Reference ID: MET-6D1B | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Reynolds-number matching is the standard method for guaranteeing dynamic similarity when a stirred-tank process is transferred from one vessel size to another. By enforcing Re1 = Re2, the engineer ensures that the ratio of inertial to viscous forces—and therefore the flow regime (laminar, transitional, or turbulent)—remains unchanged. This is essential for predictable scale-up of mixing, heat- and mass-transfer, solids suspension, emulsification, fermentation, and other rate-limited operations.
Typical applications include bench-top to pilot-plant transfers, single-use to stainless-steel conversions, and plant debottlenecking studies. The calculation is embedded in most industrial scale-up protocols (e.g., Oldshue, Nagata, Uhl & Gray) and is a prerequisite for constant power-per-volume or constant tip-speed strategies.
Methodology & Formulas
Define the Reynolds number for a rotating impeller:
\[
Re = \frac{\rho N D^{2}}{\mu}
\]
where N is expressed in rps (rev s-1) and D in m.
Impose equality of Reynolds numbers between the reference (1) and target (2) conditions:
\[
\frac{\rho_{1} N_{1} D_{1}^{2}}{\mu_{1}} = \frac{\rho_{2} N_{2} D_{2}^{2}}{\mu_{2}}
\]
Solve for the unknown. Exactly one of {N2, D2} must be specified; the other follows algebraically.
If D2 is known:
\[
N_{2} = N_{1} \left(\frac{D_{1}}{D_{2}}\right)^{2} \left(\frac{\mu_{2}}{\mu_{1}}\right) \left(\frac{\rho_{1}}{\rho_{2}}\right)
\]
If N2 is known:
\[
D_{2}^{2} = D_{1}^{2} \left(\frac{N_{1}}{N_{2}}\right) \left(\frac{\mu_{2}}{\mu_{1}}\right) \left(\frac{\rho_{1}}{\rho_{2}}\right)
\]
Evaluate power and power-per-volume ratios assuming constant impeller power number Np:
\[
\frac{P_{2}}{P_{1}} = \left(\frac{N_{2}}{N_{1}}\right)^{3} \left(\frac{D_{2}}{D_{1}}\right)^{5}, \quad
\frac{(P/V)_{2}}{(P/V)_{1}} = \left(\frac{N_{2}}{N_{1}}\right)^{3} \left(\frac{D_{2}}{D_{1}}\right)^{2}
\]
Flow-regime limits for Newtonian fluids in baffled tanks with standard impellers
Regime
Re range
Typical consequences
Laminar
Re < 10
Viscous drag dominates; power ∝ N2
Transitional
10 ≤ Re ≤ 104
Gradual shift to inertial control; mixing time sensitive
Fully turbulent
Re > 104
Inertial forces dominate; power ∝ N3; constant Np
The calculation is unit-agnostic provided consistency is maintained; diameters must be in metres if rotational speed is supplied in rpm.
Reynolds number matching means setting the full-scale plant so its Reynolds number (Re = ρvD/μ) equals the value that gave reliable pilot-plant data. When Re is the same, the ratio of inertial to viscous forces is preserved, so flow regime, velocity profiles, mixing patterns, and heat or mass-transfer coefficients stay similar. Without this match, a process that worked in the lab can suffer from poor mixing, film buildup, or unexpected pressure drop in production.
Keep the same fluid properties (ρ, μ) if possible; if not, correct for them.
Calculate the required velocity in the large line from Relarge = Repilot → vlarge = vpilot × (Dpilot / Dlarge).
Check that the resulting velocity still gives acceptable pressure drop and NPSH.
If the velocity becomes impractical, consider using a smaller full-scale pipe or installing static mixers to recreate the same turbulence intensity rather than the same Re.
Yes—use the impeller Reynolds number Rei = ρND²/μ. Maintain Rei constant when you scale-up, but also keep the same flow regime (turbulent if Rei > 10,000). Because power per volume drops as tank size rises, you may need to adjust impeller diameter, blade width, or number of impellers to keep both Rei and mixing time within target.
Prioritize the dimensionless group that governs the rate-limiting step. For fast reactions limited by mass transfer, match Re and accept a different Sc; for diffusion-controlled polymerizations, match Sc and accept a different Re. Run a sensitivity CFD or reaction model to quantify the error introduced, then add safety factors or internal recirculation loops to compensate.
Worked Example: Matching Reynolds Number for a Scale-Model Heat-Exchanger Tube
A process engineer must verify that a 1:4 scale perspex model of a new smooth heat-exchanger tube will reproduce the same flow regime as the full-size unit. Reynolds-number matching is used to ensure dynamic similarity.
