Reference ID: MET-F845 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In agitated non-Newtonian fluids the apparent viscosity depends on the local shear rate. Because the power number Po is traditionally correlated with a Reynolds number that uses a single viscosity value, engineers must correct the viscosity for the prevailing shear regime. The procedure below converts a power-law rheogram into an “effective” Reynolds number, Repseudo, and then selects the appropriate Po correlation (laminar, transition, or turbulent). The corrected power demand is essential for sizing motors, checking shaft stresses, and ensuring that heat- and mass-transfer calculations are based on realistic energy dissipation rates. Typical applications are fermenters, polymerisation reactors, slurry mixers, and any vessel where the fluid behaves as τ = K γ̇n.
Methodology & Formulas
Convert rotational speed N [rps] = Nrpm / 60
Estimate average shear rate
The Metzner–Otto concept relates the average shear rate to the impeller speed through an impeller-specific constant ks:
γ̇ = ks N
Impeller style selects ks:
Impeller
ks
Rushton turbine
11.5
45° pitched-blade turbine
9.5
High-efficiency (HE3)
5.0
Apparent viscosity for a power-law fluid
μapp = K γ̇n–1
where K is the consistency index [Pa·sn] and n is the flow index.
Pseudo-Reynolds number
Repseudo = ρ N D2 / μapp
with ρ the fluid density and D the impeller diameter.
Select power-number correlation
The correlation used for Po depends on the flow regime:
Regime
Range
Correlation
Laminar
Repseudo < 10
Po = Kp / Repseudo
Transition
10 ≤ Repseudo < 10,000
Linear interpolation between laminar and turbulent values
Turbulent
Repseudo ≥ 10,000
Po = Poturbulent
The laminar constant Kp is impeller specific; for a Rushton turbine a representative value is 70. The turbulent asymptote Poturbulent is obtained from vendor charts or literature (e.g., 5.2 for a six-blade Rushton).
Transition interpolation
Weighting factor:
w = (Repseudo – 10) / (10,000 – 10)
Corrected power number:
Po = Polam (1 – w) + Poturbw
Calculate shaft power
P = Po ρ N3 D5 [W]
Motor sizing
Allow for gear efficiency ηgear and a safety factor SF:
Pmotor = (P / ηgear) · SF
The above sequence yields a viscosity-corrected power number and a realistic motor rating for non-Newtonian mixing applications.
Apply a correction whenever the bulk Reynolds number drops below about 10,000. Below this threshold the dimensionless power number (Np) begins to rise sharply because viscous forces start to dominate over inertial forces. Ignoring the correction in this regime will under-predict shaft power by 30–300%, leading to under-sized motors and overheated couplings.
Most engineers use the Nagata or Shamlou–Edwards form:
Np,corr = Np,∞ · (A / Re) + Np,∞
Np,∞ is the fully-turbulent power number (Re > 10,000)
Re is the impeller Reynolds number (ρN D2 / μ)
A is an impeller constant (30–70 for pitched-blade turbines, 35 for Rushton turbines, 90 for anchors)
Always validate the constant against vendor data or pilot trials; small geometric differences (D/T, C/T, blade width) shift A by ±20%.
Use an effective viscosity based on the average shear rate produced by the impeller:
Estimate shear rate: γ̇avg = k·N, where k ≈ 11 for a six-blade turbine and 25–30 for an anchor
Read μeff from the fluid’s flow curve at γ̇avg
Calculate Re with μeff and apply the laminar correction described above
If the fluid exhibits a yield stress, also check the cave-to-impeller diameter ratio; un-cavitated regions reduce primary flow and further raise Np.
Only with bench-scale validation. Extensional viscosity and elasticity can raise Np by an additional 10–40% even at the same Re. Run a 1–3 L lab reactor at matched tip speed and geometric ratios, measure torque, and scale the correction factor linearly with impeller diameter. Avoid direct extrapolation beyond a viscosity of 50 Pa·s or solids > 30 wt% without experimental confirmation.
Worked Example – Viscosity Correction for Power Number
A small pilot-scale fermenter is being designed to grow a shear-thinning broth.
The vessel is a flat-bottomed cylindrical tank, 0.6 m diameter, fitted with four baffles and a single six-blade Rushton turbine, 0.3 m diameter.
The broth is modelled as a power-law fluid with consistency index K = 45 Pa·sn and flow index n = 0.38.
The density is 1050 kg m–3.
The agitator must run at 150 rpm to provide the required oxygen mass-transfer rate.
Estimate the shaft power required to operate the impeller.
Knowns
Impeller type: Rushton turbine
Impeller diameter, D = 0.3 m
Rotational speed, N = 150 rpm = 2.5 s–1
Fluid density, ρ = 1050 kg m–3
Consistency index, K = 45 Pa·sn
Flow index, n = 0.38
Turbulent power number (fully turbulent), Poturb = 5.2
Viscosity correction constant for Rushton turbine, ks = 11.5
Laminar Reynolds limit, Relam = 10
Turbulent Reynolds limit, Returb = 10,000
Step-by-step calculation
Calculate the average shear rate for a Rushton turbine
γ̇ = ks N = 11.5 × 2.5 = 28.75 s–1
Evaluate the apparent viscosity at this shear rate
