Reference ID: MET-A818 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The shear‑rate calculation for a colloid mill (rotor‑stator homogeniser) provides the
average velocity gradient that a fluid experiences as it passes through the narrow gap
between the rotating rotor and the stationary stator. This gradient is a key design
parameter in process engineering because it directly influences droplet breakup,
particle size reduction, and the overall efficiency of emulsification or dispersion
operations. Typical applications include fruit‑puree homogenisation, dairy product
processing, and fine chemical suspensions where precise control of the shear environment
is required to achieve target product quality while avoiding excessive heat or oxidation.
Methodology & Formulas
The calculation proceeds by first converting all user‑provided quantities into SI units,
then applying the fundamental kinematic and fluid‑dynamic relationships that govern the
rotor‑stator geometry.
Geometric conversion from millimetres to metres:
\[
D = \frac{D_{\text{mm}}}{1000}, \qquad
h = \frac{h_{\text{mm}}}{1000}
\]
Rotational speed conversion from revolutions per minute to revolutions per second:
\[
N = \frac{N_{\text{rpm}}}{60}
\]
Dynamic viscosity conversion from centipoise to pascal‑seconds:
\[
\mu = \mu_{\text{cP}} \times 10^{-3}
\]
Peripheral (tangential) velocity of the rotor surface:
\[
v = \pi D N
\]
Average shear rate in the gap (assuming a linear velocity profile across the gap):
\[
\gamma = \frac{v}{h}
\]
Gap Reynolds number for a laminar‑flow check:
\[
\text{Re}_{\text{gap}} = \frac{\rho \, v \, h}{\mu}
\]
The resulting dimensionless Reynolds number is compared against the conventional laminar‑flow
threshold, and the calculated shear rate is evaluated against typical process windows for
fruit‑puree emulsification.
Criterion
Expression / Limit
Interpretation
Laminar‑flow limit
\(\text{Re}_{\text{gap}} \;<\; 2000\)
Ensures flow remains laminar; values above may indicate transition to turbulence.
Minimum effective shear rate
\(\gamma \;\ge\; 1 \times 10^{4}\ \text{s}^{-1}\)
Below this level the homogeniser may not achieve fine droplet breakup.
Maximum advisable shear rate
\(\gamma \;\le\; 1 \times 10^{5}\ \text{s}^{-1}\)
Excessive shear can generate heat and promote oxidation of sensitive ingredients.
The shear rate (\( \dot{\gamma} \)) in a colloid mill is derived from the rotor-stator geometry and the rotor speed. Use the following steps:
Identify the rotor radius (R) and the clearance gap (g) between rotor and stator (both in meters).
Measure or set the rotor rotational speed (N) in revolutions per minute (rpm).
Convert the speed to radians per second: \( \omega = \frac{2\pi N}{60} \).
Apply the shear-rate equation for a concentric-gap mill: \( \dot{\gamma} = \frac{\omega \cdot R}{g} \).
Insert the values and compute \( \dot{\gamma} \) (s-1). This gives the average shear rate experienced by the material in the gap.
Rotor radius (R) – a larger radius raises the peripheral velocity.
Gap clearance (g) – a smaller gap concentrates the velocity gradient, raising \( \dot{\gamma} \).
Viscosity of the feed – while it does not change \( \dot{\gamma} \) directly, it influences power consumption and temperature rise.
Temperature – affects viscosity and may require speed adjustments to maintain target shear.
Follow this procedure:
Determine the desired shear rate (\( \dot{\gamma}_{\text{target}} \)) based on product specifications.
Gather the mill geometry: rotor radius (R) and gap (g).
Re-arrange the shear-rate equation to solve for \( \omega \): \( \omega = \frac{\dot{\gamma}_{\text{target}} \cdot g}{R} \).
Convert \( \omega \) back to rpm: \( N = \frac{\omega \cdot 60}{2\pi} \).
Check the calculated speed against the mill’s rated speed range and motor capacity.
Adjust slightly for safety margins or to accommodate viscosity variations.
Validation can be performed by:
Measuring the motor power draw and using the relationship between power, torque, and shear to back-calculate \( \dot{\gamma} \).
Installing a calibrated torque sensor on the drive shaft and applying the shear-rate formula.
Using a high-speed camera or laser-Doppler velocimetry to directly observe fluid velocity in the gap.
Conducting a rheological test on the processed sample (e.g., particle size reduction) and correlating the results with predicted shear rates.
Worked Example – Colloid Mill Shear Rate
A beverage plant uses a toothed-colloid mill to reduce the droplet size of a flavour emulsion.
The process engineer needs to verify that the mill delivers a nominal shear rate above 90,000 s-1
to guarantee the target Sauter-mean diameter. The following data were recorded during a 12,000 rpm run.
Knowns
Rotor diameter, D = 30 mm
Gap width, h = 0.2 mm
Rotational speed, N = 12,000 rpm
Continuous-phase viscosity, μ = 5 cP (25 °C)
Density, ρ = 1,050 kg m-3
Step-by-step calculation
Convert diameter and gap to metres:
\[ D = 0.03\ \text{m},\quad h = 0.0002\ \text{m} \]
Convert speed to revolutions per second:
\[ N = \frac{12,000}{60} = 200\ \text{rps} \]
Compute rotor tip speed:
\[ v = \pi D N = \pi (0.03)(200) = 18.850\ \text{m s}^{-1} \]
Estimate nominal shear rate in the gap (assuming Couette flow):
\[ \dot{\gamma} = \frac{v}{h} = \frac{18.850}{0.0002} = 94,250\ \text{s}^{-1} \]
(Optional) Check gap Reynolds number to confirm laminar shear:
\[ \text{Re}_{\text{gap}} = \frac{\rho v h}{\mu} = \frac{(1,050)(18.850)(0.0002)}{0.005} = 792 \]
Since \( \text{Re}_{\text{gap}} \ll 1,400 \), laminar Couette flow dominates.
Final Answer
The colloid mill operates at a nominal shear rate of 94,250 s-1,
comfortably exceeding the 90,000 s-1 target for the beverage emulsion.
"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
