Thermodynamics

304 | Thermodynamicshave the same efficiency as engine C. Since the heat input to engine C is thesame as the heat input to the combined engines A and B, both systems mustreject the same amount of heat.Applying Eq. 6–13 to all three engines separately, we obtainQ 1Q 2 f 1T 1 , T 2 2, Q 2Q 3 f 1T 2 , T 3 2,and Q 1Q 3 f 1T 1 , T 3 2Now consider the identityHigh-temperature reservoirat T HQ HReversibleheat engineorrefrigeratorQ LW netQ H T H=Q L T LLow-temperature reservoirat T LFIGURE 6–44For reversible cycles, the heat transferratio Q H /Q L can be replaced by theabsolute temperature ratio T H /T L .which corresponds toQ 1 Q 1 Q 2Q 3 Q 2f 1T 1 , T 3 2 f 1T 1 , T 2 2 # f 1T 2 , T 3 2A careful examination of this equation reveals that the left-hand side is afunction of T 1 and T 3 , and therefore the right-hand side must also be a functionof T 1 and T 3 only, and not T 2 . That is, the value of the product on theright-hand side of this equation is independent of the value of T 2 . This conditionwill be satisfied only if the function f has the following form:f 1T 1 , T 2 2 f 1T 12f 1T 2 2 andf 1T 2, T 3 2 f 1T 22f 1T 3 2so that f(T 2 ) will cancel from the product of f(T 1 , T 2 ) and f(T 2 , T 3 ), yieldingQ 1 f 1T 1 , T 3 2 f 1T 12Q 3 f 1T 3 2(6–14)This relation is much more specific than Eq. 6–13 for the functional form ofQ 1 /Q 3 in terms of T 1 and T 3 .For a reversible heat engine operating between two reservoirs at temperaturesT H and T L , Eq. 6–14 can be written asQ H f 1T H2Q L f 1T L 2(6–15)This is the only requirement that the second law places on the ratio of heattransfers to and from the reversible heat engines. Several functions f(T) satisfythis equation, and the choice is completely arbitrary. Lord Kelvin firstproposed taking f(T) T to define a thermodynamic temperature scale as(Fig. 6–44)a Q Hb T H(6–16)Q L rev T LThis temperature scale is called the Kelvin scale, and the temperatures onthis scale are called absolute temperatures. On the Kelvin scale, the temperatureratios depend on the ratios of heat transfer between a reversible heatengine and the reservoirs and are independent of the physical properties ofany substance. On this scale, temperatures vary between zero and infinity.The thermodynamic temperature scale is not completely defined byEq. 6–16 since it gives us only a ratio of absolute temperatures. We alsoneed to know the magnitude of a kelvin. At the International Conference onQ 3