A Numerical Primer for the Chemical Engineer, Second Edition

Edwin Zondervan

• Publisher: CRC Press
• Year: 2019
• ISBN:
  9780429851445
• Edition: 2
• Topic: Matematica
• Subtopic: Matematica applicata

Designed as an introduction to numerical methods for students, this book combines mathematical correctness with numerical performance, and concentrates on numerical methods and problem solving. It applies actual numerical solution strategies to formulated process models to help identify and solve chemical engineering problems. Second edition comes with additional chapter on numerical integration and section on boundary value problems in the relevant chapter. Additional material on general modelling principles, mass/energy balances and separate section on DAE's is also included. Case study section has been extended with additional examples.

1

The role of models in chemical engineering

1.1 Introduction

The concept of a model has been around since ancient times. Models appear in all branches of science and engineering. However, it is often said that modeling is more art than science or engineering. In this chapter we will discuss general aspects of models, and more specifically the models that describe (chemical) process systems. It is not intended as an in-depth discussion.

Ultimately, this book is about solving the developed models in a numerical fashion. We could consider Ptolemy’s Amalgest (150 BC) as one of the first recorded studies on modeling and numerical analysis in which numerical approximations to describe the motions of the heavenly bodies with accuracy matching reality sufficiently were developed (Figure 1.1). This is basically the essence of numerical analysis. Numerical analysis is concerned with obtaining approximate solutions to problems while maintaining reasonable bounds of error, because it is often impossible to obtain exact answers. Numerical analysis makes use of algorithms to approximate solutions. Model development and solving the models is important to the world, for example in astronomy, construction, agriculture, architecture, and, of course, in engineering! In chemical engineering we use models and their (numerical) solutions to describe reactors and separators (dynamic and steady state), to perform computational fluid dynamics, to solve thermodynamic equations of state, to optimize process performance, to design and synthesize processes, and to regress experimental data, e.g., isotherms, kinetics, and so forth.

FIGURE 1.1
(Left) An image of Ptolemy; (Right) Ptolemy’s model of our solar system

1.2 The idea of a model

In Figure 1.2 we can see an image by the Belgian surrealist Rene Magritte. It is a pipe, and below this pipe is a sentence in French that says, “Ceci n’est pas une pipe” (“this is not a pipe”). Actually, it is, indeed, not a pipe; it is an image of a pipe. Models are similar. Models are not the reality, they are an approximate description of reality. Eykhoff [20] defines an engineering model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in a usable form. This implies basically that a model is (always) a simplification of reality. A model as such can give insight into the behavior of the system under study, but it does not always mean that this insight is phenomenological. For example, if an engineer develops a controller for a distillation tower, he would like to know how the distillation tower behaves dynamically. Whether this knowledge is based on first principles or not is not really relevant for his purposes. In Table 1.1 the different model types are listed.

FIGURE 1.2
“The Treachery of Images” by Rene Magritte

Type of model	Criterion or classification
Mechanistic	Based on mechanisms/underlying phenomena
	(first principles)
Empirical	Based on input-output data, trials or experiments
Stochastic	Contains elements that are probabilistic in nature
Deterministic	Based on cause–effect analysis
Lumped parameter	Dependent variables not a function of spatial position
Distributed parameter	Dependent variables as a function of spatial position
Linear	Superposition applies
Nonlinear	Superposition does not apply
Continuous	Dependent variables defined over continuous space
Discrete	Only define for discrete values of time and/or space
Hybrid	Containing continuous and discrete behavior

TABLE 1.1

Model types and their classifications

The mathematical forms of the different model types can involve linear algebraic equations, nonlinear algebraic equations, ordinary differential equations, differential algebraic equations and partial differential equations. Each of the equation forms requires special techniques for solution.

1.3 Model building

Although there have been many attempts to structure the process of setting up process models to describe phenomena or systems, the general notion is that each modeling problem requires a custom-made approach. The applications and requirements are so different that general model development strategies would be extremely difficult and decisions regarding the modeling of a system can often best be made by an expert. However there is some kind of agreement on the four elementary steps in the modeling process: problem definition, design, evaluation and application.

In the problem definition phase, the modeling problem and the goal of the model are properly formulated. This formulation is based on performance...