We have employed homotopy analysis method (HAM) to evaluate the approximate analytical solution of the nonlinear equation arising in the convective straight fins with temperaturedependent 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.
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. [
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 [
Joneidi et al. [
Consider a straight fin with a temperaturedependent thermal conductivity, arbitrary constant crosssectional area
The sketch of a straight fin.
The heat transfer rate from the fin is found by using Newton’s law of cooling.
Consider
The homotopy analysis method employs the concept of the homotopy from topology to flexibly generate a convergent series solution for nonlinear systems. The HAM [
The homotopy analysis method [
Using HAM, we can obtain the dimensionless temperature (see Appendix
Using (
Joneidi and coworker [
Our work is compared with Joneidi et al.’s [

(a) 
(b) 


HAM ( 
DTM [ 
Exact  HAM ( 
DTM [ 
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 [





NS  DTM [ 
HAM ( 
NS  DTM [ 
HAM (  
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 
Equation (
Figure
Dimensionless temperature
Figure
Dimensionless temperature
Figure
Influence of thermal conductivity parameter
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 temperaturedependent 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
Consider the following differential equation [
Differentiating (
In this way, it is easily to obtain
In this appendix, we indicate how (
Consider
After three successive iterations the solutions of
The analytical solution should converge. It should be noted that the auxiliary parameter
The
Crosssectional area of the fin (m^{2})
Fin length (m)
Heat transfer coefficient (W m^{−1} K^{−1})
Thermal conductivity of the fin material (W m^{−1} K^{−1})
Thermal conductivity at the ambient fluid temperature (W m^{−1} K^{−1})
Thermal conductivity at the base temperature (W m^{−1} K^{−1})
Fin perimeter (m)
Heattransfer rate (W)
Temperature of surface
Temperature of surface
Distance measured from the fin tip (m).
The slope of the thermal conductivitytemperature curve
Dimensionless parameter describing variation of the thermal conductivity
Fin efficiency
Dimensionless coordinate
Thermogeometric fin parameter
Dimensionless temperature.
This work is supported by the University Grant Commission (UGC) Minor project no. F. MRP4122/12 (MRP/UGCSERO), 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.