The Porous Media Flow Interface Options
Darcy’s Law
The Darcy’s Law Interface () is used for modeling fluid movement through interstices in a porous medium by homogenizing the porous and fluid media into a single medium. Together with the continuity equation and equation of state for the pore fluid (or gas) this physics interface can be used to model low velocity flows, for which the pressure gradient is the major driving force. The penetration of reacting gases into a tight catalytic layer, such as a washcoat or membrane, is a classic example for the use of Darcy’s Law.
Darcy’s law can be used in porous media where the fluid is mostly influenced by the frictional resistance within the pores. It applies to very slow flows, or media where the pore size is very small.
Brinkman Equations
Where the size of the interstices are larger, and the fluid is also influenced by internal shear or shear stresses on boundaries, the viscous shear within the fluid must be considered. This is done in the Brinkman Equations interface. Fluid penetration of filters and packed beds are applications for this mode. The Brinkman Equations Interface () is used to model incompressible flow. You can also choose to model compressible flow at speeds less than Mach 0.3, but you have to maintain control over the density and any of the mass balances that are deployed to accomplish this. Furthermore, you can select the Stokes–Brinkman flow feature to reduce the influence of inertial effects (see Figure 7-1).
Figure 7-1: The Settings window for the Brinkman Equations interface. You can model compressible or incompressible flow as well as Stokes–Brinkman flow. Combinations are also possible.
The Brinkman equations extend Darcy’s law to describe the dissipation of momentum by viscous shear, similar to the Navier–Stokes equation. Consequently, they are well suited to model transitions between slow flow in porous media, governed by Darcy’s law, and fast flow in channels described by the Navier–Stokes equations.
The Brinkman Equations interface also includes the possibility to add a Forchheimer drag term, which is a viscous drag on the porous matrix proportional to the square of the flow velocity. This term accounts for an inertial turbulent drag effect that comes into play for fast flows through large pores. Adding the Forchheimer term takes into account all drag contributions that the Ergun equation covers.
Free and Porous Media Flow, Brinkman
The Free and Porous Media Flow, Brinkman Interface () is useful for modeling equipment that contains domains with both free flow and porous media flow, such as packed-bed reactors and catalytic converters. It should be noted that if the porous medium is large in comparison with the free channel, and you are not primarily interested in results in the vicinity of the interface, you can always couple a Fluid Flow interface to a Darcy’s Law interface, to make your overall model computationally cheaper.
The Free and Porous Media Flow, Brinkman interface is used on at least two different domains; a free channel and a porous medium. The physics interface adds functionality that allows the equations to be optimized according to the definitions of the material properties of the relevant domain. For example, you can select the Stokes–Brinkman flow feature to reduce the dependence on inertial effects in the porous domain, or just the Stoke’s flow feature to reduce the dependence on inertial effects in the free channel.
Compressible flow is also possible to model in this physics interface at speeds less than Mach 0.3, but you have to maintain control over the density and any of the mass balances that are deployed to accomplish this. You can also choose to model incompressible flow, and simplify the equations to be solved.
As always, the physics interface gives you provides you with options to define, either by constants or expressions, the material properties that describe the porous media flow. This includes the density, dynamic viscosity, permeability, porosity, and matrix properties.
Free and Porous Media Flow, Darcy
The Free and Porous Media Flow, Darcy Interface () is a multiphysics interface that is used to model coupled free and porous media flow. It uses a Darcy’s Law interface in the porous domain and a Laminar Flow interface in the free-flow domain, which are coupled over the free-porous interface by the Free and Porous Media Flow Coupling multiphysics boundary feature. This coupling feature provides several options for modeling the tangential velocity component of the free flow at the free-porous interface.
As the interface uses Darcy’s law, it does not take into account inertial effects in the porous domain. In addition, it is assumed that the flow in the free-flow domain is laminar, and as such the interface is suited for situations when the flow in both the porous and the free-flow regions is reasonably slow.
This interface supports laminar, compressible flow at speeds less than Mach 0.3, as well as incompressible flow.
Two-Phase Darcy’s Law
The Two-Phase Darcy’s Law Interface () has the equations and boundary conditions for modeling two-phase fluid movement through interstices in a porous medium using Darcy’s law. The two fluids are considered immiscible, and in general, have different densities and viscosities.
As for the single phase Darcy’s Law, the total velocity field is determined by the total pressure gradient and the structure of the porous medium, but the average viscosity and average density are calculated from the saturation of each immiscible phase and their fluid properties. An extra equation is computed — the fluid content of one phase — in order to calculate the saturation transport.