Geology Reference
In-Depth Information
However, any change in qz is related to a change in / through D q
ðÞ¼
qcD/
;
where c is a generalized specific capacity (capacity per unit mass) of the medium
and qc is the generalized volumetric capacity or amount of Z per unit volume that
can be accommodated in the medium per unit increase in /; thus we can write
o
q
ðÞ
ot
¼
qc
o
/
ot
ð
3
:
20
Þ
Combining (
3.18
), (
3.19
), and (
3.20
) leads to the transient-state equation
o
/
ot
¼
1
o
ox
k
o
/
ox
ð
3
:
21
Þ
qc
If k can be taken as constant over the range of / concerned, (
3.20
) becomes
2
/
ox
2
o
/
ot
¼
a
o
where a
¼
k
qc
ð
3
:
22
Þ
The parameter a can be called a ''generalized diffusivity'' in analogy to the thermal
diffusivity in transient heat flow, for which (
3.22
) will be recognized as the gov-
erning equation when we put /
¼
T . Since (
3.21
) is homogeneous in / the gen-
eralized diffusivity always has the same dimensions (SI units m
2
s
-1
) regardless of
the particular transport process concerned.
Alternatively to the above thermodynamic treatment of transport processes,
there is the empirical or experimental approach, as mentioned in
Sect. 3.2.2
. In this
approach, the flux is related to a conveniently measurable quantity as a proxy for
the gradient of the thermodynamic potential. Fick's law for the treatment of dif-
fusion (
Sect. 3.5.2
) is an example of the empirical approach; the diffusive flux
density j is related to the gradient in concentration c
N
of the diffusing species, thus:
j
¼
D
dc
N
dx
where D is the diffusion coefficient.
3.5 Atomic Diffusion
3.5.1 General
A transport process of particular importance at relatively high temperatures is the
diffusion of atoms, or small groups of strongly bonded atoms, in a matrix of the same
or different material. The topic will be treated briefly both at the phenomological
level and in terms of atomic theory. For general references, see Shewman (
1989
,
