System Design of Reuleaux Venturi Cavitation Reactors for Process Intensification

1. Introduction

Hydrodynamic cavitation arises from controlled pressure variations that generate vapor cavities in a flowing liquid and induce their rapid collapse. The collapse of these cavities concentrates energy on very short time scales and within small volumes [1,2,3]. It generates micro-jets, shock waves, intense shear and hot spots that give rise to hydroxyl radicals. These effects intensify micromixing, renew the interfacial area and enhance mass transfer, thereby accelerating physical and chemical transformations in the liquid phase [2,3,4,5,6]. The spatial distribution and intensity of cavity collapses depend on the pressure field and on the reactor geometry [2,4,5,7,8,9,10,11].

In this work, geometry is treated as a process module that can be embedded in complex systems, with requirements formulated so as to be modeled, controlled and integrated without discontinuities.

Applications of hydrodynamic cavitation are cross-sector and span multiple process industries [7,8,12,13], including water and wastewater treatment [14,15,16,17,18,19,20,21,22,23,24,25,26,27], the food and beverage sector [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42], brewing [43,44,45,46,47,48,49,50,51,52], bioenergy [53,54,55,56,57], biotechnology and pharmaceutical processing [58,59,60], fine chemicals [61,62,63,64], materials, composites and cosmetics [65,66,67], the pulp and paper, textile and leather sectors [68,69], oil and gas and the mining industry [70], cleaning-in-place and antifouling operations [71], thermal plants [72], agritech [73] and maritime applications [74].

These applications rely on combinations of oxidation phenomena, micro- and nanoemulsification, and enhanced mixing and mass transfer, underpinning broader process-intensification schemes.

Here, cavitation is interpreted as the result of geometric constraints imposed in the throat. For a given cross-sectional area, the perimeter-to-area ratio $P/A$ is the lumped parameter that controls the pressure drop $\mathsf{\Delta }P$ between contraction and pressure recovery: at constant area, a larger perimeter yields a steeper pressure profile and a higher characteristic $\mathsf{\Delta }P\left[75\right]$. In circular Venturi throats, $P/A$ is minimal; therefore, we adopt a Reuleaux triangle cross-section (VRA), a curve of constant width that, for the same area, exhibits a larger perimeter than the circle and, consequently, a higher $\mathsf{\Delta }P\left[76\right]$. In what follows, we denote by $G$ the dimensionless geometric factor, defined as the perimeter-to-area ratio of the throat normalised by that of the reference circular cross-section. Figure 1 summarises the iso-area comparison and highlights $P/A$ and the resulting difference in $\mathsf{\Delta }P$ as the only change.

On this basis, we employ the VRA configuration: the throat retains the Reuleaux triangle cross-section without torsion, and the flow remains axial along the duct centerline, as shown in the longitudinal section of Figure 2.

The twisted configuration is denoted VRAt, where t stands for twist. It introduces an axial twist of the VRA throat that imposes a co-rotating swirl: the streamlines follow a helical trajectory and the effective path length of the fluid increases with respect to the purely axial case, as shown in Figure 3.

Compared with the circular cross-section, both VRA and its swirling VRAt variant are designed to increase the density, uniformity and spatial localization of bubble collapses. The expected benefits include higher selectivity, shorter treatment times and more efficient use of energy, subject to experimental confirmation [75].

This contribution is framed within the broader context of process intensification and provides a design framework that supports the modelling, control and optimisation of unit operations. The reduction of reagent consumption and thermal loads supports clean, low-impact technologies. Retrofitting, modularity and integration into existing process lines promote circular-economy strategies and efficient use of resources. The analysis is theoretical and calls for experimental validation. Adopting a systems perspective, the work proposes design criteria that are useful for dynamic modelling and control and for integration into plant architectures.

2.1. Availability of Nucleation Sites

In this section, only geometric effects are considered, at fixed cross-sectional area, surface condition $\alpha$, and operating conditions, including the cavitation number $\sigma$. The aim is to compare the availability of nucleation sites along the wall between a circular throat (baseline) and a VRA throat.

The total number of nucleation sites available on the wall is proportional to the perimeter of the cross-section:

$$N_{sites}=\alpha P$$

(3)

where $N_{\mathrm{sites}}$ is the total number of nucleation sites on the internal surface of the throat, $\alpha$ is the linear density of active sites associated with the surface condition, and $P$ is the perimeter of the cross-section. To relate this availability to the area swept by the fluid, the surface density of nucleation sites is introduced:

$$n={\displaystyle \frac{N_{sites}}{A}}=\alpha {\displaystyle \frac{P}{A}}$$

(4)

where $n$ is the density of sites referred to the cross-sectional area and $A$ is the cross-sectional area; the ratio $P/A$ measures how much wall length is available per unit area of flow.

The relative geometric factor is defined as:

$$G_{P/A}\equiv {\displaystyle \frac{{\left(\frac{P}{A}\right)}_{VRA}}{{\left(\frac{P}{A}\right)}_{circ}}}$$

(5)

which compares the perimeter-to-area ratio of the VRA throat with that of the circular throat. For the same cross-sectional area, the VRA cross-section has a larger perimeter-to-area ratio than the circular one, $(P/A{)}_{\mathrm{VRA}}>(P/A{)}_{\mathrm{circ}}$, from which it follows that $G_{P/A}>1$. Here $G_{P/A}$ is the dimensionless geometric factor defined in (5) as the ratio between the perimeter-to-area ratio of the VRA cross-section and that of the circular reference. Under the theoretical conditions specified above, this implies that the VRA throat exhibits a greater availability of nucleation sites along the wall than the circular throat, with potential benefits in terms of treatment speed and effectiveness.

2.2. Density of Sites and Probability of Cavitation Inception

In this section, only geometric effects are considered: the cross-sectional area $A$, the surface condition described by the parameter $\alpha$, and the operating conditions, including the cavitation number $\sigma$, are kept fixed. The surface density $n$of nucleation sites increases with the geometric ratio $P/A$ according to (4). Let $P_r$ denote the probability that a fluid element undergoes cavitation inception during a single passage through the throat, and let $p_{\mathrm{inn}}$ be the corresponding single-pass inception probability associated with an individual active site on the wall. Under a simple mean-field approximation, it is assumed that

$$Pr\propto n\cdot p_{\mathrm{inn}}$$

(6)

For compact notation in the tables, we therefore use the lumped parameter $\alpha \cdot p_{\mathrm{inn}}$, where $\alpha$ is the linear density of active sites along the wall introduced in (3). Since $n$ is proportional to $P/A$ (4), and the geometric factor $G_{P/A}$ defined in (5) is greater than one for the VRA cross-section, both $n$ and $P_r$ are expected to be larger in the VRA throat than in the circular cross-section under the conditions considered. In practice, a higher density of active sites can sustain a higher frequency and a wider spatial spread of cavitation events.

2.3. Distribution of Cavitation Events

For an effective treatment, it is not sufficient to increase the number of events; cavitation inceptions must also be distributed along the entire internal wall of the Venturi throat. In this subsection, only geometric effects are considered at fixed cross-sectional area, operating conditions and surface state.

Let $\chi \left(\epsilon \right)$ denote the fraction of the cross-sectional area lying within a distance $\epsilon$from the wall. For cross-sections with a convex and smooth boundary and for sufficiently small $\epsilon$, with $\epsilon$ smaller than both the inradius and the minimum radius of curvature, the following first-order approximation holds:

$$\chi \left(\epsilon \right)\approx {\displaystyle \frac{P}{A}}\epsilon$$

(7)

where $\chi \left(\epsilon \right)$ is dimensionless, $\epsilon$is the characteristic near-wall thickness, $P$ is the perimeter of the cross-section and $A$ is its area. In what follows, $\chi \left(\epsilon \right)$ will also be used as a compact indicator of near-wall coverage in the comparative tables.

Equation (7) shows that the fraction of the cross-section close to the wall increases, to first order, with the ratio $P/A$, consistently with the higher availability and density of nucleation sites summarized in (3) and (4). Since, at equal area, the VRA geometry exhibits a higher $P/A$ than the circular section, for the same $\epsilon$ the near-wall fraction is larger in the VRA case. This implies, in purely geometric terms, a wider azimuthal coverage of potential inceptions and a lower probability of poorly treated regions. For the same throat area, Figure 4 and Figure 5 provide a visual reference for understanding the effect of geometry on cavitation and for interpreting the parameters introduced.

2.4. Localization of Inceptions with Swirl (VRAt): Kinematic Effects Near the Wall

The assumptions of equal cross-sectional area, unchanged surface condition and identical operating conditions, including the cavitation number $\sigma$, still hold. In the VRAt configuration, the flow in the throat exhibits a co-rotating swirl, that is, a tangential component that forces the streamlines to follow a helical path. Let $L_{\mathrm{ax}}$ be the axial length of the throat, $L_{\mathrm{eff}}$ the effective path length, and $\psi$ the helix angle with respect to the axis. In a purely kinematic description, one obtains

$$L_{eff}={\displaystyle \frac{L_{ax}}{\mathit{cos}\psi }}$$

(8)

For $\psi >0$, it follows that $L_{\mathrm{eff}}>L_{\mathrm{ax}}$; under the same conditions, the swirl elongates the path and increases near-wall exposure. Let $L_{\mathrm{sw}}$ be a dimensionless index of swirl-induced near-wall localisation, used qualitatively in the tables; it increases with $\psi$ (or, equivalently, with the ratio $L_{\mathrm{eff}}/L_{\mathrm{ax}}$). Swirl does not modify the cross-sectional geometry and therefore does not change $P/A$or $\chi \left(\epsilon \right)$; instead, it acts on how inceptions are distributed within the near-wall region.

Since VRAt shares the VRA cross-sectional geometry, which exhibits a higher $P/A$ than the circular section, as reflected in the geometric factor $G_{P/A}>1$ defined in (5), and since the near-wall fraction $\chi \left(\epsilon \right)$ increases with $P/A$ according to (7), the addition of swirl in VRAt can promote a sharper localisation of inception events along the perimeter. This behaviour is illustrated in Figure 5, where the near-wall swirl causes inception events to cluster at the cusps and to decay along the arcs. The combined effects of geometry and swirl are summarised in Table 1, which reports the relative geometric and inception-related metrics for the circular, VRA and VRAt configurations.

3. Future Application Perspectives of the VRA and VRAt Geometries

Building on the application areas of hydrodynamic cavitation outlined in Section 1, the VRA geometry increases the wetted perimeter and the fluid–wall contact, which can lead to cavitation collapse fields that are denser and more uniform. The VRAt variant introduces a controlled swirl that extends the effective path length and helps localise the action near the regions of interest. These design choices aim to combine selectivity with short treatment times across multiple application domains.

In water and wastewater treatment, higher density and improved uniformity of collapses can sustain advanced oxidation of micropollutants, reduction of colour and organic load, mild disinfection and conditioning of streams for coagulation, flotation and membrane processes with reduced fouling. Near-wall localisation, including on catalytic supports, can promote selective activation and synergies with oxidants [15,18,26,69].

In food processing, the combination of shear, micro-jets and shock waves can support low-temperature sanitisation, rapid extraction of bioactive compounds, stable emulsions and clarification [28,29,32,33,38,40].

In brewing and in dairy and plant-based beverages, these effects can be exploited for intensified boiling and isomerisation, homogenisation and stabilisation [43,45,46,48,50,62].

In bioenergy, there are prospects for sludge disintegration, microalgae lysis and substrate preparation for anaerobic digestion with increased biogas yield, as well as for fine reactive emulsions that accelerate continuous transesterification in biodiesel production [54,55,56,57].

In biotechnology and pharmaceutical processing, controlled lysis, micromixing and nanoemulsion formation can improve the recovery of intracellular products and the delivery of active ingredients [58,59,60].

In fine chemicals, homogeneous catalyst dispersion and the management of reactive emulsions can favour oxidations and other liquid-phase syntheses, while in materials and composites, nanofiller dispersion and rheological control can benefit from more regular collapse fields [61,62,63,66,67].

Other scenarios include the pulp and paper, textile and leather sectors, with mild bleaching treatments and reduced effluents; oil and gas, with treatment of produced water, oxidative desulfurisation and cleaning of heat exchangers; mining and hydrometallurgy, with more effective leaching and flotation; and cosmetics, with fine emulsions and well-dispersed pigments [68,69,70,71].

In cleaning-in-place and antifouling operations, the removal of biofilms and scale deposits may become faster. Hydrodynamic cavitation reactors implemented in recirculating water-treatment loops and in strongly cavitating shear flows have been shown to enhance mass- and heat-transfer phenomena; on this basis, in thermal circuits and heating, ventilation and air-conditioning (HVAC) systems, degassing and the recovery of heat-transfer performance may become more efficient [71,72].

In agritech, there are prospects for water sanitisation, extraction of biostimulants and preparation of pesticide emulsions, while in food packaging, logistics and marine applications, further opportunities can be envisaged in washing waters, edible coatings and biofouling management in recirculating systems [31,32,33,73,74].

4. Conclusions

This theoretical study links internal reactor geometry and pressure fields to cavitation behaviour. The comparison at equal cross-sectional area, surface condition and operating conditions highlights three design objectives: increasing the density of events, improving their uniformity and guiding their localisation in regions of interest.

The VRA geometry expands the effective perimeter and the fluid–wall contact. This leads to a greater availability of nucleation sites, a higher site density per unit area and a more extended near-wall coverage. The distribution of inceptions becomes more regular and the cavitation field more coherent.

The VRAt variant introduces a controlled swirl. The rotational component lengthens the effective path, increases the useful residence time near the wall and makes the pressure distribution more favourable. The action becomes more concentrated where it is needed, with potential benefits in terms of selectivity and treatment quality. These results provide guidance for the design and comparative assessment of cavitation reactors.

The next step is experimental: measuring, calibrating and benchmarking these geometries against circular throats at the same scale and with the same average power, using in-line measurements and reproducible protocols. The framework is intended to be useful to researchers, process engineers, quality and safety managers, and energy and sustainability officers.

More broadly, the proposed picture is consistent with an efficiency-oriented view of conversion processes and their integration into energy systems. It supports design, control and scalability criteria for process engineering, promotes clean technologies and favours efficient resource use and circular-economy strategies.

Within this logic, the VRA geometry aims at a higher density and greater uniformity of cavitation events, whereas the swirling variant strengthens their localisation and can shorten treatment times and reduce energy consumption at a given process target.