Thermodynamics

332 | ThermodynamicsReversiblecyclicdeviceThermal reservoirT RTSystemINTERACTIVETUTORIALSEE TUTORIAL CH. 7, SEC. 1 ON THE DVD.δ Q Rδ QCombined system(system and cyclic device)FIGURE 7–1The system considered in thedevelopment of the Clausiusinequality.δ W revδ W sys7–1 ■ ENTROPYThe second law of thermodynamics often leads to expressions that involveinequalities. An irreversible (i.e., actual) heat engine, for example, is lessefficient than a reversible one operating between the same two thermalenergy reservoirs. Likewise, an irreversible refrigerator or a heat pump has alower coefficient of performance (COP) than a reversible one operatingbetween the same temperature limits. Another important inequality that hasmajor consequences in thermodynamics is the Clausius inequality. It wasfirst stated by the German physicist R. J. E. Clausius (1822–1888), one ofthe founders of thermodynamics, and is expressed as dQT 0That is, the cyclic integral of dQ/T is always less than or equal to zero. Thisinequality is valid for all cycles, reversible or irreversible. The symbol (integralsymbol with a circle in the middle) is used to indicate that the integrationis to be performed over the entire cycle. Any heat transfer to or from a systemcan be considered to consist of differential amounts of heat transfer. Then thecyclic integral of dQ/T can be viewed as the sum of all these differentialamounts of heat transfer divided by the temperature at the boundary.To demonstrate the validity of the Clausius inequality, consider a systemconnected to a thermal energy reservoir at a constant thermodynamic (i.e.,absolute) temperature of T R through a reversible cyclic device (Fig. 7–1).The cyclic device receives heat dQ R from the reservoir and supplies heat dQto the system whose temperature at that part of the boundary is T (a variable)while producing work dW rev . The system produces work dW sys as aresult of this heat transfer. Applying the energy balance to the combinedsystem identified by dashed lines yieldsdW C dQ R dE Cwhere dW C is the total work of the combined system (dW rev dW sys ) anddE C is the change in the total energy of the combined system. Consideringthat the cyclic device is a reversible one, we havedQ RT R dQ Twhere the sign of dQ is determined with respect to the system (positive if tothe system and negative if from the system) and the sign of dQ R is determinedwith respect to the reversible cyclic device. Eliminating dQ R from thetwo relations above yieldsdQdW C T R T dE CWe now let the system undergo a cycle while the cyclic device undergoes anintegral number of cycles. Then the preceding relation becomesW C T R dQ Tsince the cyclic integral of energy (the net change in the energy, which is aproperty, during a cycle) is zero. Here W C is the cyclic integral of dW C , andit represents the net work for the combined cycle.