Civil Engineering Reference
In-Depth Information
h
(
T
T
a
)
S
3
T
T
*
S
1
S
2
q
*
Figure 7.8
Types of boundary
conditions for two-dimensional
conduction with convection.
constant value
T
S
1
=
T
∗
.
In a finite element model of such a domain, every ele-
ment node located on
S
1
has known temperature and the corresponding nodal
equilibrium equations become “reaction” equations. The reaction “forces” are
the heat fluxes at the nodes on
S
1
. In using finite element software packages, such
conditions are input data; the user of the software (“FE programmer”) enters
such data as appropriate at the applicable nodes of the finite element model (in
this case, specified temperatures).
The heat flux on portion
S
2
of the boundary is prescribed as
q
S
2
=
q
∗
.
This
is analogous to specified nodal forces in a structural problem. Hence, for all ele-
ments having nodes on
S
2
, the third of Equation 7.38 gives the corresponding
nodal forcing functions as
f
(
e
g
=−
q
∗
n
S
2
{
N
}
t
d
S
(7.39)
S
2
Finally, a portion
S
3
of the boundary illustrates an edge convection condi-
tion. In this situation, the heat flux at the boundary must be equilibrated by the
convection loss from
S
3
. For all elements having edges on
S
3
, the convection
condition is expressed as
f
(
e
g
=−
h
(
T
(
e
)
q
S
3
n
S
3
{
N
}
t
d
S
=−
−
T
a
)
{
N
}
t
d
S
(7.40)
S
3
S
3
Noting that the right-hand side of Equation 7.40 involves the nodal temper-
atures, we rewrite the equation as
f
(
e
g
=−
h
[
N
]
T
[
N
]
{
T
}
t
d
S
3
+
hT
a
{
N
}
t
d
S
3
(7.41)
S
3
S
3
and observe that, when inserted into Equation 7.36, the first integral term on the
right of Equation 7.41 adds stiffness to specific terms of the conductance matrix
