1. Introduction
Due to large-scale application of catalysts in the petroleum and petrochemical processes, the main problem concerning these processes is the deactivation of catalyst. As the catalyst activity is related to the availability of catalyst active area for the reactants, deactivation refers to reducing the activated area of the catalyst or blocking the path of the moving reactants and the products. Mathematical modeling of catalyst deactivation is the main concern of these processes during operation. One of these processes with high potential of deactivation is the disproportonation (DP) of toluene and/or transalkilation (TA) of C
9 aromatics and toluene to C
8 aromatics over zeolites in fixed-bed reactors. DP and TA involve solid-acid catalytic processes which occur over medium-pore zeolites, such as ZSM-5, and large-pore zeolites, such as mordenite, Beta, and Y [
1,
2,
3,
4]. For H-mordenite, the temperature is a dominant factor that determines the activity and coking tendency, as the undesired reaction, due to its one-dimensional channel system [
5].
The major problem for the commercialization of solid acid catalyzed processes is the rapid deactivation of the catalyst [
4,
6,
7]. Therefore, modeling of the deactivation of the catalyst is a major problem that must be taken into account for the success of such processes. Some studies have already reported the deactivation behavior of the solid acid catalysts quantitatively [
8,
9,
10,
11]. These studies have focused on the solution of the set of differential and algebraic equations that describe catalyst deactivation in various reactor systems by experimental data. In these models, however, many simplified conditions have been assumed. Most of these models have been conducted at the laboratory scale under certain conditions with applying no dynamic data [
1,
2,
3,
4,
5,
6,
7,
12]. Additionally, online investigation of the industrial reactors’ parameters (e.g., catalyst deactivation, conversion, product quality, etc.) can be of considerable significance in the identification of these parameters. In the last two decades, researchers began to make use of the vast amounts of data being measured and collected in the industrial processes by constructing predictive models based on such data. In the context of the industrial processes, these predictive models are called soft sensors [
13,
14]. In chemical plants, soft sensors are widely used to estimate a process variable which is difficult to be measured online. In general, one can broadly classify soft sensors into two types, namely, model-based (white-box) and data-based (black-box).White-box models are dependent on a prior theoretical knowledge and thus often unavailable since industrial processes are commonly too complicated to be analyzed theoretically. These models are developed primarily for the planning and the design of the processing plants and, therefore, usually focus on the description of the ideal steady states of the processes. As a solution, the data-based soft sensors gained increasing popularity in the industrial processes. A wide variety of statistical inference techniques and machine learning techniques have been employed in data-based soft sensors, among which representative examples are principal component regression (PCR), partial least squares (PLS) regression, support vector machines (SVM), and artificial neural networks (ANN) [
13,
14,
15,
16,
17,
18,
19,
20,
21,
22].
Reactors, by their nature, are non-stationary because their complexity cannot be fully described using constant-parameter models. The modeling of such systems can be represented by a two-step procedure. First, the dynamic behavior of the system is described by a deterministic mathematical model. Second, the model is simplified to utilize in conventional control theories. The first assumption causes errors in modeling, while the second suffers from the complexity and dimensionality problem. Moreover, digitization in order to use computer facilities for application of the continuous control theories provides another source of errors in modeling and control strategies [
23,
24]. Hence, soft sensor modeling of these systems requires the use of powerful statistical inference techniques.
Data-based mechanistic (DBM) modeling philosophy, as introduced by Young [
25], is an alternative to complicated modeling strategies. The major advantage of this method is balancing between order of the model (complexity) and the model efficiency while, at the same time, good explanation of data and statistical analysis is required for identification and estimation of such systems [
25,
26,
27,
28,
29,
30]. Since the resulting model is simple and directly in digital form, it avoids both the model complication and errors produced through either simplification or digitization of a complex continuous-time model. It is worth mentioning that the resulting model and its parameters are, according to DBM philosophy, meaningful and interpretable while in data-based modeling, which is not mechanistic, this is generally not the case. In this philosophy, the non-stationary aspects of the system are reflected by a simple structure-varying parameter model, whether its parameters are varying with time or as a function of some states of the system. This model is called a state-dependent parameter (SDP) model [
26,
30]. In this case, all nonlinearities of the model can be estimated in model parameters, while the structure of the model remains linear and simple. In other words, when a system is described by a nonlinear model, such as
, the SDP model is written as
which has a linear structure but its parameter is
. This model has the complexity of a nonlinear model while using a linear structure. The parameters are also identified in a non-parametric manner; i.e., when this approach is used, the functionality of the identified parameter can be seen in a graph rather than a mathematical relation.
In this study, a new SDP method for identification and modeling of the kinetics and catalyst activity has been proposed. The reactor modeling and identification have been considered for an industrial TA reactor using the input and output dynamic data. The obtained soft sensor model has a simple structure and appropriate for control purposes. This model also presents a good identification based on effective parameters of the catalyst activity.
2. Kinetic Model
In the present study, the process data of the fixed bed TA reactor of Bou-Ali-Sina Petrochemical Complex, Mahshahr, Iran, have been collected and analyzed daily for two years of reactor's operation. In the TA reactor, benzene and xylenes are produced from toluene and C
9 aromatics through the two main reactions, DP (Equation (1)) and TA (Equation (2)). The DP produces benzene and xylenes from toluene, while the TA forms xylenes from toluene and C
9 aromatics:
Different kinetic models have been proposed to describe the catalytic toluene disproportionation and C
9 aromatics transalkylation.
Table 1 summarizes the kinetic and deactivation models proposed in the literature, as well as the experimental conditions.
The reactor contains H-mordenite zeolite catalyst, which is manufactured by Sinopec Tech Company. Its characteristics are as follows (
Table 2).
4. Results and Discussion
To verify the effectiveness of the SDP modeling, data series of fixed bed industrial TA reactor of the Bou-Ali-Sina Petrochemical Complex are analyzed. A record of 623 data as daily means for two years of reactor operation are used. Overall process data for the input and output concentrations of toluene, C
9 aromatics, olefin, hydrogen, benzene, xylene, and temperature
T versus time are shown in
Figure 1.
As the figure shows, the data are recorded with high input and output oscillation, without interfering with the working conditions and the dynamics form.
By designing the soft sensor, it is expected that while all of the states or variables are changing simultaneously, the system identification is performed. Moreover, model parameters are expected to be functions of the system state variables. When a complex nonlinear model with constant parameters is replaced by a linear structure with varying parameters, it is clear that the model with linear variations of the state variables gives a better control. The following assumptions are considered:
Modeling is based on the continuity equation;
Grey-box model is considered;
Data are registered as daily average, that is, for each 24 h one data is recorded;
Since the residence time of the reactor is half an hour and the data are recorded per day, the system is considered stable and the term for the accumulation in the continuity equation is vanishes;
Due to the high volumetric flow rate of the reactor, the terms for the axial and radial diffusions are considered as part of the system error and hence removed; and
The hydrogen molar percentage in the feed is about 90%. Therefore, the internal porous mass transfer resistance is disregarded.
Using the SDP modeling and through the investigation of the considered reactor model parameters Equations (15) and (16), it has been shown that most parameters have slight changes. Therefore, the parameters are supposed to be constant and are represented in
Table 3.
The only exception is due to the variations of the term for the deactivation of catalyst
, which is considered to be a function of different variables of the system states. The changes have been found to be significant in terms of the time and hydrogen and olefin concentrations. An important issue is the high sensitivity of the catalyst activity to the olefin concentration (lower than 200ppm). Olefin causes coke formation (as the undesired reaction) and reduces the catalyst efficiency; hence, the conversion decreases. In order to compensate for this effect, the system operation needs to increase the temperature; however, increasing the reactor temperature will also increase the deactivation rate of the catalyst. Furthermore, high molar fraction of hydrogen is required to avoid undesired coking reaction and dilution of other components in order to reduce the diffusion resistance and increased temperature result in the recovery of the catalysts. However, the process is then able to perform the required conversion in a lower temperature. Thus, the effect of olefin on the catalyst poisoning, along with the high molar fraction of hydrogen and the increased temperature for the catalyst recovery, cause the catalyst activity to have a roughly monotonic trend over the two-year period of operation. One main advantage of the SDP method to design the soft sensor is that the modeling is exclusively performed by process data and, hence, exact determination of the process information (e.g., reaction mechanism, kinetics, etc.) is not needed. Therefore, in the present study, the parameters related to such problems as the order of concentration in the reaction are found to be different from those recorded in the literature (
Table 3). As can be seen, some parameters have values close to zero with high uncertainty. Thus, the related parameters are omitted and the equation is again recovered and identified as follows:
In these equations, the term for the catalyst deactivation multiplied by the pre-exponential factor is replaced by
. The results of the new identification show a reduction in the uncertainty of the estimated parameters as represented in
Table 4.
In both models, the xylene concentration plays a decisive role, compared to the other components. Therefore, it shows that with the increase in the xylene concentration in the feed, a higher conversion is obtained. As shown by
Table 4, the activation energies of the DP and TA reactions are in an acceptable range and in compliance with the previous studies (see
Table 1). Comparison of the activation energies of the DP and TA reactions shows that the TA reaction is faster. Concerning the catalyst deactivation term in the first model (
), from the investigation of different states in the two dimensions of time and concentration of olefin, it can be observed that olefin causes catalyst toxicity and significantly decreases catalytic activity (see
Figure 2). In this context, as shown in
Figure 2a–c, it is suggested that for avoiding the catalyst toxicity, olefin concentration in the feed should be lower than 20ppm. However, during the time the catalyst activity is in an oscillatory form, and as mentioned before, it shows an approximately constant activity over the period of study. As shown in
Figure 2, the catalyst deactivation term in the first model is changing by a function of olefin concentration and time. The catalyst deactivation term in the second model (
) has been identified as a function of the two states of hydrogen concentration and time. Hydrogen concentration and the increased temperature have recovered the catalyst (
Figure 3a–c).
Based on the changes in temperature and olefin concentration (
Figure 4), it can be concluded that with an increase in the amount of olefin in the inlet flow, the reactor temperature is increased by the operator with a considerable time delay. The time delay causes a reduction in the conversion of the reactor. This can be improved by employing an online prediction method. In other words, by applying the soft sensor, the temperature can be adjusted based on the catalyst activity.
It would be a good progress for the industrial processes to be identified and fault-detected using the process data with no extra cost imposed. As shown in
Figure 5, the predicted values are in good agreement with the process data which can be considered as an important result of the SDP method. It should be noted that the SDP method with reference to the reduction in the laboratory costs is a great success for petrochemical industries. As a result, it assists in the increase of prediction and fault detection ability.
5. Conclusions
A soft sensor is designed for identification of the variations in the catalyst activity using process data of an industrial reactor. Based on the DBM philosophy, the present study is developed in an innovative method for soft sensor design, which demonstrates appropriate capability for analyzing the data with drastic changes. Using the SDP method, the catalyst activity has been estimated non-parametrically as a function of time, olefin concentration, and hydrogen concentration. Other parameters are considered to be constant. Based on the obtained results, it is suggested that the amount of olefin in the inlet flow should be lower than 20 ppm. In the corresponding models, for both reactions, the order of xylene concentration is greater than that of the other components and, thus, for the increase in the conversion it is suggested that the xylene concentration in the inlet flow should be increased through the recycle flow. However, the estimated order of concentration of the components does not show a good agreement with the previous results. The activation energies for the two DP and TA reactions have been found to be 43.8 and 18 kJ/mol, respectively. The model prediction is in good agreement with the process data. The present work can be considered as an advance towards the identification, modeling, and fault detection of a complex industrial process.