Heat - The Solid Earth - page 270

Figure 7.1. Conductive
transfer of heat through
an infinitely wide and
long plate z in thickness.
Heat flows from the hot
side of the slab to the cold
side (in the negative z
direction).

Flow of Heat

cold

z T

z + δ z

T +δ T

hot

V

z

(W m − 1 ◦ C − 1 ). 1 Thermal conductivities of solids vary widely: 418 W m − 1 ◦ C − 1
for silver; 159 W m − 1 ◦ C − 1 for magnesium; 1.2 W m − 1 ◦ C − 1

for glass; 1.7-

3.3 W m − 1 ◦ C − 1 for rock; and 0.1 W m − 1 ◦ C − 1 for wood.
To express Eq. (7.2)asadifferential equation, let us assume that the tempera-
ture of the upper surface (at z )is T and that the temperature of the lower surface
(at z + z )is T + T (Fig. 7.1). Substituting these values into Eq. (7.2) then gives

k T + T − T
z

Q ( z ) =− (7.3)

In the limit as z → 0, Eq. (7.3)iswritten

Q ( z ) =− k ∂ T
∂ z

(7.4)

The minus sign in Eq. (7.4) arises because the temperature is increasing in the
positive z direction (i.e., downwards); since heat flows from a hot region to a cold
region, it flows in the negative z direction (i.e., upwards).
If we consider Eq. (7.4)inthe context of the Earth, z denotes depth beneath
the surface. Since z increases downwards, a positive temperature gradient (tem-
perature increases with depth) means that there is a net flow of heat upwards out
of the Earth. Measurement of temperature gradients and thermal conductivity in
near-surface boreholes and mines can provide estimates of the rate of loss of heat
from the Earth.
Consider a small volume of height z and cross-sectional area a (Fig. 7.2).

Any change in temperature T of this small volume in time t depends on

1. the flow of heat across the volume's surface (net flow is in or out),
2. the heat generated in the volume and
3. the thermal capacity (specific heat) of the material.

1

Until fairly recently, the c.g.s. system was used in heat-flow work. In that system, 1 hgu
(heat-generation unit) 10 − 13 cal cm − 3 s − 1 10 − 7 Wm − 3 ;1hfu (heat-flow unit)

= = 4.2 × =

10 − 6 cal cm − 2 s − 1 10 − 2 Wm − 2 ; and thermal conductivity, 0.006 cal cm − 1 s − 1 ◦ C − 1 = 4.2 × =

2.52 W m − 1 ◦ C − 1 .