Distillation-based Separation Systems Synthesis

This chapter presents two applications of MINLP methods in the area of separations. Section 9.1 provides an overall introduction to the synthesis of separation systems. Section 9.2 focuses on sharp heat-integrated distillation sequencing. Section 9.3 presents an application of nonsharp separation synthesis. The synthesis of distillation-based separation sequences has been one of the most important subjects of investigation in the area of process synthesis. This is attributed to the significant contribution of separation processes to the total capital investment and operating expenses of a chemical plant. As a result, a lot of interest has been generated in the development of systematic approaches that select optimal sequences of distillation columns. Westerberg (1985) provided a comprehensive review of the distillation-based separation synthesis approaches, as well as presented a classification of different types of separation problems along with their associated challenges. Nishida et al. (1981) and Smith and Linnhoff (1988) reviewed the general separation synthesis problem (i.e., not only distillation-based) and presented the progress made. To illustrate the nature of the distillation-based separation system synthesis problem, let us consider its generic definition shown in Figure 9.1, which is as follows: . . . Given a number of input multicomponent streams which have specified amounts for each component, create a cost-optimal configuration of distillation columns, mixers, and splitters that produces a number of multicomponent products with specified composition of their components. . . The products feature components that exist in the input streams and can be obtained by redistributing the components existing in the input streams, while the cost-optimal configuration corresponds to the least total annual cost one. Most of distillation columns or sequences can be classified as (i) Sharp, (ii) Nonsharp, (iii) Simple, (iv) Complex, (v) Heat-integrated, and (vi) Thermally coupled. In (i), a column separates its feed into products without overlap in the components. An example is the separation of a stream consisting of four components A, B, (C, and D via a distillation column, into one product consisting of only A and another product featuring B, C, and D. If all columns are sharp, then the separation sequence is termed as sharp sequence.

Heat Exchanger Network Synthesis

This chapter focuses on heat exchanger network synthesis approaches based on optimization methods. Sections 8.1 and 8.2 provide the motivation and problem definition of the HEN synthesis problem. Section 8.3 discusses the targets of minimum utility cost and minimum number of matches. Section 8.4 presents synthesis approaches based on decomposition, while section 8.5 discusses simultaneous approaches. Heat exchanger network HEN synthesis is one of the most studied synthesis/design problems in chemical engineering. This is attributed to the importance of determining energy costs and improving the energy recovery in chemical processes. The comprehensive review of Gundersen and Naess (1988) cited over 200 publications while a substantial annual volume of studies has been performed in the last few years. The HEN synthesis problem, in addition to its great economic importance features a number of key difficulties that are associated with handling: (i) The potentially explosive combinatorial problem for identifying the best pairs of hot and cold streams (i.e., matches) so as to enhance energy recovery; (ii) Forbidden, required, and restricted matches; (iii) The optimal selection of the HEN structure; (iv) Fixed and variable target temperatures; (v) Temperature dependent physical and transport properties; (vi) Different types of streams (e.g., liquid, vapor, and liquid-vapor); and (vii) Different types of heat exchangers (e.g., counter-current, noncounter-current, multistream), mixed materials of construction, and different pressure ratings. It is interesting to note that the extensive research efforts during the last three decades toward addressing these aforementioned difficulties/issues exhibit variations in their objectives and types of approaches which are apparently cyclical. The first approaches during the 1960s and early 1970s treated the HEN synthesis problem as a single task (i.e., no decomposition into sub-tasks). The work of Hwa (1965) who proposed a simplified superstructure which he denoted as composite configuration that was subsequently optimized via separable programming was a key contribution in the early studies, as well as the tree searching algorithms of Pho and Lapidus (1973). Limitations on the theoretical and algorithmic aspects of optimization techniques were, however, the bottleneck in expanding the applicability of the mathematical approaches at that time.

Fundamentals of Nonlinear Optimization

This chapter discusses the fundamentals of nonlinear optimization. Section 3.1 focuses on optimality conditions for unconstrained nonlinear optimization. Section 3.2 presents the first-order and second-order optimality conditions for constrained nonlinear optimization problems. This section presents the formulation and basic definitions of unconstrained nonlinear optimization along with the necessary, sufficient, and necessary and sufficient optimality conditions. An unconstrained nonlinear optimization problem deals with the search for a minimum of a nonlinear function f(x) of n real variables x = (x1, x2 , . . . , xn and is denoted as Each of the n nonlinear variables x1, x2 , . . . , xn are allowed to take any value from - ∞ to + ∞. Unconstrained nonlinear optimization problems arise in several science and engineering applications ranging from simultaneous solution of nonlinear equations (e.g., chemical phase equilibrium) to parameter estimation and identification problems (e.g., nonlinear least squares).

Introduction

This chapter introduces the reader to elementary concepts of modeling, generic formulations for nonlinear and mixed integer optimization models, and provides some illustrative applications. Section 1.1 presents the definition and key elements of mathematical models and discusses the characteristics of optimization models. Section 1.2 outlines the mathematical structure of nonlinear and mixed integer optimization problems which represent the primary focus in this book. Section 1.3 illustrates applications of nonlinear and mixed integer optimization that arise in chemical process design of separation systems, batch process operations, and facility location/allocation problems of operations research. Finally, section 1.4 provides an outline of the three main parts of this book. A plethora of applications in all areas of science and engineering employ mathematical models. A mathematical model of a system is a set of mathematical relationships (e.g., equalities, inequalities, logical conditions) which represent an abstraction of the real world system under consideration. Mathematical models can be developed using (i) fundamental approaches, (ii) empirical methods, and (iii) methods based on analogy. In (i), accepted theories of sciences are used to derive the equations (e.g., Newton’s Law). In (ii), input-output data are employed in tandem with statistical analysis principles so as to generate empirical or “black box” models. In (iii), analogy is employed in determining the essential features of the system of interest by studying a similar, well understood system. The variables can take different values and their specifications define different states of the system. They can be continuous, integer, or a mixed set of continuous and integer. The parameters are fixed to one or multiple specific values, and each fixation defines a different model. The constants are fixed quantities by the model statement. The mathematical model relations can be classified as equalities, inequalities, and logical conditions. The model equalities are usually composed of mass balances, energy balances, equilibrium relations, physical property calculations, and engineering design relations which describe the physical phenomena of the system. The model inequalities often consist of allowable operating regimes, specifications on qualities, feasibility of heat and mass transfer, performance requirements, and bounds on availabilities and demands. The logical conditions provide the connection between the continuous and integer variables.

Process Synthesis

This chapter provides an introduction to Process Synthesis. Sections 7.1 and 7.2 discuss the components of a chemical process system and define the process synthesis problem. Section 7.3 presents the different approaches in the area of process synthesis. Section 7.4 focuses on the optimization approach and discusses modeling issues. Finally, Section 7.5 outlines application areas which are the subject of discussion in chapters 8, 9 and 10. Process Synthesis, an important research area within chemical process design, has triggered during the last three decades a significant amount of academic research work and industrial interest. Extensive reviews exist for the process synthesis area as well as for special classes of problems (e.g., separation systems, heat recovery systems) and for particular approaches (e.g., insights-based approach, optimization approach) applied to process synthesis problems. These are summarized in the following: Overall Process Synthesis: Hendry et al. (1973) Hlavacek(1978) Westerberg (1980) Stephanopoulos (1981) Nishida et al. (1981) Westerberg (1989) Gundersen(1991) Heat Exchanger Network Synthesis: Gundersen and Naess (1988) Separation System Synthesis: Westerberg (1985) Smith and Linnhoff( 1988) Optimization in Process Synthesis: Grossmann (1985), (1989), (1990) Floquet et al. (1988) Grossmann et al. (1987) Floudas and Grossmann (1994) Prior to providing the definition of the process synthesis problem we will describe first the overall process system and its important subsystems. This description can be addressed to the overall process system or individual subsystems and will be discussed in the subsequent sections. An overall process system can be represented as an integrated system that consists of three main interactive components : (i) Chemical plant, (ii) Heat recovery system, (iii) Utility system. In the chemical plant, the transformation of the feed streams (e.g., raw materials) into desired products and possible by-products takes place. In the heat recovery system, the hot and cold process streams of the chemical plant exchange heat so as to reduce the hot and cold utility requirements. In the utility plant, the required utilities (e.g., electricity and power to drive process units) are provided to the chemical plant while hot utilities (e.g., steam at different pressure levels) are provided to the heat recovery system.

Duality Theory

Nonlinear optimization problems have two different representations, the primal problem and the dual problem. The relation between the primal and the dual problem is provided by an elegant duality theory. This chapter presents the basics of duality theory. Section 4.1 discusses the primal problem and the perturbation function. Section 4.2 presents the dual problem. Section 4.3 discusses the weak and strong duality theorems, while section 4.4 discusses the duality gap. This section presents the formulation of the primal problem, the definition and properties of the perturbation function, the definition of stable primal problem, and the existence conditions of optimal multiplier vectors.

Convex Analysis

This chapter discusses the elements of convex analysis which are very important in the study of optimization problems. In section 2.1 the fundamentals of convex sets are discussed. In section 2.2 the subject of convex and concave functions is presented, while in section 2.3 generalizations of convex and concave functions are outlined. This section introduces the fundamental concept of convex sets, describes their basic properties, and presents theoretical results on the separation and support of convex sets.

Mixed-Integer Nonlinear Optimization

This chapter presents the fundamentals and algorithms for mixed-integer nonlinear optimization problems. Sections 6.1 and 6.2 outline the motivation, formulation, and algorithmic approaches. Section 6.3 discusses the Generalized Benders Decomposition and its variants. Sections 6.4, 6.5 and 6.6 presents the Outer Approximation and its variants with Equality Relaxation and Augmented Penalty. Section 6.7 discusses the Generalized Outer Approximation while section 6.8 compares the Generalized Benders Decomposition with the Outer Approximation. Finally, section 6.9 discusses the Generalized Cross Decomposition. A wide range of nonlinear optimization problems involve integer or discrete variables in addition to the continuous variables. These classes of optimization problems arise from a variety of applications and are denoted as Mixed-Integer Nonlinear Programming MINLP problems. The integer variables can be used to model, for instance, sequences of events, alternative candidates, existence or nonexistence of units (in their zero-one representation), while discrete variables can model, for instance, different equipment sizes. The continuous variables are used to model the input-output and interaction relationships among individual units/operations and different interconnected systems. The nonlinear nature of these mixed-integer optimization problems may arise from (i) nonlinear relations in the integer domain exclusively (e.g., products of binary variables in the quadratic assignment model), (ii) nonlinear relations in the continuous domain only (e.g., complex nonlinear input-output model in a distillation column or reactor unit), (iii) nonlinear relations in the joint integer-continuous domain (e.g., products of continuous and binary variables in the scheduling/ planning of batch processes, and retrofit of heat recovery systems). In this chapter, we will focus on nonlinearities due to relations (ii) and (iii). An excellent book that studies mixed-integer linear optimization, and nonlinear integer relationships in combinatorial optimization is the one by Nemhauser and Wolsey (1988). The coupling of the integer domain with the continuous domain along with their associated nonlinearities make the class of MINLP problems very challenging from the theoretical, algorithmic,and computational point of view. Apart from this challenge, however, there exists a broad spectrum of applications that can be modeled as mixed-integer nonlinear programming problems.

Synthesis of Reactor Networks and Reactor-Separator-Recycle Systems

This chapter discusses the application of MINLP methods in the synthesis of reactor networks with complex reactions and in the synthesis of reactor-separator-recycle systems. Despite the importance of reactor systems in chemical engineering processes, very few systematic procedures for the optimal synthesis of reactor networks have been proposed. The main reason for the scarcity of optimization strategies for reactor networks is the difficulty of the problem itself. The large number of alternatives along with the highly nonlinear equations that describe these systems have led to the development of a series of heuristic and intuitive rules that provide solutions only for simple cases of reaction mechanisms. Most of the studies considered single reactors with a specified mixing pattern and focused on investigating the effect of temperature distribution, residence time distribution, or catalyst dilution profile on its performance. In the sequel, we will briefly review the approaches developed based on their classification: (i) isothermal operation and (ii) nonisothermal operation. Trambouze and Piret (1959) proposed graphical and analytical criteria for selecting the type of reactor. Levenspiel (1962) reported heuristic rules for optimal yield and selectivity in stirred tank and tubular reactors. Aris (1964, 1969) applied dynamic programming to determine the optimal amounts of by-passes and cold streams in a multistage reaction system within a fixed structure. Gillespie and Carberry (1966) studied the Van der Vusse reaction with an intermediate level of mixing and demonstrated the potential advantages of recycle reactors for such a complex reaction. Horn and Tsai (1967) studied the effects of global and local mixing using the adjoint variables of optimization theory. Jackson (1968) proposed an algebraic structure for the reactor representation consisting of parallel ideal tubular reactors that were interconnected with side streams at various sink and source points. Different flow configurations and mixing patterns could be obtained by varying the number and the positions of the sink and source points, as well as the levels of the sidestreams. By deliberate manipulation of the flow configuration, potential improvements in the reactor performance coul be investigated. Ravimohan (1971) modified Jackson's model so as to handle cases of local mixing.

Mixed-Integer Linear Optimization

This chapter provides an introduction to the basic notions in Mixed-Integer Linear Optimization. Sections 5.1 and 5.2 present the motivation, formulation, and outline of methods. Section 5.3 discusses the key ideas in a branch and bound framework for mixed-integer linear programming problems. A large number of optimization models have continuous and integer variables which appear linearly, and hence separably, in the objective function and constraints. These mathematical models are denoted as Mixed-Integer Linear Programming MILP problems. In many applications of MILP models the integer variables are 0 — 1 variables (i.e., binary variables), and in this chapter we will focus on this sub-class of MILP problems. A wide range of applications can be modeled as mixed-integer linear programming MILP problems. These applications have attracted a lot of attention in the field of Operations Research and include facility location and allocation problems, scheduling problems, and fixed-charge network problems. The excellent books of Nemhauser and Wolsey (1988), and Parker and Rardin (1988) provide not only an exposition to such applications but also very thorough presentation of the theory of discrete optimization. Applications of MILP models in Chemical Engineering have also received significant attention particularly in the areas of Process Synthesis, Design, and Control. These applications include (i) the minimum number of matches in heat exchanger synthesis (Papoulias and Grossmann, 1983; see also chapter 8) (ii) heat integration of sharp distillation sequences (Andrecovich and Westerberg, 1985); (iii) multicomponent multiproduct distillation column synthesis (Floudas and Anastasiadis, 1988); (iv) multiperiod heat exchanger network, and distillation system synthesis (Floudas and Grossmann, 1986; Paules and Floudas, 1988); flexibility analysis of chemical processes (Grossmann and Floudas, 1987); (v) structural properties of control systems (Georgiou and Floudas, 1989, 1990); (vi) scheduling of batch processes (e.g., Rich and Prokapakis, 1986, 1986; Kondili et al., 1993; Shah et al, 1993; Voudouris and Grossmann, 1992, 1993); and (vii) planning and scheduling of batch processes (Shah and Pantelides, 1991, Sahinidis et al, 1989, Sahinidis and Grossmann, 1991). In addition to the above applications, MILP models are employed as subproblems in the mixed-integer nonlinear optimization approaches which we will discuss in the next chapter. In this section, we will present the formulation of Mixed-Integer Linear Programming MILP problems, discuss the complexity issues, and provide a brief overview of the solution methodologies proposed for MILP models.