Calculations of Hydrodynamic Cavitation Devices

2 Calculation Basis and Design Workflow

This section defines the consolidated calculation and reporting framework used throughout the article. Subsequent sections apply consistent definitions for absolute pressure, cavitation number, hydraulic and electrical energy, residence time, exposure, uncertainty, and scale translation. Equations are repeated only when a specialized derivation or interpretation is required.

2.1 Required design inputs

The calculation begins with the process duty rather than a preferred device configuration. The minimum data set in Table 2.1 separates properties required for hydraulic sizing from variables required for process validation and mechanical design. A single nominal density and viscosity are insufficient when temperature, composition, gas content, or solids loading varies during operation.

Table 2.1. Minimum design inputs for hydrodynamic cavitation calculations.

Category	Required information	Calculation affected
Capacity	Minimum, normal, and maximum volumetric or mass flow; batch and loop volume; required turndown	Flow area, velocity, pump or motor size, residence time, parallel modules
Pressure	Suction pressure, upstream pressure, downstream pressure, backpressure, allowable pressure drop, relief set pressure	Local pressure screen, cavitation number, pressure boundary, pump duty
Temperature	Minimum, normal, and maximum temperature; allowable temperature rise	Density, viscosity, vapor pressure, seal compatibility, thermal growth
Liquid properties	Density, dynamic viscosity, vapor pressure, speed of sound, surface tension where relevant	Reynolds number, cavitation inception, collapse, pressure transients
Multiphase properties	Gas content, gas solubility, solids loading and size, dispersed phase fraction, rheology	Nucleation, blockage, wear, phase separation, model selection
Product objective	Reaction conversion, droplet or particle distribution, cell disruption, contaminant removal, heating, gas dispersion	Required exposure, passes, energy, sampling and acceptance limits
Geometry constraints	Pipe size, available length, cleanability, minimum passage, allowable rotor diameter and clearance	Device family, throat or hole size, diffuser length, rotor layout
Materials environment	pH, chloride, oxidants, solvents, acids, caustic, abrasive solids, hygiene requirements	Alloy, coating, elastomer, corrosion allowance, surface finish
Utilities	Electrical supply, pump curve, cooling duty, nitrogen or gas injection, instrumentation	Installed power, control strategy, heat exchanger and interlocks
Reliability	Inspection interval, permitted metal contamination, spare strategy, acceptable downtime	Replaceable inserts, erosion allowance, monitoring and lifecycle cost

Material properties should be evaluated at each relevant operating state. For water like liquids, vapor pressure changes rapidly with temperature and can dominate the cavitation margin. For viscous oils, concentrated suspensions, polymer solutions, and emulsions, apparent viscosity may be shear dependent and pressure losses outside the cavitation element may no longer be negligible. The design envelope should therefore include startup, shutdown, cleaning, upset, and maximum temperature conditions rather than only the nominal process state.

2.2 Pressure references and measurement stations

Pressure must be placed on an absolute basis before it is compared with vapor pressure:

where $p_{abs}$ is absolute pressure, $p_g$ is gauge pressure, and $p_{atm}$ is local atmospheric pressure. A vacuum gauge reading requires the corresponding sign convention. Barometric variation is usually small relative to a high pressure device, but it can be important in low pressure systems and suction calculations.

The instrumentation arrangement should establish reproducible pressure stations upstream and downstream of the cavitation element while recognizing that these taps do not normally measure the local minimum pressure. Rapid pressure pulsation may require high frequency transducers in addition to conventional process transmitters. Tap diameter, distance from the element, pipe diameter, straight length, tubing volume, transducer range, and sampling frequency should be documented [2,4–6].

Figure 2.1. Minimum instrumentation around the flow through cavitation element [32].

For an incompressible stream, volumetric flow and mass flow are:

where $Q$ is volumetric flow rate, $A$ is cross sectional area, $\bar{V}$ is mean velocity, $\dot{m}$ is mass flow rate, and $\rho$ is density.

A practical mechanical energy balance between stations 1 and 2 is:

where $g$ is gravitational acceleration, $z$ is elevation, $h_p$ is head added by a pump within the control volume, $h_L$ is irreversible head loss, and $\alpha$ is the kinetic energy correction factor. For a short horizontal element with no pump between the taps, the elevation and pump terms are omitted. The equation remains an averaged energy balance; it does not resolve the local pressure depression inside a separated, swirling, or transient flow.

Figure 2.2 illustrates the importance of local pressure history. A sharp restriction may generate a short, steep pressure minimum followed by highly dissipative recovery, whereas a well designed Venturi can provide a longer low pressure region with more controlled recovery. These conceptual profiles do not replace calibrated pressure measurements, but they inform throat length, diffuser geometry, and the placement of downstream surfaces.

Figure 2.2. Conceptual absolute pressure histories for a sharp restriction and a Venturi with controlled recovery [32].

2.3 Dimensionless groups

The Reynolds number based on a stated characteristic length $D_c$ and velocity $V_c$ is:

where $\mu$ is dynamic viscosity. The Euler number, written here as a pressure loss coefficient, is

The Weber number is:

where $\gamma$ is surface or interfacial tension. For two phase dispersions, the relevant interfacial tension and characteristic droplet or jet dimension must be specified.

The general cavitation number is:

where $p_{ref}$ and $V_{ref}$ are explicitly defined reference pressure and velocity, and $p_v$ is vapor pressure at the local temperature. Published values are not comparable unless the reference stations and velocity definition are the same. For an orifice study, $p_{ref}$ may be downstream pressure and $V_{ref}$ the jet velocity; for a Venturi it may be throat or downstream pressure and throat velocity; for a rotor it may be chamber pressure and rotor tip speed [2–6].

A pressure drop cavitation index frequently used in flow through equipment is:

where $p_1$ and $p_2$ are stated upstream and downstream absolute pressures. This form is convenient because the denominator is measured pressure drop, but it still does not resolve the local minimum pressure. It should be treated as an operating index tied to the test geometry.

The preliminary vapor formation criterion is:

and a preliminary collapse criterion is:

where $p_{\min }$ is the calculated or modeled local minimum pressure and $p_{rec}$ is the pressure in the intended recovery region. Inception can occur above equilibrium vapor pressure because of gas nuclei and unsteady pressure, while a cavity can persist after the mean pressure has recovered. Equations (2.10) and (2.11) are therefore screening conditions rather than exact phase boundary laws [1–3].

Table 2.2. Interpretation and limitations of principal similarity groups.

Group	Principal comparison	Must be defined	Important limitation
$Re$	Inertia relative to viscosity	Velocity and length scale	Does not preserve nuclei, vapor fraction, collapse location, or residence time
$Eu$	Pressure change relative to dynamic pressure	Pressure stations and velocity	Can global pressure loss match while local pressure fields differ
$We$	Inertia relative to surface or interfacial tension	Interface and length scale	Relevant to breakup, but not sufficient for recoalescence or chemistry
$\sigma$	Pressure margin above vapor pressure	Absolute pressure and velocity reference	Definition varies by device; one value cannot describe a full pressure history
$C_v$	Downstream pressure margin normalized by measured loss	Upstream and downstream taps	Does not directly measure vena contracta, core, or blade passage pressure
$St$	Unsteady frequency relative to convection	Characteristic frequency and length	Frequency similarity does not guarantee the same vapor cloud amplitude
$Po$	Shaft power is relative to rotational inertial scale	Shaft power, speed, rotor diameter	Geometry, clearance, flow coefficient, and gas fraction must remain comparable

2.4 Hydraulic power, electrical power, residence time, and dose

The hydraulic power associated with a measured pressure drop is:

For a pump driven static device, the approximate electrical input assigned to the cavitation duty is:

where ${\eta }_p$ is pump efficiency, ${\eta }_m$ is motor efficiency, and ${\eta }_d$ represents drive or transmission efficiency. For a complete system assessment, auxiliary loads, recirculation losses, cooling, and control equipment should be included. Measured electrical power is preferred over nameplate power.

The single pass hydraulic specific energy is:

and the corresponding mass specific electrical energy over a treatment interval $t$ is:

A short device volume $V_d$ has nominal residence time:

For a well mixed recirculation tank of volume $V_L$, the nominal number of loop turnovers during time $t$ is:

This turnover count does not guarantee that each fluid element has passed through the device exactly $N_{turn}$ times. A real loop has a residence time distribution, short circuiting, dead volume, and mixing effects. For comparison of process results, the report should include the tank volume, external piping volume, device hold up, flow history, sample location, and whether the product is withdrawn continuously or only after recirculation [2–5].

2.5 Collapse time and placement of the recovery zone

A useful inertial estimate for the collapse time of an initially spherical vapor cavity of radius $R_0$ under a constant pressure difference is the Rayleigh expression:

where $p_{\infty }$ is the recovered far field pressure. The corresponding convective distance is:

where $V_c$ is a representative convection velocity. These equations do not include dissolved gas, thermal effects, non spherical collapse, cloud interaction, or wall proximity, but they help place a replaceable collapse chamber, avoid a vulnerable weld, and estimate whether a long low pressure zone will carry cavities beyond the intended active region [1,2].

For example, a water vapor cavity with $R_0=0.5$ mm exposed to a recovered pressure difference of 200 kPa has an inertial collapse time of approximately 32 microseconds. At a convection velocity of 12 m/s, the corresponding distance is approximately 0.38 mm. Real clouds can persist much farther because of gas content, collective shielding, repeated growth, and spatially distributed pressure recovery. The calculation is therefore a lower order location screen, not an erosion life prediction.

2.6 Uncertainty and model hierarchy

For a result $y=f(x_1,x_2,\dots ,x_n)$ with independent input uncertainties, the first order combined standard uncertainty is:

where $u_{x_i}$ is the standard uncertainty of input $x_i$. Correlated variables require covariance terms. A design calculation should separate measurement uncertainty from model form uncertainty. The latter includes uncertainty in discharge coefficient, pressure recovery coefficient, vortex core model, power number, two phase closure, and the relationship between hydraulic state and product response.

Figure 2.3. Local sensitivity of the preliminary orifice diameter equation to representative input perturbations. The result illustrates parameter sensitivity and does not include model form or interaction uncertainty [32].

Table 2.3. Recommended model fidelity by design question.

Design question	Minimum method	Escalation trigger	Validation evidence
Required open area and pump pressure	One dimensional continuity and calibrated loss coefficient	Non Newtonian flow, gas fraction, severe choking, uncertain $C_d$	Flow pressure curve at operating temperature
Local cavitation inception	Pressure screen plus calibrated inception map	Geometry has separation, vortex core, multiple jets, or transient rotor interaction	Visualization, acoustic signature, high speed pressure, or validated CFD
Cavity extent and collapse location	Empirical map or two phase CFD	Erosion sensitive surface, reaction selectivity, scale change	Imaging, erosion coupons, hydrophone or pressure mapping
Rotor power	Calibrated $Po$, torque measurement, motor map	Clearance, flow coefficient, gas fraction, or rheology changes	Shaft torque and electrical power over full operating map
Product performance	Empirical response surface	Composition, temperature, or scale differs from test	Replicated pilot trials and product analysis
Erosion life	Comparative material tests and pilot inserts	Long service interval, toxic metal limit, corrosive abrasive fluid	Periodic mass/thickness loss and contaminant analysis

The dominant uncertainty is often not the pressure measurement itself but the assumption that a hydraulic index predicts the required process response. A pilot program should therefore map hydraulic variables and product attributes concurrently. A model should be accepted only within the region where measured flow, pressure, power, temperature, cavity signature, process response, and equipment integrity remain mutually consistent.