Thermodynamics

302 | Thermodynamics1Irrev.HEHigh-temperature reservoirat T H2Rev.HEη th,1 < η th,2 η th,2 = η th,3Low-temperature reservoirat T LFIGURE 6–40The Carnot principles.3Rev.HEWe can draw valuable conclusions from these statements. Two conclusionspertain to the thermal efficiency of reversible and irreversible (i.e., actual)heat engines, and they are known as the Carnot principles (Fig. 6–40),expressed as follows:1. The efficiency of an irreversible heat engine is always less than the efficiencyof a reversible one operating between the same two reservoirs.2. The efficiencies of all reversible heat engines operating between thesame two reservoirs are the same.These two statements can be proved by demonstrating that the violation ofeither statement results in the violation of the second law of thermodynamics.To prove the first statement, consider two heat engines operating betweenthe same reservoirs, as shown in Fig. 6–41. One engine is reversible and theother is irreversible. Now each engine is supplied with the same amount ofheat Q H . The amount of work produced by the reversible heat engine isW rev , and the amount produced by the irreversible one is W irrev .In violation of the first Carnot principle, we assume that the irreversibleheat engine is more efficient than the reversible one (that is, h th,irrev h th,rev )and thus delivers more work than the reversible one. Now let the reversibleheat engine be reversed and operate as a refrigerator. This refrigerator willreceive a work input of W rev and reject heat to the high-temperature reservoir.Since the refrigerator is rejecting heat in the amount of Q H to the hightemperaturereservoir and the irreversible heat engine is receiving the sameamount of heat from this reservoir, the net heat exchange for this reservoir iszero. Thus, it could be eliminated by having the refrigerator discharge Q Hdirectly into the irreversible heat engine.Now considering the refrigerator and the irreversible engine together, wehave an engine that produces a net work in the amount of W irrev W revHigh-temperature reservoirat T HIrreversibleHEQ H Q HW irrev W revReversibleHE(or R)CombinedHE + RW irrev – W revQ L,irrev < Q L,rev(assumed)Q L,revQ L,rev – Q L,irrevLow-temperature reservoirat T LLow-temperature reservoirat T LFIGURE 6–41Proof of the first Carnot principle.(a) A reversible and an irreversible heatengine operating between the same tworeservoirs (the reversible heat engine isthen reversed to run as a refrigerator)(b) The equivalent combined system