This chapter evaluates the advances in knowledge accumulated over the past ten years of research at the Centre for Textile Research in Copenhagen and identifies questions that still remain to be answered.

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.