Determining thermal contact resistance of the fin-to-tube attachment in plate fin-and-tube heat exchanger

Transcription

1 2 nd Interntionl Conference on Engineering Optimiztion September 6-9, 2010, Lisbon, Portugl Determining therml contct resistnce of the fin-to-tube ttchment in plte fin-nd-tube het exchnger Did Tler 1, Artur Cebul 2 1 Acdemy of Science nd Technology, Crco, Polnd, tler@imir.gh.edu.pl 2 Crco University of Technology, Crco, Polnd, cebul@pk.edu.pl Abstrct Plte fin-nd-tube het exchngers re mde by inserting ovl tubes through sheet metl strips ith stmped holes nd then expnding the tubes slightly to cuse pressure t the tube-to-strip contcts. The gp beteen the fin bse nd the outer tube surfce my be filled ith ir or corrosion products cusing decline in the bility of plte fins to trnsfer het. The contct resistnce must be ccounted for in the design nd performnce clcultions of het exchngers. In this pper, the therml contct resistnce of the fin-to-tube ttchment is estimted from the condition tht the dimensionless correltions for the Colburn j-fctors obtined from to different methods re in good greement. The first method is bsed on the experimentl dt, hile the second one on the CFD simultion of the flo nd het trnsfer in the het exchnger. Keyords: het exchngers, fin-nd-tube het exchnger, CFD, het trnsfer 1. Introduction Externlly finned tubes, or plte-nd-tube elements re used in economizers of stem poer boilers, ir-conditioning het exchngers, convectors for home heting, induced-drft cooling toers, nd ste-het recovery systems for gs turbines. Plte-nd-tube extended surfces re lso used extensively in ir-fin coolers, in hich hot fluid flos ithin the tubes, nd tmospheric ir, serving s the cooling fluid, is circulted over the fins by fns. The plte fin-nd-tube het exchngers re mde by inserting the ovl tubes through sheet metl strips ith stmped holes nd then expnding the tubes slightly to cuse pressure t the tube-to-strip contcts. If the tubes re only expnded into the pltes to produce n interference fit some contct resistnce must be ccounted for. In the design of plte fin-nd-tube het exchngers, contct resistnce hs been included in ir-side resistnce s consequence of the dt reduction methods used. The therml contct resistnce beteen fins nd tubes hs not been studied deeply oing to the complexity of het trnsfer through rough metllic interfces. The contct resistnce in fin-nd-tube het exchnger hs been studied by Drt [1]. In his method, hot ter flos through one tube ro nd cold ter through n djcent tube ro. The estimted therml contct resistnce s compred to tht in soldered fins. The Drt method s refined by Sheffield et. l.[2]. Convective het dissiption on the fin surfce nd convective het trnsfer beteen hot nd cold tubes ere eliminted by testing plte finned het exchngers in vcuum chmber. The het is trnsferred rdilly from the hot ter tubes to the cold ter tubes only by conduction through the fins. To-dimensionl temperture distribution in the fin s determined using n electriclly conducting pper model of the fin. Similr techniques ere used by Jeong t l.[3,4]. Also, the investigted fin-nd-tube het exchnger s plced in n insulted vcuum chmber, thereby improving the ccurcy of the numericl procedure for determintion of the therml contct resistnce. On the other hnd, het trnsfer conditions on the fin surfce nd in the gp beteen fin nd tube in vcuum differ from those in ctul het exchngers. In this study, ne experimentl-numericl technique is developed for the estimtion of the therml contct resistnce of plte finned tubes. The ir side het trnsfer coefficient is determined from the condition tht the ir temperture differences cross the het exchnger obtined from the CFD simultion nd from the nlyticl model of the het exchnger re equl. Bsed on the clculted het trnsfer coefficients, the dimensionless correltions for the Colburn j-fctor s function of the Reynolds number nd the therml contct resistnce re found. The therml contct resistnce of the fin-to-tube ttchment is estimted from the condition tht the dimensionless correltions for the Colburn j-fctors obtined from the first nd second method re in good greement. 2. Therml design of fin-nd-tube het exchnger The bsic eqution for the totl rte of het trnsfer Q & in cross-flo tube het exchnger is Q& = F Ao Uo Tlm (1) here F is the correction fctor bsed on logrithmic men temperture difference T lm for counterflo rrngement nd A o is the totl externl surfce re of the tubes ithout fins (Fig. 1) on hich the overll het 1

2 trnsfer coefficient U o is bsed. The overll het trnsfer coefficient referred to the surfce re A o is given by 1 Ao 1 2Ao δt 1 Ao = Rc U A h A A k h A o in in in o t g here R c is the contct therml resistnce beteen the fins nd tube, hich is clculted from 2A o g Rc = A A k o g g The symbols δ t nd k t denote the thickness nd therml conductivity of the tube, respectively nd h in the het trnsfer coefficient on the tube inner surfce. The equivlent, enhnced ir-side het trnsfer coefficient h bsed on the outer surfce re A o of the plin tube is defined s: Afη f ( h ) A h = h (4) A g (2) (3) Figure 1: Nomenclture for the nlysis of therml contct resistnce beteen the tube nd fin; () integrl fin extruded from tube (muff-type ttchment), (b) L-footed ttchment here A f is the externl re of the fins, A is the externl surfce of the tube beteen fins nd η f is the fin efficiency [5-8]. Eqution (4) is vlid for vrious fin-to-tube ttchments: tension-ound, L-footed, nd integrl fin extruded from tube. The ir-side het trnsfer coefficient h is clculted from experimentl correltions for finned or plte-finned tube rrngements. This coefficient cn be lso clculted from correltions bsed on the CFD simultion of the flo nd het trnsfer in the het exchnger. The procedure for determining contct therml resistnce R c (Eq. (3)) is presented in the folloing. 3. Chrcteristics of the tested het exchnger The tested utomotive rditor is used for cooling the sprk ignition engine ith cubic cpcity of 1,580 cm 3. The cooling liquid rmed up by the engine is subsequently cooled don by ir in the rditor. The rditor consists of 38 tubes of n ovl cross-section, 20 of them locted in the upper pss nd 10 tubes per ro (Fig. 2). In the loer pss, there re 18 tubes ith 9 tubes per ro. The rditor is 520 mm ide, 359 mm high nd 34 mm thick. The outer dimeters of the ovl tubes re d min = 6.35 mm nd d mx = mm. The thickness of the tube ll is δ t = 0.4 mm. The totl number of plte fins equls 520. The dimensions of the single tube plte re s follos: length mm, height - 34 mm nd thickness - δ f = 0.08 mm. The plte fins nd the tubes re mde of luminum. The pth of the coolnt flo is U-shped. The to ros of tubes in the first (upper) pss re fed simultneously from one heder. The ter strems from the first nd second ro re mixed in the intermedite heder. Folloing tht, the ter is uniformly distributed beteen the tubes of the first nd second ro in the second (loer) pss (Fig. 3). The inlet, intermedite nd outlet heders re mde of plstic. The pitches of the tube rrngement re s follos: perpendiculr to the ir flo direction p 1 =18.5 mm nd longitudinl p 2 =17 mm. A smooth plte fin is divided into equivlent rectngulr fins. The efficiency of the fin s clculted by mens of the Finite Volume Method. The hydrulic dimeter of n ovl tube is clculted using the formul d t = 4 A in /P in here A in is the re of the tube cross section nd P in is the inside tube perimeter. 2

3 Figure 2: Digrm of to-ro cross-flo het exchnger (utomotive rditor) ith to psses; front vie () nd horizontl section of the upper pss (b), T ter temperture, T ir temperture Figure 3: Flo digrm of to ro cross-flo het exchnger (utomotive rditor) ith to psses; 1 first tube ro in upper pss, 2 second tube ro in upper pss, 3 first tube ro in loer pss, 4 second tube ro in loer pss 4. Prediction of correltions for the ir-side Colburn j-fctor A ne experimentl-numericl technique, bsed on to different methods for determining ir-side het trnsfer coefficient h, is developed for the estimtion of the therml contct resistnce beteen the tube nd fins. To methods for determining the correltions for Colburn j-fctor re presented. The first experimentl-numericl method requires the experimentl dt. The second is bsed on the CFD simultion of flo nd het trnsfer in the het exchnger Experimentl-numericl method The het trnsfer coefficient h in on the ter-side is knon, hile the het trnsfer coefficient on the ir-side h is to be found. The folloing prmeters re knon from the mesurements: liquid volumetric flo rte V &, ir velocity 0, inlet liquid temperture f, inlet ir temperture f, outlet liquid temperture f. The construction of the het exchnger nd the mterils of hich it is mde re lso knon. The verge het trnsfer coefficient h in on the inner surfce of the tube s clculted using the Gnielinski correltion [10] for turbulent flo ( ) 2 / 3 ξ 8 Re 1000 Pr d t Nu = 1 1/ 2 (5) 2 / ( ξ 8 ) ( Pr 1 ) L t here Nu = h in d t /k, Re = d t /ν c nd Pr = µ c p / k re the ter-side Nusselt, Reynolds nd Prndtl numbers, respectively, nd ξ is the friction fctor given by 1 1 ξ = = (6) 2 2 ( 1.82 log Re 1.64 ) ( 0.79 ln Re 1.64 ) The vlue of the ir-side het trnsfer coefficient h e,i is determined from the condition tht the clculted ter outlet temperture T,i (h e,i). must be equl to the mesured temperture f,i, here i=1,...,n denotes the dt set number. The folloing non-liner lgebric eqution hs to be solved for ech dt set to determine h e,i e f T h = i = 1,..., n (7) ( ), i, i, i 0, here n denotes the number of dt sets. In order to clculte the ter outlet temperture T,i s function of the het trnsfer coefficient h e,i, the nlyticl mthemticl model of the het exchnger developed in [11] s used. The het trnsfer coefficient on the ir-side h e,i s determined by serching the preset intervl so tht the mesured outlet temperture of the ter f,i nd the computed outlet temperture T,i re equl. The outlet ter temperture T,i (h e,i) is clculted t every serching step. Next, specific form s dopted for the correltion formul for the Colburn fctor j =j (Re ) on the ir-side, contining m unknon coefficients. The coefficients x 1, x 2,..., x m, m n re determined using the lest squres method from the condition 3

4 here ( ) 2 n e, i, i 1, 2,..., m min m n i= 1 S = j j x x x = j 1/ 3 /(Re Pr ) (8) = Nu (9) denotes the Colburn fctor nd Pr = µ c p / k is the Prndtl number. The dimensionless quntities Nu = h d h / k nd Re = mx d h /ν stnd for the ir-side Nusselt nd Reynolds numbers, respectively. The velocity mx is the ir velocity in the nrroest free flo cross-section A min. The symbol j e,i denotes the experimentlly determined vlue of the fctor, nd j,i the clculted fctor vlue hich results from the dopted pproximting function for the set vlue of the Reynolds number Re,i. The Colburn fctor j s pproximted by the poer-l function Re x 2 j = x1 (10) The unknon coefficients x 1 nd x 2 re determined by the Levenberg-Mrqurdt method (Seber nd Wild, 1989) using the Tble-Curve progrm [13]. Combining Equtions (9) nd (10), one obtins ( 1 x2 ) 1/ 3 Nu = x 1 Re Pr (11) All the ir properties tht pper in the dimensionless numbers re evluted t the verge temperture tken from the inlet temperture T m nd outlet ir temperture T m Determintion of the Colburn j-fctor on the ir-side bsed on the CFD simultion In order to determine the het trnsfer coefficients nd then Colburn j-fctors, the ir temperture differences must be clculted first using the CFD simultion of flo nd het trnsfer on the ir side Numericl simultion In order to determine the 3D flo nd het trnsfer in the ir nd het conduction through the fins nd tubes, the problem ill be studied numericlly. In this pper, the ir nd het flo in the tested to-ro utomotive rditor s simulted numericlly by using the CFD progrm FLUENT [14]. The three-dimensionl (3D) flo is treted s turbulent, since the ir-side Reynolds number Re, before the het exchnger is greter thn Oing to the complicted construction of the rditor, the numericl study of the hole rditor is very difficult to perform (Fig. 3). Therefore, due to the symmetry, the 3D flo through the single nrro pssge beteen the fins s simulted. The temperture distribution in the djcent plte fins nd tube lls s lso clculted. In this y, the effect of non-uniform het trnsfer coefficient on the tube nd fin surfces is tken into ccount, s ell s the effect of the tube-to-tube het conduction through the fins on the het trnsferred from the ter to the ir. Fig. 4 shos the mesurement dt set No.7 from Tble 1 tken s the input dt. The therml contct resistnce beteen the tube nd fins s modelled by inserting thin lyer of mteril ith knon thickness of 0.01 mm (Fig. 5).Therml conductivity of the lyer s vried to djust the function j(re ) obtined from the CFD simultion to the function j(re ) bsed on experiments. The computtions ere conducted for the 10 dt sets. The uniform frontl velocity 0 nd uniform temperture T in front of the rditor ere ssumed. The boundry condition of the third kind (convection surfce condition) is specified t the inside surfce S in of the ovl tubes (Fig. 4) A-A Figure 4: Single nrro pssge beteen the fins simulted using CFD code repetble segment of the to-ro plte fin nd tube het exchnger configurtion; the dimensions re given in millimeters 4

5 ( ) ( ) in k T n = h T T (12) t t S in t S in The symbol n denotes norml direction to the inside tube surfce. The ter-side het trnsfer coefficient h in s clculted from Eqution (5). The inlet temperture of the ter T s tken s the bulk ter tempertures t the first nd second ro of tubes. here k t is the tube s therml conductivity, T t nd T re the tempertures of the tube nd ter, respectively. Figure 5: Longitudinl section of the fin-to-tube ttchment - thin lyer ith thickness of 0.01 mm nd knon therml conductivity simultes the therml contct resistnce beteen the tube nd fin The flo pssge beteen the fins is divided into tenty lyers of finite volumes, hile only to lyers constitute hlf of the thickness of the fin. Only the qurter of the pssge shon in Fig.4 s simulted. The mesh of finite volumes ith nodes is shon in Fig.6. Figure 6: Mesh of finite control volumes The simultion revels high vlues of the het flux on the fin s leding edge due to the developing ir flo. Ner the fin surfce, the temperture of the ir increses in the flo direction nd cuses corresponding reduction in the het flux. Stgntion flo on the front of the tube in the first ro produces high het trnsfer ner the bse of the fin nd t the frontl prt of the tube circumference. Behind the tubes, lo-velocity ke regions exist. In the donstrem regions of the tubes, lo ir velocities nd very lo het fluxes cn be observed in the first nd second ro. Reltively lo het fluxes re encountered on the portions of the fin tht lie donstrem of the minimum flo cross sections. The het trnsfer rtes re especilly lo in the recircultion regions behind the tubes. In the regions ith smll ir velocities, the ir tempertures re lrge. The lrger prt of the totl het trnsfer rte is trnsferred in the first tube ro. The key to het trnsfer enhncement on the fin ssocited ith the first tube ro is mjor contribution of the developing flo region, hile the portion of the fin djcent to the second tube ro hs no developing flo contribution nd only ek vortex contribution Determintion of the Colburn j-fctor The second method is bsed on the CFD simultion of fluid flo nd het trnsfer in the het exchnger. Although the temperture nd het flux distributions on the tube nd fin surfces re knon from this simultion, the locl nd verge het trnsfer coefficients re difficult to determine becuse it is uncler ht ir temperture should be ssumed s the bulk ir temperture. The presented method circumvents this problem nd llos for determining the men ir-side het trnsfer coefficient. The het trnsfer coefficient h is determined from the condition tht temperture increse of the ir T t over to ros of tubes (over the het exchnger) clculted using the nlyticl model of the het exchnger is equl to the temperture increse obtined from the numericl simultion using FLUENT. Since the temperture distribution t the inlet nd outlet of the modelled pssge is to-dimensionl, then the mss verge ir temperture difference T computed by Fluent should be equl to the ir temperture t difference T t clculted using n nlyticl model of the het exchnger T h T =, i = 1,..., n (13) ( ) t, i, i t, i 0 The temperture difference T t is given by the expression: 5

6 T = T T = T T (14) t I II here T I = T - T nd T II = T - T represent the ir temperture increse over the first nd second tube ro, respectively (Fig. 7). The men vlue of the het trnsfer coefficient h,i over to tube ros is determined by solving Eqution (13). Determining the het trnsfer coefficients cn be significntly simplified if e tke into ccount tht the ter temperture hs no lrge influence on the serched het trnsfer coefficients becuse ir properties re not influenced substntilly by temperture. Thus, the equl ter temperture T (Fig. 7) t the first nd second ro of tubes my be ssumed. If the temperture of the ter floing inside the to tube ros is constnt, then the ir temperture T in the first nd second tube ros cn be obtined from the solution of the differentil equtions: d T ( y ) ( ) I I = N T T y I d y I T yi 0 (15) = T (16) = Figure 7: Cross-flo het exchnger ith to ros of tubes d T ( y ) ( ) II II = N T T y II d y II Solution of the initil-boundry problems (15-16) nd (17-18) simplifies to: T yii 0 ( I ) ( ) ( ) ( ) (17) = T (18) = I N yi T y = T T T e (19) I II ( N N yii ) T y = T T T e (20) II here N I =U I A/( m& c p ), N II =UI I A/( m& c p ). The ir tempertures T (y I ) nd T (y II ) t the first nd second tube ros, respectively, do not depend on the x coordinte, since the fluid temperture T in both tube ros is constnt. The differences of the ir temperture over the first nd second tube ros re given by: I N ( )( ) I II N N ( ) ( ) T = T T = T T e (21) I 1 I 1 y = yi = 0 T = T T = T T e e (22) II 1 II 1 y = yii = 0 I If the ir-side het trnsfer coefficients t the first nd second ros re ssumed to be equl, i.g. N = N = N II, then the totl temperture difference T i over to ros simplifies to 1 2 N T = T T = T T e (23) t I II ( )( ) The ir temperture increse T t is the function of the unknon het trnsfer coefficient h. 5. Test fcility The mesurements ere crried out in n open erodynmic tunnel. The experimentl setup s designed to obtin het trnsfer nd pressure drop dt from commercilly vilble utomotive rditors. The test fcility (Fig. 8) follos the generl guidelines presented in ASHRAE Stndrds nd Air is forced through the 6

7 ) b) Figure 8: Open-loop ind tunnel for experimentl investigtions of the tube-nd-fin het exchnger (cr rditor); () front vie, (b) top vie; A cr rditor, B vrible speed centrifugl fn, C chmber ith cr rditor, D pipe ith outer dimeter of 315 mm nd ll thickness of 1 mm, E ter outlet pipe, F ter inlet pipe,1 mesurement of the men nd mximum ir velocity using Pitot-sttic pressure probe, 2 mesurement of the men nd mximum ir velocity using turbine velocity meter ith hed dimeter of 11 mm, 3 mesurement of the men nd mximum ir velocity using turbine velocity meter ith hed dimeter of 80 mm, 4 ir temperture mesurement before the cr rditor, 5 mesurement of pressure drop over the cr rditor, 6 ter temperture t rditor inlet, 7 mesurement of ter temperture t rditor outlet, 8 ir temperture mesurement fter the cr rditor open-loop ind tunnel by vrible speed centrifugl fn. The ir flo pssed the hole front cross-section of the rditor. A computer-bsed dt-cquisition system s used to mesure, store nd interpret the dt. The reltive difference beteen the ir-side nd liquid-side het trnsfer rte s less thn 3%. Het trnsfer mesurements under stedy-stte conditions ere conducted to find the correltion for the ir-side Colburn j-fctor number. 6. Results The temperture difference over the rditor obtined from the CFD simultion ithout the therml contct resistnce beteen the tube nd fins is lrger thn tht obtined from the nlyticl het exchnger model, in hich the experimentl correltions for het trnsfer coefficients re used. The loer vlues of the ir temperture difference cross the rditor obtined from the nlyticl model of the het exchnger, my result from the contct resistnce beteen the fins nd the tubes. In order to sho the influence of the therml contct resistnce on the temperture differences, T I, T II nd T t, CFD simultions using FLUENT ere conducted for vrious vlues of the gp effective therml conductivity k g (Fig. 9). The velocity nd temperture of the ir before the rditor s 2.12 m/s nd K, respectively. The ter temperture in both tube ros s K. The CFD simultion s performed using the commercil CFD softre FLUENT for h in = 3549 W/(m 2 K). An inspection of the results shon in Fig. 9 indictes tht for the therml conductivity k g > 1 W/(mK) (g / k g = /1 = m 2 K/W) the temperture differences re lmost independent on the gp therml resistnce g / k g. In ddition, it hs been found from the numericl CFD simultion tht the fins nd tubes in the second ro re less effective thn those in the first ro. The temperture difference over the first tube ro is lmost three times lrger thn the second tube ro (Fig. 9). The key to het trnsfer enhncement on the fin ssocited ith the first tube ro is mjor contribution of the developing boundry lyer on the inlet prt of the fins, hile the portion of the fin djcent to the second tube ro hs no boundry lyer contribution nd only smll vortex contribution. The het trnsfer t the forrd stgntion points on the first tube ro is lso intensive. The regions behind the tubes contribute very little to the performnce of the het exchnger. The therml mesurements results for the utomotive rditor re shon in Tble 1. 7

8 Figure 9: Effect of effective therml conductivity of the gp beteen the tube nd fins on ir temperture differences over the first T I, second T II, nd over the entire cr rditor T t Correltions for the ir-side Colburn j-fctors for n utomotive rditor ere determined using the method described bove. The comprisons of correltions j (Re ) for the entire het exchnger re shon in Fig. 10. The blck dots in Fig. 10 represent vlues of j,i (Re,i ), in hich the ir-side het trnsfer coefficient s determined bsed on the mesured tempertures of ter t the outlet of the het exchnger (Tble 1). The ter-side het trnsfer coefficients h in ere clculted using the Gnielinski correltion (5). Tble 1. Mesurement dt for the utomotive rditor No. 0 [m/s] V& [L/h] T [ C] T [ C] T [ C] Re Re The solid line in Fig. 10 represents the lest squres pproximtion bsed on the 10 dt series: j = Re (24) from hich the correltion for the ir-side Nusselt number results / 3 Nu = Re Pr (25) In the correltion (25) contct resistnce beteen the tube nd fin is implicitly included in the ir-side het trnsfer coefficient, hich s determined using the first method. The other plots in Fig. 10 re bsed on the CFD simultion using FLUENT. The sme 10 dt series, s in the method I, ere used in the CFD simultion. The correltions ere obtined under the ssumption tht the ir flo s lminr. Only the ir-side het trnsfer coefficient h is determined from the condition tht the ir temperture increse cross the het exchnger obtined from the CFD simultion is equl to the temperture increse clculted using Eqution (23). The ter-side het trnsfer coefficient h in is clculted using the Gnielinski correltion (5). The results presented in Fig. 10 sho significnt influence of the therml contct resistnce on the determined j (Re ) curve. The loer is the effective therml conductivity k g of the gp the smller is the Colburn fctor j. The best greement beteen the experimentl curve j (Re ) nd the curve bsed on the CFD simultion s obtined for k = 0.12 W/(mK) (Fig. 10). Thus, it cn be concluded tht the effective therml conductivity of the gp is bout 0.12 W/(mK) nd therml contct resistnce g/k g = /0.12 = m 2 K/W. 8

9 Figure 10: Colburn j - fctor for the investigted cr rditor Hving determined the het trnsfer coefficients h,i i = 1,,n, from the solution of the non-liner lgebric eqution (13), the het trnsfer correltions re derived in the sme y s in the method I. First, the j,i re clculted for i = 1,,n using eqution (9), nd then the obtined results re pproximted by the function (10), using the lest-squres method. 7. Conclusions To different methods ere used to determine correltions for the ir-side Colburn j-fctors. The first method is bsed on the experimentl dt hile the second is bsed on the CFD simultion of the flo nd het trnsfer in the het exchnger. In the first method, the ir-side het trnsfer coefficient s determined from the condition tht the clculted nd mesured liquid outlet tempertures re equl. The het trnsfer coefficient on the tube-side s clculted using the Gnielinski correltion. An nlyticl model of the het exchnger s used to clculte the ter nd ir outlet tempertures s the function of the serched het trnsfer coefficients. The second method for determining ir-side het trnsfer correltions, bsed on the CFD simultion of flo nd het trnsfer nd on the simplified nlyticl model of het trnsfer in the het exchnger, s proposed. Bsed on the clculted het trnsfer coefficients, the dimensionless correltion for the Colburn j-fctor s function of the Reynolds number cn be found. The therml contct resistnce beteen the tube nd fins s estimted by compring the experimentl nd CFD bsed j (Re ) plots. The CFD progrms cn lso be used for clculting men het trnsfer coefficients over specified tube ro. Nomenclture A re, m 2 c p specific het t constnt pressure, J/kgK d dimeter, m d in, d o inner nd outer tube dimeter, m f mesured temperture, C or K F correction fctor, dimensionless g thickness of the gp beteen the tube nd fin, m h het trnsfer coefficient, W/m 2 K h enhnced het trnsfer coefficient bsed on the tube outer surfce A o, W/m 2 K j Colburn j-fctor, Nu/Re Pr 1/3, dimensionless k therml conductivity, W/mK L t tube length in the cr rditor, m m number of unknon coefficients, dimensionless m& mss flo rte, kg/s n dt set number, dimensionless N number of trnsfer units, N =U o A/( m& c p ), dimensionless Nu Nusselt number, dimensionless fin pitch, m p f 9

10 p 1 p 2 P Pr R Re T U pitch of tubes in plne perpendiculr to flo, m pitch of tubes in direction of flo, m perimeter, m Prndtl number, dimensionless contct therml resistnce beteen tube nd fin, m 2 K/W Reynolds number, dimensionless temperture, C or K overll het trnsfer coefficient, W/m 2 K V & volume flo rte, m 3 /s velocity, m/s x, y Crtesin coordintes, m x i unknon coefficient, dimensionless y I, y II dimensionless coordinte, y I = y I /p 2, y II = y II /p 2 δ thickness, m T temperture difference, K η fin efficiency, dimensionless µ dynmic viscosity, Ns/m 2 ν kinemtic viscosity, m 2 /s ξ Drcy-Weisbch friction fctor, dimensionless Subscript ir f fin g gp in inner m logrithmic men temperture difference o outer t tube ll I, II first nd second tube ro, respectively Superscript e experimentl inlet outlet References [1] D. M. Drt, Effect on Fin Bond, ASHRAE Trns, Vol. 1, pp , [2] J. W. Sheffield, R. A. Wood, H. J. Suer, Experimentl Investigtion of Therml Conductnce of Finned Tube Contcts, Experimentl Therml nd Fluid Science, Vol. 2, pp , [3] J. Jeong, Ch. N. Kim, B. Youn, Y. S. Kim, A Study of the Correltion beteen the Therml Contct Conductnce nd Effective Fctors in Fin-Tube Het Exchngers ith 9.52 mm Tube, Int. J. Het nd Fluid Flo, Vol. 25, pp , [4] J. Jeong, Ch. N. Kim, B. Youn, A Study on the Therml Contct Conductnce in Fin-Tube Het Exchngers ith 7mm Tube, Int. J. Het nd Mss Trnsfer, Vol. 49, pp , [5] A. D. Krus, A. Aziz, J. Welty, Extended Surfce Het Trnsfer, John Wiley & Sons, Ne York,2001. [6] J. Tler, P. Dud, Solving Direct nd Inverse Het conduction Problems, Springer, Berlin, [7] G. F. Heitt, G. L. Shires, T. R. Bott, Process Het Trnsfer, CRC Press, Boc Rton, [8] F. C. McQuiston, J. D. Prker, J. D. Spitler, Heting, Ventilting, nd Air Conditioning, Sixth Edition, John Wiley nd Sons, Ne York, [9] W. M. Kys nd A. L. London, Compct Het Exchngers, 3 rd ed., McGr-Hill, Ne York, [10] V. Gnielinski, Neue Gleichungen für den Wärme- und den Stoffübergng in turbulent durchströmten Rohren und Knälen, Forschung im Ingenieuresen, Bd. 41, Nr. 1, pp. 8-16, [11] D. Tler, Determintion of het trnsfer correltions for plte-fin-nd-tube het exchngers, Het nd Mss Trnsfer, Vol. 40, pp , [12] G. A. F. Seber, C. J Wild, Nonliner Regression, John Wiley & Sons, Ne York, [13] Tble Curve, Automted Curve Fitting Softre, AISN Softre, Chicgo, 2005 [14] FLUENT 6.0., Fluent Inc., 10 Cvendish Court, Lebnon, NH03766, USA, 2003 [15] D. Tler, Mesurements of Pressure, Velocity nd Mss Flo Rte of Fluids, University of Science nd Technology Press (UWND AGH), Crco (in Polish),