INTEGRAL TRANSFORM SOLUTION OF NATURAL CONVECTION IN A SQUARE CAVITY WITH VOLUMETRIC HEAT GENERATION

Transcription

1 Brazlan Journal of Checal Engneerng ISSN Prnted n Brazl Vol. 3, No. 4, pp , October - Deceber, 23 INTEGRAL TRANSFORM SOLUTION OF NATURAL CONVECTION IN A SQUARE CAVITY WITH VOLUMETRIC HEAT GENERATION C. An,2, C. B. Vera and J. Su Nuclear Engneerng Progra, COPPE, Phone: + (55) (2) , Fax: + (55) (2) , Unversdade Federal do Ro de Janero, CP 6859, CEP: , Ro de Janero - RJ, Brazl. E-al: sujan@nuclear.ufrj.br 2 Offshore Ol/Gas Research Center, College of Mechancal and Transportaton Engneerng, Chna Unversty of Petroleu (Bejng), Bejng 2249, Chna. (Subtted: January 4, 22 ; Revsed: October 9, 22 ; Accepted: October, 22) Abstract - The generalzed ntegral transfor technque (GITT) s eployed to obtan a hybrd nuercalanalytcal soluton of natural convecton n a cavty wth voluetrc heat generaton. The hybrd nature of ths approach allows for the establshent of benchark results n the soluton of non-lnear partal dfferental equaton systes, ncludng the coupled set of heat and flud flow equatons that govern the stea natural convecton proble under consderaton. Through perforng the GITT, the resultng transfored ODE syste s then nuercally solved by akng use of the subroutne DBVPFD fro the IMSL Lbrary. Therefore, nuercal results under user prescrbed accuracy are obtaned for dfferent values of Raylegh nubers, and the convergence behavor of the proposed egenfuncton expansons s llustrated. Crtcal coparsons aganst solutons produced by ANSYS CFX 2. are then conducted, whch deonstrate excellent agreeent. Several sets of reference results for natural convecton wth voluetrc heat generaton n a b-densonal square cavty are also provded for future verfcaton of nuercal results obtaned by other researchers. Keywords: Natural convecton; Square cavty; Voluetrc heat generaton; Integral transfor; Hybrd soluton. INTRODUCTION The nvestgatons of natural convecton flow and heat transfer processes, drven by buoyancy forces due to densty dfferences caused by teperature varatons n the flud, have been otvated by ther portance n any natural and ndustral probles (Cheng, 26, 2; Kar and Murthy, 2). In all these nvestgatons, consderable work has been done on the stu of natural convecton n voluetrcally heated cavtes due to applcatons n varous technologcal areas such as nuclear reactor desgn (Baker et al., 976a,b; Qu et al., 2), geophyscs (Runcorn, 962; Mckenze et al., 974) and astrophyscs (Bethe, 968; Trtton, 975). Many experental, analytcal and nuercal approaches have been eployed to nvestgate the characterstcs of flow and heat transfer n enclosures wth voluetrc heat generaton. Lee and Goldsten (988) used a Mach-Zehnder nterferoeter to nvestgate experentally the teperature dstrbuton and the heat transfer rates wthn an nclned square enclosure contanng flud wth nternal energy sources bounded by four rgd planes of constant equal teperature. The experental results of Lee and Goldsten (988) were later eployed to copare wth the nuercal solutons obtaned by May (99). Based on a fnte-dfference procedure, Acharya and Goldsten (985) and Rahan and Sharf (23) separately studed natural convecton To who correspondence should be addressed

2 884 C. An, C. B. Vera and J. Su n an nclned and vertcal square enclosure contanng nternal energy sources and subjected to external heatng, and analyzed the flow patterns for dfferent Raylegh nubers (both nternal and external). To verfy the stea-state experental observatons of Kawara et al. (99), Fuseg et al. (992) solved nuercally natural convecton n a dfferentally heated rectangular cavty wth nternal heat generaton by a control volue-based fnte dfference technque. Usng a fnte-volue technque, D Pazza and Cofalo (2) perfored drect nuercal twodensonal sulatons of low-prandtl nuber free convecton n a voluetrcally heated rectangular enclosure wth aspect rato ( AR = 4), adabatc top/ botto walls and sotheral sde walls. Dfferent flow reges (stea-state, perodc and chaotc) were obtaned based on the values of the Grashof nuber. Extendng prevous work, Arcdacono et al. (2) and Arcdacono and Cofalo (2) predcted the flow and teperature felds at dfferent Grashof nubers n a square cavty ( AR = ), and subsequently n a shallow cavty ( AR =.25). By expressng the governng equatons n ters of streafuncton, vortcty and teperature forulaton, Josh et al. (26) conducted a se-analytcal stu of natural convecton n cavtes of dfferent aspect ratos wth unfor voluetrc heat generaton, consderng two dfferent boundary condtons, vz., all sotheral walls and only adabatc horzontal walls. Danels and Jones (998) used atched asyptotc expansons ethod to analyze natural convecton n a shallow rectangular cavty due to nternal heat generaton, where the nonlnear convectve effects were studed. Moreover, Qu et al. (2) nvestgated nuercally the local Nusselt nuber, the flow and teperature feld at dfferent condtons of twodensonal natural convecton wth non-unfor heat generaton n a confned enclosure. Recently, An and Su (2) eployed a hybrd nuercal-analytcal approach, known as GITT, to stu the nac response of claped axally ovng beas, where excellent convergence behavor of the ntegral transfor soluton was shown by coparng the vbraton dsplaceent of dfferent ponts along the bea length. Indeed, past studes have shown that GITT s a powerful and applcable ethod for solvng dffuson and convecton-dffuson probles n heat and flud flow (Cotta, 993; Cotta and Mkhalov, 997; Cotta, 998). The ost nterestng feature of ths technque s the autoatc and straghtforward global error control procedure, whch akes t partcularly sutable for bencharkng purposes, and the only ld ncrease n overall coputatonal effort wth ncreasng nuber of ndependent varables. Leal et al. (999) exaned the convergence characterstcs of the GITT soluton for the stea lanar natural convecton of a Newtonan flud nsde rectangular enclosures dfferentally heated at the vertcal walls and deonstrated the usefulness of ths ntegral transfor ethod by perforng crtcal coparsons aganst prevously reported benchark solutons. As a contnuaton of ther research, Leal et al. (2) studed the behavor of flud flow and heat transfer for transent lanar natural convecton wth varable physcal propertes nsde cavtes, and presented the advantages of utlzng the false transent technque for obtanng stea solutons wth the GITT ethod. The objectve of ths work s to extend the GITT ethod to stu natural convecton n a square cavty wth unfor voluetrc heat generaton, whch s one of the key ssues concerned n nuclear safety. Snce there s no approxaton nvolved n the analytcal dervaton of the GITT approach, and the proble soluton s represented by the su of a rapdly-convergng n fnte seres, the GITT solutons wth the desred precson ay serve as bencharks for testng other nuercal ethods. Two dfferent boundary condtons are analyzed for the cavty: (a) horzontal walls adabatc and vertcal walls sotheral, (b) all walls sotheral. In the next secton, the atheatcal forulaton of the proble of natural convecton n a voluetrcally heated cavty s presented. Then, the hybrd nuercal-analytcal soluton s obtaned by carryng out ntegral transfor, and nuercal results wth autoatc global accuracy control are presented for dfferent boundary condtons. A coparson aganst the sulaton results of square cavtes obtaned by ANSYS CFX 2. s then perfored to assess the accuracy and convergence of the present approach. MATHEMATICAL FORMULATION Consder a square cavty of sde H, contanng a flud wth unfor voluetrc heat generaton rate q. Two dfferent boundary condtons are consdered n the present stu: (a) the horzontal walls are adabatc and the vertcal walls are sotheral, antaned at a constant teperature T, and (b) all walls are sotheral, antaned at a contant teperature T as llustrated n Fg.., Brazlan Journal of Checal Engneerng

3 Integral Transfor Soluton of Natural Convecton n a Square Cavty wth Voluetrc Heat Generaton v v p v v ρ u + v 2 2 x y = +μ + +ρβ ( ) T T g, (2b) 2 2 T T T T ρ cp u + v = k q x y ''', (3) (a) where x and y are respectvely the horzontal and vertcal spatal coordnates, u and v the velocty coponents, p the pressure, T the teperature, g the gravtatonal acceleraton, ρ the densty, β the theral expanson coeffcent, c p the specfc heat, and k the theral conductvty of the flud. The boundary condtons for the two probles consdered are respectvely gven by: u = v=, T = T at x =, (4a) u = v=, T = T at x= H, (4b) T u = v=, = at y=, T u = v=, = at y= H (4c) (4d) and u = v =, T = T at x =, (5a) (b) Fgure : Scheatc of the square cavty wth voluetrc heat generaton: (a) horzontal walls are adabatc and vertcal walls are sotheral, (b) all walls are sotheral. The governng equatons for two-densonal stea-state natural convecton, under the Boussnesq approxaton, are wrtten as: u = v =, T = T at x = H, (5b) u = v =, T = T at y =, (5c) u = v =, T = T at y = H. (5d) For two-densonal ncopressble flow, the strea-functon ψ can be defned, whch satsfes dentcally the contnuty equaton: u v + =, () ψ ψ u =, v =. (6a,b) 2 2 u u p u u ρ u + v = +μ +, 2 2 x y (2a) The governng Equatons (2a), (2b), (3) can be cobned nto the strea-functon only forulaton as follows: Brazlan Journal of Checal Engneerng Vol. 3, No. 4, pp , October - Deceber, 23

4 886 C. An, C. B. Vera and J. Su 2 2 ψ ψ ψ ψ 4 T = ν ψ βg, ''' T ψ T 2 q =α T + p ψ ρc wth the followng boundary condtons: ψ ψ = =, T = T at x =, ψ ψ = =, T = T at x = H, ψ T ψ = =, = at y =, ψ T ψ = =, = at y = H and ψ ψ = =, T = T at x =, ψ ψ = =, T = T at x = H, ψ ψ = =, T = T at y =, ψ ψ = =, T = T at y = H., (7) (8) (9a) (9b) (9c) (9d) (a) (b) (c) (d) Densonless varables can be ntroduced as follows: x y u ν x =, y =, U =, V =, H H α / H α / H ψ T T c ν μ p ψ=, θ=, Pr = =, ''' 2 α q H / k α k ''' βgq H Ra = ανk 5, () where x and y dentfy the densonal spatal varables, v the kneatc vscosty and α the theral dffusvty. The densonless governng Equatons (7), (8) are wrtten as follows: 2 2 ψ ψ ψ ψ 4 θ = Pr ψ PrRa, ψ θ ψ θ 2 = θ+, whch can be further expressed by ψ ψ ψ ψ ψ ψ ψ = y x y x Pr y x y y x 3 3 ψ ψ ψ ψ θ +, 2 3 Ra y 2 2 θ θ ψ θ ψ θ + = 2 2 wth the followng boundary condtons: ψ ψ = =, ψ = at x =, ψ ψ = =, θ= at x =, ψ θ ψ = =, = at y =, ψ θ ψ = =, = at y = and ψ ψ = =, θ= at x =, ψ ψ = =, θ= at x =, ψ ψ = =, θ= at y =, ψ ψ = =, θ= at y =, respectvely. (2) (3) (4) (5) (6a) (6b) (6c) (6d) (7a) (7b) (7c) (7d) Brazlan Journal of Checal Engneerng

5 Integral Transfor Soluton of Natural Convecton n a Square Cavty wth Voluetrc Heat Generaton 887 INTEGRAL TRANSFORM SOLUTION The an procedures of the GITT-type ntegral transforaton process are: (a) to choose auxlary egenvalue probles fro hoogeneous versons of the orgnal proble; (b) to obtan egenvalues and egenfunctons whch satsfy the orthogonalty property; (c) to defne the ntegral transfor par, ncludng the transfor forula and the nverse forula; (d) to transfor the orgnal proble wth transfor forula to obtan a set of ordnary dfferental equatons wth the calculable coeffcents; (e) to solve the ordnary dfferental equaton syste by atheatcal tools software or codes, such as Matheatca, IMSL Lbrary, etc.; and (f) to obtan the orgnal proble solutons usng the nverse forula. Accordng to the prncple of the generalzed ntegral transfor technque, the auxlary egenvalue proble needs to be chosen for the densonless governng Equatons (4), (5) wth the hoogenous boundary condtons (6 or 7). The drecton s selected to be elnated through ntegral transforaton, and the egenvalue proble of the bharonctype, prevously studed n Leal et al. (999), s adopted for the streafuncton representaton as follows: 4 d X( x) 4 = μ 4 X ( x ), < x <, dx wth the followng boundary condtons X X d X () =, =, dx ( ) d X () =, =, dx () (8a) (8b,c) (8d,e) where X( x ) and μ are, respectvely, the egenfunctons and egenvalues of proble (8). The egenfunctons satsfy the followng orthogonalty property ( ) ( ) j =δ j X x X x d x N, (9) wth δ j = for j, and δ j = for = j. The nor, or noralzaton ntegral, s wrtten as 2 N = X ( x) d. x (2) The related egenfunctons of proble (8) are gven by: X ( x) cos[ μ( x / 2)] cos[ μ / 2] cosh[ μ ( x / 2)],for odd, cosh[ μ / 2] = sn[ μ ( x / 2)] sn[ μ / 2] snh[ μ ( x / 2)],for even, snh[ μ / 2] (2a,b) where the egenvalues are obtaned fro the transcendental equaton: tan tanh ( μ / 2) = tan ( μ ) ( μ ) / 2, for odd, / 2, for even, and the noralzaton ntegral s evaluated as (22a,b) N =, =,2,3, (23) Therefore, the noralzed egenfuncton concdes, n ths case, wth the orgnal egenfuncton tself,.e. X ( x) X ( x) =. (24) N /2 For the teperature expanson, the classcal second-order dffuson operator yelds a Stur- Louvlle type proble, readly solved wth the approprate boundary condtons of the frst knd at the lateral walls, to yeld the egenfunctons and related quanttes: ( x) sn x,,2,3,, φ = β = (25) where the egenvalues are gven by β = π, =,2,3, (26) and the nor evaluaton yelds M = / 2, =,2,3, (27) The noralzed egenfuncton φ then becoes ( x) φ φ ( x) = (28) M /2. Brazlan Journal of Checal Engneerng Vol. 3, No. 4, pp , October - Deceber, 23

6 888 C. An, C. B. Vera and J. Su The soluton ethodology proceeds towards the proposton of the ntegral transfor par for the potentals, the ntegral transforaton tself and the nverson forula. For the streafuncton feld: ( ) ( ) ( ) ψ y = X x ψ x, y d x, transfor, (29a) ψ ( xy, ) = X ( x) ψ( y ), nverse, (29b) = and for the teperature feld: ( ) ( ) ( ) θ y = φ x θ x, y d x, transfor, (3a) θ ( x, y) = φ ( x) θ( y ), nverse. (3b) = The ntegral transforaton process s now eployed through operaton of (4) wth X ( x) dx,, to fnd the transfored streafuncton syste: ( y) 4 ψ dψj dψj 2 ψk = A d 4 jk k Bjk y Pr ψ + 2 j= k= d d C ψ ψ μ ψ 3 dψ d ψ k j 4 jk j Bjk 3 k 2 d ψ j j d 2 y j= = 2 D + Ra E θ, =,2,3,, (3) where the related coeffcents are obtaned analytcally through sybolc coputaton packages, such as Matheatca (Wolfra, 23), and autoatcally generated n Fortran for through ther defntons: ''' ' jk = j k jk = j k A XX X d, x B XX X d, x (32a,b) = ' '' = '' jk j k j j C X X X d, x D X X d, x (32c,d) = φ ' E X d, x (32e) Slarly, Eq. (5) s operated on wth φ ( x )d x, to yeld the transfored teperature proble: 2 d θ 2 ( y) 2 dψ j = Q θ S n= j= + β θ P, =,2,3,, dθn ψ nj n nj j (33) where the assocated coeffcents, sybolcally coputed and autoatcally generated wthn the code, are obtaned fro = φ φ ' Q X d, x (34a) nj n j ' nj = φ φ n j = φ S X d, x P d. x (34b,c) The resultng coupled nfnte syste of ordnary dfferental equatons wth boundary condton at two ponts for the transfored potentals s then descrbed by Equatons (3) and (33), together wth the ntegral transfored boundary condtons: d ψ() d θ() ψ ( ) =, =, =, (35a,b,c) d ψ() d θ() ψ () =, =, =, (35d,e,f) and d ψ() ψ ( ) =, =, θ ( ) =, (36a,b,c) dψ () ψ() =, =, θ() =, (36d,e,f) respectvely. For coputatonal purposes, the expansons are truncated at suffcently large orders, NV and NT, respectvely. Ths truncated syste s then solved by a coputer progra developed n Fortran 9, based on the use of the subroutne DBVPFD fro the IMSL Lbrary (IMSL, 23) wth autoatc control 4 of the local relatve error ( s selected for ths proble). For ths purpose, the syste coposed of (3) and (33) s frst rewrtten as a frst order ODE syste,.e. Brazlan Journal of Checal Engneerng

7 ' (,y) Integral Transfor Soluton of Natural Convecton n a Square Cavty wth Voluetrc Heat Generaton 889 w = f w (37) where the soluton vector, w, s defned as dψ d ψ d ψ dψnv w = ψ,,,,,ψ NV,, 2 3 NV NV NT,,θ 2 3,,,θ NT, d ψ d ψ dθ dθ T (38) Once the transfored potentals, ψ and θ, have been nuercally evaluated under controlled accuracy, the nverson forulae (29b) and (3b) are recalled to provde explct analytcal expressons, n the y drecton, for the orgnal potentals ψ ( x, y) and θ( x, y ). The velocty coponents, U( x, y ) and V( x, y ), are analytcally expressed fro the nverson forula (29b) and the streafuncton defnton, n the for: (a) NV ( y) dx ( x) NV dψ ( ) = = =, = U X x V ψ ( y). (39a,b) dx RESULTS AND DISCUSSION We now present nuercal results for natural convecton n a cavty wth voluetrc heat generaton. Two dfferent boundary condtons are consdered: ) adabatc horzontal walls and sotheral vertcal walls, ) all walls sotheral. A Prandtl nuber of.7 s eployed n ths stu. The solutons of the systes (3, 33) are obtaned wth NV 4and NT 4 to analyze the convergence behavor, then the accuracy s verfed by the nuercal results obtaned wth ANSYS CFX 2., where a 2 2 coputatonal grd s used. The strealne and sother contour plots for an nternally heated square enclosure wth adabatc horzontal walls and sotheral vertcal walls at Ra = are llustrated n Fg. 2. The flow feld takes on a pattern possessng blateral syetry, where two counter-rotatng convectve rolls appear. The densonless teperature s axu at the top of the vertcal centerlne and decreases near the vertcal walls, whch explans why the flows rse n the central regon of the cavty, dvde close to the top adabatc wall, then ove downward separately along the vertcal walls. (b) Fgure 2: Contour plots for an nternally heated square enclosure wth adabatc horzontal walls and sotheral vertcal walls at Ra = : (a) strealne, (b) sother. The convergence behavor of the GITT soluton s exaned by coputng the densonless teperature θ and densonless vertcal velocty V wth dfferent truncaton orders NV and NT at the postons (.5,.), (.5,.5) and (.5,.),, as presented n Table. For the densonless teperature, t can be observed that convergence s essentally acheved wth reasonably low truncaton orders ( NV 2 and NT 2). Brazlan Journal of Checal Engneerng Vol. 3, No. 4, pp , October - Deceber, 23

8 89 C. An, C. B. Vera and J. Su Table : Convergence of the densonless teperature θ and densonless vertcal velocty V wth dfferent truncaton orders NV and NT for the cavty wth adabatc horzontal walls, and coparson wth the nuercal soluton by ANSYS CFX 2. at dfferent postons Ra = Ra = 25 Ra = 49 Ra = 75 Ra = θ (.5,.) NV =, NT = NV = 2, NT = NV = 3, NT = NV = 4, NT = ANSYS CFX θ (.5,.5) NV =, NT = NV = 2, NT = NV = 3, NT = NV = 4, NT = ANSYS CFX θ (.5,.) NV =, NT = NV = 2, NT = NV = 3, NT = NV = 4, NT = ANSYS CFX V (.5,.5) NV =, NT = NV = 2, NT = NV = 3, NT = NV = 4, NT = ANSYS CFX For a full convergence to fve sgnfcant dgts, ore ters (e.g., NV = 4 and NT = 4) are requred. As would be expected, convergence rates of the vertcal velocty areslower than those of the teperature, whch s due to the fact that the velocty s proportonal to the frst dervatve of the streafuncton (Eq. (6b)), whle the teperature s defned by the orgnal egenfuncton expansons (Eq. (3b)). Ths behavor becoes ore apparent as Ra ncreases, but a full convergence to four sgnfcant dgts stll can be obtaned for ost cases. The results are n excellent agreeent wth the ones calculated by ANSYS CFX 2., whch verfes the accuracy of the GITT soluton. The hghest densonless teperatures, the correspondng postons and the spatally averaged Brazlan Journal of Checal Engneerng densonless teperatures for a cavty wth adabatc horzontal walls at several values of Ra are suarzed n Table 2. The GITT yelds good agreeent wth the soluton obtaned by ANSYS CFX 2., wth concdence of four dgts for θ ax and three dgts for θ av. The peak teperature ncreases and the average teperature decreases onotoncally wth Ra ; at Ra =, the value of θ av s nearly.7% lower than that at Ra =, whch ndcates that convecton contrbutes ore to heat transport as Ra ncreases. The varatons of θ ax and θ av wth Ra are plotted n Fg. 3 on a lnear-log scale, where the GITT soluton curves are constructed by the correspondng values coputed for every ncreent of 2 n Raylegh nubers. Table 2: The values of θ ax and θ av at dfferent Ra nubers for the cavty wth adabatc horzontal walls, and coparson wth the nuercal soluton by ANSYS CFX 2. Ra = Ra = 25 Ra = 49 Ra = 75 Ra = GITT θ ax x y..... θ av ANSYS CFX 2. θ ax x y θ av

9 Integral Transfor Soluton of Natural Convecton n a Square Cavty wth Voluetrc Heat Generaton 89 Fgure 3: Varatons of the axu densonless teperature θ ax and spatally averaged densonless teperature θ av wth Ra for a cavty wth adabatc horzontal walls, and the coparson wth the results obtaned by ANSYS CFX 2.. Fgure 5: Profle of the densonless vertcal velocty V along the horzontal centerlne at Ra = 49 for a cavty wth adabatc horzontal walls. Fgs. 4 and 5 show the varaton of the densonless teperature θ and the profle of the densonless vertcal velocty V along the horzontal centerlne at Ra = 49, respectvely. Fg. 6 reports the profle of the densonless horzontal velocty U along y =.75 at Ra = 49. The results are copared wth nuercal sulatons carred out by ANSYS CFX 2., for whch a close agreeent s observed. Fgure 6: Profle of the densonless horzontal velocty U along y =.75 at Ra = 49 for a cavty wth adabatc horzontal walls. Fgure 4: Varaton of the densonless teperature θ along the horzontal centerlne at Ra = 49 for a cavty wth adabatc horzontal walls. The stea-state densonless strea functon ψ and teperature θ for a cavty wth sotheral walls at Ra = are shown n Fg. 7. The strealne exhbts flow behavor slar to that obtaned for a cavty wth adabatc horzontal walls; only one crculaton cell fors n each half of the enclosure, wth a perfect blateral syetry. It can be noted that the streafuncton values when all walls are sotheral are approxately half of the correspondng values Brazlan Journal of Checal Engneerng Vol. 3, No. 4, pp , October - Deceber, 23

10 892 C. An, C. B. Vera and J. Su (a) (b) Fgure 7: Contour plots for an nternally heated square enclosure wth sotheral walls at Ra = : (a) strealne, (b) sother. for the case wth adabatc horzontal walls, whch has been also captured by Josh et al. (26). Wth an approxate up-down syetry, the sother plot ndcates that the axu densonless teperature occurs nearly at the center of the cavty and the lowest teperature s located along the sde walls. An analyss slar to the above s perfored on the convergence behavor of the densonless teperature θ and densonless vertcal velocty V at the center pont of a cavty wth sotheral walls, as shown n Table 3. In ths case, the Raylegh nubers are set equal to, 25, 49, 75 and, respectvely. For reference purposes, parallel coputatons are pleented wth ANSYS CFX 2.. The excellent convergence rates of both θ and V gve agreeent to four sgnfcant dgts, wth values of NV 4 and NT 4. The values of θ ax and θ av for a cavty wth sotheral walls are lsted n Table 4 to corroborate the accuracy of the GITT soluton, rrespectve of the agntude of Ra. Copared to the case wth adabatc horzontal walls, a dfferent behavor s found for the axu teperature, whch declnes slghtly and the correspondng locaton oves upward contnuously as the value of Ra ncreases. The value of θ ax at Ra = s about.32% lower than that at Ra =. Table 3: Convergence of the densonless teperature θ and densonless vertcal velocty V wth dfferent truncaton orders NV and NT for the cavty wth sotheral walls, and coparson wth the nuercal soluton by ANSYS CFX 2. at center pont Ra = Ra = 25 Ra = 49 Ra = 75 Ra = θ (.5,.5) NV =, NT = NV = 2, NT = NV = 3, NT = NV = 4, NT = ANSYS CFX V (.5,.5) NV =, NT = NV = 2, NT = NV = 3, NT = NV = 4, NT = ANSYS CFX Brazlan Journal of Checal Engneerng

11 Integral Transfor Soluton of Natural Convecton n a Square Cavty wth Voluetrc Heat Generaton 893 Table 4: The values of θ ax and θ av at dfferent Ra nubers for the cavty wth sotheral walls, and coparson wth the nuercal soluton by ANSYS CFX 2. Ra = Ra = 25 Ra = 49 Ra = 75 Ra = GITT θ ax x y θ av ANSYS CFX 2. θ ax x y θ av Fgures 8 and 9 depct the varaton of the densonless teperature θ and the profle of the densonless vertcal velocty V along the horzontal centerlne at Ra = 75 for a cavty wth sotheral walls, respectvely. Fgure reports the profle of the densonless horzontal velocty U along y =.75 at Ra = 75. The plots are qualtatvely slar to the ones obtaned fro a cavty wth adabatc horzontal walls at Ra = 49. It s observed that the drecton of the horzontal velocty changes ts sgn at around x =.5, whle the vertcal velocty s postve throughout the central regon of the cavty and negatve near the boundares. Fgure 8: Varaton of the densonless teperature θ along the horzontal centerlne at Ra = 75 for a cavty wth sotheral walls. Fgure 9: Profle of the densonless vertcal velocty V along horzontal centerlne at Ra = 75 for a cavty wth sotheral walls. Brazlan Journal of Checal Engneerng Vol. 3, No. 4, pp , October - Deceber, 23

12 894 C. An, C. B. Vera and J. Su NOMENCLATURE Fgure : Profle of the densonless horzontal velocty U along y =.75 at Ra = 75 for a cavty wth sotheral walls. CONCLUSIONS The generalzed ntegral transfor technque (GITT) has been shown n ths work to be an adequate approach for the analyss of natural convecton n a square cavty contanng nternal energy sources, provdng an accurate nuercal-analytcal soluton for the teperature and velocty coponents. The nuercal results reported are n excellent agreeent wth the soluton obtaned by ANSYS CFX 2., whch supports the valdty of the present coputatons. Ths approach can be ether eployed for bencharkng purposes, yeldng sets of reference orders and exceptonal coputatonal perforances. It s portant to note that the ethodology developed s applcable to any boundary condtons, and ay also be appled to the analyss of natural convecton n a rectangular cavty wth dfferent aspect ratos or an nclned cavty. ACKNOWLEDGMENTS The authors gratefully acknowledge the fnancal support provded by CNPq, CAPES and FAPERJ of Brazl for ther research work. Chen An also would lke to acknowledge the fnancal support provded by the Chna Scholarshp Councl. g acceleraton due to s -2 gravty H cavty length and heght k theral conductvty W - K - M noralzaton ntegral N noralzaton ntegral NT truncaton order n the teperature expansons NV truncaton order n the streafuncton expansons p pressure Pa Pr Prandtl nuber q heat generaton per unt W -3 volue Ra Raylegh nuber T teperature K T o teperature at the K boundares u velocty coponent n the x- drecton s - U densonless velocty coponent n the x- drecton v velocty coponent n the y- drecton s - V densonless velocty coponent n the y- drecton x densonless space coordnate y densonless space coordnate X egenfuncton X noralzed egenfuncton Greek Letters α flud theral dffusvty 2 s - β theral expanson coeffcent K - β egenvalue θ densonless teperature θ densonless transfored teperature θ av spatally averaged densonless teperature θ ax axu densonless teperature φ egenfuncton φ noralzed egenfuncton Brazlan Journal of Checal Engneerng

13 Integral Transfor Soluton of Natural Convecton n a Square Cavty wth Voluetrc Heat Generaton 895 ψ densonless streafuncton ψ transfored streafuncton μ egenvalue ν kneatc vscosty 2 s - Superscrpt Subscrpts, densonal varables egenquanttes orders REFERENCES Acharya, S., Goldsten, R. J., Natural-convecton n an externally heated vertcal or nclned square box contanng nternal energy-sources. Journal of Heat Transfer-Transactons of the ASME, 7(4), (985). An, C., Su, J., Dynac response of claped axally ovng beas: Integral transfor soluton. Appled Matheatcs and Coputaton, 28(2), (2). Arcdacono, S., Cofalo, M., Low-Prandtl nuber natural convecton n voluetrcally heated rectangular enclosures III. Shallow cavty, AR =.25. Internatonal Journal of Heat and Mass Transfer, 44(6), (2). Arcdacono, S., D Pazza, I., Cofalo, M., Low- Prandtl nuber natural convecton n voluetrcally heated rectangular enclosures II. Square cavty, AR =. Internatonal Journal of Heat and Mass Transfer, 44(3), (2). Baker, L., Faw, R. E., Kulack, F. A., Postaccdent heat reoval - Part I: Heat transfer wthn an nternally heated, nonbolng lqud layer. Nuclear Scence and Engneerng, 6(2), (976a). Baker, L., Faw, R. E., Kulack, F. A., Postaccdent heat reoval - Part II: Heat transfer fro an nternally heated lqud to a eltng sold. Nuclear Scence and Engneerng, 6(2), (976b). Bethe, H. A., Energy producton n stars. Scence, 6(384), (968). Cheng, C. Y., Non-Darcy natural convecton heat and ass transfer fro a vertcal wavy surface n saturated porous eda. Appled Matheatcs and Coputaton, 82(2), (26). Cheng, C. Y., Natural convecton boundary layer flow of flud wth teperature-dependent vscosty fro a horzontal ellptcal cylnder wth constant surface heat flux. Appled Matheatcs and Coputaton 27(), 83-9 (2). Cotta, R. M., Integral Transfors n Coputatonal Heat and Flud Flow. CRC Press, Boca Raton, FL (993). Cotta, R. M., The Integral Transfor Method n Theral and Fluds Scence and Engneerng. Begell House, New York (998). Cotta, R. M., Mkhalov, M. D., Heat Conducton - Luped Analyss, Integral Transfors, Sybolc Coputaton. Ed. John Wley & Sons, Chchester, England (997). Danels, P. G., Jones, O. K., Convecton n a shallow rectangular cavty due to nternal heat generaton. Internatonal Journal of Heat and Mass Transfer, 4(23), (998). D Pazza, I., Cofalo, M., Low-Prandtl nuber natural convecton n voluetrcally heated rectangular enclosures I. Slender cavty, AR = 4. Internatonal Journal of Heat and Mass Transfer, 43(7), (2). Fuseg, T., Hyun, J. M., Kuwahara, K., Nuercal stu of natural-convecton n a dfferentally heated cavty wth nternal heat-generaton - Effects of the aspect rato. Journal of Heat Transfer-Transactons of the ASME, 4(3), (992). IMSL, IMSL Fortran Lbrary verson 5.. MATH/ LIBRARY Volue 2. Vsual Nuercs, Inc., Houston, TX (23). Josh, M. V., Gatonde, U. N., Mtra, S. K., Analytcal stu of natural convecton n a cavty wth voluetrc heat generaton. Journal of Heat Transfer- Transactons of the ASME, 28(2), (26). Kar, R. R., Murthy, P. V. S. N., Effect of vscous dsspaton on natural convecton heat and ass transfer fro vertcal cone n a non-newtonan flud saturated non-darcy porous edu. Appled Matheatcs and Coputaton, 27(2), 8-84 (2). Kawara, Z., Kshguch, I., Aok, N., Mchyosh, I., Natural convecton n a vertcal flud layer wth nternal heatng. In: Proceedngs of the 27 th Natonal Heat Transfer Syposu of Japan. Vol. II. pp. 5-7 (99). Leal, M. A., Machado, H. A., Cotta, R. M., Integral transfor solutons of transent natural convecton n enclosures wth varable flud propertes. Internatonal Journal of Heat and Mass Transfer, 43(2), (2). Leal, M. A., Pérez-Guerrero, J. S., Cotta, R. M., Natural convecton nsde two-densonal cavtes: The ntegral transfor ethod. Councatons n Nuercal Methods n Engneerng, 5, 3-25 (999). Lee, J. H., Goldsten, R. J., An experental-stu on natural-convecton heat-transfer n an nclned Brazlan Journal of Checal Engneerng Vol. 3, No. 4, pp , October - Deceber, 23

14 896 C. An, C. B. Vera and J. Su square enclosure contanng nternal energysources. Journal of Heat Transfer-Transactons of the ASME, (2), (988). May, H. O., A nuercal stu on natural-convecton n an nclned square enclosure contanng nternal heat-sources. Internatonal Journal of Heat and Mass Transfer, 34(4-5), (99). Mckenze, D. P., Roberts, J. M., Wess, N. O., Convecton n earth's antle - Towards a nuercalsulaton. Journal of Flud Mechancs, 62, (974). Qu, S. Z., Qan, L. B., Zhang, D. L., Su, G. H., Tan, W. X., Nuercal research on natural convecton n olten salt reactor wth non-unforly dstrbuted voluetrc heat generaton. Nuclear Engneerng and Desgn, 24(4), (2). Rahan, M., Sharf, M. A. R., Nuercal stu of lanar natural convecton n nclned rectangular enclosures of varous aspect ratos. Nuercal Heat Transfer Part A-Applcatons 44 (4), (23). Runcorn, S. K., Convecton currents n earth's antle. Nature, 95(4848), (962). Trtton, D. J., Internally heated convecton n atosphere of venus and n laboratory. Nature, 257(5522), - (975). Wolfra, S., The Matheatca Book. 5th Ed. Wolfra Meda/Cabrdge Unversty Press, Chapagn, IL, USA (23). Brazlan Journal of Checal Engneerng