Emulsification and Homogenization in Rotor-Stator Mixers

Dr. Oleg V. Kozyuk

okozyuk.com

Published November 2, 2026

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19 Nomenclature

Table 19.1. Latin symbols and abbreviations.

Symbol	Meaning	SI unit or status
$A$	Interfacial area	m²
$A_{0}$ , $A_{f}$	Initial and final interfacial area	m²
$A_{o}$	Total open area of stator passages	m²
$A_{s}$	Reference stator surface area	m²
$a$	Interfacial area per unit emulsion volume	m² m⁻³
$b (v \| v_{p})$	Daughter volume distribution from parent volume	m⁻³ or normalized distribution
$C_{b}$	Bulk emulsifier concentration	kg m⁻³ or mol m⁻³
$C_{H}$ , $C_{v}$	Empirical breakup constants	Dimensionless or correlation specific
$c_{p}$	Specific heat capacity	J kg⁻¹ K⁻¹
$d$	Droplet diameter	m
$d_{10}$	Number moment mean $d_{1,0}$ ; distinct from the tenth volume percentile	m
$d_{32}$	Sauter mean diameter	m
$d_{43}$	Volume weighted mean diameter	m
$d_{10}$ , $d_{50}$ , $d_{90}$	Cumulative volume percentile diameters	m
$d_{h}$	Hydraulic diameter of a passage	m
$D$	Rotor diameter	m
$D_{B}$	Brownian diffusivity	m² s⁻¹
$D a_{a d s}$	Adsorption Damköhler type ratio	Dimensionless
$E_{s}$	Signed interfacial coefficient; $E_{d i l}$ = − $E_{s}$	N m⁻¹
$E_{i j}$	Coalescence efficiency	Dimensionless
$e_{m}$	Specific energy per unit mass	J kg⁻¹
$e_{V}$	Energy per unit volume	J m⁻³
$f_{b}$	Blade or port passage frequency	Hz
$f_{o}$	Effective opening frequency	Hz
$f_{0}$	Fraction not yet processed in an ideal mixed loop	Dimensionless
$g$	Gravitational acceleration	m s⁻²
$g (v)$	Breakup frequency	s⁻¹
$K$	Adsorption constant	Concentration dependent
$K_{O R}$	Ostwald ripening rate constant	m³ s⁻¹
$k_{B}$	Boltzmann constant	J K⁻¹
$k_{N}$	Empirical pass constant	pass⁻¹
$m_{b}$	Batch mass	kg
$m_{e}$	Emulsifier mass	kg
$N$	Rotor rotational frequency or equivalent pass number, as defined	s⁻¹ or dimensionless
$N_{e}$	Mean head encounters	Dimensionless
$N_{p}$	Equivalent passes	Dimensionless
$n_{i}$	Number of droplets in class $i$	Count or number concentration
$n_{o}$	Effective opening events per revolution	Dimensionless
$P$	Power	W
$P_{h}$	Hydraulic power	W
$P_{n e t}$	Net process power	W
$P_{s}$	Shaft power	W
$P e_{g}$	Gravitational Péclet number	Dimensionless
$P o$	Power number	Dimensionless
$p_{v}$	Vapor pressure	Pa
$Q$	Volumetric flow rate	m³ s⁻¹
$Q_{h}$	Head circulation flow rate	m³ s⁻¹
$Q_{r}$	Recirculation loop flow rate	m³ s⁻¹
$R e_{R}$	Rotor Reynolds number	Dimensionless
$R e_{δ}$	Gap Reynolds number	Dimensionless
$R e_{p}$	Port Reynolds number	Dimensionless
$R_{i j}$	Collision rate between classes $i$ and $j$	m⁻³ s⁻¹
$T$	Torque or absolute temperature, as defined	N m or K
$t$	Time	s
$t_{s}$	Interface age	s
$u_{p}$	Port velocity	m s⁻¹
$u_{s}$	Stokes migration velocity	m s⁻¹
$u_{t}$	Rotor tip speed	m s⁻¹
$V$	Volume	m³
$V_{a}$	Active processing volume	m³
$V_{b}$	Batch volume	m³
$V_{c}$ , $V_{d}$	Continuous and dispersed phase volumes	m³
$W e$	Weber number	Dimensionless
O/W	Oil in water emulsion	Abbreviation
W/O	Water in oil emulsion	Abbreviation
W/O/W	Water in oil in water multiple emulsion	Abbreviation
DSD	Droplet size distribution	Abbreviation
CMC	Critical micelle concentration	Abbreviation
CFD	Computational fluid dynamics	Abbreviation
CIP	Clean in place	Abbreviation

Table 19.2. Greek letters and related notation.

Symbol	Letter name	Meaning	SI unit or status
$α_{o}$	Alpha	Stator open area ratio	Dimensionless
$β_{i j}$	Beta	Collision kernel	m³ s⁻¹
$Γ$	Gamma	Surface excess	kg m⁻² or mol m⁻²
$Γ_{\infty}$	Gamma infinity	Maximum surface coverage	kg m⁻² or mol m⁻²
$δ$	Delta	Rotor stator radial clearance	m
$Δ p$	Delta p	Pressure difference	Pa
$Δ p_{L}$	Delta p sub L	Laplace pressure	Pa
$Δ T$	Delta T	Temperature rise	K
$ε$	Epsilon	Local energy dissipation rate per unit mass	W kg⁻¹
$η_{A}$	Eta	Apparent interfacial energy efficiency	Dimensionless
$η_{H}$	Eta	Homogenization efficiency	Dimensionless
$η_{K}$	Eta	Kolmogorov length	m
$Θ_{A}$	Theta	Available interfacial coverage ratio	Dimensionless
$λ$	Lambda	Dispersed to continuous phase viscosity ratio	Dimensionless
$μ_{c}$ , $μ_{d}$	Mu	Continuous and dispersed phase dynamic viscosity	Pa s
$ν_{c}$	Nu	Continuous phase kinematic viscosity	m² s⁻¹
$Π_{ρ}$	Pi	Density ratio	Dimensionless
$Π_{t}$	Pi	Stress duration to capillary time ratio	Dimensionless
$ρ_{c}$ , $ρ_{d}$	Rho	Continuous and dispersed phase density	kg m⁻³
$σ$	Sigma	Interfacial tension	N m⁻¹
$σ_{c}$	Sigma sub c	Cavitation number	Dimensionless
$τ$	Tau	Time scale or stress, as defined	s or Pa
$τ_{a d s}$	Tau adsorption	Adsorption time scale	s
$τ_{K}$	Tau K	Kolmogorov time scale	s
$ϕ_{d}$	Phi	Dispersed phase volume fraction	Dimensionless
$\dot{γ}$	Gamma dot	Shear or strain rate	s⁻¹