Radiating extended surfaces are widely used to enhance heat transfer between primary surface and the environment. The present paper applies the homotopy perturbation to obtain analytic approximation of distribution of temperature in heat fin radiating, which is compared with the results obtained by Adomian decomposition method (ADM). Comparison of the results obtained by the method reveals that homotopy perturbation method (HPM) is more effective and easy to use.

Most scientific
problems and phenomena such as heat transfer occur nonlinearly. Except a
limited number of these problems, it is difficult to find the exact analytical
solutions for them. Therefore, approximate analytical solutions are searched
and were introduced [

The analysis of
space radiators, frequently provided in published literature, for example, [

Schematic of a heat fin radiating element.

Here, the fin
problem is solved to obtain the distribution of temperature of the fin by
homotopy perturbation method and compared with the result obtained by the
Adomian decomposition method, which is used for solving various nonlinear fin
problems [

A typical heat
pipe space radiator is shown in Figure

Employing the
following dimensionless parameters:

In this study,
we apply the homotopy perturbation method to the discussed problems. To
illustrate the basic ideas of the method, we consider the following nonlinear
differential equation,

Following
homotopy-perturbation method to (

Then we
have

By increasing the number of the terms in the solution, higher accuracy will be obtained. Since the remaining terms are too long to be mentioned in here, the results are shown in tables.

Solving (

The
coefficient

Tables

The dimensionless tip temperature for

Number of the terms in the
solution
( | ||||||||
---|---|---|---|---|---|---|---|---|

Method | HPM | ADM | HPM | ADM | HPM | ADM | HPM | ADM |

0.827821 | 0.819185 | 0.826552 | 0.825079 | 0.8267319 | 0.825063 | 0.82675 | 0.825052 | |

0.813389 | 0.814489 | 0.813351 | 0.813279 | 0.8133683 | 0.810866 | 0.813369 | 0.813236 | |

0.797708 | 0.80021 | 0.797712 | 0.799025 | 0.7977122 | 0.797812 | 0.797712 | 0.797809 | |

0.779177 | 0.777778 | 0.779147 | 0.777765 | 0.7791452 | 0.776554 | 0.779145 | 0.775333 | |

0.757702 | 0.752987 | 0.75698 | 0.752967 | 0.7568182 | 0.754144 | 0.756802 | 0.754132 | |

0.819743 | 0.814465 | 0.816013 | 0.813252 | 0.8132926 | 0.810831 | 0.812155 | 0.813201 | |

0.710294 | 0.715063 | 0.703219 | 0.700812 | 0.6988481 | 0.696134 | 0.696764 | 0.694918 |

The dimensionless tip temperature for

Number of the terms in the
solution
( | ||||||||
---|---|---|---|---|---|---|---|---|

Method | HPM | ADM | HPM | ADM | HPM | ADM | HPM | ADM |

0.139382 | 0.144312 | 0.135502 | 0.137711 | 0.1330265 | 0.134125 | 0.132616 | 0.132718 | |

0.138351 | 0.143412 | 0.134243 | 0.136526 | 0.1315594 | 0.132756 | 0.130091 | 0.131026 | |

0.137341 | 0.142026 | 0.133009 | 0.134937 | 0.1301168 | 0.131213 | 0.129486 | 0.129613 | |

0.136349 | 0.141055 | 0.131797 | 0.133785 | 0.1286995 | 0.129663 | 0.127504 | 0.127327 | |

0.135376 | 0.140165 | 0.13061 | 0.132654 | 0.127308 | 0.128143 | 0.125348 | 0.125431 | |

0.134422 | 0.139565 | 0.129445 | 0.131856 | 0.1259426 | 0.127342 | 0.124118 | 0.124432 | |

0.133485 | 0.138374 | 0.128303 | 0.130525 | 0.1246034 | 0.125336 | 0.122715 | 0.122936 |

Comparison of the HPM and ADM for

In this work, homotopy perturbation method has been successfully applied to a typical heat pipe space radiator. The solution shows that the results of the present method are in excellent agreement with those of ADM and the obtained solutions are shown in the figure and tables. Some of the advantage of HPM are that reduces the volume of calculations with the fewest number of iterations, it can converge to correct results. The proposed method is very simple and straightforward. In our work, we use the Maple Package to calculate the functions obtained from the homotopy perturbation method.