1. used for the space discretization of

1.      Numerical simulationThe numerical simulation is performed using the finite volume method. The governing equations converted to algebraic equations by a control-volume-based technique so that these equations can be solved numerically. The second upwind scheme is used for the space discretization of the convective terms in momentum and energy equations, and to couple the pressure and velocity, the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm is used. A second-order scheme is applied to the space discretization of pressure as well. The convergence criterions for the normalized residuals of all solved variables are restricted to be less than 10-6.

For the CFD simulations, the sub-relaxation iteration method is used to ensure the simulations convergence. Moreover, the detailed information of simulations is presented in Table 2. Table 2. Detailed information of simulations Fluid   Water      Density (kg/m3) 998    Heat capacity (J/kg K)    4182    Thermal conductivity (W/m K) 0.6 Porous   Aluminium      Density (kg/m3) 2719    Heat capacity (J/kg K)    871    Thermal conductivity (W/m K) 202    Wall heat flux (W/m2) 2000    Reynolds number 100-1900    Inlet temperature (K) 300  1.1.

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Grid size selectionTo verify the grid independence and to arrive at the least number of elements that can yield accurate computational results, the grid independency test is carried out. As shown in Figure 2, four line segments considered by G1, G2, G3 and G4, respectively indicate the number of points along the width of the flat part, length of the flat section, perimeter of the semi circles and along the length of the tube. Four different grid systems are tested: mesh I: 40×30×50×350, mesh II: 50×40×60×400, mesh III: 60×50×70×450 and mesh IV: 70×60×80×500. The four meshes were tested by comparing the axial velocity, temperature and heat transfer coefficient for laminar flow in flat tube with H = 6 mm filled with porous media with Hp=0.5.

The comparisons are presented in Figure 3. The velocity and temperature in x and y directions obtained from mesh I does not match with the results from the other three types of meshes. The maximum variation of V in Z direction between the Mesh II and other meshes are 0.03%, 0.01% and 0.

01% respectively for Mesh I, III and VI. The highest difference of h between the Mesh II and other meshes are 15.25%, 3.

77% and 3.33% respectively for Mesh I, III and VI. Therefore, the mesh II is chosen and all computations were run with this grid. Figure 4 shows generated mesh for a flat tube using the mentioned grids.  Figure 2.

Line segments used for Grid size selection  Figure 3. Grid independency test for H = 6 mm, (a) velocity distribution in x direction, (b) velocity distribution in y direction, (c) velocity distribution in z direction, (d) temperature distribution in x direction, (e) temperature distribution in y direction and (f) local heat transfer coefficient distribution in z direction.  Figure 4.

Grid layout used in the present numerical computation 1.2. Checking the validity of the programIn order to check the validity of the solution, in the absence of any previous research about flat tubes with a porous layer, two comparisons were tested between the results achieved by the present program and the results of the other researchers. First, the circular tubes with a porous layer inserted in the core of tube chosen to be compared. Figure 5 represents a comparison between the solution obtained by the present and the numerical solution of Mahmoudi and Karimi 45 (for Nusselt number of laminar flow in a tube partially filled with porous material).

The comparison shows that the maximum deviation and the average deviation from the data of Mahmoudi and Karimi 45 for average Nusselt number are 3.1% and 2.9 % respectively. In the second comparison, flat tubes without porous layer researched by Safikhani and Abbassi 9, is compared and presented in Fig. 6. The comparison of average heat transfer coefficient versus flattening that the maximum deviation and the average deviation from the data of Safikhani and Abbassi 9 for local heat transfer coefficient are 2.

8% and 1.9 % respectively. Therefore, it can be concluded that the considered computational model is available to solve the laminar heat transfer and flow problem for flat tube with porous insert.  Figure 5. Comparison between the present study and Mahmoudi and Karimi 45  Figure 5.

Comparison between the present study and Safikhani and Abbassi 9

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