# Application of Approximate Analytical Technique Using the Homotopy Perturbation Method to Study the Inclination Effect on the Thermal Behavior of Porous Fin Heat Sink

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{∞}. To simplify the formulation of fin problem, we made the following assumptions:

- Porous media is homogeneous and saturated with single-phase fluid.
- The interaction between the saturated fluid and medium is governed by Darcy’s model.
- Thermo-physical characteristics of the porous fin with that of the fluid are constant.
- Fin tip is adiabatic.

## 3. Method of Solution using HPM

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Acknowledgements

## Conflict of Interest

## Abbreviations

Terminology | |

A | Fin cross-sectional area |

A_{b} | Base area of the fin |

A_{s} | Fin surface area |

h_{eff} | Heat coefficient at fin base |

c_{p} | Specific heat of the fluid passing through the porous fin |

K | Permeability |

M | Thermo-geometric parameter |

$\dot{m}$ | Saturated fluid mass flowage |

Nu | Nusselt number |

P | Fin perimeter |

t | Fin thickness |

q | Rate of heat transfer |

X | Dimensionless length |

q | Internal heat generation |

Gr | Grashoff’s number |

β | Inclination angle. |

S_{h} | Porosity term. |

M | Convective heat parameter |

T | Temperature |

T_{a} | Ambient temperature |

T_{b} | Temperature at the base of the fin |

V | Average velocity of the fluid passing through the porous fin |

Greek Symbols | |

β | Inclination angle |

θ | Temperature (Dimensionless) |

η | Fin efficiency |

β_{th} | Coefficient of thermal expansion |

υ | Kinematic viscosity |

ρ | Fluid density |

Subscripts | |

s | Solid properties |

f | Fluid properties |

eff | Effective porous properties |

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**Figure 1.**(

**a**) Porous heat sink; (

**b**) schematic of the heat process; (

**c**) plain view of vertical fins heat sink; (

**d**) plain view of inclined porous fins heat sink.

x | Numerical Method (Runge-Kutta) | HPM (Present Study) | Absolute Error |
---|---|---|---|

0.00 | 0.863499231 | 0.863499664 | 0.000000433 |

0.05 | 0.863828568 | 0.863829046 | 0.000000478 |

0.10 | 0.864817090 | 0.864817539 | 0.000000449 |

0.15 | 0.866466182 | 0.866465743 | 0.000000439 |

0.20 | 0.868776709 | 0.868776261 | 0.000000448 |

0.25 | 0.871751555 | 0.871751104 | 0.000000451 |

0.30 | 0.875393859 | 0.875393404 | 0.000000455 |

0.35 | 0.879707472 | 0.879707010 | 0.000000462 |

0.40 | 0.884696967 | 0.884696500 | 0.000000467 |

0.45 | 0.890367650 | 0.890367181 | 0.000000469 |

0.50 | 0.896725569 | 0.896725096 | 0.000000473 |

0.55 | 0.903777531 | 0.903777060 | 0.000000471 |

0.60 | 0.911531120 | 0.911530658 | 0.000000462 |

0.65 | 0.919994710 | 0.919994259 | 0.000000451 |

0.70 | 0.929177488 | 0.929177056 | 0.000000432 |

0.75 | 0.939089476 | 0.939089079 | 0.000000397 |

0.80 | 0.949741555 | 0.949741203 | 0.000000352 |

0.85 | 0.961145491 | 0.961145189 | 0.000000302 |

0.90 | 0.973313964 | 0.973313764 | 0.000000200 |

0.95 | 0.986260599 | 0.986260549 | 0.000000005 |

1.00 | 1.000000000 | 1.000000000 | 0.000000000 |

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**MDPI and ACS Style**

Oguntala, G.; Sobamowo, G.; Ahmed, Y.; Abd-Alhameed, R.
Application of Approximate Analytical Technique Using the Homotopy Perturbation Method to Study the Inclination Effect on the Thermal Behavior of Porous Fin Heat Sink. *Math. Comput. Appl.* **2018**, *23*, 62.
https://doi.org/10.3390/mca23040062

**AMA Style**

Oguntala G, Sobamowo G, Ahmed Y, Abd-Alhameed R.
Application of Approximate Analytical Technique Using the Homotopy Perturbation Method to Study the Inclination Effect on the Thermal Behavior of Porous Fin Heat Sink. *Mathematical and Computational Applications*. 2018; 23(4):62.
https://doi.org/10.3390/mca23040062

**Chicago/Turabian Style**

Oguntala, George, Gbeminiyi Sobamowo, Yinusa Ahmed, and Raed Abd-Alhameed.
2018. "Application of Approximate Analytical Technique Using the Homotopy Perturbation Method to Study the Inclination Effect on the Thermal Behavior of Porous Fin Heat Sink" *Mathematical and Computational Applications* 23, no. 4: 62.
https://doi.org/10.3390/mca23040062