Hydrodynamic Cavitation in Emulsification and Homogenization

Nomenclature

The symbols below are used throughout this article. Unless otherwise stated, quantities are expressed in SI units.

Symbol or abbreviation	Definition	SI unit or status
$A$	Interfacial area	m²
$A_t$	Throat flow area	m²
$a_v$	Interfacial area per unit emulsion volume	m²/m³
$B_{\text{br}}\text{,}{D}_{\text{br}}$	Birth and death rates by breakup	Model dependent
$B_{\text{co}}\text{,}{D}_{\text{co}}$	Birth and death rates by coalescence	Model dependent
$b\left(v|v_p\right)$	Daughter droplet distribution for parent volume $v_p$	1/m³
$C_b$	Breakup model constant	Model dependent
$C_E$	Cavitation extent, vapor volume divided by device volume	Dimensionless
$C_H$	Hinze breakup constant	Dimensionless
$C_s$	Surfactant concentration in the continuous phase	kg/m³
$C_v$	Cavitation number	Dimensionless
$d$	Droplet diameter	m
$d_{10}\text{,}{d}_{50}\text{,}{d}_{90}$	Cumulative percentile diameters	m
$d_{32}$	Sauter mean diameter	m
$d_{43}$	Volume weighted mean diameter	m
$d_{k\text{,max}}$	Maximum stable droplet diameter in the cavitation model	m
$D_0$	Initial cavitation bubble diameter	m
$D_m$	Mean cavitation bubble diameter	m
$D_{-4}$	Parabolic cylinder function of order minus four	Special function
$D{a}_{\text{ads}}$	Adsorption or stabilization ratio	Dimensionless
DSD	Droplet size distribution	Abbreviation
$E_h\text{,}{E}_{\text{in}}$	Specific hydraulic or input energy	J/kg
$E_k^b$	Energy assigned to collapse of one bubble	J
$\text{Eu}$	Euler number	Dimensionless
$F_D$	Bubble size distribution factor in the Kozyuk and Fedotkin model	Units implied by Eq. (41)
HC	Hydrodynamic cavitation	Abbreviation
HCR	Hydrodynamic cavitation reactor	Abbreviation
HLB	Hydrophilic lipophilic balance	Dimensionless index
$I_3$	Third moment integral of bubble diameter	m⁴
$K_E\text{,}{K}_n$	Empirical constants in energy and pass correlations	Model dependent
$K_{\text{co}}$	Coalescence kernel	m³/s
$K_{\text{OR}}$	Ostwald ripening rate constant	m³/s
$L$	Characteristic device length	m
$m_a$	Approximate liquid mass affected by one collapse	kg
$m_{s\text{,min}}$	Minimum surfactant mass for interfacial coverage	kg
$n_i$	Number of droplets in size class $i$	Count or number density
$n_p$	Number of passes	Dimensionless
$N$	Bubble number concentration per liquid volume	1/m³
$\text{Oh}$	Ohnesorge number	Dimensionless
O/W	Oil in water emulsion	Abbreviation
$p_2$	Recovered downstream pressure	Pa
$p_B$	Bubble internal pressure	Pa
$p_v$	Vapor pressure	Pa
$p_{\infty }$	Far field pressure	Pa
$P$	Collapse zone pressure used in bubble field model	Pa
$P_h$	Hydraulic power	W
PDI	Polydispersity index	Dimensionless
$Q$	Volumetric flow rate	m³/s
$R$	Instantaneous bubble radius	m
$R_{\text{max}}$	Maximum bubble radius	m
$\text{Re}$	Reynolds number	Dimensionless
$S(v)$	Breakup frequency	1/s
$s_D$	Standard deviation of bubble diameter	m
$t$	Time	s
$t_c$	Bubble collapse time	s
$t_p$	Processing time	s
$t_r$	Residence time	s
$U\text{,}{U}_t$	Characteristic and throat velocities	m/s
$u_s$	Creaming or sedimentation velocity	m/s
$V_a$	Active or nominal reactor volume	m³
$V_B$	Batch or loop inventory	m³
$V_D$	Device volume	m³
$V_{\text{vap}}$	Vapor volume in a defined region	m³
$W_c$	Collapse work	J
W/O	Water in oil emulsion	Abbreviation
$W{e}_d$	Droplet Weber number	Dimensionless

Greek Letters

Symbol	Letter name	Meaning	SI unit or status
${\alpha }_j$	Alpha	Dispersed phase volume fraction in droplet population $j$	Dimensionless
${\beta }_{\text{coll}}$	Beta	Droplet collision frequency	Model dependent
$\gamma$	Gamma	Liquid-liquid interfacial tension	N/m
${\gamma }_{\text{lv}}$	Gamma	Liquid vapor surface tension	N/m
${\mathrm{\Gamma }}_{\infty }$	Capital gamma	Saturated surfactant surface excess	kg/m²
$\mathrm{\Delta }$	Delta	Difference or change; for example, pressure drop	According to quantity
$\epsilon$	Epsilon	Local energy dissipation rate per unit mass	W/kg
${\epsilon }_c$	Epsilon	Local collapse associated dissipation estimate	W/kg
${\eta }_K$	Eta	Kolmogorov length scale	m
${\eta }_b$	Eta	Droplet breakage efficiency	Dimensionless
${\eta }_{\text{th}}$	Eta	Thermodynamic emulsification efficiency	Dimensionless
$\theta$	Theta	Fractional interfacial coverage	Dimensionless
$\lambda$	Lambda	Dispersed to continuous viscosity ratio	Dimensionless
$\mu$	Mu	Dynamic viscosity	Pa s
${\mu }_c\text{,}{\mu }_d$	Mu	Continuous- and dispersed phase viscosities	Pa s
${\nu }_c$	Nu	Continuous phase kinematic viscosity	m²/s
${\mathrm{\Pi }}_b$	Pi	Breakup stress ratio	Dimensionless
$\rho$	Rho	Density	kg/m³
${\rho }_c\text{,}{\rho }_d$	Rho	Continuous- and dispersed phase densities	kg/m³
${\tau }_d\text{,}{\tau }_t$	Tau	Disruptive and turbulent stresses	Pa
${\varphi }_d$	Phi	Dispersed phase volume fraction	Dimensionless
${\dot{\gamma }}_s$	Gamma dot	Shear rate	1/s
${\text{A}}_E$	Calligraphic A	Net interfacial area generated per unit energy	m²/J