# Analytical Solution of Heat Conduction in a Symmetrical Cylinder Using the Solution Structure Theorem and Superposition Technique

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{1}and T

_{3}dominate compared to T

_{2}contributing little to the overall temperature. But at r > 0.4, all three temperature components will have the same role and less impact on the overall temperature (T).

## 1. Introduction

## 2. Formulation

#### 2.1. Hyperbolic Heat Conduction

#### 2.2. Solution Structure Theorems and Superposition Approach

#### 2.3. Formulation of the Problem

## 3. Results and Discussions

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Nomenclature | |

$c$ | Thermal Wave Propagation Speed, m/s |

${c}_{p}$ | Specific Heat, J/kg K |

$f$ | Total Internal Heat Generation in System |

${f}_{r}$ | Reference Laser Power Density, $\mathrm{W}/{\mathrm{m}}^{2}$ |

$g$ | Dimensionless Internal Heat Generation |

${g}_{0}$ | Transmitted Energy Strength |

${\dot{g}}^{m}$ | Internal Heat Generation, $\mathrm{W}/{\mathrm{m}}^{3}$ |

${I}_{0}$ | Laser peak Power Density, $\mathrm{W}/{\mathrm{m}}^{2}$ |

$k$ | Thermal Conductivity, $\mathrm{W}/\mathrm{m}\mathrm{K}$ |

$h$ | Convection Heat Transfer Coefficient, $\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ |

$m$ | Dimensionless Convection Heat Transfer Coefficient |

${q}^{*}$ | Heat Flux, c |

$q$ | Dimensionless Heat Flux, ${q}^{*}/{f}_{r}$ |

$Q$ | Dimensionless Source Term |

$R$ | Surface Reflectivity of the Solid |

$t$ | Dimensionless Time, ${c}^{2}{t}^{*}/2\alpha $ |

${t}^{*}$ | Time, s |

$T$ | Dimensionless Temperature, $kc{T}^{*}/\alpha {f}_{r}$ |

${T}_{\infty}$ | Dimensionless Ambient Temperature, $kc{T}_{\infty}{}^{*}/\alpha {f}_{r}$ |

${T}^{*}$ | Temperature, K |

${T}_{\infty}{}^{*}$ | Ambient Temperature, K |

${r}^{*}$ | r-coordinate, m |

$r$ | Dimensionless Space Coordinate, $c{r}^{*}/2\alpha $ |

Greek symbols | |

$\alpha $ | Thermal Diffusivity $k/\rho {c}_{p}$, ${\mathrm{m}}^{2}/\mathrm{s}$ |

${\gamma}_{n}$ | Eigen Value, ${\gamma}_{\mathrm{n}}=\sqrt{{\lambda}_{n}^{2}-1}$ |

$\epsilon $ | Relative Error |

$\mu $ | Dimensionless Absorption Coefficient, $2c\tau {\mu}^{*}$ |

$\rho $ | Density ($\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$) |

${\tau}_{0}$ | Relaxation Time $\alpha /{c}^{2}$, s |

$\phi $ | Dimensionless Initial Condition Function |

$\psi $ | Dimensionless Initial rate of Temperature Change Function |

$\xi $ | Dummy Index |

## References

- Dabby, F.W.; Paek, U.-C. High-intensity laser-induced vaporization and explosion of solid material. IEEE J. Quantum Electron.
**1972**, 8, 106–111. [Google Scholar] [CrossRef] - Yeung, W.K.; Lam, T.T. Thermal analysis of anisotropic thin-film superconductors. Adv. Electron. Packag.
**1999**, 26, 1261–1268. [Google Scholar] - Lundell, J.H.; Dickey, R.R. Vaporization of graphite in the temperature range of 4000 to 4500 K. In Proceedings of the NASA Ames Research Center, Moffett Field, CA, USA, 26–28 January 1976; pp. 76–166. [Google Scholar]
- Robin, J.E.; Nordin, P. Enhancement of CW laser melt-through of opaque solid materials by supersonic transverse gas flow. Appl. Phys. Lett.
**1975**, 26, 289–292. [Google Scholar] [CrossRef] - Chen, J.K.; Tzou, D.Y.; Beraun, J.E. Numerical investigation of ultrashort laser damage in semiconductors. Int. J. Heat Mass Transf.
**2005**, 48, 501–509. [Google Scholar] [CrossRef] - Cattaneo, C. Sur une former de l’equation de la chaleur elinant le paradoxe d’une propagation instance. Comptes Rendus Acad. Sci.
**1958**, 247, 431–432. [Google Scholar] - Vernotte, P. Les paradoxes de la theories continue de l’equation de la chaleur. Comptes Rendus Acad. Sci.
**1958**, 246, 3154–4155. [Google Scholar] - Wang, B.L.; Han, J.C.; Sun, Y.G. A finite element/finite difference scheme for the non-classical heat conduction and associated thermal stresses. Finite Elem. Anal. Des.
**2012**, 50, 201–206. [Google Scholar] [CrossRef] - Lee, H.L.; Chen, W.L.; Chang, W.J.; Wei, E.J.; Yang, Y.C. Analysis of dual-phase-lag heat conduction in short-pulse laser heating of metals with a hybrid method. Appl. Therm. Eng.
**2013**, 52, 275–283. [Google Scholar] [CrossRef] - Mishra, S.C.; Sahai, H. Analysis of non-Fourier conduction and radiation in a cylindrical medium using lattice Boltzmann method and finite volume method. Int. J. Heat Mass Transf.
**2013**, 61, 41–55. [Google Scholar] [CrossRef] - Qui, T.; Juhasz, T.; Suarez, C.; Bron, W.; Tien, C. Femto second laser heating of multi-layer II experiments. Int. J. Heat Mass Transf.
**1994**, 37, 2799–2808. [Google Scholar] - Ozisik, M.; Vick, B. Propagation and Reflection of Thermal Waves in a Finite Medium. Int. J. Heat Mass Transf.
**1984**, 27, 1845–1855. [Google Scholar] [CrossRef] - Jiang, F. Solution and analysis of hyperbolic heat propagation in hollow spherical objects. Heat Mass Transf.
**2006**, 42, 1083–1091. [Google Scholar] [CrossRef] - Moosaie, A. Non Fourier heat conduction in a finite medium with insulated boundaries and arbitrary initial condition. Int. Commun. Heat Mass Transf.
**2008**, 35, 103–111. [Google Scholar] [CrossRef] - Moosaie, A. Non Fourier heat conduction in a finite medium subjected to arbitrary non-periodic surface disturbance. Int. Commun. Heat Mass Transf.
**2008**, 35, 376–383. [Google Scholar] [CrossRef] - Ahmadikia, H.; Rismanian, M. Analytical solution of non-Fourier heat conduction problem on a fin under periodic boundary conditions. J. Mech. Sci. Technol.
**2011**, 25, 2919–2926. [Google Scholar] [CrossRef] - Bamdad, K.; Azimi, A.; Ahmadikia, H. Thermal performance analysis of arbitrary-profile fins with non-fourier heat conduction behavior. J. Eng. Math.
**2012**, 76, 181–193. [Google Scholar] [CrossRef] - lam, T.T.; Fong, E. Application of solution structure theorem to non-Fourier heat conduction problems: Analytical Approach. Int. J. Heat Mass Transf.
**2011**, 54, 1–11. [Google Scholar] [CrossRef] - Liu, F.; Chen, Q.; Kang, Z.; Pan, W.; Zhang, D.; Wang, L. Non-Fourier heat conduction in oil-in-water emulsions. Int. J. Heat Mass Transf.
**2019**, 135, 323–330. [Google Scholar] [CrossRef] - Daneshjou, K.; Bakhtiari, M.; Parsania, H.; Fakoor, M. Non-Fourier heat conduction analysis of infinite 2D orthotropic FG hollow cylinders subjected to time-dependent heat source. Appl. Therm. Eng.
**2016**, 98, 582–590. [Google Scholar] [CrossRef] - Ma, J.; Sun, Y.; Yang, J. Analytical solution of non-Fourier heat conduction in a square plate subjected to a moving laser pulse. Int. J. Heat Mass Transf.
**2017**, 115, 606–610. [Google Scholar] [CrossRef] - Wankhade, P.A.; Kundu, B.; Das, R. Establishment of non-Fourier heat conduction model for an accurate transient thermal response in wet fins. Int. J. Heat Mass Transf.
**2018**, 126, 911–923. [Google Scholar] [CrossRef] - Hana, S.; Peddieson, J. Non-Fourier heat conduction/convection in moving medium. Int. J. Therm. Sci.
**2018**, 130, 128–139. [Google Scholar] [CrossRef] - Liu, Y.; Li, L.; Lou, Q. A hyperbolic lattice Boltzmann method for simulating non-Fourier heat conduction. Int. J. Heat Mass Transf.
**2019**, 131, 772–780. [Google Scholar] [CrossRef] - Kalbasi, R.; Afrand, M.; Alsarraf, J.; Tran, M.D. Studies on optimum fins number in PCM-based heat sinks. Energy
**2019**, 171, 1088–1099. [Google Scholar] [CrossRef] - Li, Z.; Al-Rashed, A.A.; Rostamzadeh, M.; Kalbasi, R.; Shahsavar, A.; Afrand, M. Heat transfer reduction in buildings by embedding phase change material in multi-layer walls: Effects of repositioning, thermophysical properties and thickness of PCM. Energy Convers. Manag.
**2019**, 195, 43–56. [Google Scholar] [CrossRef] - Li, Z.; Shahsavar, A.; Al-Rashed, A.A.; Kalbasi, R.; Afrand, M.; Talebizadehsardari, P. Multi-objective energy and exergy optimization of different configurations of hybrid earth-air heat exchanger and building integrated photovoltaic/thermal system. Energy Convers. Manag.
**2019**, 195, 1098–1110. [Google Scholar] [CrossRef] - Nadooshan, A.A.; Kalbasi, R.; Afrand, M. Perforated fins effect on the heat transfer rate from a circular tube by using wind tunnel: An experimental view. Heat Mass Transf.
**2018**, 54, 3047–3057. [Google Scholar] [CrossRef] - Salimpour, M.R.; Kalbasi, R.; Lorenzini, G. Constructal multi-scale structure of PCM-based heat sinks. Contin. Mech. Thermodyn.
**2017**, 29, 477–491. [Google Scholar] [CrossRef] - Shanazari, E.; Kalbasi, R. Improving performance of an inverted absorber multi-effect solar still by applying exergy analysis. Appl. Therm. Eng.
**2018**, 143, 1–10. [Google Scholar] [CrossRef] - Yari, M.; Kalbasi, R.; Talebizadehsardari, P. Energetic-exergetic analysis of an air handling unit to reduce energy consumption by a novel creative idea. Int. J. Numer. Methods Heat Fluid Flow
**2019**, 29, 3959–3975. [Google Scholar] [CrossRef]

**Figure 3.**Temperature distributions from t = 0 to t = 0.1 (

**a**), t = 0.0 to t = 1 (

**b**), and t = 1 to t = 5 (

**c**).

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**MDPI and ACS Style**

Kalbasi, R.; Alaeddin, S.M.; Akbari, M.; Afrand, M.
Analytical Solution of Heat Conduction in a Symmetrical Cylinder Using the Solution Structure Theorem and Superposition Technique. *Symmetry* **2019**, *11*, 1522.
https://doi.org/10.3390/sym11121522

**AMA Style**

Kalbasi R, Alaeddin SM, Akbari M, Afrand M.
Analytical Solution of Heat Conduction in a Symmetrical Cylinder Using the Solution Structure Theorem and Superposition Technique. *Symmetry*. 2019; 11(12):1522.
https://doi.org/10.3390/sym11121522

**Chicago/Turabian Style**

Kalbasi, Rasool, Seyed Mohammadhadi Alaeddin, Mohammad Akbari, and Masoud Afrand.
2019. "Analytical Solution of Heat Conduction in a Symmetrical Cylinder Using the Solution Structure Theorem and Superposition Technique" *Symmetry* 11, no. 12: 1522.
https://doi.org/10.3390/sym11121522