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Irreversible heat engines strictly less efficient than reversible ones

I undertand how Canot's theorem implies that irreversible heat engines must be no more efficient than reversible one's, but it isless clear why they need to be less efficient, as I have seen stated in ...
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Heat engine efficiency, compression ratio vs temperatures

I want to understand the role or even existence of a cold reservoir in the Otto cycle gasoline engine. The efficiency of the Otto engine is given as a function of compression ratio $$ ...
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If maximal work is extracted, in what manner does the heat flow between bodies?

There are two bodies starting with different temperatures and specific heat capacities (for simplicity assume that the bodies are solid to the volumes are constant). Let the rate of heat flow from ...