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### Turbulence Characteristics in a Rushton & Dorr-Oliver Stirring Vessel: A numerical investigation

Vasileios N Vlachakis

06/16/2006

Outline of the Presentation

- Introduction/Motivation
- Background of the Flotation process
- Mechanically agitated vessels
- The Rushton Stirring Tank
- Computational Model
- Comparisons between them
- The Dorr-Oliver Stirring Tank
- Conclusions
- Future Work

Introduction/Motivation

The objectives of the thesis are to:

- study the hydrodynamics of two stirring tanks
- The Rushton mixing tank
- The Dorr-Oliver
- estimate accurately the velocity distribution
- discuss which turbulent model is the most suitable for this type of flow (validation with the experiments)
- determine the effect of the clearance of the impeller on the turbulence characteristics
- Vorticity
- Turbulent kinetic energy
- Dissipation rate

Significance of the Dissipation rate

- Dissipation rate controls:
- Collisions between particles and bubbles in flotation cells
- bubble breakup
- coalescence of drops in liquid-liquid dispersions
- agglomeration in crystallizers

Background

Flotation is carried out using

Mechanically agitated cells

Widely Used in Industries to

separate mixtures

- Mining
- Chemical
- Environmental
- Pharmaceutical
- Biotechnological

Principles of Froth-Flotation

The flotation process

- The flotation technique relies on the surface properties of the different particles
- Two types of particles:
- hydrophobic (needs to be separated and floated)
- hydrophilic
- Particles are fed from a slurry located in the bottom
- While the impeller rotates air is passing through the hollow shaft to generate bubbles
- Some particles attach to the surface of the air bubbles and some others fall on the bottom of the tank
- The floated particles are collected from the froth layer

The Rushton Stirring Tank

Cylindrical Tank

Diameter of the Tank

Diameter of the Impeller

Four equally spaced baffles with width

Thickness of the baffles

Blade height

Blade width

Liquid Height = Height of the Tank

Governing Equations

Unsteady 3D Navier-Stokes equations

Continuity

Momentum

Decomposition of the total velocity and pressure

Averaging rules

Time-averaged Navier-Stokes equations

Continuity

Momentum

Dimensionless ParametersScaling Laws

The Reynolds number:

Laminar flow: Re<50

Transitional: 50<Re<5000

Turbulent: Re>10000

The Power number:

Where a=5 and b=0.8 in the case of radial-disk impellers

In our case where

This Power number is hold for unbaffled tanks

Dimensionless ParametersScaling Laws

Froude number:

The Froude number is important for unbaffled tanks

It is negligible for baffled tanks or unbaffled with Re<300

In unbaffled tanks for Re>300

Flow number:

In the case of the radial-disk impellers

In our case (Rushton turbine) : Fl=1.07

Computational Grid

The computational grid consists of 480,000 cells

Grid surrounding the impeller (The unsteady Navier - Stokes equations are solved)

Outside grid (The steady Navier - Stokes equations are solved)

Two frames of reference:

The first is mounted on the

Impeller and the second is

stationary (MRF)

View from the top

3Dimensional View

The grid surrounding the impeller is more dense from the outside

Normalized radial velocity contours

The flow for the first two cases can be described as a radial jet with two recirculation regions in each side of the tank

In the case of the low clearance, a low speed jet and only one large

recirculation area is observed

Normalized dissipation rate contours

In the first two cases the dissipation rate has high values around and next

to the impeller’s blade while in the last is extended to the region below them too

Normalized TKE contours

Slices that pass through the middle plane of the impeller

The TKE is lower in the case of the low configuration

Normalized X-vorticity contours

Re=35000

In the first two cases the tip vortices that form at the end of the moving blades

can be observed while in the third case only one big vortex ring forms.

Y- Vorticity

Time-averaged experimental

results

Trailing vortices at the next blade

Trailing vortices at the 1st blade

Trailing Vortices at y/Dtank=0.167

(exactly at the end of the blades)

Vorticity superimposed with streamlines for Re=35000

Flow can be described as a radial jet with convecting tip vortices

Normalized Z-vorticity contours

In the first two cases the presence of the trailing vortices that form behind

the rotating blades can be seen.

In all cases small vortices also form behind the baffles

Radial Plots for Re=35000 along the centerline of the impeller

Normalized radial velocity

Normalized velocity magnitude

The velocity magnitudes consists only of the axial and radial components

in order to be validated by the experimental results where the tangential

component Is not available.

The low speed jet in the case of the low configuration is confirmed but a

strong axial component is present as it is shown in the second plot

Radial Plots for Re=35000 along the centerline of the impeller

Normalized tangential velocity

Re=35000

Normalized X-Vorticity

Experimental vorticity seems to be oscillating due to the periodicity and due

to the fact that trailing vortices are present.

Clearly none of the turbulent models can capture what is happening

Radial Plots for Re=35000 along the centerline of the impeller

Normalized Dissipation rate

Normalized Turbulent Kinetic Energy

The RNG k-e model has a superior behavior among the studied

turbulent models in predicting the Turbulent Dissipation Rate (TDR)

The apparent discrepancy in TKE is due to the periodicity that characterizes

the flow, since with every passage of a blade strong radial jet is created.

Normalized Maximum Dissipation rate

For C/T=1/2 and C/T=1/15

For C/T=1/3

As the Re number increases the maximum TDR decreases for the first two

configurations (agreement with the experimental data)

For case of the low clearance configuration the line of the maximum dissipation

levels off.

Reynolds Stresses & Isosurfaces

C/T=1/3

u’w’ normalized component of the RS

Isosurfaces of vorticity

The higher the helicity the more the vorticity

vector is closer to the velocity vector (swirl)

Helicity

Isosurfaces of helicity

Conclusions

- The turbulent kinetic energy and dissipation have the highest values in the immediate neighborhood of the impeller
- Good agreement with the experimental data is succeed
- Most of the times the Standard k-e model predicts better the flow velocities and the turbulent quantities while in some others has poor performance and the RNG k-e is better
- In the case of the low configuration model:
- there is a strong tendency to skew the contours downward
- the dominant downward flow is diverting the jet-like flow that leaves the tip of the impeller downward, and it convects with the turbulent features of the flow.
- The axial component of the velocity has high values

Future Work

- Experimental predictions for the Dorr-Oliver Flotation cell
- Comparisons of the studied cases with the experiments
- More Re numbers and clearances for the Dorr-Oliver Cell
- Higher Re numbers for both Tanks (100000-300000)
- Unsteady calculations
- Extension to two-phase or three phase flows

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