Abstract
With the rapid development of industry, coal-fired power generation accounts for a large proportion of the total power generation, emitting a large amount of harmful substances, such as PM2.5, seriously affecting human health. To investigate the PM2.5 collection efficiency of wire-plate electrostatic precipitator (ESP) at different temperatures, numerical simulations based on the multi-field coupling model of ESP were conducted. Support vector machine (SVM) model combined with particle swarm optimization (PSO) algorithm gives the PSO-SVM prediction model, and the simulated data are used as training data, PSO-SVM and back propagation neural network (BPNN) models are used to predict the temperature effect under different operating conditions. The results show that PM2.5 collection efficiency in the wire-plate ESP gradually decreases with increasing temperature, and the decreased rate becomes small constantly. Both PSO-SVM and BPNN models accurately describe the relationship between collection efficiency and temperature, the average relative errors of the two models for predicting the collection efficiency of 1.0 μm particles at different temperatures are 0.247% and 0.363%, respectively. Compared with BPNN, the overall error of PSO-SVM is 0.928% lower, suggesting that PSO-SVM model yields smaller relative error and higher prediction accuracy. The related findings can provide references for studying the collection performance and rapidly determining the operating parameters of ESP.
Introduction
In today's society, the hazards of fine particulate matter (such as PM2.5) are receiving the increasing attention from humans. Its main source is the large amount of soot generated in industrial production processes that rely on coal as an energy source (Amir et al., 2021; Pillarisetti et al., 2023; Yu et al., 2023). Although the soot is emitted into the atmosphere after being treated by electrostatic precipitator (ESP), the collection efficiency of ESP for PM2.5 is relatively low (Stapleton et al., 2020; Xu et al., 2009; Zhang et al., 2022; Zhu et al., 2021). Considering severe air pollution problem and the significant contribution of coal-fired power plants on it, there is a growing need to assess the health impacts and the corresponding economic losses induced by PM2.5 from coal-fired power plants for guiding the relevant policy development to reduce the concentration of PM2.5 to alleviate its harm to human health (Bojjagani et al., 2023; Verghese et al., 2023).
At present, in many industrial applications, flue gas must be purified at high temperatures (Aryal et al., 2023; Kermani et al., 2022). Exploring the collection characteristics of ESP at high temperatures and finding ways to rapidly determine the operational parameters to improve the operational efficiency of ESP are important issues in the field of collection efficiency.
Currently, research on high-temperature ESP technology mainly focuses on theoretical analysis and experimental verification, and the common research content mainly includes the influence of the type of ESP, working temperature, and so on, on the collection efficiency. Among them, Chen et al. (2020) established an experimental-scale ESP to investigate the influence of temperature and gas medium on collection efficiency and energy consumption. Xu et al. (2021) conducted collection experiments on a rare earth tungsten cathode tube ESP at temperatures ranging from 1073K to 1373K, exploring the effect of temperature on collection efficiency. Wang et al. (2019) proposed a computational fluid dynamics model to study the effects of ion wind on flow field and particle deposition in ESP, focusing on the coupled processes of corona discharge, airflow, particle charging, and transport. Xiao et al. (2015) did rational experiments on line tube high-temperature ESP at different temperatures and proposed operating parameters for optimal collection efficiency by comparing the experimental results.
In addition, Xu et al. (2015) utilized a high-temperature ESP experimental platform to study the characteristics of corona discharge, particle migration, and particle collection at temperatures ranging from 300K to 900K. Zhu et al. (2019) used numerical methods to investigate the capture process of fine particles in both corrugated plate and parallel plate ESP, concluding that the corrugated plate ESP exhibited greater electrostatic field characteristics and stronger resistance to current influences. Kang (2015) performed numerical simulations on the electrophoretic motion of multiple particles under an alternating electric field and analyzed the forces acting on charged particles based on the simulation results.
Research on intelligent prediction methods, particularly the application of back propagation neural network (BPNN), has been widely used in many aspects. Zhao et al. (2016) proposed an improved reverse BPNN based on a hybrid leapfrog algorithm and applied it to fault diagnosis of rolling bearings. Yu et al. (2022) proposed a BP-based method for estimating and correcting branch currents in electric vehicles, and discussed the selection of optimal inputs for dual BP and the adaptability of the method to different battery capacities and resistance differences. Islam et al. (2017) used an improved prediction method of BPNN to forecast the demand of power energy and compare with real data, which demonstrated the high predictive accuracy of the improved BPNN. Xiong et al. (2022) established a BPNN based on the Levenberg–Marquardt algorithm to predict the temperature field of welding interfaces, as well as trained and tested the NN model they built.
Mao et al. (2022) proposed a particle swarm optimization (PSO) algorithm based on Levy flight strategy to optimize the weights and thresholds of a BP for predicting the state of charge (SOC) of lithium batteries used in electric vehicles, thus improving the prediction accuracy of SOC. Zhong et al. (2012) presented an improved BPNN prediction model based on inductance input preprocessing and applied it to the prediction of the rotor position of switched reluctance motors. Liu et al. (2020) proposed a compound control framework for nonaffine nonlinear systems facing hysteresis disturbance, which consists of linear active disturbance rejection control and BPNN adaptive control. From the above research content, it can be seen that the application fields of BPNN prediction model are very extensive, and it often combined with other algorithms to achieve optimization results.
As a relatively novel prediction model in statistical theory, the support vector machine (SVM) and its combination with the PSO, forming the PSO-SVM prediction model, has been widely applied in many areas, such as predicting the growth cycle of plants and the flow of river, and is often compared with other intelligent algorithms. Nieto et al. (2015) proposed an SVM prediction model combined with PSO and multivariate adaptive regression splines to forecast the growth cycle of spirulina. Song et al. (2022) presented a settlement prediction method based on PSO optimized SVM for improving the accuracy of foundation pit settlement prediction, which used the simulated annealing algorithm to improve the traditional PSO algorithm. Samantaray et al. (2023) utilized water flow data from the Barak River in Assam, India, spanning from 1969 to 2018, and applied SVM, BPNN, and PSO-SVM models for flood prediction.
Han et al. (2018) developed an intelligent prediction model based on PSO-SVM technology for low-permeability reservoir recovery, and evaluated the accuracy and reliability of the proposed model using 34 datasets collected from publicly available literature, and compared it with PSO-BP from an oil and gas company. Wang et al. (2020) proposed the autoregressive moving average prediction model and the SVM prediction model based on the historical data of wind power, wind speed and temperature, and PSO algorithm was involved for parameter optimization of SVM model. Tian et al. (2017) proposed a predictive control method based on SVM and improved PSO algorithm, which improved the control performance of calcination zone temperature in a lime rotary kiln.
In conclusion, many researchers use intelligent algorithms to study the collection efficiency of PM2.5 in wire-plate ESP, whereas the numerical simulation research on the collection characteristics of ESP under high-temperature is rarely conducted, and the prediction of ESP temperature effect under different working conditions based on BPNN and PSO-SVM has not been reported. In this study, we developed a multi-field ESP coupling model and investigated the PM2.5 collection efficiency under the effects of temperature. Under different temperatures, the collection efficiency for different particles sizes and working voltages was numerically obtained, and it was predicted by combining simulated data with two prediction models above-mentioned. The research results can provide theoretical reference for the design of novel ESP aiming at efficiency and economy.
Theoretical Model
Multi-field coupling theoretical model
The research on high-temperature ESP involves studying the interactions among temperature field, electric field, fluid field, and particle dynamic field. Extensive research has been conducted on gas–solid two-phase flow, leading to significant advancements in understanding its underlying mechanisms (Cai et al., 2016; Vakilzadeh et al., 2023). When considering the electric field based on gas–solid two-phase flow, that is, the charged gas–solid two-phase flow, it can be treated as a separate discipline of research, known as ESP at room temperature. Taking the temperature field into account, it becomes necessary to comprehensively consider the coupling interactions among temperature field, electric field, fluid field, and particles dynamic field (Skodras et al., 2006; Zhang et al., 2020), and the coupling diagram is given in Fig. 1.

The coupling relationship between multiple fields.
It can be observed from Fig. 1 that the electric field affects the fluid field through ionic wind, whereas the reverse coupling is generally weak and is often neglected. The electric field influences the particle dynamic field by the particle charging and the electric force, and the particle dynamic field, in turn, affects the distribution of the electric field through space charge. The fluid field exerts air drag on the particle dynamic field. The reverse coupling can be ignored when the particle concentration is low, but there exists an indirect coupling: charged particles form spatial charge, which interacts with the electric field and generates an electric field force, thereby affecting the gas flow.
The temperature field affects the air parameters that concludes flue gas density ρ, dynamic viscosity ν, specific heat Cp, and thermal conductivity k, then influences the gas fluid field. By changing the viscosity of the flue gas, the particle motion velocity is changed, which affects the particle dynamic field. Furthermore, the temperature field affects the characteristics of the electric field that concludes corona field strength Es, breakdown voltage Uj, ion migration rate λ, ion charge density ρion, and molecular mean free path kion, thereby influencing the electric field. In this model, the influence of fluid field, particle dynamic field, and electric field on temperature field is not considered.
Temperature is one of the critical factors that have an impact on the performance of ESP, which directly influences the physical properties of flue gas, such as density and dynamic viscosity, thereby affecting the flow behavior of flue gas in ESP. The formulas for variation of flue gas properties with temperature can be expressed as follows:
where ρ is the density of flue gas (kg/m3), Mgk is the mass of air (kg), Vgk is the volume of air (m3), ν is the dynamic viscosity of flue gas (kg/(m·s)), bik is a coefficient to be determined, and Tk is the temperature (K). In this study, the given temperature field is assumed to be uniformly distributed and constant in the ESP, and it does not involve calculating the temperature field equations.
The working principle of a wire-plate ESP is mainly to ionize the gas under a high-voltage electric field, so that the dust particles are charged and removed from the airflow under the action of the electric field force, achieving the goal of purifying the flue gas (Liu et al., 2022). To accurately describe the distribution of electric field, fluid field, and particle dynamic field inside the wire-plate ESP, the finite volume method, SIMPLE algorithm, and Gaussian iterative algorithm are used to divide the computational domain and couple the gas in the above three fields during the calculation process, to achieve simulation of particle dust removal performance.
BPNN prediction model
The BPNN model is a valuable computational tool that is increasingly being used to solve resource-intensive complex problems as an alternative to using more traditional techniques (Saghafi et al., 2009). It enables the learning of multilayer networks. The basic BPNN algorithm, which calculates the actual output in the forward direction from input to output, consists of forward propagation of signals and backward propagation of errors and adjusts the weights and thresholds in the backward direction from output to input. In this study, a BPNN model is used to predict PM2.5 capture performance. The number of hidden layer nodes, the target error, the learning rate, and the iteration step are, respectively, set to 6, 10−3, 0.01, and 5,000. Figure 2 illustrates the structure of the BPNN model.

Structure diagram of BPNN model. BPNN, back propagation neural network.
PSO-SVM prediction model
The SVM model is a relatively new approach for classification and regression, which is specifically designed for prediction with limited samples, and it is a learning method based on statistical theory that gradually developed since the mid-1990s (Roohollah et al., 2012). SVM has been widely applied in various fields.
Basic principles
As a supervised learning model, SVM models use nonlinear functions to map samples to feature spaces with high or infinite dimensions (Singh et al., 2023). It constructs an optimal classification plane in this space, transforming the problem of linear inseparability in the original sample space into that of linear separability in the feature space.
Kernel functions of SVM
There are many articles and books that provide a detailed description of the kernel functions of SVM, and as the main factor determining the capability of SVM to handle nonlinear data, the kernel function has the following characteristics (Roohollah et al., 2009): When the input sample is multiple inputs corresponding to a single output, for any function
where
where θ is an undetermined coefficient, and γ is the parameter of the kernel function. The selection of kernel function parameter has a significant impact on the classification and prediction ability of SVM.
PSO algorithm is a kind of population-based stochastic optimization technique. The principle of PSO is to search for the target in N-dimensional space. Assume that a target population in space includes n particles, each particle i contains a unique position vector
where

The prediction flow chart of PSO-SVM model. PSO, particle swarm optimization; SVM, support vector machine.
Results and Discussion
Simulation of temperature effect under different particle sizes and working voltages
Under the fixed operating condition with flue gas velocity of 0.5 m/s and working voltage of 50 kV, the influence of particle size and temperatures on the dust-removal performance of the wire-plate ESP is investigated. The variation of collection efficiency with temperature for three particle diameters of 0.1, 2.5, and 4.9 μm is obtained through numerical simulation, as given in Table 1. It can be observed that the collection efficiency of particles with different diameters decreases with the rise of temperature, and the decreasing range gradually diminishes and tends to be stable finally. At the same temperature, the collection efficiency grows with the increase of particle size, but the reduction amplitude drops accordingly. Compared with three particle sizes, namely 0.1, 2.5, and 4.9 μm, it can be observed that, as the temperature increases, the collection efficiency of the 0.1 μm particles exhibits the greatest decrease, whereas the collection efficiency of the 4.9 μm particles experiences the smallest decrease, indicating that the collection efficiency of particles with smaller diameters is more susceptible to temperature. As the temperature enhances, the difference in collection efficiency between adjacent particles with different particle sizes continues to increase, showing that the effect of particle size on collection efficiency continuously improves with the increase of temperature.
Variation of Particle Collection Efficiency at Different Diameters with Temperature (0.5 m/s, 50 kV)
When flue gas velocity is 0.5 m/s, the curves of PM2.5 collection efficiency under different temperatures and working voltages are studied, as given in Table 2. It can be observed that under diverse voltages, the collection efficiency in the wire-plate ESP decreases with the increase of temperature, and the reduced magnitude gradually declines. At the same temperature, although the collection efficiency increases with the equal ascended amplitude of working voltage, the increased magnitude diminishes. With the rise of temperature, the lower the working voltage, the more significant the change in collection efficiency, indicating that the collection efficiency at lower working voltage is more sensitive to temperature.
Variation of Particle Collection Efficiency at Different Working Voltages with Temperature (0.5 m/s)
Prediction of temperature effect under different particle sizes and working voltages
The selection of different particle sizes (0.1, 2.5, 4.9 μm) and temperatures (298, 373, 473, 573, 673, 873, 973, 1,073 K) for collection efficiency serves as training data, namely the simulated data listed in Table 1. Based on these training data, BPNN and PSO-SVM models are used to predict the collection efficiency of 1.0 μm particles under different temperatures, and the predicted results of the two models are compared with the simulated data, as given in Table 3. It can be concluded that the two models have a high degree of agreement with the simulated data, which well describes the relationship between collection efficiency and temperature for 1.0 μm particles, proving that both models are reliable. Compared with the predicted results of BPNN model, those of PSO-SVM models are generally closer to the simulated data, and the latter is more consistent with the simulated ones, showing that PSO-SVM model is superior in predicting temperature effects under different particle sizes.
Comparison of Predicted and Simulated values of Collection Efficiency Relative Error (0.0 Tesla, 1.0 μM, 50 kV, 0.5 m/s)
BPNN, back propagation neural network; PSO, particle swarm optimization; SVM, support vector machine.
Based on the predicted and simulated values of collection efficiency, the relative errors of the predicted values of PSO-SVM and BPNN models can be calculated. It is not difficult to calculate that the average relative errors of the two models are 0.363% and 0.247%, respectively, which further confirms that PSO-SVM model has a higher accuracy.
To compare with the prediction results of BPNN and PSO-SVM models more clearly, Fig. 4 presents the comparison of collection efficiency with 45° line. In this figure, the X-axis and Y-axis represent simulated data and predicted values, respectively. The closer the data points are to the 45° line, the closer the predicted values are to the simulated data. The following can be easily concluded that the predicted values of PSO-SVM model are generally closer to the 45° line, which further depicts that PSO-SVM model has a higher accuracy in predicting temperature effects under different particle sizes.

Comparison of collection efficiency with 45° line.
From Table 3, it can be observed that PSO-SVM model exhibits an excellent prediction performance in the collection efficiency of 1.0 μm particles under different temperatures. In view of this, using the same training data given in Table 1, PSO-SVM model is used to predict the collection efficiency of 2.0 μm particles under different temperatures. The comparative results and relative errors between PSO-SVM prediction values and simulated ones are given in Fig. 5. It can be seen that PSO-SVM predicted values closely match the simulated data, and the trend of predicted curve aligns well with the simulated one, as given in Fig. 5(a), implying that PSO-SVM model has a greater accuracy. The relative errors are all within 0.6%, as given in Fig. 5(b), demonstrating PSO-SVM model applicability in predicting the temperature effect under different particle sizes.

Comparison of collection efficiency and relative error of predicted values (2.0 μm).
The simulated data in Table 2, namely overall collection efficiency at different working voltages (40, 50, 60 kV) and different temperatures (298, 473, 673, 873, 1,073 K), are used as the training data. The PSO-SVM and BPNN models are used to predict the collection efficiency at 45 kV under different temperatures, which is compared with simulated data, as given in Table 4. It is not difficult to find that the predicted curves of both models present a similar trend to the variation of simulated points, and they effectively describe the pattern of collection efficiency with temperature at 45 kV, proving that the predicted results of the two models are reasonably accurate. Compared with the predicted values of BPNN model, those of PSO-SVM model align better with the simulated data, and the PSO-SVM prediction curve generally approaches the simulated one, suggesting that PSO-SVM model performs better in predicting temperature effects under different working voltages than BPNN.
Comparison of Predicted and Simulated Values of Collection Efficiency Relative Error (0.0 Tesla, 45 kV, 0.5 m/s)
Based on the data in Table 4, the relative errors of PSO-SVM and BPNN models are calculated. It can be seen that the relative error curve of PSO-SVM model is generally below that of BPNN model, and the average relative errors of the two models are 0.226% and 0.329% separately, indicating that PSO-SVM model possesses a higher prediction accuracy than BPNN model.
The 45° line contrast results of the predicted values of PSO-SVM and BPNN models are given in Fig. 6, where the X-axis and Y-axis represent simulated data and predicted values, respectively. It is evident that the data points of PSO-SVM predicted values show a better agreement with the 45° line, which further exhibits the superiority of PSO-SVM model in temperature effect prediction under different operating voltages.

Contrast results of collection efficiency at 45° line (0.0 Tesla, 45 kV, 0.5 m/s).
Conclusions
Numerical simulations on the collection efficiency of high-temperature wire-plate ESP under different working conditions are carried out and the simulated values are used as training data in this study. The BPNN and PSO-SVM models are used to predict the temperature effect under diverse working conditions. The main conclusions are as follows:
Regardless of particle size or voltage, PM2.5 collection efficiency in wire-plate ESP shows a decrease with the increasing temperature. Furthermore, the reduced extent gradually diminishes and eventually reaches a stable level. Temperature weakens the capturing performance of the ESP for PM2.5, and the degree of temperature reduction in collection efficiency significantly decreases when large particle sizes and high voltages are involved, meaning that the negative impact of temperature on collection efficiency is mitigated to a considerable extent under specific operating conditions characterized by large particle diameters and high voltages. In the prediction of temperature effects under different working conditions, the predicted curves of BPNN and PSO-SVM models are commonly consistent with the trend of the simulated points, showing that the two models have accuracy in predicting the impact of temperature on collection efficiency. It is worth noting that PSO-SVM model exhibits a higher level of consistency between its predicted values and the simulated ones than BPNN model.
Footnotes
Authors' Contributions
J.P.Z.: Visualization, methodology, software, analysis, and writing—original draft preparation; Z.G.C.: Conceptualization, supervision, validation, writing—reviewing and editing; J.F.: Investigation, formal analysis and project administration; P.L.: Validation, data curation and supervision.
Author Disclosure Statement
No competing financial interests exist.
Funding Information
This work is sponsored by National Natural Science Foundation of China (Grant No. 12172228, 11572187), Natural Science Foundation of Shanghai (Grant No. 22ZR1444400), the Program of Foundation of Science and Technology Commission of Shanghai Municipality (Grant No. 22dz1204202, 22dz1206005), Shanghai Professional Technical Service Platform for Intelligent Operation and Maintenance of Renewable Energy (Grant No. 22DZ2291800), and Science and Technology Foundation of Shanghai Dong Hai Wind Power Co., Ltd.
