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\title{Optimization of Effectiveness for a Cylindrical Fin}
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             \author[1]{Abdullah Al  Mamun}

             \affil[1]{  Khulna University of Engineering and Technology}

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\date{\small \em Received: 11 December 2015 Accepted: 2 January 2016 Published: 15 January 2016}

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\begin{abstract}
        


A numerical study was performed to provide information about the temperature distribution of three dimensional cylindrical fin in steady state and homogeneous material properties. A brief literature review shows that much of work on fins has been carried out analytically and numerically in one dimensional and two dimensional conditions. This study is concerned about the three dimensional temperature distributions on a cylindrical fin, optimum dimensions and heat transfer from the fin, the fin efficiency and fin effectiveness of the cylindrical fin when fin base was maintained at a constant temperature. The necessary equations are solved by finite difference method and iteration method using FORTRAN code. The whole investigation was done using different material and different dimensional fins to find out the optimum effectiveness and efficiency for predefined condition.

\end{abstract}


\keywords{pin-fin, cylindrical fin, 3d fin, effectiveness of a fin, efficiency of pin-fin.}

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\let\tabcellsep& 	 	 		 
\section[{I. Introduction}]{I. Introduction}\par
he term extended surface or fin in commonly used to depict an important special case involving heat transfer by conduction within a solid and heat transfer by convection from the boundaries of the solid. Different types of fin such as rectangular fin, triangular fin, trapezoidal fin, parabolic fin, cylindrical fin, pin fin, annular fin etc are commonly used to enhance the heat dissipation rate from primary surfaces to its surrounding fluid medium in order to meet the ever-increasing demand for high performance, light weight and compact heat transfer equipments. Because of many more engineering applications heat transfer characteristics of fins of different geometry have been subject of continued research.\par
Fins are used to increase the heat transfer from a surface by increasing the effective surface area. However the fin itself represents a conduction resistance to heat transfer from the original surface. For this reason, there is no assurance that the heat transfer rate will be increased through the use of fins. An assessment of this matter may be made by evaluating the fin effectiveness. It is defined as the ratio of the fin heat transfer rate to the heat transfer rate that would exist without the fin. In general the use of fins may rarely be justified unless?\textunderscore f?2 . 
\section[{a) Objectives}]{a) Objectives}\par
The main objectives of this study are: a) To investigate the temperature distribution along the dimension of a cylindrical fin for different thermal conductivity of fin material. b) To determine the rate of heat transfer through the cylindrical fin. c) To determine the fin effectiveness and efficiency of the cylindrical fin. d) To determine the optimum dimension for the cylindrical fin. 
\section[{II. Mathematical Formulation a) Approximation}]{II. Mathematical Formulation a) Approximation}\par
The problem is solved, subjected to following assumptions: Three-Dimensional cylindrical fin, steady state conduction, constant thermal conductivity, homogeneous material, uniform cross section and convection heat transfer coefficient is uniform across the cylindrical fin surface, radiation from the surface is negligible so it is neglected. Fin base and ambient temperature also assumed to be constant. The Governing equation for the cylindrical fin is: 
\section[{b) Governing Equation}]{b) Governing Equation}1 r ? ?r ?r ?T ?r ? + 1 r 2 ? ? 2 T ?? 2 ? + ? ? 2 T ?z 2 ? + g k = ?T ??t\par
For steady state condition1 ?? ?? ???? ??? ???? ???? ? + 1 ?? 2 ? ?? 2 ?? ???? 2 ? + ? ?? 2 ?? ???? 2 ? + ð??"ð??" ?? = 0\par
Now by Finite Difference method we get: )/(1 ?? ?? ??? + 2 ??? 2 + 2 ?? ?? 2 ??? 2 + 2 ??? 2 )\par
Fig.  {\ref 3} : Grid elements of the tip surface of the fin.? ??????????? 2??? ??? ??,?? ,?? ? ?? ???1,?? ,?? ? ? ??????????? 2??? ??? ??,?? ,?? ? ?? ??+1,?? ,?? ? ? ???????? 2?????? ??? ??,?? ,?? ? ?? ??,?? +1,?? ? ? ???????? 2?????? ??? ??,?? ,?? ? ?? ??,?? ?1,?? ? ? ???????? 2 2??? ??? ??,?? ,?? ? ?? ??,?? ,???1 ? = ??????? 2 2??? (?? ??,?? ,?? ? ?? ? )\par
By simplification:?? ??,?? ,?? = ???????? 2 ??? ???1,?? ,?? + ?? ??+1,?? ,?? ? + ????? 2??? ??? ??,?? ?1,?? + ?? ??,?? +1,?? ? + ???????? 2 2??? ??? ??,?? ,???1 ? + ??????? 2 2 ?? ? ( 2???????? 2 + 2????? 2??? + ???????? 2 2??? + ???????\textbf{2}\par
2 )\par
The below energy conservation equation is only applied for elements those are after the first circle, which means from the r2 this equation applies. By simplification:T i,j,k = k?? ?? ?????? 2??? ?T i?1,j,k + T i+1,j,k ? + k?????? 2?? ?? ??? ?T i,j?1,k + T i,j+1,k ? + k?? ?? ?????? ??? ?T i,j,k?1 ? + T ? h?? ?? ?????? 2k?? ?? ?????? 2??? + 2k?????? 2?? ?? ??? + k?? ?? ?????? ??? + h?? ?? ??????\par
ii. At the Fin Surface  ? k * A1(T i,j,k ? T i,j,k?1 ) ??? ? k * A2(T i,j,k ? T i?1,j,k ) ??? ? k * A3(2T i,j,k ? T i,j?1,k ? T i,j+1,k ) r L ??? = ? * ??1?T i,j,k ? T ? ? + ? * ??2(T i,j,k ? T ? )\par
By simplification:T i,j,k = k * A1?T i,j,k?1 ? ??? + k * A2?T i?1,j,k ? ??? + k * A3?T i,j?1,k + T i,j+1,k ? r L ??? + h * A1 * T ? + h * A2 * T ? k * A1 ??? + k * A2 ??? + 2k * A3 r L ??? + h * A1 + h * A2\par
Fin convective heat transfer from the end:?? ð??"ð??" = (??????? ?????????? ??????????????? ) 1/2 (?? ? ?? ? ) tanh(????) + ( ? ???? ) ? ? ???? ? tanh(????) + 1\par
Convective heat transfer from the fins surface:?? ð??"ð??" = ? ????(?? ? ?? ? )\par
Effectiveness of the Fin:?? ð??"ð??" = ?? ð??"ð??" ???(?? ? ?? ? )\par
Efficiency of the cylindrical fin:?? ð??"ð??" = ?? ð??"ð??" ??? ð??"ð??"???? (?? ? ?? ? ) 
\section[{III. Result and Discussion}]{III. Result and Discussion}\par
The governing three dimentinal differential equation of cylindrical fin was transfomed into linear algebric equations by finite difference methods and these equations were solved by using a program written in FORTRAN language. This code was used to determine the temperatre at each node in the computatinal domain. The material Aluminium, Stainless Steel, Aluminum -Bronze (Alloy), Copper having thermal conductivity (k) 200, 14, 76, 250 and 400 w/m-k respectively were chosen for the analysis of cylindrical fin. The convective coefficient of the surrounding 10 w/sqm-k .The fin base was maintained at a constant base temperature (400°C) 673.15K and the surrounding or ambient fluid temperature was considered at (25°C)298.15K.\par
For testing the Programe a Referance\textasciicircum ([11]) temperature distribution is taken and compared with the result obtained in the figure  {\ref 8}. Similarly another comparison was done for the temperature distribution along the radius. The results are shown in the figure \hyperref[fig_0]{10}.\par
Variation of effectiveness and effciency due to the variation of material is shown in table \hyperref[tab_0]{1} and Figure \hyperref[fig_0]{12} and Figure \hyperref[fig_0]{13}. From the table 1 we can see that copperhas maximum effectiveness and efficiency so it was selected as the material of the fin.\par
The variation of effectiveness and effciency due to the variation of length and radius is shown in Section 3.1 and Section 3.2 respectively. Year 2016 A Fig.  {\ref 7} : Grid at the edge of the fin.\par
At Section 3.3 table \hyperref[tab_2]{4} shows the changes of effectiveness and efficiency due to change of length and radius simultaneously, thus providing us the optimum dimension of cylindrical fin which is .001m radius and .02m length.\par
At reference condition: k=206W/m°C, h=17W/sq-m°C, Atmosphere temp=26 °C, Fin base temp=120°C, L=.9m, R=.0127m      From this table the optimum dimensions can be easily found. The one having the maximum effectiveness and maximum effciency. Though the 3rd result has minimum effciency but it has the maximum effectiveness. The daviation of efficiency is not very large so the 3rd result is selected as the optimum dimension. 
\section[{IV. Conclusion}]{IV. Conclusion}\par
From the above information and comparison it's been observed that the optimum dimension for the conditions assumed is a fin having .02m of length and .001m of radius. This fin gave the maximum effectiveness and efficiency for the assumed condition. But the results can vary according to the change of condition, which were assumed to be constant for the purpose of the simplification of the whole process. Material having higher conductivity can be used to get more higher effectiveness and efficiency, but for simplification Copper is selected as the optimum material for the fin.  \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1222222}\includegraphics[]{image-3.png}
\caption{\label{fig_1}1 ?? 2 ( 2 ) 2 2 ) 2 + 2 +}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5}\includegraphics[]{image-5.png}
\caption{\label{fig_3}?Fig. 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{62}\includegraphics[]{image-6.png}
\caption{\label{fig_4}Fig. 6 :l ???z 2 ?}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{810911}\includegraphics[]{image-7.png}
\caption{\label{fig_5}Fig. 8 :Fig. 10 :Fig. 9 :Fig. 11 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1213}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Fig. 12 :Fig. 13 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{14153}\includegraphics[]{image-9.png}
\caption{\label{fig_7}Fig. 14 :Fig. 15 :Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.21135135135135136\textwidth}P{0.1837837837837838\textwidth}P{0.22972972972972974\textwidth}P{0.22513513513513514\textwidth}}
Year 2016\tabcellsep \tabcellsep \tabcellsep \\
32\tabcellsep Thermal conductivity, k\tabcellsep Efficiency(\%)\tabcellsep Effectivness\\
XVI Issue III Version I\tabcellsep 14 76 200 250 400\tabcellsep 38.0184 72.098 84.1667 85.98 88.8646\tabcellsep 15.5875 29.5604 34.508 35.253 36.4345\\
( ) Volume A\tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_1}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.85\textwidth}}
Optimization of Effectiveness for a Cylindrical Fin\end{longtable} \par
 
\caption{\label{tab_2}Table 4 :}\end{figure}
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\noindent\textbf{} \par 
\begin{longtable}{P{0.22854984894259817\textwidth}P{0.4853474320241692\textwidth}P{0.13610271903323262\textwidth}}
Year 2016\tabcellsep Fig. 16 :\tabcellsep \\
34\tabcellsep \tabcellsep \\
XVI Issue III Version I\tabcellsep \tabcellsep \\
( ) Volume A Journal of Researches in Engineering\tabcellsep c) Variation of Effectiveness and Efficiency for Different Fin Dimension Radius (m) Length (m) Effectivencess .02 .2 19.76635 .002 .03 29.323790 .001 .02 38.379875 .003 .04 26.2769\tabcellsep Efficiency(\%) 94.12550 94.592873 93.609459 94.977020\\
Global\tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_3}}\end{figure}
 			\footnote{© 2016 Global Journals Inc. (US)} 			\footnote{© 2016 Global Journals Inc. (US) Journal of Researches in Engineering} 		 		\backmatter   			  			  			 \par
Heat transfer from fin base ( s q -m ) 			 			  				\begin{bibitemlist}{1}
\bibitem[Ganji Davood Domirl et al.]{b4}\label{b4} 	 		‘Determination of temperature distribution for annular fins with temperature dependent Thermal Conductivity by HPM’.  		 			Ganji Ganji Davood Domirl 		,  		 			Ganji Hosain Zaman Ziabkhsh 		,  		 			Domirl 		.  	 	 		\textit{Thermal science, year 2011},  				15 p. .  	 	 (Suppl. 1) 
\bibitem[Nag]{b1}\label{b1} 	 		\textit{Heat and Mass transfer},  		 			P Nag 		.  		 	 	 (2nd edition Tata Mcgraw-hill Edition) 
\bibitem[Holman]{b0}\label{b0} 	 		\textit{Heat transfer},  		 			J Holman 		.  		 	 	 (9th edition Tata Mcgraw-hill Edition) 
\bibitem[Cengel and Yunus]{b3}\label{b3} 	 		\textit{Heat transfer a practical Approach},  		 			A Cengel 		,  		 			Yunus 		.  		 	 	 (2nd edition) 
\bibitem[Kreith]{b2}\label{b2} 	 		\textit{Principle of Heat transfer},  		 			Frank Kreith 		.  		 	 	 (6th edition) 
\end{bibitemlist}
 			 		 	 
\end{document}
