Heat input in a system is given by difference between the heat conduction coming
from outside through the surface A and distributed internal heat generation r per unit
mass:
Q input ¼
Z
A
À q Á n dA þ
Z
V
ρrdV
ð3:58Þ
Entropy was defined by dS ¼
dQ
T . Hence, entropy input rate can be given by
Z
A
À
q
T
Á n dA þ
Z
V
ρ r
T
dV
ð3:59Þ
We are assuming that the system is closed. If the system is open in a control
surface, [fixed in space], additional entropy input due to mass flux would be
accounted for by
Z
A
À ρsV Á n dA
ð3:60Þ
where s is entropy per unit mass. According to the second law of thermodynamics,
entropy increase rate in the system ! entropy Input rate.or
d
dt
Z
V
ρsdV !
Z
A
À
q
T
Á ndA þ
Z
V
r
T
ρdV
ð3:61Þ
This is the integral form of the Clausius-Duhem inequality. This equation means
that in an irreversible process, internal entropy production is always greater than
input. In this equation, outward normal is a positive sign. Equal sign ensures that the
equation holds for reversible [imaginary] process.
We can transform the surface integral to a volume integral. Since volume can be
an arbitrary value, we can write the local version of the Clausius-Duhem inequality
that must be satisfied at each point in any given volume as follows:
ds
dt
!
r
T
À
1
ρ
div
q
T
ð3:62Þ
or
γ
ds
dt
À
r
T
þ
1
ρT
divq À
q
ρT
2
Á grad T ! 0
ð3:63Þ
where γ is the internal entropy production rate per unit mass.
92
3 Thermodynamics