Global analysis of combined reaction distillation processes

doi:10.1016/j.compchemeng.2007.04.015

Computers & Chemical Engineering

Volume 32, Issues 1–2, January 2008, Pages 343-355

https://doi.org/10.1016/j.compchemeng.2007.04.015 Get rights and content

Abstract

Polyhedral relaxation is a powerful tool for determining global bounds on optimal solutions in chemical process synthesis. Combined reaction distillation processes are considered as a challenging application example. To reduce complexity of the resulting mixed integer linear optimization problems, model reduction by means of wave functions is proposed, and polyhedral relaxations of sigmoidal wave functions in two variables are derived. It is shown that these relaxations provide better approximation quality than approximating the composed functions individually. Further, the concave envelope of such functions is characterized and (nonlinear) convex underestimators are derived.

The approximation results are employed in the computation of lower bounds on the vapor flow of a combined reaction distillation process with a metathesis reaction $2 B ⇋ A + C$ . We restrict computations to a domain around a known local optimum, trading computation time for some of the globality. This still proves a bound on the vapor flow, but for restricted operating conditions of the column.

Introduction

The optimal design of chemical processes using mathematical optimization often leads to mixed-integer nonlinear programs (MINLP). Due to nonconvexity MINLP problems are usually difficult to solve. Typically either gradient based local optimization methods are used for this purpose or stochastic optimization methods like simulated annealing or genetic algorithms (see, e.g., Marriott & Sørensen, 2003). However, in both cases no guarantee can be given that the solution found by the algorithm is the global optimum.

To overcome this problem, a new global approach was proposed in Gangadwala, Kienle, Haus, Michaels, and Weismantel (2006). It is based on techniques to derive polyhedral approximations of the underlying nonlinear equations in such a way that a mixed-integer linear relaxation of the original problem is obtained, which can be solved rigorously. Since the feasible set of the original nonlinear problem lies within the feasible set of the relaxed problem, the latter provides global lower bounds for the optimal solution of the original problem. The lower bound approaches the true global optimum as the number of grid points of the relaxation is increased. Application was demonstrated for the synthesis of combined reaction distillation processes for producing 2,3-dimethylbutene-1 by isomerization from 2,3-dimethylbutene-2. With the new global approach we were able to prove which process configuration is most economical under the given constraints.

However, the complexity of the resulting mixed-integer linear relaxations in this approach is very challenging. Therefore, application is currently limited to problems of moderate complexity like the isomerization example discussed in Gangadwala et al. (2006). In the present paper, an extension to more complicated processes is proposed. Application is demonstrated for a ternary system with a metathesis reaction.

The outline of the paper is as follows: First, the global approach is briefly reviewed. Afterwards, an alternative modeling approach for combined reaction separation processes with side reactors is introduced. This is a modular approach, which is based on a description of the concentration profiles in the non reactive column sections by means of so-called wave functions introduced in Marquardt (1990) and Kienle (2000), for example. Afterwards, rules for the polyhedral relaxation of these wave functions are derived. Further, their concave envelope is characterized and (nonlinear) convex underestimators are given. The approximation results are employed in the computation of lower bounds on the vapor flow of a combined reaction distillation process with a metathesis reaction. We restrict computations to a domain around a known local optimum, trading computation time for some of the globality. This still proves a bound on the vapor flow, but for restricted operating conditions of the column.

Section snippets

Global bounds for mixed-integer nonlinear problems

For more than 40 years mathematicians and engineers have tried to attack synthesis problems in chemical process design. The challenge of synthesis problems is that all the corresponding models involve nonlinearities in both continuous and discrete variables. Determining an optimal design of a chemical engineering process results in a generic nonlinear mixed-integer optimization problem of the following form: $\begin{matrix} \min & f (x, y) \\ s.t. & h (x, y) = 0, \\ g (x, y) \leq 0, \\ x \in X \subseteq R^{n}, \\ y \in Y \subseteq Z^{d} . \end{matrix}$ where $f : R^{n} \times Z^{d} \to R$ is a real function, $h : R^{n} \times Z^{d}$

Wave models for combined reaction distillation processes

Application of the above approach was demonstrated for the synthesis of combined reaction distillation processes for producing 2,3-dimethylbutene-1 by isomerization from 2,3-dimethylbutene-2 (Gangadwala et al., 2006). In particular, we were able to prove which process configuration is most economical under the given constraints. The process models were based on rigorous tray-by-tray material balances for the distillation columns included in the different process configurations. Hence, the

Polyhedral approximations of wave functions

The low-order models presented in Sections 3.1 Application to DMB-1 synthesis, 3.2 Application to metathesis of 2-pentene have in common that the reduction of the number of component material balances comes at the expense of new nonlinear terms of the form: $w : R^{2} \to R, (x, y) \mapsto \frac{1}{1 + e^{x y}} .$

In order to compute global bounds on these models using the techniques described in Section 2, we need to discuss analytical properties and derive rules for domain decompositions that are suitable for good polyhedral

Computational test—trading computational effort for globality

In Gangadwala et al. (2006) and Haus, Michaels, Seidel-Morgenstern, and Weismantel (in press) we demonstrated that the approach of constructing a hierarchy of mixed-integer linear relaxations for the nonlinear model (MINLP) can successfully be used to derive global bounds for realistic applications, albeit at high computational cost. To fill the gap between such a computationally expensive global bound and purely local optima derived using solvers such as GAMS with the SBB and CONOPT solvers,

Conclusions

In the present paper an extended approach for deriving global bounds on optimal solutions for the design of combined reaction distillation processes was presented. The approach is based on a hierarchy of polyhedral relaxations. In comparison to other existing relaxation techniques for MINLPs as described in Section 2 this opens the possibility to reduce the search space using combinatorial substructures derived from linear and nonlinear relations of the underlying model (see, e.g., Haus et al.,

Acknowledgment

The financial support of the German Science Foundation (DFG) under grants FOR 468 and KI 417/2 is greatly acknowledged.

References (27)

C.S. Adjiman et al.
A global optimization method, $α$ BB, for general twice-differentiable constrained NLPs. I. Theoretical advanced
Computers & Chemical Engineering
(1998)
F. Chen et al.
Simulation of kinetic effects in reactive distillation
Computers & Chemical Engineering
(2000)
U.-U. Haus et al.
A method to evaluate the feasibility of TMB chromatography for reduced efficiencies and purity requirements based on discrete optimization
Computers & Chemical Engineering
(2007)
A. Kienle
Low-order dynamic models for ideal multicomponent distillation processes using nonlinear wave propagation theory
Chemical Engineering Science
(2000)
H. Marchand et al.
Cutting planes in integer and mixed integer programming
Discrete Applied Mathematics
(2002)
J. Marriott et al.
The optimal design of membrane systems
Chemical Engineering Science
(2003)
J. Viswanathan et al.
A combined penalty function and outer-approximation method for MINLP optimisation
Computers & Chemical Engineering
(1990)
F.A. Al-Khayyal et al.
Jointly constrained biconvex programming
Mathematics of Operations Research
(1983)
Ben-Tal, A., & Nemirovski, A. (2001) Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering...
M.A. Duran et al.
A mixed-integer nonlinear programming algorithm for process system synthesis
AIChE Journal
(1986)

M.A. Duran et al.

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

Mathematical Programming

(1986)

R. Fletcher et al.

Solving mixed integer nonlinear programs by outer approximation

Mathematical Programming

(1994)

J. Gangadwala et al.

Global Bounds on Optimal Solutions for the Production of 2,3-Dimethylbutene-1

Industrial & Engineering Chemistry Research

(2006)

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Global analysis of combined reaction distillation processes

Abstract

Introduction

Section snippets

Global bounds for mixed-integer nonlinear problems

Wave models for combined reaction distillation processes

Polyhedral approximations of wave functions

Computational test—trading computational effort for globality

Conclusions

Acknowledgment

Computers & Chemical Engineering

Computers & Chemical Engineering

Computers & Chemical Engineering

Chemical Engineering Science

Discrete Applied Mathematics

Chemical Engineering Science

Computers & Chemical Engineering

Jointly constrained biconvex programming

Mathematics of Operations Research

A mixed-integer nonlinear programming algorithm for process system synthesis

AIChE Journal

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

Mathematical Programming

Solving mixed integer nonlinear programs by outer approximation

Mathematical Programming

Global Bounds on Optimal Solutions for the Production of 2,3-Dimethylbutene-1

Industrial & Engineering Chemistry Research