Void Fraction Calculation for Packed Bed Bioreactor

This solution details the calculation of the void fraction ($\epsilon$) for a packed bed bioreactor using the given flow rate and velocity data.

Identify Given Parameters

• Reactor Length, $L = 1 \text{ m}$ (Note: Length is not needed for this calculation)
• Reactor Inside Diameter, $D = 10 \text{ cm}$
• Interstitial Velocity, $v_i = 1 \text{ cm s}^{-1}$
• Volumetric Flow Rate, $Q = 0.025 \text{ L s}^{-1}$

Unit Conversion for Flow Rate

Convert the volumetric flow rate ($Q$) from litres per second (L/s) to cubic centimeters per second (cm³/s) for consistency:

$Q = 0.025 \frac{\text{L}}{\text{s}} \times \frac{1000 \text{ cm}^3}{1 \text{ L}} = 25 \text{ cm}^3 \text{ s}^{-1}$

Calculate Reactor Cross-Sectional Area

Calculate the total cross-sectional area ($A$) of the packed bed using the inside diameter ($D$):

$A = \frac{\pi D^2}{4}$

$A = \frac{\pi (10 \text{ cm})^2}{4} = \frac{100\pi}{4} \text{ cm}^2 = 25\pi \text{ cm}^2$

Calculate Superficial Velocity

The superficial velocity ($v_s$) is the average velocity of the fluid based on the total reactor cross-sectional area. It is calculated as:

$v_s = \frac{Q}{A}$

$v_s = \frac{25 \text{ cm}^3 \text{ s}^{-1}}{25\pi \text{ cm}^2} = \frac{1}{\pi} \text{ cm s}^{-1}$

Calculate Void Fraction

The relationship between interstitial velocity ($v_i$), superficial velocity ($v_s$), and void fraction ($\epsilon$) is given by $v_i = \frac{v_s}{\epsilon}$. Therefore, the void fraction is:

$\epsilon = \frac{v_s}{v_i}$

Substitute the values:

$\epsilon = \frac{1/\pi \text{ cm s}^{-1}}{1 \text{ cm s}^{-1}} = \frac{1}{\pi}$

Calculating the numerical value:

$\epsilon \approx 0.318309...$

Final Rounded Answer

Rounding the void fraction to two decimal places:

$\epsilon \approx 0.32$