Parametric Characterization and Multi-Objective Optimization of Low-Pressure Abrasive Water Jets for Biofouling Removal from Net Cages Using Response Surface Methodology and the Entropy Method

Abstract

Deep-sea cages are highly susceptible to biofouling due to long-term seawater immersion, which promotes the attachment and growth of marine organisms on nets, significantly reducing fish survival. To address this issue, this study explores the use of low-pressure abrasive water jets (LPAWJs) for cage fouling removal through numerical simulation. Based on a Box-Behnken response surface design, nozzle inlet pressure X₁, nozzle outlet diameter X₂, and target distance X₃ were selected as optimization parameters. The peak jet impact force Z₁, stable jet impact force Z₂, peak abrasive water jet velocity Z₃, and peak abrasive particle velocity Z₄ were chosen as evaluation indicators to characterize the jet’s instantaneous impact ability, sustained action ability, and dynamic particle behavior. Using the entropy method, weights for each indicator were determined, and the jet’s overall removal capability was calculated. A regression model was developed by integrating numerical simulation with the response surface methodology (RSM), and the optimal parameter combination was identified as X₁ = 4.5 MPa, X₂ = 10 mm, and X₃ = 205.396 mm. Compared with the poorest experimental condition (Condition 1), the jet’s overall removal capability obtained under the optimal parameter combination increases by 101.35%. Experimental validation further confirms that the optimized parameters yield the best oyster-removal performance of the low-pressure abrasive jet, with the average removal rate improving by 100.55% relative to Condition 1. The methodology and results of this study provide a theoretical foundation and technical reference for the design and optimization of automated net-cleaning systems or net-cleaning robots equipped with low-pressure abrasive jets. By integrating the proposed model and operating parameters, future robotic systems will be able to predict and dynamically adjust jet conditions according to fouling characteristics, thereby improving the efficiency, cost-effectiveness, and sustainability of maintenance operations in marine aquaculture.

1. Introduction

Since the Reform and Opening-up, China’s mariculture industry has expanded rapidly under market and policy support. However, long-term high-density coastal farming has caused marine pollution, water-quality degradation, and production constraints in nearshore cage aquaculture. In contrast, deep-sea cage aquaculture has attracted significant attention in China due to its advantages of high productivity, low environmental impact, and high-quality aquatic products. It is regarded as an important approach for structural adjustment in fisheries, facilitating the sustainable development of deep-sea aquaculture, alleviating nearshore aquaculture pressure, and reducing environmental pollution [1].

A deep-sea cage mainly consists of a float system, a netting system, a mooring system, and an operation platform. Among them, the netting system, being constantly immersed in seawater, provides a surface for algae, shellfish, and other organisms to attach and proliferate rapidly. This fouling causes mesh clogging, restricts water exchange, and reduces dissolved oxygen and food availability [2]. Therefore, the cleanliness of the netting is directly related to fish growth and survival rate [3].

Traditional cleaning methods often rely on high-pressure water jets and mechanical friction to remove biofouling. For instance, Zhang et al. [4] designed a manifold-type underwater high-pressure water jet cleaning machine that utilized cavitation jet impact to address net cleaning issues. Zhuang et al. [5] developed a swirling-type deep-sea net cleaning device integrated with underwater robots, where manifold-type rotating water jets served as the cleaning power source. A specially designed dual-float system adjusted underwater posture and direction, enabling cleaning in multiple orientations. Song et al. [6] developed a deep-sea cage net cleaning system composed of a surface workboat and an underwater cleaning device. The workboat was equipped with a small generator and crane, while rotary brushes performed intense frictional cleaning on the net surface to remove fouling organisms. Although high-pressure water jets are efficient, their application is constrained by the pressure-bearing capacity of the netting. To address this limitation, this study incorporates abrasives into the jet, transforming continuous water impact into particle cutting and enabling strong removal capability under low-pressure conditions [7]. Xiong et al. [8] experimentally compared the paint removal performance of low-pressure water jets with that of LPAWJs, confirming the superior effectiveness of the latter. The nozzle structure is a critical factor influencing the efficiency of abrasive water jets. Related studies have employed orthogonal experimental designs and RSM to optimize nozzle parameters and enhance cleaning performance [9]. Qiu et al. [10] utilized RSM combined with numerical simulation to optimize the high-pressure water jet parameters for cleaning deep-sea mining vehicles and identified the optimal combination for maximum impact performance. Wang et al. [11] applied orthogonal design to investigate the effects of key structural parameters on spray nozzle atomization and dust suppression performance, deriving the influence laws of nozzle design on efficiency. Sun et al. [12] studied a conventional Helmholtz nozzle by adding an expansion tube structure at the outlet to enhance cavitation effects, and numerically analyzed the effects of nozzle cavity height, cavity width, expansion angle, and pump pressure on cavitation jet performance. Previous studies have often focused on optimizing single indicators such as jet impact force, jet exit velocity, or jet flushing width, or conducted simultaneous optimization of multiple indicators [13]. However, these approaches either lacked comprehensiveness or required complex experimental setups. To balance efficiency and accuracy, this study introduces the entropy method to assign objective weights to multiple evaluation indicators, enabling a comprehensive assessment of cleaning performance across experimental conditions. The entropy method, known for its objectivity and adaptability, has been widely applied in environmental and enterprise performance evaluations [14].

Compared with high-pressure water jets and mechanical brushing, the LPAWJ system offers clear cost and operational advantages. High-pressure jets require large, energy-intensive pumps and can accelerate net-pen degradation, while mechanical brushing depends heavily on labor and causes significant wear. LPAWJ operates at much lower pressures, reducing pump power demand, equipment wear, and net damage. Although quartz-sand abrasives introduce a modest consumable cost, the system’s low energy use, minimal maintenance needs, and compatibility with automated cleaning platforms make its overall operating cost highly competitive.

In this study, oysters were selected as representative fouling organisms. By combining numerical simulation, the Box-Behnken design, and the entropy method, we examined the effects of nozzle inlet pressure, outlet diameter, and target distance on the jet’s overall removal capability, established a regression model, and determined the optimal parameters. The main contributions are: (1) proposing a dimensionless evaluation index—the jet’s overall removal capability—constructed via the entropy method; (2) revealing the coupled influence mechanisms of multiple parameters through simulation and regression analysis; and (3) validating the numerical and response surface models using a self-developed LPAWJ experimental platform. These advances establish an integrated framework for optimizing LPAWJ removal in marine aquaculture.

2. Methods

2.1. CFD Method

FLUENT is a widely used commercial computational fluid dynamics (CFD) software capable of simulating turbulence, multiphase flow, and transient dynamics. In this study, FLUENT is selected as the simulation tool to perform numerical analysis of the flow field resulting from the mixing of low-pressure water jets with abrasives for the removal of oyster biofouling on cage net.

2.1.1. Geometric Model

Common nozzle types include cylindrical [15], fan-shaped [16], and custom-shaped designs [17]. In this study, based on the application of LPAWJ, a commercially available conical-straight nozzle was selected for modeling. Quartz sand was chosen as the abrasive, and a premixing method was employed to mix water and sand. Quartz sand was selected due to its chemical inertness, non-toxicity, and natural abundance in marine sediments; although it may temporarily increase turbidity, such effects dissipate rapidly through dilution and particle settling. As shown in Figure 1a, the cleaning assembly consists of a water inlet interface, sand inlet interface, mixing chamber, and nozzle. High-pressure water enters the mixing chamber and creates a negative pressure that draws in the quartz sand. The water-abrasive mixture is then expelled through the nozzle to remove oyster fouling. The nozzle structure, shown in Figure 1b, mainly comprises an inlet, contraction section, compression section, and outlet. Considering that the outlet diameter significantly affects abrasive velocity and jet impact force [18], it is selected as the key optimization parameter in this study. Other geometric parameters are kept constant: inlet diameter of 13.42 mm, contraction angle of 20°, compression section length of 24.8 mm, while the contraction section length is adjusted accordingly with the outlet diameter.

For numerical simulation, the flow field model of the nozzle was first constructed in DesignModeler, as shown in Figure 2. Based on statistical data, oysters were simplified as a cuboid measuring 60 × 40 × 3 mm. Simulations indicate that abrasives form an effective “abrasive velocity core” [19] within 100–300 mm from the nozzle exit. Therefore, the target distance is set within the 100–300 mm range and is considered a key flow parameter. To prevent divergence in the flow field, the lateral width outside the nozzle is set to 20 mm, ensuring both energy efficiency and cleaning performance.

The geometric model was imported into Meshing, where the nozzle region was discretized using a sweep method to ensure vertical continuity. Due to the steep pressure gradients inside the nozzle, additional sweep partitions were introduced for local mesh refinement, enhancing the accuracy in capturing flow characteristics. The external flow field and oyster region were meshed using hexahedral elements. The final mesh configuration is shown in Figure 3.

2.1.2. Boundary Conditions and Solution Settings

To improve the accuracy of the numerical simulation, appropriate boundary conditions were defined in this study. The nozzle inlet was set as a pressure inlet, the outlet as a pressure outlet, and all other surfaces were defined as no-slip adiabatic walls. The nozzle outlet pressure was fixed at 101,325 Pa. According to Bernoulli’s principle, the inlet pressure directly affects the jet exit velocity, which in turn influences the effectiveness of fouling removal. Therefore, inlet pressure was selected as a key parameter. Quartz sand particles were modeled as rigid spheres of uniform size, with detailed parameters listed in

Table 1. Parameters of quartz sand particles.

Particle Density (kg/m3)	Particle Size (mm)	Particle Mass Flow Rate (kg/s)
2600	0.6	0.075

. At wall boundaries, particles were assigned the “Reflect” condition. At the outlet, the “Escape” condition was applied, allowing particles to exit the flow within [20].

Since the oyster removal process by LPAWJ involves both water and air phases, the Volume of Fluid (VOF) model with transient tracking capability was employed [21], with the two phases set as immiscible. To enhance pressure calculation accuracy, the Pressure-Implicit with Splitting of Operators (PISO) algorithm was used for transient solution [22].

To validate the simulation results, a comparison was made between the theoretical and simulated jet velocities at the nozzle inlet under different inlet pressures. Based on Bernoulli’s principle, the relationship between jet velocity and inlet pressure can be simplified as:

$$v=\sqrt{{\displaystyle \frac{2P}{\rho }}}$$

(1)

where $v$ is the water jet velocity at the nozzle inlet, $P$ is the inlet pressure, and $\rho$ is the water density.

A comparison of the theoretical and simulated inlet jet velocities is shown in

Table 2. Comparison of theoretical and simulated water jet velocities at the nozzle inlet.

Inlet Pressure (MPa)	Theoretical Water Jet Velocity (m/s)	Simulated Water Jet Velocity (m/s)	Error
2.5	70.70	70.77	0.099%
3.5	83.66	83.74	0.096%
4.5	94.86	94.95	0.095%

. The results indicate that the error between the simulated and theoretical velocities ranges from 0.095% to 0.01%, demonstrating strong agreement and confirming the high accuracy of the numerical simulation.

2.1.3. Mesh Independence Verification

Generally, mesh size has a significant impact on the accuracy of flow within calculations [23]. In this study, four meshing schemes were developed by adjusting the mesh size under the conditions of a nozzle inlet pressure of 3.5 MPa, nozzle outlet diameter of 6 mm, particle mass flow rate of 0.1 kg/s, and target distance of 100 mm. As shown in

Table 3. Grid independence verification cases and results.

Solution	W1	W2	W3	W4
Mesh cell	15,547	19,413	22,586	27,306
Fluid outlet velocity (m/s)	60.2347	76.6049	76.5471	76.5427
Peak jet impact force (Pa)	3,643,625	4,145,702	4,148,243	4,149,645

, the evaluation indicators used include the fluid outlet velocity and the peak jet impact force. The simulation results from schemes W₁ to W₄ indicate that when the mesh count is 15,547 (W₁), the resolution is too coarse, leading to significant computational deviation. When the mesh count exceeds 22,586, the variations in both evaluation indicators become relatively small. Considering both computational accuracy and efficiency, mesh scheme W₃ was selected as the optimal configuration for this study.

2.2. Multi-Parameter Optimization Design Method

The Box-Behnken design is an advanced statistical experimental design method within the framework of RSM, capable of systematically evaluating the effects of multiple independent variables and their interactions on a response variable [24]. Compared to full factorial designs, the Box-Behnken approach requires fewer experimental runs, thereby saving time and resources. Moreover, it avoids extreme parameter combinations, which helps prevent divergence in simulations. Therefore, the Box-Behnken design was employed to investigate the fouling removal characteristics of LPAWJ for cage net cleaning.

2.2.1. Selection of Optimization Parameters

The optimization parameters in this study include nozzle inlet pressure X₁, nozzle outlet diameter X_2, and target distance X₃. While the parameter ranges for X₂ and X₃ have been analyzed in the preceding sections, determining the appropriate range for the nozzle inlet pressure X₁ is particularly critical.

Numerical simulations were performed on the flow field of LPAWJ removing oysters attached to cage nets. When the jet impact force exceeds the compressive strength of the oyster shell, localized stress concentration on the shell surface leads to the initiation and propagation of microcracks, resulting in a reduction in the shell’s structural stiffness. As the cracks further develop, the interfacial bonding strength between the oyster and the net substrate gradually weakens, which significantly reduces the adhesion strength, enabling the jet to achieve effective removal. Therefore, it is essential to first determine the compressive limit of the oyster shell. Due to the lack of relevant data in existing literature, compression tests were conducted using a tensile-compression testing machine on 20 oysters of varying shapes, focusing on the thickest part of each shell. The experimental setup for the oyster compression test is illustrated in Figure 4. As shown in Figure 5, the minimum critical compressive load of the oyster shell is 183 N, the maximum critical compressive load is 273 N, and the average critical compressive load is 228 N. This dispersion primarily arises from the natural heterogeneity of oyster shells, including local thickness variations, irregular curvature, and microstructural anisotropy. By uniformly applying ink to the contact surface of the indenter, the contact area between the indenter and oyster shell was determined to be approximately 7.6 × 10⁻⁵ m². This leads to an estimated compressive strength of the oyster shell of about 2.3~3.5 MPa. Based on this compressive strength and the previously determined ranges for X₂ and X₃, the suitable range for nozzle inlet pressure X₁ was established through numerical simulations. The experimental ranges and levels for all optimization parameters are presented in

Table 4. Response surface experiment parameter levels.

Optimization Parameter	Parameter Code	Unit	Lower Level (−1)	Zero Level (0)	Upper Level (1)
Nozzle inlet pressure	X1	MPa	2.5	3.5	4.5
Nozzle outlet diameter	X2	mm	6	8	10
Target distance	X3	mm	100	200	300

.

2.2.2. Selection of Target Response

In multi-scheme evaluations, multi-criteria decision-making methods play a crucial role in enabling scientifically grounded decisions. Among these, the entropy method is a widely adopted objective weighting approach, frequently used for comprehensive assessments. Its procedure involves data normalization, calculation of information entropy, determination of the divergence coefficient, computation of weights, and ultimately, the calculation of a comprehensive score [25]. Information entropy is an indicator used to measure the degree of uncertainty in information. The entropy method, based on the principle of information entropy, quantifies the dispersion of each evaluation indicator by calculating its information entropy value. The rationale for adopting the entropy method in this study lies in its ability to reflect the relative sensitivity of each indicator to variations in the operating parameters—namely nozzle inlet pressure (X₁), nozzle outlet diameter (X₂), and target distance (X₃). The more significantly an indicator varies under different experimental conditions, the greater its informational contribution to the overall evaluation of the jet’s overall removal capability. Its procedure includes data standardization, calculation of information entropy, difference coefficients, weights, and comprehensive scores. In this study, the removal capability of LPAWJ was characterized by four indicators: peak jet impact force Z₁, stable jet impact force Z₂, peak abrasive water jet velocity Z₃, and peak abrasive particle velocity Z₄. Z₁ reflects whether the jet can achieve fouling removal, while Z₂ represents the jet’s sustained removal performance. Both Z₁ and Z₃ are positively correlated, and Z₄ determines the abrasive particle’s impact effect and the associated rebound phenomenon upon contacting the fouling surface. Rebound effects alter the velocity distribution and impact frequency of subsequent particles, thereby influencing removal efficiency. Consequently, these four indicators were selected as evaluation metrics for the jet’s overall removal capability. Based on the results of the Box-Behnken design simulations, the four evaluation indicators were normalized in a dimensionless manner to determine their respective weights. The weighted summation of the four normalized indicators yielded the composite score Y for each experimental condition, representing the jet’s overall removal capability (Dimensionless). The computational process of the entropy method is illustrated in Figure 6.

After defining the optimization parameters and response objective, a total of 17 sets of numerical simulation experiments (including 5 center points) were designed using the coded values from Table 4. Each simulation was repeated three times, and the average values of the four evaluation indicators were obtained using FLUENT 2023 R1. The results are presented in

Table 5. Experimental results.

Run	X1 (MPa)	X2 (mm)	X3 (mm)	Z1 (Pa)	Z2 (Pa)	Z3 (m/s)	Z4 (m/s)	Y
1	3.5	8	200	3,772,636	2,563,217	83.436	83.873	2,131,847
2	3.5	8	200	3,807,649	2,421,355	83.400	83.815	2,092,102
3	2.5	8	300	2,791,840	2,438,962	80.937	80.836	1,772,445
4	3.5	10	100	3,487,516	2,650,094	83.885	82.853	2,071,570
5	4.5	6	200	5,034,319	2,574,198	90.678	88.586	2,622,321
6	4.5	10	200	5,950,920	4,146,674	97.620	95.164	3,399,931
7	3.5	6	300	4,073,072	3,795,392	105.510	78.577	2,671,069
8	3.5	8	200	3,810,632	2,436,794	83.396	83.815	2,098,607
9	3.5	8	200	3,810,632	2,436,794	83.396	83.815	2,098,607
10	4.5	8	100	4,491,512	3,341,128	93.687	91.643	2,642,102
11	3.5	6	100	3,659,127	2,079,154	79.694	78.368	1,921,461
12	2.5	10	200	3,300,805	2,412,436	74.987	73.879	1,926,242
13	3.5	8	200	3,810,632	2,436,794	83.396	83.815	2,098,607
14	3.5	10	300	4,281,541	3,139,775	87.091	86.871	2,502,358
15	2.5	8	100	2,870,546	2,135,317	69.499	68.141	1,688,578
16	2.5	6	200	3,938,491	2,561,902	94.761	84.247	2,184,601
17	4.5	8	300	5,158,842	3,993,358	108.082	107.420	3,090,652

. These indicator data were uploaded to the SPSS 26 platform, where the entropy method was used to compute the weighting coefficients, as shown in

Table 6. Weight coefficients of evaluation index values.

Item	Information Entropy Value	Information Utility Value	Weight Coefficient
Z1 (Pa)	0.888	0.112	32.091
Z2 (Pa)	0.875	0.125	35.937
Z3 (m/s)	0.94	0.06	17.145
Z4 (m/s)	0.948	0.052	14.828

. Results showed that the contribution order of the indicators to the jet’s overall removal capability was Z₂ > Z₁ > Z₃ > Z₄. Using the derived weights, the jet’s overall removal capability for each experimental point was calculated, with the results also listed in Table 5. Subsequently, Design-Expert 10 software was employed to analyze the experimental results, and a response surface model was constructed using the least-squares method [26], with the following expression:

$$\mathrm{Y}={\mathsf{\beta }}_0+\sum _{\mathrm{i}=1}^{\mathrm{k}}{\mathsf{\beta }}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}+\sum _{\mathrm{i}=1}^{\mathrm{k}}{\mathsf{\beta }}_{\mathrm{ii}}{\mathrm{X}}_{\mathrm{i}}^2+\sum _{\mathrm{i},\mathrm{j}}^{\mathrm{k}}{\mathsf{\beta }}_{\mathrm{i},\mathrm{j}}{\mathrm{X}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{j}}\hspace{1em}\mathrm{i}<\mathrm{j}$$

(2)

In the model expression, Y represents the response variable (target response), $X$ denotes the optimization parameters, are the coefficients of the linear terms, ${\beta }_i$ is the slope of parameter $X_i$, is the quadratic term of parameter $X_i$, and ${\beta }_{i,j}$ are the coefficients of the interaction terms between parameters $X_i$ and $X_j$.

To further verify the rationality of using the entropy method, other common multi-criteria decision-making (MCDM) approaches were considered for comparison. The Analytic Hierarchy Process (AHP) is a structured decision method that constructs a hierarchical model of the evaluation problem and determines indicator weights through expert pairwise comparisons and consistency checks. Although widely used, its reliance on subjective judgments makes it unsuitable for this study, where the four indicators (Z₁–Z₄) originate from numerical simulations and require objective, data-driven weighting without human intervention. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) ranks alternatives by calculating their Euclidean distances to a positive ideal solution (best performance across all indicators) and a negative ideal solution (worst performance). Alternatives closer to the positive ideal and farther from the negative ideal are considered superior. Therefore, TOPSIS was applied in SPSS 26 to evaluate the same four indicators and obtain the relative ranking of jet removal capability under all experimental conditions. The ranking produced by TOPSIS was fully consistent with the ranking derived from the entropy-based composite scores in Table 5, demonstrating that the weights generated by the entropy method do not bias the evaluation results. However, because TOPSIS does not perform weighting and cannot produce continuous composite scores required as response variables in the Box-Behnken response surface model, the entropy method remains the most appropriate approach for this study. The TOPSIS ranking and its comparison with the entropy-based results are provided in Appendix B.

3. Results and Discussion

3.1. Regression Equation

Using Design-Expert software to perform multiple regression analysis, the polynomial regression model between the three optimization parameters and the response was obtained as follows:

Y = 8,421,040 − 2,088,820X₁ − 1,208,700X₂ + 8930.94X₃ + 30,489.88X₁X₂ + 911.59X₁X₃

(3)

−539.53X₂X₃ + 331,175X₁² + 74,039.25X₂² − 13.67X₃²

3.2. Analysis of Variance (ANOVA)

Table 7. ANOVA for regression models.

Source	Sum of Squares	df	Mean Squares	F-Value	p-Value
Model	4.912 × 1012	9	5.458 × 1011	245.36	<0.0001
X1	3.439 × 1012	1	3.439 × 1012	1545.99	<0.0001
X2	2.042 × 1010	1	2.042 × 1010	9.18	0.0191
X3	4.371 × 1011	1	4.371 × 1011	196.49	<0.0001
X1X2	1.487 × 1010	1	1.487 × 1010	6.69	0.0362
X1X3	3.324 × 1010	1	3.324 × 1010	14.94	0.0062
X2X3	4.657 × 1010	1	4.657 × 1010	20.94	0.0026
X12	4.618 × 1011	1	4.618 × 1011	207.61	<0.0001
X22	3.693 × 1011	1	3.693 × 1011	166.02	<0.0001
X32	7.868 × 1010	1	7.868 × 1010	35.37	0.0006
Residual	1.557 × 1010	7	2.224 × 109
Lack of Fit	1.457 × 1010	3	4.856 × 109
Pure Error	0	4	0
Cor. Total	4.928 × 1012	16

presents the ANOVA results for the regression model, which examines the relationship between the three optimization parameters and the jet’s overall removal capability. The p-value is used to assess statistical significance, where p < 0.01 indicates a highly significant effect, and p < 0.05 indicates a significant effect [27]. The results show that the overall model has a p-value less than 0.0001, indicating that the response surface model is highly significant. In terms of individual factor effects, both X₁ (nozzle inlet pressure) and X₃ (target distance) have p-value below 0.0001, demonstrating extremely significant influence. X₂ (nozzle outlet diameter) has a p-value of 0.0191, indicating a relatively smaller but still significant effect. Based on the f-value, the factors’ influence on the response follows the order: X₁ > X₃ > X₂. Regarding two-factor interactions, the p-value for X₁ × X₃ and X₂ × X₃ are 0.0062 and 0.0026, respectively, indicating significant interaction effects. The interaction between X₁ and X₂ also shows a significant effect with a p-value of 0.0362, though it is comparatively weaker.

The quality of the regression model was further evaluated using the coefficient of determination (R²), adjusted R² (Adj R²), predicted R² (Pred R²), adequate precision (Adeq Precision), and coefficient of variation (C.V) [28], as shown in

Table 8. Evaluation results of the regression model.

R2	Adj R2	Pred R2	Adeq Precision	C.V
0.9968	0.9928	0.9524	52.447	2.02%

. The results indicate that R², Adj R², and Pred R² are 0.9968, 0.9928, and 0.9524, respectively, all close to 1. The difference between Adj R² and Pred R² is less than 0.2, confirming the model’s strong fit and coefficient reliability. Figure 7 presents a comparative analysis between the predicted values of the jet’s overall removal capability obtained from the regression model and the simulated values derived from numerical simulation and the entropy weighting method. It can be observed that the data points from the 17 experimental cases are almost entirely distributed along or near the y = x line, indicating that the predicted values are highly consistent with the simulated ones. This further demonstrates the strong predictive performance of the regression model. Figure 8 presents the normal probability distribution of the externally studentized residuals. Most residual points lie closely along the reference line, indicating that the residuals generally follow an approximately normal distribution. Most data points cluster around the line with no noticeable systematic curvature or S-shaped pattern, suggesting that there is no significant deviation from normality. Only a few points lie slightly away from the line, which is acceptable for a response surface model and does not indicate the presence of influential outliers. The absence of distinct trends or structure in the residual distribution implies that the Box-Behnken regression model does not exhibit obvious overfitting, even with a limited number of experimental runs. This plot supports the model’s generalizability by showing that the residuals behave randomly and independently, which is consistent with the assumptions required for ANOVA validity. The Adeq Precision is 52.447, far exceeding the threshold of 4, while the C.V is 2.02%, well below 10%, indicating low variability in prediction error [29]. In summary, the regression model demonstrated high reliability in predicting jet’s overall removal capability under varying conditions.

3.3. Single Parameter Analysis

This section examines the influence of the three optimized parameters on the jet’s overall removal capability. As illustrated in Figure 9a, the jet’s overall removal capability exhibits an approximately parabolic increase with rising nozzle inlet pressure. When the inlet pressure exceeds 3.5 MPa, the growth rate becomes notably higher, as the increased pressure enhances the kinetic energy of both water and abrasive particles within the nozzle. Consequently, the peak jet velocity and impact force rise, leading to improved overall removal capability. Figure 9b shows that the jet’s overall removal capability initially decreases and then increases with increasing nozzle outlet diameter, indicating that the nozzle type should be selected according to specific operational requirements. As seen in Figure 9c, the jet’s overall removal capability gradually increases with target distance within the experimental range, but the rate of increase diminishes beyond 200 mm and shows a downward trend when the distance approaches or exceeds 300 mm. This occurs because the jet velocity and impact force begin to decay at larger distances, reducing the kinetic energy transferred to the target surface and thus decreasing the overall removal capability.

3.4. Two-Parameter Analysis

This section analyzes the interactive effects of three parameter combinations on the jet’s overall removal capability. As shown in Figure 10, the combination of nozzle inlet pressure and outlet diameter achieves a peak removal capacity of 3,377,709 when X₁ = 4.5 MPa and X₂ = 10 mm. When the inlet pressure is held constant, the removal capability first decreases and then increases with the outlet diameter. Conversely, with a fixed outlet diameter, the removal capacity increases significantly with the inlet pressure, indicating that inlet pressure has a greater influence. The presence of both circular and elliptical contour lines in the figure suggests that the interaction between the two parameters is not significant [30].

Figure 11 illustrates the interactive effect between nozzle inlet pressure and target distance on the jet’s overall removal capability. The peak value occurs at X₁ = 4.5 MPa and X₃ = 300 mm, corresponding to a removal capacity of 3,225,590. Using the same analytical approach, it is evident that both parameters have significant effects, with inlet pressure exerting a stronger influence. Compared to the first parameter combination, the response surface in this case exhibits a steeper slope and greater gradient, with elliptical contour lines, indicating a more pronounced interaction between the two parameters and a stronger overall effect.

Figure 12 illustrates the interactive effect of nozzle outlet diameter and target distance on the jet’s overall removal capability. The peak value is observed at X₂ = 6 mm and X₃ = 300 mm, with a corresponding removal capacity of 2,621,828. Using the same analytical method, it can be concluded that both parameters have significant effects, with target distance exerting a stronger influence. Compared to the second parameter combination, this response surface exhibits a lower slope and gentler gradient; however, the presence of elliptical contour lines indicates that the interaction between these two parameters remains significant.

3.5. Multi-Parameter Optimization Design and Optimization Results

After establishing the polynomial regression model describing the relationship between the three optimization parameters and the jet’s overall removal capability, the Design-Expert software was used to maximize the objective function. The goal was to identify the optimal combination of parameters—within the defined experimental range—that yields the highest jet’s overall removal capability based on the regression model:

FindMaxY = [X₁, X₂, X₃]^T (2.5 MPa < X₁ < 4.5 MPa, 6 mm < X₂ < 10 mm, 100 mm < X₃ < 300 mm)

As shown in Table 5, experimental point 15 resulted in the lowest removal capability, with both the peak jet impact force Z₁ and the stable jet impact force Z₂ being insufficient to remove oysters. This condition is referred to as Condition 1 in this study. In contrast, experimental point 6 achieved the highest removal capability and is designated as Condition 2. By solving the regression model, the optimal parameter combination was determined to be: nozzle inlet pressure X₁ = 4.5 MPa, nozzle outlet diameter X₂ = 10 mm, and target distance X₃ = 205.396 mm. This optimal condition is referred to as Condition 3, with a predicted maximum jet’s overall removal capability of 3,408,646. A numerical simulation was conducted using this optimized parameter set to obtain the actual values of the evaluation indicators, from which the actual jet’s overall removal capability was calculated, as shown in

Table 9. Comparison of simulation results and optimization results under Condition 3.

Z1 (Pa)	Z2 (Pa)	Z3 (m/s)	Z4 (m/s)	Actual Y Value	Predicted Y Value
6,313,952	4,029,857	93.9754	93.6029	3,474,450	3,408,646

. The deviation between the predicted and actual values was only 1.89%, indicating a high level of reliability in the optimization results.

Figure 13 compares the velocity and pressure contour plots for the three operating conditions. It is evident that the trends of velocity and pressure variations of the abrasive water jet are largely consistent across different conditions. Before reaching the oyster surface, the abrasive jet maintains a relatively high velocity and low pressure. Upon impact with the oyster surface, the pressure rises sharply, generating a substantial jet impact force that facilitates the fragmentation and removal of the oysters. Simultaneously, both the water jet and abrasive particles experience rebound and splashing, resulting in a subsequent reduction in velocity.

Figure 14 compares the axial abrasive water jet velocity, axial abrasive particle velocity, and jet impact force under the three working conditions. From Figure 13 and Figure 14a,b the abrasive water jet and abrasive particle velocities in Conditions 2 and 3 are significantly higher than those in Condition 1, with peak velocities occurring near the oyster surface. Specifically, the peak abrasive water jet velocities in Conditions 2 and 3 increase by 40.46% and 35.22%, respectively, compared to Condition 1, while the peak abrasive particle velocities rise by 39.66% and 37.37%. These increases are attributed to the higher nozzle inlet pressure, which provides greater kinetic energy to the water jet. This energy enhances the entrainment and acceleration of abrasive particles, enabling more effective removal upon contact with the oyster surface. Figure 14c shows that the peak jet impact forces in Conditions 2 and 3 rise by 107.31% and 119.96%, respectively, relative to Condition 1, while their stable impact forces increase by 94.19% and 88.72%. Despite Conditions 2 and 3 sharing identical inlet pressure X₁ and nozzle diameter X₂, the peak jet and abrasive velocities in Condition 2 are slightly higher. However, its impact force is lower than that of Condition 3. This is explained by Figure 15: at a target distance of 200 mm (Condition 2), the abrasive particles have not yet reached the oyster surface when the water jet makes initial contact. In contrast, at 205.396 mm (Condition 3), the slightly reduced jet velocity is offset by the simultaneous impact of both water and abrasives. The synchronized action produces a stronger peak impact force, enhancing the overall removal capability. The jet’s overall removal capability under Condition 3 is improved by 101.35% compared with Condition 1, highlighting the superior performance of Condition 3 as the optimal parameter combination.

4. Test

4.1. Test Platform

In this study, an LPAWJ cleaning test platform was constructed, as illustrated in Figure 16. The platform consists of a mobile water jet cleaner, an abrasive tank, a two-phase flow nozzle, the fouled cage net to be cleaned, and a distance adjustment device. One end of the nozzle is connected to the spray gun of the water jet cleaner, while the other end is connected to the sand hose of the abrasive tank. The spray gun is movably fixed at the cleaning position using a custom-designed clamping device.

In the distance adjustment system, two aluminum profiles serve as the base fixed on a wooden board, while the other two profiles are movably connected to the base using angle brackets and bolts. The fouled cage net is secured between the aluminum profiles using nylon ties.

To obtain oyster-fouled netting samples, 54 nets of uniform size (400 × 300 mm) were immersed and suspended for two days at an oyster aquaculture site located in Tong’an District, Xiamen, Fujian Province, China.

To evaluate the detachment performance of the abrasive jet under the optimal parameter combination, cleaning experiments were conducted for all 17 operating conditions listed in Table 5 (three replicates per condition), and the oyster removal rates for Conditions 1–3 were compared. The average oyster removal rates for the remaining 15 conditions are provided in

Table A3. Average oyster removal rates for the remaining 15 conditions.

Run	X1 (MPa)	X2 (mm)	X3 (mm)	Y	Average Oyster Removal Rate (%)
1	3.5	8	200	2,131,847	20.05
2	3.5	8	200	2,092,102	19.80
3	2.5	8	300	1,772,445	17.73
4	3.5	10	100	2,071,570	19.46
5	4.5	6	200	2,622,321	24.04
7	3.5	6	300	2,671,069	25.02
8	3.5	8	200	2,098,607	19.76
9	3.5	8	200	2,098,607	19.54
10	4.5	8	100	2,642,102	24.80
11	3.5	6	100	1,921,461	18.85
12	2.5	10	200	1,926,242	18.71
13	3.5	8	200	2,098,607	20.69
14	3.5	10	300	2,502,358	23.29
16	2.5	6	200	2,184,601	20.24
17	4.5	8	300	3,090,652	29.39

of Appendix D. After the preparation work was completed—including setting the nozzle inlet pressure, replacing nozzles with different outlet diameters, and adjusting the target distance—the water jet cleaner and sandblasting unit were activated. Once the water and quartz sand were uniformly mixed within the nozzle and produced a stable jet, the cleaning machine was moved at a speed of 0.6 m/min. (Due to experimental constraints, the cleaning machine could only remove oysters located in the central region of the net). The test concluded when the abrasive jet fully passed over the net surface.

4.2. Experimental Results and Analysis

To objectively evaluate the removal effectiveness of the abrasive jet, the nets before and after cleaning were detached from the aluminum frame and placed on a black plastic background, which created a clear contrast between the oysters and the net for subsequent image processing. The results are shown in Figure 17, Figure 18 and Figure 19 (each figure, from top to bottom, corresponds to the first to third replicate experiments under each operating condition). The before-and-after images of the remaining 15 experimental groups (since the three repeated tests in each group showed minimal variation, only one representative set of images was selected for display) are also provided in Appendix D. The images on the left represent the nets before cleaning, while those on the right show the nets after cleaning. In MATLAB R2023a, image processing was performed by identifying the brightness and saturation differences between the pre- and post-cleaning images to extract the oyster-covered regions. The code is provided in the Appendix C. The oyster coverage rate was calculated for both cases, and the oyster removal rate was then determined using the following equation:

$$\mathrm{R}\mathrm{r}={\displaystyle \frac{\mathrm{R}\mathrm{b}-\mathrm{R}\mathrm{a}}{\mathrm{R}\mathrm{b}}}$$

(4)

where Rr represents the oyster removal rate, Rb denotes the oyster coverage rate before cleaning, and Ra denotes the oyster coverage rate after cleaning.

The oyster removal rates under the three operating conditions are presented in

Table 10. Oyster removal rates under the three operating conditions.

	Condition 1			Condition 2			Condition 3
	Rb (%)	Ra (%)	R (%)	Rb (%)	Ra (%)	R (%)	Rb (%)	Ra (%)	R (%)
Group 1	61.43	51.38	16.35	54.64	38.18	30.11	61.95	41.59	32.85
Group 2	57.26	47.94	16.27	54.35	38.01	30.05	62.31	42.14	32.37
Group 3	61.27	51.43	16.06	53.91	37.77	29.93	59.16	39.97	32.44

. The calculated average removal rates were 16.23%, 31.46%, and 32.55% for Conditions 1, 2, and 3, respectively. As shown in the images, under Condition 1, the abrasive jet could only remove smaller oysters or those with greater shell curvature (since larger curvature corresponds to a smaller contact area and weaker adhesion), demonstrating limited removal capability. However, a substantial number of oysters remained attached after cleaning, indicating a relatively low overall jet removal capability. Under Condition 2, oyster adhesion on the net was significantly reduced, and the abrasive jet could remove larger oyster individuals, exhibiting improved removal performance. Under Condition 3, corresponding to the optimized parameter combination, nearly no oysters remained on the cleaned net, and the removal efficiency of the abrasive jet was the highest. Compared with Condition 1, the average removal rate under Condition 3 increased by 100.55%. These experimental results further verified the predictions of the simulation and the response surface model.

5. Conclusions

This study integrates numerical simulation, the Box-Behnken response surface method, and the entropy method to analyze the effects of nozzle inlet pressure, nozzle outlet diameter, and target distance on the jet’s overall removal capability. A regression model was established, and the optimal parameter combination was determined. The main conclusions are as follows:

(1)

Evaluation Indicators and Weighting: Four indicators were selected to evaluate the jet’s overall removal capability: peak jet impact force Z₁, stable jet impact force Z₂, peak abrasive water jet velocity Z₃, and peak abrasive particle velocity Z₄. Using the entropy method, the weights of these indicators were determined. Results showed that their contributions to removal capability ranked as follows: Z₂ > Z₁ > Z₃ > Z₄.

(2)

Significance Analysis: Analysis of variance and parameter analysis revealed that all three optimization parameters significantly influenced the jet’s overall removal capability. The order of influence was: nozzle inlet pressure > target distance > nozzle outlet diameter. Furthermore, significant interaction effects were observed between parameters, particularly between nozzle inlet pressure and target distance, and between nozzle outlet diameter and target distance.

(3)

Optimization Results: Optimization based on the response surface regression model yielded the optimal parameter combination: nozzle inlet pressure X₁ = 4.5 MPa, nozzle outlet diameter X₂ = 10 mm, and target distance X₃ = 205.396 mm. Under this parameter combination, the water jet and abrasive particles can simultaneously act on the oyster surface, increasing the jet’s peak impact force. Consequently, the jet’s overall removal capability obtained under this combination is 101.35% higher than that of Condition 1.

(4)

Experimental Results: Along the cleaning path of the water-jet cleaning system, under condition 1, the abrasive jet could only remove small-sized oysters or those with highly curved shell surfaces. Under condition 2, the adhesion of oysters on the net was noticeably reduced after cleaning, and the abrasive jet was able to detach some larger oysters. Under Condition 3, almost no oysters remain attached to the netting after cleaning, and the average oyster removal rate is 100.55% higher than that of Condition 1.

(5)

Application Scenarios: The results demonstrated that the optimized parameter combination significantly enhances the jet’s comprehensive removal capability, and the validated response surface model can serve as a predictive tool for estimating the cleaning performance under different operating conditions. These findings provide a theoretical basis and technical reference for the design and optimization of automated net-cleaning systems or net-cleaning robots equipped with low-pressure abrasive jets. By integrating the proposed model and operating parameters, future robotic systems can predict and adaptively adjust jet conditions according to fouling characteristics, thereby improving the efficiency, economy, and sustainability of marine aquaculture maintenance operations.

(6)

Study’s Limitations and Potential Directions: Although this study offers valuable insights, certain limitations remain. The parametric model considers only three factors—nozzle inlet pressure, standoff distance, and abrasive mass flow rate—while practical net cage cleaning may also be affected by nozzle geometry, abrasive particle characteristics, jet incidence angle, and surface conditions, which were not included in the present analysis. In addition, the jet’s overall removal capability is evaluated using four indicators (peak jet impact force, stable jet impact force, peak abrasive water jet velocity, and peak abrasive particle velocity). These metrics capture the primary detachment mechanisms but may not fully reflect the complex adhesion behavior and heterogeneous fouling structures encountered in real applications. Incorporating additional hydrodynamic and particle–surface interaction parameters could further improve model completeness.