High-Shear Rotor-Stator Mixing

Reference ID: MET-BB53 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

High-shear rotor-stator mixing is a fundamental unit operation in process engineering, primarily utilized for the creation of stable emulsions, suspensions, and dispersions. The process involves a high-speed rotor spinning within a stationary stator, forcing fluid through narrow clearances. This geometry generates intense localized shear and elongational flow fields, which are critical for overcoming interfacial tension to break down droplets or particles to the micron or sub-micron scale.

In industrial applications, such as food processing (e.g., mayonnaise, dressings) or chemical manufacturing, sizing the mixer correctly is vital. Engineers must balance the dispersive shear capability—required for droplet breakup—with the pumping capacity, which ensures the entire batch is processed uniformly. Failure to properly size these parameters often leads to inconsistent product quality, excessive energy consumption, or inadequate droplet size reduction.

Methodology & Formulas

The sizing methodology relies on dimensionless numbers and mechanical parameters to characterize the fluid regime and the efficiency of the energy transfer. The following formulas define the physical behavior of the system:

1. Kinematics and Shear
The rotor tip speed is the primary driver of shear intensity, calculated as:

\[ V_{tip} = \pi \cdot D_{rotor} \cdot N \]

where \( N \) is the rotational speed in revolutions per second (RPS). The nominal shear rate, which represents the velocity gradient across the rotor-stator gap, is defined as:

\[ \dot{\gamma} = \frac{V_{tip}}{h_{gap}} \]

2. Fluid Dynamics and Power
The flow regime is determined by the Reynolds number, which dictates the applicability of power consumption correlations. For a rotor in a viscous fluid:

\[ Re = \frac{\rho \cdot N \cdot D_{rotor}^{2}}{\mu} \]

where \( N \) is in RPS. The power draw required to drive the rotor is estimated using the power number, which is typically constant in the turbulent regime:

\[ P = N_{p} \cdot \rho \cdot N^{3} \cdot D_{rotor}^{5} \]

3. Dispersion and Pumping
The Weber number assesses the feasibility of droplet breakup by comparing inertial forces to interfacial tension:

\[ We = \frac{\rho \cdot V_{tip}^{2} \cdot d_{initial}}{\sigma} \]

where \( \rho \) is the density of the continuous phase. The volumetric pumping rate and the resulting batch turnover time are calculated to ensure adequate processing frequency:

\[ Q = K \cdot N \cdot D_{rotor}^{3} \] \[ t_{turnover} = \frac{V_{batch}}{Q} \]

Parameter	Regime / Condition	Threshold / Criteria
Tip Speed	Emulsification Range	\( 10 \text{ m/s} \leq V_{tip} \leq 50 \text{ m/s} \)
Gap Clearance	Mechanical Design	\( 0.1 \text{ mm} \leq h_{gap} \leq 3 \text{ mm} \)
Flow Regime	Turbulent (for \( N_{p} \) validity)	\( Re \geq 10,000 \)
Droplet Breakup	Inertial Dominance	\( We \geq 1 \)

To determine the optimal tip speed, process engineers should evaluate the required droplet size or particle size reduction. Generally, tip speeds range from 10 to 50 meters per second. Follow these steps:

Calculate the peripheral velocity using the formula: \( V_{tip} = \pi \cdot D_{rotor} \cdot N \), ensuring \( N \) is in revolutions per second.
Perform a pilot-scale study to correlate tip speed with the desired rheological properties of the emulsion or suspension.
Monitor the temperature rise, as excessive tip speeds can lead to thermal degradation of heat-sensitive ingredients.

Mechanical wear in high-shear equipment often manifests through changes in process efficiency and physical degradation of the components. Look for the following signs:

Increased vibration levels during operation, which may indicate bearing failure or rotor imbalance.
A noticeable widening of the gap between the rotor and stator, leading to a broader particle size distribution.
Visible erosion or pitting on the stator teeth, often caused by abrasive slurries or cavitation.
An unexplained increase in motor amperage required to maintain the same setpoint speed.

Stator geometry is critical for managing flow patterns and shear intensity. For varying viscosity ranges, consider these guidelines:

Low-viscosity fluids: Use fine-screen or multi-stage stators to maximize shear rates and ensure rapid homogenization.
High-viscosity fluids: Utilize coarse-slotted or square-hole stators to facilitate better bulk circulation and prevent flow stagnation.
Non-Newtonian fluids: Ensure the stator design allows for sufficient axial flow to prevent the formation of dead zones near the vessel walls.

Worked Example: Sizing a High-Shear Mixer for Emulsification

A process engineer is sizing a batch high-shear rotor-stator mixer to produce an oil-in-water emulsion (e.g., a sauce) with target droplet size below 20 µm. The following parameters are based on a lab-scale setup.

Known Input Parameters:

Rotor diameter, \( D_{rotor} = 50.0 \, \text{mm} \)
Rotor-stator gap, \( h_{gap} = 0.5 \, \text{mm} \)
Rotor speed, \( N = 6000 \, \text{RPM} \)
Density of continuous phase, \( \rho = 1000.0 \, \text{kg/m}^3 \)
Viscosity of continuous phase, \( \mu = 10.0 \, \text{cP} \)
Interfacial tension, \( \sigma = 0.01 \, \text{N/m} \)
Initial droplet size, \( d_{initial} = 500.0 \, \mu\text{m} \)
Power number, \( N_p = 1.0 \) (assumed)
Pumping constant, \( K = 0.75 \) (dimensionless)
Batch volume, \( V_{batch} = 50.0 \, \text{L} \)

Step-by-Step Calculation:

Convert rotational speed to revolutions per second:
\( N = \frac{6000 \, \text{RPM}}{60} = 100.0 \, \text{rev/s} \)
Calculate rotor tip speed:
\( V_{tip} = \pi \cdot D_{rotor} \cdot N \)
With \( D_{rotor} = 0.05 \, \text{m} \),
\( V_{tip} = \pi \cdot 0.05 \, \text{m} \cdot 100.0 \, \text{s}^{-1} = 15.708 \, \text{m/s} \)
Check: \( 10 \, \text{m/s} \leq 15.708 \, \text{m/s} \leq 50 \, \text{m/s} \) — within emulsification range.
Calculate nominal shear rate in the gap:
\( \dot{\gamma} = \frac{V_{tip}}{h_{gap}} \)
With \( h_{gap} = 0.0005 \, \text{m} \),
\( \dot{\gamma} = \frac{15.708 \, \text{m/s}}{0.0005 \, \text{m}} = 3.142 \times 10^{4} \, \text{s}^{-1} \)
Determine flow regime using Reynolds number:
\( Re = \frac{\rho \cdot N \cdot D_{rotor}^2}{\mu} \)
With \( \mu = 0.01 \, \text{Pa} \cdot \text{s} \),
\( Re = \frac{1000.0 \, \text{kg/m}^3 \cdot 100.0 \, \text{s}^{-1} \cdot (0.05 \, \text{m})^2}{0.01 \, \text{Pa} \cdot \text{s}} = 2.500 \times 10^{4} \)
Since \( Re \geq 10^{4} \), flow is highly turbulent, validating the use of a constant \( N_p \).
Estimate power draw:
\( P = N_p \cdot \rho \cdot N^3 \cdot D_{rotor}^5 \)
\( P = 1.0 \cdot 1000.0 \, \text{kg/m}^3 \cdot (100.0 \, \text{s}^{-1})^3 \cdot (0.05 \, \text{m})^5 = 312.5 \, \text{W} \)
Assess droplet breakup feasibility with Weber number:
\( We = \frac{\rho \cdot V_{tip}^2 \cdot d_{initial}}{\sigma} \)
With \( d_{initial} = 0.0005 \, \text{m} \),
\( We = \frac{1000.0 \, \text{kg/m}^3 \cdot (15.708 \, \text{m/s})^2 \cdot 0.0005 \, \text{m}}{0.01 \, \text{N/m}} = 1.234 \times 10^{4} \)
Since \( We \gg 1 \), droplet breakup is inertia-dominated.
Estimate batch turnover time for processing:
Volumetric flow rate: \( Q = K \cdot N \cdot D_{rotor}^3 \)
\( Q = 0.75 \cdot 100.0 \, \text{s}^{-1} \cdot (0.05 \, \text{m})^3 = 9.375 \times 10^{-3} \, \text{m}^3/\text{s} \)
With \( V_{batch} = 0.05 \, \text{m}^3 \),
Turnover time: \( t_{turnover} = \frac{V_{batch}}{Q} = \frac{0.05 \, \text{m}^3}{9.375 \times 10^{-3} \, \text{m}^3/\text{s}} = 5.333 \, \text{s} \)
For effective emulsification, process for 10-20 turnovers (≈1-2 minutes).

Final Answer:

The high-shear mixer operates with a tip speed of \( 15.708 \, \text{m/s} \), a nominal shear rate of \( 3.142 \times 10^{4} \, \text{s}^{-1} \), and a turbulent flow regime (\( Re = 2.500 \times 10^{4} \)). Power draw is \( 312.5 \, \text{W} \), and the Weber number (\( 1.234 \times 10^{4} \)) confirms sufficient energy for droplet breakup. With a volumetric flow rate of \( 9.375 \times 10^{-3} \, \text{m}^3/\text{s} \), the batch turnover time is \( 5.333 \, \text{s} \), indicating a total processing time of about 1-2 minutes for complete emulsification.

"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