Constructal design applied to elliptic tubes in convective heat transfer cross-flow of viscoplastic fluids

The objective of this work was to obtain geometries that maximize the heat transfer and minimize pressure drop for viscoplastic fluids in cross flow around elliptical section tubes. The Constructal Design associated with exhaustive search was employed to obtain the ellipse aspect ratios that maximize the Nusselt number and to minimize dimensionless pressure drop, for fixed parameters such as ellipse area, Reynolds (Re = 1), Prandtl (Pr = 1) and Herschel-Bulkley (HB = 1) numbers. The viscoplastic fluid behavior was modeled using the Herschel-Bulkley constitutive equation. The power-law index, n, was a parameter to predict fluid shear thinning. The system of differential equations for flow and heat transfer was solved numerically by the finite volume method. The aspect ratio to maximize Nu‾, rq,opt, and the one to minimize Δp̃, rp,opt, were searched for different values of n, ranging from 0.4 to 1. The rq,opt was found to be very close to 1 for all n's, which corresponds to the tube of circular cross section. Low aspect ratio tubes lead to the formation of greater unyielded zones as compared to high aspect ratio tubes. This may be a key to understand the increase in heat transfer from the tube as the aspect ratio increases to one. The aspect ratio that led to the lowest pressure drop was not the same for all fluids, contrarily to the behavior noticed for thermal purpose. It was noticed that as n decreases, the rp,opt, also decreases, meaning that to reduce pressure drop for more shear thinning fluids the tube must be slender. This may lead to the conclusion that elliptic tubes might be a good alternative in applications of heat transfer with mild viscoplastic fluids (low HB) only when there is the need to reduce pressure drop or pumping power.