Bernoulli’s Theorem

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Bernoulli’s Theorem

  1. 1. S M Mozakkir Quadri 10CES-54(5th SEM)JAMIA MILLIA ISLAMIA
  2. 2. INDRODUCTIONDaniel BernoulliA Swiss scientist born in1700’s that is most famousfor his work in fluidpressure. He died in 1782.
  3. 3. BERNOULLI’S THEOREMBernoulli’s theorem which is also known asBernoulli’s principle, states that an increase in thespeed of moving air or a flowing fluid isaccompanied by a decrease in the air orfluid’s pressure or sum of the kinetic (velocityhead), pressure(static head) and Potential energyenergy of the fluid at any point remains constant,provided that the flow is steady, irrotational, andfrictionless and the fluid is incompressible.
  4. 4. BERNOULLI’S EQUATIONIf a section of pipe is as shown above,then Bernoulli’s Equation can be written as;
  5. 5. BERNOULLI’S EQUATIONWhere (in SI units)P= static pressure of fluid at the cross section;ρ= density of the flowing fluid in;g= acceleration due to gravity;v= mean velocity of fluid flow at the cross section in;h= elevation head of the center of the cross section with respect to a datum.
  6. 6. HOW TO VERIFY?The converging-diverging nozzle apparatus (Venturimeter) is used to show the validity of Bernoulli’sEquation. The data taken will show the presence of fluidenergy losses, often attributed to friction and theturbulence and eddy currents associated with aseparation of flow from the conduit walls.
  7. 7. APPARATUSES USED Arrangement of Venturi meter apparatus(fig.1) Hydraulic bench(fig. 2). Stop watch(fig.3). fig. 2 fig. 3
  8. 8. PROCEDURE1. Note down the inlet, throat and outlet section areas.2. Measure the distances of inlet, throat ant outlet section from origin.3. Switch on the motor attached to hydraulic bench.4. If there any water bubble is present in tube remove it by using air bleed screw.5. Fully open the control valve.6. Note down the reading of piezometer corresponding to the section, simultaneously note down the time required to a constant rise of water in volumetric tank(say of 10).7. Varying the discharge and take at least six readings.
  9. 9. OBSERVATIONS1. Volume = 1000 cm32.Distance of inlet section from origin= 5.5 cm3. Distance of throat section from origin= 8.1 cm4.Distance of outlet section from origin= 15.6 cm5.Area of inlet section= 4.22 cm26.Area of throat section= 2.01 cm27.Area of outlet section= 4.34 cm2
  10. 10. OBSERVATIONS PIEZOMETRIC S NO. TIME(sec) HEAD(cm) INLET THROAT OUTLET SECTION SECTION SECTION 1 90.40 19.8 17.5 19.3 2 92.90 19.9 17.3 19.2 3 101.26 19.4 17.4 18.8 4 106.00 19.4 17.4 18.8 5 110.00 19.6 17.6 19.0 6 115.65 19.2 17.6 18.8
  11. 11. CALCULATED VALUES VELOCITY TOTAL LOSS OF VELOCITY(v) HEAD ENERGY HEAD ENERGY (cm/sec) Q (cm) (cm) (cm)(cm3/s) v1 v2 v3 v12/2g v22/2g v32/2g E1 E2 E3 E1-E2 E1-E3110.62 26.2 55.0 25.4 0.35 1.54 0.31 20.1 19.0 19.6 1.10 0.51107.64 25.5 53.5 24.8 0.33 1.46 0.31 20.2 18.7 19.5 1.47 0.7198.76 23.4 49.1 22.7 0.27 1.23 0.26 19.6 18.6 19.0 1.04 0.6194.34 22.3 46.9 21.7 0.25 1.12 0.24 19.6 18.5 19.0 1.13 0.6190.91 21.5 45.2 20.9 0.23 1.04 0.22 19.8 18.6 19.2 1.19 0.6186.47 20.4 43.0 19.9 0.21 0.94 0.20 19.4 18.5 19.0 0.87 0.41
  12. 12. RESULTSIt is observed from the calculated value that at sectionwhere area is less velocity is high and pressure is lowwhich validates the Bernoulli’s Equation. Graphs areplotted between distance v/s piezometric head anddistance v/s total energy but from the graph (B) we canobserve that there is dissipation in energy at last pointthis is because to achieve an ideal condition practically isnot possible.
  13. 13. GRAPHS obs no. 1 obs no. 1 21 21piezometric head(cm) Total energy (E)(cm) 19 19 17 17 0 5 10 15 20 0 5 10 15 20 Distance x(cm) Distance (x) (cm)
  14. 14. APPLICATIONSThe Bernoulli’s equation forms the basis for solving awide variety of fluid flow problems such as jets issuingfrom an orifice, jet trajectory flow under a gate and overa weir, flow metering by obstruction meters, flowaround submerged objects, flows associated with pumpsand turbines etc.Apart from this Bernoulli’s equation is very useful indemonstration of various aerodynamic properties likeDrag and Lift.
  15. 15. APPLICATIONSDRAG AND LIFT Fast Moving Air; Low Air Pressure Air travels fartherLeading edge airfoil Trailing edge Slow Moving Air; High Air Pressure
  16. 16. APPLICATIONS
  17. 17. CONCLUSIONFrom the result obtained, we canconclude that the Bernoulli’s equation isvalid for flow as it obeys the equation. Asthe area decreases at a section (throatsection) velocity increases, and thepressure decreases.
  18. 18. THANK YOU

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