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This book is interdisciplinary, involving two relatively new fields of human endeavor. The two fields are: Chemical/Biological^1 Engineering and Numerical Mathematics. How do these two disciplines meet? They meet through mathematical modeling.

The history of human mathematical computations goes back for several millenia. The need for numerical computations has increased since the age of enlightenment and the industrial revolution three centuries ago. For the last 50 years, the human race has become more and more dependent on numerical computations and digital computers. Computational techniques have developed from early hand computations, through table look up, mechanical adding and multiplying devices, the slide rule etc, to programmable electronic computers, mainframes, PCs, laptops, and notebooks.

After the introduction of the basic building blocks of MATLAB and the fundamentals of numerical analysis with an eye on solving scalar and differential equations in Chapter 1, we now introduce mathematical models in chemical and biological engineering^1. Our subsequent chapters will combine these two areas by introducing models for specific chemical and biological processes and finding detailed numerical solutions via MATLAB. This chapter is quite short and condensed. Any reader or student who wants to gain a deeper insight into model formulation should consult the literature on math modeling in the Resources appendix.

It is important to introduce the reader at an early stage to simple examples of nonlinear models. We will first present cases with bifurcation behavior as the more general case, followed by special cases without bifurcation. Note that this is deliberately the reverse of the opposite and more common approach. We take this path because it sets the important precedent of studying chemical and biological engineering systems first in light of their much more prevalent multiple steady states rather than from the rarer occurrence of a unique steady state.

Many chemical/biological engineering problems can be described by differential equations with known initial conditions, i.e., with known or given values of the state variables at the start of the process.

A distributed model is usually described by differential equations. Such a model differs from a lumped model that is generally described by transcendental equations. In chemical and biological engineering distributed systems often arise with tubular equipment. When a one-dimensional model is used for a distributed system there are two types of models: 1. If mixing, diffusion, and conduction are neglected, then the system is described by the so called plug flow model , expressed in terms of initial value ODEs, i.e., by initial value problems, or IVPs. 2. If the model accounts for the effects of axial dispersion, then the system is described by an axial dispersion model in terms of two-point boundary ODEs, i.e., by boundary value problems, or BVPs.

Many chemical and biological systems include multistage processes rather than only continuous contact ones. The most common multistage systems are absorption and distillation columns. Most of these systems involve more than one phase and they therefore fall under the category of heterogeneous multistage systems. Multistage systems can be cocurrent or countercurrent.

In the previous chapters we have shown how to solve many types of problems that occur in Chemical and Biological Engineering through mathematical modeling, standard numerical methods, and MATLAB.