Biomedical Engineering Reference
In-Depth Information
because it does not describe the relationship between two substates. Given two
thermodynamic substates {
E
,
r
(
a
)
} and {
E*
,
r
(
a
)
*
}, the knowledge is not sufficient
to specify whether a transition {
E
,
r
(
a
)
*
} is possible or not. This
ordering of thermodynamic substates is accomplished by the introduction of
entropy; the steps of this introduction we develop in the following paragraph.
A transition {
E
,
r
(
a
)
}
!
{
E*
,
r
(
a
)
}
!
{
E*
,
r
(
a
)
*
} between thermodynamic substates is simply
an ordered pair {
E
,
r
(
a
)
}, {
E*
,
r
(
a
)
*
} of thermodynamic substates. If the transitions
{
E
,
r
(
a
)
} are both possible, the transi-
tion is said to be
reversible
. If the transition {
E
,
r
(
a
)
}
!
{
E*
,
r
(
a
)
*
} is possible, but
{
E*
,
r
(
a
)
*
}
!
{
E
,
r
(
a
)
} is not, then the transition is said to be
irreversible
. The
directionality of transitions may be expressed in the following axiom of ordering:
Given two thermodynamic substates {
E
,
r
(
a
)
}
!
{
E*
,
r
(
a
)
*
} and {
E*
,
r
(
a
)
*
}
!
{
E
,
r
(
a
)
} and {
E*
,
r
(
a
)
*
}, it is possible to
decide whether the transition {
E
,
r
(
a
)
*
} is possible or not.
The existence of an ordering of substates is analogous to the existence of the
“greater than” relation “
r
(
a
)
}
!
{
E*
,
” for real numbers. For example, given two distinct real
numbers
a
and
a
* we have either
a
>
a
* while no such ordering
holds for, say, the complex numbers. This permits the construction of a homeomor-
phism between the ordering of thermodynamic states and the ordering of real
numbers. This is done by introducing a real-valued function
>
a
*,
a
*
>
a
or
a
¼
which assigns a
real number to each thermodynamic substate
¼
(
E
,
r
(
a
)
), in such a way that
r
(
a
)
*
} is irreversible.
The substates are thereby ordered and labeled by means of the real-valued function
of the substate,
(
E*
,
r
(
a
)
*
)
>
(
E
,
r
(
a
)
) if the transition {
E
,
r
(
a
)
}
!
{
E*
,
. In any particular physical situation such a function is empirically
determinable, and if
)
, where
f
is a monotoni-
cally increasing function of its argument. Assuming a particular function
is one such function, then so is
f(
to have
been chosen, it is called the empirical entropy of the thermodynamic system.
The thermodynamic state of a particle
X
in an object is completely specified by
the thermodynamic substate {
E
,
r
(
a
)
}and the entropy
of the (RVE associated with
the) particle. The basic assumption of thermodynamics is that the thermodynamic
state completely determines the (specific) internal energy
independent of time,
place, motion and stress, thus
e ¼ e
(
,
E
,
r
(
a
)
,
X
). Choice of the exact functional
form of
e
defines different thermodynamic substances. If
X
does not appear in the
form of
e
chosen, the substance is said to be thermodynamically simple.
In order to develop a rationale for a differential equation in entropy production
e
,
the time rate of change of internal energy
e
in the conservation of energy (3.52) is
expressed in two different ways:
D
s
e
D
t
¼
r ¼
P
E
þ
Q
E
¼
P
I
þ
Q
I
;
(10.43)
where
P
E
and
Q
E
denote the contribution of the external mechanical and nonme-
chanical power and
P
I
and
Q
I
denote the division into internal mechanical and
nonmechanical power, respectively. The external quantities are defined, using
(3.50) and (3.52), by
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