Cavitation-Assisted Particle Dispersion

Dr. Oleg V. Kozyuk

okozyuk.com

Published October 1, 2026

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Nomenclature

Symbol	Definition	SI unit or status
$a$	Particle radius	m
$a_{s}$	External particle area per unit mass	m² kg⁻¹
$A_{H}$	Hamaker constant	J
$b (d \| d^{'})$	Daughter size distribution from parent size $d^{'}$	Model dependent
$B r$	Breakage number	Dimensionless
$c_{l}$ , $c_{s}$	Sound speed in liquid and solid	m s⁻¹
$C_{d}$	Discharge coefficient	Dimensionless
$C_{R}$	Structural coefficient in agglomerate strength relation	Dimensionless
$c_{p}$	Specific heat capacity	J kg⁻¹ K⁻¹
$d_{a}$	Agglomerate characteristic size	m
$d_{b}$	Load bearing building block size	m
$d_{h}$	Hydrodynamic particle diameter	m
$d_{p}$	Particle diameter	m
$D_{10}$	Diameter below which 10 percent of the stated distribution occurs	m
$D_{43}$	Volume weighted mean diameter	m
$D_{50}$	Median diameter	m
$D_{90}$	Diameter below which 90 percent of the stated distribution occurs	m
$D_{B}$	Brownian diffusion coefficient	m² s⁻¹
$e_{1}$	Specific hydraulic energy for one pass	J kg⁻¹
$e_{c u m}$	Cumulative specific energy input	J kg⁻¹
$e$	Elementary charge	C
$F_{A}$	van der Waals attractive force	N
$F_{b}$	Characteristic interparticle bond force	N
$F_{c a p}$	Capillary force	N
$g$	Gravitational acceleration	m s⁻²
$h$	Surface separation distance	m
$I$	Ionic strength	mol m⁻³ or stated concentration basis
$k$	Polytropic exponent or pass index, as defined	Dimensionless
$k_{B}$	Boltzmann constant	J K⁻¹
$L$	Characteristic length or settling height	m
$m_{s}$	Mass of solid	kg
$m_{d i s p}$	Estimated dispersant mass	kg
$n (d, t)$	Number density distribution	Model dependent
$N_{A}$	Avogadro constant	mol⁻¹
$N_{p}$	Nominal number of passes	Dimensionless
$p_{B}$	Pressure inside bubble	Pa
$p_{g}$	Noncondensable gas pressure	Pa
$p_{l s}$	Liquid solid impact pressure	Pa
$p_{r e f}$	Reference pressure for cavitation number	Pa
$p_{v}$	Saturation vapor pressure	Pa
$p_{\infty}$	Far field or recovered pressure	Pa
$P_{h}$	Hydraulic power	W
$P_{m o t o r}$	Motor power transferred to process	W
$P e_{s}$	Sedimentation Peclet number	Dimensionless
$Q$	Volumetric flow rate	m³ s⁻¹
${\dot{Q}}_{H X}$	Heat exchanger duty	W
$R$	Instantaneous bubble radius	m
$R_{0}$	Initial or maximum bubble radius	m
$R_{D}$	Percent reduction in selected diameter	%
$S (d)$	Breakage frequency	s⁻¹
$S_{P S D}$	Particle size distribution span	Dimensionless
$S t$	Stokes number	Dimensionless
$t$	Time	s
$t_{R}$	Rayleigh collapse time	s
$T$	Absolute temperature	K
$u_{j}$	Liquid microjet velocity	m s⁻¹
$U$	Characteristic mean flow velocity	m s⁻¹
$V_{b}$	Batch volume	m³
$V_{A}$ , $V_{R}$ , $V_{S}$ , $V_{T}$	Interaction potentials	J
$v_{s}$	Settling velocity	m s⁻¹
AFM	Atomic force microscopy	Abbreviation
CFC	Controlled flow cavitation	Abbreviation
CFD	Computational fluid dynamics	Abbreviation
CMP	Chemical mechanical planarization	Abbreviation
DLS	Dynamic light scattering	Abbreviation
ICP	Inductively coupled plasma analysis	Abbreviation
PSD	Particle size distribution	Abbreviation
SEM	Scanning electron microscopy	Abbreviation
TEM	Transmission electron microscopy	Abbreviation
$F_{d}$	Deagglomeration factor, initial selected diameter divided by final selected diameter	Dimensionless
f(d)	Relative probability density of plant particle diameter in Equation (8.2)	mm⁻¹
$d_{m}$	Fitted mean plant particle diameter in Equations (8.2) and (8.3)	mm
$d_{x, 0}$ ; $d_{x, f}$ ; $d_{x, \infty}$	Initial, final, and asymptotic value of selected particle size metric	m or stated unit
$k_{x}$	Empirical pass response constant in Equation (8.5)	Pass⁻¹
Re	Reynolds number	Dimensionless
IPA	Isopropyl alcohol	Abbreviation

Greek Letters

Symbol	Letter name	Meaning	SI unit or status
$α_{c}$	Alpha	Empirical coefficient relating jet momentum to effective collapse stress	Dimensionless
$Γ_{A}$	Gamma	Adsorbed dispersant mass per unit area	kg m⁻²
$γ$	Gamma	Liquid surface tension	N m⁻¹
$Δ p$	Delta p	Pressure drop across cavitation device	Pa
$Δ T_{a d}$	Delta T	Adiabatic temperature rise	K
${\dot{γ}}_{K}$	Gamma dot	Kolmogorov strain rate	s⁻¹
$ε_{0}$	Epsilon zero	Vacuum permittivity	F m⁻¹
$ε_{r}$	Epsilon r	Relative permittivity	Dimensionless
$ε_{a}$	Epsilon a	Agglomerate porosity	Dimensionless
$ε_{t}$	Epsilon t	Turbulent energy dissipation rate per unit mass	m² s⁻³
$η_{h}$	Eta h	Overall hydraulic efficiency	Dimensionless
$η_{K}$	Eta K	Kolmogorov microscale	m
$θ$	Theta	Contact angle	rad or degree
$κ$	Kappa	Inverse Debye screening length	m⁻¹
$κ^{- 1}$	Kappa inverse	Debye screening length	m
$μ$	Mu	Dynamic viscosity	Pa s
$ν$	Nu	Kinematic viscosity	m² s⁻¹
$ρ_{l}$ , $ρ_{p}$ , $ρ_{s}$	Rho	Liquid, particle, and solid density	kg m⁻³
$σ_{c}$	Sigma	Cavitation number	Dimensionless
$σ_{a g g}$	Sigma	Agglomerate tensile strength	Pa
$τ_{c}$	Tau	Effective collapse induced stress	Pa
$τ_{f}$	Tau	Characteristic flow time	s
$τ_{K}$	Tau	Kolmogorov time scale	s
$τ_{p}$	Tau	Particle relaxation time	s
$τ_{e x t}$	Tau	Applied external stress	Pa
$τ_{i n}$	Tau	Inertial turbulent stress	Pa
$τ_{v i s}$	Tau	Viscous turbulent stress	Pa
$ψ_{0}$	Psi	Surface potential	V
λ	Lambda	Cavitation stage used in the plant raw material study	Dimensionless
$σ_{d}$	Sigma d	Standard deviation of plant particle diameter distribution	mm
τ	Tau	Cavitation processing time in the plant raw material model	min