MIXED CONVECTION HEAT TRANSFER IN TRUNCATED CONE ENCLOSURE AS SOLAR CONTAINER WITH INTERNAL CENTERED TRIANGLE OBSTACLE

ABSTRACT In this study, the mixed convection heat transfer of the air inside a truncated cone enclosure with aspect ratio of (0.75, 1.75, 2, 2.45 and 2.65) with centered triangle obstacle height varying by (0, 2.5, 5, 7.5, 10, 12.5 and 20 cm) and the heated wall inclination angles of (20, 30, 40, 50, 60), the Richardson number in the range of (7 to 11) was investigated numerically. The results are addressed to automotive a suggested solar container with titled solar collector. The heat transfer from the heat source (inclined solar collector) of the enclosure walls is investigated for mixed convection as interaction of the forced convection flow between the inlet and outlet port in the bottom wall. The parameters of heat source, Reynolds number, obstacle height, enclosure aspect ratio, and left and right walls titled angles are considered in this work. The numerical simulation of the problem is carried out using commercial CFD code. The results are given in terms of the streamlines, isothermal, and the enclosure Nusselt number that characterizes the heat transfer from the heat source and from the interior fluid to the enclosure walls, respectively. The results show that the interaction of the main flow and the flow at the heated walls and the buoyancy force at the heated walls increased by using a triangular obstacle and by increasing the obstacle heights, it increased by 20% when using obstacle at position of (h=5 cm). Also, it is found that the Nu increased with increasing Re and the wall heat flux. The Nu increased with increasing Ri in the case of using (h=0 and 5 cm) but it decreased slightly in the other cases and showed that the minimum value of Nu present at a heated wall inclination of θ=50 but the maximum value at θ=30.


INTRODUCTION
The natural convection heat transfer problem in simple geometries enclosures is extensively studied in the previous literatures in the recent years. The most cases are the Rayleigh Bernard cell, vertically heated walls enclosures, and solar container Sridhar and Reddy, (2007). A review of natural convection in cavity with various shapes has been presented in several studies such as (but not limited to) Wang and Li, (1999); He et al., (2005); and Dixit and Babu, (2006).
A various shapes as cavities with inclination angles and wavy walls studied by Aounallah et al., (2007), and various trapezoidal cavities presented by Varol et al., (2009). The study of natural convection was presented for triangular enclosures mostly performed by Fuad et al., (2007). Jan, et al., (2007) published a theoretical paper on convective heat transfer in an enclosure with titled wall to study the automotive headlights containing Light Emitting Diodes (LED). They showed that the source convective heat transfer of the wall increases with increasing Reynolds number and also increased with increasing the aspect ratio but it increasing with decreasing the titled angles of the enclosure wall. Merk and Prins, (1953); Merk and Prins, (1954) studied the iso-thermal axi-symmetric forms of the vertical cones and they showed that the vertical cone has such a solution in steady state. Furthermore, Hossain and Paul , (2001) studied the effects of the transpiration velocity on the laminar free convection flow from a vertical non-isothermal cone due to increase the temperature gradient, the velocity and the surface temperature will decrease. Ramanaiah and Kumaran, (1992) have performed an investigated study on…the natural convection about a permeable cone and a cylinder subjected to boundary condition of radiation. Alamgir, (1989) studied the natural convection with laminar flow from vertical cones by using the integral method. Pop and Takhar, (1991) studied the compressibility effects in laminar natural convection from a vertical cone. Papanicolaou and Jaluria, (1995) investigated the effect of varying the position of the heat source inside an enclosure with a constant shape and using a finite thickness of the enclosure sidewalls. Bapuji, et al., (2008) presented a numerical analysis on the natural convection with unsteady laminar flow from an incompressible viscous fluid flows past a vertical cone with uniform surface heat flux. Finally, Elsayed, et al., (2016) studied the effect of heat generation or absorption and thermal radiation on free convection flow and heat transfer over a truncated cone. Comparisons with previously published works, the present study focused on the effect of presence of triangular obstacle inside a truncated cone enclosure with tilted vertical walls on the mixed convection heat transfer coefficients.

Geometry Specification
To study the effect of buoyancy-induced flow regime inside a truncated cone enclosure with heated inclined vertical walls, a two-dimensional physical model and required boundary conditions are displayed in Fig. 1a. a) Enclosure dimensions B) Numerical boundary conditions

Governing Equations
The presentation of the buoyancy-induced regime inside an enclosures performed by the same basic equations of the general fluid motion, namely the conservation of mass, momentum, and energy. The free convection heat transfer scenarios require at least one fluid parameter, density, to vary within the domain of interest. The equation of the buoyancy-induced flows inside a vertical truncated cone for incompressible flow presented by Ganguli et al., (2009) The stress tensor in the momentum expression, , can be calculated by: By making these simplifications and expansions, Eqs. 1 and 2 can be rewritten for a steady state as ANSYS: The fluid density can be simplified at all points within the employed domain, by employing the Oberbeck-Boussinesq approximation to linearize the temperature dependency of density. The buoyancy force is redefined as Ganguli et al., (2009): Where represent a fluid compressibility found via: The air can be considered as an ideal gas at nominal atmospheric pressures and temperatures, the air compressibility found by: β = 1 T f 10 A necessary simplification was made to the mass and momentum equations present in Eqs. 4 and 5 allows for the development of the final forms for incompressible flow:

Boundary Conditions
Obstacle Surface: the fluid velocity at the surface of the solid must be zero in all directions:

Computational Grids and Numerical Method
To study the problem of the heat transfer by mixed convection parameters of the air inside a truncated cone enclosure with heated left and right inclined walls and a triangular obstacle in the middle of the enclosure. Also, the finite-volume computational nodes were generated for the present two dimensional enclosure geometry. In an effort to simplify the construction of each of this geometry and mesh files, the Auto-CAD and Gambit v. 2.3.16 software are used.
The air space modeled using a quadrilateral mesh by 20170 nodes and Fig. 2 displays an example of this computational domain. The 2D computational domain exported to the FLUENT, in this study, to couple the inlet and outlet pressure boundary conditions the SIMPLE algorithm was performed. The simulation iteration was terminated when the mass, momentum, and energy residuals drop below 10 -9 for each simulation iteration. By using under-relaxation factors of 0.70 and 0.30 for the pressure and momentum equations respectively, but the density and energy relaxation factors were left at the default values of unity.

RESULTS AND DISCUSSION
The present numerical work has been presented in mixed convection system where the Richardson number is in the range of Ri =7 to 11 for two different inlet Reynolds number Re=600 to 1600. The mixed convection problem presented in the truncated cone enclosure, top and bottom walls Were assumed as adiabatic. In this section, the laminar steady flow results have been achieved. Firstly, the streamline and isotherm contours were presented for different obstacle height varying for six different positions within the geometry (h= 0 or without obstacle, 2.5, 5, 7.5, 10, 12.5 and 20 cm (see Fig. (1b)) from the inlet port.

Kadhum A. Jehhef and Mohamed A. Siba
The present calculations were performed at constant Pr=0.712 for air [18], and constant position and length of the heated wall of the left and right surfaces of the cone. In addition, its length is about Lp=45 cm to simulate a solar container model of truncated cone enclosure geometry. The heated wall with different inclination angles of θ= 20o, 30o, 40o, 50o and 60o, in order to obtain various aspect ratio in the range of A= 0.75, 1.75, 2, 2.45 and 2.65. At each A, the Reynolds number, Richardson number, Rayleigh number was varied in order to obtain the flow conditions.

Effect of obstacle Position
In all cases of position of the obstacle chosen in this study the cases of h=0, 5, 10, and 20 presented in Fig

Effect of enclosure angle
The effect of increasing the enclosure inclination angles on the isothermal lines is plotted in Fig. 6. The results show that increasing inclination angles lead to decrease the thermal layers near the heated walls due to transfer the walls from inclined to vertical position and decreasing the buoyancy ratio values near the heated walls. Also, the increasing angles will decrease the temperature upper the obstacle. For a various enclosure shape that caused by various titled angles, the streamline were plotted, as shown in Fig. 7. The results obtained for a constant Ra=2.36×107, Richardson number Ri=11, Reynolds number Re=1600, and obstacle position h=7.5 cm. The results show that there are no significant changes in the streamlines between the obstacle height and the heated walls. If the cavity size is increased strongly there is a small effect on the streamlines in the area above the obstacle. In the zone between heated walls and the outlet port which show that the flow affected by the enclosure inclination angle.

Local and overall Nusselt number
The local Nusselt number Nu along the right side of the enclosure is presented in Fig. 12. It shows also a dependence of the source power of the heated wall for high Reynolds number