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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| Format: | Article |
| Language: | English |
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MDPI AG
2014-04-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/16/4/2291 |
| _version_ | 1798038172282650624 |
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| author | Joachim Pelkowski |
| author_facet | Joachim Pelkowski |
| author_sort | Joachim Pelkowski |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-04-11T21:36:35Z |
| format | Article |
| id | doaj.art-68d0fbe14f0f4e9ca74d0ca136c2b001 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T21:36:35Z |
| publishDate | 2014-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-68d0fbe14f0f4e9ca74d0ca136c2b0012022-12-22T04:01:43ZengMDPI 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 |
| 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 |
| title | 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_short | 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 |
| 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 |
| work_keys_str_mv | AT joachimpelkowski ontheclausiusduheminequalityandmaximumentropyproductioninasimpleradiatingsystem |