Introduction & Context
Constant Tip Speed Scale-Up is a fundamental methodology in Process Engineering used to maintain consistent shear conditions when transitioning a mixing process from laboratory-scale equipment to larger pilot or production-scale vessels. In shear-sensitive applications, such as bioprocessing or the handling of delicate emulsions, the impeller tip speed is the primary indicator of the maximum shear stress exerted on the fluid. By keeping the tip speed constant across different scales, engineers ensure that the hydrodynamic environment remains comparable, thereby preserving product integrity and process performance.
Methodology & Formulas
The calculation relies on the principle that the tangential velocity at the impeller tip must remain invariant during scale-up. The following algebraic expressions define the relationship between rotational speed, impeller diameter, and fluid dynamics:
1. Lab-scale rotational speed conversion:
\[ N_{1,rev\_s} = \frac{N_{1,rpm}}{60.0} \]
2. Impeller tip speed calculation:
\[ U_{tip} = \pi \cdot N_{1,rev\_s} \cdot D_{1} \]
3. Pilot-scale rotational speed calculation:
\[ N_{2,rev\_s} = \frac{U_{tip}}{\pi \cdot D_{2}} \]
4. Pilot-scale rotational speed conversion:
\[ N_{2,rpm} = N_{2,rev\_s} \cdot 60.0 \]
5. Reynolds number calculation for regime verification:
\[ Re_{2} = \frac{\rho \cdot N_{2,rev\_s} \cdot D_{2}^{2}}{\mu} \]
| Parameter |
Validity Constraint |
| Tip Speed (Utip) |
0.2 ≤ Utip ≤ 5.0 m s-1 |
| Reynolds Number (Re2) |
0.0 < Re2 < 20000.0 |
Constant tip speed is widely used because it maintains a consistent shear environment at the impeller blade, which is critical for processes sensitive to hydrodynamic stress. By keeping the tip speed constant, process engineers can:
- Preserve the local shear rate experienced by cells or delicate particles.
- Ensure consistent droplet breakup or agglomeration kinetics.
- Minimize the risk of mechanical damage to shear-sensitive biological products.
Worked Example: Constant Tip Speed Scale-Up for a Mixing Process
A bioprocess engineer is scaling up a mixing operation for a shear-sensitive aqueous solution from laboratory to pilot scale. The goal is to maintain identical impeller tip speed to preserve the shear environment, which is critical for product integrity.
Known Input Parameters and Units:
- Lab-scale impeller diameter, D₁ = 0.050 m
- Lab-scale rotational speed, N₁ = 100 rpm
- Fluid density, ρ = 998 kg/m³
- Fluid dynamic viscosity, μ = 0.001 Pa·s
- Pilot-scale impeller diameter, D₂ = 0.150 m
Step-by-Step Calculation:
- Convert the lab-scale rotational speed from rpm to revolutions per second:
\[ N_{1,\text{rev/s}} = \frac{N_{1,\text{rpm}}}{60} = \frac{100}{60} = 1.6667 \text{ rev/s} \]
- Calculate the tip speed at lab scale using the formula \( U_{\text{tip}} = \pi N D \):
\[ U_{\text{tip}} = \pi \times N_{1,\text{rev/s}} \times D_{1} = \pi \times 1.6667 \times 0.050 = 0.2618 \text{ m/s} \]
- For constant tip speed scale-up, set the pilot-scale tip speed equal to the lab-scale value and solve for the pilot-scale speed in rev/s:
\[ N_{2,\text{rev/s}} = \frac{U_{\text{tip}}}{\pi D_{2}} = \frac{0.2618}{\pi \times 0.150} = 0.5556 \text{ rev/s} \]
- Convert the pilot-scale speed back to rpm for practical operation:
\[ N_{2,\text{rpm}} = N_{2,\text{rev/s}} \times 60 = 0.5556 \times 60 = 33.33 \text{ rpm} \]
- Compute the Reynolds number for the pilot scale to assess the flow regime:
\[ \text{Re}_{2} = \frac{\rho N_{2,\text{rev/s}} D_{2}^{2}}{\mu} = \frac{998 \times 0.5556 \times (0.150)^{2}}{0.001} = 12476.0 \]
Final Answer: To maintain a constant tip speed of 0.2618 m/s, the pilot-scale impeller should be operated at
33.33 rpm. Under these conditions, the Reynolds number is 12476.0.
