# Performance Features of a Stationary Stochastic Novikov Engine

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Novikov Engine with Fluctuating Temperature of the Hot Bath

#### 2.1. Classical Novikov Engine

#### 2.2. Stochastic Novikov Engine

#### 2.3. Reference Example

- The reference example is a heat engine working between a hot bath with temperature ${T}_{\mathrm{H}}$ and a cold heat bath with temperature ${T}_{\mathrm{L}}$.
- ${T}_{\mathrm{H}}$ has a mean value of $670\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$.
- ${T}_{\mathrm{L}}$ has a fixed value of $300\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$.
- $\kappa =0.68\phantom{\rule{0.222222em}{0ex}}\frac{\mathrm{MW}}{\mathrm{K}}$, so the power output is approximately $50\phantom{\rule{0.166667em}{0ex}}\mathrm{MW}$ at ${T}_{\mathrm{H}}=670\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$.

## 3. Influence of the Temperature Distribution

#### 3.1. The Considered Distributions

#### 3.2. Performances Measures for the Uniform Distribution

#### 3.3. Comparison of the Performance Measures for Different Distribution Shapes

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Reitlinger, H.B. Sur L’utilisation de la Chaleur Dans Les Machines á feu; Vaillant-Carmanne: Liége, Belgium, 1929. [Google Scholar]
- Chambadal, P. Le choix du cycle thermique dans une usine generatrice nucleaire. Rev. Gén. Électr.
**1958**, 67, 332–345. [Google Scholar] - Novikov, I.I. The Efficiency of Atomic Power Stations. At. Energy
**1957**, 3, 409–412. (In Russian) [Google Scholar] [CrossRef] - Novikov, I.I. The Efficiency of Atomic Power Stations. J. Nuclear Energy
**1958**, 7, 125–128. [Google Scholar] - Curzon, F.L.; Ahlborn, B. Efficiency of a Carnot Engine at Maximum Power Output. Am. J. Phys.
**1975**, 43, 22–24. [Google Scholar] [CrossRef] - Hoffmann, K.H.; Burzler, J.M.; Fischer, A.; Schaller, M.; Schubert, S. Optimal Process Paths for Endoreversible Systems. J. Non-Equilib. Thermodyn.
**2003**, 28, 233–268. [Google Scholar] [CrossRef] - Hoffmann, K.H.; Burzler, J.M.; Schubert, S. Endoreversible Thermodynamics. J. Non-Equilib. Thermodyn.
**1997**, 22, 311–355. [Google Scholar] - De Vos, A. Endoreversible Thermodynamics of Solar Energy Conversion; Oxford University Press: Oxford, UK, 1992. [Google Scholar]
- Fischer, A.; Hoffmann, K.H. Can a quantitative simulation of an Otto engine be accurately rendered by a simple Novikov model with heat leak? J. Non-Equilib. Thermodyn.
**2004**, 29, 9–28. [Google Scholar] [CrossRef] - Bădescu, V. On the Theoretical Maximum Efficiency of Solar-Radiation Utilization. Energy
**1989**, 14, 571–573. [Google Scholar] [CrossRef] - Bejan, A.; Kearney, D.W.; Kreith, F. Second Law Analysis and Synthesis of Solar Collector Systems. J. Sol. Energy Eng.
**1981**, 103, 23–28. [Google Scholar] [CrossRef] - Carmago, M.C. Bases fisicas del aprovechamiento de la energia solar. Rev. Geophys.
**1976**, 35, 227–239. (In Spanish) [Google Scholar] - Chen, J.; Andresen, B. The Maximum Coefficient of Performance of Thermoelectric Heat Pumps. Int. J. Ambient Energy
**1996**, 17, 22–28. [Google Scholar] [CrossRef] - Chen, J.; Andresen, B. New bounds on the performance parameters of a thermoelectric generator. Int. J. Power Energy Syst.
**1996**, 16, 23–27. [Google Scholar] - Chen, J. Thermodynamic Analysis of a Solar-Driven Thermoelectric Generator. J. Appl. Phys.
**1996**, 79, 2717–2721. [Google Scholar] [CrossRef] - Gordon, J.M. Generalized Power Versus Efficiency Characteristics of Heat Engines: The Thermoelectric Generator as an Instructive Illustration. Am. J. Phys.
**1991**, 59, 551–555. [Google Scholar] [CrossRef] - Müser, H. Thermodynamische Behandlung von Elektronenprozessen in Halbleiterrandschichten. Z. Phys.
**1957**, 148, 380–390. (In German) [Google Scholar] [CrossRef] - Schwalbe, K.; Fischer, A.; Hoffmann, K.H.; Mehnert, J. Applied endoreversible thermodynamics: Optimization of powertrains. In Proceedings of the 27th International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, Turku, Finland, 15–19 June 2014; Zevenhouen, R., Ed.; Åbo Akademi University: Turku, Finland, 2014; pp. 45–55. [Google Scholar]
- De Vos, A. Reflections on the power delivered by endoreversible engines. J. Phys. D Appl. Phys.
**1987**, 20, 232–236. [Google Scholar] [CrossRef] - De Vos, A. Endoreversible Thermodynamics and Chemical Reactions. J. Chem. Phys.
**1991**, 95, 4534–4540. [Google Scholar] [CrossRef] - De Vos, A. Is a solar cell an edoreversible engine? Sol. Cells
**1991**, 31, 181–196. [Google Scholar] [CrossRef] - Wu, C. Performance of solar-pond thermoelectric power generators. Int. J. Ambient Energy
**1995**, 16, 59–66. [Google Scholar] [CrossRef] - Wu, C. Analysis of Waste-Heat Thermoelectric Power Generators. Appl. Therm. Eng.
**1996**, 16, 63–69. [Google Scholar] [CrossRef] - Kojima, S. Maximum Work of Free-Piston Stirling Engine Generators. J. Non-Equilib. Thermodyn.
**2017**, 42, 169–186. [Google Scholar] [CrossRef] - Kojima, S. Theoretical Evaluation of the Maximum Work of Free-Piston Engine Generators. J. Non-Equilib. Thermodyn.
**2017**, 42, 31–58. [Google Scholar] [CrossRef] - Páez-Hernández, R.T.; Portillo-Díaz, P.; Ladino-Luna, D.; Ramírez-Rojas, A.; Pacheco-Paez, J.C. An analytical study of the endoreversible Curzon-Ahlborn cycle for a non-linear heat transfer law. J. Non-Equilib. Thermodyn.
**2016**, 41, 19–27. [Google Scholar] [CrossRef] - Özel, G.; Açıkkalp, E.; Savaş, A.F.; Yamık, H. Comparative Analysis of Thermoeconomic Evaluation Criteria for an Actual Heat Engine. J. Non-Equilib. Thermodyn.
**2016**, 41, 225–235. [Google Scholar] [CrossRef] - Wagner, K.; Hoffmann, K.H. Endoreversible modeling of a PEM fuel cell. J. Non-Equilib. Thermodyn.
**2015**, 40, 283–294. [Google Scholar] [CrossRef] - Zhang, Y.; Guo, J.; Lin, G.; Chen, J. Universal Optimization Efficiency for Nonlinear Irreversible Heat Engines. J. Non-Equilib. Thermodyn.
**2017**, 42, 253–263. [Google Scholar] [CrossRef] - Kowalski, G.J.; Zenouzi, M.; Modaresifar, M. Entropy production: Iintegrating renewable energy sources into sustainable energy solutions. In Proceedings of the 12th Joint European Thermodynamics Conference, JETC 2013, Brescia, Italy, 1–5 July 2013; Pilotelli, M., Beretta, G.P., Eds.; Cartolibreria SNOOPY s.n.c.: Brescia, Italy, 2013; pp. 25–32. [Google Scholar]
- Birnbaum, J.; Feldhoff, J.F.; Fichtner, M.; Hirsch, T.; Jöcker, M.; Pitz-Paal, R.; Zimmermann, G. Steam temperature stability in a direct steam generation solar power plant. Sol. Energy
**2011**, 85, 660–668. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Scheme of a Novikov engine. It consists of a reversible Carnot engine coupled to two heat baths. The heat transport from the hot reservoir to the engine is irreversible.

**Figure 2.**Fluctuating steam temperature as a function of time for a solar power plant, taken from [31]. Note that the temperature can vary by nearly as much as 80 K.

**Figure 4.**$\widehat{P}$ as a function of $\langle {T}_{\mathrm{H}}\rangle $ and s for a Novikov engine with Newtonian heat transport and uniformly distributed ${T}_{\mathrm{H}}$.

**Figure 5.**$\widehat{\eta}$ as a function of $\langle {T}_{\mathrm{H}}\rangle $ and s for a Novikov engine with Newtonian heat transport and uniformly distributed ${T}_{\mathrm{H}}$.

**Figure 6.**$\widehat{\sigma}$ as a function of $\langle {T}_{\mathrm{H}}\rangle $ and s for a Novikov engine with Newtonian heat transport and uniformly distributed ${T}_{\mathrm{H}}$.

Degree | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Uniform | ${\left(\sqrt{\langle {T}_{\mathrm{H}}\rangle}-\sqrt{{T}_{\mathrm{L}}}\right)}^{2}$ | 0 | $\frac{1}{4}\sqrt{\frac{{T}_{\mathrm{L}}}{{\langle {T}_{\mathrm{H}}\rangle}^{3}}}$ | 0 | $\frac{9}{64}\sqrt{\frac{{T}_{\mathrm{L}}}{{\langle {T}_{\mathrm{H}}\rangle}^{7}}}$ |

Triangle | ${\left(\sqrt{\langle {T}_{\mathrm{H}}\rangle}-\sqrt{{T}_{\mathrm{L}}}\right)}^{2}$ | 0 | $\frac{1}{4}\sqrt{\frac{{T}_{\mathrm{L}}}{{\langle {T}_{\mathrm{H}}\rangle}^{3}}}$ | 0 | $\frac{3}{16}\sqrt{\frac{{T}_{\mathrm{L}}}{{\langle {T}_{\mathrm{H}}\rangle}^{7}}}$ |

Quadratic | ${\left(\sqrt{\langle {T}_{\mathrm{H}}\rangle}-\sqrt{{T}_{\mathrm{L}}}\right)}^{2}$ | 0 | $\frac{1}{4}\sqrt{\frac{{T}_{\mathrm{L}}}{{\langle {T}_{\mathrm{H}}\rangle}^{3}}}$ | 0 | $\frac{75}{448}\sqrt{\frac{{T}_{\mathrm{L}}}{{\langle {T}_{\mathrm{H}}\rangle}^{7}}}$ |

Pareto | ${\left(\sqrt{\langle {T}_{\mathrm{H}}\rangle}-\sqrt{{T}_{\mathrm{L}}}\right)}^{2}$ | 0 | $\frac{1}{4}\sqrt{\frac{{T}_{\mathrm{L}}}{{\langle {T}_{\mathrm{H}}\rangle}^{3}}}$ | $-\frac{1}{4}\sqrt{\frac{{T}_{\mathrm{L}}}{{\langle {T}_{\mathrm{H}}\rangle}^{5}}}$ | $-\frac{3}{64}\sqrt{\frac{{T}_{\mathrm{L}}}{{\langle {T}_{\mathrm{H}}\rangle}^{7}}}$ |

**Table 2.**Taylor coefficients of $\widehat{\eta}$ for small values of s. The asterisk (*) indicates that the expression is to lengthy to be shown here.

Degree | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Uniform | $1-\sqrt{{\displaystyle \frac{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}}}$ | 0 | $\frac{{T}_{\mathrm{L}}+\sqrt{{T}_{\mathrm{L}}\langle {T}_{\mathrm{H}}\rangle}}{8\left({\langle {T}_{\mathrm{H}}\rangle}^{3}-\sqrt{{T}_{\mathrm{L}}{\langle {T}_{\mathrm{H}}\rangle}^{5}}\right)}$ | 0 | $\frac{9\sqrt{{T}_{\mathrm{L}}}\langle {T}_{\mathrm{H}}\rangle -2{T}_{\mathrm{L}}\sqrt{\langle {T}_{\mathrm{H}}\rangle}-11{T}_{\mathrm{L}}^{3/2}}{128{\langle {T}_{\mathrm{H}}\rangle}^{9/2}\left({T}_{\mathrm{L}}+\langle {T}_{\mathrm{H}}\rangle -2\sqrt{{T}_{\mathrm{L}}\langle {T}_{\mathrm{H}}\rangle}\right)}$ |

Triangle | $1-\sqrt{{\displaystyle \frac{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}}}$ | 0 | $\frac{{T}_{\mathrm{L}}+\sqrt{{T}_{\mathrm{L}}\langle {T}_{\mathrm{H}}\rangle}}{8\left({\langle {T}_{\mathrm{H}}\rangle}^{3}-\sqrt{{T}_{\mathrm{L}}{\langle {T}_{\mathrm{H}}\rangle}^{5}}\right)}$ | 0 | $\frac{6\sqrt{{T}_{\mathrm{L}}}\langle {T}_{\mathrm{H}}\rangle -{T}_{\mathrm{L}}\sqrt{\langle {T}_{\mathrm{H}}\rangle}-7{T}_{\mathrm{L}}^{3/2}}{64{\langle {T}_{\mathrm{H}}\rangle}^{9/2}\left({T}_{\mathrm{L}}+\langle {T}_{\mathrm{H}}\rangle -2\sqrt{{T}_{\mathrm{L}}\langle {T}_{\mathrm{H}}\rangle}\right)}$ |

Quadratic | $1-\sqrt{{\displaystyle \frac{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}}}$ | 0 | $\frac{{T}_{\mathrm{L}}+\sqrt{{T}_{\mathrm{L}}\langle {T}_{\mathrm{H}}\rangle}}{8\left({\langle {T}_{\mathrm{H}}\rangle}^{3}-\sqrt{{T}_{\mathrm{L}}{\langle {T}_{\mathrm{H}}\rangle}^{5}}\right)}$ | 0 | $\frac{75\sqrt{{T}_{\mathrm{L}}}\langle {T}_{\mathrm{H}}\rangle -14{T}_{\mathrm{L}}\sqrt{\langle {T}_{\mathrm{H}}\rangle}-89{T}_{\mathrm{L}}^{3/2}}{896{\langle {T}_{\mathrm{H}}\rangle}^{9/2}\left({T}_{\mathrm{L}}+\langle {T}_{\mathrm{H}}\rangle -2\sqrt{{T}_{\mathrm{L}}\langle {T}_{\mathrm{H}}\rangle}\right)}$ |

Pareto | $1-\sqrt{{\displaystyle \frac{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}}}$ | 0 | $\frac{{\langle {T}_{\mathrm{H}}\rangle}^{2}\left({T}_{\mathrm{L}}^{2}{\langle {T}_{\mathrm{H}}\rangle}^{3}+\sqrt{{T}_{\mathrm{L}}{\langle {T}_{\mathrm{H}}\rangle}^{9}}-{T}_{\mathrm{L}}\left({\langle {T}_{\mathrm{H}}\rangle}^{4}+\sqrt{{T}_{\mathrm{L}}{\langle {T}_{\mathrm{H}}\rangle}^{7}}\right)\right)}{8{\left({\langle {T}_{\mathrm{H}}\rangle}^{3}-\sqrt{{T}_{\mathrm{L}}{\langle {T}_{\mathrm{H}}\rangle}^{5}}\right)}^{3}}$ | * | * |

Degree | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Uniform | $\frac{{\left(\sqrt{\langle {T}_{\mathrm{H}}\rangle}-\sqrt{{T}_{\mathrm{L}}}\right)}^{2}}{\sqrt{\langle {T}_{\mathrm{H}}\rangle {T}_{\mathrm{L}}}}$ | 0 | $\frac{3{T}_{\mathrm{L}}-\langle {T}_{\mathrm{H}}\rangle}{8\sqrt{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}^{5/2}}$ | 0 | $\frac{9\left(7{T}_{\mathrm{L}}-\langle {T}_{\mathrm{H}}\rangle \right)}{128\sqrt{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}^{9/2}}$ |

Triangle | $\frac{{\left(\sqrt{\langle {T}_{\mathrm{H}}\rangle}-\sqrt{{T}_{\mathrm{L}}}\right)}^{2}}{\sqrt{\langle {T}_{\mathrm{H}}\rangle {T}_{\mathrm{L}}}}$ | 0 | $\frac{3{T}_{\mathrm{L}}-\langle {T}_{\mathrm{H}}\rangle}{8\sqrt{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}^{5/2}}$ | 0 | $\frac{3\left(7{T}_{\mathrm{L}}-\langle {T}_{\mathrm{H}}\rangle \right)}{32\sqrt{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}^{9/2}}$ |

Quadratic | $\frac{{\left(\sqrt{\langle {T}_{\mathrm{H}}\rangle}-\sqrt{{T}_{\mathrm{L}}}\right)}^{2}}{\sqrt{\langle {T}_{\mathrm{H}}\rangle {T}_{\mathrm{L}}}}$ | 0 | $\frac{3{T}_{\mathrm{L}}-\langle {T}_{\mathrm{H}}\rangle}{8\sqrt{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}^{5/2}}$ | 0 | $\frac{75\left(7{T}_{\mathrm{L}}-\langle {T}_{\mathrm{H}}\rangle \right)}{896\sqrt{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}^{9/2}}$ |

Pareto | 0 | $\frac{-5{T}_{\mathrm{L}}+\langle {T}_{\mathrm{H}}\rangle}{8\sqrt{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}^{7/2}}$ | $\frac{3\left(25{T}_{\mathrm{L}}+\langle {T}_{\mathrm{H}}\rangle \right)}{128\sqrt{{T}_{\mathrm{L}}}{\langle {T}_{\mathrm{H}}\rangle}^{9/2}}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schwalbe, K.; Hoffmann, K.H. Performance Features of a Stationary Stochastic Novikov Engine. *Entropy* **2018**, *20*, 52.
https://doi.org/10.3390/e20010052

**AMA Style**

Schwalbe K, Hoffmann KH. Performance Features of a Stationary Stochastic Novikov Engine. *Entropy*. 2018; 20(1):52.
https://doi.org/10.3390/e20010052

**Chicago/Turabian Style**

Schwalbe, Karsten, and Karl Heinz Hoffmann. 2018. "Performance Features of a Stationary Stochastic Novikov Engine" *Entropy* 20, no. 1: 52.
https://doi.org/10.3390/e20010052