## Objective

The objective of this practice is to explain how to predict Darcy and Forchheimer coefficients for perforated plates through an 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 with 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 a 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 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 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 the 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**:

\[d_{h} = \frac{4 \cdot Open\; Area[m^2]}{Wet\; perimeter\;[m]}\]

**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:

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/d_{h})→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/d_{h}) » 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”**.

*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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