MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation83136210.1155/2009/831362831362Research ArticleSolution of Temperature Distribution in a Radiating Fin Using Homotopy Perturbation MethodHosseiniM. J.GorjiM.GhanbarpourM.RagabSaad A.Department of Mechanical EngineeringNoshirvani University of TechnologyP.O. Box 484, BabolIrannit.ac.ir200929042009200927112008220120092009Copyright © 2009This 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.

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.

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 , among which homotopy perturbation method (HPM)  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, , 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.

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 .

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 : 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 K(T)=Kb[1+λ(TTb)], 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: θ=TTb,ψ=εσb2Tb3kw,ξ=xb,β=λTb, the formulation of the fin problem reduces to d2θdξ2+β(dθdξ)2+βθd2θdξ2ψθ4=0, with boundary conditions dθdξ=0atξ=0,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, A(θ)f(r)=0, where A(θ) is defined as follows: 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: H(θ,P)=(1p)[L(θ)L(θ0)]+p[A(θ)f(r)]=0 Obviously, using (3.3) we have 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 asθ=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 L(θ)=d2θdξ2,N(θ)=β(dθdξ)2+βθd2θdξ2ψθ4, with the boundary condition given in (2.4b), θ(0) is any arbitrary constant, C.

Then we have 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

2ξ2θ0(ξ)=0,dθ0dξ=0atξ=0,θ0=Catξ=0;

(2ξ2θ1(ξ))+βθ0(ξ)(2ξ2θ0(ξ))+β(ξθ0(ξ))2ψθ04(ξ)=0,dθ1dξ=0atξ=0,θ1=0atξ=0;

(2ξ2θ2(ξ))+βθ0(ξ)(2ξ2θ1(ξ))+βθ1(ξ)(2ξ2θ0(ξ))+2β(ξθ0(ξ))(ξθ1(ξ))4ψθ03(ξ)θ1(ξ)=0,dθ2dξ=0atξ=0,θ2=0atξ=0;

(2ξ2θ3(ξ))+βθ2(ξ)(2ξ2θ0(ξ))+βθ0(ξ)(2ξ2θ2(ξ))+β(ξθ1(ξ))2+βθ1(ξ)(2ξ2θ1(ξ))+2β(ξθ0(ξ))(ξθ2(ξ))4ψθ03(ξ)θ2(ξ)6ψθ02(ξ)θ12(ξ)=0,dθ3dξ=0atξ=0,θ3=0atξ=0;

(2ξ2θ4(ξ))+βθ3(ξ)(2ξ2θ0(ξ))+βθ0(ξ)(2ξ2θ3(ξ))4ψθ03(ξ)θ3(ξ)+βθ2(ξ)(2ξ2θ1(ξ))+2β(ξθ0(ξ))(ξθ3(ξ))12ψθ02(ξ)θ2(ξ)θ1(ξ)+β(2ξ2θ1(ξ))(ξθ2(ξ))+βθ1(ξ)(2ξ2θ2(ξ))4ψθ0(ξ)θ13(ξ)=0,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 p1, we have θ(ξ)=C+12ψC4ξ2+16ψ2C7ξ412βC5ψξ2+13180ψ3C10ξ61124ψ2C8βξ4+12β2C6ψξ2+23720ψ4C13ξ81960ψ3C11βξ6+78β2C9ψ2ξ412β3C7ψξ2.

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 , 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.

The dimensionless tip temperature for ψ=1.

Number of the terms in the solution (M)M=5M=7M=10M=14

β=0.60.8278210.8191850.8265520.8250790.82673190.8250630.826750.825052
β=0.40.8133890.8144890.8133510.8132790.81336830.8108660.8133690.813236
β=0.20.7977080.800210.7977120.7990250.79771220.7978120.7977120.797809
β=00.7791770.7777780.7791470.7777650.77914520.7765540.7791450.775333
β=0.20.7577020.7529870.756980.7529670.75681820.7541440.7568020.754132
β=0.40.8197430.8144650.8160130.8132520.81329260.8108310.8121550.813201
β=0.60.7102940.7150630.7032190.7008120.69884810.6961340.6967640.694918

The dimensionless tip temperature for ψ=1000.

Number of the terms in the solution (M)M=5M=7M=10M=14

β=0.60.1393820.1443120.1355020.1377110.13302650.1341250.1326160.132718
β=0.40.1383510.1434120.1342430.1365260.13155940.1327560.1300910.131026
β=0.20.1373410.1420260.1330090.1349370.13011680.1312130.1294860.129613
β=00.1363490.1410550.1317970.1337850.12869950.1296630.1275040.127327
β=0.20.1353760.1401650.130610.1326540.1273080.1281430.1253480.125431
β=0.40.1344220.1395650.1294450.1318560.12594260.1273420.1241180.124432
β=0.60.1334850.1383740.1283030.1305250.12460340.1253360.1227150.122936

Comparison of the HPM and ADM for β=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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