Focus on the Clausius Inequalities as a consequence of modeling thermodynamic systems as a series of open Carnot cycles

Various results in thermodynamics developed recently are brought into focus by further refinement to set in less ambiguous form topics connected to irreversibility and the so called “Clausius Inequality”. This singular “Clausius Inequality” for both closed and open systems was traditionally deduced from the Riemann “integration” of closed Carnot cycle loops for irreversible transitions. Evidently topological problems might be expected to arise concerning boundary conditions when “open” and “closed” systems exists simultaneously in such a scheme. It has hitherto been assumed that in this scheme, only one central Clausius Inequality can exist coupling all processes. Based on a new recent development of open system Carnot cycles, it is shown that other analogous inequalities can be derived, due to the presence of another fundamental entropy state function derived in the recent development, implying non-singularity. Their properties are such as to indicate that no new non-equilibrium entropy can arise from the inequalities as has been proposed over the decades. It is shown that a sequence of points along a non-equilibrium state space must have excess variables augmenting those for the equilibrium situation, which demonstrates that the often used Principle of Local Equilibrium (PLE) is only an approximation, implying that far-from-equilibrium theories should be developed ab initio from irreversible dynamical laws rather than from PLE. Examples presented from actual computations for both systems in equilibrium and non-equilibrium appears to support this deduction. Large scale and extensive thermodynamical theories have been created based on the assumption of a single Clausius-like inequality, such as those stemming from the very influential and extensive Truesdale school, and so such pervasive developments are also open to question.