Engineering Calculations for Rotor-Stator Mixers

Dr. Oleg V. Kozyuk

okozyuk.com

Published October 25, 2026

Updated October 28, 2026

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15 Nomenclature

Table 15.1. Latin symbols

Symbol	Definition	SI unit
$A$	Reference flow area	m²
$A_{o}$	Geometric simultaneous opening area, including stated leakage paths	m²
$A_{s}$	Total geometric stator slot area	m²
$b_{r}$ , $b_{s}$	Rotor and stator slot width	m
$c$	Wave speed in liquid or compliant fluid system	m/s
$c_{p}$	Specific heat capacity	J/(kg K)
$C_{d}$	Discharge coefficient	1
$C_{p}$	Positive pressure depression coefficient used in the minimum-pressure screen	1
$D$	Active rotor diameter	m
$e_{a}$	Total acoustic wave energy per unit mass	J/kg
$e_{p, h}$	Hydraulic specific work of a pressure pulse	J/kg
$E_{b}$	Characteristic cavity collapse work	J
$E_{V}$	Volumetric specific energy	J/m³
$f_{A}$	Integrated overlap symmetry repeat frequency	Hz
$f_{p}$	Selected pressure or feature passage frequency	Hz
$f_{r}$	Local rotor feature passage frequency	Hz
$h$	Slot height or small channel dimension	m
$I_{a}$	Cycle averaged acoustic intensity for a progressive plane wave	W/m²
$I_{h}$	Hydraulic inertance	kg/m⁴
$K$	Local loss coefficient	1
$L$ , $L_{g}$	Characteristic length or active gap length	m
$l_{e}$	Effective channel length	m
$m$	Batch mass	kg
$m_{A}$	Area modulation index	1
$N$	Rotational frequency	$s^{- 1}$
$N_{p}$	Nominal number of pulse events	1
$n$	Rotor speed	rpm
$p$	Pressure; pressure difference in the hydraulic covariance relation	Pa
$p_{a}$	Peak pressure amplitude; positive negative-pulse magnitude in cavitation screening	Pa
$p_{v}$	Vapor pressure	Pa
$P_{h}$	Hydraulic power	W
$P_{s}$	Shaft power	W
$P o$	Power number	1
$p_{r}$ , $p_{s}$	Rotor and stator pitch	m
$q$	Instantaneous flow through one modeled passage	m³/s
$Q$	Mean volumetric flow	m³/s
$R$	Bubble radius or generic radius, as defined	m
$R_{h}$	Linear hydraulic resistance	Pa s/m³
$R_{r}$ , $R_{s}$	Rotor and stator radii	m
$R e$	Reynolds number	1
$S t$	Strouhal number	1
$T$	Torque or temperature, as defined by context	N m or K
$t$	Time	s
$t_{a}$	Nominal active zone residence time	s
$t_{c}$	Rayleigh collapse time	s
$t_{o}$	Nominal opening duration	s
$U$	Reference velocity	m/s
$U_{o}$	Discharge equivalent velocity based on coefficient times opening area	m/s
$U_{t}$	Rotor tip speed	m/s
$u_{r}$ , $u_{θ}$	Radial and tangential velocity components	m/s
$V_{a}$	Active internal volume	m³
$V_{T}$	Tank or loop inventory volume	m³
$z_{r}$ , $z_{s}$	Number of rotor and stator features	1
$A_{o, m i n}, A_{o}$	Minimum and cycle averaged simultaneous opening area	m²
$D_{r}, D_{s}$	Active rotor and stator diameters	m
$b_{e}$	Effective circumferential width defining an exposure event	m
$t_{r}$	Circumferential rotor tooth or bridge width	m
$h_{s}$	Stator slot active height	m
$H_{g}$	Active height of the radial throughflow region	m
$w$	Width of the parallel plate channel	m
$g (\cdot)$	Zero mean periodic overlap shape function	1
$𝐠$	Gravitational acceleration vector	m/s²
$𝐮$	Fluid velocity vector	m/s
$f_{s}$	Stator passage frequency observed in the rotating frame	Hz
$Q_{g}$	Mean radial flow crossing the gap on the declared area basis	m³/s
$T_{g}$	Laminar cylindrical gap torque magnitude	N m
$P_{g}$	Laminar cylindrical gap power	W
$E u$	Euler number on the stated pressure and velocity basis	1
$E_{m}$	Shaft energy input per processed mass	J/kg
$E_{V, b}$	Gross volumetric energy input to a recirculating batch	J/m³
$E_{v, a}$	Cycle averaged total acoustic wave energy density	J/m³
$p_{r e f}, p_{min}$	Absolute reference and minimum local pressures	Pa
$p_{u}, p_{d}$	Pressures upstream and downstream of the modeled restriction	Pa
$Δ p_{d} (t)$	Net time dependent driving pressure in the lumped passage balance	Pa
$Δ p_{p}$	Pressure difference used to define hydraulic specific pulse work	Pa
$p_{B}, p_{\infty}$	Bubble internal pressure and imposed far field liquid pressure	Pa
${p^{'}}_{r m s}$	Root mean square pressure fluctuation	Pa
$R_{max}$	Maximum bubble radius at the beginning of ideal collapse	m
$\dot{m}$	Mass flow rate through the inline unit	kg/s
$q_{a}$	Peak amplitude of sinusoidal flow fluctuation	m³/s
$P_{p u m p}$	Power exported as hydraulic transport	W
$P_{g a p, r}$	Power assigned to the radial gap	W
$P_{g a p, a}$	Power assigned to the axial gap	W
$P_{s l o t}$	Power assigned to passage losses	W
$P_{t u r b}$	Residual turbulent dissipation not counted elsewhere	W
$P_{c a v}$	Residual cavitation related dissipation not counted elsewhere	W
$P_{m e c h}$	Mechanical loss power within the declared boundary	W

Table 15.2. Greek symbols

Symbol	Definition	SI unit
$α$	Womersley number	1
$δ$	Radial rotor stator gap	m
$δ_{ν}$	Oscillatory viscous penetration depth	m
$Δ p$	Pressure difference	Pa
${\dot{γ}}_{g}$	Nominal gap shear rate	$s^{- 1}$
$γ$	Surface tension	N/m
$η_{h}$	Hydraulic efficiency	1
$η_{t h}$	Fraction of shaft power retained as sensible heat	1
$μ$	Dynamic viscosity	Pa s
$ν$	Kinematic viscosity	m²/s
$ϕ_{A}$	Phase of area waveform	rad
$ϕ_{o}$	Effective open fraction	1
$ϕ_{Q}$	Flow coefficient	1
$ρ$	Density	kg/m³
$σ$	Cavitation number	1
$τ_{g}$	Nominal gap shear stress	Pa
$ω$	Angular speed	rad/s
$ω_{p}$	Angular pressure pulse frequency	rad/s
$ψ$	Head or pressure coefficient	1
$φ$	Phase difference between pressure and flow fluctuations	rad
$ϕ_{s}$	Geometric circumferential stator open fraction	1