Thermodynamics

while exchanging heat with a single reservoir—a violation of the Kelvin–Planck statement of the second law. Therefore, our initial assumption thath th,irrev h th,rev is incorrect. Then we conclude that no heat engine can bemore efficient than a reversible heat engine operating between the samereservoirs.The second Carnot principle can also be proved in a similar manner. Thistime, let us replace the irreversible engine by another reversible engine thatis more efficient and thus delivers more work than the first reversibleengine. By following through the same reasoning, we end up having anengine that produces a net amount of work while exchanging heat with asingle reservoir, which is a violation of the second law. Therefore, we concludethat no reversible heat engine can be more efficient than a reversibleone operating between the same two reservoirs, regardless of how the cycleis completed or the kind of working fluid used.6–9 ■ THE THERMODYNAMICTEMPERATURE SCALEA temperature scale that is independent of the properties of the substancesthat are used to measure temperature is called a thermodynamic temperaturescale. Such a temperature scale offers great conveniences in thermodynamiccalculations, and its derivation is given below using some reversibleheat engines.The second Carnot principle discussed in Section 6–8 states that allreversible heat engines have the same thermal efficiency when operatingbetween the same two reservoirs (Fig. 6–42). That is, the efficiency of areversible engine is independent of the working fluid employed and itsproperties, the way the cycle is executed, or the type of reversible engineused. Since energy reservoirs are characterized by their temperatures, thethermal efficiency of reversible heat engines is a function of the reservoirtemperatures only. That is,orh th,rev g 1T H , T L 2Q HQ L f 1T H , T L 2(6–13)since h th 1 Q L /Q H . In these relations T H and T L are the temperatures ofthe high- and low-temperature reservoirs, respectively.The functional form of f(T H , T L ) can be developed with the help of thethree reversible heat engines shown in Fig. 6–43. Engines A and C are suppliedwith the same amount of heat Q 1 from the high-temperature reservoirat T 1 . Engine C rejects Q 3 to the low-temperature reservoir at T 3 . Engine Breceives the heat Q 2 rejected by engine A at temperature T 2 and rejects heatin the amount of Q 3 to a reservoir at T 3 .The amounts of heat rejected by engines B and C must be the same sinceengines A and B can be combined into one reversible engine operatingbetween the same reservoirs as engine C and thus the combined engine willChapter 6 | 303High-temperature reservoirat T H = 1000 KA reversibleHEFIGURE 6–42η th,Aη th, A = η th,B = 70%AnotherreversibleHEη th,BLow-temperature reservoirat T L = 300 KAll reversible heat engines operatingbetween the same two reservoirs havethe same efficiency (the second Carnotprinciple).Thermal energy reservoirat T 1Q 1Rev. HEAINTERACTIVETUTORIALSEE TUTORIAL CH. 6, SEC. 9 ON THE DVD.W AQ 2W CT 2 Rev. HEQ 2CRev. HE W BBQ 3Q 3Thermal energy reservoirat T 3Q 1FIGURE 6–43The arrangement of heat engines usedto develop the thermodynamictemperature scale.