Sonochemical Effect Induced by Hydrodynamic Cavitation

Nomenclature

Symbols are listed in the form used in this article. Where a symbol can carry more than one conventional meaning, the applicable definition is stated immediately after the equation in which it appears.

Symbol	Definition	SI unit
A	Pre exponential factor in the Arrhenius equation	varies
$A_{\lambda }$	Optical absorbance at wavelength λ	dimensionless
$A_s$	Radiating area of an acoustic source	m²
b	Optical path length	m or cm
C	Concentration; context specific	mol m⁻³ or kg m⁻³
C*	Equilibrium solubility	mol m⁻³ or kg m⁻³
$C_A$	Concentration of target species A	mol m⁻³
$C_{{\mathrm{I}}_3^{-}}$	Triiodide concentration	mol m⁻³
Dc	Conceptual collapse exposure density	J m⁻³
E(t)	Residence time distribution function	s⁻¹
$E_a$	Activation energy	J mol⁻¹
$E_{\mathrm{collapse}}$	Mean energy associated with one active collapse	J
$E_{\mathrm{input}}$	Energy supplied within the stated system boundary	J
$E_m$	Mass specific energy dose	J kg⁻¹
$E_V$	Volumetric energy dose	J m⁻³
EEO	Electrical energy per order	kWh m⁻³ order⁻¹
Eu	Euler number	dimensionless
$f_p$	Fractional removal per pass	dimensionless
g	Gravitational acceleration	m s⁻²
HA	Henry law constant under the stated convention	varies
h	Effective hard core radius of compressed gas	m
Iac	Nominal acoustic intensity	W m⁻²
J	Nucleation rate	m⁻³ s⁻¹
J₀	Nucleation pre exponential factor	m⁻³ s⁻¹
$K_{\mathrm{bg}}$	Bubble to liquid distribution factor	dimensionless
k	Reaction rate constant	varies
$k_{\mathrm{app}}$	Apparent rate constant	s⁻¹ for first order form
$k_B$	Boltzmann constant	J K⁻¹
$k_{\mathrm{obs}}$	Observed pseudo first order rate constant	s⁻¹
m	Treated liquid or slurry mass	kg
M	Third body in gas phase reaction	-
n	Polytropic exponent or amount of substance, as defined locally	dimensionless or mol
$n_g$	Moles of noncondensable gas in a bubble	mol
$n_v$	Moles of vapor in a bubble	mol
$n_j$	Moles of chemical product j	mol
N	Number of passes or events, as defined	dimensionless
$N_{\mathrm{active}}$	Number of chemically active collapses	dimensionless
$N_{\mathrm{pass}}$	Mean number of loop passes	dimensionless
p	Static pressure	Pa
$p_b$	Total pressure inside a bubble	Pa
$p_d$	Selected downstream or reference pressure	Pa
$p_g$	Noncondensable gas partial pressure	Pa
$p_v$	Vapor pressure or vapor partial pressure	Pa
$p_{\infty }$	Far field liquid pressure	Pa
P	Power on the stated basis	W
$P_{\mathrm{ac}}$	Acoustic power delivered to the liquid	W
$P_{\mathrm{el}}$	Electrical power	W
$P_h$	Hydraulic power dissipated across the cavitating element	W
Q	Volumetric flow rate	m³ s⁻¹
${\dot{Q}}_g$	Heat transfer rate into the bubble gas	W
R	Instantaneous bubble radius	m
R₀	Reference or initial bubble radius	m
$R_{\max }$	Maximum bubble radius	m
$R_{\min }$	Minimum bubble radius	m
$R_g$	Universal gas constant	J mol⁻¹ K⁻¹
r	Reaction rate or radial coordinate, as defined locally	varies
$r_{\mathrm{core}}$	Effective gas core reaction rate	mol m⁻³ s⁻¹
$r_{\mathrm{interface}}$	Effective interfacial reaction rate	mol m⁻³ s⁻¹
$r_{\mathrm{bulk}}$	Effective bulk liquid reaction rate	mol m⁻³ s⁻¹
$S_c$	Supersaturation ratio	dimensionless
S	Surface tension when used in the bubble equation	N m⁻¹
$S_y$	Hybrid process synergy factor	dimensionless
t	Time or treatment time	s
$t_c$	Rayleigh collapse time	s
T	Temperature	K
T₀	Initial or reference temperature	K
$T_{\max }$	Peak idealized bubble temperature	K
u	Characteristic liquid velocity	m s⁻¹
U	Internal energy of bubble contents	J
V	Volume	m³
$V_{\mathrm{loop}}$	Working liquid volume in a recirculating loop	m³
$V_R$	Selected reactor volume	m³
$V_{\mathrm{treated}}$	Treated liquid volume	m³
$v_m$	Molecular volume in nucleation theory	m³ molecule⁻¹
W	Compression work	J
$x_A$	Liquid phase mole fraction of species A	dimensionless
$y_i$	Gas phase mole fraction of species i	dimensionless
$Y_{\mathrm{cav}}$	Cavitational chemical yield	mol J⁻¹ or stated product mass J⁻¹
z	Elevation	m
$A_t$	Throat or minimum passage flow area	m²
CE	Cavitation extent, defined on the stated vapor volume and power basis	W⁻¹ or stated dimensionless form
$d_t$	Characteristic throat diameter	m or mm
Eu	Euler number on the stated pressure and velocity basis	dimensionless
$f_{\mathrm{cz}}$	Characteristic turbulence or pressure fluctuation frequency in the cavitation zone	s⁻¹ or kHz
$P_1$	Inlet pressure to the hydrodynamic cavitation device	Pa
$P_2$	Downstream or outlet pressure from the device	Pa
$R{e}_t$	Reynolds number based on throat diameter and velocity	dimensionless
$u_t$ or $v_t$	Characteristic throat velocity	m s⁻¹
$v_{\theta ,\max }$	Maximum tangential velocity in a vortex device	m s⁻¹
$V_{\mathrm{vap}}$	Vapor volume within the defined cavitation zone	m³
[7OHC]	Concentration of 7-hydroxycoumarin used as a hydroxyl radical indicator	mol m⁻³ or µM

Units shown are the preferred SI form. Analytical conventions such as L mol⁻¹ cm⁻¹ may be retained when standard for the method.

Greek Letters

Symbol	Name	Definition	SI unit
${\alpha }_g$	alpha	Gas thermal diffusivity	m² s⁻¹
γ	gamma	Heat capacity ratio c_p/c_v or polytropic exponent when specified	dimensionless
${\gamma }_{\mathrm{mix}}$	gamma	Effective heat capacity ratio of a gas mixture	dimensionless
${\gamma }_{\mathrm{sl}}$	gamma	Solid liquid interfacial energy	J m⁻²
$\mathrm{\Delta }{G}_j$	delta	Gibbs energy change associated with product or useful reaction j	J mol⁻¹
$\mathrm{\Delta }{H}_r$	delta	Enthalpy change of reaction r	J mol⁻¹
Δp	delta	Pressure drop or collapse driving pressure	Pa
${\epsilon }_{\lambda }$	epsilon	Molar absorptivity at wavelength λ	m² mol⁻¹ or L mol⁻¹ cm⁻¹
${\eta }_{\mathrm{chem}}$	eta	Chemical energy efficiency	dimensionless
${\eta }_m$	eta	Motor efficiency	dimensionless
${\eta }_p$	eta	Pump efficiency	dimensionless
λ	lambda	Optical wavelength	m or nm
μ	mu	Dynamic viscosity	Pa s
ρ	rho	Liquid density	kg m⁻³
σ	sigma	Cavitation number	dimensionless
Σ	sigma	Summation over species or reactions	-
${\chi }_g$	chi	Mole fraction of noncondensable gas in gas plus vapor bubble contents	dimensionless
${\alpha }_v$	alpha	Local vapor volume fraction	dimensionless
ΔT	delta	Temperature change relative to a reference condition	K
${\epsilon }_{\mathrm{vap}}$	epsilon	Dimensionless vapor volume or vapor volume scaling quantity	dimensionless
θ	theta	Characteristic vibrational temperature when used in molecular heat capacity models	K
${\omega }_{\mathrm{cz}}$	omega	Volume averaged specific dissipation frequency or modeled turbulence frequency in the cavitation zone	s⁻¹

A dot above a variable denotes a time derivative. A superscript dot before a chemical species denotes an unpaired electron and therefore a radical.