# Analytical Solution of Nonlinear Boundary Value Problem for Fin Efficiency of Convective Straight Fins with Temperature-Dependent Thermal Conductivity

Authors: K. Saravanakumar; V. Ananthaswamy; M. Subha; L. Rajendran
Journal: ISRN Thermodynamics
 (2013)

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## Abstract

We have employed homotopy analysis method (HAM) to evaluate the approximate analytical solution of the nonlinear equation arising in the convective straight fins with temperature-dependent thermal conductivity problem. Solutions are presented for the dimensionless temperature distribution and fin efficiency of the nonlinear equation. The analytical results are compared with previous work and satisfactory agreement is noted.


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## Body

## 1. Introduction

The discipline of heat transfer, typically considered an aspect of mechanical engineering and chemical engineering, deals with specific applied methods by which thermal energy in a system is generated, or converted, or transferred to another system. Heat transfer includes the mechanisms of heat conduction, thermal radiation, and mass transfer. The analysis of extended surface heat transfer is extensively presented by Kraus et al. [1]. Arslanturk [2] used decomposition method to evaluate the temperature distribution and analytical expression for the fin efficiency.In the study of heat transfer, a fin is a surface that extends from an object to increase the rate of heat transfer to or from the environment by increasing convection. Incropera and Dewitt [3] presented a series of studies on the topic of heat transfer. Mokheimer [4] discussed a series of fin-efficiency curves for annular fins of rectangular, constant heat flow area, triangular, concave parabolic, and convex parabolic profiles for a wide range of radius ratios, and the dimensionless parameter based on the locally variable heat transfer coefficient. The optimum dimensions of circular fins with variable profile and temperature-dependent thermal conductivity have been obtained by Zubair et al. [5]. A new approach to calculate thermal performance of a singular fin with variable thermal properties has been presented by Kou et al. [6].Joneidi et al. [7] studied an analytical solution of fin efficiency of convective straight fins with temperature-dependent thermal conductivity by the DTM. In this present paper, we first apply homotopy analysis method to obtain an approximation of analytical expression of fin efficiency of convective straight fins with temperature-dependent thermal conductivity. This problem is compared with Joneidi et al. [7].


## 2. Mathematical Formulation of the Boundary Value Problem

Consider a straight fin with a temperature-dependent thermal conductivity, arbitrary constant cross-sectional areaAC; perimeter P and length b  (see Figure 1). The fin is attached to a base surface of temperature Tb and extends into a fluid of temperature Ta, and its tip is insulated. The one-dimensional energy balance equation is given as follows
(1)ACddx[k(T)dTdx]-Pħ(Tb-Ta)=0,
where T is the temperature, k(T) is the temperature-dependent thermal conductivity of the fin material, P is the fin perimeter, and ħ is the heat transfer coefficient. The thermal conductivity of the fin material is assumed as follows:
(2)k(T)=ka[1+λ(T-Ta)],
where ka is the thermal conductivity at the ambient fluid temperature of the fin and λ is the parameter describing the variation of the thermal conductivity. The following dimensionless parameters are introduced [4]:
(3)θ=T-TaTb-Ta,ξ=xb,β=λ(Tb-Ta),ψ=(ħPb2kaAC)1/2.
Using the previous dimensionless variables, the dimensionless form of (1) becomes as follows:
(4)d2θdξ2+βθd2θdξ2+β(dθdξ)2-ψ2θ=0,
where θ is the dimensionless temperature, ξ is the dimensionless coordinate, β is the dimensionless parameter describing thermal conductivity, and ψ is the thermogeometric fin parameter. The boundary conditions are given as follows:
(5)dθdξ=0whenξ=0,θ=1whenξ=1.Figure 1
The sketch of a straight fin.


## 3. Fin Efficiency

The heat transfer rate from the fin is found by using Newton’s law of cooling.Consider(6)Q=∫0bP(T-Ta)dx.
The ratio of the fin heat transfer rate to the heat transfer rate of the fin if the entire fin was at the base temperature is commonly called as the fin efficiency:
(7)η=QQideal=∫0bP(T-Ta)dxPb(Tb-Ta)=∫ξ=01θ(ξ)dξ.


## 4. Approximation of Analytical Solution

### 4.1. Homotopy Analysis Method (HAM)

The homotopy analysis method employs the concept of the homotopy from topology to flexibly generate a convergent series solution for nonlinear systems. The HAM [8–13] was devised by Shijun Liao which is a powerful analytical method for solving nonlinear problems. The greater generality of this method often allows for strong convergence of the solution over larger spacial and parameter domains. This method provides an analytical solution in terms of an infinite power series. However, there is a practical need to evaluate this solution and to obtain numerical values from the infinite power series. In order to investigate the accuracy of the homotopy analysis method (HAM) solution with a finite number of terms, the system of differential equations was solved.The homotopy analysis method [14] is a good technique comparing to another perturbation method. The HAM gives excellent flexibility in the expression of the solution and how the solution is explicitly obtained. Different from all reported perturbation and nonperturbative techniques, the homotopy analysis method itself provides us with a convenient way to control and adjust the convergence region and rate of approximation series, when necessary. Briefly speaking, the homotopy analysis method has the following advantages: it is valid even if a given nonlinear problem does not contain any small/large parameters at all; it can be employed to efficiently approximate a nonlinear problem by choosing different sets of base functions. In this paper, we employ HAM (see Appendix A) to solve the nonlinear equation.


### 4.2. Solution of Dimensionless Temperature

Using HAM, we can obtain the dimensionless temperature (see AppendixB) as follows:
(8)θ(ξ)=(1cosh(ψ)-h(h+2)βcosh(2ψ)3cosh3(ψ)+2h2β2cosh2(2ψ)9cosh5(ψ)-3h2β2cosh(3ψ)16cosh4(ψ)-h2β2ψsinh(ψ)12cosh4(ψ))cosh(ψξ)+(h(h+2)β3cosh2(ψ)-2h2β2cosh(2ψ)9cosh4(ψ))cosh(2ψξ)+3h2β2cosh(3ψξ)16cosh3(ψ)+h2β2ψξsinh(ψξ)12cosh3(ψ).
When β→0 from (4), the temperature is as follows:
(9)θ(ξ)=cosh(ψξ)cosh(ψ).
This is the exact solution of (4) when β=0.


### 4.3. Determination of Fin Efficiency

Using (7) and (8) we can obtain the fin efficiency given as follows:
(10)η=(1cosh(ψ)+2h2β2cosh2(2ψ)9cosh5(ψ)-h(h+1)βcosh(2ψ)3cosh3(ψ)-3h2β2cosh(3ψ)16cosh4(ψ)-h2β2ψsinh(ψ)12cosh4(ψ))sinh(ψ)ψ+h(h+7)βsinh(2ψ)6ψcosh2(ψ)+h2β2sinh(3ψ)16ψcosh3(ψ)+h2β2(ψcosh(ψ)-sinh(ψ))12ψcosh3(ψ)-hβcosh(2ψ)sinh(ψ)3ψcosh3(ψ)-h2β2cosh(2ψ)sinh(2ψ)9ψcosh4(ψ).


## 4.1. Homotopy Analysis Method (HAM)

The homotopy analysis method employs the concept of the homotopy from topology to flexibly generate a convergent series solution for nonlinear systems. The HAM [8–13] was devised by Shijun Liao which is a powerful analytical method for solving nonlinear problems. The greater generality of this method often allows for strong convergence of the solution over larger spacial and parameter domains. This method provides an analytical solution in terms of an infinite power series. However, there is a practical need to evaluate this solution and to obtain numerical values from the infinite power series. In order to investigate the accuracy of the homotopy analysis method (HAM) solution with a finite number of terms, the system of differential equations was solved.The homotopy analysis method [14] is a good technique comparing to another perturbation method. The HAM gives excellent flexibility in the expression of the solution and how the solution is explicitly obtained. Different from all reported perturbation and nonperturbative techniques, the homotopy analysis method itself provides us with a convenient way to control and adjust the convergence region and rate of approximation series, when necessary. Briefly speaking, the homotopy analysis method has the following advantages: it is valid even if a given nonlinear problem does not contain any small/large parameters at all; it can be employed to efficiently approximate a nonlinear problem by choosing different sets of base functions. In this paper, we employ HAM (see Appendix A) to solve the nonlinear equation.


## 4.2. Solution of Dimensionless Temperature

Using HAM, we can obtain the dimensionless temperature (see AppendixB) as follows:
(8)θ(ξ)=(1cosh(ψ)-h(h+2)βcosh(2ψ)3cosh3(ψ)+2h2β2cosh2(2ψ)9cosh5(ψ)-3h2β2cosh(3ψ)16cosh4(ψ)-h2β2ψsinh(ψ)12cosh4(ψ))cosh(ψξ)+(h(h+2)β3cosh2(ψ)-2h2β2cosh(2ψ)9cosh4(ψ))cosh(2ψξ)+3h2β2cosh(3ψξ)16cosh3(ψ)+h2β2ψξsinh(ψξ)12cosh3(ψ).
When β→0 from (4), the temperature is as follows:
(9)θ(ξ)=cosh(ψξ)cosh(ψ).
This is the exact solution of (4) when β=0.


## 4.3. Determination of Fin Efficiency

Using (7) and (8) we can obtain the fin efficiency given as follows:
(10)η=(1cosh(ψ)+2h2β2cosh2(2ψ)9cosh5(ψ)-h(h+1)βcosh(2ψ)3cosh3(ψ)-3h2β2cosh(3ψ)16cosh4(ψ)-h2β2ψsinh(ψ)12cosh4(ψ))sinh(ψ)ψ+h(h+7)βsinh(2ψ)6ψcosh2(ψ)+h2β2sinh(3ψ)16ψcosh3(ψ)+h2β2(ψcosh(ψ)-sinh(ψ))12ψcosh3(ψ)-hβcosh(2ψ)sinh(ψ)3ψcosh3(ψ)-h2β2cosh(2ψ)sinh(2ψ)9ψcosh4(ψ).


## 5. Comparison with Previous Work (Joneidi et al. [7])

Joneidi and coworker [7] obtained the closed form of the solution of (1) using DTM (Tables 1 and 2) as follows:
(11)θ(ξ)=a+ψ2a2(1+βa)ξ2-ψ4a(2βa-1)24(1+βa)3ξ4+ψ6a(2βa-1)(14βa-1)720(1+βa)5ξ6-ψ8a(2βa-1)(25592β3a3-7152β2a2+330βa-1)40320(1+βa)7ξ8+⋯.
The value of a can be obtained from the following equation:
(12)θ(1)=a+ψ2a2(1+βa)-ψ4a(2βa-1)24(1+βa)3+ψ6a(2βa-1)(14βa-1)720(1+βa)5-ψ8a(2βa-1)(25592β3a3-7152β2a2+330βa-1)40320(1+βa)7+⋯=1.
The solution of fin efficiency is
(13)η=a+ψ2a6(1+βa)-ψ4a(2βa-1)120(1+βa)3+ψ6a(2βa-1)(14βa-1)5040(1+βa)5-ψ8a(2βa-1)(25592β3a3-7152β2a2+330βa-1)362880(1+βa)7+⋯.Table 1
Our work is compared with Joneidi et al.’s [7] work and exact solution of dimensionless temperature θ(ξ) (linear case).
ξ
(a)β=0,ψ=0.5
(b)β=0,ψ=1
HAM (8)(this work)
DTM [7](11)
Exact
HAM (8)(this work)
DTM [7](11)
Exact
0
0.88681
0.88681
0.88681
0.64805
0.64805
0.64805
0.05
0.88709
0.88709
0.88709
0.64886
0.64886
0.64886
0.10
0.88792
0.88792
0.88792
0.65129
0.65129
0.65129
0.15
0.88931
0.88931
0.88931
0.65535
0.65535
0.65535
0.20
0.89125
0.89125
0.89125
0.66105
0.66105
0.66105
0.25
0.89375
0.89375
0.89375
0.66841
0.66841
0.66841
0.30
0.89681
0.89681
0.89681
0.67743
0.67743
0.67743
0.35
0.90043
0.90043
0.90043
0.68815
0.68815
0.68815
0.40
0.90461
0.90461
0.90461
0.70059
0.70059
0.70059
0.45
0.90936
0.90936
0.90936
0.71478
0.71478
0.71478
0.50
0.91467
0.91467
0.91467
0.73076
0.73076
0.73076
0.55
0.92056
0.92056
0.92056
0.74856
0.74856
0.74856
0.60
0.92702
0.92702
0.92702
0.76824
0.76824
0.76824
0.65
0.93406
0.93406
0.93406
0.78984
0.78984
0.78984
0.70
0.94169
0.94169
0.94169
0.81341
0.81341
0.81341
0.75
0.94990
0.94990
0.94990
0.83902
0.83902
0.83902
0.80
0.95871
0.95871
0.95871
0.86673
0.86673
0.86673
0.85
0.96812
0.96812
0.96812
0.89660
0.89660
0.89660
0.90
0.97813
0.97813
0.97813
0.92871
0.92871
0.92871
0.95
0.98875
0.98875
0.98875
0.96315
0.96315
0.96315
1.00
1.00000
1.00000
0.99999
1.00000
1.00000
1.00000Table 2
Our work is compared with Joneidi et al.’s [7] work and numerical solution of dimensionless temperature θ(ξ) (nonlinear case).
ξ
β
=
0.4
,
ψ
=
1
β
=
0.2
,
ψ
=
0.5
NS
DTM [7](11)
HAM (8)(this work)
NS
DTM [7](11)
HAM (8)(this work)
0
0.71604
0.71604
0.71604
0.90344
0.90344
0.90344
0.05
0.71674
0.71674
0.71674
0.90368
0.90368
0.90368
0.10
0.71883
0.71883
0.71883
0.90440
0.90440
0.90440
0.15
0.72231
0.72231
0.72231
0.90559
0.90559
0.90559
0.20
0.72718
0.72718
0.72718
0.90727
0.90727
0.90727
0.25
0.73346
0.73346
0.73346
0.90942
0.90942
0.90942
0.30
0.74114
0.74114
0.74114
0.91206
0.91206
0.91206
0.35
0.75022
0.75022
0.75022
0.91517
0.91517
0.91517
0.40
0.76072
0.76072
0.76072
0.91877
0.91877
0.91877
0.45
0.77264
0.77264
0.77264
0.92285
0.92285
0.92285
0.50
0.78599
0.78599
0.78599
0.92741
0.92741
0.92741
0.55
0.80077
0.80077
0.80077
0.93246
0.93246
0.93246
0.60
0.81699
0.81699
0.81699
0.93799
0.93799
0.93799
0.65
0.83467
0.83467
0.83467
0.94402
0.94402
0.94402
0.70
0.85381
0.85381
0.85381
0.95053
0.95053
0.95053
0.75
0.87442
0.87442
0.87442
0.95753
0.95753
0.95753
0.80
0.89652
0.89652
0.89652
0.96503
0.96503
0.96503
0.85
0.92011
0.92011
0.92009
0.97302
0.97302
0.97302
0.90
0.94522
0.94522
0.94518
0.98151
0.98151
0.98151
0.95
0.97184
0.97184
0.97181
0.99050
0.99050
0.99050
1.00
1.00000
0.99999
1.00000
0.99999
0.99999
0.99999


## 6. Discussion

Equation (8) (this work) and (Equation (11)) (previous work) represent the analytical expressions of temperature for all values of parameters. Among these two expressions our result is the simplest one.Figure2 represents dimensionless temperature distribution θ versus the dimensionless coordinate ξ for various values of ψ using (8). From this figure, it is clear that when ψ increases (i.e., the fin length b increases or the cross-sectional area of the fin AC decreases), the dimensionless temperature θ decreases.Figure 2
Dimensionless temperatureθ versus dimensionless coordinate ξ for various values of ψ when β=0  and  h=-0.84. The key to the graph (solid line) represents our approximate solution presented in this work (8), and (+) represents Joneidi et al.’s [7] work (Equation (11)).Figure3 illustrates the temperature distribution of the fin for different values of ψ and β. Figure 3(a) depicts the effect of the thermal conductivity β on the temperature  θ. According to (8) the temperature increases when the thermal conductivity of the fin β increases for all values of ψ. As an observation, in (10), it can be concluded that the fin efficiency also increases. It is noted that the temperature on fin θ increases when β increases (see Figure 3(b)) for the fixed value of ψ=1.Dimensionless temperatureθ versus dimensionless coordinate ξ for various values of β when (a) ψ=0.5,  h=-0.84  and (b) ψ=1,  h=-0.84. The key to the graph (solid line) represents our approximate solution presented in this work (8), and (+) represents Equation (11) [7].
(a)
(b)Figure4 shows the fin efficiency η as a function of thermogeometric fin parameter ψ for different values of the thermal conductivity parameter β using (10). From this figure, it can be seen that the fin efficiency η decreases as ψ increases. However, the fin efficiency η increases as the thermal conductivity β increases.Figure 4
Influence of thermal conductivity parameterβ on the fin efficiency η with the thermogeometric fin parameter ψ.


## 7. Conclusions

There are two main goals that we aimed for this work. The first is to employ the powerful homotopy analysis method to investigate nonlinear differential equation arising in convective straight fins with temperature-dependent thermal conductivity problem. The second is to achieve good results in predicting the solution of the heat transfer equations in engineering. The two goals are achieved successfully. In HAM, we can chooseh in an appropriate way which controls the convergence of the series.



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*Source: 282481-2013-09-08.xml*