Civil Engineering Reference
In-Depth Information
substitute the nodal temperatures to obtain estimates of heat transfer properties
of individual elements as well.
The heat flux components for a two-dimensional element, per Fourier's law,
are
M
T
(
e
)
∂
k
x
∂
∂
N
i
∂
q
(
e
)
x
T
(
e
)
i
=−
=−
k
x
x
x
i
=
1
(7.47)
M
T
(
e
)
∂
k
y
∂
∂
N
i
∂
q
(
e
)
y
T
(
e
)
i
=−
=−
k
y
y
y
i
=
1
where we again denote the total number of element nodes as
M
. With the excep-
tion of the three-node triangular element, the flux components given by Equa-
tion 7.47 are not constant but vary with position in the element. As an example,
the components for the four-node rectangular element are readily computed
using the interpolation functions of Equation 6.56, repeated here as
1
4
(1
N
1
(
r
,
s
)
=
−
r
)(1
−
s
)
1
4
(1
N
2
(
r
,
s
)
=
+
r
)(1
−
s
)
(7.48)
1
4
(1
N
3
(
r
,
s
)
=
+
r
)(1
+
s
)
1
4
(1
=
−
+
N
4
(
r
,
s
)
r
)(1
s
)
Recalling that
∂
∂
1
a
∂
∂
∂
∂
1
b
∂
∂
x
=
and
y
=
r
s
we have
4
k
x
a
∂
N
i
∂
q
(
e
)
x
T
(
e
)
i
=−
r
i
=
1
4
a
(
s
s
)
T
(
e
4
k
x
1)
T
(
e
)
1
s
)
T
(
e
)
2
s
)
T
(
e
)
3
=−
−
+
(1
−
+
(1
+
−
(1
+
(7.49)
4
k
y
b
∂
N
i
∂
q
(
e
)
y
T
(
e
)
i
=−
s
i
=
1
4
b
(
r
r
)
T
(
e
4
k
y
1)
T
(
e
)
1
r
)
T
(
e
)
2
r
)
T
(
e
3
=−
−
−
(1
+
+
(1
+
+
(1
−
