Journal of Naval Science and Engineering 2015, Vol.11, No.1, pp FINITE DIFFERENCE MODEL OF A CIRCULAR FIN WITH RECTANGULAR PROFILE.

Transcription

1 Jounal o Naval Scence and Engneeng 05 Vol. No. pp FINIE DIFFERENCE MODEL OF A CIRCULAR FIN WIH RECANGULAR PROFILE İbahm GİRGİN Cüneyt EZGİ uksh Naval Academy uzla Istanbul ukye ggn@dho.edu.t cezg@dho.edu.t Abstact Numecal methods ae commonly used n engneeng whee the analytcal esults ae not eached o as a suppot o expemental studes. Vaous technques ae beng used as a numetcal method as nte deence nte volume o nte elements etc. In ths study numecal solutons ae obtaned o a ccula n o ectangula pole usng nte deence method and the esults ae compaed to the analytcal solutons. It s seen that the analytcal soluton and numecal esults ae ound to be compatble. DİKDÖRGEN KESİLİ BİR DAİRESEL KANAÇIĞIN SONLU FARK MODELİ Özetçe Analtk çözümün mümkün olmadığı duumlada veya deneysel çalışmalaa destek olmak amacıyla sayısal yöntemle mühendslkte yaygın olaak kullanılmaktadı. Sayısal yöntemle olaak sonlu akla sonlu hacm sonlu eleman metodlaı gb çeştl yöntemle kullanılmaktadı. Bu çalışmada sonlu ak yöntem kullanılaak dkdötgen kestl daesel b kanatçık çn sayısal çözüm elde edlmş ve hesaplanan sonuçla analtk çözümle kaşılaştıılmıştı. Analtk ve sayısal sonuçlaın bbleyle oldukça uyumlu olduklaı göülmüştü. Keywods: Ccula Fn Numecal Methods Fnte Deence Method. Anahta Kelmele: Daesel Kanatçık Sayısal Yöntemle Sonlu Fak Yöntem. 53

2 İbahm GİRGİN Cüneyt EZGİ. INRODUCION Numecal analyss s the combnaton o mathematcs and compute pogammng that ceates and mplements algothms o solvng the poblems o contnuous mathematcs. hese poblems occu thoughout the natual scences socal scences engneeng and the othe elds. he gowth n powe and avalablty o dgtal computes has led to an nceasng use o numecal soluton o the models n scence and engneeng. Numecal methods ae commonly used n engneeng whee the analytcal esults ae not eached o as a suppot o expemental studes. Vaous technques ae used to solve the tough patal deental equatons whch cannot be solved analytcally. Most common used numecal method o solvng a patal deental equaton s the nte deence appoach. In ths study nte deence method s used to get the numecal soluton o heat tanse nsde a ccula n. he tempeatue dstbuton nsde a ccula n s govened by the geneal heat conducton equaton. hs equaton s a thee dmensonal equaton that has both a souce tem and a tansent component. But a n can be assumed steady the base tempeatue ambent lud tempeatue and combned convecton-adaton heat tanse coecent ae constant. heeoe a one dmensonal steady smpled conducton equaton s used wth no heat souce.. FINIE DIFFERENCE MEHOD Fnte-deence methods ae numecal methods o solvng deental equatons by appoxmatng them wth deence equatons. he devatves ae appoxmated by nte deences so nte deence methods ae dscetzaton methods. oday these methods ae the most used appoach n numecal solutons o patal deental equatons []. he nte deence appoach s based upon convetng the deental equatons to nte deence equatons usng the numecal expessons o the devatves. 54

3 Numecal Model o a Ccula Fn wth Rectangula Pole he eo n an appoxmaton s dened as the deence between the appoxmaton and the exact analytcal soluton. he two souces o eo n nte deence methods ae ound-o eo and the dscetzaton eo. he ound-o eo s the loss o pecson due to compute oundng o decmal quanttes whee the dscetzaton eo s the deence between the exact soluton o the nte deence equaton and the exact quantty assumng peect athmetc. he nte deence omulas o the st and second devatves can be obtaned om aylo sees expanson. Fgue he uncton y(x) he aylo Sees Expanson o the pont x and x - om the Fgue : ı ıı 3 ııı ( x ) ( x ) h. ( x ) h. ( x ) h. ( x )... ()! 3! ı ıı 3 ııı ( x ) ( x ) h. ( x ) h. ( x ) h. ( x )... ()! 3! the st devatves at x ae expessed om the equatons above: 55

4 İbahm GİRGİN Cüneyt EZGİ ( x ) ( x ) ( ) ı ıı. ( ) ııı x h x h. ( x )... h! 3! (3) o ı ( ) ( x ) ( x ) x o( h) h (4) and the second expesson can be dened as: ( x ) ( x ) ( ) ı ıı. ( ) ııı x h x h. ( x )... (5) h! 3! o ı ( x ) ( x ) ( x ) h o( h) he o(h) tem on the ght hand sde s the tuncaton eo. he nte deence equatons o the st devatve ae called owad deence expesson wth eo o ode h: (6) ı ( x ) ( x ) ( x ) h (7) and backwad deence expesson wth eo o ode h: ( x ) ( x ) ( x ) (8) h ı I equaton s subtacted om equaton and the st devatve s deved om the esult the cental deence equaton o the st devatve wth eo ode o h : ( x ) ( x ) ı ( x ). h (9) s obtaned. 56

5 Numecal Model o a Ccula Fn wth Rectangula Pole I the equatons and ae added togethe the second devatve nte deence equaton can be wtten as: ıı ( x ) ( x ) ( x ) ( x ). h (0) wth eo ode o h. he eo ode o h means you decease h to hal the eo also be expected to decease to hal. But the eo s ode o h t means that you decease the h to hal the eo s expected to decease to /h tmes. heeoe to use the expessons wth eo o hgh ode should be peeed. But such expessons may be moe complcated and they can ncease the calculaton tme. he nte deence expessons wth eo ode oh h h h 4 can be ound n the lteatue. In ths study t has been avoded usng the nte deence expessons wth eo ode o h because t s needed much lage gd ponts to decease the tuncaton eo nto the acceptable lmts. hus the st ode owad and backwad deence expessons wth eo ode o h : ( x ) ( x ) 3 ( x ) ı 4 ( x ) () h ı 3 ( ) ( x ) 4 ( x ) ( x ) x () h has been peeed to the equatons 7 and 8 []. o use a nte deence method to nd a soluton to a poblem at st the poblem's doman must be dscetzed. hs s usually done by dvdng the doman nto a unom gd (Fg.). 57

6 İbahm GİRGİN Cüneyt EZGİ Fgue he dscetzed poblem doman he gd may be o 3 dmensonal wth espect to the natue o the poblem. A -dmensonal gd s used n ths study because tempeatue change wth espect to φ axs s zeo. φ 3. GEOMERY In themal engneeng ccula ns ae wdely used to enhance the heat tanse om the suaces. Addng a ccula n to an obect nceases the amount o suace aea n contact wth the suoundng lud whch nceases the convectve and adatve heat tanse between the obect and suoundng lud and the suaces. he adatve heat tanse usually can be neglected the convecton s oced convecton. Because the suace aea nceases as length om the obect nceases a ccula n tanses moe heat than a smla pn n at any gven length. Ccula ns ae oten used to ncease the heat tanse n lqud gas heat exchange systems. A schematc dagam o a ccula n o ectangula pole s gven n Fgue 3 [3]: 58

7 Numecal Model o a Ccula Fn wth Rectangula Pole Fgue 3 Schemetc Dagam o a Ccula Fn wth ectangula pole 4. GOVERNING EQUAION he geneal heat conducton equaton n a medum can be expessed n ectangula cylndcal and sphecal coodnate systems. Cylndcal coodnates conducton equaton s used n ths study snce the poblem s - dmensonal ths coodnate system s chosen. I ectangula o Catesan coodnate system s would be chosen the poblem would be 3-dmensonal ant t would be much moe complcated to solve the poblem. he geneal heat conducton equaton n cylndcal coodnates s gven as: k k φ φ k z z e& gen ρc t (3) whee k s themal conductvty ρ s densty c s specc heat and gen e& s the heat geneated n a unt volume. 59

8 İbahm GİRGİN Cüneyt EZGİ 60 he base tempeatue ambent lud tempeatue the combned convectonadaton coecent and themal conductvty o the n mateal ae assumed as constant. he poblem s a steady and thee s no heat geneaton nsde the n. Unde these assumptons the govenng equaton becomes: 0 z (4) Fo the govenng equaton the cental deence expessons can be wtten as:.. (5) ( ) (6) ( ) z z (7) Usng these nte deence equatons o the ntenal gd ponts the govenng equaton (4) s dscetzed as: z z z (8)

9 Numecal Model o a Ccula Fn wth Rectangula Pole Bounday Condtons At the base o the ne the tempeatue s constant and t the base tempeatue. So the bounday condton o the base s: (9) b as seen n Fgue 4. Fgue 4 he bounday condton o the base o the n Fo the uppe sde the heat conducted om the lowe nodes should be equal to the convecton to outsde k h( ) (0) z as seen n Fgue 5 whee s the ambent lud tempeatue. 6

10 İbahm GİRGİN Cüneyt EZGİ Fgue 5 he bounday condton o the uppe sde o the n As ths equaton s dcetzed by equaton the bounday condton becomes: k k h z z. () 3 k h z Fo the lowe sde the heat conducted om the uppe nodes should be equal to the convecton to outsde: k h( ) () z as seen n Fgue 6. 6

11 Numecal Model o a Ccula Fn wth Rectangula Pole Fgue 6 he bounday condton o the lowe sde o the n As ths equaton s dcetzed by equaton the bounday condton becomes: k k h z z. (3) 3 k h z Fo the tp o the n the heat conducted om the nne nodes should be equal to the convecton to outsde: k h( ) (4) as seen n Fgue 7. 63

12 İbahm GİRGİN Cüneyt EZGİ Fgue 7 he bounday condton o the tp o the n As ths equaton s dcetzed by equaton the bounday condton becomes: k k h. (5) 3 k h 5. SUDY A Matlab code has been wtten to calculate the tempeatue dstbuton nsde the n. Gauss-Sedel teatve method was used o teaton wth an oveelaxaton paamete w between and to speed up convegence: w ( w) (6) ( ) new ( )old Ate ndng the tempeatue dstbuton the heat tanseed to ambent a om the n has been calculated om the equaton: 64

13 Numecal Model o a Ccula Fn wth Rectangula Pole d Q& n ka (7) d base snce the heat tanseed to ambent lud s equal to the heat that s conducted om the base o the n. he n ecency s dened as: Q& n η n (8) Q& nmax whee Q & n max s the heat tanse om a peect n wth an nnte themal conductvty whch has a suace tempeatue equal to the base tempeatue. So Q & s dened as: n max ( ) Q & ha (9) n max n base Heat tanse om the n was calculated numecally and compaed to the analytcal soluton exst n the lteatue. he analytcal soluton o ecency s gven as: K( m ) I( m c ) I( m ) K( m c ) η n C (30) I ( m ) K ( m ) K ( m ) I ( m ) 0 c 0 whee m h kt m C and I 0 I K 0 K ae moded c Bessel unctons o the st and second knd. c 65

14 İbahm GİRGİN Cüneyt EZGİ 6. RESULS he tempeatue dstbuton nsde the n was calculated numecally o 3 cases: c c 3 and c 4. he heat tanse om the n was calculated numecally om the Eq.7 and n ecency was calculated om Eq.8 o deent ξ values / 3/ h ξ L c (3) kap whee Lc s coected length L c L t / and A p Lct. he esults ae gven n Fgue 8. he squae damond and tangle values ae the numecal esults whle the contnuous sold lnes ae analytcal values om Equaton 30. hee s a good ageement between the numecal soluton and the analytcal soluton as t s seen n the gue. It s seen n the gue that n ecency appoaches to as the dmensonless vaable ξ goes to zeo. As n length L o convecton coeent h goes to zeo o themal conductvty k o the n s vey lage n Eq. 3 ξ appoaches to 0 whch means that the tempeatue o the n s close to the tempeatue o the base whch means the ecency s vey close to as expected. 66

15 Numecal Model o a Ccula Fn wth Rectangula Pole Fgue 8 Numecal Results vs. Analytcal Soluton REFERENCES [] Gossmann C. Roos H. and Stynes M. Numecal eatment o Patal Deental Equatons. Spnge Scence & Busness Meda. pp.3. [] James M. L. Smth G. M. and Wolod J. C. Appled Numecal Methods o Dgtal Computaton. Hape Collns College Publshes. [3] Çengel Y. A. and Ghaa A. J. Heat and Mass anse. McGaw Hll. 67