Generalized models for predicting the drag coefficient and settling velocity of rigid spheres in viscoelastic and viscoinelastic power-law fluids

Abstract

The design and optimization of several fluid-particle transport systems often require the determination of particle settling velocity in non-Newtonian fluids. These fluid-particle transport systems are encountered in a wide variety of industrial processes including drill cuttings transport in oil and gas well drilling operations and proppant transport in hydraulic fracturing operations. Current conventional approaches to estimate settling velocity in these non-Newtonian fluids often exclude the effect of elasticity on particle settling which can lead to erroneous estimation. It is therefore imperative to devise an accurate means to mathematically account for the fluid elasticity in particle settling in viscoelastic fluids. An experimental study was conducted to measure the settling velocity of spherical particles in viscoelastic and viscoinelastic power-law type fluids. Using a semi-mechanistic approach, explicit mathematical models were developed for estimating drag coefficient and particle settling velocity in viscoelastic fluids and viscoinelastic fluids. The main objectives of the study were: i.) To measure the terminal settling velocity of particles in various viscoelastic and viscoinelastic power-law type fluids intended at augmenting the present corpus of experimental data in literature. ii.) To develop new drag coefficient and particle Reynolds number correlations that are applicable to viscoelastic and viscoinelastic power-law type fluids and, iii.) To present a general explicit approach for predicting settling velocities of particles in viscoelastic and viscoinelastic fluids.Particle Image Shadowgraphy (PIS) was used to measure the settling velocities of the spherical particles (Specific gravity 2.5–7.7; Diameters: 0.71–4.00 mm) in Hydrolyzed Polyacrylamide and Tylose solutions of varying elastic properties. Rheological characterization of the fluids was carried out using the C-VOR Bohlin Rheometer. Experimental results from this study were combined with various data published in the literature to create an extensive database and widen the scope for empirical analysis. The database had a consistency index (k) range of 0.012–6.773 Pa.sn and a fluid behavior index (n) range of 0.370–0.940. New drag coefficient and Reynold number correlations for particles settling in viscoelastic and viscoinelastic power-law fluids were developed by using a semi-mechanistic approach, which also included the fluid elasticity effect quantified in terms of the longest relaxation time. The newly developed correlations and an advanced statistical modeling program (OriginPro 9.0) were used to mathematically create explicit models that can be used for predicting particle settling velocity in viscoelastic and viscoinelastic fluids.Comparative analysis showed that an increase in the fluid elasticity gave a corresponding decrease in the particle Reynolds number and dampened the particle settling velocity in viscoelastic fluids. This dampening effect can be attributed to the unique ability of elastic fluids to partially or fully regain their original structure after deformation. Furthermore, statistical analysis showed that the presented new models predict settling velocity accurately with a very low with a Percentage Mean Absolute Error of about 7.5% and 30% for viscoinelastic and viscoelastic fluids respectively. The proposed models also exhibited Root Mean Square Error (RMSE) values of 0.04 m/s and 0.009 m/s for viscoinelastic and viscoinelastic fluids respectively.