www.mdpi.org/entropy/ Free Convection and Irreversibility Analysis inside a Circular Porous Enclosure

We investigate the nature of heat transfer and entropy generation for natural convection in a two-dimensional circular section enclosure. The enclosure is assumed to fill with porous media. The Darcy momentum equation is used to model the porous media. The full governing differential equations are simplified with the Boussinesq approximation and solved by a finite volume method. Whereas the Prandtl number Pr is fixed to 1.0. Results are presented in terms of Nusselt number, entropy generation number, and Bejan number.


Introduction
Horizontal cylinders filled with fluids are commonly encountered in the world around us.This type of geometry and flow configuration are commonly observed in the field of electronics, cooling system, heat exchanger, etc.When their side wall temperature is non-uniform, natural convection motion develops inside the cylinder and many efforts have been devoted, over the last decades, to understand the flow structure and related heat transfer mechanism under various heating conditions.Articles by Sierra [1], Ostrach and Hantman [2], Ostrach [3], Xin et al. [4] are some of them.Two configurations have been extensively studied: the configuration heated from below (Rayleigh-Benard convection) and that heated from the side.Most of the previous work deals with non-porous media and none of them considered a Second-law (of thermodynamics) analysis.
Therefore, in the present work, we study the entropy generation characteristics along with the nature of heat transfer inside a porous circular cavity by solving numerically the fully nonlinear momentum and energy equations in a two-dimensional Cartesian frame.More specifically, the cavity is divided into two symmetrical parts by the vertical centerline and both of the parts are perfectly isothermal, but differentially heated.Results are presented for different Rayleigh numbers (Ra=10 to 5000).

Equations and Numerical Methods
Figure 1 shows the domain to be analyzed and the adopted coordinate system.All asterisked quantities in this paper are in dimensional form.The left symmetrical part of the cylinder is cold and the right part is hot as indicated in Figure 1.It is assumed that the cavity is completely filled with the fluid.Uneven density of fluid originating from the temperature difference of the walls produces buoyancy.The saturated porous medium is assumed to be isotropic in thermal conductivity and follows the Darcy model (see Bejan [5]).Finally, the set of non-dimensional governing equations in terms of the stream function ψ and temperature Θ are subjected to the following boundary conditions Equations ( 1) and ( 2) along with the boundary conditions given in Eq. ( 4) are solved using control volume based Finite-Volume method.A non-staggered and non-uniform grid system is used with a higher mesh density near the walls.TDMA solver solves discretized and linearized equation systems.The whole computational domain is subdivided by an unequally spaced mesh with a size of 116×128.

Entropy Generation
For the porous media, which follows the Darcy model, the dimensionless form of the local rate of entropy generation (N S ) can be calculated from the following equation: where gen S ′ ′ ′ and 0 S ′ ′ ′ are local entropy generation rate and the characteristics transfer rate (see Bejan [6]), respectively.The detailed derivation of the above equation is available in Bejan [5].Equation ( 5) consists of two parts.The first part (first square bracketed term at the right-hand side of Eq. ( 5)) is the irreversibility due to finite temperature gradient and generally termed as heat transfer irreversibility (HTI).The second part is the contribution of fluid friction irreversibility (FFI) to entropy generation, which can be calculated from the second square bracketed term.Bejan number (Be) can be mathematically expressed as

Results and Discussion
We first present the flow and thermal fields' behavior in terms of streamlines and isothermal lines.Isothermal lines inside the cavity are shown in Figure 2   where s is the distance along the circular wall, S is equal to πR, and ∀ is representing the volume of the cavity.Figure 6 shows the distribution of average Nusselt number and entropy generation number as a function of Rayleigh number.

Conclusions
The nature of heat transfer, entropy generation, and heat transfer irreversibility inside a differentially heated circular cylinder is presented in this paper.In conduction regime, both average Nusselt number and the entropy generation number are independent of Rayleigh number variation.In convection

Figure 6 .
Figure 6.Average Nusselt and entropy generation numbers as a function of Rayleigh number Ra