• Angle Resolved
• energy dissipation
• SGS
• PBT
• PIV

CHAPTER 1

Introduction



Continuously stirred vessels and batch stirred vessels are the most common mixing operations in the chemical industry, frequently encountered in chemical processes which involve liquid blending, chemical reaction, gas dispersion in liquids, solid suspensions in liquids, as well as heat and mass transfer.
For this reason the study and the comprehension of the mechanisms behind mixing are of primary importance. For fast and competitive reactions, which occurs in liquid phase, the effect of an improvement in the mixing of the system, directly leads to higher selectivity and yield of the reactions, thus to a lower by-product formation. Mixing has direct effects on the economy of the production. The complex nature of the flow field generated, in the widely used stirred tanks reactors; ensure that its effects on process operations are being widely studied.
This work is concerned with the characterization of the hydrodynamic of stirred baffled vessel, studying the differences in the flow due to the change in the pumping configuration. The use of impellers in the up-pumping configuration rather than the down-pumping configuration, changes drastically the dynamics within the vessel, this radical choice in the flow configuration, leads to significant advantage if well thought (Chapman C.M et al., 1983).
In the present work, mixing is achieved through the use of a 45° 6 blades Pitched Blade Turbine (PBT), focusing the attention on the differences in the flow, due to the up-pumping or down-pumping modes. The power draw together with the power number, have been evaluated to provide the same power input per unit of mass in both the up-pumping and down pumping configurations, in order to obtain comparable proprieties form the two set ups.
The flow field on which are based the observations has been obtained with the use of two-dimensional particle image velocimetry (PIV), in the next chapters is fully described the principles behind this technique and its limits.
The most effective areas where to obtain a complete mixing in short time are located in the most energy dissipative areas. As is further explained in the next chapters, since is not possible to calculate the energy dispersion as an exact value, few methods exist for its estimation, and so a comparative study on the energy dissipation calculation and its effects on mixing have been carried out, also to the extent of understanding the reasons behind the different results of level dissipation estimated by the different methods.
The use of a synchronizer together with the use of a trigger, has made possible to acquire images with PIV, at certain blade angles. The angle resolved analysis technique leads to fully evaluate the actual flow around the impeller, highlighting the information that otherwise would have been hidden by the time averaged analysis. The chosen degree of separation, gives higher spatial resolution compared to the previous work (Kahn F.R, 2005).
Angle resolved vorticity has been evaluated as function of the angle with the blade, to measure the distance of the center of the first loop vortex from the centre of the impeller.
















CHAPTER 2

2.1 Theory on stirred vessels

A measure of the average turbulence in the system mechanically agitated is obtained by the Reynolds number of the impeller:

The condition of complete turbulent flow is achieved for Reynolds numbers higher than 104.
- Impeller












- Baffles
2.2 Flow Measurements

The flow mapping techniques are intended to determine experimentally . Flow mapping techniques can be summarized in two main categories, single-point and whole flow field measurement. For what concerns the whole flow field measurements, in every acquisition, information about the flow field are obtained over a t.

2.2.1 Single point measurements
This measurement technique is commonly used in turbulence obtained by flow fields generated by free jets (W.R. Quinn, 2005).

Figure 2.2: Removing directional ambiguity with frequency shifting. Whole flow field measurements

- Particle image Velocimetry (PIV)
The PIV is based on the images of tracer particles suspended in the flow. Figure 2.4: Schematic representation of the cross – correlation

2.3 Power number

The impeller is characterized by its power number, . The low impellers, compared with high impellers, operating at same power input P and same geometry, require less torque ( ) and are operated at higher speeds. Impellers widely used like the Ruston turbine, are high torque impellers, so they require lower speeds.
2.4 Flow number

The flow number (Fl) is the measure of the pumping capacity of an impeller. Fl is a dimensionless group, defined as the ratio between the effective non entrained pumped flow, and an impeller characteristic flow rate:

This surface would be circular for an axial flow impeller and a section of a cylinder wall for a radial flow impeller.
2.5 Turbulent kinetic energy

The kinetic energy is the energy associated with the flow, and does not give a measure of the effective mixing. Local energy dissipation rates

The resolutions in terms of time space and velocity needed to fully measure the energy dissipation are the Kolmogorov scales. Figure 2.7: Schematic model of production and dissipation of turbulent
kinematic energy, (J. Sheng et. Al, 2000)
In other words the turbulent characteristics of the flow remain frozen in time.
Taylor’s hypothesis is applied specially to point measurements, to solve the problem of measuring the spatial covariance needed to evaluate the turbulent energy dissipation. Locking the dynamic features of the turbulent flow. In the energy dissipation estimation, the PIV together with others full-field techniques, offers the advantage of obtaining spatial resolved instantaneous velocity in a plane.











2.6.1 Energy dissipation and PIV

- Estimating the local energy dissipation from dimensional analysis
Wu and Patterson, (1989), proposed the following equation to calculate the energy dissipation directly from the turbulent kinetic energy:
Where the is the integral length scale, which represents the correlation distance between to velocity points in the flow field. The length scales inside a stirred vessel depend on the homogeneity of the flow field. Direct measurement of the fluctuating velocity gradient terms in the definition of energy dissipation
Considering a time averaged turbulence energy dissipation, it can be written (Hinze, 1975):

In figure 6, the graph of the dissipation rate spectrum, shows the rate of dissipated energy that can be measured, increasing the spatial resolution until Kolmogorov length scale.


The term of the dissipation of the turbulent kinetic energy contains information on the unresolved scales:

is the strain rate tensor computed from the PIV velocity fields.
Therefore the turbulent dissipation rate can be approximately computed by the Reynolds averaged SGS dissipation rate, . Vorticity

Chapter 3

3.1 Vessel and Impellers

Figure 3.1: Perspex impellers a) PBTd b) PBTu

Figure 3.2: Schematic representation of the vessel


3.2 Measure of the power number
3.3 Vessel configuration and PIV

Figure 3.4: Vessel during acquisitions of the images.
Figure 3.5: Baffle-laser configuration

Image analysis and vector fields
3.4 Angle resolved measurements

Figure 3.6

Figure 3.6: Trigger to obtain acquisitions at the same angle
4.1 Comparison of power number and power consumption for the two flow configurations

The investigation on the energy input passed through the measurement of the power number for the two impellers. Figure 4.2 Power Number profile at variable RpM


















4.2 Power characteristics and time-averaged flow field for a PBT in up-pumping and down pumping configurations

The time averaged analysis, were obtained as described in the chapter on materials and methods from 500 instantaneous flow fields (1000 frames), fluctuations in the velocity that can be seen comparing different instantaneous images of the flow, are averaged out in the time average (H.S. Yoon et al., 2001).

4.2.1 Time averaged flow field and instantaneous flow profiles
Figure 4.3: a), b) and c) Are typical instantaneous velocity fields, and d) is the average velocity field obtained.

4.2.2 Comparison between time averaged flow field configurations
In the following graphs the time-average flow fields are compared. The structures of the two flow configurations are evident.

Figure 4.4: Time average flow field. In figure 4.4 is clear how the two flow fields generated from the two flow configurations differ from each other. Dissipation of turbulent kinetic energy
The levels of turbulent kinetic energy differ between the two configurations.




Figure 4.5: Turbulent kinetic energy in time averaged configuration


The information evinced in Figure 4.5, describe an overall high values of turbulent kinetic energy in the down pumping configuration, and a more localized area of high turbulent kinetic energy in the up pumping mode. Energy dissipations rates evaluated through dimensional analysis
In the analysis of the dissipation rates of the turbulent energy, the same energy distribution pattern is observed because a constant integral length scale as been adopted.



Figure 4.6: Time averaged turbulent kinetic energy dissipation rate evaluated with dimensional analysis.

The discharge zone near the impeller, in the up pumping configuration, has a larger zone where Ewp/N3D2 exceeds 300. Form Figure 4.7, the area where the maximum values of the energy dissipation rate are present and their extension result very clear, with clearly an advantage in terms of dissipation rate, for the up pumping configuration. Figure 4.7: Time averaged turbulent kinetic energy dissipation rate evaluated from the direct calculation from the definition.

4.2.6 Large eddy PIV method
In Figure 4.8 is shown the results of the application of the large eddy PIV method over the time average flow field. The values of Esgs/N3D2 higher than 0.02, define the area where highest is the dissipation rate.



Figure 4.8: Time averaged turbulent kinetic energy dissipation rate evaluated with the Large eddy PIV method.
4.2.7 Vorticity
Figure 4.9: Time averaged Vorticity in the two flow configuration.
4.2.8 Flow number evaluation
The flow number (NQ) is a dimensionless number that of the pumping capacity of the impeller Eq .2.9. The distances from the blade decided, where chosen after analyzing the flow field, in order to measure the fully developed flow without taking in account the entrained flow, as discussed in the second chapter.






4.3 Angle resolved PIV measurements for a PBT impeller in up-pumping and down-pumping configurations

Triggering the blade passage reduces the noise due to the macro velocity fluctuation, as blade passage, which can be identified in the time average measurement, from any energy spectrum analysis, or a frequency analysis (Yianneskis et al., 1987). Angle resolved flow field
The flow near the impeller analyzed function of the angle of the blade, in other words, function of time, gives an actual continuous flow.

0°) 15°)






30°) 45°)


Figure 4.10: Up pumping flow field at different angles


The flow around the impeller has a periodicity of 1/6 of the impeller revolution, as observed above. As shown from the flow field in figure 4.10.








0°) 15°)

30°) 45°)


Figure 4.11: Down pumping flow field at different angles

In the discharge region near the impeller, is found the maximum normalized velocity, clearly its value, varies with the blade position. The flow induced by the impeller in the main loop, maintains constant characteristics varying the blade angle. 4.3.2 Turbulent kinetic energy
The angle resolved study of the turbulent kinetic energy, is shown in the following figures.
Figure 4.12: Down pumping turbulent kinetic energy

For what concerns the up-pumping configuration, the different flow profile, determines significantly different values of the kt*. As observed in Figure 4.13.
0°) 15°)

30°) 45°)


Figure 4.13: Up pumping turbulent kinetic energy

From the set of images in Figure 4.13, the periodicity of the turbulent kinetic energy fluctuation is highlighted. Comparing the results from the two configurations, described by the Figure 4.12 and Figure 4.13, is clear that in the up pumping configuration, significantly higher values are observed.

4.3.3 Energy dissipations rates evaluated through dimensional analysis
Figure 4.14: Energy dissipation evaluated with the dimensional analysis. Down pumping configuration

The results proposed by this method of the energy dissipation are obtained from the results of the calculation of the turbulent kinetic energy. Lower values are observed in the down pumping configuration, coherently with the values of kt* previously measured.

0°) 15°)

30°) 45°)


Figure 4.15: Energy dissipation evaluated with the dimensional analysis. Up pumping configuration

The results obtained with the dimensional analysis do not give any substantially new information to the analysis of the two flow configurations. For these reasons has been adopted another method for the energy dissipation estimation; estimating the turbulent energy dissipation rate, from its fundamental definition, as explained in the second chapter.

According to previous works (P. Saarenrinne et al., 2000), has been calculated the energy dissipation rate spectrum, and its distribution function Figure 2.10, need a spatial resolution of to resolve 90% of the dissipation rate, and with a resolution of only 65% of the total energy dissipation rate is achievable. In Figure 4.16, are illustrated the results obtained for the two flow configurations.

Figure 4.16: Turbulent kinetic energy dissipation rate calculated from the definition.

Exact measurements of the turbulent kinetic energy dissipation rates, should be done to the Kolmogorov length scale. The rates of dissipation of the turbulent kinetic energy are higher in up-pumping configuration, also with this method of evaluation.
Values of the turbulent kinetic energy dissipation rate obtained from the direct calculation in this paragraph are in accord with the results obtained from previous works (Sheng J. el. al., 2000).
4.3.5 Large eddy PIV method
By overlapping and comparing the results of the flow field (Figure 4.10 and Figure 4.1 1), with the energy dissipation contour plots in Figure 4.17, it evidences that the area where the main energy dissipations occur is not exactly on the discharge of the impeller, but in the area, where is present the highest shear stress. While the second area with the sensibly higher energy dissipation rate, is found in the up pumping configuration where the upper recirculation loop, comes down towards the impeller. Figure 4.17: Turbulent kinetic energy dissipation rate evaluated with Sub-Grid-Scale model.
-Energy evaluation comparison
Is evident that the energy dissipation levels for both up and down configurations, obtained with the Sub-Grid-Scale Smagorinsky model, are the closer to the value of energy input evaluated previously.













Table 4.2: Energy comparison results

The information obtained from the angle resolved vorticity plots give a better understanding of the flow in the vessel. In Figure 4.19 is clear how the high energy dissipations zones move away from the blade for every angle.
0°) 15°)

30°) 45°)


Figure 4.19: Vorticity function of the blade angle in the down pumping configuration

Figure 4.20: Vorticity function of the blade angle in the down pumping configuration

For instance in the 30° angle can be observed that above the impeller the outward moving flow, rotating clockwise, pushes away the further fluid, creating an entrained flow with counter clockwise rotation. This can explain the presence above and under the impeller, respectively for the two flow configurations, of two separated different areas of high energy dissipation rate.






4.3.4 Flow number as a function of the blade angle
The minimum flow number is obtained for either configurations between 15° and 18°. Figure 4.21: Flow number at different blade angles
The flow number was also investigated.

The first observable difference when studying the flow for the two configurations, obtained by a 6-PBT, is the structure of the flow field. In the down pumping configuration, the flow off the blade tip is pumped down, towards the bottom of the vessel, and flows back to the top of the tank, creating one main counterclockwise loop, and a small clockwise loop beneath the impeller.

The turbulent kinetic energy was measured in the vessel for every operating condition. The dissipation of the turbulent energy, which has seen, occurs at lower length scales. The low spatial resolution of the system made impossible to measure satisfactorily the energy dissipation rates. The same periodicity was found either for the turbulent kinetic energy and the energy dissipation rates.

The flow number also has been evaluated function of the blade angle and compared between the configurations. To achieve a change in flow configuration, was used a second impeller, with out changing the rotational direction of the impeller, thus for the two flow configurations, the flow number versus the blade angle, followed the same profile.