Exact 3-D Solution for System with Rectangular Fin, Part 1

Transcription

1 th WSEAS Int. Conf. on APPLIED MATHEMATICS Cro Egpt Deceer Exct 3-D Soluton for Sste th Rectngulr Fn Prt MARGARITA BUIKE ANDRIS BUIKIS Insttute of Mthetcs nd Coputer Scence Unverst of Ltv Rn ulv. 9 Rg LV459 LATVIA lz.lv/scentsts/uks.ht Astrct: - In ths pper e construct severl exct nltcl three-densonl solutons for the dstruton of the teperture feld n the ll th rectngulr fn. We ssue tht the het trnsfer process n the ll nd the fn s sttonr. These exct solutons re otned the Green functon ethod n the for of the nd knd Fredhol ntegrl equton. The generlze trdtonl stteents n severl senses e.g. e consder 3-D stteent dfferent oundr condtons nd the het exchnge tke plce t non-hoogeneous envronentl teperture. Ke-Words: - sted-stte three-densonl het exchnge rectngulr fn non-hoogeneous envronent exct nltcl solutons. Introducton Sstes th extended surfces (fns spnes) re relted to refrgertors rdtors engnes nd odern electroncs (PC) etc. Usull ther thetcl odelng s relzed one densonl sted-stte ssuptons []-[5]. In our prevous ppers e hve constructed vrous to densonl nltcl pproxte [6] [] nd exct [] solutons. In ths pper e concentrte our ttenton on one eleent of fn ssel the hole sste (sseled nto rrs of fns) ll e consdered n the second pper. Such stteent essentll generlzes the prole consdered erler n lterture e.g. n pper []. In these to prts of our pper e otn severl ne exct nltcl solutons the Green functon ethod [3]-[6]. L B h ( B R) l B R B R k hb ( R) h ( B R) k k nd tepertures: W B R Mthetcl Forulton of 3-D Prole In ths prt e ll consder full thetcl three-densonl forulton of sted-stte prole for one eleent of sste th rectngulr fn (ths one eleent s depcted th drker color n ttched fgure). Ths thetcl forulton s essentll roder s n our ppers [6]-[]. We ll use follong densonless rguents preters: x z Δ x z B R B R B R B R V ( x z) T V( x z) T T V ( x z) T V ( x z) T T

2 th WSEAS Int. Conf. on APPLIED MATHEMATICS Cro Egpt Deceer Θ ( x z ) T Θ ( xz ) T T Θ ( z ) T Θ ( z ). T T We hve ntroduced follong densonl therl nd geoetrcl preters: k( k ) - het conductvt coeffcent for the fn (ll) h( h ) - het exchnge coeffcent for the fn (ll) B fn dth (thckness) L fn length Δ - thckness of the ll W lls dth (length) R dstnce eteen to fns (fn spcng). Further Θ ( z ) s the surroundng (envronent) teperture on the left (hot) sde (the het source sde) of the ll Θ ( x z )- the surroundng teperture on the rght (cold - the het snk sde) of the ll nd the fn. Fnll V ( x z) ( V ( x z) ) re the densonl tepertures n the fn (ll) here T ( T ) re ntegrl verged envronent tepertures over pproprte edges: W ( ) Δ L W W B T B R L W [ dz Θ( Δ z) d Δ B R W ( ) dx Θ ( x B z) dz dz Θ( Δ L z) d] T W B R d Θ ( z) dz. B R The one eleent of the ll (se) s plced n the don { x [ ] [ ] z [ ] }. The rectngulr fn n densonless rguents occupes the don { x [ l] [ ] z [ ] }. We descre the densonless teperture feld functon V ( x z ) ( V ( x z ) ) n the ll (fn). The fulfll the Lplce equtons: V V V V V V. At frst e consder the three densonl stteent th gven het fluxes fro the flnk surfces (edges) nd fro the top nd the otto edges: B Q ( x ) Q ( x ) 3 z z Q ( x ) Q ( x ). 3 z z () Such tpe of oundr condtons (BC) llos us to ke the exct reducng of ths three-densonl prole to to-densonl prole for Posson equton conservtve vergng ethod [7]- []. Let us ntroduce follong ntegrl verge vlues: Θ V ( x ) V ( x z) dz ϑ ( ) ( z) dz V( x ) V( x z) dz ϑ( x ) Θ( x z) dz. () It rens to relze the ntegrton of n equton usge of the oth BC () (correspondng one pr) nd e otn: V V Q ( x ) V V ( ). Qx (3) Here Q( x ) ( Q3( x ) Q( x ) ) Qx ( ) ( Q3( x ) Q( x )). We dd to n prtl dfferentl equtons (3) needed BC s follo: [ ϑ( ) V] x () (4) [ V ϑ( x ) ] ( ) x (5) Q( x) (6) Q ( x). (7)

3 th WSEAS Int. Conf. on APPLIED MATHEMATICS Cro Egpt Deceer We llo the terl of the fn to e dfferent fro the lls terl. It ens e ust forulte the conugtons condtons on the surfce eteen the ll nd the fn. We ssue the s del therl contct - there s no contct resstnce: V V x x (8). x x (9) We hve follong BC for the fn: [ V ϑ( x ) ] x l [ ] () Q ( x) () [ V ϑ( x ) ] x [ l].() We ssue tht ll condtons hch ensure exstence nd unqueness of clssc soluton of the prole (3)-() e.g. contnut of envronent tepertures consstenc condtons on the sdes of edges etc. re fulflled. Let s enton tht lost ll of the uthors neglgle the het trnsfer trough flnk surfce z (s ell s fro edge z ). We ssue gven (prescred) het fluxes on oth. 3 Exct Soluton of -D Prole 3. Soluton of the Splfed Prole We ould lke to expln the n de of soluton for the -D cse of perodcl sste th constnt densonless envronentl tepertures ϑ ( Θ T ) nd ϑ ( Θ T ). We neglect ddtonll the het fluxes fro flnk edges. In ths prtculr cse e hve follong n equtons for the teperture U ( ) x of the ll respectvel teperture U( x ) of the fn: U U { x [ ] [ ]} (3) U U { x [ l] [ ] }. (4) The BC (6) (7) nd () re ssued to e hoogeneous: U U U. (5) Insted of BC (4) (5) () nd () e hve: U ( U) x x () (6) U ( ) x x (7) U x l [ ] (8) U x [ l]. (9) The conugtons condtons on the lne eteen the ll nd the fn re stll stndng n the for (8) (9) for the functons U( x ) nd U ( ) x. The lner conton of the equtons (8) (9) together th BC (7) llo us rerte the s follong BC on the rght hnd sde of the ll: U x x F( ) () here U U F ( x ) () < x [ l]. On the ssupton tht the functon F ( x ) s gven e cn represent soluton for the ll n ver ell knon for the Green functon: U( x ) G( x υ) () F ( υ ) G ( x υ ). Tkng n the ccount forul () e rerte the soluton for the ll s follo: U ( x ) G ( x υ) U U G ( x υ ) d υ. ζ ζ (3) The expresson of the Green functon n () (3) hs the for (see e.g. [5]): x G ( x ζ ) G n( υ) G ( x ζυ ) (4) n ( πn) μ

4 th WSEAS Int. Conf. on APPLIED MATHEMATICS Cro Egpt Deceer x ϕ( x) ϕ( ζ) G ( x ζ ) ϕ G n ( υ) cos nπ ( υ) cos nπ ( υ). We hve for the frst one-densonl Green functon n (4) the follong expresson for the egenfunctons: ϕ( x) cos( μx) sn( μx) ϕ μ μ ( ) ( ). μ μ μ ( ) μ Here μ re the roots of the trnscendentl equton: μ ( ) tg( μ ). μ Unfortuntel the representton () s unusle s soluton for the ll ecuse of unknon functon F ( x ).e. teperture n the fn U( x ). Tht s h e ll p ttenton to the soluton for the fn no. In the se s for () e cn rerte the conugtons condtons n the for of BC on the left sde of the rectngulr fn: U F( ). (5) x Here the rght hnd sde functon of BC (5) hs the for: U F( x ) U (6) x [ ] [ ]. Then slr s for the ll e cn represent soluton for the fn n follong for: U( x ) F( η ) G( x η ) dη. (7) ( x) φ Here Gx ( ξη ) G ( x ξ ) G ( η) G ( x ) G ( x) ( ) λ κ φ ( x) φ ( ξ) ( x ξ ) G ( ) ψ ( η) ( η) ψ ( x ) φ( x) cos λ sn λ ( x ) λ l φ λ λ ψ ( η) ( ) ( ) cos κ η cos κ η ψ. κ Here λ ( κ ) re the roots of the trnscendentl equtons: λ tn( λl) tn( κ ). λ κ Usng notton () nd representton (7) e cn es otn the follong equton: F ( x ) F( η) Γ( x η) dη (8) Γ ( x ξ η) Gx ( ξ η). Fro () e otn edtel slr representton for the F( ) : F( ) ( υ) Γ (9) F ( υ ) Γ ( υ ). Here e hve ntroduced notton slr to the second equton of the forul (8): Γ ( x ζ υ) G( x ζ υ). (3) No e susttute the representton (9) n the rght hnd sde of forul (8) nd e otn follong second knd Fredhol ntegrl equton regrdng the functon F ( ) : F ( ) ϒ ( ) K( υ) F ( υ). (3) Here e hve ntroduced follong shorter denontons:

5 th WSEAS Int. Conf. on APPLIED MATHEMATICS Cro Egpt Deceer K( υ) Γ ( ηυ ) Γ( η ) dη ϒ ( ) Γ( η) dη Γ ( η υ). (3) When solved ntegrl equton (3) e edtel cn otn the teperture feld n the ll fro the representton (). In ts turn the representton (7) gves the teperture feld n the fn. B the n ll our ppers [6]-[] e restrct ourselves th hoogeneous oundr condtons (5) (7)-(9). 3. Soluton of the Generl Prole (3)-(9) Here no ll e consdered generl cse of non-hoogeneous envronentl teperture: dfferentl equtons (3) th oundr condtons (4)-(). The soluton for the ll nsted of () hs for: V( x ) Ψ ( x ) (33) F ( υ ) G ( x υ ). Here the knon ters re oned together: Ψ ( x ) Q ( ζ ) G ( x ζ) dζ Q ( ζ) G ( x ζ) dζ ϑ ( υ) G ( x υ) ϑ( υ) G ( x υ) dζ Q ( ζ υ) G ( x ζ υ). (34) In the slr for e cn represent soluton for the fn. It looks s follo: V( x ) Ψ( x ) F( η ) G( x η ) dη. The knon functon Ψ ( x ) hs the for: l Ψ ( x ) Q( ξ ) Gx ( ξ) dξ (35) ϑ ( l η) G( x l η) dη l ϑ( ξ Gx ) ( ξ d ) ξ l dξ Q( ξ η) G( x ξ η) dη. (36) We otn nsted of forule (8) nd (9) follong representtons: F( x ) Ψ ( x ) F ( υ) Γ ( x υ) F ( ) Ψ ( ) F( η) Γ( η) dη. (37) We hve ntroduced follong nottons n (37): Ψ ( x ) Ψ( x ) Ψ ( x ) Ψ( x ). We otn follong non-hoogeneous Fredhol ntegrl equton of nd knd n the se s equton (3) n su-secton 3.: F ( ) Φ ( ) K( υ) F ( υ). (38) Here Φ ( ) Ψ ( ) Ψ ( η ) Γ( η ) dη. Ths Fredhol ntegrl equton of nd knd hs contnuous kernel nd t hs unque soluton see e.g. []. Agn hen solved ntegrl equton (38) e cn otn edtel fro (33) the teperture feld n the ll. Then frst forul (37) llos fndng the conton F( x ). In ts turn forul (35) gves the teperture feld n the fn. We fnsh ths prt of our pper th the follong to rerks. Frstl the lst prole (th non-hoogeneous envronent tepertures) nd ts soluton llo conugtng teperture feld th hdrodnc (oton of flud or gs eteen to fns nd long the left edge of the ll). Secondl f e hd 3 rd tpe

6 th WSEAS Int. Conf. on APPLIED MATHEMATICS Cro Egpt Deceer BC nsted of the BC () e ould hve hd full three-densonl prole. 4 Conclusons We hve constructed severl exct three densonl nltcl solutons for one eleent of perodcl sste th rectngulr fn here the ll nd the fn consst of terls hch hve dfferent therl propertes. These solutons re n the for of Fredhol ntegrl equton of nd knd nd hs contnuous kernel. The re spler then the one otned n our pper []. The llo pssng over fro proles for ndvdul fns to proles for fns rrs hch ll e consdered n the prt of ths pper. Acknoledgeents: Reserch s supported Unverst of Ltv (proect Y-ZP9-) nd Councl of Scences of Ltv (grnt 5.55). References: [] Kern D.Q Krus A.D. Extended Surfce Het Trnsfer. McGr-Hll Book Copn. 97. [] Krus A.D. Anlss nd Evluton of Extended Surfce Therl Sstes. Hesphere Pulshng Corporton 98. [3] Mnzoor M. Het Flo through Extended Surfce Het Exchngers. Sprnger-Verlg: Berln nd Ne York 984. [4] Wood A.S. Tuphole G.E. Bhtt M.I.H. Heggs P.J. Perfornce ndctors for sted-stte het trnsfer through fn sseles Trns. ASME Journl of Het Trnsfer pp [5] Wood A.S. Tuphole G.E. Bhtt M.I.H. Heggs P.J. Sted-stte het trnsfer through extended plne surfces Int. Coun. n Het nd Mss Trnsfer No. 995 pp [6] Buks A. To-densonl soluton for het trnsfer n regulr fn ssel. Ltvn Journl of Phscs nd Techncl Scences No pp [7] Buks A. Buke M. Approxte nltcl to-densonl soluton for longtudnl fn of rectngulr profle. Act Unverstts Ltvenss vol pp [8] Buks A. Buke M. Closed to-densonl soluton for het trnsfer n perodcl sste th fn. Proceedngs of the Ltvn Acde of Scences. Secton B Vol.5 Nr pp. 8-. [9] Buke M. Sulton of sted-stte het process for the rectngulr fn-contnng sste Mthetcl Modellng nd Anlss 999 vol. 4 pp [] Mlk M.Y. Wood A.S. Buks A. An pproxte nltcl soluton to flr conugte het trnsfer prole Interntonl ournl of Pure nd Appled Mthetcs Vol. Nr. 4 pp [] Buks A. Buke M. Gusenov S. Anltcl to-densonl solutons for het trnsfer n sste th rectngulr fn. Advnced Coputtonl Methods n Het Trnsfer VIII. WIT press 4 pp [] Lehtnen A. Krvnen R. Anltcl threedensonl soluton for het snk teperture Proceedngs of IMECE p. [3] Crsl H.S. Jeger C.J. Conducton of Het n Solds. Oxford Clrendon Press 959. [4] Őzşk M. Nect. Boundr Vlue Proles of Het Conducton. Dover Pulctons Inc. Mneol Ne York 989. [5] Polnn A.D. Hndook of Lner Prtl Dfferentl Equtons for Engneers nd Scentsts. Chpn&Hll/CRC. (Russn edton ) [6] Stkgold I. Boundr Vlue Proles of Mthetcl Phscs Vol.. SIAM Phldelph. [7] Buks A. Aufgenstellung und Loesung ener Klsse von Proleen der thetschen Phsk t nchtklssschen Zustzedngungen. Rostock. Mth. Kolloq pp (In Gern) [8] Buks A. Proles of thetcl phscs th dscontnuous coeffcents nd ther pplctons. Rg p. (In Russn unpulshed ook) [9] Buks A. Conservtve vergng s n pproxte ethod for soluton of soe drect nd nverse het trnsfer proles. Advnced Coputtonl Methods n Het Trnsfer IX. WIT Press 6. p [] Vlus R. Buks A. Conservtve vergng ethod for prtl dfferentl equtons th dscontnuous coeffcents. WSEAS Trnsctons on Het nd Mss Trnsfer. Vol. Issue 4 6 p [] Guenther R.B. Lee J.W. Prtl Dfferentl Equtons of Mthetcl Phscs nd Integrl Equtons. Dover Pulctons Inc. Ne York 996.