Civil Engineering Reference
In-Depth Information
3
.
14(10
−
4
)m
2
. And the peripheral dimension (circumference) of each element is
P
=
(20
/
1000 )
=
6
.
28(10
−
2
)m
. The stiffness matrix for each element (note that all
elements are identical) is computed via Equation 7.69 as follows:
1
−
1
−
11
1
−
1
−
11
k
(
e
c
=
0
.
156(3
.
14)(10
−
4
)
0
k
x
A
L
=
.
025
1
.
9594
(10
−
3
)
−
1
.
9594
=
−
1
.
9594
1
.
9594
21
12
21
12
0
.
157
k
(
e
h
=
300(6
.
28)(10
−
2
)(0
.
025)
6
hPL
6
0
.
0785
=
=
0
.
0785
0
.
157
−
11
−
11
−
11
−
11
−
3
.
138
k
(
e
)
˙
m
=
mc
2
(0
.
2)(60)(0
.
523)
2
3
.
138
=
=
−
3
.
138
3
.
138
−
2
.
9810
k
(
e
)
=
3
.
2165
−
3
.
0595
3
.
2950
At this point, note that the mass transport effects dominate the stiffness matrix and we an-
ticipate that very little heat is dissipated, as most of the heat is carried away with the flow.
Also observe that, owing to the relative magnitudes, the conduction effects have been
neglected.
Assembling the global stiffness matrix via the now familiar procedure, we obtain
−
2
.
9810
3
.
2165
0
0
0
−
3
.
0595
0
.
314
3
.
2165
0
0
[
K
]
=
0
−
3
.
0595
0
.
314
3
.
2165
0
0
0
−
3
.
0595
0
.
314
3
.
2165
0
0
0
−
3
.
0595
3
.
2950
The convection-driven forcing function for each element per Equation 7.18 is
1
1
1
1
3
.
5325
3
.
5325
f
(
e
h
=
300(6
.
28)(10
−
2
)(15)(0
.
025)
2
hPT
a
L
2
=
=
As there is no internal heat generation, the per-element contribution of Equation 7.16 is
zero. Finally, we must examine the boundary conditions. At node 1, the temperature is
specified but the heat flux
q
1
=
F
1
is unknown; at node 5 (the exit), the flux is also un-
known. Unlike previous examples, where a convection boundary condition existed, here
we assume that the heat removed at node 5 is strictly a result of mass transport. Physi-
cally, this means we define the problem such that heat transfer ends at node 5 and the
heat remaining in the flow at this node (the exit) is carried away to some other process.
Consequently, we do not consider either a conduction or convection boundary condition
at node 5. Instead, we compute the temperature at node 5 then the heat removed at this
node via the mass transport relation. In terms of the finite element model, this means
