Effect of concentric and eccentric porous layer on forced convection heat transfer and uid ow around a solid cylinder

In this paper, heat transfer and fluid flow around a solid cylinder wrapped with a porous layer in the channel were studied numerically by computational fluid dynamics (CFD). The homogeneous concentric and eccentric porous medium round a rigid, solid cylinder are supposed at local thermal equilibrium. The transport phenomena within the porous layer, volume averaged equations were employed, however the conservation laws of mass, momentum and energy were applied in the channel. This current numerical analysis, the effects of eccentricity ( ), the variable diameter of porous layer (d=0.07,0.08,0.09), permeability, as well as the different Reynolds number and Darcy number on the heat transfer parameters and fluid flow was investigated. The main purpose of this study is analyzed and compared the heat flux of concentric and eccentric porous layer in Reynolds number range of 1 to 40 and Darcy numbers of to . It is found that with the decline of Darcy number, the vortex length is increased behind the solid cylinder surface. In addition, the heat flux rate of the cylinder is raised with the increase of Reynolds number. Finally, The results have demonstrated that with raising Reynolds and Darcy numbers, the increase of the average Nusselt numbers in the eccentric porous layer is higher than the concentric porous layer.


Introduction
In the past few decades, the variety of numerical studies have been investigated on the convective heat transfer and fluid flow in the porous-coated channel and cylinder surfaces for on the porous layer around solid cylinder surface could be decreased. Bhattacharyya and Singh [2] also numerically offered heat transfer from a porous layer with high conductivity coefficient and considered porosity around a solid cylinder. The optimal amount of porous layer thickness to improve the heat transfer rate was obtained. Their results stated that high porosity layer with high conductivity coefficient ( ) could increase the heat transfer rate, even at low permeability (Da = ). They also found that by increasing Reynolds number,the formation of periodic vortex could be formed. Similarity, Rashidi et al. [3] conducted numerical studies of fluid flow and heat transfer around a cylinder covered with a porous layer. They found that the thermal performance of heat exchanger with high permeability and thermal conductivity of the porous layer was improved. Valipour and Zare Ghadi [4] also numerically simulated the forced convective heat transfer around and through the porous cylinder in Reynolds number range of 1 to 40 and . Their numerical results showed that with the rise of Darcy number, the local and average Nusselt numbers were increased. Rong et al. [5] perfomed numerically study a square cylinder covered by a porous layer. They showed that by increasing the porous layer thickness in , the amount of drag and lift coefficient was increased and the flow fluid could penetrate into the porous layer easily. Alvandifar et al. [6] numerically investigated a tube bank with 5 rows of tubes wrapped by metal foam layers inside air-cooled heat exchangers. Their results showed that heat transfer in the tubes covered with a thin metal foam layer was much more larger than bare tube bank. Another numerically investigation was done by Odabaee et al. [7] for heat transfer rate from a single metal-foam-wrapped solid cylinder in crossflow. Their results showed that heat transfer rate was increased, as a reasult, they realized that pressure drop was increased by raising the porous layer thickness. Rahmati et al. [8] also numerically studied heat transfer rate from a solid cylinder covered with metal foam. They assumed that the local thermal equilibrium between the fluid flow and porous layer surface in the cross-flow of turbulent regime. Their achievement showed that Nusselt number of solid cylinder with a metal porous layer was almost 10 times larger than bar tube without wrapping a metal porous layer. Ait Saada et al. [9] investigated the effects of natural convection on the heat losses and its improvement a horizontal cylinder with fibrous or porou coating. In their reaserch, for the flow movement and heat transfer coefficient were analyzed. According to their numerical results, when the permeability of a limited porous coating have an equivalent to a Darcy number greater than and then the conduction and convection theory combined with the non-Darcian model should be applied. Al-Sumaily et al. [10] carried out numerically study to investigate the forced convection heat transfer from packed bed of spherical particles were accommodated in circular cylinder. They concluded that using a porous medium with large particle diameters were increased heat transfer and decreased the pressure drop. Pedras and Lemos [11] developed turbulent models for fluid flow in a porous medium in two various ways. The first one was using the macroscopic equations employing the extended Darcy-Forchheimer model and the second one was Reynolds-averaged equations. Chen et al. [12] investigated heat transfer of fluctuating tube bundles wrapped with metallic foam, experimentally. According to this experimental study, they concluded that Nusselt number was raised gradually with increasing the amount of fluid flow permeability because of high of porosity in the porous layer. In another experimental study, Al-Salem et al. [13] investigated the effect of porosity and thickness of aluminum porous layer in order to improving heat transfer performance of heated horizontal cylinder with constant heat flux. In this experimental research, they concluded that, the average Nusselt number was increased with raising the porosity of aluminum layer and also with addition of porous layer, the amount of pressure loss was decreased. Calmidi and Mahajan [14] as well as Zhao et al. [15] carried out numerical and experimental analysis on the forced convection in high porosity metal foams. Thermal spread affected in increasing heat transfer. In additionally, this study had shown that pore density and various materials could affect on heat transfer and pressure drop. Recently, Ebrahimi et al. [16] analyzed heat transfer characteristics of elastically-mounted solid cylinder covered with concentric porous layer numerically. In their research, the range of various diameter (d=0.20,0.5,1) and Darcy numbers ( ) were considered, respectively.

Physical model
the geometry of problem consists of long, rigid, solid cylinder was covered with a concentric and eccentric porous layer in the channel shows in Fig. 1, respectively. In this current research, The problem was modeled two-dimensional (2-D). To decrease the effects of channel walls on the flow near the solid cylinder, the ratio of channel height (W) to the diameter (D) of solid cylinder must be high ( , therefore, we ignored it in this study. We considered the constant temperature of solid cylinder surface at 313 K and the temperature of inlet fluid flow was fixed at 293 K. Also, the diameter of solid cylinder is 0.05 m and the characteristic of channel is 0.5×1 m. The boundary conditions included no-slip at adjacent surface of channel (upper and lower surface) and the solid cylinder surface. A laminar flow was considered to run around the isothermal cylinder. The porous zone around the cylinder was assumed to be homogeneous and isotropic. All fluid properties were considered to be constant. The temperature of solid phase was equal to the fluid phase (local thermal equilibrium (LTE)).

Governing equations
Equations governing the clear fluid zones are Navier-Stokes equations and those that govern the porous medium zones are the volume averaged equations. The clear fluid and porous medium domains are indicated by the subscripts (1) and (2), respectively.
The mass conservation equation in the region 1 (clear fluid) [3]: (1) And the momentum equations in the r and direction [3]: The energy equation in the region 1 can be written as [3]: In the region 2 (the porous medium region), the volume-averaged mass conservation equation And the momentum equations in the r and direction can be written as [3]: The energy equation with the local thermal equilibrium assumption [3]: The volume-averaged fluid velocity, inside the porous region with the porosity ( is related to the fluid velocity ( ), through the Dupuit-Forchheimer relationship, as . Here we used the Ergun correlation to calculate the Forchheimer coefficient, The permeability of the porous medium is calculated from the Kozeny-Carmen relation, as shown below [19]: where K is the permeability and is the average particle size in the porous layer.

Boundary conditions
Dimensions of the computational domain should be selected in a way to minimize the effects of the outer boundaries. The size of channel to the two-dimensional computational domain is 0.5m 1m. The governing equations (1) -(8) are subjected to the following boundary conditions.
The hydrodynamic and thermal boundary condition of inlet fluid flow at the entrance of channel: The fluid flow is both hydraulically and thermally fully developed in the r direction. The hydrodynamic and thermal boundary condition at the upper and lower boundaries are symmetric: (12) no-slip wall boundary condition with a fixed wall temperature on the solid cylinder surface: Along the upstream boundary (uniform flow):

Coupling conditions
In the current study, we assumed the continuity of velocity, shear stress, temperature and heat flux here at the interface between the porous layer and the fluid area:

Numerical method
In this paper, numerical investigations were obtained by using the finite volume methodbased solver, The steady, pressure-based solver was chosen to solve the thermo-fluid-dynamic problem discussed in this study. In the porous zone, to solve the fluid flow, the energy equations, Darcy-Brinkmann-Forchheimer model and local thermal equilibrium (LTE) was used and activated. SIMPLE algorithm was used to couple the pressure and the velocity terms for the pressure correction equation. The least squares cell-based method was applied to construct the values of a scalar at the cell faces. A second order discretization scheme for pressure and a second order upwind discretization scheme for momentum and energy were considered.

Grid independent study
In order to ensure the simulation results are the independent of the mesh size, the values of the heat flux on the solid cylinder surface wrapped with a eccentric porous layer were calculated,so, we found that the independent of the mesh size was reasenable. Then we choosed the number of cells (76360) Table 1 Comparison of heat flux on the cylinder surface for different grids.

Validation
In the first of validation, The numerical data had been compared with the distinct experimental data. The Nusselt number distribution passing through the solid cylinder without a porous layer had been studied experimentally by many reaseches. Fig. 2 shows the investigations of other studies and considered the numerical and experimental data for Nusselt number distribution.
Then, in the present study, three various diameters of the rigid layers covered around the solid cylinder in both concentric and eccentric states were considered and compared with the numerical data [20]. Table 2   . Fig. 3. The comparison of tangential velocity distribution on the porous layer with numerical results.

Temperature distribution
Comparison of temperature distribution on the eccentric porous layer with the eccentric rigid layer was conducted and has been shown in Figs Fig. 9 (a) shows the temperature distribution around the solid cylinder covered by an eccentric porous layer with Da= . Fig.9 (b), displays temperature distribution around the solid cylinder wrapped with an eccentric rigid layer. In both cases, temperature distribution in distinct Reynolds numbers and the same diameter was analyzed. We can obtain that the temperature distribution of inner and outer the rigid layer surface was more much more than inner and outer of the porous layer surface. In fact, if the layer surrounding the solid cylinder is considered rigid, the more volume of the fluid flow passing through the layer surface can under the influence of the cylinder temperature. Fig. 10 has been shows temperature distribution on the different of rigid layer diameter and porous layer diameter with the same Reynolds number (Re=20). it could be concluded that the temperature distribution on the rigid layer surface was higher than the temperature distribution on the porous layer surface, this means that, the conduction heat transfer in the rigid state was much more than the porous state.

Heat transfer flux
The effects of Darcy and Reynolds numbers on the heat transfer flux from the cylinder surface to the diameters of concentric and eccentric porous layer states were accomplished by the simulated methods. Table 3 and Table 4, show the the heat transfer flux from the solid cylinder surface in the various Darcy numbers and Reynolds numbers with different diameters, respectively. The results were obtained by the simulated methods in Table 3 and Table 4 which are as follows: 1-In the solid cylinder with constant wall temperature, by raising the Darcy number, the heat transfer flux through the cylinder could be increased. 5-the main reasult was the amount of the heat transfer flux of the solid cylinder into fluid field in eccentric porous layer state was five percent (5%) higher than that concentric porous layer state. Table 3 Simulated results for the heat transfer flux (w / m 2 ) in Re=40.

Table 4
Simulated results for the heat transfer flux (w / m2) in Da= .
Distribution of the heat transfer flux on the solid cylinder surface in the two types of concentric and eccentric porous layer was analyzed. Fig. 11     Also, we found that for Darcy numbers greater than , the average Nusselt number was increased more fast. On the other hand, the effects of Darcy numbers higher than was so much than the average Nusselt number lower than . Another remarkable point in Fig. 13 is that in a constant diameter and a fixed Darcy number, the effects of the eccentric porous layer on the Nusselt number would greater than that on the concentric porous layer. In the lower Darcy numbers, the porous layer behaves like that a solid rigid layer, this is because in the lower Darcy numbers, penetration was declined. According to Fig.14

Conclusion
In this study, the numerical simulationsbofbheat transfer and fluid flow around the two-