Engineering Calculations for Rotor-Stator Mixers

13 Scale up and design limits

For geometrically similar equipment operating in a comparable flow regime at constant flow, pressure and power coefficients, the standard similarity relations are

These relations require similarity of geometry and operating regime. In practice, clearances, tooth thickness, motor speed and manufacturing tolerance frequently prevent exact geometric scaling. Constant tip speed requires $N\propto D^{-1}$, which changes feature passage frequency if feature count is fixed. Preserving nominal gap shear additionally requires $U_t/\delta$ to remain constant. Matching cavitation number requires a compatible absolute pressure and vapor pressure, while matching specific energy requires coordinated changes in power and throughput. These requirements impose distinct constraints and should be selected according to the process mechanism.

Figure 13.1. Illustrative scaling at constant tip speed, flow coefficient and power number. Under these assumptions, throughput and power both vary with the square of rotor diameter.

Table 13.1. Scale up criteria and quantities preserved

Criterion held constant	Approximately preserves	Does not necessarily preserve
Tip speed	Rotor surface velocity	Pulse frequency, power density, cavitation number
Gap shear $U_t/\delta$	Nominal viscous deformation rate	Turbulence, residence time, opening jets
Power per volume	Mean power input per specified volume	Local peaks, pressure, event frequency
Specific energy $P/(\rho Q)$	Gross energy per processed mass	Stress distribution and thermal history
Cavitation number	Relative pressure margin	Nuclei population, gas content, pulse duration
Flow coefficient	Mean throughput similarity	Slot velocity if open area does not scale
Feature passage frequency	Number of periodic events per time	Tip speed and pumping capacity

13.1 Selection of scaling criteria

Scale selection should follow the required process endpoint. Hall's original doctoral research compared emulsification at three mixer scales and across thirteen rotor and stator configurations. The author reported that tip speed combined with constant residence time was the most suitable basis for correlating mean droplet size across the investigated laboratory and factory equipment, including repeated passes. Stator opening area also affected power and emulsification within a single scale. These findings provide an experimental basis for selecting a practical scaling strategy, while retaining its formulation and geometry limits. Matching this pair of variables does not automatically preserve pulse frequency, gap shear, hydraulic pressure, cavitation margin or temperature rise, which require separate assessment [27].

Table 13.2. Selected experimental and computational evidence relevant to engineering scale up

Study	Quantity investigated	Engineering implication and limit
James et al. [25]	Torque and power constants	Constants depend on the tested workhead and regime; verify corrected torque.
John et al. [15]	Batch and inline circulation	External throughput differs from internal exchange; match the flow definition.
John et al. [12]	Screen and inlet geometry	Controlled geometry changes support calibrated power and flow relations.
Hall et al. [23]	Emulsion droplet size	Formulation and local stresses limit transfer of specific energy correlations.
Håkansson et al. [24]	Local fragmenting stresses	Breakup location depends on the spatial and transient stress field.
Ayas et al. [11]	Viscoplastic liquid mixing	Constitutive behavior and local shear need explicit treatment.
Michael et al. [26]	CFD power and flow	A simplified preprint model retains substantial prediction uncertainty.
Hall [27]	Emulsification across scales	Tip speed with residence time was useful within the investigated systems.

Mechanical and service requirements constrain the hydraulic scaling. Centrifugal stress and shaft deflection can affect the maintained gap; thermal expansion can reduce clearance; and cavitation or abrasive wear can change opening edges and discharge behavior. Cleaning requirements may also impose minimum accessible passage dimensions. Scale up should therefore proceed through hydraulic characterization, representative fluid testing, verification of heat removal and wear, and confirmation of the product endpoint over the intended operating envelope.

13.2 Workhead selection and geometric documentation

The photographs in Figure 13.2 and Figure 13.3 illustrate the range of screen forms that can be fitted around a rotor. Their relevance to engineering calculation lies in the opening dimensions, edge thickness, accessible clearance and inlet arrangement. A screen designation should therefore be accompanied by its measured geometry when comparing power, circulation and product response [28].

Figure 13.2. Representative laboratory mixer and a close view of its working head. Photographs from Silverson technical literature [28].

Figure 13.3. Examples of commercial stator screens with general purpose, square hole, slotted and fine perforated configurations. Photographs and screen designations from Silverson technical literature [28].