Design and Control of Thermally Coupled Reactive Distillation for the

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Design and Control of Thermally Coupled Reactive Distillation for the Production of Isopropyl Acetate Hao-Yeh Lee,† I-Kuan Lai,‡ Hsiao-Ping Huang,‡ and I-Lung Chien*,‡ †

Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan



ABSTRACT: Isopropyl acetates are important organic solvents that are widely used in the production of varnishes, ink, synthetic resins, and adhesive agents. Previous studies developed a process for the production of isopropyl acetate incorporating a reactive distillation (RD) column, a decanter, and a stripper. According to the previous study, the rectifying section of the RD column has a prominent remixing phenomenon. Furthermore, the overhead compositions of RD column and the stripper are all within the liquid−liquid equilibrium envelope. Based on the above observations, a thermally coupled design of this process is established. The key points in the thermally coupled design are: to move the location of the decanter to the stripper side, to totally reflux the organic phase outlet stream, and to sidedraw a liquid stream from the stripper to the RD column. Simulation result shows that 23.14% energy savings can be realized using the proposed thermally coupled design. The control strategy of the proposed design flowsheet is also investigated using tray temperature control loops to indirectly control the product composition. The proposed control strategy is capable of maintaining high-purity product, despite changes in feed composition and throughput.

1. INTRODUCTION Process intensification represents an important trend in process technology and attracts more and more attention in industry and the research community. Because the price of the crude oil is constantly increasing, the exploitation of energy savings becomes one of the recent focuses in research. Utilizing an integrated multifunctional unit to replace some single function units is one effective approach toward this goal. Reactive distillation (RD) is one of the important applications and has demonstrated its potential for capital productivity improvements, selectivity improvements, and reduced energy usage by combining reaction and separation into a single column (cf., Malone and Doherty1). From the book of Luyben and Yu,2 there were 1105 related publications and 814 U.S. patents between 1971 and 2007. Luyben and Yu2 also highlighted 236 reaction systems, which can be designed with RD configuration. Also, another book of Sundmacher and Kienle3 surveyed more than 100 industrially or potentially important reactions for RD applications. The above references show the importance of the RD technology in industrial applications. Process intensification for energy reduction on a distillation column can be classified in the following two types. One is to integrate the energy with heat transfer by increasing pressure at one part of the column. Robinson and Gilliand4 summarized this type of heat integration method, which includes “multi-effect” distillation columns and “vapor recompression” column in a distillation system. The vapor recompression column is a prototype of internally heat-integrated distillation column (HIDiC). The other type is to use thermally coupled design (also known as divided wall column, DWC) in a distillation system for two columns under similar operating pressures. The concept of complex distillation columns such as thermally coupled column was first proposed by Wright5 in 1949. He addressed the possibility that a distillation column with an © 2012 American Chemical Society

internal wall can have energy savings for separating the feed stream by using an additional side stream. Previous studies6−10 of thermally coupled column showed that this configuration can have 30% energy savings, compared to the conventional design. Furthermore, Schultz et al.11 also demonstrated that the DWC configuration can reduce investment costs, operating costs, and space requirements by up to 25%, 35%, and 40%, respectively. An important phenomenon of the remixing effect can be observed in the conventional distillation sequences, and it causes larger demand of energy in order to repurify the mixture in the next column.12 DWC processes can decrease energy consumption because remixing is eliminated or reduced in contrast to conventional distillation sequences. Kaibel13 noted that the DWC is thermodynamically equivalent to the thermally coupled Petlyuk distillation which was studied by Petlyuk et al.14 However, this equivalence is only valid when there is no heat transfer across the dividing wall. To control these integrated distillations, Daoutidis and his coworkers15−17 proposed a concept of energy flow to analyze the process dynamics and control performance. They provided some case studies of heat-integrated distillations with good control performance, especially for double effect distillations and vapor recompression distillation. However, this approach is yet to be demonstrated for complex processes such as an integrated distillation column with azeotrope or two liquid phases and thermally coupled reactive distillation with more complex thermodynamic behavior. Reactive distillation (RD) with thermally coupled design combines two promising technologies in order to achieve Received: Revised: Accepted: Published: 11753

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Table 1. NRTL Model Coefficients for IPAc Process component i component j

HAc IPOH

HAc IPAc

HAc H2O

IPOH IPAc

IPOH H2O

IPAc H2O

bij (K) bji (K) cij

−141.644 40.962 0.305

70.965 77.900 0.301

−110.580 424.060 0.299

191.086 157.103 0.3

20.057 833.042 0.325

415.478 1373.462 0.3

Table 2. Compositions and Temperatures of the Azeotropes for the IPAc Processa

a

component i

experimental components

experimental temperature (°C)

computed components

computed temperature (°C)

IPOH−IPAc IPOH−H2O IPAc−H2O IPOH−IPAc−H2O

(0.6508, 0.3492) (0.6875, 0.3125) (0.5982, 0.4018) (0.1377, 0.4938, 0.3885)

80.1 82.5 76.6 75.5

(0.5984, 0.4016) (0.6691, 0.3309) (0.5981, 0.4019) (0.2377, 0.4092, 0.3531)

78.54 80.06 76.57 74.22

Experimental data from Horsley.38 Data for heterogeneous azeotropes shown in boldface.

2. PHASE EQUILIBRIUM AND REACTION KINETICS The IPAc process exhibit nonideal phase behavior and has four azeotropes. In order to accurately represent the phase equilibriums of the process, the selection of the form of the thermodynamic model and the determination of the parameters are essential. To account for the nonideal vapor−liquid equilibrium (VLE) and vapor−liquid−liquid equilibrium (VLLE) for the quaternary system, the NRTL (nonrandom two-liquid) activity coefficient model is adopted by Aspen Plus. The NRTL model parameter set shown in Table 1 is taken from the literature of Tang et al.26 The vapor phase nonideality, such as the dimerization of acetic acid, is also considered. The second viral coefficients of the Hayden−O’Connell29 literature are used to account for vapor-phase association of acetic acid. The Aspen Plus built-in association parameters are used to compute the fugacity coefficients. According to the study by Tang et al.,26 the thermodynamic model predicts three binary minimum boiling azeotropes and one ternary minimum boiling azeotrope. Table 2 shows that the temperatures and the compositions of these azeotropes are in good agreements between model prediction and experimental data. Notice that the ternary minimum boiling azeotrope is the lowest temperature of the system. This ternary azetrope is shown in the residue curve map (RCM) of Figure 1. By reacting iso-propanol (IPOH) with acetic acid (HAc), the esterification reaction will produce IPAc and water. The reactions are reversible and the stoichiometric balance equation is shown as follows:

substantial economical benefits from process intensification. Mueller et al.18 was the first open literature to combine the RD with DWC configuration, using a reactive divided wall column (RDWC) for the simultaneous esterification of methanol and butanol with acetic acid to yield methyl acetate, butyl acetate, and byproduct water. Kenig and his co-workers19,20 studied the transesterification of carbonates process with three types of configurations. They concluded that the RDWC configuration resulted in only half of the total cost, in contrast to others. Wang et al.21 studies another transesterification reaction of methyl acetate in the RD with thermally coupled design; both energy savings and dynamic performance are better, compared with the conventional RD configuration. As for the experimental validation of RDWC configuration, Sander et al.22 investigated the laboratory and pilot-scale experiments for the hydrolysis of methyl acetate. BarrosoMuñoz et al.,23 Hernández et al.,24 and Delgado−Delgado et al.25 implemented the esterification of ethyl acetate with RDWC configuration in a pilot-plant test. All of the above papers showed that experimental data and simulation results are quite consistent. However, the purities of the products do not seem high enough and need further treatment for industrial usage. Tang et al.26 developed different process flowsheets for the esterification of acetic acid with C1−C5 alcohols. These flowsheets show that high-purity ester products can be produced using RD technology. Lai et al.27 demonstrated that industrialquality ethyl acetate product can be obtained in a pilot-plant test using the flowsheet developed by Tang et al.26 According to the studies by Tang et al.26 and Lai et al.,28 energy requirements of the ethyl acetate (EtAc) and isopropyl acetate (IPAc) processes, which were designed as type II configuration, were much higher than other types. Among all the RD processes studied in Tang et al.,26 type II configuration has the potential to be integrated as RD with thermally coupled design, because of two columns in the flowsheet. In this paper, we try to combine the benefit of reactive distillation with thermally coupled design to produce IPAc. This improved design will be compared with the original design by Lai et al.28 Most of the RDWC studies in the open literature did not include the decanter in the original design flowsheet. Therefore, the proposed study will be an important investigation to establish thermally coupled design for this type of process. After the proposed design flowsheet is developed, it is also desirable to know if this flowsheet can be properly controlled, despite various feed disturbances. Thus, the overall control strategy will also be studied.

CH3COOH + C3H 7OH ⇄ CH3COOC3H 7 + H 2O (HAc)

(IPOH)

(IPAc)

(H 2O)

The solid catalysts in use are the acidic ion-exchange resins of Amberlyst 15.30 Table 3 displays the reaction rates expressed in the Langmuir−Hinshelwood model form. Notice that the kinetic model is represented in terms of activity and the reaction is based on catalyst weight (mcat). The catalyst weight is computed by assuming that the solid catalyst occupies 50% of the tray holdup and a catalyst density of 770 kg/m3 is used to convert the volume to catalyst weight. Before leaving this section, it should be noted here that a FORTRAN subroutine is written in the Aspen Plus to compute the extent of reaction of each tray.

3. DESIGN OF THERMALLY COUPLED REACTIVE DISTILLATION 3.1. Process Description and Basic Design Concept. The IPAc process is classified as type II system in Tang et al.26 11754

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Figure 1. Four ternary residue curve maps (RCMs) and liquid−liquid equilibria (LLEs) of the IPAc process. Figure 2. Conventional RD configuration for the IPAc process.

The characteristics of the Type II system contains one RD column with a decanter and an additional stripper. Based on the azeotrope and boiling point ranking, the heaviest component is reactant HAc, so that no bottom outlet stream of the RD column is designed. Top composition of the RD column is quite close to the minimum temperature azeotrope of IPOH−IPAc−H2O. In the RCM of Figure 1, it is found that the liquid−liquid envelope (LLE) is quite large, and the ternary minimum boiling azeotrope lies well inside the envelope. It can also be seen that the tie lines all point toward the pure water end and, consequently, relatively pure water can be recovered from the aqueous phase of a decanter designed to be located at the top of the RD column. The tie lines can also travel across the distillation boundary, so that high-purity IPAc can be obtained by further purification of the organic phase outlet stream in a stripper. Part of the organic phase material should also be refluxed back to the RD column in order to carry water out of the system. Notice that, according to the research of Tang et al.,26 it is found that both reactant feeds should be fed in the bottom of the RD column, so that larger reaction holdup can be utilized. Because of economical reasons, Lai et al.28 studied the IPAc process of this type II system by feeding industrial compositions of reactants instead of pure feeds. Based on the minimum total annual cost, Lai et al.,28 demonstrated that the optimal result of the conventional IPAc RD process is shown in Figure 2. In this study, the same feed conditions and product specifications are

considered. The IPOH feed is assumed to be close to the azeotrope composition with water. The feed composition of HAc is set to be 95 mol % and that of alcohols are set to be 64.91 mol % for IPOH, respectively. The specifications include the following: IPAc production with 99 mol % while keeping HAc purity below 0.01 mol %. Also, the tray numbers of the rectifying section, reactive section, and stripper are all fixed to be the same as those given in the optimal result by Lai et al.,28 so that the benefit, in terms of energy savings in the configuration of thermally coupled RD system, can be investigated. By observing composition profiles of the IPAc process in Lai et al.28 (see Figure 3), the IPAc compositions in the upper section of the RD column go through a maximum value and then decrease. This is the so-called “remixing effect”. This phenomenon suggests that there is potential for energy savings by using thermally coupled design. The second observation is that the top compositions of the RD column and the stripper are all within the LLE. It is easy to imagine that the liquid−liquid separation task can still be performed if the two overhead streams are mixed before feeding to the decanter. Because of the above observations, the RD with thermally coupled design is proposed in Figure 4. Since, from the previously shown Figure 2, the feed to the stripper is from the organic outlet stream of the decanter, it was decided to place the decanter on the stripper (now called the product column in

Table 3. Kinetic Equations for IPAc Processes kinetic model (catalyst)a

k1 (T = 363 K)

Keq (T = 363 K)

2.26 × 10−4 [kmol/(kgcat s)]

8.7

Langmuir−Hinshelwood/Hougen−Watson model (Amberlyst 15 by Gadewar et al.30)

r = mcat

k1(aHAcaIPOH − aIPAcaH2O / K eq) (1 + KHAcaHAc + KIPOHaIPOH + KIPAcaIPAc + K H2Oa H2O)2

(

k1 = 7.667 × 10−5exp 23.81 −

68620.43 RT

)

Keq = 8.7, KHAc = 0.1976, KIPOH = 0.2396, KIPAc = 0.147, KH2O = 0.5079 Assumption: mol H+/kgcat = 4.6 × 10−3 a

R is the universal gas constant (R = 8.314 kJ/(kmol K)), T the temperature (given in Kelvin), r the reaction rate (given in units of kmol/s), mcat the catalyst mass (given in units of kgcat), and xi the mole fraction of component i. 11755

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the vapor stream from the overhead of the RD column should go to the same sidedraw location at the product column. Based on the design of thermally coupled RD in Figure 4, the reboiler duty of the right side product column is an operating variable used to meet the purity of IPAc, and the liquid sidedraw flow rate (FL) instead of the organic reflux in the original RD configuration is used to maintain the HAc impurity in the product stream. Because of the characteristics of the configuration in Figure 4, there will be only one design variable as the “vapor−liquid exchange location” (NF). The NF value is then used to procure the minimum energy requirement. The optimization procedure is shown as follows: (1) Guess the NF value. (2) Adjust the FL and reboiler duty of the right side column until the product specification is met. (3) Go back to step (1) and change NF until the reboiler duty consumption of the overall process is minimized. 3.2. Result and Discussion. Following the optimization procedure, the information of reboiler duties versus NF is shown in Figure 5. According to this figure, the best vapor−liquid

Figure 3. (a) RD column and (b) stripper liquid composition profiles of conventional RD configuration.

Figure 5. Minimum energy requirement of the thermally coupled RD configuration.

exchange locations are at the fifth tray. Notice in this figure, it is found that the reboiler duty of the RD remained about the same when the vapor−liquid exchange location varies. However, the reboiler duty of the right side product column would go through a minimum at the fifth tray with the varying of the vapor−liquid exchange location. More-detailed results of the IPAc process with thermally coupled RD are shown in Figure 6. The reboiler duty of the RD column is 3947.42 kW, which is almost the same as the case of the conventional RD configuration shown in Figure 2. However, the reboiler duty of the right side product column is only 265.67 kW, which is significantly lower than the stripper reboiler duty of the conventional RD configuration (1558.95 kW). The optimal total reboiler duty of the thermally coupled RD is 23.14% less than that of the conventional RD system. The liquid composition profile of the thermally coupled RD for this process is shown in Figure 7. Comparing the composition of IPAc in the upper section of the RD column in the thermally coupled RD with that of the conventional RD configurations, the remixing effect is clearly eliminated. Another interesting observation in composition profile is to project the three-

Figure 4. Proposed process configuration of the thermally coupled RD configuration.

Figure 4) side. However, a liquid stream is needed to go down to the RD column to act as entrainer to carry more water toward the top of the column; thus, a liquid sidedraw from the product column is designed. To complete the thermally coupled design, 11756

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Figure 6. Optimal results of the thermally coupled RD for IPAc processes.

Figure 8. Composition trajectories of the IPAc process: transformed liquid and vapor composition in the quaternary system (XA = XHAc + XIPAc, XB = XTPOH + XIPAc) for (a) conventional RD configuration and (b) thermally coupled RD configuration.

Figure 7. (a) RD column and (b) product column liquid composition profiles of the thermally coupled RD configuration.

in this figure. In Figure 8, the curves with triangular symbols represent the composition trajectory of RD column composition profile, and the curves with circle symbols represent the composition trajectory of the product column. In Figure 8a, the vapor and liquid composition trajectories in the conventional RD column show an obvious turn in the two composition profiles. This is the remixing effect that we have mentioned in the previous section. By observing Figure 8b, it is found that the obvious turn in the composition trajectories in the thermally coupled RD is eliminated. Another observation in Figure 8 is that the composition trajectories of the vapor and liquid for the product column with thermally coupled design all show a more linear behavior than the stripper in the conventional RD system. This more-linear behavior of the composition trajectory in this transformation coordinate is indicative of the energy savings in the product column. The reason for the resulting energy savings should be related to the product column following another path of distillation line with more-effective separation.

dimensional (3D) space of a tetrahedron as two-dimensional (2D) coordinates, using the composition variable transformation method.31 The distillation path in Figure 8 shows the tray compositions of liquid and vapor phase in the coordinate of composition variable transformation space. The tie lines and the compositions of the organic and aqueous phases are also shown

4. OVERALL CONTROL STRATEGY In this section, a systematic approach is used to design the control structure for the IPAc RD with thermally coupled design. The control objective is to maintain the acetate purity at 99 mol % in product stream while keeping the HAc impurity below 0.01 mol %. In this system, there are 11 control degrees of freedom. 11757

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Figure 9. Overall control strategy of the thermally coupled RD configuration.

Figure 11. Row sums of nonsquare relative gain (NRG) for RD column and the product column.

Table 4. Process Gain Matrices, Relative Gain Array, and Tuning Parametersa process

IPAc

controlled variables manipulated variables steady state gain

Q R,PC

RGA

⎡ 0.9628 Λ=⎢ ⎣ 0.0372

tuning parameters

Figure 10. Process sensitivities of tray temperatures for ±0.1% changes in (a) FR and (b) QR,PC manipulated variables.

a

TPC,4, TRD,7 QR,PC, FR ⎡TPC,4 ⎤ ⎡ 0.06395 0.31445⎤ ⎡Q ⎤ R,PC ⎢ ⎥=⎢ ⎥ ⎥⎢ ⎢⎣TRD,7 ⎥⎦ ⎣ − 0.027 3.4342 ⎦ ⎣ FR ⎦ FR

0.0372 ⎤ TPC,4 ⎥ 0.9628 ⎦TRD,7

QR,PC − TPC,4: KC = 23.53, τI = 39.6 (min) FR − TRD,7: KC = 98.03, τI = 52.8 (min)

FR is the molar ratio of acetic acid over alcohol.

product flow from the bottom of the product column, two reboiler duties (i.e., RD column and product column), the HAc feed flow, the IPOH feed flow, the top vapor flow of the product

They are: liquid side streamflow (FL), top vapor flow of the RD column (FV), organic reflux flow of the organic phase and aqueous outlet flow of the aqueous phase from the decanter, 11758

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Figure 12. Process responses for ±20% throughput changes.

column, and the condenser duty. The fresh IPOH feed flow is chosen as the throughput manipulator. Production rate changes

are handled using this flow which is considered as a load disturbance in the subsequent study. Another degree of freedom 11759

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Figure 13. Process responses for HAc feed composition changes. 11760

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Figure 14. Process responses for −10 mol % IPOH feed composition changes.

(condenser duty) is used to maintain the decanter temperature at 50 °C. Six other degrees of freedoms are used for inventory

control purposes, to keep the total materials in balance. The six inventory loops are the RD column base level, the RD column 11761

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pressure, decanter aqueous phase level, decanter organic phase level, product column base level, and product column pressure. 4.1. Inventory Control. The pairing of the six inventory loops followed the previous studies by Hung et al.32 and Lai et al.28 for the IPAc RD system and are shown in Figure 9. They are arranged as follows. For the four-level loops, the decanter organic level is controlled by manipulating the organic reflux flow, the decanter aqueous level is controlled by manipulating the aqueous outlet flow, the product column base level is controlled by the product flow, and the RD column bottom level is controlled by manipulating the reboiler duty, because there is no outflow in the bottoms of the RD column. For the two pressure loops, the RD column pressure is controlled by manipulating the top vapor flow of the RD column via the compressor brake power in the simulation, and the product column pressure is controlled by manipulating the top vapor rate of the product column. Notice that Ling and Luyben33 also used such types of virtual compressors, which are implemented in the top of the RD column to have enough pressure drop for feeding vapor stream to the product column. Small temperature increase through compressor is ignored in the simulation. The selection of the pairing of the above inventory loops is based on the effectiveness of the manipulated variable to the controlled variable. We are left with three degrees of freedom: reboiler duty of the product column (QR,PC), liquid side streamflow rate (FL), and feed ratio of HAc/IPOH. Two of them, in theory, should be used to maintain the two product specifications of IPAc composition and HAc impurity at the bottoms of the product column in the face of the various disturbances. Notice that the feed ratio of HAc/IPOH should be selected as one of the two manipulated variables in order to maintain stochiometric feed balance under feed composition changes. 4.2. Selection of Temperature Control Trays. Recall the remaining manipulated variables, which are the reboiler duty of the product column (QR,PC), the feed ratio of HAc/IPOH (FR), and liquid side streamflow rates (FL). To simulate the process operation as divided wall column (DWC), the ratio of FL and the internal flow rate down to the product column is set as a constant in the side stream location. The two remaining manipulated variables, QR,PC and FR, are used for the temperature control loops. Sensitivity analyses are performed and shown in Figure 10. Note that in Figure 10 the temperatures in the RD column and the product column are combined and are labeled in the x-axis. In order to find the steady-state gains of tray temperature in the linear region, extremely small step changes (±0.1%) in the manipulated variables are made. As shown in Figure 10, an increase in QR,PC leads to a decrease in the tray temperatures of the RD column. An increase in QR,PC results in a larger vapor rate from the product column and subsequently condensed back to the decanter and this implies a larger liquid recycle flow for the RD column. As for the acid feed changes, an increase in the heavy reactant results in tray temperature increases throughout the RD column while the product column temperatures show little variation. The nonsquare relative gain (NRG)34 is used to find the temperature control trays in each column. The largest row sum of the NRG in each column is selected as the temperature control trays in Figure 11 for this process. 4.3. Quality Control Scheme and Control Performance. For quality control loops, we have two control objectives: (1) controlling acetate product purity, and (2) maintaining stoichiometric balance. Relative gain array (RGA) analysis in Table 4 shows the pairing result of the two selected control points (TRD,7 and TPC,4) to the two manipulated variables (QR,PC

and FR). Small loop interactions can be observed from Table 4. Thus, decentralized control is used for the product quality control and maintaining the feed stoichiometric balance. The relay feedback test35 and the Tyreus−Luyben36 PI tuning rule are selected to determine the controller tuning parameters of these two control loops. The sequential iterative tuning procedure37 is used to find the final controller settings. Table 4 also summarizes the controller settings of the two PI loops. It can be seen that a large reset time is associated with the feed ratio (FR) loop, and the reset time for the reboiler duty loop is relatively small. This implies that the two loops are dynamically decoupled with the feed stoichiometric balance maintaining in a slow and smooth manner while the product quality is tightly controlled. Three types of disturbancesthroughput changes, HAc feed composition variation, and IPOH feed composition variation are used to evaluate the control performance of the proposed control strategy. The dynamic responses for the process are shown in Figures 12, 13, and 14. For ±20% throughput changes, the steady-state deviation of the product purity is quite small for the ester product. The settling time of the thermally coupled RD process is also fast. Moreover, symmetrical responses are observed in the temperatures controlled tray temperatures (TRD,7 and TPC,4) and manipulated inputs (FR and QR,PC). For HAc feed composition changing from 0.95 to 1 and from 0.95 to 0.9, the system gives good closed-loop control performance (see Figure 13). The feed ratio is correctly adjusted to maintain feed stoichiometric balance. Finally, a −10 mol % IPOH feed composition disturbance is shown in Figure 14. It demonstrates that the ester product composition moved toward higher purity for this disturbance. The feed ratio is decreased to maintain feed stoichiometric balance, because lesser-purity IPOH feed enters into the system.

5. CONCLUSIONS In this paper, the potential energy savings of the thermally coupled reactive distillation (RD) is investigated for a Type II reactive distillation system. The thermally coupled design includes moving the location of the decanter to the stripper side, totally refluxing the organic phase outlet stream, and adding a sidedraw liquid stream from the stripper to the RD column. Simulation result shows that an energy savings of 23.14% can be realized using the proposed thermally coupled design. Overall control strategy of this thermally coupled RD is also proposed with only one tray temperature control loop in each column. Large variations in the feed composition and also throughput changes are investigated. All product compositions are maintained at high purity, despite feed disturbances.



AUTHOR INFORMATION

Corresponding Author

*Tel: +886-2-3366-3063. Fax: +886-2-2362-3040. E-mail: [email protected]. Notes

The authors declare no competing financial interest.



ACKNOWLEDGMENTS Financial support from the National Science Council and the Ministry of Economic Affairs of the R.O.C. is greatly appreciated. 11762

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