Аннотации:
We address the problem of two-dimensional heat conduction in a solid slab whose
upper and lower surfaces are subjected to uniform convection. In the midsection of the
slab there is a periodic array of isothermal pipes of general cross section. The main
objective of this work is to find the optimum shapes of the pipes that maximize the
Shape Factor (heat transport rate). The Shape Factor is obtained by transforming the
periodic array of pipes into a periodic array of strips, using the generalized Schwarz-
Christoffel transformation, and applying the collocation boundary element method on
the transformed domain. Subsequently we pose the inverse problem, i.e. finding the
shape that maximizes the Shape factor given the perimeter of the pipes. For large Biot
number the optimum shapes are in agreement with the isothermal case, i.e. circular for
sufficiently small perimeters/heat transfer, and elongated towards the surfaces of the
slab for larger perimeters/heat transfer. Furthermore, for the isothermal case, we were
able to discover a new family of optimum shapes for large thickness of the slab and large perimeters, which do not have their maximum width on the horizontal axis of symmetry. For small Biot number the optimum pipes are flatter than the isothermal ones for a given perimeter. The flatness becomes more apparent for larger perimeters. Most important, for large perimeters there exists a critical thickness which is characterized by maximum heat transfer rate. This is further investigated using the finite element method to obtain the critical thickness of a slab and the critical depth of the periodic array of circular pipes