  Exergy represents the maximum useful work possible when a system at a
specific state reaches equilibrium with the environmental dead state at
temperature To. Correspondingly, the exergy difference between two states is
the maximum work output when the system changes from one state to the other,
assuming that during the processes, the system exchanges heat reversibly with
the environment. If the process involves irreversibility, the Guoy-Stodola
theorem states that the exergy destruction equals the entropy generated during
the process multiplied by To. The exergy concept and the Gouy-Stodola theorem
are widely used to optimize processes or systems, even when they are not
directly connected to the environment. In the past, questions have been raised
on if To is the proper temperature to use in calculating the exergy
destruction. Here, we start from the first and the second laws of
thermodynamics to unambiguously show that the useful energy loss (UEL) of a
system or process should equal to the entropy generation multiplied by an
equivalent temperature associated with the entropy rejected out of the entire
system. For many engineering systems and processes, this entropy rejection
temperature can be easily calculated as the ratio of the changes of the
enthalpy and entropy of the fluid stream carrying the entropy out, which we
call the state-change temperature. The UEL is unambiguous and independent of
the environmental dead state, and it should be used for system optimization
rather than the exergy destruction.
