Response Surface Methodology for Wear Optimization of Irrigation Centrifugal Pumps in High-Sediment Water Conditions of Southern Xinjiang: Design and Experimental Validation

Abstract

This study investigates the wear characteristics and optimization of a centrifugal pump (Q = 25 m³/h, H = 50 m, n = 2900 r/min) applied in sediment-laden waters of Southern Xinjiang irrigation systems. A numerical framework integrating the Realizable $k-\epsilon$ turbulence model, Discrete Phase Model (DPM), and Oka erosion model was established to analyze wear patterns under varying parameters (particle size, density, and mass flow rate). Results indicate that the average erosion rate peaks at 0.92 kg/s mass flow rate. Subsequently, a Response Surface Methodology (RSM)-based optimization was implemented: (1) Plackett–Burman (PB) screening identified the inlet placement angle (A), inlet diameter (C), and outlet width (E) as dominant factors; (2) Full factorial design (FFD) revealed significant interactions (e.g., A × C, C × E); (3) Box–Behnken Design (BBD) generated quadratic regression models for head, efficiency, shaft power, and wear rate (R² > 0.94). Optimization reduced the average erosion rate by 31.35% (from 1.550 × 10⁻⁴ to 1.064 × 10⁻⁴ kg·m⁻²·s⁻¹). Experimental validation confirmed the numerical model’s accuracy in predicting wear localization (e.g., impeller outlet). This work provides a robust methodology for enhancing the wear resistance of centrifugal pumps for agricultural irrigation in water with high fine sediment concentration environments.

1. Introduction

As power equipment, pumps are widely utilized in fields such as hydropower generation, agricultural power infrastructure, and marine engineering. Their performance (e.g., flow rate, head, and efficiency), reliability, and wear resistance directly influence the stability and long-term operation of energy systems and hydraulic systems [1,2]. Centrifugal pumps operating in sediment-laden irrigation systems suffer severe erosion, compromising efficiency and lifespan. Field studies [3,4,5] confirm that fine sand (0.05~0.5 mm) dominates wear in agricultural pumps, with impeller outlets and volute tongues as critical zones. Numerical frameworks combining Eulerian–Lagrangian models with Oka erosion theory [6,7,8,9,10,11] accurately predict wear patterns, validated by high-speed imaging [12,13,14]. Key parametric analyses reveal that 1~2 mm particles maximize erosion at 15~30° impact angles [15,16,17], while concentrations >5% accelerate wear via secondary flows [18,19,20]. Moreover, sediment not only exacerbates wear but also significantly influences cavitation characteristics; studies have shown that in sediment-laden water, both the cavitation value and cavitation range are significantly higher than in clear water, with sediment concentration being the primary influencing factor [21]. Material hardening coatings [22,23,24] reduce wear by 40~60%, yet structural optimization remains underdeveloped for high-sediment environments.

Response Surface Methodology (RSM) enables systematic hydraulic optimization. Plackett–Burman designs efficiently screen dominant factors: blade outlet width [25] and inlet diameter [26] govern efficiency, while wrap angle [27] controls head. Full factorial designs quantify interactions; blade number and volute geometry [28,29] jointly affect cavitation, with center points detecting curvature [30]. Box–Behnken [31,32] and Central Composite Design [33,34] have been employed to establish quadratic models for vibration reduction [35] and the efficiency-head trade-off [36]. Multi-objective genetic algorithms [37,38] can derive Pareto fronts, while the integration of CFD and RSM [39,40,41] has led to a 25–38% reduction in cavitation erosion. However, these and other related studies have primarily focused on hydraulic performance, largely neglecting the mitigation of sediment-induced wear. Nevertheless, systematic optimization approaches, such as meta-heuristic algorithms, have demonstrated their effectiveness in parameter inversion within related fields [42,43].

Critical research gaps persist in wear-specific optimization. Turbulence models (Realizable $k-\epsilon$ and SST $k-\omega$) resolve particle–fluid interactions [44,45,46] but ignore irregular particle shapes [47,48]. Eulerian–Lagrangian tracking [49,50,51] overlooks inter-particle collisions in dense flows [52,53]. The Oka model outperforms generic erosion theories [19,54,55] yet lacks material-hardening adaptations [56,57]. Field validations [58,59,60] confirm wear localization but offer no geometric solutions. Crucially, RSM has not been applied to establish anti-wear design rules for sediment environments, as noted in [61,62]. Furthermore, there are also studies on the structural optimization and energy conversion of multiphase flow pumps [63,64].

As critical infrastructure in systems such as hydropower generation and agricultural irrigation, pump stations play an indispensable role, with various pumps installed therein being irreplaceable [65]. This study focuses on optimizing the wear resistance of irrigation pumps in the Nanjiang irrigation district under sand-laden water conditions. To address existing research gaps, a comprehensive CFD-RSM framework is employed to enhance the wear resistance of these pumps. A validated Oka-DPM is adopted to quantify the effects of sediment parameters, while a multi-stage RSM (PB-FFD-BBD) is implemented to identify optimal geometric configurations. Experimental validation is subsequently conducted to verify the achieved wear reduction. Specifically, the PB-FFD-BBD method adopted in this study adheres to a data-driven principle: first, a parameter interval is preset, and Full Factorial Design (FFD) experiments are conducted; after confirming significant curvature through statistical testing, Box–Behnken Design (BBD) modeling is then carried out within this interval. This process replaces subjective preset center points with data derived from FFD experiments, thereby ensuring the accuracy and reliability of the subsequent BBD model. Compared to the traditional PB-BBD approach, this method overcomes the blindness in selecting design centers, and in comparison to optimization methods such as genetic algorithms, it significantly reduces computational costs while maintaining optimization effectiveness. This study aims to develop a replicable methodology for optimizing wear reduction in centrifugal pumps operating long-term in sand-laden water environments, with the goal of extending service life in abrasive fluid systems.

2. Modeling and Numerical Computation Methods

2.1. Model Construction

A PGW 65-200-7.5 G (The pump is manufactured by Puxuante Pump Industry Co., Ltd., Taizhou, China) centrifugal pump is utilized in the irrigation areas of Southern Xinjiang as the research object, with a flow rate Q = 25 m³/h, head(H) = 50 m, and a rated speed (n) = 2900 r/min. The parameters of the centrifugal pump impeller are shown in

Table 1. Geometric parameters of model pump impeller.

	Parameter	Value
Impeller	Inlet diameter/mm	65
	Hub diameter/mm	25
	Outlet width/mm	10
	Number of blades/pcs	5
	Outlet diameter/mm	200
	Wrap angle/(°)	120
	Inlet placement angle/(°)	15
	Outlet placement angle/(°)	25

.

The computational fluid domain of this centrifugal pump primarily consists of an inlet extension, impeller, volute, and outlet extension. The 3D model is shown in Figure 1.

2.2. Grid Generation

The centrifugal pump model was meshed using tetrahedral elements in ANSYS (2024R1). Local refinement was applied to the volute tongue region, where flow complexity is significant, in order to enhance computational accuracy. In Fluent software, a minimum orthogonal quality of 0.15 satisfies the acceptable criteria. The grid quality inspection results show that the minimum orthogonal quality stands at 0.26, the maximum at 0.99, and the average grid quality achieves 0.75. This demonstrates that the generated grid is of high quality and can meet the accuracy requirements for subsequent numerical simulations. The resulting grid is presented in Figure 2.

2.3. Grid Independence Verification and y+ Evaluation

To determine the optimal grid scheme, six distinct grid configurations with varying cell counts (

Table 2. Grid scheme.

Grid Scheme	Number of Grids	H (m)	Average Wear Rate (kg·m−2·s−1)
1	860,054	52.50	1.0547 × 10−4
2	1,204,147	52.45	1.0887 × 10−4
3	1,543,867	52.34	1.0974 × 10−4
4	1,887,081	52.09	1.1259 × 10−4
5	2,405,394	52.11	1.126 × 10−4
6	3,038,054	52.09	1.1261 × 10−4

) were evaluated. By calculating the pump head under rated operating conditions and the average wear rate of the impeller, and analyzing the fluctuations of both the pump head and the average impeller wear rate with variations in mesh number, the optimal mesh configuration was ultimately determined.

Figure 3 simultaneously plots the pump head and the average wear rate of the impeller as a function of the grid number.

As observed in Figure 3, when the number of grids increased from Schemes 1–4, the pump head gradually decreased, while the average wear rate progressively rose. When the number of grids exceeded that of Scheme 4, both the head and average wear rate stabilized. To balance computational accuracy and efficiency, Grid Scheme 4 (grid count: 1.887 × 10⁶) was ultimately selected for subsequent numerical simulations.

To accurately capture the distribution characteristics of velocity gradients and strong shear effects in the flow field near the walls of key flow passage components such as the centrifugal pump impeller and volute, six layers of prismatic boundary layer grids were specifically arranged in the wall regions of these components. A refined control scheme was adopted for the grids: the grid growth rate was set to 1.2, the transition ratio was set to 0.27, and the dimensionless wall distance y+ was strictly controlled within the reasonable range of 30–300, ensuring high compatibility with the selected turbulence model and wall functions. The y+ values on the walls of the blade and volute components are shown in Figure 4.

2.4. Numerical Computation Methods

Throughout the study, the Realizable k-ε model was employed for turbulence modeling, steady-state simulation was conducted, the Discrete Phase Model (DPM) for particle tracking, and the Oka erosion model for wear prediction. The primary objective of this study is to enhance the wear resistance of the impeller by optimizing its geometric configuration. Therefore, the following simplifications were applied to the DPM discrete phase model used in this work: particles are assumed to be spherical, while their rotation, inter-particle collisions, and breakup upon impact with walls are neglected.

2.4.1. Turbulence Model

Compared to the standard k-ε model, the Realizable k-ε model demonstrates superior performance in simulating typical complex flow scenarios such as rotating flows, flow separation, and complex secondary flows. Its key advantage lies in the more accurate capture of rotational flow dynamics and detailed separation flow characteristics. The internal flow field of centrifugal pumps represents a typical rotating flow problem, exhibiting significant rotational effects and flow separation phenomena. Therefore, adopting the Realizable k-ε model enables more accurate simulation of the fluid flow process within centrifugal pumps, providing more reliable computational results for flow field analysis. The governing equations for k, turbulent dissipation rate ε, and turbulent viscosity v_t are given in Equations (1), (2) and (3), respectively.

$$\begin{array}{l}{\displaystyle \frac{\partial (\rho k)}{\partial t}}+\rho u_j{\displaystyle \frac{\partial k}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left(v+{\displaystyle \frac{v_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}+G_k-\rho \epsilon \end{array}$$

(1)

$$\rho {\displaystyle \frac{\partial \epsilon }{\partial t}}+{\displaystyle \frac{\partial (\rho \epsilon u_j)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(v+{\displaystyle \frac{v_t}{{\sigma }_{\epsilon }}}\right)+{\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]+\rho C_{1}S\epsilon -\rho C_2{\displaystyle \frac{{\epsilon }^2}{k+\sqrt{v\epsilon }}}$$

(2)

$$v_t=\rho C_{\mu }{\displaystyle \frac{k^2}{\epsilon }}$$

(3)

where v is the kinematic viscosity; ρ is the fluid density; u_j is the velocity component; x_j is the spatial coordinate component; σ_k is the Prandtl number for turbulent kinetic energy k; σ_ε is the Prandtl number for turbulent dissipation rate ε; G_k is the turbulent kinetic energy generation term produced by the mean velocity gradient; C₁ and C₂ are model constants; and S is the mean strain rate tensor.

2.4.2. Particle Motion Model

The DPM within an Eulerian–Lagrangian framework was implemented. In this approach, the fluid phase is solved as a continuous medium in the Eulerian reference frame, while solid particles are tracked as a discrete phase in the Lagrangian framework. The governing equations are presented as follows:

$$\begin{array}{l}{\displaystyle \frac{d{v}_p}{dt}}={\displaystyle \frac{18\mu }{{\rho }_p{d^2}_p}}{\displaystyle \frac{C_d\mathrm{Re}}{24}}(v_f-v_p)+{\displaystyle \frac{g({\rho }_p-{\rho }_f)}{{\rho }_p}}\end{array}$$

(4)

where v_f represents the liquid phase velocity (m/s); v_p represents the particle velocity (m/s); ρ_p represents the particle density (kg/m³); ρ_f represents the liquid phase density (kg/m³); d_p represents the particle diameter (m); g represents the gravitational acceleration (m/s²); C_d is the drag coefficient; Re is the relative Reynolds number; and μ is the dynamic viscosity of the continuous phase (Pa·s).

2.4.3. Wear Model

The Oka erosion model was selected to predict wear, accounting for both the inherent flow characteristics within the centrifugal pump and the trajectories of solid particles. The governing equations are presented as follows:

$$E=E_{90}{\left({\displaystyle \frac{V}{V_{ref}}}\right)}^{k_2}{\left({\displaystyle \frac{d}{d_{ref}}}\right)}^{k_3}f\left(\alpha \right)$$

(5)

$$f\left(\alpha \right)={\left(\sin \alpha \right)}^{n_1}{\left(1+H_v\left(1-\sin \alpha \right)\right)}^{n_2}$$

(6)

where E₉₀ represents the reference erosion rate at an impact angle of 90°; V is the particle impact velocity; V_ref is the reference velocity; d and d_ref are the particle size and reference particle size, respectively; k₂ and k₃ are the velocity exponent and particle size exponent, respectively; f(α) is the impact angle function, where α is the wall impact angle in radians; H_v is the Vickers hardness of the wall material in GPa; and n₁ and n₂ are constants for the angle function. Among these, the reference particle size d_ref and reference flow velocity v_ref are normalization reference parameters in the model. They are used to non-dimensionalize the particle size and flow velocity effects, respectively, ensuring the model’s applicability across different operating conditions, as shown in

Table 3. Table of key parameters for the Oka erosion model.

$H_v(\mathrm{GPa})$	$n_1$	$n_2$	$d_{ref}(\mathrm{mm})$	$V_{ref}(\mathrm{m}/\mathrm{s})$
1.8	0.8	1.3	0.326	104

.

3. Experiment Verification and Wear Patterns

3.1. Boundary Conditions

Boundary conditions were defined with a velocity inlet at the centrifugal pump’s intake and a pressure outlet at the discharge. The tests were conducted under rated operating conditions with a flow rate of Q = 25 m³/h and a rotational speed of n = 2900 r/min. Flow rate and velocity at the inlet were mutually converted based on the cross-sectional area. All wetted surfaces employed no-slip walls, with near-wall flow resolved via non-equilibrium wall functions. Wall roughness was modeled using the standard wall roughness approach. The roughness constant is set to 0.5. Particles are injected into the Discrete Phase Model (DPM) from the inlet surface and tracked under steady-state conditions. The residual is set to 10⁻⁴, and convergence is considered achieved when the rate of change in the average wear rate of the impeller between adjacent iterations is less than 0.1%. The inlet/outlet boundaries were set as escape for particles, while other hydraulic surfaces were designated as reflect. To ensure numerical stability and physical validity, solid particles were initialized with identical hydrodynamic conditions to the fluid. The particle density was specified as 2650 kg/m³. The coupled algorithm was implemented for pressure-velocity coupling, with spatial discretization utilizing the second-order upwind scheme. Data transfer between the inlet extension–impeller, impeller–volute, and volute–outlet extension domains was achieved through shared topology.

To validate the applicability of the numerical approach adopted in this study—including turbulence modeling, particle motion modeling, and erosion modeling—for analyzing wear characteristics of centrifugal pumps in southern Xinjiang irrigation systems, experiments on head and efficiency under different flow rates in clear water conditions, as well as qualitative wear tests, were conducted.

3.2. Experimental System and Process

The experimental system comprises a centrifugal pump, submersible pump, water tank, stirrer, inlet valve, outlet valve, electromagnetic flowmeter, console, and pressure gauge, as illustrated in Figure 5. The measurement range and accuracy of the primary instruments are as follows:

Inlet pressure sensor: measurement range (0–0.6 MPa), accuracy ±0.5%.

Outlet pressure sensor: measurement range (0–0.6 MPa), accuracy ±0.5%.

Electromagnetic flowmeter: measurement range (0.1–80 m³/h), accuracy ±2%.

Torque sensor: measurement range (0–50 N·m), accuracy ±0.5%.

To investigate the influence of flow rate on the performance of the centrifugal pump, seven distinct flow rate conditions were selected for both experimental testing and numerical simulation.

As can be seen from Figure 6a and Figure 7a, the numerical simulations show the same increasing or decreasing trends in head and efficiency as the experimental results. Further quantitative analysis through Figure 6b and Figure 7b reveals that the maximum relative errors between the numerical and experimental values of head and efficiency are within reasonable limits, fully meeting acceptable standards for engineering applications. This adequately validates the reasonableness and reliability of the numerical simulation method and model simplification scheme adopted in this study.

In subsequent experiments, we further conducted a qualitative analysis of its wear through experimental methods. During testing, centrifugal pumps were subjected to a 10 h wear test using sediment particles with a 1 mm diameter at 0.92 kg/s mass flow rate. Post-test examination focused on wear patterns at the pump inlet and impeller outlet regions.

3.3. Comparison of Results

The wear characteristics of the centrifugal pump were analyzed using the previously described numerical methodology, with results compared against experimental findings. Comparative data for the pump inlet and impeller outlet regions are presented in Figure 8 and Figure 9.

As evidenced in the preceding two figures, the numerical simulations exhibited remarkable spatial correspondence with experimental wear patterns. This validates the methodological robustness and physical fidelity of the adopted numerical approach, demonstrating its effectiveness for analyzing wear characteristics of centrifugal pumps in Southern Xinjiang irrigation systems and supporting optimization of hydraulic surfaces.

3.4. Analysis of Wear Characteristics of Prototype Pumps

To thoroughly investigate the influence of different parameters (particle size, density, and mass flow rate) on the wear characteristics of the centrifugal pump impeller, a one-variable-at-a-time approach was employed. This analysis incorporated six distinct particle size groups, four density variations, and four mass flow rate levels, as detailed in

Table 4. Sediment parameters.

No.	Parameter	Unit	Value
1	particle size	mm	0.5, 1, 1.5, 2, 2.5, 3
2	particle density	kg/m3	1350, 2000, 2650, 3300
3	mass flow rate	kg/s	0.46, 0.92, 1.38, 1.84

.

The relationship between varying sediment parameters and the average erosion rate on the impeller surface is demonstrated in Figure 10.

As indicated in Figure 10, the average erosion rate on the impeller surface increased with the growth of each independent variable, yet exhibited distinct variation patterns in its growth rate. Analysis of Figure 10a revealed that increasing the mass flow rate initially induced a sharp rise followed by a rapid decline in the erosion growth rate, peaking at 0.92 kg/s mass flow rate phenomenon potentially attributed to significant strong secondary flows or separation vortices develop within the impeller flow passage, transporting more particles toward the wall regions, where both the frequency and intensity of impacts increase simultaneously. For density variations (Figure 10b), the growth rate first increased rapidly, then decreased, and subsequently experienced a minor rebound; the initial increase may correlate with enhanced particle deposition due to elevated density, while subsequent trends likely resulted from density-driven amplification of particle impact energy, thereby altering the dominant wear mechanism. Regarding particle size (Figure 10c), the growth rate surged rapidly to a maximum at 1 mm before declining steeply and stabilizing, providing critical insights for predicting wear risks and optimizing impeller design in centrifugal pumps operating with specific particle sizes.

4. Experimental Design Based on RSM

4.1. Plackett–Burman Screening Design

The impeller serves as the core component for energy transfer and flow guidance in centrifugal pumps. Its structure directly determines the pump’s flow field characteristics, performance characteristics, and operational stability. Therefore, subsequent structural optimization efforts are focused on the design of the impeller. Eight key geometric parameters of the impeller were systematically analyzed: inlet placement angle, exit placement angle, inlet diameter, hub diameter, outlet width, number of blades, wrap angle, and outlet diameter. A Plackett–Burman (PB) screening design was conducted to identify significant factors influencing the response variables among these eight parameters. The value levels of these key impeller parameters are specified in

Table 5. Control variables for key parameters of the impeller.

Variable	Parameters	Low Level	High Level
X1	Inlet diameter D1/mm	60	70
X2	Outlet diameter D2/mm	196	204
X3	Hub diameter D3/mm	20	30
X4	Outlet width b/mm	7	13
X5	Number of blades/pcs	4	6
X6	$\mathrm{Wrap} \mathrm{angle} \alpha$/(°)	110	130
X7	$\mathrm{Inlet} \mathrm{placement} \mathrm{angle} {\beta }_1$/(°)	15	25
X8	$\mathrm{Outlet} \mathrm{placement} \mathrm{angle} {\beta }_2$/(°)	20	30

.

From an engineering practice perspective, efficiency and wear rate are generally negatively correlated, making it challenging to simultaneously achieve the dual optimization objectives of reducing wear rate and improving efficiency. Therefore, the wear resistance optimization work in this study is conducted under the core precondition of ensuring system efficiency stability. Based on this principle, efficiency is selected as the key metric for the PB screening experiments to identify the significant factors influencing the optimization objectives. In this study, the PB experimental design was executed using Minitab 22 software. Relevant data were obtained through numerical simulations, and computational analysis ultimately identified the three most influential factors on the impeller wear response variables. The PB experimental design matrix and computational outcomes are presented in

Table 6. PB experimental design and calculation results.

No.	Inlet Placement Angle (°)	Outlet Placement Angle (°)	Inlet Diameter (mm)	Hub Diameter (mm)	Outlet Width (mm)	Number of Blades	Wrap Angle (°)	Outlet Diameter (mm)	H (m)	$\mathit{\eta }$ (%)
1	15	30	70	20	13	4	110	196	54.25	64.85
2	15	30	60	20	7	6	130	204	52.78	75.14
3	15	20	70	30	13	4	130	204	57.96	63.29
4	25	30	70	20	13	6	110	204	62.93	73.24
5	15	30	70	30	7	6	130	196	48.76	76.93
6	25	20	70	20	7	4	130	204	44.81	76.32
7	15	20	60	20	7	4	110	196	42.13	80.71
8	25	20	60	20	13	6	130	196	57.39	67.62
9	25	20	70	30	7	6	110	196	48.63	78.29
10	15	20	60	30	13	6	110	204	61.98	63.47
11	25	30	60	30	13	4	130	196	53.76	64.65
12	25	30	60	30	7	4	110	204	46.35	77.48

.

Analysis of the computational results in Table 6 yielded a standardized Pareto chart of effects for efficiency, presented in Figure 11. As revealed by Figure 11, the most influential factors were outlet width (E), inlet diameter (C), and inlet placement angle (A). Consequently, these three parameters-outlet width, inlet diameter, and inlet blade angle-were selected for subsequent impeller optimization design to mitigate wear on the impeller surface.

4.2. Full Factorial Design

Key performance parameters of centrifugal pumps (e.g., erosion rate, head, efficiency, and shaft power) are influenced not only by individual factors but also by interactions among multiple parameters. Given that models constructed via full factorial design (FFD) inadequately characterize such complex nonlinear relationships, this study proposes a two-phase optimization strategy: First, FFD identifies critical inflection intervals of key parameters; subsequently, a higher-order RSM is built within these intervals to locate the optimal parameter combination. This approach significantly enhances experimental efficiency when dealing with limited factors by avoiding unnecessary resource allocation in non-critical regions. The “PB-FFD-BBD” method enhances optimization accuracy by utilizing FFD to capture interaction effects between different structural parameters, whereas the PB-BBD method lacks this FFD component and consequently demonstrates lower precision due to insufficient modeling of parameter interactions. For this section, a two-level FFD was applied to three factors-outlet width (E), inlet diameter (C), and inlet placement angle (A)-with high/low levels specified in

Table 7. Parameter range values.

Coding Proficiency	Factor
	A	C	E
−1	15	60	7
0	20	65	10
1	25	70	13

.

A three-factor two-level full factorial design was constructed using Minitab 22 software, and numerical simulations were performed. This experimental design comprised eight corner-point conditions. To mitigate the influence of stochastic variations and outliers while ensuring result reliability and robustness, two fully randomized replicate experiments were conducted at each corner point. Additionally, three center points were incorporated to detect significant curvature effects in the derived model. In total, 19 experimental runs were executed. The factorial design matrix and computational results are provided in

Table 8. Factorial design matrix and computational results.

No.	Inlet Placement Angle (°)	Inlet Diameter (mm)	Outlet Width (mm)	Average Wear Rate (kg·m−2·s−1)
1	15	60	7	1.374 × 10−4
2	25	60	7	9.76 × 10−5
3	15	70	7	1.733 × 10−4
4	25	70	7	1.168 × 10−4
5	15	60	13	1.288 × 10−4
6	25	60	13	9.63 × 10−5
7	15	70	13	1.448 × 10−4
8	25	70	13	8.33 × 10−5
9	15	60	7	1.388 × 10−4
10	25	60	7	9.97 × 10−5
11	15	70	7	1.815 × 10−4
12	25	70	7	1.293 × 10−4
13	15	60	13	1.340 × 10−4
14	25	60	13	9.95 × 10−5
15	15	70	13	1.470 × 10−4
16	25	70	13	8.14 × 10−5
17	20	65	10	1.370 × 10−4
18	20	65	10	1.345 × 10−4
19	20	65	10	1.365 × 10−4

.

Model validity was assessed via analysis of variance (ANOVA). The constructed model required at least one significant main or interaction effect to be deemed adequate. Significant curvature terms indicated optimal parameter combinations within the design space without necessitating steepest ascent experiments. Non-significant lack-of-fit (p > 0.05) confirmed model reliability. Terms with p < 0.05 were considered statistically significant, with lower p-values indicating stronger effects on response variables. ANOVA results for the full factorial design are presented in

Table 9. ANOVA table for average erosion rate.

Type	Degree of Freedom	Adj SS	Adj MS	Value of F	Value of p
Model	7	1.3566 × 10−8	1.938 × 10−9	107.44	<0.001
Linear	3	1.1669 × 10−8	3.890 × 10−9	215.64	<0.001
A	1	9.1030 × 10−9	9.103 × 10−9	504.67	<0.001
C	1	9.8000 × 10−10	9.800 × 10−10	54.32	<0.001
E	1	1.5860 × 10−9	1.586 × 10−9	87.92	<0.001
Factor interaction	3	1.5540 × 10−9	5.180 × 10−10	28.72	<0.001
A × C	1	5.0600 × 10−10	5.060 × 10−10	28.03	<0.001
A × E	1	3.0000 × 10−12	3.000 × 10−12	0.15	0.706
C × E	1	1.0460 × 10−9	1.046 × 10−9	57.99	<0.001
Bend	1	3.4300 × 10−10	3.430 × 10−10	19.03	0.001
Error	11	1.9800 × 10−10	1.800 × 10−11
Lack-of-fit	1	5.7000 × 10−11	5.700 × 10−11	4.08	0.071
Pure error	10	1.4100 × 10−10	1.400 × 10−11
Total	18	1.3765 × 10−8

, while the regression model for average erosion rate is provided in

Table 10. Regression model for average erosion rate.

S	R-Sq	R-Sq (Adjustment)	R-Sq (Prediction)
4.2 × 10−6	98.56%	97.64%	95.47%

.

Analysis of Table 8 and Table 9 demonstrated that all main and interaction effects exhibited p-values < 0.001, confirming model validity. The significant curvature term (p < 0.05) indicated the existence of stationary points within the design space. Non-significant lack-of-fit (p > 0.05) verified model adequacy, though the detected curvature rendered the first-order regression model from the full factorial design insufficient for multi-parameter optimization. Nevertheless, this experiment established the optimal parametric domain, enabling rigorous implementation of a higher-order RSM in subsequent stages.

4.3. Response Surface Experiment

Design-Expert 13 software was utilized to implement a Box–Behnken Design (BBD) for the three key factors within their predefined ranges. Seventeen experimental runs were executed, comprising five center points and twelve cube points. The spatial configuration of experimental points is illustrated in Figure 12, with factor level coding consistent with Table 7.

Numerical simulations for the BBD experimental matrix were performed using ANSYS Fluent (2024R1). The BBD matrix and corresponding computational results are presented in

Table 11. BBD matrix and corresponding calculation results.

Model	Outlet Width (mm)	Inlet Diameter (mm)	Inlet Placement Angle (°)	H (m)	P (kW)	η (%)	Average Wear Rate (kg·m−2·s−1)
1	20	65	10	54.21	5.733	64.26	1.34 × 10−4
2	25	65	13	43.75	4.521	65.78	1.04 × 10−4
3	20	60	13	52.07	5.591	63.29	1.14 × 10−4
4	25	70	10	53.32	5.815	62.31	9.51 × 10−5
5	15	60	10	58.78	6.499	61.46	1.36 × 10−4
6	20	60	7	45.11	4.499	68.14	1.23 × 10−4
7	15	70	10	53.82	5.844	62.59	1.33 × 10−4
8	25	60	10	53.51	5.793	62.77	9.06 × 105
9	15	65	13	53.57	5.799	62.78	1.32 × 10−4
10	20	65	10	46.25	4.560	68.92	1.18 × 10−4
11	20	70	7	45.88	4.558	68.40	1.5 × 10−4
12	20	65	10	59.60	6.663	60.79	1.18 × 10−4
13	20	65	10	60.37	6.731	60.95	1.21 × 10−4
14	20	65	10	53.64	5.790	62.96	1.19 × 10−4
15	25	65	7	58.91	6.496	61.63	1.21 × 10−4
16	15	65	7	53.46	5.780	62.85	1.72 × 10−4
17	20	70	13	53.53	5.794	62.78	1.1 × 10−4

.

4.4. Fitted Regression Equation and Significance Testing

A second-order response surface model was employed to analyze the nonlinear effects of multiple factors on the response variables. The generalized second-order polynomial model takes the following form:

$$y={\beta }_0+{\displaystyle \sum _{i=1}^k{\beta }_i}{x}_i+{\displaystyle \sum _{i=1}^k{\beta }_{ii}}{x_i}^2+{\displaystyle \sum _{i<j}{\beta }_{ij}}{x}_i{x}_j+\epsilon$$

(7)

where y denotes the response variable, x_i and x_j represent coded independent variables, β₀ is the constant term, β_i signifies linear effect coefficients, β_ii indicates quadratic effect coefficients, β_ij corresponds to interaction effect coefficients, and ɛ designates random error.

The experimental data sets were substituted into Equation (7) to fit regression equations for the response variables, establishing relationships between independent variables x and responses y. The established fitting regression equations for head, shaft power, efficiency, and average erosion rate are presented in dimensionless coded form, as shown in Equations (8)–(11).

(1)

Regression equation for head:

$$\begin{array}{l}H=53.66+0.0488\times A+0.4575\times C+7.08\times E-0.2675\times AC+0.99\times AE+0.0125\times CE\\ -0.233\times A^2-0.2105\times C^2-1.10\times E^2\end{array}$$

(8)

(2)

Regression equation for shaft power:

$$\begin{array}{l}P=5.78+0.0593\times A+0.0564\times C+1.03\times E-0.0425\times AC+0.0685\times AE+0.0263\times CE\\ +0.0012\times A^2-0.021\times C^2-0.204\times E^2\end{array}$$

(9)

(3)

Regression equation for efficiency:

$$\begin{array}{l}\eta =63.09-0.6462\times A-0.0575\times C-3.30\times E+0.1525\times AC+0.615\times AE-0.2325\times CE\\ -0.3390\times A^2+0.0385\times C^2+1.57\times E^2\end{array}$$

(10)

(4)

Regression equation for average erosion rate:

$$\begin{array}{ll}{R}_c=& 0.0001-0.000020403\times A+0.000003034\times C-0.00001314\times E+0.000005847\times \mathit{AE}\\ & -0.00000769\times \mathit{CE}-0.000007591\times C^2+0.00001039\times E^2\end{array}$$

(11)

Residual plots were employed to validate the regression equation. As shown in Figure 13, the samples are distributed on both sides of the x-axis with significant randomness, indicating that the response function is acceptable. However, reliance on residual examination alone is insufficient for assessing the goodness-of-fit of the response surface model and may fail to detect all potential issues. Therefore, the R-criterion was supplemented to comprehensively evaluate the model’s performance and applicability.

The R criterion comprises $R^2$, adjusted $R_{adj}^2$, and predicted $R_{pred}^2$. $R^2$ is typically used to evaluate the degree of deviation around the model mean; $R^2$ closer to 1 indicates a better goodness of fit of the model. However, as the number of parameters in the model increases, $R^2$ will also increase with the addition of parameters, even if the influence of some parameters is minimal. Hence, the adjusted $R_{adj}^2$ is introduced. $R_{adj}^2$ is posed as a supplementary measure, and its formula is as follows:

$$R^2=1-{\displaystyle \frac{S{S}_E}{S{S}_T}}(0\le R^2\le 1)$$

(12)

$$R_{adj}^2=1-{\displaystyle \frac{S{S}_E/d_E}{S{S}_T/d_T}}=1-{\displaystyle \frac{d_T}{d_E}}(1-R^2)$$

(13)

d_T represents the total degrees of freedom for the model. When both $R^2$ and adjusted $R_{adj}^2$ exceed 0.9 and their difference is minimal, it indicates an excellent model fit. When both $R^2$ and adjusted $R_{adj}^2$ are below 0.9, it suggests removing parameters with low sensitivity to the model response to improve model performance.

Predicted $R_{pred}^2$ is used to assess the predictive performance of the response surface model for unknown samples, evaluating this by quantifying the model’s prediction error on new samples. The formula for predicted $R_{pred}^2$ is as follows:

$$R_{pred}^2=1-{\displaystyle \frac{PRESS}{S{S}_T}}$$

(14)

$R_{pred}^2$ represents the sum of squared prediction residuals, used to measure the model’s goodness of fit within the sample design space. When the difference between $R_{pred}^2$ and $R_{adj}^2$ is less than 0.2, it indicates that the model has good predictive capability for new samples.

Table 12. Metrics for assessing response surface accuracy.

	Average Wear Rate (kg·m−2·s−1)	H (m)	H (%)	P (kW)
$R^2$	0.9591	0.9961	0.9709	0.9995
$R_{adj}^2$	0.9065	0.9910	0.9335	0.9990
$R_{pred}^2$	0.7804	0.9508	0.7686	0.9976

lists the numerical values of $R^2$, $R_{adj}^2$, and $R_{pred}^2$ to evaluate the accuracy of the response surface.

As can be seen from Table 12, the values of both $R^2$ and $R_{adj}^2$ are greater than 0.9, and the difference between $R_{adj}^2$ and $R_{pred}^2$ is less than 0.2. These results indicate that the response surface model exhibits high fitting accuracy and possesses good predictive capability. Therefore, this model will be employed for the subsequent optimization work.

5. Results and Discussion

5.1. Main Effect Analysis of Variables

Figure 14, Figure 15 and Figure 16 illustrate the variation trends of centrifugal pump performance versus inlet placement angle, inlet diameter, and outlet width, respectively.

As indicated in Figure 14, Figure 15 and Figure 16, the average erosion rate decreased with the increasing inlet placement angle, which initially increased and then decreased with the inlet diameter, and, conversely, decreased initially and then increased with outlet width. The head increased initially, then decreased with inlet placement angle; meanwhile, it increased with both inlet diameter and outlet width. Efficiency increased initially, then decreased with inlet placement angle, but also decreased with inlet diameter and outlet width. Shaft power consistently increased with all three parameters: inlet placement angle, inlet diameter, and outlet width. These results demonstrated that variations in centrifugal pump performance parameters were significantly influenced by complex interactions among multiple factors.

5.2. Analysis of Interaction Effects Between Variables

Interaction effects among the three significant variables ((A): inlet placement angle, (C): inlet diameter, (E): outlet width) are presented in Figure 17a through Figure 17l.

As depicted in Figure 17a–c, the response surfaces for control variables versus average impeller erosion rate demonstrated that reduced wear was achieved by the following: (1) larger inlet placement angles coupled with smaller inlet diameters, (2) larger inlet placement angles combined with larger outlet widths, and (3) larger outlet widths paired with larger inlet diameters. Figure 17d–f illustrated head response surfaces, revealing head enhancement through: (1) larger inlet diameters with smaller inlet placement angles, (2) larger inlet placement angles with larger outlet widths, and (3) larger inlet diameters with larger outlet widths. The efficiency response surfaces (Figure 17g–i) indicated efficiency improvement via (1) smaller inlet diameters with smaller inlet placement angles, (2) smaller outlet widths with smaller inlet placement angles, and (3) smaller outlet widths with larger inlet diameters. Finally, shaft power response surfaces (Figure 17j–l) showed power increase through (1) larger inlet diameters with larger inlet placement angles, (2) larger outlet widths with larger inlet placement angles, and (3) larger outlet widths with larger inlet diameters.

5.3. Comparison of Pump Performance Before and After Optimization

The optimal parameter combination for the impeller was predicted using Design-Expert 13 software. A comparison of design parameters between the prototype and optimized pumps is provided in

Table 13. Comparison of design parameters between the prototype and optimized pumps.

Model	Inlet Placement Angle (°)	Inlet Diameter (mm)	Outlet Width (mm)
Prototype pump	15	65	10
Optimized pump	21	60	9.9

, with all other parameters unchanged from the prototype.

The performance comparison of the centrifugal pump before and after optimization is shown in

Table 14. Comparison of pump performance.

Model	Average Wear Rate(kg·m−2·s−1)	H (m)	η (%)	P (kW)
Prototype pump	1.550 × 10−4	53.00	63.21	5.698
Optimized pump	1.064 × 10−4	52.41	62.84	5.668
Variation	−4.86 × 10−5	−0.59	−0.37	−0.03

.

According to Table 14, the optimized pump exhibited reductions of 0.59 m in head, 0.37% in hydraulic efficiency, and 0.03 kW in shaft power-all within acceptable tolerance limits. Crucially, the erosion rate decreased significantly by approximately 31.35% compared to the prototype, achieving the objective of enhancing wear resistance through impeller structural optimization.

The loss in head and efficiency is acceptable within actual irrigation systems. In engineering, the allowable deviation for centrifugal pump efficiency is typically 3–5%, and the allowable head deviation falls within a similar range. In field practice, irrigation systems generally possess a certain degree of operational flexibility, allowing such minor performance variations to be easily compensated for through minor adjustments to operational parameters. Most importantly, the approximately 31.35% reduction in wear rate will significantly extend the pump’s service life in sediment-laden water flow, decrease maintenance downtime and component replacement costs, thereby substantially enhancing the equipment’s economic viability and operational reliability from a lifecycle perspective. This optimization has effectively achieved the primary goal of improving wear resistance while maintaining essentially stable performance.

Figure 18, Figure 19 and Figure 20 compare the wear conditions at the same location under identical operating conditions for the original pump and the optimized pump, as shown below.

A comparison of the wear rate nephograms in Figure 18, Figure 19 and Figure 20 leads to the following conclusion: at corresponding identical locations, the proportion of severe wear areas in the optimized pump is significantly lower than that in the original pump. This confirms the effectiveness of the optimization scheme in reducing wear.

Figure 21, Figure 22 and Figure 23 present comparison curves of centrifugal pump performance before and after optimization under varying particle sizes, particle densities, and mass flow rate.

As indicated in the preceding Figures, the head of the optimized pump remained marginally lower than the prototype across increasing particle sizes, while its shaft power demonstrated greater stability, and the erosion rate was significantly reduced. When particle density increased, the prototype exhibited substantial declines in head and shaft power; although efficiency improved with density, erosion rate rose sharply. In contrast, the optimized pump showed smaller performance variations in response to density changes, maintaining more stable efficiency and erosion characteristics. Under rising mass flow rate, both configurations experienced head and efficiency reductions alongside increased shaft power and erosion. Comparative analysis revealed superior wear resistance and performance stability in the optimized pump.

6. Conclusions

(1) This study established a reliable numerical model for sediment-induced wear in centrifugal pumps, which has been validated to show high agreement with experimental results. The research reveals the influence patterns of key sediment parameters and, through comparison with previous studies, clarifies the academic value and engineering significance of these findings: the wear rate exhibits nonlinear growth with increasing particle density (1350~3300 kg/m³), consistent with the direct influence mechanism of particle inertia on impact energy. This aligns with fundamental fluid dynamics theories and conclusions from most wear studies, while further providing a clear growth trend within this density range. The wear intensity reaches its maximum at a particle size of 1 mm, which corresponds to the widely recognized phenomenon of the most critical particle size in wear research. Compared to previous studies that primarily noted this phenomenon on a macroscopic level, this paper specifies the exact critical particle size for centrifugal pumps as a typical hydraulic machinery, thereby offering precise guidance for engineering protection.

(2) Through systematic optimization using Response Surface Methodology (RSM), three key geometric parameters—the inlet placement angle (A), inlet diameter (C), and outlet width (E)—were identified as critically influencing both wear and performance. The optimized parameter combination achieved a significant reduction in wear rate by 31.35%. These findings demonstrate that multi-objective collaborative optimization can effectively enhance the wear resistance of centrifugal pumps while maintaining their hydraulic performance. This provides a practical engineering reference for the design and modification of pumps operating in high-sediment environments.

(3) Parametric effect analysis demonstrated that higher inlet placement angles reduced wear but increased shaft power; intermediate inlet diameters minimized wear-rate growth; narrower outlet widths improved efficiency but elevated wear at low sediment densities. Increasing the inlet placement angle while reducing the inlet diameter maximizes the wear resistance of the equipment, whereas simultaneously enlarging the outlet width and the inlet diameter effectively enhances its head performance. This optimized combination provides guidance for relevant engineering design and performance balancing. Under varying sediment conditions, the optimized pump maintained superior stability in head, efficiency, and wear rate compared to the prototype.

(4) The methodology provides a replicable workflow for wear-resistant hydraulic design: CFD-DPM erosion analysis identifies critical wear zones; RSM-based multi-stage screening and optimization derive anti-wear geometric rules; experimental validation ensures field applicability. For Southern Xinjiang’s high-sediment irrigation environments, this approach extends pump service life while retaining hydraulic performance within acceptable thresholds.

Limitations and future work include the following: the DPM assumes spherical particles, neglecting irregular sediment shapes common in field conditions; the Oka model does not account for material hardening effects during prolonged erosion; interactions between cavitation and sediment wear were not investigated. Future studies should incorporate non-spherical particle tracking, evaluate hardened coatings via coupled fluid–structure erosion models, and explore multi-objective optimization balancing wear resistance with cavitation performance under transient operations.