Exact 3-D Solution for System with Rectangular Fins, Part 2

Transcription

1 1th WSEAS Int. Conf. on APPLIED MATHEMATICS Caro Egypt Deceber Exact 3-D Soluton for Syste th Rectangular Fns Part ANDRIS BUIKIS MARGARITA BUIKE Insttute of Matheatcs and Coputer Scence Unversty of Latva Rana bulv. 9 Rga LV1459 LATVIA lza.lv/scentsts/bus.ht Abstract: - In ths paper e construct exact analytcal three-densonal soluton for the dstrbuton of the teperature feld n the all th arrays of rectangular fns. We assue that the heat transfer process n the all and the fn s statonary. Ths exact soluton s obtaned by the Green functon ethod. It s obtaned n the for of the syste of nd nd Fredhol ntegral equatons th the nubers of equatons equal to the nuber of the fns. Key-Words: - Steady-state Three-densonal Heat exchange Rectangular fns Array Non-hoogeneous envronent Exact analytcal soluton Fredhol ntegral equatons. 1 Introducton In part 1 of ths paper e have consdered atheatcal three-densonal forulaton of steady-state proble for one eleent of syste th rectangular fn (as usually n the lterature e.g. [1]- [6]). In ths paper e concentrate our attenton on the hole syste assebled nto array of fns. Such stateent essentally generalzes the probles consdered earler n lterature e.g. n paper [7] and doctoral thess [8]. In ths part of e obtan exact analytcal soluton by the Green functon ethod. The soluton has the for of the syste of nd nd Fredhol ntegral equatons. In other ords e have reduced the orgnal proble for the Laplace equaton n doan th extended surfaces (all th the array of fns) to the syste of Fredhol ntegral equatons. The order of ths syste s equal the nuber of the fns. x y z Δ x = y = z = = B R B R B R B R L l = B b = B R B R h ( B R) β = hb ( R) h ( B R) W β = β = = B. R Matheatcal Forulaton of 3-D Proble In ths part of our paper e are odelng the hole syste th N ( = 1 N) fns as t s depcted n fgure on ths page. We ll use the sae as n part 1 all densonless arguents and paraeters addng to the total densonal length = ( B R). Here the densonless B length = ( N 1) B R here N the total nuber of the fns: The densonless teperatures are ntroduced n the sae for as earler but n the contradcton to the part 1 n dfferent ay ll be ntroduced the averaged envronental teperatures:

2 1th WSEAS Int. Conf. on APPLIED MATHEMATICS Caro Egypt Deceber V ( x y z) Ta V( x y z) = Tb Ta V ( x y z) T V a ( x y z) = Tb Ta Θ ( x yz ) Ta Θ ( xyz ) = T T Θ ( yz ) T Θ = a ( yz ). Tb Ta b a Here Θ ( yz ) s the densonal surroundng (envronent) teperature on the left (hot) sde (the heat source sde) of the all Θ ( x yz )- the surroundng teperature on the rght sde of the all and the fns (on the heat sn sde). Further V ( x y z) ( V ( x y z) ) are the densonal teperature n the fn (all). The all (base) no occuped bounded serrated x y z [ ]. The N doan { [ ] [ ] } rectangular fns occupy the doans ( = 1 N ): x l y [ y y ] z [ ]. { [ ] } Here: y = ( b r)( 1) y = y b = 1 N. Fnally here Ta( T b) are ntegral averaged envronent teperatures over edges orthogonal to x drecton: N 1 y 1 1 Ta = ( ) dy Θ ( y z) dz = 1 y N y dz Θ ( l y z) dy ī = 1 ( ) y 1 T = dy Θ ( y z) dz. b 3 Reducton of 3-D Model to -D Proble and ts Full Matheatcal Forulaton We descrbe the densonless teperature feld by V ( x y z) V ( x y z) = 1 N n the functon ( ) all (fns). They fulfll the equatons: V V V = y V V V = = 1 N. y As the odel n ths part of our paper e consder agan the three densonal stateent th gven (prescrbed) heat fluxes fro the flan surfaces (edges): = Q( x y) = Q3 ( x y) z (1) = z= = Q ( x y) = Q ( x y). 3 z= z= Such type of boundary condtons (BC) allos us n slar to part 1 ay ae the exact reducng of ths three-densonal proble for Laplace equatons to to-densonal proble for Posson equatons. Ths can be done by conservatve averagng ethod [1] [13]. For ths goal e ntroduce follong ntegral average values: 1 = 1 = Θ 1 = 1 V ( x y) V ( x y z) dz ϑ ( y) ( y z) dz V ( x y) V ( x y z) dz ϑ( x y) = Θ( x y z) dz. () It reans to realze the ntegraton of an equaton by usage of the both BC (correspondng one par) and e obtan detals see n [13]: V V Q ( x y) = y (3) V V Q ( ) 1. x y = = N y Here 1 Q( x y) = ( Q3( x y) Q( x y) ) 1 Q ( x y) = Q ( x y) Q ( x y). ( 3 ) Agan e ust add to an partal dfferental equatons (3) needed BC as follo:

3 1th WSEAS Int. Conf. on APPLIED MATHEMATICS Caro Egypt Deceber β [ ϑ( y) V] = y ( ) 1 x= β[ V ϑ( x y) ] = y ( y y ) = 1 N 1 y y = Q ( x) = Q ( ). 1 x x= (4) (5) (6) (7) We assue the sae as n part 1 conugaton condtons on the surface beteen the all and the fns (for y [ y y ] = 1 N ). They descrbe the deal theral contact beteen the all and the fns: V = V (8) x= x= β = β. (9) x= x= Rear: e consder here the stuaton here all fns have the sae theral propertes but physcal propertes of the fns could be dfferent fro the propertes of the all. In ost general case (hen dfferent fns have dfferent theral propertes) nstead of the sae densonless heat exchange coeffcent β for all fns e ould have had dfferent β = β. We underlne that the proposed here ethod ors n ths general case thout prncpal changes. We have follong BC for the fns: β[ V ϑ( x y) ] = (1) y [ y y ] = 1 N x= l β[ V ϑ( x y) ] = y y β[ V ϑ( x y) ] = y y (11) x [ l] = 1 N. As n the part 1 e assue agan that all condtons hch ensure exstence and unqueness of classc soluton of the proble (3)-(11) e.g. contnuty of envronent teperatures consstency condtons on the sdes of edges etc. are fulflled. 4 Exact Soluton of -D Proble The cobnaton of the equatons (8) (9) and (5) allo us rerte the as follong BC: βv x x= = βf( y) (1) here F ( x y) y y y = 1 N; F ( x y) = ϑ( x y) = ϑ( y) y < y < y 1 (13) = 1 N 1; 1 F ( x y) = V ; β x [ l]. By the gven (non) functon F ( x y ) e can represent the soluton for the all n for: V ( x y) = Ψ ( x y) N β F ( υ ) G ( x y υ ) dυ l 1 In forula (14) e have denoted: Ψ ( x y) = Q ( ζ ) G ( x y ζ ) dζ 1 Q ( ζ) G ( x y ζ) dζ β ϑ ( υ) G ( x y υ) dυ N 1 1 β ϑυ ( ) G ( x y υ ) dυ 1 dζ Q ( ζ υ) G ( x y ζ υ) dυ. (14) The Green functon n (14) has slar th part 1 for see e.g. [11]. Dfference s only n the total length of the all n the y drecton: ( x) ( y) G ( x ζ ) G n ( y υ) G ( x y ζυ ) = (15) n = 1 nπ μ

4 1th WSEAS Int. Conf. on APPLIED MATHEMATICS Caro Egypt Deceber G ϕ ( x) ϕ ( ζ) ( x ζ ) = ϕ ( x) ( y) G n ( y υ) = (16) nπ nπ cos ( y υ) cos ( yυ). The egenfunctons have follong expresson for the frst one-densonal Green functon: β ϕ( x) = cos( μx) sn( μx) ϕ = μ μ ( ) ( ) β β (17) β β = 1. μ μ ( ) μ μ β Here μ are the roots of the transcendental equaton: μ ( β β ) tg( μ ) =. (18) μ ββ The representaton (14) for the soluton n the all s under explotable as soluton because of unnon functons F ( x y ).e. teperature felds V ( x y ) n the fns. In the sae ay as for (1) e can rerte the conugatons condtons n the for of BC on the left sde of each rectangular fn: βv F( y) x = β (19) x= y [ y y ] = 1 N. Here the rght hand sde functon of BC (19) has the for: 1 F( x y) = V β () x [ ] y [ y y ] = 1 N. Then slar as n forula (14) for the all e can represent soluton for the th fn n follong for: V ( x y) =Ψ( x y) y β F( η ) G( x y y η y ) dη y (1) The non functon Ψ ( x y) has the for (on the three boundares here the tradtonal boundary condtons are gven): Ψ ( xy ) = l β ϑξ ( y ) G( x y y ξ) dξ β ϑ( ξ y ) G( x y y ξ b) dξ y β ϑ ( l η) G( x y y l η y ) dη y l l y dξ Q ( ξη ) G( x y y ξη y ) dη. y Further see [11]: Gxy ( ξ η) = φ ( x) φ( ξ) ψ( y) ψ( η) = 1 φ ψ ( λ κ ) φ( x) = cos λ ( x ) β sn λ ( x ) λ l β βl φ = 1 λ β ψ( y) = cos( κy) sn ( κy) κ β ψ = b ( 1 βb). κ () (3) Here λ ( κ ) are the roots of the transcendental equatons: λ β κβ tan( λl) = tan( ) =. λ β κ β bκ (4) Usng notatons (13) () and representaton (1) e obtan easy the follong equaton: F ( x y) =Ψ ( x y) y y F( η) Γ( x y y η y ) dη (5) here Γ ( xy ξ η) = β Gxy ( ξ η) 1 Ψ ( xy ) = β Ψ( xy ). β Fro (14) e obtan edately slar representaton for the F( x y ):

5 1th WSEAS Int. Conf. on APPLIED MATHEMATICS Caro Egypt Deceber F( x y) =Ψ ( x y) N 1 F ( υ) Γ ( x y υ) dυ. l Here e have ntroduced follong notatons: Γ ( xy ζ υ) = β G( xy ζ υ) (6) 1 Ψ ( xy ) = β Ψ( xy ). β On the lnes beteen the all and fns the functon F( x y) taes the for: F( y) =Ψ ( y) N 1 F ( υ) Γ ( y υ) dυ. l (7) No e substtute the representaton (7) n the rght hand sde of forula (5): F ( y) =Ψ ( y) y y Ψ ( η ) Γ( y y η y ) dη F ( υ ) dυ y y N 1 l Γ ( ηυ ) Γ( y y η y ) dη. Fnally e obtan follong syste of the second nd Fredhol ntegral equatons regardng the functons F ( y) = 1 N : F ( y) =Φ ( y) N 1 K( y υ) F ( υ ) dυ. l (8) Here e have ntroduced follong shorter denonatons: Φ ( y) = Ψ ( y) y y Ψ ( η ) Γ( y y η y ) dη K( y υ) = y y Γ ( ηυ ) Γ( y y η y ) dη. After s solved syste of ntegral equatons (8) th contnuous ernels fro the representaton (14) e can obtan edately the teperature feld n the all and the functon F( y). In ts turn the representaton (1) gves the teperature felds n all fns. Ths proble (th non-hoogeneous envronent teperatures) and ts exact soluton allo conugatng teperature feld th hydrodynac (oton of flud or gas by the sde of fns and along the left edge of the all). Secondly f e had 3 rd type BC nstead of the BC (1) e ould have had full threedensonal proble. 5 Conclusons We have constructed exact three densonal analytcal soluton for the syste th rectangular fns here the all and the fns consst of aterals th dfferent theral propertes. The soluton has the for of the syste of nd nd Fredhol ntegral equatons. The order of ths syste s equal the nuber of the fns. Acnoledgeents: Research as supported by Unversty of Latva (proect -ZP9-1) and Councl of Scences of Latva (grant 5.155). References: [1] Kern D.Q Kraus A.D. Extended Surface Heat Transfer. McGra-Hll Boo Copany [] Kraus A.D. Analyss and Evaluaton of Extended Surface Theral Systes. Hesphere Publshng Corporaton 198. [3] Manzoor M. Heat Flo through Extended Surface Heat Exchangers. Sprnger-Verlag: Berln and Ne or [4] Wood A.S. Tuphole G.E. Bhatt M.I.H. Heggs P.J. Perforance ndcators for steady-state heat transfer through fn assebles Trans. ASME Journal of Heat Transfer pp [5] Bus A. Bue M. Closed to-densonal soluton for heat transfer n a perodcal syste th a fn. Proceedngs of the Latvan Acadey of Scences. Secton B Vol.5 Nr pp [6] Bus A. Bue M. Gusenov S. Analytcal to-densonal solutons for heat transfer n a syste th rectangular fn. Advanced

6 1th WSEAS Int. Conf. on APPLIED MATHEMATICS Caro Egypt Deceber Coputatonal Methods n Heat Transfer VIII. WIT press 4 pp [7] Lehtnen A. Karvnen R. Analytcal threedensonal soluton for heat sn teperature Proceedngs of IMECE p. [8] Lehtnen A. Analytcal Treatent of Heat Sns Cooled by Forced Convecton. Doctoral Thess. Tapere Unversty of Technology. Publcaton 579. Tapere 5. [9] Carsla H.S. Jaeger C.J. Conducton of Heat n Solds. Oxford Clarendon Press [1] Őzş M. Necat. Boundary Value Probles of Heat Conducton. Dover Publcatons Inc. Mneola Ne or [11] Polyann A.D. Handboo of Lnear Partal Dfferental Equatons for Engneers and Scentsts. Chapan&Hall/CRC. (Russan edton 1) [1] Bus A. Conservatve averagng as an approxate ethod for soluton of soe drect and nverse heat transfer probles. Advanced Coputatonal Methods n Heat Transfer IX. WIT Press 6. p [13] M.Bue A.Bus Approxate Solutons of Heat Conducton n Mult- Densonal Cnder Type Doan by Conservatve Averagng Method Part I Proc. of the Heat Transfer Theral Engneerng and Envronent Conference Voulagen Athens August pp. 15.