5 Data Analysis

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5 Data Analysis

5.1 Multiphase System

5.1.1 Comparison With Previuos Works

It is interesting to compare data collected in the present work with those reported by Guiraud et al. ( 1997) manly because of the similar geometry adopted and to figure out what could be the influence of few different parameters.

Differences are summarized in table 5.1.1.1 :

Present Work Guiraud et al. ( 1997)

Impeller Type 45° Pitched Blade Turbine Mixel TT

Particle Diameter ₅₀₀_µ_{m 253}_µ_m

Particle Density 2855 kg/m3 _{2230 kg/m}3

Table 5.1.1.1: Differences with Guiraud et al. ( 1997)

The coordinate system in the following figures is the same of Guiraud et al. ( 1997) with lengths normalized with impeller diameter and the zero for the axial coordinate taken at the impeller plane.

The concentration chosen for comparison is the highest one, i.e. 0.7% by weight, while in Guiraud et al’s work was around 1.1%.

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r/D=0.464 z/D -1.0 -0.5 0.0 0.5 1.0 1.5 Vz/Vtip -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 liquid only continuous phase solid phase r/D=0.464 z/D -1.0 -0.5 0.0 0.5 1.0 1.5 Vz/Vtip -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 liquid only continuous phase solid phase

Table 5.1.1.2: Axial profiles of the mean axial velocity, down pumping: present work (left) and Guiraud et al. ( 1997 ) (right)

Table 5.1.1.3: Axial profiles of the mean axial velocity, down pumping: present work (left) and Guiraud et al. ( 1997) (right)

Table 5.1.1.4: Axial profiles of the mean axial velocity, down pumping: present work (left) and Guiraud et al. ( 1997) (right)

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Observation of the figures suggests that the shape of the curve in the lower part of the vessel ( r/D=0.464 and z/D<0 ) is closely associated with the flows generated by the two different impellers.

With a PBT velocities in this zone were always greater than those obtained with a Mixel TT, whether in the presence of solid or not. The same trend can be observed in the upward flow: Pitched Blade had a greater pumping capacity and in the upper part both phases were still going upward at r/D=0.679. In the same way closed to wall, higher velocities were found for the PBT.

For both impellers the inertial force of the solid particles accelerates the liquid which is, in downward flow, faster than the single phase case.

5.1.2 Settling Velocity in Turbulent Flow

The technique adopted to evaluate settling velocity depended on the software employed for data processing. Each contour plot generated by the software was created from a grid of points, where the distance from one point to a contiguous one was around 0.5 mm. The two components of the slip velocity, radial and axial, were then calculated by subtracting the velocity values of the liquid phase at one point from the velocity values at the same point of the solid phase.

The subtraction was done between two average contour plots for the same conditions thus only one plot for the relative velocity was available and even a small error in the data processing was enhanced.

In section 4.1.2 it has been pointed out that an increase in solid amount, at these low concentrations, had no influence on solid and liquid, axial and radial, velocity, except for the impeller discharge region in downward flow.

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displacements in high scattering zone, we decided to average data at different concentrations with the same flow and the same impeller speed. Instead of looking at the whole vessel particular attention was focused to the main recirculation loop generated by the impeller.

The loop was divided into 17 smaller zones chosen on the strength of close velocity values as it is shown in figure 5.1.2.1.

Figure 5.1.2.1: Regions of interest

In was not possible to follow the loop more closely between zones 11-15 because of a reflection caused by the high laser power required, that obstructed signal from the solid phase (see figure 5.1.2.2).

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Figure 5.1.2.2: light reflection

The same zones with the same limitations were chosen in up pumping flow but the light reflection caused much more problems because it influenced also the first part of downward flow closed to the vertical wall.

The coordinate system adopted referred to an angular coordinate defined as the angle from the centre of the blade to the centre of each zone (see figure 5.1.2.3).

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Positive values of α were in counter-clockwise direction for downward flow, vice-versa in clockwise direction for upward flow.

Settling velocity was calculated in each zone for all the impeller speeds and plotted as a function of the angle α; positive values referred to a relative axial velocity going upward, negative values to a relative axial velocity going downward.

For comparison with the static case, settling velocity in a still fluid, calculated with equation 2.1.1.7 is also reported.

Angle 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 axia l slip vel ocity -0.200 -0.175 -0.150 -0.125 -0.100 -0.075 -0.050 -0.025 0.000 483 rpm 609 rpm 696 rpm 767 rpm

settling velocity in still fluid

Figure 5.1.2.4: axial slip velocity down pumping

Figure 5.1.2.4 has not an obvious interpretation. Some explanations can be proposed based on physical consideration.

Zone 1 (12°) is the zone closest to the discharge region: relative axial velocity is negative, thus, because the flow is going downward, the solid phase led the liquid phase. Values are greater than the settling velocity in

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still fluid (0.085 m/s) and it could be due to the inertial force acting on the particles. This inertial force is directly dependent on the impeller speed, in fact it enhances with increasing rpm.

Between 12° and 45° (zones 2-3-4-5-6-7) the inertial force decreases with moving away from the blade and, as a consequence, particles velocity decreases to values near the settling velocity in a still fluid. Because the turbulence levels are quite high in these zones for the lower impeller speeds (VZS - VZL) tends to values less than 0.085 ( Magelli et al. 1990). In zone 8 (45°) the flow direction is changing to upward, thus negative values of relative axial velocity means that the solid phase lags behind the fluid.

Near the wall in upward flow (zones 9-10-11-12) the liquid is always faster than the solid. It is worth of note that because energy dissipation rate ε enhances with increasing rpm, at higher values of impeller speed correspond higher value of ε, then lower value of the integral length scale λ , therefore lower dynamic settling velocities (in absolute value).

Around 140° (zones 13-14) axial velocities both for solid and liquid are almost zero for the lowest impeller speeds. Thus it is like in a still fluid and relative velocities tend to 0.085 m/s.

In zone 15-16-17 (beyond 140°) the flow is again downward and particles lead the fluid. Relative velocity tends to decrease because the solid accelerate less than the liquid because of particles inertia.

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5.2 Study of Turbulence

In Chapter 4 many data concerning radial and axial location of trailing vortices have been presented, both for Rushton and for Pitched Blade Turbine. The technique employed to figure out such data has never been employed before thus a deep comparison with what is already present in the literature is necessary and this is the aim of the following paragraphs.

5.2.1 Rushton Turbine

Many authors reported the trailing vortex axis location both in radial and axial coordinates but only for the first 60° degree behind the blade.

Some experiments were performed by using LDA or LDV (Yianneskis & Whitelaw, 1993; Stoots & Calabrese, 1995; Lee & Yianneskis, 1998), other by using a photographic technique (Van’t Riet and Smith, 1975). All these authors found the shape of the axis by determining the location where the axial velocity component perpendicular to the axis was zero.

Sharp and Adrian ( 2001) employed a 3D PIV system and measured the mean velocity field in the r-z and θ-z planes. The trailing vortices were detected after the subtraction of the jet and circumferential components. In figure 5.2.1.1 are shown the results of these authors compared with those of the present work.

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0.00 0.25 0.50 0.00 0.25 0.50 0.00 0.25 0.50 0.00 0.25 0.50 0 30 60 90 120 150 180 210 240 270 300 330 337 rpm 425 rpm 535 rpm

Theory (Sharp, Adrian, 2001) Theory (Lee, Yianneskis, 1998) Theory (Stoots, Calabrese, 1994) Theory (Van't Riet, Smith, 1975)

Blades and Baffles

Figure 5.2.1.1: Rushton, upper trailing vortex, radial location

It can be seen from fig. 5.2.1.1 that the axis found in the present work is very similar to those reported by Lee and Yianneskis ( 1998) and Sharp and Adrian ( 2001), and to a small extent to that reported by Stoots and Calabrese ( 1995).

In the former the system geometry differed from that of the present work for the clearance which was C=T/3 and for the top of the vessel that was closed by a lid.

Sharp and Adrian ( 2001) adopted an open vessel without baffles with clearance C=T/2. It can be seen that the presence of the baffles do not influence vortex behaviour.

In the latter the geometry was exactly the same of the present work. Reynolds number, 37500, was also close to the regimes here adopted.

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In our experiments the axis remained within r/T=0.25 until 85° from the blade then moved away reaching r/T= 0.35 at 140°.

Concerning the axial location only few authors reported some data. Sharp and Adrian ( 2001) correlated their data with a linear fit:

z_± =±

(

0.06+0.08θ

)

(5.2.1.1) where the positive and negative signs refers to the upper and lower vortices respectively. Therefore they found both vortices moving away from the impeller plane in opposite directions.

On the contrary Escudiè et al. ( 2004) observed both vortices going upward with increasing the angle from the blade but they reported only qualitative figures without any specific data, thus it was not possible to make a comparison. Angle 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 z/H -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 337 rpm upper 337 rpm lower 425 rpm upper 425 rpm lower 535 rpm upper 535 rpm lower

Theory (Sharp, Adrian, 2001)

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It can be seen from figure 5.2.1.2 that Sharp and Adrian’s ( 2001) correlation, even though proposed for the first 60° only, fitted quite well following angles for the upper vortex. It has to be reminded that from 0° until 50° measurements are not correct because of the presence of the blade that obstructed the view.

The lower vortex seemed to move parallel to the upper one but with a constant difference.

As reported on previous page Escudiè et al. ( 2004) observed a similar trend (see figure 5.2.1.3) but they didn’t quantify it.

Figure 5.2.1.3: Rushton, trailing vortices, axial location (Escudiè et al. 2004)

A possible explanation for the differences between these works could be that Sharp and Adrian ( 2001) carried out their experiments in an unbaffled tank, and it could have influenced the vortex axis.

However, concerning the radial position, their work was very close to the present one suggesting that the presence of baffles shouldn’t have any effect in the impeller swept region.

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5.2.2 Pitched Blade Turbine

For Pitched Blade Turbine data available for comparison were only those of Schafer et al. ( 1998). Radial and axial location of the vortex axis were reported for vessel with H=T and clearance C=T/3. The impeller diameter was D=T/3 and a lid was located above the liquid surface.

0.00 0.25 0.50 0.00 0.25 0.50 0.00 0.25 0.50 0.00 0.25 0.50 0 30 60 90 120 150 180 210 240 270 300 330 337 rpm 425 rpm 535 rpm

Theory (Schafer et al. 1998) Blades and Baffles

Figure 5.2.2.1: PBT, trailing vortex, radial location

It can be seen from the figure above that the clearance difference didn’t affect the radial profile of the vortex.

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Concerning the axial location the same comparison is reported in figure 5.2.2.2. Zero on the axial coordinate refers to the impeller plane.

The slightly lower slope of Schafer et al.’s ( 1998) data could be related to the different number of impeller blades, i.e. four instead the six of the present work. More blades have a greater downward pumping capacity that could have forced the vortex to move down.

Angle 0 10 20 30 40 50 60 70 80 90 100 110 120 z/H -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 534 rpm 673 rpm 848 rpm

Theory (Schafer et al. 1998)

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Another comparison is reported on figure 5.2.2.3. Vortex radius divided by distance from the blade along the vortex axis is plotted with increasing the angle. 45° 6-PBT angle 0 10 20 30 40 50 60 70 80 90 100 110 120 r/Y 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 574 rpm 673 rpm 848 rpm

Theory (Shafer, Yianneskis, 1998)

Figure 5.2.2.3: PBT vortex radius

Even in this case data collected in the present work are in really good agreement with those reported by Schafer et al. ( 1998) and confirm that a difference in the impeller clearance do not have a remarkable effect on the structure of the trailing vortex.

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5.2.3 Rushton Turbine: Reynolds Stress

The Reynolds stress is a measure of the anisotropy of the turbulent velocity fluctuations, which produce a stress on the mean flow.

The Navier-Stokes equation, for an incompressible fluid, can be written as:

( )

_       ∂ ∂ + − − ∂ ∂ + = ∂ ∂ j i j i ij i i i x v v v p x F v t ρ ρ δ ρ µ (5.2.3.1)

where ρ is the density of the fluid

viis the velocity in the I direction

Fi is the force exerted on the fluid in the I direction xi is the Cartesian coordinate

p is the pressure

δij is the Kronecker delta

Breaking up all the quantities into mean and fluctuating parts and taking an average all the terms linear in fluctuating variables gives zero, thus:

( )

_       ∂ ∂ + − − − ∂ ∂ + = ∂ ∂ j i j i j i ij i i i x v v v v v p x F v t ρ ρ δ ρ ρ µ ' ' _(5.2.3.2)

Eq. 5.2.3.2 is often called Reynolds equation and is very similar to the Navier-Stokes equation apart from the additional term ' '

j

iv

v that is known as the Reynolds stress.

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PIV enables the measurement of instantaneous velocities over an area, and therefore the instantaneous radial and axial velocities can be calculated and a direct estimation of the radial-axial component of the Reynolds stress can be performed.

The latter is reported normalised by 2 TIP

V in the following figures.

337 rpm R12/Vtip2 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 z/H -0.3 -0.2 -0.1 0.0 0.1 0.2 r/T=0.1 r/T=0.2 r/T=0.3

Figure 5.2.3.1: Axial profiles of the radial-axial component of the Reynolds stress tensor

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Figure 5.2.3.3: Axial profiles of the radial-axial component of the Reynolds stress tensor

It can be seen that as the impeller speed increases the Reynolds stress decreases in the impeller swept region.

Escudiè et al. ( 2003) employed a two-dimensional PIV system to study the flow generated by a 6 blades Rushton turbine. The impeller diameter D was equal to the clearance C=T/3. The blade width was tb=D/75.

Derksen et al. ( 1999) studied a similar geometry with a Laser Doppler Anemometry. The blade width in their impeller was tb=D/48.

In a recent work, carried out at the Department of Chemical Engineering of the University of Pisa, Galletti et al. ( 2004) adopted a geometry similar to that of the present work with C=T/2 and D=T/3 and the blade width

tb=D/100.

A comparison with these works is reported in figure 5.2.3.4 for the radial position r/T=0.2.

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r/T=0.2

R12/Vtip2 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 z/H -0.3 -0.2 -0.1 0.0 0.1 0.2 present work (337 rpm) Galletti et al. (2004) Derksen et al. (1999) Escudiè et al. (2003)

Figure 5.2.3.4: Axial profiles of the radial-axial component of the Reynolds stress tensor, comparison

It can be observed a good agreement in the shape of the curve with a negative peak below the impeller plane and a positive peak above it.

However, the magnitude of the former peak is significantly lower than those reported by Escudiè et al. ( 2003) and Galletti et al. ( 2004), while it’s very close for the latter peak.