Hydrodynamic Cavitation in Pharmaceutical Particle Engineering, Crystallization, and Drug Delivery

Dr. Oleg V. Kozyuk

okozyuk.com

Published October 1, 2026

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Nomenclature

Symbol	Definition	Typical unit
A	Accessible particle surface area	m²
B(L, t)	Particle birth term in the population balance	number m⁻¹ s⁻¹
C	Bulk or actual solute concentration	kg m⁻³ or mol m⁻³
$C_{e q}$	Equilibrium solubility concentration	kg m⁻³ or mol m⁻³
Cₛ	Concentration at the particle surface	kg m⁻³ or mol m⁻³
Cₛ(r)	Saturation concentration for a particle of radius r	kg m⁻³ or mol m⁻³
Cₛ,∞	Saturation concentration at a planar interface	kg m⁻³ or mol m⁻³
d	Particle or droplet diameter used in the Weber number	m
dₕ	Hydrodynamic diameter	m
dₚ	Diameter of a spherical particle	m
D	Translational diffusion coefficient	m² s⁻¹
D10, D50, D90	Percentile diameters of the selected particle size distribution	m, µm, or nm
D(L, t)	Particle disappearance term in the population balance	number m⁻¹ s⁻¹
DL	Drug loading	%
eₚ	Ideal specific pressure energy	J kg⁻¹
EE	Encapsulation efficiency	%
E(t)	Exit age or residence time distribution density	s⁻¹
G(L, t)	Particle growth rate	m s⁻¹
J	Nucleation rate	m⁻³ s⁻¹
J₀	Nucleation preexponential factor	m⁻³ s⁻¹
k	Overall dissolution mass transfer coefficient	m s⁻¹
k₁	Apparent first order release rate constant	s⁻¹
$k_{B}$	Boltzmann constant	J K⁻¹
$k_{H}$	Higuchi release constant	mass time⁻¹ᐟ²
L	Characteristic particle length or size coordinate	m
m	Particle or formulation mass	kg
$m_{e n c a p s u l a t e d}$	Mass of drug retained in the carrier fraction	kg or mass unit
$m_{l i p i d}$	Lipid mass	kg or mass unit
$m_{t o t a l}$	Total drug mass in the formulation	kg or mass unit
M	Dissolved drug mass	kg or mass unit
$M_{t}$	Cumulative released mass at time t	kg or mass unit
M∞	Asymptotic released mass	kg or mass unit
n(L, t)	Particle number density distribution	number m⁻¹
P	Power dissipated in the selected volume	W
p₁	Upstream pressure	Pa
p₂	Downstream or recovery pressure	Pa
$p_{B}$	Pressure inside a bubble	Pa
p∞	Far field liquid pressure	Pa
pᵥ	Liquid vapor pressure	Pa
Q	Volumetric flow rate	m³ s⁻¹
r	Particle radius	m
R	Gas constant or bubble radius, as defined locally	J mol⁻¹ K⁻¹ or m
R₀	Initial bubble or cavity radius	m
dR/dt	Bubble wall velocity	m s⁻¹
d²R/dt²	Bubble wall acceleration	m s⁻²
Re	Reynolds number	dimensionless
S	Supersaturation ratio	dimensionless
span	Particle size distribution span	dimensionless
t	Time	s
$t_{c}$	Ideal bubble collapse time	s
T	Absolute temperature	K
u	Characteristic fluid velocity	m s⁻¹
$u_{e}$	Electrophoretic mobility	m² V⁻¹ s⁻¹
V	Process or active volume	m³
Vₘ	Molar volume	m³ mol⁻¹
We	Weber number	dimensionless

Greek Letters

Symbol	Definition	Typical unit
γ	Interfacial energy or surface tension, as defined locally	J m⁻² or N m⁻¹
ΔG*	Critical nucleation free energy barrier	J
ΔGᵥ	Volumetric free energy change for solid formation	J m⁻³
Δp	Pressure drop	Pa
ε	Energy dissipation rate	W kg⁻¹
ε₀	Vacuum permittivity	F m⁻¹
$ε_{r}$	Relative permittivity	dimensionless
$η_{e}$	Dynamic viscosity used in the electrophoretic calculation	Pa s
$η_{K}$	Kolmogorov length scale	m
μ	Dynamic viscosity	Pa s
ν	Kinematic viscosity	m² s⁻¹
ρ	Liquid density	kg m⁻³
ρₚ	Particle density	kg m⁻³
σ	Cavitation number	dimensionless
τ	Nominal residence time	s
ζ	Zeta potential	V or mV