NAURAL CONVECION HEA RANFER FROM FIN ARRAY- EXPERIMENAL AND HEOREICALAL UDY ON EFFEC OF INCLINAION OF BAE ON HEA RANFER. V. Naidu 1 V. Dharma Rao 1 B. Govinda Rao A. ombabu 3 and B. reenivasulu 4 1 Department o Chemical Engineering Andhra University Visakhapatnam India Department o Mechanical Engineering G.V.P College o Engineering Visakhapatnam India 3 Nagarjuna Agrichem Limited Arinama Akkivalasa rikakulam India 4 Department o Chemical Engineering G.V.P College o Engineering Visakhapatnam India E-Mail: svnayudu@yahoo.com ABRAC he problem o natural convection heat transer rom in arrays ith inclination is studied experimentally and theoretically to ind the eect o inclination o the base o the in array on heat transer rate. A numerical model is developed by taking an enclosure hich is ormed by to adjacent vertical ins and horizontal base. Results obtained rom this enclosure are used to predict heat transer rate rom the in array. All the governing equations related to luid in the enclosure together ith the heat conduction equation in both the ins are solved by using Alternate Direct Implicit method. Numerical results are obtained or temperature along the length o the in and in the luid in the enclosure. he experimental studies have been also carried out on to geometric orientations viz. (a) vertical base ith vertical ins (vertical in array) and (b) horizontal base ith vertical ins (horizontal in array) ith the ive dierent inclinations like 3 4 6 and 9. he experimental results are compared ith the numerical results computed by the theoretical analysis shos the good agreement. Keyords: natural convection heat transer vertical in array inclined in array. 1. INRODUCION Fins are extensively used in air cooled automobile engines air crat engines cooling o generators motors transormers rerigerators cooling o computer processors and other electronic devices etc. Previously a great number o experimental and numerical orks has been carried out to study the eect o in parameters like in height in spacing etc. on heat transer rate rom in array by the investigators. Elenbaas [] conducted an experimental study on heat dissipation o parallel plates by ree convection or ide range o Rayliegh numbers viz..< Ra <. He determined that in the limit o small gap idth Nusselt number varies proportional to the channel Rayliegh number (Ra). Experimental ork on horizontal in arrays as studied by [ 13-17 18]. Numerical ork on horizontal in arrays as studied by [4 19]. Experimental ork on vertical in arrays as studied by [6 8]. Numerical ork on vertical in arrays as studied by [9 1]. Experimental ork on donard acing in arrays as studied by Dyan et al. [3]. Experimental ork on vertical base and idth-ise horizontal ins as studied by Bilitzky [1]. tarner and McManus [16] did an experimental investigation o ree convection heat transer rom rectangular in arrays. Average heat transer coeicients ere presented or our in arrays positioned ith the base vertical 4 degrees and horizontal hile dissipating heat to surrounding luid. parro and Vemuri [] did an experimental study on Natural Convection/ Radiation Heat ranser rom highly populated pin in Arrays. Experiments ere perormed to determine the combined mode o natural convection/ radiation heat transer characteristics o highly populated arrays o rod like cylindrical ins i.e. pin ins. hey ound that i the number o ins ere increased or ixed values o the other parameters the heat transer increased at irst attained a maximum and then decreased. A survey o literature indicates that very e experimental and theoretical studies are available on inclined in arrays. Hence an experimental and theoretical investigation o the natural convection heat transer rom to geometric orientations viz. (a) vertical base ith vertical in arrays and (b) horizontal base ith vertical in array ith the ive dierent inclinations are presented in this paper.. PHYICAL MODEL AND FORMULAION A horizontal in array contains ins arranged on a horizontal plate hich is called the base o the in array. he considered in arrays are consisting o seventeen eight and our ins. In our in array irst and ourth ins are the end ins and the remaining are inner ins. Hoever the end ins are exposed to an ininite ambient medium on one side. he to adjacent inner ins having a common base ith the in spacing hich is shon in Figure-1. he base is maintained at a constant temperature and > here is ambient air temperature. he length and idth o each in are L and W respectively and the ratio (W/L) is by ar greater than unity. t is the hal thickness o the in. Heat is transerred rom the base to the ins by conduction and rom the ins to the ambient air by convection. 7
ρ C p t = k - P k A y = (6) P = t F W and A = t F W (7) he u- and v- momentum balance equations i.e. Eqs. () and (3) are coupled making use o vorticity ζ hich is deined as ollos. u v ζ = (8) he terms in Eq. () are dierentiated ith respect to y and those in Eq. (3) are dierentiated ith respect to x. he resulting equations are subtracted one rom the other to yield the olloing vorticity equation. Figure-1. Physical model o to in enclosure. he problem is ormulated considering the to vertical ins and the horizontal base together as a todimensional enclosure [4]. he to ins and the base are the let right and bottom boundaries o the enclosure respectively. he top hich is open is considered to be the ourth boundary. he velocity and temperature ields in the to-in enclosure are governed by the mass momentum and energy balance equations or the luid in conjugation ith the one-dimensional heat conduction equation or each in hich are given belo. hus the equations considered are as ollos. Fluid medium: u v = (1) ρ ρ u u u p u v = - - t 1 β COφ [ ( )] g u u µ () v v v ρ p u v = - t v v µ - ρ [ 1 β( )] g inφ (3) ρ C p u v t (4) Heat conduction in let in: ρ C p t = k Heat conduction in right in: P k A y= () ζ ζ ζ ρ u v = t ρ gβ Cosφ inφ ζ ζ µ he stream unction ψ is deined as ψ ψ u = v = - () y he vorticity equation in terms o stream unction is as ollos: ψ ψ ζ = (11) he stream unctions ψ can be evaluated using Eq. (11) i the vorticities ζ are knon. he velocity components u and v are to be computed rom the values o ψ through equation (). 3. BOUNDARY CONDIION he boundary conditions at the let right bottom and top boundaries o the enclosure are given belo. Let boundary (y = ): = = constant at x = (in base); u = v = ψ = or x L = at x = L (in tip) ; ζ = ψ y= y or x L (let in) (1) y = ( y) Right boundary (y = ): = = constant at x = (in base) = at x =L (in tip); u = v = ψ = or x L (9) 8
ζ = = y ψ y y = or x L (right in) (13) ( y) Base or bottom boundary (x = ): u = v = ψ = and = =constant or y ζ = ψ x = x or y (14) x = ( x) op boundary at x = L: he top boundary is open or no boundary. Roache [11] described it as an "open light case" and suggested the olloing boundary conditions. u v = = x ζ = ψ = and x = ; x at x = L or y (top) () All the equations are normalized by making use o the olloing variables. x = x u u = y = 1 1/ ν ψ 1 ψ = ν 1/4 = y 1/4 v v = 1 1/4 ν ν t 1/ t = ζ 1 3/4 ν ζ = ri q = k q ri ( ) = α α = α 1 1/ 4 he dimensionless system parameters are namely M the conduction-convection ratio parameter A R aspect ratio and γ temperature ratio parameter. k P 1/4 γ = k A M = A R = L he energy balance and vorticity equations or the luid heat conduction equations or the let and right ins and the boundary conditions or the inner as ell as end ins are ritten in normalized orm but are not shon here to conserve space. he method o solution and accuracy and convergence criteria are explained in Dharma Rao et al. [4] 1 q 3 = ; Q 3 = ( W) q 1/4 3 (17) x x = q = d x y ; Q = (L W) q (18) y = q1 and q 3 are heat luxes rom the in and the base respectively. Q1 and Q 3 are the heat lo rates rom the in and the base taking into consideration the respective surace areas. Q reers to the heat lo rate rom the outer ace o the end in (either irst or the last in the array) o hal-thickness tf. It may be noted that Q1 reers to the heat lo rate rom one ace o the in only. Further the heat transer rates rom the let and right ins are assumed to be equal. he heat transer rate rom the in array (Q ) is given by Q = (N-1) ( Q 1 Q 3 ) Q (19) 4.1 AVERAGE NUEL FROM HE FIN ARRAY otal heat transer area or the N-in array is given by A = (B - N t F ) W N L W B = (N - 1) N t F () Where B is the breadth o the plate and N is the number o ins. he average heat transer coeicient h m or the N-in array or combined convection is deined by the olloing equation h m = Q ( ) (1) A he average Nusselt number (Nu m ) or the N-in array is deined as Nu m = h m / k (). EXPERIMEN PROCEDURE A photographic vie o the experimental setup is shon in Figure-. 4. HEA RANFER RAE FROM A FIN ARRAY he heat luxes in dimensionless notation rom the let in and base are given by the olloing equations. q = 1 L y y = d x ; Q1 = (L W) q 1 (16) Figure-. Photographic vie o an experimental setup. 9
An experimental investigation o the natural convection heat transer rom in arrays is carried out or to dierent orientations a) Horizontal base ith vertical in arrays and b) Vertical base ith vertical in arrays ith the ive dierent inclinations like 3 4 6 and 9. emperatures are measured at dierent positions on the inner in o the array during the experiments. he heat transer coeicients or the in array are calculated by making use o the measured temperatures by using the calculation procedure o ujatha and obhan [17]. Experimental heat transer rates are obtained or dierent number o ins present in the in array. Four dierent in arrays are abricated containing 4 8 and 17 ins respectively. o o the in arrays are having mm long ins and another to in arrays are having 4 mm long ins ith the breadth o 3. mm base plate and the idth is mm or all the our in arrays. he range o parameters is listed in able-1. 6. REUL AND DICUION he numerical results are obtained or vertical in array o having 14 ins ith the parameters L=.4mm W =4mm =7.9mm t F =1.16mm o 4 inclination by varying base temperature rom 9 to C. he obtained numerical results are compared ith the experimental data o tarner and McManus [16] in Figure-3. able-1. Range o parameters. Fin length (L) mm and 4 Fin thickness (t F ) mm 3 Fin array length (B) mm 3. Fin spacing () mm 47 19 and 7 Number o ins (N) 48 and 17 Fin idth (W) mm Ambient temperature 8 C Base temperature ranges( W ) 4-13 Inclinations 3 4 6 9 Average heat transer coeicient o ain array hw/m C 4. 3.8 3.6 3.4 3. 3..8.6 ARNER AND McMANU Experimental data heoretical data Ambient temperature - 8 C 4 inclination.4 8 1 14 16 18 Base temperature o the in array W Figure-3. Comparison o present theoretical data ith the experimental data available in literature (tarner and Mcmanus 1963). he numerical results sho good agreement ith the experimental data o tarner and McManus [16]. Numerical results are obtained or horizontal in array or mm and 4mm lengths in spacing 7.719 and 47mm having B = 3. mm and W = mm. Nusselt numbers are calculated ith respect to Rayleigh number and the variation o average Nusselt number ith Rayleigh number shon in Figure-4.
Average Nusselt number or an- in arraynu m 1 3 inclination 4 inclination 6 inclination -1 L=mm and 4 mmb=3.mm W=mm=7.719 and 47 mm 3 4 6 Rayleigh numberra Figure-4. Variation o average Nusselt number ith Rayleigh number. he experimental results are obtained or a in array o horizontal base ith vertical ins ith the parameters N=17 W=mm L=mm =7 mm B=3. mm and N=8 W=mm L=4mm =19 mm B= 3. mm. he convection heat transer rates are calculated rom the experimental data and shon the eect o inclination in Figures and 6 respectively. Convection heat transer rate rom a in arrayqc W 4 3 o inclination 3 o inclination 4 o inclination 6 o inclination N=17 =7mm B=3.mm L=mm W=mm. Ambient temperature - 8 C 4 6 8 1 14 Base temperature o the in array W Figure-. Eect o inclination on convection heat transer rate rom a in array. It is observed rom Figures and 6 that the convection heat transer rates are increasing rom to3 decreasing rom 3 to 4 and again increasing rom 4 to 6 and 6 to 9 inclination or orientation o horizontal base ith vertical ins. 11
Convection heat transer rate rom a in array Q C W 3 inclination 3 inclination 4 inclination 6 inclination N=8 =19mm B=3.mm L=4mm W=mm. 4 6 7 8 9 1 Base temperature o the in array W Figure-6. Eect o inclination on convection heat transer rate rom a in array. he experimental results are also obtained or in array o vertical base and vertical ins ith the parameters N=17 W=mm L=mm =7 mm B= 3. mm and N=8 W=mm L=mm =19 mm B= 3. mm. he convection heat transer rates are calculated rom the experimental data and shon the eect o inclination in Figure-7 and in Figure-8 respectively. Convection heat transer rate rom a in array Q C W 4 3 3 inclination 4 inclination 6 inclination VERICAL BAE N=17=7 mm L= mm B=3. mmw=mm Ambient temperature - 8 C 4 6 7 8 9 1 Base temperature o the in array W Figure-7. Eect o inclination on convection heat transer rate rom a in array. 1
Convection heat transer rate rom ain array Q C W 3 3 inclination 4 inclination 6 inclination VERICAL BAE N=8=19mm L=mm W=mm 4 6 7 8 9 Base temperature o the in array W Figure-8. Eect o inclination on convection heat transer rate rom a in array. It is observed rom Figures 7 and 8 that the convection heat transer rates are increasing rom to 3 decreasing rom 3 to 4 and again increasing rom 4 to 6 and 6 to 9 inclination or orientation o vertical base ith vertical ins. Experimental results are compared ith our numerical results or the in array having 17 ins shon in Figure-9. Convection heat transer rates rom a in array Q C 3 heoritical Experimental 3 inclination Ambient temperature - 8 C 4 6 8 1 14 Base temperature rom the in array W Convection heat transer rate rom a in array Q C W 4 3 heoritical Experimental 6 inclinatoin Ambient temperature - 8 C 4 6 7 8 9 1 1 Basse temperature rom the in array W Figure-9. Comparison o experimental data ith theoretical results or L= mm =7 mm N =17 B =3. mm. It shos good agreement beteen experimental and numerical results. he heat transer rates are calculated rom the experimental data or to dierent orientation o in array is compared in Figure- or the inclinations o 3 and 6. 13
Convection heat transer rate rom ain array Q C W 4 3 3 inclination 6 inclination 3 inclination 6 inclination horizontal base vertical base Ambient temperature - 8 C 4 6 8 1 14 Base temperature o the in array W Figure-. Comparison o experimental data or to dierent positions o the in array. It shos that the convection heat transer rates are less or the position o horizontal base and vertical ins ith length basis than horizontal base and vertical ins ith idth basis or the same inclination o the in array. 7. CONCLUION A theoretical model is ormulated to tackle the problem o heat transer rom a in array ith inclination. According to the model the in array is assumed to be ormed by joining successive to in enclosures. he problem or the case o a to in enclosure is theoretically ormulated and solved considering heat transer by natural convection. heoretical studies have been carried out on natural convection heat transer or Horizontal base ith vertical ins ith ive dierent inclinations like 3 4 6 and 9 Experimental data are obtained or natural convection heat transer rom a in array by placing it in to dierent orientations a) Horizontal base ith vertical ins and b) Vertical base ith vertical ins ith the ive dierent inclinations like 3 4 6. A comparison ith the experimental data existing in literature indicates satisactory agreement. REFERENCE [1] Bilitzky A. 1986. he eect o geometry on heat transer by ree convection rom in arrays. M hesis. Dept. o Mechanical Engineering Ben-Gurion University o the Negev Beer heva Israel. [] Charles D. Jones. and Lester. F. mith. 197. Optimum arrangement o rectangular ins on horizontal suraces or ree convection heat transer. AME Journal o heat transer. 9: 6-. [3] Dayan A. Kushnir R. Mittelman G. and Ullmann A. 4. Laminar ree convection underneath a donard acing hot in array. Int. J. Heat and Mass ranser. 47: 849-86. [4] Dharma Rao V. Naidu.V. Govinda Rao B. and harma K.V. 6. Heat transer rom horizontal in array by natural convection and radiation-a conjugate analysis. Int. J. Heat and Mass ranser. 49: 3379-3391. [] Elenbaas W. 194. Heat dissipation o parallel plates by ree convection. Physica. 9: 1-8 [6] Guvenc A. and Yuncu H. 1. An experimental investigation on perormance o ins on a vertical base in ree convection heat transer. prigler-verlag Heidelberg Heat and Mass ranser. 37: 49-416. [7] Incropera F.P and Deitt D.P. 1996. Fundamental o Heat and Mass ranser. 4 th Ed. John Wiley and ons. [8] Leung C.W and Probert.D. 1997. Heat-exchanger perormance: inluence o gap idth beteen consecutive vertical rectangular in arrays. Applied Energy. 6: 1-8. [9] Nancy D. Fitzroy. 1971. Optimum spacing o ins cooled by ree convection. ransactions o the AME Journal o Heat ranser. pp. 46-466. [] Rammohan Rao and Venkateshan. P. 1996. Experimental study o ree convection and radiation in horizontal in arrays. Int. J. Heat Mass ranser. 39: 779-789. 14
[11] Roache P.J. 198. Computational Fluid Dynamics. Hermosa Albuquerque N.M. [1] aikhedkar N.H. and ukhatme. P. 1981. Heat transer rom rectangular cross-sectioned vertical in arrays. Proceedings o the sixth National Heat and Mass ranser Conerence. HM: 9-81. [13] obhan C.B. Venkateshan.P and eetharamu K.N. 1989. Experimental analysis o unsteady ree convection heat transer rom horizontal in arrays. Warme-und toubertragung pringer-verlag. 4: -16. [14] obhan C.B. Venkateshan.P and eetharamu K.N. 199. Experimental studies on steady ree convection heat transer rom ins and in arrays. Warme-und toubertragung pringer-verlag. : 34-. [] parro E.M and Vemuri.B. 198. Natural convection - radiation heat transer rom highly populated pin-in arrays. AME J. Heat ranser. 7: 19-197. [16] tarner K.E. and McManus Jr. 1963. An experimental investigation o ree convection heat transer rom rectangular in arrays. J. Heat ranser. 8: 73-78. [17] ujatha K.. and obhan C.B. 1996. Experimental studies on natural convection heat transer rom in arrays. J. o Energy Heat and Mass ranser. 18: 1-7. [18] Yuncu H. and Anbar G. 1998. An experimental investigation on perormance o ins on a horizontal base in ree convection heat transer.prigler-verlag Heidelberg Heat and Mass ranser. 33: 7-14. [19] Yalcin H.G. Baskaya and ivrioglu M. 8. Numerical analysis o natural convection heat transer rom rectangular shrouded in arrays on a horizontal surace. Int. J. Heat Mass ranser. : 99-311. Nomenclature A B area o the horizontal base plate (B W) A R aspect ratio or to-in enclosure (L/) A urace area o the in (LW) m A hal the cross-sectional area o the in (t F W) m B breadth o the horizontal base plate m C p speciic heat J kg -1 K -1 g acceleration due to gravity m s - 3 asho number ( ) gβ /ν h heat transer coeicient W m - K -1 k thermal conductivity W m -1 K -1 L length o the in m N number o ins in the in array Nu Nusselt number p pressure Pascals P hal-perimeter o the in ( t F W) m Pr Prandtl number c p µ / k q heat lux W m - q normalized heat lux q / [k ( ) 1/4 Q heat transer rate W Ra Rayleigh number g β ( - ) 3 / (ν ) spacing beteen adjacent ins m t F hal-thickness o the in m t time s temperature K u velocity component in x-direction m s -1 v velocity component in y-direction m s -1 W idth o the in (also that o base plate) x position coordinate along the in measured rom the base o the in m y position coordinate normal to the in measured rom the let in eek symbols φ Inclination o the in array thermal diusivity o luid (k/ρc) m /s α / α β isobaric coeicient or thermal expansion coeicient o luid K -1. - γ temperature ratio parameter ( - )/ µ dynamic viscosity kg m -1 s -1 ν kinematic viscosity m s -1 ρ density kg m -3. ψ stream unction. σ tean-boltzman constant (.67 x -8 W m - K -4 ) θ - θ / ζ vorticity unction s -1 ubscripts 1 in in a to-in enclosure 3 base in a to-in enclosure end ins B base o the in array B base plate in the absence o ins E end ins o the in array I internal ins o the in array c convection luid. L in tip m average M middle o the in total in surace base o the in ambient medium