Civil Engineering Reference
In-Depth Information
Btu/(hr-
◦
F
), and the complete element conductance matrix is
0
.
6327
−
0
.
1003
−
0
.
2585
−
0
.
1003
k
(
e
)
=
−
.
.
−
.
−
.
0
1003
0
6327
0
1003
0
2585
Btu/(hr-
◦
F
)
−
0
.
2585
−
0
.
1003
0
.
6327
−
0
.
1003
−
0
.
1003
−
0
.
2585
−
0
.
1003
0
.
6327
EXAMPLE 7.5
Figure 7.10a depicts a two-dimensional heating fin. The fin is attached to a pipe on its
left edge, and the pipe conveys water at a constant temperature of
180
◦
F
. The fin
is surrounded by air at temperature
68
◦
F
. The thermal properties of the fin are as given
in Example 7.4. Use four equal-size four-node rectangular elements to obtain a finite
element solution for the steady-state temperature distribution in the fin.
■
Solution
Figure 7.10b shows four elements with element and global node numbers. Given the
numbering scheme selected, we have constant temperature conditions at global nodes
1, 2, and 3 such that
180
◦
F
T
1
=
T
2
=
T
3
=
while on the other edges, we have convection boundary conditions that require a bit of
analysis to apply. For element 1 (Figure 7.10c), for instance, convection occurs along
element edge 1-2 but not along the other three element edges. Noting that
s
=−
1
and
6
3
9
2 in.
4
3
5
180
F
2 in.
68
F
2
8
1
2
1
7
4
(a)
(b)
2
5
5
8
1
2
1
4
4
7
(c)
(d)
Figure 7.10
Example 7.5:
(a) Two-dimensional fin. (b) Finite element model.
(c) Element 1 edge convection. (d) Element 2 edge
convection.
