipeline conveying 0.7m3 /s of water contains a 45° bend in the horizontal plane. If the pressure in the bend is 150 KN/m2 , calculate the magnitude and direction of the force on the bend. [Answer: 13.42 kN, 67.5⁰]
68 Editor: Anuar Bin Abdul Wahab Radat Bin Tasim