Biomedical Engineering Reference
In-Depth Information
fl uid is subject to convective motion when there is a temperature gap at its
extremes. The phenomenology considered is more complex, as appears in
Table 4.2.
The previous results justify the form of equation [4.11] as coupled
phenomena have a specifi c role and their entity depends on the value
of the mixed coeffi cient
L
lk
. However, a large number of coeffi cients is
inconvenient and much experimental work would be necessary to estimate
them independently. This may be avoided by using the symmetry law of
Onsager: 'The entire set of phenomenological coeffi cients forms a matrix
where reciprocity relations are valid'. in mathematical terms it can be
expressed thus:
L
lk
=
L
kl
[4.12]
This law (1931) was the fi rst attempt to use the thermodynamics of
irreversible processes as a useful instrument for the rationalisation of
physical-chemical systems. an example of the application of the above is the
Seebeck effect in the thermoelectric fi eld. If the extremities of a bimetallic
couple reach different temperatures, there will be a fl ow of electrical
current
I
caused by the potential difference.
Considering Table 4.1 and the
discontinuous system gives:
Dy
JL
T
T
D
[4.13]
L
+
JL
JJ
Q1
JL
J
Q1
JL
J
Q1
JL
1
Q1
12
2
T
Dy
IL
T
T
D
[4.14]
L
+
IL
IL
21
22
2
T
From these relations, the following may be obtained:
Ê
Á
ˆ
˜
Ê
Ë
ˆ
¯
J
I
D
D
L
L
L
LT
2
LT
22
y
Q
12
21
Ê
Ë
Ê
Ë
Ê
Á
Ê
Ë
and -
=
=
Ê
Ë
Ê
Ë
Ê
ˆ
¯
and -
Ë
Á
Ë
Ë
D
T
T
22
D
T=0
22
i=0
22
Using the Onsager law defi ned above (
L
12
=
L
21
):
Ê
Ê
ˆ
ˆ
J
I
Ê
ˆ
D
D
L
L
y
Q
21
[4.15]
Ê
T
= -
T
Ê
Á
Ê
Ê
ˆ
=
Ê
Ê
Ê
Ê
Á
Ê
Ê
ˆ
ˆ
Ë
Ë
Ë
Ë
Á
Ë
Ë
Ë
¯
¯
Ë
Ë
¯
T
22
i=0
D
T= 0
There exists therefore, a well-defi ned proportional relation between the
temperature and potential differences in the absence of an electric current.
Table 4.2
Phenomena involved for energy and mass fl ows
D
T
D
C
Energy fl ow
Mass fl ow
Heat conduction (Fourier)
Thermal diffusion (Soret)
Thermo diffusion (Dufour)
Diffusion (Fick)
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