FLUID MECHANICS AND MACHINERY FORMULA BOOK
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This document contains a formula book for fluid mechanics and machinery prepared by three professors at R.M.K College of Engineering and Technology. It includes formulas for fluid properties like density, specific volume, specific weight, viscosity, and surface tension. Formulas are also provided for continuity equation, Bernoulli's equation, and coefficient of discharge. The book is intended as a reference for students in the Department of Mechanical Engineering taking the course CE6451 - Fluid Mechanics and Machinery.
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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 = 𝑚 𝑉 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ m Mass Kg V Volume m3 SPECIFIC VOLUME (v): 𝑣 = 𝑉 𝑚 = 1 𝜌 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ m Mass Kg V Volume m3 𝑣 Specific Volume 𝑚3 𝑘𝑔⁄ SPECIFIC WEIGTH or WEIGTH DENSITY (w): 𝑤 = 𝑊 𝑉 = 𝑚𝑔 𝑉 = 𝜌𝑔 𝑆𝑖𝑛𝑐𝑒 𝑊 = 𝑚𝑔 𝑎𝑛𝑑 𝜌 = 𝑚 𝑉⁄ Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 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 𝑤 Specific Weight 𝑁 𝑚3⁄ g Acceleration due to gravity 𝑚 𝑠2⁄ SPECIFIC GRAVITY (S): 𝑆 = 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑢𝑖𝑑 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑙𝑢𝑖𝑑 𝑆 = 𝑀𝑎𝑠𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑢𝑖𝑑 𝑀𝑎𝑠𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑙𝑢𝑖𝑑 Symbol Description Unit 𝑆 Specific Gravity No unit 𝜌 Density or Mass Density 𝑘𝑔 𝑚3⁄ 𝑤 Specific Weight 𝑁 𝑚3⁄ 𝑤 𝑤𝑎𝑡𝑒𝑟 Specific Weight of Standard Fluid (Water) = 9.81 𝑁 𝑚3⁄ 𝜌 𝑤𝑎𝑡𝑒𝑟 Mass Density of Standard Fluid (Water) = 1000 𝑘𝑔 𝑚3⁄ VISCOSITY (μ): 𝜏 𝛼 𝑑𝑢 𝑑𝑦 𝜏 = 𝜇 𝑑𝑢 𝑑𝑦 Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝑑𝑢 Change in Velocity 𝑚 𝑠⁄ 𝑑𝑦 Change in Distance 𝑚
- 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 (μ): 𝜇 = 𝜏 𝑑𝑢 𝑑𝑦⁄ Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ 𝜇 Dynamic Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝑑𝑢 Change in Velocity 𝑚 𝑠⁄ 𝑑𝑦 Change in Distance 𝑚 𝑑𝑢 𝑑𝑦⁄ Rate of Shear Strain 1 𝑠⁄ Unit Conversion: 1 𝑁𝑠 𝑚2 = 10 𝑝𝑜𝑖𝑠𝑒 1 𝐶𝑒𝑛𝑡𝑖𝑝𝑜𝑖𝑠𝑒 = 1 100 𝑝𝑜𝑖𝑠𝑒 1 𝑝𝑜𝑖𝑠𝑒 = 0.1 𝑁𝑠 𝑚2 KINEMATIC VISCOSITY (γ): 𝛾 = 𝜇 𝜌 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝜇 Dynamic Viscosity 𝑁 − 𝑠 𝑚2⁄ γ Kinematic Viscosity 𝑚2 𝑠⁄ Unit Conversion: 1 𝑠𝑡𝑜𝑘𝑒 = 10−4 𝑚2 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 𝐶𝑒𝑛𝑡𝑖𝑠𝑡𝑜𝑘𝑒 = 1 100 𝑠𝑡𝑜𝑘𝑒 VISCOSITY PROBLEMS FOR PLATE TYPE: FORCE (F): 𝜏 = 𝐹 𝐴 Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ F Force N A Area of the plate 𝑚2 POWER (P): 𝑃 = 𝐹 ∗ 𝑑𝑢 Symbol Description Unit 𝑃 Power 𝑊 F Force N 𝑑𝑢 Change in Velocity 𝑚 𝑠⁄ VISCOSITY PROBLEMS FOR SHAFT TYPE: VELOCITY OF SHAFT (u): 𝑢 = 𝜋𝐷𝑁 60 Symbol Description Unit 𝐷 Diameter of Shaft 𝑚 N Speed of Shaft Rpm 𝑢 Velocity 𝑚 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 = 𝜋𝐷𝐿 Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ F Force N A Circumference of Shaft 𝑚2 𝐷 Diameter of Shaft 𝑚 𝐿 Length of Shaft 𝑚 TORQUE ON SHAFT (T): 𝑇 = 𝐹 ∗ 𝐷 2 Symbol Description Unit 𝑇 Torque 𝑁 − 𝑚 F Force N 𝐷 Diameter of Shaft 𝑚 POWER ON SHAFT (P): 𝑃 = 2𝜋𝑁𝑇 60 Symbol Description Unit 𝑃 Power 𝑊 𝑇 Torque 𝑁 − 𝑚 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 (ω): 𝜔 = 2𝜋𝑁 60 Symbol Description Unit 𝜔 Angular Velocity 𝑟𝑎𝑑 𝑠𝑒𝑐⁄ N Speed of Shaft Rpm ANGLE (θ): 𝑡𝑎𝑛𝜃 = 𝑟1 − 𝑟2 𝐻 Symbol Description Unit 𝑟1 Outer Radius 𝑚 𝑟2 Inner Radius 𝑚 𝐻 Height 𝑚 POWER (P): 𝑃 = 2𝜋𝑁𝑇 60 Symbol Description Unit 𝑃 Power 𝑊 𝑇 Torque 𝑁 − 𝑚
- 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 = 𝜋𝜇𝜔 2ℎ𝑠𝑖𝑛𝜃 ( 𝑟1 4 − 𝑟2 4) Symbol Description Unit 𝜇 Dynamic Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝑇 Torque 𝑁 − 𝑚 𝜔 Angular Velocity 𝑟𝑎𝑑 𝑠𝑒𝑐⁄ ℎ Thickness of Oil 𝑚 𝑟1 Outer Radius 𝑚 𝑟2 Inner Radius 𝑚 CAPILLARITY: HEIGHT OF LIQUID IN TUBE (h): ℎ = 4𝜎𝑐𝑜𝑠𝜃 𝜌𝑔𝑑 Symbol Description Unit ℎ Height of Liquid in Tube 𝑚 𝜎 Surface Tension 𝑁 𝑚⁄ 𝜃 Angle of Contact between Liquid and Tube 𝑟𝑎𝑑 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝑑 Diameter of Tube 𝑚 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 𝑚2⁄ 𝜎 Surface Tension 𝑁 𝑚⁄ 𝑑 Diameter of Droplet 𝑚 PRESSURE IN BUBBLE (P): 𝑃 = 8𝜎 𝑑 Symbol Description Unit 𝑃 Pressure 𝑁 𝑚2⁄ 𝜎 Surface Tension 𝑁 𝑚⁄ 𝑑 Diameter of Bubble 𝑚 PRESSURE IN LIQUID JET (P): 𝑃 = 2𝜎 𝑑 Symbol Description Unit 𝑃 Pressure 𝑁 𝑚2⁄ 𝜎 Surface Tension 𝑁 𝑚⁄ 𝑑 Diameter of Jet 𝑚 CONTINUITY EQUATION: 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 + 𝜕𝑤 𝜕𝑧 = 0 [ 𝐹𝑜𝑟 3 − 𝐷 𝑓𝑙𝑜𝑤] 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 + = 0 [ 𝐹𝑜𝑟 2 − 𝐷 𝑓𝑙𝑜𝑤] 𝜕 𝜕𝑟 ( 𝑟𝑢 𝑟) + 𝜕 𝜕𝜃 ( 𝑢 𝜃) = 0[ 𝐹𝑜𝑟 𝑝𝑜𝑙𝑎𝑟 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠]
- 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: 𝜕𝑃 𝜌 + 𝑣. 𝑑𝑣 + 𝑔. 𝑑𝑧 = 0 𝑃1 𝜌𝑔 + 𝑣1 2 2𝑔 + 𝑧1 = 𝑃2 𝜌𝑔 + 𝑣2 2 2𝑔 + 𝑧2 + ℎ 𝑓 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁 𝑚2⁄ 𝑣1 & 𝑣2 Velocity at Section 1 & 2 𝑚 𝑠⁄ 𝑧1 & 𝑧2 Datum Head at Section 1 & 2 𝑚 ℎ 𝑓 Head Loss 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ COEFFICIENT OF DISCHARGE: 𝐶 𝑑 = 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 COEFFICIENT OF VELOCITY: 𝐶𝑣 = 𝑣 𝐴𝑐𝑡𝑢𝑎𝑙 𝑣 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 DISCHARGE OF VENTURIMETER AND ORIFICEMETER: 𝑄 = 𝐶 𝑑 𝑎1 𝑎2 √( 𝑎1 2 − 𝑎1 2) √2𝑔ℎ Symbol Description Unit 𝑎1 & 𝑎2 Area at Section 1 & 2 𝑚2 ℎ Pressure Difference between Section 1 & 2 ( 𝑃1− 𝑃2 𝜌𝑔 ) 𝑚
- 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 𝐶 𝑑 Coefficient of Discharge 𝑥 Difference in Mercury Level 𝑚 ℎ = 𝑥 (1 − 𝑆 𝑚 𝑆 ) [ 𝑤ℎ𝑒𝑛 𝑆 > 𝑆 𝑚] ℎ = 𝑥 ( 𝑆 𝑚 𝑆 − 1) [ 𝑤ℎ𝑒𝑛 𝑆 𝑚 > 𝑆] ℎ = ( 𝑃1 𝜌𝑔 + 𝑍1) − ( 𝑃2 𝜌𝑔 + 𝑍2) [ 𝐼𝑛𝑐𝑙𝑖𝑛𝑒𝑑 𝑉𝑒𝑛𝑡𝑢𝑟𝑖𝑚𝑒𝑡𝑒𝑟] MOMENTUM EQUATION: 𝐹 = 𝑑 (𝑚𝑣) 𝑑𝑡 FORCE ACTING IN X – DIRECTION: 𝐹𝑥 = 𝜌𝑄 ( 𝑣1 − 𝑣2 𝑐𝑜𝑠𝜃) + 𝑃1 𝐴1 − 𝑃2 𝐴2 𝑐𝑜𝑠𝜃 FORCE ACTING IN Y – DIRECTION: 𝐹𝑦 = 𝜌𝑄 (− 𝑣2 𝑠𝑖𝑛𝜃) − 𝑃2 𝐴2 𝑠𝑖𝑛𝜃 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁 𝑚2⁄ 𝑣1 & 𝑣2 Velocity at Section 1 & 2 𝑚 𝑠⁄ 𝐴1 & 𝐴2 Area at Section 1 & 2 𝑚 𝜃 Angle of the Bend 𝐷𝑒𝑔𝑟𝑒𝑒 𝑄 Discharge 𝑚3 𝑠⁄ RESULTANT FORCE: 𝐹𝑅 = √𝐹𝑥 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: 𝑡𝑎𝑛𝜃 = 𝐹𝑦 𝐹𝑥 MOMENT OF MOMENTUM EQUATION: 𝑇 = 𝜌𝑄 ( 𝑣2 𝑟2 − 𝑣1 𝑟1) Symbol Description Unit 𝑇 Torque 𝑁 − 𝑚 𝑣1 & 𝑣2 Velocity at Section 1 & 2 𝑚 𝑠⁄ 𝑟1 & 𝑟2 Radius of Curvature at Section 1 & 2 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄
- 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 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 + 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐻𝑒𝑎𝑑 + 𝐷𝑎𝑡𝑢𝑚 𝐻𝑒𝑎𝑑 𝑇𝐸𝐿 = 𝑃 𝜌𝑔 + 𝑣2 2𝑔 + 𝑍 Symbol Description Unit 𝑃 Pressure 𝑁 𝑚2⁄ 𝑣 Velocity 𝑚 𝑠⁄ 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑍 Datum Head 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ HYDRAULIC ENERGY LINE (HEL): 𝐻𝐸𝐿 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 + 𝐷𝑎𝑡𝑢𝑚 𝐻𝑒𝑎𝑑 𝑇𝐸𝐿 = 𝑃 𝜌𝑔 + 𝑍 HAGEN POISEUILLE’S EQUATION: SHEAR STRESS: 𝜏 = − 𝜕𝑝 𝜕𝑥 ∗ 𝑟 2 Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝑟 Radius of pipe 𝑚 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: 𝑢 = − 1 4𝜇 ∗ 𝜕𝑝 𝜕𝑥 ∗ (𝑅2 − 𝑟2 ) Symbol Description Unit 𝑢 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝑟 Radius of pipe 𝑚 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ MAXIMUM VELOCITY: 𝑢 = − 1 4𝜇 ∗ 𝜕𝑝 𝜕𝑥 ∗ (𝑅2 ) Symbol Description Unit 𝑢 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ AVERAGE VELOCITY: 𝑢̅ = − 1 4𝜇 ∗ 𝜕𝑝 𝜕𝑥 ∗ (𝑅2 ) Symbol Description Unit 𝑢̅ Average Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ RATIO BETWEEN MAXIMUM VELOCITY AND AVERAGE VELOCITY: 𝑢 𝑚𝑎𝑥 𝑢̅ = 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: 𝑢 = − 1 8𝜇 ∗ 𝜕𝑝 𝜕𝑥 ∗ 𝜋 ∗ 𝑅4 Symbol Description Unit 𝑢 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ PRESSURE DIFFERENCE: 𝑃1 − 𝑃2 = 32𝜇𝑢̅𝐿 𝐷2 Symbol Description Unit 𝑢̅ Average Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝐿 Length of Pipe 𝑚 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝐷 Diameter of Pipe 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LOSS OF HEAD: ℎ 𝑓 = 𝑃1 − 𝑃2 𝜌𝑔 = 32𝜇𝑢̅𝐿 𝜌𝑔𝐷2 [ 𝑓𝑜𝑟 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝑓𝑙𝑜𝑤] DARCY WEISBACH EQUATION: ℎ 𝑓 = 𝑃1 − 𝑃2 𝜌𝑔 = 4𝑓𝐿𝑣2 2𝑔𝑑 [ 𝑓𝑜𝑟 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑓𝑙𝑜𝑤] Symbol Description Unit 𝑣 Velocity of Fluid in Pipe 𝑚 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 𝑚 𝑓 Friction Factor 𝑑 Diameter of Pipe 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ REYNOLD’S NUMBER: 𝑅 𝑒 = 𝜌𝑣𝑑 𝜇 𝑓 = 0.079 𝑅 𝑒 0.25 [ 𝐹𝑜𝑟 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝐹𝑙𝑜𝑤] 𝑓 = 16 𝑅 𝑒 [ 𝐹𝑜𝑟 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝐹𝑙𝑜𝑤] 𝑅 𝑒 < 2000 𝑇ℎ𝑒𝑛 𝑡ℎ𝑒 𝐹𝑙𝑜𝑤 𝑖𝑠 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝑅 𝑒 > 2000 𝑇ℎ𝑒𝑛 𝑡ℎ𝑒 𝐹𝑙𝑜𝑤 𝑖𝑠 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 Symbol Description Unit 𝑣 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝑑 Diameter of Pipe 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ MAJOR LOSS IN PIPES: ℎ 𝑓 = 32𝜇𝑢̅𝐿 𝜌𝑔𝑑2 [ 𝑓𝑜𝑟 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝑓𝑙𝑜𝑤] ℎ 𝑓 = 4𝑓𝐿𝑣2 2𝑔𝑑 [ 𝑓𝑜𝑟 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑓𝑙𝑜𝑤]
- 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 𝑢̅ & 𝑣 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝑑 Diameter of Pipe 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝑙 Length of Pipe 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝑓 Friction Factor MINOR LOSS IN PIPES: LOSS DUE TO SUDDEN ENLARGEMENT: ℎ 𝑒 = ( 𝑣1 − 𝑣2)2 2𝑔 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Fluid in Pipe at Inlet and Outlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LOSS DUE TO SUDDEN CONTRACTION: ℎ 𝑐 = 𝐾𝑣2 2𝑔 𝐾 = ( 1 𝐶𝑐 − 1) 2 ℎ 𝑐 = 0.5𝑣2 2𝑔 [ 𝐼𝑓 𝐶𝑐 𝑛𝑜𝑡 𝑔𝑖𝑣𝑒𝑛] Symbol Description Unit 𝑣 Velocity of Fluid at Outlet 𝑚 𝑠⁄ 𝐶𝑐 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 𝑚 𝑠2⁄ LOSS AT ENTRANCE OF PIPE: ℎ𝑖 = 0.5𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Fluid at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LOSS AT EXIT OF PIPE: ℎ 𝑜 = 𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Fluid at Outlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LOSS DUE TO GRADUAL CONTRACTION: ℎ 𝑒 = 𝐾( 𝑣1 − 𝑣2)2 2𝑔 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Fluid in Pipe at Inlet and Outlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝐾 Coefficient of Contraction LOSS AT BEND OF PIPE: ℎ 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 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝐾 Coefficient of Bend LOSS AT DUE TO VARIOUS FITTINGS: ℎ 𝑣 = 𝐾𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Flow 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝐾 Coefficient of Fittings LOSS AT DUE TO OBSTRUCTION: ℎ 𝑣 = 𝑣2 2𝑔 ( 𝐴 𝐶𝑐 ( 𝐴 − 𝑎) − 1) 𝐶𝑐 = 𝐴 𝑐 ( 𝐴 − 𝑎) Symbol Description Unit 𝑣 Velocity of Flow 𝑚 𝑠⁄ 𝐴 Area of Pipe 𝑚2 𝑎 Area of Obstruction 𝑚2 𝐴 𝑐 Area of Vena Contraction 𝑚2 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: ℎ 𝑓 = ℎ 𝑓1 + ℎ 𝑓2 ℎ 𝑓 = 4𝑓𝑙1 𝑣1 2 2𝑔𝑑1 + 4𝑓𝑙2 𝑣2 2 2𝑔𝑑2 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Flow at Pipe 1 & 2 𝑚 𝑠⁄ 𝐴1& 𝐴2 Area of Pipe 1 & 2 𝑚2 𝑑1& 𝑑2 Diameter of Pipe 1 & 2 𝑚 𝑙1& 𝑙2 Length of Pipe 1 & 2 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝑓 Friction Factor WHEN PIPES ARE CONNECTED IN PARALLEL: DISCHARGE: 𝑄 = 𝑄1 + 𝑄2 𝑄 = 𝐴1 𝑣1 + 𝐴2 𝑣2 HEAD LOSS: ℎ 𝑓 = ℎ 𝑓1 = ℎ 𝑓2 ℎ 𝑓 = 4𝑓𝑙1 𝑣1 2 2𝑔𝑑1 = 4𝑓𝑙2 𝑣2 2 2𝑔𝑑2 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Flow at Pipe 1 & 2 𝑚 𝑠⁄ 𝐴1& 𝐴2 Area of Pipe 1 & 2 𝑚2 𝑑1& 𝑑2 Diameter of Pipe 1 & 2 𝑚 𝑙1& 𝑙2 Length of Pipe 1 & 2 𝑚
- 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 𝑚 𝑠2⁄ 𝑓 Friction Factor EQUIVALENT PIPE: 𝐿 𝐷5 = 𝐿1 𝐷1 5 + 𝐿2 𝐷2 5 + 𝐿3 𝐷3 5 + ⋯ + 𝐿 𝑛 𝐷 𝑛 5 Symbol Description Unit 𝐷 Diameter of Pipe 𝑚 𝐿 Length of Pipe 𝑚 BOUNDARY LAYER: DISPLACEMENT THICKNESS: 𝛿∗ = ∫ (1 − 𝑢 𝑈 ) 𝛿 0 𝑑𝑦 MOMENTUM THICKNESS: 𝜃 = ∫ 𝑢 𝑈 (1 − 𝑢 𝑈 ) 𝛿 0 𝑑𝑦 MOMENTUM THICKNESS: 𝛿∗∗ = ∫ 𝑢 𝑈 (1 − 𝑢2 𝑈2 ) 𝛿 0 𝑑𝑦 Symbol Description Unit 𝑢 𝑈 Velocity Distribution 𝛿 Boundary layer thickness SHEAR STRESS: 𝜏0 𝜌𝑈2 = 𝜕𝜃 𝜕𝑥
- 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 𝜃 = ∫ 𝑢 𝑈 (1 − 𝑢 𝑈 ) 𝛿 0 𝑑𝑦 DRAG FORCE: 𝐹 𝐷 = ∫ 𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑒𝑠𝑠 ∗ 𝐴𝑟𝑒𝑎 𝐿 0 𝐹 𝐷 = ∫ 𝜏0 ∗ 𝑏 ∗ 𝑑𝑥 𝐿 0 LOCAL COEFFICIENT OF DRAG: 𝐶 𝐷 ∗ = 𝜏0 1 2 𝜌𝑈2 AVERAGE COEFFICIENT OF DRAG: 𝐶 𝐷 = 𝐹 𝐷 1 2 𝜌𝐴𝑈2 Symbol Description Unit 𝜏0 Shear Stress 𝑁 𝑚2⁄ 𝑏 Width of Plate 𝑚 𝑈 Free Stream Velocity 𝑚 𝑠⁄ 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝐴 Area 𝑚2 𝐹 𝐷 Drag Force 𝑁 BLASIUS’S SOLUTION: BOUNDARY LAYER THICKNESS: 𝛿 = 4.91𝑥 √ 𝑅 𝑒𝑥
- 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: 𝐶 𝐷 ∗ = 0.664 √ 𝑅 𝑒𝑥 AVERAGE COEFFICIENT OF DRAG: 𝐶 𝐷 = 1.328 √ 𝑅 𝑒𝐿 Symbol Description Unit 𝑅 𝑒𝑥 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 ω rad/sec T-1 Acceleration a m/s2 LT-2 Angular Acceleration α rad/sec2 T-2 Speed N Rpm T-1 Discharge Q m3 /s L3 T-1 Kinematic Viscosity γ cm2 /s L2 T-1 Dynamic Viscosity μ 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 τ N/m2 ML-1 T-2 Surface Tension σ 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 η No Unit Dimensionless SIMILARITY: GEOMETRIC SIMILARITY: 𝐿 𝑝 𝐿 𝑚 = 𝑏 𝑝 𝑏 𝑚 = 𝐷 𝑝 𝐷 𝑚 = 𝐿 𝑟 𝐴 𝑝 𝐴 𝑚 = 𝐿 𝑝 𝐿 𝑚 ∗ 𝑏 𝑝 𝑏 𝑚 = 𝐿 𝑟 ∗ 𝐿 𝑟 = 𝐿 𝑟 2 𝑉𝑝 𝑉𝑚 = 𝐿 𝑝 𝐿 𝑚 ∗ 𝑏 𝑝 𝑏 𝑚 ∗ 𝑡 𝑝 𝑡 𝑚 = 𝐿 𝑟 ∗ 𝐿 𝑟 ∗ 𝐿 𝑟 = 𝐿 𝑟 3 Symbol Description Unit 𝐿 𝑝&𝐿 𝑚 Length of Prototype & Model 𝑚 𝑏 𝑝&𝑏 𝑚 Breadth of Prototype & Model 𝑚 𝐷 𝑝&𝐷 𝑚 Diameter of Prototype & Model 𝑚 𝑡 𝑝&𝑡 𝑚 Thickness of Prototype & Model 𝑚 𝐴 𝑝&𝐴 𝑚 Area of Prototype & Model 𝑚2
- 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&𝑉𝑚 Volume of Prototype & Model 𝑚3 𝐿 𝑟 Length Ratio KINEMATIC SIMILARITY: 𝑣 𝑝 𝑣 𝑚 = 𝑣𝑟 𝑎 𝑝 𝑎 𝑚 = 𝑎 𝑟 Symbol Description Unit 𝑣 𝑝&𝑣 𝑚 Velocity of Prototype & Model 𝑚 𝑠⁄ 𝑎 𝑝&𝑎 𝑚 Acceleration of Prototype & Model 𝑚 𝑠2⁄ 𝑣𝑟 Velocity Ratio 𝑎 𝑟 Acceleration Ratio DYNAMIC SIMILARITY: ( 𝐹𝑖) 𝑝 ( 𝐹𝑖) 𝑚 = ( 𝐹𝑣) 𝑝 ( 𝐹𝑣) 𝑚 = (𝐹𝑔) 𝑝 (𝐹𝑔) 𝑚 = 𝐹𝑟 Symbol Description Unit ( 𝐹𝑖) 𝑝& ( 𝐹𝑖) 𝑚 Inertia Force of Prototype & Model 𝑁 ( 𝐹𝑣) 𝑝& ( 𝐹𝑣) 𝑚 Viscous Force of Prototype & Model 𝑁 (𝐹𝑔) 𝑝 & (𝐹𝑔) 𝑚 Gravity Force of Prototype & Model 𝑁 𝐹𝑟 Force Ratio DIMENSIONLESS NUMBER: REYNOLD’S NUMBER: 𝑅 𝑒 = 𝜌𝑣𝐷 𝜇 (𝑜𝑟) 𝜌𝑣𝐿 𝜇
- 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 𝑘𝑔 𝑚3⁄ 𝑣 Velocity 𝑚 𝑠⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝐷 Diameter 𝑚 𝐿 Length 𝑚 FROUDE’S NUMBER: 𝐹𝑒 = 𝑣 √ 𝐿𝑔 Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐿 Length 𝑚 FROUDE’S NUMBER: 𝐹𝑒 = 𝑣 √ 𝐿𝑔 Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐿 Length 𝑚 EULER’S NUMBER: 𝐸 𝑢 = 𝑣 √ 𝑝 𝜌⁄ Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄
- 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 𝑚2⁄ WEBER’S NUMBER: 𝑊𝑒 = 𝑣 √ 𝜎 𝜌𝐿⁄ Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝐿 Length 𝑚 𝜎 Surface Tension 𝑁 𝑚⁄ MACH’S NUMBER: 𝑊𝑒 = 𝑣 √ 𝐾 𝜌⁄ Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝐾 Elastic Stress 𝑁 𝑚2⁄ REYNOLD’S MODEL LAW: TIME RATIO: 𝐹𝑟 = 𝑚 𝑟 𝑎 𝑟 𝐹𝑟 = 𝑚 𝑟 𝑣𝑟 𝑇𝑟 DISCHARGE RATIO: 𝑄 𝑟 = 𝜌𝑟 𝐿 𝑟 2 𝑣𝑟
- 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 𝐹𝑟 Force Ratio 𝑚 𝑟 Mass Ratio 𝑣𝑟 Velocity Ratio 𝑇𝑟 Time Ratio 𝐿 𝑟 Length Ratio 𝜌𝑟 Density Ratio FROUDE’S MODEL LAW: TIME RATIO: 𝑇𝑟 = √ 𝐿 𝑟 ACCELERATION RATIO: 𝑎 𝑟 = 1 DISCHARGE RATIO: 𝑄 𝑟 = ( 𝐿 𝑟) 5 2⁄ FORCE RATIO: 𝐹𝑟 = ( 𝐿 𝑟)3 PRESSURE RATIO: 𝐹𝑟 = 𝐿 𝑟 ENERGY RATIO: 𝐸𝑟 = ( 𝐿 𝑟)4 MOMENTUM RATIO: 𝑀𝑟 = ( 𝐿 𝑟)3 ∗ √ 𝐿 𝑟 TORQUE RATIO: 𝑇𝑟 = ( 𝐿 𝑟)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: 𝑃𝑟 = ( 𝐿 𝑟) 7 2⁄ Symbol Description Unit 𝐿 𝑟 Length Ratio DISTORTED MODELS: ( 𝐿 𝑟) 𝐻 = 𝐿𝑖𝑛𝑒𝑎𝑟 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑃𝑟𝑜𝑡𝑜𝑡𝑦𝑝𝑒 𝐿𝑖𝑛𝑒𝑎𝑟 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑀𝑜𝑑𝑒𝑙 ( 𝐿 𝑟) 𝐻 = 𝐿 𝑝 𝐿 𝑚 = 𝐵𝑝 𝐵 𝑚 ( 𝐿 𝑟) 𝑉 = 𝐿𝑖𝑛𝑒𝑎𝑟 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑃𝑟𝑜𝑡𝑜𝑡𝑦𝑝𝑒 𝐿𝑖𝑛𝑒𝑎𝑟 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑀𝑜𝑑𝑒𝑙 ( 𝐿 𝑟) 𝑉 = ℎ 𝑝 ℎ 𝑚 VELOCITY RATIO: 𝑣𝑟 = √(𝐿 𝑟) 𝑉 AREA RATIO: 𝐴 𝑟 = ( 𝐿 𝑟) 𝐻 ∗ ( 𝐿 𝑟) 𝑉 DISCAHRGE RATIO: 𝑄 𝑟 = ( 𝐿 𝑟) 𝐻 ∗ [( 𝐿 𝑟) 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 𝑢1&𝑢2 Tangential Velocity of Impeller at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣1&𝑣2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 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 𝜃 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 𝜙 Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree TANGENTIAL VELOCITY AT INLET: 𝑢1 = 𝜋𝑑1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) Internal Diameter of Impeller 𝑚 𝑁 Speed of Impeller 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝑑2 𝑁 60 Symbol Description Unit 𝑑2 Oulet (or) External Diameter of Impeller 𝑚 𝑁 Speed of Impeller 𝑟𝑝𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: 𝑡𝑎𝑛𝜃 = 𝑣𝑓1 𝑢1 Symbol Description Unit 𝑢1 Tangential Velocity of Impeller at Inlet 𝑚 𝑠⁄ 𝑣1 Absolute Velocity at Inlet 𝑚 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 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree ∵ 𝛼 = 90° 𝑣1 = 𝑣𝑓1 FROM OUTLET VELOCITY TRIANGLE DIAGRAM: 𝑡𝑎𝑛𝜙 = 𝑣𝑓2 𝑢2 − 𝑣 𝑤2 𝑣2 = √𝑣𝑓2 2 + 𝑣 𝑤2 2 𝑡𝑎𝑛𝛽 = 𝑣𝑓2 𝑣 𝑤2 Symbol Description Unit 𝑢2 Tangential Velocity of Impeller at Outlet 𝑚 𝑠⁄ 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑣2 Absolute Velocity at Outlet 𝑚 𝑠⁄ 𝑣𝑓2 Flow Velocity at Outlet 𝑚 𝑠⁄ 𝜙 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 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ WORK DONE BY AN IMPELLER PER SECOND: 𝑊 = 𝜌𝑔𝑄 𝑔 𝑣 𝑤2 𝑢2 Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ WORK DONE BY AN IMPELLER PER UNIT WEIGHT OF WATER: 𝑊 = 𝑣 𝑤2 𝑢2 𝑔 Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 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 𝑚 𝑠2⁄ MANOMETRIC EFFICIENCY: 𝜂 𝑚 = 𝑔𝐻 𝑣 𝑤2 𝑢2 Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ 𝐻 Manometric Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ POWER REQUIRED BY THE PUMP: 𝑃 = 𝜌𝑄𝑣 𝑤2 𝑢2 Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑃 Power 𝑘𝑊 MINIMUM SPEED TO START THE PUMP: 𝑁 𝑚𝑖𝑛 = 120 ∗ 𝜂 𝑚 ∗ 𝑣 𝑤2 ∗ 𝑑2 𝜋 (𝑑2 2 − 𝑑1 2 ) Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝜂 𝑚 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: 𝜂 𝑜 = 𝐼𝑚𝑝𝑒𝑙𝑙𝑒𝑟 𝑃𝑜𝑤𝑒𝑟 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 = 𝜌𝑔𝑄𝐻 𝑆. 𝑃 𝜂 𝑜 = 𝜂 𝑚𝑎𝑛𝑜 ∗ 𝜂 𝑚𝑒𝑐ℎ ∗ 𝜂 𝑣𝑜𝑙 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Manometric Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ MECHANICAL EFFICIENCY: 𝜂 𝑚𝑒𝑐ℎ = 𝜌𝑔𝑄𝐻 𝑆. 𝑃 ∗ 𝑣 𝑤2 𝑢2 𝑔𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Manometric Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑆. 𝑃 Shaft Power 𝑊 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ POWER OF PUMP: 𝑃 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 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 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝐴𝑐𝑡𝑢𝑎𝑙 𝐿𝑖𝑓𝑡 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐿𝑖𝑓𝑡 = 𝐴𝑐𝑡𝑢𝑎𝑙 𝐻𝑒𝑎𝑑 𝐼𝑑𝑒𝑎𝑙 𝐻𝑒𝑎𝑑 IDEAL HEAD: 𝑃𝐼 = 𝜌𝑔(𝑄 + 𝑞)𝐻𝑖 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑞 Leakage of Water 𝑚3 𝑠⁄ 𝐻𝑖 Ideal Head 𝑚 𝑃𝐼 Power at Input 𝑊 TORQUE EXERTED BY IMPELLER: 𝑇 = 𝜌𝑔𝑄 𝑔 ∗ 𝑣 𝑤2 ∗ 𝑅2 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑅2 Radius of Impeller at Outlet 𝑚
- 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 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 𝑁𝑠 Specific Speed SPEED RATIO: 𝐾 𝑢 = 𝑢2 √2𝑔𝐻 𝐾 𝑢 = 0.95 − 1.25 Symbol Description Unit 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓2 √2𝑔𝐻 𝐾𝑓 = 0.1 − 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 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾𝑓 Flow Ratio RECIPROCATING PUMP: DISCHARGE: 𝑄 = 𝐴𝐿𝑁 60 𝐴 = 𝜋 4 𝐷2 [ 𝐹𝑜𝑟 𝑆𝑖𝑛𝑔𝑙𝑒 𝐴𝑐𝑡𝑖𝑛𝑔 𝑃𝑢𝑚𝑝] 𝐴 = [ 𝜋 4 𝐷2 + 𝜋 4 ( 𝐷2 − 𝑑2)] [ 𝐹𝑜𝑟 𝐷𝑜𝑢𝑏𝑙𝑒 𝐴𝑐𝑡𝑖𝑛𝑔 𝑃𝑢𝑚𝑝] Symbol Description Unit 𝐴 Area of Cylinder 𝑚2 𝐿 Stroke Length 𝑚 𝑁 Speed 𝑟𝑝𝑚 𝐷 Diameter of Cylinder or Bore 𝑚 𝑑 Diameter of Piston Rod 𝑚 WEIGHT OF THE WATER DELIVERED PER SECOND: 𝑊 = 𝜌𝑔𝑄 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑊 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: 𝑊 = 𝜌𝑔𝑄𝐻 𝐻 = ℎ 𝑠 + ℎ 𝑑 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑊 Work Done 𝑊 ℎ 𝑠 Suction Head 𝑚 ℎ 𝑑 Delivery Head 𝑚 POWER DEVELOPED BY RECIPROCATING PUMP: 𝑃 = 𝜌𝑔𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 POWER REQUIRED TO DRIVE THE PUMP: 𝑃 = 𝜌𝑔𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚
- 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 𝐴𝑐𝑡𝑢𝑎𝑙 − 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Symbol Description Unit 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ COEFFICENT OF DISCHARGE: 𝐶 𝑑 = 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Symbol Description Unit 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ PERCENTAGE OF SLIP IN RECIPROCATING PUMP: % 𝑜𝑓 𝑆𝑙𝑖𝑝 = 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 − 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 % 𝑜𝑓 𝑆𝑙𝑖𝑝 = 1 − 𝐶 𝑑 Symbol Description Unit 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ 𝐶 𝑑 Coefficient of Discharge VOLUMETRIC EFFICIENCY: 𝜂 𝑉𝑜𝑙 = 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 = 𝐶 𝑑 Symbol Description Unit 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ 𝐶 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: 𝜂 𝑚𝑒𝑐ℎ = 𝑃𝑜𝑤𝑒𝑟 𝐷𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑 𝑏𝑦 𝑃𝑢𝑚𝑝 𝑃𝑜𝑤𝑒𝑟 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑡𝑜 𝐷𝑟𝑖𝑣𝑒 𝑡ℎ𝑒 𝑃𝑢𝑚𝑝 𝜂 𝑚𝑒𝑐ℎ = 𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 𝑃𝑢𝑚𝑝 𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 𝑀𝑜𝑡𝑜𝑟 𝜂 𝑚𝑒𝑐ℎ = 𝜌𝑔𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝐻 𝜌𝑔𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 ACCELERATION HEAD: ℎ 𝑎𝑠 = 𝑙 𝑠 𝑔 ∗ 𝐴 𝑎 𝑠 ∗ 𝜔2 ∗ 𝑟 ∗ 𝑐𝑜𝑠𝜃 [ 𝐴𝑡 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑆𝑡𝑟𝑜𝑘𝑒] ℎ 𝑑𝑠 = 𝑙 𝑑 𝑔 ∗ 𝐴 𝑎 𝑑 ∗ 𝜔2 ∗ 𝑟 ∗ 𝑐𝑜𝑠𝜃 [ 𝐴𝑡 𝐷𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑆𝑡𝑟𝑜𝑘𝑒] 𝐴 = 𝜋 4 𝐷2 𝑎 𝑠 = 𝜋 4 𝑑 𝑠 2 𝑎 𝑑 = 𝜋 4 𝑑 𝑑 2 𝜔 = 2𝜋𝑁 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 𝑟 = 𝐿 2 Symbol Description Unit 𝑙 𝑠 Length of Suction Pipe 𝑚 𝑙 𝑑 Length of Delivery Pipe 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐴 Area of Cylinder 𝑚2 𝑎 𝑠 Area of Suction Pipe 𝑚2 𝑎 𝑑 Area of Delivery Pipe 𝑚2 𝜔 Angular Speed 𝑟𝑎𝑑 𝑠⁄ 𝑟 Radius of Crank 𝑚 𝜃 Angle of Crank 𝑑𝑒𝑔𝑟𝑒𝑒 𝐷 Diameter of Cylinder or Bore 𝑚 𝑑 𝑠 Diameter of Suction Pipe 𝑚 𝑑 𝑑 Diameter of Delivery Pipe 𝑚 𝑁 Speed of Crank 𝑟𝑝𝑚 𝐿 Stroke Length 𝑚 PRESSURE HEAD: 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = ℎ 𝑠 + ℎ 𝑎𝑠 [ 𝐹𝑜𝑟 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑆𝑡𝑟𝑜𝑘𝑒] 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = ℎ 𝑑 + ℎ 𝑎𝑑 [ 𝐹𝑜𝑟 𝐷𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑆𝑡𝑟𝑜𝑘𝑒] Symbol Description Unit ℎ 𝑠 Suction Head 𝑚 ℎ 𝑑 Delivery Head 𝑚 ℎ 𝑎𝑠 Acceleration Head at Suction 𝑚
- 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 ℎ 𝑎𝑑 Acceleration Head at Delivery 𝑚 ABSOLUTE PRESSURE HEAD: 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = 𝐻 𝑎𝑡𝑚 − (ℎ 𝑠 + ℎ 𝑎𝑠) [ 𝐹𝑜𝑟 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑆𝑡𝑟𝑜𝑘𝑒] 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = 𝐻 𝑎𝑡𝑚 + (ℎ 𝑑 + ℎ 𝑎𝑑 ) [ 𝐹𝑜𝑟 𝐷𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑆𝑡𝑟𝑜𝑘𝑒] Symbol Description Unit ℎ 𝑠 Suction Head 𝑚 ℎ 𝑑 Delivery Head 𝑚 ℎ 𝑎𝑠 Acceleration Head at Suction 𝑚 ℎ 𝑎𝑑 Acceleration Head at Delivery 𝑚 𝐻 𝑎𝑡𝑚 Atmospheric Pressure Head 𝑚 SEPARATION HEAD: 𝑃𝑠𝑒𝑝 = 𝜌𝑔ℎ 𝑆𝑒𝑝 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ ℎ 𝑠𝑒𝑝 Separation Head 𝑚 𝑃𝑠𝑒𝑝 Separation Pressure 𝑁 𝑚2⁄ HEAD LOSS WITHOUT AIR VESSEL: ℎ 𝑓𝑊𝑂𝐴 = 4𝑓𝑙 𝑑 𝑣2 2𝑔𝑑 𝑑 Symbol Description Unit 𝑓 Friction Factor 𝑙 𝑑 Length of Delivery Pipe 𝑚
- 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 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑑 𝑑 Diameter of Delivery Pipe 𝑚 VELOCITY WITHOUT AIR VESSEL: 𝑣 = 𝐴 𝑎 𝑑 ∗ 𝜔 ∗ 𝑟 𝐴 = 𝜋 4 𝐷2 𝑎 𝑑 = 𝜋 4 𝑑 𝑑 2 𝜔 = 2𝜋𝑁 60 𝑟 = 𝐿 2 Symbol Description Unit 𝐴 Area of Cylinder 𝑚2 𝑎 𝑑 Area of Delivery Pipe 𝑚2 𝜔 Angular Speed 𝑟𝑎𝑑 𝑠⁄ 𝑟 Radius of Crank 𝑚 𝐷 Diameter of Cylinder or Bore 𝑚 𝑑 𝑑 Diameter of Delivery Pipe 𝑚 𝑁 Speed of Crank 𝑟𝑝𝑚 𝐿 Stroke Length 𝑚
- 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: ℎ 𝑓𝑊𝐴 = 4𝑓𝑙 𝑑 𝑣2 2𝑔𝑑 𝑑 Symbol Description Unit 𝑓 Friction Factor 𝑙 𝑑 Length of Delivery Pipe 𝑚 𝑣 Velocity with Air Vessel 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑑 𝑑 Diameter of Delivery Pipe 𝑚 VELOCITY WITH AIR VESSEL: 𝑣 = 𝐴 𝑎 𝑑 ∗ 𝜔 ∗ 𝑟 𝜋 𝐴 = 𝜋 4 𝐷2 𝑎 𝑑 = 𝜋 4 𝑑 𝑑 2 𝜔 = 2𝜋𝑁 60 𝑟 = 𝐿 2 Symbol Description Unit 𝐴 Area of Cylinder 𝑚2 𝑎 𝑑 Area of Delivery Pipe 𝑚2 𝜔 Angular Speed 𝑟𝑎𝑑 𝑠⁄ 𝑟 Radius of Crank 𝑚 𝐷 Diameter of Cylinder or Bore 𝑚
- 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 𝑚 𝑁 Speed of Crank 𝑟𝑝𝑚 𝐿 Stroke Length 𝑚 POWER SAVED BY AIR VESSEL: 𝑃 = 𝜌𝑔𝑄 ( 2 3 ℎ 𝑓𝑊𝑂𝐴 − ℎ 𝑓𝑊𝐴) Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ ℎ 𝑓𝑊𝑂𝐴 Head Loss Without Air Vessel 𝑚 ℎ 𝑓𝑊𝐴 Head Loss With Air Vessel 𝑚 POWER REQUIRED TO DRIVE THE PUMP: 𝑃 = 𝜌𝑔𝑄 (ℎ 𝑠 + ℎ 𝑑 + 2 3 ℎ 𝑓𝑠𝑊𝑂𝐴 + ℎ 𝑓𝑑𝑊𝐴) Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ ℎ 𝑠 Suction Head 𝑚 ℎ 𝑑 Delivery Head 𝑚 ℎ 𝑓𝑠𝑊𝑂𝐴 Head Loss Without Air Vessel at Suction 𝑚 ℎ 𝑓𝑑𝑊𝐴 Head Loss With Air Vessel at Delivery 𝑚
- 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 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛽 Angle made by Absolute Velocity at Outlet with the Direction of Motion of Vane Degree 𝜙 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: 𝑢 = 𝜋𝐷𝑁 60 Symbol Description Unit 𝐷 Diameter of Runner 𝑚 𝑁 Speed of Impeller 𝑟𝑝𝑚 VELOCITY OF JET: 𝑉1 = 𝐶𝑣√2𝑔𝐻 𝐶𝑣 = 0.97 − 0.99 Symbol Description Unit 𝐶𝑣 Coefficient of Velocity 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 VELOCITY OF WHEEL: 𝑢 = 𝑘 𝑢√2𝑔𝐻 𝑘 𝑢 = 0.43 − 0.45 Symbol Description Unit 𝑘 𝑢 Speed Ratio 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: 𝑉 𝑤1 = 𝑉1 𝑉 𝑤1 = 𝑢1 + 𝑉𝑟1
- 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 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ FROM OUTLET VELOCITY TRIANGLE DIAGRAM: cos 𝜙 = 𝑢2 + 𝑣 𝑤2 𝑣 𝑟2 tan 𝜙 = 𝑣𝑓2 𝑢2 + 𝑣 𝑤2 sin 𝜙 = 𝑣𝑓2 𝑣 𝑟2 tan 𝛽 = 𝑣𝑓2 𝑣 𝑤2 Symbol Description Unit 𝑢2 Tangential Velocity of Runner at Outlet 𝑚 𝑠⁄ 𝑣 𝑟2 Relative Velocity at Outlet 𝑚 𝑠⁄ 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑣𝑓2 Flow Velocity at Outlet 𝑚 𝑠⁄ WORK DONE BY JET PER SECOND: 𝑊 = 𝜌𝑄 [ 𝑣 𝑤1 + 𝑣 𝑤2] 𝑢 Symbol Description Unit 𝑢 Tangential Velocity of Runner 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 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: 𝜂ℎ𝑦𝑑 = 2[ 𝑣 𝑤1 + 𝑣 𝑤2] 𝑢 𝑉1 2 Symbol Description Unit 𝑢 Tangential Velocity of Runner 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ OVERALL EFFICIENCY: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊 DISCHARGE OF SINGLE JET: 𝑞 = 𝜋 4 ∗ 𝑑2 ∗ 𝑉1 Symbol Description Unit 𝑑 Diameter of Jet 𝑚 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑞 Discharge of Single Jet 𝑚3 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 𝑞 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝑞 Discharge of Single Jet 𝑚3 𝑠⁄ NUMBER OF BUCKET: 𝑍 = 15 + 𝐷 2𝑑 Symbol Description Unit 𝑑 Diameter of Jet 𝑚 𝐷 Diameter of Runner 𝑚 DIMENSIONS OF BUCKET: 𝐴𝑥𝑖𝑎𝑙 𝑊𝑖𝑑𝑡ℎ 𝐵 = 4.5𝑑 𝑅𝑎𝑑𝑖𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ 𝐿 = 2.5𝑑 𝐷𝑒𝑝𝑡ℎ 𝑜𝑓 𝐵𝑢𝑐𝑘𝑒𝑡 𝑇 = 𝑑 Symbol Description Unit 𝑑 Diameter of Jet 𝑚 KINETIC ENERGY OF JET: 𝐾. 𝐸 𝑜𝑓 𝐽𝑒𝑡 = 1 2 𝑚 𝑉1 2 𝑆𝑖𝑛𝑐𝑒 𝑚 = 𝜌𝐴𝑉 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐾. 𝐸 𝑜𝑓 𝐽𝑒𝑡 = 1 2 𝜌 ∗ 𝐴 ∗ 𝑉1 ∗ 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 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐾. 𝐸 𝑜𝑓 𝐽𝑒𝑡 = 1 2 𝜌 ∗ 𝑄 ∗ 𝑉1 2 POWER LOST IN NOZZLE: 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 = 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝑖𝑛 𝑁𝑜𝑧𝑧𝑙𝑒 POWER LOST IN RUNNER: 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 = 𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 𝑆ℎ𝑎𝑓𝑡 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝑖𝑛 𝑁𝑜𝑧𝑧𝑙𝑒 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝑖𝑛 𝑅𝑢𝑛𝑛𝑒𝑟 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝐷𝑢𝑒 𝑡𝑜 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 RESULTANT FORCE ON BUCKET: 𝐹 = 𝜌𝑄 [ 𝑣 𝑤1 + 𝑣 𝑤2] Symbol Description Unit 𝐹 Resultant Force on Bucket 𝑁 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ TORQUE: 𝑇 = 𝐹 ∗ 𝐷 2 Symbol Description Unit 𝐹 Resultant Force on Bucket 𝑁 𝐷 Diameter of Runner 𝑚 𝑇 Torque 𝑁 − 𝑚
- 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 = 2𝜋𝑁𝑇 60 Symbol Description Unit 𝑃 Power 𝑊 𝑇 Torque 𝑁 − 𝑚 N Speed of Shaft Rpm SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 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 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝜙 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: 𝑢1 = 𝜋𝑑1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) External Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝑑2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) Internal Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑤1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑤1 sin 𝜃 = 𝑣𝑓1 𝑣 𝑟1 cos 𝜃 = 𝑣 𝑤1 − 𝑢1 𝑣 𝑟1
- 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 𝜃 = 𝑣𝑓1 𝑣 𝑤1 − 𝑢1 Symbol Description Unit 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 𝑟1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑤1 − 𝑢1)2 Symbol Description Unit 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ DISCHARGE: 𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 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 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ 𝐴 Area of Runner 𝑚2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet 𝑚 𝑠⁄ MASS OF WATER FLOWING THROUGH THE RUNNER: 𝑚 = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ HEAD AT INLET OF TURBINE: 𝐻 = 1 𝑔 ∗ 𝑣 𝑤1 ∗ 𝑢1 + 𝑣𝑓1 2 2𝑔 Symbol Description Unit 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 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 𝑚 𝑠2⁄ 𝐻 Head 𝑚 POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝑣 𝑤1 𝑢1 𝑔𝐻 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 Symbol Description Unit 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 OVERALL EFFICIENCY: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻
- 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 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊 SPEED RATIO: 𝐾 𝑢 = 𝑢 √2𝑔𝐻 𝐾 𝑢 = 0.6 − 0.9 Symbol Description Unit 𝑢 Tangential Velocity 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 − 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 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 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 𝑁𝑠 Specific Speed OUTWARD FLOW REACTION TURBINE: Symbol Description Unit 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 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 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝜙 Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree TANGENTIAL VELOCITY AT INLET: 𝑢1 = 𝜋𝑑1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) Internal Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝑑2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) External Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 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 𝑤1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑤1 sin 𝜃 = 𝑣𝑓1 𝑣 𝑟1 cos 𝜃 = 𝑣 𝑤1 − 𝑢1 𝑣 𝑟1 tan 𝜃 = 𝑣𝑓1 𝑣 𝑤1 − 𝑢1 Symbol Description Unit 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 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 𝑟1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑤1 − 𝑢1)2 Symbol Description Unit 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ DISCHARGE: 𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ 𝐴 Area of Runner 𝑚2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet 𝑚 𝑠⁄ MASS OF WATER FLOWING THROUGH THE RUNNER: 𝑚 = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄
- 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 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝑣 𝑤1 𝑢1 𝑔𝐻 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 Symbol Description Unit 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚
- 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: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊 SPEED RATIO: 𝐾 𝑢 = 𝑢 √2𝑔𝐻 𝐾 𝑢 = 0.6 − 0.9 Symbol Description Unit 𝑢 Tangential Velocity 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 − 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 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾𝑓 Flow Ratio SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 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 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝜙 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: 𝑢1 = 𝜋𝑑1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) External Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝑑2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) Internal Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑤1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑤1 sin 𝜃 = 𝑣𝑓1 𝑣 𝑟1 cos 𝜃 = 𝑣 𝑤1 − 𝑢1 𝑣 𝑟1 tan 𝜃 = 𝑣𝑓1 𝑣 𝑤1 − 𝑢1
- 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 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 𝑟1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑤1 − 𝑢1)2 Symbol Description Unit 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ DISCHARGE: 𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚
- 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 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ 𝐴 Area of Runner 𝑚2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet 𝑚 𝑠⁄ CIRCUMFERENTIAL AREA OF RUNNER: 𝐴 = 𝜋𝑑1 𝑏1 = 𝜋𝑑2 𝑏2 Symbol Description Unit 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet 𝑚 𝐴 Circumferential Area of Runner 𝑚2 MASS OF WATER FLOWING THROUGH THE RUNNER: 𝑚 = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚
- 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 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝑣 𝑤1 𝑢1 𝑔𝐻 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 Symbol Description Unit 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 OVERALL EFFICIENCY: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 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 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊 SPEED RATIO: 𝐾 𝑢 = 𝑢 √2𝑔𝐻 𝐾 𝑢 = 0.6 − 0.9 Symbol Description Unit 𝑢 Tangential Velocity 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 − 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 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 − 0.4 Symbol Description Unit 𝑏1 Width of Runner at Inlet 𝑚 𝑑1 Diameter of Runner at Inlet 𝑚 𝑛 Breadth Ratio SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 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 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝜙 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: 𝑢1 = 𝜋𝐷 𝑜 𝑁 60 Symbol Description Unit 𝐷 𝑜 Inlet (or) External Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝐷 𝑏 𝑁 60 = 𝜋𝐷ℎ 𝑁 60 Symbol Description Unit 𝐷 𝑏 𝑜𝑟 𝐷ℎ Outlet (or) Boss (or) Hub Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑤1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑤1 sin 𝜃 = 𝑣𝑓1 𝑣 𝑟1 cos 𝜃 = 𝑣 𝑤1 − 𝑢1 𝑣 𝑟1 tan 𝜃 = 𝑣𝑓1 𝑣 𝑤1 − 𝑢1
- 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 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 𝑟1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑤1 − 𝑢1)2 Symbol Description Unit 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ DISCHARGE: 𝑄 = 𝜋 4 [𝐷0 2 − 𝐷 𝑏 2 ]𝑣𝑓1 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝐷0 Inlet (or) External Diameter 𝑚 𝐷 𝑏 𝑜𝑟 𝐷ℎ Outlet (or) Boss (or) Hub Diameter 𝑚
- 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 𝑚3 𝑠⁄ CIRCUMFERENTIAL AREA OF RUNNER: 𝐴 = 𝜋 4 [𝐷0 2 − 𝐷 𝑏 2 ] Symbol Description Unit 𝐷0 Inlet (or) External Diameter 𝑚 𝐷 𝑏 𝑜𝑟 𝐷ℎ Outlet (or) Boss (or) Hub Diameter 𝑚 𝐴 Circumferential Area of Runner 𝑚2 MASS OF WATER FLOWING THROUGH THE RUNNER: 𝑚 = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 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 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝑣 𝑤1 𝑢1 𝑔𝐻 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 Symbol Description Unit 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 OVERALL EFFICIENCY: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊
- 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 𝑢 = 𝑢 √2𝑔𝐻 𝐾 𝑢 = 0.6 − 0.9 Symbol Description Unit 𝑢 Tangential Velocity 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 − 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾𝑓 Flow Ratio SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 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 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 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 𝑚 𝑠⁄ 𝐻𝑠 Vertical Height of Draft Tube Above Tail Race 𝑚 𝑦 Distance of Bottom of Draft Tube from Tail Race 𝑚 FROM BERNOULLI’S EQUATION: 𝑃1 𝜌𝑔 + 𝑉1 2 2𝑔 + 𝑧1 = 𝑃2 𝜌𝑔 + 𝑉2 2 2𝑔 + 𝑧2 + ℎ 𝑓 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Inlet & Outlet of Draft Tube 𝑁 𝑚2⁄ 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube 𝑚 𝑠⁄ 𝑧1 & 𝑧2 Datum Head Inlet & Outlet of Draft Tube 𝑚 ℎ 𝑓 Head Loss 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LENGTH OF DRAFT TUBE: 𝐿 = 𝐻𝑠 + 𝑦 Symbol Description Unit 𝐿 Length of Draft Tube 𝑚 𝐻𝑠 Vertical Height of Draft Tube Above Tail Race 𝑚 𝑦 Distance of Bottom of Draft Tube from Tail Race 𝑚 EFFICIENCY OF DRAFT TUBE: 𝜂 𝑑 = ( 𝑉1 2 2𝑔 − 𝑉2 2 2𝑔 ) − ℎ 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 𝑚 𝑠⁄ ℎ 𝑓 Head Loss 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ HYDRAULIC EFFICIENCY OF DRAFT TUBE: 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝑈𝑡𝑖𝑙𝑖𝑧𝑒𝑑 𝑏𝑦 𝑇𝑢𝑟𝑏𝑖𝑛𝑒 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 𝑜𝑓 𝑇𝑢𝑟𝑏𝑖𝑛𝑒 𝜂ℎ𝑦𝑑 = 𝐻 − ℎ 𝑓𝑡 − ℎ 𝑓𝑑 − 𝑉2 2 2𝑔 𝑃1 𝜌𝑔 + 𝑉1 2 2𝑔 + 𝑧1 Symbol Description Unit 𝑃1 Pressure at Inlet of Draft Tube 𝑁 𝑚2⁄ 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube 𝑚 𝑠⁄ 𝑧1 Datum Head Inlet of Draft Tube 𝑚 ℎ 𝑓 Head Loss 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄