Free convection heat transfer from two equal triangular cylinders confined in triangular enclosure

This paper studied the natural convection heat transfer from two equal-sized cylinders of triangular cross-section confined in triangular enclosure. The inner cylinders have hot surfaces and the outer enclosure has a cold surfaces. The annular space between inner cylinders and the outer one was filled with incompressible Newtonian fluid. The work examined the behavior of fluid flow in the process of transferring heat energy. The work was achieved numerically using the commercial code ANSYS-CFX. The adopted initial conditions were selected for these ranges: Prantl number ( Pr = 1 to 10 3 ), Rayleigh number ( Ra = 10 3 to 10 5 ). It was found that the increment of Ra number increases the rate of heat transfer. However, the effect of Pr number on heat transfer is almost negligible.


Introduction
The buoyancy-driven flow in annular space has many industrial applications such as heat exchangers, nuclear reactors, solar collector and so on. Indeed many researchers have studied the heat transfer of free convection between two cylinders.
Laidoudi [1] determined the average Nusselt number of four cylinders located in circular enclosure. The obstacles are hot whereas, the enclosure is cold. The cylinders are arranged crossly. Yoo and Han [2] numerically studied the free convection in unsteady state. The studied space is horizontal and circular cross-section. The Nu was determined as function of time. Sheremet and Pop [3] numerically examined the free convection between two circular obstacles of horizontal arrangement. The nanofluid was filled between those cylinders. The values of Nu number of hot cylinder are computed and discussed. Also, the behavior of flow is discussed according to the pertinent parameters. Kuhen and Goldstein [4] experimentally studied the heat transfer of natural convection between two spaces. The outer one is cold and the inner one is heated at constant temperature. The exact quantity of heat transfer was measured. Matin, and Khan [5] used the same geometry and the under the same conditions for the non-Newtonian power-law fluids. The Nu number was calculated and studied for deferent situations based on the nature of the fluid. Other recent studies are also reported for the same principle [6][7][8][9][10].
This work is reserved for exploring the natural convection from two identical triangular cylinders confined in triangular enclosure. We present here the roles of Rayleigh number and Prandtl number on the Nu number.

Mathematical formulation and boundary conditions
The mathematical formulation of governing equations describing the thermal buoyancy in laminar flow and constant thermo-physical proprieties can be expressed in non-dimensions as: Pr .Pr.
where Pr and Ra indicates Prandtl number and Rayleigh number, respectively and they are given as: The other parameters of governing equations are obtained after considering the following changes: The physical proprieties of working fluid are: kinematic viscosity ν, density ρ, thermal diffusivity α and the expansion coefficient β.
The local Nussselt number around the surfaces of inner cylinder and the average value of each cylinder are calculated by following expressions: For the cold cylinder:

Numerical details
The present simulations were accomplished using the package ANSYS-CFX. The code converts the partial form of governing equations into algebraic system then it solves them via finite volume method. This section presents the choice of adequate grid and the validation test. Fig. 2 shows the form of elements of the mesh. Indeed, a triangular form with nonuniform distribution of elements was generated for the mesh. The grid has concentration of elements around the inner cylinders. Table 1 shows the results of grid independency test; three grids of different size were generated. It is clear that when the number of mesh elements increases the variation of Nu number decreases and the variation becomes negligible between M2 and M3. Therefore, the grid M2 with 71900 elements can be adequate for this investigation.

Results and discussions
The free convection heat transfer in annular space of triangular cross-section is the main

Conclusion
Numerical simulations were performed to study the free convection heat transfer in triangular enclosure containing two hot triangular cylinders. The effects of buoyancy strength (Ra) and the thermo-proprieties of the fluid on the flow structure and heat transfer rate were the main points of the study. At the end of the work some points were drawn: • The heat transfer rate depends positively on Ra number; • The thermo-physical properties of fluid have a limited effect on the performance of heat transfer; • As usual, the structure of flow in annular space forms a two symmetrical bubbles and the center of each bubbles shifts upwardly with increasing the thermal buoyancy strength.