Design and Operating Principles of Rotor-Stator Mixers

16. Nomenclature

Symbols follow the definitions used in the article's equations. Whether a value is local, averaged over the active zone, or averaged over the complete vessel must be stated explicitly.

Table 16.1. Latin symbols

Symbol	Meaning	SI unit
a	Interfacial area per unit mixture volume	m²·m⁻³
$A_k$	Amplitude of the kth pressure harmonic	Pa
c	Speed of sound in the liquid	m·s⁻¹
$c_p$	Specific heat capacity	J·kg⁻¹·K⁻¹
$C_H$	Empirical coefficient in maximum stable droplet diameter correlation	dimensionless
$C_i$, $C_v$, $C_{\mu }$	Coefficients in inertial stress, viscous stress, and micromixing models	dimensionless
Ca	Capillary number	dimensionless
D	Outside rotor diameter	m
d	Diameter of a droplet, particle, or agglomerate	m
$d_{32}$	Sauter mean diameter	m
$E_m$	Accumulated energy per unit mass	J·kg⁻¹
$E_{V,B}$	Nominal energy per unit volume for batch treatment	J·m⁻³
$E_{V,FT}$	Energy per unit volume for inline treatment	J·m⁻³
f	Frequency	Hz
$f_p$	Rotor element passage frequency	Hz
g	Rotor-stator clearance or effective gap	m
I	Sound intensity	W·m⁻²
L	Characteristic length	m
m	Mass of processed product	kg
n	Rotational speed in revolutions per minute	rpm
$n_i$	Number of droplets or particles in size class i	count
$n_p$	Mean equivalent number of passes	dimensionless
N	Rotational frequency	s⁻¹
Oh	Ohnesorge number	dimensionless
p	Pressure	Pa
$p_B$	Pressure inside a bubble	Pa
$p_{ref}$	Reference absolute pressure	Pa
$p_{rms}$	Root-mean-square acoustic pressure	Pa
$p_v$	Vapor pressure	Pa
Po	Power number	dimensionless
$P_{diss}$	Power dissipated in the liquid	W
$P_{loc}$	Power dissipated in the active region	W
$P_s$	Shaft power	W
Q	Volumetric flow rate	m³/s
R	Bubble radius	m
$R_0$	Initial bubble radius	m
$R{e}_g$	Gap Reynolds number	dimensionless
$R{e}_R$	Rotor Reynolds number	dimensionless
Sc	Schmidt number	dimensionless
St	Strouhal number	dimensionless
t	Time	s
$t_a$	Nominal residence time in the active zone	s
$t_c$	Rayleigh collapse time	s
$t_{\mu }$	Micromixing time	s
T	Torque	N m
U	Characteristic velocity	m·s⁻¹
$U_{rel}$	Relative velocity in the rotor-stator gap	m·s⁻¹
$U_t$	Rotor tip speed	m·s⁻¹
$u_{\eta }$	Kolmogorov velocity scale	m·s⁻¹
V	Averaging volume	m³
$V_a$	Volume of the active zone	m³
$V_T$	Tank or batch volume	m³
$W{e}_d$	Droplet Weber number	dimensionless
z	Number of repeating rotor elements	count
k; K	Harmonic index; number of harmonics in the signal representation	dimensionless
$p_{\infty }$	Liquid pressure far from a bubble	Pa
$p_1$; $p_2$	Static pressures at two reference sections	Pa
$U_1$; $U_2$	Mean velocities at two reference sections	m·s⁻¹
$U_{ref}$	Reference velocity for cavitation number	m·s⁻¹
$u_x$; x	Axial velocity component; axial coordinate	m·s⁻¹; m

Table 16.2. Greek symbols

Symbol	Meaning	SI unit
γ̇	Shear rate	s⁻¹
${\dot{\gamma }}_{nom}$	Nominal gap shear rate	s⁻¹
Δp	Pressure rise or difference	Pa
$\mathrm{\Delta }{p}_L$	Irreversible pressure loss term	Pa
ΔT	Temperature increase	K
ε	Local energy dissipation rate per unit mass	W·kg⁻¹
ε̄	Mean energy dissipation rate per unit mass	W·kg⁻¹
${\dot{\epsilon }}_x$	Extensional strain rate	s⁻¹
${\epsilon }_{V,loc}$	Local energy dissipation rate per unit volume	W·m⁻³
η	Kolmogorov length scale	m
${\eta }_B$	Batchelor scalar diffusion scale	m
${\eta }_h$	Hydraulic efficiency	dimensionless
μ	Dynamic viscosity of the continuous phase	Pa·s
${\mu }_d$	Dynamic viscosity of the dispersed phase	Pa·s
ν	Kinematic viscosity	m²·s⁻¹
ρ	Density of the continuous phase	kg·m⁻³
${\rho }_d$	Density of the dispersed phase	kg·m⁻³
σ	Surface tension in bubble dynamics	N·m⁻¹
${\sigma }_c$	Cavitation number	dimensionless
${\sigma }_i$	Interfacial tension	N·m⁻¹
τ	Shear stress	Pa
${\tau }_i$	Turbulent inertial stress	Pa
${\tau }_v$	Viscous microscale stress	Pa
${\tau }_{\eta }$	Kolmogorov time scale	s
${\phi }_d$	Dispersed-phase volume fraction	dimensionless
${\phi }_k$	Phase angle of the kth harmonic	rad
Φ	Flow coefficient	dimensionless
Ψ	Pressure coefficient	dimensionless
ω	Angular velocity	rad·s⁻¹