FUNDAMENTALSFUNDAMENTALS OFOF FLUIDFLUID MECHANICSMECHANICS ChapterChapter 1212 PumpsPumps andand TurbinesTurbines JyhJyh--CherngCherng ShiehShieh Department of Bio-Industrial Mechatronics Engineering National Taiwan University 1 MAINMAIN TOPICSTOPICS IntroductionIntroduction BasicBasic EnergyEnergy ConsiderationsConsiderations BasicBasic AngularAngular MomentumMomentum ConsiderationsConsiderations TheThe CentrifugalCentrifugal PumpPump DimensionlessDimensionless ParametersParameters andand SimilaritySimilarity LawsLaws AxialAxial--FlowFlow andand MixedMixed--FlowFlow PumpsPumps FansFans TurbinesTurbines CompressibleCompressible FlowFlow TurbomachinesTurbomachines 2 PumpsPumps andand TurbinesTurbines PumpsPumps andand turbines:turbines: FluidFluid machines.machines. Pumps:Pumps: AddAdd energyenergy toto thethe fluidfluid –– theythey dodo workwork onon thethe fluid.fluid. Turbines:Turbines: ExtractExtract energyenergy fromfrom thethe fluidfluid –– thethe fluidfluid doesdoes workwork onon them.them. 3 FluidFluid MachinesMachines PositivePositive displacementdisplacement machinesmachines (denoted(denoted asas thethe staticstatic type)type) TurbomachinesTurbomachines (denoted(denoted asas thethe dynamicdynamic type).type). 4 PositivePositive DisplacementDisplacement MachinesMachines ForceForce fluidfluid intointo oror outout ofof aa chamberchamber byby changingchanging thethe volumevolume ofof thethe chamber.chamber. TheThe pressurepressure developeddeveloped andand thethe workwork donedone areare aa resultresult ofof essentiallyessentially staticstatic forcesforces ratherrather thanthan dynamicdynamic effects.effects. Typical positive displacement pumps: (a) tire pump, (b) human heart, (c) gear pump. 5 TurbomachinesTurbomachines TurbomachinesTurbomachines involveinvolve aa collectioncollection ofof blades,blades, buckets,buckets, flowflow channels,channels, oror passagespassages arrangedarranged aroundaround anan axisaxis ofof rotationrotation toto formform aa rotor.rotor. TurbomachinesTurbomachines areare mechanicalmechanical devicesdevices thatthat eithereither extractextract energyenergy fromfrom aa fluidfluid (turbine)(turbine) oror addadd energyenergy toto aa fluidfluid (pump)(pump) asas aa resultresult ofof dynamicdynamic interactionsinteractions betweenbetween thethe devicedevice andand thethe fluid.fluid. TheThe fluidfluid usedused cancan bebe eithereither aa gasgas oror aa liquid.liquid. 6 OperatingOperating PrinciplesPrinciples ofof TurbomachinesTurbomachines TheThe basicbasic operatingoperating principlesprinciples areare thethe samesame whetherwhether thethe fluidfluid isis aa liquidliquid oror aa gas.gas. CavitationCavitation maymay bebe anan importantimportant designdesign considerationconsideration whenwhen liquidsliquids areare involvedinvolved ifif thethe prepressuressure atat anyany pointpoint withinwithin thethe flowflow isis reducedreduced toto vaporvapor pressure.pressure. CompressibilityCompressibility effectseffects maymay bebe importantimportant whenwhen gasesgases areare involvedinvolved ifif thethe MachMach numbernumber becomesbecomes largelarge enough.enough. 7 StructureStructure ofof TurbomachinesTurbomachines ManyMany turbomachinesturbomachines containcontain somesome typetype ofof housinghousing oror castingcasting thatthat surroundssurrounds thethe rotatingrotating bladesblades oror rotor,rotor, thusthus formingforming aa nn internalinternal flowflow passagewaypassageway throughthrough whichwhich thethe fluidfluid flows.flows. SomeSome turbomachinesturbomachines includeinclude stationarystationary bladesblades oror vanesvanes inin additionaddition toto rotorrotor blades.blades. TheseThese stationarystationary vanesvanes cancan bebe arrangedarranged toto accelerateaccelerate thethe flowflow andand thusthus serveserve asas anan nozzles.nozzles. TheseThese vanesvanes cancan bebe setset toto diffusediffuse thethe flowflow andand actact asas diffusers.diffusers. 8 ClassificationClassification ofof TurbomachinesTurbomachines AxialAxial--flowflow machines:machines: TheThe fluidfluid maintainsmaintains aa significantsignificant axialaxial--flowflow directiondirection componentcomponent fromfrom thethe inletinlet toto outletoutlet ofof thethe rotor.rotor. MixedMixed--flowflow machines:machines: ThereThere maymay bebe significantsignificant radialradial-- andand axialaxial--flowflow velocityvelocity componentscomponents forfor thethe flowflow throughthrough thethe rotorrotor row.row. RadialRadial--flowflow mahcinesmahcines:: TheThe flowflow acrossacross thethe bladsblads involvesinvolves aa substantialsubstantial radialradial--flowflow componentcomponent atat thethe rotorrotor inlet,inlet, exit,exit, oror both.both. 9 BasicBasic EnergyEnergy ConsiderationsConsiderations ByBy consideringconsidering thethe basicbasic operationoperation ofof HouseholdHousehold fanfan (pump).(pump). WindmillWindmill (turbine).(turbine). 10 HouseholdHousehold FanFan 1/21/2 Consider a fan blade driven at constant angular velocity by the motor. Absolute velocity is the vector sums of relative and blade velocities. r The blade velocity U = ωr The absolute velocity VV seen by a person sitting stationary at the table on which the fan rests. The relative velocity seen by a person riding on the fan blade W r r r V = W + U 11 HouseholdHousehold FanFan 2/22/2 Idealized flow through a fan: (a) fan blade geometry: (b) absolute velocity, V; relative velocity, W, and blade velocity, U at the inlet and exit of the fan blade section. 12 WindmillWindmill Consider the windmill. Rather than the rotor being driven by a motor, it is rotated in the opposite direction by the wind blowing through the rotor. Idealized flow through a windmill: (a) windmill blade geometry; (b) absolute velocity, V; relative velocity, W, and blade velocity, U at the inlet and exit of the windmill blade section. 13 ExampleExample 12.112.1 DragDrag fromfrom PressurePressure andand ShearShear StressStress DistributionsDistributions z The rotor shown in Fig. E12.1a rotates at a constant angular velocity of ω= 100 rad/s. Although the fluid initially approaches the rotor in an axial direction, the flow across the blades is primarily radial. Measurements indicate that the absolute velocity at the inlet and outlet are V1 = 12 m/s and V2 = 15 m/s, respectively. Is this device a pump or a turbine? 14 ExampleExample 12.112.1 SolutionSolution1/21/2 If the tangential component f the force of the blade on the fluid is in the direction of the blade motion (a pump) or opposite to it (a turbine). The inlet and outlet blade U1 = ωr1 = 10m /s U2 = ωr2 = 10m /s 15 ExampleExample 12.112.1 SolutionSolution2/22/2 The inlet velocity triangle The outlet velocity triangle At the inlet there is no component of absolute velocity in the direction of rotation; at the outlet this component is not zero. That is, the blade pushes and turns the fluid in the direction of the blade motion, thereby doing work on the fluid. Æ This device is a pump. 16 BasicBasic AngularAngular MomentumMomentum ConsiderationsConsiderations 17 AngularAngular MomentumMomentum ConsiderationsConsiderations 1/6 WorkWork transferredtransferred toto oror fromfrom aa fluidfluid flowingflowing throughthrough aa pumppump oror aa turbineturbine occursoccurs byby interactioninteraction betweenbetween movingmoving rotorrotor bladesblades andand thethe fluid.fluid. DDPump:Pump: TheThe shaftshaft toquetoque (the(the totorquerque thatthat thethe shaftshaft appliesapplies toto thethe rotor)rotor) andand thethe rotationrotation ofof thethe rotorrotor areare inin thethe samesame direction,direction, energyenergy isis transferredtransferred fromfrom thethe shaftshaft toto thethe rotorrotor andand fromfrom thethe rotorrotor toto thethe fluid.fluid. DDTurbine:Turbine: TheThe torquetorque exertedexerted byby thethe shaftshaft onon thethe rotorrotor isis oppositeopposite toto thethe directiondirection ofof rotation,rotation, thethe energyenergy transfertransfer isis fromfrom thethe fluidfluid toto thethe rotor.rotor. 18 AngularAngular MomentumMomentum ConsiderationsConsiderations 2/6 AllAll ofof thethe turbomachinesturbomachines involveinvolve thethe rotationrotation ofof anan impellerimpeller oror aa rotorrotor aboutabout aa centcentralral axis,axis, itit isis appropriateappropriate toto discussdiscuss theirtheir performanceperformance inin termsterms ofof torquetorque andand angularangular momentummomentum.. 19 AngularAngular MomentumMomentum ConsiderationsConsiderations 3/6 InIn aa turbomachineturbomachine aa seriesseries ofof particlesparticles (a(a continuum)continuum) passespasses throughthrough thethe rotor.rotor. ForFor steadysteady flow,flow, thethe momentmoment ofof momentummomentum equationequation appliedapplied toto aa controlcontrol volumevolume r r r (rr × F) = (rr × V)ρV ⋅ nrdA ∑ ∫CS Net rate of flow of moment-of- Sum of the external torques momentum (angular momentum) through the control volume 20 AngularAngular MomentumMomentum ConsiderationsConsiderations 4/6 Applied to the one-dimensional simplification of flow through a turbomachine rotor, the axial component Tshaft = −m& 1(r1Vθ1) + m& 2 (r2Vθ2 ) (2) Shaft work applied to the contents Euler turbomachine equation of the control volume “+” : in the same direction as rotation “-” : in the opposite direction as rotation Euler turbomachine equation : the shaft torque is directly proportional to the mass flowrate. The torque also depends on the tangential component of the absolute velocity, Vθ. 21 AngularAngular MomentumMomentum ConsiderationsConsiderations 5/6 (2) W& shaft=Tshaft