Civil Engineering Reference
In-Depth Information
The model assembly procedure for a transient heat transfer problem is ex-
actly the same as for a steady-state problem, with the notable exception that we
must also assemble a global capacitance matrix. The rules are the same. Element
nodes are assigned to global nodes and the element capacitance matrix terms are
added to the appropriate global positions in the global capacitance matrix, as
with the conductance matrix terms. Hence, on system assembly, we obtain the
global equations
{
T
(7.116)
where we must recall that the gradient force vector
{
F
g
}
is composed of either
(1) unknown heat flux values to be determined (unknown reactions) or (2) con-
vection terms to be equilibrated with the flux at a boundary node.
[
C
]
}+
[
K
]
{
T
}={
F
Q
}+{
F
g
}
7.8.1 Finite Difference Methods for the Transient
Response: Initial Conditions
The finite element discretization procedure has reduced the one-dimensional
transient heat transfer problem to algebraic terms in the spatial variable via the
interpolation functions. Yet Equation 7.116 represents a set of ordinary, coupled,
first-order differential equations in time. Consequently, as opposed to the steady-
state case, there is not
a
solution but multiple solutions as the system responds
to time-dependent conditions. The boundary conditions for a transient problem
are of the three types discussed for the steady-state case: specified nodal tem-
peratures, specified heat flux, or convection conditions. However, note that the
boundary conditions may also be time dependent. For example, a specified nodal
temperature could increase linearly with time to some specified final value. In
addition, an internal heat generation source
Q
may also vary with time.
A commonly used approach to obtaining solutions for ordinary differential
equations of the form of Equation 7.116 is the
finite difference method
. As dis-
cussed briefly in Chapter 1, the finite difference method is based on approximat-
ing derivatives of a function as incremental changes in the value of the function
corresponding to finite changes in the value of the independent variable. Recall
that the first derivative of a function
f
(
t
)
is defined by
d
f
d
t
=
f
(
t
+
t
)
−
f
(
t
)
f
=
lim
t
→
0
(7.117)
t
Instead of requiring
t
to approach zero, we obtain an approximation to the
value of the derivative by using a small, nonzero value of
t
to obtain
f
(
t
+
t
)
−
f
(
t
)
f
=
(7.118)
t
and the selected value of
t
is known as the
time step
.
To apply the procedure to transient heat transfer, we approximate the time
derivative of the nodal temperature matrix as
{
T
(
t
+
t
)
} − {
T
(
t
)
}
{
T
} =
(7.119)
t
