Correction: Albanese, L. The Venturi Reuleaux Triangle: Advancing Sustainable Process Intensification Through Controlled Hydrodynamic Cavitation in Food, Water, and Industrial Applications. Sustainability 2025, 17, 6812

The author would like to make the following corrections in the published paper [1]. The changes are as follows:

(1)

Replacing the phrase “a significant” with the word “an”

In the abstract, the phrase “a significant” should be replaced with the word “an” in the following sentence:

• However, the effectiveness of cavitation critically depends on internal geometry—particularly the perimeter-to-area ratio (P/A), which influences both pressure gradient distribution and the density of nucleation sites. In this context, an innovative configuration based on the Reuleaux triangle is proposed, allowing for a significant increase in the P/A ratio compared to conventional circular-section devices.
• However, the effectiveness of cavitation critically depends on internal geometry—particularly the perimeter-to-area ratio (P/A), which influences both pressure gradient distribution and the density of nucleation sites. In this context, an innovative configuration based on the Reuleaux triangle is proposed, allowing for an increase in the P/A ratio compared to conventional circular-section devices.

(2)

Replacing the word “α” with the word “P/A”

On page 3, the word “α” should be replaced with word “P/A” in the following sentence:

• Circular sections, having lower α values compared to rectangular or elliptical ones, tend to limit the intensity of cavitation phenomena.
• Circular sections, having lower P/A values compared to rectangular or elliptical ones, tend to limit the intensity of cavitation phenomena.

(3)

Replacing the word “$A_{crlx}$’’ with the word “$A_{rlx}$”

On page 5, the word “$A_{crlx}$” should be replaced with the word “$A_{rlx}$” in the following sentence:

• When the areas are equal $A_{circ}=A_{crlx}$, this yields
• When the areas are equal $A_{circ}=A_{rlx}$, this yields

(4)

Replacing the equations in “Section 2. The Role of the Perimeter-to-Area Ratio (P/A) in the Design of Cavitating Devices” on page 5:

Equation (5),

$${\displaystyle \frac{P_{rlx}}{A_{rlx}}}={\displaystyle \frac{4\pi }{3\left(\pi -3\right)L}}$$

(5)

with

$${\left({\displaystyle \frac{P}{A}}\right)}_{rlx}={\displaystyle \frac{\pi L}{\left(\frac{\pi -\sqrt{3}}{2}\right)L^2}}={\displaystyle \frac{2\pi }{\left(\pi -\sqrt{3}\right)L}}$$

(5)

Equation (6),

$$r=L\sqrt{{\displaystyle \frac{\pi -3}{2\pi }}}$$

(6)

with

$$r=L\sqrt{{\displaystyle \frac{\pi -\sqrt{3}}{2\pi }}}$$

(6)

Equation (7),

$${\displaystyle \frac{∆P_{rlx}}{∆A_{circ}}}={\displaystyle \frac{{\left(\frac{P}{A}\right)}_{rlx}}{{\left(\frac{P}{A}\right)}_{circ}}}\gtrsim 2.1$$

(7)

with

$${\displaystyle \frac{∆P_{rlx}}{∆P_{circ}}}={\displaystyle \frac{{\left(\frac{P}{A}\right)}_{rlx}}{{\left(\frac{P}{A}\right)}_{circ}}}={\displaystyle \frac{\pi }{\sqrt{2\pi (\pi -\sqrt{3)}}}}\approx 1.06$$

(7)

(5)

The author would like to change two parts of the table content, so we need to replace the original Table 1:

Table 1. Predicted theoretical parameters for various Venturi configurations, expressed as relative ratios with respect to baseline circular device (baseline = 1).

Configuration	Section Shape	Swirl	Swirl Direction	$∆\mathit{p}$
Circular	Circular	No	-/-	1
VRA	Reuleaux	No	-/-	≈2.1
VRAt-CR	Reuleaux	Yes	↑/↑	>2.1
VRAt-RR	Reuleaux	Yes	↑/↓	≈3

Note: All values, except those for the baseline circular device, are theoretical estimates derived from the analytical model (Equations (4)–(22)), which incorporates the effects of the increased perimeter-to-area ratio and the centrifugal contribution generated by swirl in the VRAt configurations. Swirl directions are denoted as ↑ (clockwise) and ↓ (counterclockwise) for the converging and diverging sections, respectively; the symbol “-” indicates the absence of rotation. This arrow-based notation is adopted to ensure maximum clarity in graphical representations. For the VRAt-RR configuration, the reported values represent the upper limit predicted by the model under conditions of maximum pressure gradient intensity, ideally resulting from an optimal reversal of the rotational component. These estimates should be validated and quantified through targeted experimental investigations and dedicated numerical simulations.

Table 1. Predicted theoretical parameters for various Venturi configurations, expressed as relative ratios with respect to baseline circular device (baseline = 1).

Configuration	Section Shape	Swirl	Swirl Direction	$∆\mathit{p}$
Circular	Circular	No	-/-	1
VRA	Reuleaux	No	-/-	≈2.1
VRAt-CR	Reuleaux	Yes	↑/↑	>2.1
VRAt-RR	Reuleaux	Yes	↑/↓	≈3

Note: All values, except those for the baseline circular device, are theoretical estimates derived from the analytical model (Equations (4)–(22)), which incorporates the effects of the increased perimeter-to-area ratio and the centrifugal contribution generated by swirl in the VRAt configurations. Swirl directions are denoted as ↑ (clockwise) and ↓ (counterclockwise) for the converging and diverging sections, respectively; the symbol “-” indicates the absence of rotation. This arrow-based notation is adopted to ensure maximum clarity in graphical representations. For the VRAt-RR configuration, the reported values represent the upper limit predicted by the model under conditions of maximum pressure gradient intensity, ideally resulting from an optimal reversal of the rotational component. These estimates should be validated and quantified through targeted experimental investigations and dedicated numerical simulations.

with

Table 1. Predicted theoretical parameters for various Venturi configurations, expressed as relative ratios with respect to baseline circular device (baseline = 1).

Configuration	Section Shape	Swirl	Swirl Direction	$∆\mathit{p}$
Circular	Circular	No	-/-	1
VRA	Reuleaux	No	-/-	≈1.06
VRAt-CR	Reuleaux	Yes	↑/↑	>1.06
VRAt-RR	Reuleaux	Yes	↑/↓	≈3

Note: All values, except those for the baseline circular device, are theoretical estimates derived from the analytical model (Equations (4)–(22)), which incorporates the effects of the increased perimeter-to-area ratio and the centrifugal contribution generated by swirl in the VRAt configurations. Swirl directions are denoted as ↑ (clockwise) and ↓ (counterclockwise) for the converging and diverging sections, respectively; the symbol “-” indicates the absence of rotation. This arrow-based notation is adopted to ensure maximum clarity in graphical representations. For the VRAt-RR configuration, the reported values represent the upper limit predicted by the model under conditions of maximum pressure gradient intensity, ideally resulting from an optimal reversal of the rotational component. These estimates should be validated and quantified through targeted experimental investigations and dedicated numerical simulations.

Table 1. Predicted theoretical parameters for various Venturi configurations, expressed as relative ratios with respect to baseline circular device (baseline = 1).

Configuration	Section Shape	Swirl	Swirl Direction	$∆\mathit{p}$
Circular	Circular	No	-/-	1
VRA	Reuleaux	No	-/-	≈1.06
VRAt-CR	Reuleaux	Yes	↑/↑	>1.06
VRAt-RR	Reuleaux	Yes	↑/↓	≈3

Note: All values, except those for the baseline circular device, are theoretical estimates derived from the analytical model (Equations (4)–(22)), which incorporates the effects of the increased perimeter-to-area ratio and the centrifugal contribution generated by swirl in the VRAt configurations. Swirl directions are denoted as ↑ (clockwise) and ↓ (counterclockwise) for the converging and diverging sections, respectively; the symbol “-” indicates the absence of rotation. This arrow-based notation is adopted to ensure maximum clarity in graphical representations. For the VRAt-RR configuration, the reported values represent the upper limit predicted by the model under conditions of maximum pressure gradient intensity, ideally resulting from an optimal reversal of the rotational component. These estimates should be validated and quantified through targeted experimental investigations and dedicated numerical simulations.

(6)

Replacing the word “2” with the word “1.06”

On page 8, the word “2” should be replaced with word “1.06” in the following sentence:

• 2. Reuleaux Venturi (VRA), with a pressure drop $∆p\approx 2{∆p}_{circ}$;
• 2. Reuleaux Venturi (VRA), with a pressure drop $∆p\approx 1.06{∆p}_{circ}$;

(7)

Replacing the following sentence in “Implosion Performance: Introduction and Overview” on page 9:

The model specifically predicts that the collapse time could decrease from 1 to approximately 0.67 $t_{coll}$ compared to traditional circular devices, corresponding to a preliminary energy savings estimate in the range of 30–35%.

with

The model specifically predicts that the collapse time could decrease from 1 to approximately 0.97 $t_{coll}$ compared to traditional circular devices.

(8)

On page 9, the word “values” should be replaced with the word “estimates” in the following sentence:

• These values are indicative and should be verified through dedicated experimental validation.
• These estimates are indicative and should be verified through dedicated experimental validation.

(9)

On page 9, the phrase “more than twice that” should be replaced with the phrase “higher than that” in the following sentence:

• In VRA devices, the increased perimeter-to-area ratio leads to a pressure drop more than twice that of the circular configuration, resulting in a corresponding de-crease in collapse time.
• In VRA devices, the increased perimeter-to-area ratio leads to a pressure drop higher than that of the circular configuration, resulting in a corresponding decrease in collapse time.

(10)

Replacing the equations in “Section Implosion Performance: Introduction and Overview” on page 9:

$$t_{coll,VRA}={\displaystyle \frac{t_{coll,circ}}{\sqrt{2.1}}}\approx 0.67t_{coll,circ}$$

with

$$t_{coll,VRA}={\displaystyle \frac{t_{coll,circ}}{\sqrt{1.06}}}\approx 0.97t_{coll,circ}$$

Equation (14)

$$v_{impl}=1093\sqrt{{\displaystyle \frac{∆p}{\rho }}}$$

(14)

with

$$v_{impl}=1.093\sqrt{{\displaystyle \frac{∆p}{\rho }}}$$

(14)

(11)

Replacing the following paragraphs in “Section 5. Conclusions” on page 12:

Although direct experimental data are not yet available, the model indicates that the reduced collapse time in VRA devices could yield energy savings in the range of 30–35% compared to conventional circular geometries, due to more than double the pressure drop. These estimates are preliminary and will require confirmation through future experimental studies.

The use of dimensionless parameters such as Φ, axial vorticity ${\omega }_z$, shear layer intensity ${\Gamma }_S$, and the modified cavitation number ${\sigma }_{mod}$ enables the design of cavitating devices that are scalable, selective, and efficient. From an industrial standpoint, this opens the door to integrating VRA and VRAt devices into existing process lines, offering increased operational flexibility, precise control, and potentially enhanced energy efficiency.

with

Although direct experimental data are not yet available, the model indicates that the reduced collapse time in VRA devices could improve energy efficiency compared to conventional circular geometries. These estimates are indicative and should be verified through dedicated experimental validation. The use of dimensionless parameters such as Φ, axial vorticity ${\omega }_z$, shear layer intensity ${\Gamma }_S$, and the modified cavitation number ${\sigma }_{mod}$ enables the design of cavitating devices that are scalable, selective, and efficient. From an industrial standpoint, this opens the door to integrating VRA and VRAt devices into existing process lines, offering increased operational flexibility, precise control, and potentially enhanced energy efficiency.

The author states that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.