Fluid Mechanics-Finite Control Volume Analysis

Many practical problems in fluid mechanics require analysis of the behavior of the contents
of a finite region in space 1a control volume2. For example, we may be asked to calculate the
anchoring force required to hold a jet engine in place during a test. Or, we could be called
on to determine the amount of time to allow for complete filling of a large storage tank. An
estimate of how much power it would take to move water from one location to another at a
higher elevation and several miles away may be sought. As you will learn by studying the
material in this chapter, these and many other important questions can be readily answered
with finite control volume analyses. The bases of this analysis method are some fundamental
principles of physics, namely, conservation of mass, Newton’s second law of motion, and
the first and second1 laws of thermodynamics. Thus, as one might expect, the resultant techniques
are powerful and applicable to a wide variety of fluid mechanical circumstances that
require engineering judgment. Furthermore, the finite control volume formulas are easy to
interpret physically and thus are not difficult to use.
The control volume formulas are derived from the equations representing basic laws
applied to a collection of mass 1a system2. The system statements are probably familiar to
you presently. However, in fluid mechanics, the control volume or Eulerian view is generally
less complicated and, therefore, more convenient to use than the system or Lagrangian
view. The concept of a control volume and system occupying the same region of space at an
instant 1coincident condition2 and use of the Reynolds transport theorem 1Eqs. 4.19 and 4.232
are key elements in the derivation of the control volume equations.
Integrals are used throughout the chapter for generality. Volume integrals can accommodate
spatial variations of the material properties of the contents of a control volume. Control
surface area integrals allow for surface distributions of flow variables. However, in this
chapter, for simplicity we often assume that flow variables are uniformly distributed over
cross-sectional areas where fluid enters or leaves the control volume. This uniform flow is
called one-dimensional flow. In Chapters 8 and 9, when we discuss velocity profiles and
other flow variable distributions, the effects of nonuniformities will be covered in more detail.
Mostly steady flows are considered. However, some simple examples of unsteady flow
analyses are introduced.
Although fixed, nondeforming control volumes are emphasized in this chapter, a few
examples of moving, nondeforming control volumes and deforming control volumes are also
included.