In this work, the laminar natural convection in high aspect ratio three-dimensional
enclosures has been numerically studied. The enclosures studied here were heated with
uniform heat flux on a vertical wall and cooled at constant temperature on the opposite
wall. The remaining walls were considered adiabatic. Fluid properties were assumed
constant except for the density change with temperature on the buoyancy term. The
governing equations were solved using the finite volumes method and the dimensionless
form of these equations has the Prandtl number and the modified Rayleigh number as
parameters. The influences of the Rayleigh number and of the cavity aspect ratio on the
Nusselt number, for a Prandtl number of 0.7, were analyzed. Results were obtained for
values of the modified Rayleigh number up to 106 and for aspect ratios ranging from 1 to
20. The results were compared with two-dimensional results available in the literature and
the variation of the average Nusselt number with the parameters studied were discussed

Dias Jr., T.

Milanez, L. F.

English

Biblioteca Digital de Periódicos da UFPR (Universidade Federal do Paraná)

96 Engenharia Térmica (Thermal Engineering), Vol. 3 · No. 2 · December 2004 · p. 96-99NATURAL CONVECTION IN HIGH ASPECT RATIO THREE-DIMENSIONAL ENCLOSURES WITH UNIFORM HEAT FLUXON THE HEATED WALLT. Dias Jr.,and L. F. MilanezUniversidade Estadual de CampinasFaculdade de Engenharia MecânicaDepartamento de Energia13081-970Campinas - São PauloBrasiltitodiasjr@uol.com.brABSTRACTIn this work, the laminar natural convection in high aspect ratio three-dimensionalenclosures has been numerically studied. The enclosures studied here were heated withuniform heat flux on a vertical wall and cooled at constant temperature on the oppositewall. The remaining walls were considered adiabatic. Fluid properties were assumedconstant except for the density change with temperature on the buoyancy term. Thegoverning equations were solved using the finite volumes method and the dimensionlessform of these equations has the Prandtl number and the modified Rayleigh number asparameters. The influences of the Rayleigh number and of  the cavity aspect ratio on theNusselt number, for a Prandtl number of 0.7, were analyzed. Results were obtained forvalues of the modified Rayleigh number up to 106 and for aspect ratios ranging from 1 to20. The results were compared with two-dimensional results available in the literature andthe variation of the average Nusselt number with the parameters studied were discussed.Key words: natural convection, numerical study.INTRODUCTIONNatural convection heat transfer hasincreasingly attracting the interest of researchers.This is due to its importance on many engineeringapplications, such as solar collectors, environmentalengineering and electronic packaging. Many studiesconsidering natural convection on plates, channelsand enclosures with heated walls have beenperformed (Ganzarolli and Milanez, 1995). De VahlDavis (1983) studied numerically a two-dimensionalsquare cavity with isothermal walls, and the resultsare frequently used as a benchmark to validate newsimulation softwares. Another example isOosthuizen (2000), who studied numerically a two-dimensional enclosure with uniform heat flux on avertical wall and constant temperature on the othervertical wall while the horizontal walls wereconsidered adiabatics.However, two-dimensional models are stilllimited compared to the complexity of many realsituations, where three-dimensional effects arefrequently present. Some of these effects have beenincluded in the work of Jaluria (1976), whoconsidered an axisymmetric plume flow overhemispherical sources, and in the paper ofKurdyumov and Liñán (1999), who studied flowover spherical sources.Oosthuizen and Paul (1998) studied ahorizontal three-dimensional cavity heated frombelow and cooled from above with uniformtemperatures. They analyzed the aspect ratioinfluence on the critical Rayleigh number, the valueof this parameter where the conductive regimebecomes a convective regime.Tric et al. (2000) obtained accurate results,with the pseudo-spectral Chebyshev algorithmbased on the projection-diffusion approach, for avertical cubical cavity with isothermal oppositewalls. According to them, the global relative errorof the results is in the range of 0.03-0.05% forRayleigh number varying from 103 to 107.In this work we studied three-dimensionalenclosures and compared the results with thoseavailable for two-dimensional geometries. In orderto simulate a geometric configuration usedfrequently in two-dimensional experimentalapproximation, we considered three-dimensionalenclosures with uniform heat flux on a vertical walland the opposite one cooled with uniformtemperature. The remaining walls were consideredadiabatic. According to Oosthuizen (2000), theenclosure width (W) was taken as the characteristiclength, air was the working fluid (Pr=0.7) and thedepth aspect ratio (Ax) was taken as equal to theaspect ratio Az. The study was carried out by varyingAz from 1 to 20 and the modified Rayleigh numberfrom 103 to 106.PROBLEM FORMULATIONLaminar flow and constant fluid properties97the SIMPLE algorithm, since the flow was assumedincompressible. The interpolation of gradients ofvelocity and temperature used the QUICKalgorithm, implemented on Fluent 5.4 softwarepackage.The resulting algebraic system was solvedby the point-to-point Gauss-Seidel algorithm,which is known to have a slow convergence. Toaccelerate this convergence the algebraic multigridmethod was used.The governing equations were solved in thedimensional form and then the results were changedto the dimensionless form. The fluid propertieswere taken from Bejan (1994).RESULTS AND DISCUSSIONThe influence of the aspect ratio Az on theheat transfer rate in vertical three-dimensionalenclosures has been obtained and compared to two-dimensional results obtained by Oosthuizen (2000).Figure 1 shows one of the results obtainedby Oosthuizen (2000). It is possible to observe theaverage Nusselt number variation with the aspectratio Az and the modified Rayleigh number, and acomparison between the two and three-dimensionalcases. It can be seen that the average Nusseltnumber increases with the modified Rayleighnumber for both cases and the difference betweenthese cases increases with the modified Rayleighnumber as the aspect ratio Az increases. Thereforethe two-dimensional configuration frequently useddeviates from the experimental results obtained forthree-dimensional geometry as the modifiedRayleigh number increases.Figure 1. Average Nusselt number variation withaspect ratio Az.T. Dias Jr. and L. F. Milanez  Natural Convection in High...Engenharia Térmica (Thermal Engineering), Vol. 3 · No. 2 · December 2004 · p. 96-99except for the density change with temperature onthe buoyancy term were assumed. The finitevolume control method was used for thediscretization of the elliptic governing equations,the steady three-dimensional versions of continuity,momentum and energy equations (Dias-Jr, 2003).Following Oosthuizen (2000), the dimensionlessparameters considered have been defined as below.Dimensionless Temperature)//()( kqWTT c−=θ (1)where Tc is the temperature of the cooled wall, q isthe heat flux, k is the thermal conductivity and Wis the enclosure width.Aspect Ratio Az(2)where H is the enclosure height.Modified Rayleigh Number(3)where b is thermal expansion coefficient, a is thethermal diffusivity and n is the kinematicviscosity.Average Nusselt Number(4)where hT  is the mean temperature over the heatedwall and hθ is the mean dimensionlesstemperature.The boundary conditions have beenestablished to simulate a geometric configuration usedfrequently in two-dimensional experimentalapproximation. Three-dimensional enclosures wereconsidered with uniform heat flux on a vertical walland the opposite one cooled with uniform temperature.The remaining walls were considered adiabatic.NUMERICAL PROCEDUREThe numerical solution was obtained using98 Engenharia Térmica (Thermal Engineering), Vol. 3 · No. 2 · December 2004 · p. 96-99T. Dias Jr. and L. F. Milanez  Natural Convection in High...curve. This is important since this point ofmaximum could lead the reader to look for a changeon the flow which could justify this point, an artifactarisen from the dimensionless approach.Figure 2. Variation of average Nusselt numbernormalized by the modified Rayleigh number as afunction of the modified Rayleigh numberFigure 3. Normalized average Nusselt number variationwith aspect ratio for the three-dimensional case.CONCLUSIONSThe results obtained in this work indicatethat the two-dimensional approximation, frequentlycompared to experimental results, deviates fromthe three-dimensional results as the modifiedRayleigh number increases.It was shown that the point of maximum,pointed out by Oosthuizen (2000), has disappearedwith a normalization process revealing the realbehavior of the average Nusselt number with theaspect ratio Az and with the modified Rayleighnumber. This is important since that point ofmaximum could lead the reader to look for a changeon the flow which could justify this point, an artifactarisen from the dimensionless approach.It can also be noticed a point where theaverage Nusselt number is a maximum, and thispoint and the corresponding average Nusseltnumber change with the modified Rayleigh numbertoo. This has been pointed out by Oosthuizen (2000),who derived an average Nusselt number functiondepending only on the aspect ratio Az. This curvewas obtained by dividing each Nu x Az by theircorresponding Numax x Azmax, resulting in a functionindependent of the modified Rayleigh number.Independently of the numerical results, itis important to analyze the dependence betweenthe dimensionless parameters used. As previouslydefined by Eq. (4), the average Nusselt number isinversely proportional to the dimensionless meantemperature of the heated wall. And the dimen-sionless temperature is inversely proportional tothe heat flux (q) and to the enclosure width (W);thus, increasing the modified Rayleigh number, byincreasing q or W, the dimensionless temperaturedecreases and the average Nusselt numberincreases. Although the Oosthuizen’s study is basedon the average Nusselt number dependence on themodified Rayleigh number and on the aspect ratioAz, these parameters are not independent variables.Considering only the influence of themodified Rayleigh number, it increases with the heatflux, leading to a decreasing of the dimensionlesstemperature, since the heat flux has been used in thetemperature dimensionless approach. To eliminate thisdependence we have analyzed the relation betweenthe average Nusselt number and the modifiedRayleigh number, as shown in Fig. 2. It can be seenthe curves almost coinciding in the plot.Considering now only the aspect ratioinfluence, since this is a ratio between the heightand the width of the enclosure, its increasing isequivalent to a decreasing of the characteristiclength (W), and this results in an increasing of thedimensionless temperature and consequent averageNusselt number decreasing. A decreasing of Wleads also to a decreasing in the modified Rayleighnumber, since this number depends on W4, thismeaning that Raq is not independent of the aspectratio. Therefore, to eliminate the aspect ratioinfluence we have divided the average Nusseltnumber by the aspect ratio to the third power andby the modified Rayleigh number, as shown inFig. 3. It can be noticed that the point of maximumhas disappeared revealing the real behavior of the99Engenharia Térmica (Thermal Engineering), Vol. 3 · No. 2 · December 2004 · p. 96-99ACKNOWLEDGEMENTSThis work was supported by FAPESP (99/12588-3).REFERENCESBejan, A., 1994, Convection Heat Transfer,2nd ed., John Wiley & Sons, New York, 656p.De Vahl Davis, G., 1983, NaturalConvection in a Square Cavity: a BenchmarkNumerical Solution, International Journal ofNumerical Methods in Fluids, Vol. 3, pp.249-264.Dias-Jr, T. Convecção NaturalTridimensional Sobre Fontes de Calor Discretas.Tese (Doutorado) - Faculdade de EngenhariaMecânica, Universidade de Campinas, 2003. (inPortuguese)Ganzarolli, M. M. and Milanez, L. F., 1995,Natural Convection in Rectangular EnclosuresHeated from Below and Symmetrically Cooledfrom the Sides, International Journal of Heat andMass Transfer, Vol. 38, No. 6, pp. 1063-1073.Jaluria, Y., 1976, Natural Convection FlowInteraction Above a Heated Body, Letters in Heatand Mass Transfer, Vol. 3, pp. 457-466.Kurdyumov, V. N. and Liñán, A., 1999, FreeConvection from a Point Source of Heat, and HeatTransfer from Spheres at Small Grashof Numbers,International Journal of Heat and Mass Transfer,Vol. 42, pp. 3849-3860.Oosthuizen, P. H. and Paul, J. T., 1998, ANumerical Study of Three-dimensional NaturalConvection in a Horizontal Enclosure with aUniform Heat Flux on the Lower Surface,Proceedings of the 11th International Heat  TransferConference, Vol. 3, pp. 391-396.Oosthuizen, P. H., 2000, NaturalConvective Flow in a High Aspect RatioRectangular Enclosure with a Uniform Heat Fluxon the Heated Wall, Proceedings of the 3rdEuropean Thermal Sciences Conference, Vol. 1, pp.159-164.Tric, E., Lambrosse, G. and Betrouni, M.,2000, A First Incursion into the 3D Structure ofNatural Convection on air in a Differentially HeatedCubic Cavity, from Accurate Numerical Solutions,International Journal of Heat and Mass Transfer,Vol. 43, pp. 4043-4056.T. Dias Jr. and L. F. Milanez  Natural Convection in High...

NATURAL CONVECTION IN HIGH ASPECT RATIO THREE-DIMENSIONAL ENCLOSURES WITH UNIFORM HEAT FLUX ON THE HEATED WALL

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NATURAL CONVECTION IN HIGH ASPECT RATIO THREE-DIMENSIONAL ENCLOSURES WITH UNIFORM HEAT FLUX ON THE HEATED WALL

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