Section 4: Bioprocess Engineering and Process Biotechnology
Bioreactor types, scale-up
Subtopic: Fundamentals of Bioreactor types, scale-up
Question Statement:
A packed bed bioreactor with length 1 m and inside diameter 10 cm has liquid
flowing at an interstitial velocity of 1 cm s -1. Given a volumetric flow rate of
Q.025 L s -1, the void frac tion of the packed bed is _____ _. ( rounded off to two
decimal places)
END OF QUESTION PAPER
Official Answer Key:
0.31 to 0.32
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2026 Q.65)**
**1. Governing Concept & Definition**:
- **Topic**: Bioprocess Fluid Mechanics & Packed Bed Hydraulics
- The superficial velocity ($u_0$) and interstitial velocity ($u_i$) in a packed bed are related by bed porosity / void fraction ($\epsilon$):
$u_i = \frac{u_0}{\epsilon} \implies \epsilon = \frac{u_0}{u_i}$
where superficial velocity is $u_0 = \frac{Q}{A}$.
**2. Detailed Step-by-Step Calculation**:
- **Given**:
- Column inside diameter $D = 10\text{ cm} = 0.1\text{ m}$
- Cross-sectional area $A = \frac{\pi D^2}{4} = \frac{\pi \times (0.1)^2}{4} = \frac{0.01 \pi}{4} \approx 0.007854\text{ m}^2 = 78.54\text{ cm}^2$
- Volumetric flow rate $Q = 0.025\text{ L s}^{-1} = 25\text{ cm}^3\text{ s}^{-1}$
- Interstitial velocity $u_i = 1\text{ cm s}^{-1}$
- **Step 1: Calculate superficial velocity ($u_0$)**:
$u_0 = \frac{Q}{A} = \frac{25\text{ cm}^3\text{ s}^{-1}}{78.54\text{ cm}^2} \approx 0.3183\text{ cm s}^{-1}$
- **Step 2: Calculate void fraction ($\epsilon$)**:
$\epsilon = \frac{u_0}{u_i} = \frac{0.3183\text{ cm s}^{-1}}{1.0\text{ cm s}^{-1}} \approx 0.318 \approx 0.32$
**3. Final Answer**:
- **Calculated Value**: **0.32**
- **Official GATE Accepted Range**: **0.31 to 0.32**
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