Heat transfer enhancement of a periodic array of isothermal pipes

Abstract

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 SchwarzChristoffel 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 * E-mail: t.leontiou@frederick.ac.cy † E-mail: magzhan.ikram@nu.edu.kz ‡ E-mail: kenes.beketayev@nu.edu.kz § Corresponding Author, e-mail: m.fyrillas@gmail.com 1 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. KeywordsSolid slab with periodic array of pipes; Convective heat transfer; Laplace equation, generalized Schwarz-Christoffel transformation; Shape Optimization, Optimum shapes of pipes;Critical Thickness of slab, Critical depth of the pipes.

Shape optimization of conductive-media interfaces using an IGA-BEM solver

“…For example, the objective function can be the difference between the computed temperature and the measured/prescribed temperature and the unknowns can be the boundary conditions, physical properties or the geometry of the configuration [10,11,12,13,14,15]. Another example is to have the heat flux as the objective function and obtain the shapes that optimize it [16,17,18,19,20,21,9].…”

“…Hence the solver gets trapped in local minima or diverges. Regarding inverse shape optimization problems, remedies to address the ill-possing include mapping the physical domain onto a fixed computational domain [16,17,18,19,20,21] and redistribution of ill-ordered nodal points [12,13,14,15], Tikhonov regularization [22], homogenization [23], or transforming the problem into a parameter estimation by expanding the shape in terms of a small number of parameters, e.g. adaptive mesh [24], eigenfunction expansion [10], and mesh-morphing [25].…”

“…In addition, accuracy is very significant due to the sensitivity of the solution on the parameters of the optimization. Hence, where applicable, the boundary element method [29,30,31] provides a natural selection for the numerical solution of the PDE due to its superiority of handling, with high accuracy, a large number of elements and intricate wall geometries [16,17,18,4,19,20,21,9].…”