Page 352 - Bird R.B. Transport phenomena

336 Chapter 11 The Equations of Change for Nonisothermal Systems
opposed to the gravitational field. For terrestrial problems, where the gravitational field is
independent of time, we can write
p(v-g) = -(p (11.1-8)
= -(V + Ф(У • pv) Use vector identity in Eq. A.4-19
dp
= -(V - Ф Use Eq. 3.1-4
dt
Use Ф independent of t

When this result is inserted into Eq. 11.1-7 we get

j - (\pv 2 + pU + рФ) = -(V • {\pv 2 + pU + рФ)у)

-(V-q)-(V-pv)-(V-[T-v]) (11.1-9)
Sometimes it is convenient to have the energy equation in this form.

§11.2 SPECIAL FORMS OF THE ENERGY EQUATION

The most useful form of the energy equation is one in which the temperature appears.
The object of this section is to arrive at such an equation, which can be used for predic-
tion of temperature profiles.
First we subtract the mechanical energy equation in Eq. 3.3-1 from the energy equa-
tion in 11.1-7. This leads to the following equation of change for internal energy:

U= -(V-pl/v) -(V-q)
dt P
rate of net rate of rate of internal
increase in addition of energy addition
internal internal energy by heat conduction,
energy by convective per unit
per unit transport, volume
volume per unit volume
(11.2-1)
- p(V • v) - (T:VV)
reversible rate irreversible rate
of internal of internal energy
energy increase increase per unit
per unit volume volume by
by compression viscous dissipation

It is now of interest to compare the mechanical energy equation of Eq. 3.3-1 and the in-
ternal energy equation of Eq. 11.2-1. Note that the terms p(V • v) and (T:VV) appear in
both equations—but with opposite signs. Therefore, these terms describe the intercon-
version of mechanical and thermal energy. The term p(V • v) can be either positive or
negative, depending on whether the fluid is expanding or contracting; therefore it repre-
sents a reversible mode of interchange. On the other hand, for Newtonian fluids, the
quantity — (T:VV) is always positive (see Eq. 3.3-3) and therefore represents an irreversible
degradation of mechanical into internal energy. For viscoelastic fluids, discussed in
Chapter 8, the quantity -(T:VV) does not have to be positive, since some energy may be
stored as elastic energy.
We pointed out in §3.5 that the equations of change can be written somewhat more
compactly by using the substantial derivative (see Table 3.5-1). Equation 11.2-1 can be