## S

= substrate concentration

Ks

= half-velocity constant

kd

= endogenous decay coefficient

Equation 75, however, is incorrect. Note that there is a flow Q into and out of the reactor. This means that the system is open and the full derivative, dX/dt, cannot be used. Nothing in the literature points out this mistake, however, because everything is written the same way. The Reynolds transport theorem distinguishes the difference between the total and partial derivatives, so wrong equations like the previous one will not result in any derivation if using this theorem. After the derivation below of the theorem is complete, we will come back and derive the correct material balance of the microbial kinetics of the activated sludge process.

Two general methods are used in describing fluid flow: Eulerian and Lagrangian methods. In the Eulerian method, values of the properties of the fluid are observed at several points in space and time. This is analogous to the open container above, where the value of a property varies with time and space and where results are partial derivatives. The Lagrangian method, on the other hand, involves tagging each particle of fluid and observing the behavior of the desired property as the fluid moves, independent of spatial location. For example, if velocity is the desired property, this velocity is measured as the particle moves without regard to location in space. This is analogous to the closed container, where the property is only a function of time, and where the result is a total derivative.

It is possible to convert one method of description to the other using the Reynolds transport theorem attributed to Osborne Reynolds. In other words, the theorem converts the Eulerian method of description of fluid flow to the Lagrangian method of description of fluid flow and vice versa.

To derive the theorem, invent two terms: control volume and control mass. Refer to Figure 3. This figure is composed of two intersecting solids, where the volume formed due to the intersection of the solids is indicated by the black color, Volume bcde. The volume on the left with the hatching slanting upward to the right plus Volume bcde is the control volume. The volume with the hatching is Volume I. 