New Suggested Model Reference Adaptive Controller for the Divided

Article Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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New Suggested Model Reference Adaptive Controller for the Divided Wall Distillation Column Eman M. El-Gendy,* Mahmoud M. Saafan, Mohamed S. Elksas, Sabry F. Saraya, and Fayez F. G. Areed Computers and Control Systems Engineering Department, Faculty of Engineering, Mansoura University, 35511, Mansoura, Egypt

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S Supporting Information *

ABSTRACT: Distillation columns have proved to be the most reliable separation method for separating chemical mixtures to their pure components. As the number of components to be separated increases, the number of columns and the energy required to run these columns also increase. This results in huge capital costs that are unaffordable due to limited energy resources. A most “compact” column called the divided wall column appeared to solve this difficulty. This column is capable of separating mixtures of three or more components to high purity products with energy less than that of the conventional multicolumn process. However, the control of these columns is more complicated due to the coupling effect and increased number of variables in these columns. Our focus in this paper is limited to the divided wall columns separating ternary mixtures only. There are three main objectives of this paper. First, the paper presents a survey study about the different control aspects of the divided wall columns, based on the type of control used: composition control, temperature control, or cascaded composition−temperature control. An up-to-date overview of most of the current literature is presented. Second, conventional and adaptive proportional−integral− derivative (PID) controllers are proposed to control a divided wall distillation column separating a ternary mixture of ethanol, propanol, and n-butanol. Fuzzy logic control, neural network control, and adaptive neuro-fuzzy inference systems are suggested for the tuning of the PID controllers. Particle swarm optimization technique is also applied to improve the results obtained by the adaptive PID controllers. Finally, a multi-input−multi-output neural network model reference adaptive controller based on adaptive PID controller tuned by adaptive neuro-fuzzy inference systems based particle swarm optimization is suggested. The results indicate the superiority of the adaptive PID controllers over conventional PID controllers, especially in the case of disturbances. can give outstanding performance at the same time.6 For further understanding of the distillation process and different types of distillation columns, the reader can refer to the excellent review by Kiss,7 which gives a comprehensive overview about the distillation process. The separation of a mixture of two components, i.e., binary distillation, is the main focus of most research papers. However, only limited research is concerned about separating more than two components. Logically, a cascade of binary distillation columns is used to separate a multicomponent mixture, which leads to massive energy consumption.8 If we consider a ternary feed mixture, i.e., feed mixtures of three pure components, as a case study, then in order to separate this feed using distillation columns, different column configurations are available: the direct sequence shown in Figure 1a, where light component is removed first; and the indirect sequence shown in Figure 1b,

1. INTRODUCTION Energy saving is the main topic of most of the current research. This is due to the huge increase in the price of the fuel used by almost all processes all over the world as a result of limited energy resources.1 Therefore, the term “process intensification” has been widely used in the literature. Process intensification is a process design procedure based on the combination of several components into single equipment. This gives a smaller, cleaner, safer, and more energy-efficient process technology.2 In spite of these great advantages, the interactions among the different parts of the same process result in serious problems in the control of this process.3 Distillationthe separation of a multicomponent liquid mixture into its primary components by the use of heatis the most commonly used chemical separation process.4 Vapor becomes richer in the lightmost volatile or low boiling pointcomponents, and the remaining liquid has more of the heavyleast volatile or high boiling pointcomponents.5 This process requires the use of a huge amount of energy for the heating and cooling processes. Hence, it is desirable to design distillation columns that can use the least amount of energy and © XXXX American Chemical Society

Received: April 1, 2019 Revised: April 11, 2019 Accepted: April 14, 2019

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DOI: 10.1021/acs.iecr.9b01747 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 1. Separation of a ternary mixture via (a) direct conventional distillation, (b) indirect conventional distillation, (c) Petlyuk configuration, and (d) divided wall column.

where heavy component is removed first.9 By using one of these configurations, at least two reboilers and two condensers must be used. This massively increases the total cost used by the process to provide the required amount of energy, which is a major problem. To overcome this problem, Petlyuk et al.10 suggested a fully thermally coupled distillation column design consisting of only two columns, where the vapor and liquid streams leaving the first column are directly connected to the second column. This configuration, as shown in Figure 1c, eliminates the cost needed for additional heating equipment by using only one reboiler and one condenser.11 The first column is called the prefractionator, and it is used to make a sharp split between the low and high boiling point components, while the middle point component is divided between the top and down products. The second column is called the main column. This column splits the top

product received from the prefractionator to a distillate product, which is rich in the low boiling point component, and a side stream product, which is rich in the middle boiling point component. The downstream received from the prefractionator is also split using the main column to a bottom product, which is rich in the high boiling point component, and a side stream product, which is rich in the middle boiling point component. The total side stream is the combination of the two streams resulting from the top and down separations. The final streams produced by the column are the distillate product, the side stream product, and the bottom product.12 These are the same three pure components generated using the conventional binary column sequence. Another column was introduced by Kaibel.13 This column integrated the Petlyuk column into a single column shell, by inserting a vertical wall in the vessel at an appropriate position.14 B


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Industrial & Engineering Chemistry Research Using this column, about 30% of total process energy can be saved as compared to the conventional schemes15 and, as a result, total investment costs can be reduced.16 This column is called the divided wall distillation column (DWC). A typical DWC is shown in Figure 1d. In spite of the advantages resulting from the use of the DWC, the design and control process of these columns is very complicated due to the use of more variables that results in more process interactions and coupling effect. To achieve the benefits of DWC, proper control structures must be applied.17 Based on data from Scopus provided by Elsevier B.V., the number of published materials about divided wall columns from 1975 to 2018 is shown in Figure 2a. From this figure, it is clear that an increasing number of research studies were made in the past few years, due to the growing interest in these types of columns. Parts b, c, and d of Figure 2 give the number of published materials about the DWC according to type, author, and subject area. However, these data are limited for such an important field of study. The main focus of this paper is to propose different adaptive PID controllers for the control of a DWC separating ternary mixtures, and to suggest a new multi-input−multi-output neural network model reference adaptive controller (MIMO NN MRAC) for the control of the DWC. MRAC has been used successfully for the control of the binary distillation columns, but almost no previous study applied this controller to the DWC. The reason for the use of the PID controller with adaptive parameters is that it is the most commonly used controller all over the world in the process industry. This is useful for any future practical application of this study. To indicate the novelty of our proposed controller, we discuss the recent control strategies applied to the control of the DWC. Different controllers that range from classical PID controller to the recent adaptive controllers were applied by different researchers. Also, different mixtures and different column configurations were used. There are different available reviews1,4,6,18,19 that discuss the history and operation of DWC, but the difference between these reviews and the critical survey provided by our research is the classification of the control strategies based on composition control, temperature control, or cascade composition−temperature control. This paper is organized in 11 sections. Section 1 is the introduction and review of the motivation toward the use of the DWC. Section 2 presents a controllability study with a focus on degree of freedom analysis. Section 3 presents a historical review about composition control, different temperature control methods, and cascade composition/temperature control of the DWC and general guidelines for the design and control of the DWC. The application of the DWC to reactive, extractive, and azeotropic distillations is briefed in section 4. The mathematical model of the divided wall distillation column is presented in section 5. The conventional PID controller is discussed in section 6, and the motivations toward the use of the adaptive PID controller are presented in section 7. Section 8 discusses the model reference adaptive controller. Sections 9 and 10 present the simulation and results for different suggested controllers. A detailed conclusion is given in section 11.

Figure 2. Number of published materials about divided wall columns: (a) from 1975 to 2018, (b) according to type, (c) according to author, and (d) according to subject area.

2. CONTROLLABILITY STUDY The motivation toward the use of the DWC is to produce high purity products from a ternary mixture with less energy than the conventional binary column sequence. To successfully achieve this purpose, good control design that can overcome

disturbances must be applied. Basically, there are two different control methods: composition control and temperature control. C


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Industrial & Engineering Chemistry Research Table 1. Summary of Previous References Dealing with the Control of the DWC no. of stages date

ternary mixture

pref.

main

CC or TCa

controller

software

ref

1995 1998 1999 2000 2001 2007 2009 2010 2010 2011 2011 2011 2012 2012 2013 2013 2013 2013 2013 2013 2013 2014 2015 2015 2016 2017 2018 2018

ethanol/propanol/butanol methanol/isopropanol/butanol hypothetical components hypothetical components hypothetical components ethanol/n-propanol/n-butanol benzene/toluene/xylene benzene/toluene/xylene benzene/toluene/xylene benzene/toluene/xylene n-hexanol/n-octanol/n-decanol methanol/ethanol/1-propanol n-pentane/n-hexane/n-heptane benzene/toluene/xylene pentane/hexane/heptane benzene/toluene/xylene benzene/toluene/ethylbenzeneb ethanol/n-propanol/n-butanolb hypothetical componentsc ethanol/propanol/butanolc hypothetical components ethanol/propanol/n-butanol benzene/toluene/o-xylene benzene/toluene/o-xylene ethanol/n-propanol/n-butanol benzene/toluene/o-xylene ethanol/n-propanol/n-butanol ethanol/n-propanol/n-butanol

20 14 13 13 13 65 24 24 16 16 10 36 12 24 17 24

40 32 33 33 33 116 46 46 32 32 20 62 29 46 36 46

80 48 44 44 53 46 53 60

PID PID PI, DMC PI, DMC PI, DMC PI PID PID PID, LQG, GMC, LSDP, μ-synthesis PID MPC PID PI, MPC PI tuned by pole placement PID, fuzzy-PID PID, fuzzy-PID PI PI PI PI PI PI MPC ANNPC PID PI PI MPC, PI

Aspen Aspen Matlab Matlab Matlab ChemCad Aspen Aspen Matlab/Simulink Aspen LabView, Aspen, Matlab HYSYS Aspen Aspen Aspen Aspen Aspen Aspen Aspen Aspen Matlab Matlab Matlab Matlab Aspen Aspen Aspen Matlab, Aspen

25 26 27

40 24 24 24 15 24 15 37

CC CC CC CC CC TC, cascade CC TC, TDC CC CC TC TC CC CC CC CC CC CC TC, STDC, TDC TC, STDC, TDC CC TC TC TC TC CC, DTDC, ATC TC, cascade CC, TC

29 46 31 39 24 32 41 40 34 33 35 35 36 36 42 42 37 68 43 20 44 45 47 38

a CC, composition control; TC, temperature control; TDC, temperature difference control; DTDC, double temperature difference control; cascade, cascaded composition + temperature control; STDC, simplified temperature difference control; ATC, asymmetrical temperature control. b Different configurations for benzene/toluene/ethylbenzene and ethanol/n-propanol/n-butanol. cDifferent configurations for hypothetical components and ethanol/propanol/butanol.

adjusted during operation for control purposes.20 However, different papers studied theoretically the effect of changing RV on the operation of the DWC. Six degrees of freedom are available to control the purities of the three products, i.e., regulatory control, and also the level of the tanks of the reboiler (bottom base) and condenser (reflux drum), i.e., inventory control. One remaining DOF is used for the energy minimization requirement. Different configurations from the DOFs can be used. van Diggelen et al.24 gave important guidelines for selecting the appropriate control structure using a nonlinear model. They used a ternary mixture of benzene/toluene/xylene. From the four control strategies that are based on PID loops (DB/lSV, DV/lBS, lB/DSV, and lV/DSB), the results of the dynamic simulations showed that DB/lSV and lB/DSV are the best control structures among the decentralized controllers as they can handle disturbances in a relatively short time. However, most of the current literature uses D to control the level of the reflux drum and B to control the level of the bottom base. One of the most important studies performed so far is the study by Wolff and Skogestad in 1995.25 They considered the separation of a mixture of ethanol/propanol/butanol using a Petlyuk column. First, they used a three-point control structure which means that they used l to control the purity of the distillate product, S to control the purity of the side stream product, and V to control the purity of the bottom product. Using this scheme,

In the composition control, the compositions of the three products are the controlled variables. This method has been widely used in the literature, but it needs expensive composition analyzers due to the time delay in the process.20,21 In the temperature control, specific trays are selected as the controlled variables such that controlling the temperature of these trays can maintain product purity. These sensitive trays are chosen according to their sensitivity to the manipulated variables. Different researchers used both composition and temperature controlling methods, either separately or combined. Authors also improved temperature control methods and used the temperatures of two or more trays for each manipulated variable. These papers will be discussed in section 3. To design a suitable controller for the DWC, all its variables must be studied carefully. The DWC has seven degrees of freedom (DOFs) or manipulated variables. These DOFs are the distillate flow rate D, the side stream flow rate S, the bottom flow rate B, the liquid reflux l, the vapor boil-up rate V, the liquid split ratio RL, and the vapor split ratio RV. The last two variables appeared because of the removal of the second reboiler and the second condenser.22 RL is the ratio of the liquid stream entering the prefractionator to the total liquid stream, and RV is the ratio of the vapor stream entering the prefractionator to the total vapor stream.23 However, RV is based on the cross-sectional area of each side of the bottom of the DWC, which is fixed by the location of the wall at the design stage. Hence, RV cannot be D


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suffered from longer response time, higher overshoots, and very slow convergence. Their results proved that DMC is worse than PI controller for the control of the DWC. Although the authors stated that the performance depends on the design of the DWC, they did not prove their point of view. In ref 30, a conventional PI controller and a PI controller with dynamic estimation of uncertainties were applied to three different thermally coupled distillation columns: the sequence with a side rectifier, the sequence with a side stripper, and the fully thermally coupled system (Petlyuk column). Three different mixtures were used as different case studies, and the goal of the control was to minimize the IAE criterion. The results indicate that the response of the PI controller with dynamic estimation of uncertainties was significantly better than the response obtained by the conventional PI controller, especially when the column was subjected to load disturbances. The paper by Ling and Luyben 31 implemented the composition control using the four-point control structure suggested by ref 25 to separate the ternary mixture of benzene/ toluene/xylene (BTX). They used four manipulated variables, i.e., RL, L, S, and V, with fixed RV by the physical location of the wall at the design stage. Conventional PID controllers were used for all the control loops, and disturbances of ±20% in both feed flow rate and feed compositions were applied. Improved performance was obtained from this control structure as compared to the two-conventional column sequence, but the most important result from this research is that they could prove the relation between controlling the level of the heavy impurity in the top of the prefractionator and the energy savings. In 2011, Kiss and Rewagad32 used RL as a manipulated variable to control the heavy component composition in the top of the prefractionator for energy saving purpose. They studied the response of different control structures (DB/LSV, DV/LSB, LB/DSV, LV/DSB) using conventional PID controllers, and they used the same ternary mixture used by ref 31 for the purpose of comparison. Disturbances of +10% in feed flow rate and in feed composition of the first component (xA) were applied to the four structures. Best results were obtained from DB/LSV and LB/DSV configurations in terms of short settling time and low overshoot. Their recommendations were to use DB for inventory control and LSV for regulatory control. They also found that a choice had to be made between minimum energy requirements for operation of the DWC and maximum purity of products. The tuning of the PI controller was the main topic of ref 33. Pole placement framework was used for the tuning of parameters of the PI controllers for DWC so that they can tune all the decentralized controllers simultaneously. The same model of ref 31 was used to allow fair comparison of results. Their results showed that the performance of the PI controller tuned by their method was the same as the one obtained by sequential methods, but their method is easier and straightforward. The control of the DWC is not limited to conventional PID controller only. For example, Rodriguez-Hernández et al.34 compared the results of a decentralized PI controller with model predictive controller (MPC) to control a DWC separating a ternary mixture of n-pentane, n-hexane, and n-heptane. Disturbances of ±10% in feed flow rate, feed temperature, and feed mole fraction of each component were applied. The results from both controllers indicated that disturbance in composition of n-pentane gave the largest loop interaction and longest settling time while disturbances in the composition of n-heptane gave lesser effect. Disturbances in the composition of n-hexane

the system could overcome different disturbances except for some disturbances at set point change. Therefore, the authors introduced a four-point control structure using the remaining DOF, i.e., RL, which was used to control the levels of impurities of the distillate and bottom product in the side stream. Unfortunately, the steady state feasibility space of their results showed regions with “holes”. They made another trial by using S to control the purity of the bottom product and V to control the purity of the side stream product, but this trial was also unworkable. However, a trial by Mutalib and Smith26 investigating the operation of a divided wall column for separating a ternary mixture of methanol/isopropanol/butanol using the pairing suggested by ref 25 proved by simulations that the use of S to control the purity of the bottom product and V to control the purity of the side stream product could work well. In 1999, Halvorsen and Skogestad22 focused their research on the feedback control of DWCs to overcome different possible disturbances in the feed, which are the feed flow rate, the composition of the first component in the feed, the composition of the second component in the feed, and the feed liquid fraction. They studied a hypothetical constant relative volatility system. Their focus was on the prefractionator, which should keep the heaviest component from going to the rectifying section and the lightest component from going to the stripping section. They found that it is more effective to control the level of the heavy impurity in the top of the prefractionator than the level of the light impurity in the bottom of the prefractionator, as the first affects the liquid side stream while the second only affects the vapor side stream. Therefore, RL was used as a DOF to accomplish this task.

3. DESIGN AND CONTROL OF DWC This section presents a historical review of most open literature and motivation toward our current work. 3.1. Control Methods for the DWC. As mentioned earlier, different controllers were applied to different DWC configurations in the open literature. In this section, a detailed analysis of the most important papers is presented. Based on the available results, DWC shows good controllability, short settling time, and low overshoot. Table 1 is a summary of previous references dealing with the control of the DWC. 3.1.1. Composition Control. The use of the compositions of the three products as the controlled variables ensures that using suitable and robust controllersthe products return to their reference values, and hence, the desired purity can be obtained. This advantage, however, comes with its price. The use of composition control is difficult and requires expensive composition analyzers, which increases the total cost of the process. In the series of papers by Serra et al.,27−29 they extended the study of feedback diagonal control strategies to consider the DWC control. A nonlinear model of the system was used to simulate the behavior of the DWC using PI controllers. A hypothetical system with relative volatilities (4.65, 2.15, 1) was used for their study. All possible control structures of a DWC were studied to find the best control pairing in a three-point control structure. Their results indicated that a trade-off between optimization and controllability had to be made in the case of DB inventory control. Their simulations showed that using LV for inventory control and DSB for composition control were good control structure for the DWC. They also applied dynamic matrix control (DMC) to the DWC and compared the results with PI controller. DMC had smaller deviations, but it E


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controlled variable. This is because the distillation process has huge time delays, and special expensive composition analyzers are needed to measure the compositions of the products in order to use them as controlled variables. These trays are selected based on their sensitivity to each of the manipulated variables. Several methods such as singular value decomposition (SVD)24,39 and relative gain array (RGA)26,38,46 are used to select these sensitive trays. Both of these methods apply a small step change to each of the manipulated variables to see which trays have been affected mostly. The same authors of ref 31 used temperature control with the same column configurations in ref 39. Three temperatures in the main column to control the purity of the three products and one temperature on the prefractionator to control the composition of the heaviest component at the top of the prefractionator in order to minimize energy consumption were used as the controlled variables, and PID controllers were applied. The structure responded well to ±10% disturbances in the feed flow rate and feed compositions. However, for feed composition disturbances above 10%, product purities started to deviate significantly. Therefore, they used four temperature difference control (TDC) loops to improve the performance. This structure was able handle the disturbances of 20% in feed composition with only small deviations in product purities. This was not the case for Ignat and Woinaroschy40 as they applied disturbances of ±10% only in the feed flow rate and feed compositions in a DWC separating a ternary mixture of methanol/ethanol/1-propanol. Four-point PID temperature control structure was applied, and their results indicated that the proposed control structure could handle these disturbances well. The study of the DWC is not limited to simulation study. For example, Buck et al.41 designed an MPC for controlling a divided wall column separating a mixture of n-hexanol/n-octanol/ndecanol using four-point temperature control. Performance of the MPC was investigated through experimental investigations. The performance of the controller was acceptable, especially in the case of industrially relevant disturbances. In ref 42, a simplified TDC (STDC) was suggested instead of the normal TDC using four temperature differences. Two temperature control loops were used to maintain the top and bottom product qualities, and two temperature difference control loops were used to keep a certain degree of separation in the prefractionator and to maintain the purity of the intermediate product. Three ideal ternary mixtures of hypothetical components were used first, and then their control structure was applied to separate ethanol/propanol/butanol mixture and PI controllers were used. The suggested STDC could reduce the cost of instrumentation more than the TDC scheme, and its performance was better than the performance of the temperature control structure and could be comparable with the TDC scheme. Apart from the conventional decentralized controllers, Dohare et al.43 developed a mathematical model for the DWC and used MPC to control that DWC to separate a mixture of benzene/toluene/o-xylene. They used temperature control to control the compositions of the three products and applied disturbances of ±10% in the feed flow rate and the feed compositions. Their results showed that the performance of the MPC is much better than the performance of the PI controller as compared with ref 39. Dohare et al.20 also used artificial neural network predictive control (ANNPC) to control the same DWC in ref 43, and the same disturbances were also applied. Different

barely affected the stability of DWC. However, the results from the MPC had less loop interaction and less settling times despite having some oscillation, so the MPC proved its superiority in eliminating loop coupling compared to decentralized PI controller. Tututi-Avila and Jimenez-Gutiérrez35 applied a fuzzy logic controller (FLC-PID) to control two DWCs separating pentane/hexane/heptane and benzene/toluene/o-xylene. To build the fuzzy controller and tune its scaling factors correctly, they first developed a conventional PID controller tuned by minimizing integral absolute error (IAE) criterion. Then, the resulted PID controller was replaced with a linear fuzzy logic controller. This fuzzy controller was then made nonlinear by inserting a nonlinear rule base. The final step was to optimize and tune the scaling factors. Disturbances of +10% in both feed flow rate and feed composition of the first component (xA) were applied, and the results of the fuzzy controller were compared with the results from a conventional PID controller. The responses showed that FLC was more effective in controlling the DWC and in handling the disturbance. The suitable position of the dividing wall in a DWC was the main purpose of ref 36. Three column configurations were studied: columns with the dividing wall in the lower portion of the column, columns with the dividing wall in the middle portion of the column, and columns with the dividing wall in the upper portion of the column. Two ternary mixtures were studied: benzene/toluene/ethylbenzene and ethanol/n-propanol/n-butanol. Different sets of disturbances were applied to these two systems in the three specified column configurations, and results were obtained in terms of IAE and integral time absolute error (ITAE). Their results indicated that DWC with the dividing wall in the upper portion of the column had the worst control performance of the three column types. DWC with the dividing wall in the lower portion of the column outperformed DWC with the dividing wall in the middle portion of the column in most cases, but they recommended considering the cost of the column when choosing the suitable configuration. The study by Dwivedi et al.37 to control a DWC separating a hypothetical mixture with relative volatilities close to those of ethanol, propanol, and n-butanol (4.2:2.1:1) compared four control structures, with and without RV as a DOF. D and B were used for level control, while l and V were used for composition control. Disturbances of ±20% in feed flow rate and feed compositions were applied. Two of these structureswith and without RVgave good performances, with one of the products being overpurified but with little increase in energy usage. The authors suggested the use of linear or nonlinear MPC instead of decentralized PI control for future work. Our last case study is one of the most recent papers by Wang et al.38 They applied MPC to the control of the DWC for separating ethanol/n-propanol/n-butanol ternary mixture using both composition control and temperature inferential control. Genetic algorithm (GA) was used to evaluate the parameters of the MPC. They compared the performance of this controller to its equivalent PI controller. RL was used as a manipulated variable to control the composition of the heaviest component at the top of the prefractionator, and disturbances of ±10% in feed flow rate and feed composition were applied. The results showed that MPC performed better than PI controller in maintaining product purities as it had lower values of the integral square error (ISE). 3.1.2. Temperature Control. Most of the recent research proved that temperatures of specific trays can be used as the F


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Figure 3. Application of DWC to (a) reactive DWC, (b) extractive DWC, and (c) azeotropic DWC.

compositions did not return to their corresponding desired values. They concluded that there is a trade-off between energy efficiency and controllability, and suggested using a temperature + composition cascade control scheme. The results from this cascade control showed that it could handle disturbances in all feed conditions. However, this was not the case for Jia et al.,47 who proposed four control structures to separate a ternary mixture of ethanol, n-propanol, and n-butanol using PI controllers. The first control structure was a temperature control structure, and each of the three remaining structures added a composition controller on top of the previous structure. Disturbances of +20% in feed flow rate and feed compositions were applied, and surprisingly, the results of the simulations suggesteddespite having some small steady-state deviations in the product compositionsthe use of temperature control only. 3.2. Comments on Previous Trials. From the previously discussed work, the following points are observed: 1. Most of the papers used a conventional PI or PID controller. Only a few papers used the adaptive control methods, and most of them used MPC. 2. A benzene/toluene/o-xylene mixture was more commonly used in most of the papers. 3. Disturbances were in the range of ±10 or ±20% in feed flow rate and feed composition. To solve these problems, our current work does the following: 1. It uses adaptive control methods for tuning the PID controller because PID controllers are the most commonly used controllers. 2. It uses a ternary mixture of ethanol/propanol/n-butanol to prove that the use of the DWC is not limited to the benzene/ toluene/o-xylene mixture. 3. It applies a higher disturbance in the feed flow rate and the feed compositions with only small deviations from the specified products’ purities. 3.3. General Guidelines for the Design and Control of the DWC. Kim et al.48 provided general guidelines, rules, and standards for the design and selection of the control structures for the DWC. These guidelines are as follows: • The feed composition has no effect in the selection of the manipulated variable candidates.

performance criteria parameters such as IAE, ITAE, ISE, integral time square error (ITSE), rise time, and settling time were used to compare the performance of their controller to ref 39. It was observed that the performance parameters were less for ANNPC as compared to PID controller. The paper by Qian et al.44 proved that three temperature controllers are sufficient to control the three-product DWC. They used a mixture of ethanol/n-propanol/n-butanol and applied three different control structures using PID controller: CS1 is a control structure with fixed split ratios, CS2 is a control structure with an active RL, and CS3 is a control structure with an active RV. Disturbances of ±20% feed flow rate and feed compositions were applied, and the results proved that the three control structures could handle these disturbances. For disturbances in the vapor split ratio, CS2 outperformed CS1. The performance of CS3 was not very different from CS2, except at ±20% of xA where the deviation of the side product was high. Another technique of temperature control called the asymmetrical temperature control (ATC) was suggested Yuan et al.45 They compared the performance of the proposed ATC to the composition control and to the double temperature difference control (DTDC). PI controllers were used in a DWC separating benzene/toluene/o-xylene, and disturbances of ±5 and ±10% in feed compositions were applied. Different ATC schemes were suggested, and their results showed improved steady-state responses at the expense of great deviations in the intermediate product purity. 3.1.3. Cascade Control. The combination of both composition control and temperature control was studied in some cases. For example, Wang and Wong46 studied the effects of RL and RV on the energy efficiency and controllability of a DWC separating a ternary mixture of ethanol/n-propanol/nbutanol. They proved that, for different RL and RV values, different pairing schemes were required. Three PI controllers were used to control the temperature of three sensitive trays, and disturbances of ±10% in the feed flow rate were applied. The controlled stage temperatures settled at their corresponding set points, and three product compositions returned to their desired operating values. However, this was not the same for feed composition changes. The three controlled temperatures settled at their corresponding set points, but the responses of product G


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Industrial & Engineering Chemistry Research • Use the reflux flow rate for the control of the purity of the distillate product. • Use the side stream flow rate for the control of the purity of the side product. • Use the boilup for the control of the purity of the bottom product. Additional guidelines4 can also be used: • Use the liquid split ratio to control the amount of heavy component leaving the top of the prefractionator to save energy. • Use temperature control methods rather than complex composition control. • Use cascade composition/temperature control loops to improve disturbance rejection. • Use conventional feedback control combined with feedforward control loops. • Apply MIMO control strategies, when SISO loops are not sufficient.

The DWC has six sections as shown in Figure 4, where C11 and C12 are the prefractionator sections and C21, C22, C23, and C24 are the rectifying, main column, and stripping sections of the DWC.

4. APPLICATIONS OF THE DWC The use of the DWC is not limited to ternary separations only; the DWCs can be combined with reactive distillation to get reactive DWC (R-DWC), with extractive distillation to get extractive DWC (E-DWC), and also with azeotropic distillation to get azeotropic DWC (A-DWC). 4.1. Reactive DWC. R-DWC is a highly integrated combination of a reactor and a DWC in one unit.48 This integration yields energy and cost savings. There are different published works about this type of column.49−56 The experimental application of the R-DWCs is very limited. However, further studies of the R-DWC are expected. The RDWC is shown in Figure 3a. 4.2. Extractive DWC. Extractive distillation is used to separate azeotropic or narrow boiling point mixtures. An additional substance called a solvent with a boiling point much higher than that of the two substances forming the mixture is added. This reduces the relative volatility of the heavy boiling component. By integrating both the extractive distillation and the DWC in a single vessel, the resulting configuration is called the extractive DWC (E-DWC). In this column, the light boiling point component is obtained in the left part of the column, while the high boiling point component and the solvent are separated in the right part of the column. The solvent is fed back into the left part.57−64 The E-DWC is shown in Figure 3b. 4.3. Azeotropic DWC. Azeotropic distillation can also be used to separate similar boiling point components. An additional component called the entrainer that forms an azeotrope with the components to be separated is added. By combining azeotropic distillation with the DWC, the resulting concept is the azeotropic DWC (A-DWC). Only a limited number of papers such as refs 57 and 65−67 have been published. The A-DWC is shown in Figure 3c.

Figure 4. Schematic diagram of the studied DWC.

The mass balance equation is

5. MATHEMATICAL MODEL OF DWC The model used in this paper is the same model used by ref 68, which is the same model used by ref 37 but with a lesser number of trays. Therefore, we use here the same assumptions used by ref 37 for simplification, which are (a) equilibrium on each stage (b) no vapor flow dynamics (c) linear liquid dynamics (d) total condenser

dM i = Li + 1 + Vi − 1 − Li − Vi dt

(1)

The mole fraction balance x of each component j ∈ {1, 2, 3} at each stage i is given by Mi

dxi , j dt

= Li + 1(xi + 1, j − xi , j) + Vi − 1(yi − 1, j − xi , j) − Vi (yi , j − xi , j)

(2)

The energy balance at each stage i is given by H


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Industrial & Engineering Chemistry Research d(HL, iMi)

Li = Li + 1 ,

= Li + 1(HL, i + 1 − HL, i) + Vi − 1(HV, i − 1 − HL, i)

dt

− Vi (HV, i − HL, i)

(3)

C11: Li , j = lRL

yi , j = K i , jxi , j

(18)

(4)

C12:

where

Li , j = lRL + qF

γi , jPisat ,j

Ki,j =

Pi

∑

∑

xi , j = 1,

j = 1,2,3

Li , j = l

Li , j = l(1 − RL)

(6)

Using the Antoine equation, the temperature on a stage i is calculated by B Ti = −C A − log Pi

Li , j = l(1 − RL) − S

Li , j = l + qF − S

Vi , j = VR V + (1 − q)F

d(HL,R MR ) dt

Vi , j = VR V

(9)

= L6HL,R − VHV,R − 1 − BHL,R − 1 + Q r

Vi , j = VR V + (1 − q)F + V (1 − R V ) (10)

(11)

j = 1,2,3

∑

Vi , j = V (1 − R V )

j = 1,2,3

dMD =l−V−B dt

Vi , j = V

(13)

Li = Loi +

Mi − Moi + (Vi − 1 − Voi − 1)λ τL

(30)

5.3. At the Intersection of C21 with C11 and C22. Vapor mixing is given by

(14)

The energy balance equation is = VHL,D − VHV,D − 1 − DHL,D − 1 + Q c

(29)

Linearized liquid flow dynamics means that the liquid flow dynamics are given by the following equation:

The mole fraction balance x of each component j ∈ {1, 2, 3} at the condenser is given by = lxD , j − VyD − 1, j − Dx D − 1, j

(28)

C24:

(12)

5.2. For the Condenser. The mass balance equation is

d(HL,DMD)

(27)

C23:

The summation equations are yR, j = 1

(26)

C22: Vi , j = V (1 − R V )

yR, j = KR, jx R, j

x R, j = 1,

(25)

C21:

The equilibrium is given by

dt

(24)

C12: = L6x R, j − VyR − 1, j − Bx R − 1, j

The energy balance equation is

dt

(23)

The equations for the vapor flow rates are C11:

(8)

The mole fraction balance x of each component j ∈ {1, 2, 3} at the reboiler is given by

dx D, j

(22)

C24:

(7)

dMR = L6 − V − B dt

∑

(21)

C23:

5.1. For the Reboiler. The mass balance equation is

dt

(20)

C22:

yi , j = 1

j = 1,2,3

dx R, j

(19)

C21:

(5)

The summation equations at the ith stage are

MD

(17)

The equations for the liquid flow rates in each section of the DWC are

The equilibrium is given by

MR

Vi = Vi − 1

(15)

C22 V nC21 = V nC11 −1 + Vn−1

(31)

C11 C22 C22 V nC21ynC21 = V nC11 − 1yn − 1, j + V n − 1yn − 1, j ,j

(32)

Liquid splitting is given by

The summation equation is

C21 LnC11 + 1 = R lLn

(33)

(16)

C21 LnC22 + 1 = (1 − R L)Ln

(34)

For every section of the column, the liquid and vapor flow rate on consecutive stages are given by

C22 C21 xnC11 + 1, j = xn + 1, j = xn , j

(35)

∑ j = 1,2,3

x D, j = 1

I


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Figure 5. Vmin diagram for feed composition with relative volatilities [4.2 2.1 1].

5.4. At the Intersection of C24 with C12 and C23. Vapor splitting is given by V mC12 = R VV mC24 −1

(36)

V mC23 = (1 − R V )V mC24 −1

(37)

ymC12 = ymC23 = ymC24 ,j ,j − 1, j

(38)

8. MODEL REFERENCE ADAPTIVE CONTROLLER The idea behind the model reference adaptive controller (MRAC) is to build a reference model of the system to be controlled.74,75 This model must capture all the dynamics of the system. Two types of error appear: (1) error between the output of the MRAC and the reference output of the system (2) error between the output of the MRAC and the actual output of the system The first error is used for the identification of the system and is tuned first to obtain a robust model of the system. The best identifier of the system is the neural network (NN) due to its ability to capture complex relations effectively. When the model is well-trained and tested, the second error is used to train the controllerwhich could be either conventional or adaptiveto control the actual system using the signal generated from the MRAC. Although different applications used MRAC successfully, the use of MRAC to control the DWC is limited. This was our motivation to test the effect of this controller on the DWC.

Liquid mixing is given by C12 LmC24 + LmC23 + 1 = Lm

(39)

C24 C12 C12 C23 C23 LmC24 + 1xm + 1, j = Lm xm , j + Lm xm , j

(40)

6. CONVENTIONAL PROPORTIONAL−INTEGRAL−DERIVATIVE (PID) CONTROLLER The PID controller is a simple controller with only three tunable parameters; thus it is the most commonly used controller in 90% of industrial applications.69 The control signal is given by the equation70 u(t ) = K pe(t ) + K i

∫0

t

e(t ) dt + Kd

de(t ) dt

9. SUGGESTED CONTROLLER TECHNIQUES For the steady-state design,37,68 the feed is a liquid mixture consisting of three equimolar components: ethanol, propanol, and n-butanol. The Vmin diagram shown in Figure 5 is used to calculate both the minimum energy required for operating the column, and the initial input variables to simulate the nonlinear column model. The vertical axis of this diagram shows the normalized minimum boilup (V/F), while the horizontal axis shows the net product withdrawal (D/F) in a conventional twoproduct column. It is assumed that the column is operating at the fixed maximum boilup Vmax which is 30% higher than the minimum boilup Vmin. The factors affecting the value of Vmin are the feed flow rate, feed composition, feed liquid composition, and relative volatilities of feed components. The minimum energy requirement to separate a multicomponent feed in a DWC is equal to the most difficult binary separation; this means

(41)

As seen from eq 41, the parameters of the PID controller are the proportional gain Kp, the integral gain Ki, and the derivative gain Kd. However, it is necessary to find good values for these parameters in order to get good process results.71 The Ziegler− Nichols (ZN) method72 is a well-known tuning method for the conventional PID controllers. However, this method suffers from high overshoot,70 and constant PID parameters regardless of the disturbance applied to the system. Therefore, adaptive PID controller must be used.

7. ADAPTIVE PID CONTROLLER In processes with characteristics that change dynamically, conventional PID controllers do not give acceptable performance. The automatic tuning of the PID gains using intelligent techniques is necessary for such systems in order to obtain a satisfactory performance.73 These controllers include the fuzzy logic controller, neural networks, and adaptive neuro-fuzzy inference system. The main advantage of using these controllers is that the parameters of the PID controller change online based on the conditions of the system.

Vmin = max(VAB , VBC)

(42)

Since the peak of P(BC) is the highest peak, B/C separation is the most difficult binary split in terms of energy usage, so there was excess energy in one of the sections C21 and C22 or C23 and C24 of the main column. In this paper, temperature control of the sensitive trays in the DWC is used to ensure that the purities of the three products maintains in case of disturbances. The temperatures of the trays J


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Figure 6. Response after +30% disturbance in F: (a) ethanol Composition in distillate product; (b) propanol composition in side stream product; (c) n-butanol composition in bottom product; (d) temperature of tray 10; (e) temperature of tray 38; (f) temperature of tray 59.

are the controlled variables, and the manipulated variables (degrees of freedom) are βl, l, and S. Temperature T10 is used to control βl, T38 is used to control l, and T59 is used to control S.68 As a result, there are three PID controllers in the system to control the purities of the three products. These products are ethanol as distillate product, propanol as side stream product, and n-butanol as bottom product. The details of the DWC used in this paper are the same as in ref 68. In this work, the DWC is controlled using conventional PID controller, fuzzy gain scheduling PID controller, BP neural PID tuner, ANFIS PID controller, and NN MRAC ANFIS PID controller. To improve the response, PSO is applied to find the optimal parameters of the PID controller directly, to optimize the scaling factors of the FGS PID controller, to optimize the weights for the ANN used for the tuning of the PID controller, to find the optimal parameters of the ANFIS controller that is used for the tuning of the PID controller, to tune the PID controller used in NN MRAC, and to find the optimal parameters of the ANFIS controller that is used in the NN MRAC. Different load conditions are applied to compare the performance of the suggested controllers. PI controller is also included for

comparison with ref 68. All simulations are done in Matlab/ Simulink. 9.1. Conventional PID Controller. Three classical PID controllers tuned using the conventional ZN method were used to control the temperatures at stages 10, 38, and 59. 9.2. PID Controller Parameters Optimized by PSO. The particle swarm optimization (PSO) technique is a technique for the optimization of continuous nonlinear functions, and it is one of the modern metaheuristic algorithms and one of the evolutionary computation techniques.76 At each iteration of the algorithm, a particle updates its velocity and position based on self-cognitive “local best” and social experience “global best”.77 The equations of the PSO are vki + 1 = ωki + 1vki + c1r1,i0k(pki − xki) + c 2r2,i k(pkg − xki)

(43)

xki + 1 = xki + vki + 1

(44)

In this type of controller, the parameters of the PID controller are determined using the PSO optimization algorithm for better response. Constriction factor78 is used to find the inertia factor and the social and cognitive factors. K


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Figure 7. Response after +13% disturbance in zA: (a) ethanol composition in distillate product; (b) propanol composition in side stream product; (c) n-butanol composition in bottom product; (d) temperature of tray 10; (e) temperature of tray 38; (f) temperature of tray 59.

9.3. Fuzzy Gain Scheduling PID Controller. Fuzzy logic control (FLC) is a knowledge based control strategy that is closer to human thinking and reasoning, so it is a good choice to deal with uncertainty.79 The fuzzy controller can be used for online estimation of the PID controller parameters based on the error signal and its time derivative,80 and the control signal is generated by the conventional PID controller.81 This type of control is called fuzzy gain scheduling (FGS). The membership functions and the rules used here are the same as in ref 82. 9.4. PSO Scaling Factor Tuning for Fuzzy Gain Scheduling PID Controller. In this type of control, the PSO is used for online tuning of the scaling factors of the fuzzy controller that is used to calculate the parameters of the PID controller. 9.5. PID Controller Tuned by BP Neural Network. In FGS, choosing the membership functions and the rules is a complicated and time-wasting task;83 thus another approach using the artificial neural network (ANN) is suggested for tuning PID parameters. ANN is built to mimic the way in which the human brain executes different missions without the need for complex mathematical models.84

The feedforward neural network in this paper is used as an online PID tuner to find the values of the PID controller parameters for better performance. The network consists of an input layer with four inputs of reference temperature, actual temperature, error signal, and derivative of error signal; an output layer with three neurons to represent the three parameters of the PID controller, Kp, Ki, and Kd; and one hidden layer with six neurons. Each of these neural networks is trained using the back-propagation (BP) algorithm. 9.6. PID Controller Tuned by PSO Neural Network. In this type of controller, the PSO algorithm is used to train the neural network that is used for online estimation of the parameters of the PID controller. 9.7. PID Controller Tuned by ANFIS Controller. The adaptive neuro-fuzzy inference system (ANFIS) is a multiinput−single-output (MISO) that combines both fuzzy logic and neural network, so it has the advantages of both fuzzy and neural network identification.85,86 In this paper, ANFIS is used for online PID tuning. The ANFIS model structure is a twoinput (error signal and its derivative) single-output (PID parameter Kp or Ki or Kd) feedforward structure having three hidden layers with six triangular membership functions. Since L


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Figure 8. Response after +30% disturbance in zB: (a) ethanol composition in distillate product; (b) propanol composition in side stream product; (c) n-butanol composition in bottom product; (d) temperature of tray 10; (e) temperature of tray 38; (f) temperature of tray 59.

9.11. NN MRAC ANFIS Based PSO PID Controller. In this type of control, PSO is used to train the parameters of the ANFIS controllers that are used for online calculation of the PID parameters used to control the DWC using NN MRAC.

ANFIS has only one output, there are three ANFIS controllers for each of the PID parameters. The training method for ANFIS is Hybrid. 9.8. PID Controller Tuned by PSO ANFIS Controller. In this type of controller, the PSO algorithm is used to train the ANFIS controllers that are used for online estimating the parameters of the PID controller. 9.9. NN MRAC PSO PID Controller. The NN plant is made by feedforward NN having one input layer of three neurons representing the nominal values of the three manipulated variables, one output layer of three neurons representing the three controlled temperatures, and one hidden layer having 20 neurons. NN is trained using the back-propagation algorithm. Three PID controllers are used to compare the output of the reference model with the output of the DWC. These controllers are trained using PSO with parameters. 9.10. NN MRAC ANFIS PID Controller. For each of the variables βl, l, and S, three ANFIS controllers are used to online estimate the parameters of the PID controller for controlling the temperature of the three trays of the DWC. Each PID parameter needs one ANFIS controller since ANFIS is MISO. The training method for ANFIS is Hybrid.

10. RESULTS AND DISCUSSION A comparison of all the previously discussed controllers is made in order to investigate their performance under different disturbances. Different performance indexes, i.e., the integral square error (ISE), the integral time square error (ITSE), the integral absolute error (IAE), and the integral time absolute error (ITAE), are computed to evaluate the performance of every controller, and their results are plotted in Figure 9. The most important disturbances for analyzing the control performance are feed disturbances, either in feed composition or in feed flow rate. Therefore, a disturbance of +30% in the feed flow rate, a disturbance of +13% in the composition of component A (ethanol), and a disturbance of +30% in the composition of component B (propanol) are applied to the DWC. The simulation was run for 100 min and disturbances were applied after 20 min. M


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Figure 9. Performance of different controllers against disturbances based on (a) ISE, (b) ITSE, (c) IAE, and (d) ITAE.

disturbance of +30% in feed flow rate. As seen from Figure 6, the PI controller68 gives a very high undershoot. The conventional PID controller gives a better performance. After applying

10.1. Disturbance of +30% in Feed Flow Rate. Figure 6 shows the compositions of the three products and the temperatures of the three sensitive trays after applying a N


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disturbance of +13% in zA; and ISE = 0.011, ITSE = 0.274, IAE = 0.542, and ITAE = 16.0026 for disturbance of +30% in zB. There was limited research concerning the use of the model reference adaptive controller for the control of the DWC; therefore, the NN model of the DWC was trained and used as the reference model. The results of the NN MRAC PSO PID controller were better than those of the PSO PID controller, having ISE = 0.04206, ITSE = 1.3737, IAE = 1.2324, and ITAE = 37.484 for disturbance of +30% in F; ISE = 0.0692, ITSE = 1.6906, IAE = 1.073, and ITAE = 28.0796 for disturbance of +13% in zA; and ISE = 0.0155, ITSE = 0.4004, IAE = 0.6321, and ITAE = 18.2355 for disturbance of +30% in zB. However, the use of NN MRAC with ANFIS PID controllerwith or without PSOhad nearly the same results as without the NN MRAC. As a future work, the authors suggest to apply different optimization techniques such as glowworm swarm optimization to the MRAC and to train the parameters of the PID controller.

intelligent techniques for the tuning of the PID controller, the performance becomes much better with less settling time. From the values in Figure 9, it is clear that the conventional PID controller is much better than the PI controller.68 ANFIS PID tuning is the best controller over neural and fuzzy PID tuning. After applying PSO to these controllers, the performance is enhanced. The use of NN MRAC did not improve the system for this type of disturbance. 10.2. Disturbance of +13% in Ethanol Composition. Figure 7 shows the compositions of the three products and the temperatures of the three sensitive trays after applying a disturbance of +13% in ethanol composition. As seen from Figure 7, the PI controller68 gives a bad response. The conventional PID controller improves the performance. After applying intelligent techniques for the tuning of the PID controller, the performance becomes much better with less settling time. From the values in Figure 9, it is clear that the conventional PID controller is much better than the PI controller.68 The neural PID tuning is better than the fuzzy PID tuning. ANFIS PID tuning is the best controller tuning used. After applying PSO to these controllers, the performance of all types of controllers becomes better. PSO ANFIS PID has fewer oscillations and less settling time. The use of NN MRAC for this type of disturbance could improve the ITAE, especially when applying ANFIS based PSO PID controller. 10.3. Disturbance of +30% in Propanol Composition. Figure 8 shows the compositions of the three products and the temperatures of the three sensitive trays after applying a disturbance of +30% in propanol composition. Both PI68 and conventional PID controllers have large overshoot as compared to other controllers. Intelligent PID controllers give less overshoot and fewer oscillations. The values in Figure 9 indicate that the PSO ANFIS PID controller gives the best results in comparison to other controllers. The use of NN MRAC improved the PSO PID and the ANFIS PID controllers’ results.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.9b01747. Process parameters of the DWC; ISE, ITSE, IAE, and ITAE performance indexes for +30% disturbance in F; ISE, ITSE, IAE, and ITAE performance indexes for +13% disturbance in zA; ISE, ITSE, IAE, and ITAE performance indexes for +30% disturbance in zB (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Eman M. El-Gendy: 0000-0001-7468-4087 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

11. CONCLUSION DWC is an integration of the Petlyuk column in a single shell by inserting a vertical wall to separate the prefractionator from the main column. The control of this process is difficult due to the coupling effects, so a robust control must be applied to ensure good performance against disturbances. Several trials to construct a robust controller for this decoupled process have been applied, and an overview of the most recent trials was presented in this paper. PID controller is used in almost 90% of industrial processes, so it is a must to choose its parameters correctly. Between conventional and intelligent techniques, the PID tuned by ANFIS is the best controller in terms of different performance indexes, where ISE = 0.05216, ITSE = 1.2403, IAE = 1.0259, and ITAE = 30.379 for disturbance of +30% in F; ISE = 0.0815, ITSE = 1.9723, IAE = 1.124, and ITAE = 29.403 for disturbance of +13% in zA; and ISE = 0.0131, ITSE = 0.329, IAE = 0.5903, and ITAE = 17.255 for disturbance of +30% in zB. The neural PID controller gives better results than the fuzzy gain scheduling PID controller. To enhance the performance, PSO optimization technique is used to tune the parameters of the intelligent controllers. When adding PSO, the results become much better as compared to conventional and intelligent PID controllers without PSO, where ISE = 0.04206, ITSE = 0.9873, IAE = 0.9042, and ITAE = 26.857 for disturbance of +30% in F; ISE = 0.0616, ITSE = 1.4704, IAE = 0.962, and ITAE = 25.055 for

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Dr. Mahmoud Mohamed, Mansoura University, assisted with the Matlab implementation. NOMENCLATURE

Symbols

B = bottom product flow rate (kmol/min) D = distillate product flow rate (kmol/min) l = reflux flow rate (kmol/min) F = feed flow rate (kmol/min) S = side stream flow rate (kmol/min) V = boil-up flow rate (kmol/min) RL = liquid split ratio RV = vapor split ratio Mi = total liquid holdup on ith tray (kmol) HL,i = liquid enthalpy on ith stage (J/kmol) HV,i = vapor enthalpy on ith stage (J/kmol) Li = liquid flow rate from ith tray (kmol/min) Vi = vapor flow rate from ith tray (kmol/min) xi,j = mole fraction of jth component in liquid phase at ith tray yi,j = mole fraction of jth component in vapor phase at ith tray

O


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Industrial & Engineering Chemistry Research γi,j = activity coefficient of jth component in liquid phase at ith stage q = feed condition Psat i,j = saturation pressure of jth component at ith stage (atm) Pi = pressure on ith stage (atm) Ti = temperature of ith tray (K) A, B, C = coefficients of Antoine equation to calculate temperature of each stage N = number of trays xA = top composition, component A xB = side composition, component B xC = bottom composition, component C z1 = feed composition, component A z2 = feed composition, component B i = tray number i ∈ {1, ..., N} j = component j ∈ {1, 2, 3} L6 = liquid flow rate at C24 (kmol/min) MR = liquid holdup in reboiler (kmol) MD = liquid holdup in condenser (kmol) xD,j = mole fraction of jth component in distillate xR,j = mole fraction of jth component in bottom product yD,j = Mole fraction of jth component in distillate in vapor phase yR,j = mole fraction of jth component in bottom product in vapor phase HL,D = liquid enthalpy in distillate (J/kmol) HV,D = vapor enthalpy in distillate (J/kmol) Qc = condenser duty (J/min) QR = reboiler duty (J/min) u(t) = control signal Kp = proportional gain Ki = integral gain Kd = derivative gain e(t) = error between reference and actual temperature ωk+1 = inertia factor c1 = cognitive scaling factor c2 = social scaling factor ri1,k, ri2,k = random numbers uniformly distributed in the interval [0 1] pik = best previously obtained position of ith particle pgk = best obtained position in the entire swarm at current iteration k k = iteration number vik+1 = velocity update of particle i vik = current velocity of particle i xik+1 = position update of particle i xik = current position of particle i

ANNPC = artificial neural network predictive control ATC = asymmetrical temperature control DTDC = double temperature difference control PID = proportional−integral−derivative ANFIS = adaptive neuro-fuzzy inference system FGS = fuzzy gain scheduling MISO = multi-input−single-output ISE = integral square error ITSE = integral time square error IAE = integral absolute error ITAE = integral time absolute error

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REFERENCES

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Abbreviations

DWC = divided wall column PSO = particle swarm optimization MIMO = multi-input−multi-output NN = neural network MRAC = model reference adaptive controller DOF = degree of freedom BTX = benzene/toluene/xylene FLC = fuzzy logic controller MPC = model predictive controller DMC = dynamic matrix control SVD = singular value decomposition RGA = relative gain array GA = genetic algorithm TDC = temperature difference control STDC = simplified temperature difference control P


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