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

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 int...

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Main Author: Joachim Pelkowski
Format: Article
Language:English
Published: MDPI AG 2014-04-01
Series:Entropy
Subjects:
Online Access:http://www.mdpi.com/1099-4300/16/4/2291
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spelling doaj-68d0fbe14f0f4e9ca74d0ca136c2b0012020-11-24T21:06:32ZengMDPI AGEntropy1099-43002014-04-011642291230810.3390/e16042291e16042291On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating SystemJoachim Pelkowski0KlimaCampus, Grindelberg 5, Universität Hamburg, 20144 Hamburg, GermanyA 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.http://www.mdpi.com/1099-4300/16/4/2291Clausius-Duhem inequalityentropy inequalityradiation entropyminimum and maximum entropy productionplanetary thermodynamicsEarth’s climate
collection DOAJ
language English
format Article
sources DOAJ
author Joachim Pelkowski
spellingShingle Joachim Pelkowski
On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating System
Entropy
Clausius-Duhem inequality
entropy inequality
radiation entropy
minimum and maximum entropy production
planetary thermodynamics
Earth’s climate
author_facet Joachim Pelkowski
author_sort Joachim Pelkowski
title On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating System
title_short On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating System
title_full On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating System
title_fullStr On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating System
title_full_unstemmed On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating System
title_sort on the clausius-duhem inequality and maximum entropy production in a simple radiating system
publisher MDPI AG
series Entropy
issn 1099-4300
publishDate 2014-04-01
description 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.
topic Clausius-Duhem inequality
entropy inequality
radiation entropy
minimum and maximum entropy production
planetary thermodynamics
Earth’s climate
url http://www.mdpi.com/1099-4300/16/4/2291
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