Civil Engineering Reference
In-Depth Information
7.6 HEAT TRANSFER IN THREE DIMENSIONS
As the procedure has been established, the governing equation for heat transfer
in three dimensions is not derived in detail here. Instead, we simply present the
equation as
k
x
∂
k
y
∂
k
z
∂
∂
∂
T
∂
∂
T
∂
∂
T
∂
+
+
+
Q
=
0
(7.70)
x
∂
x
y
∂
y
z
z
and note that only conduction effects are included and steady-state conditions are
assumed. In the three-dimensional case, convection effects are treated most effi-
ciently as boundary conditions, as is discussed.
The domain to which Equation 7.70 applies is represented by a mesh of finite
elements in which the temperature distribution is discretized as
M
T
(
x
,
y
,
z
)
=
N
i
(
x
,
y
,
z
)
T
i
=
[
N
]
{
T
}
(7.71)
i
=
1
where
M
is the number of nodes per element. Application of the Galerkin method
to Equation 7.70 results in
M
residual equations:
∂
∂
k
x
∂
k
y
∂
k
z
∂
Q
N
i
d
V
T
∂
∂
T
∂
∂
∂
T
∂
+
+
+
=
0
x
∂
x
y
y
z
z
V
i
=
1,
...
,
M
(7.72)
where, as usual,
V
is element volume.
In a manner analogous to Section 7.4 for the two-dimensional case, the
derivative terms can be written as
∂
∂
k
x
∂
N
i
k
x
∂
N
i
T
∂
∂
T
k
x
∂
T
∂
N
i
∂
=
−
∂
∂
∂
x
x
x
x
x
x
k
y
∂
N
i
k
y
∂
N
i
∂
∂
T
∂
∂
T
k
y
∂
T
∂
N
i
∂
=
−
(7.73)
y
∂
y
y
∂
y
∂
y
y
k
z
∂
N
i
k
z
∂
N
i
∂
∂
T
∂
∂
∂
T
∂
k
z
∂
T
∂
∂
N
i
∂
=
−
z
z
z
z
z
z
and the residual equations become
∂
∂
k
x
∂
N
i
k
y
∂
N
i
k
z
∂
N
i
d
V
T
∂
∂
∂
T
∂
∂
T
∂
+
+
+
QN
i
d
V
x
x
y
∂
y
z
z
V
V
k
x
∂
d
Vi
T
∂
N
i
∂
k
y
∂
T
∂
y
∂
N
i
∂
k
z
∂
T
∂
∂
N
i
∂
=
x
+
y
+
=
1,
...
,
M
(7.74)
∂
x
z
z
V
The integral on the left side of Equation 7.74 contains a perfect differential
in three dimensions and can be replaced by an integral over the surface of the
volume using Green's theorem in three dimensions:
If F
(
x
,
y
,
z
),
G
(
x
,
y
,
z
),
and
