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Carnot’s Theorem We introduced already the Carnot cycle with an ideal gas Now we show: Energy efficiency of the Carnot cycle is independent of the working substance 1 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 2 Remark: Tc Th heat engine reversible Tc Th heat pump Note:P>1 Textbook: coefficient of performance

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**Let’s combine a fictitious heat engine X with with a heat pump**

realized by a reversed Carnot cycle heat engine X heat pump Th Th X C Tc Tc 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**

However: with > >0 Tc Th heat engine X heat pump X C We can design the engine X such that

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False Let X be the heat pump and the Carnot cycle operate like an engine False 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. 2 Energy efficiency of the Carnot cycle is independent of the working substance. 1 Why Because: X can be a Carnot engine with arbitrary working substance

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**No engine operating between two heat reservoirs is **

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

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**Heat transferred from the cooler to the hotter reservoir **

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

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**Applications of Carnot Cycles**

We stated: 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. - gas turbine - Otto cycle Why did we calculate energy efficiencies for 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**

(Brayton or Joule cycle) Efficiency 2 3 Heating the gas (by burning the fuel) 2 3 Ph cooling 4 1 Maximum temperature: @ : 3 T3 adiabates Minimum temperature: 4 @ : T1 Pl 1 1 with

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**Efficiency of corresponding Carnot Cycle**

With Unfortunately: Gas turbine useless in the limit Because: Heat taken per cycle Work done per cycle

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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 , and are the heats exchanged by a Carnot cycle operating between reservoirs at temperatures T1 and T2. T1 T2 Measurement of Temperature ratio

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**As discussed earlier, unique temperature scale requires fixed point**

or Kelvin-scale: Tfix =Ttripel=273.16K It turns out: empirical gas temperature proportional to thermodynamic Temperature T 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 however T=0 can never be reached, because this would violate the Kelvin statement

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