Biomedical Engineering Reference
In-Depth Information
d
n
i
d
U
4.1
A globally isolated system, comprising two subsystems divided
by a moveable wall permeable to a generic
i
component.
Ê
Á
ˆ
Ê
Ê
ˆ
(
(
(
1
(
2
1
1
P
T
P
(
(
(
1
(
1
d =
d
dU
d
Ê
Á
Ê
Ë
V
+
ir
d
ir
d
Ë
-
-
(
1
()
(
(
(
1
-
()
2
(
1
(
2
TT
TT
()
TT
1
(
1
TT
1
TT
1
TT
T
Ê
Á
+-
(
(
1
(
ˆ
(
2
m
m
i
mm
+
mm
+-
+-
i
mm
+-
i
()
mm
() ()
m
()
(
)
()
1
()
()
(
()
i
(
()
(
()
[4.1]
Ê
Á
Ê
d
=
XU
XU
d
XU
U
d +d +d
+
XV
+d
V
XV
+d
+
XV
+d
+d
XV
+
Xn
+d
i
Xn
+d
+
Xn
+d
+d
Xn
++-
+-
i
Á
+-
n
U
U
XU
d
d
d
+-
(
+-
+-
(
+-
1
(
i
i
(
TT
(
TT
(
1
TT
1
TT
(
TT
1
TT
(
2
U
V
i
Ë
TT
where
S
is the irreversible entropy,
T
the temperature,
P
the pressure,
U
the
internal energy,
V
the volume and
m
the chemical potential. The
X
k
above
are defi ned as thermodynamic forces. If the time unit is considered, it is
possible to obtain:
(
(
1
(
(
(
1
(
1
d
d
d
S
t
d
U
d
V
n
ir
i
Â
X
U
X
U
X
V
X
V
X
i
X
i
X
J
+
X
VV
X
VV
X
+=
X
+=
ii
X
i
+=
i
+=
X
+=
i
+=
X
kk
X
kk
+
+
=
XJ
+
+=
+
+=
+
=
U
V
i
UU
XJ
UU
UU
VV
i
+=
i
+=
k
kk
d
d
d
t
t
t
UU
X
UU
UU
[4.2]
where
J
k
indicates the fl ow of the different extensive quantities present. In
particular, three different types of fl ows may be defi ned:
J
U
= internal energy
fl ow,
J
V
= volume fl ow,
J
i
=
i
component fl ow.
Using a pure thermal energy fl ow (heat),
J
Q
expressed as follows could
be advantageous:
[4.3]
JJHJ
QU
JJ
=
JJ
QU
JJ
=
J
QU
JJ
-
J
QU
JJ
ii
H
ii
HJ
The global fl ow of energy between two subsystems contains an associated
contribution to the molecular fl ow, that for d
n
i
transferred moles are equal to
H
i
d
n
i
. So the fl ow of pure thermal energy can be obtained by the subtraction
between the fl ows of global energy
J
U
and
H
i
d
n
i
. The speed of entropy
creation is expressed by the addition of the products of existing fl ows in the
system in respect to their associated thermodynamic forces. equation [4.2],
referring to a discontinuous system, is constituted of different subsystems,
in each of which, the values of the intensive quantities are uniform. The
extension to continuous systems in which variables differ in several points
exhibits a considerable diffi culty: the calculation of entropy in the non-
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