A REPORT ON PERFORMANCE OF ANNULAR FINS HAVING VARYING THICKNESS

VOL., NO. 8, APRIL 6 ISSN 89-668 ARPN Journal o Engineering and Applied Sciences 6-6 Asian Research Publishing Networ (ARPN). All rights reserved. A REPORT ON PERFORMANCE OF ANNULAR FINS HAVING VARYING THICKNESS Vive Kumar Gaba, Anil Kumar Tiwari and Shubhanar Bhowmic Department o Mechanical Engineering, National Institute o Technology Raipur, India E-Mail: vgaba.mech@nitrr.ac.in ABSTRACT Studying the characteristics o annular in is a ey research problem in many thermal applications. A comparison o perormance o the exponential and parabolic annular ins o varying geometry parameter is reported in the present wor. The governing dierential equation or the ins has been derived to study the temperature distribution o the ins with insulated tip. A parametric study is then carried out by varying the geometry parameters in the governing equation to investigate the eect o on in perormance and the results are presented in graphical orm. Keywords: annular ins, exponential proile, parabolic proile, aspect ratio. INTRODUCTION The main purpose o an extended surace or a in is to increase the rate o heat transer rom a heated surace to a cold luid. O these, annular ins ind numerous applications in compact heat exchangers, specialized installations o single and double pipe heat exchangers, electrical apparatus with eicient heat dissipation, cylinders o air cooled internal-combustion engines and uel cans in nuclear reactors to name a ew. The design o a in is considered to be optimum when the ins require minimum cost o manuacturing, oer the minimum resistance to the luid low, are light in weight and are easy to manuacture. A detailed review o literature on optimum design o ins has been carried out starting with Gardner (945) where upon using a set o idealizing assumptions, the eiciency o various straight ins and spines have been reported. Duin and McLain (959, 968) solved the optimization problem o straight based ins assuming that the minimum weight in had a linear temperature distribution along its length. Murry (938) presented an equation or the temperature gradient and the eectiveness o annular ins o constant thicness with a symmetrical temperature distribution around the base o the in. Brown (965) reported the optimum dimensions o uniorm annular in by relating in dimensions to the heat transer and thermal properties o the in and heat transer coeicient between the in and its surroundings. The optimized dimensions o the in can be ound in either one o the two ways: the maximum amount o heat dissipation or a given quantity o weight or the minimum weight or dissipating a given quantity o heat. Ullmann and Kalman (989) adopted the irst way and determined the eiciency and optimum dimensions o annular ins with triangular, parabolic, and hyperbolic proiles using numerical techniques. Dhar and Arora (976) described the methods o carrying out the minimum weight design o inned suraces o speciic type by irst obtaining the optimum surace proile o a in required to dissipate a certain amount o heat rom the given surace, with no restriction on the in height and then extended their study or the case when in height is given. (Duin959) gave a method or carrying out the minimum weight design o a in using a rigorous mathematical method based on Variational calculus and assumed constant thermal conductivity o a in material and a constant heat transer coeicient along the in surace. In a recent wor, (Arauzo et al 5) reported a ten-term power series method or predicting the temperature distributions and the heat transer rates o annular ins o hyperbolic proiles. Assuming ixed in volume, (Arslantur5) reported simple correlation equations or optimum design o annular ins with uniorm cross sections to obtain the dimensionless geometrical parameters o the in with maximum heat transer rates. These simple correlation equations can help the thermal design engineers or carrying out the study on optimum design o annular ins o uniorm thicness. In their recent wor, (Kundu and Das ) reported the perormance analysis and optimization o concentric annular ins with a step change in thicness using Lagrange multiplier. In a recent wor, (Iborra and Campo 9) reported that approximate analytic temperature proiles and heat transer rates are easily obtainable without resorting to the exact analytic temperature distribution and heat transer rate based on modiied Bessel unctions. Kang (9) reported the optimum perormance and in length o a rectangular proile annular in using variations separation method. Aziz and Fang () presented alternative solutions or dierent tip conditions o longitudinal ins having rectangular, trapezoidal and concave parabolic proiles and reported relationship between dimensionless heat lux, in parameter and dimensionless tip temperature or all the geometries. Aziz and Khani () presented an analytical solution or thermal perormance o annular ins o rectangular and dierent convex parabolic proiles mounted on a rotating shat, losing heat by convection to its surroundings. In their wor, convection heat transer coeicient was assumed to be a unction o radial coordinate and shat speed. In an experimental study, heat transer rate and eiciency or circular and elliptical annular ins were 5

VOL., NO. 8, APRIL 6 ISSN 89-668 ARPN Journal o Engineering and Applied Sciences 6-6 Asian Research Publishing Networ (ARPN). All rights reserved. analyzed or dierent environmental conditions by (Nagarani)and high eiciency was reported or elliptical ins as compared to circular ones. In another recent wor, (Aziz and Fang ) derived analytical expressions or the temperature distribution, tip heat low and biot number at the tip and reported thermal perormance o the annular in under both cooling and heating conditions. The present wor reports the perormance o exponential and parabolic annular in subject to heat low in radial direction with insulated tip boundary condition. The eect o geometry parameters governing the thicness variation is studied on the in perormance and a comparison is drawn amongst the two proiles. MATHEMATICAL FORMULATION Assuming the eect o external environment on the surace convection to be negligible, the second order dierential equation or the heat transer through the ins is developed to determine the temperature proile. and the in parameter h m L δ a Upon introducing the normalized variables, the governing equation becomes, d d A A dx dx Where dδ A x δ dx and R a R - m dδ δ A δ 4 dx (3) Equation (3) can be generalized or varying thicness o the in using parabolic and exponential variation along the length o the in as deined in the ollowing equations shown in Table-. Table-. Geometry proiles considered in the study. Figure-. Sectional view o in geometry. For calculating the heat balance, the details or a control volume o length dr o a in is shown in Figure-. The resulting second order dierential equation has been solved using a computational algorithm. Applying the law o conservation o energy or thermal energy balance: Q r=q r+dr+ Q conv. () Using above equation the ollowing equation can be arrived at d dθ dδ πrδ dr 4πrhθ dr dr dr dr The in geometry and physical parameters are normalized using the ollowing expressions, () In the proile equations, it is to be noted that parameter m, being an index to radius, changes the proile at any radius and is responsible o linear or non-linear variation o the proile while parameter n, being a scalar multiplier to the radius aects the thicness o the proile obtained due to m at any radius. Using Equation (3) and relations rom Table, or exponential proile, the governing equation becomes d m d x 3 dx x dx where m.5 nr m 4m4 e x.x m nmr δ =- ; =( )( )R ; =nmr (4) (4m-) m 3 (R -) R while, or parabolic proile, the governing equation becomes δ θ R r δ,, R, x δa θ R R 5

VOL., NO. 8, APRIL 6 ISSN 89-668 ARPN Journal o Engineering and Applied Sciences 6-6 Asian Research Publishing Networ (ARPN). All rights reserved. m d 4x d m dx x 5x dx (5) m.5 6 m 7.x 5x where m m m 4δ 4=nmR ; 5=nR ; 6= ; 7=( ) (R -) R Equation (4) and (5) are solved using the ollowing boundary conditions: (i) = at x = (ii) d at x R dx (insulated tip) Eiciency o the in is obtained rom the general equation as ollows: ' x η in (6) m R m Lx R x dx R Using Equation (6), the in eiciency or exponential and parabolic proile respectively are derived as ollows (Equation.7-8): ' x exp (7) m R AEdx R Here, A E = δ. a. n.. e. m x 4 R par m R Here, ' x R m m m n. x. R m R x Adx δa m m p R 4 R A.. n.m x x p RESULTS AND DISCUSSIONS The dimensionless temperature being a unction o normalized variables depends on n and m due to chosen in shape, m, R and x due to in geometry. Considering x as the only independent variable, the Equation.(4-5), are solved or a range o n and m. The system properties are taen as h=5 W/m, = W/m-K, a =. m, and R =. m. To calculate the in eiciency the irst derivative o temperature at the in base i.e. x= is evaluated by solving second order dierential equation in MATLAB (8) using bvp4c routine. A proven technique to solve boundary value problems is to choose a continuous piecewise polynomial or a spline that adjusts the boundary conditions. The remaining unnown coeicients are determined by collocating the algebraic equations at several points. Code bvp4c(shampine et. al. ) is an adaptive inite dierence code that employs three-stage Lobatto III-a collocation ormula by computing a cubic spline on each subinterval[x i, x i+ ] o a mesh. The important requirements are that the piecewise polynomial must satisy the boundary conditions and collocate the ODE at endpoints and midpoint o the subinterval. The approximate continuous solution is obtained by controlling the residual over each subinterval [x i,x i+] approximated using a ive-point Lobatto quadrature ormula. The solution is validated with benchmar results available or rectangular proile (Kreith and Bohn ).The results are ound to be in good agreement with benchmar results as shown in Figure-.The eect o geometry parameters on in perormance or exponential and parabolic proiles are studied and reported next. Figure : Validation o the present study with Kreith and Bohn () Exponential proile For exponential proile, the eect o geometry parameters m and n on the eiciency is shown in Figure-3 and 4 respectively. It is observed rom Figure-3 that eiciency decreases with increase in m and approaches the eiciency or rectangular proile i.e. n=.however, rom Figure-4, it is observed that variation o n monotonically increases eiciency. This is obvious due to the act that increasing n increases the thicness o any proile obtained at a given value o m. A contour plot o eiciency with n and m is presented in Figure-5 and can be used as a design monogram to determine the eiciencies o exponential annular in or any combination o n and m. Parabolic proile Similar studies are carried out or parabolic proile and results are plotted in Figure-6 and 7 or the identical value o aspect ratio. The eect o geometry 5

VOL., NO. 8, APRIL 6 ISSN 89-668 ARPN Journal o Engineering and Applied Sciences 6-6 Asian Research Publishing Networ (ARPN). All rights reserved. parameter m on eiciency in case o parabolic proile is more pronounced as compared to exponential proile. In Figure-8, a contour plot o eiciency with n and m is plotted and can also be used as a design monogram as the expected eiciency at any combination o parameter n and m can be predicted rom the plot i.e. Figure-8.The above results indicate that although parameter n has a monotonic eect on eiciency o both exponential and parabolic annular ins at a given value o m, the same does not apply the other way round. In act a comparison o the variation o m and its eect on the eiciencies o both proiles gives an interesting insight. As a result a comparative plot o eiciency versus geometry parameter m (n=.5) or both the proiles is shown in Figure-9. It is observed rom the plot that the selection o proile is governed by a limiting value o geometry parameter m(given by ml). For the value o m less than the ml parabolic proile yields better eiciency as compared to exponential proile while or values o m greater than ml the exponential proile yields better perormance. This implies that or the manuacturer ml is an important design parameter that decides the shape o annular in. It is, thus, evident rom Figure-9 that a limiting value o m always exists, denoted by ml, which decides the criteria or selection o in proile amongst exponential and parabolic geometries or a given value o n and R. To study the variation o ml with Ra plot between ml and n is shown in Figure. or dierent R. It is observed that initially as the n increases, m L increases. However or larger values o n, ml tend to reach an optimum value. An interesting insight obtained rom the plot is that as n directly aects the value o ml at any given aspect ratio and hence the suitability o parabolic proile increases over the exponential one, as abricating an exponential proile is more typical. However, increasing aspect ratio has an inverse eect on ml. Thus or larger aspect ratio, exponential proiles possess broader limits to have an advantage over the parabolic counterparts. Figure-4. Variation o eiciency with parameter n at dierent m values or exponential geometry (R=3). Figure-5. Contour plot o variation o eiciency with geometry parameters or exponential proile (R =3). Figure-3. Variation o eiciency with parameter m at dierent n values or exponential geometry (R=3). Figure-6. Variation o eiciency with parameter m at dierent n values or parabolic geometry (R=3). 53

VOL., NO. 8, APRIL 6 ISSN 89-668 ARPN Journal o Engineering and Applied Sciences 6-6 Asian Research Publishing Networ (ARPN). All rights reserved. Figure-7. Variation o eiciency with parameter n at dierent m values or parabolic geometry (R=3). Figure-. Plot o ml, with parameter n, or dierent aspect ratio. CONCLUSIONS The perormance o annular ins having exponential and parabolic thicness variation is reported. The study is carried out or dierent values o geometry parameters n and m and aspect ratio. It is observed that or both exponential and parabolic, as the parameter m increases, the eiciency o the in decreases and approaches to minimum. The eect o variation o n on eiciency is direct in nature. As n increases, due to more convection area available at the base o in, the eiciency also increases. A comparative study reveals that parabolic proile is having higher eiciency or the lower values o m as compared to exponential proile but as the value o m increases and exceeds a certain value the eiciency o exponential proile becomes better. REFERENCES Figure-8. Contour plot o variation o eiciency with geometry parameters or parabolic proile (R =3). [] Aziz A., Fang T., (). Alternative solutions or longitudinal ind o rectangular, trapezoidal, and concave parabolic proiles. Energy Conversion and Management, vol. 5, pp. 88-94. [] Aziz A., F. Khani, (). Analytic solution or a rotating radial in o rectangular and various convex parabolic proiles. Communication in Nonlinear Science and Numerical Simulation, vol. 5, pp 565-574. [3] Aziz A., Tiegang Fang, (). Thermal analysis o an annular in with (a) simultaneously imposed base temperature and base heat lux and (b) ixed base and tip temperatures. Energy Conversion and Management, vol. 5, pp 467-478. Figure-9. A comparison o variation o in eiciency or both proiles with parameter m at n =.5, R = 3. [4] Acosta-Iborra A.,Campo A.,(9). Approximate analytic temperature distribution and eiciency or annular ins o uniorm thicness. Int. J. o Thermal Sciences, vol. 48, pp. 773-78. 54

VOL., NO. 8, APRIL 6 ISSN 89-668 ARPN Journal o Engineering and Applied Sciences 6-6 Asian Research Publishing Networ (ARPN). All rights reserved. [5] Arauzo I., Campo A, and Cortes C., (5). Quic estimate o the heat transer characteristics o annular ins o hyperbolic proile with the power series method. J. Applied Thermal Engineering, Vol. 5, pp. 63-634. [6] Arslantur C., (5). Simple correlation equations or optimum design o annular ins with uniorm thicness. J. Applied thermal Engineering, Vol. 5, pp. 463 468. [7] Brown A., (965). Optimum dimensions o uniorm annular ins. Int.J. Heat and Mass Transer, Vol. 8, pp. 655-66. [8] Dhar P.L. and Arora C.P., (976). Optimum design o inned suraces. J. Franlin Inst. Vol. 3, pp. 379-39. [9] Duin R.J. and McLain D.T., (968). Optimum Shape o a cooling in on a convex cylinder. J. Math. Mech., Vol. 7, pp. 769 784. [] Duin R.J.,(959). A variational problem relating to cooling ins. J. Math. Mech. W.S., pp. 47-56. [] Gardner K.A., (945). Eiciency o extended suraces. Trans. ASME Vol. 67, pp. 6-63. [] Kang H.,(9). Optimization o a rectangular proile annular in based on ixed in height.journal o Mechanical Science and Technology, vol. - 3, pp 34-33. [3] Kreith F. and Bohn M. S., () Principles o Heat Transer, Thomson learning, Sixth Edition. [4] Kundu B., and Das P.K., (). Perormance analysis and optimization o annular in with a step change in thicness. ASME J. Heat Transer, Vol. 3, pp. 6 64. [5] Murray W.M., (938). Heat dissipation through an annular dis or in o uniorm thicness. J. Appl. Mech. Trans. ASME Vol. 6, p. 78. [6] Nagaran N., (). Experimental heat transer analysis on annular circular and elliptical ins. International Journal o Engineering Sciences and Technology, vol., pp 839-845. [7] Shampine L. F., Reichelt M. W. and Kierzena J., () Solving boundary value problems or ordinary dierential equations in Matlab with bvp4c. [8] Ullmann A., Kalman H., (989).Eiciency and optimized dimensions o annular ins o dierent cross-section shapes. Int. J. Heat and Mass Transer, Vol. 3, pp. 5 5. 55