Overview

Overview

PART I
Formulating the problem is perhaps the most crucial step in optimization. Problem formulation requires identifying the essential elements of a conceptual or verbal statement of a given application and organizing them into a prescribed mathematical form, namely,

1. The objective function (economic criterion)
2. The process model (constraints)

The objective function represents such factors as profit, cost, energy, and yield in terms of the key variables of the process being analyzed. The process model and constraints describe the interrelationships of the key variables. It is important to learn a systematic approach for assembling the physical and empirical relations and data involved in an optimization problem, and Chapters 1, 2, and 3 cover the recommended procedures. Chapter 1 presents six steps for optimization that can serve as a general guide for problem solving in design and operations analysis. Numerous examples of problem formulation in chemical engineering are presented to illustrate the steps.

Chapter 2 summarizes the characteristics of process models and explains how to build one. Special attention is focused on developing mathematical models, particularly empirical ones, by fitting empirical data using least squares, which itself is an optimization procedure.

Chapter 3 treats the most common type of objective function, the cost or revenue function. Historically, the majority of optimization applications have involved trade-offs between capital costs and operating costs. The nature of the tradeoff depends on a number of assumptions such as the desired rate of return on investment, service life, depreciation method, and so on. While an objective function based on net present value is preferred for the purposes of optimization, discounted cash flow based on spreadsheet analysis can be employed as well.

It is important to recognize that many possible mathematical problem formulations can result from an engineering analysis, depending on the assumptions made and the desired accuracy of the model. To solve an optimization problem, the mathematical formulation of the model must mesh satisfactorily with the computational algorithm to be used. A certain amount of artistry, judgment, and experience is therefore required during the problem formulation phase of optimization.

PART II describes modern techniques of optimization and translates these concepts into computational methods and algorithms. Because the literature on optimization techniques is vast, we focus on methods that have proved effective for a wide range of problems. Optimization methods have matured sufficiently during the past 20 years so that fast and reliable methods are available to solve each important class of problem.

Seven chapters make up Part II of this book, covering the following areas:

Mathematical concepts (Chapter 4)

One-dimensional search (Chapter 5)

Unconstrained multivariable optimization (Chapter 6)

Linear programming (Chapter 7)

Nonlinear programming (Chapter 8)

Optimization involving discrete variables (Chapter 9)

Global optimization (Chapter 10)

The topics are grouped so that unconstrained methods are presented first, followed by constrained methods. The last two chapters in Part II deal with discontinuous (integer) variables, a common category of problem in chemical engineering, but one quite difficult to solve without great effort.

As optimization methods as well as computer hardware and software have improved over the past two decades, the degree of difficulty of the problems that can be solved has expanded significantly. Continued improvements in optimization algorithms and computer technology should enable optimization of large-scale nonlinear problems involving thousands of variables, both continuous and integer, some of which may be stochastic in nature.

PART III
THIS SECTION OF the book is devoted to representative applications of the optimization techniques presented in Chapters 4 through 10. Chapters 11 through 16 include the following major application areas:

Heat transfer and energy conservation (Chapter 11)

Separations (Chapter 12)

Fluid flow (Chapter 13)

Reactors (Chapter 14)

Large-scale plant design and operations (Chapter 15)

Integrated planning, scheduling, and control (Chapter 16)

Each chapter presents several detailed studies illustrating the application of various optimization techniques. The following matrix shows the classification of the examples with respect to specific techniques. Truly optimal design of process plants cannot be performed by considering each unit operation separately. Hence, in Chapter 15 we discuss the optimization of large-scale plants, including those represented by flowsheet simulators.

We have not included any homework problems in Chapters 11 through 16. As a general suggestion for classroom use, parameters or assumptions in each example can be changed to develop a modified problem. By changing the numerical method employed or the computer code one can achieve a variety of problems.

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