Numerical Study on Incompressible Fluid Flow and Heat Transfer of Porous Media using Lattice Boltzmann Method

Abstract

A porous media is an artificial network containing solid and open or closed pores where the fluid cannot pass easily due to the complex structure. When the fluid is injected into the porous media, due to low permeability, a portion of the fluids also leaks out of the porous media. As a consequence, the mechanics of the fluid flow in porous media is playing a significant role to determine the porous dimensions. Hence, it is important to analyze the fluid flow and heat transfer characteristics of porous media, especially in solid matrix or micro-porous heat exchangers, electronic cooling, filtering, food processing, fuel cells, insulation, porous bearing, solar collectors, nuclear reactors, and many others. Over the few decades, the Lattice Boltzmann Method (LBM) has been developed based on kinetic theory for research and development in various engineering and academic applications as an alternative method of Computational Fluid Dynamics (CFD). Originally, the LBM was developed from Cellular Automata (CA). In CA, the two-dimensional hexagonal lattices applied to ensure macroscopic isotropy. The framework described that every node has the individual path which is surrounded by six closest neighboring nodes with six potential velocity distribution functions. The nodal distribution throughout the domain occurs by two principles, namely, propagation and collision. When the microscopic particles move to the nearest neighbor along their velocity direction, it is termed as propagation. Collision is the most important part; it can force particles to change direction and is decided by the collision operator. LBM has been built up on the D2Q9 (two dimensional lattice with nine velocities) model with single relaxation method called Lattice-BGK (Bhatnagar-Gross-Krook) model. In this model, instead of solving the macroscopic fluid quantities – such as velocity, density and pressure – the movement of the fluid particles is considered by using the Particle Distribution Function (PDF). The fluid flow and heat transfer of porous media are studied with this approximation. In this numerical study, the LBM was adopted to simulate the fluid and heat transfer of the porous media. The Lattice Boltzmann Equation (LBE) is paired with the Brinkman-Forcheimer Equation (BME) to predict the velocity and temperature field in the porous media, which has a forcing term and equilibrium distribution function including porosity. A modified thermal energy distribution function is applied to obtain the temperature field. Numerical analysis was carried out in a two-dimensional porous channel and porous square cavity. To analyze the fluid flow behavior and heat transfer characteristics of the porous media, a wide range of non-dimensional parameters such as Porosity, Reynolds number, Darcy number, and Rayleigh number were used. The results are visualized in terms of velocity vector and isotherms. Three geometrical cases – namely, a rectangular porous channel, a square cavity with moving wall and square cavity with adiabatic wall – were used to analyze the fluid flow and heat transfer with porous media characteristics. For the rectangular porous channel, six different cases of parameters were used to study the fluid flow characteristics. For all the six cases, the porosity was fixed at e = 0.1. For Cases 1, 2, 3, the Reynolds number was fixed at Re = 50 and the Darcy numbers were allowed to vary at 0.01, 1 and 10 respectively. For Cases 4, 5, and 6, the Darcy number was fixed at 0.00001 and the Reynolds number was varied from 1, 50 and 100 respectively. For Cases 1, 2, and 3, with varying Darcy number, the results show that for low Darcy number, the fluid cannot pass through the porous media, while for high Darcy number the permeability increases and the velocity distribution becomes non-linear. For Cases 4, 5 and 6, with fixed Darcy number, the results show that the slip flow velocity is near to the walls for low Reynolds number and for high Reynolds number, the velocity distribution is plane throughout the centerline of the channel. Also, the relative error of the comparison between the results of the LBM and FDM is less than 10 %. For the square cavity with moving wall, six different cases of parameters were used to study the fluid flow characteristics. For Cases 1, 2 and 3, the porosity is fixed at e = 0.99 and for Cases 4, 5 and 6, the porosity is fixed at e = 0.1. For Cases 1, 2 and 3, the Reynolds number are varied to Re = 100, Re = 400 and Re = 1000 respectively and for Cases 4, 5 and 6, the Reynolds number is fixed at 10. For Cases 1, 2 and 3, the Darcy number is fixed at 104, and for Cases 4, 5 and 6, the Darcy number are varied to Da = 10-2, Da = 10-3 and Da = 10-4 respectively. The shape of the velocity profiles becomes nearly circular close to both the side walls due to strong vortex generation for Cases 2 and 3. It is observed that the generation of vortex becomes weaker at low Reynolds number when compared to high Reynolds number for all the Cases. When the Darcy number increases at the fixed porosity and Reynolds number then the permeability in the porous media also increases gradually so that the pressure drop reduces. The near linear profiles of the velocity in the central core of the cavity indicates a uniform vortex region is generated in the cavity at high Reynolds number. The results show that there is no vortex generation in the cavity corner for low Darcy number; but the primary and secondary downstream vortex is generated at the bottom corner due to increase of velocity. The velocity profiles becomes almost linear for high Darcy number while it becomes curve for low Darcy number. It is also observed that the vortex layer thickness close to the moving wall reduces due to decreases of Darcy number resulting in the physical strength of the reduction of vortex layer in the cavity. For the square cavity with adiabatic wall, four different cases of parameters were used to study the fluid flow and heat transfer characteristics in the cavity. For Case 1, the porosity is fixed at e = 0.999 and the Prandtl number is fixed at Pr = 0.72, whereas for Cases 2, 3 and 4, the porosity is fixed at e = 0.4 and the Prandtl number is fixed at Pr = 1.0. The Darcy numbers and the Rayleigh numbers are varied for all the four cases differently. The four cases are simulated to compare the effects of the natural convection phenomena of the porous square cavity. The results show that the porosity and Darcy number variations significantly affect the natural convective flow structure, heat transfer characteristics and the boundary layer thickness. The convective heat transfer is also enhanced with increasing Rayleigh number at low porosity. The results show that for Cases 1, 2 and 3, the heat transfer increases with increasing Rayleigh number. In particular, the high Rayleigh number significantly affects the fluid flow and heat transfer when the porosity and Darcy number are low. For Case 4, the high Darcy number affects the fluid flow and heat transfer characteristics at fixed Rayleigh number. Overall, the fluid flow and heat transfer of Case 3 is higher than that of the other cases. For the rectangular and the square cavity cases, the results have been validated with well-documented data available in literature, and a satisfactory agreement is observed with the channel flow and lid-driven cavity filled with a porous medium. The calculated errors are below 10%. For the square cavity with the adiabatic wall case, the average Nusselt numbers obtained in this study shows good agreement with the average Nusselt numbers of previous literature study in a square cavity flow filled with a porous medium.