A bioreactor is scaled up based on equal impeller tip speed. Consider the following parameters for
small and large bioreactors:
Parameters
Small bioreactor
Large bioreactor
Impeller speed
N1
N2
Diameter of impeller
D1
D2
Power consumption
P1
P2
Assuming geometrical similarity and the bioreactors are operated in turbulent regime, what will be
P2/P1?
Options:
(A) If (iii), then (iv).
(B) If (i), then (iii).
(C) If (i), then (ii).
(D) If (ii), then (iv).
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2016 Q.59)**
**1. Governing Concept & Question Classification**:
- **Section**: BT
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Bioprocess Engineering / Scale-Up (Scale-Up on Constant Impeller Tip Speed)
**2. Detailed Step-by-Step Derivation**:
- In stirred bioreactor scale-up under geometric similarity:
- Let $D_1, N_1, P_1$ be the impeller diameter, rotational speed, and power consumption of the small bioreactor.
- Let $D_2, N_2, P_2$ be the corresponding parameters of the large bioreactor.
- **Scale-up criterion**: Equal impeller tip speed ($v_{\text{tip}} = \text{constant}$):
$v_{\text{tip}} = \pi N_1 D_1 = \pi N_2 D_2 \implies N_2 D_2 = N_1 D_1 \implies \frac{N_2}{N_1} = \frac{D_1}{D_2}$
- **Power consumption in turbulent regime** ($Re > 10^4$):
The Power number $N_p = \frac{P}{\rho N^3 D^5}$ is constant:
$P \propto \rho N^3 D^5$
For identical fluid density ($\rho_1 = \rho_2$):
$\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^3 \left(\frac{D_2}{D_1}\right)^5$
- Substitute $\frac{N_2}{N_1} = \frac{D_1}{D_2} = \left(\frac{D_2}{D_1}\right)^{-1}$:
$\frac{P_2}{P_1} = \left(\frac{D_1}{D_2}\right)^3 \left(\frac{D_2}{D_1}\right)^5 = \left(\frac{D_2}{D_1}\right)^{-3} \left(\frac{D_2}{D_1}\right)^5 = \left(\frac{D_2}{D_1}\right)^{5 - 3} = \left(\frac{D_2}{D_1}\right)^2$
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: Inverts the diameter ratio to $(D_1/D_2)^2$.
- **Option (B) [CORRECT]**: $\frac{P_2}{P_1} = \left(\frac{D_2}{D_1}\right)^2$ is the exact analytical power ratio derived from equal tip speed in the turbulent regime.
- **Option (C) [INCORRECT]**: Assumes constant speed $N_1 = N_2$ with inversion.
- **Option (D) [INCORRECT]**: Assumes constant impeller rotational speed ($N_1 = N_2$), where $P_2/P_1 = (D_2/D_1)^5$.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **B**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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