Different models can define the capacity of the porous medium to transmit flow. Some use the permeability, κ (SI unit: m
2), and the viscosity of the fluid,
μ (SI unit: Pa·s
); others use the hydraulic conductivity,
K (SI unit: m/s). Several relationships have been established to describe Darcian and non-Darcian flow through different porous materials.
The Ergun equation is an extension of Kozeny–Carman equation to higher Reynolds numbers (Re > 10), where Darcy’s linear relation between pressure drop and velocity is no longer valid. In Ergun equation the linear relation between pressure drop and velocity is augmented by a quadratic term. The pressure drop (
Ref. 6) in Ergun’s equation is given by:
Here, dp is the effective (average) particle diameter in the porous medium and
εp is the porosity. With definitions of permeability
κ and parameter
β (SI unit 1/m) as
we get Equation 2-2 and Ergun’s velocity can then be calculated as
From this expression, we observe that for low Reynolds numbers Re < 1 (or equivalently, high friction factors) the flow can be described by the linear Darcy’s Law (
Equation 2-1).
For increasing filter velocities (Re > 10), inertial effects and turbulent friction forces become significant. In the Forchheimer drag this is considered by adding an additional term to Darcy’s law:
Here, the Forchheimer parameter cF is dimensionless. The permeability
κ is specified by the user and is not computed from porosity nor particle diameter as for Ergun’s equation. If the parameter
β is defined as
The Burke-Plummer equation is applicable in flow regimes with Reynolds number Re > 1000. Here, the pressure drop is proportional to the square of the velocity:
here, L is the length of the packed bed,
Δp is the pressure drop
dp is the effective (average) particle diameter in the porous medium,
εp is the porosity, and
v is the velocity magnitude.
Using the same definition for the parameter β (SI unit 1/m) as in Ergun’s equation (
Equation 2-5) the pressure drop is related to the velocity as
here, κ∞ is the permeability of the gas at high pressure and density (Knudsen number 0.001
< Kn < 0.1),
bK is the Klinkenberg parameter (also called
Klinkenberg slip factor, or
gas slippage factor), whose default value is set to 1 kPa according to
Ref. 8, and
pA is the absolute pressure as defined in the Darcy’s Law interface (
pA =
p +
pref).