Geoscience Reference
In-Depth Information
Figure 7.2.
A volume
element of height
z
and
cross-sectional area
a
.
Heat is conducted into
and out of the element
across the shaded faces
only. We assume that
there is no heat transfer
across the other four
faces.
aQ(z)
z
z
z
+δ
aQ(z+
δ
z)
The heat per unit time entering the volume across its face at
z
is
aQ
(
z
), whereas
the heat per unit time leaving the element across its face at
z
+
z
is
aQ
(
z
+
z
).
Expanding
Q
(
z
+
z
)inaTaylor series gives
Q
(
z
+
z
)
=
Q
(
z
)
+
z
∂
Q
(
z
)
2
2
∂
2
Q
∂
z
2
∂
z
+
+ ···
(7.5)
z
)
2
term and those of higher order are very small and
can be ignored. From Eq. (7.5) the net gain of heat per unit time is
In the Taylor series, the (
heat entering across z
−
heat leaving across
z
+
z
=
aQ
(
z
)
−
aQ
(
z
+
z
)
=−
a
z
∂
Q
∂
z
(7.6)
Suppose that heat is generated in this volume element at a rate
A
per unit volume
per unit time. The total amount of heat generated per unit time is then
Aa
z
(7.7)
Radioactive heat is the main internal heat source for the Earth as a whole; however,
local heat sources and sinks include radioactive heat generation (Section 7.2.2),
latent heat, shear heating and endothermic and exothermic chemical reactions.
Combining expressions (7.6) and (7.7)gives the total gain in heat per unit time
to first order in
z
as
Aa
z
−
a
z
∂
Q
∂
z
(7.8)
The
specific heat c
P
of the material of which the volume is made determines
the rise in temperature due to this gain in heat since specific heat is defined as
the amount of heat necessary to raise the temperature of 1 kg of the material by
1
◦
C
. Specific heat is measured in units of W kg
−
1
◦
C
−
1
.
If the material has density
ρ
and specific heat
c
P
, and undergoes a temperature
increase
T
in time
t
, the rate at which heat is gained is
c
P
a
z
ρ
T
t
(7.9)
