# Dissipative Endoreversible Engine with Given Efficiency

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## Abstract

**:**

## 1. Introduction

## 2. General Formalism

#### 2.1. Subsystems

#### 2.2. Reversible and Irreversible Interactions

## 3. Leaky Interaction

## 4. Engine Setup with Leaky Interaction

## 5. Dissipative Engine with Given Efficiency

## 6. Full Model of a Dissipative Engine

## 7. Example: AC Motor

## 8. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Reversible interaction with arbitrary extensity and (

**b**) irreversible interaction with heat transfer between two subsystems.

**Figure 2.**(

**a**) Irreversible interaction between reservoirs with different intensities and (

**b**) lossy interaction between reservoirs with different intensities where a loss extensity transfer occurs towards a third reservoir. In both cases, the generated entropy is transferred to another additional reservoir.

**Figure 3.**Engine setup with lossy extensity transfer. The upper reservoir is connected to Engine 3 via a lossy interaction with loss extensity and entropy flux towards Reservoirs 4 and 5, respectively, while the lower reservoir is reversibly connected to the engine, which further has an unspecified power output.

**Figure 4.**Simplification of (

**a**) the engine with lossy extensity transfer to (

**b**) a black box dissipative engine with given efficiency. The resulting black box model (rectangle with rounded corners) operates between Reservoirs 1 and 2 with efficiency ${\eta}_{\mathrm{G}}$, while the loss extensity is reversibly transferred to Reservoir 4 and the generated entropy to Reservoir 5.

**Figure 5.**Engine setup with lossy extensity transfer for both extensities $\alpha $ and $\beta $ (

**a**) and the corresponding black box model with given efficiencies (

**b**). The engine operates between Reservoirs 1 and 2 and Reservoirs 3 and 4 with efficiency $\eta $, while loss extensity transfers of extensities $\alpha $ and $\beta $ occur towards Reservoirs 6 and 7, respectively, and the generated entropy is transferred to Reservoir 8.

**Figure 6.**Full endoreversible engine setup used to model an AC motor. Compared to Figure 5a, we now have charge and angular momentum fluxes so that $\alpha =Q$ and $\beta =L$, respectively, and there is no loss flux towards Reservoir 6.

**Figure 7.**Loss charge flux ${J}_{6}^{Q}$ (left) and loss angular momentum flux ${J}_{7}^{L}$ (right) over engine speed n and output torque $\tau $ of the AC (Alternating Current) motor investigated in [50] using the introduced dissipated engine model.

**Figure 8.**Entropy generation rates due to copper losses ${\sigma}_{\mathrm{Cu}}$ (left) and iron and mechanical losses ${\sigma}_{\mathrm{FeMec}}$ (right) over engine speed n and output torque $\tau $ of the AC motor investigated in [50] using the introduced dissipated engine model.

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Masser, R.; Hoffmann, K.H.
Dissipative Endoreversible Engine with Given Efficiency. *Entropy* **2019**, *21*, 1117.
https://doi.org/10.3390/e21111117

**AMA Style**

Masser R, Hoffmann KH.
Dissipative Endoreversible Engine with Given Efficiency. *Entropy*. 2019; 21(11):1117.
https://doi.org/10.3390/e21111117

**Chicago/Turabian Style**

Masser, Robin, and Karl Heinz Hoffmann.
2019. "Dissipative Endoreversible Engine with Given Efficiency" *Entropy* 21, no. 11: 1117.
https://doi.org/10.3390/e21111117