Civil Engineering Reference
In-Depth Information
Substituting into Equation 7.15, the element conductance matrix is
9630
1
0909
21
12
2
.
1448
k
(
e
)
=
−
1
−
1
.
8721
1
.
+
0
.
=
−
−
.
.
11
1
8721
2
1448
Following the direct assembly procedure, the system conductance matrix is
2
.
1448
−
1
.
8721
0
0
0
−
1
.
8721
4
.
2896
−
1
.
8721
0
0
[
K
]
=
0
−
1
.
8721
4
.
2896
−
1
.
8721
0
0
0
−
1
.
8721
4
.
2896
−
1
.
8721
0
0
0
−
1
.
8721
2
.
1448
As no internal heat is generated,
f
Q
=
0
. The element convection force components per
Equation 7.18 are
50
1
.
5708
12
(72)
1
12
1
1
19
Btu/hr
f
(
e
h
=
hPT
a
L
2
.
6375
=
=
19
.
6375
2
Assembling the contributions of each element at the nodes gives the system convection
force vector as
19
.
6375
39
.
2750
39
.
2750
39
.
2750
19
.
6375
{
F
h
}=
Btu/hr
Noting the cancellation of terms at nodal connections, the system gradient vector
becomes simply
Aq
1
0
0
0
−
Aq
5
Aq
1
0
0
0
−
Ah
water
(
T
5
−
40)
Aq
1
0
0
0
−
0
.
1364
T
5
+
5
.
4542
{
F
g
}=
=
=
Btu/hr
and the boundary condition at the pin-water interface has been explicitly incorporated.
Note that, as a result of the convection boundary condition, a term containing unknown
nodal temperature
T
5
appears in the gradient vector. This term is transposed in the final
equations and results in a increase in value of the
K
55
term of the system matrix. The final
assembled equations are
2
.
1448
−
1
.
8721
0
0
0
180
T
2
T
3
T
4
T
5
19
.
6375
+
Aq
1
39
.
2750
39
.
2750
39
.
2750
25
.
0917
−
1
.
8721
4
.
2896
−
1
.
8721
0
0
=
0
−
1
.
8721
4
.
2896
−
1
.
8721
0
0
0
−
1
.
8721
4
.
2896
−
1
.
8721
0
0
0
−
1
.
8721
2
.
2812
