Calculations of Hydrodynamic Cavitation Devices

Nomenclature

Unless otherwise stated, quantities are expressed in SI units. Subscripts identify the relevant station, component, or reference condition.

Symbol	Definition	SI unit or status
$A$	Cross sectional area	m²
$A_b$	Projected bluff body area	m²
$A_e$	Exposed erosion test area	m²
$A_i$	Area of one tangential inlet	m²
$A_o$	Total open area of an orifice or multihole insert	m²
$A_p$	Pipe cross sectional area	m²
$A_t$	Venturi throat area	m²
$A_{vc}$	Vena contracta area	m²
$AR$	Chamber aspect ratio	dimensionless
$B$	Bluff body blockage ratio	dimensionless
$C_c$	Contraction coefficient	dimensionless
$C_d$	Discharge coefficient	dimensionless
$C_{p,r}$	Pressure recovery coefficient	dimensionless
$C_v$	Pressure drop cavitation index defined by Equation (2.9)	dimensionless
$C{V}_Q$	Coefficient of variation of parallel module flow	dimensionless
$c$	Speed of sound in liquid	m/s
$c_p$	Specific heat capacity	J/(kg·K)
$c_{corr}$	Corrosion allowance	m
$c_{eros}$	Erosion allowance	m
$c_{mach}$	Machining or resurfacing allowance	m
$D$, $D_c$	Stated characteristic or chamber diameter	m
$D_1$, $D_2$	Upstream and downstream diameters	m
$D_h$	Hydraulic diameter	m
$D_p$	Pipe internal diameter	m
$D_r$	Rotor diameter	m
$d_b$	Bluff body characteristic dimension	m
$d_{eq}$	Equivalent circular diameter of total open area	m
$d_h$	Diameter of one orifice hole	m
$d_t$	Venturi throat diameter	m
$E$	Weld joint efficiency in pressure wall screen	dimensionless
$Eu$	Euler number or pressure loss coefficient	dimensionless
$E{u}_v$	Vortex device Euler number	dimensionless
$e_{el}$	Mass specific electrical energy	J/kg or kWh/t
$e_{h,sp}$	Single pass hydraulic specific energy	J/kg
$e_s$	Single pass shaft specific energy	J/kg
$f_s$	Wake shedding frequency	Hz
$f_{bp}$	Rotor feature passage frequency	Hz
$g$	Gravitational acceleration	m/s²
$g_c$	Rotor stator clearance	m
$g_{\min }$	Minimum required running clearance	m
$H_c$	Vortex chamber height or axial length	m
$h_L$	Irreversible head loss	m
$h_p$	Pump head added within the control volume	m
$h_{allow}$	Maximum permitted local penetration	m
$h_{meas}$	Measured local penetration	m
$K_{sys}$	System curve coefficient	Pa·s²/m⁶ for the form in Equation (7.3)
$K_v$	Calibrated vortex device capacity coefficient	dimensionless for the stated form
$L_c$	Estimated convective collapse distance	m
$L_d$	Diffuser length	m
$L_s$	Restriction t -target standoff or stage spacing	m
$M{a}_{tip}$	Rotor tip Mach number	dimensionless
$MDE$	Mean depth of erosion	m
$m_t$	Negative mill tolerance fraction used in nominal thickness selection	dimensionless
$m_{treated}$	Mass of liquid or product assigned to a treatment energy	kg
$\dot{m}$	Mass flow rate	kg/s
$MDPR$	Mean depth penetration rate	m/s or mm/h
$N_h$	Number of equal orifice holes	integer
$N_i$	Number of tangential inlets	integer
$N_m$	Number of parallel modules	integer
$N_s$	Number of cavitation stages	integer
$N_{turn}$	Nominal loop turnovers	dimensionless
$N_{rpm}$	Rotational speed	rpm
$n$	Rotational speed	s⁻¹
$p$	Absolute static pressure unless otherwise stated	Pa
$p_1$, $p_2$	Upstream and downstream absolute pressures	Pa
$p_{abs}$	Absolute pressure	Pa
$p_{atm}$	Local atmospheric pressure	Pa
$p_b$	Defined chamber or bulk pressure for rotor calculation	Pa
$p_c$	Vortex core pressure	Pa
$p_g$	Gauge pressure	Pa
$p_{\min }$	Local minimum pressure	Pa
$p_o$	Outer radius reference pressure in vortex model	Pa
$p_{rec}$	Pressure in intended recovery region	Pa
$p_t$	Venturi throat pressure	Pa
$p_v$	Vapor pressure at local temperature	Pa
$P_{active}$	Power assigned to the active cavitation volume	W
$P_{el}$	Electrical input power	W
$P_h$	Hydraulic power	W
$P_s$	Shaft power	W
$P_{loss,out}$	Other power leaving the selected heat balance boundary	W
$P_w$	Wetted perimeter	m
$Po$	Power number	dimensionless
$Q$	Volumetric flow rate	m³/s
$Q_j$	Flow through parallel module $j$	m³/s
$Q_{tot}$	Total flow through parallel modules	m³/s
${\dot{Q}}_{cool}$	Heat removal rate	W
$R_0$	Initial cavity radius in collapse time screen	m
$Re$	Reynolds number	dimensionless
$R{e}_o$	Orifice hole Reynolds number	dimensionless
$R{e}_r$	Rotational Reynolds number	dimensionless
$r$	Radius from vortex axis	m
$r_c$	Finite vortex core radius	m
$r_o$	Outer radius used in vortex pressure model	m
$S$	Allowable material stress in pressure wall screen	Pa
$St$	Strouhal number	dimensionless
$S_v$	Defined tangential to axial velocity ratio	dimensionless
$s_Q$	Standard deviation of parallel branch flows	m³/s
$T_s$	Shaft torque	N·m
$t$	Time	s
$t_c$	Rayleigh inertial collapse time	s
$t_{code}$	Code required pressure thickness	m
$t_{nom}$	Selected nominal thickness	m
$t_p$	Plate or insert thickness	m
$t_{rem}$	Screened remaining service time	s or h
$t_s$	Simplified pressure wall thickness	m
$\mathrm{\Delta }{h}_{product}$	Product specific enthalpy change across the heat balance boundary	J/kg
$u_{tip}$	Rotor tip speed	m/s
$u_y$, $u_{x_i}$	Standard uncertainty of result or input	units of the associated quantity
$V$, $\bar{V}$	Mean velocity	m/s
$V_1$, $V_2$	Mean velocity at stations 1 and 2	m/s
$V_{active}$	Defined active cavitation volume	m³
$V_c$	Representative convection velocity in collapse distance estimate	m/s
$V_d$	Device hold up volume	m³
$V_i$	Tangential inlet velocity	m/s
$V_j$	Nominal jet velocity based on physical open area	m/s
$V_L$	Loop or tank liquid volume	m³
$V_t$	Venturi throat velocity	m/s
$V_{vc}$	Vena contracta velocity	m/s
$V_{\theta }$	Tangential velocity	m/s
$\bar{Q}$	Mean flow among parallel branches	m³/s
$We$	Weber number	dimensionless
$W_{cav}$	Mechanical cavitation loss	mass, depth, or rate as stated
$W_{corr}$	Corrosion loss without cavitation	mass, depth, or rate as stated
$W_{syn}$	Erosion corrosion interaction term	mass, depth, or rate as stated
$W_{tot}$	Total combined material loss	mass, depth, or rate as stated
$z$	Elevation	m
$z_f$	Number of repeating rotor features	integer

Greek Letters

Greek letter	Name	Definition	SI unit or status
$\alpha$	alpha	Kinetic energy correction factor	dimensionless
${\beta }_{eq}$	beta	Equivalent open diameter to pipe diameter ratio	dimensionless
${\beta }_t$	beta	Venturi throat to inlet diameter ratio	dimensionless
$\mathrm{\Gamma }$	gamma, capital	Vortex circulation	m²/s
${\mathrm{\Gamma }}_{crit}$	gamma, capital	Circulation at the finite core vapor pressure screen	m²/s
$\gamma$	gamma	Surface or interfacial tension	N/m
${\dot{\gamma }}_{gap}$	gamma dot	Approximate rotor stator gap shear rate	s⁻¹
$\mathrm{\Delta }$	delta, capital	Finite difference or change in the associated quantity	units of associated quantity
${\delta }_y$	delta	Normalized measured to calculated deviation	%
${\epsilon }_V$	epsilon	Defined active zone power density	W/m³
${\eta }_d$	eta	Drive or transmission efficiency	dimensionless
${\eta }_m$	eta	Motor efficiency	dimensionless
${\eta }_p$	eta	Pump efficiency	dimensionless
$\theta$	theta	Diffuser half angle	rad or degrees
$\mu$	mu	Dynamic viscosity	Pa·s
$\rho$	rho	Liquid density	kg/m³
${\rho }_s$	rho	Solid or rotor material density	kg/m³
$\sigma$	sigma	Cavitation number	dimensionless
${\sigma }_r$	sigma	Rotor based cavitation number	dimensionless
${\sigma }_{\theta ,ring}$	sigma	Circumferential stress in thin ring screen	Pa
${\tau }_d$	tau	Nominal device residence time	s
$\varphi$	phi	Rotor flow coefficient	dimensionless
${\varphi }_A$	phi	Total open area fraction	dimensionless
$\psi$	psi	Rotor pressure or head coefficient	dimensionless
$\omega$	omega	Angular velocity	rad/s