# Controlling and Optimizing Entropy Production in Transient Heat Transfer in Graded Materials

^{1}

^{2}

^{3}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

“Evaluations of most appropriate temperature distribution, for the development of analytical models have to be studied precisely for accurate evaluation of plate deformations.”

## 2. Mathematical Modeling

#### 2.1. Initial and Boundary Conditions

#### 2.2. Grading of the Material

- Using a low thermal conductivity (${k}_{l}$) matrix to be graded with a high thermal conductivity (${k}_{u}$) material$${k}_{z}(z)={k}_{u}(1-{V}_{f})+{k}_{l}{V}_{f}.$$
- Using a high thermal conductivity matrix to be graded with a low thermal conductivity material$${k}_{z}(z)={k}_{u}{V}_{f}+{k}_{l}(1-{V}_{f}).$$

#### 2.3. Entropy Calculation

#### 2.4. Numerical Methodology

## 3. Numerical Results

#### 3.1. Validation of the Numerical Code

#### 3.2. Numerical Results for the Present Case

## 4. Discussion

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Thai, H.T.; Kim, S.E. A review of theories for the modeling and analysis of functionally graded plates and shells. Compos. Struct.
**2015**, 128, 70–86. [Google Scholar] [CrossRef] - D’Ans, P.; Degrez, M. How to minimise thermal fatigue in surface multi-treatments and coatings? Comput. Mater. Sci.
**2012**, 62, 276–281. [Google Scholar] [CrossRef] - Birman, V.; Byrd, L.W. Modeling and Analysis of Functionally Graded Materials and Structures. Appl. Mech. Rev.
**2007**, 60, 195–216. [Google Scholar] [CrossRef] - Hamza-Cherif, S.M.; Houmat, A.; Hadjoui, A. Transient heat conduction in functionally graded materials. Int. J. Comput. Methods
**2007**, 4, 603–619. [Google Scholar] [CrossRef] - Sakurai, H. Transient and steady-state heat conduction analysis of two-dimensional functionally graded materials using particle method. WIT Trans. Eng. Sci.
**2009**, 64, 45–54. [Google Scholar] [CrossRef] - Ma, C.C.; Chen, Y.T. Theoretical analysis of heat conduction problems of nonhomogeneous functionally graded materials for a layer sandwiched between two half-planes. Acta Mech.
**2011**, 221, 223–237. [Google Scholar] [CrossRef] - Zhao, N.; Cao, L.; Guo, H. Transient heat conduction in functionally graded materials by LT-MFS. Adv. Mater. Res.
**2011**, 189, 1664–1669. [Google Scholar] [CrossRef] - Rahideh, H.; Malekzadeh, P.; Haghighi, M.G. Heat conduction analysis of multi-layered FGMs considering the finite heat wave speed. Energy Convers. Manag.
**2012**, 55, 14–19. [Google Scholar] [CrossRef] - Khan, W.A.; Aziz, A. Transient heat transfer in a functionally graded convecting longitudinal fin. Heat Mass Transf.
**2012**, 48, 1745–1753. [Google Scholar] [CrossRef] - Zajas, J.; Heiselberg, P. Determination of the local thermal conductivity of functionally graded materials by a laser flash method. Int. J. Heat Mass Transf.
**2013**, 60, 542–548. [Google Scholar] [CrossRef] - Akbarzadeh, A.H.; Chen, Z.T. Dual phase lag heat conduction in functionally graded hollow spheres. Int. J. Appl. Mech.
**2014**, 6, 1450002. [Google Scholar] [CrossRef] - Li, M.; Wen, P.H. Finite block method for transient heat conduction analysis in functionally graded media. Int. J. Numer. Methods Eng.
**2014**, 99, 372–390. [Google Scholar] [CrossRef] - Li, G.; Guo, S.; Zhang, J.; Li, Y.; Han, L. Transient heat conduction analysis of functionally graded materials by a multiple reciprocity boundary face method. Eng. Anal. Bound. Elem.
**2015**, 60, 81–88. [Google Scholar] [CrossRef] - Yang, Y.C.; Wang, S.; Lin, S.C. Dual-phase-lag heat conduction in a furnace wall made of functionally graded materials. Int. Commun. Heat Mass Transf.
**2016**, 74, 76–81. [Google Scholar] [CrossRef] - Cimmelli, V.A.; Jou, D.; Sellito, A. Heat transport equations with phonons and electrons. Acta Appl. Math.
**2012**, 122, 117–126. [Google Scholar] [CrossRef] - Jou, D.; Carlomagno, I.; Cimmelli, V.A. A thermodynamic model for heat transport and thermal wave propagation in graded systems. Phys. E Low-Dimens. Syst. Nanostruct.
**2015**, 73, 242–249. [Google Scholar] [CrossRef] - Jou, D.; CarIomagno, I.; Cimmelli, V.A. Rectification of low-frequency thermal wave in graded Si
_{c}Ge_{1−c}. Phys. Lett. A**2016**, 380, 1824–1829. [Google Scholar] [CrossRef] - Swaminathan, K.; Sangeetha, D. Thermal analysis of FGM plates—A critical review of various modelling techniques and solution methods. Compos. Struct.
**2017**, 160, 43–60. [Google Scholar] [CrossRef] - Hahn, D.W.; Özisik, M.N. Heat Conduction; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2012. [Google Scholar]
- Dettori, R.; Melis, C.; Cartoixà, X.; Rurali, R.; Colombo, L. Thermal boundary resistance in semiconductors by non-equilibrium thermodynamics. Adv. Phys. X
**2016**, 1, 246–261. [Google Scholar] [CrossRef] - Machrafi, H.; Lebon, G.; Jou, D. Thermal rectifier efficiency of various bulk-nanoporous silicon devices. Int. J. Heat Mass Transf.
**2016**, 97, 603–610. [Google Scholar] [CrossRef] - Tamura, S.; Ogawa, K. Thermal rectificatioin in nonmetalic solid junctions: Effect of Kapitza resistance. Solid State Commun.
**2012**, 152, 1906–1911. [Google Scholar] [CrossRef] - Saha, B.; Koh, Y.K.; Feser, J.P.; Sadasivam, S.; Fisher, T.S.; Shakouri, A.; Sands, T.D. Phonon wave effects in the thermal transport of epitaxial TiN/(Al,Sc)N metal/semiconductor superlattices. J. Appl. Phys.
**2017**, 121, 015109. [Google Scholar] [CrossRef] - Ezzahri, Y.; Dilhaire, S.; Grauby, S.; Rampnoux, J.M.; Claeys, W. Study of thermomechanical properties of Si/SiGe superlattices using femtosecond transient thermoreflectance technique. Appl. Phys. Lett.
**2005**, 87, 103506. [Google Scholar] [CrossRef] - Vo, T.Q.; Barisik, M.; Kim, B.H. Atomic density effects on temperature characteristics and thermal transport at grain boundaries through a proper bin size selection. J. Chem. Phys.
**2016**, 144, 194707. [Google Scholar] [CrossRef] - Gonzalez-Valle, C.U.; Ramos-Alvarado, B. Spectral mapping of thermal transport across SiC-water interfaces. Int. J. Heat Mass Transf.
**2018**, 131, 645–653. [Google Scholar] [CrossRef] - Zhang, Y.; Ma, L. Optimization of ceramic strength using elastic gradients. Acta Mater.
**2009**, 57, 2721–2729. [Google Scholar] [CrossRef] [PubMed] - Zhang, Y.; Sun, M.J.; Zhang, D. Designing functionally graded materials with superior load-bearing properties. Acta Biomater.
**2012**, 8, 1101–1108. [Google Scholar] [CrossRef] - Shen, H.S. Functionally Graded Materials Nonlinear Analysis of Plates and Shells; Taylor & Francis: Abingdon, UK, 2009. [Google Scholar]
- Carrera, E.; Fazzolari, F.A.; Cinefra, M. Thermal Stress Analysis of Composite Beams, Plates and Shells: Computational Modelling and Applications; Academic Press, Elsevier: Amsterdam, The Netherlands, 2017. [Google Scholar]
- Kreuzer, H.J. Nonequilibrium Thermodynamics and Its Statistical Foundations; Clarendon Press: Oxford, UK, 1981. [Google Scholar]
- Versteeg, H.K.; Malalasekera, W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method; Longman Scientific & Technical: Harlow, UK, 1995. [Google Scholar]
- Núñez, J.; Ramos, E.; Lopez, J.M. A mixed Fourier–Galerkin–finite-volume method to solve the fluid dynamics equations in cylindrical geometries. Fluid Dyn. Res.
**2012**, 44, 031414. [Google Scholar] [CrossRef] - Alipour, S.M.; Kiani, Y.; Eslami, M.R. Rapid heating of FGM rectangular plates. Acta Mech.
**2015**, 227, 421–436. [Google Scholar] [CrossRef] - Hassanzadeh, R.; Bilgili, M. Improvement of thermal efficiency in computer heat sink using functionally graded materials. Commun. Adv. Comput. Sci. Appl.
**2014**, 2014, cacsa-00018. [Google Scholar] [CrossRef]

**Figure 3.**Dimensional temperature at the center of the material as a function of time for two different grading profiles used for validation. Solid lines correspond to the present work whereas the symbols were extracted from Alipour et al. [34].

**Figure 4.**Steady state dimensionless temperature profiles for different values of N for case ${C}_{1}$, see Equation (8). Dashed lines represent the profiles of case 1 with inverted boundary conditions (${C}_{1}^{I}$).

**Figure 5.**Steady state dimensionless temperature profiles for different values of N for case ${C}_{2}$, see Equation (9). Dashed lines represent the profiles of case 2 with inverted boundary conditions (${C}_{2}^{I}$).

**Figure 6.**Dimensionless temperature $\theta $ at the center of the graded material as a function of dimensionless time for different values of N for case ${C}_{2}$. The black points (•) mark the time needed to reach the steady state ${t}_{S}$ and the corresponding temperature.

**Figure 7.**Dimensionless time needed to reach steady state ${t}_{S}$ and corresponding dimensionless temperature $\theta $ at the center for both cases. The superscript I means inverted boundary conditions. Each symbol corresponds to a numerical experiment. The continuous lines are a guide to the eye.

**Figure 8.**Dimensionless entropy production $\dot{S}$ profiles at steady state as a function of N for case ${C}_{2}$.

**Figure 9.**Dimensionless entropy production $\dot{S}$ profiles for different instants of time. Case ${C}_{2}$ and $N=2.5$.

**Figure 10.**Dimensionless global entropy production for the whole computation time as function of N. The superscript I means inverted boundary conditions. Each symbol corresponds to a numerical experiment.

**Figure 11.**Dimensionless global entropy production until reaching steady conditions as a function of N. Each symbol corresponds to a numerical experiment.

**Table 1.**Numerical results for maximum and minimum values of entropy production during the transient and for the whole time interval.

Material | Min/Max | N | ${\dot{\mathit{S}}}_{\mathit{g}}$ | ${\dot{\mathit{S}}}_{\mathit{g}}^{\mathit{T}}$ | $\mathit{R}={\dot{\mathit{S}}}_{\mathit{g}}/{\dot{\mathit{S}}}_{\mathit{g}}^{\mathit{T}}$ |
---|---|---|---|---|---|

${C}_{1}$ | Min | 0.6 | 0.008 | 0.025 | 0.32 |

${C}_{1}^{I}$ | Max | 5.5 | 0.013 | 0.069 | 0.19 |

${C}_{2}$ | Max | 0.6 | 0.0122 | 0.053 | 0.23 |

${C}_{2}$ | Min | 5 | 0.0098 | 0.014 | 0.7 |

${C}_{2}^{I}$ | Max | 0.3 | 0.0095 | 0.062 | 0.15 |

${C}_{2}^{I}$ | Min | 2.5 | 0.0067 | 0.017 | 0.39 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pérez-Barrera, J.; Figueroa, A.; Vázquez, F. Controlling and Optimizing Entropy Production in Transient Heat Transfer in Graded Materials. *Entropy* **2019**, *21*, 463.
https://doi.org/10.3390/e21050463

**AMA Style**

Pérez-Barrera J, Figueroa A, Vázquez F. Controlling and Optimizing Entropy Production in Transient Heat Transfer in Graded Materials. *Entropy*. 2019; 21(5):463.
https://doi.org/10.3390/e21050463

**Chicago/Turabian Style**

Pérez-Barrera, James, Aldo Figueroa, and Federico Vázquez. 2019. "Controlling and Optimizing Entropy Production in Transient Heat Transfer in Graded Materials" *Entropy* 21, no. 5: 463.
https://doi.org/10.3390/e21050463