hw6_f03_soln

hw6_f03_soln - EMA 405, Fall 2003 Solutions for HW #6 1....

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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 mid-side 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 r-z 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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hw6_f03_soln - EMA 405, Fall 2003 Solutions for HW #6 1....

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