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Published byMustafa Gilday Modified over 2 years ago

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TcTc ThTh heat pump TcTc ThTh heat engine Carnot’s Theorem We introduced already the Carnot cycle with an ideal gas Now we show: 1 Energy efficiency of the Carnot cycle is independent of the working substance 2 Any cyclic process that absorbs heat at one temperature, and rejects heat at one other temperature, and is reversible has the energy efficiency of a Carnot cycle reversible Remark: Note: P >1 Textbook: coefficient of performance

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TcTc ThTh heat engine X Let’s combine a fictitious heat engine X with with a heat pump realized by a reversed Carnot cycle TcTc ThTh heat pump XC We can design the engine X such that Let’s calculate with

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If X would be a Carnot engine it would produce the work with > TcTc ThTh heat engine X TcTc ThTh heat pump XC We can design the engine X such that However: >0

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False Let X be the heat pump and the Carnot cycle operate like an engine False 1 Energy efficiency of the Carnot cycle is independent of the working substance. 2 Any cyclic process that absorbs heat at one temperature, and rejects heat at one other temperature, and is reversible has the energy efficiency of a Carnot cycle. Why Because: X can be a Carnot engine with arbitrary working substance

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Carnot’s theorem:No engine operating between two heat reservoirs is more efficient than a Carnot engine. Proof uses similar idea as before: We can design the engine X such that Again we create a composite device TcTc ThTh heat engine X TcTc ThTh heat pump XC operates the Carnot refrigerator

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My statement holds man Let’s assume that Note: this time engine X can be also work irreversible like a real engine does > Heat transferred from the cooler to the hotter reservoir without doing work on the surrounding Violation of the Clausius statement Rudolf ClausiusClausius ( )

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Applications of Carnot Cycles Any cyclic process that absorbs heat at one temperature, and rejects heat at one other temperature, and is reversible has the energy efficiency of a Carnot cycle. We stated: Why did we calculate energy efficiencies for - gas turbine - Otto cycle Because:they are not 2-temperature devices, but accept and reject heat at a range of temperatures Energy efficiency not given by the Carnot formula But:It is interesting to compare the maximum possible efficiency of a Carnot cycle with the efficiency of engineering cycles with the same maximum and minimum temperatures

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Consider the gas turbine again adiabates (Brayton or Joule cycle) Efficiency PhPh PlPl Maximum 3 Minimum 1 23 Heating the gas ( by burning the fuel ) 41 cooling : T3T3 : T1T1 with

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Efficiency of corresponding Carnot Cycle With Unfortunately:Gas turbine useless in the limit Because: Heat taken per cycle 0 Work done per cycle 0

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Absolute Temperature We showed:Energy efficiency of the Carnot cycle is independent of the working substance. Definition of temperature independent of any material property A temperature scale is an absolute temperature scale if and only if where, andare the heats exchanged by a Carnot cycle operating between reservoirs at temperatures T 1 and T 2. Measurement of Temperature ratio T1T1 T2T2

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As discussed earlier, unique temperature scale requires fixed point or Kelvin-scale: T fix =T tripel =273.16K It turns out: proportional to thermodynamic Temperature T empirical gas temperature Why Because: Calculation of efficiency of Carnot cycle based on yields With a=1

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From definition of thermodynamic temperature If any absolute temperature is positive all other absolute temperatures are positive there is an absolute zero of thermodynamic temperature when the rejected heat 0 T=0 can never be reached, because this would violate the Kelvin statement however

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