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Consider:

http://www.me.umn.edu/education/courses/me3331/modules/3331-Mod-28.pdf

more precisely "The Clausius Theorem" and the assumption entitled
"Equivalent reversible and irreversible processes" according to which
any "Irreversible Process" can be closed by a reversible process (in
this case consisting of a "Reversible Adiabat", "Reversible Isotherm"
and "Reversible Adiabat") to become a cycle. Is the assumption
correct? If it is not, what remains of the concept of entropy?

This is one of the most dangerous questions in thermodynamics. For 140
years it has been asked only once, by Jos Uffink:

http://philsci-archive.pitt.edu/archive/00000313/
p.39: "A more important objection, it seems to me, is that Clausius
bases his conclusion that the entropy increases in a nicht umkehrbar
[irreversible] process on the assumption that such a process can be
closed by an umkehrbar [reversible] process to become a cycle. This is
essential for the definition of the entropy difference between the
initial and final states. But the assumption is far from obvious for a
system more complex than an ideal gas, or for states far from
equilibrium, or for processes other than the simple exchange of heat
and work. Thus, the generalisation to all transformations occurring in
Nature is somewhat rash."

The answer to the dangerous question is implicit in Uffink's
conclusion:

http://philsci-archive.pitt.edu/archive/00000313/
Jos Uffink: "This summary leads to the question whether it is fruitful
to see irreversibility or time-asymmetry as the essence of the second
law. Is it not more straightforward, in view of the unargued
statements of Kelvin, the bold claims of Clausius and the strained
attempts of Planck, to give up this idea? I believe that Ehrenfest-
Afanassjewa was right in her verdict that the discussion about the
arrow of time as expressed in the second law of the thermodynamics is
actually a RED HERRING."

Pentcho Valev
pvalev@yahoo.com