## Fluid Mechanics

Of the fluid mechanics applications used in this book, the Reynolds transport theorem is a bit sophisticated. Hence, a review of this topic is warranted. This theorem will be used in the derivation of the amount of solids deposited onto a filter and in the derivation of the activated sludge process. These topics are discussed somewhere in the textbook. Except for these two topics, the theorem is used nowhere else. The general knowledge of fluid mechanics will be used under the unit operations part of this book.

To understand the derivation of the Reynolds transport theorem, there are some prerequisite mathematics needed. The students would have had these mathematics already in their undergraduate training before taking physical-chemical treatment, however, they are presented here as a review. They include the topics on integration symbols, vectors, the Gauss-Green divergence theorem, and partial and total derivatives. These topics are not really used in this book, but they are, however, needed, as mentioned, in the understanding of the derivation of the Reynolds transport theorem. The mathematics used in this book pertains mostly to algebra.

### Integration Symbols

The symbols that we are going to discuss are as follows: j>A and jV. The symbol j>A is used to signify integration over a closed surface area and the symbol jV is used to signify integration over a volume. But, what does this sentence mean exactly?

Refer to Figure 1. This figure shows a volume of solid, where a differential area dA is shown. This area is on the surface of the solid and, if it is extended around z a A x

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