Multiobjective Optimization for Synthesizing Compressor-Aided

Article pubs.acs.org/IECR

Multiobjective Optimization for Synthesizing Compressor-Aided Distillation Sequences with Heat Integration J. Rafael Alcántara-Avila,† Manabu Kano,† and Shinji Hasebe*,† †

Department of Chemical Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan ABSTRACT: Various column structures, configurations, and energy conservation methods have been proposed to reduce the cost and energy consumption in distillation sequences, such as thermally coupling or heat integration. In this work, a synthesis method of optimal separation sequences including thermally coupled columns is proposed. In the proposed method, vapor recompression of the flow to the condenser and pressure change within thermally coupled columns are taken into account to increase the possibility of heat integration. Furthermore, the energy requirement (ER) was adopted as the objective function as well as the total annual cost (TAC). By solving the multiobjective optimization problem, the relationship between the required annual cost and the amount of energy can be explicitly evaluated, i.e., the necessary cost increase per unit of energy input to the sequence can easily be assessed. A superstructure approach was adopted to enumerate all the candidate distillation sequences which consisted of conventional and thermally coupled columns. Rigorous simulations of each column were effectively used to assign reliable data to the optimization stage. Then, the process synthesis problem was formulated as a mixed integer linear programming (MILP) problem whose objective is to minimize TAC and ER. For two case study problems, Pareto-optimal solutions have been derived. The results of two case studies indicated that nonconventional sequences became the optimal for both cases. Vapor recompression was effective for minimizing the energy requirement.

1. INTRODUCTION Distillation is widely used to separate mixtures in the chemical and petrochemical industries; however, its construction and operation inherently entail thermodynamic inefficiency and high costs. As environmental restrictions are getting stricter, it becomes more important to increase the efficiency of distillation and to reduce the energy requirements and costs. Heat integration is one of the dominant methods for reducing the fixed and operating cost of distillation sequences. Andrecovich and Westerberg1 formulated the synthesis problem of heat-integrated distillation sequences as a mixed integer linear programming (MILP) problem. Their proposed superstructure consists of sharp split distillation sequences at several discrete pressures. The pressures of columns were adjusted so that the top vapor stream of a column can be heat integrated with the bottom liquid stream of other column. Floudas and Paules IV2 proposed a superstructure of sharp split distillation sequences and formulated the synthesis problem as a mixed integer nonlinear programming (MINLP) problem. Short-cut simulations for columns involved in the superstructure were performed at several pressures to obtain explicit expressions of the cost and heat loads as a function of pressure. The pressure selection to promote heat integration was influenced by the equilibrium constant and bubble point temperature of the streams leaving each column. In addition to superstructure-based optimization approaches, there are also heuristic and thermodynamic approaches that do not require sophisticated mathematical skills. The effective reduction of a search domain, however, greatly depends on the user’s knowledge. Aly3 proposed an approach based on several heuristic rules generating feasible matches between process streams for heat integration in predetermined distillation sequences with sharp or sloppy splits. The necessary pressure © 2012 American Chemical Society

difference between columns to generate heat exchange was estimated in advance as a function of the condenser temperature. Sobocan and Glavic4 proposed a thermodynamic approach based on short-cut methods and a graphical representation of distillation columns in a temperature−enthalpy flow rate diagram. Distillation sequences over a given pressure range were simulated to derive the sequence which minimizes the sum of the temperature difference between condenser and reboiler multiplied by its average heat flow for all the columns in the sequence. Then, rigorous simulation of the best sequence was performed to obtain the final heat-integrated solution. In these approaches1−4 only sequences consisting of conventional columns were treated, and the total annual cost (TAC) was adopted as the objective function. Sequences of conventional columns exhibit thermodynamic irreversibility owing to the mixing of streams at some stages in the column.5 At those intermediate stages the molar fraction of a middle component is higher than that in the condenser or reboiler. Petlyuk et al.5 mentioned that this inefficiency could be reduced by introducing sloppy separations and thermal coupling between columns. In thermal coupling a liquid side stream from a column is fed to the top of another column to remove the condenser, and/or a vapor side stream from a column is fed to the bottom of another column to remove the reboiler. Thermally coupled distillation columns mitigate remixing and therefore reduce energy consumption. On the other hand, Rev et al.6 showed that in most cases heat-integrated sequences are Special Issue: AMIDIQ 2011 Received: Revised: Accepted: Published: 5911

August 8, 2011 December 26, 2011 February 13, 2012 February 13, 2012 dx.doi.org/10.1021/ie2017527 | Ind. Eng. Chem. Res. 2012, 51, 5911−5921

Industrial & Engineering Chemistry Research

Article

There has been an ambiguous economic interpretation when compressors are included in distillation sequences. Therefore, this research aims to derive a meaningful interpretation of vapor recompression at top vapor streams of columns and at the vapor streams from low to high pressure section of thermally coupled columns. Both TAC and the energy requirement (ER) are selected as the objective functions and are optimized simultaneously using the multiobjective optimization technique. A superstructure-based approach is adopted to model the problem. First, the compositions between the columns are calculated using the material balance. For each column and for each discrete pressure, rigorous simulations are executed to decide the number of stages. The results of rigorous simulations are used in the synthesis problem formulated as an MILP problem, in which the TAC and ER are minimized.

more energy-efficient than the Petlyuk column. They said that the Petlyuk column was a better alternative only over a small range of relative volatility ratio, feed composition, and price structure. However, they did not include configurations that are partially coupled or thermodynamically equivalent to the Petlyuk column in their comparisons. Caballero and Grossmann7 proposed a superstructure representation based on short-cut methods to synthesize conventional (a single feed, two product streams, a condenser, and a reboiler) and complex (several feeds and product streams with or without a condenser or a reboiler) distillation sequences with heat integration. This representation was formulated as a generalized disjunctive programming (GDP) problem and then solved as a reformulated MINLP problem. Their results showed that the best sequence did not have heat integration when the pressure was kept constant; however, the best sequence had heat integration when pressure change in conventional columns was possible. In either case, the resulting sequence, which minimizes the TAC, was a combination of conventional and complex columns. Yiqing et al.8 proposed a simulated annealing approach based on short-cut methods and pinch analysis to synthesize conventional and thermally coupled distillation sequences with heat integration. The pressure selection was embedded in the short-cut methods, but columns with thermal coupling were operated at the same pressure. Their results showed that a combination of conventional and thermally coupled heat integrated columns yielded the best sequence, which minimized the TAC. Although less researched, vapor recompression is another alternative to reduce the energy consumption in distillation columns. A vapor stream is recompressed to a higher pressure to increase its temperature, and thereby heat integration becomes possible. Vapor recompression has been primarily applied to distillation of binary mixtures with close boiling points. Fitzmorris and Mah9 proposed a simulation procedure to explore the thermodynamic efficiency of vapor recompression of the top stream and the vapor stream connecting the rectifying and stripping sections in an ethylene-ethane distillation column. Their results showed that vapor recompression is a valid alternative to reduce energy requirements. Oliveira et al.10 proposed a simulation procedure for an ethanol−water distillation column with vapor recompression. Their results suggested that vapor recompression was effective in saving energy. It also could be environmentally more attractive than conventional distillation since it uses the own fluids of the column. Finally, our past research11 focused on the recompression of vapor side streams in thermally coupled columns for a ternary mixture. Our results showed that the pressure change of vapor streams between thermally coupled columns could yield energy savings. However, the use of compressors made them economically unappealing: the compressor cost exceeded the energy cost savings under some circumstances. Usually, the pressures of thermally coupled columns are the same because vapor flow exists between the columns. If the pressures of two columns can be selected arbitrarily, more flexible designs with heat integration can be realized. Although several approaches have been suggested to handle pressure change between columns to promote heat integration, the use of compressors in distillation sequences has not been addressed comprehensively, especially for multicomponent mixtures. The use of compressors expands the search domain for optimization problems of synthesizing distillation sequences.

2. PROBLEM STATEMENT In this research, a systematic procedure for synthesizing the optimal distillation sequences is proposed. The proposed procedure explicitly includes many types of thermally coupled distillation columns as candidate sequences, and the vapor recompression of streams to condensers is allowed so as to effectively use the sensible heat of the vapor. Figure 1 shows all subsequences adopted in this research. The conventional column (CC) is shown in Figure 1a, and a structure having a sloppy split (SS) is in Figure 1b. The other figures show thermally coupled columns: the Petlyuk column (PET) in Figure 1c, the partial Petlyuk column with a reboiler in the first column (PPV) in Figure 1d, the partial Petlyuk column with a condenser in the first column (PPL) in Figure 1e, the partially coupled sequence with vapor side stream to the second column (PCV) in Figure 1f, and the partially coupled sequence with liquid side stream to the second column (PCL) in Figure 1g. The feed streams in Figure 1 are not restricted to a ternary mixture. L represents the light component or a set of light components, M represents the middle component or a set of middle components, and H represents the heavy component or a set of heavy components. The final separation structure is the combination of those structures. The Petlyuk column is a fully thermally coupled column that requires lower energy than conventional distillation sequences.12,13 The dividing wall column is thermodynamically equivalent to the Petlyuk column when all the sections are operated at the same pressure. When pressure change is permitted, the Petlyuk column is more flexible than the dividing wall column. Thus, the Petlyuk column configuration is adopted as a candidate subsequence. Agrawal and Fidkowski pointed out that the thermally coupled columns become more operable by reducing the number of interconnections14 and adopting unidirectional vapor streams.15 Thus, the partially coupled sequences (PPV and PPL) and the unidirectional vapor connection sequences (PCV and PCL) are explicitly included in candidate subsequences. Complex sequences that can separate a mixture into four or more products (e.g., four-component Petlyuk column16) are not treated in this research. PPV and PCL can be expressed by the same mathematical model when the first and second columns operate at the same pressure. These two structures, however, show different separation performance and column cost when the two columns operate at different pressures. Thus, PPV and PCL are treated 5912

dx.doi.org/10.1021/ie2017527 | Ind. Eng. Chem. Res. 2012, 51, 5911−5921


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Figure 1. Candidate subsequences investigated in the research.

pressure change between columns. Both TAC and ER are selected as objective functions, which are explained in detail in section 5.3.

as different subsequences. For the same reason, PPL and PCV are also treated as different subsequences. To enhance the heat integration among the columns, the pressures of two columns in a subsequence are not necessarily the same. When the pressures of two columns are different, pumps, compressors, and/or valves are installed between them. As the fixed and operating costs of compressors cannot be neglected, they are added to the TAC. Another feature introduced in this research to enhance the heat integration is the vapor recompression of the top stream, which is shown in Figure 2. Compression of the vapor raises its

3. SUPERSTRUCTURE GENERATION The first step in solving the synthesis problem is to define all candidate distillation sequences. In this work, a superstructurebased approach is adopted to enumerate them. The complexity of the superstructure greatly depends on the number of candidate sequences and the number of components to be separated. Here, candidate sequences are limited to the combination of subsequences shown in Figure 1. The superstructure of distillation sequences separating a four-component mixture is shown in Figure 3, where each rectangle represents a subsequence in Figure 1 and ellipses are the feed, intermediate products, and final products. The mixture consists of A, B, C, and D, with A the most volatile component. A/BCD means that mixture ABCD is separated into A and BCD by a conventional column, while AB/C/D means that ABCD is separated into AB, C, and D by a complex subsequence in Figure 1. The superstructure generated in this work is a state task network representation. Each ellipse is a state having information of the composition and physical conditions. When the amount of feed and its composition as well as the composition of every product are given, the amount of each product can be calculated from material balance equations. Some states in Figure 3 have more than one inlet and/or outlet flows. It is assumed that the number of inlet to and outlet from the state is restricted to one or zero for the finally derived structure, i.e., flow split is not permitted. By introducing this assumption, the conditions of intermediate states such as ABC or BCD in Figure 3 are uniquely determined. It is assumed that each state is liquid at its saturated temperature and that the compression and decompression cost of liquid is negligible. Thus, liquids with the same composition are expressed by one state even if

Figure 2. Conventional column with vapor recompression.

condensing temperature and enhances the possibility of heat integration. The problem to be solved in this research is to derive optimal separation sequences consisting of the subsequences shown in Figure 1 by taking into account the vapor recompression and 5913



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Figure 3. Superstructure of a four-component separation problem.

Figure 4. Decomposition of aggregated tasks SS (left) and PET (right).

4. RIGOROUS OPTIMIZATION OF TASKS

the pressure is different. It is also possible to include states as saturated vapor. In such cases, as the compression cost of vapor cannot be neglected, the vapor at different pressure should be treated as different task even if the composition and the flow rate are the same. The fixed cost and the operating cost depend on the operating pressure. Thus, the columns with different pressures are treated as different tasks even if the feed and product states are the same. A thin rectangle in Figure 3 shows a set of simple tasks executed by a conventional column, SS. Tasks executed by subsequences in Figure 1b through 1g are hereafter called aggregated tasks. A set of aggregated tasks having the same inlet and outlet states is expressed by a thick rectangle in Figure 3. In this work, the pressures of the first and second columns in a subsequence are not necessarily the same. If the pressure of some columns is different, such subsequences are treated as different aggregated tasks even when the inlet and outlet states are the same. To avoid complexity, the set of aggregated tasks with the same feed and product states is expressed by one rectangle in Figure 3. An aggregated task is a compacted representation of two or three tasks and intermediate states. Suppose that SS in Figure 1 can be used to separate ABCD into AB/C/D. In this case, aggregated task AB/C/D can be decomposed into three tasks and two intermediate states as shown in the left-hand side of Figure 4. When PET in Figure 1 is adopted, aggregated task AB/C/D can be decomposed into two tasks and two intermediate states as shown in the right-hand side of Figure 4. It is assumed that each column in the sequence can take one of the predetermined discrete pressures.

4.1. Determination of the Number of Stages. Prior to the rigorous simulation and optimization of tasks, the number of stages for each column corresponding to each task has to be determined. For the tasks in Figure 3, the feed and product states are completely determined in advance by the material balance equations. When the mixture is nonideal, short-cut methods are not reliable to calculate the number of stages, especially for high pressure cases. To deal with such conditions, our presented approach is based on rigorous simulations performed by RadFrac in Aspen Plus as follows: 1 Set a large number of equilibrium stages (i.e., 100 stages) and assume 80% of tray efficiency. 2 Calculate the minimum reflux ratio, Rmin, which satisfies the product specifications at the top and the bottom flows. The NQ Curves tool in RadFrac is used in this step. 3 Allot 1.2 Rmin to the actual reflux ratio. 4 Calculate the minimum number of stages that satisfies the actual reflux ratio. The NQ Curves tool can also be used in this step. For SS in Figure 1, the number of stages of the first column can be determined in the same way by setting the maximum amount of heavy key component in the top intermediate stream and that of light key component in the bottom intermediate stream. The composition of intermediate states can be obtained by the rigorous simulation of the first column and is used for the stage calculation of the down stream two columns. 5914



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Figure 5. Analogy between SS and two thermally coupled columns.

thermally coupled columns can drastically reduce the computation time for optimization. In this work, rigorous simulations were performed with Aspen Plus by using the module RadFrac which contains the mass and energy balance equations. The thermodynamic model to calculate the equilibrium relation is Wilson’s for calculating the activity coefficients and Redlich−Kwong for calculating the fugacity coefficients. 4.3. Simulation of Vapor Recompression. A distillation module in commercial process simulators does not have vapor recompression as a possibility. Thus, the simulation of vapor recompression part, which is depicted by the solid lines in Figure 6, is executed separately from the main part, which is

For aggregated tasks made of thermally coupled columns, a simple method is used to determine the number of stages. An example of the analogy between SS and thermally coupled columns is illustrated in Figure 5. In this research, this analogy is used to decide the number of stages of thermally coupled columns, i.e., the sections having the same figure are assumed to have the same number of stages. This section analogy is applied to other aggregated tasks made of thermally coupled columns. Without losing generality, this section analogy is also applied to more than three-component cases so long as a feasible design of SS exists. 4.2. Rigorous Simulation. A short-cut simulation has been widely used to derive information on condenser and reboiler duties as well as the vapor flow rate in the column, which can be used to estimate the cost and energy requirements of each task.1,2,7,8 Since the results based on short-cut simulations are not accurate for nonideal cases, rigorous models are necessary to obtain reliable results. However, when rigorous models are directly embedded in the synthesis formulation,17 crucial difficulties in solving MINLP problems arise due to nonlinearity and nonconvexity of the formulated model. Therefore, their implementations have been mostly restricted to ideal or nearly ideal mixtures. In the proposed method, rigorous simulations are executed to derive the number of stages and to derive the column conditions, and the derived data are used in the synthesis problem formulated as an MILP problem. Thus, more accurate input data can be obtained compared with the shortcut methods, and less time is consumed compared with MINLP formulation. For simple tasks executed by CC and aggregated tasks executed by SS, data such as the reboiler and condenser heat duties, Q ireb and Q cond have been obtained at the step of deterj mining the number of stages. For an aggregated task made of thermally coupled columns, the number of stages of each column has been determined by using the analogy with SS. The flow rates of thermal linking streams are optimized at the reference pressure, e.g., 3 atm, so that total energy consumption at the reboilers is minimized. The rigorous model is also used in this step, and the optimization procedure proposed by Hernandez and Jimenez18 is adopted. Then, additional simulations at several discrete pressures are executed by fixing the flow rates of thermal linking streams at the optimal value. Fixing the flow rates of liquid and/or vapor streams between

Figure 6. Schematic representation of vapor recompression.

depicted by the dotted lines in the same figure. The flow rate and composition of a vapor stream leaving the column, which are obtained by the rigorous simulation, and the pressures before and after the compressor are used to calculate vapor recompression work duty, Wj VR, the condenser heat duty, Q cond j , the temperature after vapor recompression, Tj vap, and the temperature of the liquid leaving the condenser, Tjliq.

5. MATHEMATICAL FORMULATION The MILP formulation proposed in this work is based on the transportation problem presented by Andrecovich and Westerberg.1 The advantage of the proposed formulation is that the following data for each task can be obtained by the prior execution of rigorous simulations: 5915



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Previous researches suggested using a feasible heat exchange matrix based on a minimum temperature difference restriction.1−4,7,8 In this research, the minimum temperature difference is not used to restrict feasible matches, and the matches are possible as long as temperature of a hot stream is larger than that of a cold stream. However, the cost of a heat exchanger becomes more expensive as the temperature difference decreases. The temperature difference ΔTi,j for possible matches between cold stream i and hot stream j is given by

(1) Design parameters (condenser, reboiler, and compressor duties, number of stages, column diameter), (2) Flow rate and composition of each input and output streams, and (3) Top and bottom temperatures. Although these data can be obtained from short-cut methods, rigorous simulations supply more realistic values especially in nonideal cases. Rigorous simulations of each column implicitly deal with all the nonlinearity such as complex thermodynamics and nonconvexities such as stage-by-stage bilinear products of flow rate and composition. Thus, at the synthesis stage, the problem can be expressed by an MILP problem, which can be solved easily compared with a nonlinear, nonconvex MINLP problem which requires strong and sophisticated mathematical skills and might fail to warranty global optimality. The combination of rigorous simulations and MILP optimization is easier to perform and can derive global optimal solution in shorter computation time. 5.1. Compressor Cost. Subsequences (c) to (g) in Figure 1 have vapor streams between two columns. A compressor is required when the pressure of the feeding column is lower than that of the receiving column. Let column i and column j be first and second columns which form subsequence (c) to (g) in Figure 1. The compressor work duty Wijcomp is calculated when the pressures of two columns are different and there exists vapor flow from low pressure column to high pressure column. Otherwise, Wijcomp is set equal to zero. The derived compressor work duty is used to calculate the compressor cost Cijcomp as reported by Turton et al.19 comp

Ci , j

comp

= f (W i , j

, ϕcomp)

(i , j) ∈ CA

⎧ ⎫ ⎪ ⎪ liq ⎪ vap ⎪ out − in − − ( T T ) ( T T ) ⎪ j ⎪ i i j ΔTi , j = max⎨ , 0⎬ vap ⎛ ⎞ out ⎪ ⎪ T j − Ti ⎟ ⎪ ⎪ ln⎜ liq ⎜ T − T in ⎟ ⎪ ⎪ i ⎠ ⎝ j ⎩ ⎭ i ∈ CS , j ∈ HS

where Tlin and Tlout are the temperatures of the cold stream in task i or cold utility i entering and leaving the heat exchanger, and Tjvap and Tjliq are the temperatures of the hot stream in task j or hot utility j entering and leaving the heat exchanger, respectively. CS is the set of cold streams and CS = TK∪CU, where CU is the set of cold utilities. HS is the set of hot streams and HS = TK∪HU, where HU is the set of hot utilities. If the task i or task j does not have a cold or hot stream, the corresponding ΔTi,j is set equal to zero. Using ΔTi,j and the reboiler and condenser heat duties, Q reb i ref and Q cond j , the size reference value of heat transfer area Ai,j is calculated1

(1)

where ϕcomp is a set of parameters such as a compressor type, construction materials, and a pressure factor. CA is the set of the pair of columns which compose an aggregated task. Two types of isentropic compressors were investigated in this work: rotary-type compressors whose work duty is between 18 and 450 kW and centrifugal-type compressors whose work duty is between 450 and 3000 kW. The work duty for recompression Wi VR obtained in section 4.3 is used to calculate the compressor cost of vapor recompression for column i, CiVR. The cost function of the compressor installed for vapor recompression is generally described by CiVR = f (WiVR , ϕcomp)

i ∈ VR ⊂ TK

(3)

ref

Ai , j =

ref

Ai , j = ref Ai , j =

Q icond ,j U condΔTi , j Q ireb ,j U exΔTi , j Q ireb ,j U rebΔTi , j

i ∈ CU , j ∈ TK

i ∈ TK , j ∈ TK

i ∈ TK , j ∈ HU (4)

where Ucond, Uex, and Ureb are the overall heat transfer coefficient for matches between cold utilities and process streams, for matches between process streams, and for matches between process streams and hot utilities, respectively. The reference cost is calculated as a function of the reference size as reported by Turton et al.19

(2)

where VR is the set of tasks with vapor recompression. It is a subset of the set of all tasks, TK. 5.2. Heat Exchange Cost. Liquid streams which must be vaporized at reboilers are treated as cold streams to be heated, and vapor streams which must be condensed at condensers are treated as the hot stream to be cooled. These process streams are used for heat integration. Temperatures before and after heating/cooling are calculated through rigorous simulations as well as the condenser and reboiler heat duties. In addition to those process streams, several types of cooling and heating utilities can be used in a heat exchange network.

ref ref XCi , j = f (A i , j , ϕex)

i ∈ CS , j ∈ HS

(5)

where ϕex is a set of parameters such as an equipment type, construction materials, and a pressure factor. Three types of heat exchangers were investigated: floating head for condensers, multiple pipes for heat exchangers between process streams, and kettle for reboilers. 5916



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Finally, XCi,jex which shows the heat exchange cost per unit amount of heat transferred is calculated

variable between zero and one instead. This definition effectively works to reduce the number of binary variables. Here, two types of objective functions are introduced: the total annual cost, TAC, and the energy requirements, ER. TAC is expressed by the following equation

ref

XCies, j =

XCi , j ref

A i , j U condΔTi , j

i ∈ CU , j ∈ TK

ref

XCies, j =

XCi , j ref

A i , j U exΔTi , j

TAC =

1 (FC) + H(OC) T

FC =

∑

i ∈ TK , j ∈ TK

i ∈ TK

ref

XCies, j =

ref

A i , j U rebΔTi , j

i ∈ TK , j ∈ HU

∑

Yi −

i ∈ TKFs

Yi = 0

i ∈ TK j ∈ HU

Q iex ,j +

∑

OC =

j ∈ TK

∑

Q iex ,j +

i ∈ TK

∑

(10)

where Qi,jex is the amount of heat exchanged between the bottom liquid of task i (or cold utility i) and the top vapor of task j (or hot utility j). An infeasible match (i, j) is avoided by assigning a very large value, such as 1e6, to its heat exchange cost. Finally, the following equations are necessary to express the existence of a compressor in a vapor stream from a low pressure column to a higher pressure column

∑

comp

Yi ,j

− Yi = 0

∑ i ∈ TKFs

ER =

comp

− Yj = 0

i ∈ CU j ∈ TK comp comp Ci , j Y i , j

CicoolQ iex ,j +

∑ i ∈ TK j ∈ HU

ex C heat j Q i,j

Q iex ,j

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ comp comp⎟ elec ⎜ VR ∑ W j Yj + ∑ ∑ W i , j Y i , j + 3C ⎜ j ∈ TK ⎟ d ∈ IDc i ∈ TKFd ⎜⎜ ⎟⎟ j ∈ TKPd ⎝ ⎠

(11)

Yi ,j

∑ i ∈ TK j ∈ HU

i ∈ TKFs , s ∈ STc

j ∈ TKPs

ex XCiex ,j Q i,j

where FC is the fixed cost and OC is the operating cost. Ccol is the column cost and Ctray is the tray cost. CVR and Ccomp are the compressor cost for vapor recompression and that for thermally coupled columns, respectively. Ccool, Cheat, and Celec are the costs of cooling utilities, heating utilities, and electricity, respectively. Finally, T is the payout time and H is the annual operation hours. Since compressors are expensive, their fixed and operating costs are included in TAC. Compressors have a negative impact on the fixed and operating costs because of their expensive hardware and electricity costs, but they also have a positive impact through reducing the amount of energy used at the reboilers. Thus, a second objective function that explicitly evaluates the energy requirements of a sequence is added to evaluate the positive and negative impacts of compressors

j ∈ TK

i ∈ CU

∑

∑

(13)

(9)

cond Q iex , j = Q j Yj

ex XCiex ,j Q i,j +

⎛ ⎞ ⎜ ⎟ comp comp⎟ VR elec ⎜ ∑ Wi , j Yi , j ⎟ + C ⎜ ∑ W j Yj + ∑ s ∈ STc i ∈ TKFs ⎜ j ∈ TK ⎟ j ∈ TKPs ⎝ ⎠

i ∈ TK

j ∈ HU

∑ i ∈ CU j ∈ TK

(8)

reb Q iex , j = Q i Yi

ex XCiex ,j Q i,j

s ∈ STc i ∈ TKFs j ∈ TKPs

where Yi is a binary variable assigned to each task. If task i is selected, Yi takes one; otherwise it takes zero. FEED denotes the set of tasks whose input stream is the original feed mixture. ST is the set of states. TKFs and TKPs are the set of tasks which have state s as an output and as an input, respectively. The energy balance constraints to satisfy the energy demand at the condensers and at the reboilers are described by

∑

∑

+

s ∈ ST

i ∈ TKPs

∑

+

(7)

i ∈ FEED

+ CiVR )Yi

i , j ∈ TK (6)

Yi = 1

∑

∑

+

XCi , j

A small number, e.g., 1e-4, is added to the denominators in eqs 4 and 6 to avoid division by zero in case the temperature difference ΔTi,j is zero. 5.3. Mixed Integer Linear Programming formulation. The constraints and the objective function are explained here. The connectivity constraints between columns to generate feasible sequences are given by

∑

tray

(Cicol + Ci

j ∈ TKPs , s ∈ STc

(14)

The first term of the right-hand side of eq 14 is the amount of heat used at the reboilers and the second term is the work duty at the compressors. The work duty is converted into energy by multiplying its value by 3, which is an empirical coefficient estimated from electricity efficiency in Japan. The energy removed from tasks by cold utilities is not included in ER.

(12)

where Yi,jcomp is a variable that indicates the existence of a compressor between task i and task j. STc is the set of intermediate states of complex subsequences. Since Yi,jcomp is determined by binary variables Yi and Yj, there is no need to define Yi,jcomp as a binary variable; it can be defined as a continuous 5917



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The final formulation of the problem is to minimize eqs 13 and/or 14 under the constraint of eqs 7 to 12. The decision variables are {Yi}, {Q exi,j }, and {Yi,jcomp}. 5.4. Multiobjective Optimization. To optimize the design and operation of industrial processes, cost has been generally used as an objective function. This strategy is natural for commercial enterprises, but sometimes another objective needs to be taken into account. Minimizing an environmental stress is an example of such objectives, which usually conflict with minimizing cost. In the synthesis of compressor-aided distillation sequences investigated in this work, the use of compressors reduces ER but increases TAC. This trade-off can be evaluated through multiobjective optimization. Although there are several methods for solving multiobjective optimization problems, Marler and Aurora mentioned that the most robust approach involves normalizing each term in the objective function.20 In our research, a normalized adaptive min-max weighted sum (AMMWS) is proposed, which combines a min-max weighted sum and the adaptixve algorithm presented by Kim and de Weck.21 This objective function outperforms the typical normalized weighted sum (WS) in two aspects: 1 It can find Pareto-optimal solutions in concave regions of the Pareto front. 2 It can find Pareto-optimal solutions that are more evenly distributed along the Pareto front. In AMMWS, TAC and ER are normalized and embedded into the objective function as follows ⎧ TACi − TAC◦ ER − ER◦ ⎫ ⎬ min U = max⎨w , (1 − w) Ni N ◦ ⎩ TAC − TAC ER − ER◦ ⎭ i ∈ TK ⎛ TACi − TAC◦ ER − ER◦ ⎞ + (1 − w) Ni + ρ⎜w ⎟ N ◦ ⎝ TAC − TAC ER − ER◦ ⎠

Table 1. Data for Utilities and Heat Exchange Coefficients for All the Examples22a utility

temperature [K]

cost [$/GJ]

chilled water+, (CHW) cooling water+, (CW) 4.4 atm steam, (S1) 11.2 atm steam, (S2) 31.6 atm steam, (S3) electricity +20 K rise

278 305 420 459 511 ---

4.000 0.254 3.102 5.257 8.174 16.667

a

Heat transfer coefficients [MJ/(m2hK)]. Condenser: 2.16, reboiler: 3.6, heat exchanger: 1.8.

optimized at 3 atm, and recompression of vapor streams was done at 3 and 5 atm. The feed flow rate is 100 kmol/h, while the specifications for all the states are shown in Table 2, where T denotes the top distillate and W the bottom stream. For this example, the global optimal solution which minimizes both TAC and ER was obtained and is shown in Figure 7a. The optimal structure is a sloppy split sequence with vapor recompression in every column resulting in selfintegration. In addition, it can be observed that cooling utilities are not used. Figure 7b is the suboptimal solution with vapor recompression of a side and top streams. It adopts vapor recompression resulting in heat integration within the thermally coupled column. The best sequences without vapor recompression or heat integration was the Petlyuk column operating at 1.2 atm. The global optimal solution and two suboptimal solutions were compared with the base case (sloppy split sequence without heat integration or vapor recompression at 1.2 atm), and the results are shown in Table 3. For case study 1, a wide pressure range was assigned to the problem. Thus, the heat integration by changing the vapor pressure is effectively adopted in the optimal solution. In addition, the optimization results show that the thermally coupled columns shown in Figure 1d to 1g are more likely to realize heat integration than the Petlyuk column when pressure change is possible. 6.2. Four-Component Mixture of Acetone, Ethanol, 1-Propanol, and 1-Butanol (Case Study 2). The second case study involves a mixture of acetone (A), ethanol (B), 1-propanol (C), and 1-butanol (D). This mixture is more difficult to separate because the relative volatility between acetone and ethanol is close to one. Furthermore, there is a quaternary azeotrope at pressures higher than 4.5 atm, and a binary azeotrope between acetone and ethanol at pressures higher than 2.3 atm. To avoid azeotropic behavior, the candidate columns were simulated at three pressure levels: 1, 1.5, and 2 atm. The linking flow rates of thermally coupled subsequences were decided on the basis of the optimization results at 1.5 atm. Although the azeotropic condition is avoided, the system still shows highly nonideal behavior. The feed flow rate is 200 kmol/h. The product specifications for all the states that separate the quaternary mixture are shown in Table 4, and those that separate the ternary and binary mixtures are shown in Table 5. The original mixture and the intermediate streams were separated by a combination of the sharp and sloppy split conventional columns as well as thermally coupled columns shown in Figure 1. On the Pareto front, two optimal solutions are shown in Figure 8. Both of them consist of a combination of

(15)

where TAC° and ER° are the optimal values when eqs 13 and 14 are optimized individually. TACN is the value of TAC when eq 14 is optimized individually, and ERN is the value of ER when eq 13 is optimized individually. In addition, w is a weighting parameter to be manipulated by the decision maker, and ρ is a parameter in the range of 1e-2 through 1e-4. Furthermore, eq 15 satisfies the necessary and sufficient condition for Pareto optimality.20

6. RESULTS AND DISCUSSION This section presents two case studies to synthesize distillation sequences where heat integration and compressor addition are possible. The MILP problems were formulated with the abovementioned method and solved by IBM ILOG OPL under the following conditions: 10 years of payout time and 8000 h of operation per year. Table 1 shows a list of utilities and heat transfer coefficients used in the case studies. Comparisons among the optimal sequences on the Pareto front were addressed from the viewpoint of TAC, ER, and their savings. 6.1. Ternary Mixture of Cyclopentane, Benzene, and Toluene (Case Study 1). A ternary mixture of cyclopentane, benzene, and toluene is investigated in the first case study. This mixture has relative volatility close to 3 between adjacent components and does not exhibit azeotropic behavior or any major nonlinearity, which implies their separation is not difficult. In this case study all the sequences shown in Figure 1 were simulated at three different pressures: 1.2, 3, and 5 atm. The linking flows of thermally coupled subsequences were 5918



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Table 2. Specifications for All the States in Case Study 1 [mol %] AB/C

A/B/C

T

W

10 60 30

a

a

a

a

96

a

a

cyclopentane (A) benzene (B) toluene (C) a

A/BC

ABC

1

T

A/B

W

T

W

96

1

a

a

a

a

1

a

B/C

T

W

T

W

1

96

a

a

a

a

a

92

92

a

a

a

a

96

Depend on tasks.

Figure 7. Optimal solution for case study 1 (a) and a suboptimal solution (b).

lowest TAC. It consists of a PPL sequence that separates AB from C and D and a conventional column separating A from B. The conventional column supplies energy to the first column in the PPL sequence. Figure 8b realized vapor recompression in addition to heat integration, and this combination resulted in the lowest ER. It consists of a PPL sequence that separates A and B from CD and a conventional column separating C from D. The conventional column supplies energy to the second column in the PPL sequence. In both cases the conventional column supplies energy to the thermally coupled column.

Table 3. Economic and Energy Savings of Optimal Solutions in Case Study 1 solution

TAC [k$/y]

economic savings [%]

ER [GJ/h]

energy savings [%]

base case Figure 7a Figure 7b Petlyuk column

389.1 229.4 275.6 310.5

--41.0 29.2 20.2

5.31 1.04 2.44 3.65

--80.5 54.0 31.3

conventional and thermally coupled columns. Figure 8a shows the sequence that realized heat integration and resulted in the

Table 4. Specifications of States for the Quaternary Mixture [mol %] A/BCD T

ABC/D

ABCD

T

W

30 30 80 60

97

0.5

a

a

a

a

a

0.5

a

a

0.5

a

a

a

a

a

0.5

acetone (A) ethanol (B) 1-propanol (C) 1-butanol (D) a

AB/CD W

T

A/B/CD W

a

T

A/BC/D W

T

AB/C/D

W

T

W

a

a

0.2

a

0.2

a

a

a

a

a

a

a

a

a

0.2

a

a

0.2

a

a

a

a

a

97

a

a

0.2

a

0.2

a

Depend on tasks.

Table 5. Specifications of States for Ternary and Binary Mixtures [mol %] A/BC acetone (A) ethanol (B) 1-propanol (C) 1-butanol (D) a

AB/C T

B/CD

BC/D T

A/B

B/C

T

W

W

T

W

97

0.5

a

a

a

a

a

a

a

a

95

0.5

a

a

a

a

a

0.5

95

a

a

a

a

a

a

a

a

a

a

a

0.5

97

a

a

a

C/D

W

T

W

T

W

T

W

a

97

a

a

a

a

a

95

95

a

a

a

a

a

95

95

a

a

a

a

97

Depend on tasks. 5919



Article

Figure 8. Pareto front solutions for case study 2.

of the columns separating (A/BCD), (B/CD), and (C/D). Vapor recompression is also adopted in the last column. In the last case, all subsequences in Figure 1 are considered, but heat integration is not executed. In such condition, the combination of a conventional column and PET is selected as the best structure which minimizes TAC. In this case, the feed is separated to (A/BCD) by the conventional column, and then a Pelyuk column is used to separate (B/C/D). For case study 2, the feasible pressure range was very narrow. This feature limits the application of vapor recompression. However, vapor recompression can still realize the lowest ER.

In addition to two Pareto optimal solutions, simulation results of another three solutions are listed in Table 6. One is Table 6. Economic and Energy Savings of Optimal Solutions in Case Study 2 solution

TAC [k$/y]

economic savings [%]

ER [GJ/h]

energy savings [%]

base case Figure 8a Figure 8b CC+HI CC+PET

743.2 530.6 554.9 674.0 548.3

--28.6 25.3 9.3 26.2

18.73 10.77 9.31 13.33 13.52

--42.5 50.3 28.7 27.8

7. CONCLUSIONS We have proposed the optimal synthesis procedure of distillation sequences considering heat integration and compressor addition. A superstructure-based approach which combines rigorous simulations and mathematical programming was adopted, and the optimal synthesis problem was formulated as an MILP problem. A variety of thermally coupled columns were evaluated as candidate subsequences. To avoid computational complexity, the synthesis problem was divided into the following three steps: First, the number of stages was decided by using rigorous simulations. The analogy of the column structure was

the best distillation sequence consisting of only conventional columns and without heat integration. In such condition, the combination of SS and a conventional column minimizes TAC. In this structure, the feed is separated into (AB/C/D) by SS, and then a conventional column is used to separate A from B. This structure is expressed as base case in Table 6. The second is the case where the sequence consists of only CC. Heat integration and vapor recompression are considered to derive the optimal sequence, which minimizes TAC. The result is also shown in Table 6 as CC+HI. In this case, the structure consists 5920



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(6) Rev, E.; Emtir, M.; Szitkai, Z.; Mizsey, P.; Fonyo, Z. Energy Savings of Integrated Distillation Systems. Comput. Chem. Eng. 2001, 25, 119−140. (7) Caballero, J. A.; Grossmann, I. E. Structural Considerations and Modeling in the Synthesis of Heat-Integrated-Thermally Coupled Distillation Sequences. Ind. Eng. Chem. Res. 2006, 45, 8454−8474. (8) Yiqing, L.; Xigang, Y.; Fenglian, D. Synthesis and heat integration of thermally coupled complex distillation system. Int. J. Energy Res. 2010, 34, 626−634. (9) Fitzmorris, R. E.; Mah, R. S. H. Improving Distillation Column Design Using Thermodynamic Availability Analysis. AIChE J. 1980, 26, 265−273. (10) Oliveira, S. B. M.; Pitanga Marques, R.; Parise, J. A. R. Modelling of an ethanol-water distillation column with vapour recompression. Int. J. Energy Res. 2001, 25, 845−858. (11) Alcántara-Avila, J. R.; Kano, M.; Hasebe, S. Two-Level Approach for Synthesizing Externally and Internally Heat Integrated Distillation Sequences. 13th APCChE Congress. 2010. Taipei, Taiwan. (12) Fidkowski, Z. T.; Agrawal, R. Multicomponent Thermally Coupled Systems of Distillation Columns at Minimum Reflux. AIChE J. 2001, 47, 2713−2724. (13) Fidkowski, Z. T. Distillation Configurations and their Energy Requirements. AIChE J. 2006, 52, 2098−2106. (14) Agrawal, R.; Fidkowski, Z. T. New Thermally Coupled Schemes for Ternary Distillation. AIChE J. 1999, 45, 485−496. (15) Agrawal, R.; Fidkowski, Z. T. More Operable Arrangements of Fully Thermally Coupled Distillation Columns. AIChE J. 1998, 44, 2565−2568. (16) Christiansen, A. C.; Skogestad, S.; Lien, K. Complex Distillation Arrangements: Extending the Petlyuk Ideas. Comput. Chem. Eng. 1997, 21, S237−S242. (17) Yeomans, H.; Grossmann, I. E. Optimal Design of Complex Distillation Columns using Rigorous Tray-by-Tray Disjunctive Programming Models. Ind. Eng. Chem. Res. 2000, 39, 4326−4335. (18) Hernandez, S.; Jimenez, A. Design of energy-efficient Petlyuk systems. Comput. Chem. Eng. 1999, 23, 1005−1010. (19) Turton, R.; Bailie, R. C.; Whiting, W. B.; Shaeiwitz, J. A. Analysis, Synthesis, and Design of Chemical Processes; Prentice Hall: Upper Saddle River, NJ, 2003. (20) Marler, R. T.; Arora, J. S. Survey of multi-objective optimization methods for engineering. Struct. Multidisc. Optim. 2004, 26, 369−395. (21) Kim, I. Y.; de Weck, O. L. Adaptive weighted-sum method for bi-objective optimization: Pareto front generation. Struct. Multidisc. Optim. 2005, 29, 149−158. (22) Seider, W. D.; Seader, J. D.; Lewin, D. R.; Widagdo, S. Product and Process Design Principles; John Wiley and Sons, Inc.: Asia, 2010.

effectively used to decide the number of stages of thermally coupled columns. Second, the optimal operating condition of subsequences consisting of conventional columns or thermally coupled columns was derived. Then, using these results, the synthesis problem was formulated as an MILP problem. Usually, nonlinearity of the model arises in the former two steps, and rigorous simulations were executed for these steps. Thus, the proposed procedure can derive accurate results even for the cases in which the process has high nonlinearlity. In this research, vapor recompression at top vapor streams and pressure change in thermally coupled columns were considered. Though the compression of vapor expands the solution space, it requires an expensive compressor. To evaluate the beneficial points and drawbacks of the use of compressors in the sequences, two objectives, the total annual cost and energy requirement, were selected as objective functions. By solving multiobjective optimization problems, a more meaningful interpretation of vapor recompression and pressure change in conventional and complex distillation columns could be attained. The results of two case studies indicated that nonconventional sequences became the optimal for both cases. Vapor recompression was effective for minimizing the energy requirement. The feasible pressure range, relative volatility between adjacent components, and cost of utilities are three crucial factors to successfully implement vapor recompression or pressure change in multicomponent distillation. For the four-component case, thermally coupled columns could effectively reduce TAC and ER. Compared with the case consisting of a conventional column only, up to 29% reduction of TAC was achieved. By combining a min-max weighted sum and the adaptive algorithm, a normalized adaptive min-max weighted sum (AMMWS) was proposed to effectively solve multiobjective problems. The feature of the proposed procedure is to separate the rigorous simulations of each column and the optimization of the distillation sequence. This procedure effectively obtains reliable solutions and reduces the computational load of the MILP problem. Furthermore, it becomes easy to add new complex column structures as candidates of subsequences. Our future research plans will involve reformulating the problem to include internal heat integration (e.g., HIDiC column) and to evaluate operability of sequences with vapor recompression and pressure change in multicomponent mixtures.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

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REFERENCES

(1) Andrecovich, M. J.; Westerberg, A. W. An MILP Formulation for Heat-Integrated Distillation Sequence Synthesis. AIChE J. 1985, 31, 1461−1474. (2) Floudas, C. A.; Paules, G. E. A IV Mixed-Integer Nonlinear Programming formulation for the synthesis of Heat-Integrated distillation sequences. Comput. Chem. Eng. 1988, 12, 531−546. (3) Aly, S. Heuristic approach for the synthesis of heat-integrated distillation sequences. Int. J. Energy Res. 1997, 21, 1297−1304. (4) Sobocan, G.; Glavic, P. A simple method for systematic synthesis of thermally integrated distillation sequences. Chem. Eng. J. 2002, 89, 155−172. (5) Peltyuk, F. B.; Platonov, V. M.; Slavinskii, D. M. Thermodynamically Optimal Method of Separating Multicomponent Mixtures. Int. Chem. Eng. 1965, 5, 555−561. 5921


Multiobjective Optimization for Synthesizing Compressor-Aided

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