In the paper, Clausius and Darwin Can Both Be Right, the author claims to show an apparent paradox in the status of the heat that is not converted into work in a modified version of the Carnot Cycle. In this modified version, after one reversible cycle (at which point the transfer of heat to the cold bath is reversible since the work done by the cycle is available to reverse the heat flow), the available work is then dissipated into heat that flows into the cold bath, e.g, via driving a paddle in the cold bath causing friction.
The author claims that the entire change in entropy in this overall process is due to the increase in entropy caused by the dissipation of the work into heat, which is irreversible. But if so, this would apparently show that the heat that was not converted to work caused no increase in entropy. But since this latter heat flow can no longer be reversed since the work done has been dissipated, the author claims that this shows an irreversible heat flow with no increase in entropy.
This seems wrong to me, since presumably someone would have shown that there are irreversible processes that do not increase entropy long ago if the reasoning were correct. But I can't quite put my finger on where his argument breaks down in the paper.
Is it simply that the status of the heat that was not converted to work incrementally changes from reversible to irrevesible as the work is dissipated? If so, this feels a little strange, almost like entanglement's spooky action at a distance. The dissipation of the work produced by a local cycle in some remote system (e.g., in the lifting of a weight) can change the status of the heat that flowed into cold reservoir.
My apologies for not including the equations in my question, I'm not proficient in creating such equations.
The relevant discussion takes place in section 3.1. I am not concerned with any other part of the paper or any of the other claims made in other sections, eg claims about evolution vs entropy. Also, note that the flow of the text on p.56 is a little strange. It seems that Fig.2 should have been a page break. The right column above Fig.2 should be read after the left column above Fig.2. Then the right column below the figure should be read after the left.