FINITE ELEMENT STUDY OF DDNC IN BOTTOM HEATED ENCLOSURES WITH MASS DIFFUSIVE SIDE WALLS

Transcription

1 DOI: 98/hmt.8.7 Global Digital Cetral ISSN: -869 Frotiers i Heat ad Mass Trasfer Available at FINITE ELEMENT STUDY OF DDNC IN BOTTOM HEATED ENCLOSURES WITH MASS DIFFUSIVE SIDE WALLS Nithish Reddy*, K. Murugesa Mechaical & Idustrial Egieerig Departmet, Idia Istitute of Techology Roorkee, Roorkee , Idia ABSTRACT I this paper DDNC pheomeo i a bottom heated eclosure eposed to mass diffusio from its side walls is ivestigated umerically. The iterplay betwee thermal ad solutal buoyacy forces o fluid circulatio ad heat trasfer rates is studied i three differet eclosures of aspect ratios, ad. Fiite elemet base umerical code has used to solve the goverig equatios, here velocity ad vorticity are take as primary variables for flow field. Numerical results are well validated with that of the literature. The relative stregth of solutal to thermal buoyacy forces is defied by buoyacy ratio parameter N, umerical ivestigatios were carried out for differet values of N i the rage of to ad three differet Rayleigh umbers(ra=, 4, ). From results it is observed that solutal buoyacy forces broke dow the larger fluid cells ad promoted local circulatios of stroger stregth. Icrease i cotributio of solutal buoyacy force has resulted i a maimum of.6%, 4.% ad % icrease i average Nusselt umber for shallow, square ad deep eclosures respectively whe heat trasfer is primarily by covectio while the effect of buoyacy ratio is ot felt much o heat trasfer at diffusio domiat coditios i.e at Ra=. Keywords: atural covectio, cavity, buoyacy ratio, double diffusive. square eclosure where the mai focus was kept o uderstadig the effect of solutal buoyacy forcesfor a fied thermal Rayleigh umber (7), they gave correlatios for Nusselt umber ad Sherwood umber for the rage of parameters cosidered i their research. Later Nishimura et al. (998) came up with ewer perspective of eamiig the formatio of fluid cells uder differet buoyacy ratios. They addressed the oscillatory behaviour of the covectio cells with chage i buoyacy ratio parameter N. Hyu ad Lee (99) made a attempt to uderstad the ature of DDNC i the case of rectagular eclosure ad eposed the major discrepacies i evolutio patter betwee cooperatig ad opposig flows through stream lies,temperature ad cocetratio plots. More etesive study o the effect of differet parameters o DDNC pheomeo was carried out by Che et al. (). They have chose higher Rayleigh umber regime (6 ad 7), differet aspect ratios (-.) ad buoyacy ratios(8<n<.). Results show that at Ra=6 oly oe fluid cell is observed for differet aspect ratios but for buoyacy ratios greater tha uity, a umber of vortices were geerated ad they varied with chage i eclosure aspect ratio. Few works cocetrated o the study of DDNC pheomeo i eclosures with higher aspect ratios. Amog them Ha ad Kuh (99) figured out that at aspect ratio 4 i a differetially heated eclosure, a uicell flow was observed without ay clear demarcatio weather the case is pure solutal domiat oe or pure thermal domiat oe. Chouikh et al. (7) aalysed DDNC i a iclied glazig eclosure of aspect ratio resemblig the solar distillatio set up. It was observed that the formatio of sigle rotatig cell is foud to be favourable for allowig sufficiet time for vapour to cool dow ad better performace of solar distiller. Qi et al. (4) ivestigated double diffusive covectio i a differetially heated rectagular eclosure of aspect ratio usig a compact differece method. Alloui et.al. () cosidered the soret effect i their aalysis o DDNC i a very shallow eclosure filled INTRODUCTION Double diffusive atural covectio (DDNC) flows are widely see i may idustrial systems such as CVD chambers, food processig uits, chemical egieerig setups etc. where it s importat to study fluid ad heat flow patters for various reasos. Natural covectio pheomeo i eclosures is oe of the widely studied research subject over years due to its importace i may practical applicatios (Heather et al., ; Maoj et al., 4). Sice last decade there is a icreasig level of iterest amog the researchers i studyig the flow behavior ad heat trasfer mechaism i the eclosures uder the combied ifluece of thermal ad solutal buoyacy forces. These problems are also called as double diffusive atural covectio (DDNC) or thermo-solutal covectio problems. Such type of fluid covectio is usually ecoutered i may mechaical ad chemical egieerig systems ad i umerous geophysical pheomea, icludig covectio i the oceas, i the earth's outer core etc.(wag et al. 4; Aouachria et al., ; Kaladhar ad Sriivasacharya, 4; Maoj et al., ; Kumar et al., ).Because of their wide importace double diffusive covectio flows are studied i differet types of systems by various researchers. A iitial eperimetal work o DDNC has bee coducted i low aspect ratio rectagular eclosures whose side walls are differetially heated ad salted (kamotai et al., 98). They reported that study of flow structures i such coditios with wide differece i diffusio rates is importat for crystallie growth applicatios. Costa et al., (997) established a umerical model for double diffusive atural covectio pheomeo i a eclosure with mass diffusive wall usig moist air as medium. This model could effectively visualise heat ad mass lies throughout the eclosure. Beghei et al. (99) carried out research o DDNC i a differetially heated ad salted * Correspodig author. krimufme@iitr.ac.i

2 DOI: 98/hmt.8.7 Global Digital Cetral ISSN: -869 with biary mitures subjected to solutal flu o horizotal boudaries. Few ivestigatios (Nikbakthi ad Rahimi, ; Khodakhah ad Nikbakthi, 6; Oueslati et al., 4; Jeaa et al., ) were carried out o doublediffusive atural covectio i cavities whose side walls are partially heated ad cooled.. Nazari et al. (), Moufekkir et al. () ad Re et al. (6) umerically ivestigated double diffusive atural covectio i cavities usig Lattice Boltzma method. Corcioe et al. () gave correlatios for the ddc i square eclosures iduced by opposite temperature ad cocetratio gradiets. It has bee observed that most of the works carried out till date dealt with iteractio of thermos-solutal buoyacy forces i traditioal differetially heated eclosures, otherwise very limited outcomes are available i the literature o other models of heatig ad saltig. I the curret study a attempt is made to ivestigate the heat ad fluid flow characteristics i bottom heated eclosure subjected to mass diffusive side walls. Numerical eperimets were coducted o eclosures of three differet aspect ratios (shallow),(square) ad.(deep) ad results are discussed. The effect of iterplay betwee thermal ad solutal buoyacy forces o fluid covectio ad heat trasfer has bee eplored for <N<, <Ra<. V U Pr X Y X Y Ra Pr N X Y Velocity Poisso equatio U U X Y Y V V X Y X U U X Y Le X Y (8) where o-dimesioal umbers are defied as, Rayliegh umber Ra= gβtδth/ v, buoyacy ratio N = βcδc/βtδt = GRs/GRT, Pradtl umber, Pr = υ/α ad Le = Sc/Pr. These o-dimesioal equatios (Eqs. () to (8)) are the reduced to weak form by applyig Galerki s weighted residual fiite elemet method, itegrated over the elemets to fially obtai the matri form at elemet level as preseted below. () Vorticity trasport equatio: C ijk ( ) t Pr K ik Ra Pr tn S ijk ik () ijk ik Ra Pr t S ijk ik ik ( ) t G ijk (9) Eergy equatio: Velocity Poisso equatios: () K ijk U ik T ijk ik K ijk V ik S ijk ik Solutal cocetratio equatio: C V. C D C t C ijk t Pr K ijk t G ijk ik Velocity- Poisso equatio: T V. T T t (7) Solutal cocetratio equatio where T is temperature, C is cocetratio, V = (u, v) are the velocity compoets i - ad y-directios respectively, is the vorticity compoet. V ω (6b) U V X Y X Y Flow is cosidered i compressible ad Boussiesq approimatio is assumed for desity variatios. The followig are the goverig equatios that are solved for the preset double diffusive covectio problem. Vorticity trasport equatio: ω V ω ω g T T T t (6a) Eergy equatio GOVERNING EQUATIONS g c C C () (a) (b) Eergy equatio (4) C ijk t K ijk ik C ijk ik ik ( ) t K ijk ik t G ijk Equatios (),(),() ad (4) are the coverted i to o-dimesioal form by applyig the followig scalig parameters Spatial coordiates, X = /L, Y = y/l, velocities, U = ul/, V = vl/, vorticity, ς = ωl/, time, τ = t/l, temperature Θ= T Tc/Th Tc ad solutal () Solutal cocetratio equatio cocetratio = C Cc/Ch Cc. After substitutig the above scalig parameters the dimesio form of goverig equatios are trasformed as the followig. C ijk t K ijk ik C ijk ik () ( ) t ik K ijk ik t G ijk Le where i, j ad k are the elemet, row ad colum idices respectively ad is a weightig factor for time discretizatio with rage,, ad Vorticity trasport equatio:

3 DOI: 98/hmt.8.7 G ijk K ijk C ijk N Global Digital Cetral ISSN: -869 (g) Repeat steps to (f) util the field variables velocities, vorticity temperature ad solutal cocetratio are coverged at curret time level. N k k N j X N lu l dxdy N j Y N lv l dxdy XY e XY e N j N k N j N k X X Y Y N j N kdxdy XY e dxdy (iv) Go to step (iii) for the et time step. Computatioal domai The schematic diagram of the problem domai is show i Figure. Bottom wall is kept hot ad top wall is maitaied cold, while both the vertical walls are assumed adiabatic to heat flow.no-slip boudary coditios for velocity compoets are assumed o each of the boudaries. Adiabatic solutal boudary coditios are chose o both horizotal walls ad a liearly varyig cocetratio profile is assumed alog the vertical walls as show i Figure. I the curret work the chage i heat ad fluid flow patter with icrease i stregth of solutal buoyacy forces has bee studied for differet parameters i the eclosures of aspect ratio, ad. Iitial ad boudary coditios chose for the preset problem are give i the followig equatios. XY e T ijk N j N k dxdy Y XY e S ijk Nj XY e N k dxdy X A Isoparametric formulatio is used to trasform the global coordiates ito local coordiate system so that the Gaussia quadrature ca be employed for the umerical itegratio of all the terms i the weighted goverig equatios.. SOLUTION METHODOLOGY The above show fial equatios of vorticity, velocity poiso, eergy ad cocetratio are solved over computatio domai with appropriate iitial ad boudary coditios. The solutio domai is first discretized ito a umber of biliear iso-parametric elemets ad for each elemet the partial differetial goverig questios are approimated usig discretizatio techiques. Secod order-accurate Crak Nicolso scheme is used for discretizatio of time derivatives. To miimize the residual i the computatioal domai formed as cosequeces of approimatio a iterative computatioal procedure is followed till the error i thermal, solutal ad flow fields reduces to specified value i.e. Oce the solutio i all the flow fields is coverged at the preset time level, the iteratio procedure is repeated to solve for the et time level. The algebraic equatios obtaied at each ode are solved without assemblig usig global matri free algorithm. Cojugate gradiet iterative solver is employed for the solutio of the fial odal equatios. Computatioal algorithm The below show computatio procedure followed to obtai umerical solutio Perform the itegral operatios for all elemet matrices as show i equatios (9) to () Fig. Schematic diagram Iitial ad boudary coditios: Iitial coditios: Determie the elemets surroudig each boudary ode for vorticity, velocity, temperature ad solutal cocetratio fields based o the specified Dirichlet boudary coditios. (iii) Start computig over etire domai for solutio at t=t+ =, U=V= = = Boudary coditios for > Iitialize the vorticity field at Y= for <X< y at Y= for <X< U= V=, =, y Eforce velocity boudary coditios at elemet level ad solve for velocity field usig velocity Poisso equatios. U=V=, =, Usig the compoets of velocity obtaied i previous step over flow domai compute for vorticity values at the boudaries. U=V=,, = y/h at X= ad X= for <Y< X (d) Obtai vorticity field by solvig vorticity trasport equatio after eforcig vorticity boudary coditios at elemetal level. 4. MESH SENSITIVITY AND VALIDATION Structured o-uiform mesh has bee used i the curret work; figure gives the picture of computatioal mesh used for square cavity. Trasfiite iterpolatio(tfi) method is used to geerate computatioal grid. The computer code developed has bee tested for grid idepedece for double diffusive atural covectio i eclosures with aspect ratios, ad for N= ad Ra=4 usig three differet meshes. (e) Obtai temperature field by solvig the correspodig eergy equatios after eforcig the thermal boudary coditios. (f) Obtai for solutal field by solvig the correspodig solutal cocetratio equatios after eforcig the boudary coditios.

4 DOI: 98/hmt.8.7 Global Digital Cetral ISSN: -869 Table Mesh sesitivity results at N= ad Ra= 4 for shallow, square ad deep cavities. s.o Aspect Mesh Avg Nu Error(%) ratio AR= 4(M).67 (shallow) (M) (M).64 4 AR= 44(M).7 (square) (M) (M).78 8 AR= 4(M).774 ( deep ) (M) (M).784 Fig. Computatioal mesh. RESULTS AND DISCUSSION I the curret study umerical ivestigatios were carried out for the bottom heated eclosure with salted side walls as show i Figure. Presece of varyig stregth cocetratio source o side walls creates a secodary desity differece i the eclosure ad iitiates a solutal covectio accompayig thermal covectio. I the preset sectio results correspodig to detailed study o thermo-solutal covectio with chage i buoyacy ratio at differet operatig parameters has bee preseted. The relative stregth of solutal to thermal buoyacy forces is deoted by buoyacy ratio N. Here N= correspods to pure thermal covectio case ad at N= solutal covectio strogly domiates over thermal. Results correspodig to detailed ivestigatio o flow field, temperature, cocetratio ad Nusselt umber are illustrated i the followig sectios. The Rayleigh umber is assumed to vary from to ad buoyacy ratio from to ad the results are discussed for three aspect ratios,, ad. Nishimura et al.(998) Table a Compariso of the preset results for Nu with the bechmark results at N= case ad Pr=. Ra=4 Ra= Pr= Ra= Preset work..9 Tura et.6.94 al.() Error(%) 4 Preset Fig. Iso-therms, iso-cocetratio ad stream lie comparisos at Ra=e,N=8. Table b Compariso of the preset results for Nu with the bechmark results at N= case ad Pr= Ra=4 Ra= Pr= Ra= Preset work.9.88 Tura et al () Error (%) 8 49 Average Nusselt umber has bee take as the parameter for compariso of the performace of three meshes ad the results obtaied are show i Table. From the table oe ca observe that the error betwee two cosecutive meshes, M ad M ad M ad M keeps decreasig with refiemet of meshes. As the error betwee meshes M ad M is much less tha %, mesh M has bee used i all the simulatio for the test problems discussed i the paper. For the purpose of validatio, double diffusive atural covectio i a eclosure cosidered by Tura et al.() has bee cosidered. Results obtaied for the average Nusselt umber for Ra= to are compared with the results of Tura et al. () as show i Tables ad for Pr= ad respectively. It ca be see from these results that the preset code is able to predict the Nusselt umber with a maimum error of % compared to the results reported by Tura et al.(). Other tha the Nusselt umbers, isotherms, isococetratio ad streamlies are also compared with the results of Nishimura et al. (998) for the purpose of validatio. Figure shows the compariso of the preset results for the above cotours with the above referece. It ca be observed that the cotours preseted by the preset research resemble almost the same patters of cotours reported by Nishimura et al. (998).. Stream lie ad vorticity patters Figures 4 to 6 show the streamlie ad vorticity cotours i all the three eclosures at differet buoyacy ratios for Rayleigh umbers,, 4 ad respectively. From stream lies at Ra= (see Figure 4a) it is observed that at N= i.e uder pure thermal covectio case sigle circular fluid cell is formed i case of square eclosure (AR=).I case of shallow eclosure (AR=) where relative distace betwee hot ad cold walls is less whe compared to adiabatic side walls two such circulatio cells are oticed, while a sigle vertically elogated circulatio cell is observed with deep eclosure cofiguratio (AR=). Alog with these large fluid cells a couple of small cells are observed at oe of the diagoally opposite corers. With icrease i buoyacy ratio from to, these small circulatio cells have grow i size ad squeezed the fluid i betwee to form a quadra cell structure i case of AR=. While 4

5 DOI: 98/hmt.8.7 Global Digital Cetral ISSN: -869 for AR= case they eded up with a patter of arrow fluid passage betwee two large circulatios cells. I case of AR= flow structure has ot chaged much with icrease i buoyacy ratio. From these observatios oe ca say that icrease i stregth of solutal covectio is promotig the formatio of multiple fluid circulatio cells. Such circulatios are favorable for the local miig of reactat i chemical process but may also create thermal imbalace ad o-uiformity at global level. From vorticity results (see Figure 4b) it is observed that at N=, square eclosure (AR=) ehibited low vorticity characteristics ad shallow (AR=) domiated amog the three. With icrease i buoyacy ratio it is observed that all the three eclosures ehibited icreased level of fluid vorticity stregth. Thus oe ca say that icrease i cotributio of solutal buoyacy forces addig the fluid circulatio stregth, this is favorable for higher heat trasfer rates. Aother importat observatio is higher fluid vorticity is oticed i case of deep eclosure ad low with shallow eclosure where thermal boudaries are relatively close. Figure a ad b presets the stream lie ad vorticity cotours respectively for differet buoyacy ratios ad aspect ratios i case of Rayleigh umber=4. Uder relatively weaker buoyacy forces with decrease i Rayleigh umber to Ra=4 a differet kid of flow patters are observed for chage i cotributio of solutal buoyacy forces. At N= stream lie patter remaied same for AR= ad AR= case but for AR=. case duel elogated circular structures are oticed. At N= surprisigly Quadra oval structures have formed i case of shallow ad deep eclosures,with further icrease i N value these oval shaped structures got slightly deformed i shallow case while i case of deep eclosure two of the diagoally opposite cells tried to merge. A dual circulatio cells have oticed i case of square eclosure at N=,with further icrease i N to ad fluid tried to break dow ito Quadra cells. Iterestigly vorticity cotours(see Figure b) revealed that hardly ay circulatio stregth is oticed i case of shallow ad deep cavities at N= whe flow is purely govered by thermal covectio. With itroductio of solutal covectio i.e. at N= some amout of vorticity is oticed, which icreased with further icrease i N value. Here square eclosure showed strog circulatio stregth whe compared to other two for all buoyacy ratios. This eposed that both icrease ad decrease i aspect ratio has resulted i loss of fluid vorticity at Ra=4. Figure 6a ad 6b reports the stream lie ad vorticity respectively for shallow, square ad deep eclosures for differet buoyacy ratio for Ra=. At N= for pure thermal covectio case duel cell structures compatible to eclosure shape are oticed for each aspect ratio. At N= Quadra cell structures agai i a shape compatible to eclosure shape are recorded. Here at Ra= with weaker buoyacy forces, buoyacy ratio has ot show sigificat effect o fluid circulatio. Ay further improvemet i buoyacy ratio has ot show ay chage i fluid circulatio patter. Vorticity cotours (see Figure 6b) also reveal that ecept at N= values of vorticity are recorded very low. Thus heat trasfer mode ca be epected to be purely diffusive i ature i this case ad the shape of thus formed fluid cells have ot much to offer as they are almost still. N= N= N=. N= N= N= N= Thermal ad solutal fields Temperature distributios withi all the three eclosures are illustrated for differet N values i Figures 7 to 9. At Ra = (Figure 7), whe N=, a similar patter is observed i case of square ad deep eclosure compatible to the shape of the geometry. Isotherms are observed symmetric about mid-sectio i case of shallow eclosure but with icrease i N value the patter got upside dow, this may be due to reverse i the directio of rotatio of fluid cells. With icrease i buoyacy ratio thermal gradiets ear bottom wall became deser for square, shallow, deep eclosures. At N=, sharp ad strog temperature gradiets are observed i all the three eclosures., these cotours are observed symmetric about mid-sectio of eclosure as similar to stream lie patter. With decrease i Rayleigh umber to 4(Figure 8), at N=, the thermal buoyacy force is foud sufficiet N= Fig. 4 Effect of buoyacy ratio o Stream lie ad voritcity patters i cavities of aspect ratios ad. at Ra=

6 DOI: 98/hmt.8.7 Global Digital Cetral ISSN: -869 Fig. Effect of buoyacy ratio o Stream lie ad voritcity patters i cavities of aspect ratios ad. at Ra=4 Fig. 6 Effect of buoyacy ratio o Stream lie ad voritcity patters i cavities of aspect ratios ad. at Ra= 6

7 DOI: 98/hmt Global Digital Cetral ISSN: N= N= N= N= to set up fluid covectio i case of square eclosure, isotherms here are observed as iverted S shape. At this value, the other two eclosures ehibited oly diffusive heat trasfer with parallel temperature cotours. As far as the variatio i the shallow ad deep eclosures are cocered o sigificat chage is observed for N=, ad. At N=, isotherms are observed curvy idicatig the sig of covectio currets I the case of square eclosure, isotherms are observed curvy with differet patter for differet N values which is due to the chage i fluid flow patter. Figure 9 shows the temperature cotours at Ra=. At such low value of Rayleigh umber the desity gradiets geerated withi the eclosure are foud ot strog eough to set up fluid circulatio ad hece heat trasfer had take place purely by diffusio. This tred is clearly observed i all the three eclosures for N varyig from to. However at N=, slight fluid covectio is foud to be set up at the iterior of the square eclosure. The reaso may be that at N=, the solutal buoyacy force which is ow the order of times to the thermal buoyacy forces is strog eough to produce some fluid movemet. Figures 7, 8, 9 show the distributios of cocetratio i all the three eclosures at differet values of N for Ra=, 4 ad respectively. The cocetratio varies from zero to from the top ed to the bottom ed of the vertical walls. At Ra equal to (Figure 7), due to the effect of thermal buoyacy force decet fluid covectio is set up i the eclosure. At N=, sharp cocetratio gradiets are oticed i all the three eclosures. However, with icrease i N from to, the gradiets i the eclosures got altered due to the oset of solutal buoyacy forces. As oticed i the vorticity cotours the stregth of circulatio has ehaced with icrease i N value, for N= i all the three eclosures iso-cocetratio lies observed steep idicatig ehaced covective mass trasfer ad this has improved further whe N is icreased to Whe Ra is decreased to 4 (Figure 8) at N=, the cocetratio gradiets appeared symmetric about half sectio i shallow ad deep eclosures with a straight horizotal lie of value at mid-way, this tred idicates the absece of covective trasport of mass. But the square eclosure could ehibit covective mass trasfer which is resembled by its cocetratio distributio. With icrease i N value, shallow ad deep eclosures does ot ehibit ay chage i the gradiets of cocetratio up to N=, whereas the gradiets become very steep i the case of square eclosure. At N=, due to the strog cotributio of solutal buoyacy force, mass trasfer happeed by covectio i all the eclosures ad the iso- cocetratio lie at mid-sectio are ow altered i shallow eclosure.. I the case of deep eclosure, the covective fluid flow produced cocave ad cove type iso-cocetratio lies. As see from Figure 9 for Ra=, the effect of chage i the value of N has ot show ay sigificat variatio i the cocetratio profile. The value of N is i relatio to the stregth of thermal buoyacy force, hece at Ra= the buoyacy force due to temperature gradiet is small, hece covective flow is ot observed i the eclosure ad the cocetratio distributios correspod oly to pure diffusive mass trasfer process N= N= N= N= The covective heat trasfer that takes place withi the eclosure is better uderstood by computig the Nusselt umber, which is the odimesioal temperature gradiet at the hot wall. Figure gives the local Nusselt umber alog the bottom wall i case of square eclosure for differet Rayleigh umber ad buoyacy ratio. From Figure a i.e at Ra= it is observed that with icremet i buoyacy ratio from to. Nusselt umber has improved. Nusselt umber curve is observed symmetric about mid-sectio whe N is where solutal buoyacy forces play strog role ad whe N= peak Nusselt umber is observed at X=7, such variatios i Nusselt umber alog the bottom wall are directly related to the fluid motio ear bottom wall. With decrease i Rayleigh umber to Ra=4 a similar kid of Nusselt umber patter is observed with chage i buoyacy ratio. Here the differece is the curves ted to become flatter, ad the magitude of Nusselt umber Fig. 7 Effect of buoyacy ratio o Iso-therms ad Isococetratio lies i cavities of aspect ratios ad. at Ra= 7

8 DOI: 98/hmt.8.7 Global Digital Cetral ISSN: -869 Fig. 8 Effect of buoyacy ratio o Iso-therms ad Isococetratio lies i cavities of aspect ratios ad. at Ra=4 Fig. 9 Effect of buoyacy ratio o Iso-therms ad Isococetratio lies i cavities of aspect ratios ad. at Ra= 8

9 DOI: 98/hmt.8.7 Global Digital Cetral ISSN: N= N= N=. N= 9 8 N= N= N=. N= Nu Nu X 7 X N= N= N=. N= Nu X Fig. Effect of buoyacy ratio o Nusselt umber alog the hot bottom wall i case of square cavity(ar=) at differet Rayleigh umber Ra= Ra=4 ad Ra= Fig. Effect of buoyacy ratio o Nusselt umber alog the hot bottom wall i case of shallow cavity(ar=) at differet Rayleigh umber Ra= Ra=4 ad Ra= 9

10 DOI: 98/hmt.8.7 Global Digital Cetral ISSN: -869 Nusselt umber has decreased due to reduced stregth of buoyacy forces. With further reductio i buoyacy ratio ecept at N= for all other buoyacy ratios the curves became completely flat idicatig sig of very poor covectio. With icrease i aspect ratio (see Figure ) i.e i case of deep eclosure (AR=) Nusselt umber curve became falter eve at Ra=4, ad at Ra= for all N= also o sig of ay sigificat variatio is observed i Nusselt umber alog the bottom wall. Aother importat observatio is that peak Nusselt umber is foud lower tha that of square eclosure case for all three Rayleigh umber. I case of shallow eclosure (see Figure ) the peak Nusselt umber that has registered is observed higher tha square ad deep eclosures. At Ra= (Figure a) with icrease i buoyacy ratio from N= to heat trasfer rate has slightly icreased although flow structure remaied same. I case of Ra=4 epect at N= for all other cases o variatio i Nusselt umber is observed with alog the bottom wall ad for Ra= eve N= has ot bee effective. stregth of solutal buoyacy forces ecouraged the formatio of Quadra cell structures of greater vorticity stregth. (iv) At Ra= where vorticity of fluid cells is appreciably high, with icrease i buoyacy ratio ay variatio i stream lie patter show sigificat ifluece o bottom wall Nusselt umber. (v) Although the stream lie patter varied with icrease i buoyacy ratio o sigificat chage is observed i Nusselt umber at Ra=4 ad below especially i case of deep ad shallow eclosures. (vi) At Ra= chage i buoyacy ratio as ot show ay effect o Nusselt umber for all three aspect ratios ad four cell structures with shape compatible to aspect ratio of eclosure are observed whe N> NOMENCLATURE AR C D g H Le N Nu Pr Ra Sc Sh t T u,v U W, y X, Y Table Compariso of average Nusselt umber for differet Rayleigh umber ad buoyacy ratio. Ra N= N= N= N= N= 7 Square Deep Shallow Table gives the report of average Nusselt umber for the cases cosidered i this work. A importat observatio is miimum Nusselt umber is oticed at N= for all the cases this may happeed due to competitio betwee equally strog thermal ad solutal buoyacy forces. The percetage icrease i Nusselt umber with chage i buoyacy ratio from to is observed to be % for deep eclosure, 4.46% for square eclosure ad oly.6% for shallow eclosure at Ra=.At Ra= ad 4 the effect of buoyacy ratio o average Nusselt umber is observed relatively less. I case of Ra=4 maimum of % for deep eclosure,6.4% for square ad 4.% for shallow cavity are recorded. At Ra= % icremet is observed for deep cavity,4.9 % for square ad % icremet i case of shallow cavity are oticed. This tells the effect of buoyacy ratio is felt more i case of deep cavity where the relative distace betwee solutal boudaries are shorter tha thermal boudaries. 6. aspect ratio cocetratio of species (kg/m) biary diffusio coefficiet (m/s) gravitatioal acceleratio (m/s) height of the eclosure Lewis umber buoyacy ratio local Nusselt umber Pradtl umber Rayleigh umber Schmidt umber local Sherwood umber time (s) temperature(k) horizotal ad vertical velocity compoets, m/s lid velocity (m/s) width of the eclosure horizotal ad vertical coordiates(m) o dimesioal coordiates Greek Symbols thermal diffusivity (m/s) βc cocetratio volumetric epasiocoefficiet(m/kg) βt thermal volumetric epasio coefficiet (K-) μ dyamic viscosity(kg/s.m) υ kiematic viscosity(m/s) ϕ o dimesioal cocetratio of species Ѳ o-dimesioal temperature ρ desity (kg/m) τ o dimesioal time vorticity (s ) o dimesioal vorticity Δ differece CONCLUSIONS A umerical study has bee coducted o double diffusive atural covectio pheomeo i the bottom heated eclosure of differet aspect ratio uder mass diffusive side walls. Fiite elemet solutio procedure has bee used to solve goverig equatios ad obtai the solutio for the field variables vorticity, velocity, temperature ad cocetratio. Simulatio results have bee obtaied for covective heat trasfer uder varyig Rayleigh umber ad buoyacy ratios i the rage, Ra ad N respectively for eclosures of three aspect ratios, ad. Iteractio of thermal ad solutal buoyacy forces iside the eclosures resulted i followig observatios. Effect of solutal gradiets imposed from boudary coditios greatly felt o fluid circulatio iside the eclosure, i most of the cases solutal buoyacy forces had multiplied the covectio cells ad ehaced the fluid vorticity. The maimum icrease i the Nusselt umber with icreases i buoyacy ratio for deep eclosure, square ad shallow eclosure are observed as %, 4.% ad.6% respectively. (iii) Fluid vorticity got hampered slightly whe both thermal ad solutal buoyacy forces are of equal magitude. 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