The document describes using the homotopy perturbation method (HPM) to solve the temperature distribution in a radiating fin. The HPM is applied to the nonlinear differential equation that governs the temperature distribution. This results in a perturbation series solution for the dimensionless temperature. The HPM solution is compared to the Adomian decomposition method (ADM) solution. Results show that the HPM provides more accurate results and is easier to implement than the ADM for this problem of finding the temperature distribution in a radiating fin.

1. Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2009, Article ID 831362, 8 pages
doi:10.1155/2009/831362
Research Article
Solution of Temperature
Distribution in a Radiating Fin Using
Homotopy Perturbation Method
M. J. Hosseini, M. Gorji, and M. Ghanbarpour
Department of Mechanical Engineering, Noshirvani University of Technology, P.O. Box 484, Babol, Iran
Correspondence should be addressed to M. Gorji, gorji@nit.ac.ir
Received 27 November 2008; Accepted 22 January 2009
Recommended by Saad A. Ragab
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.
Copyright q 2009 M. J. Hosseini et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
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 1–5 , among
which homotopy perturbation method HPM 6–12 and Adomain decomposition method
ADM 13, 14 are the most effective and convenient ones for both weakly and strongly
nonlinear equations.
The analysis of space radiators, frequently provided in published literature, for
example, 15–22 , is based upon the assumption that the thermal conductivity of the fin
material is constant. However, since the temperature difference of the fin base and its tip
is high in the actual situation, the variation of the conductivity of the fin material should be
taken into consideration and includes the effects of the variation of the thermal conductivity
of the fin material. The present analysis considers the radiator configuration shown in
Figure 1. In the design, parallel pipes are joined by webs, which act as radiator fins. Heat
flows by conduction from the pipes down the fin and radiates from both surfaces.

2. 2 Mathematical Problems in Engineering
2w
x
W b
Figure 1: 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 23–25 .
2. The Fin Problem
A typical heat pipe space radiator is shown in Figure 1. Both surfaces of the fin are radiating
to the vacuum of outer space at a very low temperature, which is assumed equal to zero
absolute. The fin is diffuse-grey with emissivity ε, and has temperature-dependent thermal
conductivity k, which depends on temperature linearly. The base temperature Tb of the fin
and tube surfaces temperature is constant; the radiative exchange between the fin and heat
pipe is neglected. Since the fin is assumed to be thin, the temperature distribution within the
fin is assumed to be one-dimensional. The energy balance equation for a differential element
of the fin is given as 26 :
d dT
2w k T − 2εσT 4 0, 2.1
dx dx
where k T and σ are the thermal conductivity and the Stefan-Boltzmann constant,
respectively. The thermal conductivity of the fin material is assumed to be a linear function
of temperature according to
K T Kb 1 λ T − Tb , 2.2
where kb is the thermal conductivity at the base temperature of the fin and λ is the slope of
the thermal conductivity temperature curve.
Employing the following dimensionless parameters:
3
T εσb2 Tb x
θ , ψ , ξ , β λTb , 2.3
Tb kw b
the formulation of the fin problem reduces to
2
d2 θ dθ d2 θ
β βθ − ψθ4 0, 2.4a
dξ2 dξ dξ2

3. Mathematical Problems in Engineering 3
with boundary conditions
dθ
0 at ξ 0, 2.4b
dξ
θ 1 at ξ 1. 2.4c
3. Basic Idea of Homotopy Perturbation Method
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,
A θ −f r 0, 3.1
where A θ is defined as follows:
A θ L θ N θ , 3.2
where L stands for the linear and N for the nonlinear part. Homotopy perturbation structure
is shown as the following equation:
H θ, P 1 − p L θ − L θ0 p A θ −f r 0 3.3
Obviously, using 3.3 we have
H θ, 0 L θ − L θ0 0,
3.4
H θ, 1 A θ −f r 0,
where p ∈ 0, 1 is an embedding parameter and θ0 is the first approximation that satisfies the
boundary condition. We Consider θ and as
M
θ pi θi θ0 pθ1 p2 θ2 p3 θ3 p4 θ4 ··· . 3.5
i 0
4. The Fin Temperature Distribution
Following homotopy-perturbation method to 2.4a , 2.4b , and 2.4c , linear and non-linear
parts are defined as
d2 θ
L θ ,
dξ2
2 4.1
dθ d2 θ
N θ β βθ 2 − ψθ4 ,
dξ dξ
with the boundary condition given in 2.4b , θ 0 is any arbitrary constant, C.

4. 4 Mathematical Problems in Engineering
Then we have
dθ
0 at ξ 0, θ C at ξ 0. 4.2
dξ
Substituting 3.5 in to 4.1 and then into 3.3 and rearranging based on power of p-terms,
we have the following
p0
∂2
θ0 ξ 0, 4.3a
∂ξ2
dθ0
0 at ξ 0, θ0 C at ξ 0; 4.3b
dξ
p1
2
∂2 ∂2 ∂
θ1 ξ βθ0 ξ θ0 ξ β θ0 ξ − ψθ0 ξ
4
0, 4.4a
∂ξ2 ∂ξ2 ∂ξ
dθ1
0 at ξ 0, θ1 0 at ξ 0; 4.4b
dξ
p2
∂2 ∂2 ∂2
θ2 ξ βθ0 ξ θ1 ξ βθ1 ξ θ0 ξ
∂ξ2 ∂ξ2 ∂ξ2
4.5a
∂ ∂
2β θ0 ξ θ1 ξ − 3
4ψθ0 ξ θ1 ξ 0,
∂ξ ∂ξ
dθ2
0 at ξ 0, θ2 0 at ξ 0; 4.5b
dξ
p3
2
∂2 ∂2 ∂2 ∂
θ3 ξ βθ2 ξ θ0 ξ βθ0 ξ θ2 ξ β θ1 ξ
∂ξ2 ∂ξ2 ∂ξ2 ∂ξ
∂2 ∂ ∂ 4.6a
βθ1 ξ θ1 ξ 2β θ0 ξ θ2 ξ − 4ψθ0 ξ θ2 ξ
3
∂ξ2 ∂ξ ∂ξ
− 6ψθ0 ξ θ1 ξ
2 2
0,
dθ3
0 at ξ 0, θ3 0 at ξ 0; 4.6b
dξ

5. Mathematical Problems in Engineering 5
1
ψ 1
0.8
Dimensionless temperatrure, θ
0.6 ψ 10
0.4
ψ 100
0.2
0
0 0.2 0.4 0.6 0.8 1
Dimensionless coordinate, ξ
HPM
ADM
Figure 2: Comparison of the HPM and ADM for β 1.0 and M 14.
p4
∂2 ∂2 ∂2
θ4 ξ βθ3 ξ θ0 ξ βθ0 ξ θ3 ξ − 4ψθ0 ξ θ3 ξ
3
∂ξ2 ∂ξ2 ∂ξ2
∂2 ∂ ∂
βθ2 ξ θ1 ξ 2β θ0 ξ θ3 ξ − 12ψθ0 ξ θ2 ξ θ1 ξ
2
4.7a
∂ξ2 ∂ξ ∂ξ
∂2 ∂ ∂2
β θ1 ξ θ2 ξ βθ1 ξ θ2 ξ − 4ψθ0 ξ θ1 ξ
3
0,
∂ξ2 ∂ξ ∂ξ2
dθ4
0 at ξ 0, θ4 0 at ξ 0; 4.7b
dξ
and so forth.
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 4.3a , 4.4a , 4.5a , 4.6a , and 4.7a results in θ ξ . When p → 1, we have
1 1 2 7 4 1 5 2 13 3 10 6
θ ξ C ψC4 ξ2 ψ C ξ − βC ψξ ψ C ξ
2 6 2 180
11 2 8 4 1 2 6 2 23 4 13 8 19 3 11 6
− ψ C βξ β C ψξ ψ C ξ − ψ C βξ 4.8
24 2 720 60
7 2 9 2 4 1 3 7 2
β C ψ ξ − β C ψξ · · · .
8 2

6. 6 Mathematical Problems in Engineering
Table 1: The dimensionless tip temperature for ψ 1.
Number of the terms M 5 M 7 M 10 M 14
in the solution M
Method HPM ADM HPM ADM HPM ADM HPM ADM
β 0.6 0.827821 0.819185 0.826552 0.825079 0.8267319 0.825063 0.82675 0.825052
β 0.4 0.813389 0.814489 0.813351 0.813279 0.8133683 0.810866 0.813369 0.813236
β 0.2 0.797708 0.80021 0.797712 0.799025 0.7977122 0.797812 0.797712 0.797809
β 0 0.779177 0.777778 0.779147 0.777765 0.7791452 0.776554 0.779145 0.775333
β −0.2 0.757702 0.752987 0.75698 0.752967 0.7568182 0.754144 0.756802 0.754132
β −0.4 0.819743 0.814465 0.816013 0.813252 0.8132926 0.810831 0.812155 0.813201
β −0.6 0.710294 0.715063 0.703219 0.700812 0.6988481 0.696134 0.696764 0.694918
Table 2: The dimensionless tip temperature for ψ 1000.
Number of the terms M 5 M 7 M 10 M 14
in the solution M
Method HPM ADM HPM ADM HPM ADM HPM ADM
β 0.6 0.139382 0.144312 0.135502 0.137711 0.1330265 0.134125 0.132616 0.132718
β 0.4 0.138351 0.143412 0.134243 0.136526 0.1315594 0.132756 0.130091 0.131026
β 0.2 0.137341 0.142026 0.133009 0.134937 0.1301168 0.131213 0.129486 0.129613
β 0 0.136349 0.141055 0.131797 0.133785 0.1286995 0.129663 0.127504 0.127327
β −0.2 0.135376 0.140165 0.13061 0.132654 0.127308 0.128143 0.125348 0.125431
β −0.4 0.134422 0.139565 0.129445 0.131856 0.1259426 0.127342 0.124118 0.124432
β −0.6 0.133485 0.138374 0.128303 0.130525 0.1246034 0.125336 0.122715 0.122936
5. Results
The coefficient C representing the temperature at the fin tip can be evaluated from the
boundary condition given in 2.4c using the numerical method.
Tables 1 and 2 show the dimensionless tip temperature, that is, coefficient C, for
different thermal conductivity parameters, β. The tables state that the convergence of
the solution for the higher thermo-geometric fin parameter, ψ, is faster than the solution
with lower fin parameter. It is clear from the tables that the solution is convergent. In
order to investigate the accuracy of the homotopy solution, the problem is compared with
decomposition solution 26 , also the corresponding results are presented in Figure 2. It
should be mentioned that the homotopy results in the tables are arranged for first 5, 7, 10,
14 terms of the solution M . It is seen that the results by homotopy perturbation method,
and adomian decomposition method are in good agreement. The results of the comparison
show that the difference is 3.1% in the case of the strongest nonlinearity, that is, β 1.0 and
ψ 100.
6. Conclusions
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

7. Mathematical Problems in Engineering 7
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.
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