6-7 THE CARNOT CYCLE (Heat Engine) The Reversed Carnot Cycle
The Carnot Cycle (Heat Engine)6-7 THE CARNOT CYCLEPη C 1 TLTHPTH1TLQ 0QH2TH constWnet ,out43V4QL V2V3TL6 - 6 Reversible Processes11Q-Q-WW2V2223V32V2 at thermal equilibrium the paths coinside heat transfer due to infinitelyQ 0V2V13V34V43QLV4V34V4V11gassmall difference in temperatureIrreversible Processes T 0 heat transfer due to T 0 heat generation by friction6-8 CARNOT PRINCIPLESTL constVTLV11TH const2TL constVThe Carnot Cycle consists of 4 reversible processesgascontinuesto expand4Wnet ,inQ 0Q 0THV1TH1QHSadi Carnot (1822-1888)The Reversed Carnot Cycle (Refrigerator)insulatedinsulatedQLQHIsothermal expansionat THAdiabatic expansionheat QH is addedTemperature dropped to TLQ 0Isothermal compressionat TLAdiabatic compressionheat QL is rejectedTemperature raised to THQ 0Efficiency of two Heat Engines operating between the same two reservoirs at TL and THTHViolation of Carnot Principlesyields violation of theC1η irreversible η reversibleIR2 nd Law of ThermodynamicsC2η R1 η R 2η IR η RR1R2η reversible 1 η reversible 2TL6-9 THE THERMODYNAMIC TEMPERATURE SCALEFor reversible heat engine operating between TL and TH :QLT LQHTH6-10 THE CARNOT HEAT ENGINEKelvin Scale of Absolute TemperatureThe efficiency of heat engine operating on a reversible Carnot cycle between TL and TH :ηC 1 6-11 THE CARNOT HEAT PUMPQLT 1 LQHTHCarnot Heat Pump: COPCHP1 Q1 LQHCarnotEfficiencyηC is the highest possible efficiencyof the heat engine operating betweentwo reservoirs at TL and THCarnot Refrigerator:1T1 LTH COPCR1 QH 1QL1TH 1TL
INSTITUT NATIONAL DES SCIENCES APPLIQUEES, LYON, FRANCE
COMMENTS AND ADDITIONAL NOTES ON CARNOT CYCLEconsists of 4 reversible processes which operate between two temperature recervoirs at TH and TL .Carnot CycleWhere TH and TL are temperatures according to not specified temperature scale (empirical).We know only that ordering of this temperature scale is such that TH TLto be consistent with the 2nd Law of Thermodynamics (Clausius statement).Carnot PrinciplesCP1η any η reversibleAssume the opposite, η any η reversible , to demonstratethat violation of the CP1 yields violation of the 2nd Law.CP2η reversible,1 η reversible,2From CP1, we haveη reversible,1 η reversible,2 andη reversible,2 η reversible,1Derivation of the Thermodynamic Temperature Scale (from Carnot Cycle with intermediate temperature reservoir at T0 )for any two Carnot cyclesηC1between the same TH and TLQ1Q21 1Lη C2 1 L2 , therefore, QHQHTHQHQHQ0T0Q0QLQL f (TL ,TH )QHdepends only on TH and TL , but not on the engineQH f (T0 ,TH )Q0where T0 is any such that TH T0 TLQL f (T0 ,TL )Q0QLTLQL QHQL Q0 Q0 QH f (TL ,TH ) f (T0 ,TL )f (T0 ,TH )θ (TL ) θ (TH )Kelvinchosed TLTHDemonstrate that the temperature of the Thermodynamic Temperature Scale is the same asAbsolute Temperature of the Ideal Gas equation of statePV mRTThis absolute temperature T can be measured by the Ideal Gas thermometer by measuring V of the fixed mass at constantpressure:P T VmRor by measuring pressure P in the rigid tank of constant volume V : TV PmR(formally, then zero absolute temperature corresponds to zero volume or zero pressure (no molecules in the tank))
Consider reversible process without heat transfer (adiabatic) expansion/compression of ideal gas (process 4 1 ):differential energy balancem duδ Q PdV 4δQ 0V4 1mRTdV mcv dTV c 11dV v dTVR T 11 1dV dTVk 1 TdV1V1( k 1) 1dV V1mRTVfrom ideal gas equation of state P c p cv RcRv cv11 c p cv c pk 1 1cv1lnC T dT lnV k 1 lnT lnCT V k 1 CPVPV k 1V C , from ideal gas equation, T mRmRPV kpolytropic process with Ckn k11T V k 1 C V ( CT 1 ) k 1 CT 1 kPT 1 k C1Carnot Cycle T k 1V2 L V3 TH Carnot Cycle 2 31 T k 1V4 H V1 TL V1 V4Carnot Cycle 4 11 TL k 1 V2 V3 TH QH mRTH lnQL mRTL lnV2V1V3V4 mRTH lnV3V4V3 V2 V4 V1Carnot Cycle 1 2Carnot Cycle 3 4Divide second equation by the first, then the Thermodynamic temperature scale of Kelvin for Carnot cycle is obtained:QLT LQHTHwhere TL and TH are absolute temperature of ideal gas equation.
Dependence of the temperature of the atmosphere on the height above the sea level(application of adiabatic expansion of a gas)control massWhen air rises to the upper regions of lower pressure, it EXPANDS.This expansion can be considered ADIABATIC,because air is a poor conductor of heat.Pascal's Lawmideal gas eqndP1 ρ g gdzv1P v RTdPP gdzRTseparate variables11dP gdzPRTzmpolytropic processfor ideal gaskPT 1 k Cthen differentialk d PT 1 k use product rulek 0 T 1 k dP P T 1 k k T k2k 1 1 1 k k kkT 1 kT 1 k0 , where T dT T 1 k 1 k 1k 1dP dTP1 k T k 11dT gdz1 k TRTdT k 1 gdzk RdT gdzcpT ( z ) T (0 ) eintegrate, using initial temperature T ( 0 ) , to solve for gzcp oC dTgThen temperature gradient dz 9.8 dzcp km
Another derivation:Consider the Carnot Cycle operating between two temperature reservoirs (for which both TLtemperature of ideal gas equation of state) for the fixed mass m of ideal gas, and show thatand TH are the absoluteQLT L .QHTHEnergy balance forIsothermal process 1 - 2 :QH W12 m ( u2 u1 )QH mRTH lnV2 0V1 QL W34 m ( u4 u3 )QL mRTL lnV4 0V3QL mRTL lnQL mRTL lnAdiabatic process 2 - 3:boundary work for isothermal processV2V1QH mRTH lnIsothermal process 3 - 4 :T T TH const21T T T const34Lboundary work for isothermal processV4V3V3V4Q W23 m ( u3 u2 ) PdV m dudifferential balance PdV m cv dTideal gas 1mRTdV mcv dTV c 11dV v dTVR T 1dV Vcv1 dTc p ccv T 1dV Vcpcv 1 1 from ideal gas equation of state P from c p cv R1dTT11 1dV dTVk 1 Tk cpcvspecific heats ratio1mRTV
33111 2 V dV k 1 2 T dT( k 1) lnlnTV2 ln 3V3T2TV2k 1 ln 3k 1T2V3V2k 1T LTHV3k 1can be obtained faster for isentropic process1 T k 1V2 L V3 TH Adiabatic process 4 - 1:V4k 1T HTLV1k 11 T k 1V4 H V1 TL 1Consider TL k 1V2 V3 TH V11 1V4 TH k 1 TL thenV2 V1 V3 V4V2 V3 V1 V4ConsiderQH mRTH lnCarnot Cycle 1- 2QL mRTL lnV2V1 mRTH lnV3V4V3V4Divide second equation by the first, thenQLT LQHTHwhere both TL and TH are absolute temperature of ideal gas equation of state.Conclusion: the absolute thermodynamic scale coincides with the absolute temperature scale of the ideal gas thermometer.
Derivation of1 ηC QLT1 L QHTHwith the help of entropy:QL TL ( S4 S3 )QH TH ( S 2 S1 )Because adiabatic processes 2- 3 and 4- 1 are isentropic, S1 S4 and S2 S3 , TL ( S4 S3 )QL QH TH ( S3 S4 )Division yieldsQLT LQHTHCOP in terms of η 1 COPHP COPCR 1 Q1 LQH1 QH 1QLQL:QH1η1η 1Negative absolute temperatureDefinition of the thermodynamic temperature scale does not prohibit to choose the negative absolute temperature scale, becauseQL QH TL THTLTHShow that choice of the negative absolute temperature scale will yield violation of the 2nd law of thermodynamics.Zero absolute temperature – is it allowed by this definitions?
EntropyEquationQLT LQHTHlooks like the basic equation for derivation of the results for reversible machines.Yes, this equation, in fact, is fundamental for the further development of classical thermodynamics, if rewritten asQLQ HTLTHQ S is called entropy.TQuantity‘loose definition of entropy” kJ It is equal to transferred heat per unit temperature . K Change of entropy during the Carnot cycle is zero:QH QL 0.TH TLEntropy is a property of the system and depends on its state like other properties P, T , V ,.Nernst’s postulate: entropy of any system at zero absolute temperature is zero.Differential change of entropy during the differential part of reversible process:δQ dST2 1ratio of amount of heat transferred to the systemto temperature of the system during this differential reversible processδQT S 2 S1change of entropy does not depend on path of reversible process
6-6 Reversible Processes. Q-W. 2. 1. T0. H. H. 6-8 CARNOT PRINCIPLES T TEfficiency of two Heat Engines operating between the same two reservoirs at . L and . H. C1. ηη irreversible reversible C2 . ηη reversible 1 reversible 2 L. 6-9 THE THERMODYNAMIC and TEMPERATURE SCALE : For reversible heat engine operating between T L T H. LL . HH QT QT
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Carnot. The theoretical heat engine that operates on the Carnot cycle is called the Carnot heat engine. The Carnot cycle is composed of four reversible processes—two isothermal and two adiabatic—and it can be executed either in a closed or a steady-flow system. The four reversible processes that make up the Carnot cycle are as follows:
ervoirs. In addition, let us attach a Carnot engine between the same reservoirs. Because the Carnot cycle is reversible, the Carnot engine can be run in reverse as a Carnot heat pump as shown on the right in Figure 22.9. We match the output work of the engine to the input work of the heat pump, W 5 W C, so there is no exchange of energy by work .
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The thermal efficiency of any heat engine, reversible or irreversible, is given by η th 1 QL QH Then the efficiency of a Carnot engine, or any reversible heat engine, becomes: η th, rev 1 TL TH This relation is often referred to as the Carnot efficiency, since the Carnot heat engine is the best known reversible engine.
