Civil Engineering Reference
In-Depth Information
The element gradient boundary conditions, using Equation 5.55, described by
x
1
−
d
T
d
x
A
q
|
x
1
{
f
g
}=
k
x
A
x
2
=
(5.70)
d
T
d
x
−
q
|
x
2
are such that, at internal nodes where elements are joined, the values for the
adjacent elements are equal and opposite, cancelling mathematically. At external
nodes, that is, at the ends of the body being analyzed, the gradient values may be
specified as known heat flux input and output or computed if the specified bound-
ary condition is a temperature. In the latter case, the gradient computation is
analogous to computing reaction forces in a structural model. Also note that the
area is a common term in the preceding equations and, since it is assumed to
be constant over the element length, could be ignored in each term. However, as
will be seen in later chapters when we account for other heat transfer conditions,
the area should remain in the equations as defined. These concepts are illustrated
in the following example.
EXAMPLE 5.6
The circular rod depicted in Figure 5.9 has an outside diameter of 60 mm, length of 1 m,
and is perfectly insulated on its circumference. The left half of the cylinder is aluminum,
for which
k
x
=
200 W/m-°C and the right half is copper having
k
x
=
389 W/m-°C. The
extreme right end of the cylinder is maintained at a temperature of 80
°
C, while the left
end is subjected to a heat input rate 4000 W/m
2
. Using four equal-length elements, deter-
mine the steady-state temperature distribution in the cylinder.
■
Solution
The elements and nodes are chosen as shown in the bottom of Figure 5.9. For aluminum
elements 1 and 2, the conductance matrices are
1
−
1
−
11
1
−
1
−
11
=
2
.
26
1
−
1
−
11
W/
◦
C
200(
/
4)(0
.
06)
2
0
.
25
k
x
A
L
[
k
al
]
=
=
Al
Cu
q
in
q
out
1
1
2
2
3
3
4
4
5
0.25 m
0.25 m
0.25 m
0.25 m
Figure 5.9
Circular rod of Example 5.6.
