JP Journal of Heat and Mass Transfer Volume 6, Number 3, 01, Pages 3-34 Available online at http://pphmj.com/journals/jphmt.htm Published by Pushpa Publishing House, Allahabad, INDIA HEAT TRANSFER AND TEMPERATURE DISTRIBUTION OF DIFFERENT FIN GEOMETRY USING NUMERICAL METHOD Department of Mechanical Engineering Pondicherry Engineering College Pondicherry, India e-mail: pmathi@pec.edu c Abstract Convective heat dissipation from the metal surface can be significantly increased by the use of fins. Heat transfer from the fin involves a complex conjugate system of conduction and convection. The influence of the fin parameter, fin geometry, base and environment temperature on the fin have been considered. The purpose of this study is to describe the application of analytical method for the estimation of heat transfer and temperature distribution of different fin profiles. The performance analysis is carried out using the analytical and numerical methods. The result shows that the rate of heat transfer is high for the triangular fin, followed by pin fin and rectangular fin. To assess the correctness of the analytical method, the results are compared against finite difference method (FDM). Notation A Cross sectional area ( m ) A Surface area of fin ( m ) f 01 Pushpa Publishing House Keywords and phrases: aluminum fin, fin geometry, FDM. Received December 3, 011
4 h Convection heat transfer coefficient ( m k) k Thermal conductivity ( W mk) L Length of fin ( m ) n Nodal point P Perimeter of pin ( m ) Q Rate of heat loss from fin ( W ) T Temperature ( C) T Base temperature ( C) 0 T Ambient temperature ( C) x Axial location ( m ) W η Fin efficiency θ ( x) Excess temperature ( C) θ 0 T 0 T ( C) 1. Introduction Extended surface is used specially to enhance the heat transfer rate between a solid and an adjoining fluid. Such an extended surface is termed a fin. If thin strips (fins) of metals are attached to the basic surface, extending into one fluid, the total surface for heat transfer is thereby increased. Thus the fins find numerous applications in electrical apparatus in which generated heat must be efficiently dissipated. Recently, finned surface are widely used in compact heat exchanges that are used in many applications such as air conditioners, aircraft, chemical process, plants, etc. The fin involves combined effects of conduction and convection of heat transfer and temperature distribution along the geometry of the materials. The combined effect of heat transfer of fin is very important from a practical application point of view. Taking this point under consideration, the present work is to
Heat Transfer and Temperature Distribution of Different Fin 5 compare the heat transfer rate and temperature distribution of different fin geometry using the analytical and numerical methods (FDM)..1. Analytical solution. Method of Analysis of Fins The analytical solution of uniform cross-sectional area fins can be found in many literatures [4]. The fin is assumed to be infinitely long (i.e., the tip of the fin is at the same temperature as the adjacent fluid). The following assumptions are considered: One-dimensional conduction in the x-direction. Steady state conditions. Constant thermal conductivity. No heat generation. Constant and uniform convective heat transfer coefficient over the entire surface. Negligible radiation from the surface. Under the above assumptions, the energy equation and boundary conditions can be written as where m = ( hp ka c ). ( d T dx ) m ( T T ) = 0, (1) T ( ) = T, () 0 0 T ( L ) = T, (3) The temperature distribution in the fin is The total heat transfer by the fin T( x) = T + ( T0 T ) exp( mx). (4) Q = hpkac ( T0 T ). (5)
6 Figure 1. Schematic diagram of pin fin. The differential equation for fin of variable cross-sectional area (triangular fin) is d θ dx + ( 1 x) dθ dx Bθ x = 0. (6) Using modified Bessel function, the solution for the above equation is For long fin θ = AI ( B ( x)) + BK ( B ). (7) 0 0 x θ θ = I ( B ( x )) I ( B ( )). (8) 0 0 0 L The temperature distribution along the triangular fin is given by the equation T ( x) = T + ( T T )[ I ( B ( x)) I ( B ( ))], (9) 0 0 0 L B = Lh kt, B = 4.04. Simplifying the equation, T ( x) = 30 + 0.1853I0( 8.084 ( x)).
Heat Transfer and Temperature Distribution of Different Fin 7 Figure. Schematic diagram of triangular fin... Fin efficiency The fin efficiency, η, is defined as the ratio of the actual heat transfer through the fin to the ideal heat transfer through the fin if the entire fin surface were at fin base temperature. For the present, efficiency of the fin can be expressed by: η = 1 ml. (10).3. Finite difference method long fin The numerical method solution is employed to determine the temperature distribution of fins. The finite-difference numerical scheme is described by Chapra and Canale [5]. In this method, the differential equation of heat conduction is approximated by a set of algebraic equations for temperature at a number of nodal points. Therefore, the first step in the analysis is the transformation of the differential equation of heat conduction in the fin into a set of algebraic equations (i.e., the finite-difference representation of the differential equation). The energy balance of internal node for the fin and the temperatures at boundaries are prescribed; that is, T ( 0) = T0 and T ( L) = T. The fin is divided into N sub-regions, each Δ x = L N and the node temperature is denoted by T n, n = 0, 1,,..., N. The resulting general form of the finite-
8 difference equation for the internal node ( i.e., n = 1,,..., N 1) is: Hence Q cond, left + Q cond, right + Q convection = ΔE Δt, ΔE Δt = 0 (Steady state). According to Fourier law: Along the length, L, Q c, left + Q c, right + Q convection = 0. Q = ka( dt dx ). Q = ka( ΔT L ), Q c, left ka ( T T ) Δ, = c m 1 m x Q c, right ka ( T T ) Δ, = c m m+ 1 x Q convection ha ( T ). = s T m The general form of the finite-difference equation for the internal node ka c Δx ( T T ) + ka Δx ( T T ) + hpδx( T T ) 0. m 1 m c m+ 1 m m = The equation becomes Therefore, m 1 m c m+ 1 m m = ka ( T T ) + ka ( T T ) + hpδx ( T T ) 0. c AT AT m 1 ATm + ATm + 1 + BT BTm = ( A + B) T + AT = BT. m 1 m m+ 1 Now applying at the node points m = 1 : AT0 ( A + B ) T1 + AT = BT m = : AT1 ( A + B ) T + AT3 = BT 0,
Heat Transfer and Temperature Distribution of Different Fin 9 Triangular fin: Q = 0, m = 3 : AT ( A + B ) T3 + AT4 = BT m = 4 : AT3 ( A + B ) T4 + AT5 = BT m = 5 : AT ( A + B) T = BT. 4 5 Q c, left + Q c, right + Q convection = 0, ka Δx ( T T ) + ka Δx ( T T ) + hpδx( T T ) 0. (11) c m 1 m c m+ 1 m m = The equation becomes [ 1 ( m 0.5) Δx L]( Tm 1 Tm ) + [ 1 ( m + 0. 5) ]( Tm+ 1 Tm ) + h Δx m kl sin θ ( T T ) = 0..4. Geometries and materials The geometry which is proposed in the present work is shown in Figures 1-. The fin materials are considered aluminum alloys and the specifications are given in Table 1. Table 1. Specification Pin fin Rectangular fin Triangular fin n = 5 n = 5 n = 5 k = 180W/mK k = 180W/mK k = 180W/mK L = 900 mm L = 900 mm L = 900 mm D = 9.5 mm t = 5.43 mm, w = 9.5 mm t = 7.1mm, w = 9.5 mm 3. Result and Discussions The temperature distribution along the fin was calculated for the base temperature at 90 C. For all these conditions, the temperature at the tip of fin was the same as that of air. It was found that the temperature profile of the
30 triangular fin is having quick conduction rate when compared with other fins. The calculated values are shown in Table and Figure 3. Distance (x) (mm) Table. Temperature distribution of varying geometry Pin fin ( C) Rectangular fin ( C) Triangular fin ( C) Base(0) 90.1 90.1 90.1 100 65.63 66.86 7.3 00 51.16 5.64 56.80 300 4.51 43.91 47.37 400 37.46 38.54 40.35 500 34.43 35.5 35.88 600 3.63 33. 33.35 700 31.56 31.98 30.64 800 30.9 31.1 30.69 900 30.55 30.75 30.5 1000 30.3 30.45 30.1 1100 30.19 30.8 30.09 100 90 80 PinFin Rectangular Fin Triangular Fin 70 Temperature ( C ) 60 50 40 30 0 10 0 0 100 00 300 400 500 600 700 800 900 1000 1100 100 Distance x(mm) Figure 3. Temperature profile of different fin geometry.
Heat Transfer and Temperature Distribution of Different Fin 31 A higher slope can be observed near the base of the rectangular fin and pin fin compared to triangular fin due to the maximum temperature difference between fin surface and the surrounding medium at the base. The temperature distribution values were calculated using the analytical and numerical methods and are presented in Table 3. The values are closer to each other for all geometry fins. The rate of heat transfer by the fin and its efficiency are also calculated using equations (5) and (10), respectively. Rate of heat transfer by the fin increases linearly with respect to the base temperature as shown in Table 4 and Figure 4. Table 3. Temperature distribution at different nodal points Distance x (mm) Analytical solution pin fin ( C) Numerical solution pin fin ( C) Base (0) 90.1 90.1 180 53.49 54.47 360 41.45 41.4 540 33.60 33.14 70 31.14 31.36 900 30.55 9.98 Distance x (mm) Analytical solution rectangular fin ( C) Numerical solution rectangular fin ( C) Base(0) 90.1 90.1 180 54.963 53.74 360 40.386 39.85 540 34.3 33.94 70 31.79 31.4 900 30.75 9.77
3 Distance x (mm) Analytical solution triangular fin ( C) Numerical solution triangular fin ( C) Base(0) 90.1 90.1 180 31.06 31.455 360 34. 3.79 540 4.45 43.659 70 53.1 57.50 900 30.09 9.816 Table 4. Heat transfer rate in different fin geometry Temperature Pin fin Rectangular fin Triangular fin ( C) (W) (W) (W) 73.8.9.48 3.77 90.1 3.99 3.407 5.08 109. 5.6 4.49 6.70 13.1 6.78 5.78 8.64 10 9 8 Pin fin Rectangular fin Tri angular fin Heat transfer rate (W) 7 6 5 4 3 1 0 60 70 80 90 100 110 10 130 140 Base temperature ( C) Figure 4. Heat transfer rate of different fin geometry.
Heat Transfer and Temperature Distribution of Different Fin 33 The estimated efficiency data are shown in Table 5 and Figure 5, the result shows that the triangular fin is having better efficiency than other fins. Table 5. Efficiency of the fin Pin fin Rectangular fin Triangular fin 1.36.79 6.05 Efficiency comparison Fin of Different Geometry 30 6.05 5 0 1.36.79 Efficiency(%) 15 10 5 0 Pin Fin Rectangular Fin Triangular Fin Fin Geometry Figure 5. Comparison of fin efficiency. 4. Conclusion Analytical and numerical values are calculated for temperature distribution and heat transfer for the different geometry of the fin at the base temperature 90 C. The following results were obtained: 1. Numerical and analytical results are comparable and the values were found close to each other. The approximate error is around 1%.. Temperature distribution and heat transfer rate are higher for triangular fin when compared with other fins. 3. Main advantage of triangular fin is that it utilizes minimum material and hence this is most economical.
34 References [1] H. I. Abu-Mulaweh, Integration of a fin experiment into the undergraduate heat transfer laboratory, Inter. J. Mech. Eng. Edu. 33(1) (005), 83-9. [] Ganesh Murali and Subrahmanya S. Katte, Experiment investigation of heat transfer enhancements in radiating pin fin, Jordan J. Mech. Indust. Eng. (3) (008), 163-167. [3] N. Sahiti, F. Durst and A. Dewan, Heat transfer enhancement by pin elements, Inter. J. Heat Mass Trans. 48 (005), 4738-4747. [4] M. N. Ozisik, Heat Transfer, McGraw-Hill, New York, 1985. [5] S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, McGraw-Hill, New York, 1998. [6] D. Q. Kern and A. D. Kraus, Extended Surface Heat Transfer, McGraw-Hill, 197.