Accurate Approximate Methods for the Fully Developed Flow of Shear-Thinning Fluids in Ducts of Noncircular Cross Section

Abstract

The fully developed laminar flow of Non-Newtonian fluids in ducts has broad application in engineering. The power-law viscosity model is utilized most often in the engineering literature, but it is deficient for many fluids as it does not admit limiting Newtonian viscosities at low and high shear rates. The goal of this work is to demonstrate two approximate but accurate and efficient methods for computing the pressure gradient in ducts of noncircular cross section for shear-thinning fluids following a general viscosity curve. Both methods predict the pressure gradient to better than 1% as established by full numerical solutions for ten cross-sectional shapes, a result representing an order-of-magnitude improvement over previous approximate methods. In the first method, an approach recently proposed and demonstrated to be accurate for a circular duct is shown to be equally applicable to noncircular ducts. In the second method, a widely used approach for noncircular ducts based on a generalization of the Rabinowitsch–Mooney equation is improved through an alternate evaluation of its parameters. Both methods require one-time numerical solutions of the power-law viscosity model for a duct shape of interest, and the necessary results are tabulated for the ten cross-sectional shapes analyzed. It is additionally demonstrated that the pressure-gradient error of the second method is approximately halved by simply replacing the hydraulic diameter with a viscous diameter obtained from the Hagen–Poiseuille equation.