Shape factor and shape optimization for a periodic array of isothermal pipes

Abstract

a b s t r a c tWe address the problem of two-dimensional heat conduction in a solid slab embedded with a periodic array of isothermal pipes of general cross-section. The objective of this work is two-fold: to develop a numerical procedure through which we can obtain the shape factor associated with a given configuration and, to develop a numerical shape optimization algorithm through which we can compute shapes that extremize the transport rate. The shape factor is obtained by first transforming the periodic array of pipes into a periodic array of strips using the generalized Schwarz-Christoffel transformation and, subsequently, by developing an integral equation of the first kind for the temperature gradient using the boundary element method. The integral equation is solved both numerically and analytically/asymptotically with excellent agreement between the results. The shape optimization problems, which are formulated with respect to the parameters of the generalized Schwarz-Christoffel transformation, are solved numerically to compute the shape that maximizes the cross-sectional area and the shape that minimizes the perimeter of the cross-section, given the shape factor and the distance between two consecutive pipes. It is inferred that the problems are adjoint to the transport rate minimization and transport rate maximization problems, respectively. The optimal shapes are computed numerically and validated with available analytical and numerical results for a single pipe. Furthermore, motivated by the analytical result, we propose a parametric set of equations that describe well the optimal shapes. The versatility of the Laplace equation suggests that similar formulations have applications in continuum mechanics, electricity, hydraulics and drug reduction.

Heat transfer enhancement of a periodic array of isothermal pipes

“…The case of a periodic array of isothermal pipes was treated in [11] and [19]. In the former work, both surfaces of the slab were assumed isothermal while in the latter, the lower surface was assumed adiabatic.…”

“…The analysis of this section closely follows the definitions and notation outlined in [11], where the surfaces of the slab were assumed to be isothermal. In this work we assume that the surfaces of the slab are subjected to uniform convection with a uniform heat transfer coefficient (h), hence the work considered in [11] is the asymptotic limit of the present analysis for h → ∞ (large Biot number, Bi → ∞), i.e.…”

“…A similar problem has been addressed by Fyrillas [11] where, however, the surfaces of the slab were assumed to be isothermal. When the slab is subjected to uniform convection, Fyrillas & Stone [12] showed that there exists a critical insulation thickness associated with a slab embedded with a periodic array of isothermal strips.…”

“…Following the analysis in [11,[15][16][17][18][19], the physical domain is transformed into a rectangular channel using the generalized Schwarz-Christoffel transformation [20][21][22][23]. The heat…”

Numerical and Experimental Determination of Wavy Fin-Tube Shape Factor

“…Shape factors for simple shapes such as plates, spheres, and hollow cylinders, are readily available from textbooks (Cengel, Boles, & Kanoglu, 2007;Incropera & De Witt, 2002). However, to handle complex systems bounded by two isothermal surfaces such as an elliptic cylinder or oblate sphere, the problem is much more difficult (Al-Doori, 2011;Fyrillas, 2008Fyrillas, , 2010Yovanovich, 1973). These complex shapes were studied as early as 1973 and only limited to shapes that were predefined in their studies.…”