Introduction & Context

The average shear rate generated by an impeller is a key scaling parameter in the mixing of non-Newtonian fluids. It is used to:

Estimate the apparent viscosity of power-law fluids at process conditions.
Calculate the Generalized Reynolds number, \(Re_g\), which determines whether the flow regime around the impeller is laminar, transitional, or turbulent.
Scale-up or scale-down mixing vessels while maintaining the same average shear environment.

The Metzner–Otto concept assumes that the average shear rate in the vessel is proportional to the impeller rotational speed. The proportionality constant, \(k_s\), is geometry-dependent and has been correlated for many impeller types.

Methodology & Formulas

Convert rotational speed
\(N = \dfrac{\text{rpm}}{60}\) [s⁻¹]
Average shear rate (Metzner–Otto)
\(\bar\gamma = k_s\,N\) [s⁻¹]
Apparent viscosity for power-law fluid
\(\mu_{\text{app}} = K\,\bar\gamma^{\,n-1}\) [Pa·s]
Generalized Reynolds number
\[ Re_g = \frac{\rho\,N^{2-n}\,D^{2}}{K\,k_s^{\,n-1}} \]

Validity Regimes

Parameter	Range	Remarks
\(k_s\)	10 – 13	Typical for turbine impellers with power-law fluids
\(Re_g\)	\(\le 30\)	Laminar–transitional limit for the correlation
\(\bar\gamma\)	10 – 1000 s⁻¹	Power-law parameters valid in this shear window

Shear rate at an impeller is the velocity gradient (s⁻¹) between the fastest-moving fluid (at the blade tip) and the surrounding slower fluid. It is critical because it governs:

Micro-mixing efficiency and reaction selectivity
Cell viability in bioreactors
Drop or bubble size in dispersions
Power draw and heat-transfer coefficients

Accurate estimation prevents over-design or process failures caused by excessive shear.

Use the tip-speed–based correlation:

γ̇_max ≈ k · N · D / δ

where:

N = rotational speed (rps)
D = impeller diameter (m)
δ = boundary-layer thickness ≈ 0.1–0.3 mm for turbulent flow
k = proportionality constant ≈ 5–10 for Rushton turbines; 3–5 for pitched-blade or hydrofoils

For quick sizing, replace δ with blade thickness if δ data are unavailable.

Maintain constant tip speed (πND) and keep geometric similarity. If that is impossible:

Scale on constant power per unit volume (P/V) while checking that γ̇_max does not exceed the lab value
Use a lower-shear impeller style (e.g., hydrofoil) at larger D and lower N to keep tip speed unchanged
Validate with pilot trials measuring drop size, cell viability, or reaction selectivity as direct shear indicators

After convergence, compute the magnitude of the strain-rate tensor:

|S| = √(2S_ij S_ij) where S_ij = ½(∂u_i/∂x_j + ∂u_j/∂x_i)

Then:

Plot |S| on a plane cutting the blade tip; the 95th percentile of |S| in this plane is taken as γ̇_max
Use a mesh ≤ 0.5 mm at the blade edge to capture the steep gradient
Turn on turbulence correction (e.g., SST k-ω) to account for turbulent viscosity contribution to effective shear

Worked Example – Estimating Average Shear Rate at a Rushton Turbine

A small pilot-scale fermenter is being commissioned to grow a shear-sensitive filamentous fungus. To avoid cell damage, the process engineer must verify that the average shear rate generated by the 120 rpm Rushton impeller remains below the 1000 s⁻¹ damage threshold quoted in the strain licence. The broth behaves as a power-law fluid; calculate the shear rate and confirm acceptability.

Knowns

Impeller diameter, D = 0.133 m
Rotational speed, N = 2.0 rps (120 rpm)
Fluid density, ρ = 1080 kg m⁻³
Consistency index, K = 1.2 Pa·sⁿ
Flow behaviour index, n = 0.65
Metzner–Otto constant, k_s = 11.5

Step-by-step calculation

Calculate average shear rate from impeller speed and Metzner–Otto constant: \[ \gamma_{\text{avg}} = k_s\,N = 11.5 \times 2.0 = 23\ \text{s}^{-1} \]
Determine apparent viscosity at this shear rate for a power-law fluid: \[ \mu_{\text{app}} = K\,\gamma_{\text{avg}}^{\,n-1} = 1.2 \times 23^{(0.65-1)} = 1.2 \times 23^{-0.35} = 0.400\ \text{Pa·s} \]
Compute the generalised Reynolds number to check flow regime: \[ Re_g = \frac{\rho N^{2-n} D^2}{K\,k_s^{\,n-1}} = \frac{1080 \times 2^{1.35} \times 0.133^2}{1.2 \times 11.5^{-0.35}} = 95.4 \] Because 95.4 > 30, the impeller operates in the transitional regime, confirming the use of the Metzner–Otto approach.

Final Answer

The average shear rate at the impeller is 23 s⁻¹, well below the 1000 s⁻¹ damage limit; the chosen agitation speed is acceptable for this organism.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle