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Many chemical engineering processes involve the suspension of solid particles in a liquid. In dense systems, agitation leads to the formation of a clear liquid layer above a solid cloud. Cloud height, defined as the location of the clear liquid interface, is a critical measure of process performance. In this study, solid-liquid mixing experiments were conducted and cloud height was measured as a function operating conditions and stirred tank configuration. Computational fluid dynamics simulations were then performed using an Eulerian-Granular multiphase model. The effects of hindered and unhindered drag models and turbulent dispersion force on cloud height were investigated. A comparison of the experimental and computational data showed excellent agreement over the full range of conditions tested.

The suspension of solid particles in a liquid is a key requirement of many industrial processes. Examples relevant to this work include crystallisation, dissolution, and adsorption processes. Each of these can be characterized as complex multiphase processes that are often facilitated by mechanically stirred tanks. Maximizing contact between the solid and liquid phases facilitates mass transfer and reaction, therefore, assessing the ability of the process to suspend particles is a key objective. The quality of suspension can be quantified by three parameters: just suspension velocity, solids distribution, and cloud height. The latter is defined as the location of the interface between solids-rich liquid and clear liquid regions. Cloud height is critical because of limited mixing between the suspended solids and the upper clear liquid layer. The current study aims to develop and validate a computational model of a solids suspension process using experimental measurements of cloud height.

The hydrodynamics in stirred vessels are complex, three-dimensional, and turbulent. The interplay between solid and liquid materials, vessel design, impeller type and location, and level of agitation all determine the efficiency of the mixing process. Numerous empirical models have been developed to relate tank performance to operating conditions and geometry [

Computational fluid dynamics (CFD) methods are powerful tools for improving our understanding of mixing in stirred tanks. In the last few years, encouraging results have been obtained in the simulation of solid suspension systems [

The Granular model is an extension of the Eulerian model wherein the solids are modeled as a pseudo-fluid. The pseudo-fluid physical properties, such as viscosity, solids pressure and stress, are derived from the kinetic theory of granular flow. According to Montante et al. [

Turbulence modeling is also critical to the success of CFD analysis in solid-liquid systems. Turbulence plays an important role in developing single-phase flow structures, and it is generally accepted that it will also have some influence on the solids distribution in stirred tanks. Previous CFD investigations have highlighted the importance of turbulence-assisted solids dispersion in stirred tanks [

To address the above, this study aims to develop a CFD modeling strategy for predicting cloud height in a solid suspension process. An Eulerian-Granular multiphase model is used to model the particle suspension process. Experimental data is used to justify the selection of model inputs, specifically the drag model and turbulent dispersion force. Each of these were evaluated independently to illustrate their relative importance. Finally, a comparison of the model results with corresponding experimental data establishes the validity of the modeling approach across a wide range of experimental conditions.

The experimental work was carried out at BHR Group as part of the fluid mixing processes (FMPs) industrial research program. The tests were performed in a transparent Perspex tank with a diameter

Schematic representation of the stirred tank reactor.

The solid phase was comprised of sand particles with a nominal average diameter of 180 ^{3}. The liquid phase was water with a density of 1,000 kg/m^{3} and a viscosity of 0.001 Pa-s. Solids concentrations of 10% (w/w) and 15% (w/w) were tested. These values correspond to 4% and 6.2% by volume, respectively.

The solids cloud height was obtained by measuring the distance from the bottom of the tank to the cloud surface. Measurements were determined visually from outside the tank. Given the turbulent nature of the flow, the cloud surface was not flat. Therefore, the minimum and maximum positions of the surface at a fixed radial position on the wall of the tank were recorded. The average cloud height

For both impeller systems, the minimum impeller speed for just suspension

Figure

Fractional average cloud height (

With the single-impeller system, there is no significant difference between the two different solid concentrations. There is a small difference at high impeller speeds with the dual-impeller system. However, this could be due to experimental error (typically 5 to 10%).

The simulations were performed using the Euler-Granular multiphase model available in the commercial CFD software, ANSYS Fluent 13.0 (ANSYS Inc., Canonsburg, PA, USA). This model assumes that each phase coexists at every point in space in the form of interpenetrating continua. The continuity and momentum equations are solved for all phases and the coupling between phases is obtained through pressure and interphase exchange coefficients.

The conservation equation for each phase

In the Eulerian-Granular model, the solid phase momentum equation includes an additional solid pressure term. The solids pressure and solids stress tensor terms

In (

The geometric models and meshes were constructed in the commercial software, GAMBIT 2.4 (ANSYS, Inc., Canonsburg, PA, USA). Full hexahedral meshes were constructed for each of the two tank configurations (Figure

CFD meshes for the single- and dual-impeller stirred tanks.

A grid-dependency study was carried out to evaluate mesh suitability for the single impeller configuration. The procedure involved comparing results from the baseline mesh to a refined grid with 3.2 million hexahedral cells, representing an almost 8-fold increase over the baseline mesh. The effects on one flow parameter—impeller power number—and one multiphase model output—cloud height—were extracted and compared (see Table

Grid dependency study for the single-impeller tank. (A) 10% (w/w) solids loading and 300 RPM, (B) 15% (w/w) solids loading and 250 RPM.

Test case | Baseline | Refined mesh | Difference (%) | |
---|---|---|---|---|

Case A | Normalized Cloud Height ( |
0.655 | 0.644 | 1.68 |

Power Number | 1.762 | 1.760 | 0.11 | |

| ||||

Case B | Normalized Cloud Height ( |
0.492 | 0.501 | 1.83 |

Power Number | 1.837 | 1.767 | 3.80 |

The realizable

The multiple reference frame (MRF) approach was used to model impeller rotation. Although an approximation, this approach has been shown to produce satisfactory results, especially when the interaction between baffles and impellers is weak. The free surface was assumed to be fixed, thus a symmetry boundary condition was prescribed. A no-slip boundary condition was applied for all other wall surfaces.

High-order discretization schemes were chosen to minimize numerical diffusion. The Green-Gauss Node Based Gradient option was used for gradient calculations at cell interfaces and the QUICK discretization scheme was used for momentum, volume fraction and turbulence equations. All cases were initialized with a uniformly distributed solid phase. The steady state solver was utilized to solve for all flow variables.

Liquid velocities provide insight into the mixing processes occurring inside stirred tanks. Figures

Contour plots for 200 RPM, 10% (w/w) loading for single- and dual-impeller configurations on a midplane. (a) Velocity vector of liquid phase. (b) Volume fraction of solid phase.

Contour plots for 400 RPM, 10% (w/w) loading for single- and dual-impeller configurations on a midplane. (a) Velocity vector of liquid phase. (b) Volume fraction of solid phase.

Drag forces induce significant interphase momentum transfer and tend to dominate other interphase processes. Therefore, drag force model selection is critical to the accuracy of the solid-liquid suspension model. The Gidaspow and Schiller-Naumann drag models were investigated. The Gidaspow drag model has been specifically tailored to take into consideration particle induced hindrance. For this reason, it is expected to give good predictions for high solid volume fractions (as was observed under certain operating conditions studied here). Additional simulations with the Schiller-Naumann model were performed to help elucidate the importance of particle hindrance in this system.

Figure

Effect of drag model. Volume rendered plots with isosurface for single- and dual-impeller configurations, 15% (w/w) loading, 300 RPM. The double-headed arrows marked “EXP” represent the location of experimental observation with arrow thickness representing the observed 8% spread in the experimental data.

Effect of drag model. Plot of normalized cloud height against normalized impeller speed for 15% (w/w) loading and single-impeller.

The contribution of the turbulent dispersion force is significant when the size of the turbulent eddies is larger than the particle size [

Effect of turbulent dispersion force (TDF). Volume rendered plots with isosurface for (a) single-impeller, 10% (w/w) loading, 400 RPM and (b) dual-impeller, 15% (w/w) loading, 150 RPM. The double-headed arrows marked “EXP” represent the location of experimental observation with arrow thickness representing the observed 8% spread in the experimental data. Gidaspow drag model.

Results from the previous sections establish the model requirements based on a limited set of experiments. Applying the modeling strategy over the full range of experimental operating conditions establishes model validation. The experimental study included single- and dual-impeller tanks operating at six to seven speeds and for two different solids concentrations (10% (w/w) and 15% (w/w)), resulting in a total of 26 experimental measurements. Numerical simulations were performed for each of these experimental points.

The profiles of the normalized cloud heights are plotted in Figures

(a) Plot of normalized cloud height against normalized impeller speed for 10% (w/w) loading; (b) plot of normalized cloud height against normalized impeller speed for 15% (w/w) loading.

A CFD modeling strategy was developed to predict cloud height in stirred tanks with single- and dual-axial PBT impellers. The CFD model was based on Eulerian-Granular multiphase theory. The effects of two different drag models, Gidaspow and Schiller-Naumann, and a drift velocity-based turbulent dispersion force were analyzed. The modeling strategy was developed for a limited set of experimental conditions and then applied across the full range solids loading, agitation rates, and reactor configurations. It was determined that the inclusion of turbulent dispersion force was critical to accurate prediction of cloud height for the conditions studied. The drag model was observed to have a much lower influence. The resulting modeling strategy was then compared to the full experimental range of cloud height measurements. The model was found to be predictive of cloud height across a broad range of experimental conditions, the exception being the single-impeller configuration at low agitation rates. It is the authors’ opinion that solids concentration-based inclusion of the TDF can be studied further to improve the model prediction at lower agitation rates.

Distance of lower impeller from the tank bottom, m

Distance of upper impeller from the tank bottom, m

Drag coefficient

Impeller diameter, m

Particle diameter, m

fluid-solid turbulent dispersion coefficient,

Lift, virtual mass, and any other external forces acting on the solid phase,

Liquid fill level, m

Cloud height, m

Interphase exchange coefficient,

Fluid-solid interphase mass exchange,

Impeller rotational speed, rpm

Just suspension speed, rpm

Pressure shared by all phases,

Relative Reynolds number

Fluid-solid interphase momentum exchange,

Tank diameter, m

Velocity of fluid phase, m/s

Interphase velocity, m/s

Velocity of

Velocity of solid phase, m/s

Phasic average velocity of fluid phase, m/s

Phasic average velocity of solid phase, m/s

Drift velocity, m/s.

Volume fraction of phase

Stress-strain tensor for phase

Kinematic viscosity of liquid phase, kg/m-s

Dispersion Prandtl number

Density of phase

Fluid phase

Exchange between fluid and solid phases

Phase number

Solid phase

Turbulent.

The authors gratefully acknowledge the invaluable help provided by Prem Andrade and Mohan Srinivasa (ANSYS, Pune) for their guidance during the CFD simulations.