# Carnot-Like Heat Engines Versus Low-Dissipation Models

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## Abstract

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## 1. Introduction

## 2. Correspondence between the HE’s Variables of Both Models

## 3. Physical Space of the HE Variables

## 4. Maximum-Power Regime

#### 4.1. Low Dissipation Heat Engine

#### 4.2. Carnot-Like Model without Heat Leak (Endoreversible Model)

#### 4.3. Carnot-Like Model with Heat Leak

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) Sketch of a low dissipation heat engine characterized by entropy generation laws $\Delta {S}_{\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{h}}}$ and $\Delta {S}_{\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{c}}}$; (

**b**) Sketch of an irreversible Carnot-like heat engine characterized by generic heat transfers ${Q}_{\mathrm{h}}$, ${Q}_{\mathrm{c}}$ and ${Q}_{\mathrm{L}}$.

**Figure 2.**(

**a**) ${T}_{\mathrm{hw}}$ and ${T}_{\mathrm{cw}}$ from Equation (9). Note how, as the heat leak increases, the possible physical combinations of ${T}_{\mathrm{hw}}$ and ${T}_{\mathrm{cw}}$ become more limited; (

**b**) $\tilde{t}({\tilde{Q}}_{\mathrm{L}},{a}_{\mathrm{c}})$ according to Equation (10). The representative values $\alpha =\frac{1}{5}=\tau $, ${\tilde{\Sigma}}_{\mathrm{c}}=\frac{1}{2}$ have been used, however, the displayed behavior is similar for any other combination of values.

**Figure 3.**Possible values of ${\tilde{Q}}_{\mathrm{L}}$ as a function of the control parameters $\alpha $ and $\tilde{t}$. We used the values ${\tilde{\Sigma}}_{\mathrm{c}}=0.8$ and $\tau =0.2$.

**Figure 4.**(

**a**) upper and lower bounds of the MP efficiency in terms of the exponent of the heat transfer law k of the Carnot-like heat engine; (

**b**) the $\tilde{{\Sigma}_{\mathrm{c}}}$ values that reproduce the upper and lower bounds of the endoreversible engine.

**Figure 5.**Maximum-power efficiency for the symmetric case ${\tilde{\Sigma}}_{\mathrm{c}}=1/2$, assuming the LD condition that $\frac{{t}_{\mathrm{c}}}{{t}_{\mathrm{h}}}=\sqrt{\tau}$ and using the resulting ${\sigma}_{\mathrm{hc}}$ value that fulfills the endoreversible hypothesis. Notice that the matching with the CA efficiency is approximate for the interval $k\in \left[-1,1\right]$ and is exact for $k=\{-1,1\}$, as can be seen in the zoom of this region on the right side of the figure.

**Figure 6.**Influence of the heat leak over the optimized efficiencies appearing in Figure 4. See the text for explanation.

**Figure 7.**(

**a**) Physically well behaved region of the $\alpha $‒${\tilde{\Sigma}}_{\mathrm{c}}$ variables. The shaded areas come from the LD model and the dashed curves come from the Carnot-like model; (

**b**) The same for the $\tilde{t}$‒${\tilde{\Sigma}}_{\mathrm{c}}$ variables. Notice the agreement in both models. In these plots, we use $\tau =0.2$.

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**MDPI and ACS Style**

Gonzalez-Ayala, J.; Roco, J.M.M.; Medina, A.; Calvo Hernández, A.
Carnot-Like Heat Engines Versus Low-Dissipation Models. *Entropy* **2017**, *19*, 182.
https://doi.org/10.3390/e19040182

**AMA Style**

Gonzalez-Ayala J, Roco JMM, Medina A, Calvo Hernández A.
Carnot-Like Heat Engines Versus Low-Dissipation Models. *Entropy*. 2017; 19(4):182.
https://doi.org/10.3390/e19040182

**Chicago/Turabian Style**

Gonzalez-Ayala, Julian, José Miguel M. Roco, Alejandro Medina, and Antonio Calvo Hernández.
2017. "Carnot-Like Heat Engines Versus Low-Dissipation Models" *Entropy* 19, no. 4: 182.
https://doi.org/10.3390/e19040182