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Bas Goorden
| Speaker: |
Bas Goorden
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| Date: |
Tuesday February 15, 2011
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| Title: |
Design of
mixing tubes based on chaotic billiards (Master's thesis
presentation)
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Abstract
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Within
several
years, most of the world’s lighting are based on LEDs. Many
applications will involve mixing of light. This can be done using a
mixing tube, which is essentially just a solid plastic rod. The light
enters at the bottom, reflects against the boundaries and exits at the
top. At that point, it is assumed to be mixed.
The quality of mixing is determined by the length of the mixing tube,
the positioning of the LEDs with respect to the tube and the shape of
the tube. It is known from practice that square tubes are better than
circular ones and hexagonal tubes are better than square ones.
We have developed a new family of mixing tubes. This is done by
modeling mixing tubes as billiards and investigating the trajectories
of rays inside these billiards. We have looked for behavior that is
both chaotic and ergodic. In this context, chaotic implies that rays
starting close to each other will move apart rapidly. Ergodic means
that almost every ray will travel all over the billiard in a uniform
sense. The assumption is that billiards which exhibit these properties
mix well. We investigate the family of regular polygons with rounded
corners. These billiards are proven to be ergodic. We calculate the
rate at which neighboring rays spread, which is an indicator of speed
of mixing. Positions for which there are directions in which the light
does not spread at all turn out to be particularly bad. We develop a
shape for which such directions do not exist: the rounded right-angled
triangle. Mixing tubes of this shape seem better than traditional ones.
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