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Proceedings of Clima 2007 WellBeing Indoors Natural Convection Heat Transfer in a Saltbox Roof with Eave in Winter Day Conditions Ahmet Koca, Hakan F. Oztop and Yasin Varol Firat University, Turkey Corresponding email: ysnvarol@gmail.com SUMMARY In this study, a numerical analysis has been performed to examine the natural convection heat transfer and flow field inside a saltbox roof with eave in winter day conditions. This analysis is important for applications since it shows the effective parameters on natural convection heat transfer. The governing equations of natural convection in streamfunction-vorticity form were solved using central difference method to obtain flow and temperature fields inside the roof. Also, the Successive Under Relaxation (SUR) technique was used to solve linear algebraic equations. Results are presented by streamlines, isotherms and local and mean Nusselt number by using effective parameters as Rayleigh number, aspect ratio of the roof, and length of eave. The results indicated that both eave length and aspect ratio can be used as control parameter for heat transfer. Keywords: natural convection, saltbox roof, eave INTRODUCTION Natural convection is formed in some parts of buildings due to temperature difference between cold and hot surfaces. The roof is an important part of the building because it protects the building from the environmental effects. Natural convection heat transfer occurred inside the roof due to temperature difference which depends on the climate such as winter or summer. Natural convection heat transfer analyzed in differentially heated square or rectangular enclosures in earlier studies due to simplicity of numerical solution [1]. However, the number of study about partially heated enclosures is very limited and they investigated mostly for square cross sectional geometries [2-4]. Researchers the natural convection in roofs using different numerical techniques for triangular cross-sectioned roofs (Gable roofs) were investigated by different researchers. Asan and Namli [5] investigated the laminar natural convection heat transfer inside the triangular roof with different aspect ratio in winter day boundary conditions. They observed that, both aspect ratio and Rayleigh number affect the heat transfer inside the roof. Tzeng et al. [6], Akinsete and Coleman [7], Holtzman et al. [8] and, recently, Ridouane et al. [9] investigated the laminar natural convection inside the triangular cross-section enclosure which is applicable for roof geometry of building at different constant temperature or constant heat flux boundary conditions. On the other hand, Varol et al. [10,11] investigated the natural convection in different roof shapes including Gambrel and Saltbox roofs. They observed that natural Proceedings of Clima 2007 WellBeing Indoors convection flow fields strongly depend on the geometrical shape and thermal boundary conditions of the roof. Moukalled and Acharya [12] analyzed the natural convection heat transfer in a trapezoidal roof with baffles that located at the top and bottom of the wall. They indicated that baffles are effective parameters on thermal field inside the roof. The main purpose of the present study is to analyze the flow and thermal fields inside the saltbox roof with eave in winter day conditions. According to knowledge of the authors and literature review given above, laminar natural convection heat transfer has not been investigated yet for partially heated roofs. Thus, the present study will be the first attempt in that area and it will help the designers and constructers from the energy saving point of view. The photo and physical model of the saltbox roof with eave and the boundary conditions is shown in Figure 1 with coordinate system. It is considered that the inclined surface and eave have constant cold temperature. Also, room ceiling (bottom surface of the roof) has constant hot temperature to simulate winter day boundary conditions. The vertical boundary has adiabatic with height, H. Thus, total length of the bottom of roof (L) can be calculated by adding the length of room ceiling (B) and the length of eave (E). An aspect ratio can be defined as the ratio of the height of the roof to the length of bottom wall (AR=H/L) which can be changed according to climatical conditions. Figure 1a. Photo of saltbox roof from Ref. [14]. y TCold, u=0, v=0 H THot, u=0, v=0 x TCold Insulated wall u=0 v=0 ∂T / ∂x = 0 B E L Figure 1b. Physical model Proceedings of Clima 2007 WellBeing Indoors GOVERNING EQUATIONS AND NUMERICAL SOLUTIONS The governing equations of natural convection (Eqs. 1-3) are written in streamfunction- vorticity form for laminar regime in two-dimensional form for steady, incompressible, and Newtonian fluid with Boussinesq approximation. It is assumed that radiation heat exchange is negligible according to other modes of heat transfer and the gravity acts in vertical direction. ∂2Ψ ∂ 2Ψ −Ω = + (1) ∂X 2 ∂Y 2 ∂ 2 Ω ∂ 2 Ω 1 ⎛ ∂Ψ ∂Ω ∂Ψ ∂Ω ⎞ ⎛ ∂θ ⎞ + = ⎜ − ⎟ − Ra ⎜ ⎟ (2) ∂X 2 ∂Y 2 Pr ⎝ ∂Y ∂X ∂X ∂Y ⎠ ⎝ ∂X ⎠ ∂ 2 θ ∂ 2 θ ∂Ψ ∂θ ∂Ψ ∂θ + = − (3) ∂X 2 ∂Y 2 ∂Y ∂X ∂X ∂Y The employed non-dimensional variables are given as x y ψ Pr ω(L) 2 Pr T - Tcold X= , Y= , Ψ= , Ω= , θ= , (4) L L υ υ Thot − Tcold ∂ψ ∂ψ ⎛ ∂v ∂u ⎞ β g(Thot − Tcold )L3 Pr υ u= , v=− , ω = ⎜ − ⎟ , Ra = ⎜ ∂x ∂y ⎟ , Pr = . (5) ∂y ∂x ⎝ ⎠ υ 2 α Boundary conditions for the considered model are depicted on the physical model (Figure 1(b)). In this model, boundary conditions as On all solid walls, u=v=0 ∂T On the adiabatic wall, =0 ∂x On the bottom wall, 0 < x < B, T=THot, B ≤ x ≤ L, T= TCold On the inclined wall, T= TCold Local and mean Nusselt numbers are calculated via Eq. 6 a and b, respectively. ⎛ ∂θ ⎞ B Nu x = ⎜ − ⎟ , Nu = ∫ Nu x dx (6 a.b) ⎝ ∂Y ⎠ Y =0 0 Governing equations in streamline-vorticity form (Eqs. 1-3) are solved through finite difference method. Algebraic equations are obtained via Taylor series and they solved using Successive Under Relaxation (SUR) technique, iteratively. The central difference method is used for discretization procedure. The detailed solution technique is well described in the literature [13]. The convergence criterion, 10-4, is chosen for all depended variables and value of 0.1 is taken for under-relaxation parameter. Some grid tests are made between 34x23 and 352x235 to obtain optimum grid dimension. The test results showed that 214x143 grid dimension is enough for calculations. Proceedings of Clima 2007 WellBeing Indoors RESULTS AND DISCUSSION A numerical simulation is performed for a saltbox attic with eave for winter season. Different parameters such as Ra number, eave length and aspect ratio (AR) of the attic were tested to see the effects of these parameters on natural convection. Figure 2 shows flow field (by streamline, on the left) and temperature distribution (by isotherms, on the right) for different Rayleigh numbers and E=6%, AR=0.6. Both AR and eave length represent real values which is used in architecture. In the case of small Ra number (Ra=104), conduction mode of heat transfer is dominant to convection and almost parallel temperature distribution is formed. For both Ra=104 and 105, two circulation cells are formed in different rotation directions. However, multiple cells are obtained for the highest Ra number, namely Ra=106 (Figure 2c). At this value of Ra number, plumlike temperature distribution is observed. However, for all values of Ra number, at the intersection point of bottom wall and eave, isotherms are stored due to higher temperature difference. a) b) c) Figure 2. Streamlines (on the left) and isotherms (on the right) at E=%6, AR=0.6, a) Ra=104, b) Ra=105, c) Ra=106 Proceedings of Clima 2007 WellBeing Indoors Figure 3 is presents the flow and temperature distribution for AR=0.9 and Ra=106 to show the effects of eave length. As can be seen from the figure, length of eave affects the isotherms and streamlines due to length of heated and cooled part of the bottom wall are changed. Strong plumelike distributions are observed for isotherm. Even, they show mushroom shaped distribution near the right corner. It means that strong convection is occurred. For the highest value of eave, two cells are formed which one of them locates to the top of the triangle like region and the other forms almost at the middle of enclosure (Figure 3(c)). a) b) c) Figure 3. Streamlines (on the left) and isotherms (on the right) at AR=0.9, Ra=106, a) E=3%, b) E=6%, c) E=9% Proceedings of Clima 2007 WellBeing Indoors The results of mean Nusselt number are shown in Figures 4 (a) and (b) for different aspect ratios and eave lengths, respectively. Figure 4(a) shows that heat transfer decreases with increasing AR. However, conduction mode of heat transfer is dominant to convection at lower Ra number (Ra<104). It enhances with increasing Ra number due to domination of convection mode of heat transfer. Mean Nusselt number decreases with increasing eave length due to decreasing heater surface. It means that eave makes negative effect on heat transfer inside the roof. Variation of local Nusselt number with Rayleigh number and eave length is shown in Figure 5 and 6, respectively. In these figures, results are presented for both inclined and bottom surfaces. As can be seen from the Figure 5 (a) that results of local Nusselt numbers are close to each other due to domination of conduction mode of heat transfer along the bottom wall. However, higher values were obtained for Ra=106. At this value of Rayleigh number, local Nu number shows wavy variation. The highest Nu number was formed at the intersection point of eave and ceiling due to high temperature difference. Local Nusselt number shows similar variation along the inclined surfaces except intersection point of eave and ceiling. The variation of local Nusselt number for different eave length is shown in Figure 6 (a) (along the bottom surface) and 6(b) (along the inclined surface). As can be illustrated in Figures, length of eave directly affects the peak point of local Nusselt number due to changing of location of maximum temperature difference. Higher values were obtained due to increasing of heater surfaces. 15 15 10 10 Nu Nu AR=0.3 E=3% 5 5 E=6% AR=0.6 E=9% AR=0.9 0 0 104 4 105 5 106 6 104 4 105 5 106 6 Ra Ra a) b) Figure 4. Variation of mean Nusselt number with Rayleigh number, a) at different ARs, b) at different eave lengths 100 100 90 Ra=10E4 90 Ra=10E4 80 Ra=10E5 80 Ra=10E5 70 70 Ra=10E6 Ra=10E6 60 60 Nux Nuy 50 50 40 40 30 30 20 20 10 10 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x y a) b) Figure 5. Variation of local Nusselt number for different Rayleigh numbers a) along the bottom wall, b) along the inclined wall Proceedings of Clima 2007 WellBeing Indoors 140 100 E=3% 90 E=3% 120 E=6% 80 E=6% 100 E=9% 70 E=9% 80 60 Nux Nuy 50 60 40 40 30 20 20 10 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x y a) b) Figure 6. Variation of local Nusselt number for different eave lengths a) along the bottom wall, b) along the inclined wall, CONCLUSIONS Steady state flow field by natural convection in a saltbox roof with eave for different aspect ratios has been numerically studied. Based on the findings in this study, we conclude that heat transfer regime is conduction for lower Rayleigh numbers. Heat transfer increases with increasing of Rayleigh number with domination of convection mode of heat transfer. Due to geometrical shape of the roof at least double circulation number was formed. It is observed that number of cells is strongly depending on eave length and Rayleigh number. Higher heat transfer was formed for lower aspect ratios. Similarly, smaller eave length enhances the heat transfer. NOMENCLATURE AR aspect ratio, AR=H/L B length of room ceiling (m) E eave length (m) g gravitational acceleration (ms-2) Gr Grashof number H height of roof (m) L length of bottom wall (m) Nu Nusselt number Pr Prandtl number Ra Rayleigh number T temperature (K) u, v axial and radial velocities (ms-1) x,y cartesian coordinates (m) X, Y non-dimensional coordinates Greek Letters υ kinematic viscosity (m2s-1) Ω non-dimensional vorticity θ non-dimensional temperature β thermal expansion coefficient (K-1) α thermal diffusivity (m2s-1) Ψ non-dimensional streamfunction Proceedings of Clima 2007 WellBeing Indoors REFERENCES 1. Vahl Davis, G, Jones, De L P. 1983. Natural convection in a square cavity: A comparison exercise. Int. Num. Method in Fluids. 3, pp 227-248. 2. Turkoglu, H, Yucel, N. 1995. Effect of heater and cooler locations on natural convection in square cavities. Numerical Heat Transfer Part A. 27, pp 351-358. 3. Chu, H H S, Churchill, S W, Patterson, C V S. 1976. 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