Complete deductions of the model equations and all of the thinking used in the model development process are not presented in the case studies. You will need to consult the original papers if you want more detail than that which is provided,

Details are not given about how the various equations were originally written and manipulated in order to arrive at the equations that are presented. Chapters 12 to 20 have given the general principles for the deduction of the equations. In fact, in order to be able to write the correct equations and solve them, it is necessary to have skills in several areas:

• mass and energy balancing;

• heat and mass transfer phenomena;

• differential and integral calculus;

• numerical methods for solving differential equations;

• programming in a computer language such as FORTRAN

We have assumed that the majority of our readers do not in fact have such skills. Of course, this is not a problem since, if you want to develop or modify a mathematical model of an SSF bioreactor and you do not have the necessary skills yourself, you can interact with people who do, whom we can refer to as "modelers". The aim of this book (which has guided the detail in the previous chapters and will guide the detail presented in these modeling case study chapters) is to give you sufficient understanding to allow you to interact effectively with these modelers. For example:

• you will understand something about what modeling can and cannot do, thereby having realistic expectations about what benefits a final model can bring (i.e., after reading this book you will be better able to discuss Step 1 of the modeling process with the modeler, see Sect. 12.4.1);

• you will understand something about heat and mass transfer and kinetic phenomena, improving your ability to communicate with the modeler and understand what he or she says;

• you will be able to recognize the mathematical forms of expressions used to describe various heat and mass transfer phenomena and kinetic phenomena (even if you do not know how to derive the correct form of the expression for a particular situation). In other words, looking at an equation within the model, you will have an idea about which phenomena it describes, and how. This eases the interaction with the modeler.

Further, a modeler with whom you interact may have the necessary engineering, mathematical, and programming skills but may not be familiar with SSF systems. Such a person should also read this book, with the aim of understanding the general features of the various bioreactors and the principles of growth kinetics in SSF systems. With this information, the modeler will be better placed to participate in decisions about what level of detail to use to describe the system and the processes occurring within and between the phases in the system.

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