Mixing Scale-up Rules from Lab to Plant
Scaling a mixing process from lab to pilot to plant is trickier than it looks because there's no single "correct" way to do it. Constant tip speed, constant power-per-volume (P/V) and constant Reynolds number are different, mutually exclusive scale-up rules. Which one matters depends on your process.
Want to run your own numbers? Try the Mixing / Scale-up Calculator.
The Core Numbers
Tip Speed = π × D × N
Power = Np × ρ × N³ × D⁵
P/V = Power / Volume
Where N is impeller rotational speed (rev/s), D is impeller diameter, ρ and μ are fluid density and viscosity, and Np is the power number (a dimensionless constant depending on impeller type and Reynolds regime).
Scale-up Rules at a Glance
For geometrically similar vessels (same D/T, same impeller type, same baffling), every common rule reduces to one formula for the plant agitator speed:
| Scale-up rule | n | Use it when |
|---|---|---|
| Constant tip speed | 1 | Shear matters: crystals, flocs, fragile solids |
| Constant P/V (turbulent) | 2/3 | Blending, heat transfer, gas–liquid or liquid–liquid dispersion |
| Constant Reynolds number | 2 | Rarely practical; gives a very low plant RPM |
Worked Example
A lab reactor with a 0.1 m impeller running at 300 rpm (5 rev/s) needs to scale to a plant vessel with a 1.0 m impeller. Scaling on constant tip speed:
N (plant) = Tip speed / (π × D) = 1.57 / (π × 1.0) ≈ 0.5 rev/s (30 rpm)
Scaling the same jump on constant P/V instead (P/V scales with N³D² in the turbulent regime, so n = 2/3):
And on constant Reynolds number (n = 2): N = 300 × (0.1)² = 3 rpm, which is far too slow to mix a plant vessel. The same lab result gives 3, 30 or 65 rpm depending on the rule, so picking the right one for your chemistry matters more than the arithmetic. At constant P/V the total power also rises with the volume, about 1,000 times here for a tenfold jump in impeller diameter.
In the Plant
Constant tip speed suits shear-sensitive systems (crystals you don't want to break, some biological materials). Constant P/V suits systems where bulk blend time and heat/mass transfer matter more than shear. Constant Reynolds number is rarely workable at large scale-up ratios since it typically demands an unrealistically low large-vessel RPM.
Common Mistakes
Applying one scale-up rule "by default" without checking whether it actually protects the thing that matters for that specific product; comparing power numbers between different impeller types as if they were interchangeable; and forgetting that geometric similarity (same D/T ratio, same baffling) is an assumption behind all of these formulas. If the plant vessel's geometry doesn't match the lab vessel's, none of these scale-up rules apply cleanly.
Frequently Asked Questions
What are the main mixing scale-up rules?
For geometrically similar vessels, N2 = N1 × (D1/D2)n, with n = 1 for constant tip speed, n = 2/3 for constant P/V (turbulent) and n = 2 for constant Reynolds number.
Which scale-up rule should I use?
Constant tip speed when shear damages the product, constant P/V when blending, heat transfer or mass transfer control the result. If you are unsure, calculate both and check what each one does to the property you care about.