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Entropy 2014, 16(4), 2291-2308; doi:10.3390/e16042291

On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating System

KlimaCampus, Grindelberg 5, Universität Hamburg, 20144 Hamburg, Germany
Received: 5 January 2014 / Revised: 10 March 2014 / Accepted: 14 April 2014 / Published: 22 April 2014
(This article belongs to the Special Issue Maximum Entropy Production)
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Abstract

A black planet irradiated by a sun serves as the archetype for a simple radiating two-layer system admitting of a continuum of steady states under steadfast insolation. Steady entropy production rates may be calculated for different opacities of one of the layers, explicitly so for the radiative interactions, and indirectly for all the material irreversibilities involved in maintaining thermal uniformity in each layer. The second law of thermodynamics is laid down in two versions, one of which is the well-known Clausius-Duhem inequality, the other being a modern version known as the entropy inequality. By maximizing the material entropy production rate, a state may be selected that always fulfills the Clausius-Duhem inequality. Some formally possible steady states, while violating the latter, still obey the entropy inequality. In terms of Earth’s climate, global entropy production rates exhibit extrema for any “greenhouse effect”. However, and only insofar as the model be accepted as representative of Earth’s climate, the extrema will not be found to agree with observed (effective) temperatures assignable to both the atmosphere and surface. This notwithstanding, the overall entropy production for the present greenhouse effect on Earth is very close to the maximum entropy production rate of a uniformly warm steady state at the planet’s effective temperature. For an Earth with a weak(er) greenhouse effect the statement is no longer true. View Full-Text
Keywords: Clausius-Duhem inequality; entropy inequality; radiation entropy; minimum and maximum entropy production; planetary thermodynamics; Earth’s climate Clausius-Duhem inequality; entropy inequality; radiation entropy; minimum and maximum entropy production; planetary thermodynamics; Earth’s climate
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This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Pelkowski, J. On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating System. Entropy 2014, 16, 2291-2308.

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