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


Objective

Objective of this practice is to explain how to predict Darcy and Forchheimer coefficients for perforated plates through analytical approach, using empirical equations.

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 along a porous medium, with respect to the velocity of the flow inside the medium.

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

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

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

1st Model

First equation is a simple approach proposed by Idelchik [1]. Following range represents the availability of the Idelchiks equation:

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

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

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

\[d_{h} = \frac{4 \cdot OpenArea[m]}{Wet\; surface\; area[m]}\]

This picture shows how the perforated plate geometry looks like and what does diameter and thickness mean

Porosity Open area closed area perforated plate SimScale

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 [-].

Porosity:

\[\phi = \frac{Open\; area\; of\; the\; perforated\; plate\; [m^{2}]}{Total\; area\; of\; the\; perforated\; plate\; [m^{2}]}\]

Idelchiks equation can be modified to be used as Forchheimer coefficient as follows:

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

Since the Idelchiks 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

Second equation is slightly more complex, modeled to differentiate thin and thick perforated plates [2]:

Discharge coefficient α:

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

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:

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

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”.

[1] Idelchik, I. E., Handbook of hydraulic resistance, third edition, CRC Press Inc., Boca Raton, US (1960)

[2] Kast W., (Revised by Hermann Nirschl), Gaddis E.S., Wirth KE., Stichlmair J. (2010) L1 Pressure Drop in Single Phase Flow. In: VDI e. V. (eds) VDI Heat Atlas. VDI-Buch. Springer, Berlin, Heidelberg

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