Numerical Study of Mixed Convection Coupled to Radiation in a Square Cavity with a Lid-Driven

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1 Numercal Study of Mxed Convecton Coupled to Radaton n a Square Cavty wth a Ld-Drven Mohamed Amne Belmloud, Nord Eddne Sad Chemloul Internatonal Scence Index, Aerospace and Mechancal Engneerng waset.org/publcaton/ Abstract In ths study, we nvestgated numercally heat transfer by mxed convecton coupled to radaton n a square cavty; the upper horzontal wall s movable. he purpose of ths study s to see the nfluence of the emssvty ε and the varyng of the Rchardson number R on the varaton of average Nusselt number Nu. he vertcal walls of the cavty are dfferentally heated, the left wall s mantaned at a unform temperature hgher than the rght wall, and the two horzontal walls are adabatc. he fnte volume method s used for solvng the dmensonless Governng Equatons. Emssvty values used n ths study are ranged between 0 and 1, the Rchardson number n the range 0.1 to 10. he Raylegh number s fxed to Ra=10 and the Prandtl number s mantaned constant Pr=0.71. Streamlnes, sothermal lnes and the average Nusselt number are presented accordng to the surface emssvty. he results of ths study show that the Rchardson number R and emssvty ε affect the average Nusselt number. Keywords Numercal study, mxed convecton, square cavty, wall emssvty, ld-drven. NOMENCLAURE A Aspect rato of the cavty. Convecton. F j Forms factor «the surface S to S j». g Gravty acceleraton. Gr Grashof number. eght of the cavty. I Irradaton J,j N dmensonless radosty Interacton parameter convecton-radaton. r Nu Average total number of Nusselt. Pr Q rd Ra R rd 0 u,v Prandtl number. Radatve heat flux. Raylegh number. Rchardson number. Radaton Dmensonless flud temperature. Reference temperature. Dmensonless velocty components. x,y Dmensonless Cartesan coordnates. Greek Symbols hermal dffusvty of flud Coeffcent of thermal expanson of the flud. Knematc vscosty. hermal conductvty. Mohamed Amne Belmloud s wth the Ibn Khaldoun Unversty, aret, Algera (e-mal: belmloud_amne@yahoo.fr). I. INRODUCION E fundamental problem of heat transfer by mxed convecton coupled to radaton n a square cavty wth a ld-drven has receved consderable attenton from the researchers. hs problem s often encountered n many engneerng applcatons. hese types of problems also arse n electronc packages, mcroelectronc devces durng ther operatons. References [1] and [2] studed the mxed convecton n a cavty wth shfts of varous boundary condtons. Reference [3] studed the flow calculaton of a thermally stratfed vscous flud n a square cavty. he flud flow s from the once the movement of the upper plate and the buoyancy. References [] and [5] conducted respectvely two and three-dmensonal numercal smulaton of mxed convecton n a square cavty heated from the top movng wall. Reference [6] found solutons wth the fnte element method of the mxed convecton n a square cavty heated by the bottom wall. Reference [7] performed a numercal analyss of three-dmensonal models of mxed convecton n an arcooled cavty n order to compare the varatons n dfferent propertes wth the results of two-dmensonal models. Reference [8] studed the process of lamnar mxed convecton n shallow two-dmensonal rectangular cavtes at three dfferent Rchardson numbers representng the domnatng forced convecton, mxed convecton, and domnatng natural convecton usng the Fluent commercal code. he effects of the nclnaton of the cavty on the process result of convecton are also studed. he ncrease n the Nusselt number wth the nclnaton of the cavty s average for the forced convecton and sgnfcant for the natural convecton. Mxed convecton of power-law fluds along a vertcal wedge wth convectve boundary condton n a porous medum have been studed by [9]. Reference [10] analyzed the natural convecton n a prsmatc enclosure; they showed that the temperature decreases as the Raylegh number ncreases. he nteracton between lamnar and turbulent convecton and radaton of non-gray gas was studed by [11] and [12]. Reference [13] has studed numercally natural convecton combned wth radaton n a square cavty dfferentally heated. he enclosure s partally dvded by a thn vertcal baffle. he result shows that the radaton ncreases the crculaton and enhances the heat transfer rate. Reference [1] nvestgated the turbulent natural convecton coupled wth radaton n a square enclosure and proposed a new relatve correlaton between the Nusselt number for sothermal vertcal walls levels dependng on the Raylegh number and the optcal thckness medum. he numercal analyss of the effects of orthotropc thermal conductvty of substrates n natural 1815 scholar.waset.org/ /

2 Internatonal Scence Index, Aerospace and Mechancal Engneerng waset.org/publcaton/ convecton coolng of dscrete heat sources was gven by [15]. her results showed that the radaton s the domnant mode of heat transfer and the effect of the thermal conductvty of the substrate s mportant n the transfer of heat n the enclosure. An expermental and numercal study of combned heat transfer n a three-dmensonal enclosure wth a protrudng heat source was gven by [16]. Combned natural - convecton and volumetrc radaton n an annulus space was studed by [17] usng spectral methods and fnte volumes and by [2] usng a dscontnuous fnte-element formulaton. Y Reference [18] showed that the radaton of a gray flud affects the total Nusselt number, tends to reduce the effects of buoyancy, and accelerates the development of the temperature feld. Reference [19] studed the effects of surface radaton on conjugate mxed convecton n a vertcal channel wth a dscrete heat source n each wall. Reference [20] studed the lnear nstablty and nonlnear stablty of a model for convecton nduced by selectve absorpton of radaton and demonstrated that theory of lnear stablty does not make t possble to predct the convecton onset. Reference [21] has analyzed one -dmensonal lnear stablty for three- dmenson al Raylegh-Benard convecton wth a radatvely partcpatng medum usng spectral methods. hey used a lnear stablty and weakly nonlnear analyss to determne the crtcal Raylegh number for the onset of convecton n the combned mode of heat transfer. Reference [22] appled a multdmensonal scheme wth usng the classcal dscrete ordnates method, found that t s sutable for accurate calculatons n radatve transfer, and can mnmze the ray effect n complex geometrc stuatons. Reference [23] studed the effects of radatve surface on natural convecton n a cavty. he purpose of ths study s to see the nfluence of the emssvty and the varyng of the Rchardson number on thermal and hydrodynamc characterstcs of the flow. II. ANALYSIS For the numercal smulaton, the values of emssvty ε n the range from 0 to 1, the Raylegh number fxed at Ra = 10 and Rchardson number R n nterval the 0.1 to 10. he geometry consdered s a square cavty of whch the two horzontal walls are adabatc, the horzontal top wall s movable n the drecton Ox, and the two vertcal walls are dfferentally heated > C. he four walls are assumed gray. he aspect rato s A=1 and the propertes of ar are manly set at the reference temperature 0 = K. A. Governng Equatons he General Equatons of Conservaton that s, the Contnuty Equaton, the Equatons of Momentum along x and y and the Energy Equaton are gven n ther dmensonless form as: 2 2 u u P 1 u u u v 2 2 x y x Re x y v v P 1 v v u v R x y y Re x y u B. Radaton Equaton 1 x y Re Pr x y 2 2 v 2 2 he General Equatons of Radaton are shown below: Radosty Equaton: Net flux Equaton: J N 1 1- j. j 0 j1 Q N rd 1 - Fj. J j 0 j1 (2) (3) F J () (5) In () and (5), the dmensonless varables are defned by: x y u v P x, y, u, v, P uld uld uld - J Q J Q 0 0, j rd 0,,, j, rd -C -C 0 0 wth C 0 = s the reference temperature. he Raylegh 2 number Ra, the Reynolds number Re, the Rchardson number R and the Prandtl number Pr s defned by: 3 g -C Ra uld Ra ;R ;Re ;Pr (7) 2 Pr Re C. Boundary Condtons o solve (2)-(), the boundary condtons are consdered. hese condtons are gven under ther dmensonless form as: At the rght wall: 0.5;u v 0 At the left wall: 0.5;u v 0 In the adabatc upper horzontal wall: u 1;v 0; NrQrd 0 x In the adabatc lower horzontal wall: u v 0 ; NrQrd 0 x (6) u x v y 0 (1) 1816 scholar.waset.org/ /

3 Internatonal Scence Index, Aerospace and Mechancal Engneerng waset.org/publcaton/ Fg. 1 Schematc of the physcal problem III. NUMERICAL ANALYSIS A. Computatonal Procedure he Mass, Momentum, and Energy Equatons have been solved by a fnte dfference algorthm called the Sem- Implct Method for Pressure Lnked Equatons (SIMPLE). Detals of ths method are descrbed by [25]. he Dfferental Equatons are dscretzed over a control volume. he power law dfference scheme (PLDS) has been employed for the calculaton of scalar varables and the quadratc upstreamweghted nterpolaton for convectve knematcs (QUICK) scheme for [26] vector varables. he relatve tolerance for the 5 error crtera s consdered to be10. B. Grd Check he results are gven n the form of stream lne, sotherm contour, and average Nusselt number n the left wall. he propertes of ar are lnked to the reference temperature K. he Rchardson number used n the range 0.1 to 10 for the flow s lamnar. It can be seen from able I that the dfferences n R = 0.1 between the grd szes of (2 2) and (21 21) are 1.69 % whch s the hgher. owever, the dfferences between R = 1 and R = 10 are comparatvely the lower. he dfference n R = 1 between the grd szes of (62 62) and (2 2) s 0.022% s the lowest. owever, the dfferences between R = 0.1 and R = 10 are the greatest. he grd used n all subsequent calculatons s (2 2). ABLE I GRID EFFEC ON E OBAINED RESULS OF E OAL NUSSEL NUMBER FOR Ra = 10 AND ε=0.5 Grd y O x Average total Nusselt number Nu % Change n Nu (ABS) R = 0.1 R =1 R =10 R = 0.1 R =1 R =10 (21X21) (2X2) (62X62) L C. Valdaton of the Code he frst valdaton of the numercal code was valdated aganst the results of [27] obtaned n the case of a square cavty dfferentally heated. able II shows the comparson g u ld C between the values of the average Nusselt number for Nu, evaluated at the heated wall for Ra varyng n the range Ra 10. It s seen from the table a farly good agreement wth relatve maxmum devatons lmted to 0. % for 0 and 1.8 % for 1. ABLE II EFFEC OF RAYLEIG Ra ON E AVERAGE NUSSEL NUMBER: Nu, EVALUAED ON E EAING WALL OF A SQUARE CAVIY FOR = K AND C = K 0 1 Ra Present work Akyama and Chong [27] he second valdaton of the numercal code has been valdated aganst the results of [28], whch were reported for heat transfer lamnar by mxed convecton n a cavty heated from below ld-drven. he comparson was conducted whle employng the followng dmensonless parameters Re = 500, R = 0. and Pr = 1.Excellent agreement was acheved, as llustrated n Fg. 2, between results of the present work and the numercal results of [28] for both the stream lnes and sothermal lne nsde the cavty. Present work Stream lnes Isothermal lnes Moallem and Jang [28] Fg. 2 Comparson of the stream lnes and sotherms lnes between the present work and that of [28] usng Re = 500, R = 0. and Pr = 1 IV. RESULS AND DISCUSSION A. Stream Lne and Isothermal Dstrbuton he effect of emssvty ( ε ) on the stream lnes and sothermal lnes for Ra = 10 and R = 1 s shown n Fg. 3. he stream lnes take the same form for all three cases. owever, they are entraned by the moblty of the upper 1817 scholar.waset.org/ /

4 horzontal wall. he velocty of crculaton ncreases wth ncreasng the emssvty ε, ths shows that the radatve heat transfer affects the velocty contour, we beleve that ths s due to thermo-physcal propertes of ar. o see the nfluence of the emssvty ε on the sothermal lnes, we compared the dfference between the two vertcal walls; the remark was observed that the heat transfer by convecton s good at the hot wall relatve to the cold wall. Knowng that the heat transfer from pass the left vertcal wall towards the top horzontal wall and then the rght vertcal wall and fnally the bottom horzontal wall, and from the drecton of moblty of the upper horzontal wall, ths explans why the sothermal lnes move away from the cold vertcal wall and the convectve heat transfer decreases n ths wall. For the two horzontal walls, we notce the presence of the force of frcton between the flud and the walls, knowng that the two walls are adabatc. he frctonal force caused a decrease n exchange heat convecton. Internatonal Scence Index, Aerospace and Mechancal Engneerng waset.org/publcaton/ Fg. 3 Effect of emssvty ε on the streamlne and sothermal lnes for Ra = 10 and R = 1 Fg. shows the Rchardson number R effect on streamlnes and sothermal lnes for Ra = 10 andε =0.5, the crtcal values of the convectve heat transfer were found by [29]. When the Rchardson number s n the nterval (0.1 to 16), the convecton s mxed, for low values of Rchardson R < 0.1, forced convecton s domnant, the Rchardson number hgher R > 16, the natural convecton s domnant. As mentoned above, the two values of Ra and Pr are constants, and accordng to the Rchardson number 2 R = Ra PrRe, the value of Reynolds number Re decreases wth ncrease the number of Rchardson R. For the value of Rchardson R = 0.1, we notced a small vortex on the rght bottom part of the cavty. hs vortex was caused due to the flud velocty and velocty of shearng the largest upper horzontal wall. For sothermal lnes, we fnd that forced convecton domnates the natural convecton snce we are near the zone of the forced convecton. For R = 1, we observed that the nfluence of the movng the upper horzontal wall on the streamlnes and the sothermal lnes are reduced compared to the frst case R = 0.1, whch ndcated that both natural and forced convectons exst. We also notce a reducton n the predomnance of the forced convecton on the natural convecton. In the latter case R = 10, the streamlnes and sothermal lnes are almost centered, relatve to the mddle of the cavty. We beleve that ths s due to the exstence of the gravty whch affects the natural convecton, and natural convecton predomnates scholar.waset.org/ /

5 Internatonal Scence Index, Aerospace and Mechancal Engneerng waset.org/publcaton/ Fg. Effect of Rchardson Number R on the stream lnes and sothermal lnes for B. Varaton of eat ransfer Fgs. 5 (a) and (b) respectvely show the varaton of convectve Nusselt Nu and total Nusselt Nu accordng to dfferent emssvty ε for Ra = 10 and for dfferent numbers of Rchardson R = , knowng that all the results that we obtaned at the hot wall. We note that Nusselt number Nu decreases wth the ncrease of the Rchardson number R, ths shows that the convectve heat transfer ncreases when the convectve Nusselt number Nu R = 0. 1 R = 1 R = 1 0 Ra = 10 and ε=0.5 velocty of the cavty a ld ncreases, n the case of the emssvty n the range 0.1 to 1, we note that the convectve Nusselt number decreases wth ncreasng ε. he dfference between the Nusselt values Nu correspondng to R = 0.1 and R = 1, takes a maxmum value of hs dfference decreases to 0.55 for Nusselt values correspondng to R = 1 and R = 10. Fg. 5 (b) shows that passes through a maxmum value ε=0.1 and a mnmum value ε=0. to ncrease then. otal Nusselt number Nu R = 0. 1 R = 1 R = (a) (b) Fg. 5 Varaton of the Nusselt number as a functon of the emssvty ε for the hot wall to R = , number; (b) total Nusselt number Ra = 10 : (a) convectve Nusselt 1819 scholar.waset.org/ /

6 Internatonal Scence Index, Aerospace and Mechancal Engneerng waset.org/publcaton/ A. Varatons of the Dmensonless Velocty and emperature Fg. 6 shows the varaton of the components of u velocty and v velocty profles along the vertcal and horzontal centerlne of the cavty (.e. x =0.5 and y = 0.5), respectvely, for Ra = 10, R = 1 and for dfferent values of the emssvty ε.he u velocty profle s shown n Fg. 6 (a) and t can be seen that are nearly superposed wth the values maxmum of the u velocty obtaned to ε = 1. For the v velocty component, the same comments as those of the component u velocty can be consdered (Fg. 6 (b)). he results s showng that the ncreasng of radaton nfluence on the velocty of crculaton. y v - velocty u - v e l o c t y (a) x = 0 = 0. 5 = (b) Fg. 6 Varaton of the velocty components for Ra = 10, R = 1 and for dfferent values of the emssvty ε : (a) u velocty profle along x = 0.5 ; (b) v velocty profle along y = 0.5 Fg. 7 shows the varaton of the temperature profle along the vertcal and horzontal centerlne of the cavty, evaluated at Ra = 10, R = 1 and for dfferent values of the emssvty ε. he temperature profle as a functon of x for the poston y = 0.5, are gven by Fg. 7 (a). It can be seen that the profles are nearly superposed wth the maxmum value of the temperature for ε=1, n the range 0<x 0.19 and 0.35 < x < 1. If the value of x s n n the range ( ), we obtaned the maxmum value of for ε=0. Fg. 7 (b) s showng the varaton of the temperature profle as a functon of y for the poston x = 0.5. We remark that the profles whch are dstant from each other ntersect at y = 0.35, reman very close beyond ths value. he temperature s decreasng when the emssvty ε ncreases y x (a) (b). 5 Fg. 7 Varaton of the temperature for Ra = 10, R = 1 and for dfferent values of the emssvty ε : (a) profle along y = 0.5; (b) profle along x =0.5 V. CONCLUSION he purpose of ths study deals wth the analyss of the effect of emssvty ε, of the varaton n the number of Rchardson R on mxed convecton coupled to a radaton n a square cavty wth a ld-drven. From the results obtaned we concluded: he velocty s ncreased and the Nusselt number convectve Nu beng reduced when the emssvty ε s ncreased, whch shows that the convectve heat transfer decreases wth the ncrease of the emssvty ε. he heat transfer s the domnant mode forced convecton, when the values of the Rchardson number R are very close to those correspondng to the forced convecton R < 0.1. When the Rchardson number R ncreases, the heat transfer s the domnant mode natural convecton scholar.waset.org/ /

7 Internatonal Scence Index, Aerospace and Mechancal Engneerng waset.org/publcaton/ Fnally, the velocty ld of the cavty nfluences the thermal transfer; ths s proved by the decrease n the Nusselt number Nu when the Rchardson number R ncreases. REFERENCES [1] B. Gebhart, Y. Jalura, R. L. Mahajan and B. Sammaka, Buoyancy nduced flows and transport, emsphere, New York (1988) [2] M. asnaou, E. Blgen and P. Vasseur, Natural convecton above an array of open cavtes heated from below, AIAA J. hermophys. eat ransfer, 6 (1990) [3] R. Iwatsu, J. M. yun and K. Kuwahara, Numercal smulaton of flows drven by a torsonally oscllatng ld n a square cavty, J. Fluds Eng., 11 (1992) [] R. Iwatsu, J. M. yun and K. Kuwahara, Mxed convecton n a drven cavty wth a stable vertcal temperature gradent, Intl. J. eat Mass ransfer, (36) (1993) [5] R. Iwatsu and J. M. yun, hree-dmensonal drven cavty flows wth a vertcal temperature gradent, Intl. J. eat Mass ransfer, (38) (1995) [6] S. C. Lee and C. K. 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Wang, Convectve and Ventlaton ransfers n Greenhouses, Part 1: the Greenhouse consdered as a Perfectly Strred ank, Bosystems Engneerng (2002) 83 (1), Mohamed Amne Belmloud was born n 1988, Algera, receved hs degree master n mechancal engneerng (energy opton) at the Unversty Ibn Khaldoun, aret, Algera and he s currently a Ph.D. canddate n mechancal engneerng, Unversty of Ibn Khaldoun, aret, Algera. s research nterests nclude eat transfer and the Mass, Computatonal Flud Dynamcs (CFD) scholar.waset.org/ /