Page 351 - Bird R.B. Transport phenomena

§11.1 The Energy Equation 335

disintegrations have not been included in the energy balance. In Chapter 19,
where the energy equation for mixtures with chemical reactions is considered,
the chemical heat source S c appears naturally, as does a "diffusive source
term," S (j a • j.
g
ft
We now translate Eq. 11.1-1 into mathematical terms. The rate of increase of kinetic
and internal energy within the volume element Ax Ay Az is

Ax Ay Az — (\pv 2 + pU) (11.1-2)

Here U is the internal^ energy per unit mass (sometimes called the "specific internal en-
2
ergy"). The product pU is the internal energy per unit volume, and \pv 2 = \p{v\ + v 2 + v )
z
is the kinetic energy per unit volume.
Next we have to know how much energy enters and leaves across the faces of the
volume element Ax Ay Az.
Ay Az(e \ - e \ ) + Ax Az(e | y y - е | ) + Ax Ayfe | 2 2 - e \ ^ ) (11.1-3)
z z+/
z
x x
у у+Ду
x x+Ax
Keep in mind that the e vector includes the convective transport of kinetic and internal
energy, the heat conduction, and the work associated with molecular processes.
The rate at which work is done on the fluid by the external force is the dot product
of the fluid velocity v and the force acting on the fluid (p Ax Ay Az)g, or
p Ax Ay bz(v g + v g + v g ) (11.1-4)
x x y y z z
We now insert these various contributions into Eq. 11.1-1 and then divide by Ax Ay
Az. When Ax, Ay, and Az are allowed to go to zero, we get
де х +p{Vxgx + v + Vz8z)
dt { ч >*> (11.1-5)

This equation may be written more compactly in vector notation as
f ( > 2 + pU) = -(V • e) + p(v • g)
t 2 (11.1-6)

Next we insert the expression for the e vector from Eq. 9.8-5 to get the equation of energy:

| (W + u) = -(V • O 2 + pLDv) - (V • q)
P
rate of increase of rate of energy addition rate of energy addition
energy per unit per unit volume by per unit volume by
volume convective transport heat conduction
-(V-pv) - ( V - [ T - V ] > + p(v • g) (11.1-7)
rate of work rate of work done rate of work done
done on fluid per on fluid per unit on fluid per unit
unit volume by volume by viscous volume by external
pressure forces forces forces

This equation does not include nuclear, radiative, electromagnetic, or chemical forms of
energy. For viscoelastic fluids, the next-to-last term has to be reinterpreted by replacing
"viscous" by "viscoelastic."
Equation 11.1-7 is the main result of this section, and it provides the basis for the re-
mainder of the chapter. The equation can be written in another form to include the poten-
tial energy per unit mass, Ф, which has been defined earlier by g = — VO (see §3.3). For
moderate elevation changes, this gives Ф = gh, where h is a coordinate in the direction