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. . Arslanturk  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  presented a series of studies on the topic of heat transfer. Mokheimer  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. . A new approach to calculate thermal performance of a singular fin with variable thermal properties has been presented by Kou et al. .

Joneidi et al.  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. .

2. Mathematical Formulation of the Boundary Value Problem

Consider a straight fin with a temperature-dependent thermal conductivity, arbitrary constant cross-sectional area AC; 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 : (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.

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  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  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: (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 β0 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. [<xref ref-type="bibr" rid="B7">7</xref>])

Joneidi and coworker  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+.

Our work is compared with Joneidi et al.’s  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 (11) Exact HAM (8)(this work) DTM (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 1.00000

Our work is compared with Joneidi et al.’s  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 (11) HAM (8)(this work) NS DTM (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.

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 AC decreases), the dimensionless temperature θ decreases.

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  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.

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) .

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.

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 [<xref ref-type="bibr" rid="B15">15</xref>] Homotopy Analysis Method

Consider the following differential equation : (A.1)N[u(t)]=0, 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: (A.2)(1-p)L[φ(t;p)-u0(t)]=phH(t)N[φ(t;p)], where p[0,1] is the embedding parameter, h0 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: (A.3)φ(t;0)=u0(t),φ(t;1)=u(t), 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 (A.4)φ(t;p)=u0(t)+m=1+um(t)pm, where (A.5)um(t)=1m!mφ(t;p)pm|p=0. 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 (A.6)u(t)=u0(t)+m=1+um(t). Define the vector (A.7)un={u0,u1,,un}.

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: (A.8)L[um-χmum-1]=hH(t)m(um-1), where (A.9)m(um-1)=1(m-1)!m-1N[φ(t;p)]pm-1|p=0,χm={0,m1,1,m>1. Applying L-1 on both sides of (A.8), we get (A.10)um(t)=χmum-1(t)+hL-1[H(t)m(um-1)].

In this way, it is easily to obtain um for m1, at Mth order, we have (A.11)u(t)=m=0Mum(t). 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 . 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 (<xref ref-type="disp-formula" rid="EEq4">4</xref>) 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: (B.1)d2θdξ2+βθd2θdξ2+β(dθdξ)2-ψ2θ=0. 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 (B.2)(1-p)[d2θdξ2-ψ2θ]=ph[d2θdξ2+βθd2θdξ2+β(dθdξ)2-ψ2θ]. The approximate solution of (B.2) is as follows: (B.3)θ=θ0+pθ1+pθ2+. Substituting (B.3) into (B.1) and comparing the coefficients of like powers of p, we get (B.4)p0:d2θ0dξ2-ψ2θ0=0,(B.5)p1:d2θ1dξ2-ψ2θ1=(h+1)[d2θ0dξ2-ψ2θ0]+hβ[θ0d2θ0dξ2+(d2θ0dξ2)2],(B.6)p2:d2θ2dξ2-ψ2θ2=(h+1)[d2θ1dξ2-ψ2θ1]+hβ[θ0d2θ1dξ2+θ1d2θ0dξ2+2dθ0dξ2dθ1dξ2]. The boundary conditions are (B.7)dθ0dξ=0whenξ=0,θ0=1whenξ=1,(B.8)dθidξ=0whenξ=0,θi=0whenξ=1i=1,2,3,. Now applying the boundary conditions (B.7) in (B.4) we get (B.9)θ0(ξ)=cosh(ψξ)cosh(ψ). Substituting the value of θ0 in (B.5) and solving the equation using the boundary conditions (B.8), we obtain the following results: (B.10)θ1(ξ)=-hβcosh(2ψ)3cosh3(ψ)cosh(ψξ)+hβcosh(2ψξ)3cosh2(ψ). 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: (B.11)θ2(ξ)=(2h2β2cosh2(2ψ)9cosh5(ψ)-h(h+1)βcosh(2ψ)3cosh3(ψ)-3h2β2cosh(3ψ)16cosh4(ψ)-h2β2ψsinh(ψ)12cosh4(ψ))cosh(ψξ)+(h(h+1)β3cosh2(ψ)-2h2β2cosh(2ψ)9cosh4(ψ))cosh(2ψξ)+3h2β2cosh(3ψξ)16cosh3(ψ)+h2β2ψξsinh(ψξ)12cosh3(ψ).

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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M134"><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:math></inline-formula> 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.

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

Nomenclature A C :

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)

k a :

Thermal conductivity at the ambient fluid temperature (W m−1 K−1)

k b :

Thermal conductivity at the base temperature (W m−1 K−1)

P :

Fin perimeter (m)

Q :

Heat-transfer rate (W)

T a :

Temperature of surface a (K)

T b :

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.

Kraus A. Aziz A. Welty J. Extended Surface Heat Transfer 2001 New York, NY, USA John Wiley & Sons Arslanturk C. A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity International Communications in Heat and Mass Transfer 2005 32 6 831 841 2-s2.0-18144425459 10.1016/j.icheatmasstransfer.2004.10.006 Incropera F. P. Dewitt D. P. Introduction to Heat Transfer 1996 3rd New York, NY, USA John Wiley & Sons Mokheimer E. M. A. Performance of annular fins with different profiles subject to variable heat transfer coefficient International Journal of Heat and Mass Transfer 2002 45 17 3631 3642 2-s2.0-0036681185 10.1016/S0017-9310(02)00078-9 Zubair S. M. Al-Garni A. Z. Nizami J. S. The optimal dimensions of circular fins with variable profile and temperature-dependent thermal conductivity International Journal of Heat and Mass Transfer 1996 39 16 3431 3439 2-s2.0-0030296822 10.1016/0017-9310(96)00011-7 Kou H.-S. Lee J.-J. Lai C.-Y. Thermal analysis of a longitudinal fin with variable thermal properties by recursive formulation International Journal of Heat and Mass Transfer 2005 48 11 2266 2277 2-s2.0-17044437763 10.1016/j.ijheatmasstransfer.2004.12.038 Joneidi A. A. Ganji D. D. Babaelahi M. Differential Transformation Method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity International Communications in Heat and Mass Transfer 2009 36 7 757 762 2-s2.0-67649095208 10.1016/j.icheatmasstransfer.2009.03.020 Liao S. J. The proposed homotopy analysis technique for the solution of non linear problems [Ph.D. thesis] 1992 Shanghai Jiao Tong University Liao S.-J. An approximate solution technique not depending on small parameters: a special example International Journal of Non-Linear Mechanics 1995 30 3 371 380 2-s2.0-0000615571 Liao S. J. On the homotopy analysis method for nonlinear problems Applied Mathematics and Computation 2004 147 2 499 513 2-s2.0-0141961626 10.1016/S0096-3003(02)00790-7 Liao S. J. An optimal homotopy-analysis approach for strongly nonlinear differential equations Communications in Nonlinear Science and Numerical Simulation 2010 15 8 2003 2016 2-s2.0-76049117931 10.1016/j.cnsns.2009.09.002 Liao S. J. The Homotopy Analysis Method in Non Linear Differential Equations 2012 New York, NY, USA Springer Domairry G. Bararnia H. An approximation of the analytic solution of some nonlinear heat transfer equations: a survey by using homotopy analysis method Advanced Studies in Theoretical Physics 2008 2 11 507 518 2-s2.0-48249108438 Jafari H. Chun C. Saeidy S. M. Analytical solution for nonlinear gas dynamic Homotopy analysis method Applied Mathematics 2009 4 149 154 Liao S. J. Beyond Perturbation Introduction to the Homotopy Analysis Method 2003 1st Boca Raton, Fla, USA Chapman & Hall/CRC