Question

If the average diameter of air bubbles in a bioreactor is 2 mm and the gas hold up is 10%, then the surface area of gas bubbles per liter of reactor is:

• A. 30 cm$^2$
• B. 300 cm$^2$
• C. 3000 cm$^2$
• D. 30000 cm$^2$

Hint:

In bioreactors, the surface area of gas bubbles is a critical factor affecting mass transfer, and increasing the surface area can improve the efficiency of gas exchange in microbial cultures.

Answer 1

The Correct Option is D

Step 1: Recall the formula for the surface area of gas bubbles.
The surface area \( A \) of a single spherical gas bubble is given by the formula: \[ A = 4 \pi r^2 \] Where \( r \) is the radius of the gas bubble. 
Step 2: Convert the diameter into radius.
The average diameter of air bubbles is given as \( 2 \, {mm} \), so the radius \( r \) will be: \[ r = \frac{2}{2} = 1 \, {mm} = 0.1 \, {cm} \] Step 3: Calculate the surface area of a single bubble. 
Substituting \( r = 0.1 \, {cm} \) into the formula for surface area: \[ A = 4 \pi (0.1)^2 = 4 \pi \times 0.01 = 0.1256 \, {cm}^2 \] Step 4: Calculate the total surface area of bubbles per liter of reactor. 
The gas hold-up is 10%, meaning 10% of the reactor volume is occupied by gas bubbles. For a reactor with volume \( V \) (in this case 1 liter = 1000 cm$^3$), the total volume of gas bubbles is \( 0.1 \times 1000 = 100 \, {cm}^3 \). The total number of bubbles in the reactor, assuming they are spherical with volume \( V_{{bubble}} = \frac{4}{3} \pi r^3 \), is: \[ V_{{bubble}} = \frac{4}{3} \pi (0.1)^3 = \frac{4}{3} \pi \times 0.001 = 0.00419 \, {cm}^3 \] The number of bubbles in the reactor is: \[ {Number of bubbles} = \frac{100}{0.00419} \approx 23818 \, {bubbles} \] Step 5: Calculate the total surface area of the bubbles. The total surface area \( A_{{total}} \) of all the bubbles in the reactor is: \[ A_{{total}} = 23818 \times 0.1256 = 2999.96 \, {cm}^2 \approx 30000 \, {cm}^2 \]