NATURAL CONVECTION WITHIN TRAPEZOIDAL ENCLOSURE WITH TWO BAFFLES: EFFECT OF VARIOUS ANGLES OF INCLINATION

NATURAL CONVECTION WITHIN TRAPEZOIDAL ENCLOSURE WITH TWO BAFFLES: EFFECT OF VARIOUS ANGLES OF INCLINATION Éliton Fontana, eliton_fontana@hotmail.com Universidade Federal de Santa Catarina - UFSC Adriano da Silva, adrianodasilva.ufsj@gmail.com Universidade Federal de São João Del-Rei - UFSJ Francisco Marcondes, marcondes@ufc.br Departamento de Engenharia Mecânica Universidade Federal do Ceará - UFC Viviana Cocco Mariani, viviana.mariani@pucpr.br Departamento de Engenharia Mecânica - PPGEM Pontifícia Universidade Católica do Paraná PUCPR Departamento da Engenharia Elétrica - DELT Universidade Federal do Paraná - UFPR Abstract. In this study numerical results for natural-convection heat transfer in partially divided trapezoidal cavities representing industrial buildings is described. The left short vertical wall is heated while the right long vertical wall is cooled (buoyancy-assisting mode along the upper inclined surface of the cavity). The governing parameters in the problem are the Rayleigh number (Ra), the heights of the upper and lower baffles (Hb,l and Hb,u), and the upper surface inclination angle (θ ). Results are obtained for two baffle heights (0, H*/3), on offset baffle configurations (Lb,l = L/3, Lb,u = 2L/3), three angles of inclination (, 15 o and 20 o ) and Rayleigh number values varying between 10 3 and 10 6, with constant Prandtl number (Pr = 0.7). Results are presented in the form of representative streamlines, isotherms, and local and average Nusselt number values. The flow pattern and the isotherms are not significantly influenced by the angle of inclination, especially for intermediaries Rayleigh numbers. The Nusselt number, however, shows a significant increase as the angle of inclination growth. Keywords: natural-convection, trapezoidal cavity, angle of inclination, Nusselt number 1. INTRODUCTION Processes involving natural convection in cavities are important for rectangular and non-rectangular cavities. In recent years, an ever-increasing awareness in thermally driven flows reflects that fluid motions and transport processes generated or altered by buoyancy force are of interest due to the practical significances in many fields of science and technology. Natural convection due to thermal buoyancy effects occurs in several applications of engineering, supported by experimental studies or/and numerical simulations. For understanding of natural convection and transport processes characteristics in an enclosures, the several works is reported in literature were Buoyancy-driven flows have been extensively analyzed (Hall et al. (1988), Hyun and Lee (1989), Fusegi et al. (1992), Lage and Bejan (1991,1993) and Xia and Murthy 2002). A number of studies on natural convection within trapezoidal enclosures involve various applications. The natural convection within trapezoidal cavities have been extensively analyzed in the literature. Transient natural convection in a trapezoidal cavity with parallel top and bottom walls and inclined side walls were investigated for Karyakin (1989). Lee (1984;1991) and Peric (1993) presented numerical results of Natural convection in a trapezoidal enclosures of horizontal bottom and top walls that are insulated and inclined side walls. For up to a Rayleigh number of 10 5 Peric (1993) investigated with a series of systematically refined grids from 10 x 10 to 160 x 160 control volume and observed the convergence of results for grid independent solutions. The laminar natural convection flow in trapezoidal enclosures were investigate for Sadat and Salagnac (1995) using vorticity-stream function formulation. Steady state solutions were obtained for aspect ratios 3 and 6. A preliminary investigation on double diffusive convection in trapezoidal enclosures is carried out by Dong and Ebadian (1994). Moukalled and coworkers studied natural convection within trapezoidal enclosures for applications involving partitioned cavity or baffles on walls within cavity (Moukalled and Acharya, 1997, 2000, 2001; Maoukalled and Darwish, 2003, 2004, 2007). Natural convection flows in a trapezoidal enclosure with uniform and non-uniform heating of bottom wall were investigated for Natarajan et. Al. (2008) were a non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case.. Fontana et al. (2010) carried out numerical investigations on natural convection in trapezoidal cavities observing the effects of one and two baffles with different heights on the flow.

This work presents a study of natural convection phenomenon inside trapezoidal cavities. A two-dimensional static model was employed to evaluate a cavity heated by the left vertical wall and cooled by the right vertical wall. The parameters investigated are the Rayleigh number (10 3 to 10 6 ) and the angle of inclination of the upper wall (10 to 20 ), for two different geometry models, a baffle-free cavity and a geometry with two baffles positioned in the adiabatic horizontal walls. Numerical results are conducted to investigate the structure of the flow and the energy distribution, represented by streamlines and isotherms respectively, and the heat exchange between the fluid and the walls, represented by the local Nusselt number along the vertical walls and the average Nusselt number. 2. MATHEMATICAL FORMULATION Let us consider the general schematic configurations of the two-dimensional trapezoidal cavities with two inside baffles as seen in Fig. 1. The vertical left and right walls of the trapezoidal cavity are heated and cooled, respectively, at constant temperatures T H and T C, where T H > T C. The horizontal surface and the inclined wall remain adiabatic. The width of the cavity (L) is 4 times the height (H) of the shortest vertical wall. Tree inclination of the upper wall of the cavity is investigated, 10 o, 15 o and 20 o. Baffles are also placed at two heights (H b = 0 and H*/3) where H* denotes the height of the cavity where the baffle is located. The baffle thickness, W b = L/20, and baffle locations L b1 = L/3 and L b2 = 2L/3 are considered. Figure 1. Schematic diagram of the physical trapezoidal cavities. a) without baffles, b) with two baffles. We assume the fluid properties to be constant. The exception is the density in the buoyancy force term, in the y direction, of the momentum equation. We approximate this term using the Boussinesq approximation. The flow field is considered to be steady state, laminar, and two-dimensional. Therefore, the governing equations for the fluid flow and heat transfer are those expressing the conservation of mass, momentum, and energy. In dimensional form, the transport equations are given by: u v + = 0, (1) uu) vu) 1 p u u + = + ν + ν, (2) ρ uv) vv) 1 p v v + = + ν + ν ρ + gβ ( T TC ), (3) 2 2 ut ) vt ) T T + = α +, (4) where u (m/s) is the velocity in x-direction, v (m/s) is the velocity in y-direction, ρ (kg/m 3 ) is fluid density, ν (m 2 /s) is kinematic viscosity, α (m 2 /s) is thermal diffusivity (α = k/ρc p ), β (1/K) is the thermal expansion coefficient of air, T C (K) is the cold (and reference) temperature, T (K) is temperature, and g (m/s 2 ) is gravitational acceleration. Along the vertical wall, the following Dirichlet conditions are used: T (x = 0, y) = T H, (5) T (x = L, y) = T C. (6)

As shown in Equations (7) through (10), we assume no-slip velocities on all the walls: u ( x = 0, y) = v( x = 0, y) = 0, (7) u ( x = L, y) = v( x = L, y) = 0, (8) u ( x, y = 0) = v( x, y = 0) = 0, (9) u ( x, y = H + x * tg( θ )) = v( x, y = H + x * tg( θ )) = 0. (10) The bottom and top walls remain insulated. T T y= 0 = 0, (11) y= H + x* tg( θ ) = 0 The energy balance at the baffle-air interface can be stated as: 1 ) r k r r ( n. θ ) i = ( n. θb) i, (13) Pr Pr where n ) is a unit vector normal to the baffle-air interface, the subscript i refers to the interface, and k r is the ratio between the thermal conductivity of the baffle and the convective fluid. The Rayleigh number, for all results shown, is based on the shortest length of the vertical wall. Therefore, the Rayleigh number is defined by: 3 Ra = gβ ( TH TC ) H / να. (14) 2.1 Solution Procedure The continuity, momentum and energy balance equations, Eqs. (1) (4) are solved using the commercial computational fluid dynamics code ANSYS CFX version 12.0. In this code, the conservation equations for mass, momentum are solved together using an element based finite volume method (EbFVM). The resulting discrete system of linear equations is solved using an algebraic multigrid methodology called the additive correction multigrid method. The solution was considered converged when the sum of absolute normalized residuals for all cells in the domain solution was less than 10-12, and double precision for all variables was used. The fluid motion is displayed using the stream function ψ obtained from velocity components u and v. The relationships between stream function, ψ and velocity components for two dimensional flows are (Batchelor, 1967): ψ ψ u =, v =. (15) The heat transfer coefficient in terms of the local Nusselt number (Nu) is defined by Nu ) x= 0, x L ( TH TC ). (16) y = T = The average Nusselt number at the left and right walls are (12) 1 Nu = H * L 0 Nu y dy (17) where H* denotes the height of either the hot or cold wall. Based on this definition, the average Nusselt numbers along both cold and hot walls must be the same. 3. RESULTS AND DISCUSSION

This section presents the results for several cavity configurations, performed for Pr = 0.7 (air), with 10 3 Ra 10 6 for two height s baffle (H b, = 0 and H*/3) and three inclination angle ( ; 15 o and 20 o ). To ensure the convergence of the numerical solution to the exact solution, the mesh refinement was performed for all height s baffle, Prandtl, and Rayleigh numbers investigated and the computed results presented in this study are found to be independent of grid sizes. The grid refinement was mainly promoted at the walls of the trapezoidal cavity and next to the baffles, where the gradients are expected to be highest. Four different structured non-uniform grids composed of 31 34, 62 68, 72 84, 124 136, and 248 272 volumes were used. The results obtained with mesh refinement are not shown, however since the differences between the results obtained with grids formed by 72 84, 124 136, and 248 272 volumes were minor we chose the 72 84 non-uniform grid for all the simulations presented in this work. The fluid motion and heating patterns are studied for constant heated left wall and cooled right wall in presence of adiabatic top and bottom walls. 3.1 Flow pattern and energy distribution The thermal pattern and fluid dynamic behavior of the air inside cavities varying the angle of inclination of the upper wall and the Rayleigh number are show in this section. Figure 2 shows the streamlines for Ra=10 3 and for the H b = 0 (Fig. 2-a) and H b = H*/3 (Fig. 2-b) varying the angle of inclination. For the baffle-free cavity, a single clockwise eddy can be observed occupying all internal space, with the eye of recirculation dislocated from the cavity center due to the influence of upper wall. This pattern is observed for all angles of inclination of the upper wall. The presence of internal baffles changes drastically the fluid flow structure, as can be seen in Figure 2-b. For the case three small separated eddies appears, indicating the formation of more stratified regions. As the angle of inclination increase, two of these eddies merges in a more distorted recirculation. Figure 2. Streamlines for Ra=10 3 and H b = 0 and H b = H*/3 Figure 3 shows the lines of constant temperature (isotherms) for the same cases presented in Figure 2. For this value of Rayleigh (Ra=10 3 ) the predominant heat transfer mechanism is the conduction through the fluid and this reflect in a more pronounced vertical isotherms. Comparing the results for all the angles with and without baffles we can observe that the presence of baffles tends to reduce the convective effect, which is a reflex of the existence of stratified zones. The increase in the angle of inclination of the upper wall tends to let the flow structure more complex. For the fluid near the bottom right wall has a greater influence of the convective effect than the fluid in the upper right corner, indicating can exist regions inside the cavity with different main mechanism of heat transfer as the inclination increase. Figure 4 shows the same results that Figure 2, but for Ra=10 4. For the baffle-free cavity (Fig. 4-a), the same configuration of only one clockwise eddy is observed, but in this case the stronger convective effect causes the deformation of the recirculation. For the cavity with baffles (Fig. 4-b) a separation of the flow can be observed by the formation of two eddies between the baffles and the walls. The increase of the angle of inclination causes a reduction in the stratification, but the two eddies can be observed for all values evaluated.

Figure 3. Isotherms for Ra=10 3 and H b = 0 and H b = H*/3 Figure 4. Streamlines for Ra=10 4 and H b = 0 and H b = H*/3 The isotherms for Ra=10 4 are presented in Figure 5 for H b = 0 (Fig. 5-a) and H b = H*/3 (Fig. 5-b). For the bafflefree cavity, the isotherms have a predominant horizontal shape, indicating a greater influence of the convective heat transfer. This aspect is more significant as the angle of inclination increase, so despites the non-uniformity in the geometry added by the inclination, the increase in the right wall facilitates the convective motion. The cavities with baffles (Fig. 5-b) also show a convective predominant shape, but the influence of the baffles reduce the recirculation intensity and creates regions more stratified near the vertical walls, where we can observe a reduction in the temperature gradient compared with the baffle-free cavities, especially for the cases with minor angles of inclination. The increase in Rayleigh number causes a change in the dominant heat transfer and fluid flow mechanism inside the cavity. For Ra=10 5, a similar behavior to case where Ra=10 4 was found, since both cases represents a convective predominant flow where no significant instabilities arise. However for Ra=10 6 the changes in fluid flow pattern are more visible, as can be seen in Figure 6. For the baffle-free cavity, occurs the appearance of an instability in the bottom right corner. It seems to be a hydrodynamic instability caused by the downstream flows near the right wall. For the cavity with baffles (Fig. 6-b), a complex flow patter was obtained. For two distinct regions of recirculation can

be observed near the vertical walls. As the angle of inclination increase these principal recirculations splits in many minors parts. For the structure of the flow are extremely complex and instabilities appears in all cavity. Figure 5. Isotherms for Ra=10 4 and H b = 0 and H b = H*/3 H b = 0 H b = H*/3 Figure 6. Streamlines for Ra=10 6 and H b = 0 and H b = H*/3 The isotherms for Ra=10 6 are shown in Figure 7. The results for the baffle-free cavity (Fig. 7-a) exhibit a similar behavior that the presented for Ra=10 4 (Fig. 5-a), but the temperature gradient near the vertical walls is much more pronounced. Moreover, the instabilities in the bottom right corner reflect in a different temperature distribution. The presence of baffles causes significantly changes in the isotherms, as can be seen in Figure 7-b. In this case, the isotherms have a more deformed form and the energy distribution reflect different phenomena, principally for the greater angles of inclination.

Figure 7. Isotherms for Ra=10 6 and H b = 0 and H b = H*/3 3.2 Heat Transfer The isotherms and streamlines give us information about the nature and which phenomena control the structure of the flow, but do not quantify the heat exchange between the walls and the fluid. In this section we present the local and average Nusselt numbers for some cases, in order to understand the influence of the angle of inclination in the amount of energy exchange between the fluid and the prescript temperature walls. Figure 8 shows the local Nusselt number for Ra=10 3 and 10 4 along the cold (Fig. 8-a) and hot (Fig. 8-b) walls for H b = H*/3. As can be seen, the lines has the same format for all angles of inclination, but the increase in the angles causes a consecutive increase in the Nusselt number. The same behavior is observed for Ra=10 5 and 10 6, as presented in Figure 9. Figure 8. Local Nusselt number for H b = H*/3 and Ra=10 3 and 10 4 along the cold wall and the hot wall

Figure 9. Local Nusselt number for H b = H*/3 and Ra=10 5 and 10 6 along the cold wall and the hot wall The average value of Nusselt number found for all cases are shown in Table 1. The presence of baffles causes a significantly reduce in the heat transfer rate, which can be observed for all cases evaluated. For the mean reduction, considering all Rayleigh numbers, is approximately 30.1%, for is 24.5% and for is 23.0%, and so the reduction is more significant for small values of the angle of inclination. Despites do not affect drastically the fluid flow structure and the shape of the energy distribution, the increase in the angle of inclination of the upper wall causes a considerably increase in the average Nusselt number. The mean increase of the values found for in relation to the founds for for H b = 0 is 23.2% and for H b = H*/3 is 28.5%. This values are quite significant considering that the values of angles of inclination evaluates are not too distant. Table 1. Variation of average Nusselt number with Rayleigh number and height s baffle for Pr = 0.7 Ra H b = 0 H b = H*/3, H b = 0 H b = H*/3, H b = 0 H b = H*/3, 10 3 0.532 0.381 0.714 0.465 0.904 0.558 10 4 2.101 1.403 2.476 1.896 2.757 2.301 10 5 4.918 4.107 5.380 4.947 5.760 5.455 10 6 8.863 7.948 9.592 9.106 10.222 9.747 4. CONCLUSIONS An analysis of the natural convection phenomena within trapezoidal cavities is presented in this work, where the influence of the angle of inclination of the upper wall is investigated for two models of cavity, without baffle and with two baffles positioned in the horizontal adiabatic walls. Three values of angles of inclination are considered, θ = 10 o, and. The flow pattern and the format of the energy distribution (isotherms) are not affect so much by the angle of inclination, especially for intermediary values of Rayleigh number, as Ra=10 4 and Ra=10 5. However, the intensity of the heat exchange between the fluid and the wall, measured by the Nusselt number, demonstrate a significant influence of the angle of inclination, showing for an average increase of 23.2% for the baffle-free cavity and 28.5%. for the baffled cavity in comparison with the values obtained for. 5. ACKNOWLEDGEMENTS The authors would like to thank CAPES (Brazil) for the scholarship granted to the first author, as well as CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brazil) (processes: 568221/2008-7, 475689/2010-0, 302786/2008-2/PQ, 314789/2009-0 and 504102/2009-5) for its financial support of this work. 6. REFERENCES Dong, Z.F. and Ebadian, M.A., 1994, Investigation of double-diffusive convection in a trapezoidal enclosure, J. Heat Transfer Trans. ASME, Vol. 116, pp. 492 495.

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