Civil Engineering Reference
In-Depth Information
and for the copper portion (element 2),
=
−
81
5
.
1327
−
5
.
8660
0
.
7332
7
k
(2)
=
389(
/
4)(0
.
006)
2
3(0
.
5)
−
−
−
.
.
−
.
8 6
8
5
8660
11
7320
5
8660
1
−
87
0
.
7332
−
5
.
8660
5
.
1327
At the internal nodes of each element, the flux terms are zero, owing to the nature of the
interpolation functions
[
N
2
(
x
1
)
=
N
2
(
x
2
)
=
0]
. Similarly, at the junction between the
two elements, the flux must be continuous and the equivalent “forcing” functions are
zero. As no internal heat is generated,
Q
=
0
, that portion of the force vector is zero for
each element. Following the direct assembly procedure, the system conductance matrix
is found to be
2
.
6389
−
3
.
0159
0
.
3770
0
0
−
.
.
−
.
3
0159
6
0319
3
0159
0
0
W/
◦
C
[
K
]
=
0
.
3770
−
3
.
0159
7
.
7716
−
5
.
8660
0
.
7332
0
0
−
5
.
8660
11
.
7320
−
5
.
8660
0
0
0
.
7332
−
5
.
8660
5
.
1327
and we note in particular that “overlap” exists only at the juncture between elements. The
gradient term at node 1 is computed as
x
1
=
q
1
A
=
4000
4
(0
f
g
1
=−
k
x
A
d
T
d
x
06)
2
.
=
11
.
3097 W
while the heat flux at node 5 is an unknown to be calculated via the system equations.
The system equations are given by
11
.
3097
0
0
0
2
.
6389
−
3
.
0159
0
.
3770
T
1
T
2
T
3
T
4
80
0
0
−
.
.
−
.
3
0159
6
0319
3
0159
0
0
=
0
.
3770
−
3
.
0159
7
.
7716
−
5
.
8660
0
.
7332
0
0
−
5
.
8660
11
.
7320
−
5
.
8660
−
Aq
5
0
0
0
.
7332
−
5
.
8660
5
.
1327
Prior to solving for the unknown nodal temperatures
T
1
-
T
4
, the nonhomogeneous bound-
ary condition
T
5
=
80
◦
C
must be accounted for properly. In this case, we reduce the sys-
tem of equations to
4
×
4
by transposing the last term of the third and fourth equations
to the right-hand side to obtain
2
.
3689
−
3
.
0159
0
.
3770
11
.
3907
0
0
T
1
T
2
T
3
T
4
−
.
.
−
.
3
0159
6
0319
3
0159
0
=
0
.
3770
−
3
.
0159
7
.
7716
−
5
.
8660
−
0
.
7332(80)
0
0
−
5
.
8660
11
.
7320
5
.
8660(80)
11
.
3097
0
=
−
58
.
6560
489
.
2800
