PIV Measurements to Study the Effect of the Reynolds Number on the Hydrodynamic Structure in a Baffled Vessel Stirred by a Rushton Turbine

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3. Mathematical Formulation

To analyze different types of fluctuations induced by the Rushton turbine, an angle resolved measurement has been conducted. An encoder mounted on the shaft of the impeller enables the velocity field measurement to be synchronized. Thus, the three components of the velocity vector can be evaluated. The instantaneous velocity field is decomposed into average velocity and fluctuant velocity :

(1)

This is commonly known as Reynolds decomposition. Where is the instantaneous velocity, is the ensemble average velocity calculated for each plane as follow:

(2)

Where n is the number of acquisitions, and the over-bar denotes the time average.

The component is the fluctuant velocity directly obtained by subtracting the average velocity from the instantaneous velocity:

(3)

It is wide clear that the fluctuant velocity changes in sign with time evolution either positive or negative. To characterize turbulence strength independently whatever the sign, the term rms velocity calculated for the radial velocity are as follow:

(4)

The subscript rms stands for "root-mean-square". In mathematics, rms velocity is also known as quadratic mean. A larger indicates a higher level turbulence.

The turbulent kinetic energy field can be calculated using the formula:

(5)

With 2-D PIV data, there is no knowledge of the third rms velocity component (usually tangential) and k can only be estimated using a pseudo-isotropic assumption defined as follows:

(6)

This yields the following simplified formula:

(7)

This assumption was verified experimentally by Khan et al. ^[13] for a PBT using stereoscopic PIV data. In the later discussion. The turbulent kinetic energy is normalized using the square of the impeller tip speed.

The dissipation rate of the turbulent kinetic energy could be estimated from dimensional relations:

Where is a constant, the turbulence level in the main flow direction and a characteristic length scale.

The dissipation rate of the turbulent kinetic energy could be obtained directly from PIV output data. PIV enables the measurement of instantaneous velocities over an area, and therefore the instantaneous spatial gradients can be calculated and direct estimation of can be performed. In cylindrical coordinates it is calculated as follows:

(8)

Where is the effective viscosity, r and z are the coordinate system. This time-averaged equation contains unknown terms. In our case, 2-D PIV measurements have been conducted. The local isotropy hypothesis has been employed to overcome the lack of information of the remaining terms of the ε equation, where only twelve terms could be determined.

The use of the isotropic approximation to calculate velocity gradients evokes two concepts that must be clarified, the homogeneous turbulence and the isotropic turbulence. The distinction between these two concepts is very important. In fact, “Homogeneous turbulence” is defined as statistically invariant to shifts of the coordinate system, and “isotropic turbulence” is additionally invariant to rotations and reflections.

The Reynolds number Re is defined as follow:

(9)

Where is the fluid density, N is the impeller speed, d is the diameter of the impeller and is the fluid viscosity.

4. Measuring Technique

The flow was seeded with neutrally buoyant 20 µm polyamide particles (Figure 2). The acquisition system for the PIV measurements consists of a double-cavity 30 mJ per pulse at wavelength 532 nm Nd:YAG laser form QUANTEL and a PCO camera of 1600 x1200 pixels. The laser beam is converted to a sheet of laser of thickness 1.5 mm to provide a near two-dimensional analysis of the flow. The PIV image pairs are cross-correlated with a 32×32 interrogation window and 50 % overlap. The time between pulses was chosen to ensure that the maximum displacement does not exceed a quarter of the side of the interrogation area as described by Keane and Adrian ^[21] and Adrian ^[22]. Thus, the exposure time delay between images was set to 0.2 ms. This yielded a spatial resolution of 0.25 mm/pixels. A gray scale recorded image which represents the scattered light of embedded tracer particles in water flow mixed by a Rushton turbine derived from the experimental work at the laboratory. All distances are normalized with the impeller diameter d and the velocities are normalized with the blade tip velocity U_tip. The reference coordinate system is fixed at the bottom of the impeller axis and the impeller rotation is clockwise.

Figure 2. Gray scale recorded image of a sample

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5. Results and Discussion

Impeller speeds of N=100 rpm, N=150 rpm, N=200 rpm, N=250 rpm, N=300 rpm and N=350 rpm were used corresponding to Reynolds numbers of Re=17000, Re=25000, Re=33000, Re=42000, Re=50000 and Re=58000. In these conditions, the blade tip-velocities values vary between U_tip=0.52 m.s^-1and U_tip=1.83 m.s^-1. The measurement area covers the whole flow field along a plane over the right half of the vessel.

5.1. Velocity Field

Figure 3. Velocity vector map

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Figure 3 presents the time-averaged velocity field between two blades over 170 image pairs of instantaneous velocity fields for impeller speed varying over Reynolds numbers ranging from Re=17000 to Re=58000. According to these results, it is clear that the Reynolds number has a direct effect on the intensify values. However, the same distribution of the velocity field has been observed. In fact, the Rushton turbine generates two main vertical motions, above and below the impeller region. The jet flow coming off the impeller blades streams towards the wall and divides into two flows under the influence of the vessel wall. The first one is downward along the wall to the bottom of the vessel, then back to the impeller region. The second rises along the wall to the top of the vessel and back to the impeller region, forming double loop. The vector plot shows that the predominant direction of the stream is in the radial direction. For low Reynolds numbers, the flow motion indicates a slowdown in the upper section of the vessel. The flow tends to be strong and fully developed at high Reynolds numbers as shown in Figure 4.e and Figure 4.f. The highest values are close to the tip of the blade and decreasing with radial distance from the blade. Regions of slower flow in the upper section of the planar measurement window begin to develop as the field progresses with increased impeller speed.

5.2. Average Radial Velocity

The contour plot values normalized by U_tip as depicted in Figure 4, show maximum magnitudes located at the tip of the blade for different Reynolds numbers of Re=17000, Re=25000, Re=33000, Re=42000, Re=50000 and Re=58000. These values increase with rotational impeller speed of the stream jet in the radial direction along the impeller jet-stream. Indeed, the same distribution of the average radial velocity has been observed.

Figure 4. Contour plots of the average radial velocity

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5.3. Average Axial Velocity

Figure 5 shows the contour plots of the average axial velocity for different Reynolds numbers of Re=17000, Re=25000, Re=33000, Re=42000, Re=50000 and Re=58000. According to these results, it is clear that the Reynolds number has a direct effect on the intensify values. However, the same distribution of the average axial velocity has been observed. The above mentioned outward motion is divided into two flows under the influence of the vessel wall. One is downward (blue color) along the wall, to the bottom of the vessel, then back to the impeller region. The other rises along the wall (red color), to the top of the vessel and back to the impeller region. A comparison between radial and axial contour plots reveals the well known radial character of this type of turbines. The intensity of the axial component in the upward direction near the vessel walls may be up to 50 % of the blade tip velocity in the impeller stream region. The absence of velocity vectors at the top of the upper section of the vessel is caused by the aerated surface of the flow which leads to the loss of correlation between particle image pairs.

5.4. Axial Profiles of the Average Radial Velocity

In order to study the evolution of the radial discharge stream in the impeller region, a superposition of axial profiles of radial averaged velocities has been made. As expected, it is observable in Figure 6 that the maximum value is located at the dimensionless axial position equal to z=0.66 which corresponds to the bottom clearance normalized by the tank diameter. For the Reynolds number Re=17000, the maximum radial dimensionless velocity is equal to U=0.213 and reaches U=0.54 for Re=58000. It should be noted that the influence of the baffle facing the measurement area is unavoidable with the given configuration since it reduces significantly the velocity especially at its borders where the imaging of suspended particles is extremely difficult and consequently the correlation is affected.

Figure 5. Contour plots of the average axial velocity

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Figure 6. Axial profiles of the radial component of time-averaged velocity

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Figure 7. Evolution of the maximum value of the turbulent kinetic energy

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