“Strindberg and the Western canon” is a huge topic, and has to be restricted in one way or another. I have done this in the sense that I have put brackets around Strindberg's position in our contemporary canon formation, instead focusing on Strindberg's own relation to the canon, how he saw it and evaluated it. Despite the obvious methodological difficulties of such a maneuver, I will do it in order to make a few essential distinctions about the canon and Strindberg's relation to it. Moreover, I have chosen not to make this attempt systematically but in a series of observations.

First of all: what is a canon? Aleida Assmann, eminent scholar of Memory Studies, has defined the canon as active cultural memory that has retained its power to speak to us. In this capacity, it must be distinguished from the archive, a storehouse of works that, at least temporarily, have been disconnected from their addresses (Assmann 2010:99). I would like to add that a specifically literary canon, particularly in the modern age, is also marked by constant re-reading and re-interpretation of the heritage, making the canon ever changing and also the object of criticism. There is a considerable difference between, say, literary and religious canon formations and how they work.

But what does the canon consist of in Strindberg's case? Is it a homogenous phenomenon? This leads us immediately to our first observation.

Strindberg's wide understanding of the canon

As active cultural memory, a canon can, according to Assmann, encompass three core areas: religion, history, and art. In all of these areas the canonical works or events have survived the normal fate of oblivion and achieved “continuous repetition and re-use,” as she phrases it (Assmann 2010:99). But in Strindberg's case it is necessary to have an even wider definition. His canon is rather written in the plural, comprising not only the canons of religion, history, and the arts, but also a variety of scientific discourses mixed up with pre-scientific and esoteric traditions in bewildering ways. This wide and transgressive view of the canon is linked to Strindberg's disrespect for literature or art as an autonomous field. Therefore, fiction, drama or poetry only occupied a minor part of his library. And some of his own works can not be described as fictional or poetic at all (Balzamo 2012:367).

Introduction

In chemical engineering, mathematical modeling is crucial in order to design equipment, choose proper operating conditions, regulate processes, etc. It is almost always necessary to use experimental data for model development. Figure 7.1(a) shows a data set and linear fit for these data. From this result, is easy to see that this line describes these data. However, from the data set shown in Figure 7.1(b), this is not so clear. The solid line represents the linear fit for these data, which is derived from regression analysis. By simply observing the data, it can be seen that either of the dashed lines could be possible fits. These results clearly show that it is not possible to determine parameters for models only by which line looks a good fit, but that a detailed statistical analysis is needed.

In this chapter, we start by describing linear regression, which is a method for determining parameters in a model. The accuracy of the parameters can be estimated by confidence intervals and regions, which will be discussed in Section 7.5. Correlation between parameters is often a major problem for large mathematical models, and the determination of so-called correlation matrices will be described. In more complex chemical engineering models, non-linear regression is required, and this is also described in this chapter.

A mathematical model can never give an exact description of the real world, and the basic concept in all engineering modeling is, “All models are wrong – some models are useful.” Reformulating or simplifying the models is not tampering with the truth. You are always allowed to change the models, as long as the results are within an acceptable range. It is the objective of the modeling that determines the required accuracy: Is it a conceptual study limited to order of magnitude estimations? Or is it design modeling in which you will add 10–25% to the required size in order to allow for inaccuracies in the models and future increase in production? Or is it an academic research work that you will publish with as accurate simulations as possible?

A simulation may contain both errors and uncertainties. An error is defined as a recognizable deficiency that is not due to lack of knowledge, and an uncertainty is a potential deficiency that is due to lack of knowledge. All simulations must be validated and verified in order to avoid errors and uncertainties. Validation and verification are two important concepts in dealing with errors and uncertainties. Validation means making sure that the model describes the real world correctly, and verification is a procedure to ensure that the model has been solved in a correct way.

Differential equations play a dominant role in mathematical modeling. In practical engineering applications, only a very limited number of them can be solved analytically. The purpose of this chapter is to give an introduction to the numerical methods needed to solve differential equations, and to explain how solution accuracy can be controlled and how stability can be ensured by selecting the appropriate methods. The mathematical framework needed to solve both ordinary and partial differential equations is presented. A guideline for selecting numerical methods is presented at the end of the chapter.

Ordinary differential equations

A characteristic of a differential equation is that it involves an unknown function and one or more of the function’s derivatives. If the unknown function depends on only one independent variable, it is classified as an ordinary differential equation (ODE). The order of the differential equation is simply the order of the highest derivative that appears in the equations. Consequently, a first-order ODE contains only first derivatives, whilst a second-order ODE may contain both second and first derivatives. The ODEs can also be classified as linear or non-linear. Linear ODEs are the ones in which all dependent variables and their derivatives appear in a linear form. This implies that they cannot be multiplied or divided by each other, and they must be raised to the power of 1. An ODE has an infinite number of solutions, but with the appropriate conditions that describe systems, i.e. the initial value or the boundary value, the solutions can be determined uniquely.

Many phenomena in engineering are very complex and we do not have sufficient knowledge at the moment to develop a model from first principles; instead, we have to rely on empirical correlations. Today most process development is done using empirical or semi-empirical models. These models are usually accurate and very useful. The drawback is that they are only valid for specific equipment within an experimental domain where the parameters are determined.

In developing a correlation, we need first to identify all the variables that may have an influence on it. There are different approaches to finding important variables. One approach is to formulate the governing equations even if we do not have sufficient knowledge or computer resources to solve the equations. These equations and the corresponding boundary conditions provide information about which variables are important in formulating an empirical correlation. A second approach, used by experienced engineers, is to list all variables that are believed to be important. The final correlation is then obtained by experimenting and model fitting using experimental design to obtain reliable results and to minimize correlations between the parameters in the model.

In this introductory chapter the use of mathematical models in chemical engineering is motivated and examples are given. The general modeling procedure is described, and some important tools that are covered in greater detail later in the book are outlined.

Why do mathematical modeling?

Mathematical modeling has always been an important activity in science and engineering. The formulation of qualitative questions about an observed phenomenon as mathematical problems was the motivation for and an integral part of the development of mathematics from the very beginning.

Although problem solving has been practiced for a very long time, the use of mathematics as a very effective tool in problem solving has gained prominence in the last 50 years, mainly due to rapid developments in computing. Computational power is particularly important in modeling chemical engineering systems, as the physical and chemical laws governing these processes are complex. Besides heat, mass, and momentum transfer, these processes may also include chemical reactions, reaction heat, adsorption, desorption, phase transition, multiphase flow, etc. This makes modeling challenging but also necessary to understand complex interactions.

In this chapter mathematical models are classified by

• grouping into opposite pairs;

• mathematical complexity;

• degree of resolution.

The intention is to give the reader an understanding of differences between models as reflected by the modeling goal. Which question is the model intended to answer?

Grouping of models into opposite pairs

In this section, we will examine various types of mathematical models. There are many possible ways of classification. One possibility is to group the models into opposite pairs:

• linear versus non-linear;

• steady state versus non-steady state;

• lumped parameter versus distributed parameter;

• continuous versus discrete variables;

• deterministic versus stochastic;

• interpolation versus extrapolation;

• mechanistic versus empirical;

• coupled versus not coupled.

Linear versus non-linear

Linear models exhibit the important property of superposition; non-linear ones do not. Equations (and thus models) are linear if the dependent variables or their derivatives appear only to the first power; otherwise they are non-linear. In practice, the ability to use a linear model for a process is of great significance. General analytical methods for equation solving are all based on linearity. Only special classes of non-linear models can be attacked with mathematical methods. For the general case, where a numerical method is required, the amount of computation is also much less for linear models, and in addition error estimates and convergence criteria are usually derived under linear assumptions.

The aim of this textbook is to give the reader insight and skill in the formulation, construction, simplification, evaluation/interpretation, and use of mathematical models in chemical engineering. It is not a book about the solution of mathematical models, even though an overview of solution methods for typical classes of models is given.

Models of different types and complexities find more and more use in chemical engineering, e.g. for the design, scale-up/down, optimization, and operation of reactors, separators, and heat exchangers. Mathematical models are also used in the planning and evaluation of experiments and for developing mechanistic understanding of complex systems. Examples include balance models in differential or integral form, and algebraic models, such as equilibrium models.

The book includes model formulation, i.e. how to describe a physical/chemical reality in mathematical language, and how to choose the type and degree of sophistication of a model. It is emphasized that this is an iterative procedure where models are gradually refined or rejected in confrontation with experiments. Model reduction and approximate methods, such as dimensional analysis, time constant analysis, and asymptotic methods, are treated. An overview of solution methods for typical classes of models is given. Parameter estimation and model validation and assessment, as final steps, in model building are discussed. The question “What model should be used for a given situation?” is answered.