Geology Reference
In-Depth Information
y
T
c
q
T
h
T
T
c
T
h
Figure 5.3. Conducted heat flow,
q
, is proportional to the temperature gradient.
Heat flows from hotter things to colder things. Experiments have established that
the rate of heat flow is proportional to the local temperature gradient. For example,
in Figure 5.3, the bottom of a layer of material is at a high temperature,
T
h
,andthe
top is at a cool temperature,
T
c
. Suppose the temperature varies linearly through
the layer, as depicted in the graph of
T
versus height
y
on the right. Suppose also
the layer thickness is
h
. Then the rate at which heat flows,
q
, will be
q
=−
K
(
T
c
−
T
h
)
/h.
(5.10)
I have written the temperature difference as the temperature at the larger value of
y
, minus the temperature at the smaller value of
y
, so that a positive temperature
gradient will correspond to a positive slope in the graph. The slope is actually
negative, yet the heat flows in the positive direction. Therefore the minus sign
is required in front, to ensure that heat flows from hot to cold. The constant of
proportionality,
K
,isthe
thermal conductivity
, another material property. Equation
(5.10) expresses the earlier statement that the rate of heat flow is proportional to the
local temperature gradient. It is sometimes known as Fourier's law. In Eq. (5.10),
q
is actually a heat
flux
, the rate at which heat flows through a unit area of the surface
of the material. Heat is a form of energy, so its units are joules, J. A
rate
of heat
flow is measured in joules per second, or watts, W. Thus the units of
q
are W/m
2
.
In order to deal with thermal boundary layers, we need to be able to consider how
temperatures change with time (in Figure 5.3 the temperature is steady, even though
heat is flowing through the layer). To do this, consider a hot body placed against
a cold boundary, as depicted in Figure 5.4(a). Initially the body has a uniform
temperature
T
m
and extends downwards from depth
z
=
0. The temperature at the
surface,
z
0. You can think of this as the temperature
profile through new oceanic crust and mantle that has just formed at a mid-ocean
ridge spreading centre: hot mantle wells up and comes into contact with cold
sea water. The temperature profile after contact is shown in Figure 5.4(b) by the
short-dashed line labelled
t
=
=
0, is maintained at
T
=
0.
The question we now address is how the temperature profile will change with
time. Heat will flow from the hot body through the cold boundary, so at a later
time
t
1
the temperature profile might be something like the solid line shown in
