Fundamental Balance in Momentum Transport

In this section, concepts are discussed that one must understand to construct force balances based on momentum transfer rate processes. The fluid, the specific problem, and the coordinate system are generic at this stage of the development. If the discussion that follows seems quite vague, then perhaps it will become more concrete when specific problems are addressed. The best approach at present is to state the force balance in words and then focus on each type of momentum transfer rate process separately. 

The strategy for solving fluid dynamics problems begins by putting a control volume within the fluid that matches the symmetry of the macroscopic boundaries, and balancing the forces that act on the system. The system is defined as the fluid that is contained within the control volume V, which is completely surrounded by surface S. Since a force is synonymous with the time rate of change of momentum as prescribed by Newton s laws of motion, the terms in the force balance are best viewed as momentum rate processes. The force balance for an open system is stated without proof as l = 2- -3H-4- -5, where 

It should be emphasized that force is a vector quantity and, hence, the force balance described qualitatively above is a vector equation. A vector equation implies that three scalar equations must be satisfied. This is a consequence of the fact that if two vectors are equal, then it must be true that they have the same X-component, the same y-component, and the same z-component, for example, in rectangular coordinates. 

It is necessary to associate mathematical quantities with each type of momentum transfer rate process that is contained in the vector force balance. The fluid momentum vector is expressed as p, which is equivalent to the overall mass flux vector. This is actually the momentum per unit volume of fluid because mass is replaced by density in the vectorial representation of fluid momentum. Mass is an extrinsic property that is typically a linear function of the size of the system. In this respect, mv is a fluid momentum vector that changes magnitude when the mass of the system increases or decreases. This change in fluid momentum is not as important as the change that occurs when the velocity vector is affected. On the other hand, fluid density is an intrinsic property, which means that it is independent of the size of the system. Hence, pv is the momentum vector per unit volume of fluid that is not affected when the system mass increases or decreases. The total fluid momentum within an arbitrarily chosen control volume V is 

The rate of accumulation of fluid momentum within V involves the use of a total time derivative to detect changes in fluid momentum during a period of observation that is consistent with the time frame over which the solution to a specific problem is required. The mathematical representation of the accumulation term 1 with units of momentum per time (hence, rate of momentum) is 

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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... 

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. 

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