40th AIAA Aerospace Sciences Meeting and Exhibit January 14 17, 2002/Reno, NV

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1 AIAA Rsistnc Appoc fo Annul Fins wit Contct Conductnc nd End Cooling Susnt K. Mit nd M. Micl Yovnovic Dptmnt of Mcnicl Engining, Univsity of Wtloo, Wtloo, Cnd, N2 3G 40t AIAA Aospc Scincs Mting nd Exibit Jnuy 4 7, 2002/Rno, NV Fo pmission to copy o publis, contct t Amicn Institut of Aonutics nd Astonutics 80 Alxnd Bll Div, Suit 500, Rston, VA

2 Rsistnc Appoc fo Annul Fins wit Contct Conductnc nd End Cooling Susnt K. Mit Λ nd M. Micl Yovnovic y Dptmnt of Mcnicl Engining, Univsity of Wtloo, Wtloo, Cnd, N2 3G A simplifid mtod, bsd on contol volum, is dvlopd to obtin t sistnc of nnul fins of constnt ticknss wit bs contct nd nd cooling. Tis ppoc is fut xtndd to nnul fins wit vibl ticknss fo wic nlyticl solution is not possibl. Suc novl mtod is bl to mtc t nlyticl solution fo constnt ticknss fins wit minimum numb of contol volums nd sows xcllnt gmnt fo vious limiting css of Biot numbs. Nomncltu tip ticknss A coss-sctionl b bs ticknss C constnt wit modifid Bssl functions Bi Biot numb t tnsf cofficint I modifid Bssl function of fist kind k tml conductivity K modifid Bssl function of scond kind fin lngt m non-dimnsionl fin pmt n numb ofcontol volum Q t flow dius R fin sistnc t lf-fin ticknss T fin tmptu ff non-dimnsionl inn dius fi non-dimnsionl out dius smi-tpd ngl fin tmptu xcss of mbint ψ non-dimnsionl tmptu Ω constnt wit modifid Bssl functions ρ non-dimnsionl dil position Subscipts c bs contct nd cooling f convctiv sufc i inn j indx fo contol volum o out mbint Λ Psnt Addss: Ontio Pow Gntion, Nucl Sfty Anlysis Division, Toonto, Cnd, M5G X6. y Pincipl Scintific Adviso, Micolctonics Ht Tnsf botoy, Univsity ofwtloo, Fllow AIAA. Copyigt cfl 2002 by t Amicn Institut of Aonutics nd Astonutics, Inc. No copyigt is sstd in t Unitd Stts und Titl 7, U.S. Cod. T U.S. Govnmnt s oyltyf licns to xcis ll igts und t copyigt climd in fo Govnmntl Puposs. All ot igts svd by t copyigt own. Supscipt nd conduction Intoduction ANNUAR fins fquntly usd s xtndd sufcs to nnc t t tnsf t in vious pplictions suc s i conditioning, t xcngs nd mico-lctonic pckgs. Wn suc fins ddd to sufc in contct wit t suounding fluid, t following sistncs ply ky ol in t nti t tnsf pocss: () t contct sistnc du to t mcnicl contct btwn t fin bs nd t pviously xposd sufc, (b) t conductiv sistnc to t flow witin t fin itslf, nd (c) t sistnc to t flow toug t convctiv film of t suounding fluid. Svl nlyticl solutions fo t stdy-stt t conduction witin n nnul fin of constnt ticknss, pfct contct t t bs, nd insultd nd (o som ppoximtion fo nd cooling) ldy xist., 7 On t ot nd, t on-dimnsionl stdy-stt nlyticl solution fo nnul fin wit constnt ticknss long wit bs contct sistnc nd nd cooling 3 involvs modifid Bssl functions tt difficult to comput nd lso computtionlly intnsiv, t dtils of wic will b discussd lt. Howv, in tcsofvibl ticknss nnul fin, nlyticl solutions do not xist. Suc poblms solvd numiclly, it toug finit diffnc 4 o finit lmnt 5 mtod, o by mns of intgl contol volum ppoc. 6 T min objctiv of t psnt wok is to dvlop simplifid, yt ccut tcniqu, by mns of sistnc mtod, fo dtmining t fin sistnc of constnt ticknss nnul fin wit bs contct sistnc nd nd cooling. It is lso sown tt suc tcniqu cn b dily xtndd fo n nnul fin wit vibl ticknss. of5 Amicn Institut of Aonutics nd Astonutics Pp

3 CV CVj (2 < j< n-) CV n 2t k c Nodl Point o i 2n /n /n /n /n /n 2n Fig. 2 Subdivision into n contol volums. Fig. Scmtic digm of constnt ticknss nnul fin. Constnt Ticknss Fin Anlyticl Mtod Figu sows scmtic digm fo n nnul fin of constnt ticknss 2t, tml conductivity k, nd inn nd out dii of i nd o, spctivly. T fin is coold long t sids toug unifom film cofficint nd t t nd toug unifom film cofficint. T contct conductnc, c,t t bs is ssumd to b unifom. T stdy stt on-dimnsionl govning t tnsf qution fo t fin cn b wittn s: 7» d ρ dψ m 2 ψ =0 () ρ dρ dρ w ψ = = b, wit dfind s = T () T nd b = T ( i ) T; ρ = =t, ndm 2 is t nondimnsionl fin pmt. By intoducing t following non-dimnsionl Biot numbs: Bi = t=k (= m 2 ) (2) Bi c = c t=k (3) Bi = t=k (4) t bs nd nd boundy conditions cn b wittn s: dψ ρ = ff; dρ = Bi c[ ψ] (5) ρ = fi; dψ dρ = Bi ψ (6) w ff = i =t nd fi = o =t. T solution of t govning qution subjctd to t boundy conditions givs t fin sistnc, R fin, in t following fom: R fin =[4ßktmffC fωk (mff) I (mff)g] (7) w I nd K t modifid Bssl functions of t fist nd t scond kind of od unity, spctivly; C nd Ω constnts wic involv modifid Bssl functions, t dtils of wic povidd by Yovnovic t l. 3 Tfo, it bcoms obvious tt t nlyticl solution fo constnt ticknss nnul fin quis vlution of t Bssl functions, wic t tims my b difficult to obtin. R s R R s(i) R f(i) R c R s2 Fig. 3 Diffnt sistncs ssocitd wit c contol volum. Rsistnc Mtod As n ltntiv ppoc to t xct closd fom solution, t sistnc mtod is usd to obtin t fin sistnc ccutly nd sily wit minimum computtion ffot. T nti fin lngt (= o i )is dividd into n contol volums, s sown in Fig. 2. T nodl points loctd t t cnt of c contol volum. Figu 3 sows t tml sistncs ptinnt toccontol volum du to t conduction witin t fin mtil (R s ), t convctiv sid (R f )ndnd(r ) coolings nd t bs contct (R c ). T conduction sistncs, (R s), fo t two nd contol volums (CV & CV n ) sptly vlutd in od to simplify t tml cicuit. T gnl xpssions fo c sistnc cn b wittn s: R c = R = R f(j) = = c (4ß i t) (4ß o t) ß (» o (j ) n 2 (8) (9) )(0) o n j i 2 (fo j =;:::(n )) (» ) 2 () (n ) ß o n 2 i (fo j = n) 2of5 Amicn Institut of Aonutics nd Astonutics Pp

4 Tbl t fin. Gomticl nd pysicl pmts fo Pmts i o t k c Vlus 5 mm 0 mm mm 20 W=mK 50 W=m 2 K 20 W=m 2 K 500 W=m 2 K R s(j) =» 2no f2(i )g ln 2n o f32(i )g 4ßkt (fo j =;:::(n )) 2 3 R s = ln 4 o o 2n 5 4ßkt 2 3 R s2 = ln 4 i 2n i 5 4ßkt (2) (3) (4) In od to obtin t totl sistnc, t sistncs fo c contol volum summd, stting fom t tip nd moving towds t bs. Tfo, t quivlnt sistnc fo c contol volum cn b wittn s: 2 = R R f (R s R (5) ) 2 = (6) R 2 (R R s) R 3 =. = R n R f2 2 R f3 (R 2 R s2). 2 R fn (R n R s(n )) (7) (8) w R j (j =;:::n) is t quivlnt sistnc fo t j t contol volum. Hnc, t ovll fin sistnc is xpssd s R = R n R c R s2 (9) Rsults nd Discussions T gomticl nd t pysicl pmts usd to clcult t fin sistnc povidd in Tbl. Wit ts typicl vlus, t nlyticl solution fo t fin sistnc, givn by Eq. (7), is 7:52K=W. By using t sistnc mtod, it is found tt t nlyticl solution cn b obtind wit minimum numb ofcontol volum, s sown in Fig. 4. It is obsvd tt vn wit t contol volums (n =3), t diffnc in t solution btwn t nlyticl nd t sistnc mtods is only 0:04%. It is to b pointd out tt pvious ttmpts using sistnc ppoc 4 quid lg numb of ittions Fin Rsistnc (K/W) Rsistnc Mtod. Exct Solution No. of Contol Volums Fig. 4 Compison of t sult obtind by sistnc mtod wit xct solution. Fin Rsistnc (K/W) Rsistnc Mtod. Appox. Solution[2] No. of Contol Volums Fig. 5 Compison of t solution obtind by sistnc mtod wit ppoximt mtod. 2 (ppoximtly 50 ittions) to obtin t nlyticl solution fo t constnt ticknss fin. Tfo, it sows tt t psnt ppoc isvy fficint, yt ccut, in dtmining t fin sistnc wit minimum computtionl ffot. T psnt sistnc mtod cn b pplid to obtin vious ppoximt solutions wic involv coction fcto fo tip convction nd zo bs sistnc. 2 Fo suc css, good gmnt cn b obtind wit t ppoximt solution by tting t nd nd t sid convction cofficints to b qul, s sown in Fig. 5 wit = = 50 W=m 2 K. T fin sistncs lso clcultd fo diffnt limiting css of Biot numbs nd ty compd wit t nlyticl xpssions givn blow: R = ln( o= i ) 4ßkt R = ln( o= i ) ρ Bic! Bi! Bi! 0 4ßkt 4ß o t ρ Bic! 0 <Bi < Bi! 0 (20) (2) 3of5 Amicn Institut of Aonutics nd Astonutics Pp

5 Tbl 2 Compison of fin sistnc t diffnt limiting css. (n =3) Biot Numb Exct Rsistnc Solution Mtod (K/W) (K/W) Bi! 0 Bi! Bi c! Bi! 0 0 <Bi < Bi c! Bi! 0 0 <Bi < <Bi c < θ k o c i b Fig. 6 Scmtic digm of vibl ticknss nnul fin. R = ln( o= i ) 4ßkt 4ß o t c 4ß o t ρ 0 <Bic < 0 <Bi < Bi! 0 (22) Tbl 2 summizs t sults nd it is found tt by using only t contol volums, t sistnc mtod yilds solutions sm s tos obtind nlyticlly. Vibl Ticknss Fin Rsistnc Mtod So f, it is obsvd tt t psnt sistnc mtod vluts t fin sistnc ccutly fo constnt ticknss nnul fin. As nxt stp, tis ppoc is xtndd to stimt t sistnc of vibl ticknss fin fo wic nlyticl solutions not vilbl in littu. Figu 6 sows scmtic digm fo vibl ticknss nnul fin wos tip nd bs ticknsss nd b spctivly nd is smi-tpd ngl of t fin. By pplying t sistnc mtod to tis gomty, t plcmnt of t contol volums nd t ssocitd tml sistncs min t sm s bfo. T min difficulty wit vibl ticknss fin is to obtin xpssions fo sistncs du to conduction witin t fin, wic diffnt fom tos sttd in Eqs. (2)-(4). To ovcom tis difficulty, Foui's w of t conduction is usd, wic is s follows: Q = ka T w Q is t t flow toug coss-sction A nd ovticknss of wit tmptu diffnc of T. By using t dfinition of sistnc, it cn b sown tt R = T Q = ka (23) w R is t conduction sistnc fo t ticknss. Tis is good ppoximtion, s wit t incs in t numb of contol volums, bcoms smll nd tfo in t limiting cs, t conduction sistnc ppocs t xct vlu. Hnc, xpssions fo t sistncs du to conduction witin t fin, bs conductnc nd tip convction tk t following fom: R s(j) = R s = R s2 = R = R c = =n 4ßk o j n 2 j i (24) n tn (fo j =;:::(n )) =(2n) 4ßk o 4n i 2 4n tn (25) 4ßk i 4n 2ß o 2ß c i b =(2n) 2 4n i(26) tn (27) (28) It is to b notd tt fo sistncs du to convction, t ffctiv t tnsf is tkn s t pojction of t cosponding fo constnt ticknss nnul fin. Tfo, t convctiv sistncs obtind by multiplying t cosponding sistncs of t constnt ticknss fin, givn in Eqs. (0)-(), wit t cosin of t smi-tpd ngl of t fin. T finl xpssion fo t fin sistnc cn b obtind in simil mnn s sown in Eqs. (5)-(9). Rsults nd Discussions In cs of vibl ticknss fin, two dditionl gomticl pmts, nd b, quid long wit tos listd in Tbl. T typicl vlus usd fo nd b 2mm nd 4mm, spctivly. Figu 7 sows t vition of t fin sistnc wit t numb of contol volums, stting wit n = 3. It is obsvd tt t fin sistnc symptoticlly ppocs to constnt vlu witin 20 contol volums. It is lso found tt t diffnc btwn t symptotic vlu nd tt obtind wit t contol volums is only 4of5 Amicn Institut of Aonutics nd Astonutics Pp

6 Fin Rsistnc (K/W) N0. of Contol Volums Fig. 7 Fin sistnc fo vibl ticknss nnul fin. Rfncs W. M. Muy. Ht dissiption toug n nnul disk o fin of unifom ticknss. J. Appl. Mc., 5:A 78, F. P. Incop nd D. P. DWitt. Fundmntls of Ht nd Mss Tnsf. Jon Wily & Sons, Nw Yok, M. M. Yovnovic, J. R. Culm, nd T. F. mczyk. Simplifid nlyticl solutions nd numicl computtion of on nd two-dimnsionl cicul fins wit contct conductnc nd nd cooling. AIAA 24t Aospc Scincs Mting, AIAA : 0, A. J. Cpmn. Ht Tnsf. T Mcmilln Compny, Nw Yok, G. E. Mys. Anlyticl Mtods in Conduction Ht Tnsf. McGw-Hill Book Compny, Nw Yok, M. M. Yovnovic. Non-ittiv contol volum ppoc to on-dimnsionl stdy conduction wit convction: pplictions to xtndd sufcs. ASME Wint Annul Mting, 28:59 69, D. Q. Kn nd A. D. Kus. Extndd Sufc Ht Tnsf. McGw-Hill Book Compny, Nw Yok, 972. Tbl 3 Compison of t fin sistnc fo = 2mm; b =2mm;2t =2mm nd =0(n =30). Constnt Vibl Ticknss Ticknss (mm) (K=W) (K=W) 5 7:5208 7: :455 38:4437 0:02%. Sinc, no closd fom nlyticl solution is possibl fo tis cs, tfo t ccucy of t solution could not b climd wit ctinty. Howv, t solution fo t vibl ticknss fin cn b cckd wit t limiting cs, wn t tpd fin ppocs constnt ticknss fin. Tfo, t vibl ticknss fin solution sould ppoc t constnt ticknss sult by stting = 0 in Eqs. (24) - (26). Tbl 2 sows xcllnt gmnt btwn t constnt ticknss fin nd t vibl ticknss fin fo = 0. Hnc, tis ppoc psnts simpl wy of finding fin sistncs fo diffnt complx gomtis, fo wic only numicl solutions possibl. Conclusion A sistnc ppoc, involving contol volums, is dvlopd fo clculting t fin sistncs of nnul fins wit contct conductnc nd nd cooling. T mtod is fist pplid to constnt ticknss fin nd it is found tt n ccut nd quick solution cn b obtind wit only fw numb of contol volums. Vious limiting css of Biot numbs lso clcultd nd xcllnt gmnts obtind wit nlyticl sults. Tis ppoc is fut xtndd to vibl ticknss fins fo wic no closd fom solution is vilbl. It is obsvd tt t sistnc mtod cn lso b pplid succssfully fo suc vibl ticknss nnul fins. 5of5 Amicn Institut of Aonutics nd Astonutics Pp