Ind. Eng. Chem. Res. 2004, 43, 6441-6445
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Modeling and Optimization of a Styrene Monomer Reactor System Using a Hybrid Neural Network Model Heejin Lim, Min-gu Kang, and Sunwon Park* Department of Chemical and Biomolecular Engineering, KAIST, 373-1 Guseong-dong, Yuseong-gu, Deajeon, Korea
Jeongseok Lee Performance Polymers Research Institute, LG Chem Ltd., 70-1 Hwachi-Dong, Yosu-City, Chunranam-Do, Korea
The operation of a current styrene monomer plant requires large quantities of energy to heat and cool its processing streams. Especially, operating a styrene monomer reactor system under proper conditions is very important because this reactor system occupies a large portion of the total operating cost using a large amount of expensive high-pressure steam. The optimization of the operating conditions of this reactor system can therefore be used to significantly reduce the total cost. To predict the dehydrogenation reactor conditions and take into account the effects of catalyst deactivation, we propose in this paper an alternative hybrid model of the reactor that is composed of a mathematical model and a neural network model. The mathematical model is a first principle model that predicts the compositions and the temperature and pressure profiles from the reaction mechanism and reactor geometry. The catalyst deactivation factor used in the mathematical model is calculated with the neural network model. Actual plant data were used in this study to test the hybrid model. Using this reactor model, we were able to solve the optimization problem for this plant. The objective of the optimization was to maximize the performance of the dehydrogenation reactor. A trajectory optimization method is proposed in this study that reduces the calculation required for the optimization. In this method, the trajectory of each operating variable is optimized while the other operating variables are held constant at their average values. Empirical equations are then obtained from the optimal trajectories, and the parameters of the empirical equations for all operation trajectories are optimized simultaneously. We found that the optimal profit was greater than that currently obtained by the plant. Introduction In Korea, the styrene monomer industry operates under production-limited market conditions and the shortfall is imported. Since the import price for styrene monomer is much higher than its production cost, styrene producers want to maximize the rate at which they can produce styrene monomer. However, increasing the production rate results in problems of fast catalyst deactivation and high operating costs. Therefore, the ultimate goal of producers is to maximize their net profits by optimizing plant operating conditions. The operation of current styrene monomer plants requires large quantities of energy to heat and cool their processing streams. Operating a styrene monomer reactor system under optimal conditions is very important because the reactor system operation is responsible for a large proportion of the total operating cost. The reason for their high operating costs is that they use a large amount of expensive high-pressure steam to produce styrene monomer through a dehydrogenation reaction.1 Hence, to maximize styrene monomer plant production rates and minimize their operating costs, modeling and optimizing the performance of the reactor system are essential. * To whom correspondence should be addressed. Tel.: +82-42-869-3920. Fax: +82-42-869-3910. E-mail: sunwon@ kaist.ac.kr.
Several types of styrene monomer reactors have been used in industrial plants with various reaction schemes, flow directions, and conditions: dehydrogenation and oxidation reactors; plug flow and radial flow reactors; and adiabatic and isothermal reactors. The styrene monomer reactor system in our study consists of adiabatic radial flow dehydrogenation reactors in series. This reactor type has higher equipment and operating costs than the other types of reactors. However, it has been widely used because of its high conversion ratio of ethylbenzene and its low usage of catalyst.2,3 There have been many previous attempts to improve the productivity of the dehydrogenation reactor system. Early researchers were interested in the reaction mechanisms of ethylbenzene dehydrogenation and the mathematical modeling of industrial dehydrogenation.2,4-9 Even though their mathematical models gave a reasonable prediction for the industrial dehydrogenation reactor system, their prediction was limited due to catalyst deactivation. Many experiments have been carried out with the aim of understanding the deactivation mechanism of the dehydrogenation catalyst and in particular have focused on the effects of components in the catalyst such as potassium.6,10-13 Prediction of catalyst deactivation in industrial sites is especially difficult because observation of reactor variables is limited and so only uncertain information about catalyst deactivation is available. Thus, mathematical models using plant data
10.1021/ie049936x CCC: $27.50 © 2004 American Chemical Society Published on Web 08/27/2004
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Figure 1. Configuration of the adiabatic radial flow reactor system.
are inadequate for describing reactor dynamics and catalyst deactivation. To predict reactor dynamics and catalyst deactivation, we propose an alternative hybrid model of adiabatic radial flow. This model is composed of a mathematical model and a neural network model. The mathematical model is a first principle model based on the reaction mechanisms and reactor geometry for predicting the compositions and the temperature and pressure. The neural network model provides a catalyst deactivation factor for use in the mathematical model, as explained in the neural network model and the hybrid model sections in detail. A novel operating trajectory optimization method is proposed in order to optimize the reactor operating conditions. By reduction of the number of decision variables using operation trajectory equations, a suboptimal solution that has very reasonable trends is obtained. This optimization method and its solution are described in the optimization section. System Description Our target system has two ethylbenzene dehydrogenation reactors in series, as shown in Figure 1. The hydrocarbon feed (ethylbenzene, EB) is mixed with superheated steam prior to its entry into reactor 1. As it passes through reactor 1, the mixed stream is partially converted to styrene monomer and the temperature of the stream drops to ∼580 °C. Before entering reactor 2, this low-temperature flow passes through the heat exchanger, where it is reheated to ∼630 °C. This reheated stream is fed into reactor 2, and the remaining ethylbenzene is converted into styrene monomer. The final product then passes into the cooling and separation units. The reactor operating pressure, temperature, steamto-hydrocarbon ratio (the S/O ratio, i.e., the ratio between the feeding rate of ethylbenzene and the inlet rate of steam), and the ethylbenzene inlet flow rate are considered the important variables in dehydrogenation reactor system operation because these are the variables that most influence the conversion ratio and the selectivity of ethylbenzene. In particular, the S/O ratio has various effects on the productivity. The steam entering with EB acts not only as a heating medium but also as a diluent. Hence, a higher S/O ratio results in higher
conversion but is not preferred because the superheated high-pressure steam entering with EB is expensive. There are other conditions for obtaining high productivity in the dehydrogenation reactor system: low operating pressures and high temperatures. These conditions are highly favored because the dehydrogenation reaction is endothermic and it increases the number of molecules in the flow. Further, a high flow rate in the ethylbenzene inlet stream is preferred because it directly increases the production rate without requiring large changes in the operating conditions. However, all these requirements have high operating costs. To maximize the total profit, it is necessary to optimize the operating conditions. First Principle Model of the Dehydrogenation Reactor. A radial flow dehydrogenation reactor has a parabolic deflector in its center and a catalyst shell around the deflector. The feed stream enters at the center of the reactor and then flows through the catalyst layer in the radial directions (Figure 1). The mathematical modeling of this reactor, in particular the effects of the deflector, is difficult. To avoid these difficulties, we have simplified the reactor model by making several assumptions. First, only distributions in the radial direction are considered because the inlet stream entering into the center of the reactor is uniformly distributed in the axial direction and its flow direction is changed to radial direction by the deflector. Second, the temperature, pressure, and concentration distributions except in the catalyst layer are ignored because most of the reactions occur in the catalyst layer. Third, the flow inside the reactor is assumed to be that of an ideal gas mixture because the reactor is operated at very high temperatures and low pressures. Finally, a linear pressure drop was assumed. Generally, the momentum balance is used to express the pressure drop. But, the model takes a long time to solve due to the iterative calculation to fit the given input and output data because the model has differential algebraic equations having two-point boundary values. When we solved the model, we obtained a linear pressure drop inside the catalyst layer. Therefore, we simplified the pressure balance as a linear equation in order to reduce the computational load. Even though it is highly simplified, it gives about the same solution as the original model. Based on these assumptions, the mass and energy balances and the pressure drop equation were derived and are shown in eqs 1-3.
dgi
1 ) (2πrL dr F
∑j fijMWi), ∑i gi ) 1
2πrL dT dt
)
∑j (-∆Hj∑i fij) FCp
dP Pout - Pin ) dr Rout - Rin
(1)
(2)
(3)
For the reaction mechanisms and rate equations, Scheel and Crowe’s rate equations4 were used. To solve these differential algebraic equations, a differential algebraic equation solver DASPK (Computation Science and Engineering Program, http:// www.engineering.ucsb.edu/∼cse/) was used. A commercial software package CHEMKIN (Reaction Design,
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Figure 2. Neural network structure for calculation of the deactivation factor.
2000. http://www.reactiondesign.com) was used to calculate the physical and chemical properties such as heat capacity of flow that are necessary to apply our first principle model. Neural Network Model of Catalyst Deactivation. The feed flow rate, inlet temperature, partial pressure of steam, and partial pressure of ethylbenzene are known as important operating variables that influence the deactivation of the dehydrogenation catalyst. To predict the activity of the catalyst at the current time t, the catalyst activity with these operating variables at time t - 1 is necessary. Hence, the deactivation factor at time t, Φ(t), is calculated by the neural network model from the operating conditions at time t - 1 and the previous deactivation factor Φ(t-1) as a recurrent variable. Based on this information, we proposed a neural network model to calculate the deactivation factor in order to describe catalyst deactivation. Its structure is shown in Figure 2. This neural network model is a feed-forward network with one hidden layer having the five input (T, PTOTAL, PSTM, PEB, and Φ(t-1) in Figure 2) and one output (Φ(t) in Figure 2) variables. A sigmoid function is used as the activation function of the model. This model has three hidden nodes, which are selected by trial-and-error method. Hybrid Model of the Dehydrogenation Reactor. Even though the first principle model predicts the reactor condition very well in the early operating days, its performance deteriorates as the operating days increase due to the deactivation of the catalyst. Hence, we propose a hybrid model. This model has the first principle model as its basis. The proposed neural network model is applied to it to determine the deactivation factor. With the deactivation factor from the neural network model and the given input conditions from operating data, the first principle model can calculate the reactor outputs such as the temperature and the composition. The structure of this hybrid model, which combines the first principle model and the neural network model, is presented in Figure 3. To train this model, the back-propagation with gradient descent method was applied. Due to the differences of the absolute values of the input variables, we normalized all the input values to the range from 0 to 1. For training, 15 sets of the operating data were used, and the trained model was validated by 7 sets of data. The average relative error of the training sets is 1.3% and the error for the validation sets is 1.7%. Operation Trajectory Optimization. The objective of our study of styrene monomer reactor operation is to maximize the production rate of styrene monomer and minimize the operating costs. Increasing the production rate results in increasing operating costs, so an optimization of the reactor operating conditions is required with respect to profit. In this dehydrogenation reactor, main product is styrene monomer and byproduct is toluene. The income is made by sales of these products.
Figure 3. Schematic diagram of the hybrid model.
To get the net profit, raw material cost and the operating cost should be subtracted from the income. The production rates of styrene monomer and toluene are obtained from the simulation results of the proposed hybrid model. We formulated the profit function used in our optimization as shown in eq 4. t [FCSMgSM(t) + FCTLgTL(t) - FCEB ∫t)0 FCNTH ∑(Cpigi)(TINR2(t) - TOUTR1(t)) + i
Profit )
end
FCH2OCP,H2OgH2O(TINR1(t) - T0)] dt (4) The revenue from each product is calculated by multiplying the cost parameters (CSM and CTL) and flow rate of each product (total inlet flow rate F multiplied by gSM(t) or gTL(t)). For the raw material cost, the cost parameter (CEB) and the inlet flow rate is multiplied. Energy cost is calculated as the sum of the inlet steam cost and the cost of steam to the heat exchanger between reactors 1 and 2. Because the ethylbenzene feed flow rate F is fixed, control variables for this problem are S/O ratio (F/(FgH2O)) and inlet temperatures of reactors 1 and 2 (TINR1 and TINR2). Optimization of the operation of the dehydrogenation reactor is an optimal control problem because of catalyst deactivation, so the optimal trajectories of the three control variables need to be determined. If we were to attempt to solve this problem by piecewise constant approximation of each control variable, the computational load would be very large because the number of total decision variables becomes very large. To reduce the number of decision variables, we used parametrization of the control functions. This method has two steps. In the first step, we obtain an optimal trajectory for each control variable. When optimizing the trajectory of each control variable, the other control variables are held constant at average values between their maximum and minimum bounds. Because the optimal trajectories were found to exhibit very similar trends, we were able to
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Table 1. Optimization Problem: Maximizing the Profit of the Styrene Monomer Reactor Objective Function
Profit )
∫
tend
t)0
[FCSMgSM(t) + FCTLgTL(t) - FCEB NC
FCNTH
∑(Cp g )(T i i
INR2(t)
- TOUTR1(t)) +
i
FCH2OCP,H2OgH2O(TINR1(t) - T0)] dt Equality Constraints
TINR1 ) a11 log(t + b11) + a12t2 + a13t + a14 +
a15 t + b12
TINR2 ) a21 log(t + b21) + a22t2 + a23t + a24 +
a25 t + b22
Figure 4. Trajectories of reactor inlet temperatures: reactor 1 (TINR1) and reactor 2 (TINR2).
a35 F ) a31 log(t + b31) + a32t2 + a33t + a34 + FgH2O t + b32 dgi
1 ) (2πrL dr F
dT dt
∑f MW ), ∑g ) 1 2πrL∑(-∆H ∑f ) ij
i
i
j
)
i
j
j
ij
i
FCp
dP Pout - Pin ) dr Rout - Rin Φ ) Φ(t,F,FgH2O,FgEB,T) (neural network model) Inequality Constraints 600 e TINR2 e 650 600 e TINR1 e 650 0.5 e gH2O e 0.63 0 e gi e 1 (i ) 1, 2, ..., 9)
construct an empirical eq 5 for the control variable trajectories.
y ) a1 log(t + b1) + a2t2 + a3t + a4 +
a5 t + b2
(5)
Even though the temperature constraints are active, this empirical equation gives a good approximation within 0.2% of average relative error. In the second step of the trajectory optimization, the control variables in eq 4 are replaced with the empirical trajectory equations provided by eq 5. By determining the values of the five parameters a1-a5 in each empirical equation that maximize the profit, we obtain the optimal trajectories of the reactor inlet temperatures and the S/O ratio. The advantage of the proposed method is that it requires very small computational load to give the solution robustly for all cases. The computational load of an NLP problem increases exponentially when the number of decision variables increases. In piecewise constant approximation of the control variables for the styrene monomer reactor, the NLP problem has 135 decision variables. This optimization problem took a long time to converge and often failed to give solutions depending on the initial conditions. In parametrization of control variables, the NLP problem has 15 decision variables. Due to the reduction of the number of decision variables, the trajectory optimization method using the parametrization of control variables provides solutions robustly and reasonably fast.
Figure 5. Trajectory of the S/O ratio.
In the first step of the trajectory optimization method, we obtained the parameters a1-a5 and coefficients b1 and b2 of eq 5 for each control variable. Using these parameters as initial values, in the second step, we obtain the optimal operating conditions. The optimization problem is summarized in Table 1, and the optimized trajectories are presented in Figure 4 along with the current operating conditions of the plant. The optimization horizon of this problem is 525 days, which is given by the plant data. As shown in Figure 4, the optimized inlet temperatures exhibit a trend very similar to the current operating data but are higher. These are very reasonable results because high temperatures favor the endothermic dehydrogenation reaction. In contrast, the optimized S/O ratio trajectory in Figure 5 exhibits a trend different to that of the plant data. The optimum trajectory of the S/O ratio has a slowly increasing trend, but in the plant site, the S/O ratio is decreased as the operating time increases to reduce the operating cost. However, our results suggest that the increase in profit that results from the production rate increase at a higher S/O ratio is higher than the operating cost increase, so the S/O ratio should be increased during operation in order to counter the effects of catalyst deactivation. The maximum profit of $5.02 × 106/yr was obtained by solving the optimization problem while the calculated profit was $4.00 × 106/yr when the current operating data were used.
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Conclusion The optimization of the operating conditions of an industrial styrene monomer reactor system can significantly reduce the total operating cost. The catalyst deactivation and complex geometry of the reactor were obstacles to optimizing the reactor system. We proposed a hybrid model that couples a first principle model and a neural network model. The first principle model estimates the reactor conditions such as composition, temperature, and pressure while the neural network model predicts the catalyst deactivation factor. With this model, an optimization of the dehydrogenation operation was performed. To reduce the computational load and to obtain solutions robustly, we have proposed a trajectory optimization based on parametrization of control functions. Our simulation result shows that the calculated profit using the optimal trajectories of the operating variables is about $1million/yr greater than that using the current operation data. Acknowledgment This work was supported by the Brain Korea 21 project, LG Chem Ltd., and the Center for Ultramicrochemical Process Systems sponsored by KOSEF. All plant data and information were provided by LG Chem Ltd. Nomenclature Ci ) cost parameter of component i (i ) styrene monomer (SM), ethylbenzene (EB), toluene (TL), H2, H2O, CO2) Cp ) heat capacity of gas flow Cpi ) heat capacity of component i in the gas flow gi ) mass fraction of component i fij ) reaction rate of component i in reaction j F ) total mass flow rate of gas ∆Hi ) enthalpy of reaction j L ) axial length of catalyst bed MWi ) molecular weight of component i PTOTAL ) total pressure Pout, Pin ) outlet and inlet pressures of reactor Rout, Rin ) outer and inner radii of catalyst bed r ) radial direction length of catalyst bed t ) time T ) reactor temperature
TINR1, TINR2 ) inlet temperatures of reactors 1 and 2 in Figure 1 TOUTR1 ) outlet temperature of reactor 1 T0 ) reference temperature Φ ) catalyst deactivation factor
Literature Cited (1) Haung, W. Optimize styrene units. Hydrocarbon Process. 1983, (April), 119. (2) Lim, H. Modeling and Operation Strategy Development for a Styrene Monomer Reactor. M.S. Thesis, KAIST, Deajeon, Korea, 2000. (3) Wett, T. Monsanto/Lummus styrene process is efficient. Oil Gas J. 1981, (July 20), 76. (4) Scheel, J. G. P.; Crowe, C. M. Simulation and optimization of an existing ethylbenzene dehydrogenation reactor. The Can. J. Chem. Eng. 1969, 47, 183. (5) Clough, D. E.; Ramirez, W. F. Mathematical Modeling and Optimization of the Dehydrogenation of Ethylbenzene to Form Styrene. AIChE J. 1976, 22 (6), 1097. (6) Hirano, T. Roles of potassium in potassium-promoted iron oxide catalyst for dehydrogenation of ethylbenzene. Appl. Catal. 1986, 26, 65. (7) Sundaram, K. M.; Sardian, H.; Fernandez-Baujin, J. M.; Hildreth, J. M. Styrene plant simulation and optimization. Hydrocarbon Process. 1991, (January), 93. (8) Abdalla, B. K.; Elnashaie, S. S. E. H.; Alkhowaiter, S.; Elshishini, S. S. Intrinsic kinetics and industrial reactors modeling for the dehydrogenation of ethylbenzene to styrene on promoted iron oxide catalysts. Appl. Catal. 1994, 113, 89. (9) Savoretti, A. A.; Borio, D. O.; Bucala, V.; Porras, J. A. Nonadiabatic radial-flow reactor for styrene production. Chem. Eng. Sci. 1999, 52, 205. (10) Carra, S.; Forni, L. Kinetics of catalytic dehydrogenation of ethylbenzene to styrene. Appl. Catal. 1965, 4 (3), 281. (11) Coulter, K.; Goodman, D. W. Kinetics of the dehydrogenation of ethylbenzene to styrene over unpromoted and Kpromoted model iron oxide catalysts. Catal. Lett. 1995, 31, 1. (12) Devoldere, K. R.; Froment, G. F. Coke formation and gasification in the catalytic dehydrogenation of ethylbenzene. Ind. Eng. Chem. Res. 1999, 38, 2626. (13) Lundin, J.; Hansson, T.; Pettersson, J. B. C. Kinetics for potassium, desorption from an iron oxide catalyst studied by field reversal. Appl. Surf. Sci. 1994, 74, 343.
Received for review January 20, 2004 Revised manuscript received June 28, 2004 Accepted July 12, 2004 IE049936X
