Page 350 - Bird R.B. Transport phenomena

334 Chapter 11 The Equations of Change for Nonisothermal Systems

energy is an extension of the first law of classical thermodynamics, which concerns the
difference in internal energies of two equilibrium states of a closed system because of
the heat added to the system and the work done on the system (that is, the familiar
Ш= Q + W). 1
Here we are interested in a stationary volume element, fixed in space, through
which a fluid is flowing. Both kinetic energy and internal energy may be entering and
leaving the system by convective transport. Heat may enter and leave the system by heat
conduction as well. As we saw in Chapter 9, heat conduction is fundamentally a molecu-
lar process. Work may be done on the moving fluid by the stresses, and this, too, is a
molecular process. This term includes the work done by pressure forces and by viscous
forces. In addition, work may be done on the system by virtue of the external forces,
such as gravity.
We can summarize the preceding paragraph by writing the conservation energy in
words as follows:
rate of net rate of kinetic net rate of heat
increase of and internal addition by
kinetic and > = ,energy addition • + < molecular
internal by convective transport
energy transport (conduction)
rate of work r rate of work
done on system done on system
<by molecular by external (11.1-1)
mechanisms forces
(i.e., by stresses). Ite.g ., by gravity).
s
In developing the energy equation we will use the e vector of Eq. 9.8-5 or 6, which in-
cludes the first three brackets on the right side of Eq. 11.1-1. Several comments need to
be made before proceeding:

(i) By kinetic energy we mean that energy associated with the observable motion of
the fluid, which is \pv 2 = \p{w • v), per unit volume. Here v is the fluid velocity
vector.
(ii) By internal energy we mean the kinetic energies of the constituent molecules cal-
culated in a frame moving with the velocity v, plus the energies associated with
the vibrational and rotational motions of the molecules and also the energies of
interaction among all the molecules. It is assumed that the internal energy U for
a flowing fluid is the same function of temperature and density as that for a
fluid at equilibrium. Keep in mind that a similar assumption is made for the
thermodynamic pressure p(p, T) for a flowing fluid.
(Hi) The potential energy does not appear in Eq. 11.1-1, since we prefer instead to
consider the work done on the system by gravity. At the end of this section,
however, we show how to express this work in terms of the potential energy.
(iv) In Eq. 10.1-1 various source terms were included in the shell energy balance. In
§10.4 the viscous heat source S appeared automatically, because the mechani-
v
cal energy terms in e were properly accounted for; the same situation prevails
here, and the viscous heating term — (T:VV) will appear automatically in Eq.
11.2-1. The chemical, electrical, and nuclear source terms (S , S , and S ) do not
c e n
appear automatically, since chemical reactions, electrical effects, and nuclear

R. J. Silbey and R. A. Alberty, Physical Chemistry, Wiley, New York, 3rd edition (2001), §2.3.
1