Q.No. 22 To facilitate mass transfer from a gas to a liquid phase, a gas bubble of radius r is introduced into the liquid. The gas bubble then breaks into 8 bubbles of equal radius. Upon this change, the ratio of the interfacial surface area to the gas phase volume for the system changes from 3/r to 3n/r. The value of n is __________.
Gas Bubble Radius Mass Transfer: Surface Area to Volume Ratio
Numeric Answer: n = 2
Problem Breakdown
A single gas bubble of radius r breaks into 8 equal smaller bubbles. The interfacial surface area to gas phase volume ratio shifts from 3/r to 3n/r, where n quantifies the enhancement.
Original Bubble
Volume: V = (4/3)πr3
Surface Area: A = 4πr2
Ratio: A/V = (4πr2)/((4/3)πr3) = 3/r
After Bubble Breakage
Volume conservation:
8 × (4/3)πrs3 = (4/3)πr3
⇒ rs = r/2
For 8 Small Bubbles
Total Volume: (unchanged) (4/3)πr3
Surface Area of each small bubble:
4π(r/2)2 = 4π(r2/4) = πr2
Total Surface Area:
8 × πr2 = 8πr2
New Ratio
(8πr2)/((4/3)πr3) = (8 × 3)/(4r) = 6/r
Thus 3n/r = 6/r ⇒ n = 2
Error Checks
- Radius reduces by factor 1/2
- Area increases ×2
- Volume stays constant
Key Takeaway Table
| Parameter | Original | After Breakage | Scale Factor |
|---|---|---|---|
| Radius | r | r/2 | 1/2 |
| Total Volume | (4/3)πr3 | (4/3)πr3 | 1 |
| Total Surface Area | 4πr2 | 8πr2 | 2 |
| A/V Ratio | 3/r | 6/r | 2 (n = 2) |
Applications in Biotech
Smaller bubbles increase interfacial area, improving mass transfer in bioreactors and fermentation systems.
