1. Introduction
With the significant development of the economy and society, the utilization of energy resources by humans is gradually increasing. Centrifugal pumps are key to energy transmission and utilization systems. The fluid driven by the mechanical energy generated by the prime mover can be transported to a designated target. At present, many industries rely on energy transportation because of the huge demand for centrifugal pumps [
1], such as municipal sewage, power systems, agricultural irrigation, and chemicals. Therefore, an inevitable problem is proposed regarding the effective design of centrifugal pumps. In traditional design and manufacturing methods, a large amount of time is consumed owing to the complicated operation process, which leads to high costs [
2]. Performance prediction is one of the most effective ways to improve the optimization design of centrifugal pumps, which helps researchers quickly understand the performance of the designed pump, thereby accelerating the development of pump products and saving costs.
In recent years, numerical simulation methods based on computational fluid dynamics (CFD) have always been the main method used by researchers for performance prediction [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14]. The 3D simulation of the impeller and its stationary flow in the centrifugal pump casing were analyzed using CFD [
3,
14], and the head and efficiency of the pump under different flow rates were obtained. Yang [
4] used computational fluid dynamics (CFD) to predict the performance of a single-stage centrifugal pump in forward and reverse modes and verified it with experimental data. To improve the performance of a centrifugal pump, a new impeller structure was explored using a numerical simulation method [
5]. By calculating the head and efficiency of the centrifugal pump and comparing them with the experimental data, it was found that the overall performance of the centrifugal pump was improved based on the proposed impeller design. However, complex numerical simulation of turbulent flow has always been an unsolved difficulty in computational fluid dynamics (CFD) [
11]. High-quality grids and appropriate boundary conditions require a strong work experience for designers, and the long simulation time of CFD is also not conducive to the design of centrifugal pumps.
Theoretical models have also been used for the performance prediction of pumps [
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31], which is a prediction method based on empirical formulas and various assumptions. Among these, theoretical loss models are widely used for performance prediction [
22,
23,
24,
25]. This method, which classifies the losses in the suction chamber, impeller, and pressurized water chamber of the pumps, uses different calculation methods to calculate the losses of each part and then obtains the performance curves according to the basic equation of the pump [
25]. A theoretical model based on an Oseen vortex was proposed to optimize the performance of multistage multiphase pumps. Through verification of the three-stage multiphase pump, the optimized pump head and efficiency can be increased by an average of 0.29% and 0.19%, respectively. To explore the changing condition of the pump performance curves under different speeds, three different methods were summarized by Pedersen [
30] to predict the pump performance curves based on proportional law. The results show that the highest prediction accuracy occurs when the speed difference method is used. Although the influence of various factors, such as secondary flow and return flow, is fully considered by the loss model, it is an assumption made under certain conditions, and satisfactory results can only be obtained within a certain range. When conditions change, the model is no longer applicable and is not universal.
Over the last 10–20 years, tremendous progress has been made in the computer industry, and the computing power of computers has become very powerful. In this process, a large amount of data is accumulated, and there is an urgent need for a method that can reasonably utilize the data for analysis. In recent years, machine learning has attracted widespread attention as a method for processing large amounts of data. In the pump industry, machine learning is widely used for pump fault diagnosis [
32,
33] and performance prediction [
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44]. Deep learning methods based on neural networks are widely used for pump performance prediction [
35,
36,
37,
38]. The BP neural network is one of the most widely used deep learning methods. Considering the huge computing resources and running time required for numerical simulation, a hybrid neural network based on a theoretical loss model and a BP neural network was proposed in [
37]. By introducing the theoretical loss model, the mean-squared error of the head and the efficiency are both significantly reduced. A performance prediction method for a centrifugal pump based on the Levenberg–Marquardt training algorithm and a double-hidden-layer BP neural network [
38] is proposed to solve the shortcomings of the traditional single-hidden-layer BP neural network. Compared with the traditional single hidden layer structure, the convergence time of the improved neural network is significantly reduced, and the situation of low learning efficiency and falling into a local optimum is effectively solved. Because the isentropic efficiency plays a significant role in the system performance of the eddy current pump, a prediction model of the isentropic efficiency of the eddy current pump was constructed by Ping [
39], which was combined with experimental data and deep learning methods. In [
41], to improve the performance of a centrifugal pump, a global optimization algorithm combining an artificial neural network with an artificial bee colony was developed to redesign the geometry of the impeller, and the CFD method was used to analyze and verify all the regions inside the centrifugal pump. However, a slow learning rate and complex network topology have always been problems with neural networks [
37]. Even though the desired neural network structure can be searched using a genetic algorithm, the selection of parameters of the genetic algorithm also has a significant impact on the convergence accuracy [
43]. In addition, it is cumbersome to readjust the structure and weights of the neural network when new data dimensions need to be added to the training data to improve the prediction accuracy of pump performance [
33].
Unlike the way in which the neural network continuously approximates the true value, support vector regression (SVR) has a good prediction effect on small-scale and multidimensional data based on its solid theoretical foundation [
43]. However, the relationship between centrifugal pump performance can be fragmented when only machine-learning methods are used to make predictions. Furthermore, different model structures can yield different prediction results. Therefore, an energy performance prediction method for centrifugal pumps based on performance constraints was proposed. By combining the relationship between centrifugal pump performance, particle swarm optimization (PSO), and support vector regression, a performance prediction model that satisfies the performance constraint is found. The remainder of this paper is organized as follows. A detailed theoretical description of the proposed method is provided in
Section 2 and
Section 3, respectively.
Section 4 discusses the building process of the performance-prediction model and determines the final model structure.
Section 5 presents the effects of the model structure. In addition, a comparison of XGBoost with performance constraints and BP neural networks without performance constraints is provided. The conclusions are presented in
Section 6.
2. Geometric Features
Centrifugal pumps are regarded as important devices in drainage systems. The wide applications of centrifugal pumps benefit from their easy installation, low maintenance cost, etc. Three main components are passed by the fluid when the centrifugal pump operates: inlet tube, impeller, and volute, as shown in
Figure 1. The impeller is the core component of the centrifugal pump that converts the mechanical energy of the rotor into the kinetic energy of the fluid. Therefore, a commonly used method is to optimize the parameters of the impeller to obtain a satisfactory pump performance [
45]. The main geometrical parameters are described in
Figure 2, including the inlet diameter of the impeller
, the inlet diameter of the blade
, the hub diameter of the impeller
, the inlet angle of the blade
, the outlet diameter of the impeller
, the outlet width of the impeller
, the outlet angle of the blade
, the wrap angle of the blade
and the number of blades
.
Another important component is the volute, which plays a significant role in improving the performance of a centrifugal pump, as shown in
Figure 3. The main structure of the volute is the diameter of the base circle
, placement angle of the cut tongue
, and width of the inlet
. The structural diagram of the eighth section of the volute is shown in the upper-left corner. It is accepted that kinetic energy can be converted into pressure energy using a reasonable volute structure.
Usually, energy performance is described by head
, efficiency
, and power
. In addition, the flow rate
is usually regarded as a significant operating condition. Therefore, nine geometric parameters, namely
,
,
,
,
,
,
,
,
, and the flow rate
are considered as the input variables, which represent the major parameters of the centrifugal pump. Mathematically, it can be expressed as
, where
,
, and
are the dimensions of the variable. Furthermore, the
th sample of the centrifugal pump can be written as Equation (1).
6. Conclusions
Based on the geometric parameters of the impeller and volute of centrifugal pumps, a multi-condition performance prediction model of centrifugal pumps is proposed that incorporates the performance relationship into the particle swarm optimization algorithm. The performance (i.e., head, power and efficiency) of the centrifugal pumps can be predicted simultaneously and satisfied with the performance relationship. The performance prediction model proposed in this study can be used as a reference for the prediction method of centrifugal pump performance curves. A total of 428 samples were used to train the performance prediction model, 107 samples to test the generalization ability of the model, and 46 samples to verify the prediction effect of the model. The following conclusions were drawn:
(1) The structure of the energy performance prediction model under multi-condition operations can be effectively determined based on the particle swarm optimization and performance relationship. The penalty coefficient and kernel function coefficient of the regression are 10,000 and 1.48, respectively.
(2) The multi-condition performance is well predicted by considering the performance constraints, the maximum ARE of the head, power and efficiency of the 46 verification samples are 5.76%, 6.42%, and 5.02%, respectively, and the MARE are 0.85%, 1.53%, and 1.15%, respectively. The overall MARE is less than 3%.
(3) The MARE of the head, power and efficiency corresponding to the PSO-SVR model decreased by 58.54%, 65.38%, and 76.05%, respectively, compared with those of the PSO-XGBoost model, indicating that the SVR model is more suitable than XGBoost for the performance prediction of centrifugal pumps. When compared with the BP neural network, the MARE of the head, power, and efficiency corresponding to the SVR prediction model with performance constraints decreased by 62.21%, 82.06%, and 85.33%, respectively, indicating that the introduction of performance constraints can effectively improve the overall prediction accuracy.