EMA 405, Fall 2003
Solutions for HW #6
1.
The suggested preliminary analysis models the tapered fitting as a straight cylinder with inner
and outer radii determined by the midplane dimensions of the fitting,
r
i
=1.5 cm and
r
o
=2.25 cm.
The temperature distribution resulting from an analytic solution of the heat conduction problem
for the approximate geometry is given in the assignment,
()
−
+
=
o
i
o
o
i
o
r
r
r
r
T
T
T
r
T
ln
ln
)
(
,
where
T
i
=393 K and
T
o
=313 K for our calculation.
At
r
=1.75 cm and 2.0 cm, the predicted
temperatures are 362.59 K and 336.24 K, respectively.
For the FEA, I used quadrilateral elements with quadratic interpolation functions, i.e. PLANE77
for the ANSYS thermal analysis.
My initial mesh was a very coarse 1x4 mesh, and the
resolution was manually increased to 2x8, 3x12, and 6x24.
From Fig. 1.1, we can see that the
temperature distribution across the midplane converges very quickly (some midside node data is
plotted via “FULL” graphics), and if the quadratic interpolation of the basis functions were used
in the plots, we would see very impressive accuracy even with the coarsest mesh.
This direct
comparison with fine mesh results and with the preliminary analysis shows that at this axial
position, the temperatures are accurate to fractions of a percent in the initial coarse mesh
computation.
If we look at other axial positions, however, we can see that the convergence is not rapid
throughout the domain.
The zero heat flux boundary condition at the
z
directed top and bottom
surfaces imply that the temperature gradient should be in the
r
direction only at these locations.
This condition competes with the imposed temperature distributions along the inner and outer
surfaces, which are not oriented vertically.
The competition between adjacent boundaries leads
to temperature distributions in the corners of the
rz
cross section that are difficult to resolve, like
stress concentrations in structural analysis.
Vertical heat flux of order 3% of the radial heat flux
is evident at the top and bottom of the fitting, as shown in Fig. 1.2.
Zooming in to the corners
shows vertical heat flux values that are ten times larger, but these fluxes are confined to regions
that decrease in size as the mesh is resolved.
Again, the slow convergence is similar to what is
observed in stress concentration problems.
1/10
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T(K)
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
310
320
330
340
350
360
370
380
390
T
cyl
T
1x4
T
2x8
T
3x12
T
6x24
Figure 1.1.
Temperatures at the midplane axial position from the cylindrical analytic model
(T
cyl
) and from the finite element computations with the meshes indicated in the legend.
Figure 1.2.
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 Spring '04
 Witt
 mechanics, axial stress, Cylinder stresses, nodal diameter, tapered fitting

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