Fluid Flow and Heat Transfer Characteristics across an Internally Heated Finned Duct

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1 J. Enegy Powe Souces ol. No. 6 4 pp ceived: Augus 3 4 Published: Decembe 3 4 Jounal of Enegy and Powe Souces Fluid Flow and ea ansfe Chaaceisics acoss an Inenally eaed Finned Duc Sofiane ouahi and oufik Boufendi Enegy Physics Laboaoy Physics Dep. Consanine Univesiy Consanine Algeia Coesponding auho: oufik Boufendi (boufendi@yahoo.f Absac: We numeically sudy he fluid flow and he hea ansfe chaaceisics in a hoional pipe euipped by longiudinal and ansvesal fins aached on is inenal suface used in seveal aeas of hemal sciences. he longiudinal fins numbe is: wo veical fou and eigh while he ansvesal fins numbe is eigh. he consideed heighs fins ae. and.4 cm. he convecion in he fluid domain is conjugaed o conducion in he pipe and in he fins hickness. he physical popeies of he fluid ae hemal dependan and he hea losses o he ambien ae consideed. he model euaions ae numeically solved by a finie volume mehod wih a second ode disceiaion. As expeced fo he longiudinal fins he axial Nussel numbe incease wih inceasing of numbe and heigh of fins. Fo G 5. 5 he Nussel numbe wihou fins is eual o he inoduce of longiudinal fins gives a Nussel numbe eual o and 3.6 fo wo veical fou and eigh fins especively. he paicipaion of fins locaed in he lowe pa of he ube on he impovemen of hea ansfe is highe han he uppe fins. he longiudinal fins paicipae diecly on incease he hea ansfe; his is jusified by he lage local Nussel numbe along he fins ineface. his paicipaion is modeae in he case of ansvese fins hese lae ae used o mix he fluid fo incease he local Nussel numbe in he axial secions following he ansvesal fins. eywods: Conjugae hea ansfe mixed convecion inenal fins numeical simulaion. Nomenclaue: D Pipe diamee (m G Modified Gashof numbe G gβgd i 5 s ν s G Non-dimensional hea geneaion P eigh of he fin (m h ea ansfe coefficien (W/m hemal conduciviy (W/m Non-dimensional hemal conduciviy L Non-dimensional pipe lengh L D i Nu( Local Nussel numbe Nu Nussel numbe (dimensionless P Pessue (Pa P Non-dimensional pessue (P - P ρ P Pandl numbe ν α ea flux (W/m R Radial coodinae (m ynolds numbe D i ν Non-dimensional adial coodinae D i empeaue ( C Non-dimensional empeaue ( - (GD i s elociy (m/s Non-dimensional velociy Y Dimensionless wall disance Z Axial coodinae (m Z Non-dimensional axial coodinae D i. Inoducion Finned ubes ae ofen used in many engineeing secos fo exend he conac suface beween he ube wall and he fluid and impove he hea ansfe; he eseaches have sudied he poblem of opimiing he shape and geomey of aached fins in ode o incease hea ansfe effeciveness. Mos of he sudies pefomed on his opimiaion conside longiudinal fins which have symmeical laeal pofiles; his assumpion simplifies he eamen of he poblem wih egad o he bounday condiions and gives

2 Fluid Flow and ea ansfe Chaaceisics acoss an Inenally eaed Finned Duc 97 symmeical esuls concening velociy and empeaue pofiles. Many invesigaions boh expeimenal and numeical have been conduced fo diffeen kinds of inenally finned ubes. An analyical model fo fully developed ubulen ai flow in inenally finned ubes and annuli was pesened by []. In his sudy he longiudinal fins ae aached in he inne wall and he consan hea flux is he hemal bounday condiion a he inne suface. he esuls concen he hea ansfe and he pessue dop coefficiens. A combined numeical and expeimenal sudy of plae-fin and ube hea exchanges was examined by []. he deailed numeical esuls of pessue dop and hea ansfe coefficien ae pesened. A numeical sudy on hydodynamically fully developed hemally developing flow inside cicula ubes wih inenal longiudinal fins having apeed laeal pofiles was conduced by [3]. he esuls showed significan hea ansfe enhancemen wih he inclusion of inenal fins. Wae and engine oil wee assumed as fluids in hei numeical sudies and hey concluded wae o be a bee coolan as compaed o engine oil. In f. [4] a numeical sudy of hemally developing flow in an ellipical duc wih fou longiudinal inenal fins of eo hickness is consideed. A conol volume based on finie diffeence echniue was used and an opimum value of he local Nussel numbe was obained as a funcion of he fin heigh. f. [5] pefoms an expeimenal analysis of hea ansfe in an inenally finned ube he expeimenal esuls wee compaed wih esuls fom he smooh channel ube and a significan impovemen in hea ansfe was obseved fo inenally finned cases. Simila sudies wee also examined numeically and expeimenally by [6-]. In his wok we sudied numeically he hea ansfe by mixed convecion in hoional pipe euipped by longiudinal and ansvesal aached fins on is inenal wall. he mixed convecion is conjugae o hemal conducion in he pipe and fins walls. he physical popeies of he fluid ae hemo- dependen and he hea losses wih he exenal envionmen ae consideed. he objecive of ou wok is sudy he enhancemen gives o he hea ansfe by using diffeen shape of fins.. he Geomey and Mahemaical Model Fig. illusaes he poblem geomey. We conside a long hoional pipe having a lengh L m an ouside diamee D cm and an inside diamee D i.96 cm his lae is euipped by longiudinal and ansvesal aached fins on is inenal suface. he longiudinal fins ae fixed a ( π/4 π/ 3π/4 π 5π/4 3π/ and 7π/4 while he ansvesal fins ae aached a eigh axial saions fom o he pipe and fins ae made of Inconel having a hemal conduciviy s W m - -. An elecic cuen passing along he pipe (in he solid hickness poduced a hea geneaion by Joulean effec. his hea is ansfeed o disilled wae flow in he pipe. A he enance he flow is of Poiseuille ype wih an aveage axial velociy eual o 7. - m s - and a consan empeaue of 5 C. he densiy is a linea funcion of empeaue and he Boussines appoximaion is adoped. he physical pinciples involved in his poblem ae well modeled by he following non dimensional consevaion paial diffeenial euaions wih hei iniial and boundaies condiions. (b Fig. Pipe wih (a longiudinal fin; (b ansvesal fin. (a D i D i D e D e

3 Fluid Flow and ea ansfe Chaaceisics acoss an Inenally eaed Finned Duc 98. Modeling Euaions A ( A > Mass consevaion euaion: ( Radial momenum consevaion euaion: (cos G P (3 Angula momenum consevaion euaion: sin G P (4 Axial momenum consevaion euaion: P (5 Enegy Consevaion Euaion P G (6 whee fluid he in solid he in P / S G he viscous sess enso componens ae: (7 he hea fluxes ae: and Z (8. he Boundaies Condiions he pevious diffeenial euaions ae solved wih he following boundaies condiions: ( A he pipe enance: In he fluid domain: 5. and π (- 4 (9 In he solid domain: and π ( ( A he pipe exi: 4.7 In he fluid domain: 5. and π ( In he solid domain: and π ( (3 A he oue wall of he pipe:.58 D h h i c (3 h εσ (4 he emissiviy of he oue wall ε is abiaily chosen o.9 while h c is deived fom he coelaion of [] valid fo all P and fo Rayleigh numbes in he ange 6 Ra 9. [ ] 7 8/ 9/6 /6.559/ P / / ai ai i c Ra D h Nu (5 wih [ ] ai ai ai ai ai o o P D R g Ra α ν ν α β / - 3 (6 In E. (6 he hemophysical popeies of he ai ambien ae evaluaed a he local film empeaue given as: [ ] ( R o film. In ou calculaions we have consideed he solid as a fluid wih a dynamic viscosiy eual o 3. his vey

4 Fluid Flow and ea ansfe Chaaceisics acoss an Inenally eaed Finned Duc 99 lage viscosiy wihin he solid domain ensues ha he velociy of his pa emains null and conseuenly he hea ansfe is only by conducion deduced fom E. (6..3 he Nussel Numbes A he cylindical solid-fluid ineface he local Nussel numbe is defined as: ( h( D i.5 Nu ( (7 (.5 - ( b he axial Nussel numbe fo he cylindical ineface is: π Nu ( Nu ( d (8 π Finally we can define an aveage Nussel numbe fo he whole cylindical solid-fluid ineface: ( π( 4.7 π 4.7 Nu A Nu( d d (9 A he longiudinal fin ineface he local Nussel numbe is defined as ( ( h ( D h fin Nu( ( ( ( fin m he axial Nussel numbe fo he longiudinal fin is: Nu( Nu( d ( ( fin d ( ( ( fin m he aveage Nussel numbe fo he longiudinal fin ineface is defined as: L Nu A Nu ( d ( L A he ansvesal fin ineface he local Nussel numbe is ( ( h( Dh fin Nu( (3 ( ( fin m he axial Nussel numbe fo he ansvesal fin ineface is defined as: Nu( ( π π R i ( ( ( 3. Numeical soluion fin fin m ( d d (4 Fo he numeical soluion of modeling euaions we used he finie volume mehod well descibed by []; he using of his mehod involves he disceiaion of he physical domain ino a discee domain consiued of finie volumes whee he modeling euaions ae disceied in a ypical volume. We used a empoal disceiaion wih a uncaion eo of ode. he convecive and nonlinea ems have been disceied accoding o he Adams-Bashfoh numeical scheme wih a uncaion eo of ode he diffusive and pessue ems ae implici. gading he spaial disceiaion we used he cenal diffeences paen wih a uncaion eo of and ode. So ou spaio-empoal disceiaion is second ode. he mesh used conains poins in he adial angula and axial diecions. he consideed ime sep is 5-4 and he ime maching is pusued unil he seady sae is eached. he seady sae is conolled by he saisfacion of he global mass and enegy balances as well as he leveling off of he ime evoluion of he hydodynamic and hemal fields. he accuacy of he esuls of ou numeical code has been esed by he compaison of ou esuls wih hose of ohe eseaches. A compaison wih he esuls of [3] who sudied he non conjugae and he conjugae mixed convecion hea ansfe in a pipe wih consan physical popeies of he fluid. Some of hei esuls concen he simulaneously developing hea ansfe and fluid flow in a unifomly heaed inclined pipe ( α 4.he conolling paamees of he poblem ae: 5 P 7. G 4 and 6 L D i 9 R o D i.583 s 7. he used gid is in he and diecions especively. We epoduced he esuls of he cied efeence wih he

5 3 Fluid Flow and ea ansfe Chaaceisics acoss an Inenally eaed Finned Duc Nu Ouane and Galanis [3]: Ou esuls: G 4 non conjugae case G 6 non conjugae case G 4 conjugae case G 4 non conjugae case G 6 non conjugae case G 4 conjugae case /(P Fig. Axial evoluion of he cicumfeenially mean axial Nussel numbe; A compaison wih he esul of [3]. fis ode calculus code concening he conjugae and non conjugae mixed convecion. In Fig. we illusae he axial evoluion of he cicumfeenially aveaged Nussel numbe. I is seen ha hee is a good ageemen beween ou esuls and heis Z4.7 Fig. 3 he seconday flow vecos a he pipe exi fo longiudinal fins. 4. suls and Discussion 4. Developmen of he Seconday Flow All he esuls pesened in his pape wee calculaed fo ynolds numbe and he Pandl numbe P 8.8 while he Gashof numbe is eual o he obained flow fo he sudied cases is chaaceied by a main flow along he axial diecion and a seconday flow influenced by he densiy vaiaion wih empeaue which occus in he plane ( his flows ae pesened fo eigh fins in he longiudinal case and fou fins in he ansvesal case. In he efeence case (foced convecion he ansvese moion is nonexisen; he only flow is in axial diecion. In he pesence of volumeic heaing in he pipe and fins wall a ansvese flow exiss and explained as follows: he ho fluid moves along he ho wall fom he boom of he oue ube ( π upwads ( and moves down fom he op o he boom along he cene of ube. he veical plane passing hough he angles ( and ( π is a plane of symmey. he ansvese flow in he ( plane is epesened by coune oaing cells; he cells numbe is popoional o longiudinal fins numbe. Z4.694 Fig. 4 he seconday flow vecos a he fouh ansvesal fin secion. In Fig. 3 we pesen he seconday flow vecos a he pipe exi fo he case of longiudinal fins. Fo he case of ansvesal fins he posiion of fins is only in seleced axial secions. Fa fom hese secions he seconday moion is simila o ha of simple cylindical pipe. he Fig. 4 illusaes he seconday flow vecos a he fouh ansvesal fin secion ( he Axial Flow Developmen A he enance he axial flow is axisymmeic wih he maximum velociy in he cene of he pipe. In he pesence of volumeic heaing in he pipe and he fins walls he configuaion of he axial flow compleely changes because he seconday flow causes an angula

6 Fluid Flow and ea ansfe Chaaceisics acoss an Inenally eaed Finned Duc 3 Z4.7 Fig. 5 Axial velociy pofiles a he pipe exi fo longiudinal fins. Z4.694 Fig. 6 Axial velociy pofiles a he fouh ansvesal fin secion. vaiaion which has a diec influence on he disibuion of axisymmeic axial flow. In Fig. 5 he axial flow is pesened a he exi of he pipe fo he case of longiudinal fins (heigh.875. In Fig. 6 he axial flow is pesened a he fouh ansvesal fin secion ( hough hese figues i is clea ha he axial velociy is null in he fins walls. 4.3 Developmen of he empeaue Field In he pesence of volumeic heaing a ansvese wih flow exiss and hus changes he axisymmeic disibuion of fluid and gives i an angula vaiaion his vaiaion explained as follows: he ho fluid nea fom Z4.7 Fig. 7 he isohems a he pipe exi fo longiudinal fins. Z Fig. 8 he isohems a he fouh ansvesal fin secion. he pipe and fins walls moves upwads unde he buoyancy foce effec he elaively cold fluid descends down in de cene of he pipe. A pemanen geneaion of hea in he pipe and he fins walls imposes a coninuous incease of he empeaue of he fluid up o he exi of he pipe. he obained esuls show ha a given secion he maximum fluid empeaue is all he ime locaed a.5 and (op of solid-fluid ineface because he ho fluid is diven by he seconday moion owads he op of he pipe. he isohems ae pesened a he exi of he pipe fo he case of longiudinal fins having heigh eual o.875 in Fig. 7 and a he fouh ansvesal fin secion ( in Fig. 8.

7 3 Fluid Flow and ea ansfe Chaaceisics acoss an Inenally eaed Finned Duc 4.4 he ea ansfe he phenomenon of hea ansfe has been chaaceied in ems of Nussel numbes calculaed a he inne wall of he pipe obained by E. (7 and hose calculaed a he fins wall obained by E. ( and E. (3. he vaiaion of local Nussel numbe a cylindical ineface is pesened in Fig. 9 fo he case of longiudinal fins and in Fig. fo he case of ansvesal fins. he local Nussel numbe of longiudinal fins placed in he igh side of he pipe a ( π/4 π/ 3π/4 π is shown in Fig.. he local Nussel numbe akes a maximum value eual o on he fin placed a ( 3π/4 4.7 and.384. he compaison of axial Nussel numbes beween finned ube and smooh ube is shown in Fig.. Quaniaively hee is a lage incease in he axial Nussel when he numbe of fins is inceased. A he exi of he pipe he axial Nussel numbe is eual o and 47.3 fo he cases: Smooh ube Fig. 9 he local Nussel numbe vaiaion a cylindical ineface fo longiudinal fins case Fig. he local Nussel numbe vaiaion a cylindical ineface fo ansvesal fins case (a (b (c (d

8 Fluid Flow and ea ansfe Chaaceisics acoss an Inenally eaed Finned Duc 33 (e Fig. he local Nussel numbe vaiaion a longiudinal fins ineface: (a fin placed a ( ; (b a ( π/4; (c a ( π/; (d a ( 3π/4; (e fin placed a ( π. Axial Nussel numbe smooh ube fins.5 4 fins.5 8 fins Fig. aiaion of he axial Nussel numbe fo he diffeen longiudinal fins sudied cases. wo veical fins fou fins and eigh fins especively. he aveage Nussel numbes fo hese cases ae and Conclusions his sudy consides he numeical simulaion of he hee dimensional mixed convecion hea ansfe in a hoional pipe euipped by inenal longiudinal and ansvesal fins. he pipe and fins ae heaed by an elecical inensiy passing hough is small hickness. he obained esuls show ha he longiudinal fins paicipae diecly in impoving he hea ansfe; his is jusified by he high local Nussel numbe a he ineface of longiudinal fins. By agains he ansvese fins paicipae in an indiec way in impoving he hea ansfe hei locaion facing he flow allowed o Z eaange he sucue of he fluid fo each passage hough he fins which is used o mix he fluid and o incease he hea ansfe o he cylindical inefaces. he numbe and fins heigh ae also impoan facos in impoving he hea ansfe. feences [] S.. Paanka M. Ivanovic E.M. Spaow Analysis of ubulen flow and hea ansfe in inenally finned ubes and annuli Jounal of ea ansfe ( [] J.Y. Yang M.C. Wu W.J. Chang Numeical and expeimenal sudies of hee-dimensional plae-fin and ubulen exchanges In. Jounal of ea and Mass ansfe 39 ( [3] I. Alam P.S. Ghoshdasida A sudy of hea ansfe effeciveness of cicula ubes wih inenal longiudinal fins having apeed laeal pofiles In. Jounal of ea and Mass ansfe 45 (6 ( [4] Z.F. Dong M.A. Ebadian A numeical analysis of hemally developing flow in ellipical duc wih inenal fins In. Jounal of ea and Fluid Flow ( ( [5] A.M. u M.M. Rahman Expeimenal measuemens of hea ansfe in an inenally finned ube In. Comm. ea Mass ansfe 5 (5 ( [6]. Boufendi M. Afid hee-dimensional conjugae conducion-mixed convecion wih vaiable fluid popeies in a heaed hoional pipe v. des Enegies nouvelables 8 (5-8. [7] S. ouahi. Boufendi Numeical sudy of he conjugae hea ansfe in a hoional pipe heaed by Joulean effec hemal Science 6 ( ( [8] W.M. Yan P.J. Sheen ea ansfe and ficion chaaceisics of fin and ube hea exchanges In. Jounal of ea and Mass ansfe 43 ( [9] B. Yu J.. Nie Q. Wang W. ao Expeimenal sudy on he pessue dop and hea ansfe chaaceisics of ubes wih inenal wave-like longiudinal fins ea and Mass ansfe 35 ( [] C.C. Wang W.L Fu C.. Chang ea ansfe and ficion chaaceisics of ypical wavy fin and ube hea exchange Exp. hemal Fluid Science 4 ( [] S.W. Chuchill.S. Chu Coelaing euaion fo lamina and ubulen fee convecion fom a hoional cylinde In. Jounal of ea and Mass ansfe 8 ( [] S.. Paanka Numeical ea ansfe and Fluid Flow McGaw-ill New Yok 98. [3] M. Ouane N. Galanis Effecs of paieal conducion and hea flux epaiion on mixed convecion nea he enance of an inclined duc In. Jounal of hemal Sciences 38 ( (in Fench

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, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission