Rigorous Design of Complex Distillation Columns Using Process

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Rigorous Design of Complex Distillation Columns Using Process Simulators and the Particle Swarm Optimization Algorithm J. Javaloyes-Antón, R. Ruiz-Femenia,* and J. A. Caballero Department of Chemical Engineering, University of Alicante, Ap. 99, E-03080 Alicante, Spain ABSTRACT: We present a derivative-free optimization algorithm coupled with a chemical process simulator for the optimal design of individual and complex distillation processes using a rigorous tray-by-tray model. The proposed approach serves as an alternative tool to the various models based on nonlinear programming (NLP) or mixed-integer nonlinear programming (MINLP) . This is accomplished by combining the advantages of using a commercial process simulator (Aspen Hysys), including especially suited numerical methods developed for the convergence of distillation columns, with the benefits of the particle swarm optimization (PSO) metaheuristic algorithm, which does not require gradient information and has the ability to escape from local optima. Our method inherits the superstructure developed in 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 (11), 4326−4335, in which the nonexisting trays are considered as simple bypasses of liquid and vapor flows. The implemented tool provides the optimal configuration of distillation column systems, which includes continuous and discrete variables, through the minimization of the total annual cost (TAC). The robustness and flexibility of the method is proven through the successful design and synthesis of three distillation systems of increasing complexity.

1. INTRODUCTION The costs of a chemical process are often dominated by the costs of the separation and purification of the products. Among the various separation techniques, distillation is one of the most important and commonly used in all chemical and petrochemical industries. However, the equipment is very energetically inefficientheat is provided in the reboiler and removed afterward in the condenserand entails high capital costs. In fact, about 90% of all separation and purification operations in the United States are distillations, which represented around 3% of the total U.S. energy consumption in 2002, that is, 91 GW or 54 million tons of crude oil.1 It is therefore desirable to have robust and reliable tools with which to design optimal distillation processesto reduce the investment and operating costs of the equipment (specifically the energy consumption, which has a large economic and environmental impact). The economic optimization of complex distillation columns is a nontrivial problem due to the discrete nature of the tray-bytray column model and also because of the high degree of nonlinearity and nonconvexity of the underlying MESH equations (mass, equilibrium, summation, and energy). In the optimization, discrete decisions are related to the calculation of the number of trays and the location of feed and side product streams, whereas continuous variables are related to the operating conditions (e.g., reflux ratio, boilup ratio, distillate to feed ratio, etc.). A number of different approaches, based on mathematical programming, have been developed in the past decades in order to provide reliable rigorous tray-by-tray optimization models. Most of these methods are formulated as mixed integer nonlinear programming problems (MINLP), where the distillation column is modeled as a superstructure with variable column ends (reflux and reboil location, or even both), or as a generalized disjunctive programming (GDP) representation, © XXXX American Chemical Society

where the column is modeled as a superstructure in which the nonexisting trays are considered as bypasses of liquid and vapor flows. Methods that have addressed the solution of MINLP problems include the following: branch and bound (BB),2−5 generalized Benders decomposition (GBD),6,7 outer approximation (OA),8,9 among others. But all these methods assume convexity to guarantee convergence to the global optimum. A significant advance in the modeling and solution of design problems was the introduction of disjunctive programming,10,11 from the point of view of both modeling and solution. The logical part of a model can be represented by a set of disjunctions that allow the researcher to focus on the model itselfdecoupling, at least partially, the model formulation from the solution. The solution then can be performed either by reformulating to an MI(N)LP problem (i.e., using a convex hull or a big M approach) or by using directly a logic-based algorithm.11,12 The first successful MINLP model to optimize simultaneously the number of trays and feed location for a specified separation was published by Viswanathan and Grossmann.13 These authors proposed a superstructure with multiple tray locations for the reflux and boilup flows, and they assigned binary variables to the existence of each of those potential streams; see Figure 1a. A major difficulty of the resulting model is the fact that the vapor−liquid equilibrium conditions are enforced in all trays of the column, even in the case of inactive trays (see Figure 1b) where no mass transfer takes place. In these trays there is a zero Received: March 21, 2013 Revised: June 3, 2013 Accepted: September 26, 2013

A

dx.doi.org/10.1021/ie400918x | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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Figure 1. (a) Superstructure of Viswanathan and Grossmann for the optimal feed location, total number of trays, and optimal operation conditions. (b) Inactive trays in the Viswanathan and Grossmann model.

liquid flow in the rectifying section or a zero vapor flow in the stripping section, which can produce numerical problems as a result of the convergence of the equilibrium equations (which are highly nonlinear expressions) with zero flow of one of the two phases.14 Despite these drawbacks, different research groups have successfully applied the model by Viswanathan and Grossmann to optimize individual columns and superstructures. For example, Ciric and Gu15 used the MINLP approach to optimize a reactive distillation column. Bauer and Stichlmair16 applied the MINLP approach to the synthesis of sequences of azeotropic columns, and Dünnebier and Pantelides17 used the model to generate sequences of thermally coupled distillation columns. In order to overcome some of the difficulties with MINLP models, Yeomans and Grossmann18 proposed a GDP representation, in which the nonexisting trays are considered as simple bypasses of vapor and liquid flows without mass transfer. Therefore, mass and energy balances are trivially satisfied, and the only difference with respect to active trays is in the application of the equilibrium equations (see Figure 2). Barttfeld et al.14 showed that multiple representations of the optimization of a single distillation column are possible when it is solved by means of a GDP formulation. However, the computational results showed that the most effective alternative was the original configuration proposed by Yeomans and Grossmann18 (shown in Figure 2). The GDP formulation is as follows:

Figure 2. Superstructure of Yeomans and Grossmann for the optimal feed location, total number of trays, and optimal operation conditions.

by these algorithms and the quality of the solution strongly depends on the initialization point of the search. As a consequence of these difficulties, the resulting optimization tools are very complex, and so, only those skilled in the art are able to utilize and adapt them to their own needs. An alternative for handling these very difficult or even unsolvable problems, preserving the mathematical programming approach, is to substitute the rigorous models by simplified ones. Nevertheless, this procedure can entail the loss of good solutions.19 Caballero et al.20 proposed an algorithm that integrates a process simulator in a GDP formulation. In this way, all numerical aspects related to the convergence of a distillation column are solved at the level of the process simulator. Therefore, by taking advantage of the tailored numerical

All previous models were developed in an equation-based environment and required sophisticated initialization techniques to get a feasible solution. In addition, because of the nonconvexity of the problem, only local optima are guaranteed B



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all mass and energy balances are solved at the level of the process simulator. A considerable amount of literature supports the synergy obtained by smart integration of a process simulator with an external optimizer based on metaheuristic algorithms. Gross and Roosen19 demonstrated the suitability of a genetic algorithm coupled with the process simulator Aspen Plus to optimize arbitrary flow sheets. In particular, the results for the first example of that work, which deals with the synthesis of the distillation sequence for a five-component mixture, concluded that the position of the feed tray and number of trays (integer variables) should be considered as design variables, which can be successfully optimized by the genetic algorithm. Leboreiro and Acevedo25 also succeeded in problems where deterministic mathematical algorithms had failed, using an optimization framework for the synthesis of complex distillation sequences based on a modified GA coupled with Aspen Plus. Specifically, in case III of their work, the authors were able to optimize a complex system such as a Petlyuk column. The same combination of process simulator and metaheuristic algorithm is adopted by Vazquez-Castillo et al.26 to address the optimization of five distillation sequences, involving dividing wall columns (DWCs). The difference with the former work lies in the introduction of several objectives (the total number of stages of each column and the heat duty of the existing reboilers), which are simultaneously minimized by the multiobjective genetic algorithm with constraints developed ́ 27 Subsequent by Gutiérrez-Antonio and Briones-Ramirez. works also used the same multiobjective GA for the optimization of thermally coupled distillation systems28−30 and for the retrofit of a subcritical pulverized coal power plant with an MEA-based carbon capture and CO2 compression system.31 The design of this plant involves the optimization of two packed columns (an absorber and stripper), where the packing heights were specified as design variables. Apart from GA, other gradient free optimization algorithms are combined with process simulators. For instance, Aspelund et al.32 proposed a optimization-simulation tool based on a global Tabu Search (TS) and a local Nelder-Mead search coupled with Aspen Hysys. Moreover, Dantus and High33 investigated the performance of a stochastic annealing algorithm combined with Aspen Plus for the multiobjective optimization of a methyl chloride process. With the above in mind, the purpose of this paper is to develop an optimization tool for designing complex distillation processes that combines the advantages of using a process simulator (Aspen Hysys) with the benefits of the PSO metaheuristic algorithm, which is implemented in Matlab code. Thus, all numerical aspects related to the convergence of distillation columns, including the selection of thermodynamic models, are specified at the level of the process simulator; and the external PSO optimizer (Matlab) is interfaced with the simulator in order to solve the optimization problem. Everything is controlled by an external executive program. The rest of the paper is organized as follows. The next section presents the main aspects of our simulationoptimization framework. Then, we illustrate the application of the proposed methodology with three case studies, which show how to develop different superstructures based on the model of the single column. Finally, we discuss the conclusions that can be drawn from this work.

techniques specially developed for converging distillation columns and built into the process simulators, the difficulties related to the initialization are overcome. However, some important difficulties arise in the implementation of this approach due to the modular architecture of the process simulators, in which the different blocks (processing units) are user-supplied “black box” models; and there usually is no access to the original code, and derivative information is not available either. This fact may be significant because gradient based algorithms for solving MINLP, as previously mentioned, require accurate derivative information; and when a process simulator is used, derivatives for the design variables can only be obtained by perturbation, which introduces the following two important drawbacks: • the perturbation of a variable requires solving the entire flow sheet each time a variable is perturbed, resulting in a significant increase in the CPU computation time, and • unit operations in process simulators (“black boxes”) introduce numerical noise, preventing the calculation of accurate derivatives. This numerical noise is due to the low sensitivities of some variables to the convergence criteria used by the model. For example, in a distillation column the mass and energy balance are checked and closed with strict tolerance criteria. However, reboiler and condenser heat flows can change by orders of magnitude larger than the mass or energy tolerances. This is not relevant from the point of view of simulationthe conditioning of the system is low enough to ensure accurate valuesbut this could produce catastrophic effects if the derivatives must be calculated using perturbations. These difficulties can be overcome by derivative-free algorithms. Metaheuristic algorithms coordinate an interaction between local improvement procedures and higher level strategies to create a process capable of escaping from local optima. There are a wide variety of metaheuristics. We may cite, among others, simulated annealing (SA), tabu search (TS), variable neighborhood search (VNS), scatter search, ant colony optimization (ACO), iterated local search (ILS), particle swarm optimization21 (PSO), and genetic algorithms (GAs). For a detailed description of these solution methods, see, for example, the books of Gendreau,22 Luke,23 and Gonzalez.24 Many free-derivative search algorithms are population-based procedures, where an individual represents a particular solution to the optimization problem and a population is a set of individuals competing with each other with respect to their objective function values. Since these algorithms do not require gradient information, it is possible to treat the objective function as a black-box. This black-box evaluation can be performed by any simulation software if an interface between process simulator and optimizer is given. Although metaheuristic algorithms are not able to guarantee the optimality of the solutions found, gradient-based methods (which theoretically can provide such a certification) are often incapable of finding solutions whose quality is close to that obtained by the metaheuristics.22 This is especially true for realworld problems, which exhibit high levels of complexity. Perhaps the most serious disadvantages of metaheruistic algorithms are that the number of function evaluations to converge could be large, and as well as they exhibit poor performance in highly constrained systems. However, the last problem is easily solved when a process simulator is used, since C



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2. SIMULATION-OPTIMIZATION STRATEGY 2.1. PSO Algorithm. Figure 3 shows a basic flow chart depicting the general PSO algorithm. Two mean reasons lead

yli, k

⎧ 0 if r ≤ sig(v i ) ⎪ 3, l l ,k =⎨ ⎪1 if r > sig(v i ) 3, l l ,k ⎩

(2)

where r3 is a uniform random vector, whose elements are between 0 and 1, and sig(vil,k) is the sigmoid function: sig(vli, k) =

1 i

1 + e(−vl ,k)

(3)

The stopping criterion depends on the type of problem being solved: usually, the algorithm is run for a fixed number of function evaluations or until a specified error bound is reached. 2.2. Objective Function. As stated previously, the objective function to be minimized comprises the annualized investment cost, or capital cost of the main items (column shell, trays, and heat exchangers) and the most relevant operating costs (vapor steam and cooling water). The estimation of the capital costs, which depends on the column diameter, total number of trays, and heat exchanger areas, was performed by means of the nonlinear cost correlations given by Turton et al.36 On the other hand, the operating costs were calculated from the corresponding heat loads of the reboiler and condenser, using the steam and cooling utility costs.36 Thus, the objective function can be stated as follows: min TAC($/yr) = Cop + FCcap

(4)

where Cop is the operating cost per year and Ccap is the total cost of installed equipment, both updated by the global CEPCI cost index of 2011 (Chemical Engineering, April 2012). The annualization factor of the capital cost, F, was calculated by eq 5, as recommended by Smith,37 and takes into account the fractional interest rate per year (i) and the horizon time (n). Typical values are a fixed interest rate of 10% and a horizon time of 5 years. It is worth mentioning that changing the annualization period can lead to different optimal column configurations because of the trade-off between the capital and operation costs.

Figure 3. Flow chart of the gbest PSO algorithm for continuous and discrete variables.

F=

i(1 + i)n (1 + i)n − 1

(5)

2.3. Single Distillation Column Superstructure. The basic idea is to consider a conventional distillation column as a set of permanent trays. Among them are the feed tray, the condenser and the reboiler, and a set of conditional trays above and below the feed that can either exist or not (see superstructure proposed by Yeomans and Grossmann,18 Figure 2). For a desired separation the number and distribution of the trays above and below the feed is a function of the relative volatilities and composition of the feed. Initially the number of conditional trays in the rectification and stripping sections must be larger than the minimum needed to ensure the desired purity of the products, providing an upper bound to the optimal number of trays. After the optimization, the optimal number of trays and feed location will be defined by the number of active trays in each section. In a process simulator such as Aspen Hysys, it is possible to generate the previously described superstructure using a built-in distillation column. The existence or nonexistence of the conditional trays (equilibrium stages) can be performed by forcing the trays to behave as a simple bypass of liquid and vapor flows, without mass or heat transfer, simply by setting the

us to choose the stochastic PSO for our simulationoptimization tool: few tuning parameters and ease of implementation. Depending on the interaction scheme between the particles, two main versions of the PSO exist. In the first version, called gbest, every particle is attracted to the best ; solution found by any member of the population, PG‑global k hence, this structure is equivalent to a fully connected social network. In the second one, called lbest (g and l stand for “global” and “local”), each individual particle is affected by the best performance of its immediate neighbors PL‑global . In our k work we use the gbest model. This interaction scheme offers a faster rate of convergence34 at the expense of robustness and permits only a single best solution, called the global best particle, across all the particles in the swarm. Here, by robustness, we mean the ability of the algorithm to solve many kinds of problems with a minimum amount of special adjustments to account for special qualities of a particular problem. To handle the binary variables, our tool uses the discrete binary version of PSO,35 where each component l of the vector yi, which contains the binary variables associated with particle i, is updated according to the following: D



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Murphree tray efficiency to zero for the inactive trays.20 As can be seen in eq 6, the Murphree efficiency is calculated from the following: the mole fraction of the vapors leaving trays n and n + 1, yn and yn+1, respectively, and the composition of vapor in equilibrium with the liquid leaving the tray n, y*n . y − yn + 1 EMV = n 0 ≤ EMV ≤ 1 y* − y (6) n

is not necessary to introduce any external constraints on purity or recovery. Under any other circumstances, if two operating conditions of the column are selected, such as the reflux ratio and boilup ratio, it will be necessary to add extra constraints that take into account the specifications of purity or recovery initially imposed on the product flows. This can be accomplished by including these constraints in the PSO. However, although the PSO technique has proved to be efficient for solving nonconvex optimization problems,21 the original PSO is not as successful in solving constrained problemsthe algorithm handles the constraints by penalizing infeasible solutions. Therefore, it would be necessary to modify the objective function by adding a penalty term to account for the deviation between the desired purity or recovery specifications in the product streams, Xspc,i, and the obtained value at each iteration of the process xi, so that, when a constraint is violated it will appear as a positive contribution in the objective function, as shown in eq 7:

n+1

Note that equilibrium equations are trivially satisfied in those trays in which the efficiency is set equal to zero (yn = yn+1). To avoid equivalent solutions in each section of the column, the active trays must be consecutive. Therefore, to prevent “empty spaces” between active trays in the superstructure, we follow these criteria: in the rectification section, if a given tray exists, then all the trays below it down to the feed tray must exist. And in the stripping section, if a given tray exists, then all trays above it must exist (see Figure 4). The indexes of the

min TAC = Copt year + fc Ccap +

∑ wi|(Xspc ,i − xi)| i

(7)

where wi is a positive penalty parameter of the same magnitude as the costs. That is why it is advisable, whenever possible, to select as design variables the specifications of purity or recovery required in the distillate and bottoms products. In this way, the PSO does not have to handle the continuous variables, and we do not have to introduce any penalty term in the objective. However, that is not always possible. In some systems with recycle streams, where convergence by itself (without the optimization process) is quite complicated, it is better to select design variables that result in more robust flow sheets, such as the reflux and boilup ratios, with the aim of making an optimization stage easier. If we need to add any other constraint, such as temperature bounds for security or stability reasons, for example, a penalty term in the objective function should be used. It is worth remarking that we are using an exact penalty function. In that way we ensure that the solution of the original problem (without penalties) and the reformulated problem is the same if the penalty term is large enough, and at the same time the magnitude of the penalty is not too large (larger than or equal to the lagrange multiplier related to the constraint).38 2.5. Optimization Algorithm with Embedded Process Simulator. A scheme of the proposed approach is shown in Figure 5. The external optimization solver (PSO), as well as the objective function and all other auxiliary files were implemented in Matlab. An executive program controls all these files and establishes the connection with the process simulator, Aspen Hysys. We use the binary-interface standard component object model (COM), by Microsoft, to interact with Aspen Hysys through the objects exposed by the developers of the process simulator. We utilize Matlab as an automation client to access these objects and interact with Aspen Hysys, which works as an automation server. By writing Matlab code, it is possible to send and receive information to and from the process simulator. Thus, the exposed objects make it possible to perform nearly any action that is accomplished through the Aspen Hysys graphical user interface, allowing us to use Aspen Hyspys as a calculation engine. According to the objectoriented programming nomenclature,39 in Aspen Hysys, the functions defined within each object (e.g., reactor) are called methods (i.e., governing equations), and the variables

Figure 4. Scheme of the proposed criteria to avoid equivalent solutions.

existing top ends in the rectification and stripping sections are defined by the following two integer variables: NR and NS, respectively, where 1 ≤ NR ≤ NR and NS ≤ NS ≤ NT (NT is the total number of trays in the column). Therefore, in the optimization process, the PSO will only have to adjust these two integer variables in order to vary the total number of trays and feed tray location of the column. 2.4. Design and Process Specifications. Once the superstructure is defined, and all the basic design decisions required by the distillation column module are selected in the simulator environment (i.e., selection of the thermodynamic model, feed specification, and column pressures), there are different ways of managing the remaining degrees of freedom of the column, those that correspond to the design/operation variables (continuous variables). From a large list of “column specifications”, Hysys allows us to select as many operating conditions as degrees of freedom in the column. Note that, for the case of a conventional distillation column, with a known feed and two products streams, once the operating pressure is fixed, there remain only two degrees of freedom. If we define as design variables the purity specification or recovery of the key components required in the product flows (distillate and bottom), then the problem is completely defined. Therefore, it E



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Figure 5. Scheme of the proposed algorithm.

population size from 5 to 200 particles to show the trade-off between the objective function value and CPU time. The results show that a population size higher than 20 increases considerably the computational time but only yields a negligible improvement of the optimum value. All the examples were solved on a computer with a 1.66 GHz Intel Core Duo processor and 1 GB of RAM. 3.1. Single Distillation Column. Let us consider the optimization of a single conventional distillation column, with one feed and two products streams: the distillate and bottom. The problem can be stated as follows: given a feed of known composition, determine the optimal configuration (feed location and total number of trays) and the optimal operating conditions (e.g., distillate flow rate, molar ratio of distillate to feed, reflux ratio, boilup ratio, ...) for separating the feed into two product streams within given specifications and needed to minimize the total annualized cost of equipment and utilities. The feed for this example is a multicomponent mixture of hydrocarbons ranging from c-4 to c-6, and the objective is to obtain the c-4 hydrocarbons as top products at a minimum total annual cost (TAC). The molar flow rate and composition of the feed, and other data for the problem, are shown in Table 1. We assume that the maximum heavy impurity in the top product stream leaving the column (isopentane) must be 0.5 mol % and that the light impurity in the residue (cyclobutane) must also be 0.5 mol %. These constraints can be treated as specifications in the process simulator or as external constraints as mentioned in section 2.4. In the latter case, two new column specifications must be chosen, for instance, the reflux ratio (RR) and the boilup ratio (BR), which the process simulator will attempt to

contained in an object are known as properties (e.g., feed composition). The next step of the proposed algorithm is the initialization of the PSO parameters. Then, the values of the indices of the top active trays of each column section (NR and NS) are converted in the Matlab environment to a vector of ones and zeros according to the existence or not of the trays, respectively. This information is sent to the built-in distillation column module as a vector of Murphree efficiencies. At this point, the distillation column is automatically updated and converged, and the process simulator returns all the dependent variables needed for calculating the total annual cost (TAC). Usually, the algorithm runs until a specified stopping criterion is reached for instance, until the objective function of all the particles in the PSO has collapsed under a specified toleranceor for a fixed number of function evaluations.

3. NUMERICAL EXAMPLES Three case studies are presented to illustrate the methodology and how to proceed with the proposed method. The first example is the optimization of the single distillation column; afterward, two more complex examples are constructed based on the first example. For all the cases, the first stage in our methodology involves developing a superstructure that covers all the interesting alternatives. In this work, we use the superstructure developed by Yeomans and Grossmann,18 in which the nonexisting trays are considered as simple bypasses of liquid and vapor flows. We use a population size of 20 particles for all the examples. This key PSO parameter was tuned after a set of computational experiments varying the F



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Table 1. Data for Examples Heat Exchangers

Distillation Columns

Condenser: U = 800 W/(m2 K) Kettle reboiler U = 820 W/(m2 K) Material of construction: carbon steel (shell and tubes)

Calculation based on sieve tray (one pass) Material of construction: carbon steel (sieves and tower) Tray spacing, d, 0.609 m Column height H (m) = 3 + NT·d Tray Sizing based on design limit: flooding (85%) Utility Costs

Low pressure steam (5 barg, 160 °C) Medium pressure steam (10 barg, 184 °C) High pressure steam (41 barg, 254 °C) Cooling water (30−45 °C) Electricity (Calculation based on 8000 h/yr of operation) Example 1 Feed Composition (mole fraction) isobutane n-butane cyclobutane isopentane n-pentane cyclopentane 2-methylpentane (isohexane) n-hexane cyclohexane pressure thermodynamics (fluid package) specifications molar fraction isopentane (heavy key comp) in distillate molar fraction cyclobutane (light key comp) in bottoms

7.78 $/GJ 8.22 $/GJ 9,83 $/GJ 0.354 $/GJ 60.0 $/MWh Example 2

1000 kmol/h (85 °C) 0.17 0.12 0.06 0.13 0.09 0.07 0.09 0.15 0.12 600 kPa Soave−Redlich−Kwong

Feed Composition (mole fraction) Acetone Methanol thermodynamics (fluid package) specifications Extractive Column pressure molar fraction acetone in distillate acetone recovery Entrainer−Recovery Column Pressure molar fraction methanol in distillate molar fraction DMSO in bottoms

540 kmol/h (47 °C) 0.50 0.50 UNIQUAC

101 kPa ≥0.9999 ≥99.95% 101 kPa ≥0.9995 ≥0.9999

≤0.005 ≤0.005 Example 3 500 kmol/h (111 °C)

Feed Composition (mole fraction) Benzene Toluene p-Xylene thermodynamics (fluid package) specifications Prefractionator Column pressure benzene recovery p-xylene recovery Main Column Pressure molar fraction benzene in distillate molar fraction toluene in side stream molar fraction p-xylene in bottoms

0.30 0.40 0.30 Soave−Redlich−Kwong

120 kPa ≥99.95% ≥99.95% 101 kPa ≥0.999 ≥0.999 ≥0.999

adjust in such a way that the desired purities (added as external constraints) are achieved. In this example and the following ones, the product specifications are treated as column specifications in the simulator environment. A superstructure with an upper bound of 60 trays is initially considered (condenser and reboiler are not included). The feed tray is located at tray 30 when numbering from top to bottom. The number of conditional trays in each column section was 25, as shown in Figure 6. We also specify a minimum number of 10 permanent trays in the middle of the column (including the feed tray). Although this is not strictly necessary, it helps

during the optimization search procedure by reducing the number of alternatives. For the case where the optimal solution lies at one of these limits, 1 ≤ NR ≤ NR (25) and NS (36) ≤ NS ≤ NT(60), the upper bound of trays is increased or the minimum number of permanent trays is decreased, or even both. When presenting the optimization results, it must be taken into account that the PSO algorithm is a stochastic optimization method, and hence convergence to the same solution is not always guaranteed. In addition, although the G



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Table 2. Results of 20 Consecutive Executions of the PSO AlgorithmExample 1a Execution

Total Trays

Feed Tray

NR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

45 45 46 45 45 45 44 46 45 45 45 45 47 45 45 45 45 45 44 46

21 21 21 21 21 21 20 21 21 21 21 21 22 21 21 21 21 21 20 21

10 10 10 10 10 10 11 10 10 10 10 10 9 10 10 10 10 10 11 10

Figure 6. Single Distillation Column Superstructure.

distillation columns usually have a single global optimal solution, there may be several solutions near the best one (e.g., given a distillation column with n trays, a similar structure but with a certain number of trays more − which means a greater capital cost − can result in a similar objective function, since the energy consumption is decreased). Here the annualization factor, F, plays an important role − see eq (4). For that reason, one way to assess the statistical convergence of the proposed optimization approach is to run the algorithm for a certain number of times and to check how many times the algorithm converges to the best solution among all the executions. In our examples, we performed 20 executions for each optimization problem. Furthermore, since the optimizer can move the decision variables to a location in the search space where the convergence of the simulator fails, we use a penalty strategy to remedy this problem. Hence, we choose a large value for the penalty coefficient for the simulations ending with warnings or errors (infeasible designs). The main results of 20 consecutive executions of the optimization algorithm are summarized in Table 2. As can be seen, there are four different configurations of the distillation column; all of them are very close, not only in structure but also in value of the objective function. Among them, the configuration with the best objective function, and also the most repeated (70%), is the distillation column with 45 trays and feed tray at stage 21 (NR = 10, NS = 54). It is worth mentioning that, for a given solution, the small variations in the value of the objective function (TAC) are a consequence of the intrinsic numerical noise of the process simulator. The optimal design characteristics and computational results for the best configuration found by the algorithm are shown in Figure 7 and Table 3. 3.2. Extractive Distillation System. The second case study involves the optimization of an extractive distillation process. This sort of assisted distillation is commonly used to separate close boiling or homogeneous binary azeotropes by adding a higher-boiling component, the so-called entrainer. The new component facilitates the separation by interacting with the original mixture and attracting one of the components. The proposed case study is taken from Luyben,40 and the objective

NS

TAC (k$/yr)

CPU time (s)

Stopping Criterion

54 54 55 54 54 54 54 55 54 54 54 54 55 54 54 54 54 54 54 55

1803,4 1802,7 1803,3 1802,8 1802,7 1802,7 1803,1 1803,4 1802,7 1803,5 1803,3 1803,3 1803,0 1803,2 1802,7 1802,7 1803,5 1803,5 1802,9 1803,4

30 25 38 22 63 25 30 41 23 17 25 25 68 20 59 31 28 17 18 28

Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2 Criterion 2

a

Stopping Criterion 1: maximum number of iterations has been reached (50 major iterations). Stopping Criterion 2: below specified tolerance between best and worst particle (1 × 10−5).

Figure 7. Best solution of example 1.

Table 3. Computational Results for Best Solution Example 1 PSO Description Number of Particles Major Iterations Function Evaluations Discrete Variables Stopping Criterion CPU Time (s) TAC (k$/yr) Capital Cost (k$) Operating Cost (k $/yr)

H

20 18 380 2 below specified tolerance between best and worst particle 23 Optimal Solution 1802.7 1150.3 1499.3



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Figure 8. Acetone/methanol extractive distillation with DMSO solvent superstructure.

Table 4. Results of 20 Consecutive Executions of the PSO AlgorithmExample 2a Extractive Column

Entrainer−Recovery Column

Execution

Total Trays

DMSO Feed Tray

Feed Tray

NR

NM

NS

Total Trays

Feed Tray

NR

NS

DMSO flow rate (kmol/h)

TAC (k$/yr)

CPU time (s)

Stopping Criterion

1 2 3 4 5 6 7 8 g 10 11 12 13 14 15 16 17 18 19 20

46 46 46 46 46 47 46 45 46 46 46 46 47 45 46 46 46 46 46 46

3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

29 29 29 29 29 30 29 29 29 29 29 30 28 29 29 29 29 29 29 29

8 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

13 14 14 14 14 13 14 14 14 14 14 14 13 15 14 14 14 14 14 14

55 55 55 55 55 55 55 54 55 55 55 55 55 55 55 55 55 55 55 55

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

488 488 488 488 488 486 488 489 488 488 488 488 487 489 488 488 488 488 488 488

3042,4 3042,4 3042,4 3042,4 3042,4 3042,8 3042,4 3042,3 3042,4 3042,4 3042,2 3042,4 3042,7 3042,8 3042,4 3042,4 3042,3 3042,4 3042,4 3042,4

143 90 87 110 89 134 88 96 116 113 91 68 69 133 108 95 115 135 82 103


a Stopping Criterion 1: maximum number of iterations has been reached (70 major iterations). Stopping Criterion 2: below specified tolerance between best and worst particle (1 × 10−5).

The classical extractive distillation system comprises a set of two distillation columns: the extractive column, which has two feeds, and the entrainer-recovery column, represented in Figure 8. The entrainer is fed into the extractive column above the process feed. One of the original components, the acetone, is obtained at the top of the extractive column, while the methanol, together with the entrainer (DMSO), forms the bottoms product. In the second column, the entrainer is separated from the methanol and recycled back to the first column.

is to separate an isomolar mixture of acetone and methanol using dimethyl sulfur oxide (DMSO) as entrainer. This system has the following properties: the acetone/ methanol mixture has a binary homogeneous azeotrope at 77.6 mol % acetone and atmospheric pressure; the normal boiling points of acetone and methanol are 329 and 338 K, respectively, while the normal boiling point of the entrainer DMSO is 464 K. DMSO is much higher boiling than either acetone or methanol and is also a very effective solvent; this makes it possible to attain high product purities. I



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Figure 8 shows the proposed superstructure for the optimization of the acetone/methanol extractive distillation process with DMSO. The extractive column has two different feeds that divide the column into three different sections with conditional trays. On the other hand, the solvent-recovery column is a conventional column, and the superstructure is basically the same as in the previous example. In optimizing this system, we must take into account that the entrainer flow rate is also a variable that requires optimization and that it has a direct influence on the reflux. In this regard, the recycle stream poses an even more difficult challenge. The upper bound for the number of trays of each column, and the conditional trays of all the column sections are shown in Figure 8. As in the previous example we also specify a minimum number of permanent trays in addition to the feed trays, condensers, and reboilers. The molar fraction required for the acetone in the overhead of the first column is ≥0.9995 for a minimum recovery of 99.95%. The methanol molar fraction, the top product of the second column, must be greater than 0.9995, and the entrainer must be recovered with a minimum purity of 99.99%. These constraints are stated in the simulator environment as the column specifications. The remaining data for example 2 are shown in Table 1. Table 4 summarizes the important results of the 20 consecutive executions of the proposed approach. Notice that there are four different configurations of the extractive column (one with 45 trays, two with 46, and one with 47 trays), and only one configuration of the entrainer−recovery column with a total number of 12 trays. The minimum objective function (TAC = 3042.2 k$/yr) corresponds to one of the configurations with 46 trays inside the extractive column. All configurations and objective function values are very close; Figure 9 and Table 5 show the design and computational results of the best solution found by the algorithm, which also happens to be the most repeated (75%). 3.3. Divided Wall Column. The objective of this example is to optimize a fully thermally coupled distillation system, or

Table 5. Computational Results for Best Solution Example 2 PSO Description Number of Particles Major Iterations Function Evaluations Discrete variables Continuous variables Stopping Criterion CPU Time (s) TAC (k$/yr.) Capital Cost (k$) Operating Cost (k $/yr)

20 49 1000 5 1 below specified tolerance between best and worst particle 91 Optimal Solution 3042.2 1217.0 2721.1

Petlyuk column,41 for separating a three component mixture without azeotropes. As shown in Figure 10, thermally coupled structures result when vapor−liquid interconnections are used between two columns in the place of the condenser or reboiler of one of the columns.

Figure 10. (a) Direct sequence. (b) Thermally coupled direct sequence (side-rectifier arrangement). J



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In this particular example the reboiler of the first column is replaced by a thermal couple. The liquid from the bottom of the first column is transferred to the second as before, but now the vapor required by the first column is supplied by the second column, instead of a reboiler on the first column. The Petlyuk structure (Figure 11a) only uses two heat exchangers (one condenser and one reboiler) compared with

undesirable side effects: (a) the computation time for a single simulation considerably increases, and (b) it is relatively easy for the system to become prone to errors. In any case, if we try to use the PSO algorithm, the systems should be robust and easy to converge, and this condition is completely lost when a large number of recycles are introduced. To facilitate simulation of the dividing wall column, we make use of the novel strategy for the simulation of thermally coupled distillation sequences developed by Navarro et al.42 In this paper, the authors use two conventional distillation column modules connected by a combination of a material and an energy stream, instead of material recycle streams; this avoids the recycle structure that appears in thermally coupled distillation columns. In the rectifying section, the material stream is vapor at its dew point and the energy stream is equivalent to the energy removed if we include a partial condenser to provide reflux to the first column. In the stripping section, the material stream is liquid at its bubble point and the energy stream is equivalent to the energy added if we include a reboiler to provide vapor to the first column. Figure 12 shows the superstructure proposed for the optimization of a dividing wall column for separating a mixture of benzene, toluene, and p-xylene into pure components (≥99.95 mol %). The upper bound for the number of trays of the first distillation column, which corresponds to sections I and II of the divided wall column, was set equal to 70, and that for the second column was set equal to 122 (sections III to VI). The conditional and permanent trays of each column section, as well as the feed trays and the remaining parameters of the superstructure, can be deduced from Figure 12. All the required specifications of the divided wall column are listed in Table 1. It should be noted that, although the process simulator treats the divided wall column as two separate columns, all the column sections are in fact located inside a single shell. Thus, in the mathematical model of the superstructure, we force the number of active trays in sections I and IV and II and V to be equal, respectively. Therefore, between the active trays of these

Figure 11. (a) Fully thermally coupled distillation system or Petlyuk configuration (1965). (b) Divided wall column system.

the four heat exchangers of the direct/indirect sequence. Thus, it is possible to obtain savings in the investment costs. It is even possible to go one step further and integrate the two columns of the Petlyuk configuration into a single shell, divided by an internal wall. This is known as the dividing wall column (DWC) configuration, as illustrated in Figure 11b. The configurations in Figure 11a and b are thermodynamically equivalent if there is no heat transfer across the partition wall. The simulation of a divided wall column using a process simulator is far from straightforward: the liquid−vapor side streams connecting the two columns produce a flow sheet with a large number of recycle streams. Therefore, at each major iteration, both columns must be converged with two

Figure 12. Divided Wall Column Superstructure (Acyclic System for simulation). K



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Table 6. Results of 20 Consecutive Executions of the PSO AlgorithmExample 3a Execution

Total Trays

Feed Tray

ATI

ATII

ATIII

ATIV

ATV

ATVI

TAC (k$/yr)

CPU time(s)

Stopping Criterion

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

86 86 85 85 85 84 85 85 86 85 85 85 86 85 86 86 85 85 85 85

32 33 28 29 31 29 29 31 32 29 29 28 34 31 32 32 29 29 32 31

10 7 11 11 9 11 11 9 10 10 8 11 8 10 7 10 11 11 10 8

61 54 55 65 55 64 65 55 61 63 51 55 57 66 63 61 65 65 63 51

16 18 19 18 18 18 18 18 16 19 19 19 16 17 19 16 18 18 19 19

45 35 39 48 44 45 48 44 45 41 36 39 35 38 37 45 48 48 41 36

91 88 91 81 91 71 81 91 91 69 89 91 89 76 89 91 81 81 69 89

115 117 118 117 115 117 117 115 115 119 117 118 115 114 117 115 117 117 119 117

2131,1 2130,1 2126,8 2126,6 2127,6 2128,3 2126,6 2127,6 2131,1 2126,9 2127,4 2126,8 2131,0 2127,9 2132,9 2131,1 2126,6 2126,6 2126,9 2127,4

613 687 632 632 623 610 668 626 617 661 648 630 625 635 658 880 660 637 626 665


a Stopping Criterion 1: maximum number of iterations has been reached (150 major iterations). Stopping Criterion 2: below specified tolerance between best and worst particle (1 × 10−5).


sections, ATj, where j is the index for the set of column sections, 1 ≤ j ≤ 6 (which are obtained by means of the integer variables assigned to the top ends of each column section), we choose the greatest of each pair of sections, max{ATI,ATIV} and max{ATII,ATv}, in order to ensure that the required separation is achieved on both sides of the wall. Actually this is not strictly necessary because it is possible to use a different number of trays of different tray spacing, as well as various types of trays or packing, on each side of the wall. However, as the total height of the divided wall column is determined by the most difficult separations, we can assert that the proposed superstructure represents a good approach. In addition, because the number of

active trays in sections I and IV and II and V are, respectively, the same, the mechanic design of the column becomes easier and it also becomes possible to use conventional double-pass sieve trays, adapted for locating the internal wall. In any case, using the same number of trays on both sides of the wall will make the installation of the fixing rings easier. Of course, one of the sides of the wall will then be executing an overseparation relative to the initial specificationsbecause of the presence of extra traysbut this always enhances the separation. As in the previous examples, we summarize the main results of the 20 consecutive executions of the algorithm in a table (Table 6 below). L



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of trays. Three numerical examples were solved to illustrate how to build different superstructures using the process simulator and to assess the robustness and performance of the implemented method. In addition, this approach can be extended to other separation processes. To that end, our future work will be extended to sharp and nonsharp distillation sequences, as well as to thermally coupled distillation sequences.

An interesting feature of thermally coupled systems in general, or a Divided Wall Column (DWC) in particular, is the existence of a relatively large number of alternative configurations (similar number of total trays but with different arrangements in sections) of very similar total costs. This is true of the 20 consecutive executions, where different solutions are obtained but all of them very close to each other in terms of total cost and structure. Curiously, this fact gives the designer an extra degree of freedom to consider other aspects such as the controllability, hydrodynamics, etc. when selecting the most adequate tray distribution. Notice that there are two main types of configurations, with 85 and 86 trays, and that the mean values of the objective functions for these configurations are 2127 and 2130 k$/yr, respectively. The configuration with 85 trays has the lowest TAC value and was obtained 65% of the time. All the substructures can be considered as good solutions, but solutions 4, 7, 17, and 18 are the best among the column configurations with 85 trays. The computational and design results for these solutions are summarized in Figure 13 and Table 7.

■

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors would like to acknowledge financial support from the Spanish “Ministerio de Ciencia e Innovación” (CTQ200914420-C02-02 and CTQ2012-37039-C02-02).

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Table 7. Computational Results for Best Solution Example 3

TAC (k$/yr) Capital Cost (k$)̀ Operating Cost (k$/yr)

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PSO Description Number of Particles Major Iterations Function Evaluations Discrete Variables Stopping Criterion CPU Time (s)

AUTHOR INFORMATION

20 150 3020 6 max. number of iterations has been reached 668 Optimal Solution 2126.6 1754.4 1540.0

4. CONCLUSIONS In this work we have provided a general review of the area of optimal design and synthesis of distillation columns. We have shown that the rigorous optimization of complex distillation processes represents a challenging problem due to the large impact on the investment and operating costs involved. In the literature there are wide array of different approaches, most of them based on the mathematical programming embodied in MI(N)LP or GDP representations. However, these models suffer from important difficulties because of the high degree of nonlinearity and nonconvexity of the equations describing the separations units. This restricts the initial guess to one that has to be very close to a realistic simulation result, and strongly affects the quality of the solution. As a consequence, the resulting optimization models are far from being straightforward, and so, only those skilled in the art are able to utilize and adapt them to their own needs. In order to overcome the main difficulties that arise in the MI(N)LP and GDP approaches, we have proposed a systematic method that takes advantage of the process simulators and the free derivative optimization algorithm PSO (Particle Swarm Optimization algorithm) for solving global optimization problems: the method simultaneously optimizes the operating parameters of the distillation columns (reflux and reboiler rations, recoveries, ...), as well as the discrete design decisions of the feed and product location, and obtains the optimal number M



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Rigorous Design of Complex Distillation Columns Using Process

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