Alternative representations and formulations for the economic optimization of multicomponent distillation columns

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Abstract

This paper examines alternative models for the economic optimization of multicomponent distillation columns. Different column representations are modeled involving rigorous Mixed Integer Nonlinear Programming (MINLP) and General Disjunctive Programming (GDP) formulations. The different representations involve various ways of representing the choices for the number of trays and feed tray location. Also, alternatives are considered for modeling the heat exchange when the number of trays of the column must be determinated. A preprocessing procedure developed in a previous paper [Ind. Eng. Chem. Res. (2002a)] is extended in this work to provide good initial values and bounds for the variables involved in the economic models. This initialization scheme increases the robustness and usefulness of the optimization models. Numerical results are reported on problems involving the separation of zeotropic and azeotropic mixtures. Trends about the behavior of the different proposed alternative models are discussed.

Introduction

The economic optimization of a distillation column involves the selection of the configuration and the operating conditions to minimize the total investment and operation cost. Discrete decisions are related to the calculation of the number of trays and feed and products locations and continuous decisions are related to the operation conditions and energy use involved in the separation.

There are two major formulations for the mathematical representation of problems involving discrete and continuous variables: Mixed-Integer Nonlinear Programming (MINLP) and General Disjunctive Programming (GDP) where the logic is represented through disjunctions and propositions (Grossmann, 2001). Both approaches have been employed in the literature to model distillation columns. The MINLP formulation has been used with economic objective functions (Viswanathan & Grossmann, 1990, Viswanathan & Grossmann, 1993, Bauer & Stilchmair, 1998, Aguirre, Corsano & Barttfeld, 2001, Dunnebier & Pantelides, 1999). Two different representations arise from this formulation according to the way the discrete decisions related to the tray optimization are modeled. In one, a binary variable with a value of ‘1’ is assigned to each tray of the column denoting its existence, and with a value of ‘0’ its absence (Viswanathan & Grossmann, 1990). In the other representations, binary variables are used for the discrete decisions related to the location of the reflux, reboil or both (Viswanathan & Grossmann, 1993, Bauer & Stilchmair, 1998; Aguirre et al., 2001). MINLP problems can be solved for instance with the computer code DICOPT (Viswanathan & Grossmann, 1990), which is an implementation of the Outer Approximation/Equality Relaxation (OA/ER) algorithm (Kocis & Grossmann, 1989). The computational expense in solving these models depends largely on the problem structure. There is also the computational difficulty that each constraint must be solved even if the stage ‘disappears’ from the column. It would be desirable to eliminate these constraints, not only to reduce the size of the NLP subproblems, but also to avoid singularities that are due to the linearization at zero flows.

Motivated by the potential of using logic to improve the modeling and solution of network systems, a logic-based MINLP algorithm has been developed by Turkay and Grossmann (1996). This algorithm has been successfully applied for solving GDP models of conventional distillation columns (Yeomans & Grossmann, 2000a, Yeomans & Grossmann, 2000b), as well as reactive distillation columns (Jackson & Grossmann, 2001). Different approaches can be used with this formulation depending on which trays are defined as permanent in the configuration. It is this issue that we will be analyzing in depth in this work.

Another major difficulty that arises in the MINLP and GDP approaches is dealing with the nonlinearities that are involved in distillation models, which complicates the convergence of solvers and often leads to infeasible solutions. Therefore, developing methods for the initialization and bounding of the variables involved in the problem is an essential part for the successful application of optimization formulations and algorithms for distillation columns.

Fletcher and Morton (2000) examined the infinite reflux case for generating good initial values for the NLP solution of general distillation columns. Bruggemann and Marquardt (2001) have proposed a short cut method based on the Rectification Body Method (RBM) that provides qualitative insights for rigorous simulations. The method gives information on the minimum energy demand involved in a separation by a trial and error procedure. Given the products and feed compositions as well as the operating pressure, an estimate of the energy demand is determinated to calculate the pinch points to construct the rectification bodies related to both column sections. The energy involved in the separation under minimum reflux is achieved when the bodies intersect in exactly one point. An automatic initialization scheme based on the successive solution of NLP and MINLP optimization problems was presented in a previous paper (Barttfeld & Aguirre, 2002a). These authors developed rigorous and robust optimization models that approach reversible conditions in order to initialize and bound zeotropic distillation models. No external parameters have to be tuned in the model to achieve convergence.

The main objective of this paper is to study the different representations and models that can be used for the optimization of a single distillation column. General models comprising different column configurations will be presented for the MINLP and GDP formulations involving the separation of zeotropic and azeotropic mixtures. In order to increase the robustness of convergence of the proposed models, a general preprocessing phase is adapted and extended to GDP formulations. In this preliminary phase, thermodynamics is combined with mathematical programming. In a second step, a particular initialization scheme is derived for each mathematical formulation.

This paper is organized as follows. We first examine in Section 3 the different representations to model the economic optimization problem for a given separation task. In Section 4, the general models for both the MINLP and GDP formulations comprising all the alternative column representations are presented. In Section 5, the solution procedures for the proposed formulations are presented. Preprocessing techniques are included to reduce difficulties related to the economic optimization task. In this section, reversible distillation theory is combined with mathematical programming tools. In Section 6, several case studies are analyzed. Finally, several extensions and conclusions of this work are discussed.

Section snippets

Problem definition

The problem addressed in this paper can be stated as follows. Given is a multicomponent feed with known flow and composition, and given are the desired products specifications. The problem then consists in selecting the number of trays, feed location, condenser and reboiler duties and areas of a distillation column so as to minimize the total annualized investment and operating cost. In order to tackle this problem we examine different representations for the column and their formulations in

Representation and formulation alternatives

In this section the different distillation column representations are presented for the MINLP and GDP models.

Single columns models

In this section, the general MINLP and GDP models are presented for all the representations discussed in Section 3.

A distillation column model can be formulated in two different ways:

  • 1

    Employing total flows, e.g. Ln, and mole fractions related to each flow, e.g. xn,i.

  • 2

    Employing individual flows, e.g. LIQn,i defined as LIQn,i=Lnxn,i.

The first alternative has the advantage of providing a convenient framework for the evaluation of the thermodynamic properties and bounds can be expressed in a more

Solution of models

In this section, the algorithms and procedures employed for solving the models presented in the previous section are described.

As was previously mentioned, independently of the formulation or the column representation employed, the initialization procedure of distillation models is a relevant point that must be considered. For that reason, in this work we extend the preprocessing phase as a preliminary solution phase to the economic optimization. The general preprocessing procedure that is

Numerical examples

The MINLP and GDP models derived in Section 5 are tested with three ternary mixtures. The low boiler-rich distillate separation is specified for cases involving zeotropic mixtures, while the reversible product compositions are specified for azeotropic mixtures. A constant pressure of 1.01 bar is considered. In all cases, a feed flow of 10 mol/s and saturated liquid products are considered. The minimum number of trays nmin is 5. The thermodynamic properties are taken form Reid, Prausnitz and

Conclusions

This paper has presented different alternatives for representing and formulating the economic optimization problem of a single multicomponent distillation column. The different alternatives involve different ways of representing the choices for the number of trays in the column and the energy demand. Rigorous MINLP and GDP formulations were developed in each of the cases. In order to increase the robustness in the solution of these formulations, a general automatic preprocessing phase was

Acknowledgements

The authors want to thank for the financial support from Consejo Nacional de Investigaciones Cientı́ficas y Técnicas (CONICET), Agencia Nacional de Promoción Cientı́fica y Técnica (ANPCYT), Universidad Nacional del Litoral (UNL) from Argentina and the Center for Advanced Process Decision-making at Carnegie Mellon.

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