Reference ID: MET-4B58 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
For pseudoplastic (shear-thinning) fluids the viscosity measured in a Brookfield or cup test is meaningless at the high shear rates created by an impeller. The impeller instead “sees” an apparent viscosity that is a function of the local shear rate. Correcting the Reynolds number with this apparent viscosity lets the process engineer select the proper Power number and therefore size the motor, predict heat-of-agitation, and avoid under- or over-designing industrial fermenters, ketchup kettles, or slurry storage tanks.
Methodology & Formulas
All relations below are algebraic duplicates of the Python code; no empirical constants beyond those embedded in the code are introduced.
Representative shear rate generated by a Rushton turbine
\(\dot\gamma = k_s\,N\)
Apparent viscosity for a power-law fluid (with consistency index \(K\) and flow index \(n\))
\(\mu_{\text{app}}=K\bigl(\dot\gamma\bigr)^{\,n-1}\)
Impeller Reynolds number corrected for shear-thinning behaviour
\[\text{Re}' = \frac{\rho\,N\,D^{2}}{\mu_{\text{app}}}\]
Mechanical power draw
\[P=N_p\;\rho\;N^{3}\;D^{5}\]
Validation regime limits (from code)
Parameter
Acceptable range
Consequence if violated
Power-law index \(n\)
\(0.2 < n < 1\)
Code raises ValueError
‘Flat’ Power-number assumption (turbulent regime)
\(\text{Re}' \geq 10,000\)
Code raises ValueError; Power number may vary below this threshold
Units used in every expression are:
Consistent engineering units
Quantity
Unit
\(K\)
Pa·sn
\(n\)
—
\(N\)
rev s-1
\(\rho\)
kg m-3
\(D\)
m
\(\mu_{\text{app}}\)
Pa·s
\(P\)
W
Constants embedded in the algorithm are:
\(k_s=11.5\) (Rushton turbine, fully-baffled standard tank)
\(N_p=5.5\) (turbulent regime flat value, valid only for \(\text{Re}' > 10,000\))
A correction is required once the impeller Reynolds number (Re) drops below about 10,000–30,000 because laminar and transitional flow fields reduce the pumping capacity of the impeller and raise the apparent power number (Np). For Re > 30,000 the bulk fluid behaves as turbulent and the standard Np for water can be used without correction. In borderline cases, apply the Metzner-Otto method (calculate shear rate from N and Ks, then apparent viscosity) and correct Np with published correlations or vendor data.
Use the Metzner-Otto method for pseudoplastic and yield-stress fluids to estimate the apparent viscosity and corrected Reynolds number. Required inputs: consistency index (K, Pa·sn), flow behaviour index (n), impeller shear-rate constant (ks), rotational speed (N, s⁻¹), density (ρ, kg/m³), and impeller diameter (D, m). Calculate the apparent viscosity μapp = K (ks N)n-1, then the corrected Reynolds number Re' = ρ N D² / μapp. Use standard Power number vs. Reynolds number correlations for the impeller type to find Np. For turbulent flow (Re' > 10,000), Np is approximately constant.
Create a cell for apparent viscosity (µapp) calculated with Metzner-Otto parameters: µapp = K (ks N)n-1.
Calculate the corrected Reynolds number: Re' = ρ N D² / µapp.
Add a lookup table that maps Re' to Np based on impeller type; linearly interpolate between tabulated values. For Re' > 10,000, use the constant turbulent Np.
Compute the mechanical power: P = Np ρ N³ D⁵. Program a low-Re alarm so operators are warned when Re' < 10,000 or the shaft power approaches the motor limit.
High-solidity, low-pumping impellers (helical ribbons, anchors) and radial turbines display steep power-number rises at low Re. To minimise the penalty:
Increase D/T ratio to raise shear rate and Re'
Use multiple-pitched-blade stages or retreat-curve impellers to improve axial flow and shift the effective Re' upward
For shear-thinning fluids, operate in the shear-thinning plateau by choosing a speed where apparent viscosity is <50% of the zero-shear value
If heat transfer demands high viscosity, supply the missing mixing energy with larger impellers rather than higher speed to avoid excessive power draw
Worked Example: Viscosity Correction for Power Number in a Mixing Tank
A process engineer needs to size the agitator motor for mixing concentrated tomato paste, modeled as a shear-thinning power-law fluid, in a fully baffled tank with a standard Rushton turbine. The objective is to compute the mechanical power required by correcting the impeller Reynolds number using the fluid's apparent viscosity, ensuring the flow is turbulent for a constant Power number.
Validate the flow regime: Re' = 12360 > Remin = 10000.000. Therefore, the assumption of a constant power number Np = 5.500 in the turbulent regime is applicable.
