AR Theory

Onuki, Ar Theory of Phase Transition in Polymer Gels. Vol. 109, pp. 63-120. 

Fig. 7.26. Positronium total scattering cross sections for helium and argon gases. Experiment Garner, Laricchia and Ozen (1996) , He A, Ar. Theory...

Due to the geometric nature of attainable region (AR) theory, as well as the complexity of the systems considered, we often need software tools to help us interpret and visualize our problems. Rather than attempting to populate a conventional CD-ROM with software and additional examples that cannot be adapted over time, we have decided to release this material on a companion website (http //attainableregions.com), which has been developed for this book. We hope that this approach will allow us to cater to the changing needs of the reader and AR community as a whole, where these software resources can be tailored accordingly over time. 

This book is concerned with a field of study called attainable regions (ARs), which is a set of ideas intended to address a generalized problem, often encountered in chemical reactor and process design. Although the problem can become quite detailed if we allow it, the basic idea is simple to understand. This chapter serves to articulate the type of problems AR theory could help address. To gain a sense of the scientific discipline that we are interested in, many (but not all) of the problems we are concerned with can be represented by Figure 1.1. 

AR theory is hence a set of ideas in chemical reactor design that aims to understand the reactor network synthesis problem. But to understand this problem will ultimately require us to understand a broader problem of what it means to be the best, for various designs within the block may achieve the same outcome. This is our primary concern in AR theory. It is related to the reactor network synthesis problem, but it is also distinct. 

To motivate why AR theory is useful, let us begin with a story, involving a team of three young engineers, who are interested in the production of toluene. We would like to gain a sense of some of the typical design considerations encountered by the team, which may also be addressed with AR theory. Sections 1.2.2-1.2.5 describe the team s story, whereas Section 1.2.6 explains the story in relation to AR theory. Sections 1.3 and 1.4 are intended for readers who are already familiar with the rudiments of chemical reactor design. 

Could we have foreseen these challenges from the beginning How do we find these limits And what tasks must be done in order to achieve them There is a theme of attainability that runs through these questions, and AR theory provides... 

AR theory also helps us to understand what equipment is needed to achieve these targets. In relation to reactive equipment, this means providing insight into what type of reactor should be used, and how different reactor types should be... 

Figure 1.6 proposes a number of general approaches to the reactor synthesis problem, starting from the simplest and most constrained approach, to the most general (and difficult) approach. In AR theory, we are ultimately concerned with problems involving reactor structures that produce the best performance, and thus AR theory falls into Approach 3 of Figure 1.6. 

We find that methods such as AR theory are required not because of problems involving multiple reactors, but because of problems involving multiple reactions. Many of these methods are unnecessary when the reactions are inherently simple. The reactor network synthesis problem arises often as a result of complexities in the system from multiple reactions. 

When the reaction is complex, the best performance is often achieved in a reactor network (a combination of reactors). AR theory deals with reactor problems involving more than one reactor. 

In AR theory, we have an interest in the reactor network synthesis problem, but it is not our chief interest. It is sometimes not possible to generate an optimal reactor structure using AR theory, but it is still possible to generate important information that may help to understand what the optimal reactor structure might be. Let us refer back to the following two points from Section 1.4.1.2 ... 

This ordering of tasks is depicted graphically in Figure 1.9. In a classical (non-AR theory) approach, reactor structures are first formulated, solved, and then optimized to produce a particular output. An output exists for a particular reactor structure, and the exact output obtained... 

As a precursor to the following chapters, we wish to leave with a number of scenarios that AR theory could help to address. These scenarios are related to the BTX problem discussed earlier. It is important to keep these questions in mind, for they shape the approach we would like to adopt with AR theory. 

In the first (non-AR theory) approach, the answer is obtained by proposing a structure. 

In the second (AR theory) approach, it is first assumed that a certain answer is achievable, and then we determine what is required to achieve it. 

We write this book out of a need to understand a very basic question—when do we know we have achieved the best AR theory is an approach that seeks to help answer this question for chemical reactor networks (reactor structures). In this chapter, we described how problems of this nature arise when attempting to optimally produce toluene in the BTX reaction. In general, three approaches to the solution of this problem emerged, which are summarized pictorially in Figure 1.11. 

From this perspective, AR theory should not be viewed as a competing method to optimization or superstructure methods, but rather as complementary tool used to benchmark current designs. Superstructures can be used in conjunction with AR theory to both set reactor performance targets and design the reactors needed to achieve these targets. 

We do not wish to stumble in the dark, only identifying our position in the room once we have reached it. Rather, we hope that the ideas of AR theory will act as a flashlight, illuminating aspects of the unknown so that it is easier to advance forward. It is the awareness that these ideas are important that eventually promotes improvement. In short, we cannot fix what we do not know. The most useful feature that we would like to generate with AR theory is hindsight. 

The specific reactor type used, and why it is needed, is described in Chapter 4. A number of worked problems, using AR theory, are given in Chapter 5. 

In this chapter, we wish to focus on laying the foundations for AR theory, with the view of eventually employing reactor structures to improve performance in the BTX system. As a first step, the data given in Chapter 1 will be revisualized and related to a number of simple geometric concepts related to concentration and mixing. These topics are crucial to the proper understanding of AR theory. 

We have therefore gained some insight into how the BTX reaction should be carried out. And although this result is simple, it highlights an important approach in AR theory In AR theory, we will seek to interpret all concentration data graphically. Visualizing data in this manner allows us to adopt a different perspective of reactors but to understand why, we must first introduce a number of fundamental concepts. 

Revisualizing data in the phase plane is the first important step to understanding AR theory. Viewing data in this manner often provides a different perspective on the system under investigation. 

Viewing concentrations as coordinates in concentration space is useful in representing the state of a system. Moreover, when this coordinate is interpreted as a vector with a unique magnitude and direction, the geometric properties of vectors may be exploited. This representation will prove to be highly useful in AR theory. Coordinate values will be enclosed in square brackets (in plaee of parentheses) to indicate veetors. 

Both matrices and vectors are common in AR theory, and hence it is important that we are comfortable with the associated notation. To display vectors compactly, column vectors will often be expressed as the matrix transpose of the equivalent row vector. The superscript T, as used previously, therefore represents the transpose operation from linear algebra. 

Entry c, is therefore the concentration of component i in solution. We call the vector C the concentration vector. C is significant in AR theory because it may be generalized to represent a number of contexts. For example, C could indicate... 

In AR theory, C might be used to represent the concentration of various mixtures within a process. C might not necessarily even contain concentration data only. 

C is hence used as a representation of the state of the system, and thus the values contained within C must also represent physically realizable states. (For example, since concentrations can never be negative, C must always contain positive values.) AR theory will help us to generate a set of values for C that represent physically achievable states. The concentration vector concept therefore lies at the heart of AR theory. 

When density is assumed constant, then mixing has a special geometric property. Mixtures lie on a straight line joining the two concentrations being mixed in concentration space. Mixing is therefore a linear process. This has important consequences in AR theory, as will be seen later. 

Mixture concentrations are commonly viewed as a linear combination of vectors. In AR theory, the vector difference C2 - Cl is frequently referred to as the mixing vector... 

This exercise has highlighted important points about concentration and mixing. Mixing allows us to achieve new points and expand from an initially small fixed set. This idea is central to AR theory—mixing allows one to achieve many solution states from a smaller set of achievable points. Let us extend on the ideas presented so far and formalize them into concrete terminology and definitions. 

Convex hulls are an important property of AR theory, and so they will be used often throughout this book. We will denote the convex hull of a set of points X by conv(X). It will not be our goal to formalize the convex hull mathematically. It is more important to develop an appreciation for what the convex hull is geometrically, and the meaning with regard to attainability in chemical systems. Let us therefore strengthen the idea of convex hulls with a number of simple (everyday) examples. 

We will frequently refer to data as being from a geometric viewpoint throughout the book. However, from the examples discussed so far, the same meaning could be applied to from a graphical viewpoint instead. We make a distinction between graphical and geometric viewpoints because there is a subtle difference in AR theory ... 

For this reason, we will refer to AR theory as being a geometric method as opposed to a graphical one, as the ideas are applicable to higher dimensional problems as well. 

This chapter discussed concentration and mixing from a geometric viewpoint. These concepts are easy to understand intuitively from a physical perspective. They are also easily expressed mathematically, and hence easily measured and calculated given the necessary data. The geometric perspective, however, is ultimately more interesting and useful in AR theory. When concentration and mixing can be... 

The geometric perspective of a system is an important aspect of AR theory, for it allows us to utilize the fundamental concepts of concentration vectors, mixing, and convex hulls. In Chapter 3, we will return to the BTX beaker experiment and use the graphical concepts described in this chapter to improve the maximum toluene concentration (larger than that obtained in Chapter 1). 

These profiles suggest that there are no other starting concentrations in region A that could be used to increase the concentration of toluene— in Chapter 4, we shall describe why this behavior is always true, which is a useful property used in AR theory. With this, let us rather investigate the concave portion of the region, designated by region Bj. 

In Chapter 4, we shall expand our knowledge of the geometric viewpoint to include reaction, and describe the function of three fundamental reactor types that are used in AR theory. With this knowledge, we will be in a position to generate candidate ARs efficiently, and use this information to form optimal reactor networks, which may be applied to optimize complex reactive systems. 

In this chapter, we wish to describe how reaction may be viewed as a geometric process, similar to that of concentration and mixing. This viewpoint will allow us to describe three important reactor models used in AR theory. Importantly, this theory will also assist us in transitioning our early ideas from batch reaction to continuous operation. ... 

In Chapter 7, we describe how AR theory may be applied to batch reactors. 

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