Study on the Heat Transfer Performance of the Annular Fin under Condensing Conditions Abdenour Bourabaa, Malika Fekih, Mohamed Saighi

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1 Study n the Heat Tranfer Perfrmance f the Annular Fin under Cndening Cnditin Adenur Buraaa, Malika Fekih, Mhamed Saighi Atract A numerical invetigatin f the fin efficiency and temperature ditriutin f an annular fin under dehumidificatin ha een preented in thi paper. The nn-hmgeneu ecnd rder differential equatin that decrie the temperature ditriutin frm the fin ae t the fin tip ha een lved uing the central finite difference methd. The effect f variatin in parameter including relative humidity, air temperature, air face velcity n temperature ditriutin and fin efficiency are invetigated and cmpared with the under fully dry fin cnditin. Al, the effect f fin pitch n the dimeninle temperature ha een tudied. Keywrd Annular fin, Dehumidificatin, Fin efficiency, Heat and ma tranfer, Wet fin. I. INTRODUCTION XTENDED urface r fin are frequently deigned t add Eadditinal urface area in rder t enhance heat tranfer etween the urface and the amient air. They can e ued in variety applicatin uch a in air cnditining ytem, cndener and evapratr. In uch applicatin a evapratr, the urface temperature i generally elw the dew pint temperature f the incming air. A a reult, cmined heat and ma tranfer ccur ver the cled urface. In general, the fin perfrmance i uually expreed in term f heat tranfer cefficient and fin efficiency. In the cae when dehumidificatin f mit air ccur, the temperature ver the fin urface change imultaneuly with the humidity rati. Thi make the wet fin perfrmance tudy mre cmplicated. In the ther wrd, the tudy f the perfrmance f the wet fin i cnideraly different frm the tudy f the ame fin under dry cnditin. Under dry fin cnditin, Brwn [] derived an equatin t determine the ptimum dimenin f unifrm annular fin. Hi ecnd rder differential equatin wa lved y the ue f the Beel functin. Thi apprach ha een later ued y Ullmann and Kalman [] in which the ecnd rder differential equatin ha een lved t hw the efficiency and ptimized fin dimenin. Under dehumidificatin, the reult frm Cney et al [3] hw that cndenatin ha a utantial influence n the fin perfrmance. The imprtance f th heat and ma tranfer n the fin perfrmance f different gemetrie are preented in Adenur.Buraaa wa with Thermdynamic and Energetically Sytem Laratry; Faculty f Phyic, USTHB, Ba Ezzuar, Al Alia 6, Algier, Algeria. He i nw with the Centre de Dévelppement de Energie Renuvelale,634, Algier, Algeria (phne: ; fax: ; adenururaaa@yah.fr). Malika. Fekih and Mhamed. Saighi are with Thermdynamic and Energetically Sytem Laratry; Faculty f Phyic, Univerity f Science and Technlgy Huari Bumediene, Ba Ezzuar, Al Alia 6, Algier, Algeria ( fekih.mali5@yah.fr, aighimhamed@yah.fr). a large numer f paper [4]-[3]. Kazeminejad et al. [4] carried ut a numerical tudy t analyze the cnjugate cnvectin and cnductin heat tranfer characteritic f humid air flw ver a vertical rectangular fin. The effect f variu pychrmetric and flw cnditin n the fin efficiency ha een dicued. Theretical analyi f the rectangular fin perfrmance with cmined heat and ma tranfer ha een carried ut y Sunden [4]. Under dehumidificatin, a relatin etween the lcal change in temperature with humidity rati i neceary in rder t determine fin efficiency. The temperature prfile alng the fin urface i gverned y nn-hmgeneu ecnd rder differential equatin. Several tudie have een perfrmed t lve thi equatin. Fr intance, Cney et al. [3], tudied numerically the fin temperature ditriutin, the cndenate film thickne and the fin effectivene fr a vertical fin in a laminar humid air cr flw. The value f the humidity rati i related t temperature y a ecnd-degree plynmial relatinhip. Chen [5] prped a mdel t analyze the fin perfrmance with imultaneu heat and ma tranfer. He ued the ame humidity rati/temperature relatinhip a Cney et al. [3]. Elmahdy and Bigg [6] preented an algrithm t determine the efficiency f lngitudinal and circular fin with a unifrm thickne. They prped a linear relatinhip etween the pecific humidity f the aturated air n the fin urface and it crrepnding temperature. Their reult hwed a decreae in fin efficiency and al temperature ditriutin with the increae f relative humidity. The temperature ditriutin ver a circular partially wet r ttally wet fin ha een invetigated y Liang et al. [7]. They ued a third-degree plynmial crrelatin fr the relatinhip etween the pecific humidity and the fin urface temperature. Sharqawy and Zuair [8] tudied analytically a wet annular fin in which they prped anther linear relatinhip etween the humidity rati and it crrepnding temperature. They ugget that the maximum temperature at the fin tip fr wet urface cnditin i the dew pint temperature f the air tream. Naphn [9] carried ut a numerical analyi t invetigate the perfrmance f an annular fin under ttally dry, ttally wet and partially wet fin cnditin. He ued a third degree plynmial develped y Liang et al. [7] t expre the relatinhip etween the aturated pecific humidity n the fin urface and it crrepnding temperature. Liang et al. [] ued al the crrelatin prped y Liang et al. [7] t preent their nedimeninal and tw-dimeninal fin efficiency f a plate finand-tue heat exchanger under cndening cnditin. The nn-hmgeneu ecnd rder differential equatin that decrie the temperature ditriutin frm the fin tip t the 53

2 fin ae ha een lved y Naphn [9] uing the central finite difference cheme. Kazeminejad [] tudied a rectangular fin aemly under dehumidificatin. When nly enile heat tranfer ccur Kazeminejad develped a lutin t the differential equatin that decrie the temperature ditriutin. Hwever, when dehumidificatin f mit air ccur, he lved thee equatin numerically y the hting methd which cmine the Runge-Kutta methd and the Newtn-Raphn iteratin. Salah El-Din [] tudied analytically a rectangular fin aemly under fully wet and partially wet fin urface cnditin. He aumed al that the aturated humidity rati varie linearly with the lcal fin urface temperature. Rari and Rahman [3] tudied the wet radial fin aemly. The effect f the cld fluid and air temperature, relative humidity, fin thermal cnductivity and al the effect f the fin aemly dimenin n the temperature ditriutin ha een tudied. Hi mdel can e ued in predictin f partially wet fin aemly. In the preent wrk th the temperature ditriutin and the fin efficiency ver a circular fin urface are invetigated. Fllwing the ame prcedure preented y Buraaa et al. [4] fr rectangular wet fin, the ecnd rder differential frmulatin that decrie the temperature ditriutin ver circular fin ha een implified y aviding the ue f Beel functin. II. MATHEMATICAL PROCEDURE A. Temperature Ditriutin-Analytical Methd When the urface i dry the driving ptential fr heat tranfer i imply the urface t air temperature difference. Hwever, when the urface i wet the heat rate tranferred t the urface i gverned y the dual driving ptential f temperature and water vapr cncentratin. It i hwn frm Fig. that a heat alance n an element f a circular fin urface lead t the fllwing exprein: h c i dq h da T + w w m pa fg a a hc cpa, m i the heat tranfer urface, J kg - K - i the pecific heat f the incming air tream, W m - K - and kg m - - are the enile heat and ma tranfer cefficient; repectively, C and C are the ulk air and the urface temperature; repectively, and kg w /kg a and kg w /kg a are the humidity rati f the incming air and the humidity rati f aturated air at aturated temperature; repectively. The dimeninle gruping in the previu equatin i a cnvective Lewi numer, thu () Fig. Schematic Diagram f Circular fin Equatin (), therefre, can e rewritten a i fg wa w dq h da + T T cpale Ta T a The effect f the preence f miture n the fin urface can e decried y the ecnd term n the right ide f (3). The heat tranfer equatin cannt e expreed in term f the air t urface temperature difference alne. A humidity rati difference al appear in thi equatin. Therefre, the humidity rati f aturated air and it crrepnding temperature at any lcatin n the fin urface are required. T d thi, it i neceary t lve the fllwing ecnd rder differential equatin that derived frm a imple energy alance n a differential element f a circular fin f unifrm thickne: d T dt h i fg + + Ta + wa w dr r dr kt cpale The humidity rati f aturated air at fin temperature can e calculated frm the crrelatin y Sharqawy and Zuair [8] (3) (4) w a + T (5) wdp w a w T dp w dp Tdp w Here, and are the ae and the dew pint temperature; repectively, and, are the humidity rati f aturated air evaluated at the ae temperature and at the dew pint temperature, repectively. The ne-dimeninal temperature ditriutin decried y (4) can e tranfrmed t the fllwing equatin: (6) (7) Le h () hmcpa d d + m m BC (8) dr rdr T T a a (9) 53

3 i B () c Le m fg pa h kt () ( + ) () m m B wa a T C T a a (3) In the ave equatin, W m - K - and m are the fin material cnductivity and the fin thickne, repectively. Auming that the heat tranfer thrugh the fin tip i negligile, the fllwing undary cnditin are ued t lve the nn-hmgeneu ecnd rder differential equatin. at thefin ae r r ; (4) d at the Fin tip r re ; (5) dr Suject t the undary cnditin f (4) and (5), the aciated analytical lutin fr circular fin give u the temperature ditriutin alng the fin urface, and are the mdified Beel functin f firt and ecnd kind. + I mr K mr + I mr K mr p e e + p I mr K mre + I mre K mr (6) BC p (7) + B B. Fin Efficiency By definitin, the fin efficiency i the rati f the actual heat tranferred t fin urface t the maximum pile heat tranfer rate if the entire fin urface wa at the ae temperature. The heat tranfer rate tranferred t the entire fin urface i calculated frm Furier law f heat cnductin: dt d qfin ka π tkr ( Ta ) (8) dr dr r r r r Uing (6), the ave equatin ecme: qfin π tkmr Ta + p I mr K mr I mr K mr I mr K mr I mr K mr ( ) ( e) ( e) ( ) + e e (9) Equatin () i, nw, ued t calculate the maximum pile heat tranfer rate in term f radiu and integrating ver the entire fin urface: re max 4π ( a ) + ( a ) () r q h T T B w w rdr Cmining (9), (), (), (), (3), (7) and (), the fllwing equatin that decrie the maximum pile heat tranfer rate i tained: max π p a e q kt m + T T r r () Thu, the exprein f the fin efficiency can e written a: q q r fin max q max m r r ( e ) ( e) ( ) ( ) ( e) + I mr K mr I mr K mr I mr K mr I mr K mr e e () C. Temperature Ditriutin-Numerical Methd In thi ectin we perfrmed a numerical methd in rder t implify the lutin t the nn-hmgeneu ecnd rder differential equatin that decrie the temperature ditriutin alng the wet circular fin urface y aviding the ue f mdified Beel functin. Amng all the apprximatin, the finite difference cheme i the implet ued ne [4]. It i prped t ue a dicrete patial increment that can e cmputed y r r N e Δ r (3) Here, N i the numer f the ndal pint the dimeninle temperature i t e determined, ee Fig.. Fig. Finite difference cheme fr -D circular fin The ecnd rder differential equatin, (8), can e dicretized uing the central difference fr ecnd derivative a fllw: + + m i i i+ i+ i Δr ri Δr i mbc r Δr r ( ) Δ + Δ + + ri Δ i m r i i+ mbc r ri i ( ) (4) (5) r r + i Δ r (6) In thi apprximatin the temperature ditriutin at firt 53

4 nde and at the end nde mut e knwn a pecial cae. Fr i we have and (5) can e ued directly, thu: Δr Δr + m Δ r + + m BC Δr r r Fr the end nd, at i (7) N, we can generate a ackward d / dr ", a apprximatin and, t d thi, let write fllw: ' ' " " d ' N N N N dr N Δr The firt derivative at pint N can e written a: Δr ' N N N (8) (9) Equatin (9) and (8) can e rearranged and uject t undary cnditin (5), the fllwing equatin can e derived: Δr Δr " N N Tgether, (8), (5) and (3) give u fr i N m r N mbc r N : (3) + Δ Δ (3) Equatin (5), (7) and (3) cntitute a ytem f N equatin, in which the temperature ditriutin alng a wet circular fin can e eaily fund y lving thi ytem. III. RESULTS AND DISCUSSION The dimeninle fin urface temperature i pltted againt the nn-dimeninal fin radiu meaured frm the fin ae. The preent finite difference methd fr temperature ditriutin ver wet circular fin ha een cmpared with the cmmnly ued Beel equatin. The reult fr 5% and % relative humidity (RH) value are hwn in Fig. 3. The variatin f the fin efficiency with relative humidity i hwn in Fig. 4 fr tw value f frntal velcity (u fr and 3 m/). A can e een, there are tw main regin. In the dryurface fin n cndenatin f miture i ccurring. Thu, n variatin f the fin efficiency with the relative humidity i een. The reanale explanatin f the effect f the relative humidity n the fin efficiency i a fllw. The fin efficiency i defined a the rati etween the actual fin heat tranfer rate t maximum pile heat tranfer rate if the entire fin i at the ae temperature. In the dry fin regin, hwever, the actual fin heat tranfer and the maximum pile heat tranfer increae y the ame amunt. In the weturface fin, the fin efficiency deceae rapidly with the increae in relative humidity. (Ta-T)/(Ta-T). RH% RH5% Tai8 C T5 C D.7mm Fp.3mm tmm ufr3m/. equuatin (6) preent wrk. R(r-r)/(re-r) Fig. 3 Fin urface temperature cmparin Thi ituatin crrepnd t the partially-wet fin in which the actual heat tranfer rate thrugh the fin urface and the maximum pile heat tranfer de nt increae y the ame amunt. On the ther wrd, the maximum pile heat tranfer increae mre rapidly than the actual fin heat tranfer. Hwever, the variatin f the fin efficiency with the relative humidity diminihe a relative humidity increae. By increaing relative humidity the wet prtin n the fin urface increae until the fin ecme in fully wet ituatin. There, the rati f the actual heat tranfer rate t the maximum pile heat tranfer rate i cmparatively cntant fr fully wet fin urface. η. Dry fin Tai 8 C T 5 C D.7mm Fp.3mm Wet fin ufrm/ ufr3m/ RH (%) Fig. 4 Effect f frntal velcity n fin efficiency The effect f frntal velcity n the fin efficiency i al illutrated in Fig. 4. A can e een, fr a cntant relative humidity, the fin efficiency fr u fr 3m/ i lwer. When air velcity i increaed the amunt f cndenate flwing n the fin urface increaed. Therefre, the fin efficiency i affected 533

5 y the preence f miture n the fin urface. On the ther hand, the preence f miture reduce cnideraly the fin efficiency. A a reult, lwer fin efficiency can e een fr higher air velcity. The influence f relative humidity n the fin urface temperature i illutrated in Fig. 5. The value f fin temperature gradient are pltted againt the dimeninle radiu fr three value f relative humidity (RH6%, 8% and %) and cmpared with a dry fin. A can e een, the curve under dehumidificatin lie elw that f dry fin. The wet prtin n the fin urface increaed with the increae f relative humidity. A reprted y Elmahdy and Bigg [6] and Kazeminejad [], when the inlet relative humidity increae the departure f the temperature prfile frm the dry urface curve ecme greater. Thu, the rie in relative humidity invlve water vapr cndening n the fin urface. Thi reult in higher amunt f ma tranfer and larger amunt f latent energy and, y cnequence, higher fin urface temperature. Fr a given et f cnditin, relative humidity, ae temperature, fin pitch, utide diameter and fin thickne, Fig. 6 depict the effect f dry ul temperature n the dimeninle fin urface temperature a a functin f the dimeninle radiu. Lwer dimeninle temperature curve i een fr higher dry air temperature. Fr a cntant relative humidity, an increae in the urrunding temperature increae the heat tranfer whatever the ituatin dry/r wet f the fin urface. It i ntewrthy that the dew pint temperature increae a the dry ul temperature increae under the ame relative humidity. Hence, higher dry air temperature can caue a larger ma f miture, which cnequently lead t a higher amunt f latent energy... Tai 8 C T 5 C ufr 3m/ Fp.3mm t mm dry fin RH 6% RH 8% RH %.. R (r-r)/(re-r) Fig. 5 Effect f relative humidity n fin urface temperature Fig. 7 preent the fin efficiency a a functin f relative humidity with tw value f dry air temperature, 4 and 8 C. The fin efficiency decreae with the increae f dry air temperature in partially wet and fully wet regin. The amunt f dehumidificatin increae with increaing dry temperature. It i imprtant t nte that the wet prtin n the fin urface increae with dry ul temperature. η.. T 5 C RH 7% Fp.3mm D.7mm tmm Ta 4 C Ta 8 C. R Fig. 6 Effect f air temperature n fin urface temperature. T 5 C ufr 3m/ Fp.3mm D.7mm Ta 4 C Ta 8 C RH (%) Fig. 7 Effect f air temperature n fin efficiency Fig. 8 hw the effect f fin pitch n the annular fin urface temperature fr m/ and 4 m/ air frntal velcitie and 6% relative humidity. The dimeninle temperature decreae a the fin pitch decreae. When the fin pitch i mall enugh the cndenate water i eaier t adhere etween the fin. The preence f the cndenate water n the fin urface enhance the latent and enile heat tranfer rate and hence lwer fin urface temperature curve are fr lwer fin pitch value. Thi effect tend t diminih a air face velcity decreae. Thi i in cnnectin with the cndenate lw ff phenmenn. When air velcity and fin pitch are increaed jut a maller amunt f cndenate flw alngide the fin a a larger amunt f cndenate may upend etween fin. 534

6 .. ufr4m/ RH 6% Tai8 C T5 C D.7mm tmm ufrm/ Fp.3mm Fp.3mm Fp 3.mm. R (r-r)/(re-r) Fig. 8 Effect f fin pitch n the temperature ditriutin IV. CONCLUSION In the freging analyi, the nn-hmgeneu ecnd rder differential equatin, derived frm the heat alance n a circumferential cntrl vlume f an annular fin, ha een lved numerically. The reult preented here hw the effect f dehumidificatin n th fin urface temperature and fin efficiency. The reult here hw the imprtance f the latent energy prtin n the wet circular fin perfrmance when dehumidificatin f mit air ccur. REFERENCES [] A. Brwn, Optimum dimenin f unifrm annular fin, Internatinal Jurnal f Heat and Ma Tranfer, Vl. 8, pp , 965. [] A. Ullmann and H. Kalman, Efficiency and ptimized dimenin f different cr-ectin hape, Internatinal Jurnal f Heat and Ma Tranfer, Vl. 3, N. 6, pp. 5-, 989. [3] J. E. R. Cney, C. G.W. Sheppard, and E. A. M. El-Shafei, Fin perfrmance with cndenatin frm humid air: a numerical invetigatin, Internatinal Jurnal f Heat and Fluid Flw. vl., N, 3, pp. 4 3, 989. [4] H. Kazeminejad, M. A. Yaghui and F. Bahri, Cnjugate frced cnvectin-cnductin analyi f the perfrmance f cling and dehumidifying vertical rectangular fin, Internatinal Jurnal f Heat and Ma Tranfer, vl. 36, N. 4, pp , 993. [5] D. Beednjak and A. Pred, Efficiency f cled extended urface, Internatinal Jurnal f Refrigeratin, vl., N. 5, pp , 998. [6] C. N. Lin and J. Y. Jang, A tw dimeninal fin efficiency analyi f cmined heat and ma tranfer in elliptic fin, Internatinal Jurnal f Heat and Ma Tranfer, vl. 45, pp ,. [7] H. T. Chen and J. C. Chu, Invetigatin f natural-cnvectin heat tranfer cefficient n a vertical quare fin f finned-tue heat exchanger, Internatinal Jurnal f Heat and Ma Tranfer, vl. 49, pp , 6. [8] B. Kundu, D. Barman and S. Denath, An analytical apprach fr predicting fin perfrmance f triangular fin uject t imultaneu heat and ma tranfer, Internatinal Jurnal f Refrigeratin, vl. 3, pp. 3-, 8. [9] B. Kundu, Apprximate analytic lutin fr perfrmance f wet fin with a plynmial relatinhip etween humidity rati and temperature, Internatinal Jurnal f Thermal Science, vl.48, pp. 8-8, 9. [] M. H. Sharqawy and S. M. Zuair, Efficiency and ptimizatin f traight fin with cmined heat and ma tranfer An analytical lutin, Applied Thermal Engineering, vl. 8, pp , 8. [] H. T. Chen and J. C. Chu, Invetigatin f natural-cnvectin heat tranfer cefficient n a vertical quare fin f finned-tue heat exchanger, Internatinal Jurnal f Heat and Ma Tranfer, vl.49, pp , 6. [] H. T. Chen and W. L. Hu, Etimatin f heat-tranfer characteritic n a vertical annular circular fin f finned-tue heat exchanger in frced cnvectin, Internatinal Jurnal f Heat and Ma Tranfer, vl. 5, pp. 9-93, 8. [3] H. T. Chen and H. C. Wang, Etimatin f heat-tranfer characteritic n a fin under wet cnditin, Internatinal Jurnal f Heat and Ma Tranfer, vl. 5, pp. 3-38, 8. [4] B. Sunden, The effect f Prandtl numer n cnjugate heat tranfer frm rectangular fin, Int. Cmm. Heat Ma Tranfer, vl., pp. 5-3, 985. [5] L. T. Chen, Tw-dimeninal fin efficiency with cmined heat and ma tranfer etween water-wetted fin urface and mving mit airtream, Internatinal Jurnal f Heat and Fluid Flw. vl., N,, pp. 4 3, 99. [6] A. H. Elmahdy, and R. C. Bigg, Efficiency f extended urface with imultaneu heat and ma tranfer, ASHRAE Tran. vl. 89, Part A, pp , 983. [7] S. Y. Liang, M. Liu, T. N. Wang and G. K. Nathan, Analytical tudy f evapratr cil in humid envirnment, Applied Thermal Engineering. vl. 9, pp. 9 45, 999. [8] M. H. Sharqawy, and S. M. Zuair, Efficiency and ptimizatin f an annular fin with cmined heat and ma tranfer-an analytical lutin, Internatinal Jurnal f Refrigeratin. vl. 3, pp , 7. [9] P. Naphn, Study n the heat tranfer characteritic f the annular fin under dry-urface, partially wet-urface, and fully wet-urface cnditin, Internatinal Cmmunicatin in Heat and Ma Tranfer. vl. 33, pp., 6. [] S. Y. Liang, T. N. Wng and G. K. Nathan, Cmparin f nedimeninal and tw-dimeninal mdel fr wet-urface fin efficiency f a plate-fin-tue heat exchanger, Applied Thermal Engineering. vl., pp ,. [] H. Kazeminejad, Analyi f ne-dimeninal fin aemly heat tranfer with dehumidificatin, Internatinal Jurnal f Heat and Ma Tranfer. vl. 38, pp , 995. [] M. M. Salah El-Din, Perfrmance analyi f partially-wet fin aemly, Applied Thermal Engineering. vl. 8, N. 5, pp , 998. [3] L. Rari and M. M. Rahman, Analyi f heat tranfer in a partially wet radial fin aemly during dehumidificatin, Internatinal Jurnal f Heat and Fluid Flw. vl., pp , 999. [4] A. Buraaa, M. Saighi, M. Fekih and B. Belal, Study n the heat tranfer f the rectangular fin with dehumidificatin: Temperature ditriutin and fin efficiency, Internatinal Review f Mechanical Engineering (I.RE.M.E), Vl. 7, N. 5, pp ,