# Performance Features of a Stationary Stochastic Novikov Engine

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## 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

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

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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.
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**Chicago/Turabian Style**

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