Fabrication of High Performance Catalysts by Controlled Flow Cavitation

Nomenclature

The following symbols and abbreviations are used in this article. Unless otherwise stated, quantities are expressed in SI units.

Symbol	Definition	SI unit or status
$a_m$	Geometric mass specific surface area	m2 kg−1
$C$	Concentration or empirical constant as defined	mol m−3 or dimensionless
$C^{\ast }$	Equilibrium solubility at the same temperature	mol m−3
$C_d$	Restriction discharge coefficient	dimensionless
$D$	Crystallite size from diffraction	m or nm
$D_h$	Hydraulic diameter	m
$d_c$	Characteristic crystallite diameter	m
$d_o$	Restriction diameter	m
$E_s$	Specific hydraulic energy referred to isolated solids	J kg−1
$g$	Gravitational acceleration	m s−2
$J$	Nucleation rate	m−3 s−1
$J_0$	Nucleation preexponential factor	m−3 s−1
$K$	Scherrer shape factor	dimensionless
$k_d$	Apparent deactivation constant	s−1
$m_s$	Mass of isolated solids	kg
$N_p$	Nominal number of loop passes	dimensionless
$P_1$	Upstream static pressure	Pa
$P_b$	Back pressure	Pa
$P_r$	Recovered pressure used in cavitation number	Pa
$P_t$	Static pressure at the throat	Pa
$P_v$	Liquid vapor pressure	Pa
$p_B$	Pressure inside a bubble	Pa
$p_{\infty }$	Far field liquid pressure	Pa
$Q$	Volumetric flow rate	m3 s−1
$R$	Instantaneous bubble radius	m
$R_g$	Universal gas constant	J mol−1 K−1
$R_{\max }$	Maximum bubble radius before collapse	m
$r^{\ast }$	Critical nucleus radius	m
$S$	Supersaturation ratio	dimensionless
$S_r$	Relative supersaturation	dimensionless
$t$	Processing time	s
$t_c$	Idealized bubble collapse time	s
$T$	Absolute temperature	K
$u$	Characteristic liquid velocity	m s−1
$u_t$	Throat velocity	m s−1
$V_L$	Liquid inventory in recycle loop	m3
$V_R$	Reactor volume used for productivity basis	m3
$v_m$	Molar volume of solid phase	m3 mol−1
$W_c$	Idealized mechanical collapse work	J
$z$	Elevation	m

Greek Letters

Symbol	Letter name	Meaning	SI unit or status
$\beta$	Beta	Corrected diffraction peak breadth	radian
$\gamma$	Gamma	Liquid surface tension	N m−1
${\gamma }_{sl}$	Gamma	Solid liquid interfacial energy	J m−2
$\mathrm{\Delta }{G}^{\ast }$	Delta G	Critical nucleation free energy barrier	J mol−1 or J per nucleus
$\mathrm{\Delta }P$	Delta P	Pressure difference across processor	Pa
$\mathrm{\Delta }{P}_L$	Delta P sub L	Lumped irreversible pressure loss	Pa
$ϵ$	Epsilon	Microstrain or turbulent energy dissipation as defined	dimensionless or W kg−1
${\epsilon }_h$	Epsilon	Mass specific turbulent energy dissipation rate	W kg−1
${\eta }_K$	Eta	Kolmogorov length scale	m
${\eta }_p$	Eta	Pump efficiency	dimensionless
$\lambda$	Lambda	X ray wavelength	m or nm
$\mu$	Mu	Dynamic viscosity	Pa s
$\nu$	Nu	Kinematic viscosity	m2 s−1
${\mathrm{\Pi }}_q$	Pi	Dimensionless material quality response	dimensionless
$\rho$	Rho	Liquid density	kg m−3
${\rho }_s$	Rho	Solid density	kg m−3
$\sigma$	Sigma	Cavitation number	dimensionless
${\tau }_K$	Tau	Kolmogorov time scale	s
${\tau }_L$	Tau	Mean loop residence time	s
$\theta$	Theta	Bragg angle	radian or degree

Abbreviations

Abbreviation	Definition
BET	Brunauer Emmett Teller gas adsorption surface area
CFC	Controlled flow cavitation
CFD	Computational fluid dynamics
DoE	Design of experiments
DLS	Dynamic light scattering
HDS	Hydrodesulfurization
HC	Hydrodynamic cavitation
ICP MS	Inductively coupled plasma mass spectrometry
ICP OES	Inductively coupled plasma optical emission spectrometry
PAT	Process analytical technology
PZT	Lead zirconate titanate
RSM	Response surface methodology
SAED	Selected area electron diffraction
SEM	Scanning electron microscopy
STEM	Scanning transmission electron microscopy
STY	Space time yield
TEM	Transmission electron microscopy
TOF	Turnover frequency
TPR	Temperature programmed reduction
TPO	Temperature programmed oxidation
XAS	X ray absorption spectroscopy
XPS	X ray photoelectron spectroscopy
XRD	X ray diffraction