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

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.

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 area 𝐴_𝐶; perimeter 𝑃 and length 𝑏  (see Figure 1). The fin is attached to a base surface of temperature 𝑇_𝑏 and extends into a fluid of temperature 𝑇_𝑎, and its tip is insulated. The one-dimensional energy balance equation is given as follows 𝐴_𝐶𝑑𝑘(𝑇)𝑑𝑥𝑑𝑇𝑇𝑑𝑥−𝑃ℏ_𝑏−𝑇_𝑎=0, (1) 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: 𝑘(𝑇)=𝑘_𝑎1+𝜆𝑇−𝑇_𝑎, (2) where 𝑘_𝑎 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]: 𝜃=𝑇−𝑇_𝑎𝑇_𝑏−𝑇_𝑎𝑥,𝜉=𝑏,𝛽=𝜆𝑇_𝑏−𝑇_𝑎,𝜓=ℏ𝑃𝑏²𝑘_𝑎𝐴_𝐶^1/2. (3) Using the previous dimensionless variables, the dimensionless form of (1) becomes as follows: 𝑑²𝜃𝑑𝜉²𝑑+𝛽𝜃²𝜃𝑑𝜉²+𝛽𝑑𝜃𝑑𝜉²−𝜓²𝜃=0, (4) 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: 𝑑𝜃𝑑𝜉=0when𝜉=0,𝜃=1when𝜉=1. (5)

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 𝑄=^𝑏₀𝑃𝑇−𝑇_𝑎𝑑𝑥. (6) 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: 𝑄𝜂=𝑄_ideal=∫^𝑏₀𝑃𝑇−𝑇_𝑎𝑑𝑥𝑃𝑏𝑇_𝑏−𝑇_𝑎=¹_𝜉=0𝜃(𝜉)𝑑𝜉. (7)

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 Appendix B) as follows: 𝜃(𝜉)=1𝜓−ℎ()coshℎ+2𝛽cosh2𝜓3cosh³𝜓+2ℎ²𝛽²cosh²2𝜓9cosh⁵𝜓−3ℎ²𝛽²cosh3𝜓16cosh⁴𝜓−ℎ²𝛽²𝜓sinh𝜓12cosh⁴𝜓+ℎ()𝛽cosh𝜓𝜉ℎ+23cosh²𝜓−2ℎ²𝛽²cosh2𝜓9cosh⁴𝜓+cosh2𝜓𝜉3ℎ²𝛽²cosh3𝜓𝜉16cosh³𝜓+ℎ²𝛽²𝜓𝜉sinh𝜓𝜉12cosh³𝜓. (8) When 𝛽→0 from (4), the temperature is as follows: 𝜃(𝜉)=cosh𝜓𝜉cosh𝜓. (9) 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: 1𝜂=𝜓+cosh2ℎ²𝛽²cosh²2𝜓9cosh⁵𝜓−ℎ(ℎ+1)𝛽cosh2𝜓3cosh³𝜓−3ℎ²𝛽²cosh3𝜓16cosh⁴𝜓−ℎ²𝛽²𝜓sinh𝜓12cosh⁴𝜓sinh𝜓𝜓+ℎ(ℎ+7)𝛽sinh2𝜓6𝜓cosh²𝜓+ℎ²𝛽²sinh3𝜓16𝜓cosh³𝜓+ℎ²𝛽²𝜓cosh𝜓−sinh𝜓12𝜓cosh³𝜓−ℎ𝛽cosh2𝜓sinh𝜓3𝜓cosh³𝜓−ℎ²𝛽²cosh2𝜓sinh2𝜓9𝜓cosh⁴𝜓. (10)

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: 𝜃(𝜉)𝜓=𝑎+²𝑎2𝜉1+𝛽𝑎²−𝜓⁴𝑎2𝛽𝑎−1241+𝛽𝑎³𝜉⁴+𝜓⁶𝑎2𝛽𝑎−114𝛽𝑎−17201+𝛽𝑎⁵𝜉⁶−𝜓⁸𝑎2𝛽𝑎−125592𝛽³𝑎³−7152𝛽²𝑎²+330𝛽𝑎−1403201+𝛽𝑎⁷𝜉⁸+⋯. (11) The value of 𝑎 can be obtained from the following equation: 𝜃(1)𝜓=𝑎+²𝑎2−𝜓1+𝛽𝑎⁴𝑎2𝛽𝑎−1241+𝛽𝑎³+𝜓⁶𝑎2𝛽𝑎−114𝛽𝑎−17201+𝛽𝑎⁵−𝜓⁸𝑎2𝛽𝑎−125592𝛽³𝑎³−7152𝛽²𝑎²+330𝛽𝑎−1403201+𝛽𝑎⁷+⋯=1. (12) The solution of fin efficiency is 𝜓𝜂=𝑎+²𝑎6−𝜓1+𝛽𝑎⁴𝑎2𝛽𝑎−11201+𝛽𝑎³+𝜓⁶𝑎2𝛽𝑎−114𝛽𝑎−150401+𝛽𝑎⁵−𝜓⁸𝑎2𝛽𝑎−125592𝛽³𝑎³−7152𝛽²𝑎²+330𝛽𝑎−13628801+𝛽𝑎⁷+⋯. (13)

Table 1 

Our work is compared with Joneidi et al.’s [7] work and exact solution of dimensionless temperature 𝜃(𝜉) (linear case).

Table 2 

Our work is compared with Joneidi et al.’s [7] work and numerical solution of dimensionless temperature 𝜃(𝜉) (nonlinear case).

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.

Figure 2 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 𝐴_𝐶 decreases), the dimensionless temperature 𝜃 decreases.

Figure 2 

Dimensionless temperature 𝜃 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)).

Figure 3 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.

(a)

(b)

Figure 3 

Dimensionless temperature 𝜃 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].

Figure 4 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 parameter 𝛽 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 choose ℎ in an appropriate way which controls the convergence of the series.

Appendices

A. Basic Idea of Liao’s [15] Homotopy Analysis Method

Consider the following differential equation [15]: 𝑁[𝑢(𝑡)]=0, (A.1) where 𝑁 is a nonlinear operator, 𝑡 denotes an independent variable, and 𝑢(𝑡) is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can be treated in the similar way. By means of generalizing the conventional homotopy method, Liao constructed the so-called zero-order deformation equation as follows: 1−𝑝𝐿𝜑𝑡;𝑝−𝑢₀(𝑡)=𝑝ℎ𝐻(𝑡)𝑁𝜑𝑡;𝑝, (A.2) where 𝑝∈[0,1] is the embedding parameter, ℎ≠0 is a nonzero auxiliary parameter, 𝐻(𝑡)≠0 is an auxiliary function, 𝐿 is an auxiliary linear operator, 𝑢₀(𝑡) is an initial guess of 𝑢(𝑡), and 𝜑(𝑡∶𝑝) is an unknown function. It is important that one has great freedom to choose auxiliary unknowns in HAM. Obviously, when 𝑝=0 and  𝑝=1, it holds: 𝜑(𝑡;0)=𝑢₀(𝑡),𝜑(𝑡;1)=𝑢(𝑡), (A.3) respectively. Thus, as 𝑝 increases from 0 to 1, the solution 𝜑(𝑡;𝑝)  varies from the initial guess 𝑢₀(𝑡) to the solution 𝑢(𝑡). Expanding 𝜑(𝑡;𝑝) in Taylor series with respect to 𝑝, we have 𝜑𝑡;𝑝=𝑢₀(𝑡)+^+∞_𝑚=1𝑢_𝑚(𝑡)𝑝^𝑚, (A.4) where 𝑢_𝑚(𝑡)=1𝜕𝑚!^𝑚𝜑𝑡;𝑝𝜕𝑝^𝑚_𝑝=0. (A.5) If the auxiliary linear operator, the initial guess, the auxiliary parameter ℎ, and the auxiliary function are so properly chosen, the series equation (A.4) converges at 𝑝=1, then we have 𝑢(𝑡)=𝑢₀(𝑡)+^+∞_𝑚=1𝑢_𝑚(𝑡). (A.6) Define the vector 𝑢_𝑛=𝑢₀,𝑢₁,…,𝑢_𝑛. (A.7)

Differentiating (A.2) for 𝑚 times with respect to the embedding parameter 𝑝, then setting 𝑝=0, and finally dividing them by 𝑚!, we will have the so-called 𝑚th-order deformation equation as follows: 𝐿𝑢_𝑚−𝜒_𝑚𝑢_𝑚−1=ℎ𝐻(𝑡)ℜ_𝑚𝑢_𝑚−1, (A.8) where ℜ_𝑚𝑢_𝑚−1=1()!𝜕𝑚−1^𝑚−1𝑁𝜑𝑡;𝑝𝜕𝑝^𝑚−1_𝑝=0,𝜒_𝑚=0,𝑚≤1,1,𝑚>1. (A.9) Applying 𝐿⁻¹ on both sides of (A.8), we get 𝑢_𝑚(𝑡)=𝜒_𝑚𝑢_𝑚−1(𝑡)+ℎ𝐿⁻¹𝐻(𝑡)ℜ_𝑚𝑢_𝑚−1. (A.10)

In this way, it is easily to obtain 𝑢_𝑚 for 𝑚≥1, at 𝑀th order, we have 𝑢(𝑡)=^𝑀_𝑚=0𝑢_𝑚(𝑡). (A.11) When 𝑀→+∞, we get an accurate approximation of the original (A.1). For the convergence of the above method we refer the reader to Liao [15]. If (A.1) admits unique solution, then this method will produce the unique solution. If (A.1) does not possess unique solution, the HAM will give a solution among many other (possible) solutions.

B. Solution of (4) Using Homotopy Analysis Method

In this appendix, we indicate how (8) in this paper is derived. The homotopy analysis method was constructed to determine the solution of (4)-(5) as follows: 𝑑²𝜃𝑑𝜉²𝑑+𝛽𝜃²𝜃𝑑𝜉²+𝛽𝑑𝜃𝑑𝜉²−𝜓²𝜃=0. (B.1) In order to solve (B.1) by means of the HAM, we first construct the zeroth-order deformation equation by taking 𝐻(𝑡)=1.

Consider 𝑑1−𝑝²𝜃𝑑𝜉²−𝜓²𝜃𝑑=𝑝ℎ²𝜃𝑑𝜉²𝑑+𝛽𝜃²𝜃𝑑𝜉²+𝛽𝑑𝜃𝑑𝜉²−𝜓²𝜃. (B.2) The approximate solution of (B.2) is as follows: 𝜃=𝜃₀+𝑝𝜃₁+𝑝𝜃₂+⋯. (B.3) Substituting (B.3) into (B.1) and comparing the coefficients of like powers of 𝑝, we get 𝑝⁰∶𝑑²𝜃₀𝑑𝜉²−𝜓²𝜃₀𝑝=0,₁∶𝑑²𝜃₁𝑑𝜉²−𝜓²𝜃₁=(𝑑ℎ+1)²𝜃₀𝑑𝜉²−𝜓²𝜃₀+ℎ𝛽𝜃₀𝑑²𝜃₀𝑑𝜉²+𝑑²𝜃₀𝑑𝜉²²,𝑝²∶𝑑²𝜃₂𝑑𝜉²−𝜓²𝜃₂=(𝑑ℎ+1)²𝜃₁𝑑𝜉²−𝜓²𝜃₁+ℎ𝛽𝜃₀𝑑²𝜃₁𝑑𝜉²+𝜃₁𝑑²𝜃₀𝑑𝜉²+2𝑑𝜃₀𝑑𝜉²𝑑𝜃₁𝑑𝜉². (B.4)(B.5)(B.6) The boundary conditions are 𝑑𝜃₀𝑑𝜉=0when𝜉=0,𝜃₀=1when𝜉=1,𝑑𝜃_𝑖𝑑𝜉=0when𝜉=0,𝜃_𝑖=0when𝜉=1∀𝑖=1,2,3,…. (B.7)(B.8) Now applying the boundary conditions (B.7) in (B.4) we get 𝜃₀(𝜉)=cosh𝜓𝜉cosh𝜓. (B.9) Substituting the value of 𝜃₀ in (B.5) and solving the equation using the boundary conditions (B.8), we obtain the following results: 𝜃₁(𝜉)=−ℎ𝛽cosh2𝜓3cosh³𝜓cosh𝜓𝜉+ℎ𝛽cosh2𝜓𝜉3cosh²𝜓. (B.10) Upon solving (B.6) by substituting the values of 𝜃₀ and 𝜃₁ and using the boundary conditions equation (B.8), we can find the following results: 𝜃₂(𝜉)=2ℎ²𝛽²cosh²2𝜓9cosh⁵𝜓−ℎ(ℎ+1)𝛽cosh2𝜓3cosh³𝜓−3ℎ²𝛽²cosh3𝜓16cosh⁴𝜓−ℎ²𝛽²𝜓sinh𝜓12cosh⁴𝜓+ℎ()𝛽cosh𝜓𝜉ℎ+13cosh²𝜓−2ℎ²𝛽²cosh2𝜓9cosh⁴𝜓+cosh2𝜓𝜉3ℎ²𝛽²cosh3𝜓𝜉16cosh³𝜓+ℎ²𝛽²𝜓𝜉sinh𝜓𝜉12cosh³𝜓. (B.11)

After three successive iterations the solutions of 𝜃 reach the better approximation. Adding (B.9) to (B.11), we get (8) in the text.

C. Determining the Region of ℎ for Validity

The analytical solution should converge. It should be noted that the auxiliary parameter ℎ controls the convergence and accuracy of the solution series. The analytical solution represented by (8) contains the auxiliary parameter ℎ, which gives the convergence region and rate of approximation for the homotopy analysis method. In order to define the region such that the solution series is independent of ℎ, a multiple of ℎ curves are plotted. The region where the dimensionless temperature 𝜃(𝜉) and 𝜃^′(𝜉) versus ℎ is a horizontal line known as the convergence region for the corresponding function. The common region among 𝜃(𝜉) and its derivatives are known as the overall convergence region. To study the influence of ℎ on the convergence of solution, ℎ-curves of 𝜃(0.75) and 𝜃^′(0.75) are plotted in Figures 5(a) and 5(b), respectively, for 𝛽=0.2,𝜓=0.1. These figures clearly indicate that the valid region of ℎ  is about −0.86 to −0.82. Similarly, we can find the value of the convergence control parameter ℎ for different values of the constant parameters.

(a)

(b)

Figure 5 

The ℎ curves to indicate the convergence region for 𝛽=0.2 and𝜓=0.1.

Nomenclature

𝐴𝐶:	Cross-sectional area of the fin (m2)
𝑏:	Fin length (m)
ℏ:	Heat transfer coefficient (W m−1 K−1)
𝑘:	Thermal conductivity of the fin material (W m−1 K−1)
𝑘𝑎:	Thermal conductivity at the ambient fluid temperature (W m−1 K−1)
𝑘𝑏:	Thermal conductivity at the base temperature (W m−1 K−1)
𝑃:	Fin perimeter (m)
𝑄:	Heat-transfer rate (W)
𝑇𝑎:	Temperature of surface 𝑎 (K)
𝑇𝑏:	Temperature of surface 𝑏 (K)
𝑥:	Distance measured from the fin tip (m).

Greek Letters

𝜆:	The slope of the thermal conductivity-temperature curve
𝛽:	Dimensionless parameter describing variation of the thermal conductivity
𝜂:	Fin efficiency
𝜉:	Dimensionless coordinate
𝜓:	Thermogeometric fin parameter
𝜃:	Dimensionless temperature.

Acknowledgments

This work is supported by the University Grant Commission (UGC) Minor project no. F. MRP-4122/12 (MRP/UGC-SERO), Hyderabad, Government of India. The authors are thankful to Shri. S. Natanagopal, Secretary at the Madura College Board, and Dr. R. Murali, Principal at the Madura College (Autonomous), Madurai, Tamil Nadu, India, for their constant encouragement.