Enter a value or expression for the Saturation fluid 1 s1 (a dimensionless number between 0 and 1). The default values are 0.

Outflow

The Outflow feature adds a boundary condition for the outflow perpendicular (normal) to the boundary:

–n Dc c1 = 0

where Dc is the capillary diffusion (SI unit: m2/s) and c1 = s1 1 is the fluid 1 content (SI unit: kg/m3). This means that the normal gradient of fluid saturation does not change through this boundary.

B O U N D A R Y S E L E C T I O N

From the Selection list, choose the boundaries to apply an outflow boundary.

P R E S S U R E

Enter a value or expression for the Pressure p. The default value is 0.

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T h e o r y f o r t h e D a r c y ’ s L a w I n t e r f a c e

In a porous medium, the global transport of momentum by shear stresses in the fluid is often negligible because the pore walls impede momentum transport to the fluid outside the individual pores. A detailed description, down to the resolution of every pore, is not usually practical. A homogenization of the porous and fluid media into a single medium is a common alternative approach. Darcy’s law together with the continuity equation and equation of state for the pore fluid (or gas) has a complete mathematical model suitable for a wide variety of applications involving porous media flows, for which the pressure gradient is the major driving force.

Darcy’s Law—Equation Formulation

Darcy’s law states that the velocity field is determined by the pressure gradient, the fluid viscosity, and the structure of the porous medium:

In this equation, denotes the permeability of the porous medium (m2), the dynamic viscosity of the fluid (SI unit: kg/(m·s)), p the pressure (SI unit: Pa), and u the Darcy velocity (SI unit: m/s). The Darcy’s Law Interface combines Darcy’s law with the continuity equation:

In the above equation, is the density of the fluid (SI unit: kg/m3), is the porosity, and Qm is a mass source term (kg/(m3·s)). Porosity is defined as the fraction of the control volume that is occupied by pores. Thus, porosity can vary from zero for pure solid regions to unity for domains of free flow.

If the Darcy’s Law interface is coupled with an energy balance, then the fluid density can be a function of the temperature, pressure, and composition (for mixture flows). For gas flows in porous media, the dependence is given by the ideal gas law:

where R= 8.314 J/(mol·K) is the universal gas constant, M is the molecular weight of the gas (kg/mol), and T is the temperature (K).

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T h e o r y f o r t h e B r i n k m a n E q u a t i o n s I n t e r f a c e

The Brinkman Equations Interface theory is described in this section:

•About the Brinkman Equations

•Brinkman Equations Theory

•References for the Brinkman Equations Interface

About the Brinkman Equations

The Brinkman equations describe fluids in porous media for which the momentum transport within the fluid due to shear stresses is of importance. This mathematical model extends Darcy’s law to include a term that accounts for the viscous transport in the momentum balance, and it treats both the pressure and the flow velocity vector as independent variables. Use the Free and Porous Media Flow interface for modeling combinations of porous media and free flow domains. These types of problems are often encountered in applications such as monolithic reactors and fuel cells.

• The Free and Porous Media Flow Interface

See Also

In porous domains, the flow variables and fluid properties are defined at any point inside the medium by means of averaging of the actual variables and properties over a certain volume surrounding the point. This control volume must be small compared to the typical macroscopic dimensions of the problem, but it must be large enough to contain many pores and solid matrix elements.

Porosity is defined as the fraction of the control volume that is occupied by pores. Thus, porosity can vary from zero for pure solid regions to unity for domains of free flow.

The physical properties of the fluid, such as density, viscosity, and pressure, are defined as intrinsic volume averages that correspond to a unit volume of pores. Defined this way, they present the relevant physical parameters that can be measured experimentally,

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and they are assumed to be continuous with the corresponding parameters in the adjacent free flow.

The flow velocities are defined as superficial volume averages, and they correspond to a unit volume of the medium including both pores and matrix. They are sometimes called Darcy velocities, defined as volume flow rates per unit cross section of the medium. Such a definition makes the velocity field continuous across the boundaries between porous regions and regions of free flow.

Brinkman Equations Theory

The dependent variables in the Brinkman equations are the directional velocities and pressure. The flow in porous media is governed by a combination of the continuity equation and momentum balance equation, which together form the Brinkman equations:

In these equations, denotes the dynamic viscosity of the fluid (SI unit: kg/(m·s)), u is the velocity vector (SI unit: m/s), is the density of the fluid (SI unit: kg/m3), p is the pressure (SI unit: Pa), p is the porosity, is the permeability of the porous medium (SI unit: m2), and Qbr is a mass source or mass sink (SI unit: kg/(m3·s)). Influence of

gravity and other volume forces can be accounted for via the force term F (SI unit: kg/ (m2·s2)).

When the Neglect inertial term (Stokes-Brinkman) check box is selected, the term (u · u p) is disabled on the left-hand side of Equation 10-5.

The mass source, Qbr, accounts for mass deposit and mass creation in domains, and the mass exchange is assumed to occur at zero velocity.

The Forchheimer drag option adds a viscous force proportional to the square of the fluid velocity, FF F u u, to the right-hand side of Equation 10-5. The

Forchheimer term F has SI units of kg/m4.

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