AE 1350 Lecture Notes #6We have studied...Topics to be studiedContinuityMomentum EquationMomentum Equation (Contd..)Slide 7Slide 8Slide 9Slide 10Bernoulli’s EquationApplications of Bernoulli’s EquationSlide 13The Venturi MeterHow does the Venturi Meter work?That’s all folks!!!AE 1350Lecture Notes #6We have studied...•Meaning of Incompressible Flow•How and why speed of the flow affects compressibility•Streamlines and stream tubes•Continuity EquationTopics to be studied•Conservation of Momentum–Euler’s equation•Conservation of Energy–Bernoulli’s equation (even though Euler thought of it first.)•Practical Applications of Bernoulli’s EquationContinuityStation 1Density 1Velocity V1Area A1Station 2Density 2Velocity V2Area A2Mass Flow Rate In = Mass Flow Rate Out1 V1 A1 = 2 V2 A2Momentum EquationBased on Newton’s Second Law:Rate of Change of Momentum of a particle= Forces acting on itConsider an infinitesimally small slice of stream tube in space.Rate of change of momentum of the fluid particles within thisstream tube must be due to forces acting on it.Momentum Equation (Contd..)Density velocity VArea =ADensity dvelocity V+dVArea =A+dAMass Flow Rate in = Mass Flow rate out VA = (+d)(V+dV)(A+dA)Momentum rate in=Mass flow rate times velocity= V2AMomentum Rate out=Mass flow rate times velocity= VA (V+dV)Momentum Equation (Contd..)Density velocity VArea =ADensity dvelocity V+dVArea =A+dAMomentum rate in=Mass flow rate times velocity= V2AMomentum Rate out=Mass flow rate times velocity= VA (V+dV)Rate of change of momentum within this element = Momentum rate out - Momentum rate in = VA (V+dV) - V2A = VA dVMomentum Equation (Contd..)Density velocity VArea =ADensity dvelocity V+dVArea =A+dARate of change of momentum as fluid particlesflow through this element= VA dVBy Newton’s law, this momentum change must be caused byforces acting on this stream tube.Forces acting on the Stream tube Pressuretimesarea=pA(p+dp)(A+dA)Horizontal Force = Pressure times area of the ring=(p+dp/2)dAArea of this ring = dANet force = pA + (p+dp/2)dA-(p+dp)(A+dA)=- Adp - dp • dA/2 -AdpProduct of two small numbersMomentum EquationFrom the previous slides,Rate of change of momentum when fluid particles flowthrough the stream tube = AVdVForces acting on the stream tube = -AdpWe have neglected all other forces - viscous, gravity, electricaland magnetic forces.Equating the two factors, we get: VdV+dp=0This equation is called the Euler’s EquationBernoulli’s EquationEuler equation: VdV + dp = 0For incompressible flows, this equation may be integrated:ConstpVOrdpVdV 221,0Kinetic Energy + Pressure Energy = ConstantBernoulli’sEquationApplications of Bernoulli’s Equation•See examples 4.1 through 4.3 in the text.•We will do more worked out examples at our next lecture.•Important Applications include:–Pitot Tube–Venturi Meter–Flow over airfoilsPitot tubes are used on aircraft as a speedometer.The Venturi MeterIt is used to measure Flow rates. Gas companies, Water works, and aircraft fuel monitors all use this device.How does the Venturi Meter work?2222111221212221112121:,:_VpVpBernoulliAAVVThusFlowibleIncompressAVAV 11222112111222212112212221122221_:rate flow Compute12:Vfor Solve21212121AVrateFlowAAppVppAAVppVVVppVVThat’s all
View Full Document