Page 349 - Bird R.B. Transport phenomena

Nonisothermal Systems

§11.1 The energy equation
§11.2 Special forms of the energy equation

§11.3 The Boussinesq equation of motion for forced and free convection
§11.4 Use of the equations of change to solve steady-state problems
§11.5 Dimensional analysis of the equations of change for nonisothermal systems

In Chapter 10 we introduced the shell energy balance method for solving relatively sim-
ple, steady-state heat flow problems. We obtained the temperature profiles, as well as
some derived properties such as average temperature and energy fluxes. In this chapter
we generalize the shell energy balance and obtain the equation of energy, a partial differ-
ential equation that describes the transport of energy in a homogeneous fluid or solid.
This chapter is also closely related to Chapter 3, where we introduced the equation
of continuity (conservation of mass) and the equation of motion (conservation of mo-
mentum). The addition of the equation of energy (conservation of energy) allows us to
extend our problem-solving ability to include nonisothermal systems.
We begin in §11.1 by deriving the equation of change for the total energy. As in
Chapter 10, we use the combined energy flux vector e in applying the law of conserva-
tion of energy. In §11.2 we subtract the mechanical energy equation (given in §3.3) from
the total energy equation to get an equation of change for the internal energy. From the
latter we can get an equation of change for the temperature, and it is this kind of energy
equation that is most commonly used.
Although our main concern in this chapter will be with the various energy equa-
tions just mentioned, we find it useful to discuss in §11.3 an approximate equation of
motion that is convenient for solving problems involving free convection.
In §11.4 we summarize the equations of change encountered up to this point. Then
we proceed to illustrate the use of these equations in a series of examples, in which we
begin with the general equations and discard terms that are not needed. In this way we
have a standard procedure for setting up and solving problems.
Finally, in §11.5 we extend the dimensional analysis discussion of §3.7 and show
how additional dimensionless groups arise in heat transfer problems.

111.1 THE ENERGY EQUATION
The equation of change for energy is obtained by applying the law of conservation of en-
ergy to a small element of volume Ax Ay Az (see Fig. 3.1-1) and then allowing the dimen-
sions of the volume element to become vanishingly small. The law of conservation of

333