Civil Engineering Reference
In-Depth Information
so that we, in effect, look back in time to approximate the derivative during the
previous time step. Substituting this relation into Equation 7.116 gives
[
C
]
{
T
(
t
)
}−{
T
(
t
−
t
)
}
+
[
K
]
{
T
(
t
)
}={
F
Q
(
t
)
}+{
F
g
(
t
)
}
(7.123)
t
In this method, we evaluate the nodal temperatures at time
t
based on the state of
the system at time
t
−
t
, so we introduce the notation
t
=
t
i
,
t
i
−
1
=
t
−
t
,
i
=
1, 2, 3,
....
Using the described notation and rearranging, Equation 7.123
becomes
...
(7.124)
If the nodal temperatures are known at time
t
i
−
1
, Equation 7.124 can be solved
for the nodal temperatures at the next time step (it is assumed that the forcing
functions on the right-hand side are known and can be determined at
t
i
). Noting
that the time index is relative, Equation 7.125 can also be expressed as
([
C
]
+
[
K
]
t
)
{
T
(
t
i
)
}=
[
C
]
{
T
(
t
i
−
1
)
}+
F
Q
(
t
i
)
t
+
F
g
(
t
i
)
t
i
=
1, 2, 3,
([
C
]
+
[
K
]
t
)
{
T
(
t
i
+
1
)
}
...
(7.125)
If we compare Equation 7.125 with Equation 7.121, we find that the major dif-
ference lies in the treatment of the conductance matrix. In the latter case, the
effects of conductance are, in effect, updated during the time step. In the case of
the forward difference method, Equation 7.121, the conductance effects are held
constant at the previous time step. We also observe that Equation 7.125 cannot be
solved at each time step by “simply” inverting the capacitance matrix. The coef-
ficient matrix on the left-hand side changes at each time step; therefore, more
efficient methods are generally used to solve Equation 7.125.
Another approach to approximation of the first derivative is the central differ-
ence method. As the name implies, the method is a compromise of sorts between
forward and backward difference methods. In a central difference scheme, the
dependent variable and all forcing functions are evaluated at the center (midpoint)
of the time step. In other words, average values are used. In the context of transient
heat transfer, the time derivative of temperature is still as approximated by Equa-
tion 7.119 but the other terms in Equation 7.120 are evaluated at the midpoint of
the time step. Using this approach, Equation 7.120 becomes
=
[
C
]
{
T
(
t
i
)
} +
F
Q
(
t
i
)
t
+
F
g
(
t
i
)
t
i
=
0, 1, 2,
[
K
]
T
(
t
[
C
]
{
T
(
t
+
t
)
} − {
T
(
t
)
}
+
t
)
+
T
(
t
)
+
t
2
F
Q
(
t
F
g
(
t
+
t
)
+
F
Q
(
t
)
+
t
)
+
F
g
(
t
)
=
+
(7.126)
2
2
The forcing functions on the right-hand side of Equation 7.126 are either known
functions and can be evaluated or “reactions,” which are subsequently computed
