Darcy–Brinkman–Forchheimer Model for Film Boiling in Porous Media

Abstract

The paper presents results of the modelling of heat transfer at film boiling of a liquid in a porous medium on a vertical heated wall bordering with the porous medium. Such processes are observed at cooling of high-temperature surfaces of heat pipes, microstructural radiators etc. Heating conditions at the wall were the constant wall temperature or heat flux. The outer boundary of the vapor film was in contact with moving or stationary liquid inside the porous medium. An analytical solution was obtained for the problem of fluid flow and heat transfer using the porous medium model in the Darcy–Brinkman and Darcy–Brinkman–Forchheimer approximation. It was shown that heat transfer at film boiling in a porous medium was less intensive than in the absence of a porous medium (free fluid flow) and further decreased with the decreasing permeability of the porous medium. Significant differences were observed in frames of both models: 20% for small Darcy numbers at Da < 2 for the Darcy–Brinkman model, and 80% for the Darcy–Brinkman–Forchheimer model. In the Darcy–Brinkman model, depending on the interaction conditions at the vapor–liquid interface (no mechanical interaction or stationary fluid), a sharp decrease in heat transfer was observed for the Darcy numbers lower than five. The analytical predictions of heat transfer coefficients qualitatively agreed with the data of Cheng and Verma (Int J Heat Mass Transf 24:1151–1160, 1981) though demonstrated lower values of heat transfer coefficients for the conditions of the constant wall temperature and constant wall heat flux.

A Stabilizer Free Weak Galerkin Finite Element Method for Brinkman Equations

“…The Brinkman equations (1.1)-(1.3) can be seen as a modified version of Darcy's law obtained by adding viscous forces to the Navier-Stokes equations [5]. This model has been applied in many fields, such as power engineering, petroleum industry, geology, geophysics, and so on [4,9,10,20]. Mathematically speaking, the Brinkman equations have different properties due to the varying permeability tensor κ.…”

Numerical approximation of convective Brinkman-Forchheimer flow with variable permeability

“…It is widely used as the basis for studying fluid flow in a wide range of applications, including chemical engineering and pharmaceutical and cosmetic industries. The Forchheimer law accounts for inertial effects in faster flows by including a nonlinear term, but does not account for the Brinkman term [1]. The Brinkman-Forchheimer model adequately models viscous flow in a porous medium and accounts for both inertial and viscous effects.…”

A conforming discontinuous Galerkin finite element method for Brinkman equations

“…These equations extend Darcy's law to describe the dissipation of kinetic energy caused by viscous forces, similarly to the Navier-Stokes equations [5]. Brinkman equations are also applied in many other fields, such as environmental science, geophysics, petroleum engineering, biotechnology, and so on [3,9,10,14,18].…”