#
Fractal Representation of Exergy^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Toy Model

**Figure 1.**Algorithm for the building of a correlon (

**a**) and diagram of a typical trajectory (

**b**). (

**a**) Probabilities of diffusion of the heat quantum for j > 0 (p

_{out}and p

_{back}are inverted otherwise); (b) Diagram of a correlon obtain for θ = 0.15 and N

_{Q}= 60.

## 3. Correlon Characteristics

**Figure 2.**Correlons obtained for a given heat load N

_{Q}at four different temperatures. The central line (j = 0) and the hypothetical piston are represented by a black dashed line and a red one respectively. (

**a**) Correlon for θ = 0.2 and N

_{Q}= 1000, N

_{X}= 189, L

_{A}/l

_{c}= 800; (

**b**) Correlon for θ = 0.4 and N

_{Q}= 1000, N

_{X}= 377, L

_{A}/l

_{c}= 600; (

**c**) Correlon for θ = 0.6 and N

_{Q}= 1000, N

_{X}= 583, L

_{A}/l

_{c}= 400; (

**d**) Correlon for θ = 0.8 and N

_{Q}= 1000, N

_{X}= 813, L

_{A}/l

_{c}= 200.

**Table 1.**Characteristic quantities of the correlons shown on Figure 2 and their corresponding analogous energetic values.

Figure | θ | Num. of Quantum Movement (N_{Q}) | Energy (Q) | Horizontal Length (L_{X}) | Exergy (X) | Duct Length (L_{A}) | Anergy (A) |
---|---|---|---|---|---|---|---|

2a | 0.2 | 1000 | $1000\phantom{\rule{0.166667em}{0ex}}q$ | $189\phantom{\rule{0.166667em}{0ex}}{l}_{c}$ | $200\phantom{\rule{0.166667em}{0ex}}q$ | $800\phantom{\rule{0.166667em}{0ex}}{l}_{c}$ | $800\phantom{\rule{0.166667em}{0ex}}q$ |

2b | 0.4 | 1000 | $1000\phantom{\rule{0.166667em}{0ex}}q$ | $377\phantom{\rule{0.166667em}{0ex}}{l}_{c}$ | $400\phantom{\rule{0.166667em}{0ex}}q$ | $600\phantom{\rule{0.166667em}{0ex}}{l}_{c}$ | $600\phantom{\rule{0.166667em}{0ex}}q$ |

2c | 0.6 | 1000 | $1000\phantom{\rule{0.166667em}{0ex}}q$ | $583\phantom{\rule{0.166667em}{0ex}}{l}_{c}$ | $600\phantom{\rule{0.166667em}{0ex}}q$ | $400\phantom{\rule{0.166667em}{0ex}}{l}_{c}$ | $400\phantom{\rule{0.166667em}{0ex}}q$ |

2d | 0.8 | 1000 | $1000\phantom{\rule{0.166667em}{0ex}}q$ | $813\phantom{\rule{0.166667em}{0ex}}{l}_{c}$ | $800\phantom{\rule{0.166667em}{0ex}}q$ | $200\phantom{\rule{0.166667em}{0ex}}{l}_{c}$ | $200\phantom{\rule{0.166667em}{0ex}}q$ |

#### 3.1. Fractal Behaviour

**Figure 3.**Fractal dimensions as functions of θ. The curves are obtained by averaging over 50 correlons for each temperature. (

**a**) Internal and shape fractal dimensions of correlons as functions of θ. (

**b**) Internal fractal dimension ${\Delta}_{\mathrm{int}}$ as a function of θ for various heat loads N

_{Q}in the scale range ${[{l}_{c},{l}_{0}]}_{750}$.

#### 3.2. Entropies

**Figure 4.**Two Gibbs-Shannon based entropies as functions of θ. The curves are obtained by averaging over 50 correlons for each temperature. (

**a**) Internal and shape entropies as functions of θ and their difference. (

**b**) Shape entropy S

_{sh}as a function of θ for various heat loads N

_{Q}.

- a strongly oriented energy type, efficient to produce work,
- a diffusive form of energy that tends to fill the space.

## 4. Conclusions

- the box-counting method gives a dimension related to the overall fractal shape of the correlons, regardless of their internal details;
- the Richardson’s method, sensible to the internal diffusion, allows a characterisation of the overlapping.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Beck, C.; Schlögl, F. Thermodynamics of Chaotic Systems; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Le Méhauté, A.; Nigmatoullin, R.-R.; Nivanen, L. Flèches du Temps et Géométrie Fractale; Hermes: Paris, France, 1998. (In French) [Google Scholar]
- Gosselin, L.; Bejan, A. Constructal heat trees at micro and nanoscales. J. Appl. Phys.
**2004**, 96, 5852–5859. [Google Scholar] [CrossRef] - Bejan, A.; Moran, M.J. Thermal Design and Optimization; Wiley: New York, NY, USA, 1995. [Google Scholar]
- Bejan, A. Constructal-theory network of conducting paths for cooling a heat generating volume. Int. J. Heat Mass Transf.
**1997**, 40, 799–816. [Google Scholar] [CrossRef] - Bejan, A. Shape and Structure, from Engineering to Nature; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Queiros-Conde, D. Le modèle des Peaux Entropiques en Turbulence développée. Comptes Rendus de l’Académie des Sci.
**2000**, 328, 541–546. (In French) [Google Scholar] [CrossRef] - Queiros-Condé, D. A diffusion equation to describe scale- and time-dependent dimensions of turbulent interfaces. Proc. Royal Soc. Lond. A Math. Phys. Eng. Sci.
**2003**, 459, 3043–3059. [Google Scholar] [CrossRef] - Queiros-Condé, D. Dynamique des Peaux Entropiques Dans les Systèmes Intermittents et Multi-échelles; Habilitation à Diriger des Recherches, Université Henri Poincaré: Nancy, France, 2006. (In French) [Google Scholar]
- Chen, G.Q. Exergy consumption of the earth. Ecol. Model.
**2005**, 184, 363–380. [Google Scholar] [CrossRef] - Wu, X.; Chen, G.; Wu, X.; Yang, Q.; Alsaedi, A.; Hayat, T.; Ahmad, B. Renewability and sustainability of biogas system: Cosmic exergy based assessment for a case in China. Renew. Sustain. Energy Rev.
**2015**, 51, 1509–1524. [Google Scholar] [CrossRef] - Dincer, I.; Cengel, Y.A. Energy, entropy and exergy concepts and their roles in thermal engineering. Entropy
**2001**, 3, 116–149. [Google Scholar] [CrossRef] - Leff, H.S. Entropy, its language, and interpretation. Found. Phys.
**2007**, 37, 1744–1766. [Google Scholar] [CrossRef] - Nottale, L.; Chaline, J.; Grou, P. Les Arbres de L’évolution; Hachette: Paris, France, 2000. (In French) [Google Scholar]
- Mandelbrot, B.B. The Fractal Geometry of Nature; W.H. Freeman: New York, NY, USA, 1982. [Google Scholar]
- Lawler, G.F. The dimension of the frontier of planar Brownian motion. Electron. Commun. Probab.
**1996**, 1, 29–47. [Google Scholar] [CrossRef] - Queiros-Condé, D.; Chaline, J.; Dubois, J. Le Monde des Fractales—La Nature Trans-échelles; Ellipses: Paris, France, 2015. (In French) [Google Scholar]
- Feidt, M. Thermodynamique et Optimisation Énergétique des Systèmes et Procédés; Technique et Documentation Lavoisier: Paris, France, 1996. (In French) [Google Scholar]
- Neveu, P. Apports de la Thermodynamique Pour la Conception et L’intégration des Procédés. Ph.D. Thesis, Université de Perpignan, Perpignan, France, December 2002. [Google Scholar]

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Canivet, Y.; Queiros-Condé, D.; Grosu, L.
Fractal Representation of Exergy. *Entropy* **2016**, *18*, 56.
https://doi.org/10.3390/e18020056

**AMA Style**

Canivet Y, Queiros-Condé D, Grosu L.
Fractal Representation of Exergy. *Entropy*. 2016; 18(2):56.
https://doi.org/10.3390/e18020056

**Chicago/Turabian Style**

Canivet, Yvain, Diogo Queiros-Condé, and Lavinia Grosu.
2016. "Fractal Representation of Exergy" *Entropy* 18, no. 2: 56.
https://doi.org/10.3390/e18020056