Hybrid rheological model for non‐Newtonian turbulent pipe flow

Abstract

Many slurry rheograms do not follow the Bingham model, but are curved (convex upwards) at low-strain rates. Alternate models such as the yield-power law (YPL), provide a good fit to the low-strain rate data, but they tend to underestimate apparent viscosities at high-strain rates. The current paper considers a hybrid rheological model consisting of a cubic-spline fit to the low-strain rate data merging into a Bingham linear model above a limiting strain rate. This model predicts turbulent flow well, depending only on the area difference between the Bingham rheogram and the cubic spline.Un bon nombre de rhéogrammes de boues liquides ne suivent pas le modèle de Bingham mais sont incurvés (convexes vers le haut) aux faibles vitesses de dilatation. Des modèles de remplacement tels que la loi rendement-puissance offrent une bonne correspondance avec les données de faibles vitesses de dilatation mais elles tendentà sous-estimer les viscosités apparentes aux vitesses de dilatationélevées. L'article présent considère un modèle rhéologique hybride composé d'une spline cubique adapté aux données de faibles vitesses de dilatation, ce systèmeétant fusionné avec un modèle linéaire de Bingham au-dessus d'une vitesse de dilatation limite. Ce modèle bien lesécoulements turbulents, dépendant uniquement de la différence de surface entre le rhéogramme de Bingham et la spline cubique. Keywords: non-Newtonian turbulent flow, rheology PROBLEMS WITH RHEOLOGICAL MODEL SELECTIONI n calculating pressure drops in pipelines carrying nonNewtonian fluids, it is often necessary to extrapolate to values of shear stress considerably in excess of those on the rheogram. Although rheograms are very often curved (convex upward) at low-strain (or shear) rates, they often tend to approximate straight lines at high-strain rates as is illustrated later in Figure 1. The Wilson-Thomas prediction method for turbulent flow of Bingham plastic fluids, first published in 1985, has been extended recently to prediction of the transition velocity (Wilson and Thomas, 2006) and from smooth to rough walled pipes (Thomas and Wilson, 2007). These previous analyses are based on an assumed true Bingham rheogram with a yield stress and a linear section of slope equal to the plastic viscosity. That is, they do not allow for the curved sections of the rheograms at low-strain rates. Whilst alternate models, such as the YPL, provide a good fit to the low-strain rate data they tend to physically unrealistic apparent viscosities (often less than water at high-strain rates).In turbulent pipe flow, the effect of fluid viscosity is confined to a thin sub-layer, which extends from the wall to a distance ı given by:where and are viscosity and density, respectively, and U* is the shear velocity at the wall. In the main flow (y > ı), momentum transfer takes place by turbulent mixing, which is an inertial process rather than a viscous one. It follows that the velocity profile in the main flow is logarithmic, as opposed to the linear