2 ème Congrès Français de Mécanique Bordeaux, 26 au 3 août 23 Natural eat Transfer Convection in a Square Cavity Including a Square eater K. RAGUI, Y.K. BENKALA, N. LABSI, A. BOUTRA University of Science and Technology ouari Boumediene (USTB), B.P. 32, El-Alia Bab-Ezzouar, 6 ALGERIA. Résumé : L'objectif du présent travail est d'analyser, par le biais d'une méthode numérique basée sur les volumes finis, l'effet du nombre de Rayleigh ainsi que la présence d'un obstacle isotherme sur les caractéristiques hydrodynamiques et thermiques de l'écoulement d'un fluide newtonien. Ce dernier se trouve emprisonné dans une cavité carrée, de parois verticales isothermes et froides, d'une paroi inférieure isotherme, maintenue à une température chaude et d'une paroi supérieure adiabatique. Le code de calcul basé sur l algorithme SIMPLER a été validé par comparaison des résultats avec ceux de la littérature. Les résultats montrent que le nombre de Nusselt moyen est une fonction croissante du nombre de Rayleigh et de la taille de l obstacle également. Des corrélations permettant de prédire la taille optimale, offrant un meilleur transfert thermique au sein de la cavité, sont proposées. Abstract : This paper reports a numerical study of natural heat transfer convection in square enclosure, filled with a Newtonian fluid, having a centrally-placed heated block. The cavity consists of adiabatic upper wall and hot bottom wall while the vertical walls are maintained at a cold temperature. The parametric study covers the range 3 Ra 6 is done at Prandtl number equal to 7.. Additionally, the effect of increasing the size of the heater is investigated as well. The finite volume method and the SIMPLER algorithm are used to solve the governing equations. The validity of the numerical code used was ascertained by comparing our results with previously published ones. The results show that the mean Nusselt number is an increasing function of the Rayleigh number. Moreover, with higher Rayleigh number, the cavity heat transfer increases with the width of the heater until it reaches a critical value, where the heat transfer reaches its maximum. Summarizing the numerical study, useful correlations predicting this optimum width as a function of Ra are proposed. Key words : Newtonian fluid, square enclosure, isothermally heated block, finite volume method. Nomenclature C p Specific heat transfer at constant w Width of the block (m) Pressure (J kg - K - ) x, y orizontal and vertical coordinates (m) Gr Grashof number, = g β ρ 2 T 3 / µ 2 X, Y orizontal and vertical dimensionless Cavity height (m) coordinates k Fluid thermal conductivity (W m - K - ) Greek symbols Nu i Local Nusselt number β Thermal expansion coefficient (K - ) Nu avg Mean Nusselt number, = Nu i(walls) /4 µ Dynamic viscosity of the fluid (kg m - s - ) Nu* eat enhancement parameter, Nu' avg /Nu avg ρ Fluid density (kg m -3 ) p Pressure (Pa) ρ Fluid density at a reference temperature P Dimensionless pressure (kg m -3 ) Pr Prandtl number, = C p µ / k θ Dimensionless temperature Ra 2 Rayleigh number, = g β ρ T 3 C p /µ k Subscripts T Temperature (K) ' Case 2 u, v orizontal and vertical velocity avg Average
2 ème Congrès Français de Mécanique Bordeaux, 26 au 3 août 23 components (m s - ) c Cold value U, V orizontal and vertical dimensionless h ot value velocity components opt Optimal value Introduction For many years, the natural heat transfer convection phenomenon in the rectangular cavities is a topic of great interest as it is frequently encountered in numerous industrial applications such as heat exchangers, home ventilation, electronic cooling devices, and solar power collectors. The understanding of the recirculation flow and heat transfer within the enclosure is considered as one of the fundamental challenge of computational fluid research. Even though there have been numerous investigations conducted on natural convection in empty cavities under various configurations and boundary conditions [-3], relatively few studies are conducted for the case of natural convection in a cavity having an internal heated square partition [4, 5]. In the present work, comprehensive numerical investigation is performed on laminar natural convection in a square cavity, completely filled with Newtonian fluid, having vertical cold walls, and including a central square heater of different sizes. A large temperature gradient is caused by the hot bottom wall and the heater inside the cavity. Keeping the Prandtl number fixed, a wide range of Rayleigh number and blockage size are considered. We add that two cases are studied in this paper, case which consists in a square cavity filled completely only with the Newtonian fluid. Case 2, the same cavity with the heater placed at the centre. 2. Problem statement The problem under investigation is a laminar, two dimensional natural heat transfer convection in a square cavity, with vertical cold walls and hot bottom one, while the upper wall is kept adiabatic. The physical problem as well as its boundary conditions is shown in figure. Two cases are considered: Case which is the square enclosure of figure without the heater, filled completely with the convection fluid. Case 2, shown in figure, is an enclosure with an isothermally heated square block placed at the centre, the remaining area filled completely with the convecting fluid, its thermo-physical properties are assumed constant, except for the density in the buoyancy term in the momentum equations which is treated according to Boussinesq model. U = V =, θ/ = g U = V =, θ = Y O X θ = w U = V =, θ = U = V =, θ = FIG. Schematic of the simulation domain with its boundary conditions. 3. Governing equations The dimensionless equations of continuity (), momentum (2, 3) and energy (4) are respectively: U + = () U U P U U U + V = + 2µ + µ + (2) Gr 2
2 ème Congrès Français de Mécanique Bordeaux, 26 au 3 août 23 P U U + V = + 2 + θ X Y Y Gr X µ + Y X + µ (3) where the non-dimensional variables used in () to (4) are defined as: X = x, Y = y, U = 2 2 θ θ θ θ U + V = + (4) 2 2 Pr Gr u gβ T, V = v gβ T P = ρ For the cavity walls, the following generic boundary conditions are used: For the heater: p ( gβ T), θ = ( T Tc ) ( T T ) X = < Y < U = V= θ = (6) X = < Y < U = V= θ = (7) Y = < X < U = V= θ = (8) Y = < X < U = V= θ / = (9) U = V = θ = () The heat transfer across the walls of the enclosure can be quantified by using a wall surface averaged Nusselt number, based on the enclosure length scale () which is given as: θ Nu left wall = Nu right wall = dy () Nu wall θ = dx (2) bottum wall Y = 4. Numerical procedure and validation The governing equations are discretized by means of the finite volume method. The resulting algebraic equations and the associated boundary conditions are solved using the line by line method and the SIMPLER algorithm. The 8 2 uniform grid was chosen for the computational purpose of the present work. The present numerical code was validated against the Turan problem [6] of natural convection in a square cavity at various values of the Rayleigh number, where the same hypotheses are maintained. Figure 2 demonstrates a comparison of the average Nusselt number evaluated at the hot wall, between the present simulations and those reported by Turan et al. [6]. As it is observed, very good agreements exist between results of the two studies with a maximum discrepancy of about 2%. h c (5) 2 5 Turan et al. [6] Ra = 6 Ra = 5 Ra = 4 Ra = 3 Nu h 5,2,4,6,8, Y FIG. 2 Nusselt number evaluated at the hot wall. Pr =7. 3
..5. 2 ème Congrès Français de Mécanique Bordeaux, 26 au 3 août 23 5. Results and discussion We start our investigation with the case of a square enclosure completely filled with a Newtonian fluid, in the absence of the heated square blockage (Case). The effect of Rayleigh number lying between 3 and 6 on the streamline and isotherm plots is shown in figure 3. For lower Rayleigh numbers, e i. conduction dominant heat transfer regime, streamlines show two symmetrical counter-rotating eddies, established in the cavity. With increasing the Rayleigh number, the streamlines become more packed adjacent to the side cold walls. As Rayleigh number increases to 6, the convection mechanism becomes more pronounced and consequently the streamlines move upward. owever, two secondary vortices develop along the bottom hot surface of the cavity, these secondary vortices are generated by the fluid that remains confined inside a small region, created by the two large vortices. Regarding the isotherms as shows figure 3-b, for Ra = 3 and 4, they are in some ways evenly distributed, and demonstrating a conduction dominated heat transfer regime. With higher values of Rayleigh number, Ra > 4, the natural convection effect start to dominate, and then thin thermal boundary layers are formed along the sides of the enclosure. (a)..5...3.3.3.5.7.9.3.5.7.9.3.7.9.3..7.5.9 (b) Ra = 3 Ra = 4 Ra = 5 Ra = 6 FIG. 3 Streamlines (a) and isotherms (b) in the cavity filled only with the convecting fluid Pr = 7..5.7 Through the Table, it is interesting to note that the mean Nusselt number occurs in the enclosure is an increasing function with increase Rayleigh number. Table. Mean Nusselt number of the empty cavity for different Rayleigh numbers Ra 3 4 5 6 Nu avg 8.37 9.748.44 3.883 At Ra = 6, figure 4 displays streamline and isotherm plots for an enclosure having a centrally heated square block inside (case 2). We present in this section three different sizes of the block as: w =,.5 and.7. Liken the second case with the empty cavity (figure 3), the proposed scheme does affect the flow structure of the fluid and so, the heat distribution. When the natural convection become dominant, the second gradient developed by the heater makes the streamlines more densely packed inside the cavity, when the eyes of the counter-rotating move downwards consequently. In addition to this, the increasing in width w to.5 causes a rise of two secondary vortices, 4
.7. 2 ème Congrès Français de Mécanique Bordeaux, 26 au 3 août 23 located in the upper surface of the heater, while the primary eddies move to the side walls. For w =.7, the primary counter-rotating eddies are quite weak as the fluid volume is replaced by the heated square blockage. The isotherms for each width are adjusted according to the presented plots. As we can see, for w =, some distinct thermal boundary layers are formed around the square heater as well as along the cold vertical walls. With increasing the width of the heater w >.5, the isotherms show a conduction-dominated heat transfer regime. Moreover, a conduction effect is plotted near the bottom wall of the square cavity, which is going to die with increasing the width as it is illustrated in the case of w =.7. (a).7.7.9.9.7.9.9.7.7.5.9.9.5.9.9.5.5..3.7.3...3.3..3.5.5.3..9.9 (b) FIG. 4 Streamlines (a) and isotherms (b) in the cavity with central blockage placement Ra = 6 ; Pr = 7. Figure 5 shows the effect of different values of w on heat transfer enhancement inside the cavity, for a range of Ra taken between 3 and 6. It must be mentioned here that, for Ra = 6, a width named w opt of the heater is required to get an optimum (maximum) heat transfer. Using figure 6 which summarizes the results of the numerical simulations, predictive correlations relating w opt to the Ra and Nu*as well are proposed of the form: w opt = [ Log( Ra ) A 2 ] (3) A 2 w opt + B2w opt B3 Nu * = B + (4) where the coefficients of each equation are listed in Table 2, along with the R 2 values to indicate the goodness of the curve-fit employed in figure 6. The correlations in (3) and (4) are found to predict the numerical results to within ±3%. 5
2 ème Congrès Français de Mécanique Bordeaux, 26 au 3 août 23 3,5 Nu' avg 22 2 8 6 4 Ra = 6 Ra = 5 Ra = 4 Ra = 3 Log (Ra) 7,5 7, 6,5 Log (Ra) = f(w opt ) w opt, Eq. 3 Nu* = f(w opt ) Nu*, Eq. 4 3, 2,5 2, Nu* 2 6,,5 5,5 8,,,2,3,4,5,6,7,8 w FIG. 5 Variation of mean Nusselt number with the heater block width, for different Ra.,,4,5,6,7,8 w opt FIG. 6 Variation of the optimum block width w opt and maximum heat enhancement Nu*with Ra. Table 2. Values of curve-fit constants of equations(3) and (4). A A 2 B B 2 B 3 R 2 for Eq. 3 R 2 for Eq. 4 5.5364 3.2445 2.2-8.793 2.9249.9954.9989 5. Conclusion the natural convection phenomenon inside a square cavity, having an isothermally internal heated square block mounted at the centre has been studied numerically. A parametric study involving the effect of the Rayleigh number and the heater size on the fluid and temperature fields is conducted. The obtained results may resume as follows: In both cases (with or without the heater), the heat transfer in the cavity is an increasing function of Rayleigh number. Moreover it is found that the cavity heat transfer increases, with increase in the width of the square heater, until it reaches a critical value w = w opt, where the heat transfer is getting its maximum. A further increase in the heater width beyond w opt, reduces heat transfer, as the heater size becomes larger and reduces the thermal mass of the convecting fluid on one hand, and reduces the temperature gradient caused by the bottom wall, on the other hand. Useful correlations predicting this optimum heater width and the corresponding maximum heat transfer as a function of Ra are proposed; these predict within ±3%, the numerical results. References [] de Vahl Davis G., Natural convection of air in a square cavity: a benchmark numerical solution, Int. J. Numer. Methods Fluids, 3, 227-248, 983. [2] Freitas C J., Street R L., Findikakis A N., Koseff J R., Numerical simulation of three-dimensional flow in a cavity, Int. J. Numer. Meth. Fluids, 5, 56-575, 985. [3] Paolucci S, Chenoweth D R., Natural convection in shallow enclosures with differentially heated end walls, J. eat Transfer,, 625-634, 988. [4] Yeong a M, Mi J J., A numerical study on three-dimensional conjugate heat transfer of natural convection and conduction in a differentially heated cubic enclosure with a heat-generating cubic conducting body, Int. J. eat Mass Transfer, 43, 4229-4248, 2. [5] Bouafia M., Daude O., Natural convection for large temperature gradients around a square solid body within a rectangular cavity, Int. J. eat Mass Transfer, 5, 3599-365, 27. [6] Turan O., Nilanjan C., Poole R J., Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls, J. Non-Newtonian Fluid Mech, 65, 9-93, 2. 6