Chapter (6)

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  CHAPTER (VI) ENTROPY   1  6.1: Clausius inequality:  2 To demonstrate the validity of the Clausius inequality, consider a system connected to a thermal energy reservoir at a constant thermodynamic temperature of    T  R   (absolute) through a reversible cyclic device (Fig. 6-1). The cyclic device receives heat  δ Q R   from the reservoir and supplies heat  δ Q to the system whose temperature at that part of the boundary is T    (a variable), while producing work   δ W  rev  . The system produces work δ W  sys   as a result of this heat transfer. FIGURE 6-1   Applying the energy balance to the combined system identified by dashed lines yields: δ W  c  = δ Q R   –   dE  c Eliminating δ Q R  from the two relations above yields: We now let the system undergo a cycle while the cyclic device undergoes an integral number of cycles. Then the preceding relation becomes: where  δ W  c  is the total work of the combined system ( δ W  rev   + δ W  sys   ) and dE  c  is the change in the total energy of the combined system. Considering that the cyclic device is a reversible one, we have: 3   It appears that the combined system is exchanging heat with a single thermal energy reservoir while involving (producing or consuming) work W  c during a cycle. On the basis of the Kelvin-Planck statement of the second law, which states that no system can produce a net amount of work while operating in a cycle and exchanging heat with a single thermal energy reservoir  , we reason that W  c  cannot be a work output, and thus it cannot be a positive quantity. Since T  R  is the thermodynamic temperature and thus a positive quantity, we must have: (6-1) which is the Clausius inequality. “This   inequality   is valid for all thermodynamic cycles, reversible or irreversible, including the refrigeration cycles”  .   4
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