Environmental Engineering Reference
In-Depth Information
Heat-Conduction Equations
We now derive the differential equation of heat conduction for a stationary, homo-
geneous, isotropic solid with heat generation within the body. Heat generation may
be due to unclear, electrical, chemical, gamma-ray, or other sources that may be
a function of time and/or position. The heat generation rate in the medium, gener-
ally specified as heat generation per unit time, per unit volume, is denoted by the
symbol
g
.
Consider an arbitrary volume element
V
of boundary surface
S
in a heat conduc-
tion domain
(
M
,
t
)
Ω
and a heat conduction process within an arbitrary time period
[
0
,
T
]
.
The first law of thermodynamics states that
heat entering
through
S
energy generation
in
V
change in storage
energy in
V
+
=
.
Various terms in this equation are evaluated as
T
T
First term in the left side
=
q
·
n
d
S
d
t
=
−
∇
·
q
d
V
d
t
0
0
S
V
k
T
0
=
Δ
u
d
V
d
t
,
(1.18)
V
T
Second term in the left side
=
g
(
M
,
t
)
d
V
d
t
,
(1.19)
0
V
T
c
∂
u
(
M
,
t
)
Term in the right side
=
ρ
d
V
d
t
.
(1.20)
∂
t
0
V
Here
n
is the outward-drawn normal unit vector to the surface element d
S
,and
ρ
and
c
are the density and the specific heat of material respectively. Note also that we
have used the divergence theorem to convert the surface integral to volume integral.
Substituting those into the energy-balance equation yields
T
k
d
V
d
t
c
∂
u
Δ
u
+
g
(
M
,
t
)
−
ρ
=
0
.
∂
t
0
V
By the continuity of the integrand and the localization theorem, we obtain
c
∂
u
ρ
t
=
k
Δ
u
+
g
(
M
,
t
)
,
M
∈
Ω
,
t
>
0
∂
or
a
2
u
t
=
Δ
u
+
f
,
(1.21)
where
a
2
=
k
/
(
ρ
c
)
,
f
=
g
/
(
ρ
c
)
.
Search WWH ::
Custom Search