## The System

As discussed in Chapter 1, the basic principles that apply to the analysis and solution of flow problems include the conservation of mass, energy, and momentum in addition to appropriate transport relations for these conserved quantities. For flow problems, these conservation laws are applied to a system, which is defined as any clearly specified region or volume of fluid with either macroscopic or microscopic dimensions (this is also sometimes referred to as a ''control volume''), as illustrated in Fig. 5-1. The general conservation law is

Rate of X Rate of X Rate of accumulation into the system out of the system of X in the system where X is the conserved quantity, i.e., mass, energy, or momentum. In the case of momentum, because a ''rate of momentum'' is equivalent to a force (by Newton's second law), the ''rate in'' term must also include any (net) forces acting on the system. It is emphasized that the system is not the "containing vessel'' (e.g., a pipe, tank, or pump) but is the fluid contained within the designated boundary. We will show how this generic expression is applied for each of the these conserved quantities.

Figure 5-1 A system with inputs and outputs.

II. CONSERVATION OF MASS A. Macroscopic Balance

For a given system (e.g., Fig. 5-1), each entering stream (subscript i) will carry mass into the system (at rate mi), and each exiting stream (subscript o) carries mass out of the system (at rate mo). Hence, the conservation of mass, continuity," equation for the system is or

. m - £ mo = in out where ms is the mass of the system. For each stream, m = dm =

0 0