Elementary Mechanics of FluidsEnergySlide 3Energy EquationWorkFlow WorkSlide 7Kinetic Energy Correction FactorEnergy Equation for Pipe FlowEx (7.35)EGL & HGL for a Pipe SystemSlide 12Slide 13Slide 14Slide 15Slide 16HW (7.8)HW (7.16)HW (7.27)Slide 20HW (7.41)Elementary Mechanics of FluidsCE 319 FDaene McKinneyEnergyEquationEnergy•First Law of Thermodynamics–Where •E = energy of a system (extensive property)•Q = Heat added to the system•W = Work done by the system–Where •Eu = Internal energy of the system•Ek = Kinetic energy of the system•Ep = Potential energy of the system–Intensive property (energy per unit mass)pkuEEEE dtdWdtdQdtdEpkeeue Energy of a system changes as heat is added to the system or work is done on the systemEnergy22/22VMMVmassmassofenergykineticek•Kinetic energy per unit mass•Potential energy per unit massgzzmassmassofenergypotentialepEnergy Equation•Reynolds Transport TheoremCSCVsysdbdbdtddtdBAV•b = e ; Bsys = ECSCVdededtddtdEAVCSpkCVpkdueedueedtddtdWdtdQAV)()(•Rate of Work•Shaft work work done by a mechanical device which crosses the CS•Flow work work done by pressure forces on the CS•Viscous work work done by viscous stresses at the CSWorkviscousflowshaftdtdWdtdWdtdWdtdWshaftdtdWflowdtdWviscousdtdWFlow Work•Work occurs at the CS when a force associated with the normal stress of the fluid acts over a distance. The normal stress equals the negative of the fluid pressure.22222222222222AVpVApdtdWtVAplApW1111AVpdtdWCSCSflowdppdtdWAVAVEnergy EquationCSCVshaftdugzVpdugzVdtddtdWdtdQAV)2()2(22Kinetic Energy Correction FactorCSCSCSCVshaftdVugzpdugzVpdugzVdtddtdWdtdQAVAVAV2)()2()2(2222222VmdVCS AVFor nonuniform flows, this term requires special attentionWe can modify this kinetic energy term by a dimensionless factor, , so that the integral is proportional to the square of the average velocityAdAV AV1where AdAVVA31and2211AVAVAVmso22222211121122ugzpVugzpVwqshaftshaftshaftdtdWmwdtdQmq1;1Kinetic energy correction factorEnergy Equation for Pipe Flowpumpturbineshaftwww 22222211121122ugzpVugzpVwqshaft)(22122222211211quuwgzpVwgzpVturbinepump222221121122zpgVhhhzpgVlossturbinepumpgquuhgwhgwhlossturbineturbinepumppump12;;Ex (7.35)•Given: PA= 10 psi (12 in dia. pipe), PB= 40 psi (6 in dia. pipe), Q=3.92 cfs•Find: Horsepower of pump•Solution: sftAQVsftAQVBBAA/96.194/)5.0(92.3;/99.44/)1(92.322hphQPftgppgVVhzpgVhhhzpgVABABpumpBBBBlossturbinepumpAAAA4.3355004.75*4.62*92.355004.754.62144*)4010(2)19.6()99.4(222222222EGL & HGL for a Pipe System•Energy equation•All terms are in dimension of length (head, or energy per unit weight)•HGL – Hydraulic Grade Line•EGL – Energy Grade Line•EGL=HGL when V=0 (reservoir surface, etc.)•EGL slopes in the direction of flow222221121122zpgVhzpgVLzpHGL gVHGLzpgVEGL2222EGL & HGL for a Pipe System•A pump causes an abrupt rise in EGL (and HGL) since energy is introduced hereEGL & HGL for a Pipe System•A turbine causes an abrupt drop in EGL (and HGL) as energy is taken out•Gradual expansion increases turbine efficiencyEGL & HGL for a Pipe System•When the flow passage changes diameter, the velocity changes so that the distance between the EGL and HGL changes•When the pressure becomes 0, the HGL coincides with the systemEGL & HGL for a Pipe System•Abrupt expansion into reservoir causes a complete loss of kinetic energy thereEGL & HGL for a Pipe System•When HGL falls below the pipe the pressure is below atmospheric pressureHW (7.8)HW (7.16)HW (7.27)HW (7.27)HW