# Optimization, Stability, and Entropy in Endoreversible Heat Engines

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## Abstract

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## 1. Introduction

## 2. A Quick Look at the Endoreversible Model

- The Newton case ($k=1$) gives an efficiency at maximum power, ${\eta}_{MP}={\eta}_{CAN}\equiv 1-\sqrt{\tau}$, the well-known Curzon–Ahlborn–Novikov (CAN) efficiency, which does not depend on the ${\sigma}_{hc}$ ratio and appears in a large variety of contexts linked to the maximization of power output, work, and kinetic energy [37].
- The $k=-1$ case, frequently called the phenomenological law, referring to the natural results arising in the linear irreversible thermodynamics framework. It allows to obtain the same limits of efficiency as in the low-dissipation model, where the self-optimization property has been studied. In this case ($k=-1$), ${\eta}_{MP}$ is ${\sigma}_{hc}$-dependent, bounded by ${\eta}_{MP}\in \left(\frac{{\eta}_{C}}{2},\frac{{\eta}_{C}}{2-{\eta}_{C}}\right)$, according to whether ${\sigma}_{hc}$ varies from 0 to ∞ [7].

## 3. The Relevant Region for Optimization: The Pareto Front

- In the phase space (${T}_{hw}$, ${T}_{cw}$), the region of physical relevance is defined (${T}_{h}\ge {T}_{hw}\ge {T}_{cw}\ge {T}_{c}$).
- A random set of points in the phase space is obtained and the thermodynamic functions are evaluated (energetic space).
- A set of non-dominated points in the energetic space is obtained, giving a provisional Pareto front.
- From the corresponding Pareto optimal set (phase space), a convex region is defined and extended in order to cover a larger region for searching new points in the Pareto front. Details on the definition of the extended region are given below.
- From the new region, a new set of random points is proposed and a new set of non-dominated points in the energetic space is obtained.

## 4. Stability Dynamics and Relaxation Times

## 5. Thermodynamics of the Relaxation Trajectories

## 6. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of an endoreversible heat engine. The working fluid realizes a Carnot cycle operating between the isothermal processes at effective temperatures ${T}_{hw}$ and ${T}_{cw}<{T}_{hw}$. The working fluid is irreversibly coupled to external reservoirs at temperatures ${T}_{h}$ and ${T}_{c}<{T}_{h}$.

**Figure 2.**Parabolic behavior of the $\eta $, P, and $\dot{S}$ curve typical of the endoreversible model for (

**a**) the $k=1$ case and (

**b**) the $k=-1$ case. The values ${T}_{h}$ = 500 K, ${\sigma}_{hc}=1$, and $\tau =0.4$ weer fixed. Additionally, ${\sigma}_{c}$ was chosen (for representation/comparison purposes) in such a way that the maximum power (MP) was 70 W in both cases, so that $\eta $, P, and $\dot{S}$ ranged in similar intervals.

**Figure 3.**Isocontours of the relaxation velocity for (

**a**) the $k=1$ case and (

**b**) the $k=-1$ case. The values ${T}_{h}=500$ K, ${\sigma}_{hc}=1$, and $\tau =0.4$ were fixed. As in Figure 2, ${\sigma}_{c}$ and ${t}_{h}$ were chosen (for representation/comparison purposes) in such a way that the MP was 70 W and the entropy at MP was 70 J/K in both cases. By fixing S at MP conditions, a time scale for ${t}_{h}$ was established, and therefore, a scale for the relaxation time. The qualitative behavior for the other values of S was similar to the one presented here.

**Figure 4.**Trajectories in the ${T}_{hw}$–${T}_{cw}$ space and the mapping over the $\eta $–P, $\dot{S}$–P, and $\dot{S}$–$\eta $ spaces for $k=1$. (

**a**) the ${\sigma}_{hc}={10}^{-6}$ case; (

**b**) the ${\sigma}_{hc}=1$ case; (

**c**) the ${\sigma}_{hc}={10}^{6}$ case. The values ${T}_{h}=500$ K and $\tau =0.4$ were fixed. As in Figure 2, ${\sigma}_{c}$ and ${t}_{h}$ were chosen (for representation/comparison purposes) in such a way that the MP was 70 W and the entropy at MP was 70 J/K in both cases.

**Figure 5.**Trajectories in the ${T}_{hw}$–${T}_{cw}$ space and mapping over the $\eta $–P, $\dot{S}$–P, and $\dot{S}$–$\eta $ spaces for $k=-1$. In column (

**a**) the ${\sigma}_{hc}={10}^{-6}$ case; (

**b**) the ${\sigma}_{hc}=1$ case; (

**c**) the ${\sigma}_{hc}={10}^{6}$ case. The values ${T}_{h}=500$ K and $\tau =0.4$ were fixed. As in Figure 2, ${\sigma}_{c}$ and ${t}_{h}$ were chosen (for representation/comparison purposes) in such a way that the MP was 70 W and the entropy at MP was 70 J/K in both cases.

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**MDPI and ACS Style**

Gonzalez-Ayala, J.; Mateos Roco, J.M.; Medina, A.; Calvo Hernández, A.
Optimization, Stability, and Entropy in Endoreversible Heat Engines. *Entropy* **2020**, *22*, 1323.
https://doi.org/10.3390/e22111323

**AMA Style**

Gonzalez-Ayala J, Mateos Roco JM, Medina A, Calvo Hernández A.
Optimization, Stability, and Entropy in Endoreversible Heat Engines. *Entropy*. 2020; 22(11):1323.
https://doi.org/10.3390/e22111323

**Chicago/Turabian Style**

Gonzalez-Ayala, Julian, José Miguel Mateos Roco, Alejandro Medina, and Antonio Calvo Hernández.
2020. "Optimization, Stability, and Entropy in Endoreversible Heat Engines" *Entropy* 22, no. 11: 1323.
https://doi.org/10.3390/e22111323