Page 90 - Bird R.B. Transport phenomena

Chapter 3

The Equations of Change

for Isothermal Systems

§3.1 The equation of continuity
§3.2 The equation of motion
§3.3 The equation of mechanical energy
§3.4° The equation of angular momentum

§3.5 The equations of change in terms of the substantial derivative
§3.6 Use of the equations of change to solve flow problems
§3.7 Dimensional analysis of the equations of change

In Chapter 2, velocity distributions were determined for several simple flow systems by
the shell momentum balance method. The resulting velocity distributions were then
used to get other quantities, such as the average velocity and drag force. The shell bal-
ance approach was used to acquaint the novice with the notion of a momentum balance.
Even though we made no mention of it in Chapter 2, at several points we tacitly made
use of the idea of a mass balance.
It is tedious to set up a shell balance for each problem that one encounters. What we
need is a general mass balance and a general momentum balance that can be applied to
any problem, including problems with nonrectilinear motion. That is the main point of
this chapter. The two equations that we derive are called the equation of continuity (for the
mass balance) and the equation of motion (for the momentum balance). These equations
can be used as the starting point for studying all problems involving the isothermal flow
of a pure fluid.
In Chapter 11 we enlarge our problem-solving capability by developing the equa-
tions needed for nonisothermal pure fluids by adding an equation for the temperature.
In Chapter 19 we go even further and add equations of continuity for the concentra-
tions of the individual species. Thus as we go from Chapter 3 to Chapter 11 and on to
Chapter 19 we are able to analyze systems of increasing complexity, using the com-
plete set of equations of change. It should be evident that Chapter 3 is a very important
chapter—perhaps the most important chapter in the book—and it should be mastered
thoroughly.
In §3.1 the equation of continuity is developed by making a mass balance over a
small element of volume through which the fluid is flowing. Then the size of this ele-
ment is allowed to go to zero (thereby treating the fluid as a continuum), and the desired
partial differential equation is generated.

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