Agitating cylindrical particles in laminar liquid flow

Highlights

• Particle-resolved simulations with cylindrical particles in complex flow.

• Dense agitated suspension with frequent collisions between particles and impeller.

• Kinetic energy in particle translation ten times higher than in rotation.

• Particle orientation strongly related to their aspect ratio.

Abstract

Three-dimensional, time-dependent simulations of dense agitated solid-liquid suspensions involving particles of cylindrical shape in a Newtonian liquid have been performed. The liquid flow is resolved by the lattice-Boltzmann method at length scales finer than the size of the particles, which implies particle-resolved simulations. The flow solution includes the hydrodynamic forces and moments on each particle that are used to integrate their linear and rotational equations of motion. No-slip at the particle surfaces is imposed by an immersed boundary method (IBM). The marker points of the IBM are also used to detect and carry out collisions between particles. This numerical procedure has been applied to systems contained in a rectangular box and agitated by a revolving disk as well as by a pitched-blade turbine with an impeller-based Reynolds number of 87, which indicates laminar flow. The overall solids volume fraction has been fixed to 15%; the number of particles is of the order of one thousand. We study the effect of impeller type and particle shape (in terms of the length over diameter ratio of the cylinders that has been varied between 1 and 4) on the extent to which the solids are suspended and on the way the cylinders orient themselves.

Introduction

There are numerous challenges when it comes to predictive simulations of liquid flows carrying solid particles, specifically for industrially relevant solid-liquid systems. To name some important ones: resolving turbulence in the fluid phase and accurately accounting for how flow turbulence interacts with the particulate phase; representing (wide) particle size distributions; dealing with dense, collision-dominated suspensions and accounting for the shape of the particles. This paper focuses on the latter two aspects: dense suspension of non-spherical particles. At the same time, it does not consider turbulence and size distributions. The reason for not considering turbulence at this stage is that we aim for simulations that accurately represent the shape of particles as well as the close-range interactions between particles in dense suspensions, i.e. we aim for particle-resolved simulations. This needs three-dimensional grids that are finer (by at least one order of magnitude in each coordinate direction) than the particle size so that we need to restrict ourselves to small systems that are hardly able to develop turbulence.

Computational research on solid particles suspended in liquids has so far largely focused on particles of spherical shape. For example, models for solid-liquid interaction are usually formulated in terms of drag force correlations for spheres (Khandai et al., 2003; Van der Hoef et al., 2005; Tenneti et al., 2011); constitutive models for solids derived from kinetic theory of granular matter normally assume the granular equivalent of monoatomic gases, which implies spherical particles (Gidaspow, 1994); and also particle-resolved, direct simulations mostly involve spheres (Kidanemariam and Uhlmann, 2017; Vowinckel et al., 2014; Derksen and Sundaresan, 2007). Particle shape, however, is expected to have significant impact on the individual behavior of particles and therefore also on the collective behavior of solids-liquid mixtures (Richardson and Zaki, 1954). In order to elucidate particle-shape effects in a computational manner, we need a simulation procedure that explicitly accounts for the shape of the particles.

With this in mind we have devised a method for particle-resolved simulations of dense solid-liquid suspensions involving non-spherical particles (Shardt and Derksen, 2012; Derksen, 2019). Given the high solids volume fractions we are interested in, collision handling and close-range interactions between particles are an important feature of the numerical procedure. So far, we have applied this method to fully periodic systems and studied solid-liquid fluidization with cylindrical particles as well as settling of solid particles having the shape of red blood cells. These are homogeneous systems without walls. In order to increase the flow’s complexity, to study interactions with walls and moving solid objects (such as a revolving impeller), and also to simulate flow systems that can be easily replicated experimentally, we here report the behavior of solid-liquid suspensions with non-spherical particles in small agitated tanks. As we will see, these are strongly inhomogeneous and often only partially suspended systems. In such cases, part of the particles form a granular bed on the bottom of the container. Mobilizing the bed critically depends on how the bed is packed with the structure of the packing strongly influenced by the shape of the particles. Granular bed erosion has implications well beyond agitated tanks in areas such as sediment transport in pipelines, rivers, and coastal regions (Uijttewaal, 2014; Ramesh et al., 2011). One advantage of working with small, confined agitated systems is that they can be built easily and are amenable to quantitative (flow) visualization experiments, ideally with refractive index matching between solids and liquid so that one can look deep inside the system. As an example, we have reported refractive-index-matched Particle Image Velocimetry (PIV) experiments for measuring flow velocities of the interstitial liquid in an agitated solids suspension involving spherical particles (Li et al., 2018).

The aim of this paper is to study the way particles of cylindrical shape are entrained by laminar liquid flow and how this depends on the way the liquid is agitated, and on the shape (here aspect ratio, length over diameter) of the particles. From a computational methods perspective, we show in this paper that the immersed boundary method is an elegant way of not only dealing with imposing no-slip boundary conditions on moving solid surfaces but is also instrumental when it comes to collision detection and handling between particles and between particles and a revolving impeller.

This paper is organized in the following manner: First the flow geometries and conditions are defined, mostly in dimensionless terms. We then briefly describe the numerical procedure which ― compared to previous papers (Shardt and Derksen, 2012; Derksen, 2019) ― has been extended with particle-wall and particle-impeller interactions. Subsequently parameter settings and parameter ranges are discussed. In the Results section we begin by qualitatively showing particle suspension levels for different mixing configurations. These observations are then quantified in terms of the average elevation of particles as a function of process conditions as well as how solids and their kinetic energy are distributed over the height of the tank under dynamically steady conditions. Also the orientation of the particles and their alignment with the flow field have been investigated. The final section presents conclusions and suggestions for future work.