# Solution of Temperature Distribution  in a Radiating Fin Using Homotopy  Perturbation Method

Authors: M. J. Hosseini; M. Gorji; M. Ghanbarpour
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

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.


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## Body

## 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.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 Figure1. 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]:
(2.1)2wddx[k(T)dTdx]−2εσT4=0,
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
(2.2)K(T)=Kb[1+λ(T−Tb)],
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:(2.3)θ=TTb,ψ=εσb2Tb3kw,ξ=xb,β=λTb,
the formulation
of the fin problem reduces to
(2.4a)d2θdξ2+β(dθdξ)2+βθd2θdξ2−ψθ4=0,
with boundary conditions
(2.4b)dθdξ=0atξ=0,(2.4c)θ=1atξ=1.


## 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,(3.1)A(θ)−f(r)=0,
where A(θ) is defined as follows:
(3.2)A(θ)=L(θ)+N(θ),
where L stands for the linear and N for the nonlinear part. Homotopy perturbation
structure is shown as the following equation:
(3.3)H(θ,P)=(1−p)[L(θ)−L(θ0)]+p[A(θ)−f(r)]=0
Obviously, using (3.3) we have
(3.4)H(θ,0)=L(θ)−L(θ0)=0,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(3.5)θ=∑i=0Mpiθi=θ0+pθ1+p2θ2+p3θ3+p4θ4+⋯.


## 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
(4.1)L(θ)=d2θdξ2,N(θ)=β(dθdξ)2+βθd2θdξ2−ψθ4,
with the
boundary condition given in (2.4b), θ(0) is any arbitrary constant, C.Then we
have(4.2)dθdξ=0atξ=0,θ=Catξ=0.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)
(4.3a)∂2∂ξ2θ0(ξ)=0,(4.3b)dθ0dξ=0atξ=0,θ0=Catξ=0;(p1)
(4.4a)(∂2∂ξ2θ1(ξ))+βθ0(ξ)(∂2∂ξ2θ0(ξ))+β(∂∂ξθ0(ξ))2−ψθ04(ξ)=0,(4.4b)dθ1dξ=0atξ=0,θ1=0atξ=0;(p2)
(4.5a)(∂2∂ξ2θ2(ξ))+βθ0(ξ)(∂2∂ξ2θ1(ξ))+βθ1(ξ)(∂2∂ξ2θ0(ξ))+2β(∂∂ξθ0(ξ))(∂∂ξθ1(ξ))−4ψθ03(ξ)θ1(ξ)=0,(4.5b)dθ2dξ=0atξ=0,θ2=0atξ=0;(p3)
(4.6a)(∂2∂ξ2θ3(ξ))+βθ2(ξ)(∂2∂ξ2θ0(ξ))+βθ0(ξ)(∂2∂ξ2θ2(ξ))+β(∂∂ξθ1(ξ))2+βθ1(ξ)(∂2∂ξ2θ1(ξ))+2β(∂∂ξθ0(ξ))(∂∂ξθ2(ξ))−4ψθ03(ξ)θ2(ξ)−6ψθ02(ξ)θ12(ξ)=0,(4.6b)dθ3dξ=0atξ=0,θ3=0atξ=0;(p4)
(4.7a)(∂2∂ξ2θ4(ξ))+βθ3(ξ)(∂2∂ξ2θ0(ξ))+βθ0(ξ)(∂2∂ξ2θ3(ξ))−4ψθ03(ξ)θ3(ξ)+βθ2(ξ)(∂2∂ξ2θ1(ξ))+2β(∂∂ξθ0(ξ))(∂∂ξθ3(ξ))−12ψθ02(ξ)θ2(ξ)θ1(ξ)+β(∂2∂ξ2θ1(ξ))(∂∂ξθ2(ξ))+βθ1(ξ)(∂2∂ξ2θ2(ξ))−4ψθ0(ξ)θ13(ξ)=0,(4.7b)dθ4dξ=0atξ=0,θ4=0atξ=0;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
(4.8)θ(ξ)=C+12ψC4ξ2+16ψ2C7ξ4−12βC5ψξ2+13180ψ3C10ξ6−1124ψ2C8βξ4+12β2C6ψξ2+23720ψ4C13ξ8−1960ψ3C11βξ6+78β2C9ψ2ξ4−12β3C7ψξ2⋯.


## 5. Results

The
coefficientC representing the temperature at the fin tip can be evaluated from
the boundary condition given in (2.4c) using the numerical method.Tables1 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.Table 1
The dimensionless tip temperature forψ=1.
Number of the terms in the
solution
(M)M=5M=7M=10M=14MethodHPMADMHPMADMHPMADMHPMADMβ=0.60.8278210.8191850.8265520.8250790.82673190.8250630.826750.825052β=0.40.8133890.8144890.8133510.8132790.81336830.8108660.8133690.813236β=0.20.7977080.800210.7977120.7990250.79771220.7978120.7977120.797809β=00.7791770.7777780.7791470.7777650.77914520.7765540.7791450.775333β=−0.20.7577020.7529870.756980.7529670.75681820.7541440.7568020.754132β=−0.40.8197430.8144650.8160130.8132520.81329260.8108310.8121550.813201β=−0.60.7102940.7150630.7032190.7008120.69884810.6961340.6967640.694918Table 2
The dimensionless tip temperature forψ=1000.
Number of the terms in the
solution
(M)M=5M=7M=10M=14MethodHPMADMHPMADMHPMADMHPMADMβ=0.60.1393820.1443120.1355020.1377110.13302650.1341250.1326160.132718β=0.40.1383510.1434120.1342430.1365260.13155940.1327560.1300910.131026β=0.20.1373410.1420260.1330090.1349370.13011680.1312130.1294860.129613β=00.1363490.1410550.1317970.1337850.12869950.1296630.1275040.127327β=−0.20.1353760.1401650.130610.1326540.1273080.1281430.1253480.125431β=−0.40.1344220.1395650.1294450.1318560.12594260.1273420.1241180.124432β=−0.60.1334850.1383740.1283030.1305250.12460340.1253360.1227150.122936Figure 2
Comparison of the HPM and ADM forβ=1.0 and M=14.


## 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 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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*Source: 831362-2009-04-29.xml*