Multi-Objective Optimization and Design for Industrial Vinyl Chloride Reactor by Hybrid Model

Abstract

The acetylene conversion rate and vinyl chloride production capacity are the main economic indexes for vinyl chloride synthesis, and the reaction temperature is an important operating parameter to prevent the Hg active component from being loss. These three factors have not been taken into consideration simultaneously in the traditional optimization process, making it difficult to achieve optimization targets perfectly for industrial application. To overcome this problem, an efficient strategy framework was proposed based on a hybrid model. Compared with conventional paradigms, the proposed framework could not only reduce computational expense but optimize these two economic indexes with a constrained reaction temperature simultaneously. In addition, a machine learning method was used to conduct a feature analysis, which can reveal the potential interaction between different variables so key variables of this reactor could be identified. To demonstrate and verify this framework, multi-objective optimization involving multiple variables with constrained conditions for the industrial reactor was conducted from design and operation perspectives, respectively. The proposed strategy could provide optimal operational direction for the industrial reactor from these design and operation aspects, which contribute to the sustainable and highly efficient process development in this field. For the first section of an industrial vinyl chloride reactor, this strategy could realize significant improvement in the acetylene conversion rate from 81.34% to over 95.00% and in the vinyl chloride production capacity from 2.60 to above 3.40 mol/h in the operation scenarios, which can meet production requirements. So, the second section of the traditional reactor system is not needed at all.

1. Introduction

Polyvinyl chloride (PVC) is widely used in various fields for its excellent physical and chemical properties, including mechanical properties, electrical properties, and easy processability [1,2]. It is mainly produced from vinyl chloride monomer (VCM) by the ethylene method or the acetylene method [3]. In addition, VCM can also be used as a refrigerant to form copolymers via copolymerization with vinyl acetate, vinyl chloride, butadiene, acrylonitrile, acrylic esters, and other monomers. In China, the acetylene method is the most common process due to its abundant coal resource [4], and the most widely used catalysts in plants are mercury catalysts loaded on carbon support [5]. Because of its poor thermal stability, the HgCl₂ component will gradually lose at elevated temperatures, reducing catalyst activity and increasing mercury consumption, leading to environmental and health problems [6]. At present, though some studies [7,8,9] have reported mercury-free catalysts, most of them are still in the laboratory stage, and there are few public reports about their industrial application. To optimize a vinyl chloride reactor (VCR), some mathematical models, such as the one-dimensional and two-dimensional pseudo-homogeneous model [10,11], were established to investigate the relations between input parameters and output values, showing that the two models could approximately reflect the industrial reactor. Liang [10] established one-dimensional (1D) and two-dimensional (2D) mathematical models, which showed certain accuracy and reliability compared with the actual industrial data from a factory. The effects of operational parameters on the reactor temperature and acetylene conversion rate (ACR) were also analyzed without optimization analysis and design. Chen [11] established mathematical models based on different types of reactors and used the verified models to conduct a simulation analysis on different factors (space velocity (SV), wall temperature (T_w), and tube diameter (d_t)) and system optimization. Yang et al. [12] analyzed the effects of SV, coolant temperature (T_c) on the temperature distribution in the reactor, and ACR. Huang et al. [13] optimized VCR operational parameters via analyzing their effects on the ACR with Aspen Plus. This analysis was a univariate analysis during which the ACR was less than 85% under the optimal condition. These simulations and optimizations are mostly based on a single-variable sensitivity analysis or single-objective function, which could not reveal the underlying influential mechanisms for the multiple-input multiple-output (MIMO) system. In other words, the optimized results may be not the real optimal solutions. To enhance VCR performance, mercury-free catalysts and related reaction mechanisms were also investigated [14,15,16,17,18], including g-C₃N₄/BiOCl catalysts, single-atom AuI-N3, and Au/CeO₂-based catalysts. Though those studies showed good performance, they are still in the laboratory stage and it will take a period of time to achieve industrialization. Currently, the acetylene method based on mercury catalysts is still a common process; thus, the multi-objective optimization of VCR is essential for the industrial production. The models used for the simulation of VCR are summarized in

Table 1. Models used for the simulation of VCR.

Activate Component	Types of Models	Operational Parameter	Ref.
Mercury	1D and 2D	Tw~95 °C, RMI~1.05	[10]
Mercury	2D	SV~30 h−1, Tw~98 °C, dt	[11]
Mercury	1D	SV, Tw	[12]
Gold-based catalyst	1D	RMI, SV, Tin, Tw	[13]

.

The modeling of the operation unit in the chemical engineering process includes a mechanistic model based on the specific knowledge of the process, a data-driven model (or artificial neural network (ANN), black-box model) based on the historical data from the process, and a gray model integrating mechanistic knowledge with the data-driven model. A mechanistic model, also known as the first principle model (FPM), is derived from the principles of mass, energy, and momentum balance and shows good interpretable and extrapolative performance, but it poses significant challenges to the knowledge of system-specific characteristics and computational capacity involving complex mathematical equations with nonlinearity and multidimensionality characteristics [19]. Data-driven models, such as ANN, can cope with the flaws of the FPM, whose accuracy depends on the quantity and quality of the training dataset. Presently, research on the ANN model for a VCM reactor is still in its infancy. To reduce computational load, this work first proposed a methodology of using a surrogate model instead of FPM to conduct multi-objective optimization for the industrial VCM reactor, where the trained data are obtained from simulation by the FPM as few industrial data are available.

In recent years, the development of machine learning (ML) and artificial intelligence (AI) and their powerful advantages in solving complex system problems have attracted the attention of chemical researchers and engineers [20,21,22,23]. The chemical engineering process, especially chemical reactors, usually involves MIMO coupled with nonlinear problems, which cannot be solved with high precision using traditional methods. With the aid of AI and ML, engineers are exploring new paradigms. Zhao et al. [24] conducted a multi-objective optimization of a radically stirred tank based on CFD and machine learning, which confirmed the accuracy and reliability of the machine learning-based optimization method. Rahimpour et al. [25] used a multilayer perception (MLP) neural network to determine the optimum production of ethylene dichloride, and the error of simulation was found to be less than 5%. Bhakte et al. [26] used a deep neural network (DNN) based on process alarms to assist operators in understanding DNN’s prediction during online process monitoring.

The annual production of vinyl chloride in the world is huge because of its market demand, and the investment and energy cost are very high, so even a small improvement in plant operation can lead to considerable benefits. Thus, there is a requirement of optimal design and operation for the vinyl chloride reactor. Particle swarm optimization (PSO) is a simple and yet powerful metaheuristic search algorithm, widely used to solve various optimization problems [27].

The monitoring, control, and optimization of industrial VC reactors are essential for the plant, but in general, optimization is based on a single-objective optimization, even ignoring the constrained maximum reactor temperature (T_max) during the optimization process, which may be insufficient for this scenario. In addition, FPM needs more computational time compared with the ANN model, but prediction in advance is essential to the production, based on which timely management could be achieved. To cope with the above obstacles, we propose a new prediction model and optimization strategy for the industrial VC reactor. Firstly, a historical dataset obtained from actual production is used to conduct feature analysis based on machine learning, based on which a surrogate model for the reactor is built, trained, and verified. This surrogate model can be used to conduct the prediction, optimization, control, etc. Here, a surrogate model coupled with an artificial intelligent algorithm is used to conduct multi-objective optimization with constrained conditions. Detailed information on the optimization strategy is shown in Figure 1, which mainly includes three parts: data preprocessing (datasets collection, pretreatment, and feature analysis), predictive model selection, and calibration (the prediction model may be FPM, ANN, or a hybrid model, depending on the specific process and purposes). The development of this optimization strategy for vinyl chloride industrial reactors has important practical applications: (1) a safety production plan based on the calibrated model; (2) operational optimization based on a reliable surrogate and multi-objective optimization algorithm; and (3) the prediction of the reactor performance for designing effective control strategies.

2. Materials and Methods

Due to the fact that large and high-quality training data are needed and yet it is difficult to obtain sufficient historical data from the actual industrial production in machine learning, we used FPM to generate a large dataset replacing industrial historical data, which could reflect the effectiveness of this proposed idea. The methodology in this research consists of two main sections. In the first section, first principle models (FPMs) were verified using industrial data, and then a dataset generated with the calibrated FPMs was used to train and test the relevant surrogate models. In the second section, the calibrated models coupled with PSO were used to optimize the VCM reactor from design and operation aspects, respectively.

2.1. Mechanistic Model and Data Generation

At present, the acetylene method is most commonly used in China for the production of vinyl chloride monomer [28]. The main reaction of the acetylene method is as below [29].

$$\mathrm{C}\mathrm{H}\equiv \mathrm{C}\mathrm{H}+\mathrm{H}\mathrm{C}\mathrm{l}\stackrel{\mathrm{C}\mathrm{a}\mathrm{t}}{\to }{\mathrm{C}\mathrm{H}}_2=\mathrm{C}\mathrm{H}\mathrm{C}\mathrm{l} \mathsf{\Delta }H=-124.8 \mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$$

(1)

This study is based on a real industrial reactor, in which the catalyst is mercury chloride loaded on carbon support. Reactors for acetylene hydrochlorination are usually fixed beds with multiple tubes. Because the tube diameter (d_t) is small and gas velocity is high, the axial dispersion inside the tubes can be neglected. By merging the effect of industrial catalysts and pore diffusion into the reaction kinetics, a homogeneous model is used instead of a heterogeneous model. In this study, one-dimensional and two-dimensional homogeneous mathematical models are developed, where reaction kinetics [30] is adopted.

One-dimensional model:

$${\displaystyle \frac{{\mathrm{d}x}_{\mathrm{A}}}{\mathrm{d}z}}={\displaystyle \frac{(-r_A)(1-\epsilon )}{F_{\mathrm{A}0}}}{\displaystyle \frac{\mathsf{\pi }{d}_t^2}{4}}$$

(2)

$${\displaystyle \frac{\mathrm{d}T}{\mathrm{d}Z}}=\left({\displaystyle \frac{\mathsf{\Delta }H}{4}}\left(-r_A\right)\left(1-\epsilon \right)+U\left(T_w-T\right)\right){\displaystyle \frac{\mathsf{\pi }{d}_t}{F_t{C}_{\mathrm{p}}}}$$

(3)

Two-dimensional model:

$${\displaystyle \frac{\partial x_A}{\partial z}}=a_1\left({\displaystyle \frac{{\partial }^2{x}_A}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial x_A}{\partial r}}\right)+b_1\left(-r_A\right)$$

(4)

$${\displaystyle \frac{\partial T}{\partial z}}=a_2({\displaystyle \frac{{\partial }^{2}T}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T}{\partial r}})+b_2(-r_A)$$

(5)

where:

$$a_1={\displaystyle \frac{E_r}{u}},b_1={\displaystyle \frac{{\rho }_B}{u{C}_{A0}}}$$

$$a_2={\displaystyle \frac{{\lambda }_{er}}{G\overline{{\mathrm{C}}_P}}},b_2={\displaystyle \frac{{\rho }_B\left(-\mathsf{\Delta }{H}_R\right)}{G\overline{{\mathrm{C}}_P}}}$$

$$r={\displaystyle \frac{k{k}_H{P}_{A0}^{2}a}{3.6}}{\displaystyle \frac{(1-x_A)(RMI-x_A)}{(P-P_{A0}{x}_A{)}^2+k_H{P}_{A0}(P-P_{A0}{x}_A\left)\right(RMI-x_A) }\text{ mol/(s·kg-cat)}}$$

(6)

(RMI represents the mole ratio of HCl to C₂H₂)

where:

$$\eta ={\displaystyle \frac{57.0}{(T-300)}}-0.12\times e^{0.1x_A}$$

$$k=25.6\times \eta \times e^{{\displaystyle \frac{-4065}{RT}}}$$

$$k_H=3.31{\times 10}^6\times e^{{\displaystyle \frac{-10630}{RT}}}$$

The industrial reactor data [11] were used to verify the models’ accuracy.

Firstly, based on the above 1D and 2D mathematical models, two types of VCR simulation framework were built with the Python program. To compare with industrial data, the input parameters of the above two models should agree with the industrial VCR (d_t = 0.044 m; RMI = 1.1; SV = 34.12 h⁻¹; a = 0.209; T_in = 362.8 K), based on which VCR simulation calculations were conducted, where differential equations were solved using Runge–Kutta and difference methods.

Table 2. Comparison of simulation results and industrial data.

Items	SV, h−1	Tmax, K	Tout, K	xout, %
Industrial data	34.12	393.20	372.20	81.48
One-dimensional model	34.15	392.08	374.30	81.58
Two-dimensional model	34.12	391.56	374.29	81.35

shows that the results of the two FPMs agree well with the real industrial data, demonstrating that these two models are both credible; the values of T_max, T_out, and x_out are average values for the radial direction position at the axial position in the tube reactor. To simplify the simulation and reduce the calculation cost, the following work was based on the one-dimensional model. About 39,000 sets of data were generated via simulation, and feature analysis was firstly conducted based on Pearson correlation [31].

$$R={\displaystyle \frac{\sum _{i=1}^n (x_i-\overline{X})(y_i-\overline{Y})}{\sqrt{\sum _{i=1}^n (x_i-\overline{X}{)}^2}{\sqrt{\sum _{i=1}^n (x_i-\overline{Y}{)}^2}}^{\leftarrow }}}$$

(7)

Figure 2 shows the R between different variables, demonstrating that input variables have different effects on all three target variables. The absolute value of R represents the strength of the relationship between two variables, and the sign represents the effect tendency (positive or negative). For example, as to ACR(x), the effect order of input variables is a, l, T_w (temperature of tube wall), RMI, SV, d_t, and T_in (inlet temperature), while as to vinyl chloride production capacity (VCPC, Pro), the order of influence is l, d_t, SV, a, T_w, RMI, and T_in. Meanwhile, there exists a contrary correlation between the same input variables and different target variables, for example, there is a negative correlation (R = −0.14) between SV and ACR(x), and a positive correlation (R = 0.24) between SV and VCPC(Pro). Because SV decides the retention time of reactants in the reactor, an elevated SV means a shorter reaction period, and thus SV has a negative effect on the ACR, whereas SV contributes to Pro based on Equation (10), and thus the effect of SV on Pro from Figure 2 is a comprehensive result based on multiple input variables. Therefore, multi-objective optimization involving MIMO is necessary for the industrial reactor.

2.2. Surrogate Prediction Model

In this study, an ANN was employed to accurately predict target functions; its architecture contains an input layer, multiple hidden layers, and an output layer (Figure 3).

Before the ANN training phase, data pre-processing is needed. Here, min-max normalization [32] was used to deal with inputs, scaling the range of each input to [0,1], since the order of magnitude difference among these features will affect the prediction performance.

$$x^{\star }=(x-min)/(max-min)$$

(8)

where min and max represent the minimum and maximum value corresponding to input variables, respectively.

During the ANN training process, the acetylene conversion rate (x_A), the maximum temperature in the reactor (T_max), and vinyl chloride production (Pro) were selected as target functions. In the design process, feature variables were RMI, SV (/h), a, T_w (K), T_in (K), d_t (m), and l (m), while in the operation process, d_t and l were removed from the above feature variables. The optimization algorithm and loss function were Adam (Adaptive Moment Estimation) [33] and MSE (mean square error).

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n(y_i-y_{pre\_i}{)}^2$$

(9)

In the operation scenarios, except for the feature variables, the others were similar to those in the design scenarios. The inputs consisted of RMI, SV, a, T_w, and T_in. In this study, vinyl chloride production is expressed as shown below:

$$\mathrm{P}\mathrm{r}\mathrm{o}=C_{A0} * x_A * {dt}^2 * {\displaystyle \frac{\pi }{4}} * SV * l \text{(mol/h)}$$

(10)

where $C_{A0}$ represents acetylene mole concentration (mol/m³), Pro represents vinyl chloride production capacity per tube, and x_A represents the conversion rate of acetylene.

2.3. Multi-Objective Optimization

To optimize the vinyl chloride industrial reactor, multi-objective optimization algorithm coupled with surrogate prediction model was constructed. Because the HgCl₂ component will lose at elevated temperature, T_max should be constrained in a range; in this study, the value was set to 423.15 K. PSO mimics the foraging behavior of a group of birds, where the birds are treated as particles and the foraging behavior is treated as a process of merit-seeking in a particle swarm. Each particle has its own direction and velocity, and the particle position is the decision variable for the optimization problem. The velocity and position of the particles are updated as follows [34]:

$$V_i=w\times V_i+c1\times rand\left(\mathrm{0,1}\right)\times \left(P_{best}-X_i\right)+c1\times rand\left(\mathrm{0,1}\right)\times \left(g_{best}-X_i\right)$$

(11)

$$X_i=X_i+V_i$$

(12)

where $P_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ and $g_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ represent the particle’s historical optimal position and the optimal position sought by all particles, respectively, both of which control the direction of motion and position of the particles. PSO can also be used for multi-objective optimization problems, for which not a single optimal solution but a series of optimal solutions exist. A set of nondominant solutions, not dominated by others in the solution, is called the Pareto fronts solution. In the optimization process for the vinyl chloride reactor, the optimized goals involve two variables, which are multi-objective optimization problems; the detailed expression is as shown below:

max f(X) = [x_A,Pro]s.t T_max ≤ 423.15

(13)

The numerical solution and the optimization algorithm were both implemented through Python programming language.

3. Results and Discussion

3.1. Surrogate Model Training and Validation

A dataset containing 390,625 sets of data was obtained from the simulation based on the one-dimensional model. Given that the dataset came from simulation, data cleansing was ignored, which is necessary for industrial datasets. First, data preprocessing was conducted, where the values of all feature variables were scaled to the range of 0 to 1. Then, the dataset was divided into training and testing sets; here, the testing set size was 20% of the dataset. This is a MIMO problem, and thus a parallel ANN structure is needed. Here, a parallel ANN consisting of three sub-ANNs was constructed, where all sub-ANNs share inputs. The number of ANN layers, activation functions, and nodes are essential parameters, directly influencing prediction accuracy.

Based on Section 2, the surrogate prediction models of the VCM reactor in respect to the design and operation scenarios were trained using the training dataset. In the design scenarios, for x_A, the ANN structure consisted of ReLU of activation function, one hidden layer, 100 neural nodes, and a learning_rate of 0.005, while those for T_max and Pro were ReLu, one, 600, and 0.0001; and ReLU, one, 200, and 0.0001, respectively. Similarly, in the operation scenarios, those for x_A, T_max, and Pro were ReLu, one, 80, and 0.0005; ReLu, one, 600, and 0.0008; and ReLu, one, 300, and 0.00008. To further test the prediction accuracy of these two models, the outputs between FPM and the surrogate model were compared (

Table 3. Comparison between FPM and surrogate model in design scenarios.

NO.	RMI	SV	a	dt	Tw	l	Tin	FPM			Surrogate Model
								xA	Tmax	Pro	xA	Tmax	Pro
1	1.03	35.2	0.23	0.041	375.1	3.52	366.2	0.876	397.674	2.904	0.876	397.831	2.939
2	1.06	38.93	0.31	0.05	382.2	3.88	368.2	0.950	422.940	5.774	0.951	423.510	5.787
3	1.11	34.93	0.41	0.045	385.1	4.7	368.2	0.995	438.166	5.324	0.994	437.394	5.324
4	1.15	39.93	0.48	0.042	389.2	4.85	371.2	0.997	443.006	5.373	0.998	443.333	5.493
5	1.19	37.93	0.21	0.042	377.6	3.13	362.4	0.828	397.156	2.760	0.831	397.557	2.804

and

Table 4. Comparison between FPM and surrogate model in operation scenarios.

NO.	RMI	SV	a	Tw	Tin	FPM			Surrogate Model
						xA	Tmax	Pro	xA	Tmax	Pro
1	1.19	37.93	0.21	377.6	362.4	0.820	398.911	2.906	0.821	398.709	2.916
2	1.03	35.2	0.23	375.1	366.2	0.845	400.100	2.792	0.848	399.906	2.793
3	1.06	38.93	0.31	382.2	368.2	0.924	418.103	3.345	0.920	417.923	3.351
4	1.11	34.93	0.41	385.1	368.2	0.970	436.538	3.169	0.975	436.408	3.189
5	1.15	35.93	0.43	388.1	370.2	0.981	441.386	3.297	0.983	441.345	3.306

).

Five sets of inputs were chosen randomly to further verify the accuracy of the surrogate models in design and operation scenarios (Table 3 and Table 4). Table 3 shows that the prediction data from the surrogate models agreed well with those from FPM, and all relative deviations for these three outputs were below 1.5%. In addition, the accuracy of the surrogate models in operation scenarios was also verified, where all relative deviations for these three outputs were lower than 1%. Therefore, the trained models above can be used to optimize the VCM reactor with high fidelity.

3.2. Sensitivity Analysis of Input Variables

In this section, the single variable method was adopted to determine the range of input variables. For example, to determine the range of RMI, all other variables were kept constant, and SV, a, d_t, T_w, l, and T_in were set to 34.13 h⁻¹, 0.209, 0.044 m, 372.2 K, 3.0 m, and 362.2 K, respectively. The results of RMI, SV, a, d_t, T_w, T_in, and l are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10.

Figure 4 shows that the ultimate mole conversion rate of acetylene (Pro_xa) and Pro first increase and then decrease with increasing RMI values (see the left graph). The reason for this is that the term of (m-x_A) dominates the value of r_A when RMI is less than 1.2, while the term of P_A0 dominates r_A when RMI is greater than 1.2 (see Equation (6)). In addition, T_max decreases with RMI increment (see the right graph), because more hydrogen chloride will share the effect of reaction heat on the reaction temperature. Thus, in view of operation cost and production efficiency, the range of RMI was chosen to be from 1.01 to 1.20.

Figure 5 shows that Pro_xa decreases with SV increasing, while Pro trends downward (see the left graph). An increasing SV means a rising linear velocity of C₂H₂ and an increasing mass flux of C₂H₂, leading to increased Pro. The reason for the different variation trend in Pro is the shortened reaction time at an elevated SV. In addition, an elevated linear velocity will improve the total thermal conductive coefficient, and thus T_max will decrease with increasing SV (see the right graph). Thus, considering Pro_xa and production efficiency, the range of SV was chosen to be 33−40 h⁻¹.

Catalysis activity directly affects the catalytic reaction rate; its sensitivity analysis is shown in Figure 6. It was found that Pro_xa, Pro, and T_max all increase with catalysis activity increasing, because a directly contributes to the reaction rate, which will accelerate the exothermic reaction, thus leading to a reactor temperature increase. Due to the dependance of catalyst properties on the reaction temperature, the range of a was chosen as 0.2 to 0.5.

Figure 7 shows that Pro_xa, Pro, and T_max all increase with increasing d_t. Because a rising d_t reduces the total thermal conductivity and weakens the heat removal via coolants, T_max increases (right graph). Now that the reaction rate correlates with T_max, Pro_xa increases with increasing d_t whereas a high T_max will lower the catalyst performance, so the range of d_t was set as 0.04 to 0.05 m.

As T_w represents the heat removal capacity via the tube wall, a higher T_w means less heat which can be removed and an increased reaction temperature. As shown in Figure 8 regarding the effect of T_w on Pro_xa, Pro, and T_max, the reliance of catalyst properties on temperature corresponds with a range of T_w from 371 to 393 K.

The reactor tube length affects the residence time of the reactants, and thus increasing the tube length (l) contributes to Pro_xa and Pro. Though the heat generated by the reaction increases with l increasing, the heat removal area also expands. The overall result is that T_max changes little when l is varied in the range of 2.6 to 5.0 m. Taking the operation cost and equipment occupation space into consideration, the range of l was chosen to be from 3 to 5 m. In addition, the result from the sensitivity analysis indicates that T_in varying from 360 to 380 K has a small effect on Pro_xa, Pro, and T_max. Taking into account the industrial condition, the range of T_in was chosen to be from 362.2 to 373.2 K.

3.3. Optimization in the Design Scenarios

In the design scenarios, equipment parameters and process parameters are adjustable and they all can be selected as feature variables except for the operation pressure, according to Section 2.1. The ranges of RMI, a, SV, T_w, d_t, l, and T_in are listed in

Table 5. The ranges of different feature variables in the design scenarios.

Items	Minimum Value	Maximum Value
RMI	1.01	1.20
a	0.20	0.50
SV, h−1	33.00	40.00
Tw, K	371.00	393.00
dt, m	0.04	0.05
l, m	3.00	5.00
Tin, K	362.20	373.20

.

The Pareto fronts obtained using the PSO algorithm for the vinyl chloride industrial reactor in the design scenarios are marked in blue in Figure 11. There is a contrary changing trend between x_A and Pro with constrained T_max in the tube reactor. In the analysis ranges for all variables, Pro can approach 6.81 mol h⁻¹ when x_A is about 0.986, while a Pro value of about 5.99 mol h⁻¹ gives rise to an x_A value of 0.990. In practice, a trade-off between Pro and x_A is essential according to the commodity market, and appropriate input conditions from the Pareto fronts should be chosen. For example, if 0.98 satisfies the upstream and downstream requirements, the input condition can choose the leftmost set, where Pro is highest in the range. Some Pareto fronts and their corresponding inputs are shown in

Table 6. Some Pareto fronts and their input conditions in design scenarios.

NO.	RMI	SV	a	dt	Tw	l	Tin	x	Pro	Tmax
1	1.186	35.972	0.327	0.045	384.94	4.96	366.24	0.990	5.763	422.95
2	1.186	36.127	0.331	0.046	383.53	4.99	365.68	0.989	6.038	422.93
3	1.190	36.591	0.313	0.048	383.84	5.00	365.33	0.988	6.569	422.37

.

3.4. Optimization in the Operation Scenarios

In the operation scenarios, equipment parameters are fixed (d_t = 0.044 m; l = 3.0 m), and thus they should be removed from the feature variables. The ranges of RMI, a, SV, T_w, and T_in are shown in

Table 7. The ranges of different feature variables in the operation scenarios.

Items	Min Value	Max Value
RMI	1.01	1.20
a	0.20	0.50
SV, h−1	33.00	40.00
Tw, K	371.00	393.00
Tin, K	362.20	373.20

.

The Pareto fronts obtained using the PSO algorithm for the vinyl chloride industrial reactor in the operation scenarios are marked in blue in Figure 12. There is a contrary changing trend between x_A and Pro with constrained T_max in the tube reactor. In the analysis ranges for all variables, Pro can approach 3.47 mol h⁻¹ when x_A is about 0.952, while x_A can approach 0.953 when Pro is about 3.42 mol h⁻¹. In practice, industries should make a trade-off between Pro and x_A according to market requirement, choosing the appropriate input conditions from the Pareto fronts. For example, if 0.95 satisfies the upstream and downstream needs, the leftmost set can be chosen for the input condition, where Pro is highest in the range. Some Pareto fronts and their corresponding inputs are shown in

Table 8. Some Pareto fronts and their input conditions in operation scenarios.

NO.	RMI	SV	a	Tw	Tin	x	Pro	Tmax
1	1.20	38.728	0.336	385.05	369.54	0.9526	3.447	423.14
2	1.20	38.201	0.337	384.70	369.55	0.9533	3.406	423.12
3	1.20	38.857	0.334	385.29	369.49	0.9521	3.456	423.10

.

3.5. Comparison Between Single-Objective and Multi-Objective Optimization

As mentioned in the introduction section, the optimization of VCR mainly focuses on single-objective optimization without taking all related variables into consideration simultaneously. To further verify the advantages of the proposed multi-objective optimization frame over conventional paradigms, single-objective optimization involving multiple inputs with a constrained condition of T_max was conducted, in which the ranges of different feature variables were the same as in multi-objective optimization. The results of the design scenario are shown in Figure 13. It can be seen that x_A could approach 99.4% with the following parameters: dt 0.045 m, l 5.0 m, RMI 1.20, SV 33.0 h⁻¹, a 0.417, T_w 372.3 K, and T_in 362.8 K. In order to verify the optimum performance, the just-mentioned parameters were substituted into the model, showing a Pro of 5.36 mol/h. In contrast, Pro could approach 5.76 mol/h or above with an x_A of 98.5% using the multi-objective optimization proposed in this work. This comparison demonstrates the obvious advantage of multi-objective optimization over single-objective optimization for industrial VCR, and the former adapts better to industrial demand.

4. Conclusions

The optimization of vinyl chloride industrial reactors faces multi-objective optimization issues, and to maintain its optimal operational performance, an efficient optimization control strategy based on machine learning and hybrid model was proposed. The trained and verified surrogate model could be used to monitor and predict the reactor operation performance timely and accurately. A multi-objective optimization algorithm coupled with a prediction model could provide optimal operational direction for the industrial reactor, which contributes to sustainable development in this field and can even be extended to relative fields.

Taking the commonly used first section of a VC reactor as an example, the adoption of this optimization strategy could realize significant improvement for ACR and VCPC, rising from 81.34% to over 95% for ACR and from 2.60 to over 3.40 mol h⁻¹ for VCPC in the operation scenarios based on the reference value obtained from industrial data. For example, with the parameters of the first section being d_t 0.044 m, l 3.0 m, RMI 1.20, SV 38.73 h⁻¹, a 0.336, T_w 385 K, and T_in 369.5 k, ACR and VCP could approach 95.26% and 3.44 mol/h, respectively, which are exactly the same as those of the traditional two-section reactor. So, in the future, the first section could be used alone for industrial production, with large savings in device investment.

The surrogate model can be a data-driven model or hybrid model. This work chooses a data-driven model owing to insufficient available industrial data, thus only demonstrating the practical application for the proposed strategy. In addition, an ML model can also be used to substitute FPM to compensate for the drawbacks of the latter in monitoring and control, fault diagnosis, etc., in complex chemical engineering industry. Finally, it is generally true that a pure ML model only has sufficient fidelity in the range of the training dataset, which is reliable for industrial processes. If prediction accuracy and interpretability are to be further enhanced, a hybrid model is necessary. The combination of ANN with FPMs becomes powerful as ANN can compensate for the gap between FPMs and real industry while FPMs can enhance the interpretability and extrapolation of surrogate ANN models.