Natural Convective Heat Transfer from Horizontal Isothermal Surface of Polygons of Octagonal and Hexagonal Shapes

Name, first author1 Ahmad Kalendar Dept. of Mechanical Power and Refrigeration Technology College of Technological Studies-PAAET, Shuwaikh, Kuwait a_kalendar@yahoo.com  Name, second author Abdulrahim Kalendar Dept. of Mechanical Power and Refrigeration Technology College of Technological Studies-PAAET, Shuwaikh, Kuwait ay.kalendar1@paaet.edu.kw  Name, third author Yousuf Alhendal Dept. of Mechanical Power and Refrigeration Technology College of Technological Studies-PAAET, Shuwaikh, Kuwait ya.alhendal@paaet.edu.kw  Name, fourth author Sayed Karar


Introduction
The phenomenon of convective heat transfer, which naturally occurs in elements with relatively complex geometrical shapes, have found numerous applications in the cooling of electronic components and devices. Consequently, the study of natural convective heat transfer has attracted significant attention, in particular, for simple two-dimensional shapes, which could be used to in deriving generalized equations pertinent to those practical cooling applications.
The literature is abundant with studies of natural convective heat transfer from upward and downward two-dimensional facing horizontal plates, albeit being of relatively simple shapes and in various situations. Insights from these prior studies, offered numerous empirical and analytical correlations, which have been reviewed in many textbooks, e.g., see [1][2][3][4][5][6]. In most simple configurations, the analytical expression for the mean Nusselt A significant body of work on the theoretical studies of natural convective heat transfer between horizontally placed surfaces can be found, e.g., see [7][8][9][10][11]. While the earlier experimental studies of the same phenomenon exist, e.g., see [12][13][14][15][16][17][18][19][20][21][22]. A verity of situations has been considered, in the literature, for natural convective heat transfer between three-dimensional bodies, e.g., see [1][2][3][4][5]. The aforementioned studies analyzed a variety of inclination angles in relation to the vector of gravity. In general, a unique, yet different relationship between Nu and Ra for natural convective heat transfer can be found for specific shapes and aspect ratios. However, introducing the concept of the characteristic length, allows the derivation of a single relationship between Nu and Ra, which can accommodate a variety of these situations, albeit being chosen somewhat arbitrarily as discussed by Yovanovich and Jafarpur [23]. If the definition of Nu and Ra incorporates the ratio of the surface area of the element to its perimeter, then a unique relationship between the Nusselt and Rayleigh numbers arise at a given Prandtl number.
In such situations, the variation of Nusselt and Rayleigh numbers is effectively identical for all element shapes, e.g., see [1][2][3][4][5]. It is found that if the length scale based on A/P where A and P are the surface area of the element and its perimeter, respectively, then the variation of Nu and Ra for a given Pr is effectively the same for all horizontal element shapes. Yovanovich et al, e.g., see [23][24][25][26], showed that utilizing the characteristic length h to describe the body gravity function reveals the sensitivity of the area mean Nusselt number on the element orientation and geometry. However, the authors found that defining the characteristic length as √ , is a superior choice for the length scale as the it renders the body-gravity function insensitive to the orientation and geometry of the said element.
Although countless analytical correlation equations are presented in the literature for natural convective heat transfer, they do not account for the shrinking element sizes.
In particular, as Ra and the size of the element's surface area becomes smaller, it becomes increasingly imperative to consider the effects of thermal diffusion and three-dimensional flow near the edges of the elements. The inadequacy of the generalized empirical formulas, available in the literature, becomes fully manifested, when three-dimensional flow exists, and the size of the surface area of the elements becomes smaller.
Consequently, utilized these formulas underestimate the rate of heat transfer. For these situations, the numerical solution of the full three-dimensional governing equations is warranted. For example, numerical solutions for complex narrow plane surfaces with different aspect ratios and inclination angles in a variety of flow regions have been discussed in [25][26][27][28][29][30][31][32][33][34][35], where some empirical correlations have been developed.
The work presented in references [36,37]

Numerical Solution Procedure
The assumed flow in this work has been steady and laminar. The Boussinesq approach used in this work, the density of the fluid varies with temperature, which causes the buoyancy force, while other fluid properties are assumed constants. The full threedimensional form of the governing equations, which being written in terms of dimensionless variables has been solved numerically. These governing equations and its dimensionless variables are being described in our previous work of Kalendar et al. [38].
The numerical simulation domain used in modeling the various shapes is as shown in Fig.   4. The boundary conditions imposed on the numerical solutions for the different surface shapes shown in Fig. 2, are in terms of a set of dimensionless variables.
The following dimensionless variables were defined: , , where ϴ is the dimensionless temperature. The x-coordinate is measured in the direction normal to the heated element surface, the y-coordinate and the z-coordinate are measured in the horizontal direction in the plane of the heated element surface (see Fig. 4).
In terms of these dimensionless variables, the governing equations are: The assumed boundary conditions are as follows; for heated element surface, the dimensionless velocity components are nulled (set to zero), while the dimensionless temperature is set to unity. In the case of the adiabatic base, the imposed boundary conditions are set to zero for both dimensionless velocities perpendicular to all surfaces and dimensionless temperature gradient perpendicular to the surface. On the outer planes of the numerical solution, both dimensionless temperature and velocity components are set to zero in the plane of the surface. Furthermore, both dimensionless temperature and pressure components at the surface are also set to zero, albeit on the upper outer plane of the numerical solution domain as shown in Eq. (7).

Fig. 4 Numerical solution domain for upward facing heated element
The mean Nusselt number has been expressed in terms of mean heat transfer rate from the heated surface as follows: where the Nu is the mean Nusselt numbers for the heated element surface based on temperature differences, and z is the characteristic lengths used in this study of w or m or √ and TF is the undisturbed fluid temperature. The commercial finite volume method modelling software FLUENT 14.5, has been used for solving the dimensionless governing equations while imposing the aforementioned boundary conditions discussed above. In the present study, hexahedral cells were created with a fine mesh near the heated surface wall and the heated surface edges using GAMBIT. A non-uniform grid distribution was used in the planes perpendicular and parallel to the main flow direction.
Close to the heated surface, the number of grid points or control volumes was increased to enhance the resolution and accuracy.
A fine mesh was rendered near the walls and the edges of the heated surface using hexahedral cells. To improve fidelity of the numerical simulations, detailed grid and convergence criterion were independently tested. The total grid of 10 6 nodes was found to be sufficient as further refinement did not change the predicted mean mean Nusselt number results. In fact the actual number of nodes utilized was found to depend on the Rayleigh number and size of the heated surface. Consequently, the heat transfer predictions in this work are within 1% independence of both the number of grid points and of the convergence-criterion. Furthermore, the effect of locating the outer surfaces of the solution domain (i.e., surfaces THRS, ABRH, AHTF, BRSN, and ABNF in Fig. 4) from the heated surface was minimized to ensure that the heat transfer results were independent to within 1%. The average Nusselt number was stable within 1% for residual values below 10 -4 . In particular, each equation for mass and momentums have been iterated until the residuals fall below 10 -4 , while the energy equation has been iterated until the residual falls below 10 -6 . The pressure-based solver utilized a second-order upwind scheme for convective terms in the mass, momentums and energy equations. In the case of pressure discretization, the Presto scheme has been employed while the SIMPLE-algorithm has been used for pressure-velocity coupling discretization.   Rohsenow et al. [6] are in a good agreement.  Table 1 Figure 9 shows that at the lower values of Ra<2×10 3 , the numerical mean Nusselt number has a higher value than that obtained from the empirical equation of Fishenden and Saunders [12], while at the higher value of This is due to edge effects near the surface edges and its corresponding corners in a region of temperature flow differences which cause three dimensional flows near the edges and corners. It is worth noting that the edge effects is more pronounced at lower values of Rayleigh numbers due to thicker boundary layer than that at higher values of Rayleigh numbers.     Rohsenow et al. [6] and higher values than the mean Nusselt numbers obtained from the empirical correlation equation of Fujii and Imura [16] for all values of W considered. Figure 11, for hexagonal h=d, shows that at the lowest values of Ra, the numerical mean Nusselt numbers have higher values than those obtained from the empirical equations of Rohsenow et al. [6] and Fujii and Imura [16], while at the highest values of Ra, the numerical mean Nusselt numbers have a lower values than those obtained from the empirical equation of Rohsenow et al. [6]. For all other values of Ra there is a good agreement between the numerical mean Nusselt numbers and those obtained from the empirical equation of Rohsenow et al. [6] for all values of W considered in this study. Figure 12 shows that at the lower values of Ra<2×10 5 , the numerical mean Nusselt number has a higher value than that obtained from the empirical equation of Rohsenow et.al. [6], while at the higher value of Ra>8×10 6 , the numerical mean Nusselt number has a lower value than that obtained from the empirical equation of Rohsenow et al. [6].
For all other intermediate values of 2×10 5 <Ra<8×10 6 there is a good agreement between the numerical mean Nusselt numbers and those obtained from the empirical equation of Rohsenow et al. [6], for all values of W considered in this study.    This comparison between the numerical mean Nusselt numbers and Ra have been carried out for hexagonal and octagonal surface shapes facing upward and facing downward as in ( Fig. 13(b, d)) and ( Fig. 13(a, c)) respectively. This characteristic length was selected to show the effect of the dimensionless width W of octagonal and hexagonal surface shapes on the heat transfer rates. It will be seen that the mean Nusselt number tends to increase Therefore for constant Prandtl number; For downward facing heated surfaces, i.e., by: Comparisons of the mean Nusselt numbers given by the correlation equations Eqs. (21) and (22) (21) and (22) for isothermal upward and downward heated surfaces respectively.

Acknowledgments
This work was supported by the Public Authority for Applied Education and Training of Kuwait (PAAET).