Simulations of mobilization of Bingham layers in a turbulently agitated tank

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Abstract

Numerical simulations were used to study mobilization and mixing of a bottom layer of Bingham liquid by agitating a Newtonian liquid above the Bingham layer. The agitation is done by a pitched-blade impeller at a Reynolds number of 6000. The Bingham liquid and the Newtonian liquid are miscible. The parameter space of the simulations has a yield stress number and a Richardson number as dimensionless variables. The yield stress number quantifies the importance of the yield stress relative to inertial stresses, the Richardson number the role of the density difference between the two liquids. The simulation procedure is based on the lattice-Boltzmann method for the flow dynamics, and a finite volume scheme to solve for the local and time dependent composition of the liquid mixture. Flow dynamics and liquid composition are intimately coupled. The moderate Reynolds number tentatively allows us to directly simulate the transitional flow, without a need for a turbulence closure model. The results quantify the increase of mixing time with increasing yield stress and (to a weaker extent) density difference.

Graphical abstract

Vertical and horizontal cross section through the mixing tank showing the liquid mixture composition (red: Newtonian, blue: Bingham).

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Highlights

► Proposed a procedure for detailed simulations of Newtonian/Bingham mixtures. ► Simulated the interaction of turbulent Newtonian flow and Bingham liquids. ► Quantified mobilization of the Bingham liquid as a function of its yield stress. ► Monitored homogenization as a function of yield stress and density differences.

Introduction

In this paper a specific – though not uncommon – situation is considered: a mixing tank that has been left un-agitated for some time has its contents segregated into a thick, paste-like layer on the bottom, and a lighter and less viscous portion of liquid above. The bottom layer is considered a Bingham liquid; the liquid above is supposed to be Newtonian. The research question is if turbulent agitation of the less viscous Newtonian phase is an efficient means to mobilize the Bingham liquid, and subsequently homogenize the tank contents. Obviously, many variables determine the answer to this question. The focus of the present study is on how the mixing process depends on the yield stress of the bottom layer. Also the effect of the density difference between the two liquids has been considered.

The situation as sketched above has been approached in a computational manner: we have attempted to mimic the mixing process by performing three-dimensional, time-dependent numerical simulations starting from a zero velocity and fully segregated state. The flow system poses a few interesting numerical challenges. In the first place we deal with turbulent flow, at least in the portion of the tank with Newtonian liquid. We do not, however, want to revert to turbulence modeling given the presence of the Bingham liquid and its (most likely) laminar flow character. Turbulence models usually are not designed and tested for accurately capturing (spatial or temporal) laminar-turbulent transitions. Not using turbulence models means that the flow needs to be fully resolved (“direct” simulations) which puts strong resolution demands. We mitigate these by considering a modest Reynolds number (Re = 6000 with Re precisely defined in the next section), and we check for grid effects, i.e. a significant number of cases that are physically identical have been performed on grids with different resolution. Also the discontinuous behavior of Bingham liquids (upon reaching the yield stress, zero deformation in the liquid switches to non-zero deformation) is a challenge for numerical methods.

Interesting flow physics is expected; not in the least given the turbulent nature of the flow “attacking” the Bingham layer. The average flow may be too weak to erode away the bottom Bingham layer, the intermittency of the turbulent flow emerging from the impeller may from time to time be sufficiently strong to locally overcome the yield stress and chip away some of the Bingham liquid. This will change the surface topology of the Bingham layer, making it rougher (more undulated) and therefore more susceptible for further erosion. With the gradual removal of the Bingham layer and associated changing bottom topology also the global structure of the flow of the Newtonian liquid gradually changes with time making this an interesting problem with a broad spectrum of time and length scales. Again adequate resolution of flow structures with high viscous and/or inertial stress is a key issue to realistically capture this process. It is anticipated that the density difference of the two liquids is not as important to the erosion process as the yield stress. For mixing the liberated Bingham material into the bulk flow, however, we do expect slower homogenization with increased density differences.

The nature of this work is quite specific, and I am not aware of similar studies in the literature. Closest comes the work by Frigaard and co workers [1], [2], [3], [4] who studied displacement of wall layers consisting of yield stress liquid by a Newtonian liquid. In their experiments and modeling work they mainly considered laminar flow in channels and pipes.

The aim of this paper in the first place is to quantify the homogenization of the initially segregated system. As in former papers [5], [6], we use the decay of scalar variance in the stirred vessel as a means to monitor the mixing process and the time it takes to reach a certain low level of scalar variance as a measure for the mixing time. A wider ranging goal is to enhance our understanding as to how turbulent flow interacts with yield stress material. This goes beyond mixing tanks only. Examples of this nature are also encountered in fouling and removal of fouling in process equipment such as heat exchangers [7], or food processing devices, flow over biofilms [8], and sediment transport in slurry pipelines [9], to name only a few.

This paper is organized in the following manner: First the flow system and the liquid properties are described. Based on these, dimensionless numbers are defined, and the parameter range covered in this paper is identified. Subsequently the simulation procedure is outlined schematically with references to the literature for further details. We then present results. The emphasis will be on the effect of the yield stress on the mobilization of the Bingham layer. Conclusions are summarized in the last section.

Section snippets

Flow systems

There are two liquids present in the tank: a Newtonian liquid with density ρN and dynamic viscosity ρNν, and a Bingham liquid with density ρB (with ρBρN), yield stress τY and plastic (dynamic) viscosity ρBν. This implies that the two liquids share the same kinematic viscosity ν; we made this choice to limit the dimensions of the parameter space that we want to cover in this study.

The tank and agitator, and the coordinate system as used in this work are shown in Fig. 1. The tank is cylindrical

Modeling approach

The lattice-Boltzmann method (LBM) has been applied to numerically solve the incompressible flow equations [10], [11]. Lattice-Boltzmann fluids can be viewed as collections of (fictitious) fluid particles moving over a regular lattice, and interacting with one another at lattice sites. These interactions (collisions) give rise to viscous behavior of the fluid, just as colliding/interacting molecules do in real fluids. The main reasons for employing the LBM for fluid flow simulations are its

Flow and scalar field impressions

The qualitative discussion of the results of our simulations first focuses on the vertical plane through the center of the tank, in between two baffles (the xz-plane in the coordinate system defined in Fig. 1). We show velocity vector fields, contours of liquid composition c, and contours of liquid deformation γ˙ in this plane. As the default Richardson number we chose Ri = 0.25.

The way the concentration field c in the xz-plane evolves in time is shown in Fig. 2 for four different values of Y. It

Summary

This paper discussed the practically relevant, though quite specific situation of mobilization and mixing of a layer of Bingham liquid at the bottom of a mixing tank through agitation with an impeller of Newtonian liquid above. The two liquids have been considered miscible. The main dependencies that have been investigated relate to the yield stress of the Bingham liquid, and the density difference between the Bingham liquid and the Newtonian liquid. The yield stress has been

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