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How to Predict Darcy and Forchheimer Coefficients for Perforated Plates Using Analytical Approach?

The objective of this article is to explain how to predict Darcy and Forchheimer coefficients for perforated plates through an analytical approach, using empirical equations. Note that empirical equations are only suitable for single-phase flow.

Analytical Approach

Darcy–Forchheimer equation is an empirical equation, which relates the pressure loss, due to friction across a porous medium, with respect to the velocity of the flow inside the medium.

$$\Delta P = \mu \cdot d \cdot L \cdot u + \frac{\rho }{2} \cdot f \cdot L \cdot u^{2} \tag{1}$$

Friction is occurred by a combination of shear forces and pressure forces. While the shear forces are represented by a linear equation (Darcy equation), pressure forces are represented by a quadratic equation (Forchheimer equation). Therefore \(d\) is a linear resistance coefficient (Darcy coefficient) and \(f\) is a quadratic resistance coefficient (Forchheimer coefficient).

Porosity

Porosity is the measure of empty space or the void percentage in a porous material. For a perforated plate of unit thickness porosity can be mathematically described as:

\[\phi = \frac{Open\: area\: of\: the\: perforated\: plate\: }{Total\: area\: of\: the\: perforated\: plate\: }\]

For circular holes, hydraulic diameter is the diameter of the hole. For non-circular conduit, the following equation can be used to calculate hydraulic diameter:

\[d_{h} = \frac{4 \cdot Open\: Area }{Wet\: perimeter\: }\]

or

\[d_{h} = \frac{4 \cdot Open\: Volume }{Wet\: surface\: area\: }\]

This picture shows how the perforated plate geometry looks like and what does diameter and thickness mean
Figure 1: Perforated plate acting as a porous media

Where \(d_h\) is the hydraulic diameter and \(L\) is the thickness of the perforated plate.

Unconsolidated porous medium

An unconsolidated medium is a pile of solid particles packed inside a bed. Fluid flows around these particles.

unconsolidated porous medium darcy and forchheimer
Figure 2: Unconsolidated porous medium. The spherical beads are packed into a bed creating small voids for the fluid to pass through.

If the medium is composed of uniform spherical particles, mean particle diameter (spherical particles) can be found as follows:


\[d_{m} = \frac{3 \cdot \left ( 1-\phi \right ) \cdot d_{h}}{2 \cdot \phi }\]

Darcy and Forchheimer coefficients can be calculated as follows:

\[d = 150 \cdot \frac{(1-\phi )^{2}}{\left ( \phi^{3} \cdot d_{m}^{2} \right )} \tag{2}\]

\[f = 3.5 \cdot \frac{\left ( 1-\phi \right )}{\left ( \phi^{3} \cdot d_{m} \right )} \tag{3}\]

Various studies were conducted to correlate pressure drop versus flow rate and geometry. With literature research, a convenient empirical equation should be found for specific perforated plate geometry and flow conditions. Here, we will show only two methods:

1st Model

The First equation is a simple approach proposed by Idelchik\(^1\). The following range represents the availability of Idelchik’s equation:

\[0< \frac{L}{d_{h}}< 0.015\; and\; Re> 10^{5}\]

Resistance coefficient:

\[k = \left [ 0.707 \cdot \left ( 1-\phi \right )^{0.375} +1 – \phi^{2} \right ] \cdot \frac{1}{\phi^{2}}\]

Where, \(\phi \) is the porosity (open area ratio) and \(k\) is the pressure resistance factor.

Idelchik’s equation can be modified to be used as Forchheimer coefficient as follows:

\[f = \frac{k}{L} \tag{4}\]

Since Idelchik’s pressure coefficient is a quadratic resistance coefficient, one can assume the Darcy coefficient as zero:

\[\Delta P = \frac{\rho }{2} \cdot f \cdot L \cdot u^{2}\]

2nd Model

The second equation is slightly more complex, modeled to differentiate thin and thick perforated plates\(^2\):

Discharge coefficient \(\alpha\):

\[\alpha = 0.6 + 0.4 \cdot \phi^{2} \]

Thin plates are assumed to have relatively large holes. For thin plates (L/dh)→0, resistance coefficient:

\[\alpha = 0.6 + 0.4 \cdot \phi^{2} \]

\[k = \left ( \frac{1}{\alpha }- \phi \right )^{2}\]

Thick plates are assumed to have relatively small holes. For thin plates (L/dh) » 0, resistance coefficient:

\[k = \left ( \frac{1}{\alpha }- 1 \right )^{2}+\left ( 1-\phi \right )^{2}\]

Resistance coefficient can be modified to be used as Forchheimer coefficient as follows:

\[f = \frac{k}{\phi^{2} \cdot L } \tag{5}\]

Pressure loss with this method is also a quadratic function, therefore Darcy coefficient should be defined as zero:

\[\Delta P = \frac{\rho }{2} \cdot f \cdot L \cdot u^{2}\]

You can use this spreadsheet to calculate \(d\) and \(f\). Please use the page “Empirical Method”.

References

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Last updated: April 1st, 2021

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