Automatic design and optimization of column sequences and column

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Process Systems Engineering

Automatic design and optimization of column sequences and column stacking using a process simulation automation server Yuan-Wei Ni, and Jeffrey Daniel Ward Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03943 • Publication Date (Web): 02 May 2018 Downloaded from http://pubs.acs.org on May 5, 2018

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Industrial & Engineering Chemistry Research

Automatic design and optimization of column sequences and column stacking using a process simulation automation server Yuan-Wei Ni and Jeffrey D. Ward* Dept. of Chemical Engineering, National Taiwan University, Taipei 106-07, Taiwan

*

Correspondence concerning this article should be addressed to Jeffrey. D. Ward at [email protected]. ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Simulation-optimization is applied to determine nearly optimal process designs in column sequencing and stacking problems using a simulated annealing algorithm (SAA) in the client (MATLAB) and a single distillation column in the server (Aspen Plus). Each column in every sequence is optimized separately and the total cost of each sequence is calculated. This method benefits both from the rigor of column modeling in a process simulator and the flexibility of implementing sequential procedures and derivative-free optimization in a procedural programming language. Using this method, a single column can be optimized within six minutes of computer time. Furthermore, when stacking rules and available utilities are specified, a nearly optimal heat-integrated distillation column sequence can be designed automatically. Results are illustrated using two case studies of column sequencing problems.

Keywords: distillation column sequence, column stacking, heat integration, simulated annealing algorithm, optimization

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1. Introduction Optimization is the selection of the best set of variables from numerous alternatives. Chemical engineering process design optimization problems are often complicated, time-consuming, discrete and non-convex. If a near-optimal conceptual design can be obtained automatically and efficiently for further detailed study, a great deal of time and effort on the part of process designers can be saved. Synthesis of the separation systems is one of the most important tasks in chemical process design. An important issue in the synthesis of the separation system is the choice of distillation column sequence (DCS). The problems can be defined as follows. Given a known feed stream and desired product compositions, determine the column sequence that minimizes the total annual cost (TAC). Many researchers [1–6] have proposed different methods for solving this problem. Optimization algorithms, including the simulated annealing algorithm, have been (hydrodealkylation process [7-8], process synthesis [9-11], heat exchanger network [12], pressure swing distillation [13], reactive distillation [14]), genetic algorithm (dividing wall columns [15], process synthesis [16-18]) and tabu search (hydrodealkylation process [7], synthesis of reaction/separation systems [19]) have successfully been used in process design and optimization problems. Simulation-optimization [20-23] is a new type of optimization technique that replaces deterministic mathematical formulations (usually based on simplified process models) with the use of simulation. Simulation-optimization has been applied to chemical engineering processes such as a turbo-expander process [24], pressure swing distillation [13], reactive distillation process [14], dividing wall columns [15], distillation of ternary mixtures and extractive distillation [18]. In distillation column sequences, it is possible that the condenser of one column provides some or all of heat duty to the reboiler operating at a lower temperature in the other column [25]. The heat integrated distillation synthesis problems can be defined as follows: Given available utilities and a



distillation column sequence, determine whether column stacking is feasible and if so what heat—exchange connections should be made between columns and how much money can be saved by stacking. This problem has been studied by a number of authors [10, 18,26–32]. Although the problems of column sequencing and stacking have been studied by many researchers, all existing methods have certain limitations. Approaches that rely on shortcut column design methods are limited by the accuracy of those methods, and approaches that use rigorous process models but rely on heuristics for column sequencing and stacking may fail to find the best process design. In the present work, simulation-optimization is applied to column sequencing and stacking problems. For the optimization of each column, the simulated annealing algorithm (SAA) frequently used in the literature to obtain the optimum design using a rigorous column model in Aspen Plus and a process simulation automation server.


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2. Column modeling, sequencing and stacking 2.1. Single column simulation Simple distillation columns are those with a single feed stream and two product streams. Sharp separation is separation in which no component is distributed (i.e. present in a significant amount) in both product streams. Therefore, for simple sharp distillation, separation of an n-component mixture requires n-1 columns. Thompson and King [33] developed an equation for the number of sequences (Sn) for simple sharp separation of n components: S =

2 − 1 ! ! − 1 !

(1)

The number of column sequences increases dramatically as the number of components increases. Therefore, it becomes difficult to obtain overall column sequences conditions and optimization results. In this research, we make an assumption to simplify column sequencing and stacking problems. Figure (1) shows the concept and illustration. Each column is assumed to have only the components that must be separated. The pure or key components that are separated in previous columns do not appear in the later columns. With this assumption, a process simulator flowsheet containing only a single distillation column can be used to simulate all possible column sequences. With simple arrangements and combinations, all column sequences can be modelled. This saves a great deal of time compared to constructing a separate flowsheet simulation for every column sequence. In the single column, we can specify the inlet conditions, set column specifications or change any other variable from an external application and use the single column to model each column in every sequence separately.



2.2. Column sequencing procedure Using single column simulation discussed previously, the flowsheet is shown in Figure (3). The feed is specified to be saturated liquid at the pressure of the previous column. If the column is the first column, the feed pressure is set to be 1 bar. The pump duty is determined by comparing the inlet flow pressure and feed stage pressure. The column condenser pressure is 1 bar. After feed properties and pump duty are specified, the simulated annealing algorithm is applied to minimize the total annual cost (TAC). The design variables in this work are number of trays (NT) and feed stage (NF) for each column. All the above procedures are complete with Aspen Plus ActiveX automation server. The complete procedure is as follows: Step 1. Determine the feed stream pressure. The feed stream pressure of the first column is set to be 1 bar. For all other columns, the feed pressure is the pressure at the top or bottom of the previous column (whichever is the feed to the present column). Step 2. If the feed stream pressure is less than the column feed stage pressure, a pump is added to raise the pressure as necessary. Step 3. Proceed SAA to obtain the optimization Step 4. Proceed the next column until all columns are optimized.


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2.3. Column stacking procedure Separation of chemicals by distillation consumes a great deal of energy and reducing this energy demand is often a priority in separation system design. If two or more distillation columns are present in a process, and the condenser temperature in one column is higher than the reboiler temperature in the other column, then the higher-temperature condenser can be used to power the lower-temperature reboiler. Furthermore, it is possible to adjust column pressures to match temperatures in order to achieve heat integration. The basic idea was proposed by King [39] and the conceptual process design was implemented by Andrecovich and Westerberg et al. [27]. The approach is also called column stacking heat integration. Hence, given the available utilities, columns can be stacked by adjusting their pressures to appropriate values. Figure (4) shows the flowchart for column stacking with SAA used in this work. First, the average of condenser and reboiler temperatures is calculated for each column. Then columns are arranged and numbered in order of ascending average temperature. Next, for the column with the lowest average temperature (i=1) the condenser pressure corresponding to saturated liquid at a temperature is 10K higher than cooling water temperature is determined using a flash module in Aspen Plus. After the condenser pressure is determined, the column is designed using the SAA. The algorithm then determines the condenser pressure for the second column so that liquid in the condenser will be saturated at a temperature 10K higher than the reboiler of the first column. The procedure is repeated until no more stacking is possible. All the above statements are complete with Aspen Plus ActiveX automation server.

3. Simulated annealing algorithm (SAA) Many methods have been proposed and developed to solve optimization problems. Many process design optimization problems are non-convex and discrete. Derivative-free methods may perform better than derivative-based methods for such problems. Derivative-free, stochastic algorithms such as genetic algorithm (GA), simulated annealing algorithm and tabu search (TS) are



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commonly applied to such problems. SAA is a probabilistic algorithm for estimating global optimum. The concept of simulated annealing was first proposed by Metropolis et al. [34]. Kirkpatrick et al. [35] applied the concept of simulated annealing to optimization problem and successfully solved the problem of traveling salesman. SAA has the advantages that it involves fewer loops than a sequential optimization procedure and is relatively easy to program. The concept of simulated annealing is analogous to metallurgical annealing, in which metals are cooled in such a manner as to reduce crystal defects. At first, molecules are disordered at high temperature and gradually adopt a lower energy, ordered configuration as the metal is cooled. Therefore, they can be regarded as being in thermodynamic equilibrium at each moment and at each temperature. If the system is in thermodynamic equilibrium at given temperature (T), the probability of the system adopting an energy state (E) is given by the Boltzmann energy distribution: PE =

1 E exp − Zt

K T

(2)

Where Kb is the Boltzmann’s constant and 1/Z(t) is a normalization factor. Note that molecules are allowed to move randomly from original energy state to new energy state depending on the current temperature and energy state. Therefore, we can calculate the ratio of the probability between a possible new energy state and the old (current) energy states according to Boltzmann energy distribution, and define it as the acceptance probability: P =

PE!" E!" − E#$% ∆E = exp − = exp − PE#$% KT T

(3)

Where ∆E is the normalized energy difference between new and old energy state. Based on the Metropolis criterion, the new energy state is accepted if one of the following two statements is true. (1) If new energy state is less than old energy state, the system accepts the new state. (2) If new energy state is greater than old energy state but the acceptance probability, exp(−∆E/T), is greater than a value generates from uniformly distributed random numbers, the system accepts the new state. ACS Paragon Plus Environment

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Otherwise, the original energy state is retained. From above statements, we can see that the system is more likely to accept a high energy state as current solution at higher temperature, and less likely to do so at lower temperature. Therefore, SAA can avoid becoming trapped in local minima since it may accept higher energy states over lower energy states, especially at the beginning, and still achieves a low-energy state at the end. The complete procedure is described below and a flowchart for the SAA used in this work is shown in Figure (2). Step 1. Generate a set of variables dnew, which is nearby original design variables d. Step 2. Execute process simulator, and calculate ∆E (∆E = E(dnew)− E(d) ) from external application. Step 3. (a) If ∆E0 and exp(-∆E/T)>U(0,1), accept dnew as the new solution. U(0,1) is a uniformly distributed random numbers on the interval between 0 and 1. (c) Otherwise, set dnew =d. (The solution is unchanged.) Step 4. Return to step 1 and continue until thermodynamic equilibrium at the current temperature is achieved. Step 5. Reduce the system temperature based on the temperature decrement or cooling schedule. Step 6. Return to step 1 and continue until the freezing temperature (Tf) or the termination criterion is achieved. There are some annealing schedule parameters in SAA procedures, such as initial temperature (T0), freezing temperature, temperature decrement and equilibrium detection. Parameter settings have an important impact on the convergence of the algorithm and CPU execution time. Aarts et al. [36], Patel et al. [37], Painton et al. [38] and Cheng et al. [14] have discussed parameter settings for SAA. The definition of parameters and the move generator are discussed in the following sections.



3.1. Move generator Move generator is the generation of new design variables. The next design variables depend on the step sizes defined by users. Move generator can be categorized into two sections: integer variables and continuous variables. (a) Integer variables: dnew=d+INT{[2×U(0,1)-1]Nstep}, where Value_LB≤ dnew ≤ Value_UB (b) Continuous variables: dnew=d+{[2×U(0,1)-1]Nstep}, where Value_LB≤ dnew ≤ Value_UB d and dnew mean the original and new set of variables respectively. Value_LB and Value_UB are the lower and upper bound of dnew. INT means the value is rounded to the nearest integer, U(0,1) is a uniformly distributed random number on the interval between 0 and 1, and Nstep is the step size for each move. In this work, Nstep is set to be 1. 3.2. Initial Temperature (T0) Initial temperature is the start point of the system temperature. The purpose of initial temperature is to ensure that there is a high enough acceptance ratio at the start of SAA. If the acceptance ratio at the beginning of the simulation is too low, the system may not escape a local minimum. The initial temperature should be large enough to ensure that the algorithm has the possibility of finding the global optimum. However, a very large CPU execution time will be required if initial temperature is too high. In this work, an arbitrary value is specified and several hundred moves are made. If the acceptance is greater than 80%, we set that value as initial temperature for the system. Otherwise, the initial temperature is doubled and the procedure is repeated until an acceptance ratio greater than 80% is achieved. 3.3. Freezing Temperature The freezing temperature, also called the final temperature, is the termination point of the SAA. If the freezing temperature is close to initial temperature, the result may not be correct since only a few executions may occur. However, if freezing temperature is too small in comparison with the ACS Paragon Plus Environment

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initial temperature, computation time will be unnecessarily large. In this work, the total number of simulation executions is fixed, and the freezing temperature is determined by the calculation given the other parameters (i.e. initial temperature, temperature decrement and equilibrium detection criterion). The algorithm terminates when the system temperature reaches the given value. 3.4. Temperature Decrement The temperature decrement affects the convergence and execution performance of SAA. If system temperature is reduced too quickly, the energy state may not escape from a local minimum. If system temperature is reduced too slowly, CPU execution time will be longer than necessary. In this work, the expression of temperature decrement is as follows. T!() = αT , where 0.8≦α≦0.9 (α is temperature decrement factor and n is the counter) 3.5. Equilibrium Detection Equilibrium detection is a quasi-equilibrium at given temperature throughout the SAA. System temperature decreases when equilibrium is detected. NE should be large enough to ensure that the algorithm explores the neighborhood of a given set of variables in all directions at a given temperature. If there are n optimization variables and each variable can remain the same, increase or decrease, the equilibrium detection variable could be set equal to 3n. In this work, the equilibrium detection parameter is set to be 15 to ensure that equilibrium is achieved at each temperature. 3.6. Recovery after special cases While the simulated annealing is running, there are two special cases that must be handled. One is that the Aspen Plus flowsheet fails to converge, and the other is that the Aspen Plus program crashes. If the Aspen Plus flowsheet fails to converge, that point is discarded and the algorithm proceeds to try another point. The discarded point is not counted toward the equilibrium detection. To handle the case where Aspen Plus crashes, a second instance of Matlab is run in the background. If it is found that Aspen Plus has not responded after a certain amount of time (about fifteen ACS Paragon Plus Environment


seconds), the Aspen Plus task is killed (using the Windows command taskkill), Aspen Plus is restarted, the simulation file is reloaded and the optimization resumes from the point at which Aspen Plus became unresponsive.

4. Case Studies 4.1. Alcohol case study The case study used in this example is taken from King (1971) [39]. Boozarjomehry et al. [16-17] proposed a genetic algorithm to obtain an optimal column sequence using short-cut column models. Andrecovich and Westerberg [27-28] have discussed the heat-integrated distillation using both constant Q∆T and non-constant Q∆T approaches to calculate minimum utility use within utility bound. In this work, the procedure described previously is applied to determine an optimal column sequence and column stacking configuration. Five components are considered in this study: ethanol, isopropanol, n-propanol, isobutanol and n-butanol. Those are labeled A, B, C, D and E respectively according to their relative volatilities. The total flowrate is 500 koml/hr and component compositions are 0.25/0.15/0.35/0.10/0.15 respectively. The feed is saturated liquid at 1 bar. The purity of key components in each column is specified to be 0.98. UNIFAC property methods are used in Aspen Plus simulations. TAC calculations are taken from Luyben [40] and the detailed information is shown in appendix. Table (1) shows the 14 possible column sequences for a five-component separation problem. Table (S1) in the supplementary material shows optimization results including the optimal number of stages and feed stage for each column. The execution time is about five to six minutes per column. Therefore, it takes about 3 hours to determine the optimal design of all 34 columns in all 14 column sequences. After every column is optimized, the TAC for all fourteen column sequences can be determined by summation. Table (S2) shows the TAC for the fourteen column sequences and Figure (5) shows the sequence TAC results in the form of a bar chart. Sequence 1, which is the direct sequence, has the lowest total annual cost ($10.296×106). Figure (6) shows SAA results for


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the four columns in the direct sequence (sequence 1). As was discussed previously, SAA may accept solutions with a higher TAC at the beginning to avoid being trapped at extreme but should converge to a solution with a lower TAC at the end to ensure that the answer is near to the lowest cost. Therefore, the curve fluctuates significantly at the beginning but becomes flat by the end. For each column, the optimization is repeated three times using different initial conditions and SAA parameter values and similar results were found for each run. Figure (7) shows the column stacking T-Q diagram for fourteen sequences and Table (S3) shows the optimization results for number of trays and feed stage for each column in each stacked sequence. The operating pressure of each column is also shown in the table. The procedure stacks the columns in order by the average column temperature at the optimal pressure for that column. While this is likely to be the optimal stacking configuration, the possibility that a better configuration exists is not checked by the algorithm. Increasing pressure to achieve a given temperature allows for heat integration and reduces energy consumption. However, it may increase capital costs and energy costs due to operation at higher pressure or with higher-grade utility. Therefore, column stacking is not guaranteed to reduce total cost. Therefore, for each column sequence, the total cost of the stacked sequence is compared with the total cost of the unstacked sequence in Table (S4). Because of the reasons mentioned above, the percentage savings is quite different for different sequences, from 20 to 38 percent. Some sequences have enough space (temperature driving force between the hot utility and the hottest reboiler) that further stacking can be accomplished by dividing one of the columns into two columns operated at different pressures (double-effect distillation). Therefore, the previous column stacking procedure is modified to permit such splitting. Figure (8) shows the flowchart allowing column splitting if possible. A criterion is applied to determine whether column splitting can be implemented. If the high pressure steam temperature minus the highest reboiler temperature (Tavail) is greater than the temperature difference between condenser and reboiler of the highest heat duty



column (Tdifference) plus the driving force temperature difference (∆Tdriving force), further stacking by column splitting is possible. In that case, the column with the highest duty is split into two columns and the stacking procedure is repeated. This process is repeated until Tavail is less than Tdifference, at which point the algorithm is terminated. Figure (9) shows the results of further column sequence for sequences 1 to 3. Table (6) shows the comparison between column stacking without splitting and column stacking with column splitting. Column splitting can further reduce the TAC by about 8 to 10 percent. 4.2. Styrene case study Vasudevan et al. (2009) [41] have proposed several process design and control structure for the styrene process using integrated configurations. Luyben (2011) [42] has provided different process design for the styrene process, and developed the plantwide optimization and control structure. In this work, the procedure described previously is applied to determine an optimal column sequence and column stacking configuration. Four components are used in this study: benzene, toluene, ethylbenzene and styrene. Those are labeled A, B, C and D respectively according to their relative volatilities. It is assumed that total flowrate is 400 kmol/hr and each component has the same molar flowrate. The feed is assumed to be saturated liquid at 1 bar. The desired purity of key components in each column is set to be 0.999. Peng-Robinson physical properties are used in the Aspen Plus simulations. TAC calculations are taken from Luyben [40] and additional details are given in the appendix. Column sequences and labels are shown in Table (7). Table (8) shows the optimization results: The optimal number of trays and feed stage for each column and the minimum TAC. The computer time required is again about 5–6 minutes per column. Therefore, 1 to 1.5 hours was required to determine the minimum cost for all 13 columns in all five column sequences. Figure (10) shows SAA results for all 13 columns. With this data, the TAC of all five column sequences can be determined by summation. Table (9) and Figure (11) show the TAC results for all five column


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sequences. Sequence 1, the direct sequence, again has the lowest total annual cost ($6.0807 × 102 ). In order to verify the results, the entire direct sequence was constructed in Aspen Plus and the SAA was used to perform optimization. In this case, SAA changes six variables simultaneously (the number of trays and feed tray for each column). Table (10) shows the results and deviations between the simulation of the entire sequence and column-by-column simulations. The errors (less than about 0.3%) are quite small if the purity specifications are high. The flowsheets for the individual columns and the combined column sequence are shown in Figure S1 in the supplementary material. Column stacking was also studied for the styrene example. Figure (12) shows column stacking T-Q diagrams for five sequences and Table (11) shows the optimization results of number of trays and feed stage. The operating pressure at given temperature is also shown in the table. The number of trays required in a given column increase as pressure increases because of the difficulty of separation increases. Table (12) shows TAC comparison between column sequences with and without stacking. The percentage savings varies from 3 to 20 percent. The best stacked sequence, the direct sequence, is the same as the best unstacked sequence. From Figure (12), we can see that the separation of ethylbenzene and styrene dominates the majority of utility consumption as a result of the difficult separation. Further, the separation of ethylbenzene and styrene could be conducted under vacuum to prevent styrene polymerization if there are no inhibitors [42]. Therefore, the stacking procedure is modified so that the ethylbenzene/styrene separation occurs at lowest pressure. The new stacking flowchart is shown in Figure (13). It is almost the same flowchart as previous stacking flowchart except that the stacking sequence is modified. Figure (14) shows T-Q diagram results for five sequences. The third column cannot be stacked upon the second column because of the larger temperature range in the first two column. However, the second and third columns can both be stacked upon the first column. Therefore, the flowchart is modified again to achieve this goal. The further modified stacking flowchart is shown in Figure (15). The condenser temperature of columns 2 and 3 are set to be 10K



higher than the reboiler of column 1. Figure (16) shows further T-Q diagram results for five sequences and Table (13) shows TAC and comparison of two methods for stacking ethylbenzene/styrene first. When the second and third columns are stacked upon the first column, the reboiler and condenser duty is reduced but the energy cost of the third column is increased because of the use of higher-grade utility. The cost savings on a percentage basis vary from 5 to 13 percent.


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5. Conclusion Simulation-optimization is a useful optimization technique that replaces deterministic mathematical formulations with the use of simulation. In this work, simulation-optimization is applied to the problems of column sequencing and stacking. A simulated annealing algorithm written in Matlab interfaces with a flowsheet simulator, Aspen Plus, with a model of a single distillation column. The flowsheet is used to model and optimize every column in every sequence, after which the total cost (TAC) of each sequence can be determined. The results demonstrate that the optimal sequence can be obtained quickly, with each column being optimized within 5–6 minutes. For both case studies considered, the direct sequence was found to have the lowest cost. A column stacking algorithm is also implemented. Given utility temperatures, heat integration between columns can be achieved by adjusting the operating pressure of each column. Overall cost savings of 20–30% can be achieved compared to the unstacked sequences. Because a rigorous column model is employed, the results are more accurate than results based on shortcut column design methods. Furthermore, the simulated annealing algorithm is more likely to achieve a results closer to the global minimum than either a heuristic-based method or a sequential optimization method.



Nomenclature NT

Number of trays

NF

Feed stage

P

Pressure

T

Temperature

Q

Heat duty

Kb

Boltzmann’s constant

E

Energy state

T0

Initial temperature

Tf

Frozen temperature

Nstep

Step size

d

Design variables

new

d

New design variables

∆Tdriving force

Driving force temperature

INT

Rounded to the nearest integer

TAC

Total annual cost

SAA

Simulated annealing algorithm

Greek symbols α

Temperature decrement factor


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Figures

Figure (1) The concept and illustration of using single column to simulate entire systems Start To, Tf, NE, d No

T > Tf

End Optimization

Yes i=1 T=αT

Yes

i 3 NE

i=i+1

No Generate dnew Execute Aspen Plus Calculate E(dnew) and ∆4=E(dnew)-E(d) Metropolis Criteria ∆4 ≤ 5 67 89:5, < ≤ =>? −∆4/A

Yes

d=dnew E(d)=E(dnew)

No

Figure (2) The SAA flowchart


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P=Last case output pressure Vapor fraction=0 Pump pressure (1) If feed stage pressure is higher than input pressure, P= feed stage pressure (2) Else, P=inlet pressure

Figure (3) Single column simulation flowsheet



Start (n-columns need to be stacked) A =

Arrange columns by temperature with

Automatic design and optimization of column sequences and column

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