Buoyancy Induced Heat Transfer in a Trapezoidal Enclosure with Offset Baffles

Transcription

1 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes F. Mokaed and M. Darwish Department of Mechanica Engineering American University of Beirt P.O.Box Riad E Soh, Beirt Lebanon ABSTRACT A nmerica stdy has been condcted to examine the effects on heat transfer of monting two offset baffes onto the pper incined and ower horizonta srfaces of trapezoida cavities. Two therma bondary conditions are considered. In the first, the eft short vertica wa is heated whie the right ong vertica wa is cooed (boyancy assisting mode aong the pper incined srface of the cavity). In the second, the right ong vertica wa is heated whie the eft short vertica wa is cooed (boyancy opposing mode aong the pper incined srface of the cavity). For both bondary conditions, comptations are performed for: severa offset baffe heights, for Rayeigh nmber vaes, three Prandt (Pr) nmber vaes, and two baffe positions (Position I and Position II). In Position I, the ower baffe is offset toward the short vertica wa and the pper baffe is offset toward the ong vertica wa of the encosre, whereas in Position II, the ower and pper baffes are offset toward the ong and short vertica was, respectivey. Rests revea a decrease in heat transfer in the presence of baffes with its rate generay increasing with increased baffe height and Pr. At a given baffe height and Ra, N vaes are ower in the boyancy opposing mode. For both bondary conditions, the highest decrease is achieved in fy partitioned encosres. Athor to whom a correspondence shod be addressed. Emai: memok@ab.ed.b

2 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 2 NOMENCLATURE A c P g H area specific heat of fid gravitationa acceeration height of the short vertica wa H height of the cavity at the ocation of the baffe H b i j k k b k r L L b n N N p,p Pr Q & Ra height of baffe nit vector in x-direction nit vector in y-direction therma condctivity baffe therma condctivity condctivity ratio (k b /k) encosre width distance between short wa and baffe norma nit vector at baffe-air interface oca Nsset nmber average Nsset nmber dimensioness and dimensiona pressre Prandt nmber (=μc P /k) heat fx Rayeigh nmber (=gβ(t h - T c )H 3 /να)

3 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 3 T dimensiona temperatre, U dimensioness and dimensiona horizonta veocity component dimensioness veocity vector v, V dimensioness and dimensiona vertica veocity component W b baffe thickness x, X dimensioness and dimensiona coordinate aong the horizonta direction y, Y dimensioness and dimensiona coordinate aong the vertica direction GREEK SYMBOLS β μ ν θ ρ coefficient of therma expansion viscosity kinematic viscosity (μ/ρ) dimensioness temperatre density SUBSCRIPTS b c h i max baffe cod wa hot wa condition at baffe-air interface ower maximm vae pper

4 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes INTRODUCTION Boyancy indced heat transfer in encosres is sti attracting the attention of many researchers de to its reevance to many engineering appications invoving among others cooing of eectronic eqipment, soar coectors, soidification, ncear waste disposa, and natra convection in attics. The hydrodynamics and heat transfer characteristics being highy dependent on the geometry and bondary conditions of the encosre rest in fndamenta sotions specific to the configration at hand and necessitate obtaining new sotions for new configrations. This artice reports on a nmerica investigation condcted to expore the effects of attaching two offset baffes to the ower horizonta base and pper incined srface of a trapezoida encosre. Extensive work on natra convection heat transfer in regar shaped encosres was reported in the iteratre, a review of which can be fond in [1,2]. However in many practica probems the boyancy-indced fows are say srronded by compicatedshaped bondaries. Since it is diffict to envisage sotions a priori from those obtained in regar encosres, attention has recenty been directed towards stdying boyancyindced heat transfer in encosres of irregar shapes. Experimenta and theoretica investigations of natra convection heat transfer in an incined trapezoida cavity formed from parae cyindrica top and bottom was and adiabatic side was were reported by Iyican et a. [3,4]. Simiary, Lam et a. [5] reported experimenta and nmerica natra convection rests in a trapezoida cavity formed from two vertica adiabatic side was, a horizonta hot bottom wa, and an incined cod top wa. Experimenta measrements reveaed that the two-dimensiona nmerica mode

5 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 5 sed in [5] is capabe of predicting heat transfer rates to an acceptabe eve of accracy. Nmerica rests for aminar natra convection in trapezoida encosres with insated horizonta bottom and top was and incined hot and cod sidewas were reported by severa researchers [6-]. The Lee [6,7] and Peric [8] predictions were imited to a Rayeigh nmber vae of 5, the Sadat and Saagnac [9] rests were for Rayeigh nmber vaes ranging from 3 to 2x 5, and the Kyper and Hoogendoorn [] comptations were for Rayeigh nmbers between 4 and 8. Transient rests in the encosre were recorded by Karyakin [11]. Ridoane et a. [12] reported on natra convection heat transfer in an attic space with soped roofs and horizontay sspended ceiings for smmer-ike and winter-ike bondary conditions. Depending on the sope of the roof, the shape varied from an isoscees triangar encosre to a trapezoida encosre. Their stdy reveaed that the presence of insated side was provides a sizabe saving in energy to keep the attic at the desired temperatre dring both smmer and winter. Stdies on boyancy-indced heat transfer in partiay divided trapezoida cavities are imited to the ones reported by Mokaed and Acharya [13-15], and Mokaed and Darwish [15,16]. Mokaed and Acharya [13-15] investigated nmericay, for smmerike and winter-ike bondary conditions, the effect on natra convection heat transfer of partiay dividing a trapezoida encosre. In [13,14] the partia dividers were attached to the ower horizonta base [13] and pper incined srface [14] of the cavity. In [15] however, two offset partia vertica dividers attached to the pper incined srface and the ower horizonta base of the cavity, were empoyed. The stdies reported by Mokaed and Darwish [16,17] differ from the previos ones in the geometry and bondary

6 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 6 conditions. The encosre sed in this work, simiar to that empoyed in [16,17], is haf the one stdied in [13-15] and the eft vertica bondary is a wa bondary (symmetry bondary in [13-15]). However nike the configrations in [16,17] where a baffe was either attached to the ower horizonta base [16] or the pper incined srface of the cavity [17], two offset partia dividers attached to the ower and pper srfaces are sed. PHYSICAL MODEL AND GOVERNING EQUATIONS Figre 1a is a schematic of the encosre examined in this stdy. The baffes protrde from the ower horizonta base and pper incined srface of the trapezoida cavity. Their finite width, W b, is fixed at 5% of the encosre width, L, in a comptations. Their heights, H b, and H b,, are independenty assigned for different vaes (0, H /3, 2H /3, and H, where H is the height of the cavity at the ocation of baffe). In addition, two offset baffe positions (L b, =L/3, L b, =2L/3 (Position I) and L b, =2L/3, L b, =L/3 (Position II)) are considered. As shown in Figre 1a, L b, (L b, ) represents the distance from the short vertica wa to the ower (pper) baffe. Moreover, the height of the short vertica wa of the cavity, H, is ¼ the width, L (H=L/4). With the incination of pper srface fixed at 15 o, the height of the ta vertica wa is 2.072H. The effects of the presence of the partia dividers on the hydrodynamics and heat transfer characteristics in the cavity are anayzed nder boyancy-aiding and boyancy-opposing bondary conditions. For both conditions, the ower horizonta base and pper incined pane of the cavity are insated. The difference is in the conditions maintained on the vertica was. For the boyancy-aiding mode, the eft short vertica wa of the cavity is maintained at the niform hot temperatre T h and the right ong vertica wa is

7 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 7 maintained at the niform cod temperatre T c. For the boyancy opposing mode, the eft wa is cod (T c ) whie the temperatre of the right wa is T h. The eqations governing the fow and heat transfer are those expressing the conservation of mass, momentm, and energy. The fow is assmed to be aminar, steady, and twodimensiona with constant fid properties, except for the indced variations in the body force term. The transport eqations are non-dimensionaized sing the foowing dimensioness variabes: x X, y Y, U, v V, p P + ρgy T T c = = = = = θ = (1) 2 H H ν / H ν / H ρ( ν / H) Th Tc With the stated assmptions and the Bossinesq approximation, the dimensioness governing transport eqations of mass, momentm, and energy are, respectivey, written as, = 0 (2) = i p + (3) Ra v = j p + v + θ (4) Pr 1 θ = θ (5) Pr In the baffe region, the ony conservation eqation needed is the Lapace eqation and is given by k b / k ( θb ) = 0 (6) Pr where k b and θ b denote the therma condctivity and non-dimensiona temperatre in the baffe, respectivey. The energy baance at the baffe-air interface can be stated as

8 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 8 1 k b / k [( θ) n] i = [( θ b ) n] Pr Pr i (7) where n is a nit vector in the direction norma to the baffe-air interface and the sbscript i refers to the interface. The hydrodynamic and therma bondary conditions needed to sove the above system of eqations (Eqs. (2)-(5)) are the no-sip condition on the encosre was, non-dimensiona niform temperatres of 1 and 0 aong the hot and cod was, and the zero temperatre gradient aong the insated was. After cacating the veocity and temperatre fieds, the oca and average Nsset nmbers aong the hot or cod vertica wa are cacated as 1 N = h / k N = NdX (8) 0 where is the height of the hot or cod wa. Based on this definition, the average Nsset nmber vaes aong both was are eqa. Moreover, the heat transfer coefficient h is defined as dt Q & = ha( Th Tc ) = ka dx (9) Using dimensioness qantities, the foowing reation for the heat transfer coefficient is obtained: k dθ h = () H dx SOLUTION PROCEDURE A coocated Finite Vome Method (FVM) is sed to sove the coped system of eqations governing the fow and temperatre fieds (Eqs. (2)-(5)). Checkerboard

9 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 9 pressre and veocity fieds are eiminated throgh the se of the Momentm Weighted Interpoation Method (MWIM) for the cacation of the mass fxes across the contro vome faces [18]. Pressre-veocity coping is accompished throgh the se of the SIMPLE agorithm of Patankar [19]. Sotions are obtained by sbdividing the physica space into a nmber of contro vomes with grid points paced at their geometric centers (Figre 1b). The discretized eqations are obtained throgh a two-step procedre. In step 1, the conservation eqations are integrated over a contro vome (Figre 1a) to obtain a discretized description of the conservation aw. In step 2, an interpoation profie is sed to redce the integrated eqations to agebraic eqations by expressing the variation in the dependent variabe and its derivatives in terms of the grid point vaes. The approximation scheme prodces an expression for the face vae which is dependent on the noda vaes in the vicinity of the face. The diffsion fx is discretized aong each srface of the contro vome sing the method described in Zwart et a. [20], whie the convective fx is cacated sing the third order SMART scheme [21] appied within the context of the NVSF methodoogy [22]. In addition, the integra vae of the sorce term over the contro vome P (Figre 1b) is evaated by assming the estimate of the sorce at the contro vome center to represent the mean vae over the whoe contro vome. Then, the set of agebraic eqations is soved iterativey sing the Tri-Diagona Matrix Agorithm (TDMA) [19]. Moreover, the grid (Figre 1c) is generated sing the transfinite interpoation techniqe [23]. Frthermore, the presence of the baffe in the cacation domain is acconted for by the specia treatment sggested by Patankar [19]. Finay, since a conservative scheme is sed, arranging the contro vome face to

10 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes coincide with the divider interface ensres energy baance at the baffe-air interface and forces eqation (7) to be impicity satisfied. NUMERICAL ACCURACY Grid-independent sotions were estabished by comparing sotions generated on different grid sizes. A fina non-niform mesh of size 122x122 grid points was sed in generating a rests presented in this paper. The grid points were concentrated cose to soid bondaries where arge gradients are expected (Figre 1c). The accracy of the cacations was verified by comparing representative compted profies of veocity, temperatre, and oca Nsset nmber vaes sing the 122x122 non-niform grid with those obtained on a 240x240 neary niform grid. The maximm difference between the two sotions in the varios qantities predicted was smaer than 0.1%. Conservation of the varios physica qantities was satisfied to within -5 % in each contro vome. RESULTS AND DISCUSSION The governing parameters in the probem are the Prandt nmber (Pr), the Rayeigh nmber (Ra), the condctivity ratio (k r ), the heights of the pper and ower baffes (H b, and H b, ), and the reative position of the offset baffes. For both bondary conditions, rests are obtained for for baffe heights (0, H /3, 2H /3, and H ), two offset baffe configrations (L b, =L/3, L b, =2L/3 (Position I) and L b, =2L/3, L b, =L/3 (Position II)), three Prandt nmbers (Pr=0.7,, and 130), and Rayeigh nmber vaes varying between 3 and 6. Moreover, the condctivity ratio is fixed at 2 to simate a poory

11 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 11 condcting divider. Rests are presented in the form of representative streamines, isotherms, and oca and average Nsset nmber vaes. BUOYANCY-AIDING MODE Streamines and isotherms Representative streamine and isotherm pots for Pr=0.7 are dispayed in Figres 2-4. For Position I, the fow patterns and temperatre distribtions are depicted in Figre 2 for the case where the ower and pper baffes are at heights Hb, / 3 and Hb, / 3, respectivey. Rests indicate that at ow and moderate Ra vaes (Figres 2a and 2b) the fow consists of three contercockwise rotating vortices commnicating throgh a thin overa rotating eddy. The cockwise rotation of the fow indicates that it moves p aong the eft hot vertica and insated incined was, then trns arond the pper baffe, down aong the cod wa, and then to the eft aong the horizonta base of the cavity and arond the ower baffe. As Ra increases to 5 (Figre 2c), the fid rising aong the eft hot vertica wa becomes more boyant aowing deeper penetration into the right portion of the domain, the interaction between the vortices increases, the eyes of the circating eddies eongate, and the midde core disintegrates in favor of the right and eft vortices. With frther increase in Ra to 6 (Figre 2d), cearer separation of the vortices occr with formation of two jet-ike fows. The first one, which is directed from the hot wa to the eft baffe tip, impinges on the eft face of the pper baffe and retrns back to the eft portion of the cavity. The second jet-ike fow, which is directed from the cod wa to the right baffe tip, impinges on the right face of the baffe attached to the ower horizonta base of the

12 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 12 cavity and retrns back to the right portion of the cavity. This behavior is de to the increase of the stratification eve in the encosre with increasing Ra vae as ceary seen by the isotherms presented in Figres 2e-2h. At ow Ra (Ra= 3, Figre 2e), isotherms are niformy distribted between the hot and cod was showing dominant condction heat transfer. As Ra vaes increase, the distribtion of isotherms impies higher stratification eves within the encosre (compare Figres 2e-2h) and conseqenty higher convection contribtion. In addition, the bondary-ayer-type fow aong the hot and cod was becomes cearer. For Position II, streamine and isotherm maps are dispayed in Figre 3. As depicted, for Ra= 3 and 4 the hydrodynamic (Figres 3a and 3b) and thermodynamic (Figres 3e and 3f) featres of the fow are simiar to those presented in Figres 2a and 2b (Position I) with the fow fied being composed of three vortex cores rotating in a cockwise direction. As Ra increases (Figres 3c and 3d), commnication between the inner vortices increases and streamines revea that increasing amonts of the fow, moving down the right cod vertica wa, is defected off it near the ower baffe tip. This is de to therma stratification (Figres 3g and 3h) in the ower right portion of the domain between the divider and the cod wa, which inhibits fow penetration into these regions. Representative streamines and isotherms showing the effects of baffe height on the veocity and temperatre fieds are presented in Figre 4 in an encosre with offset baffes (Position I) of heights 0, H/3, 2H/3, and H for a Ra of 5. The maximm strength of the fow is in the baffe-free encosre (Figre 4a), which is composed of a singe vortex rotating cockwise. As the offset baffe height increases, a weaker fow is observed in the cavity. For a fy partitioned encosre (Figre 4d), three

13 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 13 simiar cockwise rotating eddies are apparent with their strength ower than the singe vortex fow in the non-partitioned cavity. This is de to smaer convective area in each part combined with a decrease in the avaiabe temperatre difference. These findings are refected on the maps presented in Figres 4e-4h, which ceary show the decrease in convection heat transfer throgh the spread of isotherms. Nsset nmbers Typica oca Nsset nmber variations ( N) aong the hot and cod vertica was are presented in Figre 5. Vaes are potted as a fnction of Y/Y max where Y max is the height of the hot or cod vertica wa. The effect of Ra on Nsset nmber vaes is dispayed aong the hot and cod was in Figres 5a and 5b, respectivey, for an encosre with baffe heights of Hb, /3 and Hb, /3 paced in Position I. The oca vae of the Nsset nmber decreases as the fid moves pward aong the hot wa and downward aong the cod wa de to a decrease in the temperatre difference between the fid and the wa. As depicted, the Nsset nmber eves increase with increasing vaes of Ra indicating higher convection contribtion to the tota heat transfer. Moreover, N peaks at the pper section of the cod wa at the ocation of impingement of the hot rising fid. Simiar peaks occr in both the ower and pper parts of the hot wa bt are not as sharp. The steepness of the peak as the hot fid impinges on the cod wa is de to the aiding effects of boyancy aong the pper incined pane of the encosre, which frther increases the veocity of the hot fid before striking the cod wa. The Nsset nmber distribtions aong the hot and cod was, dispayed respectivey in Figres 5c and 5d, in an encosre with offset baffes in Position I at a Ra vae of 5 ceary revea the decrease in heat transfer as the offset baffe heights

14 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 14 increase. This decrease is de to a redction in the convection contribtion to tota heat transfer cased by the presence of the partia dividers, which interrpt the convective motion and redce the convection heat transfer coefficient. The average Nsset nmber ( N ) vaes in the encosre are presented in Tabe 1. In comparison with a baffe free encosre, the presence of the baffes generay redces the amont of heat transported across the cavity with the percentage of redction increasing with increasing baffes heights and Rayeigh nmber vaes. Moreover, the heat transfer is normay ower for a cavity with offset baffes in Position I. This is de to the fact that in Position I the pper baffe is ocated coser to the cod wa than in Position II, which interrpts the convective motion at a distance coser to the cod wa before impinging it. As expected, N increases with increasing Pr de to a decrease in the therma bondary ayer thickness aong the was with a conseqent increase in the temperatre gradient. The rate of increase sows down as the Prandt nmber increases with the vaes for Pr= and Pr=130 being very cose. BUOYANCY-OPPOSING MODE Streamines and isotherms For the boyancy-opposing heat transfer mode, streamines and isotherms are presented in Figres 6-8 with a vae of Pr=0.7, representing air as the working fid. Figres 6 and 7 revea the effects of Ra on the fow and the heat transfer characteristics for a partitioned encosre with offset baffes of heights Hb, /3and Hb, /3 paced in positions I and II, respectivey. Opposite to the boyancy-aiding sitation (Figre 2), the fid in the encosre moves in a contercockwise direction. By comparing rests in Figres 6

15 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 15 and 7 against rests in Figres 2 and 3 it is easy to notice that for the boyancy-opposing mode the fow strctre over a domain with baffes in Position I (Figre 6) are simiar to those for the boyancy-aiding mode with baffes paced in Position II (Figre 3), and vice versa (i.e. simiarity of fow strctre in Figres 7 and 2). For both baffe positions, the fow characteristics are simiar at ow Ra vaes (Figres 6a and b and Figres 7a and b) with the fow fied being composed of three recircation zones. As Ra increases, baffe position becomes important and the fow characteristics change with the midde recircation zone increasing in size for Position I whie spitting and forming a jet-ike fow for baffes paced in Position II. In both positions, as Ra increases the eyes of the recircation zones eongate and move towards the bottom of the domain (opposite to the boyancy-aiding mode). No separation of the fid on either side of the dividers was noted. By comparing streamines in Figre 6 against those reported in Figre 7 it can be inferred that, for Hb, /3 and Hb, /3, pacing the pper divider cose to the hot wa intensifies the midde recircation zone, which is expected to increase convection heat transfer across the cavity. At Ra= 3 the temperatre niformy varies over the domain. As Ra increases, isotherms become more distorted reveaing an increase in convection contribtion to heat transfer in the cavity. Moreover, at high Ra vaes (Figres 6g, 6h, 7g, and 7h) isotherms revea high stratification eves on the top right side of the domain where the rising hot fid has to descend aong the incined top srface of the encosre. Figre 8 reveas the effects of baffe height in an encosre with offset baffes in Position I and for Ra= 5. In the absence of baffes (Figre 8a), the fow in the encosre is composed of a singe contercockwise rotating ce. At this vae of Ra, the eye of the

16 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 16 recircation is eongated and is abot to separate into two smaer vortices one cose to the hot wa and the other cose to the cod wa. As the height of baffes increases, the fow becomes weaker (Figres 8a-8d) and a decrease in convection effects is observed which is manifested by a ighter cstering of isotherms aong the hot and cod was (Figres 8e-8h). Nsset nmbers Representative pots showing the distribtion of Nsset nmber vaes aong the vertica was of the encosre are dispayed in Figre 9. The effects of Ra on heat transfer are depicted in Figres 9a and 9b where variations in oca N estimates are dispayed aong the cod and hot was of a cavity with offset baffes of heights Hb, /3 and Hb, /3 paced in Position I. It is easiy inferred from the figres that convection heat transfer increases with increasing vaes of Ra. Aong both the cod (Figre 9a) and hot (Figre 9b) was, the Nsset nmber decreases in the direction of fid motion and rests in the variations shown by the figres. This decrease is de to the decrease in the temperatre difference between the fid and the wa (the fid coos aong the cod wa and heats aong the hot wa). Moreover, the high N vae near Y/Y max =1 (Figre 9a), is de to the arge temperatre difference between the hot fid and the cod wa. Frthermore, the peak at the eading edge in Figre 9b is cased by the impingement of the cod fid on the hot wa whie trying to negotiate the corner. The effect of baffe height on heat transfer is presented in Figres 9c and 9d for Ra= 5. When baffe heights are increased, pots revea a decrease in N vaes aong the hot and cod was as a conseqence of the decrease in convection heat transfer.

17 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 17 The average Nsset nmber ( N ) vaes for a cases stdied are dispayed in Tabe 2. As expected, vaes are ower than their conterparts dispayed in Tabe 1 for the boyancy-aiding mode. The average Nsset nmber increases with increasing Ra vaes. Frther, N in a partitioned cavity is ower than its vae in a baffe free cavity indicating a redction in convection heat transfer de to the presence of baffes. At Ra= 3, there exists a baffe height at which the decrease in heat transfer is maximized. This optimm height is dictated by the competing effects of convection and diffsion which decrease and increase, respectivey, with the increase in either of the offset baffe heights. At Ra 4, convection is the dominant heat transfer mode and N decreases with increasing offset baffe heights with the owest vae obtained for the fy partitioned cavity (i.e. when H = and b, H H = ). In both baffe positions, the ower b, H baffe is more effective in redcing heat transfer than the pper baffe with this effectiveness being higher for offset baffes in Position I (e.g. at Ra= 6, N = for Hb, /3 and Hb, = H / 3 whie N = for Hb, = H / 3 and Hb, / 3 ). As in the boyancy-assisting case, N increases with increasing Pr de to an increase in the temperatre gradient aong the was. Again, the rate of increase goes down as the Prandt nmber increases with the vaes for Pr= and Pr=130 being amost identica. CLOSING REMARKS Natra convection in a trapezoida encosre with offset baffes monted onto its pper incined and ower horizonta srface has been stdied nmericay. For the two offset baffe positions considered (Positions I and II), the effects of the Rayeigh nmber,

18 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 18 Prandt nmber, and baffe height on heat transfer were investigated. Two bondary conditions were stdied representing boyancy assisting and boyancy opposing modes aong the pper incined srface of the cavity. Rests reveaed a redction in heat transfer in the presence of baffes with its rate increasing with increasing Pr and/or offset baffe heights. The heat transfer eve was fond to be ower in the boyancy opposing case. ACKNOWLEDGMENTS The financia spport provided by the Lebanese Nationa Conci for Scientific Research (LNCSR) throgh Grant No is gratefy acknowedged.

19 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 19 REFERENCES 1. Ostrach, S., "Natra Convection in Encosres," Jorna of Heat Transfer, vo. 1, pp , Jaria, Y., Natra Convection, in A. Bejan and A.D. Kras (eds.), Heat Transfer Handbook, chap. 7, Wiey, New York, Iyican, L., Bayazitog, Y., and Witte, L., "An Anaytica Stdy of Natra Convective Heat Transfer within a Trapezoida Encosre," Jorna of Heat Transfer, vo. 2, pp , Iyican, L., Witte, L.C., and Bayazitog, Y., "An Experimenta Stdy of Natra Convection in Trapezoida Encosres", Jorna of Heat Transfer, vo. 2, pp , Lam, S.W., Gani, R., and Simons, J.G., "Experimenta and Nmerica Stdies of Natra Convection in Trapezoida Cavities, Jorna of Heat Transfer, vo. 111, pp , Lee, T.S., "Nmerica Experiments with Fid Convection in Tited Nonrectangar Encosres," Nmerica Heat Transfer, Part A, vo. 19, pp , Lee, T.S., "Comptationa and Experimenta Stdies of Convective Fid Motion and Heat Transfer in Incined non-rectangar encosres," Internationa Jorna of Heat and Fid Fow, vo.5, pp , Peric, M., "Natra Convection in Trapezoida Cavities," Nmerica Heat Transfer, Part A, vo. 24, pp , 1993.

20 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes Sadat, H. and Saagnac, P., "Frther rests for Laminar Natra Convection in a Two-dimensiona Trapezoida Encosre," Nmerica Heat Transfer, Part A, vo. 27, pp , Kyper, R.A. and Hoogendoorn, C.J., "Laminar Natra Convection Fow in Trapezoida Encosres," Nmerica Heat Transfer, Part A, vo. 28, pp , Karyakin, Y.E. "Transient Natra Convection in Prismatic Encosres of Arbitrary cross-section," Internationa Jorna of Heat and Mass Transfer, vo. 32, no. 6, pp , Ridoane, H., Campo, A., and Hasnaoi, M.," Benefits Derivabe From Connecting the Bottom and Top Was of Attic Encosres with Insated Vertica Wa," Nmerica Heat Transfer, Part A, vo. 49, pp , Mokaed, F. and Acharya, S., "Boyancy-Indced Heat Transfer in Partiay Divided Trapezoida Cavities," Nmerica Heat Transfer, Part A, vo. 32, pp , Mokaed, F. and Acharya, S., "Natra Convection in Trapezoida Cavities with Baffes Monted on The Upper Incined Srfaces," Nmerica Heat Transfer, Part A, vo. 37, no. 6, pp , Mokaed, F. and Acharya, S., "Natra Convection in a Trapezoida Encosre with Offset Baffe," AIAA Jorna of Thermophysics and Heat Transfer, vo. 15, no. 2, pp , Mokaed, F. and Darwish, M., "Natra Convection in a Partitioned Trapezoida Cavity Heated from the Side," Nmerica Heat Transfer, Part A, vo. 43, pp , 2003.

21 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes Mokaed, F. and Darwish, M., "Natra Convection in a Trapezoida Encosre Heated from the Side with a Baffe Monted on Its Upper Incined Srface," Heat Transfer Engineering, vo. 25, no. 8, pp , Peric, M., "A Finite Vome Method for the Prediction of Three Dimensiona Fid Fow in Compex Dcts," Ph.D. Thesis, Imperia Coege, Mechanica Engineering Department, London, Patankar, S.V., Nmerica Heat Transfer and Fid Fow, New York, Hemisphere Pbishing Corporation, Zwart, P.J., Raithby, G.D., and Raw, M.J., "An Integrated Space-Time Finite-Vome Method for Moving-Bondary Probems," Nmerica Heat Transfer, Part B, vo. 34, pp , Gaske, P.H. and La, A.K.C., "Crvatre Compensated Convective Transport: SMART, A New Bondedness Preserving Transport Agorithm," Internationa Jorna for Nmerica Methods in Fids, vo. 8, pp , Darwish, M. and Mokaed, F., "Normaized Variabe and Space Formation Methodoogy for High-Resotion Schemes," Nmerica Heat Transfer, Part B, vo. 26, pp , Gordon, W.J. And Thei, L.C., "Transfinite Mappings and Their Appications to Grid Generation," in Thompson J.F. (ed.), Nmerica Grid Generation, North Hoand, New York, pp , 1982.

22 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 22 LIST OF TABLES Tabe 1 Average Nsset nmber vaes N for hot eft wa and cod right wa (boyancy assisting bondary condition). Tabe 2 Average Nsset nmber vaes N for cod eft wa and hot right wa (boyancy opposing bondary condition).

23 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 23 FIGURE CAPTIONS Figre 1 (a) Physica domain; (b) a typica contro vome; (c) comptationa domain and an istrative grid network. Figre 2 Streamine and isotherm pots ( H b, /3, H b, /3, Position I) for the boyancy assisting bondary condition. Figre 3 Streamine and isotherm pots ( H b, /3, H b, /3, Position II) for the boyancy assisting bondary condition. Figre 4 Streamine and isotherm pots (Ra= 5, Position I) at different baffes height for the boyancy assisting bondary condition. Figre 5 Loca Nsset nmber distribtion aong (a,c) hot was and (b,d) cod was for the boyancy assisting bondary condition, Position I; (a,b) effects of Ra ( H b, = H /3, H b, = H /3); (c,d) effects of baffe heights (Ra= 5 ). Figre 6 Streamine and isotherm pots ( H b, /3, H b, /3, Position I) for the boyancy opposing bondary condition. Figre 7 Streamine and isotherm pots ( H b, /3, H b, /3, Position II) for the boyancy opposing bondary condition. Figre 8 Streamine and isotherm pots (Ra= 5, Position I) at different baffes height for the boyancy opposing bondary condition. Figre 9 Loca Nsset nmber distribtion aong (a,c) hot was and (b,d) cod was for the boyancy assisting bondary condition, Position I; (a,b) effects of Ra ( H b, = H /3, H b, = H /3); (c,d) effects of baffe heights (Ra= 5 ).

24 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 24 Tabe 1 Average Nsset nmber vaes N for hot eft wa and cod right wa (boyancy assisting bondary condition). Position I (L b, =L/3, L b, =2L/3) H b, H / 3 2 H / 3 H H / 3 2 H / 3 H H / 3 2 H / 3 H Ra No Baffe H b, = H / 3 H b, = 2 H / 3 H b, = H Pr= Pr= Pr= Position II (L b, =2L/3, L b, =L/3) Pr= Pr= Pr= Tabe 2 Average Nsset nmber vaes N for cod eft wa and hot right wa (boyancy opposing bondary condition).

25 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 25 Position I (L b, =L/3, L b, =2L/3) H b, H / 3 2 H / 3 H H / 3 2 H / 3 H H / 3 2 H / 3 H Ra No Baffe H b, = H / 3 H b, = 2 H / 3 H b, = H Pr= Pr= Pr= Position II (L b, =2L/3, L b, =L/3) Pr= Pr= Pr=

26 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes ο L b, H b, H H L b, H b, L (a) N Un W U w P Ue E Us S (b) (c) Figre 1 (a) Physica domain; (b) a typica contro vome; (c) comptationa domain and an istrative grid network.

27 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 27 Streamines Isotherms (a) Ra= 3 (e) (b) Ra= 4 (f) (c) Ra= 5 (g) (d) Ra= 6 (h) Figre 2 Streamine and isotherm pots ( H b, /3, H b, /3, Position I) for the boyancy assisting bondary condition.

28 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 28 Streamines Isotherms (a) Ra= 3 (e) (b) Ra= 4 (f) (c) Ra= 5 (g) (d) Ra= 6 (h) Figre 3 Streamine and isotherm pots ( H b, /3, H b, /3, Position II) for the boyancy assisting bondary condition.

29 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 29 Streamines Isotherms (a) No baffe (e) (b) H b, = H /3 H b, = H /3 (f) (c) H b, /3 H b, /3 (g) (d) H = b, H H = (h) b, H Figre 4 Streamine and isotherm pots (Ra= 5, Position I) at different baffes height for the boyancy assisting bondary condition.

30 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes Ra Ra h N 4 c N Y/Y max Y/Y max 8 (a) (b) 9 h N H b, =0 H b, =0 H =2H /3 H =2H /3 b, b, H =H /3 H =H /3 b, b, H b,=h H b, =H c N H b, =0 H b, =0 H =2H /3 H =2H /3 b, b, H =H /3 H =H /3 b, b, H b,=h H b, =H Y/Y max Y/Y max (c) (d) Figre 5 Loca Nsset nmber distribtion aong (a,c) hot was and (b,d) cod was for the boyancy assisting bondary condition, Position I; (a,b) effects of Ra ( H b, = H /3, H b, = H /3); (c,d) effects of baffe heights (Ra= 5 ).

31 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 31 Streamines Isotherms (a) Ra= 3 (e) (b) Ra= 4 (f) (c) Ra= 5 (g) (d) Ra= 6 (h) Figre 6 Streamine and isotherm pots ( H b, /3, H b, /3, Position I) for the boyancy opposing bondary condition.

32 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 32 Streamines Isotherms (a) Ra= 3 (e) (b) Ra= 4 (f) (c) Ra= 5 (g) (d) Ra= 6 (h) Figre 7 Streamine and isotherm pots ( H b, /3, H b, /3, Position II) for the boyancy opposing bondary condition.

33 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes 33 Streamines Isotherms (a) No baffe (e) (b) H b, = H /3 H b, = H /3 (f) (c) H b, /3 H b, /3 (g) (d) H = b, H H = (h) b, H Figre 8 Streamine and isotherm pots (Ra= 5, Position I) at different baffes height for the boyancy opposing bondary condition.

34 Boyancy Indced Heat Transfer in a Trapezoida Encosre with Offset Baffes c N 12 8 Ra h N 8 Ra Y/Y max Y/Y max c N (a) (b) H b, =0 H b, =0 H =2H /3 H =2H /3 b, b, H =H /3 H =H /3 b, b, H b,=h H b, =H Y/Y max h N Y/Y max H b, =0 H b, =0 H =2H /3 H =2H /3 b, b, H =H /3 H =H /3 b, b, H b,=h H b, =H (c) (d) Figre 9 Loca Nsset nmber distribtion aong (a,c) hot was and (b,d) cod was for the boyancy opposing bondary condition, Position I; (a,b) effects of Ra ( H b, = H /3, H b, = H /3); (c,d) effects of baffe heights (Ra= 5 ).