Mathematical Problems in Engineering
Volume 2007 (2007), Article ID 57360, 28 pages
Heat Conduction in Lenses
Corporate Technology and Innovation Center, Leica Geosystems AG, Heerbrugg 9435, Switzerland
Received 7 November 2006; Accepted 9 March 2007
Academic Editor: Semyon M. Meerkov
Copyright © 2007 Beat Aebischer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider several heat conduction problems for glass lenses with different
boundary conditions. The
problems dealt with in Sections sec:1 to sec:3 are motivated by the problem of
an airborne digital camera that is
initially too cold and must be heated up to reach the required image quality.
The problem is how to
distribute the heat to the different lenses in the system in order to reach acceptable
operating conditions as
quickly as possible. The problem of Section sec:4 concerns a space borne
laser altimeter for planetary
exploration. Will a coating used to absorb unwanted parts of the solar spectrum
lead to unacceptable
heating? In this paper, we present analytic solutions for idealized cases that help
in understanding the essence of the
problems qualitatively and quantitatively, without having to resort to finite element
computations. The use
of dimensionless quantities greatly simplifies the picture by reducing the number
of relevant parameters.
The methods used are classical: elementary real analysis and special functions. However,
conditions dictated by our applications are not usually considered in classical works on
the heat equation,
so that the analytic solutions given here seem to be new. We will also show how
energy conservation leads
to interesting sum formulae in connection with Bessel functions.
The other side of the story, to determine the deterioration of image quality by given
temperature distributions in the optical system, is not dealt with here.