Civil Engineering Reference
In-Depth Information
70
T
4
60
50
T
3
40
T
2
30
20
5
10
15
20
25
t
, sec
Figure 7.21
Time histories of the nodal
temperatures.
where [
C
] now represents the reduced
3
×
3
capacitance matrix. Utilizing Equation 7.121
and multiplying by
[
C
]
−
1
yields
i
+
1
=
i
−
i
t
+
t
T
2
T
3
T
4
T
2
T
3
T
4
0
.
4988
−
0
.
3521
0
.
0880
T
2
T
3
T
4
18
.
0456
−
0
.
3521
0
.
5869
−
0
.
3521
−
4558
7
.
7762
6
.
0
.
0880
−
0
.
3521
0
.
4988
as the two-point recurrence relation.
Owing to the small matrix involved, the recurrence relation was programmed into a
standard spreadsheet program using time step
1
sec. Calculations for nodal tem-
peratures
T
2
,
T
3
, and
T
4
are carried out until a steady state is reached. Time histories of
each of the nodal temperature are shown in Figure 7.21. The figure shows that steady-
state conditions
T
2
=
t
=
0
.
5
◦
C
are attained in about 30 sec.
Interestingly, the results also show that the temperatures of nodes 3 and 4 initially
decrease. Such phenomena are physically unacceptable and associated with use of a con-
sistent capacitance matrix, as is discussed in Chapter 10.
67
.
5
◦
C
,
T
3
=
55
◦
C
, and
T
4
=
42
.
7.8.2 Central Difference and Backward
Difference Methods
The forward difference method discussed previously and used in Example 7.11
is but one of three commonly used finite difference methods. The others are the
backward difference method and the central difference method. Each of these is
discussed in turn and a single two-point recurrence relation is developed incor-
porating the three methods.
In the
backward difference method,
the finite approximation to the first
derivative at time
t
is expressed as
−
−
T
(
t
)
T
(
t
t
)
T
(
t
)
=
(7.122)
t
