Chapter 9- Vapor Power Systems - College of … Ideal Rankine Cycle An ideal Rankine cycle consists of four reversible processes. Process 1à2 Isentropic expansion through the turbine

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Chapter9:VaporPowerSystems

TableofContents

Introduction........................................................................................................................................2

AnalyzingtheRankineCycle...............................................................................................................4

RankineCyclePerformanceParameters............................................................................................5

IdealRankineCycle.............................................................................................................................6Example........................................................................................................................................................7

RankineCycleIncludingIsentropicEfficiencies.................................................................................8Example........................................................................................................................................................9Example......................................................................................................................................................10

RankineCycleImprovements...........................................................................................................11Example......................................................................................................................................................12

NumericalAnswerstoExamples......................................................................................................13

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IntroductionVaporpowersystems,orvaporpowerplants,convertaprimaryenergysourceintoelectricitybyalternatelyvaporizingandcondensingaworkingfluid(usuallywater).TheRankinecycleisthebasicbuildingblockofvaporpowersystems.Theprimaryenergysource(e.g.,fossil-fuel,nuclear,solar,orgeothermal)supplies𝑄",theenergyneededtovaporizetheworkingfluidintheboiler.Theturbinepowershaftisconnectedtoanelectricgenerator,whichgeneratestheelectricitythatisthentransmittedanddistributedtoconsumersviatheelectricalgrid.

Forfossil-fueledvaporpowerplants,𝑄" issuppliedbycombustionofthefossilfuel(e.g.,coal).1

1(Moran,Shapiro,Boettner,&Bailey,2011)

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Fornuclearvaporpowerplants,𝑄" issuppliedbyacontrollednuclearreaction.2

Forsolarpowerplants,𝑄" issuppliedbycollectedandconcentratedsolarradiation.3

2(Moran,Shapiro,Boettner,&Bailey,2011)3(Moran,Shapiro,Boettner,&Bailey,2011)

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Forgeothermalpowerplants,𝑄" issuppliedbyhotwaterand/orsteamdrawnfrombelowtheearth’ssurface.Ofnote,theworkingfluidinageothermalpowerplantisanorganicsubstance,suchasisobutene,thathasalowerboilingpointthanwater.4

AnalyzingtheRankineCycleWewillanalyzethecomponentsoftheRankinecyclebyapplyingthe1stLawtoeachdeviceinthecycle(turbine,condenser,pump,boiler).Foralldeviceswewillassumesteadystate,steadyflow(SSSF),one-dimensional(1D)flow,uniformflow,andaquasiequilibriumprocess.Wewillalsoneglectanychangesinkineticandpotentialenergy.

4(Moran,Shapiro,Boettner,&Bailey,2011)

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Neglectingheattransferwiththesurroundings,therateatwhichworkisdevelopedperunitmassofvaporpassingthroughtheturbineis:

𝑊$%&'()* =𝑊$%&'()*

𝑚= ℎ. − ℎ0

Thesignof𝑊$%&'()* willbepositive,followingoursignconventionthatworkispositivewhenitisdonebythesystemandonthesurroundings.

Theonlyworkinteractionforthecondenserisflowwork.Therefore,therateatwhichheatistransferredperunitmassofvaporpassingthroughthecondenseris:

𝑄12)3*)4*& =𝑄12)3*)4*&

𝑚= ℎ5 − ℎ0 = −𝑄6

Thesignof𝑄12)3*)4*& willbenegative,followingoursignconventionthatheattransferispositivewhenheatistransferredtothesystemfromthesurroundings.

Neglectingheattransferwiththesurroundings,therateatwhichworkisdevelopedperunitmassofvaporpassingthroughthepumpis:

𝑊7%87 =𝑊7%87𝑚

= ℎ5 − ℎ9

Thesignof𝑊7%87willbenegative,followingoursignconventionforwork.

Theonlyworkinteractionfortheboilerisflowwork.Therefore,therateatwhichheatistransferredperunitmassofvaporpassingthroughtheboileris:

𝑄'2(:*& =𝑄'2(:*&𝑚

= ℎ. − ℎ9 = 𝑄"

Thesignof𝑄'2(:*& willbepositive,followingoursignconventionforheattransfer.

RankineCyclePerformanceParametersThethermalefficiencyoftheRankinecyclewillbeacomparisonofwhatweget(thenetworkofthecycle)towhatwehavetopay(theheatsuppliedtothecycle).

𝜂 =𝑊$%&'()* + 𝑊7%87

𝑄'2(:*&=

ℎ. − ℎ0 + ℎ5 − ℎ9ℎ. − ℎ9

= 1 +ℎ5 − ℎ0ℎ. − ℎ9

Thebackworkratiocomparestheworkrequiredbythepumptotheworkdevelopedbytheturbine.

𝑏𝑤𝑟 =𝑊7%87𝑊$%&'()*

=ℎ5 − ℎ9ℎ. − ℎ0

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IdealRankineCycleAnidealRankinecycleconsistsoffourreversibleprocesses.

Process1à2 Isentropicexpansionthroughtheturbinefromsaturatedvaportothecondenserpressure2à3 Constantpressureheatrejectionthroughthecondensertosaturatedliquid3à4 Isentropiccompressionthroughthepump4à1 Constantpressureheatadditionthroughtheboiler

Thepumpworkcanbeevaluatedusingthe1stLaworbyusingtheexpressionformechanicalworkforsteady-flowreversibleprocesses,wherechangesinkineticandpotentialenergyhavebeenneglected.

𝑊𝑚 &*A

= − 𝑣𝑑𝑝2%$

()

Thespecificvolumeofthefluidflowingthroughthepumpisapproximatelyconstant.Therefore,pumpworkcanbeapproximatedas:

𝑊7%87𝑚 &*A

≈ −𝑣5 𝑝9 − 𝑝5

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ExampleWateristheworkingfluidinanidealRankinecycle.Thecondenserpressureis6kPa.Theboilerpressureis10MPa.FindthethermalefficiencyofthecycleandcomparetoCarnotcycleefficiency.

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RankineCycleIncludingIsentropicEfficienciesAnalysisofanidealRankinecycleassumedallprocessesoperatedreversibly.WecanalsoanalyzeRankinecycleswhengivenisentropicefficienciesoftheturbineandofthepump.

ThethermalefficiencyofaRankinecycleincludingisentropicefficienciesoftheturbineandofthepumpcanbecalculatedby:

𝜂 =ℎ. − ℎ0 + ℎ5 − ℎ9

ℎ. − ℎ9=𝜂$%&'()* ℎ. − ℎ04 + ℎ5 − ℎ94

𝜂7%87ℎ. − ℎ9

=𝜂$%&'()* ℎ. − ℎ04 + 𝑣5 𝑝5 − 𝑝9𝜂7%87

ℎ. − ℎ9

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ExampleSteamistheworkingfluidinaRankinecycle.Superheatedvaporenterstheturbineat10MPa,480°C.Condenserpressureis6kPa.Theturbineandpumphaveisentropicefficienciesof80%and70%.Findtheheatadditionthroughtheboiler(kJ/kg),thethermalefficiencyofthecycle,andtheheatrejectionthroughthecondenser(kJ/kg).

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ExampleWateristheworkingfluidinaRankinecycle.Superheatedvaporenterstheturbineat10MPa,480°Cwithamassflowrateof7.8kg/sandexitsat8kPa.Theisentropicefficiencyoftheturbineis88%andtheisentropicefficiencyofthepumpis82%.FindthenetpowerdevelopedinkWandthethermalefficiencyofthecycle.

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RankineCycleImprovementsSuperheatandReheataretwooptionsforimprovingthethermalefficiencyofaRankinecycle.ARankinecyclewithsuperheatallowstheturbineinlettobesuperheatedvaporratherthansaturatedvapor.

ARankinecyclewithreheatincludesatwo-stageturbine.Steamexpandsthroughafirststageturbine(1à2),returnstotheboilertobereheated(2à3),andthenexpandsthroughthesecondstageoftheturbine(3à4)beforemovingthroughthecompressorandpump.

ThethermalefficiencyoftheRankinecyclewithreheatisagainacomparisonofwhatweget(thenetworkofthecycle)towhatwehavetopay(theheatsuppliedtothecycle).Forthis

𝜂 =ℎ. − ℎ0 + ℎ5 − ℎ9 + ℎF − ℎG

ℎ. − ℎG + ℎ5 − ℎ0

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ExampleSteamat10MPa,600°Centersthefirst-stageturbineofanidealRankinecyclewithreheat.Steamleavesthereheatsectionoftheboilerat500°C.Thecondenserpressureis6kPa.Thequalityattheexitofthesecond-stageturbineis90%.FindthethermalefficiencyofthecycleandcompareittotheCarnotefficiency.

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NumericalAnswerstoExamplesPage Answer(s)7 38.5%,47.1%9 3155kJ/kg,32.8%,2120kJ/kg10 7800kW,32%12 52.5%,64.5%