Reynolds Number Calculation for Filter Bed

This solution details the calculation of the Reynolds number for fluid flow through a filter bed, using the provided parameters for anthracite media and filtration rate.

Filter Bed Parameters and Unit Conversion

To accurately calculate the Reynolds number, it's essential to work with consistent units. We will convert all given parameters to feet and seconds, aligning with the provided kinematic viscosity.

• Particle Size ($d_p$): $1.6 \text{ mm}$
• Particle Shape Factor ($\phi_s$): $0.50$
• Filtration Rate ($v_s$): $4 \text{ gal/ft}^2/\text{min}$
• Kinematic Viscosity ($\nu$): $1.091 \times 10^{-5} \text{ ft}^2/\text{s}$

Unit Conversions:

Particle Size Conversion:

We convert the particle size from millimeters (mm) to meters (m), and then to feet (ft).

$$ d_p = 1.6 \text{ mm} = 0.0016 \text{ m} $$

Using the conversion factor $1 \text{ ft} = 0.3048 \text{ m}$:

$$ d_p = \frac{0.0016 \text{ m}}{0.3048 \text{ m/ft}} \approx 0.005249 \text{ ft} $$

Filtration Rate Conversion:

The filtration rate, representing the superficial velocity ($v_s$), needs to be converted from gallons per square foot per minute (gal/ft²/min) to feet per second (ft/s).

Using the conversion factor $1 \text{ gal} = 0.133681 \text{ ft}^3$:

$$ v_s = 4 \frac{\text{gal}}{\text{ft}^2 \cdot \text{min}} = 4 \times \frac{0.133681 \text{ ft}^3}{\text{ft}^2 \cdot \text{min}} = 0.534724 \frac{\text{ft}}{\text{min}} $$

Converting minutes to seconds ($1 \text{ min} = 60 \text{ s}$):

$$ v_s = \frac{0.534724 \text{ ft}}{60 \text{ s}} \approx 0.008912 \text{ ft/s} $$

Reynolds Number Formula for Packed Beds

For flow through packed beds, especially with non-spherical particles like anthracite, the Reynolds number ($Re$) is often calculated using an effective diameter ($d_{eff}$). This $d_{eff}$ is derived by incorporating the particle's shape factor ($\phi_s$) into the particle size ($d_p$).

The formula for the effective diameter is:

$$ d_{eff} = d_p \times \phi_s $$

The Reynolds number ($Re$) is then determined using the superficial velocity ($v_s$), the effective diameter ($d_{eff}$), and the kinematic viscosity ($\nu$) with the following equation:

$$ Re = \frac{v_s \times d_{eff}}{\nu} $$

Packed Bed Reynolds Number Calculation Steps

First, we calculate the effective diameter ($d_{eff}$) using the given particle size and shape factor:

$$ d_{eff} = 1.6 \text{ mm} \times 0.50 = 0.8 \text{ mm} $$

Next, we convert this effective diameter from millimeters to feet:

$$ d_{eff} = 0.8 \text{ mm} = 0.0008 \text{ m} $$

Using the conversion factor $1 \text{ ft} = 0.3048 \text{ m}$:

$$ d_{eff} = \frac{0.0008 \text{ m}}{0.3048 \text{ m/ft}} \approx 0.002625 \text{ ft} $$

Now, we substitute the converted values ($v_s$, $d_{eff}$, and $\nu$) into the Reynolds number formula:

$$ Re = \frac{(0.008912 \text{ ft/s}) \times (0.002625 \text{ ft})}{1.091 \times 10^{-5} \text{ ft}^2/\text{s}} $$

Calculate the numerator (product of velocity and effective diameter):

$$ \text{Numerator} = 0.008912 \times 0.002625 \approx 2.340 \times 10^{-5} \text{ ft}^2/\text{s} $$

Finally, perform the division to find the Reynolds number:

$$ Re = \frac{2.340 \times 10^{-5} \text{ ft}^2/\text{s}}{1.091 \times 10^{-5} \text{ ft}^2/\text{s}} \approx 2.145 $$

Final Result

The calculated Reynolds number for the filter bed is approximately $2.15$.