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Mathematical and Computer Modelling of Dynamical Systems
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Model reduction and optimization of a reactive dividing wall batch distillation column inspired by response surface methodology and differential evolution
Maysam Safe , Seyed Masoom Khazraee , Payam Setoodeh & Abdolhosein H.
Jahanmiri
To cite this article: Maysam Safe , Seyed Masoom Khazraee , Payam Setoodeh & Abdolhosein H. Jahanmiri (2013) Model reduction and optimization of a reactive dividing wall batch distillation column inspired by response surface methodology and differential evolution, Mathematical and Computer Modelling of Dynamical Systems, 19:1, 29-50, DOI: 10.1080/13873954.2012.691521 To link to this article: https://doi.org/10.1080/13873954.2012.691521
Published online: 30 Jul 2012.
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Model reduction and optimization of a reactive dividing wall batch distillation column inspired by response surface methodology
and differential evolution
Maysam Safea, Seyed Masoom Khazraeeb, Payam Setoodehaand Abdolhosein H. Jahanmiria*
aDepartment of Chemical Engineering, School of Chemical and Petroleum Engineering, Shiraz University, Shiraz 71345, Iran;bEngineering Research Institute, Fars Engineering Research Center,
Shiraz, Iran
(Received 7 November 2011; final version received 3 May 2012)
Carrying out reaction and separation simultaneously in a reactive dividing wall batch distillation column batch RDWC in the case of ethyl acetate synthesis provides the pos- sibility of separating both products and increasing the equilibrium reaction conversion.
Overcoming the known azeotrope conditions, high purity for ethyl acetate and decreas- ing the batch time compared to simple reactive batch distillation are the advantages of this configuration. The corresponding dynamic simulation is carried out by simul- taneously solving the model-associated system of differential and algebraic equations.
In this study, the optimal values of the vapour and liquid split ratios are considered as the decision variables in order to maximize the amount of ethyl acetate accumulated during batch time. The optimization strategy is implemented inspired by response sur- face methodology in which an optimal surface is fitted to the collected data set using differential evolution (DE). The optimal surface relevant algebraic equation is then con- sidered as the reduced form of the complex model and the optimal values are obtained using the DE method.
Keywords: reactive distillation; dividing wall; ethyl acetate; differential evaluation method; response surface methodology
1. Introduction
Although distillation has low thermodynamic efficiency, this system is still used in chem- ical industries. Finding new methods to improve distillation efficiency is in progress.
Thermal coupling of distillation columns is a novel method in the separation of multi- component mixtures in which the thermal linking of distillation columns reduces relevant equipment and energy costs. Energy savings of up to 30% has been reported for this system compared to the conventional sequence of distillation columns [1]. By applying the heat- ing vapour and cooling liquid-splitted streams from the main column to the side column in this configuration, there would be no need to employ a condenser as well as a reboiler for the side column. Side rectifier, side striper, Petlyuk column and dividing wall column (DWC) are examples of proposed thermally coupled distillation configurations. Petlyuk et al. introduced a fully thermally coupled two-column structure for ternary mixture
*Corresponding author. Email: [email protected]
© 2013 Taylor & Francis
Figure 1. (a) Petlyuk distillation column; (b) Dividing wall distillation column.
separation, which is shown in Figure 1(a). It should be mentioned that in the conventional multi-component distillation, remixing occurs in the first column. This effect is responsible for wasting a portion of the consumed energy that is employed to purify the component with the intermediate boiling point from the light component. However, when it is added to a mixture in which the heaviest component is the dominant component, this consumed energy would be wasted. Because the condenser and the reboiler of the prefractionator have been omitted due to the thermal linking of the columns, and the remixing effect in the pre- fractionator column has been minimized, Petlyuk columns have become known as energy- and cost-saving structures. The possibility of locating Petlyuk’s columns in a single-shell column is achievable using the DWC configuration (Figure 1(b)). Besides, the mentioned advantage of the Petlyuk configuration, DWC in short, is an adequate substitution of two distillation towers in a single shell. If the heat transfer from the dividing wall is negligible, the Petlyuk column and DWC are thermodynamically equivalent, which is possible because of the studies conducted by Petlyuk et al. [2] as well as Hernandez and Segovia [3]. The combination of simultaneous reaction and separation in a DWC results in a novel process intensification called reactive dividing wall column (RDWC). This configuration is very useful and applicable especially when the system involves reversible reactions, azeotropes and undesired product formations. High selectivity and capital investment reduction are the most important advantages of this intensification.
The process of ethyl acetate synthesis from ethanol and acetic acid is equilibrium limited. Separation of both products from the reaction zone displaces the chemical equi- librium, which will increase the reaction conversion [4,5]. The usage of a batch RDWC in the synthesis of ethyl acetate provides this possibility. Water will leave the reaction zone as the side product and the ethyl acetate will be separated by the distillate stream. The esteri- fication reaction rate is low; therefore, in continuous processes sulphuric acid is utilized as a catalyst to increase the rate [6]. This substance has high corrosive properties. As a result, the continuous RDWC must be immunized against this effect, as it increases the operation and investment costs. However, in a batch RDWC the retention time is suitable enough to
achieve the desired conversion for the mentioned equilibrium esterification reaction; con- sequently, there is no need to use sulphuric acid. This is the most important advantage of the batch RDWC in comparison to continuous operation, which reduces the investment costs.
The first theoretical basis for thermally coupled distillation system (TCDS) has been presented by Brugma [7]. After that Petlyuk et al. [2] extended this theory and introduced a fully thermally coupled two-column structure. Despite all the advantages this system has, the associated theory has remained an academic theory for many years. The main barrier for practical use was the lack of a suitable controlling system. Reintroduction of TCDS to the distillation world was done by Kaible [8]. Triantafyllou and Smith [9] presented advanced modelling and designing methods for DWC simulation and operation. Since then, many researchers have worked on designing, simulation, controlling and optimization of DWC. A formal procedure based on mathematical programming for detecting the optimal design of integrated distillation columns has been reported by Dunnebier and Pantelides [10]. Designing a control system for TCDS has been the aim of many researches such as Abdul Mutalib et al. [11] and Serra et al. [12]; also, simulation and optimization of a DWC have been covered in some publications such as the studies conducted by Amminudin et al.
[13] and Muralikrishna et al. [14].
Reactive distillation (RD) is another attractive subject of the process intensification scheme involving the combined operations of reaction and separation in a single unit.
In fact, RDWC is a further development of RD in which simultaneous reaction and separa- tion occur in DWC. Thus, having a clear background of RD can help the reader understand the advantages of RDWC. Many Researchers such as Suzuki et al. [15], Alejski et al. [16], Simandl and Svrcek [17] and Komatsu [18] have worked on RD, including esterification of acetic acid with ethanol.
Published research focusing on the dynamic simulation of RD and its optimization is limited [19–22]. The introduction of material balance equations and ideal plates with a constant molar hold-up with heat balance equations and the cell model of trays in the form Welch et al. have proposed was done by Alejski and Dupart. They have investigated a dynamic mathematical model of RD in the case of a kinetically controlled chemi- cal reaction [22–24]. The model is formulated using four basic assumptions: negligible vapour hold-up compared to molar liquid hold-up, perfect mixing of phases, corrections of equilibrium values by plate efficiency and corrections of conversion for mixing effects by conversion efficiency. Schneider et al. [25] have developed a rigorous dynamic rate- based approach, including heat and mass transfer, coupled with the chemical reaction.
Diffusion phenomena in multi-component mixtures resulting from molecular interactions are considered.
With regard to RDWC, published studies are few, and limited assumptions are usually considered in these publications. Mueller et al. have studied a reactive dividing wall col- umn using the rate-based approach in the process of synthesis of diethyl carbonate from dimethyl carbonate and ethanol. Their rate-based model was applied to Aspen Plus so that the prefractionator acted as a simple RD column and the main column separated unreacted materials from products. Their analysis shows that a reactive dividing wall leads to signifi- cant savings in energy consumption. It should be mentioned that heat transfer through the wall has been considered in their study [26].
Hernandez et al. studied the controllability of a reactive Petlyuk distillation column in which the chemical reaction was conducted in the reboiler of the main column. The com- mercial process simulator (Aspen PlusTM, Aspen Technology, Inc., Burlington, MA, USA)
was used for the purpose of simulation. It was found that the reactive Petlyuk column can achieve set point changes in two control loops of temperature. Besides, for load rejection, the control loops can eliminate the effect of the disturbances in the feed composition [27].
Recently, Khazraee et al. considered the application of the adaptive neuro-fuzzy inference system instead of the highly non-linear model of a reactive batch distillation column for achieving optimization.
In this study, the authors focus on model reduction and optimization of a reactive divid- ing wall batch distillation column in ethyl acetate synthesis process inspired by response surface methodology and differential evolution (DE). First, the case study process is described. Then the basic assumptions of the static model are represented. Because of the strictly non-linear nature of this model, finding the optimal values of the optimization variables is done based on the reduced model. Optimization is achieved by maximizing the amount of accumulated distillate ethyl acetate at final batch time. Applying response sur- face methodology with respect to the generated data from the strict model is described in the next section. The most appropriate parameter values for fitting the response surface are determined using an optimization algorithm employing DE method. Then, a similar opti- mization algorithm is applied to find the best operation conditions. This problem includes both equality (reduced model) and inequality (decision variable bonds) constraints. Note that in this research, it is assumed that chemical reaction can occur wherever reactants are available including the reboiler, the trays and the condenser.
2. The chemical reaction kinetics
Reactants including acetic acid and ethanol are fed to the reboiler of the proposed system.
The equilibrium esterification reaction is:
CH3COOH+ C2H5OH←−−−−−−−−−−−−→kk12 CH3COOC2H5+ H2O (1) Kaelo [28] has determined the kinetic model for this chemical reaction, which is repre- sented by Equation (2):
r= k1CCH3COOHCC2H5OH− k2CCH3COOC2H5CH2O
k1= 4.76 × 10−4, k2 = 1.63 × 10−4(l/gmol min) (2) The reaction mixture exhibits azeotrope due to the formation of ethyl acetate-ethanol, acetic acid-water and ethyl acetate-acetic acid-water azeotropes. Ethyl acetate is the main product formed and has the lowest boiling temperature in the mixture. The removal of ethyl acetate and water by distillation shifts the chemical equilibrium further to the right and consequently improves the conversion. As a case study, a reactive Petlyuk batch distillation system is simulated, having 20 and 10 trays in the main and the side columns, respectively.
The vapour and liquid streams to the side column are extracted from stages 5 and 16, respectively. In this study, trays are counted from the bottom. A schematic representation of the proposed reactive Petlyuk system is illustrated in Figure 2.
3. Mathematical modelling and simulation
This process is modelled using a tray-by-tray description, resulting in 161 differential non- linear equations. The model is solved by explicit Euler’s method according to the following four basic assumptions based on the study by Khazraee et al. [4,5]:
Figure 2. Schematic representation of the proposed reactive Petlyuk system.
(1) The molar vapour hold-up is negligible compared to the molar liquid hold-up.
(2) Vapour and liquid phases on each plate are perfectly mixed. Thus, based on this assumption, the leaving liquid phase from each plate has the same composition as the liquid phase on that plate.
(3) The theoretical equilibrium compositions are corrected for mixing effects, flow configuration and mass transfer limitations by introducing plate efficiency.
(4) Simple or complex reactions proceed in the liquid phase only, and their course can be described by appropriate kinetic equations.
Khazraee et al. [4] used these assumptions to validate the mathematical model of their experimental reactive batch distillation column.
The total and the species mass and energy balance equations for specified sections of the reactive Petlyuk system shown in Figure 3 are classified by the equations mentioned in Appendix A.
Here, it is assumed that a total condenser is used and vapour and liquid are in thermo- dynamic equilibrium with each other. There are also similar equations for the side column that are not represented here.
The hold-ups of the trays are calculated using Equation (3) [29]:
LV= L × MwAve
DensityAve
, MV =
LV
999× WLS
0.6667
+WHS 12
× πd2 4× 144 M = MV ×DensityAve
MwAve
(3)
The reboiler hold-up at any given time is calculated from an algebraic combination of the initial charge, the amount of material in the column and the total material removed up to that instant in time:
Figure 3. The reactive Petlyuk column used for mathematical modelling.
MB= MB 0+
Nt i=1
Mi 0−
Nt i=1
Mi−batch time
0 D dt−batch time
0 S dt (4)
The thermodynamic model used in this study is a non-random two-liquid (NRTL) model based on Lee et al. [30]. The parameters of NRTL model are employed to predict the composition and the temperature of the mixture. The complete set of NRTL parameters is given in Appendix B. The mathematical model was simulated by solving the system of differential and algebraic equations simultaneously, using integration with the Euler’s method.
4. Model reduction and optimization 4.1. Optimization strategy
The aim of this section is to find the optimum values of the vapour and liquid split ratios as the decision variables in order to maximize the amount of accumulated distillate ethyl acetate at final batch time. Besides, minimizing the amount of ethyl acetate in side stream contributes to maximum amount of water in side stream. The dynamic optimization problem is set-up as:
Objective function:
PEtAc=batch time
0 D× XEtAcdt−batch time
0 S× XEtAcdt (5)
Subject to:
Mathematical model (Appendix A) & 0.15 < RV& RL < 0.85 (6)
4.2. Model reduction
Because the mathematical model is non-differentiable, gradient-based optimization tech- niques are not applicable to solve the problem. Moreover, because of the large size and strictly non-linear nature of the mathematical model, random search-based optimization algorithms such as genetic algorithm and DE are also unsuitable to handle the problem due to the long computational time required. Thus, it seems that a kind of model reduc- tion algorithm would be useful. To achieve this, a set of data points are collected through frequent runs of the mathematical model for several RVand RL values in their physically feasible region. Then, for each pair of RVand RL, the optimal location of the side stream tray has been found considering the maximum amount of extracted water.
4.2.1. Data generation
As already mentioned, we have examined RDWC model for many different cases. In these cases, characteristics of the column hardware are considered as presented in Table 1. The column specifications are like those of the conventional RD column which has been con- sidered by Khazraee et al. [4] with a difference that a wall divided the column into two vertical sections through trays 6 to 16. Table 2 includes some selected simulation runs that represent the effect of changing design variables on RDWC system.
In all the examined cases, tray 7 (trays counted from bottom) contains a mixture in which water is the dominant component. For instance, Table 3 shows the effect of changing side stream tray on the amount of produced EtAc in one case study in which the authors consider RV, RL, RSand R equal to 0.3, 0.2, 0.3 and 8, respectively.
Obviously, this system is extremely complex. With regard to the optimization case, the reflux ratio, the side stream location and the ratio of side stream molar flow to interstage liquid flow (RS) are considered to be fixed. Authors take the vapour split ratio (RV) and the liquid split ratio (RL) into account as optimization decision variables.
4.2.2. Response surface methodology
Based on response surface methodology, a response surface is then fitted to the collected data set using DE as a robust and efficient optimization method [31]. The optimal surface
Table 1. Specifications of the considered RDWC.
No. of plates 20
Number of divided stages 10 (6–16)
Initial charge (kmol) 0.5
Initial mole fraction CH3COOH(1) 0.35
C2H5OH(2) 0.45
CH3COOC2H5(3) 0
H2O(4) 0.2
Condenser pressure (kpa) 101.325
Column diameter (m) 0.08
WHS (m) 0.013
WLS (m) 0.0236
QB(kJ/hr) 3587.2
Table 2. Examples of different case studies have been checked with the batch model.
R RV RL RS XEtAc,D XEtAc,S XWater,S T (min) S.P (lb mol) D.P (lb mol)
4 0.3 0.2 0.5 0.3529 0.1202 0.4005 540 0.6091 0.3859
4 0.25 0.4 0.3 0.3900 0.1799 0.3995 834 0.5830 0.4250
4 0.3 0.2 0.3 0.3865 0.1836 0.3765 726 0.4927 0.5094
4 0.2 0.3 0.5 0.3421 0.090 0.4434 582 0.5748 0.4175
6 0.2 0.3 0.5 0.3717 0.1164 0.4206 630 0.6672 0.3211
6 0.2 0.3 0.3 0.4003 0.1754 0.4027 882 0.5620 0.4414
6 0.5 0.5 0.5 0.4200 0.3973 0.2339 822 0.6232 0.3998
8 0.4 0.2 0.3 0.4382 0.3783 0.2339 906 0.6843 0.3436
8 0.25 0.3 0.3 0.4381 0.2447 0.3486 972 0.3734 0.6424
8 0.25 0.2 0.3 0.4253 0.2182 0.3626 888 0.3435 0.6706
8 0.25 0.4 0.3 0.4522 0.2787 0.3269 1074 0.4097 0.6091
8 0.25 0.5 0.3 0.4656 0.3198 0.2982 1200 0.4543 0.5674
Table 3. Effect of changing side stream tray on the amount of produced EtAc.
Side stream tray XEtAc,D XEtAc,S XWater,S T (min) S.P (lb mol) D.P (lb mol)
6 0.4226 0.2613 0.3217 894 0.6751 0.3436
7 0.4348 0.2610 0.3219 894 0.6752 0.3435
8 0.4325 0.2623 0.3189 888 0.6706 0.3712
9 0.4310 0.2628 0.3177 888 0.6706 0.3413
10 0.4294 0.2640 0.3156 888 0.6706 0.3415
11 0.4278 0.2658 0.3129 888 0.6706 0.3415
12 0.4259 0.2681 0.3096 888 0.6706 0.3416
13 0.4237 0.2710 0.3057 888 0.6706 0.3416
14 0.4210 0.2747 0.3011 888 0.6706 0.3417
15 0.4175 0.2794 0.2955 888 0.6706 0.3417
relevant algebraic equation is then considered as the reduced form of the complex model.
Afterwards, the optimal values for RVand RL are again obtained using the DE method.
The lower and upper bounds for RVand RL are 0.15 and 0.85, respectively. It should be mentioned that it is possible to select a smaller part of the feasible region, which includes the optimum in order to obtain more successive curve-fitting with higher accuracy.
In this study, second- to fifth-order polynomial forms are employed and checked to find the most appropriate algebraic model for the surface. For instance, Equation (7) represents a second-order model containing quadratic terms that are commonly used in experiment designs using response surface methodology [32].
u= b0+ b1z1+ b2z2+ b12z1z2+ b11z21+ b22z22 (7) where z1 and z2represent the factors (independent variables), u the response (dependent variable) and bi, bj and bij the model coefficient parameters that should be estimated through optimization procedure with regard to the available data set.
Employing the mentioned algebraic form, the amount of produced ethyl acetate is presented as a function of two variables RVand RL.
Consequently, the optimization problem is reformed to a new procedure in which the parameters of the quadratic function (six parameters) should be obtained in order to mini- mize the error between the data points and the values calculated by the function. Also, for the fourth-order function, there would be 15 parameters for minimizing the error.
In this procedure, the mean absolute relative error is considered as the objective function of the procedure, which is calculated from the following equation:
%MARE= 1 n
n k=1
|EtAccal− EtAcdata| EtAcdata
× 100 (8)
where n is the number of data points and EtAccaland EtAcdataare the amount of ethyl acetate values obtained by curve fitting and mathematical modelling calculations, respectively.
As mentioned before, the DE method is used to find the optimal parameters.
4.3. Differential evolution method
DE is a heuristic approach based on optimization evolutionary algorithms and random search methods. This technique was presented by Storn and Price for the first time and soon it has been paid much attention as a fast, easy to use, robust and efficient method of optimization, and consequently employed for several kinds of problems with computation- intensive cost functions. DE is used for minimization of non-linear, non-differentiable, multimodal and continuous space functions and is able to handle optimization problems involving complex mathematical models [33]. DE is a self-organizing and stochastic direct search approach that is compatible for numerical optimization and is more likely to find a function’s true global optimum. These advantages make DE a useful and an applicable technique for optimization problems such as chemical engineering relevant problems. DE has already been successfully applied for solving several complex problems and is now being identified as a potential source for more accurate and faster optimization. Very little input is required from the user when using the DE method. The main input variables chosen by the user are NP, number of populations, F, scaling factor and CR, crossover constant.
The basic strategy of the DE algorithm consists of four main parts as shown in Figure 4 as a six-step procedure:
(1) In the first part, an initial vector population is randomly chosen that covers the entire parameter space. The number of population chosen by the user remains constant during the optimization procedure.
Figure 4. A schematic flowchart representing one stage in DE algorithm.
(2) In the second part, for each point (called target vector) of the population, a mutant vector is generated (mutation/perturbation). The first four steps shown in Figure 4 are related to this part of the algorithm.
(3) In the third part, a trial vector for each point is generated using the crossover pro- cess (crossover/recombination). It should be mentioned that the perturbation and recombination processes are employed to increase the diversity of the generated vectors in the feasible region and consequently the possibility of the random search method to find the optimal point in the whole feasible region increases.
(4) In the fourth and the last part, each point is compared to its associated trial vec- tor utilizing objective function evaluations, and the better vector is selected as a member of the next generation. Thus, continuing the procedure new populations are generated which approach the global optimum through iteration to iteration.
Complete information about the mentioned steps and the relevant mathematical formula- tions is available in Price and Storn’s paper and website [33,34]. Different strategies can be adopted in the DE algorithm depending on the type of problem for which DE is employed.
The strategies can vary based on the vector to be perturbed, number of difference vectors considered for perturbation and, finally, the type of crossover used. Ten different strategies have been proposed by Price and Storn that are represented in the form DE/m/n/l:
(1) DE/best/1/exp (2) DE/rand/1/exp (3) DE/rand-to-best/1/exp (4) DE/best/2/exp (5) DE/rand/2/exp
(6) DE/best/1/bin (7) DE/rand/1/bin (8) DE/rand-to-best/1/bin (9) DE/best/2/bin (10) DE/rand/2/bin
where m stands for a vector to be perturbed that could be the best member of the previous generation or could be randomly chosen; n is the number of difference vectors consid- ered for perturbation of m; and l is the type of crossover process (exp: exponential; bin:
binomial) [34].
A simple pseudo code for the sixth strategy of the DE method is given below:
Input D, NP≥ 4; F ∈ (0, 1); CR ∈ [0, 1] and initial bounds: lower (ui); upper (ui);
i= 1, . . . D
Initialize P= {−→u1 . . . ,−→uNP} For each individual j∈ P
−
→uij= lower(ui)+ randi[0, 1]× (upper(ui)− lower(ui)); i= 1, . . . D End for each
Evaluate P
While the stopping iteration is not satisfied do for all j≤ NP
Set r1= rbestrandomly select r2,r3∈ (1, . . . NP) j= r1= r2= r3
randomly select irand∈ (1, . . . D) for all i≤ D
ui=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
uri3+ F × uri1− uri2
if (random [0, 1]< CR
i= irand
ui otherwise end for all
if f ( u) ≤ f uj Then,−→
u will be a member of the next iteration, Else, −→ujwill be a member of the next iteration end for all
end while Print the result
In addition, several modified versions of DE such as MDE, TDE and CDE have been proposed in recent years, which make the traditional DE algorithm faster, more efficient and more applicable. Because the mathematical model considered in this study is not too complex to be handled, traditional DE algorithm would be useful and applicable. Also, the sixth strategy is employed due to its efficiency in spite of the simple form. Figure 4 also illustrates the sixth strategy schematically.
Choosing NP, F and CR depends on the specific problem applied, and is often dif- ficult. However, some general guidelines are available. Normally, NP should be about 5–10 times the number of parameters in a vector. As for F, it lies in the range 0.4–1.0.
Initially, F= 0.5 can be tried, then F and/or NP is increased if the population converges
Table 4. Strategy and parameters used for DE.
Strategy DE/best/1/bin
Population size (NP) 20× ND
Scaling factor (F) 0.8
Crossover constant (CR) 0.8
prematurely. A good first choice for CR is 0.1, but in general CR should be as large as pos- sible. More details of the basic version of DE (pseudo code), its strategies and choosing the operating parameters have been reported by Babu, Angira and Munawar [35–37]. In this study, the strategy and parameters used for DE are presented in Table 4. It should be men- tioned that simulations are coded in MATLAB on a PC with AMD PhenomTMII X4 965, 3.41 GHz processor, 3.25 GB RAM.
5. Results and discussion
In the following subsections, the results of the optimization procedure are discussed.
Furthermore, the performance of the reaction–separation system with optimal values of RVand RLare discussed in detail.
5.1. Optimization results
The three-dimensional surface representing the variation of ethyl acetate production rate versus RVand RL is illustrated in Figure 5. As mentioned in the previous section, these values are obtained through frequently solving the mathematical model for different values of RVand RLin the feasible region.
Representing the surface shown in Figure 5 entirely by a polynomial equation may change the location of the optimum considerably. Therefore, a smaller region is considered for more accurate curve fitting. Figure 6 more clearly illustrates the region including the optimum.
0.2 0.4 0.6 0.8 0.4 0.2
0.6 0.8 0.14 0.16 0.18 0.2
RV RL
EtAc (lbmol)
Figure 5. Variation of ethyl acetate production versus RVand RL.
0.2
0.4 0.6
0.2 0.1 0.4 0.3
0.6 0.5 0.16 0.17 0.18 0.19 0.2 0.21
RV RL
EtAc (lbmol)
Figure 6. The smaller part of the response surface which includes the optimum.
Table 5. The number of parameters and the objective function evaluation for the fitted polynomials.
Order of fitted polynomial Number of parameters %MARE
Second order 6 1.102
Third order 10 0.393
Fourth order 15 0.309
Fifth order 21 0.332
Second- to fifth-order polynomial forms are employed and checked to find the most appropriate algebraic model for the surface. The related objective function evaluations (%MARE) obtained for each form as well as the relevant number of parameters are presented in Table 5.
The best value of %MARE is obtained as 0.309%. As can be seen, the fourth-order model with the following form is the best choice to fit the considered data set:
y= b0+ b1z1+ b2z2+ b12z1z2+ b11z21+ b22z22+ b112z12z2+ b122z1z22+ b111z31 + b222z32+ b1122z12z22+ b1112z31z2+ b1222z1z32+ b1111z41+ b2222z42 (9)
Table 6 represents the optimal parameters determined for Equation (9) and Figure 7 illustrates the modelled surface.
Figure 8 depicts the best values for objective function versus the iterations with regard to the fourth-order surface. The trend of approach clearly demonstrates the good con- vergence properties of DE method. For instance, as shown in Figure 8, it can be clearly visualized that the speed of DE to approach the optimum especially for the first itera- tions is very high. It should be noted that because of the large domain of variations during the first iterations, the results shown in Figure 8 are related to more than 1000 itera- tions. Furthermore, the authors conducted the same procedure in the same conditions with genetic algorithm (MATLAB Toolbox) to compare the DE results. The %MARE for genetic algorithm reached to 1.5688 after 30,000 iterations. This demonstrates the efficiency of DE algorithm and its more rapid convergence ability.
Table6.Theoptimalparametersforthefittedfourth-orderpolynomial. Parameterb0b1b2b12b11b22b112B122b111b222b1122b1112b1222b1111b2222 Value0.09020.05260.50890.82270.8255−1.8463−3.1634−0.5587−1.58383.07721.73722.0983−0.33510.7256−1.7893
0.2 0.4 0.2 0.1
0.4 0.3 0.6 0.5
0.16 0.17 0.18 0.19 0.2 0.21
RV RL
EtAc (lbmol)
Figure 7. Fourth-order surface fitted to the data set.
0 0.5 1 1.5 2 2.5 3
×104 0
0.5 1 1.5 2 2.5
Iteration
Objective function evaluation
Figure 8. The objective function (%MARE) evaluations versus the iterations.
Table 7. Optimal values of RL, RVand the ratio of product to feed.
Optimal parameter Optimal value
RV 0.4038
RL 0.2702
Objective function evaluation 0.2046
The next step is to find the maximum value of ethyl acetate production applying the polynomial as the model using another model-based optimization procedure. For this approach too the DE algorithm is employed and the optimal values of RL, RVas well as the ratio of product to feed are determined (Table 7).
5.2. Column performance
In this subsection, the performance of the reaction–separation system at optimal condition is discussed. Figure 9 shows the composition profile in the condenser with respect to time for the reflux ratio of 4. As EtAc is the lightest compound, withdrawing it will shift the chemical equilibrium to the right. This fact results in an ascending time-varying concentration profile.
Figure 10 demonstrates the composition profile of EtAc in the reboiler with respect to time for the reflux ratio of 4. First, the composition of EtAc increases. As the chemical reaction starts up, the amount of produced EtAc is more than the capacity of the separa- tion system. When the reaction rate due to the consumption of the reactants is decreased, more EtAc can be separated. Therefore, the composition profile of EtAc shows a descend- ing behaviour. Water and acetic acid have higher boiling points in the mixture, and large amounts of these two components are expected to be accumulated at the end of the opera- tion. After approximately 600 minutes of the operation start-up, when all EtAc and EtOH are removed from the column a separation just between water and AcOH is expected. Also, as water is the lighter component than AcOH, first it will decrease in the bottom of the col- umn. This expectation conforms to the result of the simulation as shown in Figure 11.
Figure 11 shows the water composition profile of the side stream tray (the seventh tray) with respect to time. The start-up of the batch distillation system comes with an ascending behaviour for the water composition profile because the chemical reaction proceeds and the composition of water as a by-product increases. When the reaction rate, due to the lack of the reactants, decreases, the production rate of the water decreases and the water composition profile shows a descending behaviour. Near the end of the operation, heavy components such as water can reach the upper part of the distillation system. As a result, the composition of the water increases and the ascending behaviour for the water composition profile can be observed again.
Moreover, as mentioned before, the aim is to maximize the ethyl acetate amount in distillate. Besides, authors consider the ethyl acetate in side stream to be minimized in the objective function. Thus, more water can be obtained as a by-product in side stream.
It seems that the RDWC is capable of separating two different components in multi- component reactive separation systems. Obviously, taking apart numbers of trays in RD
0 100 200 300 400 500 600 700
0 0.2 0.4 0.6 0.8 1
Time (minute) Condencer composition (mole fraction)
AcOHEtOH EtAcWater
Figure 9. Condenser composition profile of RDWC for reflux ratio of 4.
0 100 200 300 400 500 600 700 0
0.2 0.4 0.6 0.8 1
Time (minute) Boiler composition (mole fraction)
AcOHEtOH EtAc Water
Figure 10. Reboiler composition profile of EtAc for reflux ratio of 4.
0 100 200 300 400 500 600 700
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time (minute) Water composition (mole fraction)
Figure 11. Composition profile of water in side stream tray (tray 7).
column just by a wall is really more efficient in case of energy and cost in comparison to using two conventional RD columns with reboilers.
6. Conclusion
In this study, the production of ethyl acetate in a reactive dividing wall batch distillation column (RDWC) is studied. The effect of the side stream tray position on the amount of produced ethyl acetate and extracted water is considered. The optimum side stream tray is the seventh tray. The effect of the vapour and liquid split ratio variations on production is studied and the optimal values are obtained. For optimization, first, a model reduction procedure is employed due to the strictly non-linear nature of the model. For this, a set of data points are collected through frequent running of the mathematical model for several RVand RLvalues. Inspired by response surface methodology, a surface is then fitted to this data set using DE method to determine the optimal surface parameters. The optimal sur- face relevant algebraic equation is then considered the reduced form of the complex model
and the optimal values are also obtained using the DE method. Furthermore, the system performance with the optimal values of RVand RLhas been discussed in detail. It should be mentioned that this work is a simulation study in which the obtained results demon- strate that the recommended methodology is perfectly capable for model reduction and optimization of chemical processes containing complex non-linear mathematical models.
Nomenclature
Symbol Dimension Description
AcOH − Acetic acid
EtOH − Ethanol
EtAc − Ethyl acetate
DC cm Column diameter
D kmol/h Distillate flow rates
DensityAve kg/m3 Average density
H J/mol Molar enthalpy
L kmol/h Liquid flow rates
Lside 1 kmol/h Liquid flow rate from side column to tray P
Lside top kmol/h Liquid flow rate from tray M to the top of side column
LV cm3/h Liquid volumetric flow rate
M kmol Molar liquid hold-up on tray
MV cm3 Volumetric liquid hold-up on tray
%MARE − Mean absolute relative error percentage MwAve kg/kmol Average molecular weight
Nc − Number of components in mixture
QR kJ/h Reboiler heat input
Rij 1/h Chemical reaction rates for jth component on ith tray
R − Reflux ratio
RV − Ratio of interlink molar flow to interstage flow for vapour phase RL − Ratio of interlink molar flow to interstage flow for liquid phase Rs − Ratio of side stream molar flow to interstage flow for liquid
phase
V kmol/h Vapour flow rates
Vside 1 kmol/h Vapour flow rate from tray P to the bottom of side column
Vside top kmol/h Vapour flow rate from the top of side column to tray M
WHS cm Weir height
WLS cm Weir length
XEtAc − Ethyl acetate composition
XEtAc,D − Average mole fraction of EtAc in distillate product XEtAc,S − Average mole fraction of EtAc in side stream product
XWater,S − Average mole fraction of water in side stream product
S.P lb mol Total mole of side stream
D.P lb mol Total mole of distillate
T min Batch time
x − Liquid mole fraction
y − Vapour mole fraction
γi − Activity coefficient
NP − Number of population
ND − Number of decision variable
F − Mutation factor
CR − Crossover
References
[1] S. Hernandez and A. Jimenez, Design of energy-efficient Petlyuk systems, Comput. Chem. Eng.
23 (1999), pp. 1005–1010.
[2] F.B. Petlyuk, V.M. Platonv, and D.M. Slavinskii, Thermodynamically optimal methods for separating multicomponent mixtures, Int. Chem. Eng. 5 (1965), pp. 555–561.
[3] S. Hernandez and J.G. Segovia, Thermodynamically equivalent distillation schemes to the Petlyuk column for ternary mixtures, Energy 31 (2003), pp. 2176–2176.
[4] S.M. Khazraee, S.A. Ghorayshi, and A.H. Jahanmiri, Model reduction and optimization of reactive batch distillation based on the Adaptive Neuro-Fuzzy inference system and Differential evolution, Neural Comput. Appl. 20 (2011), pp. 239–248.
[5] S.M. Khazraee and A.H. Jahanmiri, Composition estimation of reactive batch distillation by using adaptive neuro-fuzzy inference system. Product engineering and chemical technology, Chin. J. Chem. Eng. 18 (4) (2010), pp. 703–710.
[6] J. Kolena, J. Lederer, P. Moravek, J. Hanika, Q. Smejkal, and D. Skala (2004), Method of pro- ducing ethyl acetate and equipment for carrying out this method. US patent, No. 6693213 B1.
[7] A.J. Brugma, Fractional Distillation of Liquid Mixtures, Holanda, Patente 41 (1937), p. 850.
[8] G. Kaible, Rectification column for the continuous distillative separation of a multi-substance mixture, European Patent Application (1987), EP0804951.
[9] C. Triantafyllou and R. Smith, The design and optimization of fully thermally coupled distillation columns, Trans. IChemE A 70 (1992), pp. 118–132.
[10] G. Dunnebier and C. Pantelides, Optimal design of thermally coupled distillation Columns, Ind. Eng. Chem. Res. 38 (1999), pp. 162–176.
[11] A. Mutalib, Operation and control of dividing wall distillation columns, Trans. IchemE 76 (1998), pp. 308–318.
[12] M. Serra and M. Perrie, Study of the divided wall column controllability influence of design and operation, Comput. Chem. Eng. 24 (2000), pp. 901–907.
[13] K.A. Amminudin and R. Smith, Design and optimization of fully thermally coupled distillation columns. Part 2: application of dividing wall columns in retrofit, Chem. Eng. Res. Des. 79 (2001), pp. 716–724.
[14] M. Muralikrishna, V.K.P. Madhavan, and S.S. Shah, Development of dividing wall distillation column design space for a specified separation, Chem. Eng. Res. Des. 80 (2002), pp. 155–166.
[15] I. Suzuki, H. Yagi, M. Hirata, and H. Komatsu, Calculation of multicomponent distillation accompanied by a chemical reaction, J. Chem. Eng. Jpn. 4 (1971), pp. 26–33.
[16] K. Alejski, M. Bogacki, and J. Szymanowski, Synthesis of ethyl acetate in a reacting distillation column, Pol. J. Appl. Chem. 33 (1989), pp. 509–520.
[17] J. Simandl and W.Y. Svrcek, Extension of the simultaneous solution and inside outside algorithms to distillation with chemical reactions, Comput. Chem. Eng. J. 15 (1991), pp. 337–348.
[18] H. Komatsu, Application of the relaxation method for solving reacting distillation problems, J.
Chem. Eng. Jpn. 10 (1977), pp. 200–205.
[19] H. Komatsu and C.D. Holland, A new method of convergence for solving reacting distillation problems, J. Chem. Eng. Jpn. 4 (1977), pp. 292–297.
[20] J. Grosser, F. Doherty, and M. Malone, Modeling of reactive distillation systems, Ind. Eng.
Chem. Res. 26 (1987), pp. 983–989.
[21] M. Bogacki, K. Alejski, and J. Szymanowski, The fast method of the solution of a reacting distillation problem, Comput. Chem. Eng. 13 (1989), pp. 1081–1085.
[22] K. Alejski, Computation of the reacting distillation column using a liquid mixing model on the plates, Comput. Chem. Eng. 15 (1991), pp. 313–323.
[23] N.E. Welch, L.D. Durbin, and C.D. Holland, Mixing on valve trays and in downcomers of a distillation column, AIChE J. 10 (1964), pp. 373–379.
[24] K. Aljeski and F. Dupart, Dynamic simulation of the multi component reactive distillation, Chem. Eng. Sci. 51 (1996), pp. 4252–4252.
[25] R. Schneider, C. Noeres, L.U. Kreul, and A. Gorak, Dynamic modeling and simulation of reactive batch distillation, Comput. Chem. Eng. J. 25 (2001), pp. 169–176.
[26] I. Mueller, C. Pech, D. Bhatia, and E.Y. Kenig, Rate based analysis of reactive distillation sequence with different degrees of integration, Chem. Eng. Sci. 62 (2007), pp. 7327–7335.
[27] S. Hernandez and R.F. Vergara, Reactive dividing wall distillation columns: simulation and implementation in a pilot plant, Chem. Eng. Proces. 48 (2009), pp. 250–258.
[28] P. Kaelo, A numerical study of some modified differential evolution algorithms, Eur. J. Operat.
Res. 169 (2006), pp. 1176–1184.
[29] M. Rovaglio and M.F. Doherty, Dynamics of heterogeneous azeotropic distillation columns, AIChE J. 36 (1990), pp. 39–52.
[30] H.Y. Lee, H.P. Huang, and I.L. Chien, Control of reactive distillation process for production of ethyl acetate, J. Proc. Cont. 17 (2007), pp. 363–377.
[31] M.A. Bezerra, R.E. Santelli, E.P. Oliveira, L.S. Villar, and L.A. Escaleira, Response surface methodology (RSM) as a tool for optimization in analytical chemistry, Talanta 78 (2008), pp. 965–977.
[32] D.C. Montgomery, Design and Analysis of Experiments, 3rd ed., John Wiley & Sons, New York, 1991.
[33] R. Storn and K. Price, Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim. 11 (1997), pp. 341–359.
[34] K. Price and R. Storn, Web site of DE as on July 2006. (2006). Available at http://www.ICSI.
Berkeley.edu/~storn/code.html
[35] B.V. Babu and R. Angira, Modified Differential Evolution (MDE) for optimization of non-linear chemical processes, Comput. Chem. Eng. 30 (2006), pp. 989–1002.
[36] B.V. Babu and S.A. Munawar, Differential evolution strategies for optimal design of shell and- tube heat exchangers, Chem. Eng. Sci. 62 (2007), pp. 3720–3739.
[37] R. Angira and B.V. Babu, Optimization of process synthesis and design problems: a modified differential evolution approach, Chem. Eng. Sci. 61 (2006), pp. 4707–4721.
Appendix A
Total and species mass and energy balance equations for specified sections of the reactive Petlyuk system: Reboiler (subscript B):
dMB
dt = L1− VB+ MBRB (A1)
d MBxB, j
dt = L1x1, j− VByB, j+ MBRB, j, j= 1, . . . , Nc (A2) d(MBHL,B)
dt = L1HL,1− VBHV ,B+ QB (A3)
First plate- main column (subscript 1):
dM1
dt = VB+ L2− V1− L1+ M1R1, (A4) d(M1x1, j)
dt = VByB, j+ L2x2, j− V1y1, j− L1x1, j+ M1R1, j, j= 1, . . . , Nc (A5) d(M1HL,1)
dt = VBHV ,B− V1HV ,B+ L2HL,2− L1HL,1. (A6) Plate P (subscript P):
dMP
dt = VP−1+ LP+1+ Lside1− VP− Vside1− LP+ MPRP, (A7)
d(MPxi, j)
dt = VP−1yP−1, j− VPyP, j− Vside1yP, j+ LP+1xP+1, j− LPxP, j
+ Lside1xside1, j+ MPRP, j, j= 1, . . ., Nc (A8)
d(MPHL,P)
dt = VP−1HV ,P−1− VPHV ,P− Vside1HV ,side1
+ LP+1HL,P+1− LPHL,P+ Lside1HL,side1
. (A9)
Plate S – side stream plate (subscript S):
dMS
dt = LS+1− LS− S + VS−1− VS+ MSRS, (A10)
d(MSxi, j)
dt = LS+1xS+1, j− LSxs, j− S.xS, j+ VS−1ys−1, j
− VSyS, j+ MsRS, j, j= 1, . . . , Nc (A11) d(MSHL,S)
dt = VS−1HV ,S−1− VSHV ,S+ LS+1HL,S+1− LSHL,S+ SHL,S. (A12) Intermediate plates (subscript i):
dMi
dt = Vi−1+ Li+1− Vi− Li+ M1R1, (A13) d(Mixi, j)
dt = Vi−1yi−1, j− Viyi, j− Li+1xi+1, j− Lixi, j+ MiRi, j, j= 1, . . ., Nc, (A14) d(MixL,1)
dt = Vi−1HV ,i−1− ViHV ,i− Li+1HL,i+1− LiHL,i. (A15) Plate m (subscript m):
dMm
dt = Vm−1+ Lm+1− Lside.top− Vm+ Vside.top− Lm+ MmRm, (A16)
d(Mmxm, j)
dt = Vm−1ym−1, j− Vmym, j+ Vside.topyside.top, j+ Lm+1xm+1, j− Lmxm, j
− Lside.topxside.top, j+ MmRm, j, j= 1, . . ., Nc (A17)
d(MmHL,m)
dt = Vm−1HV ,m−1− VmHV ,m+ Vside.topHV ,side.top
+ Lm+1HL,m+1− LmHL,m+ Lside.topHL,side.top
(A18)
Top plate-main column (subscript Nt):
dMNt
dt = VNt−1+ L0− LNt− LNt+ MiRii, (A19) d(MNtxNt, j)
dt = VNt−1yNt−1, j− L0xD, j− LNtxNt, j− VNtyNt, j+ MNtRNt, j (A20) d(MNtHL,Nt)
dt = VNt−1HV ,Nt−1− VNtHV ,Nt+ L0HL,D− LNtHL,Nt. (A21)
Condenser (subscript D):
dMD
dt = VNt− L0− D + MDRD (A22)
d(MDxD, j)
dt = VNtyNt, j− DxD,j− L0xD, j+ MDRD, j, j= 1, . . ., Nc (A23) d(MDHL,D)
dt = VNtHV ,Nt− L0HL,D− DHL,D+ QD (A24)
L0= VNt− D (A25)
Appendix B
The parameters of a non-random two-liquid (NRTL) model are presented in Table A1:
lnγi=
nc j=1τjiGjiXj
nc k=1GkiXk
+
nc j=1
GjiXj
nc j=1GkjXk
τij−
nc
k=1XkτkiGkj
nc k=1GkjXk
,
Gij= exp(−αijτij),τij= aij+ bij
T (K),αij= αji, τii= τjj= 0.(35)
Table A1. NRTL model parameters [4].
Component i HAc(1) HAc(1) HAc(1) EtOH(2) EtOH(2) EtAc(3)
Component j EtOH(2) EtAc(3) H2O(4) EtAc(3) H2O(4) H2O(4)
aij 0 0 −1.9763 1.817306 0.806535 −2.34561
aji 0 0 3.3293 −4.41293 0.514285 3.853826
bij −252.482 −235.279 609.8886 −421.289 −266.533 1290.464
bji 225.4756 515.8212 723.888 1614.287 444.8857 −4.42868
αij 0.3 0.3 0.3 0.1 0.4 0.364313
