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Turbine wheel
It is crucial for some turbines to be able to accelerate quickly. The best
acceleration is obtained by the turbine with the smallest possible mass moment
of inertia. The shape of the blades is determined by the
hydrodynamic properties which are very important to the efficiency of the
turbine, so to improve the structural performance, we focus our attention on the hub. We want to find the shape
of the hub that minimizes the mass moment of inertia and still is able to carry
the load. The function of the hub is to fix the blades. This requires a certain strength, because the centrifugal forces are very high. In fact, the centrifugal forces are the limiting factor when the maximum speed of the turbine is determined. A faster turbine means more power at a lower cost. The self weight of the hub also contributes to the centrifugal load. The dual function of the material in the hub as a load carrier and a contributor to the centrifugal force makes it very difficult to determine manually just where to add material and where to remove it. The turbine disk is subjected to a temperature field. It keeps the edge of the disk at a higher temperature than the center. This contributes to the tension in the disk because the relatively warm rim tries to expand more than the center will allow. When we change the design, we also change the way the heat flows from the rim to the center, and this again changes the stresses in a very unpredictable way. The thermal stresses are not the only complication. The strength of the material depends on the temperature as shown in this figure. The allowable stress in each point depends on the temperature which in turn depends on the design. The fact that the temperature curve has a kink and is mathematically non-differentiable is an additional serious complication for the optimization system. In short, the physics of this problem is so complicated that optimization is required not only to obtain an optimal design, but to obtain an acceptable design at all.
The figures to the right show the initial and final designs. The colors represent the stresses normalized by the local strength at each point, depending on the temperature level at that point. A value of 1.0 means that the material is fully loaded. Values above 1 are not acceptable. Notice that the initial design is infeasible. It overloads the material in some parts by 29.7% while other parts are not exploited well. The final design is feasible and the material is almost uniformly used to its maximum. The mass moment of inertia is reduced by 25%. |
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Sprocket | Slide gate | Turbine wheel | Electricity Pole | Truck Floor Services | References | Contact | Links Last modified: november 15, 2001 |
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This
is an artificial-example. Geometry, material and all other data are completely
fictitious. We have solved quite a few industrial problems of this type, but our
customers prefer to keep the results confidential. This is quite understandable,
because the ability to do optimization of thermo-elastic problems and to handle
"funny" criteria are two of the things that make our capability and
the designs of our customers unique. This example is particularly nasty because
the material strength at each point depends on the local temperature,
and the temperature field depends on the design.
