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Energy Optimization in Process Systems
By Stanislaw Sieniutycz Jacek Jezowski
Elsevier
Copyright © 2009 Elsevier Ltd.
All right reserved. ISBN: 978-0-08-091442-8
Chapter One
Brief review of static optimization methods
1.1. INTRODUCTION: SIGNIFICANCE OF MATHEMATICAL MODELS
All rational human activity is characterized by continuous striving for progress and development. The tendency to search for the best solution under defined circumstances is called optimization—in the broad sense of the word. In this sense, optimization has always been a property of rational human activity. However, in recent decades, the need for methods which lead to an improvement of the quality of industrial and practical processes has grown stronger, leading to the rapid development of a group of optimum-seeking mathematical methods, which are now collectively called methods of optimization. Clearly, what brought about the rapid development of these methods was progress in computer science, which made numerical solutions of many practical problems possible. In mathematical terms, optimization is seeking the best solution within imposed constraints.
Process engineering is an important area for application of optimization methods. Most technological processes are characterized by flexibility in the choice of some parameters; by changing these parameters it is possible to correct process performance and development. In other words, decisions need to be made which make it possible to control a process actually running. There are also decisions that need to be made in designing a new process or new equipment. Thanks to these decisions (controls) some goals can be reached. For example, it may be possible to achieve a sufficiently high concentration of a valuable product at the end of a tubular reactor at minimum cost; or in another problem, to assure both a relatively low decrease of fuel value and a maximum amount of work delivered from an engine. How to accomplish a particular task is the problem of control in which some constraints are represented by transformations of the system's state and others by boundary conditions of the system. If this problem can be solved, then usually a number of solutions may be found to satisfy process constraints. Therefore, it is possible to go further and require that a defined objective function (process performance index) should be reached in the best way possible, for example, in the shortest time, with the least expenditure of valuable energy, minimum costs, and so on. An optimization problem emerges, related to the optimal choice of process decisions.
In testing a process it is necessary to quantify the related knowledge in mathematical terms; this leads to a mathematical model of optimization which formulates the problem in the language of functions, functionals, equations, inequalities, and so on. The mathematical model should be strongly connected with reality, because it emerges from and finds its application in it. However, the mathematical model often deals with very abstract ideas; thus, finding an optimal solution requires a knowledge of advanced methods. We shall present here only selected methods, suited to the content of this book.
In technology, practically every problem of design, control, and planning can be approached through an analysis leading to the determination of the least (minimum) or the greatest (maximum) value of some particular quantity – physical, technological or economic – which is called the optimization criterion, performance index, objective function or profit function. The choice of decisions (also of a physical, technological or economic nature), which can vary in a defined range, affects the optimization criterion, the criterion being a measure of the effectiveness of the decisions. The task of optimization is to find decisions to assure the minimum or maximum value of the optimization criterion.
The existence of decisions as quantities whose values are not prescribed, but rather chosen freely or within certain limits, makes optimization possible. Optimization – understood as an activity leading to the achievement of the best result under given conditions, always an inevitable part of human activity – only acquired a solid scientific basis when its meanings and methods were described mathematically. Thanks to recent computational techniques and the use of high-speed computing, optimization research has gained economic ground, and the range of problems solved has increased enormously. Apart from the use of digital computers, many optimization problems have been solved by using analog or hybrid computers.
In this book we assume that each optimization problem can be represented by a suitable mathematical model. Clearly, the mathematical model can simulate the behavior of a real system in a more or less exact way. Whenever good agreement is observed between the behavior of a real system and its model, optimization results can be used to improve the performance of the system. However, cases may exist where the process data are not reliable enough and over simplifications may occur in construction of the model; in these cases the results of optimization cannot be accepted without criticism. Clearly, where there is high data inaccuracy or model invalidity, optimization results will not be reliable. However, the models and the data which are now used for optimization are, in fact, the same as those used in design and process control. In many cases these models are well established, so the related optimization is desirable.
The technical implementation of a correct optimization solution may often prove to be difficult. In these cases optimization results can still be useful to expose extremal or limiting possibilities of the system from the viewpoint of an accepted optimization criterion. For example, an obtained limit can be represented by an upper (lower) bound on the amount of electrical or mechanical energy delivered from (supplied to) the system. Real system characteristics, which lie below (above) the limits predicted by the optimal solution, can sometimes be taken into account by considering suboptimal solutions. The latter may be easier to accomplish than the optimal solution.
The mathematical model of optimization is the system of all the equations and inequalities that characterize the process considered, including the optimization criterion. The model makes it possible to determine how the optimization criterion changes with variations in the decisions. In principle, mathematical models can be obtained in two ways. On the basis of physical laws so-called analytical models are formulated. After identification of the system experimental models are determined, often based on regression analysis. Sometimes they are represented by polynomial equations linking outputs and inputs of the system.
In design, analytical models are usually used because only they can make possible the wide extrapolation of data that is necessary when the process scale is changed. In analytical models the number of unknown coefficients to be determined is usually much lower than in empirical models. However, when controlling existing processes, empirical models are still, quite frequently, applied.
If an optimization is associated with planning and doing experiments and its partial purpose is finding the data which help to determine optimal decisions, we are dealing with experimental optimization. If a mathematical description is used, which takes into account the process, its environment and a control action, we are dealing with analytical optimization. This book deals with analytical optimization. The models we use in most of the chapters are deterministic ones. Yet, since some results in the field of energy limits may be linked with random processes, uncertainty and simulated annealing criteria (Nulton and Salamon, 1988; Harland and Salamon, 1988; Andresen and Gordon, 1994), the final part of this chapter discusses basic techniques of stochastic optimization.
In the working state of a technological process the problem of adaptation of the mathematical model plays an important role. Adaptation should always be made whenever variations are observed in uncontrolled variables of the system, which normally should remain constant. For slowly varying changes, adaptation of the continuous type is possible. Fast varying changes require periodic adaptation for the averaged values of changes which are regarded as noise. Optimization on line is carried out simultaneously with adaptation of the model; in this optimization a control action is accomplished directly by the computer. Yet, a computer-aided control involves optimization off line.
The optimization criterion (performance index) is an important quantity that appears in the mathematical model in the form of a function or a functional. Usually this is an explicit and analytical form. The choice of optimization criterion in an industrial process must be the subject of very accurate analysis, which often involves both technological and economic terms. This is because the definition of the criterion has an important effect on the problem's solution and expected improvements.
Along with the performance index there appear in the mathematical model some equality and inequality equations (algebraic, differential, integral, etc.) which characterize constraints imposed on the process. Both constraints and performance index can contain decision variables or controls—adjustable variables that an engineer or a researcher can influence.
Some other variables can also appear both in the constraining equations and in the performance index. These are called uncontrolled variables. These variables are determined by certain external factors independent of the observer (the composition of a raw material, for example). They cannot be controlled but they can often be measured. For optimization, controlled variables are the most important, as they characterize external action performed on the process and are of utmost importance to the optimization criterion. Important also is sensitivity analysis of the optimization criterion with respect to the decisions, because their number determines to a large extent the difficulty of the optimization problem. Leaving out decisions which affect the optimization criterion in an insignificant way should be treated as a natural procedure which contributes to increased transparency of the results and often facilitates the problem solving.
Imposing constraints on decision variables is typical for all practical problems of optimization. There are, for example, constraints on consumption of some resources, process output, product purity, concentrations of contaminants, and so on. Constraints can also be formulated for thermodynamic parameters of the process in order to specify allowable ranges of temperatures and pressures, intervals of catalyst activities, reaction selectivities, and so on. Constraints which assure reliability and safety of the equipment are important, for example, those imposed on reagent concentrations in combustible mixtures (preventing an explosion), on gas and fluid flows in absorbers (prevention of flood), on the superficial velocity of fluidization (prevention of material blowing out), and so on. Constraints may also be imposed on construction parameters, for example, there are constraints imposed on the size of apparatus in enclosed residential areas or constraints on the lengths of pipes in heat exchangers, which arise from standardization. Although requiring some experience, it is important to leave these constraints out of the optimization formulation because they have a negligible effect on the optimal solution. This enables one to use easier problem-solving techniques yet still maintain precision of the optimization result.
(Continues...)
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