Biomedical Engineering Reference
In-Depth Information
D
s
N
ð
i
Þ
D
t
and the internal entropy production
is simply related to a similar specific
internal entropy production quantity
D
s
s
D
t
,
ð
O
r
D
s
N
ð
i
Þ
D
t
¼
D
s
s
D
t
d
v
;
(10.51)
where
s
is the specific internal entropy. From experiment and experience it is
known that, at constant temperature, the excess of external power over internal
power (
P
E
-
P
I
) must be greater than zero; that is to say, at constant temperature,
one cannot recover from an object more power than was supplied to the object. It is
also a fact of experience and experiment that heat flows from the hotter to the colder
parts of an object and not in the reverse way. In equation form we note this last
assertion by
q
ry
0 for
P
E
P
I
¼
0
;
(10.52)
while the former assertion is summarized in the statement (
P
E
-
P
I
)
0.
Guided by these results it is postulated that the internal entropy production is
always greater than or equal to zero
D
s
D
t
0. This postulate is a form of the second
law of thermodynamics; the postulate of irreversibility. The second law applied to
an object occupying a volume
O
may be stated as
D
s
N
ð
i
Þ
D
t
D
s
N
ð
e
Þ
D
t
D
s
N
¼
D
t
0. In terms
D
s
N
ð
e
Þ
D
t
D
s
N
ð
i
Þ
D
t
D
s
N
D
t
of
,
,
and
, the following terminology is customary: an equilibrium
D
s
N
ð
i
Þ
D
t
D
s
N
state is defined by
D
t
¼
0, a reversible process is characterized by
¼
0, an
D
s
N
ð
i
Þ
D
t
characterizes an
adiabatic process and only in the case of an adiabatically insulated system does
the second law of thermodynamics in the form
D
s
D
t
D
s
N
ð
i
Þ
D
t
>
D
s
N
irreversible process is characterized by
0
;
D
t
¼
0 apply. It should be noted
that, in general, the various entropies might satisfy the three inequalities
D
s
N
D
t
>
0,
D
s
N
ð
i
Þ
D
t
>
0
,
and
D
s
N
ð
e
Þ
D
t
0.
In terms of the specific or continuum variables, the second law of thermody-
namics
D
s
N
ð
i
Þ
D
t
<
D
s
N
ð
e
Þ
D
t
D
s
N
¼
D
t
0 may be written as
ð
O
r
þ
ð
D
s
D
t
h
n
r
r
y
d
v
þ
d
a
d
v
0
(10.53)
y
@O
O
using (
10.48
) and (
10.50
). In order to convert the integral equation (
10.53
) to a field
or point equation the divergence theorem (A183) is employed as well as the
argument that was used to convert the integral equation (3.4) to the field equation
(3.5). Recall that this was an argument employed four times in Chap.
3
. Applying
these arguments to (
10.53
) it follows that
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