Page 474 - Bird R.B. Transport phenomena

Chapter 1,

Nonisothermal Systems

§15.1 The macroscopic energy balance
§15.2 The macroscopic mechanical energy balance
§15.3 Use of the macroscopic balances to solve steady-state problems with flat velocity
profiles
§15.4 The d-foims of the macroscopic balances

§15.5° Use of the macroscopic balances to solve unsteady-state problems and problems
with nonflat velocity profiles

In Chapter 7 we discussed the macroscopic mass, momentum, angular momentum, and
mechanical energy balances. The treatment there was restricted to systems at constant
temperature. Actually this restriction is somewhat artificial, since in real flow systems
mechanical energy is always being converted into thermal energy by viscous dissipation.
What we really assumed in Chapter 7 is that any heat so produced is either too small to
change the fluid properties or is immediately conducted away through the walls of the
system containing the fluid. In this chapter we extend the previous results to describe
the overall behavior of nonisothermal macroscopic flow systems.
For a nonisothermal system there are five macroscopic balances that describe the re-
lations between the inlet and outlet conditions of the stream. They may be derived by in-
tegrating the equations of change over the macroscopic system:

(eq. of continuity) dV = macroscopic mass balance

(eq. of motion) dV = macroscopic momentum balance
V(t)
(eq. of angular momentum) dV = macroscopic angular momentum balance
V(t)
(eq. of mechanical energy) dV = macroscopic mechanical energy balance
J V(t)
I (eq. of (total) energy) dV = macroscopic (total) energy balance
Jvu)
The first four of these were discussed in Chapter 7, and their derivations suggest that
they can be applied to nonisothermal systems just as well as to isothermal systems. In
this chapter we add the fifth balance—namely, that for the total energy. This is derived
in §15.1, not by performing the integration above, but rather by applying the law of con-
servation of total energy directly to the system shown in Fig. 7.0-1. Then in §15.2 we re-
visit the mechanical energy balance and examine it in the light of the discussion of the

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