Optimization of Hydraulic Efficiency and Internal Flow Characteristics of a Multi-Stage Pump Using RBF Neural Network

Abstract

In order to improve the hydraulic efficiency and internal flow pattern of a multi-stage pump under multiple flow conditions, an intelligent optimization design was proposed for its hydraulic components. Sensitivity analysis was used to select the key parameters influencing the hydraulic efficiency of a multi-stage pump. The optimal Latin hypercube sampling and non-dominated sorting genetic algorithm Ⅱ were employed to build a multi-objective optimization system. Moreover, a radial basis function neural network was adopted as the surrogate model of hydraulic efficiency. The research results showed that the impeller outlet width, impeller blade wrap angle, impeller outlet blade angle, and diffuser inlet width were the key factors affecting the hydraulic efficiency. The efficiency of the optimized model increased by 4.35% under the design condition and the matching of the internal flow between the optimized impeller and diffuser was significantly enhanced under the nominal condition. The improved flow pattern could be clearly observed in the flow passage of both the pump impeller and the diffuser. After optimization, the wear performance of the model was also improved compared to the original design. The wear area decreased in size and was distributed more evenly, resulting in a noticeable decrease in the maximum amount of wear.

1. Introduction

According to the report published by the International Conference on Fluid Machinery and Engineering in recent years [1], the power consumption of the pumping industry accounts for approximately 20% of the total power generation in China. As one of the most widely used pumps, multi-stage centrifugal pumps are widely employed in many industrial processes, such as coal and mining, chemical engineering, agriculture, and marine engineering. They consume a lot of power every year. An improved efficiency of multi-stage centrifugal pumps can effectively reduce greenhouse gas emissions and, therefore, enhancing the energy efficiency of multi-stage centrifugal pumps is a necessary [2]. With the increase in in-depth research on multi-stage centrifugal pumps worldwide, performance optimization has always been a research hotspot. Improving the optimized design and performance of multi-stage pumps can not only contribute to the advancement of energy systems in China, but will also provide a reliable research basis for developing the global renewable energy market [3,4,5].

Currently, the optimized design of a pump’s hydraulic performance is regarded as one of the most common research aims by many researchers. Hydraulic performance is strongly related with both the external characteristics and the internal flow characteristics of pumps. On the basis of previous studies and the analyses of the internal flow losses, Neumann et al. [6] established a correlation between the external characteristic parameters, internal flow characteristic parameters, and geometric parameters of hydraulic components in centrifugal pumps. Benturki et al. [7] established a mathematical model for the optimal design of hydraulic performance of a low-specific-speed centrifugal pump with the highest pump efficiency as the objective function and the geometric parameters of hydraulic components as the optimized variables. Wang et al. [8] proposed the optimized design of a typical multi-stage centrifugal pump based on an energy loss model and computational fluid dynamics (CFD) to evaluate the pump performance by calculating various energy losses in the multi-stage centrifugal pump. Shahram et al. [9] used Artificial Neural Networks (ANNs) and the Artificial Bee Colony (ABC) algorithm to optimize the centrifugal pump; the results showed an efficiency improvement of 3.59% at only 6.89 m increase in total pressure, and indicated reasonable improvement in the optimal design of pump impeller. Westra RW et al. [10] optimized the centrifugal pump impeller using the Differential Evolution algorithm; the cavitation characteristics of the optimized impeller have been significantly improved. Gao et al. [11] took the five-stage centrifugal pump as the research object, used the principle of Tesla valve, and added a set of auxiliary blades on the impeller, which reduced the leakage of the impeller ring by changing the direction and size of the leaked fluid’s flow, and reduced the volumetric loss to improve the efficiency of the multi-stage centrifugal pump.

Compared to single-objective optimization, multi-objective optimization can obviously and simultaneously improve several pump performance indicators. Based upon multiple constraint conditions, Chen et al. [12] constructed a multi-objective optimization model to improve the hydraulic performance of a centrifugal pump, and obtained the optimized parameters of the impeller’s inlet and outlet. Tong et al. [13] used the entropy theory to quantitatively analyze and accurately locate the energy loss of an axial flow pump under different working conditions, and analyzed the effect of groove flow control technology on the energy performance of an axial flow pump. The energy loss characteristics before and after adding the grooves were compared at different flow rates. The mechanism of the grooved flow control technology to improve the energy performance of axial flow pumps was also revealed. Han et al. [14] proposed a two-step strategy for multi-operating condition optimization. In this optimization strategy, the first step is to optimize pump performance under the design condition, and the second step is to optimize pump performance under various other operating conditions. In the application of multi-parameter, multi-operating condition optimizations of axial flow pumps, there are eight parameters: two cascading stability parameters, two blade load parameters, two blade angle parameters, blade quantity, and axial length. Alawadhi et al. [15] used a multi-objective genetic algorithm (MOGA) and response surface methodology (RSM) to optimize the centrifugal pump. Six geometric parameters were design variables: number of vanes, inlet beta shroud, exit beta shroud, hub inlet blade draft, Rake angle, and the impeller’s rotational speed. The objective functions employed are pump power, hydraulic efficiency, volumetric efficiency, and pump efficiency. Nourbakhsh and Safikhani et al. [16,17] established an approximate model through a neural network that relates NPSHr and efficiency to parameters such as blade exit angle, blade wrap angle, and volute throat area. Optimization using the particle swarm optimization method (MOPSO) and the multi-objective genetic algorithm (NSGA-II) resulted in a 1.08 percentage point improvement in the model’s efficiency. Shahram et al. [18] optimized the geometric parameters of a centrifugal pump’s impeller and verified the optimization results by comparing the optimized impeller with the prototype through CFD calculations. The results showed a significant improvement in the pump’s hydraulic performance after the optimization. Benturki et al. [7] dealt with the optimization of a two-stage centrifugal pump using the Non-dominated Sorting Genetic Algorithm II (NSGA-II), coupled with a three-dimensional Reynoldsaveraged Navier-Stokes (3D-RANS) flow solver. The overall efficiency as well as the head of the optimized pump were increased by 9.8% and 15.7%, respectively, at best efficiency point. Thakkar et al. [19] presented an effective approach for the enhancement of the performance of a sanitary centrifugal pump by adopting CFD and response surface methodology with a multi-objective optimization algorithm. Three input design parameters, namely blade outlet angle, blade wrap angle, and blade outlet width, are investigated for the simultaneous optimization of pump head and efficiency. The results showed a 9.154% and 10.15% improvement in head and efficiency at the design point as compared to initial pump.

The fluid conveying in a multi-stage centrifugal pump usually carries solid particles, which can cause significant wear on the surfaces of hydraulic components. Therefore, the impact of the optimization of pump hydraulic efficiency on the wear characteristics of the optimized pump should be paid special attention. Early research on pump wear relied mainly on experimental methods. With the development of computers, CFD methods began to be applied to multiphase flow inside the pumps. For multiphase flow inside pumps, early studies mainly used the Mixture model, which assumes that the fluid and solid particles are a homogeneously mixed continuous medium. Zhao et al. [20] and Zhang et al. [21] studied the effects of particle concentration and size on the performance of mixed flow pumps using the Mixture model. The results showed that particle concentration and size have certain effects on pump’s head and efficiency. Liu et al. [22] simulated the solid–liquid two-phase flow inside a dual-channel pump using the Mixture model and found that the concentration and size of particles had the greatest influence on the distribution of particles inside the pump. However, because the Mixture model cannot accurately simulate the motion of solid particles inside the pump, researchers began to use the Discrete Phase Model (DPM), which directly solves the motion of particles coupled with the flow field. Nemitallah et al. [23] used a DPM model to simulate the solid–liquid two-phase flow behind an orifice plate. The results showed that wear is more severe in the area around the orifice plate and is strongly correlated with particle size and flow velocity. Pagalthivarthi et al. [24] used the DPM model to study the erosion wear trends in centrifugal pump casing when pumping dilute slurries, the effect of several operational parameters on pump flow rate, pump speed (RPM), particle diameter, and various geometry conditions including tongue curvature, slope of the discharge pipe, and casing width. The results showed that, with an increase in pump flow rate, wear rates tend to even out whereas with increased casing width, wear rates are found to decrease. According to Sommerfeld’s research [25], when the particle volume fraction exceeds 1 × 10⁻⁶, the two-way coupling between particles and fluid must be considered, and the fluid (particle) volume fraction should be calculated in the fluid governing equations. Dai et al. [26] used the CFD-DEM coupling method to numerically simulate the vertical pipeline and compared it with experimental results. The results show that the internal flow velocity and particle concentration affect the stress and transport performance of the pipeline. Zhao et al. [27] simulated the effects of particle mass load and various interphase forces on the wear of 90° bend based on the CFD-DEM model. The research results indicate that the mass loading of particles is the most significant parameter to cause the erosion rate in the bend and the lift force exerted on the particles by the fluid is insignificant in causing the bend erosion. Liu et al. [28] studied the hydraulic performance and wear characteristics of centrifugal pumps using the CFD-DEM algorithm. The results showed that, with the increase in impeller blade angle, the head, efficiency, and average wear rate all first increased, and then decreased. Deng et al. [29] evaluated the improvement of particle transport characteristics and hydraulic performance of deep-sea mining pumps through CFD-DEM simulation. They restricted and modified most of the pre-swirling flow and solved the problem of particle accumulation in the impeller suction area. Zhao et al. [30], based on the CFD-DEM method and considering the influence of particles on turbulence modulation, validated the accuracy of the computational model by comparing it with existing experimental results. They studied the effect of solid particle concentration on pump performance and wear. The results showed that the addition of low-concentration particles weakened the turbulent kinetic energy inside the pump, resulting in more stable flow and improved head and efficiency. Under high-concentration conditions, particle aggregation inside the pump became severe, leading to an increase in turbulent kinetic energy and a decrease in head and efficiency. The frequent collision between particles and the leading edge and working surface of the impeller blades resulted in significant wear at these locations. Particle accumulation occurred at the tongue of the volute casing, causing the most severe wear at this location.

In short, many researchers have focused on using CFD and optimization algorithms to improve the hydraulic performance of various single-stage pumps. According to the literature listed above, the combination of neural networks and genetic algorithms not only greatly improves optimization efficiency, but also enhances the accuracy of optimization, making the optimization results more accurate. However, there are relatively fewer cases of optimizing the hydraulic performance of multi-stage centrifugal pumps, especially for considering the matching of internal flow between the pump’s impeller and diffuser. In this paper, a multi-objective optimization method is used to improve the efficiency of a multi-stage centrifugal pump using geometric parameters such as the impeller’s outlet blade angle, impeller’s outlet width, impeller’s wrap angle, and diffuser’s inlet width as the optimized variables. A model pump that meets the design conditions with sufficient hydraulic performance was first designed, and then the optimized model was presented and numerically analyzed using CFD. The optimal solution was obtained by using a radial basis function (RBF) neural network and non-dominated sorting genetic algorithm II (NSGA-II) through comparison. Furthermore, previous optimization efforts have primarily focused on hydraulic performance, with little research conducted on the correlation between hydraulic optimization and changes in the wear characteristic, which is also discussed in the current work.

2. Methodology

2.1. Geometric Model

A multi-stage pump was designed using the software package CFturbo 9.2.5, and the three-dimensional (3D) model of the whole calculation domain was generated in the UG 11 platform. The numerical simulation solved the flows in all domains between the inlet and outlet of the pump. The assembly diagram of the fluid domain for the multi-stage pump model is shown in Figure 1. The hydraulic components are composed of an inlet pipe, two sets of centrifugal impellers and diffusers, and an outlet pipe. The basic design and main geometric parameters are presented in

Table 1. Basic design and main geometric parameters.

Parameters	Value
Design flow (Qd)	0.025 m3/s
Head of delivery (H)	11 m
Rated speed (n)	2900 r/min
Specific speed (ns)	277
Impeller inlet diameter (D1)	100 mm
Impeller outlet diameter (D2)	135 mm
Impeller hub diameter (D3)	45 mm
Impeller outlet width (b2)	23 mm
Impeller wrap angle (θ)	125°
Impeller inlet blade angle (β1)	32°
Impeller outlet blade angle (β2)	23°
Impeller blade number (Z)	5
Diffuser inlet width (b1′)	27 mm
Diffuser outlet width (b2′)	27.5 mm
Diffuser inlet blade angle (β1′)	20°
Diffuser outlet blade angle (β2′)	92°
Diffuser wrap angle (θ′)	70°
Diffuser’s vane number (ZS)	7

.

2.2. Numerical Simulation Setting

The flow inside a pump is typically complex 3D turbulent flow with a Reynolds number greater than 10⁶. This article performs quasi-steady numerical simulations to resolve the flow in all pump components using ANSYS CFX 17.2. The k-ε turbulence model was used to solve the Reynolds time-averaged equation, whereas the reference pressure was set to be 1 atm. The inlet and outlet were set as translational periodic boundary interfaces, while the dynamic–static interface was set as the Frozen-rotor type. It is consistent with the numerical simulation method of the mixed flow pump by Ji et al. [31,32]. The scalable wall functions were used for all boundary layers near the wall and the wall surfaces were set as the “no-slip” condition. The discretization scheme of the convective term was set as “High Resolution”, which allowed the variation of a blend factor between 0 and 1 in flow regions for both robustness and accuracy.

2.3. Mesh Generation and Irrelevant Argument

In order to solve the governing equation in the computational domain, the generation of mesh was conducted in ICEM 17.2 software to spatially discretize each calculation sub-region [33]. A hybrid mesh scheme was employed for the model discretization. Considering both the calculation accuracy and the computational resources, the structured mesh strategy was adopted for the impeller and diffuser, while the unstructured mesh was used for the rest of the parts. The impeller and diffuser are discretized from a hexahedral mesh structure. The flow channels near a single blade are meshed by using J/O type grid topology, and then the overall grid partitioning is achieved through the periodic array method. In order to ensure the calculation accuracy of turbulence, the mesh near the wall was refined. Figure 2 and Figure 3 display the generated meshes for the impeller and diffuser, respectively.

An insufficient number of meshes may result in relatively low accuracy, while more computational resources are required due to an excessive number of meshes. In order to balance this confliction, the present study carried out the mesh independence analysis for the computational domain. The steady simulation under the rated flow condition was performed for the computational domain with the total mesh numbers of 1.72 million, 2.28 million, 2.77 million, 3.27 million, 3.58 million, and 4.04 million. The variations in the pump’s head and efficiency under different mesh numbers were obtained, as shown in Figure 4. The results show that, with the increase in the number of meshes, the efficiency fluctuates slightly. As for the change in head (or total pressure), it first decreases with the increase in the number of meshes to 3.27 million, and then goes up again slowly beyond 3.27 million meshes. Therefore, the total number of meshes for the calculation domain was eventually determined to be 3.58 million.

2.4. Optimization of Design Parameters

The design parameters of the diffuser have a significant impact on its matching with the impeller. Therefore, five main design parameters of the diffuser were selected as the initial optimization variables and included diffuser’s blade number Z, diffuser’s wrap angle θ′, diffuser’s inlet width b₁′, diffuser’s inlet blade angle β₁′, and diffuser’s outlet blade angle β₂′. Meanwhile, four design parameters of the impeller were selected as variables to further consider the matching between diffuser and impeller: impeller’s outlet width b₂, impeller’s inlet blade angle β₁, impeller’s outlet blade angle β₂, and impeller’s wrap angle θ. The selected design parameters are graphically shown in Figure 5.

2.5. Sensitivity Test Design

According to Section 2.4, 4 geometric parameters of the impeller and 5 geometric parameters of the diffuser were selected as the initial optimization variables. Considering that the 9 variables will consume a lot of computational resources and reduce the efficiency of the optimization process, a sensitivity test was used to screen out the parameters that have a relatively smaller effect on pump performance.

Orthogonal experimental design is a sensitivity experimental method to study the effect of multiple factors and multiple levels on targeted performance [34]. Based on orthogonality, it selects representative points from the design space. These representative points possess the characteristics of uniform dispersion, uniformity, and comparability. It is an efficient, fast, and economical experimental design method. There were 9 factors influencing the performance of multi-stage centrifugal pumps and each factor contributes to the target at different weights. Therefore, the workload and the amount of calculations can be effectively reduced by using the orthogonal experimental method.

The high and low levels of each factor were determined and 27 schemes were designated according to L16 orthogonal table. CFD calculations were carried out to obtain the corresponding pump performances.

Table 2. Design Factors and levels.

Factors	Low Level (−)	High Level (+)
Impeller outlet width b2/mm	19	25
Impeller outlet blade angle β2/(°)	20	28
Impeller wrap angle θ/(°)	110	140
Impeller inlet blade angle β1/(°)	28	36
Diffuser blades number ZS	7	9
Diffuser wrap angle θ′/(°)	60	80
Diffuser inlet width b1′/mm	26	32
Diffuser inlet blade angle β1′/(°)	17	23
Diffuser outlet blade angle β2′/(°)	82	102

presents the detailed design factors and levels.

The 27 design scenarios obtained from the orthogonal experimental design were numerically calculated with focus on the hydraulic efficiency of the pump. Based on the simulation results, a regression equation, with efficiency as the evaluation index, was established. The normalized Pareto plot between the efficiency at design point and the 9 design parameters is shown in Figure 6, where the larger value denotes a greater correlation and more significant impact on the hydraulic efficiency of the pump.

Figure 6 shows that the impeller’s outlet width has the most significant impact on the hydraulic efficiency of the multi-stage pump, followed by the impeller’s wrap angle. The impeller’s outlet blade angle and the diffuser’s inlet width also have a significant impact on the efficiency of the pump. Therefore, these four design parameters were selected as the final variables for the optimized design in the present study.

2.6. Sample Test Sampling

In order to establish a high-precision approximation model, it is necessary to select appropriate sample points in the sample design space. The optimal Latin hypercube sampling is a stratified sampling method that can fully reflect the characteristics of the samples in the design space with fewer sampling times. In the current investigation, 56 sets of sample points were arranged for the four variables in the design space, which is far more than the required number of samples to construct an approximate model. The sampling space range is presented in

Table 3. Scope of the sampling space.

Optimized Design Variables	Lowerlimit	Upperlimit	Original Value
Impeller wrap angle θ/(°)	106.25	143.75	125
Impeller outlet width b2/mm	19.55	26.45	23
impeller outlet blade angle β2/(°)	19.55	26.45	23
Diffuser inlet width b1′/mm	22.95	31.05	27

.

2.7. Numerical Calculations and the Optimization of Platform Design

CFturbo, ICEM, CFD, and CFX were integrated through batch commands in the Insight platform. Then, an automatic numerical simulation procedure for optimizing the multi-stage centrifugal pump was built to ensure the accuracy and consistency of numerical simulation and shorten the optimization cycle. The multi-objective optimization function with the hydraulic efficiencies of 1.0 Q_d and 0.75 Q_d as the optimization objectives is calculated using Equation (1). The software integration framework is shown in Figure 7.

$$\{\begin{array}{l}F\mathrm{ind}X=\left[b_2,\theta ,{b_1}^{\prime },{\beta }_2\right]\\ \{\begin{array}{l}\max {\eta }_1=f_1\left(b_2,\theta ,{b_1}^{\prime },{\beta }_2\right)\\ \max {\eta }_2=f_2\left(b_2,\theta ,{b_1}^{\prime },{\beta }_2\right)\end{array}\\ \mathrm{Subject}\mathrm{to}:\\ 19.55\le b_2\le 26.45\\ 106.25\le \theta \le 143.75\\ 22.95\le {b_1}^{\prime }\le 31.05\\ 19.55\le {\beta }_2\le 26.45\end{array}$$

(1)

Fifty-six sets of arrangements and corresponding optimized objective values were provided by the automatic optimization platform. Based upon the CFturbo software for turbomachinery’s design, the design of the diffusers and volute of the multi-stage centrifugal pump can be parameterized. The 3D models of the diffuser and the volute were separately imported into ICEM CFD. The scripting program was enabled to automatically perform the generation of mesh. Finally, the numerical simulation settings and calculations were automatically conducted by importing them into the CFX 17.2 software.

2.8. RBF Neural Network

In this paper, a radial basis function (RBF) neural network was selected to construct a functional mapping between the input and output parameters. When fitting with multi-objectives and multi-parameters, RBF neural networks have the advantages of high accuracy, rapid convergence, and adaptive adjustment. A typical RBF artificial neural network is shown in Figure 8. The first layer is the input layer, the second layer is the hidden layer of the transformation function, while the third layer is the output layer.

Based on the 56 sets of sampled data, RBF neural network was used to construct the nonlinear relationship between the optimization variables and the objective function. The input layer contained 4 neurons, while the output layer contained 2 neurons with 10 neurons used in the hidden layer. To test the fitting accuracy, 85% of the data was used to train the neural network and 15% of the data was adopted to conduct the regression analysis in terms of the coefficient of determination (R²). As can be seen from Figure 9, the R² for the output was calculated to have the target values of greater than 0.95 at both 1.0 Q_d and 0.75 Q_d, indicating a high degree of agreement between the trained RBF neural network and the CFD calculations. This indicates that the model can accurately predict the nonlinear relationship between the hydraulic performance and the geometric parameters of the centrifugal pump.

2.9. Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

NSGA-Ⅱ, shown in Figure 10, is known as a fast non-dominant ranking genetic algorithm with elite retention strategy, and has been proposed based on the first generation of genetic algorithms [35]. Compared with the NSGA algorithm, the NSGA-Ⅱ has been improved in the following aspects.

(1) Elite retention strategy

By setting the elite retention strategy, the population with good parent performance is directly inherited to the offspring, which avoids the loss of optimal solutions in the Pareto frontier. This significantly improves the performance of the genetic algorithm for finding the best solution.

(2) Fast non-dominated ranking algorithm

The time complexity of the NSGA-Ⅱ algorithm was reduced to O(MN²), where M is the number of objective functions and N is the size of population. This significantly reduces the computational burden of the algorithm.

(3) Crowding degree operator

The NSGA selected excellent individuals by setting shared functions, and the individuals that met the parametric requirements were retained. Figure 11 shows the NSGA-Ⅱ algorithm procedure. The trained RBF neural network was used as the fitness evaluation response model for the NSGA-Ⅱ genetic algorithm to carry out the optimization of multi-stage centrifugal pumps under multiple operating conditions.

The initial population, number of generations, and crossover probability were set to be 100, 500, and 0.90, respectively. After 50,000 iterations of optimization, the global Pareto optimal solution was obtained, from which some individuals were selected to form the corresponding Pareto optimal frontier.

2.10. Wear Model

This article selected the wear prediction model for the liquid–solid two-phase flow obtained through extensive experimental measurements [36]. The wear rate is calculated using Equations (1) and (2).

$$ER=C{\left(BH\right)}^{-0.59}{F}_s{\mathrm{v}}_p^{n}F\left(\theta \right)$$

(2)

$$F\left(\theta \right)=\sum _{i=1}^5{A}_i{\theta }^i$$

(3)

where ER represents the wear rate in mm/year, indicating the loss in weight of the material caused by particle collisions on the surface of the material, BH denotes the Brinell hardness of the wall material, F_s = 0.2 represents the particle shape coefficient, v_p is the particle collision velocity, θ is the collision angle of the particles in radians, n = 2.41 is the velocity exponent, and C = 2.17 × 10⁻⁷ is an empirical constant.

2.11. Settings of the Coupled Algorithm

The software ANSYS FLUENT 17.2 was used to conduct the calculations regarding fluid flow. Considering the interactive forces between the fluid and particles, a two-way coupled transient simulation was adopted. To be consistent with the rated and 0.75 Q_d flow rates simulated above, the inlet flow rates were specified as 25 kg/s and 18.75 kg/s, respectively. The outlet was set as a pressure outlet with the static pressure value equal to the standard atmospheric pressure. To ensure adequate development of the fluid and solid components, a straight pipe segment with a length of five times the pipe diameter was defined before the impeller’s inlet. No-slip wall boundary conditions were assumed. Turbulence and swirling flow existed inside the pump, so the RNG k-ε turbulence model was selected. Moreover, the SIMPLEC algorithm was used to couple the velocity and pressure fields in the flow domain.

The software EDEM 2018 was used for particle phase simulation. The properties of the particles and the wall material are listed in

Table 4. Properties of the particles.

Physical Property	Unit	Value
Density	Kg/m3	2650
Diameter	mm	2
Poisson’s Ratio		0.23
Young’s Modulus	Pa	5.9 × 1010
Restitution Coefficient		0.9

and

Table 5. Properties of the wall.

Physical Property	Unit	Value
Density	Kg/m3	8200
Hardness	GPa	3.43
Poisson’s Ratio		0.3
Young’s Modulus	Pa	2.07 × 1011
Particle-Wall Restitution Coefficient		0.8
Particle-Wall Friction Coefficient		0.2

. The rotational speed of the blades was synchronized with that in the CFD simulation. The contact model between the particles was set as Hertz–Mindlin (no-slip). The wear of particles on the wall was calculated using a wear model. Particles were generated at the inlet with a velocity of 4 m/s and a volume concentration of 5%, which represented 290,000 particles per second for the 1.0 Q_d operating condition. The velocity of 3 m/s and a volume concentration of 5% represented 220,000 particles per second for the 0.75 Q_d operating condition.

The FLUENT time step was set to be 2.874 × 10⁻⁴ s, which corresponded to one calculation at every 5 degrees of the impeller’s rotation. The maximum number of iterations was set to be 40, with a total of 3600 steps. The time step of EDEM is one-tenth of that in FLUENT, which was set to be 2.874 × 10⁻⁵ s. The data was saved every 5 revolutions of the impeller. The total calculation time was 1.03 s, which was equivalent to 50 revolutions of the impeller. The analysis focused on the wear distribution within the last 20 revolutions, corresponding to a period of 0.4 s.

3. Results and Discussion

3.1. Optimization of Hydraulic Performance

After a specified number of iterations using the NSGA-II algorithm, the optimized values for the variables were obtained. These optimized variables were then used to re-design the pump’s impeller and diffuser. Then, numerical calculations were performed to obtain hydraulic performance for the model under optimal design conditions.

Table 6. Comparison of various parameters before and after optimization.

Category	b2/mm	β2/(°)	b1′/mm	θ/(°)	η/%		H/m
					0.75 Qd	1.0 Qd	0.75 Qd	1.0 Qd
Prototype	23	23	27	125	73.9	79.29	33.19	22.74
Optimized model	19.551	22.143	23.217	107.23	74.76	83.64	32.52	23.33

presents a comparison between the design parameters and the numerically predicted performances before and after optimization. It can be observed that, after optimization, both the inlet width of the diffuser and the outlet width of the impeller decreased. That is to say, the gap that connects the impeller’s exit and the diffuser’s entrance calls for high matching to shift the flow pattern from the impeller’s blade training edge to the diffuser’s vane leading edge. In addition, the impeller’s outlet blade angle and the impeller’s wrap angle also decreased. The efficiency of the optimized model at design flow rate reached 83.64%, which was 4.35% higher than the prototype pump. Meanwhile, the hydraulic efficiencies of the pump at 1.0 Q_d and 0.75 Q_d predicted by the RBF neural network are 83.96% and 75.23%, which differed from the predicted values of the numerical simulations by only 0.32% and 0.47%., further proving the high accuracy of the neural network model. The improved performance of the optimized model at the rated and low flow rates may be due to the increased matching of the internal flow field between the impeller and the diffuser.

3.2. Comparison of Internal Flow Field before and after the Optimization

The streamlines on the unfolded cross-section plane of the impeller and diffuser at three typical spanwise locations (span = 0.05, span = 0.5, and span = 0.95) before and after the optimization under two operating conditions are shown in Figure 12. The results showed that the streamlines in the impeller before and after the optimization were relatively uniform, while large-scale vortices were formed in the flow passage of the diffuser, leading to the highly non-uniform flow regimes and unwanted hydraulic losses. After the optimization, the vortices in the diffuser were basically eliminated and the flow pattern significantly improved to be smoother and more uniform, especially at the rated flow rate.

The 3D velocity streamlines before and after the optimization in the diffuser domain are shown in Figure 13. After optimization, the vortices in the diffuser flow passage were significantly weakened and the flow pattern improved to some degree.

3.3. Comparison of Pressure Distribution before and after the Optimization

The pressure distribution at three typical spanwise locations (span = 0.05, span = 0.5, and span = 0.95) of the impeller and diffuser before and after the optimization under two operating conditions are shown in Figure 14. It is seen that the pressure gradually increased to show a stratified contour in the blade passage from the leading edge towards the trailing edge of the blade. Under the condition of 0.75 Q_d, the pressure distribution is more uniform along the circumferential direction in the optimized model while the pressure is varied more obviously at the inlet of the impeller and the space between impeller and diffuser of the prototype. Under the condition of 1.0 Q_d, low pressure regions were found at the leading edge of the impeller’s blade and high-pressure regions were seen at the suction side of the diffuser’s vane in the prototype, which may cause large pressure pulsation, reversed flow, and cavitation. However, these areas with extreme pressure were basically diminished in the optimized model, indicating an improved state of flow after the optimization. It should be noted that a negative value in pressure is the relative pressure referred to the reference pressure, i.e., 1 atm. Since the simulation of cavitation was not the research of interests in this work, the absolute pressure was not employed in the calculation.

3.4. Comparison of Velocity Distribution before and after the Optimization

One of the key goals of this work is to optimize the internal flow at the interface between the diffusers of the first stage and the impeller’s blades of the second stage. Figure 15 shows the velocity field around the interface between the first stage and second stage at the nominal flow condition. Three typical spanwise locations are presented. It is very clear that the area and strength of wakes right after the outlet of the diffusers are weakened in the optimized model at the position of 0.5 span where the bulk flow moves through. In the optimized model at 0.5 span, the flow leaving from the outlets of the diffusers in the first stage can effectively attach the leading edge of the impeller’s blades in the second stage. It is seen that there is no significant flow separation at the pressure side of the impeller’s blades. The improved flow field indicates that the optimization on the geometrical parameters achieves a better match at the interface between two stages and increases the hydraulic efficiency at the nominal flow rate.

3.5. Comparison of Wear Distribution before and after the Optimization

According to the contours of the predicted wear distribution on the impeller blades of first stage (Figure 16), the wear mainly occurred at the blade inlets and blade outlets and at the junction of the working surfaces with the back shroud. The most severe wear occurred at the leading edge because there were fierce collisions between the particles and the blade head with high circumferential velocity. The annual wear at the blade head exceeded 3000 mm/y. Under the condition of 0.75 Q_d, the wear mainly occurred in the posterior region after the blade inlet, reaching 50 mm/y. When the flow rate increased from 0.75 Q_d to 1.0 Q_d, the wear extended towards the middle portion of the working surface, exceeding 150 mm/y. The wear at the trailing edge significantly increased, with an annual wear exceeding 300 mm/y. By comparing Figure 16c,d, it is evident that the distribution of wear in the optimized model was more uniform than in the prototype. Moreover, the wear at the trailing edge of the blades was notably reduced, with an annual wear rate not exceeding the value of 150 mm/y. This improvement is attributed to the optimization on the geometrical parameters of the impeller blade outlet.

The wear on the working surface of the impeller blades of the second stage showed very similar patterns to those in the first stage. However, the wear rate was reduced on all eroded regions. The reason is attributed to the particles possessing a circumferential velocity after leaving the diffuser of the first stage and matching better with the impeller blades of the second stage.

The wear on the back surface of the impeller blades were also similar between the two stages while the overall wear rate on the backside of the secondary impeller blades intensified, with a maximum of 30 mm/y except for the area of leading edge. Herein, the wear on the back surface of the blades of the second stage is shown in Figure 17. Due to the improved flow characteristics in the optimized model, where the flow lines inside the impeller passages were more uniform and smoother, the contact between the particles and the backside of the blades was greatly reduced, resulting in a significant reduction in the wear on the backside. Under the condition of 0.75 Q_d, the wear mainly occurred in the anterior region of the backside, with a maximum wear of 15 mm/y. After optimization, most areas of the trailing region had wear that did not exceed the value of 6 mm/y. Under the condition of 1.0 Q_d, the wear mainly occurred in the anterior region of the backside, with a maximum value of 30 mm/y. After optimization, most areas of the anterior region had wear that did not exceed the value of 12 mm/y.

Regarding the wear distribution on the concave surfaces of the diffuser’s vanes (shown in Figure 18), particles leaving the impeller with a relatively high tangential velocity entered the vane passages and accumulated near the front cover plate, flowing towards the outlet. With the accumulation of wear, a band-shaped wear area was formed. Under the condition of 0.75 Q_d, the annual wear amounted to approximately 100 mm/y, while under the condition of 1.0 Q_d, it reached around 200 mm/y. Due to the optimized model having smoother flow lines inside the vane sections, which resulted in a more uniform distribution, its wear rate was slightly reduced compared to the original model, while the distribution was also more even.

Although the wear performance of the optimized model has greatly been improved compared to the original model, the annual wear rate at the head and middle sections near the back shroud of the primary impeller still exceeded 150 mm/y. Additionally, the concave surfaces of the guide vanes formed a band-shaped wear area near the front cover plate with an annual wear rate of more than 150 mm/y. Therefore, further optimization and improvements are required to ensure stable operation even under harsh conditions.

4. Conclusions

Based on orthogonal experiments, the sensitivity analysis revealed that the impeller’s outlet width had the greatest influence on the efficiency of the pump, whereas the impeller blade’s wrap angle worked as the secondary factor. The impeller blade’s outlet angle and the diffuser’s inlet width also had considerable influence on the performance of the pump.

The RBF neural network was used to train the approximate model, and the R² values for the efficiency were greater than 0.9500 under both flow conditions. The predicted efficiency obtained from the RBF neural network was only 0.32% different from the CFD result, showing a good agreement. Therefore, the RBF neural network model was able to accurately predict the nonlinear relationship between the hydraulic efficiency and the geometric parameters of the multi-stage centrifugal pump.

After analyzing the external characteristics, the optimized diffuser’s inlet width and the impeller’s outlet width were both reduced to achieve better matching of the internal flow between impeller and diffuser. The hydraulic efficiency of the optimized model under multiple conditions was improved, especially under the rated flow condition where it reached the value of 83.64% and was 4.35% higher than the prototype. The enhancement in efficiency was attributed to the improved compatibility between the diffuser and the impeller, which had a significant impact on the performance of the multi-stage centrifugal pump.

The matching between the impeller and the diffuser of the optimized model was significantly improved by looking at the internal flow field, which became smoother with reduced energy dissipation. The streamlines and pressure distribution in the flow passage of the impeller and diffuser were evenly distributed, whereas the unstable flow patterns such as flow separations and vortices were significantly improved after optimization. The wear rate on the surfaces of both the impeller blades and diffuser’s vanes were simultaneously reduced in the optimized model, in which the improved internal flow field dispersed the particle trajectories and attenuated the collisions with the wall surfaces.