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FLUID MECHANICS AND MACHINERY FORMULA BOOK

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ASHOK KUMAR RAJENDRAN
ASHOK KUMAR RAJENDRAN

CE6451 - FLUID MECHANICS AND MACHINERY FORMULA BOOK

Engineering
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R.M.K COLLEGE OF ENGINEERING AND TECHNOLOGY RSM NAGAR, PUDUVOYAL-601206 DEPARTMENT OF MECHANICAL ENGINEERING CE6451 – FLUID MECHANICS & MACHINERY III SEM MECHANICAL ENGINEERING Regulation 2013 FORMULA BOOK PREPARED BY C.BIBIN / R.ASHOK KUMAR / N.SADASIVAN
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 2 PROPERTIES OF FLUID: MASS DESNITY (ρ): 𝜌 = π‘š 𝑉 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ m Mass Kg V Volume m3 SPECIFIC VOLUME (v): 𝑣 = 𝑉 π‘š = 1 𝜌 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ m Mass Kg V Volume m3 𝑣 Specific Volume π‘š3 π‘˜π‘”β„ SPECIFIC WEIGTH or WEIGTH DENSITY (w): 𝑀 = π‘Š 𝑉 = π‘šπ‘” 𝑉 = πœŒπ‘” 𝑆𝑖𝑛𝑐𝑒 π‘Š = π‘šπ‘” π‘Žπ‘›π‘‘ 𝜌 = π‘š 𝑉⁄ Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ m Mass Kg V Volume m3 UNIT – I – FLUID PROPERTIES AND FLOW CHARACTERISTICS
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 3 𝑀 Specific Weight 𝑁 π‘š3⁄ g Acceleration due to gravity π‘š 𝑠2⁄ SPECIFIC GRAVITY (S): 𝑆 = 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 π‘Šπ‘’π‘–π‘”β„Žπ‘‘ π‘œπ‘“ 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑒𝑖𝑑 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 π‘Šπ‘’π‘–π‘”β„Žπ‘‘ π‘œπ‘“ π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ 𝑓𝑙𝑒𝑖𝑑 𝑆 = π‘€π‘Žπ‘ π‘  𝐷𝑒𝑛𝑠𝑖𝑑𝑦 π‘œπ‘“ 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑒𝑖𝑑 π‘€π‘Žπ‘ π‘  𝐷𝑒𝑛𝑠𝑖𝑑𝑦 π‘œπ‘“ π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ 𝑓𝑙𝑒𝑖𝑑 Symbol Description Unit 𝑆 Specific Gravity No unit 𝜌 Density or Mass Density π‘˜π‘” π‘š3⁄ 𝑀 Specific Weight 𝑁 π‘š3⁄ 𝑀 π‘€π‘Žπ‘‘π‘’π‘Ÿ Specific Weight of Standard Fluid (Water) = 9.81 𝑁 π‘š3⁄ 𝜌 π‘€π‘Žπ‘‘π‘’π‘Ÿ Mass Density of Standard Fluid (Water) = 1000 π‘˜π‘” π‘š3⁄ VISCOSITY (ΞΌ): 𝜏 𝛼 𝑑𝑒 𝑑𝑦 𝜏 = πœ‡ 𝑑𝑒 𝑑𝑦 Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝑑𝑒 Change in Velocity π‘š 𝑠⁄ 𝑑𝑦 Change in Distance π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 4 DYNAMIC VISCOSITY (ΞΌ): πœ‡ = 𝜏 𝑑𝑒 𝑑𝑦⁄ Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ πœ‡ Dynamic Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝑑𝑒 Change in Velocity π‘š 𝑠⁄ 𝑑𝑦 Change in Distance π‘š 𝑑𝑒 𝑑𝑦⁄ Rate of Shear Strain 1 𝑠⁄ Unit Conversion: 1 𝑁𝑠 π‘š2 = 10 π‘π‘œπ‘–π‘ π‘’ 1 πΆπ‘’π‘›π‘‘π‘–π‘π‘œπ‘–π‘ π‘’ = 1 100 π‘π‘œπ‘–π‘ π‘’ 1 π‘π‘œπ‘–π‘ π‘’ = 0.1 𝑁𝑠 π‘š2 KINEMATIC VISCOSITY (Ξ³): 𝛾 = πœ‡ 𝜌 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ πœ‡ Dynamic Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ Ξ³ Kinematic Viscosity π‘š2 𝑠⁄ Unit Conversion: 1 π‘ π‘‘π‘œπ‘˜π‘’ = 10βˆ’4 π‘š2 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 5 1 πΆπ‘’π‘›π‘‘π‘–π‘ π‘‘π‘œπ‘˜π‘’ = 1 100 π‘ π‘‘π‘œπ‘˜π‘’ VISCOSITY PROBLEMS FOR PLATE TYPE: FORCE (F): 𝜏 = 𝐹 𝐴 Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ F Force N A Area of the plate π‘š2 POWER (P): 𝑃 = 𝐹 βˆ— 𝑑𝑒 Symbol Description Unit 𝑃 Power π‘Š F Force N 𝑑𝑒 Change in Velocity π‘š 𝑠⁄ VISCOSITY PROBLEMS FOR SHAFT TYPE: VELOCITY OF SHAFT (u): 𝑒 = πœ‹π·π‘ 60 Symbol Description Unit 𝐷 Diameter of Shaft π‘š N Speed of Shaft Rpm 𝑒 Velocity π‘š 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 6 FORCE (F): 𝜏 = 𝐹 𝐴 𝜏 = πœ‹π·πΏ Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ F Force N A Circumference of Shaft π‘š2 𝐷 Diameter of Shaft π‘š 𝐿 Length of Shaft π‘š TORQUE ON SHAFT (T): 𝑇 = 𝐹 βˆ— 𝐷 2 Symbol Description Unit 𝑇 Torque 𝑁 βˆ’ π‘š F Force N 𝐷 Diameter of Shaft π‘š POWER ON SHAFT (P): 𝑃 = 2πœ‹π‘π‘‡ 60 Symbol Description Unit 𝑃 Power π‘Š 𝑇 Torque 𝑁 βˆ’ π‘š N Speed of Shaft Rpm
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 7 VISCOSITY PROBLEMS FOR CONICAL BEARING: ANGULAR VELOCITY (Ο‰): πœ” = 2πœ‹π‘ 60 Symbol Description Unit πœ” Angular Velocity π‘Ÿπ‘Žπ‘‘ 𝑠𝑒𝑐⁄ N Speed of Shaft Rpm ANGLE (ΞΈ): π‘‘π‘Žπ‘›πœƒ = π‘Ÿ1 βˆ’ π‘Ÿ2 𝐻 Symbol Description Unit π‘Ÿ1 Outer Radius π‘š π‘Ÿ2 Inner Radius π‘š 𝐻 Height π‘š POWER (P): 𝑃 = 2πœ‹π‘π‘‡ 60 Symbol Description Unit 𝑃 Power π‘Š 𝑇 Torque 𝑁 βˆ’ π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 8 N Speed of Shaft Rpm THICKNESS OF OIL (h): 𝑇 = πœ‹πœ‡πœ” 2β„Žπ‘ π‘–π‘›πœƒ ( π‘Ÿ1 4 βˆ’ π‘Ÿ2 4) Symbol Description Unit πœ‡ Dynamic Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝑇 Torque 𝑁 βˆ’ π‘š πœ” Angular Velocity π‘Ÿπ‘Žπ‘‘ 𝑠𝑒𝑐⁄ β„Ž Thickness of Oil π‘š π‘Ÿ1 Outer Radius π‘š π‘Ÿ2 Inner Radius π‘š CAPILLARITY: HEIGHT OF LIQUID IN TUBE (h): β„Ž = 4πœŽπ‘π‘œπ‘ πœƒ πœŒπ‘”π‘‘ Symbol Description Unit β„Ž Height of Liquid in Tube π‘š 𝜎 Surface Tension 𝑁 π‘šβ„ πœƒ Angle of Contact between Liquid and Tube π‘Ÿπ‘Žπ‘‘ 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝑑 Diameter of Tube π‘š SURFACE TENSION: PRESSURE IN LIQUID DROPLET (P): 𝑃 = 4𝜎 𝑑
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 9 Symbol Description Unit 𝑃 Pressure 𝑁 π‘š2⁄ 𝜎 Surface Tension 𝑁 π‘šβ„ 𝑑 Diameter of Droplet π‘š PRESSURE IN BUBBLE (P): 𝑃 = 8𝜎 𝑑 Symbol Description Unit 𝑃 Pressure 𝑁 π‘š2⁄ 𝜎 Surface Tension 𝑁 π‘šβ„ 𝑑 Diameter of Bubble π‘š PRESSURE IN LIQUID JET (P): 𝑃 = 2𝜎 𝑑 Symbol Description Unit 𝑃 Pressure 𝑁 π‘š2⁄ 𝜎 Surface Tension 𝑁 π‘šβ„ 𝑑 Diameter of Jet π‘š CONTINUITY EQUATION: πœ•π‘’ πœ•π‘₯ + πœ•π‘£ πœ•π‘¦ + πœ•π‘€ πœ•π‘§ = 0 [ πΉπ‘œπ‘Ÿ 3 βˆ’ 𝐷 π‘“π‘™π‘œπ‘€] πœ•π‘’ πœ•π‘₯ + πœ•π‘£ πœ•π‘¦ + = 0 [ πΉπ‘œπ‘Ÿ 2 βˆ’ 𝐷 π‘“π‘™π‘œπ‘€] πœ• πœ•π‘Ÿ ( π‘Ÿπ‘’ π‘Ÿ) + πœ• πœ•πœƒ ( 𝑒 πœƒ) = 0[ πΉπ‘œπ‘Ÿ π‘π‘œπ‘™π‘Žπ‘Ÿ π‘π‘œπ‘œπ‘Ÿπ‘‘π‘–π‘›π‘Žπ‘‘π‘’π‘ ]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 10 BERNOULLI’S EQUATION: πœ•π‘ƒ 𝜌 + 𝑣. 𝑑𝑣 + 𝑔. 𝑑𝑧 = 0 𝑃1 πœŒπ‘” + 𝑣1 2 2𝑔 + 𝑧1 = 𝑃2 πœŒπ‘” + 𝑣2 2 2𝑔 + 𝑧2 + β„Ž 𝑓 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁 π‘š2⁄ 𝑣1 & 𝑣2 Velocity at Section 1 & 2 π‘š 𝑠⁄ 𝑧1 & 𝑧2 Datum Head at Section 1 & 2 π‘š β„Ž 𝑓 Head Loss π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ COEFFICIENT OF DISCHARGE: 𝐢 𝑑 = 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ COEFFICIENT OF VELOCITY: 𝐢𝑣 = 𝑣 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑣 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ DISCHARGE OF VENTURIMETER AND ORIFICEMETER: 𝑄 = 𝐢 𝑑 π‘Ž1 π‘Ž2 √( π‘Ž1 2 βˆ’ π‘Ž1 2) √2π‘”β„Ž Symbol Description Unit π‘Ž1 & π‘Ž2 Area at Section 1 & 2 π‘š2 β„Ž Pressure Difference between Section 1 & 2 ( 𝑃1βˆ’ 𝑃2 πœŒπ‘” ) π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 11 𝐢 𝑑 Coefficient of Discharge π‘₯ Difference in Mercury Level π‘š β„Ž = π‘₯ (1 βˆ’ 𝑆 π‘š 𝑆 ) [ π‘€β„Žπ‘’π‘› 𝑆 > 𝑆 π‘š] β„Ž = π‘₯ ( 𝑆 π‘š 𝑆 βˆ’ 1) [ π‘€β„Žπ‘’π‘› 𝑆 π‘š > 𝑆] β„Ž = ( 𝑃1 πœŒπ‘” + 𝑍1) βˆ’ ( 𝑃2 πœŒπ‘” + 𝑍2) [ 𝐼𝑛𝑐𝑙𝑖𝑛𝑒𝑑 π‘‰π‘’π‘›π‘‘π‘’π‘Ÿπ‘–π‘šπ‘’π‘‘π‘’π‘Ÿ] MOMENTUM EQUATION: 𝐹 = 𝑑 (π‘šπ‘£) 𝑑𝑑 FORCE ACTING IN X – DIRECTION: 𝐹π‘₯ = πœŒπ‘„ ( 𝑣1 βˆ’ 𝑣2 π‘π‘œπ‘ πœƒ) + 𝑃1 𝐴1 βˆ’ 𝑃2 𝐴2 π‘π‘œπ‘ πœƒ FORCE ACTING IN Y – DIRECTION: 𝐹𝑦 = πœŒπ‘„ (βˆ’ 𝑣2 π‘ π‘–π‘›πœƒ) βˆ’ 𝑃2 𝐴2 π‘ π‘–π‘›πœƒ Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁 π‘š2⁄ 𝑣1 & 𝑣2 Velocity at Section 1 & 2 π‘š 𝑠⁄ 𝐴1 & 𝐴2 Area at Section 1 & 2 π‘š πœƒ Angle of the Bend π·π‘’π‘”π‘Ÿπ‘’π‘’ 𝑄 Discharge π‘š3 𝑠⁄ RESULTANT FORCE: 𝐹𝑅 = √𝐹π‘₯ 2 + 𝐹𝑦 2
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 12 ANGLE MADE BY RESULTANT FORCE: π‘‘π‘Žπ‘›πœƒ = 𝐹𝑦 𝐹π‘₯ MOMENT OF MOMENTUM EQUATION: 𝑇 = πœŒπ‘„ ( 𝑣2 π‘Ÿ2 βˆ’ 𝑣1 π‘Ÿ1) Symbol Description Unit 𝑇 Torque 𝑁 βˆ’ π‘š 𝑣1 & 𝑣2 Velocity at Section 1 & 2 π‘š 𝑠⁄ π‘Ÿ1 & π‘Ÿ2 Radius of Curvature at Section 1 & 2 π‘š 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 13 TOTAL ENERGY LINE (TEL): 𝑇𝐸𝐿 = π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ + 𝐾𝑖𝑛𝑒𝑑𝑖𝑐 π»π‘’π‘Žπ‘‘ + π·π‘Žπ‘‘π‘’π‘š π»π‘’π‘Žπ‘‘ 𝑇𝐸𝐿 = 𝑃 πœŒπ‘” + 𝑣2 2𝑔 + 𝑍 Symbol Description Unit 𝑃 Pressure 𝑁 π‘š2⁄ 𝑣 Velocity π‘š 𝑠⁄ 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑍 Datum Head π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ HYDRAULIC ENERGY LINE (HEL): 𝐻𝐸𝐿 = π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ + π·π‘Žπ‘‘π‘’π‘š π»π‘’π‘Žπ‘‘ 𝑇𝐸𝐿 = 𝑃 πœŒπ‘” + 𝑍 HAGEN POISEUILLE’S EQUATION: SHEAR STRESS: 𝜏 = βˆ’ πœ•π‘ πœ•π‘₯ βˆ— π‘Ÿ 2 Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ π‘Ÿ Radius of pipe π‘š UNIT – II – FLOW THROUGH CIRCULAR CONDUITS
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 14 VELOCITY: 𝑒 = βˆ’ 1 4πœ‡ βˆ— πœ•π‘ πœ•π‘₯ βˆ— (𝑅2 βˆ’ π‘Ÿ2 ) Symbol Description Unit 𝑒 Velocity of Fluid in Pipe π‘š 𝑠⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ π‘Ÿ Radius of pipe π‘š πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ MAXIMUM VELOCITY: 𝑒 = βˆ’ 1 4πœ‡ βˆ— πœ•π‘ πœ•π‘₯ βˆ— (𝑅2 ) Symbol Description Unit 𝑒 Velocity of Fluid in Pipe π‘š 𝑠⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ AVERAGE VELOCITY: 𝑒̅ = βˆ’ 1 4πœ‡ βˆ— πœ•π‘ πœ•π‘₯ βˆ— (𝑅2 ) Symbol Description Unit 𝑒̅ Average Velocity of Fluid in Pipe π‘š 𝑠⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ RATIO BETWEEN MAXIMUM VELOCITY AND AVERAGE VELOCITY: 𝑒 π‘šπ‘Žπ‘₯ 𝑒̅ = 2
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 15 DISCHARGE: 𝑒 = βˆ’ 1 8πœ‡ βˆ— πœ•π‘ πœ•π‘₯ βˆ— πœ‹ βˆ— 𝑅4 Symbol Description Unit 𝑒 Velocity of Fluid in Pipe π‘š 𝑠⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ PRESSURE DIFFERENCE: 𝑃1 βˆ’ 𝑃2 = 32πœ‡π‘’Μ…πΏ 𝐷2 Symbol Description Unit 𝑒̅ Average Velocity of Fluid in Pipe π‘š 𝑠⁄ 𝐿 Length of Pipe π‘š πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝐷 Diameter of Pipe π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS OF HEAD: β„Ž 𝑓 = 𝑃1 βˆ’ 𝑃2 πœŒπ‘” = 32πœ‡π‘’Μ…πΏ πœŒπ‘”π·2 [ π‘“π‘œπ‘Ÿ πΏπ‘Žπ‘šπ‘–π‘›π‘Žπ‘Ÿ π‘“π‘™π‘œπ‘€] DARCY WEISBACH EQUATION: β„Ž 𝑓 = 𝑃1 βˆ’ 𝑃2 πœŒπ‘” = 4𝑓𝐿𝑣2 2𝑔𝑑 [ π‘“π‘œπ‘Ÿ π‘‡π‘’π‘Ÿπ‘π‘’π‘™π‘’π‘›π‘‘ π‘“π‘™π‘œπ‘€] Symbol Description Unit 𝑣 Velocity of Fluid in Pipe π‘š 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 16 𝐿 Length of Pipe π‘š 𝑓 Friction Factor 𝑑 Diameter of Pipe π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ REYNOLD’S NUMBER: 𝑅 𝑒 = πœŒπ‘£π‘‘ πœ‡ 𝑓 = 0.079 𝑅 𝑒 0.25 [ πΉπ‘œπ‘Ÿ π‘‡π‘’π‘Ÿπ‘π‘’π‘™π‘’π‘›π‘‘ πΉπ‘™π‘œπ‘€] 𝑓 = 16 𝑅 𝑒 [ πΉπ‘œπ‘Ÿ πΏπ‘Žπ‘šπ‘–π‘›π‘Žπ‘Ÿ πΉπ‘™π‘œπ‘€] 𝑅 𝑒 < 2000 π‘‡β„Žπ‘’π‘› π‘‘β„Žπ‘’ πΉπ‘™π‘œπ‘€ 𝑖𝑠 πΏπ‘Žπ‘šπ‘–π‘›π‘Žπ‘Ÿ 𝑅 𝑒 > 2000 π‘‡β„Žπ‘’π‘› π‘‘β„Žπ‘’ πΉπ‘™π‘œπ‘€ 𝑖𝑠 π‘‡π‘’π‘Ÿπ‘π‘’π‘™π‘’π‘›π‘‘ Symbol Description Unit 𝑣 Velocity of Fluid in Pipe π‘š 𝑠⁄ 𝑑 Diameter of Pipe π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ MAJOR LOSS IN PIPES: β„Ž 𝑓 = 32πœ‡π‘’Μ…πΏ πœŒπ‘”π‘‘2 [ π‘“π‘œπ‘Ÿ πΏπ‘Žπ‘šπ‘–π‘›π‘Žπ‘Ÿ π‘“π‘™π‘œπ‘€] β„Ž 𝑓 = 4𝑓𝐿𝑣2 2𝑔𝑑 [ π‘“π‘œπ‘Ÿ π‘‡π‘’π‘Ÿπ‘π‘’π‘™π‘’π‘›π‘‘ π‘“π‘™π‘œπ‘€]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 17 Symbol Description Unit 𝑒̅ & 𝑣 Velocity of Fluid in Pipe π‘š 𝑠⁄ 𝑑 Diameter of Pipe π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝑙 Length of Pipe π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝑓 Friction Factor MINOR LOSS IN PIPES: LOSS DUE TO SUDDEN ENLARGEMENT: β„Ž 𝑒 = ( 𝑣1 βˆ’ 𝑣2)2 2𝑔 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Fluid in Pipe at Inlet and Outlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS DUE TO SUDDEN CONTRACTION: β„Ž 𝑐 = 𝐾𝑣2 2𝑔 𝐾 = ( 1 𝐢𝑐 βˆ’ 1) 2 β„Ž 𝑐 = 0.5𝑣2 2𝑔 [ 𝐼𝑓 𝐢𝑐 π‘›π‘œπ‘‘ 𝑔𝑖𝑣𝑒𝑛] Symbol Description Unit 𝑣 Velocity of Fluid at Outlet π‘š 𝑠⁄ 𝐢𝑐 Coefficient of Contraction
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 18 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS AT ENTRANCE OF PIPE: β„Žπ‘– = 0.5𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Fluid at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS AT EXIT OF PIPE: β„Ž π‘œ = 𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Fluid at Outlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS DUE TO GRADUAL CONTRACTION: β„Ž 𝑒 = 𝐾( 𝑣1 βˆ’ 𝑣2)2 2𝑔 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Fluid in Pipe at Inlet and Outlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝐾 Coefficient of Contraction LOSS AT BEND OF PIPE: β„Ž 𝑏 = 𝐾𝑣2 2𝑔
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 19 Symbol Description Unit 𝑣 Velocity of Flow π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝐾 Coefficient of Bend LOSS AT DUE TO VARIOUS FITTINGS: β„Ž 𝑣 = 𝐾𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Flow π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝐾 Coefficient of Fittings LOSS AT DUE TO OBSTRUCTION: β„Ž 𝑣 = 𝑣2 2𝑔 ( 𝐴 𝐢𝑐 ( 𝐴 βˆ’ π‘Ž) βˆ’ 1) 𝐢𝑐 = 𝐴 𝑐 ( 𝐴 βˆ’ π‘Ž) Symbol Description Unit 𝑣 Velocity of Flow π‘š 𝑠⁄ 𝐴 Area of Pipe π‘š2 π‘Ž Area of Obstruction π‘š2 𝐴 𝑐 Area of Vena Contraction π‘š2 WHEN PIPES ARE CONNECTED IN SERIES: DISCHARGE: 𝑄 = 𝑄1 = 𝑄2 𝑄 = 𝐴1 𝑣1 = 𝐴2 𝑣2
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 20 HEAD LOSS: β„Ž 𝑓 = β„Ž 𝑓1 + β„Ž 𝑓2 β„Ž 𝑓 = 4𝑓𝑙1 𝑣1 2 2𝑔𝑑1 + 4𝑓𝑙2 𝑣2 2 2𝑔𝑑2 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Flow at Pipe 1 & 2 π‘š 𝑠⁄ 𝐴1& 𝐴2 Area of Pipe 1 & 2 π‘š2 𝑑1& 𝑑2 Diameter of Pipe 1 & 2 π‘š 𝑙1& 𝑙2 Length of Pipe 1 & 2 π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝑓 Friction Factor WHEN PIPES ARE CONNECTED IN PARALLEL: DISCHARGE: 𝑄 = 𝑄1 + 𝑄2 𝑄 = 𝐴1 𝑣1 + 𝐴2 𝑣2 HEAD LOSS: β„Ž 𝑓 = β„Ž 𝑓1 = β„Ž 𝑓2 β„Ž 𝑓 = 4𝑓𝑙1 𝑣1 2 2𝑔𝑑1 = 4𝑓𝑙2 𝑣2 2 2𝑔𝑑2 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Flow at Pipe 1 & 2 π‘š 𝑠⁄ 𝐴1& 𝐴2 Area of Pipe 1 & 2 π‘š2 𝑑1& 𝑑2 Diameter of Pipe 1 & 2 π‘š 𝑙1& 𝑙2 Length of Pipe 1 & 2 π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 21 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝑓 Friction Factor EQUIVALENT PIPE: 𝐿 𝐷5 = 𝐿1 𝐷1 5 + 𝐿2 𝐷2 5 + 𝐿3 𝐷3 5 + β‹― + 𝐿 𝑛 𝐷 𝑛 5 Symbol Description Unit 𝐷 Diameter of Pipe π‘š 𝐿 Length of Pipe π‘š BOUNDARY LAYER: DISPLACEMENT THICKNESS: π›Ώβˆ— = ∫ (1 βˆ’ 𝑒 π‘ˆ ) 𝛿 0 𝑑𝑦 MOMENTUM THICKNESS: πœƒ = ∫ 𝑒 π‘ˆ (1 βˆ’ 𝑒 π‘ˆ ) 𝛿 0 𝑑𝑦 MOMENTUM THICKNESS: π›Ώβˆ—βˆ— = ∫ 𝑒 π‘ˆ (1 βˆ’ 𝑒2 π‘ˆ2 ) 𝛿 0 𝑑𝑦 Symbol Description Unit 𝑒 π‘ˆ Velocity Distribution 𝛿 Boundary layer thickness SHEAR STRESS: 𝜏0 πœŒπ‘ˆ2 = πœ•πœƒ πœ•π‘₯
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 22 πœƒ = ∫ 𝑒 π‘ˆ (1 βˆ’ 𝑒 π‘ˆ ) 𝛿 0 𝑑𝑦 DRAG FORCE: 𝐹 𝐷 = ∫ π‘†β„Žπ‘’π‘Žπ‘Ÿ π‘†π‘‘π‘Ÿπ‘’π‘ π‘  βˆ— π΄π‘Ÿπ‘’π‘Ž 𝐿 0 𝐹 𝐷 = ∫ 𝜏0 βˆ— 𝑏 βˆ— 𝑑π‘₯ 𝐿 0 LOCAL COEFFICIENT OF DRAG: 𝐢 𝐷 βˆ— = 𝜏0 1 2 πœŒπ‘ˆ2 AVERAGE COEFFICIENT OF DRAG: 𝐢 𝐷 = 𝐹 𝐷 1 2 πœŒπ΄π‘ˆ2 Symbol Description Unit 𝜏0 Shear Stress 𝑁 π‘š2⁄ 𝑏 Width of Plate π‘š π‘ˆ Free Stream Velocity π‘š 𝑠⁄ 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝐴 Area π‘š2 𝐹 𝐷 Drag Force 𝑁 BLASIUS’S SOLUTION: BOUNDARY LAYER THICKNESS: 𝛿 = 4.91π‘₯ √ 𝑅 𝑒π‘₯
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 23 LOCAL COEFFICIENT OF DRAG: 𝐢 𝐷 βˆ— = 0.664 √ 𝑅 𝑒π‘₯ AVERAGE COEFFICIENT OF DRAG: 𝐢 𝐷 = 1.328 √ 𝑅 𝑒𝐿 Symbol Description Unit 𝑅 𝑒π‘₯ Reynold’s Number at distance x 𝑅 𝑒𝐿 Reynold’s Number at distance L
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 24 UNITS: Physical Quantity Symbol Unit Dimensions Length L m L Mass M Kg M Time T Sec T Area A m2 L2 Volume V m3 L3 Diameter D m L Head H m L Roughness k M L Velocity v m/s LT-1 Angular Velocity Ο‰ rad/sec T-1 Acceleration a m/s2 LT-2 Angular Acceleration Ξ± rad/sec2 T-2 Speed N Rpm T-1 Discharge Q m3 /s L3 T-1 Kinematic Viscosity Ξ³ cm2 /s L2 T-1 Dynamic Viscosity ΞΌ N-s/m2 ML-1 T-1 Force F N MLT-2 Weight W N MLT-2 Thrust T N MLT-2 Density ρ Kg/ m3 ML-3 Pressure P N/m2 ML-1 T-2 UNIT – III – DIMENSIONAL ANALYSIS
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 25 Physical Quantity Symbol Unit Dimensions Specific Weight w N/m3 ML-2 T-2 Young’s Modulus E N/m2 ML-1 T-2 Bulk Modulus K N/m2 ML-1 T-2 Shear Stress Ο„ N/m2 ML-1 T-2 Surface Tension Οƒ N/m MT-2 Energy / Work W/E J = N-m ML2 T-2 Torque T N-m ML-2 T-2 Power P W=J/s ML-2 T-3 Momentum M Kg m/s MLT-1 Efficiency Ξ· No Unit Dimensionless SIMILARITY: GEOMETRIC SIMILARITY: 𝐿 𝑝 𝐿 π‘š = 𝑏 𝑝 𝑏 π‘š = 𝐷 𝑝 𝐷 π‘š = 𝐿 π‘Ÿ 𝐴 𝑝 𝐴 π‘š = 𝐿 𝑝 𝐿 π‘š βˆ— 𝑏 𝑝 𝑏 π‘š = 𝐿 π‘Ÿ βˆ— 𝐿 π‘Ÿ = 𝐿 π‘Ÿ 2 𝑉𝑝 π‘‰π‘š = 𝐿 𝑝 𝐿 π‘š βˆ— 𝑏 𝑝 𝑏 π‘š βˆ— 𝑑 𝑝 𝑑 π‘š = 𝐿 π‘Ÿ βˆ— 𝐿 π‘Ÿ βˆ— 𝐿 π‘Ÿ = 𝐿 π‘Ÿ 3 Symbol Description Unit 𝐿 𝑝&𝐿 π‘š Length of Prototype & Model π‘š 𝑏 𝑝&𝑏 π‘š Breadth of Prototype & Model π‘š 𝐷 𝑝&𝐷 π‘š Diameter of Prototype & Model π‘š 𝑑 𝑝&𝑑 π‘š Thickness of Prototype & Model π‘š 𝐴 𝑝&𝐴 π‘š Area of Prototype & Model π‘š2
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 26 𝑉𝑝&π‘‰π‘š Volume of Prototype & Model π‘š3 𝐿 π‘Ÿ Length Ratio KINEMATIC SIMILARITY: 𝑣 𝑝 𝑣 π‘š = π‘£π‘Ÿ π‘Ž 𝑝 π‘Ž π‘š = π‘Ž π‘Ÿ Symbol Description Unit 𝑣 𝑝&𝑣 π‘š Velocity of Prototype & Model π‘š 𝑠⁄ π‘Ž 𝑝&π‘Ž π‘š Acceleration of Prototype & Model π‘š 𝑠2⁄ π‘£π‘Ÿ Velocity Ratio π‘Ž π‘Ÿ Acceleration Ratio DYNAMIC SIMILARITY: ( 𝐹𝑖) 𝑝 ( 𝐹𝑖) π‘š = ( 𝐹𝑣) 𝑝 ( 𝐹𝑣) π‘š = (𝐹𝑔) 𝑝 (𝐹𝑔) π‘š = πΉπ‘Ÿ Symbol Description Unit ( 𝐹𝑖) 𝑝& ( 𝐹𝑖) π‘š Inertia Force of Prototype & Model 𝑁 ( 𝐹𝑣) 𝑝& ( 𝐹𝑣) π‘š Viscous Force of Prototype & Model 𝑁 (𝐹𝑔) 𝑝 & (𝐹𝑔) π‘š Gravity Force of Prototype & Model 𝑁 πΉπ‘Ÿ Force Ratio DIMENSIONLESS NUMBER: REYNOLD’S NUMBER: 𝑅 𝑒 = πœŒπ‘£π· πœ‡ (π‘œπ‘Ÿ) πœŒπ‘£πΏ πœ‡
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 27 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑣 Velocity π‘š 𝑠⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝐷 Diameter π‘š 𝐿 Length π‘š FROUDE’S NUMBER: 𝐹𝑒 = 𝑣 √ 𝐿𝑔 Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐿 Length π‘š FROUDE’S NUMBER: 𝐹𝑒 = 𝑣 √ 𝐿𝑔 Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐿 Length π‘š EULER’S NUMBER: 𝐸 𝑒 = 𝑣 √ 𝑝 πœŒβ„ Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 28 𝑝 Pressure 𝑁 π‘š2⁄ WEBER’S NUMBER: π‘Šπ‘’ = 𝑣 √ 𝜎 πœŒπΏβ„ Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝐿 Length π‘š 𝜎 Surface Tension 𝑁 π‘šβ„ MACH’S NUMBER: π‘Šπ‘’ = 𝑣 √ 𝐾 πœŒβ„ Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝐾 Elastic Stress 𝑁 π‘š2⁄ REYNOLD’S MODEL LAW: TIME RATIO: πΉπ‘Ÿ = π‘š π‘Ÿ π‘Ž π‘Ÿ πΉπ‘Ÿ = π‘š π‘Ÿ π‘£π‘Ÿ π‘‡π‘Ÿ DISCHARGE RATIO: 𝑄 π‘Ÿ = πœŒπ‘Ÿ 𝐿 π‘Ÿ 2 π‘£π‘Ÿ
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 29 Symbol Description Unit πΉπ‘Ÿ Force Ratio π‘š π‘Ÿ Mass Ratio π‘£π‘Ÿ Velocity Ratio π‘‡π‘Ÿ Time Ratio 𝐿 π‘Ÿ Length Ratio πœŒπ‘Ÿ Density Ratio FROUDE’S MODEL LAW: TIME RATIO: π‘‡π‘Ÿ = √ 𝐿 π‘Ÿ ACCELERATION RATIO: π‘Ž π‘Ÿ = 1 DISCHARGE RATIO: 𝑄 π‘Ÿ = ( 𝐿 π‘Ÿ) 5 2⁄ FORCE RATIO: πΉπ‘Ÿ = ( 𝐿 π‘Ÿ)3 PRESSURE RATIO: πΉπ‘Ÿ = 𝐿 π‘Ÿ ENERGY RATIO: πΈπ‘Ÿ = ( 𝐿 π‘Ÿ)4 MOMENTUM RATIO: π‘€π‘Ÿ = ( 𝐿 π‘Ÿ)3 βˆ— √ 𝐿 π‘Ÿ TORQUE RATIO: π‘‡π‘Ÿ = ( 𝐿 π‘Ÿ)4
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 30 POWER RATIO: π‘ƒπ‘Ÿ = ( 𝐿 π‘Ÿ) 7 2⁄ Symbol Description Unit 𝐿 π‘Ÿ Length Ratio DISTORTED MODELS: ( 𝐿 π‘Ÿ) 𝐻 = πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π»π‘œπ‘Ÿπ‘–π‘§π‘œπ‘›π‘‘π‘Žπ‘™ π·π‘–π‘šπ‘’π‘›π‘ π‘–π‘œπ‘›π‘  π‘œπ‘“ π‘ƒπ‘Ÿπ‘œπ‘‘π‘œπ‘‘π‘¦π‘π‘’ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π»π‘œπ‘Ÿπ‘–π‘§π‘œπ‘›π‘‘π‘Žπ‘™ π·π‘–π‘šπ‘’π‘›π‘ π‘–π‘œπ‘›π‘  π‘œπ‘“ π‘€π‘œπ‘‘π‘’π‘™ ( 𝐿 π‘Ÿ) 𝐻 = 𝐿 𝑝 𝐿 π‘š = 𝐡𝑝 𝐡 π‘š ( 𝐿 π‘Ÿ) 𝑉 = πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π·π‘–π‘šπ‘’π‘›π‘ π‘–π‘œπ‘›π‘  π‘œπ‘“ π‘ƒπ‘Ÿπ‘œπ‘‘π‘œπ‘‘π‘¦π‘π‘’ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π·π‘–π‘šπ‘’π‘›π‘ π‘–π‘œπ‘›π‘  π‘œπ‘“ π‘€π‘œπ‘‘π‘’π‘™ ( 𝐿 π‘Ÿ) 𝑉 = β„Ž 𝑝 β„Ž π‘š VELOCITY RATIO: π‘£π‘Ÿ = √(𝐿 π‘Ÿ) 𝑉 AREA RATIO: 𝐴 π‘Ÿ = ( 𝐿 π‘Ÿ) 𝐻 βˆ— ( 𝐿 π‘Ÿ) 𝑉 DISCAHRGE RATIO: 𝑄 π‘Ÿ = ( 𝐿 π‘Ÿ) 𝐻 βˆ— [( 𝐿 π‘Ÿ) 𝑉] 3 2⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 31 CENTRIFUGAL PUMP: VELOCITY TRIANGLE DIAGRAM: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Impeller at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣1&𝑣2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree UNIT – IV – PUMPS
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 32 πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝛽 Angle made by Absolute Velocity at Outlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π‘‘1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) Internal Diameter of Impeller π‘š 𝑁 Speed of Impeller π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π‘‘2 𝑁 60 Symbol Description Unit 𝑑2 Oulet (or) External Diameter of Impeller π‘š 𝑁 Speed of Impeller π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: π‘‘π‘Žπ‘›πœƒ = 𝑣𝑓1 𝑒1 Symbol Description Unit 𝑒1 Tangential Velocity of Impeller at Inlet π‘š 𝑠⁄ 𝑣1 Absolute Velocity at Inlet π‘š 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 33 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree ∡ 𝛼 = 90Β° 𝑣1 = 𝑣𝑓1 FROM OUTLET VELOCITY TRIANGLE DIAGRAM: π‘‘π‘Žπ‘›πœ™ = 𝑣𝑓2 𝑒2 βˆ’ 𝑣 𝑀2 𝑣2 = βˆšπ‘£π‘“2 2 + 𝑣 𝑀2 2 π‘‘π‘Žπ‘›π›½ = 𝑣𝑓2 𝑣 𝑀2 Symbol Description Unit 𝑒2 Tangential Velocity of Impeller at Outlet π‘š 𝑠⁄ 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑣2 Absolute Velocity at Outlet π‘š 𝑠⁄ 𝑣𝑓2 Flow Velocity at Outlet π‘š 𝑠⁄ πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree 𝛽 Angle made by Absolute Velocity at Outlet with the Degree
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 34 Direction of Motion of Vane DISCHARGE: 𝑄 = πœ‹π‘‘1 𝑏1 𝑣𝑓1 = πœ‹π‘‘2 𝑏2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝑄 Discharge π‘š3 𝑠⁄ WORK DONE BY AN IMPELLER PER SECOND: π‘Š = πœŒπ‘”π‘„ 𝑔 𝑣 𝑀2 𝑒2 Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ WORK DONE BY AN IMPELLER PER UNIT WEIGHT OF WATER: π‘Š = 𝑣 𝑀2 𝑒2 𝑔 Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 35 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ MANOMETRIC EFFICIENCY: πœ‚ π‘š = 𝑔𝐻 𝑣 𝑀2 𝑒2 Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ 𝐻 Manometric Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ POWER REQUIRED BY THE PUMP: 𝑃 = πœŒπ‘„π‘£ 𝑀2 𝑒2 Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑃 Power π‘˜π‘Š MINIMUM SPEED TO START THE PUMP: 𝑁 π‘šπ‘–π‘› = 120 βˆ— πœ‚ π‘š βˆ— 𝑣 𝑀2 βˆ— 𝑑2 πœ‹ (𝑑2 2 βˆ’ 𝑑1 2 ) Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š πœ‚ π‘š Manometric Efficiency
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 36 OVERALL EFFICIENCY: πœ‚ π‘œ = πΌπ‘šπ‘π‘’π‘™π‘™π‘’π‘Ÿ π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ = πœŒπ‘”π‘„π» 𝑆. 𝑃 πœ‚ π‘œ = πœ‚ π‘šπ‘Žπ‘›π‘œ βˆ— πœ‚ π‘šπ‘’π‘β„Ž βˆ— πœ‚ π‘£π‘œπ‘™ Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Manometric Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ MECHANICAL EFFICIENCY: πœ‚ π‘šπ‘’π‘β„Ž = πœŒπ‘”π‘„π» 𝑆. 𝑃 βˆ— 𝑣 𝑀2 𝑒2 𝑔𝐻 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Manometric Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝑆. 𝑃 Shaft Power π‘Š 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ POWER OF PUMP: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 37 𝐻 Manometric Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = π΄π‘π‘‘π‘’π‘Žπ‘™ 𝐿𝑖𝑓𝑑 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ 𝐿𝑖𝑓𝑑 = π΄π‘π‘‘π‘’π‘Žπ‘™ π»π‘’π‘Žπ‘‘ πΌπ‘‘π‘’π‘Žπ‘™ π»π‘’π‘Žπ‘‘ IDEAL HEAD: 𝑃𝐼 = πœŒπ‘”(𝑄 + π‘ž)𝐻𝑖 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ π‘ž Leakage of Water π‘š3 𝑠⁄ 𝐻𝑖 Ideal Head π‘š 𝑃𝐼 Power at Input π‘Š TORQUE EXERTED BY IMPELLER: 𝑇 = πœŒπ‘”π‘„ 𝑔 βˆ— 𝑣 𝑀2 βˆ— 𝑅2 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑅2 Radius of Impeller at Outlet π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 38 SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed SPEED RATIO: 𝐾 𝑒 = 𝑒2 √2𝑔𝐻 𝐾 𝑒 = 0.95 βˆ’ 1.25 Symbol Description Unit 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓2 √2𝑔𝐻 𝐾𝑓 = 0.1 βˆ’ 0.25
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 39 Symbol Description Unit 𝑣𝑓2 Flow Velocity at Outlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio RECIPROCATING PUMP: DISCHARGE: 𝑄 = 𝐴𝐿𝑁 60 𝐴 = πœ‹ 4 𝐷2 [ πΉπ‘œπ‘Ÿ 𝑆𝑖𝑛𝑔𝑙𝑒 𝐴𝑐𝑑𝑖𝑛𝑔 π‘ƒπ‘’π‘šπ‘] 𝐴 = [ πœ‹ 4 𝐷2 + πœ‹ 4 ( 𝐷2 βˆ’ 𝑑2)] [ πΉπ‘œπ‘Ÿ π·π‘œπ‘’π‘π‘™π‘’ 𝐴𝑐𝑑𝑖𝑛𝑔 π‘ƒπ‘’π‘šπ‘] Symbol Description Unit 𝐴 Area of Cylinder π‘š2 𝐿 Stroke Length π‘š 𝑁 Speed π‘Ÿπ‘π‘š 𝐷 Diameter of Cylinder or Bore π‘š 𝑑 Diameter of Piston Rod π‘š WEIGHT OF THE WATER DELIVERED PER SECOND: π‘Š = πœŒπ‘”π‘„ Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ π‘Š Weight of Water 𝑁 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 40 WORK DONE BY RECIPROCATING PUMP: π‘Š = πœŒπ‘”π‘„π» 𝐻 = β„Ž 𝑠 + β„Ž 𝑑 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š π‘Š Work Done π‘Š β„Ž 𝑠 Suction Head π‘š β„Ž 𝑑 Delivery Head π‘š POWER DEVELOPED BY RECIPROCATING PUMP: 𝑃 = πœŒπ‘”π‘„ π΄π‘π‘‘π‘’π‘Žπ‘™ 𝐻 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š POWER REQUIRED TO DRIVE THE PUMP: 𝑃 = πœŒπ‘”π‘„ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ 𝐻 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 41 SLIP OF RECIPROCATING PUMP: 𝑆 = 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ βˆ’ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Symbol Description Unit 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ COEFFICENT OF DISCHARGE: 𝐢 𝑑 = 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Symbol Description Unit 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ PERCENTAGE OF SLIP IN RECIPROCATING PUMP: % π‘œπ‘“ 𝑆𝑙𝑖𝑝 = 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ βˆ’ 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ % π‘œπ‘“ 𝑆𝑙𝑖𝑝 = 1 βˆ’ 𝐢 𝑑 Symbol Description Unit 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ 𝐢 𝑑 Coefficient of Discharge VOLUMETRIC EFFICIENCY: πœ‚ π‘‰π‘œπ‘™ = 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ = 𝐢 𝑑 Symbol Description Unit 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ 𝐢 𝑑 Coefficient of Discharge
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 42 MECHANICAL EFFICIENCY: πœ‚ π‘šπ‘’π‘β„Ž = π‘ƒπ‘œπ‘€π‘’π‘Ÿ π·π‘’π‘£π‘’π‘™π‘œπ‘π‘’π‘‘ 𝑏𝑦 π‘ƒπ‘’π‘šπ‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘…π‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘ π‘‘π‘œ π·π‘Ÿπ‘–π‘£π‘’ π‘‘β„Žπ‘’ π‘ƒπ‘’π‘šπ‘ πœ‚ π‘šπ‘’π‘β„Ž = π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘œπ‘“ π‘ƒπ‘’π‘šπ‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘œπ‘“ π‘€π‘œπ‘‘π‘œπ‘Ÿ πœ‚ π‘šπ‘’π‘β„Ž = πœŒπ‘”π‘„ π΄π‘π‘‘π‘’π‘Žπ‘™ 𝐻 πœŒπ‘”π‘„ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ 𝐻 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š ACCELERATION HEAD: β„Ž π‘Žπ‘  = 𝑙 𝑠 𝑔 βˆ— 𝐴 π‘Ž 𝑠 βˆ— πœ”2 βˆ— π‘Ÿ βˆ— π‘π‘œπ‘ πœƒ [ 𝐴𝑑 π‘†π‘’π‘π‘‘π‘–π‘œπ‘› π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] β„Ž 𝑑𝑠 = 𝑙 𝑑 𝑔 βˆ— 𝐴 π‘Ž 𝑑 βˆ— πœ”2 βˆ— π‘Ÿ βˆ— π‘π‘œπ‘ πœƒ [ 𝐴𝑑 π·π‘’π‘™π‘–π‘£π‘’π‘Ÿπ‘¦ π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] 𝐴 = πœ‹ 4 𝐷2 π‘Ž 𝑠 = πœ‹ 4 𝑑 𝑠 2 π‘Ž 𝑑 = πœ‹ 4 𝑑 𝑑 2 πœ” = 2πœ‹π‘ 60
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 43 π‘Ÿ = 𝐿 2 Symbol Description Unit 𝑙 𝑠 Length of Suction Pipe π‘š 𝑙 𝑑 Length of Delivery Pipe π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐴 Area of Cylinder π‘š2 π‘Ž 𝑠 Area of Suction Pipe π‘š2 π‘Ž 𝑑 Area of Delivery Pipe π‘š2 πœ” Angular Speed π‘Ÿπ‘Žπ‘‘ 𝑠⁄ π‘Ÿ Radius of Crank π‘š πœƒ Angle of Crank π‘‘π‘’π‘”π‘Ÿπ‘’π‘’ 𝐷 Diameter of Cylinder or Bore π‘š 𝑑 𝑠 Diameter of Suction Pipe π‘š 𝑑 𝑑 Diameter of Delivery Pipe π‘š 𝑁 Speed of Crank π‘Ÿπ‘π‘š 𝐿 Stroke Length π‘š PRESSURE HEAD: π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ = β„Ž 𝑠 + β„Ž π‘Žπ‘  [ πΉπ‘œπ‘Ÿ π‘†π‘’π‘π‘‘π‘–π‘œπ‘› π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ = β„Ž 𝑑 + β„Ž π‘Žπ‘‘ [ πΉπ‘œπ‘Ÿ π·π‘’π‘™π‘–π‘£π‘’π‘Ÿπ‘¦ π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] Symbol Description Unit β„Ž 𝑠 Suction Head π‘š β„Ž 𝑑 Delivery Head π‘š β„Ž π‘Žπ‘  Acceleration Head at Suction π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 44 β„Ž π‘Žπ‘‘ Acceleration Head at Delivery π‘š ABSOLUTE PRESSURE HEAD: π΄π‘π‘ π‘œπ‘™π‘’π‘‘π‘’ π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ = 𝐻 π‘Žπ‘‘π‘š βˆ’ (β„Ž 𝑠 + β„Ž π‘Žπ‘ ) [ πΉπ‘œπ‘Ÿ π‘†π‘’π‘π‘‘π‘–π‘œπ‘› π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] π΄π‘π‘ π‘œπ‘™π‘’π‘‘π‘’ π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ = 𝐻 π‘Žπ‘‘π‘š + (β„Ž 𝑑 + β„Ž π‘Žπ‘‘ ) [ πΉπ‘œπ‘Ÿ π·π‘’π‘™π‘–π‘£π‘’π‘Ÿπ‘¦ π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] Symbol Description Unit β„Ž 𝑠 Suction Head π‘š β„Ž 𝑑 Delivery Head π‘š β„Ž π‘Žπ‘  Acceleration Head at Suction π‘š β„Ž π‘Žπ‘‘ Acceleration Head at Delivery π‘š 𝐻 π‘Žπ‘‘π‘š Atmospheric Pressure Head π‘š SEPARATION HEAD: 𝑃𝑠𝑒𝑝 = πœŒπ‘”β„Ž 𝑆𝑒𝑝 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ β„Ž 𝑠𝑒𝑝 Separation Head π‘š 𝑃𝑠𝑒𝑝 Separation Pressure 𝑁 π‘š2⁄ HEAD LOSS WITHOUT AIR VESSEL: β„Ž π‘“π‘Šπ‘‚π΄ = 4𝑓𝑙 𝑑 𝑣2 2𝑔𝑑 𝑑 Symbol Description Unit 𝑓 Friction Factor 𝑙 𝑑 Length of Delivery Pipe π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 45 𝑣 Velocity without Air Vessel π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝑑 𝑑 Diameter of Delivery Pipe π‘š VELOCITY WITHOUT AIR VESSEL: 𝑣 = 𝐴 π‘Ž 𝑑 βˆ— πœ” βˆ— π‘Ÿ 𝐴 = πœ‹ 4 𝐷2 π‘Ž 𝑑 = πœ‹ 4 𝑑 𝑑 2 πœ” = 2πœ‹π‘ 60 π‘Ÿ = 𝐿 2 Symbol Description Unit 𝐴 Area of Cylinder π‘š2 π‘Ž 𝑑 Area of Delivery Pipe π‘š2 πœ” Angular Speed π‘Ÿπ‘Žπ‘‘ 𝑠⁄ π‘Ÿ Radius of Crank π‘š 𝐷 Diameter of Cylinder or Bore π‘š 𝑑 𝑑 Diameter of Delivery Pipe π‘š 𝑁 Speed of Crank π‘Ÿπ‘π‘š 𝐿 Stroke Length π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 46 HEAD LOSS WITH AIR VESSEL: β„Ž π‘“π‘Šπ΄ = 4𝑓𝑙 𝑑 𝑣2 2𝑔𝑑 𝑑 Symbol Description Unit 𝑓 Friction Factor 𝑙 𝑑 Length of Delivery Pipe π‘š 𝑣 Velocity with Air Vessel π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝑑 𝑑 Diameter of Delivery Pipe π‘š VELOCITY WITH AIR VESSEL: 𝑣 = 𝐴 π‘Ž 𝑑 βˆ— πœ” βˆ— π‘Ÿ πœ‹ 𝐴 = πœ‹ 4 𝐷2 π‘Ž 𝑑 = πœ‹ 4 𝑑 𝑑 2 πœ” = 2πœ‹π‘ 60 π‘Ÿ = 𝐿 2 Symbol Description Unit 𝐴 Area of Cylinder π‘š2 π‘Ž 𝑑 Area of Delivery Pipe π‘š2 πœ” Angular Speed π‘Ÿπ‘Žπ‘‘ 𝑠⁄ π‘Ÿ Radius of Crank π‘š 𝐷 Diameter of Cylinder or Bore π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 47 𝑑 𝑑 Diameter of Delivery Pipe π‘š 𝑁 Speed of Crank π‘Ÿπ‘π‘š 𝐿 Stroke Length π‘š POWER SAVED BY AIR VESSEL: 𝑃 = πœŒπ‘”π‘„ ( 2 3 β„Ž π‘“π‘Šπ‘‚π΄ βˆ’ β„Ž π‘“π‘Šπ΄) Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ β„Ž π‘“π‘Šπ‘‚π΄ Head Loss Without Air Vessel π‘š β„Ž π‘“π‘Šπ΄ Head Loss With Air Vessel π‘š POWER REQUIRED TO DRIVE THE PUMP: 𝑃 = πœŒπ‘”π‘„ (β„Ž 𝑠 + β„Ž 𝑑 + 2 3 β„Ž π‘“π‘ π‘Šπ‘‚π΄ + β„Ž π‘“π‘‘π‘Šπ΄) Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ β„Ž 𝑠 Suction Head π‘š β„Ž 𝑑 Delivery Head π‘š β„Ž π‘“π‘ π‘Šπ‘‚π΄ Head Loss Without Air Vessel at Suction π‘š β„Ž π‘“π‘‘π‘Šπ΄ Head Loss With Air Vessel at Delivery π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 48 PELTON WHEEL: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛽 Angle made by Absolute Velocity at Outlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree UNIT – V – TURBINES
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 49 TANGENTIAL VELOCITY AT INLET AND OUTLET (OR) VELOCITY OF WHEEL: 𝑒 = πœ‹π·π‘ 60 Symbol Description Unit 𝐷 Diameter of Runner π‘š 𝑁 Speed of Impeller π‘Ÿπ‘π‘š VELOCITY OF JET: 𝑉1 = πΆπ‘£βˆš2𝑔𝐻 𝐢𝑣 = 0.97 βˆ’ 0.99 Symbol Description Unit 𝐢𝑣 Coefficient of Velocity 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š VELOCITY OF WHEEL: 𝑒 = π‘˜ π‘’βˆš2𝑔𝐻 π‘˜ 𝑒 = 0.43 βˆ’ 0.45 Symbol Description Unit π‘˜ 𝑒 Speed Ratio 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: 𝑉 𝑀1 = 𝑉1 𝑉 𝑀1 = 𝑒1 + π‘‰π‘Ÿ1
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 50 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ FROM OUTLET VELOCITY TRIANGLE DIAGRAM: cos πœ™ = 𝑒2 + 𝑣 𝑀2 𝑣 π‘Ÿ2 tan πœ™ = 𝑣𝑓2 𝑒2 + 𝑣 𝑀2 sin πœ™ = 𝑣𝑓2 𝑣 π‘Ÿ2 tan 𝛽 = 𝑣𝑓2 𝑣 𝑀2 Symbol Description Unit 𝑒2 Tangential Velocity of Runner at Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ2 Relative Velocity at Outlet π‘š 𝑠⁄ 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑣𝑓2 Flow Velocity at Outlet π‘š 𝑠⁄ WORK DONE BY JET PER SECOND: π‘Š = πœŒπ‘„ [ 𝑣 𝑀1 + 𝑣 𝑀2] 𝑒 Symbol Description Unit 𝑒 Tangential Velocity of Runner π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 51 HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 2[ 𝑣 𝑀1 + 𝑣 𝑀2] 𝑒 𝑉1 2 Symbol Description Unit 𝑒 Tangential Velocity of Runner π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š DISCHARGE OF SINGLE JET: π‘ž = πœ‹ 4 βˆ— 𝑑2 βˆ— 𝑉1 Symbol Description Unit 𝑑 Diameter of Jet π‘š 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ π‘ž Discharge of Single Jet π‘š3 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 52 NUMBER OF JET: 𝑛 = 𝑄 π‘ž Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ π‘ž Discharge of Single Jet π‘š3 𝑠⁄ NUMBER OF BUCKET: 𝑍 = 15 + 𝐷 2𝑑 Symbol Description Unit 𝑑 Diameter of Jet π‘š 𝐷 Diameter of Runner π‘š DIMENSIONS OF BUCKET: 𝐴π‘₯π‘–π‘Žπ‘™ π‘Šπ‘–π‘‘π‘‘β„Ž 𝐡 = 4.5𝑑 π‘…π‘Žπ‘‘π‘–π‘Žπ‘™ πΏπ‘’π‘›π‘”π‘‘β„Ž 𝐿 = 2.5𝑑 π·π‘’π‘π‘‘β„Ž π‘œπ‘“ π΅π‘’π‘π‘˜π‘’π‘‘ 𝑇 = 𝑑 Symbol Description Unit 𝑑 Diameter of Jet π‘š KINETIC ENERGY OF JET: 𝐾. 𝐸 π‘œπ‘“ 𝐽𝑒𝑑 = 1 2 π‘š 𝑉1 2 𝑆𝑖𝑛𝑐𝑒 π‘š = πœŒπ΄π‘‰ π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ 𝐾. 𝐸 π‘œπ‘“ 𝐽𝑒𝑑 = 1 2 𝜌 βˆ— 𝐴 βˆ— 𝑉1 βˆ— 𝑉1 2
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 53 𝑆𝑖𝑛𝑐𝑒 𝑄 = 𝐴𝑉 π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ 𝐾. 𝐸 π‘œπ‘“ 𝐽𝑒𝑑 = 1 2 𝜌 βˆ— 𝑄 βˆ— 𝑉1 2 POWER LOST IN NOZZLE: 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ = 𝐾𝑖𝑛𝑒𝑑𝑖𝑐 πΈπ‘›π‘’π‘Ÿπ‘”π‘¦ + π‘ƒπ‘œπ‘€π‘’π‘Ÿ πΏπ‘œπ‘ π‘‘ 𝑖𝑛 π‘π‘œπ‘§π‘§π‘™π‘’ POWER LOST IN RUNNER: 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ = π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘œπ‘“ π‘†β„Žπ‘Žπ‘“π‘‘ + π‘ƒπ‘œπ‘€π‘’π‘Ÿ πΏπ‘œπ‘ π‘‘ 𝑖𝑛 π‘π‘œπ‘§π‘§π‘™π‘’ + π‘ƒπ‘œπ‘€π‘’π‘Ÿ πΏπ‘œπ‘ π‘‘ 𝑖𝑛 π‘…π‘’π‘›π‘›π‘’π‘Ÿ + π‘ƒπ‘œπ‘€π‘’π‘Ÿ πΏπ‘œπ‘ π‘‘ 𝐷𝑒𝑒 π‘‘π‘œ π‘€π‘’π‘β„Žπ‘Žπ‘›π‘–π‘π‘Žπ‘™ π‘…π‘’π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’ RESULTANT FORCE ON BUCKET: 𝐹 = πœŒπ‘„ [ 𝑣 𝑀1 + 𝑣 𝑀2] Symbol Description Unit 𝐹 Resultant Force on Bucket 𝑁 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ TORQUE: 𝑇 = 𝐹 βˆ— 𝐷 2 Symbol Description Unit 𝐹 Resultant Force on Bucket 𝑁 𝐷 Diameter of Runner π‘š 𝑇 Torque 𝑁 βˆ’ π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 54 POWER: 𝑃 = 2πœ‹π‘π‘‡ 60 Symbol Description Unit 𝑃 Power π‘Š 𝑇 Torque 𝑁 βˆ’ π‘š N Speed of Shaft Rpm SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 55 REACTION TURBINE: INWARD FLOW REACTION TURBINE: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Degree
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 56 Direction of Motion of Vane TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π‘‘1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) External Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π‘‘2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) Internal Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑀1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑀1 sin πœƒ = 𝑣𝑓1 𝑣 π‘Ÿ1 cos πœƒ = 𝑣 𝑀1 βˆ’ 𝑒1 𝑣 π‘Ÿ1
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 57 tan πœƒ = 𝑣𝑓1 𝑣 𝑀1 βˆ’ 𝑒1 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 π‘Ÿ1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑀1 βˆ’ 𝑒1)2 Symbol Description Unit 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ DISCHARGE: 𝑄 = πœ‹π‘‘1 𝑏1 𝑣𝑓1 = πœ‹π‘‘2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 58 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝑄 Discharge π‘š3 𝑠⁄ 𝐴 Area of Runner π‘š2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet π‘š 𝑠⁄ MASS OF WATER FLOWING THROUGH THE RUNNER: π‘š = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ HEAD AT INLET OF TURBINE: 𝐻 = 1 𝑔 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 + 𝑣𝑓1 2 2𝑔 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 59 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 βˆ— 𝑄 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 𝑣 𝑀1 𝑒1 𝑔𝐻 πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 βˆ’ π»π‘’π‘Žπ‘‘ πΏπ‘œπ‘ π‘  π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π»
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 60 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š SPEED RATIO: 𝐾 𝑒 = 𝑒 √2𝑔𝐻 𝐾 𝑒 = 0.6 βˆ’ 0.9 Symbol Description Unit 𝑒 Tangential Velocity π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 βˆ’ 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 61 SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed OUTWARD FLOW REACTION TURBINE: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 62 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π‘‘1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) Internal Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π‘‘2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) External Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM:
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 63 sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑀1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑀1 sin πœƒ = 𝑣𝑓1 𝑣 π‘Ÿ1 cos πœƒ = 𝑣 𝑀1 βˆ’ 𝑒1 𝑣 π‘Ÿ1 tan πœƒ = 𝑣𝑓1 𝑣 𝑀1 βˆ’ 𝑒1 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 64 RELATIVE VELOCITY AT INLET: 𝑣 π‘Ÿ1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑀1 βˆ’ 𝑒1)2 Symbol Description Unit 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ DISCHARGE: 𝑄 = πœ‹π‘‘1 𝑏1 𝑣𝑓1 = πœ‹π‘‘2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝑄 Discharge π‘š3 𝑠⁄ 𝐴 Area of Runner π‘š2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet π‘š 𝑠⁄ MASS OF WATER FLOWING THROUGH THE RUNNER: π‘š = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 65 INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 βˆ— 𝑄 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 𝑣 𝑀1 𝑒1 𝑔𝐻 πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 βˆ’ π»π‘’π‘Žπ‘‘ πΏπ‘œπ‘ π‘  π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 66 OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š SPEED RATIO: 𝐾 𝑒 = 𝑒 √2𝑔𝐻 𝐾 𝑒 = 0.6 βˆ’ 0.9 Symbol Description Unit 𝑒 Tangential Velocity π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 βˆ’ 0.3
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 67 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 68 FRANCIS TURBINE: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 69 TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π‘‘1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) External Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π‘‘2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) Internal Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑀1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑀1 sin πœƒ = 𝑣𝑓1 𝑣 π‘Ÿ1 cos πœƒ = 𝑣 𝑀1 βˆ’ 𝑒1 𝑣 π‘Ÿ1 tan πœƒ = 𝑣𝑓1 𝑣 𝑀1 βˆ’ 𝑒1
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 70 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 π‘Ÿ1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑀1 βˆ’ 𝑒1)2 Symbol Description Unit 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ DISCHARGE: 𝑄 = πœ‹π‘‘1 𝑏1 𝑣𝑓1 = πœ‹π‘‘2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 71 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝑄 Discharge π‘š3 𝑠⁄ 𝐴 Area of Runner π‘š2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet π‘š 𝑠⁄ CIRCUMFERENTIAL AREA OF RUNNER: 𝐴 = πœ‹π‘‘1 𝑏1 = πœ‹π‘‘2 𝑏2 Symbol Description Unit 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝐴 Circumferential Area of Runner π‘š2 MASS OF WATER FLOWING THROUGH THE RUNNER: π‘š = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 72 POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 βˆ— 𝑄 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 𝑣 𝑀1 𝑒1 𝑔𝐻 πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 βˆ’ π»π‘’π‘Žπ‘‘ πΏπ‘œπ‘ π‘  π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 73 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š SPEED RATIO: 𝐾 𝑒 = 𝑒 √2𝑔𝐻 𝐾 𝑒 = 0.6 βˆ’ 0.9 Symbol Description Unit 𝑒 Tangential Velocity π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 βˆ’ 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 74 BREADTH RATIO: 𝑛 = 𝑏1 𝑑1 𝑛 = 0.1 βˆ’ 0.4 Symbol Description Unit 𝑏1 Width of Runner at Inlet π‘š 𝑑1 Diameter of Runner at Inlet π‘š 𝑛 Breadth Ratio SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 75 KAPLAN TURBINE: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 76 TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π· π‘œ 𝑁 60 Symbol Description Unit 𝐷 π‘œ Inlet (or) External Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π· 𝑏 𝑁 60 = πœ‹π·β„Ž 𝑁 60 Symbol Description Unit 𝐷 𝑏 π‘œπ‘Ÿ π·β„Ž Outlet (or) Boss (or) Hub Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑀1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑀1 sin πœƒ = 𝑣𝑓1 𝑣 π‘Ÿ1 cos πœƒ = 𝑣 𝑀1 βˆ’ 𝑒1 𝑣 π‘Ÿ1 tan πœƒ = 𝑣𝑓1 𝑣 𝑀1 βˆ’ 𝑒1
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 77 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 π‘Ÿ1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑀1 βˆ’ 𝑒1)2 Symbol Description Unit 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ DISCHARGE: 𝑄 = πœ‹ 4 [𝐷0 2 βˆ’ 𝐷 𝑏 2 ]𝑣𝑓1 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐷0 Inlet (or) External Diameter π‘š 𝐷 𝑏 π‘œπ‘Ÿ π·β„Ž Outlet (or) Boss (or) Hub Diameter π‘š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 78 𝑄 Discharge π‘š3 𝑠⁄ CIRCUMFERENTIAL AREA OF RUNNER: 𝐴 = πœ‹ 4 [𝐷0 2 βˆ’ 𝐷 𝑏 2 ] Symbol Description Unit 𝐷0 Inlet (or) External Diameter π‘š 𝐷 𝑏 π‘œπ‘Ÿ π·β„Ž Outlet (or) Boss (or) Hub Diameter π‘š 𝐴 Circumferential Area of Runner π‘š2 MASS OF WATER FLOWING THROUGH THE RUNNER: π‘š = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 βˆ— 𝑄 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 79 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 𝑣 𝑀1 𝑒1 𝑔𝐻 πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 βˆ’ π»π‘’π‘Žπ‘‘ πΏπ‘œπ‘ π‘  π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 80 SPEED RATIO: 𝐾 𝑒 = 𝑒 √2𝑔𝐻 𝐾 𝑒 = 0.6 βˆ’ 0.9 Symbol Description Unit 𝑒 Tangential Velocity π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 βˆ’ 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 81 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed DRAFT TUBE:
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 82 Symbol Description Unit 𝑉1&𝑉2 Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝐻𝑠 Vertical Height of Draft Tube Above Tail Race π‘š 𝑦 Distance of Bottom of Draft Tube from Tail Race π‘š FROM BERNOULLI’S EQUATION: 𝑃1 πœŒπ‘” + 𝑉1 2 2𝑔 + 𝑧1 = 𝑃2 πœŒπ‘” + 𝑉2 2 2𝑔 + 𝑧2 + β„Ž 𝑓 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Inlet & Outlet of Draft Tube 𝑁 π‘š2⁄ 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube π‘š 𝑠⁄ 𝑧1 & 𝑧2 Datum Head Inlet & Outlet of Draft Tube π‘š β„Ž 𝑓 Head Loss π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LENGTH OF DRAFT TUBE: 𝐿 = 𝐻𝑠 + 𝑦 Symbol Description Unit 𝐿 Length of Draft Tube π‘š 𝐻𝑠 Vertical Height of Draft Tube Above Tail Race π‘š 𝑦 Distance of Bottom of Draft Tube from Tail Race π‘š EFFICIENCY OF DRAFT TUBE: πœ‚ 𝑑 = ( 𝑉1 2 2𝑔 βˆ’ 𝑉2 2 2𝑔 ) βˆ’ β„Ž 𝑓 𝑉1 2 2𝑔
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 83 Symbol Description Unit 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube π‘š 𝑠⁄ β„Ž 𝑓 Head Loss π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ HYDRAULIC EFFICIENCY OF DRAFT TUBE: πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ π‘ˆπ‘‘π‘–π‘™π‘–π‘§π‘’π‘‘ 𝑏𝑦 π‘‡π‘’π‘Ÿπ‘π‘–π‘›π‘’ π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 π‘œπ‘“ π‘‡π‘’π‘Ÿπ‘π‘–π‘›π‘’ πœ‚β„Žπ‘¦π‘‘ = 𝐻 βˆ’ β„Ž 𝑓𝑑 βˆ’ β„Ž 𝑓𝑑 βˆ’ 𝑉2 2 2𝑔 𝑃1 πœŒπ‘” + 𝑉1 2 2𝑔 + 𝑧1 Symbol Description Unit 𝑃1 Pressure at Inlet of Draft Tube 𝑁 π‘š2⁄ 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube π‘š 𝑠⁄ 𝑧1 Datum Head Inlet of Draft Tube π‘š β„Ž 𝑓 Head Loss π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄

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FLUID MECHANICS AND MACHINERY FORMULA BOOK

  • 1. R.M.K COLLEGE OF ENGINEERING AND TECHNOLOGY RSM NAGAR, PUDUVOYAL-601206 DEPARTMENT OF MECHANICAL ENGINEERING CE6451 – FLUID MECHANICS & MACHINERY III SEM MECHANICAL ENGINEERING Regulation 2013 FORMULA BOOK PREPARED BY C.BIBIN / R.ASHOK KUMAR / N.SADASIVAN
  • 2. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 2 PROPERTIES OF FLUID: MASS DESNITY (ρ): 𝜌 = π‘š 𝑉 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ m Mass Kg V Volume m3 SPECIFIC VOLUME (v): 𝑣 = 𝑉 π‘š = 1 𝜌 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ m Mass Kg V Volume m3 𝑣 Specific Volume π‘š3 π‘˜π‘”β„ SPECIFIC WEIGTH or WEIGTH DENSITY (w): 𝑀 = π‘Š 𝑉 = π‘šπ‘” 𝑉 = πœŒπ‘” 𝑆𝑖𝑛𝑐𝑒 π‘Š = π‘šπ‘” π‘Žπ‘›π‘‘ 𝜌 = π‘š 𝑉⁄ Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ m Mass Kg V Volume m3 UNIT – I – FLUID PROPERTIES AND FLOW CHARACTERISTICS
  • 3. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 3 𝑀 Specific Weight 𝑁 π‘š3⁄ g Acceleration due to gravity π‘š 𝑠2⁄ SPECIFIC GRAVITY (S): 𝑆 = 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 π‘Šπ‘’π‘–π‘”β„Žπ‘‘ π‘œπ‘“ 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑒𝑖𝑑 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 π‘Šπ‘’π‘–π‘”β„Žπ‘‘ π‘œπ‘“ π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ 𝑓𝑙𝑒𝑖𝑑 𝑆 = π‘€π‘Žπ‘ π‘  𝐷𝑒𝑛𝑠𝑖𝑑𝑦 π‘œπ‘“ 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑒𝑖𝑑 π‘€π‘Žπ‘ π‘  𝐷𝑒𝑛𝑠𝑖𝑑𝑦 π‘œπ‘“ π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ 𝑓𝑙𝑒𝑖𝑑 Symbol Description Unit 𝑆 Specific Gravity No unit 𝜌 Density or Mass Density π‘˜π‘” π‘š3⁄ 𝑀 Specific Weight 𝑁 π‘š3⁄ 𝑀 π‘€π‘Žπ‘‘π‘’π‘Ÿ Specific Weight of Standard Fluid (Water) = 9.81 𝑁 π‘š3⁄ 𝜌 π‘€π‘Žπ‘‘π‘’π‘Ÿ Mass Density of Standard Fluid (Water) = 1000 π‘˜π‘” π‘š3⁄ VISCOSITY (ΞΌ): 𝜏 𝛼 𝑑𝑒 𝑑𝑦 𝜏 = πœ‡ 𝑑𝑒 𝑑𝑦 Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝑑𝑒 Change in Velocity π‘š 𝑠⁄ 𝑑𝑦 Change in Distance π‘š
  • 4. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 4 DYNAMIC VISCOSITY (ΞΌ): πœ‡ = 𝜏 𝑑𝑒 𝑑𝑦⁄ Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ πœ‡ Dynamic Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝑑𝑒 Change in Velocity π‘š 𝑠⁄ 𝑑𝑦 Change in Distance π‘š 𝑑𝑒 𝑑𝑦⁄ Rate of Shear Strain 1 𝑠⁄ Unit Conversion: 1 𝑁𝑠 π‘š2 = 10 π‘π‘œπ‘–π‘ π‘’ 1 πΆπ‘’π‘›π‘‘π‘–π‘π‘œπ‘–π‘ π‘’ = 1 100 π‘π‘œπ‘–π‘ π‘’ 1 π‘π‘œπ‘–π‘ π‘’ = 0.1 𝑁𝑠 π‘š2 KINEMATIC VISCOSITY (Ξ³): 𝛾 = πœ‡ 𝜌 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ πœ‡ Dynamic Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ Ξ³ Kinematic Viscosity π‘š2 𝑠⁄ Unit Conversion: 1 π‘ π‘‘π‘œπ‘˜π‘’ = 10βˆ’4 π‘š2 𝑠⁄
  • 5. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 5 1 πΆπ‘’π‘›π‘‘π‘–π‘ π‘‘π‘œπ‘˜π‘’ = 1 100 π‘ π‘‘π‘œπ‘˜π‘’ VISCOSITY PROBLEMS FOR PLATE TYPE: FORCE (F): 𝜏 = 𝐹 𝐴 Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ F Force N A Area of the plate π‘š2 POWER (P): 𝑃 = 𝐹 βˆ— 𝑑𝑒 Symbol Description Unit 𝑃 Power π‘Š F Force N 𝑑𝑒 Change in Velocity π‘š 𝑠⁄ VISCOSITY PROBLEMS FOR SHAFT TYPE: VELOCITY OF SHAFT (u): 𝑒 = πœ‹π·π‘ 60 Symbol Description Unit 𝐷 Diameter of Shaft π‘š N Speed of Shaft Rpm 𝑒 Velocity π‘š 𝑠⁄
  • 6. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 6 FORCE (F): 𝜏 = 𝐹 𝐴 𝜏 = πœ‹π·πΏ Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ F Force N A Circumference of Shaft π‘š2 𝐷 Diameter of Shaft π‘š 𝐿 Length of Shaft π‘š TORQUE ON SHAFT (T): 𝑇 = 𝐹 βˆ— 𝐷 2 Symbol Description Unit 𝑇 Torque 𝑁 βˆ’ π‘š F Force N 𝐷 Diameter of Shaft π‘š POWER ON SHAFT (P): 𝑃 = 2πœ‹π‘π‘‡ 60 Symbol Description Unit 𝑃 Power π‘Š 𝑇 Torque 𝑁 βˆ’ π‘š N Speed of Shaft Rpm
  • 7. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 7 VISCOSITY PROBLEMS FOR CONICAL BEARING: ANGULAR VELOCITY (Ο‰): πœ” = 2πœ‹π‘ 60 Symbol Description Unit πœ” Angular Velocity π‘Ÿπ‘Žπ‘‘ 𝑠𝑒𝑐⁄ N Speed of Shaft Rpm ANGLE (ΞΈ): π‘‘π‘Žπ‘›πœƒ = π‘Ÿ1 βˆ’ π‘Ÿ2 𝐻 Symbol Description Unit π‘Ÿ1 Outer Radius π‘š π‘Ÿ2 Inner Radius π‘š 𝐻 Height π‘š POWER (P): 𝑃 = 2πœ‹π‘π‘‡ 60 Symbol Description Unit 𝑃 Power π‘Š 𝑇 Torque 𝑁 βˆ’ π‘š
  • 8. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 8 N Speed of Shaft Rpm THICKNESS OF OIL (h): 𝑇 = πœ‹πœ‡πœ” 2β„Žπ‘ π‘–π‘›πœƒ ( π‘Ÿ1 4 βˆ’ π‘Ÿ2 4) Symbol Description Unit πœ‡ Dynamic Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝑇 Torque 𝑁 βˆ’ π‘š πœ” Angular Velocity π‘Ÿπ‘Žπ‘‘ 𝑠𝑒𝑐⁄ β„Ž Thickness of Oil π‘š π‘Ÿ1 Outer Radius π‘š π‘Ÿ2 Inner Radius π‘š CAPILLARITY: HEIGHT OF LIQUID IN TUBE (h): β„Ž = 4πœŽπ‘π‘œπ‘ πœƒ πœŒπ‘”π‘‘ Symbol Description Unit β„Ž Height of Liquid in Tube π‘š 𝜎 Surface Tension 𝑁 π‘šβ„ πœƒ Angle of Contact between Liquid and Tube π‘Ÿπ‘Žπ‘‘ 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝑑 Diameter of Tube π‘š SURFACE TENSION: PRESSURE IN LIQUID DROPLET (P): 𝑃 = 4𝜎 𝑑
  • 9. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 9 Symbol Description Unit 𝑃 Pressure 𝑁 π‘š2⁄ 𝜎 Surface Tension 𝑁 π‘šβ„ 𝑑 Diameter of Droplet π‘š PRESSURE IN BUBBLE (P): 𝑃 = 8𝜎 𝑑 Symbol Description Unit 𝑃 Pressure 𝑁 π‘š2⁄ 𝜎 Surface Tension 𝑁 π‘šβ„ 𝑑 Diameter of Bubble π‘š PRESSURE IN LIQUID JET (P): 𝑃 = 2𝜎 𝑑 Symbol Description Unit 𝑃 Pressure 𝑁 π‘š2⁄ 𝜎 Surface Tension 𝑁 π‘šβ„ 𝑑 Diameter of Jet π‘š CONTINUITY EQUATION: πœ•π‘’ πœ•π‘₯ + πœ•π‘£ πœ•π‘¦ + πœ•π‘€ πœ•π‘§ = 0 [ πΉπ‘œπ‘Ÿ 3 βˆ’ 𝐷 π‘“π‘™π‘œπ‘€] πœ•π‘’ πœ•π‘₯ + πœ•π‘£ πœ•π‘¦ + = 0 [ πΉπ‘œπ‘Ÿ 2 βˆ’ 𝐷 π‘“π‘™π‘œπ‘€] πœ• πœ•π‘Ÿ ( π‘Ÿπ‘’ π‘Ÿ) + πœ• πœ•πœƒ ( 𝑒 πœƒ) = 0[ πΉπ‘œπ‘Ÿ π‘π‘œπ‘™π‘Žπ‘Ÿ π‘π‘œπ‘œπ‘Ÿπ‘‘π‘–π‘›π‘Žπ‘‘π‘’π‘ ]
  • 10. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 10 BERNOULLI’S EQUATION: πœ•π‘ƒ 𝜌 + 𝑣. 𝑑𝑣 + 𝑔. 𝑑𝑧 = 0 𝑃1 πœŒπ‘” + 𝑣1 2 2𝑔 + 𝑧1 = 𝑃2 πœŒπ‘” + 𝑣2 2 2𝑔 + 𝑧2 + β„Ž 𝑓 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁 π‘š2⁄ 𝑣1 & 𝑣2 Velocity at Section 1 & 2 π‘š 𝑠⁄ 𝑧1 & 𝑧2 Datum Head at Section 1 & 2 π‘š β„Ž 𝑓 Head Loss π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ COEFFICIENT OF DISCHARGE: 𝐢 𝑑 = 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ COEFFICIENT OF VELOCITY: 𝐢𝑣 = 𝑣 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑣 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ DISCHARGE OF VENTURIMETER AND ORIFICEMETER: 𝑄 = 𝐢 𝑑 π‘Ž1 π‘Ž2 √( π‘Ž1 2 βˆ’ π‘Ž1 2) √2π‘”β„Ž Symbol Description Unit π‘Ž1 & π‘Ž2 Area at Section 1 & 2 π‘š2 β„Ž Pressure Difference between Section 1 & 2 ( 𝑃1βˆ’ 𝑃2 πœŒπ‘” ) π‘š
  • 11. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 11 𝐢 𝑑 Coefficient of Discharge π‘₯ Difference in Mercury Level π‘š β„Ž = π‘₯ (1 βˆ’ 𝑆 π‘š 𝑆 ) [ π‘€β„Žπ‘’π‘› 𝑆 > 𝑆 π‘š] β„Ž = π‘₯ ( 𝑆 π‘š 𝑆 βˆ’ 1) [ π‘€β„Žπ‘’π‘› 𝑆 π‘š > 𝑆] β„Ž = ( 𝑃1 πœŒπ‘” + 𝑍1) βˆ’ ( 𝑃2 πœŒπ‘” + 𝑍2) [ 𝐼𝑛𝑐𝑙𝑖𝑛𝑒𝑑 π‘‰π‘’π‘›π‘‘π‘’π‘Ÿπ‘–π‘šπ‘’π‘‘π‘’π‘Ÿ] MOMENTUM EQUATION: 𝐹 = 𝑑 (π‘šπ‘£) 𝑑𝑑 FORCE ACTING IN X – DIRECTION: 𝐹π‘₯ = πœŒπ‘„ ( 𝑣1 βˆ’ 𝑣2 π‘π‘œπ‘ πœƒ) + 𝑃1 𝐴1 βˆ’ 𝑃2 𝐴2 π‘π‘œπ‘ πœƒ FORCE ACTING IN Y – DIRECTION: 𝐹𝑦 = πœŒπ‘„ (βˆ’ 𝑣2 π‘ π‘–π‘›πœƒ) βˆ’ 𝑃2 𝐴2 π‘ π‘–π‘›πœƒ Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁 π‘š2⁄ 𝑣1 & 𝑣2 Velocity at Section 1 & 2 π‘š 𝑠⁄ 𝐴1 & 𝐴2 Area at Section 1 & 2 π‘š πœƒ Angle of the Bend π·π‘’π‘”π‘Ÿπ‘’π‘’ 𝑄 Discharge π‘š3 𝑠⁄ RESULTANT FORCE: 𝐹𝑅 = √𝐹π‘₯ 2 + 𝐹𝑦 2
  • 12. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 12 ANGLE MADE BY RESULTANT FORCE: π‘‘π‘Žπ‘›πœƒ = 𝐹𝑦 𝐹π‘₯ MOMENT OF MOMENTUM EQUATION: 𝑇 = πœŒπ‘„ ( 𝑣2 π‘Ÿ2 βˆ’ 𝑣1 π‘Ÿ1) Symbol Description Unit 𝑇 Torque 𝑁 βˆ’ π‘š 𝑣1 & 𝑣2 Velocity at Section 1 & 2 π‘š 𝑠⁄ π‘Ÿ1 & π‘Ÿ2 Radius of Curvature at Section 1 & 2 π‘š 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄
  • 13. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 13 TOTAL ENERGY LINE (TEL): 𝑇𝐸𝐿 = π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ + 𝐾𝑖𝑛𝑒𝑑𝑖𝑐 π»π‘’π‘Žπ‘‘ + π·π‘Žπ‘‘π‘’π‘š π»π‘’π‘Žπ‘‘ 𝑇𝐸𝐿 = 𝑃 πœŒπ‘” + 𝑣2 2𝑔 + 𝑍 Symbol Description Unit 𝑃 Pressure 𝑁 π‘š2⁄ 𝑣 Velocity π‘š 𝑠⁄ 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑍 Datum Head π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ HYDRAULIC ENERGY LINE (HEL): 𝐻𝐸𝐿 = π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ + π·π‘Žπ‘‘π‘’π‘š π»π‘’π‘Žπ‘‘ 𝑇𝐸𝐿 = 𝑃 πœŒπ‘” + 𝑍 HAGEN POISEUILLE’S EQUATION: SHEAR STRESS: 𝜏 = βˆ’ πœ•π‘ πœ•π‘₯ βˆ— π‘Ÿ 2 Symbol Description Unit 𝜏 Shear Stress 𝑁 π‘š2⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ π‘Ÿ Radius of pipe π‘š UNIT – II – FLOW THROUGH CIRCULAR CONDUITS
  • 14. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 14 VELOCITY: 𝑒 = βˆ’ 1 4πœ‡ βˆ— πœ•π‘ πœ•π‘₯ βˆ— (𝑅2 βˆ’ π‘Ÿ2 ) Symbol Description Unit 𝑒 Velocity of Fluid in Pipe π‘š 𝑠⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ π‘Ÿ Radius of pipe π‘š πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ MAXIMUM VELOCITY: 𝑒 = βˆ’ 1 4πœ‡ βˆ— πœ•π‘ πœ•π‘₯ βˆ— (𝑅2 ) Symbol Description Unit 𝑒 Velocity of Fluid in Pipe π‘š 𝑠⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ AVERAGE VELOCITY: 𝑒̅ = βˆ’ 1 4πœ‡ βˆ— πœ•π‘ πœ•π‘₯ βˆ— (𝑅2 ) Symbol Description Unit 𝑒̅ Average Velocity of Fluid in Pipe π‘š 𝑠⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ RATIO BETWEEN MAXIMUM VELOCITY AND AVERAGE VELOCITY: 𝑒 π‘šπ‘Žπ‘₯ 𝑒̅ = 2
  • 15. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 15 DISCHARGE: 𝑒 = βˆ’ 1 8πœ‡ βˆ— πœ•π‘ πœ•π‘₯ βˆ— πœ‹ βˆ— 𝑅4 Symbol Description Unit 𝑒 Velocity of Fluid in Pipe π‘š 𝑠⁄ πœ•π‘ πœ•π‘₯ Pressure Gradient 𝑁 π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ PRESSURE DIFFERENCE: 𝑃1 βˆ’ 𝑃2 = 32πœ‡π‘’Μ…πΏ 𝐷2 Symbol Description Unit 𝑒̅ Average Velocity of Fluid in Pipe π‘š 𝑠⁄ 𝐿 Length of Pipe π‘š πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝐷 Diameter of Pipe π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS OF HEAD: β„Ž 𝑓 = 𝑃1 βˆ’ 𝑃2 πœŒπ‘” = 32πœ‡π‘’Μ…πΏ πœŒπ‘”π·2 [ π‘“π‘œπ‘Ÿ πΏπ‘Žπ‘šπ‘–π‘›π‘Žπ‘Ÿ π‘“π‘™π‘œπ‘€] DARCY WEISBACH EQUATION: β„Ž 𝑓 = 𝑃1 βˆ’ 𝑃2 πœŒπ‘” = 4𝑓𝐿𝑣2 2𝑔𝑑 [ π‘“π‘œπ‘Ÿ π‘‡π‘’π‘Ÿπ‘π‘’π‘™π‘’π‘›π‘‘ π‘“π‘™π‘œπ‘€] Symbol Description Unit 𝑣 Velocity of Fluid in Pipe π‘š 𝑠⁄
  • 16. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 16 𝐿 Length of Pipe π‘š 𝑓 Friction Factor 𝑑 Diameter of Pipe π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ REYNOLD’S NUMBER: 𝑅 𝑒 = πœŒπ‘£π‘‘ πœ‡ 𝑓 = 0.079 𝑅 𝑒 0.25 [ πΉπ‘œπ‘Ÿ π‘‡π‘’π‘Ÿπ‘π‘’π‘™π‘’π‘›π‘‘ πΉπ‘™π‘œπ‘€] 𝑓 = 16 𝑅 𝑒 [ πΉπ‘œπ‘Ÿ πΏπ‘Žπ‘šπ‘–π‘›π‘Žπ‘Ÿ πΉπ‘™π‘œπ‘€] 𝑅 𝑒 < 2000 π‘‡β„Žπ‘’π‘› π‘‘β„Žπ‘’ πΉπ‘™π‘œπ‘€ 𝑖𝑠 πΏπ‘Žπ‘šπ‘–π‘›π‘Žπ‘Ÿ 𝑅 𝑒 > 2000 π‘‡β„Žπ‘’π‘› π‘‘β„Žπ‘’ πΉπ‘™π‘œπ‘€ 𝑖𝑠 π‘‡π‘’π‘Ÿπ‘π‘’π‘™π‘’π‘›π‘‘ Symbol Description Unit 𝑣 Velocity of Fluid in Pipe π‘š 𝑠⁄ 𝑑 Diameter of Pipe π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ MAJOR LOSS IN PIPES: β„Ž 𝑓 = 32πœ‡π‘’Μ…πΏ πœŒπ‘”π‘‘2 [ π‘“π‘œπ‘Ÿ πΏπ‘Žπ‘šπ‘–π‘›π‘Žπ‘Ÿ π‘“π‘™π‘œπ‘€] β„Ž 𝑓 = 4𝑓𝐿𝑣2 2𝑔𝑑 [ π‘“π‘œπ‘Ÿ π‘‡π‘’π‘Ÿπ‘π‘’π‘™π‘’π‘›π‘‘ π‘“π‘™π‘œπ‘€]
  • 17. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 17 Symbol Description Unit 𝑒̅ & 𝑣 Velocity of Fluid in Pipe π‘š 𝑠⁄ 𝑑 Diameter of Pipe π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝑙 Length of Pipe π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝑓 Friction Factor MINOR LOSS IN PIPES: LOSS DUE TO SUDDEN ENLARGEMENT: β„Ž 𝑒 = ( 𝑣1 βˆ’ 𝑣2)2 2𝑔 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Fluid in Pipe at Inlet and Outlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS DUE TO SUDDEN CONTRACTION: β„Ž 𝑐 = 𝐾𝑣2 2𝑔 𝐾 = ( 1 𝐢𝑐 βˆ’ 1) 2 β„Ž 𝑐 = 0.5𝑣2 2𝑔 [ 𝐼𝑓 𝐢𝑐 π‘›π‘œπ‘‘ 𝑔𝑖𝑣𝑒𝑛] Symbol Description Unit 𝑣 Velocity of Fluid at Outlet π‘š 𝑠⁄ 𝐢𝑐 Coefficient of Contraction
  • 18. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 18 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS AT ENTRANCE OF PIPE: β„Žπ‘– = 0.5𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Fluid at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS AT EXIT OF PIPE: β„Ž π‘œ = 𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Fluid at Outlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LOSS DUE TO GRADUAL CONTRACTION: β„Ž 𝑒 = 𝐾( 𝑣1 βˆ’ 𝑣2)2 2𝑔 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Fluid in Pipe at Inlet and Outlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝐾 Coefficient of Contraction LOSS AT BEND OF PIPE: β„Ž 𝑏 = 𝐾𝑣2 2𝑔
  • 19. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 19 Symbol Description Unit 𝑣 Velocity of Flow π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝐾 Coefficient of Bend LOSS AT DUE TO VARIOUS FITTINGS: β„Ž 𝑣 = 𝐾𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Flow π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝐾 Coefficient of Fittings LOSS AT DUE TO OBSTRUCTION: β„Ž 𝑣 = 𝑣2 2𝑔 ( 𝐴 𝐢𝑐 ( 𝐴 βˆ’ π‘Ž) βˆ’ 1) 𝐢𝑐 = 𝐴 𝑐 ( 𝐴 βˆ’ π‘Ž) Symbol Description Unit 𝑣 Velocity of Flow π‘š 𝑠⁄ 𝐴 Area of Pipe π‘š2 π‘Ž Area of Obstruction π‘š2 𝐴 𝑐 Area of Vena Contraction π‘š2 WHEN PIPES ARE CONNECTED IN SERIES: DISCHARGE: 𝑄 = 𝑄1 = 𝑄2 𝑄 = 𝐴1 𝑣1 = 𝐴2 𝑣2
  • 20. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 20 HEAD LOSS: β„Ž 𝑓 = β„Ž 𝑓1 + β„Ž 𝑓2 β„Ž 𝑓 = 4𝑓𝑙1 𝑣1 2 2𝑔𝑑1 + 4𝑓𝑙2 𝑣2 2 2𝑔𝑑2 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Flow at Pipe 1 & 2 π‘š 𝑠⁄ 𝐴1& 𝐴2 Area of Pipe 1 & 2 π‘š2 𝑑1& 𝑑2 Diameter of Pipe 1 & 2 π‘š 𝑙1& 𝑙2 Length of Pipe 1 & 2 π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝑓 Friction Factor WHEN PIPES ARE CONNECTED IN PARALLEL: DISCHARGE: 𝑄 = 𝑄1 + 𝑄2 𝑄 = 𝐴1 𝑣1 + 𝐴2 𝑣2 HEAD LOSS: β„Ž 𝑓 = β„Ž 𝑓1 = β„Ž 𝑓2 β„Ž 𝑓 = 4𝑓𝑙1 𝑣1 2 2𝑔𝑑1 = 4𝑓𝑙2 𝑣2 2 2𝑔𝑑2 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Flow at Pipe 1 & 2 π‘š 𝑠⁄ 𝐴1& 𝐴2 Area of Pipe 1 & 2 π‘š2 𝑑1& 𝑑2 Diameter of Pipe 1 & 2 π‘š 𝑙1& 𝑙2 Length of Pipe 1 & 2 π‘š
  • 21. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 21 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ 𝑓 Friction Factor EQUIVALENT PIPE: 𝐿 𝐷5 = 𝐿1 𝐷1 5 + 𝐿2 𝐷2 5 + 𝐿3 𝐷3 5 + β‹― + 𝐿 𝑛 𝐷 𝑛 5 Symbol Description Unit 𝐷 Diameter of Pipe π‘š 𝐿 Length of Pipe π‘š BOUNDARY LAYER: DISPLACEMENT THICKNESS: π›Ώβˆ— = ∫ (1 βˆ’ 𝑒 π‘ˆ ) 𝛿 0 𝑑𝑦 MOMENTUM THICKNESS: πœƒ = ∫ 𝑒 π‘ˆ (1 βˆ’ 𝑒 π‘ˆ ) 𝛿 0 𝑑𝑦 MOMENTUM THICKNESS: π›Ώβˆ—βˆ— = ∫ 𝑒 π‘ˆ (1 βˆ’ 𝑒2 π‘ˆ2 ) 𝛿 0 𝑑𝑦 Symbol Description Unit 𝑒 π‘ˆ Velocity Distribution 𝛿 Boundary layer thickness SHEAR STRESS: 𝜏0 πœŒπ‘ˆ2 = πœ•πœƒ πœ•π‘₯
  • 22. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 22 πœƒ = ∫ 𝑒 π‘ˆ (1 βˆ’ 𝑒 π‘ˆ ) 𝛿 0 𝑑𝑦 DRAG FORCE: 𝐹 𝐷 = ∫ π‘†β„Žπ‘’π‘Žπ‘Ÿ π‘†π‘‘π‘Ÿπ‘’π‘ π‘  βˆ— π΄π‘Ÿπ‘’π‘Ž 𝐿 0 𝐹 𝐷 = ∫ 𝜏0 βˆ— 𝑏 βˆ— 𝑑π‘₯ 𝐿 0 LOCAL COEFFICIENT OF DRAG: 𝐢 𝐷 βˆ— = 𝜏0 1 2 πœŒπ‘ˆ2 AVERAGE COEFFICIENT OF DRAG: 𝐢 𝐷 = 𝐹 𝐷 1 2 πœŒπ΄π‘ˆ2 Symbol Description Unit 𝜏0 Shear Stress 𝑁 π‘š2⁄ 𝑏 Width of Plate π‘š π‘ˆ Free Stream Velocity π‘š 𝑠⁄ 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝐴 Area π‘š2 𝐹 𝐷 Drag Force 𝑁 BLASIUS’S SOLUTION: BOUNDARY LAYER THICKNESS: 𝛿 = 4.91π‘₯ √ 𝑅 𝑒π‘₯
  • 23. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 23 LOCAL COEFFICIENT OF DRAG: 𝐢 𝐷 βˆ— = 0.664 √ 𝑅 𝑒π‘₯ AVERAGE COEFFICIENT OF DRAG: 𝐢 𝐷 = 1.328 √ 𝑅 𝑒𝐿 Symbol Description Unit 𝑅 𝑒π‘₯ Reynold’s Number at distance x 𝑅 𝑒𝐿 Reynold’s Number at distance L
  • 24. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 24 UNITS: Physical Quantity Symbol Unit Dimensions Length L m L Mass M Kg M Time T Sec T Area A m2 L2 Volume V m3 L3 Diameter D m L Head H m L Roughness k M L Velocity v m/s LT-1 Angular Velocity Ο‰ rad/sec T-1 Acceleration a m/s2 LT-2 Angular Acceleration Ξ± rad/sec2 T-2 Speed N Rpm T-1 Discharge Q m3 /s L3 T-1 Kinematic Viscosity Ξ³ cm2 /s L2 T-1 Dynamic Viscosity ΞΌ N-s/m2 ML-1 T-1 Force F N MLT-2 Weight W N MLT-2 Thrust T N MLT-2 Density ρ Kg/ m3 ML-3 Pressure P N/m2 ML-1 T-2 UNIT – III – DIMENSIONAL ANALYSIS
  • 25. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 25 Physical Quantity Symbol Unit Dimensions Specific Weight w N/m3 ML-2 T-2 Young’s Modulus E N/m2 ML-1 T-2 Bulk Modulus K N/m2 ML-1 T-2 Shear Stress Ο„ N/m2 ML-1 T-2 Surface Tension Οƒ N/m MT-2 Energy / Work W/E J = N-m ML2 T-2 Torque T N-m ML-2 T-2 Power P W=J/s ML-2 T-3 Momentum M Kg m/s MLT-1 Efficiency Ξ· No Unit Dimensionless SIMILARITY: GEOMETRIC SIMILARITY: 𝐿 𝑝 𝐿 π‘š = 𝑏 𝑝 𝑏 π‘š = 𝐷 𝑝 𝐷 π‘š = 𝐿 π‘Ÿ 𝐴 𝑝 𝐴 π‘š = 𝐿 𝑝 𝐿 π‘š βˆ— 𝑏 𝑝 𝑏 π‘š = 𝐿 π‘Ÿ βˆ— 𝐿 π‘Ÿ = 𝐿 π‘Ÿ 2 𝑉𝑝 π‘‰π‘š = 𝐿 𝑝 𝐿 π‘š βˆ— 𝑏 𝑝 𝑏 π‘š βˆ— 𝑑 𝑝 𝑑 π‘š = 𝐿 π‘Ÿ βˆ— 𝐿 π‘Ÿ βˆ— 𝐿 π‘Ÿ = 𝐿 π‘Ÿ 3 Symbol Description Unit 𝐿 𝑝&𝐿 π‘š Length of Prototype & Model π‘š 𝑏 𝑝&𝑏 π‘š Breadth of Prototype & Model π‘š 𝐷 𝑝&𝐷 π‘š Diameter of Prototype & Model π‘š 𝑑 𝑝&𝑑 π‘š Thickness of Prototype & Model π‘š 𝐴 𝑝&𝐴 π‘š Area of Prototype & Model π‘š2
  • 26. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 26 𝑉𝑝&π‘‰π‘š Volume of Prototype & Model π‘š3 𝐿 π‘Ÿ Length Ratio KINEMATIC SIMILARITY: 𝑣 𝑝 𝑣 π‘š = π‘£π‘Ÿ π‘Ž 𝑝 π‘Ž π‘š = π‘Ž π‘Ÿ Symbol Description Unit 𝑣 𝑝&𝑣 π‘š Velocity of Prototype & Model π‘š 𝑠⁄ π‘Ž 𝑝&π‘Ž π‘š Acceleration of Prototype & Model π‘š 𝑠2⁄ π‘£π‘Ÿ Velocity Ratio π‘Ž π‘Ÿ Acceleration Ratio DYNAMIC SIMILARITY: ( 𝐹𝑖) 𝑝 ( 𝐹𝑖) π‘š = ( 𝐹𝑣) 𝑝 ( 𝐹𝑣) π‘š = (𝐹𝑔) 𝑝 (𝐹𝑔) π‘š = πΉπ‘Ÿ Symbol Description Unit ( 𝐹𝑖) 𝑝& ( 𝐹𝑖) π‘š Inertia Force of Prototype & Model 𝑁 ( 𝐹𝑣) 𝑝& ( 𝐹𝑣) π‘š Viscous Force of Prototype & Model 𝑁 (𝐹𝑔) 𝑝 & (𝐹𝑔) π‘š Gravity Force of Prototype & Model 𝑁 πΉπ‘Ÿ Force Ratio DIMENSIONLESS NUMBER: REYNOLD’S NUMBER: 𝑅 𝑒 = πœŒπ‘£π· πœ‡ (π‘œπ‘Ÿ) πœŒπ‘£πΏ πœ‡
  • 27. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 27 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑣 Velocity π‘š 𝑠⁄ πœ‡ Viscosity 𝑁 βˆ’ 𝑠 π‘š2⁄ 𝐷 Diameter π‘š 𝐿 Length π‘š FROUDE’S NUMBER: 𝐹𝑒 = 𝑣 √ 𝐿𝑔 Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐿 Length π‘š FROUDE’S NUMBER: 𝐹𝑒 = 𝑣 √ 𝐿𝑔 Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐿 Length π‘š EULER’S NUMBER: 𝐸 𝑒 = 𝑣 √ 𝑝 πœŒβ„ Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄
  • 28. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 28 𝑝 Pressure 𝑁 π‘š2⁄ WEBER’S NUMBER: π‘Šπ‘’ = 𝑣 √ 𝜎 πœŒπΏβ„ Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝐿 Length π‘š 𝜎 Surface Tension 𝑁 π‘šβ„ MACH’S NUMBER: π‘Šπ‘’ = 𝑣 √ 𝐾 πœŒβ„ Symbol Description Unit 𝑣 Velocity π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝐾 Elastic Stress 𝑁 π‘š2⁄ REYNOLD’S MODEL LAW: TIME RATIO: πΉπ‘Ÿ = π‘š π‘Ÿ π‘Ž π‘Ÿ πΉπ‘Ÿ = π‘š π‘Ÿ π‘£π‘Ÿ π‘‡π‘Ÿ DISCHARGE RATIO: 𝑄 π‘Ÿ = πœŒπ‘Ÿ 𝐿 π‘Ÿ 2 π‘£π‘Ÿ
  • 29. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 29 Symbol Description Unit πΉπ‘Ÿ Force Ratio π‘š π‘Ÿ Mass Ratio π‘£π‘Ÿ Velocity Ratio π‘‡π‘Ÿ Time Ratio 𝐿 π‘Ÿ Length Ratio πœŒπ‘Ÿ Density Ratio FROUDE’S MODEL LAW: TIME RATIO: π‘‡π‘Ÿ = √ 𝐿 π‘Ÿ ACCELERATION RATIO: π‘Ž π‘Ÿ = 1 DISCHARGE RATIO: 𝑄 π‘Ÿ = ( 𝐿 π‘Ÿ) 5 2⁄ FORCE RATIO: πΉπ‘Ÿ = ( 𝐿 π‘Ÿ)3 PRESSURE RATIO: πΉπ‘Ÿ = 𝐿 π‘Ÿ ENERGY RATIO: πΈπ‘Ÿ = ( 𝐿 π‘Ÿ)4 MOMENTUM RATIO: π‘€π‘Ÿ = ( 𝐿 π‘Ÿ)3 βˆ— √ 𝐿 π‘Ÿ TORQUE RATIO: π‘‡π‘Ÿ = ( 𝐿 π‘Ÿ)4
  • 30. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 30 POWER RATIO: π‘ƒπ‘Ÿ = ( 𝐿 π‘Ÿ) 7 2⁄ Symbol Description Unit 𝐿 π‘Ÿ Length Ratio DISTORTED MODELS: ( 𝐿 π‘Ÿ) 𝐻 = πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π»π‘œπ‘Ÿπ‘–π‘§π‘œπ‘›π‘‘π‘Žπ‘™ π·π‘–π‘šπ‘’π‘›π‘ π‘–π‘œπ‘›π‘  π‘œπ‘“ π‘ƒπ‘Ÿπ‘œπ‘‘π‘œπ‘‘π‘¦π‘π‘’ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π»π‘œπ‘Ÿπ‘–π‘§π‘œπ‘›π‘‘π‘Žπ‘™ π·π‘–π‘šπ‘’π‘›π‘ π‘–π‘œπ‘›π‘  π‘œπ‘“ π‘€π‘œπ‘‘π‘’π‘™ ( 𝐿 π‘Ÿ) 𝐻 = 𝐿 𝑝 𝐿 π‘š = 𝐡𝑝 𝐡 π‘š ( 𝐿 π‘Ÿ) 𝑉 = πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π·π‘–π‘šπ‘’π‘›π‘ π‘–π‘œπ‘›π‘  π‘œπ‘“ π‘ƒπ‘Ÿπ‘œπ‘‘π‘œπ‘‘π‘¦π‘π‘’ πΏπ‘–π‘›π‘’π‘Žπ‘Ÿ π‘‰π‘’π‘Ÿπ‘‘π‘–π‘π‘Žπ‘™ π·π‘–π‘šπ‘’π‘›π‘ π‘–π‘œπ‘›π‘  π‘œπ‘“ π‘€π‘œπ‘‘π‘’π‘™ ( 𝐿 π‘Ÿ) 𝑉 = β„Ž 𝑝 β„Ž π‘š VELOCITY RATIO: π‘£π‘Ÿ = √(𝐿 π‘Ÿ) 𝑉 AREA RATIO: 𝐴 π‘Ÿ = ( 𝐿 π‘Ÿ) 𝐻 βˆ— ( 𝐿 π‘Ÿ) 𝑉 DISCAHRGE RATIO: 𝑄 π‘Ÿ = ( 𝐿 π‘Ÿ) 𝐻 βˆ— [( 𝐿 π‘Ÿ) 𝑉] 3 2⁄
  • 31. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 31 CENTRIFUGAL PUMP: VELOCITY TRIANGLE DIAGRAM: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Impeller at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣1&𝑣2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree UNIT – IV – PUMPS
  • 32. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 32 πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝛽 Angle made by Absolute Velocity at Outlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π‘‘1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) Internal Diameter of Impeller π‘š 𝑁 Speed of Impeller π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π‘‘2 𝑁 60 Symbol Description Unit 𝑑2 Oulet (or) External Diameter of Impeller π‘š 𝑁 Speed of Impeller π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: π‘‘π‘Žπ‘›πœƒ = 𝑣𝑓1 𝑒1 Symbol Description Unit 𝑒1 Tangential Velocity of Impeller at Inlet π‘š 𝑠⁄ 𝑣1 Absolute Velocity at Inlet π‘š 𝑠⁄
  • 33. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 33 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree ∡ 𝛼 = 90Β° 𝑣1 = 𝑣𝑓1 FROM OUTLET VELOCITY TRIANGLE DIAGRAM: π‘‘π‘Žπ‘›πœ™ = 𝑣𝑓2 𝑒2 βˆ’ 𝑣 𝑀2 𝑣2 = βˆšπ‘£π‘“2 2 + 𝑣 𝑀2 2 π‘‘π‘Žπ‘›π›½ = 𝑣𝑓2 𝑣 𝑀2 Symbol Description Unit 𝑒2 Tangential Velocity of Impeller at Outlet π‘š 𝑠⁄ 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑣2 Absolute Velocity at Outlet π‘š 𝑠⁄ 𝑣𝑓2 Flow Velocity at Outlet π‘š 𝑠⁄ πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree 𝛽 Angle made by Absolute Velocity at Outlet with the Degree
  • 34. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 34 Direction of Motion of Vane DISCHARGE: 𝑄 = πœ‹π‘‘1 𝑏1 𝑣𝑓1 = πœ‹π‘‘2 𝑏2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝑄 Discharge π‘š3 𝑠⁄ WORK DONE BY AN IMPELLER PER SECOND: π‘Š = πœŒπ‘”π‘„ 𝑔 𝑣 𝑀2 𝑒2 Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ WORK DONE BY AN IMPELLER PER UNIT WEIGHT OF WATER: π‘Š = 𝑣 𝑀2 𝑒2 𝑔 Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄
  • 35. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 35 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ MANOMETRIC EFFICIENCY: πœ‚ π‘š = 𝑔𝐻 𝑣 𝑀2 𝑒2 Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ 𝐻 Manometric Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ POWER REQUIRED BY THE PUMP: 𝑃 = πœŒπ‘„π‘£ 𝑀2 𝑒2 Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑃 Power π‘˜π‘Š MINIMUM SPEED TO START THE PUMP: 𝑁 π‘šπ‘–π‘› = 120 βˆ— πœ‚ π‘š βˆ— 𝑣 𝑀2 βˆ— 𝑑2 πœ‹ (𝑑2 2 βˆ’ 𝑑1 2 ) Symbol Description Unit 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š πœ‚ π‘š Manometric Efficiency
  • 36. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 36 OVERALL EFFICIENCY: πœ‚ π‘œ = πΌπ‘šπ‘π‘’π‘™π‘™π‘’π‘Ÿ π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ = πœŒπ‘”π‘„π» 𝑆. 𝑃 πœ‚ π‘œ = πœ‚ π‘šπ‘Žπ‘›π‘œ βˆ— πœ‚ π‘šπ‘’π‘β„Ž βˆ— πœ‚ π‘£π‘œπ‘™ Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Manometric Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ MECHANICAL EFFICIENCY: πœ‚ π‘šπ‘’π‘β„Ž = πœŒπ‘”π‘„π» 𝑆. 𝑃 βˆ— 𝑣 𝑀2 𝑒2 𝑔𝐻 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Manometric Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝑆. 𝑃 Shaft Power π‘Š 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ POWER OF PUMP: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
  • 37. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 37 𝐻 Manometric Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = π΄π‘π‘‘π‘’π‘Žπ‘™ 𝐿𝑖𝑓𝑑 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ 𝐿𝑖𝑓𝑑 = π΄π‘π‘‘π‘’π‘Žπ‘™ π»π‘’π‘Žπ‘‘ πΌπ‘‘π‘’π‘Žπ‘™ π»π‘’π‘Žπ‘‘ IDEAL HEAD: 𝑃𝐼 = πœŒπ‘”(𝑄 + π‘ž)𝐻𝑖 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ π‘ž Leakage of Water π‘š3 𝑠⁄ 𝐻𝑖 Ideal Head π‘š 𝑃𝐼 Power at Input π‘Š TORQUE EXERTED BY IMPELLER: 𝑇 = πœŒπ‘”π‘„ 𝑔 βˆ— 𝑣 𝑀2 βˆ— 𝑅2 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑅2 Radius of Impeller at Outlet π‘š
  • 38. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 38 SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed SPEED RATIO: 𝐾 𝑒 = 𝑒2 √2𝑔𝐻 𝐾 𝑒 = 0.95 βˆ’ 1.25 Symbol Description Unit 𝑒2 Tangential Velocity at Outlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓2 √2𝑔𝐻 𝐾𝑓 = 0.1 βˆ’ 0.25
  • 39. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 39 Symbol Description Unit 𝑣𝑓2 Flow Velocity at Outlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio RECIPROCATING PUMP: DISCHARGE: 𝑄 = 𝐴𝐿𝑁 60 𝐴 = πœ‹ 4 𝐷2 [ πΉπ‘œπ‘Ÿ 𝑆𝑖𝑛𝑔𝑙𝑒 𝐴𝑐𝑑𝑖𝑛𝑔 π‘ƒπ‘’π‘šπ‘] 𝐴 = [ πœ‹ 4 𝐷2 + πœ‹ 4 ( 𝐷2 βˆ’ 𝑑2)] [ πΉπ‘œπ‘Ÿ π·π‘œπ‘’π‘π‘™π‘’ 𝐴𝑐𝑑𝑖𝑛𝑔 π‘ƒπ‘’π‘šπ‘] Symbol Description Unit 𝐴 Area of Cylinder π‘š2 𝐿 Stroke Length π‘š 𝑁 Speed π‘Ÿπ‘π‘š 𝐷 Diameter of Cylinder or Bore π‘š 𝑑 Diameter of Piston Rod π‘š WEIGHT OF THE WATER DELIVERED PER SECOND: π‘Š = πœŒπ‘”π‘„ Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ π‘Š Weight of Water 𝑁 𝑠⁄
  • 40. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 40 WORK DONE BY RECIPROCATING PUMP: π‘Š = πœŒπ‘”π‘„π» 𝐻 = β„Ž 𝑠 + β„Ž 𝑑 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š π‘Š Work Done π‘Š β„Ž 𝑠 Suction Head π‘š β„Ž 𝑑 Delivery Head π‘š POWER DEVELOPED BY RECIPROCATING PUMP: 𝑃 = πœŒπ‘”π‘„ π΄π‘π‘‘π‘’π‘Žπ‘™ 𝐻 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š POWER REQUIRED TO DRIVE THE PUMP: 𝑃 = πœŒπ‘”π‘„ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ 𝐻 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š
  • 41. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 41 SLIP OF RECIPROCATING PUMP: 𝑆 = 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ βˆ’ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Symbol Description Unit 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ COEFFICENT OF DISCHARGE: 𝐢 𝑑 = 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Symbol Description Unit 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ PERCENTAGE OF SLIP IN RECIPROCATING PUMP: % π‘œπ‘“ 𝑆𝑙𝑖𝑝 = 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ βˆ’ 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ % π‘œπ‘“ 𝑆𝑙𝑖𝑝 = 1 βˆ’ 𝐢 𝑑 Symbol Description Unit 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ 𝐢 𝑑 Coefficient of Discharge VOLUMETRIC EFFICIENCY: πœ‚ π‘‰π‘œπ‘™ = 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ = 𝐢 𝑑 Symbol Description Unit 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ 𝐢 𝑑 Coefficient of Discharge
  • 42. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 42 MECHANICAL EFFICIENCY: πœ‚ π‘šπ‘’π‘β„Ž = π‘ƒπ‘œπ‘€π‘’π‘Ÿ π·π‘’π‘£π‘’π‘™π‘œπ‘π‘’π‘‘ 𝑏𝑦 π‘ƒπ‘’π‘šπ‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘…π‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘ π‘‘π‘œ π·π‘Ÿπ‘–π‘£π‘’ π‘‘β„Žπ‘’ π‘ƒπ‘’π‘šπ‘ πœ‚ π‘šπ‘’π‘β„Ž = π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘œπ‘“ π‘ƒπ‘’π‘šπ‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘œπ‘“ π‘€π‘œπ‘‘π‘œπ‘Ÿ πœ‚ π‘šπ‘’π‘β„Ž = πœŒπ‘”π‘„ π΄π‘π‘‘π‘’π‘Žπ‘™ 𝐻 πœŒπ‘”π‘„ π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ 𝐻 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 π΄π‘π‘‘π‘’π‘Žπ‘™ Actual Discharge π‘š3 𝑠⁄ 𝑄 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘‘π‘–π‘π‘Žπ‘™ Theoretical Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š ACCELERATION HEAD: β„Ž π‘Žπ‘  = 𝑙 𝑠 𝑔 βˆ— 𝐴 π‘Ž 𝑠 βˆ— πœ”2 βˆ— π‘Ÿ βˆ— π‘π‘œπ‘ πœƒ [ 𝐴𝑑 π‘†π‘’π‘π‘‘π‘–π‘œπ‘› π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] β„Ž 𝑑𝑠 = 𝑙 𝑑 𝑔 βˆ— 𝐴 π‘Ž 𝑑 βˆ— πœ”2 βˆ— π‘Ÿ βˆ— π‘π‘œπ‘ πœƒ [ 𝐴𝑑 π·π‘’π‘™π‘–π‘£π‘’π‘Ÿπ‘¦ π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] 𝐴 = πœ‹ 4 𝐷2 π‘Ž 𝑠 = πœ‹ 4 𝑑 𝑠 2 π‘Ž 𝑑 = πœ‹ 4 𝑑 𝑑 2 πœ” = 2πœ‹π‘ 60
  • 43. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 43 π‘Ÿ = 𝐿 2 Symbol Description Unit 𝑙 𝑠 Length of Suction Pipe π‘š 𝑙 𝑑 Length of Delivery Pipe π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐴 Area of Cylinder π‘š2 π‘Ž 𝑠 Area of Suction Pipe π‘š2 π‘Ž 𝑑 Area of Delivery Pipe π‘š2 πœ” Angular Speed π‘Ÿπ‘Žπ‘‘ 𝑠⁄ π‘Ÿ Radius of Crank π‘š πœƒ Angle of Crank π‘‘π‘’π‘”π‘Ÿπ‘’π‘’ 𝐷 Diameter of Cylinder or Bore π‘š 𝑑 𝑠 Diameter of Suction Pipe π‘š 𝑑 𝑑 Diameter of Delivery Pipe π‘š 𝑁 Speed of Crank π‘Ÿπ‘π‘š 𝐿 Stroke Length π‘š PRESSURE HEAD: π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ = β„Ž 𝑠 + β„Ž π‘Žπ‘  [ πΉπ‘œπ‘Ÿ π‘†π‘’π‘π‘‘π‘–π‘œπ‘› π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ = β„Ž 𝑑 + β„Ž π‘Žπ‘‘ [ πΉπ‘œπ‘Ÿ π·π‘’π‘™π‘–π‘£π‘’π‘Ÿπ‘¦ π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] Symbol Description Unit β„Ž 𝑠 Suction Head π‘š β„Ž 𝑑 Delivery Head π‘š β„Ž π‘Žπ‘  Acceleration Head at Suction π‘š
  • 44. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 44 β„Ž π‘Žπ‘‘ Acceleration Head at Delivery π‘š ABSOLUTE PRESSURE HEAD: π΄π‘π‘ π‘œπ‘™π‘’π‘‘π‘’ π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ = 𝐻 π‘Žπ‘‘π‘š βˆ’ (β„Ž 𝑠 + β„Ž π‘Žπ‘ ) [ πΉπ‘œπ‘Ÿ π‘†π‘’π‘π‘‘π‘–π‘œπ‘› π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] π΄π‘π‘ π‘œπ‘™π‘’π‘‘π‘’ π‘ƒπ‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ π»π‘’π‘Žπ‘‘ = 𝐻 π‘Žπ‘‘π‘š + (β„Ž 𝑑 + β„Ž π‘Žπ‘‘ ) [ πΉπ‘œπ‘Ÿ π·π‘’π‘™π‘–π‘£π‘’π‘Ÿπ‘¦ π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’] Symbol Description Unit β„Ž 𝑠 Suction Head π‘š β„Ž 𝑑 Delivery Head π‘š β„Ž π‘Žπ‘  Acceleration Head at Suction π‘š β„Ž π‘Žπ‘‘ Acceleration Head at Delivery π‘š 𝐻 π‘Žπ‘‘π‘š Atmospheric Pressure Head π‘š SEPARATION HEAD: 𝑃𝑠𝑒𝑝 = πœŒπ‘”β„Ž 𝑆𝑒𝑝 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ β„Ž 𝑠𝑒𝑝 Separation Head π‘š 𝑃𝑠𝑒𝑝 Separation Pressure 𝑁 π‘š2⁄ HEAD LOSS WITHOUT AIR VESSEL: β„Ž π‘“π‘Šπ‘‚π΄ = 4𝑓𝑙 𝑑 𝑣2 2𝑔𝑑 𝑑 Symbol Description Unit 𝑓 Friction Factor 𝑙 𝑑 Length of Delivery Pipe π‘š
  • 45. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 45 𝑣 Velocity without Air Vessel π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝑑 𝑑 Diameter of Delivery Pipe π‘š VELOCITY WITHOUT AIR VESSEL: 𝑣 = 𝐴 π‘Ž 𝑑 βˆ— πœ” βˆ— π‘Ÿ 𝐴 = πœ‹ 4 𝐷2 π‘Ž 𝑑 = πœ‹ 4 𝑑 𝑑 2 πœ” = 2πœ‹π‘ 60 π‘Ÿ = 𝐿 2 Symbol Description Unit 𝐴 Area of Cylinder π‘š2 π‘Ž 𝑑 Area of Delivery Pipe π‘š2 πœ” Angular Speed π‘Ÿπ‘Žπ‘‘ 𝑠⁄ π‘Ÿ Radius of Crank π‘š 𝐷 Diameter of Cylinder or Bore π‘š 𝑑 𝑑 Diameter of Delivery Pipe π‘š 𝑁 Speed of Crank π‘Ÿπ‘π‘š 𝐿 Stroke Length π‘š
  • 46. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 46 HEAD LOSS WITH AIR VESSEL: β„Ž π‘“π‘Šπ΄ = 4𝑓𝑙 𝑑 𝑣2 2𝑔𝑑 𝑑 Symbol Description Unit 𝑓 Friction Factor 𝑙 𝑑 Length of Delivery Pipe π‘š 𝑣 Velocity with Air Vessel π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝑑 𝑑 Diameter of Delivery Pipe π‘š VELOCITY WITH AIR VESSEL: 𝑣 = 𝐴 π‘Ž 𝑑 βˆ— πœ” βˆ— π‘Ÿ πœ‹ 𝐴 = πœ‹ 4 𝐷2 π‘Ž 𝑑 = πœ‹ 4 𝑑 𝑑 2 πœ” = 2πœ‹π‘ 60 π‘Ÿ = 𝐿 2 Symbol Description Unit 𝐴 Area of Cylinder π‘š2 π‘Ž 𝑑 Area of Delivery Pipe π‘š2 πœ” Angular Speed π‘Ÿπ‘Žπ‘‘ 𝑠⁄ π‘Ÿ Radius of Crank π‘š 𝐷 Diameter of Cylinder or Bore π‘š
  • 47. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 47 𝑑 𝑑 Diameter of Delivery Pipe π‘š 𝑁 Speed of Crank π‘Ÿπ‘π‘š 𝐿 Stroke Length π‘š POWER SAVED BY AIR VESSEL: 𝑃 = πœŒπ‘”π‘„ ( 2 3 β„Ž π‘“π‘Šπ‘‚π΄ βˆ’ β„Ž π‘“π‘Šπ΄) Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ β„Ž π‘“π‘Šπ‘‚π΄ Head Loss Without Air Vessel π‘š β„Ž π‘“π‘Šπ΄ Head Loss With Air Vessel π‘š POWER REQUIRED TO DRIVE THE PUMP: 𝑃 = πœŒπ‘”π‘„ (β„Ž 𝑠 + β„Ž 𝑑 + 2 3 β„Ž π‘“π‘ π‘Šπ‘‚π΄ + β„Ž π‘“π‘‘π‘Šπ΄) Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ β„Ž 𝑠 Suction Head π‘š β„Ž 𝑑 Delivery Head π‘š β„Ž π‘“π‘ π‘Šπ‘‚π΄ Head Loss Without Air Vessel at Suction π‘š β„Ž π‘“π‘‘π‘Šπ΄ Head Loss With Air Vessel at Delivery π‘š
  • 48. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 48 PELTON WHEEL: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛽 Angle made by Absolute Velocity at Outlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree UNIT – V – TURBINES
  • 49. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 49 TANGENTIAL VELOCITY AT INLET AND OUTLET (OR) VELOCITY OF WHEEL: 𝑒 = πœ‹π·π‘ 60 Symbol Description Unit 𝐷 Diameter of Runner π‘š 𝑁 Speed of Impeller π‘Ÿπ‘π‘š VELOCITY OF JET: 𝑉1 = πΆπ‘£βˆš2𝑔𝐻 𝐢𝑣 = 0.97 βˆ’ 0.99 Symbol Description Unit 𝐢𝑣 Coefficient of Velocity 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š VELOCITY OF WHEEL: 𝑒 = π‘˜ π‘’βˆš2𝑔𝐻 π‘˜ 𝑒 = 0.43 βˆ’ 0.45 Symbol Description Unit π‘˜ 𝑒 Speed Ratio 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: 𝑉 𝑀1 = 𝑉1 𝑉 𝑀1 = 𝑒1 + π‘‰π‘Ÿ1
  • 50. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 50 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ FROM OUTLET VELOCITY TRIANGLE DIAGRAM: cos πœ™ = 𝑒2 + 𝑣 𝑀2 𝑣 π‘Ÿ2 tan πœ™ = 𝑣𝑓2 𝑒2 + 𝑣 𝑀2 sin πœ™ = 𝑣𝑓2 𝑣 π‘Ÿ2 tan 𝛽 = 𝑣𝑓2 𝑣 𝑀2 Symbol Description Unit 𝑒2 Tangential Velocity of Runner at Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ2 Relative Velocity at Outlet π‘š 𝑠⁄ 𝑣 𝑀2 Whirl Velocity at Outlet π‘š 𝑠⁄ 𝑣𝑓2 Flow Velocity at Outlet π‘š 𝑠⁄ WORK DONE BY JET PER SECOND: π‘Š = πœŒπ‘„ [ 𝑣 𝑀1 + 𝑣 𝑀2] 𝑒 Symbol Description Unit 𝑒 Tangential Velocity of Runner π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
  • 51. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 51 HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 2[ 𝑣 𝑀1 + 𝑣 𝑀2] 𝑒 𝑉1 2 Symbol Description Unit 𝑒 Tangential Velocity of Runner π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š DISCHARGE OF SINGLE JET: π‘ž = πœ‹ 4 βˆ— 𝑑2 βˆ— 𝑉1 Symbol Description Unit 𝑑 Diameter of Jet π‘š 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ π‘ž Discharge of Single Jet π‘š3 𝑠⁄
  • 52. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 52 NUMBER OF JET: 𝑛 = 𝑄 π‘ž Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ π‘ž Discharge of Single Jet π‘š3 𝑠⁄ NUMBER OF BUCKET: 𝑍 = 15 + 𝐷 2𝑑 Symbol Description Unit 𝑑 Diameter of Jet π‘š 𝐷 Diameter of Runner π‘š DIMENSIONS OF BUCKET: 𝐴π‘₯π‘–π‘Žπ‘™ π‘Šπ‘–π‘‘π‘‘β„Ž 𝐡 = 4.5𝑑 π‘…π‘Žπ‘‘π‘–π‘Žπ‘™ πΏπ‘’π‘›π‘”π‘‘β„Ž 𝐿 = 2.5𝑑 π·π‘’π‘π‘‘β„Ž π‘œπ‘“ π΅π‘’π‘π‘˜π‘’π‘‘ 𝑇 = 𝑑 Symbol Description Unit 𝑑 Diameter of Jet π‘š KINETIC ENERGY OF JET: 𝐾. 𝐸 π‘œπ‘“ 𝐽𝑒𝑑 = 1 2 π‘š 𝑉1 2 𝑆𝑖𝑛𝑐𝑒 π‘š = πœŒπ΄π‘‰ π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ 𝐾. 𝐸 π‘œπ‘“ 𝐽𝑒𝑑 = 1 2 𝜌 βˆ— 𝐴 βˆ— 𝑉1 βˆ— 𝑉1 2
  • 53. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 53 𝑆𝑖𝑛𝑐𝑒 𝑄 = 𝐴𝑉 π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’ 𝐾. 𝐸 π‘œπ‘“ 𝐽𝑒𝑑 = 1 2 𝜌 βˆ— 𝑄 βˆ— 𝑉1 2 POWER LOST IN NOZZLE: 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ = 𝐾𝑖𝑛𝑒𝑑𝑖𝑐 πΈπ‘›π‘’π‘Ÿπ‘”π‘¦ + π‘ƒπ‘œπ‘€π‘’π‘Ÿ πΏπ‘œπ‘ π‘‘ 𝑖𝑛 π‘π‘œπ‘§π‘§π‘™π‘’ POWER LOST IN RUNNER: 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ = π‘ƒπ‘œπ‘€π‘’π‘Ÿ π‘œπ‘“ π‘†β„Žπ‘Žπ‘“π‘‘ + π‘ƒπ‘œπ‘€π‘’π‘Ÿ πΏπ‘œπ‘ π‘‘ 𝑖𝑛 π‘π‘œπ‘§π‘§π‘™π‘’ + π‘ƒπ‘œπ‘€π‘’π‘Ÿ πΏπ‘œπ‘ π‘‘ 𝑖𝑛 π‘…π‘’π‘›π‘›π‘’π‘Ÿ + π‘ƒπ‘œπ‘€π‘’π‘Ÿ πΏπ‘œπ‘ π‘‘ 𝐷𝑒𝑒 π‘‘π‘œ π‘€π‘’π‘β„Žπ‘Žπ‘›π‘–π‘π‘Žπ‘™ π‘…π‘’π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’ RESULTANT FORCE ON BUCKET: 𝐹 = πœŒπ‘„ [ 𝑣 𝑀1 + 𝑣 𝑀2] Symbol Description Unit 𝐹 Resultant Force on Bucket 𝑁 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ TORQUE: 𝑇 = 𝐹 βˆ— 𝐷 2 Symbol Description Unit 𝐹 Resultant Force on Bucket 𝑁 𝐷 Diameter of Runner π‘š 𝑇 Torque 𝑁 βˆ’ π‘š
  • 54. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 54 POWER: 𝑃 = 2πœ‹π‘π‘‡ 60 Symbol Description Unit 𝑃 Power π‘Š 𝑇 Torque 𝑁 βˆ’ π‘š N Speed of Shaft Rpm SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed
  • 55. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 55 REACTION TURBINE: INWARD FLOW REACTION TURBINE: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Degree
  • 56. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 56 Direction of Motion of Vane TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π‘‘1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) External Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π‘‘2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) Internal Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑀1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑀1 sin πœƒ = 𝑣𝑓1 𝑣 π‘Ÿ1 cos πœƒ = 𝑣 𝑀1 βˆ’ 𝑒1 𝑣 π‘Ÿ1
  • 57. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 57 tan πœƒ = 𝑣𝑓1 𝑣 𝑀1 βˆ’ 𝑒1 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 π‘Ÿ1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑀1 βˆ’ 𝑒1)2 Symbol Description Unit 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ DISCHARGE: 𝑄 = πœ‹π‘‘1 𝑏1 𝑣𝑓1 = πœ‹π‘‘2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit
  • 58. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 58 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝑄 Discharge π‘š3 𝑠⁄ 𝐴 Area of Runner π‘š2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet π‘š 𝑠⁄ MASS OF WATER FLOWING THROUGH THE RUNNER: π‘š = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ HEAD AT INLET OF TURBINE: 𝐻 = 1 𝑔 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 + 𝑣𝑓1 2 2𝑔 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
  • 59. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 59 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 βˆ— 𝑄 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 𝑣 𝑀1 𝑒1 𝑔𝐻 πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 βˆ’ π»π‘’π‘Žπ‘‘ πΏπ‘œπ‘ π‘  π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π»
  • 60. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 60 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š SPEED RATIO: 𝐾 𝑒 = 𝑒 √2𝑔𝐻 𝐾 𝑒 = 0.6 βˆ’ 0.9 Symbol Description Unit 𝑒 Tangential Velocity π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 βˆ’ 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio
  • 61. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 61 SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed OUTWARD FLOW REACTION TURBINE: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄
  • 62. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 62 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π‘‘1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) Internal Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π‘‘2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) External Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM:
  • 63. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 63 sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑀1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑀1 sin πœƒ = 𝑣𝑓1 𝑣 π‘Ÿ1 cos πœƒ = 𝑣 𝑀1 βˆ’ 𝑒1 𝑣 π‘Ÿ1 tan πœƒ = 𝑣𝑓1 𝑣 𝑀1 βˆ’ 𝑒1 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree
  • 64. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 64 RELATIVE VELOCITY AT INLET: 𝑣 π‘Ÿ1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑀1 βˆ’ 𝑒1)2 Symbol Description Unit 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ DISCHARGE: 𝑄 = πœ‹π‘‘1 𝑏1 𝑣𝑓1 = πœ‹π‘‘2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝑄 Discharge π‘š3 𝑠⁄ 𝐴 Area of Runner π‘š2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet π‘š 𝑠⁄ MASS OF WATER FLOWING THROUGH THE RUNNER: π‘š = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄
  • 65. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 65 INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 βˆ— 𝑄 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 𝑣 𝑀1 𝑒1 𝑔𝐻 πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 βˆ’ π»π‘’π‘Žπ‘‘ πΏπ‘œπ‘ π‘  π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š
  • 66. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 66 OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š SPEED RATIO: 𝐾 𝑒 = 𝑒 √2𝑔𝐻 𝐾 𝑒 = 0.6 βˆ’ 0.9 Symbol Description Unit 𝑒 Tangential Velocity π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 βˆ’ 0.3
  • 67. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 67 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed
  • 68. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 68 FRANCIS TURBINE: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree
  • 69. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 69 TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π‘‘1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) External Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π‘‘2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) Internal Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑀1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑀1 sin πœƒ = 𝑣𝑓1 𝑣 π‘Ÿ1 cos πœƒ = 𝑣 𝑀1 βˆ’ 𝑒1 𝑣 π‘Ÿ1 tan πœƒ = 𝑣𝑓1 𝑣 𝑀1 βˆ’ 𝑒1
  • 70. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 70 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 π‘Ÿ1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑀1 βˆ’ 𝑒1)2 Symbol Description Unit 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ DISCHARGE: 𝑄 = πœ‹π‘‘1 𝑏1 𝑣𝑓1 = πœ‹π‘‘2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š
  • 71. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 71 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝑄 Discharge π‘š3 𝑠⁄ 𝐴 Area of Runner π‘š2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet π‘š 𝑠⁄ CIRCUMFERENTIAL AREA OF RUNNER: 𝐴 = πœ‹π‘‘1 𝑏1 = πœ‹π‘‘2 𝑏2 Symbol Description Unit 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet π‘š 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet π‘š 𝐴 Circumferential Area of Runner π‘š2 MASS OF WATER FLOWING THROUGH THE RUNNER: π‘š = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š
  • 72. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 72 POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 βˆ— 𝑄 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 𝑣 𝑀1 𝑒1 𝑔𝐻 πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 βˆ’ π»π‘’π‘Žπ‘‘ πΏπ‘œπ‘ π‘  π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
  • 73. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 73 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š SPEED RATIO: 𝐾 𝑒 = 𝑒 √2𝑔𝐻 𝐾 𝑒 = 0.6 βˆ’ 0.9 Symbol Description Unit 𝑒 Tangential Velocity π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 βˆ’ 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio
  • 74. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 74 BREADTH RATIO: 𝑛 = 𝑏1 𝑑1 𝑛 = 0.1 βˆ’ 0.4 Symbol Description Unit 𝑏1 Width of Runner at Inlet π‘š 𝑑1 Diameter of Runner at Inlet π‘š 𝑛 Breadth Ratio SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed
  • 75. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 75 KAPLAN TURBINE: Symbol Description Unit 𝑒1&𝑒2 Tangential Velocity of Runner at Inlet & Outlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1&𝑣 π‘Ÿ2 Relative Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣 𝑀1&𝑣 𝑀2 Whirl Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree πœ™ Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree
  • 76. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 76 TANGENTIAL VELOCITY AT INLET: 𝑒1 = πœ‹π· π‘œ 𝑁 60 Symbol Description Unit 𝐷 π‘œ Inlet (or) External Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š TANGENTIAL VELOCITY AT OUTLET: 𝑒2 = πœ‹π· 𝑏 𝑁 60 = πœ‹π·β„Ž 𝑁 60 Symbol Description Unit 𝐷 𝑏 π‘œπ‘Ÿ π·β„Ž Outlet (or) Boss (or) Hub Diameter π‘š 𝑁 Speed of Turbine π‘Ÿπ‘π‘š FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑀1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑀1 sin πœƒ = 𝑣𝑓1 𝑣 π‘Ÿ1 cos πœƒ = 𝑣 𝑀1 βˆ’ 𝑒1 𝑣 π‘Ÿ1 tan πœƒ = 𝑣𝑓1 𝑣 𝑀1 βˆ’ 𝑒1
  • 77. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 77 Symbol Description Unit 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑉1 Absolute Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree πœƒ Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 π‘Ÿ1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑀1 βˆ’ 𝑒1)2 Symbol Description Unit 𝑣 π‘Ÿ1 Relative Velocity at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ DISCHARGE: 𝑄 = πœ‹ 4 [𝐷0 2 βˆ’ 𝐷 𝑏 2 ]𝑣𝑓1 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐷0 Inlet (or) External Diameter π‘š 𝐷 𝑏 π‘œπ‘Ÿ π·β„Ž Outlet (or) Boss (or) Hub Diameter π‘š
  • 78. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 78 𝑄 Discharge π‘š3 𝑠⁄ CIRCUMFERENTIAL AREA OF RUNNER: 𝐴 = πœ‹ 4 [𝐷0 2 βˆ’ 𝐷 𝑏 2 ] Symbol Description Unit 𝐷0 Inlet (or) External Diameter π‘š 𝐷 𝑏 π‘œπ‘Ÿ π·β„Ž Outlet (or) Boss (or) Hub Diameter π‘š 𝐴 Circumferential Area of Runner π‘š2 MASS OF WATER FLOWING THROUGH THE RUNNER: π‘š = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝜌 Density π‘˜π‘” π‘š3⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 βˆ— 𝑄 βˆ— 𝑣 𝑀1 βˆ— 𝑒1 Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄
  • 79. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 79 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ HYDRAULIC EFFICIENCY: πœ‚β„Žπ‘¦π‘‘ = 𝑣 𝑀1 𝑒1 𝑔𝐻 πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 βˆ’ π»π‘’π‘Žπ‘‘ πΏπ‘œπ‘ π‘  π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 Symbol Description Unit 𝑒1 Tangential Velocity of Runner at Inlet π‘š 𝑠⁄ 𝑣 𝑀1 Whirl Velocity at Inlet π‘š 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š OVERALL EFFICIENCY: πœ‚ π‘œ = π‘†β„Žπ‘Žπ‘“π‘‘ π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝐼𝑛𝑝𝑒𝑑 π‘ƒπ‘œπ‘€π‘’π‘Ÿ πœ‚ π‘œ = 𝑆. 𝑃 πœŒπ‘”π‘„π» Symbol Description Unit 𝜌 Density π‘˜π‘” π‘š3⁄ 𝑄 Discharge π‘š3 𝑠⁄ 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐻 Head π‘š 𝑆. 𝑃 Shaft Power π‘Š
  • 80. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 80 SPEED RATIO: 𝐾 𝑒 = 𝑒 √2𝑔𝐻 𝐾 𝑒 = 0.6 βˆ’ 0.9 Symbol Description Unit 𝑒 Tangential Velocity π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾 𝑒 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 βˆ’ 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet π‘š 𝑠⁄ 𝐻 Head π‘š 𝑔 Acceleration due to Gravity π‘š 𝑠2⁄ 𝐾𝑓 Flow Ratio SPECIFIC SPEED: 𝑁𝑠 = π‘βˆš 𝑄 𝐻 3 4⁄ 𝑁𝑠 = π‘βˆš 𝑃 𝐻 5 4⁄
  • 81. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 81 Symbol Description Unit 𝑄 Discharge π‘š3 𝑠⁄ 𝐻 Head π‘š 𝑃 Power π‘˜π‘Š 𝑁 Speed π‘Ÿπ‘π‘š 𝑁𝑠 Specific Speed DRAFT TUBE:
  • 82. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 82 Symbol Description Unit 𝑉1&𝑉2 Velocity at Inlet & Outlet π‘š 𝑠⁄ 𝐻𝑠 Vertical Height of Draft Tube Above Tail Race π‘š 𝑦 Distance of Bottom of Draft Tube from Tail Race π‘š FROM BERNOULLI’S EQUATION: 𝑃1 πœŒπ‘” + 𝑉1 2 2𝑔 + 𝑧1 = 𝑃2 πœŒπ‘” + 𝑉2 2 2𝑔 + 𝑧2 + β„Ž 𝑓 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Inlet & Outlet of Draft Tube 𝑁 π‘š2⁄ 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube π‘š 𝑠⁄ 𝑧1 & 𝑧2 Datum Head Inlet & Outlet of Draft Tube π‘š β„Ž 𝑓 Head Loss π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ LENGTH OF DRAFT TUBE: 𝐿 = 𝐻𝑠 + 𝑦 Symbol Description Unit 𝐿 Length of Draft Tube π‘š 𝐻𝑠 Vertical Height of Draft Tube Above Tail Race π‘š 𝑦 Distance of Bottom of Draft Tube from Tail Race π‘š EFFICIENCY OF DRAFT TUBE: πœ‚ 𝑑 = ( 𝑉1 2 2𝑔 βˆ’ 𝑉2 2 2𝑔 ) βˆ’ β„Ž 𝑓 𝑉1 2 2𝑔
  • 83. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 83 Symbol Description Unit 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube π‘š 𝑠⁄ β„Ž 𝑓 Head Loss π‘š 𝑔 Acceleration due to gravity π‘š 𝑠2⁄ HYDRAULIC EFFICIENCY OF DRAFT TUBE: πœ‚β„Žπ‘¦π‘‘ = π»π‘’π‘Žπ‘‘ π‘ˆπ‘‘π‘–π‘™π‘–π‘§π‘’π‘‘ 𝑏𝑦 π‘‡π‘’π‘Ÿπ‘π‘–π‘›π‘’ π»π‘’π‘Žπ‘‘ 𝐼𝑛𝑙𝑒𝑑 π‘œπ‘“ π‘‡π‘’π‘Ÿπ‘π‘–π‘›π‘’ πœ‚β„Žπ‘¦π‘‘ = 𝐻 βˆ’ β„Ž 𝑓𝑑 βˆ’ β„Ž 𝑓𝑑 βˆ’ 𝑉2 2 2𝑔 𝑃1 πœŒπ‘” + 𝑉1 2 2𝑔 + 𝑧1 Symbol Description Unit 𝑃1 Pressure at Inlet of Draft Tube 𝑁 π‘š2⁄ 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube π‘š 𝑠⁄ 𝑧1 Datum Head Inlet of Draft Tube π‘š β„Ž 𝑓 Head Loss π‘š 𝜌 Density of Liquid π‘˜π‘” π‘š3⁄ 𝑔 Acceleration due to gravity π‘š 𝑠2⁄