Process Synthesis of Batch Distillation Systems - Industrial

Article pubs.acs.org/IECR

Process Synthesis of Batch Distillation Systems Santosh Jain,† Jin-Kuk Kim,*,‡ and Robin Smith† †

Centre of Process Integration, The University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom Department of Chemical Engineering, Hanyang University, 222 Wangshimni-ro, Seongdong-gu, Seoul, 133-791, Republic of Korea

‡

S Supporting Information *

ABSTRACT: A novel synthesis framework is presented for evaluating different configurations and sequences of batch distillation processes and identifying the most appropriate flowsheet and cost-effective operating conditions simultaneously. A superstructure for batch distillation systems is constructed by employing a generic unit, which consists of segments of trays with top and bottom vessels attached to it, and accommodating three different types of integrated network design options for feed connections, sequencing connections, and configuration connections. This systematic formulation of batch distillation sequencing design problems allows not only the optimization of conventional flowsheets, but also the generation of novel configurations and sequences. The optimization framework is based on an mixed-integer nonlinear programming (MINLP) problem, which is effectively solved with a hybrid optimization method using simulated annealing and sequential quadratic programming (SA-SQP). Case studies are presented to demonstrate how the proposed approach can be applied in practice to improve cost-effectiveness of batch distillation systems.

1. INTRODUCTION Batch distillation is frequently used for the production of fine and specialty chemicals, and various configurations for batch distillation systems can be arranged. Although the batch rectifier is the most common configuration, nonconventional configurations can also be explored. Other batch distillation arrangements, such as the inverted column (batch stripper), middle vessel column, and multivessel column (Figure 1) have

from the results of many simulation runs for three component mixtures of different compositions. Three column configurations (i.e., rectifier, stripper, and middle vessel column) were compared for various performance indices, including product purity and recovery, flexibility and feasibility of the design, and thermodynamic efficiency. Various design experiments were carried out for a three-component feed at different compositions and product purity requirements. The middle vessel column was found to be superior when compared with the rectifier and stripper configuration. The rectifier configuration is advantageous if the feed is rich in the more-volatile component and high purity of the more-volatile component is required. The stripper is preferred if the feed is rich in the lessvolatile component and high purity of the less-volatile component is required. Although the work provides heuristic rules for column selection at high and low composition of more- and less-volatile components, no guidelines can be provided for intermediate compositions. The sequence of the rectifier, stripper, and middle vessel columns was not considered, which might have provided better process performance. Heuristic rules are often contradictory, and their applicability is often limited for the narrow operating range. Also, heuristic rules are not flexible to consider operational changes during the operation, since the heuristics are applicable at the start of the operation. The batch distillation synthesis problem was solved, considering only the batch rectifier.2 A superstructure was presented that embedded combinations of direct and indirect separations. A shortcut model of batch distillation was used, and an optimization model was solved using a nonlinear programming method. Cut locations for various distillation

Figure 1. Different configurations for batch distillation separation.

the potential to improve the performance, depending on the separation. To synthesize batch distillation systems, decisions on the configuration of process flowsheet should be made, including identifying the best sequence. The synthesis of batch distillation systems is not a trivial task, because a batch distillation flowsheet may include several configurations. A systematic way is needed to screen different configurations and sequences and identify the most economic and operable flowsheet. Seeking novel configurations is also an important feature in the process synthesis of batch distillation. This allows designers to explore new opportunities in the design for achieving better productivity and/or energy-efficient operation of batch distillation systems. Kim and Diwekar1 presented some heuristics rules for the selection of the best configuration. Heuristics were developed © XXXX American Chemical Society

Received: January 1, 2013 Revised: May 7, 2013 Accepted: May 28, 2013

A

dx.doi.org/10.1021/ie400003p | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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maximized. A general problem statement for such design problems is given as follows: For a given feed, the thermodynamic and physical property data, desired composition and flow rate of product composition, and given process performance measures determine the design and operational options of batch distillation systems that maximizes the process performance. 2.1. Superstructure. The representation uses a stack of trays (tray segments) with top and bottom vessels attached as a generic unit to represent different batch distillation designs. Figure 2 shows a superstructure proposed for three products,

fractions and reflux ratios were optimized. However, design of multiple columns and complex columns was not addressed. The batch distillation sequencing problem was considered for a batch rectifier−stripper combination.3 Their design framework considers rectifier−stripper combinations, and used a distillation unit consisting of top and bottom vessels, so that a column can be operated either as a rectifier or as a stripper. A mixed logic dynamic optimization approach was proposed for batch distillation design. Complex columns and multiple columns were not addressed in the formulation. The approach proposed needs a good initialization of the primal subproblem, because the design changes by the master problem provide complete new process dynamics. The approach faces feasibility problems for each new design generated. Also, it requires more time to search for a feasible solution than to optimize it. The middle vessel batch distillation has been modeled and simulated in conjunction with setting up control strategies, as an alternative option to the conventional batch distillation, in which two products are simultaneously produced.4 The heat-integrated design concept has been further considered to the middle vessel batch distillation for reducing energy consumption by applying a heat pumping system, in which part of the vapor product is first compressed with the aid of a variable-speed compressor, and then is used to provide the heat in the reboiler before remixing it with the other condensed vapor product stream.5,6 On the other hand, improving energy efficiency of the batch distillation column had been attempted through the introduction of a thermally coupled column design in which internal heat integration between the rectifying section and a concentric reboiling jacket is made, and the produced vapor stream in the top of the jacket is compressed and introduced at the bottom of the rectifying section.7 This internally heat-integrated distillation column was investigated further to assess two typical distribution options, namely, uniform heat-transfer area and uniform heat-distribution schemes.8 The study was also made to improve conventional operation strategies and to identify feasible and cost-effective control schemes, leading to savings of energy and operating times for batch distillation systems. One such example is to exploit the column operating in a cyclic manner by transitioning between rectification and stripping operation.9 Overall, previous work for batch distillation synthesis is limited to certain configurations and a fixed number of columns. Also, the formulation and strategy proposed by Oldenburg et al.3 faces feasibility problems. Therefore, in this work, a more general design framework is proposed for the batch distillation synthesis. A superstructure is proposed to include all the possible configurational options for the batch distillation. This allows not only systematic investigation of conventional design for achieving significantly improved performance, but also effective identification of novel design and sequences for stimulating new development in batch distillation systems. The optimization framework is formulated as an MINLP problem, which is solved by a simulated annealing− sequential quadratic programming (SA-SQP) algorithm. With the new optimization and synthesis model, a set of batch distillation networks is proposed for the given objective function and design constraints. The new approach proposed in this paper optimizes the operating profile of batch operation simultaneously with structural configurations, which leads to reliable and practical design.

Figure 2. Superstructure representation for batch distillation systems.

which accounts for all the possible configurations, sequences, and operating policies. The superstructure enables the direct task, indirect task, and intermediate tasksor all combinations of these tasksto be generated in new column sequences. The superstructure consists of three types of connections: (i) Feed connection, which distributes the feed among the different vessels. Changing the feed connection in the superstructure provides new links between the existing columns. Figure 3 shows how new designs are obtained by changing the feed connections.

Figure 3. Different designs resulting from changing feed connections.

2. PROCESS SYNTHESIS FRAMEWORK This paper presents the automatic synthesis of batch distillation separation processes, such that the process performance is

(ii) Configuration connection, which combines different segments to form a new complex column (multivessel column) configuration. A configuration connection determines B



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Direct, indirect, intermediate, and hybrid separation tasks are considered in this work. Rather than a network connected between unit operations through process streams, task-based representation is used in the modeling and synthesis framework by defining sequencing information as separation tasks. This approach assumes a number of components to be separated into a predefined set of products with specified product purity and recovery. The use of separation tasks was first introduced by Hendry and Hughes,10 who addressed the simple distillation sequencing problem. For a mixture of n components, they identified discrete tasks (TS) on the product subgroup (SS) to represent all sequences (SQ).

whether the product is drawn from the top or bottom of a single segment unit. Thus, it converts a rectifier to a stripper and vice versa. A configuration connection also changes vapor flow connections between different segments to determine whether vapor should pass through or bypass intermediate vessels. (iii) Sequencing connection, which connects different segments to each other, and to the products. A sequencing connection is dependent on the byproducts to be withdrawn and separation tasks that are allowed to be performed. New designs configurations and sequences are generated by manipulating the above three types of connections. A middle vessel column and a multivessel column can be generated by combining different generic units. The combination of two units occurs when the bottom vessel of the top unit is merged with the top vessel of the bottom unit. If the units are arranged in a sequence, the superstructure allows the opportunity to pass the intermediate feed from one column to other columns in different ways. Intermediate feed can be charged to a single vessel of the unit, or it can be distributed among the different vessels of the same unit. 2.2. Separation Tasks. Batch distillation processes require intermediate byproducts to achieve the requirement of product purity. For a batch rectifier (stripper) operation, first the lightest (heaviest) product is withdrawn from the reflux drum (reboiler) until the composition of product is satisfied in the product tank. As the lightest (heaviest) components are withdrawn continuously, the reboiler (reflux drum) becomes leaner for the composition of the lighter (heavier) components. Furthermore, operation with the charge remaining in the reboiler (ref lux drum) cannot meet the product specification. Also, since the charge remaining in the reboiler (ref lux drum) still contains some amount of lighter (heavier) components, the next product cannot be withdrawn without removing the lighter (heavier) components. Therefore, a byproduct or off-cut or slop-cut or waste product is withdrawn from the reflux drum (reboiler) to remove the remaining lighter (heavier) components, and the next product is withdrawn. The operation is continued until either the residue remaining in the reboiler (reflux drum) meets the heaviest (lightest) product specification, or no further products can be withdrawn from the residue. Generally, for NP desired products, NP − 1 byproducts are withdrawn. Byproducts lie between two desired products. Therefore, along with the desired products, byproducts are also included in the task representation. It is not necessary in this representation for all byproducts to be features in all of the design cases. Only active byproducts are considered. It is a degree of freedom in the optimization to activate or deactivate a byproduct. Each desired product must be defined with the key component recovery and purity. Byproducts have low recovery of the key component, and purity lower than the previous desired product. In this work, the desired product purity is taken as a hard design constraint. Also, the recovery of the desired products, and purity and recovery of byproducts are applied to determine whether the operation of the column is feasible. These are soft constraints for the problem and can be changed to obtain a minimum purity level of the desired products. How feasibility in the batch distillation operation should be managed in the synthesis of batch distillation systems will be discussed in detail in section 2.4.

SSn =

n(n − 1) 2

TSn =

(n − 1)n(n + 1) 6

SQ n =

[2(n − 1)]! n! (n − 1)!

(1)

For a five-product (ABCDE) system, there are 10 subgroups of products: ABCDE, ABCD, BCDE, ABC, BCD, CDE, AB, BC, CD, and DE. Discrete splits in each subgroup lead to the tasks shown in Figure 4. All discrete simple tasks consist of a single

Figure 4. Separation tasks for a five-product system.

feed and two products. Choosing the appropriate combination of tasks can generate any sequence of splits. For example, a direct split of a five-product system will include tasks 1, 8, C



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Figure 5. Generating a hybrid task by combining two simple tasks.

15, and 20, while an indirect split will perform tasks 4, 7, 12, and 17. A task adjacency matrix is used to provide the sequence connection of columns in the synthesis framework. The task adjacency matrix shows which tasks are adjacent to each other. A task i is adjacent to a task j if one of the products of task j is a potential feed to task i. Consider a separation task sequence for a five-product system involving tasks 2, 17, 15, and 20. Task 2 provides feed for tasks 17 and 15, so tasks 17 and 15 are adjacent to task 2. Similarly, task 20 receives feed from task 15, and task 20 is adjacent to task 15. The task adjacency matrix for this separation sequence is given by

while, if the same task is formed by combining ABC/DE and A/BC, the mix task vector will be

ITmix = [3 11]T

The task mix vector keeps inverse mapping from being unique. If the batch distillation sequence contains hybrid tasks, a new hybrid task adjacency matrix (THadj) determines the connectivity in the sequence. THadj is obtained from the task adjacency matrix Tadj of simple tasks associated with the sequence. From the task adjacency matrix defined in eq 2, if a hybrid task is present containing tasks 2 and 17, then the hybrid task adjacency matrix THadj can be obtained from the Boolean summation of rows and columns related to tasks 2 and 17. If any diagonal entry in THadj is 1, then it signifies that the task is hybrid. For example, if tasks 2 and 17 are combined, THadj for eq 2 is defined as

2 17 15 20 2 Tadj = 17 15 20

⎡0 ⎢ ⎢1 ⎢1 ⎢⎣ 0

0 0 0 0

0 0 0 1

0⎤ ⎥ 0⎥ 0⎥ ⎥ 0⎦

⎡1 0 0 ⎤ ⎢ ⎥ THadj = ⎢1 0 0 ⎥ ⎣0 1 0⎦

(2)

The task adjacency matrix provides a mathematical representation of the separation sequence. Such a mathematical representation is helpful to determine the new sequence by changing the existing sequence. For example, if a task is removed from the sequence, then the task adjacency matrix can easily provide the connectivity of the remaining sequence. Complex tasks (hybrid tasks) are an ordered combination of simple distillation tasks. Complex batch distillation columns consist of hybrid tasks. The number of simple tasks associated with each hybrid task is the order of the hybrid task. An nthorder hybrid task contains n splits and draws n + 1 products. Figure 5 shows a hybrid task for simple tasks ABC/DE and A/BC. When combining two tasks, feasibility is considered: two tasks are only combined if the product of one task is a potential feed of the other task. Thus, ABC/DE can be combined with D/E, A/BC, AB/C, A/B, and B/C but not with BC/DE or BC/D. Although a combination of two simple tasks provide a unique hybrid task, the inverse mapping is not true. For example, a hybrid A/BC/DE can be formed by combining A/BCDE and BC/DE or ABC/DE and A/BC. To make the inverse mapping unique, each hybrid task is associated with the mix task vector ITmix. From Figure 4, if A/BC/DE is formed by combining A/BCDE and BC/DE, then, in that case, the mix task vector will be ITmix = [1 9]T

(4)

(5)

2.3. Modeling of a Batch Distillation Column. A wide range of mathematical models for capturing the dynamics of batch distillation are available, from the short-cut models2,11−13 to rigorous stage-to-stage models.14−16 In this work, a semirigorous model developed by the current authors17 is used to simulate the dynamics of the column. The model of batch distillation is based on the compartment analysis which considers the cumulative holdup of each tray at one sensitive tray such that the dynamics of the tray segment is assigned to one tray only.18 Modeling assumptions and detailed descriptions of mathematical model are given in Appendix A. The model is solved by a backward difference method, using a FORTRAN routine DASSL.19 2.4. Feasibility of Batch Distillation Systems. The successful synthesis for batch distillation systems is dependent on the procedure for ensuring feasible flowsheets and their operation. For the batch distillation column, feasibility should be attained by achieving the desired purity of the products. The selection of the operating policy and the design to provide a feasible design and operation is not straightforward in batch distillation processes. Previous studies in batch distillation have shown that the synthesis and optimization of batch distillation always faces this problem. Also, during optimization, most of the computational effort is spent on searching the feasible operation and design. In particular, when the novel sequences

(3) D



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are the reflux ratio, the vapor flow rate, and the product flow rate. For the optimal operation of a column, the control variables should follow the optimal profiles. The optimal operation of a batch distillation has been studied by various researchers. Rigorous models and control vector parametrization for the optimal operation of batch rectification were used.22−24 Furlonge et al.25 developed a rigorous model for the multivessel column and studied the optimal operation at different operating modes. Farhat et al.14 converted a dynamic optimization problem into a nonlinear programming problem using a linear and an exponential reflux policy for batch rectification. In this work, parameters for determining the shape and size of the profile are taken as optimization variables, rather than obtaining the profile through dynamic simulation and enforcing constraints related to the operational characteristics throughout the optimization. In other words, rather than generating a profile from the solution of differential equations in the model, a profile is imposed on the model, and an optimal profile is obtained by the optimization of profile parameters. Applicability of the proposed concept is dependent on the versatility of the profile generation algorithm. A novel profile generation tool20 can generate a wide range of practically realizable profiles for the dynamic operation. Six parameters are required to specify a profile, and details of their method are given in Appendix B. They used the profile generation tool for the optimization of batch crystallization processes. In this work, the same profile generation tool is adopted for the parametrization of control variables.

and configurations of batch distillation columns are sought, there is no systematic design method available for finding feasible operation of such new designs. When rigorous and semirigorous models are used in the synthesis framework, it becomes more time-consuming to deal with complex mathematical models within the optimization. In dynamic optimization problems for a path-constrained variable, it is difficult to meet constraints of paths, since the value of the end-point constrained variable is only available at the end of the period. A profile generation tool was introduced to meet the path constraints.20,21 They used this approach, for example, to avoid the dry-up problem in batch crystallization unit by selecting the holdup of the solvent as a control variable. They applied a profile generation algorithm for the control variable, and imposed bounds on the profile parameters, to meet the path constraints. For end-point constraints, this is not possible, since, with the profile generation tool, bounds can be set at the end point but feasibility of operating bounds cannot be considered for intermediate values. In this work, a limiting gradient approach is applied, which was proposed by the current authors,17 as illustrated in Figure 6.

3. OPTIMIZATION OF BATCH DISTILLATION SYSTEMS 3.1. Hybrid Optimization Approach. The superstructure generated in this work contains both discrete decisions (i.e., selecting different configurations, separation tasks, etc.) and continuous decisions (i.e., time, operating variables, feed distribution, etc.). It formulates the optimization problem as a mixed-integer nonlinear programming (MINLP) problem. The problem is solved using a hybrid optimization approach. Simulated annealing (SA) is used to provide structure perturbations, while sequential quadratic programming (SQP) optimization is used to provide optimal operating variables for the designs obtained from the SA. A set of optimal designs are selected from SA, which are further treated by SQP and can be tested further for practicality constraints. SA is a robust stochastic optimization technique that was proposed.26 The SA algorithm resembles a physical metal annealing process. In physical annealing, during the cooling process, metal atoms try to achieve a minimum energy state. An analogy can be drawn with the mathematical minimization of an objective function. A control parameter called the annealing temperature is used to guide the optimization process. At each level of the annealing temperature, the search of the solution space constitutes a series of Markov chain events. Large and infinite Markov chain events allow the optimization to converge to the equilibrium. The new state generated by SA is simulated, which is then accepted or rejected by using the criteria proposed by Metropolis et al.27 After the Markov chain event, the annealing temperature is reduced according to the predetermined cooling schedule. The cooling schedule employed in this work was that proposed by Aarts and van Laarhoven.28 A slow cooling schedule permits the optimization to converge at a global minimum. For successful optimization, a large Markov chain length and slow cooling schedule is recommended.

Figure 6. Limiting gradient for an end-point constrained variable.11

If the value of the end-point constrained variable is Xo at any point To and the value is Xf at end point Tf, then the limiting gradient is calculated according to Figure 6. Therefore, at any point T, the limiting gradient is calculated, and it is multiplied by a factor greater than 1.0 (a safety factor) to obtain the new gradient at time T. The multiplication factor is taken as the optimization variable (controlled variable). With the calculated gradient, operating variables are determined using the corresponding model equations. The underlying concept of this approach is to achieve the target value before the end point is reached. Also, the assumption is made that a potential available at point T to meet the target is greater than the next point. This means that extra work can be done in the beginning, rather than doing it later. For the batch distillation, each column segment has a maximum allowable vapor and liquid flow rate. At the start of the operation, when the feed is rich in the desired component, compared to the later stage of operation (i.e., in the reboiler of a rectifier column, if there are more-volatile components in the beginning compared to the later stage), a lower reflux rate is required. Therefore, to avoid high reflux requirements in the later operation stage, the maximum use of the initial potential is desired. For the batch distillation, the end-point constrained variables are chosen to be product purity and recovery. Using the above approach, operating variables (liquid and vapor flow rates) are calculated. 2.5. Optimal Operation of a Batch Distillation Column. The control variables in the batch distillation column E



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Figure 7. Superstructure moves of a batch distillation network.

Different design configurations can be generated from SA by providing probability biases for structure moves. The success of SA is dependent on the cooling schedule and the Markov chain length at any temperature level. The selection of SA intrinsic parameters, such as Markov chain length, cooling parameter, and initial and final annealing temperatures must be specified before the optimization. In the approach used here, SA provides the structural changes and initial feasible operation for the structure. After the SA run, the few best structures and corresponding operation are obtained. SQP technique is then used for solving NLP optimization and provides the optimal operation of the design. 3.2. Perturbation Moves. Moves are provided to the current structure by altering any of the three connections (feed connection, configuration connection, and sequence connection) provided in the superstructure and by changing the operation of the existing design. Randomly selected moves are performed to provide a new design and sequence for the batch distillation. Figure 7 presents the tree structure for batch distillation moves. There are four primary moves: the configuration move, the sequence move, the feed distribution move, and the

operation move. These primary moves are further divided into submoves. Move probabilities for structure perturbation are given in Table S1 in the Supporting Information, while Table S2 in the Supporting Information provides move probabilities for the profiles of the multiplication factor. 3.2.1. Sequence Move. A sequence move changes the existing batch distillation sequence. The sequence move is performed by changing the separation tasks. Separation tasks can be changed either with the fixed number of products, or by adding a new byproduct or by deleting an existing byproduct. Sequence moves are divided into submoves of product move and separation task move. Product Move. Production of undesired byproducts can be activated or deactivated from the design if necessary. Byproducts production is required if the product purity constraint is restrictive. If a byproduct is to be separated, it is removed as a distillate in a rectifier from the next-heavier desired product. The following steps are taken after adding a byproduct in the structure: (i) Select the columns in which the activated byproduct is considered. F



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(ii) Select the column (j) from which the byproduct will be separated, along with the next-heavier desired product. (iii) Add a rectifier in the design, which separates the activated byproduct from the next-heaviest desired product. (iv) Add a row and a column in the task adjacency matrix, and complex task adjacency matrix, if required. (v) All the entries in the new added column of the task adjacency matrix and complex task adjacency matrix are zero. (vi) All the entries in the new added row of the task adjacency matrix and complex task adjacency matrix are zero, except the jth element of the new added row. Figure 8 shows a move in which byproduct D is added. If all the tasks are direct tasks, then the initial task adjacency and complex task adjacency matrices are

exists in the sequence, then the complex task adjacency matrix is removed. Figure 9 shows a move in which byproduct D is deleted. If the first two tasks are indirect tasks and the next two tasks are

Figure 9. New separation tasks after deleting byproduct D.

direct tasks, then the initial task adjacency and complex task adjacency matrices are

⎡0 ⎢ 1 Tadj = ⎢ ⎢0 ⎢⎣ 0

0 0 1 0

0 0 0 1

(7)

0⎤ ⎥ 0⎥ 0⎥ ⎥ 0⎦

⎡1 0 0 ⎤ ⎢ ⎥ THadj = ⎢1 0 0 ⎥ ⎣0 1 0⎦

(10)

(11)

⎡0 0 0⎤ ⎢ ⎥ Tadj = ⎢1 0 0 ⎥ ⎣0 1 0⎦

After adding a byproduct, the new task adjacency and complex task adjacency matrices are

⎡0 ⎢ 1 Tadj = ⎢ ⎢0 ⎢⎣ 0

0⎤ ⎥ 0⎥ 0⎥ ⎥ 0⎦

After deleting byproduct D, the complex column is converted to a simple column. Hence, no complex task adjacency matrix is required. The new task adjacency matrix is

(6)

⎡1 0 ⎤ THadj = ⎢ ⎣1 0 ⎥⎦

0 0 0 1

⎡1 0 0 ⎤ ⎢ ⎥ THadj = ⎢1 0 0 ⎥ ⎣0 1 0⎦

Figure 8. New separation tasks after adding byproduct D.

⎡0 0 0⎤ ⎢ ⎥ Tadj = ⎢1 0 0 ⎥ ⎣0 1 0⎦

0 0 1 0

(12)

In this way, by adding or deleting a byproduct from the existing separation sequence, a new separation sequence is generated. The configuration of a distillation column is also changed through this move if, by deleting a byproduct, an existing complex column is converted to a simple column. Separation Task Move. A combination of direct, indirect, and intermediate tasks is allowed to be selected during the optimization. A new sequence is generated by arranging the order of separation tasks for the columns. Figure 10 shows a separation move allowed in the algorithm. In the first sequence, both the top product and the bottom product of the first rectifier are fed to the following columns. In the second sequence, the first rectifier is connected to the next middle vessel column only. The total batch distillation time is changed as the separation task and configuration changes. The fraction of total batch distillation time assigned to each column is changed, which is dependent on the throughput, withdrawal from the column, according to the task assigned to that column, and difficulty of the separation. The separation factor of the column is defined as follows.

(8)

(9)

The following steps are taken after deactivating a byproduct: (i) Select the columns in which the deactivated byproduct lies. (ii) Select the column (j) from which the byproduct is separated from the system. If column j is not a complex column, remove the column from the system; otherwise, convert the column to a less-complex column. (iii) Delete the jth row and the jth column in the task adjacency matrix, and complex task adjacency matrix, if required. (iv) If column j is a middle vessel column (complex column with an order of 2), then removing the byproduct will make the column a simple column. In this case, the simple column is a rectifier. After changing the complex column to a simple column, if no other complex column

itmicol

SFicol =

∑ Prodit , icol × it = 1

αhk αlk

(13)

The fraction of total batch time for each column is assigned as

ticol = G

SFicol ncol ∑icol = 1 SFicol

(14)



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only. Complex column moves include the change of complex column order and the change of vessel type. Using complex column order moves, either the order of a complex column is increased or decreased (0.5 probability for each case), or a new complex column is generated if the sequence does not have any. Complex columns can be created by combining two rectifiers, or two strippers, or a rectifier with a stripper. Two columns are only combined if one column provides the intermediate feed to the second. The order of a complex column represents the number of separation tasks or segments. Complex column order can be increased by combining a complex column with a simple column. A simple column can only be combined to the complex column if either it provides feed to the complex column or it receives feed from the complex column. Complex column order can be decreased by removing one task and the related segment from the column. The segment removed is designed as a rectifier. Intermediate vessel type of complex column can also be changed. Moves are allowed to provide outlet stream connections from intermediate vessels or changing vapor stream connections through the intermediate vessels. Segment Move. Segment moves are performed on existing segments in the sequence. Using segment moves, a segment from the sequence can be deleted, or a new segment can be added into the sequence. Two segments can be swapped in the sequence. The size of a segment can also be modified. By deleting a segment from the sequence, any of other segments are used to perform its separation task. A segment is not deleted if its deletion affects the order of a complex column (i.e., if the sequence contains two segments and a middle vessel column, then a segment cannot be deleted from the sequence). 3.3.3. Feed Distribution Move. A feed distribution move is only applied for a complex column, or if a total reflux column or a column with a large holdup of distillate in the receiver is selected. The feed to the complex column can be either charged to the reboilers only, or can be distributed equally to the vessels. For a total reflux column, feed is distributed in the vessels according to the product recovery. 3.3.4. Operational Moves. During SA optimization, the operating parameters for time and multiplication factors are changed. Also, the recovery of a product and a byproduct, and the purity of the byproduct, are also changed in this move. The multiplication factor follows a profile (six parameters for each profile), so probabilities are provided to the profile parameters (see Table S2 in the Supporting Information). 3.4. Nonlinear Optimization. Structural and operational moves in an existing batch distillation sequence were discussed above. The new sequence generated from these moves is simulated to measure its performance. SA provides a set of best designs selected stochastically. Since the optimal operation of a batch distillation contains various optimization variables, a large Markov chain length is required to obtain their optimal values from SA. This requires a large amount of computational time. However, the problem can be reduced if nonlinear programming (NLP) optimization is used after SA to obtain optimal operation for the designs. First, SA optimal designs are obtained with low probability to operation moves. A set of designs are obtained from SA, which are further refined for optimal operation using NLP. Various designs possible from the superstructure are shown in Figure 11. Various objective functions can be employed for batch distillation optimization, e.g., maximization of profit, minimization of batch distillation time, minimization of energy

Figure 10. Change of separation sequence by changing separation tasks using same configurations.

For the new design generated, batch time is calculated according to the change in the separation factor. ncol

(∑icol = 1 SFicol)new tnew = ncol told (∑icol = 1 SFicol)old

(15)

3.2.2. Configuration Move. Configuration moves provide changes in a batch distillation column design. Different configurations are allowed to be designed in the optimization. In this work, rectifier, stripper, and multivessel columns with and without vapor bypass configuration are considered. Configuration moves are divided into simple column moves, complex column moves, and segment moves. Simple Column Move. Simple column moves are performed on the simple columns (rectifier, stripper, total reflux column) of an existing batch distillation flowsheet. These moves are divided into rectifier moves, stripper moves, total reflux moves, and middle vessel moves. If a rectifier is converted to a stripper or vice versa, the feed distribution to the column is arranged accordingly. For example, in a rectifier, for example, 99% of the feed is charged to the reboiler and the remainder to the condenser, while in a stripper, 90% of the feed is charged in the top vessel (reflux drum) and the remainder to the reboiler. In a stripper, the reboiler is charged with 10% of the feed to provide sufficient vapor flow in the column. However, it should be noted that the distribution of the feed is also an optimization variable that can be either optimized using the feed distribution move or using NLP. When a rectifier or a stripper is converted to a total reflux column, then feed is randomly distributed into the column vessels (reboiler and reflux drum). Middle vessel moves are performed when the existing sequence contains at least one complex column and two simple columns. Also, two simple columns must be adjacent (i.e., one column should provide intermediate feed to the next column). The new design generated for the middle vessel column may or may not have the connection of a vapor stream through the intermediate vessel. The probability to select vapor connection through the intermediate vessel is 0.5. Complex Column Move. Complex column moves are performed either on existing complex columns or to generate a complex column if the sequence consists of simple columns H



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Figure 11. Possible designs from the superstructure.

Figure 12. A hybrid optimization framework.

consumption, maximizing revenue, etc. The overall optimization framework employed for the proposed synthesis framework is given in Figure 12, which shows how a hybrid optimization algorithm works.

PI =

total products (mol) batch time (h)

(16)

The problem data are given in Tables 1 and 2. The superstructure consists of two segments of trays, with attached top and bottom vessels. Two byproducts are allowed to be withdrawn to maintain the desired purity. Table S3 in the Supporting Information provides the SA annealing parameters used in this case study. The initial structure provided to the

4. CASE STUDIES 4.1. Case Study 1. This case study is taken from Noda et al.16 The objective function is the performance index (PI), as defined below: I



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Table 1. Problem Data for Case Study 1 parameter

value/remark

total number of stages feed feed composition (A/B/C) maximum vapor flow rate liquid flow rate rectifier, multivessel column stripper number of components number of products number of byproducts maximum number of segments

10 103 kmol (0.3/0.4/0.3) 50.0 kmol/h 20−50 kmol/h 50−80 kmol/h 3 (A/B/C) 3 2 2

Table 2. Product Data for Case Study 1 P1 P2 P3 W1a W2a a

desired component

purity (%)

recoveryb (%)

A B C A B

95 95 95 50 b 50 b

95 95 95 5 5

Byproducts. bSoft bound. Figure 13. Optimal designs for case study 1.

optimization is a sequence of four batch rectifiers, using a single unit. The minimum value of the multiplication factor is 1.0, and the maximum value is 5.0. Bounds for the total batch time are 2.0−5.0 h. A maximum of two tray segments are used with a constant number of trays. For this case study, a traditional sequence of four batch rectifiers with a direct task provides a PI value of 27.8, while novel designs are identified from the optimization. The best few designs obtained are given in Figure 13, and its performance is compared with the conventional design as shown in Table 3. The best design obtained contains a rectifier and a subsequent middle vessel column. 4.2. Case Study 2. Case 2 is a retrofit study. A batch distillation plant has three existing batch distillation columns. The objective function is set to maximize revenue using the available distillation columns in the plant. The case study assumes that all the available columns can be configurationally rearranged in different ways. The distillation column can be operated either as a rectifier or as a stripper. Also, two or more columns can be combined together to design a multivessel distillation column. Column data are given in Table 4. The separation of an alcohol mixture is studied. Feed and product specifications are given in Tables 5 and 6. Ideal gas and ideal solutions are assumed. Vapor pressure data are given in Table S4 in the Supporting Information.29 A maximum of three byproducts are allowed to be withdrawn to maintain purity of the desired products. SA parameters are given in Table S5 in the Supporting Information. A superstructure is generated using three tray segments, and seven products (four desired products and three byproducts). A total of 10 optimization runs were performed. All the cases in this paper were calculated on an AMD Athlon XP system with 1.83 GHz processor, and the average run time for this case study was ∼4 h. SA results are given in Table 7. During the SA optimization, the multiplication factor to the limiting gradient is assumed to follow a constant linear profile. The minimum value

Table 3. Results for Case Study 1 Value parameter

design C1-1

design C2-2

conventional rectifier sequence

time byproducts performance index, PI

2.18 h 0.8 kmol 45.5

2.54 h 0.0 kmol 40.7

3.4 h 8.5 kmol 27.8

Table 4. Distillation Column Data for Case Study 2 segment

number of trays

maximum vapor flow (kmol/h)

1 2 3

20 15 10

250 250 250

Table 5. Problem Data for Case Study 2 parameter feed cost of feed components composition of feed (mole fraction) number of products maximum byproducts bounds on total batch distillation time holdup of tray cleaning time setup time

value 100 kmol $2.0/kmol methanol, ethanol, n-propanol, n-butanol 0.45/0.35/0.1/0.1 4 3 1.0−8.0 h 0.01 kmol 0.5 h 0.5 h

of the multiplication factor is 1.0 and the maximum value is 5.0. SA provides moves to the multiplication factor with increments of 0.5. Thus, during SA, the multiplication factor takes values from 1.0, 1.5, 2.0, 2.5, ..., 5.0. After the multiple runs, the best J



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Table 6. Product Data for Case Study 2 product

desired component

purity (%)

recoveryb (%)

price ($/kmol)

P1 P2 P3 P4 W1a W2a W3a

methanol ethanol n-propanol n-butanol methanol ethanol n-propanol

95 95 99 99 50b 50b 50b

95 95 99 99 5 5 5

7.7 17.2 52.8 81.4

a

Byproduct. bLight bound.

Table 7. Simulated Annealing (SA) Results for Case Study 2 parameter

value

number of runs maximum objective mean objective standard deviation

10 $7.63 MM/yr $7.1 MM/yr 0.06

Figure 16. Best selected designs for case study 2 (III).

design containing only a rectifier is also proposed, which uses only one segment. The conventional design is also capable of producing desired products, without producing any of the byproducts, but requires a large batch distillation time. Table 8 provides a comparison of the best designs selected with that of a conventional rectifier design. All the

few solutions were selected and optimized using SQP to obtain the optimal operating policy. Figures 14, 15, and 16 provide the best sequences and configurations obtained from the optimization study. The best

Table 8. Results for Case Study 2 (without Setup Time and Cleaning Time) Value parameter

design C2-1

batch time annual revenue increase from base case number of segments used complexity

design C2-2

design C2-3

base case

1.73 h 2.4 h 2.68 h 6.6 h $7.3 MM/yr $6.8 MM/yr $6.04 MM/yr $3.2 MM/yr 130% 110% 90% 3

2

3

1

very high

high

medium

low

designs perform better than the conventional design (i.e., a 90%−130% increase in revenue, relative to the conventional design). The above designs have been selected without accounting for cleaning and setup time. In design C2-2, segment 1 is used in the middle vessel column and in the rectifier. Thus, it requires additional setup and cleaning time. If time required for cleaning and setup is added for design C2-2, only a 15% increase in revenue is obtained, compared to the conventional design. Table 9 provides the results when cleaning time and setup time are considered in the calculation of revenue.

Figure 14. Best selected designs for case study 2 (I).

Table 9. Results for Case Study 2 (Setup Time and Cleaning Time Included) Value

Figure 15. Best selected designs for case study 2 (II).

parameter

design C2-2

design C2-3

batch time annual revenue increase from base case

4.4 h $3.7 MM/yr 15%

3.7 h $4.6 MM/yr 45%

4.3. Case Study 3. This case study is a synthesis problem for a ternary separation system. A ternary feed of cyclohexane, n-heptane, and toluene mixture is separated into three relatively pure products. The problem specifications are given in Table 10. The composition of the feed and products, and their cost data,

configuration was a multivessel column using three tray segments, while the second-best sequence used two tray segments arranged in different configurations. In the second sequence, segment 1 is used twice, to reduce the distillation time. The third sequence uses all of the segments. A conventional K



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Table 10. Problem Data for Case Study 3a

a

parameter

value

amount of feed condenser holdup total trays holdup total plant operation time number of byproducts allowed cost of byproducts

29.3 kmol 0.035 kmol 0.056 kmol 8000 h/yr 2 $1.0/kmol

Data taken from ref 22.

Table 11. Feed and Product Composition and Cost Data for Case Study 3 feed cyclohexane n-heptane toluene cost

0.407 0.394 0.199 $2.0/kmol

product 1

product 2

Figure 18. Selected design (design 2) of case study 3.

product 3

chain length is used and more complex thermodynamic calculations are needed. Optimal designs for this case study are given in Figures 17 and 18. Both designs use a middle vessel column. Conventional design (base case) is based on a batch rectifier. Performances of these designs are given in Table 13. These two new designs

0.895 0.863 $30.0/kmol

$26.0/kmol

0.99 $24.0/kmol

are presented in Table 11. Thermodynamic properties are determined using the Soave−Redlich−Kwong (SRK) method, and critical properties and vapor pressure data for the system are provided in Table S6 in the Supporting Information. The objective of the case study is to maximize the profit when a new batch distillation flowsheet is designed. The calculation of the profit function is given in Appendix C. First, a superstructure is generated using four segments of different sizes. The size and capacity of these segments can be changed during the optimization. A maximum of two byproducts is allowed in the system. Annealing parameters are given in Table S7 in the Supporting Information. SA results are given in Table 12. Twenty (20) SA runs were carried out with an average runtime of 8 h. A longer run time is required in this case, compared with case study 2, because a longer Markov

Table 13. Comparison of Batch Distillation Designs for Case Study 3 Value

parameter

value 20 $1.08 MM/yr $0.98 MM/yr 0.08

design C3-1

design C3-2

base case

total distillation time profit byproduct production increase from base case

3.6 h $1.09 MM/yr 0.02 kmol 75%

4.3 h $0.9 MM/yr 0.12 kmol 45%

7.59 h $0.62 MM/yr 0.23 kmol

provide significant improvement in profit and produce a small amount of byproducts, while satisfying higher product purity requirements and low relative volatility of the systems.

5. CONCLUSIONS A novel superstructure-based optimization framework is presented for the synthesis of batch distillation processes. The framework is formulated to accommodate all the possible structural options that exist in the design of batch distillation processes and to identify the most appropriate flowsheet, subject to an objective function and design constraints. A tray segment with top and bottom vessel is considered as the generic unit of the superstructure. A separation task-based approach is used to determine the separation sequence. Hybrid tasks are used to generate complex columns (middle vessel, multivessel columns). The thermodynamic performance and separation characteristics of each configuration generated from the optimization framework are evaluated with a semirigorous model during the optimization. Feasibility and operability of new designs generated from the superstructure is evaluated and the optimization of operational strategy for batch processes is carried out simultaneously with the limiting gradient approach and profile generation algorithm. A hybrid optimization employing a stochastic method using simulated annealing (SA) and sequential quadratic programming (SQP) is applied to effectively deal with a complex optimization model, which includes a large number of decision variables and highly nonlinear equations. SA has been used to provide a set of best configurations of batch distillation systems, which are further optimized using SQP to provide an optimal operating policy.

Table 12. Simulated Annealing Results for Case Study 3 number of runs maximum objective mean objective standard deviation

parameter

Figure 17. Selected design (design 1) of case study 3. L



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Three case studies have been presented to demonstrate the applicability of the approach. The results from the case studies suggest that significantly improved designs with high process performance can be obtained from the systematic consideration of design options available for batch distillation (e.g., using multiple tray segments) with the aid of synthesis methods proposed in this paper.

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APPENDIX A. MATHEMATICAL MODEL OF A BATCH DISTILLATION COLUMN17 The following assumptions are made in the modeling of batch distillation in this work: • Constant liquid holdup of the trays • Vapor holdup is negligible • Separate compartments for condenser and reboiler are used • Constant molar overflow • The vapor and liquid leaving a stage are in equilibrium (i.e., ideal stages) • Pressure drop effect is neglected • Constant holdup for the sensitive tray The followings are model equations for a middle-vessel batch distillation column shown in Figure A1, which results in a set of differential algebraic equations. Model equations for the sensitive tray:

Figure A1. Batch distillation unit with an intermediate vessel.17

Material balances for the intermediate vessel: dHiv = (Viseg = iv − Viseg = iv − 1) + (Liseg = iv − 1 − Liseg = iv) − Div dt dHivxi , iv = (Viseg = ivyi , in + 1 − Viseg = iv − 1yi , iv ) dt + (Liseg = iv − 1xi , in − Liseg = ivxi , iv) − Divxi , iv

ntray

∑

Hc =

hitray

itray = 1

Hc

= Viseg(yi , s + 1 − yi , s ) + Liseg (xi , s − 1 − xi , s) dt i = 1, 2 , ..., NC − 1

APPENDIX B. A PROFILE GENERATION TOOL20,21 Profiles are divided into two fundamental profiles, exponential and asymptotic profiles as shown in Figure B1.

yi , s = f (T , P , xi , s)

∑ xi , s = 1 i

(A5)

■

dxi , s

(A1)

Model equations for the jth tray (which is not the sensitive tray): 0 = Viseg(yi , j + 1 − yi , j ) + Liseg (xi , j − 1 − xi , j) i = 1, 2 , ..., NC − 1 yi , j = f (T , P , xi , j)

∑ xi ,j = 1 i

(A2)

Material balances for the condenser: dHiv = 1 = V1 − L1 − D1 dt dHiv = 1xi ,iv =1 = V1yi ,1 − L1xi ,iv =1 − D1xi ,iv =1 dt

(A3)

Material balances for the reboiler: dHiv = nvessel = Vnseg − Lnseg − Dnseg dt dHiv = nvesselxi ,iv =nvessel = V1yi , iv = nvessel − L1xi , n − D1xi ,iv =nvessel dt Figure B1. Basic types of profiles: Type I and Type II.20,21

(A4) M



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Figure B2. Type I + Type II and Type II + Type I profiles.20,21

present numerical difficulties, since they cannot be differentiated. According to the profile generation algorithm, only six parameters are needed to generate a profile; z1, z2, z3, tinter, A1, and A 2 . Various profiles can be generated with the combinations of Type I and Type II, and different values of profile parameters. Also, the governing equation is polynomial in nature, with respect to the independent variable t, so it can be easily differentiated.

Type I profile (exponential curve): ⎛ t ⎞ A1 z(t ) = z1 − (z1 − z 2)⎜ ⎟ ⎝ t total ⎠

(B1)

Type II profile (asymptotic curve): ⎛t − z(t ) = z 2 − (z 2 − z1)⎜ total ⎝ t total

t⎞ ⎟ ⎠

A2

■

(B2)

APPENDIX C The profit function for case study 3 is calculated as

Both profiles depend on three parameters: the initial value of the profile (z1), the final value of the profile (z2), and the curvature of the profile (A1 for an exponential curve, and A2 for an asymptotic curve). Other profiles (Figure B2) can be generated by the combination of the above two profiles.

profit = (Rev − OC)NB − ACC

where Rev is the revenue generated per batch, which is determined from the price of products and the cost of feed: Revb =

Type I + II profile: ⎛ t ⎞ A1 z(t ) = z1 − (z1 − z 2)⎜ ⎟ ⎝ t inter ⎠

∑ PC i i − FCf i

NB =

A2

t inter < t ≤ t total

Type II + I profile:

⎛ t − t inter ⎞ A1 z(t ) = z 2 − (z 2 − z 3)⎜ ⎟ ⎝ t total − t inter ⎠

H yr t tot

(C3)

ttot is the total batch distillation time, which includes operation time of batch distillation, setup time to for column setup, and cleaning time.

(B3)

A2 ⎛t − t⎞ z(t ) = z 2 − (z 2 − z1)⎜ inter ⎟ ⎝ t inter ⎠

(C2)

NB is the total number of batches produced per year:

0 < t ≤ t inter

⎛ t −t ⎞ z(t ) = z 3 − (z 3 − z 2)⎜ total ⎟ ⎝ t total − t inter ⎠

(C1)

t tot = tb + ts + tcl 0 < t ≤ t inter

(C4)

Column cleaning is required prior to using a column for separation. Additional setup time is required if a column is used as different configurations in a flowsheet. OC represents the operating cost per batch, which includes the cost of hot utility, cold utility, and cleaning cost:

t inter < t ≤ t total (B4)

OC = CUhot + CUcold + Cclean

Combinations of Type I + Type I, and Type II + Type II (Figure B3) were not considered, because such types of profiles can create significant discontinuities where the curves meet and cannot be readily implemented in practice. Discontinuities also

(C5)

For a batch distillation flowsheet, capital investment is required for distillation columns, heat exchangers (including the condenser, and the reboiler), storage vessels, and pressure N



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Figure B3. Nondifferentiability in Type I + Type I and Type II + Type II profiles at tinter.20,21

vessels (reflux drum, intermediate vessels of distillation column). Additional capital investment is also required for piping, instrumentation, site preparation, etc. The cost of the distillation column is dependent on the height of the column and the diameter of the column, which, in turn, is a function of the number of trays and the vapor flow rate in the column shell. Thus, a column cost correlation can be developed in terms of the number of trays and vapor flow rate. The following correlation is developed for determining the cost of the batch distillation column: 0.5 costcol = 1400 × V max × Nt 0.85

cost svess = 11500 ×

where Vol is the volume of the vessel (in m ). Parameters for pressure vessel cost and storage vessel cost are regressed from the detailed costs of various designs. All the correlations provided above determine the cost as of January 2000. Apart from the equipment costs, other costs (e.g., piping, utility, instrumentation, and building) are also required. Factors were provided to accommodate these costs.30 We used a factor of 4.8 to determine the additional cost required for the equipment. Thus, the total capital cost for batch distillation flowsheet is

(C6)

Captotal = 5.8 × (costcol + costHE + cost pvess + cost svess) (C11)

The capital cost can be annualized using a suitable interest rate and payback period. We used a factor of 0.5 to annualize the capital cost. A batch distillation flowsheet can contain different configurations. If a distillation column is configured in different manners in a flowsheet, then additional setup cost is required in terms of piping. The piping cost of a distillation column is usually a factor of 1.0 of column cost.30 Thus, setup cost is

Csetup = costcol The annualized capital cost is calculated as ACC = 0.5 × (cost setup + Captotal )

(C13)

The operating cost is calculated using hot utility and cold utility cost data. For the hot utility, low-pressure steam (up to 5 bar) with a cost of $5.3/t is used.31 Cooling water with a cost of $0.066/t is used as the cold utility.25 The utility cost data are based on 1990 data (CE index = 357.6). The cleaning cost is taken to be zero in this work. All costs are calculated in U.S. dollars.

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ASSOCIATED CONTENT

S Supporting Information *

(C8)

This material is available free of charge via the Internet at http://pubs.acs.org.

■

The vapor term in the above correlation is the average vapor flow. Vapor flow is given in units of kmol/h. The cost of the pressure vessel is calculated as cost pvess = 1000 × H 0.67

(C12)

(C7)

where C1 is the equipment cost in year 1, C2 the equipment cost in year 2, INDEX1 the cost index in year 1, and INDEX2 the cost index in year 2. The cost of the heat exchanger is dependent on the area and type of the heat exchanger. The area of a heat exchanger is calculated using total heat load, temperature difference, and heat-transfer coefficient. For a given feed distillation, condenser and reboiler costs are functions of vapor flow, which usually determines the heat load. The following correlation is regressed for condenser and reboiler cost, using detailed costs of various rigorous designs. costHE = 900 × V 0.68

(C10) 3

Vmax (kmol/h) is used in the column correlation, as the maximum vapor flow rate determines the diameter of the column. The parameters of the column costing are regressed from detailed costing of various rigorous column designs. The detailed cost calculates the cost of the column based on the weight of the shell and information about the trays (number of trays, diameter, and type). For detailed cost calculations, cost data are taken from ref 30. The above correlation for column cost determines the cost on the basis of January 2000 data. Cost index (CE index) for January 2000 is 435.8. Cost can be updated using the present CE index (489.4 in June 2004). CE index data are published in Chemical Engineering Magazine. The following correlation is used to update the cost: C1 INDEX1 = C2 INDEX 2

⎛ Vol ⎞0.53 ⎜ ⎟ ⎝ 5.0 ⎠

AUTHOR INFORMATION

Corresponding Author

*Tel.: +82 2 2220 2331. E-mail: [email protected].

(C9)

Notes

H is the holdup of the vessel, given in units of kmol. The cost of the storage vessel is determined as

The authors declare no competing financial interest. O



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ACKNOWLEDGMENTS The authors wish to express their gratitude to the Process Integration Research Consortium (PIRC) for providing the research funding. This research was also supported by a grant of “Floating Production Platform Topside Systems and Equipment Development” from the Ministry of Trade, Industry and Energy of the Korean government, and by the International Research & Development Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning of Korea (Grant No. 2011-0031290).

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NOMENCLATURE A = relative volatility A1 = curvature parameters for profile 1 A2 = curvature parameters for profile 2 D = outlet flow rate from a vessel (kmol/h) H = holdup of vessel (kmol) H = holdup of tray (kmol) ITmix = task mix vector for hybrid task L = liquid flow rate (kmol/h) N = number of products NC = total number of components Ncol = total number of columns in a sequence Nvessel = total number of vessels in a column P = pressure Prod = product out from a separation task PI = performance index SF = separation factor SSn = product subgroup of n products SQn = total sequences of n products T = temperature t = dependent value of a profile or time (h) Tadj = task adjacency matrix THadj = hybrid task adjacency matrix TSn = discrete separation tasks of n products V = vapor flow rate (kmol/h) X = liquid composition Y = vapor composition Z = dependent value of a profile

Subscripts

1 = inlet index of profile (eqs B1−B4) 2 = intermediate index of profile (eqs B1−B4) 3 = final index of profile (eqs B1−B4) i = component index inter = intermediate index of independent variable of profile (eqs B1−B4) Hk = heavy key icol = column index (eq 6) iseg = segment index it = task index (eq 6) iv = vessel index j = tray index lk = light key s = sensitive tray index total = final index of independent variable of profile (eqs B1−B4)

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REFERENCES

(1) Kim, K. J.; Diwekar, U. M. Comparing batch column configurations: Parametric study involving multiple objectives. AIChE J. 2000, 46, 2475−2488. P



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(27) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. Equation of state calculations for fast computing machines. J. Chem. Phys. 1953, 21, 1087−1092. (28) Aarts, E. H. L.; van Laarhoven, P. G. M. Statistical cooling: A general approach to combinatorial optimization problems. Philips J. Res. 1985, 40, 193−226. (29) Reid, R. S.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th Edition; McGraw−Hill: New York, 1987. (30) Smith, R. Chemical Process Design; John Wiley and Sons, Ltd.: Chichester, 2005. (31) Peters, M. S.; Timmerhaus, K. D. Plant Design and Economics for Chemical Engineers, 4th Edition; McGraw−Hill: New York, 1999.

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Process Synthesis of Batch Distillation Systems - Industrial

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