On the suspension of solid particles in turbulent stirred vessels

Suspending solids in liquids is a highly common and industrially relevant process. Agitators must be designed to suspend solid particles in liquids, or the vessels will fill with settled solids and cease to operate, which is known as sanding out. The minimum operating condition for solids suspension is when all the particles are in motion. The minimum design impeller speed, the just suspended impeller speed, has been previously determined by the Zwietering model. However, recent analysis and research has determined the Zwietering model to be flawed in the inclusion of a viscosity term in turbulent flow, a scale-dependent proportionality constant, and a scale-up methodology that does not agree with experimental evidence (see Ayranci and Kresta 2014, Grenville et al. 2015). 

A new correlation by Grenville et al., proposed in 2012 (published in 2015), utilizes an energy balance approach coupled with turbulence theory. While this model is attractive due to its mechanistic approach, only the turbulent and particle length scales are considered in its derivation. Yet to fully describe turbulence, time scales are also necessary. This thesis further investigates the energy balance approach and seeks to improve the understanding of the interaction between turbulent eddies and particles by also considering the fluid time scales instead of just length scales. This endeavor also leads us into studying what happens when particles, due to their density and size, interact with other regions of the turbulent spectrum to become suspended.

This research utilizes both a computational and an experimental approach to studying the solid-liquid mixing and suspension process. Large Eddy Simulation (LES), coupled with Laser Doppler Velocimetry Eulerian turbulence analysis techniques, are used to produce estimates of the turbulent characteristics above the base of a stirred vessel. This region is of interest as it is the location where suspension and deposition of particles occurs. Our LESs are found to behave to a satisfactory degree for our system and were conducted at similar corresponding experimental conditions in which particles are just suspended. Though our LESs are single phase, the corresponding experiments are dilute (𝑥𝑣 ≤ 1% v/v) and particle effects on turbulence considered negligible. We proposed that only considering the particle and eddy length scales was not a complete picture of the suspension process and subsequently showed through matching simulation to experiment that the suspending eddy time scale is on the order of 1.5 to 2 times and its length scale on the order of 1.2 to 1.6 times the particle diameter. With respect to the diameter-eddy relationship, this is in agreement by multiphase simulation results obtained by Ten Cate et al. 2004 which show turbulence modulation occurring at 1.2 times the particle diameter.

These simulated and experimental comparisons also revealed the potential for a mechanism change when particles interact with eddies larger than the integral scale and outside the inertial subrange of the turbulence spectrum. This region on the turbulent spectrum could be considered the anisotropic subrange as these eddies are on the scale of the impeller and vessel geometry and thus are affected by mean flow dynamics and may have directional preference. Experiments were conducted to determine if a third suspension regime (viscous subrange and inertial subrange being the first and second regimes) existed. These experiments also expanded the validated range of Archimedes number, 𝐴𝑟, beyond that of the original dataset which the Grenville et al. correlation was originally based upon from a maximum of 103 to a new maximum of 107, which is an important result for industry. We also determined through LESs the local 𝜀 above the base of the vessel is on the order of 2.3 times the mean 𝜀 for an axial flow hydrofoil which may explain why the mean 𝜀, with the aid of minor geometric factors, tends to correlate suspension datasets to a satisfactory degree. Thus, we conclude that the use of the mean or average power per mass, 𝜀, in suspension correlations is reasonable. Regardless, it is the local turbulence on the base of the vessel doing the suspension and so the minor geometric factors (impeller size and position) mentioned above are necessary to adjust the mean dissipation to reflect the local dissipation.

To continue the theme of studying the interaction between particles and eddies above the base of the vessel, different vessel base topologies were investigated. From an agitator vendors’ perspective, there is no one ‘standard’ vessel geometry in which solid-liquid mixing is confined to, it depends on the specific unit operation and scale. A useful suspension correlation needs to be applicable across all vessel geometries. We measured the just suspended impeller speed in four common vessel geometries: cylindrical with a flat base, cylindrical with a dished base, square cross section with a flat base, and a horizontally oriented cylindrical vessel with flat end caps. We discovered that we could correlate just suspended datasets obtained from the four vessel geometries through the introduction of a new parameter, the vessel sphericity, 𝜓. We then showed that this term could correct the Grenville et al. suspension correlation and generalize across the four tested vessel geometries. The new correlation following the Baldi et al. energy balance approach and scaling arguments we developed appears robust enough that extrapolation/interpolation to other possible and common vessel geometries seems reasonable.

Through this research, not only has the mechanistic understanding of the suspension process been furthered, the precision, validity, and applicability of the turbulent eddy-particle energy balance approach has been significantly advanced. Thus, both academic and industrial interests have been satisfied in that a better understanding is gained, and a newly developed, more industrially useful correlation has resulted. Because of this research, other impellers can be added to the new correlation’s database with a much simpler design of experiment and the vessel geometry in which the data collected is now less critical. We have included a commonly tested pitched blade turbine and wide blade hydrofoil, which has not previously been studied in solid liquid mixing, at least not to a degree from which a suspension correlation could be developed. The newly developed correlation for the just suspended speed applies currently to concentrations from 1 to 30% v/v of freely settling particles (particle Archimedes number greater than 10-2) for agitators operating in the turbulent flow regime (Reynolds number greater than 10,000). Future advancements should strive to improve the concentration model beyond that of the simple power law model.