1. Introduction
Ethylbenzene (EB) is an important chemical material to produce textile fabric, plastics, detergents, etc. Meanwhile, it is also an intermediate in the production of styrene [
1]. The output and consumption of global ethylbenzene continue to increase with the improvement of industrial production [
2]. Almost all ethylbenzene is converted from the alkylation of benzene and ethylene.
In order to maximize the use of refinery resources, ethylene in dry gas from fluid catalytic cracking (FCC) process has been one of the main raw materials of ethylbenzene. The FCC process produces a large amount of dry gas, which contains 10–30% ethylene [
3,
4]. The recovery of ethylene in dry gas is of great significance to resource utilization and environmental protection. Some studies were carried out on the process development and optimization of dry gas to ethylbenzene. Chen et al. [
5] has developed five generations process technologies of dry gas to ethylbenzene. With each generation of technology update, different processes or technical combinations are adopted to improve the life of the catalyst and the selectivity of ethylbenzene. Zhu et al. [
6] designed a new process for producing ethylbenzene from FCC dry gas by combining gas phase alkylation and liquid phase transalkylation. Through optimizing the operating conditions of the reactor, the ethylbenzene selectivity can be increased from about 90% to more than 99%. Liu et al. [
7] used dilute ethylene, benzene, and transalkylation streams as reactor feeds to improve ethylbenzene selectivity. In the reactor, liquid benzene and gaseous benzene can coexist, so that the alkylation process and the transalkylation process can occur simultaneously. Qian et al. [
8] have studied catalyst deactivation in dry gas to ethylbenzene reactor. A catalyst deactivation model was established based on the on-site data and those from laboratory analysis. The real-time online dynamic simulation of the reactor is carried out, which can provide timely and reasonable guidance for the equipment operators.
However, the reactor models in the process optimization have been simplified, which may lead to deviations in temperature and composition. Therefore, it is necessary to study the model and optimal operation of the alkylation reactor, which is the critical equipment in the ethylbenzene manufacturing process [
9]. Hamid et al. [
10] developed a one-dimensional alkylation and transalkylation reactor model for the factory, which provided a basis for subsequent optimization. Ivashkina et al. [
11] established a mathematical model for the liquid-phase benzene alkylation batch reactor to study the influence of heavy hydrocarbon concentration on catalyst activity. Additionally, through optimization, the impact of catalyst deactivation can be offset by adjusting operating conditions.
All aforementioned studies on the reactor consider the case of pure ethylene as a raw material. However, the open literature lacks studies on dry gas based ethylbenzene reactors. The dry gas comes from the catalytic cracking unit, which contains many impurities. Additionally, the composition of the dry gas can vary depending on the cracking process. Uncertainty in dry gas composition can lead to unexpected hot spots, which can result in catalyst deactivation. Hot spot constraint violations can be avoided by optimizing the operating conditions. To address hot spot constraint violations under feed uncertainty, a back-off to the hot spot constraints has been introduced and validated [
12,
13]. In order to address the above problems, in this work, a two-dimensional alkylation multistage reactor model is developed to simulate the process of benzene with dry gas to ethylbenzene. A two-dimensional reactor model can more accurately describe the hotspots within the reactor [
14]. The model consists of energy and mass balances considering the radial transfer and hyperbolic reaction kinetics. In addition, the Soave–Redlich–Kwong equation of state (SRK-EoS) is used to calculate the physical properties of the mixture to improve the model accuracy. Considering the hot spot constraint violations, a robust multi-objective optimization framework is proposed: the strategy of introducing back-off [
15] in constraints is combined with the multi-objective optimization algorithm to hedge against the worst case, and then multi-criteria decision-making (MCDM) [
16] is introduced to select the optimal operating point. The reactor optimization objectives considered are maximizing selectivity of ethylene and conversion of ethylbenzene. Additionally, the distribution ratios of dry gas are defined as decision variables, which are important operating conditions for the performance of the reactor.
The key contributions of this paper are listed below:
A two-dimensional homogeneous alkylation reactor model is established to describe a dry gas-based ethylbenzene production process. This model can obtain the temperature distribution in the reactor, and then observe the hot spots.
A robust multi-objective optimization framework is proposed by combining back-off in constraints, multi-objective optimization algorithm and multi-criteria decision-making. The proposed framework can effectively handle the hotspot temperature violation caused by uncertain dry gas composition.
The effectiveness of the proposed robust multi-objective optimization framework is verified through an industrial case study.
The rest of the paper is organized as follows.
Section 2 presents the process description and detailed information on the process simulation. In
Section 3, model equations, as well as physical properties, parameter calculation methods are introduced.
Section 4 illustrates the optimization problem and the robust multi-objective optimization method. The reactor model validation and robust multi-objective optimization results are provided in
Section 5. Finally, the conclusions are drawn in
Section 6.
2. Industrial Process
The reactor studied is based on a practical alkylation reactor consisting of a five-stage adiabatic fixed bed. The first four stages are used as the bed of the main reaction zone. The fifth stage is used to ensure the complete conversion of ethylene at the outlet of the reactor. Dry gas enters the reactor from the inlet of the first four stages according to certain ratios. Therefore, the high concentrations of ethylene and ethylbenzene will not coexist, which will avoid the progress of side reactions. The alkylation reaction is a strongly exothermic reaction. However, a high temperature will cause the catalyst to deactivate [
17]. The staged entry of dry gas can keep the temperature in the reactor within a certain range.
Figure 1 shows a simple diagram of the alkylation reactor of EB production.
The gas-phase alkylation of benzene with ethylene for ethylbenzene manufacture is studied in this paper: the main reaction of benzene (BZ) with ethylene (ET) alkylation into ethylbenzene (EB), and the side reactions producing diethylbenzene (DEB). The other side effects are much smaller and, therefore, neglected in this study [
18]. Two reactions in Equations (
1) and (
2) are considered, which describe the overall process.
The reaction rates [
10] are presented in Equations (
3) and (
4). Kinetic parameters are obtained by fitting real production data. For non-linear parameter optimization, a pattern search method has been applied. The kinetic coefficients follow the Arrhenius equation with parameters presented in
Table 1.
is reaction rate for the reaction j in , R is universal gas constant in , T is the absolute temperature in K. , and represent, respectively, ethylene, benzene, and ethylbenzene concentrations in .
3. Mathematical Model
The model of the alkylation reactor is composed of three parts, namely physical property calculation, transfer parameters and balance equations. The physical behavior of the mixture includes the residual enthalpy, the residual heat capacity, and the viscosity, which is obtained by the SRK-EoS. The transfer parameters used to describe the radial transfer process are the radial effective thermal conductivity and the effective radial diffusion coefficient. Balance equations are composed of material balance equation and energy balance equation to describe the energy and material change process inside the reactor. In addition to the equations of the reactor, there are also the material balance equations to calculate mixing process between the reactor stages.
3.1. Mixture Behavior with SRK-EoS
The dry gas contains many impurities, and the composition is also uncertain. Therefore, ideal gas behavior is not accurate to describe the thermodynamic properties of the mixture. The SRK-EoS, mainly used in gas and refining processes, is appropriate to describe its thermodynamic properties, which was proposed by Soave to improve the RK-EoS [
19]. The equation is expressed as follows.
Meanwhile, the SRK-EoS can also be expressed in the form of a compression factor.
where
A and
B are given by Equations (
7) and (
8).
Residual variables are calculated to modify the behavior of the mixture by adding residual items in the model. The residual variables used are residual enthalpy and residual heat capacity [
20]. The residual enthalpy of mixture,
, can be calculated as:
where
is the mole fraction of component
i.
. is the same as defined by Soave. The correlation between
and the eccentricity factor
is given by Equation (
10).
The residual heat capacity of mixture,
, can be calculated as:
where
is calculated as:
where
is given by Equation (
13),
3.2. Transfer Parameter
The radial effective thermal conductivity of fixed reactor is affected by the convection heat exchange of particle and fluid, the heat conduction of particle and fluid and the heat exchanged by radiation. Thermal conductivity of the mixture can be calculated by Equation (
14) [
21]:
Here, is thermal conductivity of component i in . is molar mass of component i in .
The effective thermal conductivity can be calculated by Equation (
15) [
22]:
where
is thermal conductivity of catalyst in
.
The molecular diffusion coefficient for each component
i in the multicomponent gas mixture can be obtained by Equation (
16) [
23]:
Here,
is the binary molecular diffusion coefficient for component
i in component
j. The binary molecular diffusion coefficient
can be calculated using Equation (
17) [
24]:
where
is the atomic diffusion volumes in
. The effective radial diffusion coefficient of the component
i is calculated as [
14]:
Here, is the bed void fraction.
3.3. Balances for the Reactor
The continuity equations and energy balance equations are defined considering the system operating in steady-state. The continuity equations for each component follow the basic format presented by Equation (
19).
Here, u is the fluid velocity only in the flow direction in . is concentration of component i. l is the length of the reactor in m. is the effective radial diffusion coefficient of component i in . r is the radius of the reactor in m. is the rate of generation of component i in reaction j in .
There is a velocity gradient in the reactor. The relationship between the velocity gradient and the radius is as follow [
25].
where
is the maximum flow rate at the center of the reactor.
is the reactor radius.
Due to the addition of the residual heat capacity, the energy balance equation will add a term
. The energy balance is expressed by Equation (
21).
where
is the molar flow rate of component
i in
.
is ideal isobaric heat capacity of component
i in
.
T is the reactor temperature in K.
is reactor cross-sectional area in
.
is effective diffusion coefficient of the reactor in
.
is reaction heat of reaction
j in
.
The boundary conditions are rewritten as follows:
Here, is reactor diameter in m.
Between the stages, the cold shock heat exchange method is used by directly mixing the dry gas with the export materials of the previous stage. The material balance formula is as follows.
, , is the molar flow rate, respectively, of previous stage, dry gas raw material and current stage of component i, is the dry gas ratio of stage k.
Because the temperature of the dry gas feed is lower than the outlet temperature of the stage, the energy balance between stages expressed by Equation (
26) is used to find the inlet temperature of the next stage.
3.4. Numerical Methods
The alkylation reactor model is composed of Equations (
19)–(
26), which are a set of coupled, linear partial differential and algebraic equations. The partial differential equations (PDEs) are solved with the method of lines. The radial coordinate is discretized by applying orthogonal collocation. Ordinary differential equations are solved using Runge–Kutta method (ode45) in MATLAB R2020a.
6. Conclusions
In this paper, a two-dimensional homogeneous model is developed and implemented in Matlab, for steady state simulation of an industrial multi-stage catalytic reactor for ethylbenzene. Through the validation procedure, it is proven that the developed model is accurate. The establishment of a two-dimensional reactor model can calculate the hot spots inside the reactor. The hot spots appear in the center of the reactor. A robust multi-objective optimization method is proposed by introducing a back-off to the hot spot constraints, which can drastically reduce constraint violations and provide a trade-off of conversion and selectivity. The distribution ratios of dry gas are defined as decision variables, which is related to the temperature inside the reactor and affects the performance of the reaction. After robust multi-objective optimization, the robust result is obtained. The ratios of the first four stages of dry gas are, respectively, 0.7438, 0.2234, 0.0325, and 0.0003. The conversion and selectivity can be, respectively, achieved as 99.94% and 90.88%. In addition, robust optimization methods reduce the impact of dry feed composition uncertainty. Compared with the general optimization method, the robust optimization results show that the proportion of temperature constraint violations decreases from 13.7% to 3.8%.