Experimental Research and Parameter Optimization of High-Pressure Abrasive Water Jet Machining

Abstract

Machining of No. 45 steel (AISI 1045) becomes more vital due to its widespread use. In this study, machining performances of abrasive water jet machining (AWJM) of No. 45 steel, including material removal rate, notch depth, and nozzle wear rate, were obtained by experimental and computational results. The Taguchi L16 orthogonal array design was used to study the influence of process parameters on machining performance. The optimal material removal rate and notch depth were achieved when abrasive particle size, operating pressure, and abrasive feed rate were 80 #, 400 MPa, and 840 g/min, respectively. The optimal nozzle wear rate was achieved when abrasive particle size, operating pressure, and abrasive feed rate were 80 #, 400 MPa, and 260 g/min, respectively. When the abrasive particle size is 80 # (namely the mesh number is 80), the particle diameter is usually between 0.18 and 0.25 mm according to the corresponding relationship between the international standard mesh number and particle diameter. Analysis of Variance was conducted to evaluate the statistical significance of the experimental results. Using regression analysis, an empirical model was developed to predict the response values of the AWJM process. Multi-response optimization was then carried out using the Decision Engineering Analysis and Resolution method. The optimal parameter solution for a higher material removal rate, a bigger notch depth, and a smaller nozzle wear rate was achieved when abrasive particle size, operating pressure, and abrasive feed rate were 120 #, 400 MPa, and 870 g/min, respectively.

1. Introduction

The growing demand for sustainable and high-precision machining technologies has driven significant advancements in abrasive water jet machining (AWJM), a non-conventional process that utilizes a high-velocity water jet mixed with abrasive particles to cut a wide range of materials [1,2]. Unlike traditional machining methods, AWJM eliminates heat-affected zones, thermal stress, and tool wear, making it particularly suitable for machining difficult-to-cut materials such as composites, ceramics, and high-strength steels [3]. Among these materials, carbon structural steels, particularly No. 45 steel (AISI 1045), have garnered significant attention due to their widespread use in industries such as automotive, aerospace, and machinery manufacturing. No. 45 steel is characterized by its high strength, excellent wear resistance, and good machinability, making it a preferred material for critical components [4,5]. However, its high hardness and deformation resistance pose challenges for conventional machining processes, necessitating the development of advanced techniques such as AWJM [6]. A comprehensive understanding of the interplay between process parameters and their effects on machining performance is essential.

In recent years, extensive experimental and computational studies have been conducted to investigate the AWJM process, with a focus on understanding the effects of key process parameters—such as abrasive particle size, operating pressure, abrasive feed rate, and traverse speed—on machining performance metrics like material removal rate (MRR), kerf geometry, surface roughness, and nozzle wear. Numerous experimental studies have investigated the AWJ machining of various steel grades. Loschner P [7] studied the effect of cutting speed on surface roughness across the cut surface in AWJ cutting of 316 L austenitic stainless steel. It was found that with the decrease in cutting speed, cut surface quality visibly improved. M. Chithirai Pon Selvan [8] conducted experiments to assess the influence of AWJ cutting process parameters on the depth of cut of stainless steel. Results indicated that by keeping the other parameters considered as constant, an increase in water pressure and abrasive mass flow rate would result in an increase in depth of cut, while the increase in traverse speed would decrease the depth of cut. Then, an empirical model for the prediction of depth of cut was developed using regression analysis. However, the interaction between these parameters is complex, and their combined effects on machining performance are not yet fully understood. To address this, researchers have employed advanced experimental design methods, such as Taguchi orthogonal arrays and response surface methodology (RSM), to systematically analyze the influence of process parameters. Reddy D [9] optimized the AWJM process parameters, such as water pressure, focusing tube size, traverse speed, and abrasive flow rate, for the material removal rate and surface roughness. The optimum response values of material removal rate and surface roughness were, respectively, 5.87 g/min and 2.8 μm. Fuse K [10] combined RSM and a heat-transfer search (HTS) algorithm to optimize the process parameters of AWJM. A maximum material removable rate of 0.2304 g/min, a minimum surface roughness of 2.99 µm, and a minimum kerf taper angle of 1.72 were obtained. Using the scanning electron microscope, the surface morphology revealed that the material-removal mechanism in AWJM was due to ploughing, particle disintegration, and embedding of fractured abrasive particles in the machined surface. Kawecka E [11] applied the Whale Optimization Algorithm to AWJ machining of tool steel. Then, the optimal combination of cutting parameters for achieving the greatest depth of cut was obtained. The depth of cut reached a value of 28.0419 mm. These approaches have provided valuable insights into the optimization of AWJM for specific materials and applications.

Computational modeling has also played a critical role in advancing the understanding of AWJM. Techniques such as computational fluid dynamics (CFD) and finite element analysis (FEA) have been used to simulate the abrasive water jet flow, particle dynamics, and wear mechanisms [12]. Changjiang Chen [13] studied the energy transfer rate in AWJs based on the VOF-DEM method. With an increase in abrasive volume fraction and standoff distance, the kinetic energy of the overall particles decreases. It was concluded that the efficiency could be significantly enhanced by increasing the kinetic energy density of the abrasive group and the ratio of the kinetic energy of the abrasive group to the total energy of the jet. Narayanan [14] analyzed, in detail, the formation mechanism of AWJ. Results showed that abrasive particle breakage had a significant impact on the energy transfer process. The size distribution of the abrasive particles was indispensable in predicting the correct energy flux. A detailed mathematical model was established to predict the energy of abrasive particles leaving the outlet of the focusing tube. The use of the broken probability density function provided better predictions for the energy flux. Zou X [15] employed a CFD-DEM coupling numerical approach to investigate the wear inside the HP-AWJ nozzle. It was concluded that particle kinetic energy, acceleration, and stress concentration variations affected the particle erosion rate on the nozzle wall. These studies have revealed important physical phenomena, including the interaction between abrasive particles and the workpiece, the distribution of kinetic energy, and the wear mechanisms of the nozzle. Despite these advancements, the development of predictive models that can accurately simulate the AWJM process under varying conditions remains a challenge [16].

Recent studies have highlighted the potential of integrating experimental data with machine learning algorithms to enhance the predictive accuracy of AWJM models. For example, Artificial Neural Networks (ANNs) and Genetic Algorithms (GAs) have been used to optimize process parameters and predict machining outcomes with high precision. Andrzej Perec [17] presented the use of ANNs in the modeling of the AWJ cut process of brass. The research confirmed that the ANN was a practical tool for choosing the optimum AWJ machining parameters. Through experimental investigation, Jani S P [18] varied the water jet pressure, nozzle traverse speed, and standoff distance to predict the optimal process condition. It was found that jet pressure had a higher contribution towards kerf wall inclination. Additionally, the application of digital image processing techniques has enabled the real-time monitoring and analysis of kerf characteristics and surface quality [18]. Alrasheed M R A [19] utilized five evolutionary techniques, namely Artificial Neural Network (ANN), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Nonlinear Least Square Error (LSE), to minimize surface roughness, maximize kerf width and maximize material removal rate in the AWJM. This research suggested that nonlinear LSE, SA, and PSO were promising optimization techniques to closely predict optimum machining parameters. These studies have significantly improved the efficiency and reliability of AWJM, paving the way for its adoption in high-precision manufacturing industries.

Despite these advancements, several challenges remain [20,21]. First, most studies have focused on individual process parameters, with limited attention to their combined effects on machining performance. Second, there is limited research on the sustainable optimization of AWJ cutting processes, particularly for widely used materials like No.45 steel. Third, the development of sustainable AWJM processes that minimize energy consumption and environmental impact is an emerging area of research. So this study aims to address these gaps by conducting a comprehensive experimental investigation and parameter optimization of abrasive water jet machining (AWJM) for No. 45 steel. Three process parameters, namely abrasive particle size, operating pressure, and abrasive feed rate, were used to study the machining performances like material removal rate, notch depth, and nozzle wear rate. Furthermore, the experimental research on the three process parameters and optimized analysis of the processing efficiency have not been conducted. In this study, the effects of process parameters on machining performance were studied using the Taguchi orthogonal array. Predictive models that combined these effects were developed to further improve the processing efficiency. Through the Decision Engineering Analysis and Resolution (DEAR) method, the optimal process parameters were obtained. The corresponding machining performances were verified by experiments. The findings provide valuable insights into the optimization of AWJM for sustainable and efficient machining of high-strength materials, contributing to the advancement of green manufacturing technologies.

2. Cutting Experiment with AWJ Nozzle

2.1. Experimental Method

The experiments were carried out on No. 45 steel workpieces with a thickness of 23 mm. The chemical composition and mechanical properties of No. 45 steel are listed in Table 1 and Table 2, respectively. The structure of the CNC abrasive water jet cutting equipment used in the experiments is illustrated in Figure 1a. The equipment model is APW2015BA-16, which is a post-mixing abrasive water jet machining system. The equipment mainly includes a CNC abrasive water jet cutting processing system, high-pressure booster system, abrasive feed control system, equipment control platform, cutting table, waste collection and treatment system. The high-pressure booster system uses a low-pressure plunger pump to supply water to produce 1–23 MPa of low-pressure water, which is then transported to the high-pressure reciprocating plunger pump for secondary pressurization. Then high-pressure pure water is provided. The high-pressure reciprocating plunger pump (the booster ratio is 1:20) can theoretically stabilize the output of 20–450 MPa ultra-high pressure pure water. The abrasive feed control system uses an electromagnetic unit and pneumatic regulation to control the abrasive particle flow, including the start and stop of abrasive sand inlet control. The control range of abrasive flow rate is 70–880 g/min.

During the experiments, the standoff distance between the nozzle and the workpiece was maintained at 2 mm, with a constant nozzle feed speed of 200 mm/min. The workpieces used in the experiments had dimensions of 46 mm × 46 mm × 23 mm. The abrasive water jets were produced at room temperature by mixing filtered pure water with garnet abrasives in an appropriate proportion. In this study, three response parameters—abrasive particle size m_d, abrasive feed rate ṁ_a, and operating pressure P—were varied at four levels, as detailed in Table 3. Figure 1b depicts the composition of the AWJ nozzle cutting experimental setup. The structure of the AWJ nozzle is shown in Figure 2, with its corresponding structural parameters listed in Table 4. An L16 orthogonal array was designed based on the Taguchi method. To ensure complete cutting of each workpiece, a 50 mm straight cut was performed. Each experiment was repeated four times to obtain average values for key parameters, including material removal rate MRR, notch depth w_h, and nozzle wear rate E.

2.2. Measurement Methods

In the experiment, the material removal rate MRR was calculated by Equation (1):

$$MRR={\displaystyle \frac{W_i-W_f}{t}}$$

(1)

In this study, W_i and W_f represent the initial weight and final weight of the material, respectively, measured in grams (g). t denotes the machining time, recorded in seconds (s) [22].

Table 1. Chemical composition list for 45 steel [23].

Material Composition	Value
C	0.42–0.5%
Cr	≤0.25%
Mn	0.6–0.9%
Ni	≤0.25%
P	≤0.035%
S	0.17–0.37%
Si	0.17–0.37%

Table 1. Chemical composition list for 45 steel [23].

Material Composition	Value
C	0.42–0.5%
Cr	≤0.25%
Mn	0.6–0.9%
Ni	≤0.25%
P	≤0.035%
S	0.17–0.37%
Si	0.17–0.37%

Table 2. Mechanical and physical properties of 45 steel [24,25,26].

Mechanical Property	Unit	Value	Physical Property	Unit	Value
Tensile strength	MPa	600	Density	g/cm3	7.85
Yield strength	MPa	355	Poisson’s ratio	/	0.24–0.28
Elongation	%	16	Modulus of elasticity	GPa	196–206
Shrinkage of section	%	40	Thermal conductivity(20 °C)	W/(mgK)	85.29
Hardness	HB	197	Specific heat(20–40 °C)	J/(GgK)	0.939

Table 2. Mechanical and physical properties of 45 steel [24,25,26].

Mechanical Property	Unit	Value	Physical Property	Unit	Value
Tensile strength	MPa	600	Density	g/cm3	7.85
Yield strength	MPa	355	Poisson’s ratio	/	0.24–0.28
Elongation	%	16	Modulus of elasticity	GPa	196–206
Shrinkage of section	%	40	Thermal conductivity(20 °C)	W/(mgK)	85.29
Hardness	HB	197	Specific heat(20–40 °C)	J/(GgK)	0.939

Table 3. Machining parameters of AWJ.

Parameter	Unit	Value	Annotation
Abrasive particle size	#	60, 80, 100, 120	md
Operating pressure	MPa	250, 300, 350, 400	P
Abrasive feed rate	g/min	260, 430, 630, 870	ṁa

Table 3. Machining parameters of AWJ.

Parameter	Unit	Value	Annotation
Abrasive particle size	#	60, 80, 100, 120	md
Operating pressure	MPa	250, 300, 350, 400	P
Abrasive feed rate	g/min	260, 430, 630, 870	ṁa

Table 4. Design parameters of experimental AWJ nozzle model.

Parameter	Value
Particle incidence angle θ (°)	30
Converging angle α (°)	20
Diameter of gem orifice d0 (mm)	0.33
Diameter of mixing chamber d1 (mm)	4
Diameter of focusing tube d2 (mm)	1.02
Diameter of abrasive inlet Da (mm)	3
Length of mixing chamber l1 (mm)	9
Length of focusing tube l2 (mm)	76.2

Table 4. Design parameters of experimental AWJ nozzle model.

Parameter	Value
Particle incidence angle θ (°)	30
Converging angle α (°)	20
Diameter of gem orifice d0 (mm)	0.33
Diameter of mixing chamber d1 (mm)	4
Diameter of focusing tube d2 (mm)	1.02
Diameter of abrasive inlet Da (mm)	3
Length of mixing chamber l1 (mm)	9
Length of focusing tube l2 (mm)	76.2

The notch depth, denoted as w_h, which represents the depth of the slit incision in the material after cutting, was determined by using a digital image processing technique. The measurement setup is illustrated in Figure 3. A high-definition CCD micro-camera was employed to calibrate the pixel dimensions of the captured images, with each pixel corresponding to a precision of 0.01 mm. The images were further grayscale processed. The pixel values at the initial position and the lowest position of the slit incision were obtained, and then the difference between them was calculated to determine the pixel depth. The notch depth was obtained by multiplying the pixel depth data by the pixel accuracy of the camera. The notch depth w_h was ascertained with a measurement accuracy of ±0.01 mm.

In a previous study [15], a CFD-DEM coupling method was applied to study the wall wear phenomenon caused by realistic abrasive particles in AWJ. The nozzle wear rate E of AWJ was extracted from the simulation results. By integrating the erosion rate across the focusing tube of the CFD results, the nozzle wear rate was then obtained by statistical analysis.

Due to the wide range of the obtained response data, the output response parameters were normalized by converting them into Signal-to-Noise Ratio (S/N ratio) values. The corresponding S/N ratio conversion formulas are provided in Equations (2) and (3):

$$S/N_{\mathrm{Larger}\mathrm{the}\mathrm{better}}=10\times \log (y^2)$$

(2)

$$S/N_{\mathrm{Smaller}\mathrm{the}\mathrm{better}}=-10\times \log (1/y^2)$$

(3)

Here y denotes the response variable of the data to be transformed. S/N _{Larger the better} and S/N _{Smaller the better} represent the two criteria of “bigger is better” and “smaller is better” in the output response value, respectively.

3. Results and Discussions

Three response factors, including abrasive particle size m_d, abrasive feed rate ṁ_a, and operating pressure P, were each varied at four levels. Based on the Taguchi experimental method and orthogonal experimental design theory, an L16 orthogonal experiment table was constructed to evaluate the effects of these factors. Finally, experimental results, including material removal rate MRR, notch depth w_h and nozzle wear rate E were statistically analyzed and summarized in

Table 5. Orthogonal experiment results of the green process for AWJM.

Number	Machining Parameters			Response Results
	md (#)	P (MPa)	ṁa (g/min)	MRR (g/s)	wh (mm)	E (-)
1	60	250	260	0.0937	2.5750	2.2618 × 10−4
2	60	300	430	0.1645	3.8025	6.2449 × 10−4
3	60	350	630	0.2417	5.3300	1.3908 × 10−3
4	60	400	870	0.2847	6.1000	7.2890 × 10−4
5	80	250	430	0.1717	4.5125	6.2503 × 10−4
6	80	300	630	0.2345	5.6850	1.3836 × 10−3
7	80	350	870	0.2935	6.8575	2.5774 × 10−3
8	80	400	260	0.2345	6.0800	6.2387 × 10−5
9	100	250	630	0.1696	4.2750	1.4235 × 10−3
10	100	300	870	0.2297	5.7370	2.6862 × 10−3
11	100	350	260	0.1786	4.8350	2.4037 × 10−4
12	100	400	430	0.2705	6.4900	1.9950 × 10−4
13	120	250	870	0.1590	4.4500	2.7186 × 10−3
14	120	300	260	0.1568	4.3270	2.3940 × 10−4
15	120	350	430	0.2285	6.1630	6.3828 × 10−4
16	120	400	630	0.2863	7.3950	4.3591 × 10−4

.

Based on the response results presented in Table 5, Equations (2) and (3) were applied to transform the quantitative relationships of the corresponding results into numerical values with reduced data amplitude variations.

Table 6. S/N ratio of the green processing response results for AWJM.

Number	Machining Parameters			Conversion Results of S/N Ratio
	md (#)	P (MPa)	ṁa (g/min)	MRR (g/s)	wh (mm)	E (-)
1	60	250	260	−20.5637	8.2155	−72.9110
2	60	300	430	−15.6758	11.6014	−64.0895
3	60	350	630	−12.3357	14.5345	−57.1346
4	60	400	870	−10.9128	15.7066	−62.7466
5	80	250	430	−15.3071	13.0883	−64.0820
6	80	300	630	−12.5965	15.0946	−57.1801
7	80	350	870	−10.6469	16.7233	−51.7763
8	80	400	260	−12.5971	15.6781	−84.0981
9	100	250	630	−15.4132	12.6187	−56.9330
10	100	300	870	−12.7755	15.1737	−51.4171
11	100	350	260	−14.9624	13.6879	−72.3823
12	100	400	430	−11.3583	16.2449	−74.0011
13	120	250	870	−15.9730	12.9672	−51.3130
14	120	300	260	−16.0931	12.7237	−72.4176
15	120	350	430	−12.8216	15.7958	−63.8998
16	120	400	630	−10.8631	17.3788	−67.2121

summarizes the Signal-to-Noise Ratio (S/N ratio) conversion results for the responses and green parameters associated with the abrasive water jet machining (AWJM) process. Subsequently, regression analysis was conducted on the transformed data, and the influences of various factors on the response results were further evaluated using Analysis of Variance (ANOVA).

3.1. Effect of Processing Parameters on Material Removal Rate MRR

Based on the above research, the influence of the S/N ratio on MRR was analyzed using the range analysis method, as presented in

Table 7. Range analysis of material removal rate.

	md (#)	P (MPa)	ṁa (g/min)
KM1	−59.4879	−67.2569	−64.2163
KM2	−51.1476	−57.1409	−55.1628
KM3	−54.5093	−50.7665	−51.2084
KM4	−55.7508	−45.7312	−50.3081
kM1	−14.8720	−16.8142	−16.0541
kM2	−12.7869	−14.2852	−13.7907
kM3	−13.6273	−12.6916	−12.8021
kM4	−13.9377	−11.4328	−12.5770
Range R	2.0851	5.3814	3.4770

. In the table, K_Mi denoted the sum of all data at the i-th level for each factor. k_Mi was the average value of K_Mi. The range R is defined as the difference between the maximum and minimum value of K_Mi for each factor. Consequently, the magnitude of range R reflected the relative importance of the corresponding factor. A larger range R indicated a greater influence of the factor on the response variable. It was observed that the range R of operating pressure P was the largest, followed by abrasive feed flow ṁ_a and abrasive particle size m_d. This indicated that operating pressure P had the most significant effect on material removal rate MRR, followed by abrasive feed rate ṁ_a and abrasive particle size m_d.

Figure 4 shows the effect of abrasive particle size, operating pressure, and abrasive feed rate on the material removal rate, which was one of the response parameters in the AWJM process. The ordinate represented the average sum of all results of each factor at the i-th level, corresponding to k_Mi in Table 7. It was observed that as the abrasive size m_d increased, the material removal rate MRR initially increased and then decreased. This trend can be attributed to the impact of particle size on the material removal efficiency. Smaller particles exert less kinetic energy on the workpiece, resulting in reduced surface damage and lower MRR. Conversely, larger particles, when entrained into the high-pressure nozzle under the Venturi effect, may cause blockages at the abrasive inlet of the AWJ nozzle. This leads to inefficient mixing of the abrasive particles with the high-pressure water jet, further reducing the MRR. The lowest material removal rate was observed at an abrasive particle size of 60 #.

As the operating pressure P increased, the material removal rate MRR gradually increased. This is attributed to the higher kinetic energy of the pure water jet, which enhanced the kinetic energy and material removal capacity of the abrasive particles upon mixing, thereby improving the MRR of the jet on the workpiece. When the abrasive feed rate ṁ_a increased, the material removal rate MRR initially increased and then became stable. The increase in abrasive flow rate augmented the kinetic energy of the water jet, enabling the abrasive particles to penetrate deeper into the workpiece and initiate micro-cracks. The material was subsequently removed in the form of chips, leading to an improved MRR. In addition, it was also observed that when the abrasive flow rate exceeded a critical value, the material removal rate tended to be stable. This phenomenon can be explained by the fact that excessive abrasive flow rates resulted in complete mixing of particles with the high-pressure jet within the mixing chamber of the AWJ nozzle, limiting further improvements in MRR. For the material removal rate MRR, the optimization criterion was “Larger-the-Better”. Based on the range analysis, the optimal parameter combination was an abrasive particle size m_d of 80 #, an operating pressure P of 400 MPa, and an abrasive feed rate ṁ_a of 840 g/min. It should be noted that the optimal combination was not included in the L16 orthogonal array experimental design presented in Table 5. This discrepancy arises from the multi-factor nature of Taguchi’s orthogonal experimental design, which would require 3⁴ = 81 experiments to cover all possible combinations.

3.2. Effect of Machining Parameters on Notch Depth w_h

The influence of the S/N ratio on the notch depth w_h was analyzed using the range analysis method, as shown in

Table 8. Range analysis table of notch depth.

	md (#)	P (MPa)	ṁa (g/min)
KH1	50.0581	46.8898	50.3053
KH2	60.5843	54.5934	56.7305
KH3	57.7252	60.7416	59.6266
KH4	58.8655	65.0083	60.5708
kH1	12.5145	11.7225	12.5763
kH2	15.1461	13.6484	14.1826
kH3	14.4313	15.1854	14.9067
kH4	14.7164	16.2521	15.1427
Range R	2.6316	4.5296	2.5664

. In the table, K_Hi was the sum of all data at the i-th level for each factor. k_Hi was the average value of K_Hi. The range R represented the difference between the maximum and minimum value of k_Hi for each factor. From the analysis, it was observed that the range R for operating pressure P was the largest, followed by abrasive particle size m_d and abrasive feed flow ṁ_a. This indicated that operating pressure P had the most significant effect on the notch depth w_h, followed by abrasive particle size m_d and abrasive feed rate ṁ_a.

Figure 5 shows the influence of abrasive particle size m_d, operating pressure P, and abrasive feed rate ṁ_a on the notch depth w_h in the AWJM process. The ordinate represented the average sum of all results for each factor at the i-th level, corresponding to k_Hi in Table 8. It was observed that when the abrasive size m_d increased, the notch depth w_h reached its maximum cutting level at an abrasive size of 80#. The diameter of the abrasive particle significantly influenced the effective sand inlet capacity of the AWJ nozzle and the mixing efficiency within the nozzle. Smaller particles (120 #) had a smaller impact on the material due to their lower kinetic energy. However, compared to 100 # particles, 120 # particles were more effectively entrained into the mixing chamber, enabling better particle acceleration and thus enhancing the cutting performance on No.45 steel. Conversely, when larger particles were introduced into the high-pressure nozzle under the Venturi effect, blockages at the abrasive inlet and focusing tube of the AWJ nozzle reduced the mixing efficiency of the abrasive particles and the high-pressure water jet. This resulted in a minimized notch depth at an abrasive particle size of 60 #. The evaluation criterion for the notch depth was “Larger-the-Better”. Based on the range analysis, the optimal parameter combination was an abrasive particle size m_d of 80 #, an operating pressure P of 400 MPa, and an abrasive feed rate ṁ_a of 840 g/min. Notably, this combination was consistent with the optimal levels identified for maximizing the material removal rate MRR.

3.3. Effect of Machining Parameters on Nozzle Wear Rate E

The influence of the S/N ratio on the nozzle wear rate E was analyzed using the range analysis method, as shown in

Table 9. Range analysis of nozzle wear rate.

	md (#)	P (MPa)	ṁa (g/min)
KE1	−256.8818	−245.2390	−301.8090
KE2	−257.1364	−245.1043	−266.0724
KE3	−254.7336	−245.1930	−238.4598
KE4	−254.8424	−288.0579	−217.2530
kE1	−64.2204	−61.3097	−75.4523
kE2	−64.2841	−61.2761	−66.5181
kE3	−63.6834	−61.2982	−59.6149
kE4	−63.7106	−72.0145	−54.3132
Range R	0.6007	10.7384	21.1390

. Here, K_Ei was the sum of all data at the i-th level for each factor. k_Ei was the average of K_Ei. The range R represented the difference between the maximum and minimum values of k_Ei for each factor. The results showed that the range R for abrasive feed rate ṁ_a was the largest, followed by that of abrasive particle size m_d. The range R for operating pressure P was only 0.6007, indicating its relatively minor influence. Therefore, abrasive feed rate ṁ_a had the most significant effect on the nozzle wear rate E, followed by operating pressure P and abrasive particle size m_d.

Figure 6 shows the influence of abrasive particle size m_d, operating pressure P, and abrasive feed rate ṁ_a on the nozzle wear rate E in the AWJM process. The ordinate represented the average sum of all results for each factor at the i-th level, corresponding to the k_Ei values in Table 9. It was observed that changes in abrasive particle size m_d had little influence on the nozzle wear rate. In the actual production, the wear of the nozzle caused by abrasive particles was unavoidable. However, during the simulation, the nozzle wall wear rate was calculated over a short duration of 15 s. Extending the calculation time would likely yield more pronounced differences in results. As the operating pressure P increased, the nozzle wear rate decreased gradually. This trend can be attributed to the improved mixing of particles within the nozzle’s mixing chamber at higher pressures, reducing particle congestion in the focusing tube and thereby lowering the wear rate. However, increasing the abrasive feed rate led to a higher number of particles within the nozzle, causing particle aggregation and an increase in wear rate at the focusing tube wall surface. The evaluation criterion for nozzle wear rate was “Smaller-the-Better”. Based on the range analysis, the optimal parameter combination was an abrasive particle size m_d of 80 #, an operating pressure P of 400 MPa, and an abrasive feed rate ṁ_a of 260 g/min. This combination corresponded to the sixth experimental design in the L16 orthogonal experiment in Table 5. The corresponding response results were as follows: material removal rate MRR of 0.2345 g/s, notch depth w_h of 6.08 mm, and nozzle wear rate E of 6.2387 × 10⁻⁵.

3.4. Regression Analysis of Experimental Results

Regression analysis and Analysis of Variance (ANOVA) were conducted to evaluate the statistical significance of the experimental results. An empirical model was developed using SPSS 27 software to predict the response values of the AWJM process, and to assess the effect of machining parameters on the response parameters, including material removal rate MRR, notch depth w_h, and nozzle wear rate E. The regression equations for the S/N ratios of these three response parameters were provided in Equations (4), (5) and (6), respectively.

$$S/N_{MRR}=-29.187+0.01\times m_d+0.035\times P+0.005\times {\dot{m}}_a$$

(4)

$$S/N_{w_h}=-0.476+0.029\times m_d+0.03\times P+0.004\times {\dot{m}}_a$$

(5)

$$S/N_E=-62.73+0.011\times m_d-0.064\times P+0.034\times {\dot{m}}_a$$

(6)

In the regression Equations (4) and (5), abrasive particle size m_d, operating pressure P, and abrasive feed rate ṁ_a exhibited a positive correlation with the S/N ratios of material removal rate MRR and notch depth w_h. In Equation (6), abrasive particle size and abrasive feed rate showed a positive correlation with the S/N ratio of the nozzle wear rate, while operating pressure demonstrated a negative correlation.

The significance of the statistical model and its applicability were further validated through Analysis of Variance (ANOVA) at a 95% confidence level. In the ANOVA, the statistical validity of the developed model was obtained using the coefficient R² and the adjusted R². The most influential parameters and their statistical significance on the response variables were studied by statistical tests (F-values) and probability values (p-values). A higher F-value indicated that the corresponding parameter had greater significance. A parameter is considered statistically significant when its p-value is less than 0.05. The coefficient of R² represented the ratio of the regression sum of squares to the total sum of squares, quantifying the variability of the response variables. R² ranged from 0 to 1, with values closer to 1 indicating a higher degree of statistical significance for the model. In order to fit the adjusted R² of the regression model, a backward elimination method was used to exclude non-significant terms. It is important to note that non-significant items should not be removed from the model, as it may not only distort the relationship between input and output variables but also reduce the model’s predictive accuracy.

Based on the S/N ratio data of material removal rate MRR in Table 6 above, the ANOVA results were obtained, as shown in

Table 10. ANOVA results of the S/N ratio of the material removal rate MRR.

Source of Variance	Sum of Squares of Deviation	Degrees of Freedom	Mean Square	F-Value	p-Value	Significance	R2
md (#)	8.8964	3	2.9655	11.2237	0.0071	Significant	0.0844
P (MPa)	64.6120	3	21.5373	81.5142	<0.0001	Significant	0.6132
ṁa (g/min)	30.2888	3	10.0963	38.2122	0.0003	Significant	0.2874
Error	1.5853	6	0.2642	/	/	/	0.0150
Total	105.3825	15	/	/	/	/	/

. Similarly, the ANOVA results for the S/N ratios of notch depth w_h and nozzle wear rate were shown in

Table 11. ANOVA results of the S/N ratio of the notch depth w_h.

Source of Variance	Sum of Squares of Deviation	Degrees of Freedom	Mean Square	F-Value	p-Value	Significance	R2
md (#)	16.2243	3	5.4081	33.8901	0.0004	Significant	0.2034
P (MPa)	46.4984	3	15.4995	97.1282	<0.0001	Significant	0.5828
ṁa (g/min)	16.0987	3	5.3662	33.6278	0.0004	Significant	0.2018
Error	0.9575	6	0.1596	/	/	/	0.0120
Total	79.7788	15	/	/	/	/	/

and

Table 12. ANOVA results of the S/N ratio of the nozzle wear rate E.

Source of Variance	Sum of Squares of Deviation	Degrees of Freedom	Mean Square	F-Value	p-Value	Significance	R2
md (#)	1.2429	3	0.4143	5.2206	0.0414	Significant	0.0009
P (MPa)	344.7439	3	114.9146	1448.0982	<0.0001	Significant	0.2556
ṁa (g/min)	1002.2179	3	334.0726	4209.8207	<0.0001	Significant	0.7431
Error	0.4761	6	0.0794	/	/	/	0.0004
Total	1348.6808	15	/	/	/	/	/

, respectively. From Table 10, Table 11 and Table 12, it is evident that the p-values of S/N ratios of material removal rate MRR, notch depth w_h, and nozzle wear rate E were all less than 0.05, indicating that these factors are statistically significant.

3.5. Optimization of Multi-Response Variables

In manufacturing and industrial production environment scenarios, AWJM optimization modeling is carried out by using the multi-criteria Decision Engineering Analysis and Resolution (DEAR) method. The DEAR method is a systematic method for solving multi-criteria decision problems. It analyzes and evaluates multiple options by synthesizing multiple criteria or guidelines to find the optimal solution. In engineering and manufacturing, DEAR methods are often used to optimize process parameters, select the best technical solution, or evaluate the feasibility of a project. The DEAR method usually consists of the following steps:

(a)

Define decision-making objectives and criteria

First, it is necessary to clarify the overall goal of the decision and determine all relevant criteria. Each criterion should be able to reflect one aspect of the decision. When optimizing AWJM process parameters under the condition of two or more responses, the material removal rate MRR and notch depth w_h are first considered to determine the conditions for optimal manufacturing parameters. Second, the nozzle wear rate E was integrated into the analysis to refine parameter selection and identify superior operational conditions. In the optimization of multi-response variable results, response variables such as material removal rate, notch depth, and nozzle wear rate are considered criteria. The experimental design incorporated input parameters, including particle size, operating pressure, and abrasive feed rate;

(b)

Establish a decision matrix

The decision matrix is a table that lists how each option performs under different criteria. The rows in the table represent the choices, and each column represents each criterion. Throughout the optimization process, a decision matrix is developed that includes multiple criteria as response parameters, as well as multiple operational and input parameter settings as alternatives. In the whole optimization process, a decision matrix was developed by the experimental results presented in Table 6;

(c)

Weight determination

The importance of each criterion is usually different, so it is necessary to determine the weight of each criterion. The weights can be determined by expert opinion, statistical analysis, or subjective assignment. The weight reflects the degree of influence of each criterion on the final decision. The commonly used weight determination methods include: Analytic Hierarchy Process (AHP), entropy weight method, and Delphi method. The decision problem is decomposed into criteria and sub-criteria by choosing the analytic hierarchy process, and the weight of each criterion is determined by pair comparison. Based on the Taguchi S/N ratio, the weights for each AWJM process parameter were determined by Equations (7)–(9).

$$W_{MMR}={\displaystyle \frac{MMR}{{\displaystyle \sum MMR}}}$$

(7)

$$W_{w_h}={\displaystyle \frac{w_h}{{\displaystyle \sum w_h}}}$$

(8)

$$W_E={\displaystyle \frac{(1/E)}{{\displaystyle \sum (1/E)}}}$$

(9)

(d)

Calculating the comprehensive score: Multi-Performance Response Index (MPRI)

The MPRI score is used to evaluate the overall performance of each choice. It is a performance evaluation index used in multi-objective optimization. The overall performance of the AWJM process system is measured by combining the results of multiple response variables. MPRI is also used in multi-objective decision making to evaluate and compare performance under different experimental conditions, further helping to determine the optimal processing parameters. Its numerator is a positive indicator (such as material removal rate and notch depth), and the denominator is a negative indicator (nozzle wear rate). So multiple criteria can be considered comprehensively. The above Equations (7)–(9) are further formulated by the weighting decision matrix, and the formulas are merged into Equations (6)–(10). The computed MPRI values for each AWJM parameter are systematically presented in

Table 13. Optimization results of the DEAR method for AWJM process.

Number	Weight Ratio of Each Process			MPRI
	WMMR	${\mathit{W}}_{{\mathit{w}}_{\mathit{h}}}$	WE
1	0.0276	0.0304	0.0972	3683.6347
2	0.0484	0.0532	0.0352	9571.3534
3	0.0711	0.0634	0.0158	16,167.7031
4	0.0838	0.0744	0.0301	21,726.1119
5	0.0505	0.0491	0.0352	10,483.9840
6	0.0690	0.0593	0.0159	16,089.8951
7	0.0864	0.0703	0.0085	23,082.8767
8	0.0690	0.0661	0.3522	19,024.1464
9	0.0499	0.0553	0.0154	11,135.6462
10	0.0676	0.0662	0.0082	17,986.0083
11	0.0526	0.0620	0.0914	14,071.1945
12	0.0796	0.0717	0.1102	22,153.0171
13	0.0468	0.0621	0.0081	12,914.0484
14	0.0461	0.0579	0.0918	11,735.2633
15	0.0673	0.0676	0.0344	19,660.1467
16	0.0843	0.0778	0.0504	27,286.1648

.

$$MPRI={\displaystyle \frac{M+W}{E_e}}$$

(10)

Here M, W, and E_e represent the weighted product of AWJM process parameters (W_MMR,$W_{w_h}$, and W_E) and their corresponding experimental parameters (MMR, W_h, and E) in Table 5, as mathematically expressed in Equations (11)–(13). Then, Equations (4)–(6) were used to formalize the weighted decision matrix and determine the MPRI value for each AWJM output parameter. The results were listed in Table 13, providing quantitative insights into process performance optimization.

$$M=W_{MMR}\times MMR$$

(11)

$$W=W_{w_h}\times W_h$$

(12)

$$E_e=W_E\times E$$

(13)

(e)

Sort and select the best scheme

All options are sorted according to the overall score, and the option with the highest score is the best option. Through the comparison of MPRI values, we can clearly judge the advantages and disadvantages of each choice, and make decisions on this basis. Range analysis was carried out on the MPRI value in the optimization results of the DEAR method for the AWJM process. The analytical results are shown in

Table 14. Range analysis of multi-performance response index based on the DEAR method.

	md (#)	P (MPa)	ṁa (g/min)
KD1	51,148.8031	38,217.3133	48,514.2389
KD2	68,680.9023	55,382.5201	61,868.5012
KD3	65,345.8661	72,981.9210	70,679.4092
KD4	71,595.6232	90,189.4402	75,709.0454
kD1	12,787.2008	9554.3283	12,128.5597
kD2	17,170.2256	13,845.6300	15,467.1253
kD3	16,336.4665	18,245.4803	17,669.8523
kD4	17,898.9058	22,547.3601	18,927.2614
Range R	5111.7050	12,993.0317	6798.7016

. K_Di was the sum of all data at the i-th level of each factor. k_Di was the average value of K_Di. Range R represented the difference between the maximum and minimum value of the result k_Di of each factor. Results showed that the range of operating pressure P was the largest, followed by abrasive feed flow ṁ_a and abrasive particle size m_d. This indicated that operating pressure P had the most significant impact on the comprehensive performance evaluation of multi-response capability in the AWJM process, while abrasive feed rate ṁ_a and particle size m_d demonstrated progressively lesser influence.

The contribution of each process parameter was quantitatively assessed through the ANOVA results of the MPRI, with detailed statistical results presented in

Table 15. ANOVA results of MPRI based on the DEAR method.

Source of Variance	Sum of Squares of Deviation	Degrees of Freedom	Mean Square	F-Value	p-Value	Significance	R2
md (#)	6.1605 × 107	3	2.0535 × 107	14.0364	0.0001	Significant	0.1114
P (MPa)	3.7636 × 108	3	1.2545 × 108	85.7504	<0.0001	Significant	0.6803
ṁa (g/min)	1.0648 × 108	3	3.5493 × 107	24.2609	<0.0001	Significant	0.1925
Error	8.7779 × 106	6	1.4630 × 106	/	/	/	0.0159
Total	5.5322 × 108	15	/	/	/	/	/

. It was known that operating pressure P emerged as the dominant factor, accounting for 68.03% of the total variation, followed by abrasive feed rate ṁ_a at 19.15% and abrasive particle size m_d at 11.14%. Through the final optimization using the DEAR method, the optimal AWJM process parameters were determined to be an abrasive particle size m_d of 120 #, an operating pressure P of 400 MPa, and an abrasive feed rate ṁ_a of 870 g/min. It was noteworthy that this optimal parameter combination was not included in the original L16 orthogonal array experimental design (Table 5). This discrepancy arises from the inherent limitations of the Taguchi orthogonal design, which considers only a fraction 3⁴ = 81 of the full factorial experimental space.

Under the validated working conditions with optimal parameter settings with an abrasive particle size m_d of 120 #, operating pressure P of 400 MPa, and abrasive feed flow ṁ_a of 870 g/min, the AWJM process demonstrated superior machining performances: material removal rate MRR of 0.3297 g/s, notch depth w_h of 8.40 mm, and nozzle wear rate E of 1.3443 × 10⁻³. These experimental results confirm the effectiveness of the optimized parameter combination in achieving enhanced machining performance.

4. Conclusions

This study provides an investigation into the optimization of abrasive water jet machining for No.45 steel, integrating experimental analysis, computational modeling, and multi-objective optimization using the Decision Engineering Analysis and Resolution (DEAR) method. The investigation focused on three machining performance—material removal rate MRR, notch depth w_h, and nozzle wear rate E—in relation to three key process parameters: abrasive particle size m_d, operating pressure P, and abrasive feed rate ṁ_a. The following conclusions were drawn:

(1)

Through parametric influence analysis, it was known that operating pressure emerged as the most influential parameter in the machining of No. 45 steel, accounting for 68.03% of the total variation in machining performance. This finding provides critical insights into the dominant role of kinetic energy transfer in material removal;

(2)

According to the results of single-factor parameter analysis, the maximization of material removal rate and notch depth was predominantly controlled by operating pressure. The minimization of nozzle wear rate was predominantly controlled by abrasive feed rate. This finding contributes to the fundamental understanding of energy-material interactions in AWJM;

(3)

By range analysis method, the optimal solution for material removal rate and notch depth was an abrasive particle size of 80 #, an operating pressure of 400 MPa, and an abrasive feed rate of 840 g/min. The optimal solution for nozzle wear rate was an abrasive particle size of 80 #, an operating pressure of 400 MPa, and an abrasive feed rate of 260 g/min;

(4)

By using the DEAR method, the optimal parameter solution was obtained when the abrasive particle size is 120 #, the operating pressure is 400 MPa, and the abrasive feed rate is 870 g/min. Experimental validation confirmed the above machining performances: material removal rate of 0.3297 g/s, notch depth of 8.40 mm, and nozzle wear rate of 1.3443 × 10⁻³;

(5)

Using the optimal solution provided by the DEAR method, the AWJM process provides a higher material removal rate and a larger incision depth, while reducing the nozzle wall wear rate under the same working conditions, making the process parameters more suitable for the machining of 45 # steel.

By addressing the limitations of conventional approaches and providing a systematic framework for multi-objective optimization, this research contributes to both the scientific understanding and practical application of AWJM technologies. More parameters will be studied in the future, including the feed speed of abrasive water jet nozzles and the standoff distance.