Linear Mathematical Models in Chemical Engineering

The following sections are included:

• Preface
• Contents

Engineers write mathematical models for numerous reasons. To design processes, to scale them up or down, to optimize process performance and financial gain, to minimize formation and release of undesirable or hazardous compounds, to design control strategies and so on. There hardly seems a need to justify why engineers need to learn some mathematics and develop the basic skills needed for writing mathematical models.

In this chapter we will discuss some common tools for simplifying and editing models. The first topic, dimensionless models, is a tool for simplifying models without introducing any approximations. All the remaining topics are tools that give rise to a simpler but also approximate model.

Probably everyone who has ever taught mathematical modelling of physical systems will have amassed a collection of common errors that students make when writing models. Many of these common errors can be avoided by being mindful of common pitfalls while developing the model and by doing a brief preliminary model analysis to check the validity of the model. The purpose of this deliberately short chapter is to point out some of these pitfalls and some of the tools that can be used for preliminary model analysis. And, by presenting this material after the reader has had a chance to work through the modelling problems in the first two chapters, to convey an appreciation of these pitfalls while writing the model and to examine a model to detect errors and to develop a feel for the solution.

This chapter covers various ordinary differential equations that can be solved by quadrature or for which a theory exists for analyzing and possibly solving the equation. A particular differential equation may match several of the types listed below and either of the possible solution methods can then be used. Pick the method that is the easiest to use. The treatment below is sketchy and is intended as a summary and review of solution methods and not as a rigorous analysis of these methods.

One usually first encounters a vector depicted as a directed line segment in Euclidean space, or what amounts to the same thing, as an ordered n-tuple of numbers of the form {a_i}_i=1,…,n. The former is a geometrical description of a vector, whereas the latter can be interpreted as an algebraic representation with respect to a given coordinate system. However, the interplay between geometry and algebra is not transparent in more general vector spaces that could be infinite dimensional.

It the previous chapter, matrices were viewed as fixed objects, sets of numbers indexed by two counters. The main new idea in this chapter is that matrices, just like vectors, can be viewed as coordinate representations of some underlying object. This underlying object is a linear operator, a concept that was briefly alluded to in Example 8 on page 150.

Difference equations are recurrence relationships that relate the values in a sequence of numbers to one another. Most readers are probably familiar with these equations from the study of numerical integration of ODEs. For instance, the ODE initial value problem

An ODEs of Nth order is an equation of the form

Most of the algebraic concepts and insights obtained from the development of finite dimensional spaces transfer over to infinite dimensional spaces. However, there are important differences between finite dimensional and infinite dimensional vector spaces. Arguably the most important difference is that the Hamel basis, as defined previously for finite dimensional spaces, is of little use in infinite dimensional spaces. The reason for this is that the Hamel basis is an algebraic concept and therefore does not permit infinite sums which is a concept that belongs to calculus. The way out of this dilemma is to impose additional structure on the vector space so that infinite sums can be well-defined, and for this we need the topological notion of convergence. This is less important in finite dimensions because finite dimensional vector spaces of the same dimension are isomorphic with the same topology. The isomorphism property between infinite dimensional spaces is only meaningful if it also incorporates the topology. There may be a variety of topologies that can be used in a given context, but the one with the most useful properties is the main topic of this chapter, and is called a Hilbert space. We first loosely describe some of the issues and problems that complicate the study of infinite dimensional spaces, and then introduce normed spaces. The concept of a Hilbert space is a type of normed space that is suitable for many applications, mainly because it admits easy-to-use bases. A particularly important example of a Hilbert space involves Fourier series, which is discussed in some detail.

Partial Differential Equations (PDEs) are almost always solved by converting the problem to solution of one or more ODEs. This chapter describes a solution method for solving various types of partial differential equations by summing solutions of self-adjoint ODEs. These sums are Fourier series and the solution method rely on identifying a linear operator in the PDE and on solving the eigenvalue problem associated with this operator. The method is primarily used when the operators are self-adjoint and the associated eigenvalue problem therefore is easy to solve. Typically, this means that Fourier series methods can be used for PDEs of even order over a finite domain.

First order linear partial differential equations (PDEs) are equations of the form

A similarity transformations in finite dimensional vector spaces is essentially a variable transformation resulting from a change of basis. In particular, the transformation can change a problem in N dimensions to N uncoupled problems, each in one dimension. When the term is applied to PDEs, it indicates a similar idea; the idea that the dimensionality of the problem can be reduced by some clever variable transformation. Most often, that PDE in two free variables can be simplified to an ODE.

The following sections are included:

• Model formulation
• ODEs
• Finite dimensional vector spaces
• Tensors
• Linear difference equations
• Linear ODEs
• Hilbert spaces
• PDEs

The following section is included:

• Complex numbers

The following section is included:

• Index