Environmental Engineering Reference
In-Depth Information
3.3.2 Fourier's Law and Heat Transport
In this chapter energy is understood as heat energy throughout. The main aim is to
derive the fundamental equations for heat transport, which are based on the energy
conservation principle. Other forms of energy, for example the energy consumed or
produced in reactions or by phase transitions, may be relevant in certain cases, but
will not be treated here.
In analogy to mass conservation, the conservation of thermal energy can be
stated in the following form:
ðrCÞ
@
@t
T ¼r
j
e
þ q
e
(3.21)
where (
temperature)](often:
[J (
K)
1
m
3
]) and j
e
the heat flux in [energy/(area
time)] (often [Watt/area]).
The energy sink or source
q
e
, as well as the entire differential equation, measure the
volumetric energy rate in the physical unit [energy/(volume
rC
) denotes heat capacity in [energy/(volume
time)].
5
Some values of
heat capacities are listed in Table
3.1
.
The left side describes the storage of heat, while in the first term on the right side
mass differences are expressed through the spatial change of heat fluxes. The coeffi-
cient on the left side relates energy storage in form of heat due to temporal tempera-
ture change to the mass. The heat capacity
C
is the expression of the energy - mass
relationship, while specific heat capacity
rC
is the expression for energy - volume
relation. At first instance, (
3.21
) is a formulation for pure phases. In porous media as
a two-phase system, either storage and fluxes can be added weighted by their relative
volumetric share, expressed in terms of porosity:
yrð
f
@
Þ rð
s
@
@t
T
f
þ
ð
1
y
@t
T
s
¼r
j
e
þ q
e
(3.22)
In (
3.22
) both phases may have different temperatures:
T
f
in the pore water,
T
s
in
the sediment. Usually heat transport is slow in relation to interphase heat transfer,
i.e. the heat exchange between fluid and solid phase is fast and as a result
temperatures in the two phases are the same:
T
s
¼ T
f
¼ T
. Concerning the long-
term development of non-oscillating thermal regimes, as it is mostly met in field
situations, the assumption of one temperature level is true. Then holds:
h
i
@
@t
T ¼r
yrð
f
þ
ð
1
y
Þ rð
s
j
e
þ q
e
(3.23)
5
The given formulation is a simplified version already. The left side can be derived by using the
following formula for internal energy:
e ¼ e
0
þ
R
c
v
dT
with specific heat
c
v
. The given formulation
is also valid for incompressible media under nearly constant pressure (H
afner et al.
1992
).
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