Fluid Machinery As per Revised Syllabus of Leading Universities including APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY Dr. S. Ramachandran Prof. R. Devaraj Professors School of Mechanical Engineering Sathyabama University Chennai - 600 119 Mr. YVS. Karthick AIR WALK PUBLICATIONS (Near All India Radio) 80, Karneeshwarar Koil Street Mylapore, Chennai - 600 004. Ph.: 466 1909, 94440 81904 Email: aishram006@gmail.com, airwalk800@gmail.com www.airwalkpublications.com
First Edition : 0-0-017 ISBN : 978-93-84893-66-8
ME06 FLUID MACHINERY APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY SYLLABUS Module Contents Hours I II III Impact of jets: Introduction to hydrodynamic thrust of jet on a fixed and moving surface (flat and curve), - Series of vanes - work done and efficiency. Hydraulic Turbines: Impulse and Reaction Turbines - Degree of reaction - Pelton Wheel - Constructional features - Velocity triangles - Euler s equation - Speed ratio, jet ratio and work done, losses and efficiencies, design of Pelton wheel - Inward and outward flow reaction turbines - Francis Turbine - Constructional features - Velocity triangles, work done and efficiencies Axial flow turbine (Kaplan) - Constructional features - Velocity triangles - work done and efficiencies - Characteristic curves of turbines - Theory of draft tubes - Surge tanks - Cavitation in turbines - Governing of turbines - Specific speed of turbine, Type Number - Characteristic curves, Scale Laws - Unit speed - Unit discharge and Unit power. First Internal Exam Rotary motion of liquids - free, forced and spiral vortex flows Rotodynamic pumps - centrifugal pump impeller types, - velocity triangles - manometric head - work, efficiency and losses, H-Q characteristic, typical flow system characteristics, operating point of a pump. Cavitation in centrifugal pumps - NPSH required and available - Type number - Pumps in series and parallel operations. Performance characteristics - Specific speed - Shape numbers - Impeller shapes based on shape numbers. Syllabus S.1 Sem. Exam Marks 7 15 % 7 15 % 7 15 %
S. Fluid Machinery - www.airwalkpublications.com Module Contents Hours IV Positive displacement pumps - Reciprocating pump - Single acting and double acting - Slip, negative slip and work required and efficiency - Indicator diagram - Acceleration head - Effect of acceleration and friction on Indicator diagram - Speed calculation - Air vessels and their purposes, saving in work done to air vessels multi cylinder pumps. Multi stage pumps - Selection of pumps - Pumping devices - Hydraulic ram, Accumulator, Intensifier, Jet pumps, gear pumps, vane pump and lobe pump. Second Internal Exam V Compressors: classification of compressors, reciprocating compressor-single stage compressor, equation for work with and without clearance volume, efficiencies, multistage compressor, intercooler, free air delivered (FAD). VI Centrifugal compressor - working, velocity diagram, work done, power required, width of blades of impeller and diffuser, isentropic efficiency, slip factor and pressure coefficient, surging and chocking. Axial flow compressors: working, velocity diagram, degree of reaction, performance. Roots blower, vane compressor, screw compressor. End Semester Exam Sem. Exam Marks 7 15 % 7 0 % 7 0 %
Contents C.1 CONTENTS Module 1 / Part 1 Impact of Jets 1.1 Impact of Jets..................................... 1.1 1. Hydrodynamic Thrust of Jet on A Fixed Surfaces...... 1.1 1..1 Impact of jet on a stationary (fixed) vertical plate 1.1 1.. Impact of jet on fixed curved plate......... 1.3 1.3 Impact of Jet on A Hinged Plate..................... 1.9 1.4 Hydrodynamic Thrust of Jet on A Moving Surface (Flat and Curved Plates).......................... 1.13 1.4.1 Thrust of jet on a flat vertical plate moving in the direction of jet.................. 1.13 1.4. Thrust on the inclined plate moving in the direction of jet (in x direction)......... 1.14 1.4.3 Impact of the jet on moving curved plate in the direction of jet.................. 1.19 1.4.4 Force exerted by a jet of water on moving curved vane, when it strikes tangentially at one end... 1.5 1.5 Thrust of Jet of Water on Series of Vanes........... 1.31 1.5.1 Workdone per second (or) Power of jet on a series of a radial curved vanes........... 1.3 1.5. Efficiency of the Radial curved vane........ 1.33 Module 1 - Part Hydraulic Turbines 1.6 Hydraulic Turbines - Introduction................... 1.39 1.7 Classification of Hydraulic Turbines................. 1.39 1.8 Euler s Equation.................................. 1.40 1.9 Velocity Triangles................................. 1.44 1.10 Degree of Reaction............................... 1.47 1.11 Pelton Turbine (or) Pelton Wheel................... 1.49
C. Fluid Machinery - www.airwalkpublications.com 1.11.1 Working of a Pelton Wheel............. 1.50 1.11. Velocity triangle and work done for a Pelton wheel..................... 1.51 1.11.3 Efficiencies of a Turbine.............. 1.54 1.11.4 Points to be remembered for the design of Pelton wheel turbine................ 1.55 1.1 Governing of Pelton Wheel........................ 1.77 1.13 Solved University Problems on Pelton Wheel......... 1.79 Module 1 - Part 3 Reaction Turbines 1.14 Reaction Turbines Introduction................... 1.9 1. Radial Flow Turbines.................. 1.9. Axial Flow Turbines................... 1.9 3. Mixed Flow Turbines.................. 1.9 1.15 Francis Turbine.................................. 1.95 1.15.1 Working of a Francis Turbine........... 1.95 1.15. Velocity Triangles and work done by water in Francis Turbine................. 1.96 1.15.3 Hydraulic Efficiency h for Francis Turbine.. 1.97 1.15.4 Points to be remembered in Francis Turbine.. 1.97 1.15.5 Solved Problems on Francis Turbine....... 1.98 Module Axial Flow Reaction Turbines.1 Axial Flow Reaction Turbines.........................1.1.1 Working Principle of a Kaplan Turbine........ Velocity Diagram for Kaplan Turbine..................3.3 Specific Speed of Turbine.............................8.4 Draft Tube........................................19.4.1 Functions of a Draft Tube..............19.4. Types of Draft Tubes..................19
Contents C.3.4.3 Theory of Draft tube to find Pressure Head at the Exit of a Turbine.................0.4.4 Efficiency of a Draft Tube d............1.5 Cavitation in Reaction Turbines......................3.5.1 Effects of Cavitation..................4.5. Precaution against Cavitation.............4.6 Scale Laws and Models.............................5.6.1 Similitude........................5 A. Geometric Similarity...................6 B. Kinematic Similarity...................6 C. Dynamic Similarity....................7.6. Specific Quantities...................7.6.3 Unit Quantities.....................9.7 Characteristic Curves of Turbines....................36.7.1 Types of Characteristic Curves............36 (a) Main Characteristic Curves...............36 (b) Operating Characteristic Curves............38 (c) Constant Efficiency Curves or Iso Efficiency Curves.38.8 Selection of Turbines...............................40.9 Governing of Turbine...............................41.9.1 Working of oil pressure governor...........4.10 Surge Tank.......................................43 Questions and Problems...................44
C.4 Fluid Machinery - www.airwalkpublications.com Module 3 Centrifugal Pumps 3.1 Rotary Motion of Liquids............................ 3.1 3.1.1 Forced Vortex flow.................. 3.1 3.1. Free Vortex flow................... 3.1 3.1.3 Cylindrical Vortex flow............... 3. 3.1.4 Spiral Vortex flow.................. 3. 3. Pumps: Definition.................................. 3. 3.3 Classifications of Pumps............................. 3.3 3.4 Centrifugal Pump.................................. 3.3 3.4.1 Main Components of a Centrifugal Pump..... 3.3 3.4. Working Principle of a Centrifugal Pump..... 3.4 3.4.3 Classification..................... 3.5 3.4.4 Velocity triangle and work done by the Centrifugal Pump................... 3.8 3.4.5 Efficiencies of a Centrifugal Pump......... 3.11 3.4.6 Minimum Starting Speed.............. 3.1 3.4.7 Centrifugal Pump Proportions............ 3.13 3.5 H-Q Characteristics of a Centrifugal Pump........... 3.5 3.6 Typical Flow System Characteristics................. 3.54 3.6.1 System characteristics curve............. 3.54 3.6. Pump characteristics curve............. 3.55 3.6.3 Operating point.................... 3.56 3.7 Priming.......................................... 3.57 3.8 Cavitation........................................ 3.58 3.8.1 Effects of Cavitation................. 3.58 3.8. Precaution against Cavitation............ 3.58 3.8.3 Thoma s cavitation factor for centrifugal pumps. 3.59 3.9 Net Positive Suction Head (NPSH).................. 3.59 3.9.1 NPSH Required NPSH R.............. 3.60 3.9. NPSH Available NPSH A.............. 3.61
Contents C.5 3.10 Type Number.................................... 3.6 3.11 Multi Stage Centrifugal Pumps.................... 3.6 3.11.1 Multi Stage Pumps for High Heads (Pumps in Series).................. 3.6 3.11. Pumps in Parallel (Multistage pump for more discharge).................... 3.63 3.1 Performance Curves.............................. 3.68 3.1.1 Main Characteristic curves............. 3.68 (i) Q v/s H Curve...................... 3.68 (ii) Q v/s Curve...................... 3.68 (iii) Q v/s P Curve..................... 3.69 3.1. Operating Characteristics Curves......... 3.69 3.13 Model Testing of Centrifugal Pumps................ 3.70 3.14 Specific Speed of Centrifugal Pump................. 3.7 3.15 Shape Numbers N q.............................. 3.80 Module 4 Positive Displacement Pumps 4.1 Reciprocating Pumps................................ 4.1 4. Classification of Reciprocating Pumps................. 4.1 4.3 Main Parts of a Reciprocating Pump.................. 4. 4.4 Working of a Reciprocating Pump.................... 4.3 4.5 Discharge, Workdone and Power required to drive a Single Acting Reciprocating Pump................... 4.3 4.6 Discharge, Work Done and Power required to drive a Double Acting Pump.............................. 4.4 4.7 Slip of Reciprocating Pump.......................... 4.6 4.7.1 Negative Slip..................... 4.7 4.8 Indicator Diagram................................. 4.11 4.8.1 Effect of acceleration of piston in suction and delivery pipes on indicator diagram........ 4.13
C.6 Fluid Machinery - www.airwalkpublications.com 4.8. Effect of acceleration in the suction pipe and delivery pipe...................... 4.16 4.8.3 Effect of friction in the suction and delivery pipes on the Indicator Diagram.............. 4.19 4.8.4 Separation....................... 4.30 4.9 Maximum Speed of a Reciprocating Pump............ 4.40 4.10 Air Vessels...................................... 4.46 4.10.1 Work saved by the air vessels........... 4.47 4.10. Work saved in a double acting Pump with Air Vessels...................... 4.50 4.11 Pump Selection.................................. 4.66 4.1 Hydraulic Ram................................... 4.67 4.13 Hydraulic Accumulator............................ 4.7 4.14. Hydraulic Intensifiers (or) Boosters................. 4.8 4.15 Various Types of Pumps.......................... 4.85 4.16 Jet Pump....................................... 4.86 4.17 Advantages of Positive Displacement Pumps Over Dynamic Pumps................................. 4.88 4.18. Some Positive Displacement Pumps................ 4.88 I. Gear pumps....................... 4.88 II. Vane pumps....................... 4.88 III. Piston pumps..................... 4.88 4.19 Rotary Pumps................................... 4.89 4.19.1 Working Principle of External Gear Pump.... 4.89 4.19. Volumetric displacement and Theoretical flow rate...................... 4.90 4.19.3 Volumetric Efficiency................ 4.91 4.19. Working Principle of Internal Gear Pump.... 4.9 4.19.3 Lobe Pumps..................... 4.93 4.19.4 Screw Pump..................... 4.93 4.19.5 Vane Pump..................... 4.94 4.0 Piston Pumps.................................... 4.96
Contents C.7 Module 5 Reciprocating Air Compressors 5.1 Introduction....................................... 5.1 5. Classification of Air Compressors..................... 5.1 5.3 Single Acting Reciprocating Air Compressor........... 5. 5.4 Double Acting Air Compressor....................... 5.3 5.5 Single Stage Compressor............................ 5.3 5.6 Multi Stage Compressor............................. 5.4 5.7 Working Principle of Reciprocating Air Compressors.... 5.4 5.7.1 Workdone during Isothermal compression PV c without clearance volume.............. 5.6 5.7. Workdone during Polytropic compression [ PV n constant ] without clearance volume.... 5.6 5.7.3 Workdone during Isentropic compression PV constant without clearance Volume.... 5.8 5.8 Minimum Workdone................................ 5.9 5.9 Power required to run the Compressor................ 5.9 5.9.1 Clearance Volume................... 5.1 5.10 Workdone by Reciprocating Air Compressor with Clearance Volume................................ 5.13 5.11 Isothermal Efficiency of a Reciprocating Air Compressor..................................... 5.14 5.1 Volumetric Efficiency in a Reciprocating Air Compressor..................................... 5.15 5.1.1 Factors affecting volumetric efficiencies...... 5.16 5.13 Important Technical Terms........................ 5.16 5.14 Single Stage Compression......................... 5.19 5.15 Two Stage Compression........................... 5.50 5.15.1. Complete (or) Perfect Intercooling......... 5.51 5.15. Incomplete (or) Imperfect Intercooling....... 5.51
C.8 Fluid Machinery - www.airwalkpublications.com 5.15.3 Workdone when perfect and imperfect........ intercooling...................... 5.51 5.15.4 Minimum Work Required for Stages and multi stages..................... 5.5 5.16 Multistage Reciprocating Compressors............... 5.5 5.16.3 Conditions for Minimum Workdone........ 5.55 5.16.4 Intermediate Pressures for Z Stage compressor Running under Ideal Condition.......... 5.57 5.16.5 Heat Rejected per stage per kg of air...... 5.57 5.16.6 The Change in Entropy in First Stage Compression..................... 5.57 5.16.7 For Two Stage Compression............ 5.57 Module 6 Centrifugal Compressors and Axial Flow Compressors 6.1. Centrifugal Compressor............................. 6.1 6.1 Principle of Operation................. 6.3 6. Velocity and Pressure Variation...................... 6.4 6.3 Static Temperature and Total Head (or) Stagnation Temperature....................................... 6.4 6.3.1 Stagnation State and Stagnation Properties.... 6.6 6.3. Stagnation Enthalpy................. 6.6 6.4 Steady-flow Energy Equation......................... 6.7 6.5 Euler s Equation - (Energy Transfer).................. 6.8 6.6 Impeller Blade Shape............................. 6.1 6.7 Velocity Triangle Work Done, Power Required by the Centrifugal Compressor........................ 6.14 6.8 Important Formulae............................... 6.16 6.9 Axial Flow Compressors............................ 6.36 6.9.1 Working Principles of a Compressor Stage.... 6.37 6.9. Stage Velocity Triangles............... 6.38
Contents C.9 6.9.3 Blade Loading, Flow Coefficients and Specific Work..................... 6.40 6.9.4 Static Pressure Rise in a Stage........... 6.40 6.10 Degree of Reaction............................... 6.43 6.11 Infinitesimal Stage Efficiency (or) Polytropic Efficiency........................................ 6.45 6.1 Finite Stage Efficiency............................ 6.47 6.13 Important Formulae.............................. 6.48 6.14 Losses in Axial Flow Compressor Stage............. 6.50 6.15 Surging and Choking............................. 6.51 6.16 Stalling......................................... 6.53 6.17 Comparison between Reciprocating and Centrifugal Compressors..................................... 6.54 6.18 Comparison between Reciprocating and Rotary Air Compressors...................................... 6.55 6.19 Comparison between Centrifugal and Axial Flow Compressors.................................................... 6.55 6.0 Various Types of Compressors Rotary Positive Displacement Compressors........................ 6.7 6.0.1 Different Types of Rotary Positive Displacement Compressor............. 6.7 6.0. Different Types of Rotary Non-positive Displacement Compressors............. 6.7 6.1 Roots Blower.................................... 6.7 6.1.1 Back Flow of Air.................. 6.7 6. Vane Type Blower of Compressor.................. 6.76 6.3 Screw Compressor................................ 6.8 Twin - Screw compressor................. 6.8 Single - Screw compressor................. 6.84
Module 1 / Part 1 Impact of Jets Impact of jets: Introduction to hydrodynamic thrust of jet on a fixed and moving surface (flat and curve), - Series of vanes - work done and efficiency. Hydraulic Turbines : Impulse and Reaction Turbines - Degree of reaction - Pelton Wheel - Constructional features - Velocity triangles - Euler s equation - Speed ratio, jet ratio and work done, losses and efficiencies, design of Pelton wheel - Inward and outward flow reaction turbines - Francis Turbine - Constructional features - Velocity triangles, work done and efficiencies 1.1 IMPACT OF JETS When pressurized water jet strikes on the plate, a force is exerted on the plate. This is called impact of jets on plates. 1. Force exerted (Impact of jet) on a fixed plate (a) Plate may be vertical (perpendicular to the jet) (b) Plate may be inclined to jet (c) Plate may be curved. Force exerted (Impact of jet) on a moving plate 1. HYDRODYNAMIC THRUST OF JET ON A FIXED SURFACES 1..1 Impact of jet on a stationary (fixed) vertical plate 1. Impact of jet on vertical plate V Velocity of jet; d dia of jet Area of jet A 4 d Initially the jet is moving horizontally (in x direction) with velocity V 1. Finally, it moves with velocity V in y direction ie perpendicular direction. Initial velocity in x direction V
1. Fluid Machinery - www.airwalkpublications.com V Nozzle d y Plate x Jet of Water V Fig:1.1 Impact of Jet on vertical Plate Final velocity in x direction 0 Force is obtained from the Newton s second law of motion or from impulse-momentum equation. Force exerted by the jet on the plate in the direction of jet F x F x Rate of change of momentum in x direction Mass Initial velocity Mass Final velocity Time Mass Initial velocity Final velocity Time A V [V 0] Mass Time Mass Sec A V F x A V [Final velocity = 0]. Impact of jet on fixed inclined plate Angle between the jet and plate F n Force exerted by the jet on the plate in the direction normal to the plate F x Component of F n in x direction (in the direction of flow)
Impact of Jets 1.3 F y V V sin y V (90 - ) F x x Jet d o F n n Plate Fig: 1. Impact of jet on inclined Plate F n cos 90 F n sin Now F n Mass Sec [Initial velocity of jet in n direction Final velocity F n A V [V sin 0] A V sin F n F n sin A V sin F x F n sin A V sin F y F n sin A V sin cos of jet in n direction] 1.. Impact of jet on fixed curved plate (a) Jet striking the curved plate at centre Here it is assumed that there is no friction in the plate and hence there is no loss of energy while jet striking the curved plate and leaving the plate. Initial velocity of jet striking on plate V
1.4 Fluid Machinery - www.airwalkpublications.com y V V cos V sin Final Velocity Intrial Velocity V Fixed Curved Plate x d Fig: 1.3 Impact of jet on a fixed curved plate at centre Final velocity of jet leaving the curved plate in x direction V cos in y direction V sin Force exerted by jet on curved plate in x direction A V V V cos A V V V cos in y direction A V 0 V sin A V sin [ sign indicates that force is acting downwards] 180 Angle of deflection of jet (b) Jet striking the curved plate tangentially at one end Assume the curved plate is symmetrical about x - axis Angle of jet with x axis at inlet and outlet V Inlet velocity = Outlet velocity (Assuming the plate is smooth and loss of energy due to impact is zero) Mass/sec of jet A V
Impact of Jets 1.5 Force exerted by jet on plate F x in x direction A V V cos V cos A V V cos A V cos F y in y direction A V V sin V sin 0 V V cos y V sin V sin x (c) Jet striking the curved plate (not symmetrical) tangentially at one end Jet V V cos Angle of jet with x axis at inlet d Angle of jet with x axis at exit Hence, force exerted by jet on plate F x in x direction A V V cos V cos F x A V cos cos F y in y direction A V V sin V sin F y A V sin sin Problem 1.1: Find the force exerted by a jet of water of diameter 50 mm on a stationary flat plate when the jet strikes the plate normally with velocity of 15 m/s. Given: Diameter of jet, d 50 mm 0.05 m Solution Area, A 4 d 4 0.05 1.9634 10 3 m
1.6 Fluid Machinery - www.airwalkpublications.com Velocity of jet, V 15 m/s The impact of jet of water on a fixed vertical plate F F A V where 1000 kg/m 3 F 1000 1.9634 10 3 15 441.8 N Problem 1.: A jet of water of diameter 40 mm strikes a fixed plate in such a way that the angle between the plate and the jet is 30. The force exerted in the direction of the jet is 1500 N. Determine the velocity of jet and the rate of flow of water. Given: Diameter of jet, d 40 mm 0.04 m Solution Area, A 4 0.04 1.6 10 3 m Angle, 30 Force in the direction of jet, F x 1500 N A V sin 1500 1000 1.6 10 3 V sin 30 V 4774.6 V 69.1 m/s Discharge, (rate of flow) Q Q Area Velocity 1.6 10 3 69.1 0.087 m /s Q 87.06 liters/s Problem 1.3: A jet of water of diameter 60 mm moving with a velocity of 30 m/s, strikes on fixed symmetrical plate at the centre, Find the force exerted by the jet of water in the direction of jet, if the jet is deflected through an angle of 10 at the outlet of the curved plate. Given: Diameter of the jet, d 0 mm 0.06 m
Impact of Jets 1.7 Solution V Area, A 4 0.06.87 10 3 Velocity of jet, V 30 m/s Angle of deflection 10 The angle of deflection 180 180 10 or 180 10 60 Force exerted by the jet on the curved plate in the direction of the jet F x V Angle of Deflection F x A V 1 cos 1000.87 10 3 30 1 cos 60 3817 N Problem 1.4: A 10 cm diameter jet of water exerts a force of kn in the direction of flow against a stationary flat plate which is inclined at 30 with the axis of the stream. Find (i) Force normal to the plate (ii) Velocity of the jet (iii) Mass flow rate of water in kg/s. (May 01 - Calicut University) Given: d 10 cm 0.1 m; F x kn 000 N; 30 F y V V s in y V F x x Jet d (90 - ) F n o n Plate Fig: Impact of jet on inclined Plate
1.8 Fluid Machinery - www.airwalkpublications.com Area of jet 4 0.1 7.854 10 3 We know F x AV sin 000 1000 7.854 10 3 V sin 30 V 1018.6 V 31.9 m/s Velocity of jet V 31.9 m/s F x Force normal to the plate A V sin Mass flow rate of water in kg/s 1000 7.854 10 3 31.9 sin 30 4000 N 4 kn m A V 1000 7.854 10 3 31.9 50.7 kg/s Problem 1.5 Find the force exerted by a 75 mm dia jet on a stationary flat plate. Jet moves with a velocity of 40 m/s. (June 009 - C58184 - Calicut University) Given: d 75 mm 0.075 m; V 40 m/s Force exerted on the plate by the jet A V [V 0] A V 1000 4 0.075 40 7068.6 N 7.07 kn
Impact of Jets 1.9 1.3 IMPACT OF JET ON A HINGED PLATE A vertical plate is hinged at one end O. h height of plate A The centre point at which jet strikes the plate Consider the jet of water striking the hinged vertical plate at point A. Due to the thrust, the plate with swing through some angle. B The point at which jet strikes after it is inclined to an equilibrium position. Before striking O O x h x x A d A d A F n W B Fig: 1.5 (a) Force on a hinged plate Fig: 1.5 (b) Force on a hinged plate After striking x Distance of centre of jet from hinge O. OA OA Half height of plate The weight of plate acts at centre A of the plate. After striking, when the plate is in equilibrium, the moment of all forces about hinge should be zero. There are forces acting on plate. 1. Force F n due to jet of water, normal to the plate. F n A V sin 90
1.10 Fluid Machinery - www.airwalkpublications.com F n A V cos Moment due to force F n F n OB A V cos OB. Weight of plate, W A V cos OA cos AV x Moment due to W about hinge W OA sin W x sin When the plate is in equilibrium, equate both moments. A V x W x sin sin A V W Problem 1.6: A jet of water of.5 cm diameter, moving with a velocity of 0 m/s strikes on hinged square plate of weight 00 N at the centre of the plate. The plate is of uniform thickness. Find the angle through which the plate will swing. Solution Given: Diameter of jet, d.5 cm 0.05 m Velocity of jet, V 0 m/s ; Weight of plate, W 00 N Area of jet, A 4 d 4 0.05 4.9 10 4 m The angle through which the plate will swing Q sin A V W 1000 4.9 10 4 0 00 0.98 78.5
Impact of Jets 1.11 Problem 1.7: A jet of water of 5 mm diameter strikes a hinged square plate at its centre with a velocity of 5 m/s. The plate is deflected through an angle of 0. Find the weight of plate. If the plate is not allowed to swing, what will be the force required at the lower edge to keep the plate in vertical position. Given: Diameter of the jet, d 5 mm 0.05 m Solution Area, A 4 d 4 0.05 4.9 10 4 O Hing e Velocity of jet, V 5 m/s ; Angle of swing, 0 sin A V W V Jet h sin 0 1000 4.9 10 4 5 W W 306.8 sin 0 897 N 306.8 W P If the plate is not allowed to swing, a force P will be applied at the lower edge of the plate as shown in Fig. The weight of the plate is acting vertically downward through the C.G. of the plate. Let F Force exerted by jet of water A V 1000 4.9 10 4 5 3065 N h Height of plate Distance of P from the hinge O Since jet strikes at the centre of the plate and hence distance of the centre of the jet from hinge h. Take moments about the hinge, O, P h F h. P F h h F 3065 153.13 N
1.1 Fluid Machinery - www.airwalkpublications.com Problem 1.8: A square plate weighing 110 N and of uniform thickness and 30 cm edge is hung so that horizontal jet 3 cm diameter and having a velocity of 1 m/s impinges on the plate. The centre line of the jet is 15 cm below the upper edge of the plate, and when the plate is vertical, the jet strikes the plate normally and at its centre. Find what force must be applied at the lower edge of the plate in order to keep the plate vertical. If the plate is allowed to swing freely, find the inclination to vertical which the plate will assume under the action of jet. (May 014 - MGU) Given: W 110 N; h 30 cm 0.3 m; d 3 cm 0.03 m; V 1 m/s; x 15 cm 0.15 m; O Hinge O W 15cm V d=3cm F 15cm A 30cm 15cm D A G B F n P Let P Force applied at the lower edge to keep the plate verified. F Force exerted by the jet of water at the centre of the plate. A V 1000 4 0.03 1 101.79 N This force F is acting at centre ie 15 cm from upper edge O. P is acting at lower end ie 30 cm from upper edge O.
Impact of Jets 1.13 Taking moment about hinge O F 15 P 30 P 101.79 15 30 50.89 N When the plate in allowed to swing freely about hinge sin A V W 101.79 110 0.954 67.7 1.4 HYDRODYNAMIC THRUST OF JET ON A MOVING SURFACE (FLAT AND CURVED PLATES) We have discussed about the hydrodynamic thrust of jet on fixed plate. Now we can discuss the impact of jet on moving plates - Flat and curved plates. 1.4.1 Thrust of jet on a flat vertical plate moving in the direction of jet V (Absolute) velocity of the jet A Area of cross section of jet u (Absolute) Velocity of the plate (V-u) y V Jet F u x Fig: 1.6 Thrust on a vertical moving plate (V-u)
1.14 Fluid Machinery - www.airwalkpublications.com Here the jet does not strike, the plate with absolute velocity v. Instead, it strikes the plate with relative velocity V u. Mass of water/sec striking the plate m m A V u Thrust (force) of jet on moving plate in the direction of jet (x - direction) F x m [Initial velocity with which it strikes final velocity] A V u [V u 0] [... Final velocity = 0 in the direction of x] A V u Workdone/sec by the jet on the plate Distance moved by the plate Force Time F x u A V u u in watts N m s J s watts 1.4. Thrust on the inclined plate moving in the direction of jet (in x direction) (V-u) F n y V u x (V-u) Fig: 1.7 Thrust of jet on an inclined moving plate
Impact of Jets 1.15 V Absolute velocity of jet u Absolute velocity of plate in the direction of jet A Cross sectional area of the jet Angle of inclination of the plate with jet x axis Relative velocity of jet with respect to plate V u velocity of jet with which it strikes Mass of water striking/sec m A V u Here it is assumed that, the plate is smooth and the loss of energy due to impact of jet is zero. Hence the velocity of jet leaving the plate will be same as V u. Thrust of jet perpendicular to plate F n F n m [Initial velocity perpendicular (normal) to plate Final velocity perpendicular normal to plate] m [V u sin 0] F n A V u sin [Final velocity V u is moving along the plate. Hence final velocity perpendicular (normal) to plate is zero] This normal force F n is resolved into F x and F y components. F n in x (in the jet) direction F x F n in y (perpendicular to jet) direction F y F x F n sin A V u sin F y F n cos A V u sin cos
1.16 Fluid Machinery - www.airwalkpublications.com Workdone/sec by the jet on the plate in x direction F x Distance moved by plate in x direction time F x u A V u u sin watts Workdone/sec in y direction is zero, since the plate velocity in y direction is zero. Problem 1.9: A 7.5 cm diameter jet having a velocity f 30 m/s strikes a flat plate, the normal of which is inclined at 45 to the axis of the jet. 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/s and away from the jet. Also determine the power and efficiency of the jet when the plate is moving. [May 013 - MGU] Solution Given: Diameter of the jet, d 7.5 cm 0.075 m Area, A 4 0.075 0.004418 m Angle between the jet and plate 90 45 45 Velocity of jet, V 30 m/s (i) When the plate is stationary, the normal force on the plate F n F n A V sin 1000 0.004418 30 sin 45 81 N, (ii) When the plate is moving with a velocity 15 m/s and away from the jet, the normal force on plate is given by equation as F n A V u sin where u 15 m/s 1000 0.004418 30 15 sin 45 70.8 N. Ans. (iii) The power and efficiency of the jet when plate is moving: Workdone per second by the jet = Power Force in the direction of jet Distance moved by the plate in the direction of jet F x u, where F x F n sin 70.8 sin 45 496.95 N
Impact of Jets 1.17 Power F x u 496.95 15 745.43 watts 7.454 kw Efficiency of the jet Output Input 7454.3 1 A V V 0.149 1.49% 7454.3 1 A V3 Power output Kinetic energy of the jet 7454.3 1 A V3 7454.3 1 1000 0.004418 303 Problem 1.0: A jet of water of diameter 8 cm strikes a flat vertical plate normally with a velocity of 5 m/s. The plate is moving with a velocity of 10 m/s in the direction of the jet and away from jet. Find (i) Hydro dynamic thrust by the jet on the plate (ii) Power of the jet on the plate (iii) Efficiency of the jet Given: Diameter of the jet, d 8 cm 0.08 m Solution Area, A 4 d 4 0.08 5.07 10 3 Velocity of jet, V 5 m/s Velocity of the plate, u 10 m/s (i) Thrust of the jet on a moving flat vertical plate F x (ii) Power of the jet F x A V u 1000 5.07 10 3 5 10 N 1131 N F x u 1131 10 11310 Watts 11.31 kw (iii) Efficiency of the jet Power output of the jet in watts Input of the jet in watts Power output of jet = Work done by jet per second = 11310 watts Input of jet in watts Kinetic energy of the jet/sec
1.18 Fluid Machinery - www.airwalkpublications.com 1 mass sec V 1 A V V 1 A V3 1 1000 5.07 10 3 5 3 Nm/s 3973.44 watts of the jet 11310 0.879 8.79% 3973.44 Problem 1.1: A nozzle of size 10 cm diameter issues a jet of water with a velocity of 50 m/s. The jet strikes a moving plate perpendicularly at centre. The plate is moving with a velocity of 15 m/s in the direction of the jet. Calculate, (i) the force exerted on the plate (ii) the workdone (iii) efficiency of the jet. (June 011 - C15640 - MGU) Given: d 10 cm 0.1 m; V 50 m/s; u 15 m/s (i) Force exerted by the jet on the plate F x A 4 0.5 7.853 10 3 m F x A V u 1000 7.853 10 3 50 15 961.13 N F x 961.13 N (ii) The workdone per second by jet = Power of jet F x u 961.13 15 144317 Watts 144.317 kw (iii) Efficiency of jet Output power in watts Input Kinetic Energy in watts Kinetic energy in watts 1 m V 1 A V V 1 A V3 1 1000 7.853 10 3 50 3 49081.5 watts 490.813 kw
144.317 0.94 9.4% 490.813 Problem 1.: A jet of water 5 cm in dia having velocity of 0 m/s strikes normally on a flat plate. Determine the thrust on the plate, if (i) The plate is at rest (ii) The plate is moving in the same direction of the jet with a velocity of 8 m/s. (Jun 008 - C48863 - CUSAT) Given: d 5 cm 0.05 m; V 0 m/s; A 4 0.05 1.963 10 3 m (i) Thrust F on plate if the plate is at rest, F A V 1000 1.963 10 3 0 785.4 N (ii) Thrust F x on plate if plate velocity u 8 m/s F x A V u 1000 1.963 10 3 0 8 8.74 N 1.4.3 Impact of the jet on moving curved plate in the direction of jet Assume jet is striking at the centre of moving curved plate Relative velocity of jet with respect to curved plate V u (V-u) Exit velocity (V-u) sin It is assumed that the plate is smooth and loss of energy due to impact of jet is zero. Hence Velocity of jet leaving curved vane V u Inlet velocity (V-u) cos Inlet Velocity V This exit velocity is resolved into components, one in the direction of jet and other perpendicular to direction of jet. Fig. 1.8 Jet of Water Impact of Jets 1.19 Moving curved vane
1.0 Fluid Machinery - www.airwalkpublications.com V x in direction of jet V u cos [ sign indicates that it moves in the opposite direction of jet]. V y in perpendicular to direction of jet V u sin m A V u and force exerted in direction of jet F x F x [Initial velocity Final velocity] A V u V u [ V u cos A V u V u V u cos A V u [1 cos ] Workdone/sec on the plate F x u A V u u 1 cos Problem 1.3: A jet of water of diameter 10 cm strikes a curved plate at its centre with a velocity of 5 m/s. The curved plate is moving with a velocity of 8 m/s in the direction of the jet. The jet is deflected through an angle of 160. Assuming the plate is smooth, find: (i) Force exerted on the plate in the direction of jet, (ii) Power of the jet, and (iii) Efficiency of the jet. [Similar type of Feb 01 and Feb 013 of CUSAT] Given: Diameter of the jet, d 10 cm 0.1 m Solution. Area, A 4 0.1 7.854 10 3 Velocity of the jet, V 5 m/s Velocity of the plate, u 8 m/s Angle of deflection of the jet, 160 Angle made by the relative velocity at the outlet of the plate, 180 160 0
Impact of Jets 1.1 (i) Force exerted by the jet on the plane in the direction of the jet F x F x A V u 1 cos 1000 7.854 10 3 5 8 1 cos 0 440.7 N (ii) Work done by the jet on the plate per second, Power of the jet F x u 440.7 8 351.8 watts (iii) Efficiency of the jet Output Input Power 1 A V V 0.5740 57.4% 35. kw Work done by jet/sec Kinetic energy of jet/sec 351.8 1 1000 7.854 10 3 5 3 351.8 61,359 Problem 1.4: A jet of water of diameter 6 cm strikes a curved vane at its centre. The curved vane is moving with a velocity of 10 m/s in the direction of jet. If the velocity of the jet is m/s and it is deflected through an angle of 160, determine (i) Force exerted on the vane in the direction of jet (ii) Power of jet (iii) Efficiency of the jet. (Apr 01 - CUSAT) Given: d 6 cm 0.06 cm; u 10 m/s; V m/s; 180 160 0 (i) Force exerted on the vane in the direction of jet F x F x A V u 1 cos F x 1000.83 10 3 10 1 cos 0 F x 789.75 N A 4 0.06.83 10 3 m (ii) Power of jet F x u 789.75 10 7897.5 watts 7.8975 kw (iii) Efficiency of the jet Output power Input kinetic energy/sec
1. Fluid Machinery - www.airwalkpublications.com 7897.5 7897.5 1 1 A V V 1000.83 10 3 0.54 5.4% Problem 1.5 A jet of water is deflected through 50 from its original direction in a fixed curved plate which it enters tangentially without shock with a velocity of 40 m/s and leaves with a mean velocity of 35 m/s. If the discharge from the nozzle is 0.8 kg/s, calculate the magnitude and direction of the resultant force on the vane, if the vane is stationary. Solution Given: Inlet velocity, V 1 40 m/s Outlet velocity, V 35 m/s F y F x 35 sin 50 35 m /s 50 o 35 cos 50 Mass per second, m 0.8 kg/s Force in the direction of jet, F x m V 1x V x 40 m/sec 50 o Orig inal Direction of Jet where, V 1x Initial velocity in the direction of x 40 m/s V x Final velocity in the direction of x 35 cos 50.5 m/s F x 0.8140.51 14 N Similarly, Force normal to the jet, F y m V 1y V y 0.8 [0 35 sin 50] 1.5 N ve sign means that F y is acting downward. Resultant force on the vane F x Fy 14 1.5 5.66 N The angle made by the resultant with x - axis tan F y F x 1.5 14.0 1.536
Impact of Jets 1.3 56.93 ve sign means the angle is in the clockwise direction with x - axis as shown in Fig. Problem 1.6: A jet of water 60 mm dia strikes a curved vane at its centre with a velocity of 18 m/s. The curved vane is moving with a velocity of 6 m/s in the direction of the jet. The jet is deflected through an angle of 165. Assuming the plate to be smooth, find: (i) Thrust on the plate in the direction of the jet (ii) Power of the jet (iii) Efficiency of the jet Given: d 60 m 0.06 m; V 18 m/s; u 6 m/s; 180 165 15 Solution (i) Thrust on the plate F x F x A V u 1 cos [A 4 0.06.83 10 7 m ] 1000.83 10 3 18 6 1 cos 15 800.43 N (ii) Power of jet F x u 800.43 6 480.6 watts (iii) of the jet Output power Input kinetic Energy of jet/sec 480.6 480.6 1 1 A V V 1000.83 10 3 18 3 0.5819 58.19% Problem 1.7: A stationary vane having an inlet angle of zero degree and an outlet angle of 5 receives water at a velocity of 50 m/s. Determine the components of force acting on it in the direction of the jet velocity and normal to it. Also find the resultant force in magnitude and direction per kg of flow. (b) If the vane stated above is moving with a velocity of 0 m/s in the direction of the jet, calculate the force components in the direction of the vane velocity and across it, also the resultant force in magnitude and
1.4 Fluid Machinery - www.airwalkpublications.com direction. Calculate the work done and power developed per kg of flow. [Feb 01 - CUSAT] Given: Velocity of jet, V 50 m/s ; Angle at outlet, 5 ; Mass flow rate m A V 1 kg/s Case (i) For the stationary vane, the force in the direction of jet is given as F x m V 1x V x V 1x 50 m/s V x 50 cos 5 45.315 m/s Force in the direction of jet per kg of flow 5 o 50 cos 5 o y F y 50 m /sec V = 50 m/sec F R F x 50 sin 5 o Fixed Stationary vane x 1 [50 45.315] 95.315 N Force exerted by jet in the direction perpendicular to the direction of the jet per kg of flow Case F y m V 1y V y 1 [0 50 sin 5] 1.13 N [ - sign indicates F y in downward direction] Resultant force F x Fy 95.35 1.13 97.66 N Angle made by the resultant with the x - axis, tan F y F x 1.13 95.35 0.16 1.49 Velocity of the vane = 0 m/s. If the vane is moving in the direction of the jet, then force exerted by the jet on the plate in the direction of jet, F x m V 1x V x Initial velocity of the striking water V 1 x
Impact of Jets 1.5 V u 50 0 30 m/s (V -u) (V-u) sin 5 o Final velocity in the direction of x V x V u cos 5 30 cos 5 7.189 m/s F x 1 30 7.189 57.189 N Force exerted by the jet in the direction perpendicular to direction of jet, per kg of flow F y m V 1y V y where V 1y 0; V y V u sin 5 50 0 sin 5 30 sin 5 F y 1 0 30 sin 5 1.68 N Resultant force 57.189 1.68 58.57 N The angle made by the resultant with x - axis, tan 1.68 57.69 0.17 Work done per second 1.5 Power developed F x u 57.189 0 1144 watts 1.144 kw 1.4.4 Force exerted by a jet of water on moving curved vane, when it strikes tangentially at one end. Let V 1 Velocity of the jet at inlet. u 1 Velocity of the vane at inlet. V = 50m/s (V-u) 5 o (V-u) cos 5 o V r1 Relative velocity between jet and vane at inlet Angle between the direction of the jet and direction of motion of the vane (or) guide blade angle. y F y F x u = 0m/s x
1.6 Fluid Machinery - www.airwalkpublications.com u V w G F H Vr V V f E u B V 1 V r1 V f1 JET A C D u 1 V w1 Fig: 1.9 Jet striking on a moving curved vane at one of the tips Angle made by the inlet relative velocity V r1 with the direction of motion of jet (or) vane angle at inlet. V w1 and V f1 The components of the velocity of the jet V 1, in the direction of motion and perpendicular to the direction of motion of the vane respectively. V w1 is also known as velocity of whirl at inlet. V f1 is also known as velocity of flow at inlet. V Velocity of the jet, leaving the vane or velocity of jet at outlet of the vane. u Velocity of the vane at outlet. V r Relative velocity of the jet with respect to the vane at outlet.
Impact of Jets 1.7 Angle made by the velocity V with the direction of motion of the vane at outlet. Angle made by the relative velocity V r with the direction of motion of the vane at outlet and also called vane angle at outlet. V w and V f Components of the velocity V, in the direction of motion of vane and perpendicular to the direction of motion of vane respectively at outlet. V w is also called the velocity of whirl at outlet. V f is also called velocity of flow at outlet. The triangles ABD and EGH are called the velocity triangles at inlet and outlet. 1. Velocity Triangle at Inlet Take any point A and draw a line AB V 1 at an angle with the horizontal line AD. Next draw a line AC u. Join C and B. Then CB represents V f1. If the loss of energy at inlet due to impact is zero, then CB must be in the tangential direction to the vane at inlet. Now resolve V 1 into V f1 and V w1 Then BD Velocity of flow at inlet V f1 AD Velocity of whirl at inlet V w1 BCD Vane angle at inlet.. Velocity Triangle at Outlet Assuming the vane surface is very smooth and hence loss of energy due to friction will be zero. The water will be gliding over the surface of the vane with V r V r1. And also the V r is in tangential direction to the vane is at outlet.
1.8 Fluid Machinery - www.airwalkpublications.com Draw EG in the tangential direction of the vane at outlet and cut EG V r. From G, draw a line GF u in the direction of vane. Join EF which represents V. Now resolve V into V f and V w components. EH Velocity of flow at outlet V f FH Velocity of whirl at outlet V w Angle of vane at outlet. Angle made by V with the direction of motion of vane at outlet. Since vane is smooth and is having velocity in the direction of motion at inlet and outlet equal, then u 1 u u Velocity of vane in the direction of motion and V r1 V r Mass of water striking vane per sec m A V...(i) A Area of jet of water, Force exerted by the jet in the direction of motion F x Mass of water striking per sec [Initial velocity with which jet strikes in the direction of motion of vane Final velocity of jet in the direction of motion of vane]...(ii) When the jet strikes tangentially, the loss of energy due to impact of jet will be zero. Since the vane is moving, the relative velocity should be taken for striking velocity. m A V r1 F x m V w1 u 1 V w u m V w1 u 1 V w u Here u 1 u m V w1 V w General equation of F x m V w1 V w [Because sometimes V w will be as same direction of V w1 ;
Impact of Jets 1.9 sign when V w is in opposite direction of V w1 and sign when V w is as same direction of V w1 ] Workdone per sec F x u m V w1 V w u watts of jet Output power Input kinetic Energy/sec Problem 1.8: A jet of water having a velocity of 5 m/s strikes a curved vane which is moving with velocity of 10 m/s. The vane is symmetrical and is so shaped that the jet is deflected through 10. Find the angle of the jet at inlet of the vane so that there is no shock. What is the absolute velocity of the jet at outlet in magnitude and direction and the work done per sec per kg of water. Assume the vane to be smooth. Solution E u V w F G V f V r V * H u=5 m/sec Angle of Deflection 10 o o (30 - ) B A V 1 C V r1 V f1 D u 1 V w1
1.30 Fluid Machinery - www.airwalkpublications.com Given: Velocity of jet, V 1 5 m/s ; Velocity of vane, u 1 10 m/s As vane is symmetrical. Hence angle Angle of deflection of the jet 10 180 Let the angle of jet at inlet Absolute velocity of jet at outlet V 60 or each angle, 30 Angle made by V at outlet with direction of motion of vane. u 1 u u 10 m/s V r1 V r as vane is smooth Applying the sine rule to ACB, AB sin 180 AC sin 30 or V 1 sin 180 30 u 1 sin 30 u 1 sin 30 5 sin 30 10 10 sin 30 or sin 30 0. sin 30 5 30 11.31 or 30 11.31 18.69 Also Applying sine rule to ACB, we have AB sin 180 30 CB sin or V 1 sin 30 V r1 sin 18.69 V r1 16.0 m/s V r V r1 16.0 m/s From velocity HEG at outlet, V r cos u V w or 16.0 cos 30 10 V w V w 3.88 m/s Also, we have V r sin V f or V f 16.0 sin 30 8.01 m/s
Impact of Jets 1.31 In HFG, V V f Vw 8.01 3.88 8.9 m/s tan V f V w 8.01 3.88.06 64.15 Work done per sec per kg of the water striking m V w1 V w u (... ve sign taken as is an acute angle) 1 V 1 cos 3.88 10... Vw1 V 1 cos 5 [5 cos 18.69 3.88] 137.8 Watts 1.5 THRUST OF JET OF WATER ON SERIES OF VANES It is not practicably possible to strike by a jet on a single plate (flat or curved) continuously. Hence, the plates in the form of vanes are fixed on the circumference of a wheel and now the jet is striking on these vanes to Plates (Vanes) Wheel Jet of Water V u Fig: 1.10 (a) Jet Striking a Series of Vanes
1.3 Fluid Machinery - www.airwalkpublications.com O y F u V r G V V w H V f 130 o R x E R 1 Tangent at E Wheel B V 1 V r1 V f1 Tangent at B A u 1 C D get power continuously. With the power of jet, the wheel starts moving at a constant speed as shown in Fig. 1.5.1 Workdone per second (or) Power of jet on a series of a radial curved vanes Power of jet can be derived in the same way as previously explained. Here Fig: 1.10 (b) Series of Radial Curved Vanes Mounted on a Wheel R 1 Radius of wheel at inlet of vane R Radius of wheel at outlet of vane Angular velocity of the wheel
Impact of Jets 1.33 Vane velocity u 1 at inlet R 1 Vane velocity u at outlet R From the velocity triangles, the force exerted by jet in the direction of motion of the vane F x F x m [Initial velocity component in x direction] F x A V 1 V w1 V w [... m A V 1 ] sign, if is acute angle 90 ; sign if is obtuse angle 90 [Final velocity component in x direction] And V w 0, if 90 ie if the outlet velocity is radial. Now, workdone/sec (or) power of jet F x u Here u 1 u A V 1 V w1 u 1 V w u 1.5. Efficiency of the Radial curved vane Power of jet in watts Kinetic energy of jet in watts A V 1 V w1 u 1 V w u 1 m V 1 A V 1 V w1 u 1 V w u 1 A V 1 V 1 V w1 u 1 V w u V 1 m V w1 u 1 V w u Problem 1.9 A jet of water having a velocity of 30 m/s impinges on a series of vanes with a velocity of 15 m/s. The jet makes an angle of 30 to the direction of motion of vanes when entering and leaves at an angle of
1.34 Fluid Machinery - www.airwalkpublications.com 10. Draw the velocity triangle at inlet and outlet and find: (a) The angles of vanes tips so that water enters and leaves without shock,(b) The work done per sec per kg of water entering the vanes, and (c) The efficiency. (Apr 015 MGU) ; June 009 - Calicut University Solution Given: Velocity of jet, V 1 30 m/s ; Velocity of vane, u 1 u 15 m/s Angle of jet at inlet, 30 Angle made by the jet at outlet with the direction of motion of vanes 10 Angle 180 10 60 (a) Angle of vanes tips. From inlet velocity triangle V w1 V 1 cos 30cos 30 5.98 m/s V f1 V 1 sin 30sin 30 15 m/s Outlet Velocity Triangle 10 o tan V f1 15 V w1 u 1 5.98 15 1.37 53.8 V f1 15 V w1 u 1 5.98 15 1.37 By sine rule, Inlet Velocity Triangle V r1 sin 90 V f1 sin or V r1 1 15 sin 53.8 Now, V r1 18.59 m/s V r V r1 18.59 m/s From outlet velocity triangle, by sine rule V r sin 10 u 18.59 or sin60 0.886 15 sin 60
Impact of Jets 1.35 15 0.866 sin 60 0.6988 18.59 60 44.33 60 44.33 15.67 (b) Work done per sec per kg of water entering 1 V w1 V w u 1 V w1 5.98 m/s and u 1 15 m/s The value of V w is obtained from outlet velocity triangle V w V r cos u 18.59 cos 15.67 15.9 m/s Power of jet per kg 1 [5.98.9] 15 346. Watts (c) Efficiency Output power per kg Energy supplied per kg 346. 1 346. m 1 V 1 1 30 0.7694 76.94% Problem 1.30: A jet of water with a velocity of 35 m/s strikes a series of radial curved vanes of a wheel rotating at 50 r.p.m. The jet makes an angle of 0 with the tangent to the wheel at inlet and leaves the wheel with a velocity of 10 m/s at an angle of 130 to the tangent to the wheel at outlet. Water is flowing from outward in a radial direction. The outer and inner radii of the wheel are 0.5 m and 0.5 m respectively. Determine: (i) Vane angles at inlet and outlet, (ii) Work done per sec per kg of water, and (iii) Efficiency of the wheel. Solution Given: Velocity of jet, V 1 35 m/s ; Speed of wheel, N 50 r.p.m. Angular speed, Angle of jet at inlet, N 60 0 10 60 1.99 rad/s Velocity of jet at outlet, V 10 m/s
1.36 Fluid Machinery - www.airwalkpublications.com Angle made by the jet at outlet with the tangent to wheel 130 Angle, 180 130 50 Inlet radius, R 1 0.5 m Outlet radius, R 0.5 m Velocity u 1 R 1 1.99 0.5 11 m/s u R 1.99 0.5 5.5 m/s O F u V r G V V w H V f 130 o R Tan gent to W heel at E E R 1 V 1 V r1 B V f1 Tan gent to Wheel at B A u 1 C D (i) Vane angles at inlet and outlet (angle and ) respectively. From ABD, V w1 V 1 cos 35 cos 0 3.89 m/s V f1 V 1 sin 35 sin 0 11.97 m/s
Impact of Jets 1.37 In CBD, tan V f1 11.97 V w1 u 1 3.89 11 0.547 8.7 From outlet velocity, V w V cos 10 cos 50 6.43 m/s In EFH, tan 3.7 V f V sin 10 sin 50 7.66 m/s V f 7.66 u V w 5.5 6.43 0.64 (ii) Work done per second per kg of water (iii) Efficiency, m V w1 u 1 V w u ( ve sign is taken since 50 90) 1 [3.89 11 6.43 5.5] 397. Watts V w1 u 1 V w u V 1 [3.89 11 6.43 5.5] 35 0.6484 64.84% Problem 1.31: A jet of water having a velocity of 45 m/s impinges without shock on a series of vanes moving at 15 m/s. The direction of motion of the vanes is inclined at 0 to that of the jet, the relative velocity at outlet is 0.9 of that at inlet; and absolute velocity of water at exit is to be normal to the motion of vanes. Find (i) Vane angles at inlet and outlet (ii) Work done on vanes per newton of water supplied by the jet and (iii) Hydraulic efficiency (Apr - 014 - Calicut University C61576) V 1 45 m/s ; u 15 m/s ; 0 ; V r 0.9 V r1 V w1 V 1 cos 0 45 cos 0 4.9 m/s V f1 V 1 sin 0 45 sin 0 15.4 m/s Vane angle at inlet V f1 tan V w1 u 1 15.4 4.9 15 tan 0.5639 9.4 Vane angle at inlet