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Entropy Rates and Efficiency of Convecting-Radiating Fins^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Entropy Rate Due to Heat Exchange

## 3. An Entropy-Based Indicator for the Efficiency of the Fin in Its Steady State

## 4. The Pure Convective Case

## 5. The Convecting-Radiating Fin and Its Efficiency

## 6. An Example with Convecting-Radiating Aluminum Fins.

## 7. Conclusions

- (i)
- The idea underlying Equation (36) is the following: to measure the dissipation, through entropy rates, of the steady state of the fin with respect to that of an ideal fin with the highest dissipation possible. If the real fin is close to this ideal fin, the corresponding efficiency will be higher. The reference state ${T}_{b}$, as it has been shown at the end of Section (Section 3), is also the state expected to possess the maximum entropy rate.
- (ii)
- The classical definition of the efficiency $\eta $ is given by the ratio of the actual heat transfer over the ideal heat transfer for a fin at temperature equal ${T}_{b}$. Since we take the reference state for the entropic efficiency to be the same, we have the possibility to make a direct comparison between the two definitions.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The longitudinal fin: a given profile is shown, described by a suitable ${f}_{0}\left(x\right)$ together with the coordinate system, the cross-sectional area, and the geometrical properties.

**Figure 2.**Plots of the relative error $|\frac{I\left(\u03f5\right)-{I}_{N}\left(\u03f5\right)}{I\left(\u03f5\right)}|$ for $N=10$, $N=20$, and $N=50$. The scale is logarithmic. The peaks at the level of 10${}^{-9}$ are due to the numerical round-off error, since the calculation has been performed with 10 significant digits.

**Figure 4.**The plot of efficiency ${\eta}_{s}$ as a function of $\beta $ for four different values of $\alpha $ and for ${\theta}_{0}=0.5$.

**Figure 5.**The plot of efficiency ${\eta}_{s}$ as a function of $\beta $ for four different values of $\alpha $ and for ${\theta}_{0}=0.1$.

**Figure 6.**The plot of classical efficiency $\eta $ (from [4]) as a function of $\beta $ for ${\theta}_{0}=0.1$ and ${\theta}_{0}=0.5$ and four different values of $\alpha $.

**Figure 7.**The classical (circles) and entropic (diagonal crosses) efficiency values vs. the convective heat transfer coefficient for the example of an aluminum fin.

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**MDPI and ACS Style**

Giorgi, C.; Zullo, F.
Entropy Rates and Efficiency of Convecting-Radiating Fins. *Energies* **2021**, *14*, 1643.
https://doi.org/10.3390/en14061643

**AMA Style**

Giorgi C, Zullo F.
Entropy Rates and Efficiency of Convecting-Radiating Fins. *Energies*. 2021; 14(6):1643.
https://doi.org/10.3390/en14061643

**Chicago/Turabian Style**

Giorgi, Claudio, and Federico Zullo.
2021. "Entropy Rates and Efficiency of Convecting-Radiating Fins" *Energies* 14, no. 6: 1643.
https://doi.org/10.3390/en14061643