Geoscience Reference
In-Depth Information
Thus equating the expressions (7.8) and (7.9) for the rate at which heat is
gained by the volume element gives
c
P
a
z
ρ
T
Aa
z
−
a
z
∂
Q
∂
z
t
=
c
P
ρ
T
t
=
A
−
∂
Q
∂
z
(7.10)
In the limiting case when
z
,
0, Eq. (7.10) becomes
c
P
ρ
∂
T
t
→
∂
t
=
A
−
∂
Q
(7.11)
∂
z
Using Eq. (7.4) for
Q
(heat flow per unit area), we can write
c
P
ρ
∂
T
k
∂
2
T
∂
z
2
∂
t
=
A
+
(7.12)
or
∂
T
∂
t
=
ρ
c
P
∂
k
2
T
∂
z
2
A
ρ
c
P
+
(7.13)
This is the one-dimensional heat-conduction equation.
In the derivation of this equation, temperature was assumed to be a func-
tion solely of time
t
and depth
z
.Itwas assumed not to vary in the
x
and
y
directions. If temperature were assumed to be a function of
x
,
y
,
z
and
t
,athree-
dimensional heat-conduction equation could be derived in the same way as this
one-dimensional equation. It is not necessary to go through the algebra again: we
can generalize Eq. (7.13)toathree-dimensional Cartesian coordinate system as
∂
T
∂
t
=
∂
∂
z
2
2
T
∂
x
2
2
T
∂
y
2
2
T
k
ρ
c
P
+
∂
+
∂
A
ρ
c
P
+
(7.14)
Using differential-operator notation (see Appendix 1), we write Eq. (7.14)as
∂
T
∂
t
=
k
ρ
c
P
∇
A
ρ
c
P
2
T
+
(7.15)
Equations (7.14) and (7.15) are known as the
heat-conduction equation
. The
term
k
/(
. Thermal diffusivity expresses
the ability of a material to lose heat by conduction. Although we have derived this
equation for a Cartesian coordinate system, we can use it in any other coordinate
system (e.g., cylindrical or spherical), provided that we remember to use the
definition of the Laplacian operator,
ρ
c
P
)isknown as the
thermal diffusivity
κ
2
∇
(Appendix 1), which is appropriate for
the desired coordinate system.
Forasteady-state situation when there is no change in temperature with time,
Eq. (7.15) becomes
A
k
2
T
∇
=−
(7.16)
In the special situation when there is no heat generation, Eq. (7.15) becomes
∂
T
∂
t
=
k
ρ
c
P
∇
2
T
(7.17)
This is the
diffusion equation
(Section 7.3.5).
