Numerical Modelling of Mixing and Separating of Fluid Flows through Porous Media

Abstract

In present finite element study, the dynamics of incompressible isothermal flows of Newtonian and two generalised non–Newtonian models through complex mixing–separating planar channel and circular pipe filled with and without porous media, including Darcy’s term in momentum equation, is presented. Whilst, in literature this problem is solved only for planar channel flows of Newtonian and viscoelastic fluids. The primary aim of this study is to examine the laminar flow behaviour of Newtonian and inelastic non–Newtonian fluids, and investigate the robustness of the numerical algorithm. The rheological properties of non–Newtonian fluids are defined utilising a range of constitutive equations, for inelastic non–Newtonian fluids non–linear viscous models, such as Power Law and Bird–Carreau models are used to capture the shear thinning behaviour of fluids. To simulate such complex flows, steady–state solutions are sought employing time–dependent finite element algorithm. Temporal derivatives are discretised using second order Taylor series expansion, while, spatial discretisation is achieved through Galerkin approximation in combination to deal with incompressibility a pressure–correction scheme adopted. In order to achieve the algorithm of semi-implicit form Darcy’s–Brinkman equation is utilized for the conversion in Darcy’s terms and diffusion, while Crank–Nicolson approach is adopted for stability and acceleration. Simple and complex flows for various complex flow bifurcations of the combined mixing–separating geometries, for both two–dimensional planar channel in Cartesian coordinates, as well as axisymmetric circular tube in cylindrical polar coordinates system are investigated. These geometries consist of a two-inverted channel and pipe flows connected through a gap in common partitions, initially filled with non-porous materials and later with homogeneous porous materials. Computational domain is having variety it has been investigated with many configurations. These computational domains have been appeared in industrial applications of combined mixing and separating of fluid flows both for porous and non-porous materials. Fully developed velocity profile is applied on both inlets of the domain by imposing analytical solutions found during current study for porous materials. Numerical study has been conducted by varying flow rates and flow direction due to a variety in the domain. The influence of varying flow rates and flow directions are analysed on flow structure. Also the impact of increasing inertia, permeability and power law index on flow behaviour and pressure difference are investigated. From predicted solution of present numerical study, for Newtonian fluids a close agreement is realised between numerical solutions and experimental data. During simulations, it has been noticed that enhancing fluid inertia (flow rates), and permeability has visible effects on the flow domains. When the Reynolds number value increases the size and power of the vortex for recirculation increases. Under varying flow rates an early activity of vortex development was observed. During change in flow directions reversed flow showed more inertial effects as compared with unidirectional flows. Less significant influence of inertia has been observed in domains filled with porous media as compared with non-porous. The power law model has more effects on inertia and pressure as compared with Bird Carreau model. Change in the value of permeability gave significant impact on pressure difference. Numerical simulations for the domain and fluids flow investigated in this study are encountered in the real life of mixing and separating applications in the industry. Especially this purely quantitative numerical investigation of flows through porous medium will open more avenues for future researchers and scientists.