Momentum transport, fundamental balance

For a fluid flow, of course, one uses the Reynolds transport theorem to establish the relationship between a system (where the momentum balance applies directly) and a control volume (through which fluid flows). In terms of Eq. 3.2, the extensive variable N is the momentum vector P = mV and the intensive variable tj is the velocity vector V. Thus the fundamental approach yields the following vector equation... 

To establish the integral form of the basic conservation laws for mass, momentum and energy, the fundamental approach is to start out from a system analysis and then transform the balance equations into a control volume analysis by use of the transport theorem. However, to achieve a more compact presentation of this theory it is customary to start out from a generic Eulerian form of the governing equations. That is, the material control volume analysis is disregarded. 

Expressions of the conservation of mass, a particular chemical species, momentum, and energy are fundamental principles which are used in the analysis and design of any separation device. It is appropriate to formulate these laws first without specific rate expressions so that a clear distinction between conservation laws and rate expressions is made. Some of these laws contain a source or generation term, for example, for a particular chemical species, so that the particular quantity is not actually conserved. A conservation law for entropy can also be formulated which contributes to a useful framework for a generalized transport theory. Such a discussion is beyond the scope of this chapter. The conservation expressions are first presented in their macroscopic forms, which are applicable to overall balances on energy, mass, and so on, within a system. However, such macroscopic formulations do not provide the information required to size equiprrwnt. Such analyses usually depend on a differential formulation of the conservation laws which permits consideration of spatial variations of composition, temperature, and so on within a system. 

---