(256b) 3D-PTV Measurements in an Agitated Vessel with Newtonian and Non-Newtonian Fluids

Mixing of liquids in agitated vessels is one of the most common and yet complex operations in the process industry⁽¹⁾. It is influenced by numerous factors, including the rheology of the fluid. Flow measurements are crucial for a better understanding of the mixing phenomena, predictive purposes and the design, optimisation and scale-up of the equipment^{(1, 2, 3)}. 3D Particle Tracking Velocimetry (3D-PTV) is a flow visualization technique based on the individual tracking of tracers within the flow, enabling all the three dimensions of the Lagrangian velocity field to be obtained. The use of relatively large tracers (200 – 1000 µm) eliminates the need for expensive laser illumination. Single-camera setups can cut the costs even further⁽⁴⁾, at the expense of a reduction in spatial resolution and accuracy. For these two reasons, 3D-PTV is becoming an attractive alternative to the traditional laser-based velocimetry techniques, such as Particle Image Velocimetry (PIV) and Laser Doppler Anemometry (LDA). However, there is still a lack of research where 3D-PTV is applied to mixing in stirred vessels⁽⁴⁾. This is because the volumes and the large range of velocities involved represent a big challenge.

In this work, 3D-PTV with a single camera setup has been used to measure the velocity fields of Newtonian and non-Newtonian fluids in a 4.5 L cylindrical vessel (T = 180 mm, H/T = 1), equipped with a 60 mm Rushton turbine (D/T = 1/3, c/T = 1/3) and operated in the transitional flow regime.

Synchronous image sequences of the flow at 1,024×1,024 pixel resolution were recorded from two viewing orientations, using a high-speed camera and a mirror arrangement. The data processing involved the following steps:

1. The tracers in the two image planes were detected through an image analysis algorithm and their centroids were determined with subpixel precision.
2. The stereo-calibration of the two viewing orientations allowed, at each time step, the correspondences between the tracers in the two image planes to be established and the determination of their coordinates in the 3D space. The uncertainty in the 3D positions of the tracers depends greatly on the quality of the calibration process. Typical values were in the order of 100 µm.
3. The trajectories were extended step by step by finding the correct link between the current position and that at the next time step. For each particle at a certain time step, a search volume in the following step was centred at the predicted new position. The correct link among a list of candidates was found by minimizing the Lagrangian acceleration.
4. The velocity data along the trajectories were obtained by discrete differentiation of the coordinate vectors with time. A Savitzky-Golay low-pass filter was applied to the coordinates before the differentiation to limit the error amplification. The filter was applied again to the velocity data to further enhance the signal-to-noise ratio.
5. Finally, the 3D time-resolved Lagrangian data were grouped in a 2D cylindrical grid along the radial and vertical directions. The ensemble average velocity was calculated in each cell, obtaining a 2D averaged Eulerian velocity field.

The figures below show the case of a Herschel-Bulkley fluid (Carbopol 940 solution in water, 0.2% wt) agitated at 400rpm. Extending Metzner-Otto’s concept⁽⁵⁾ to the low transitional regime, the Reynolds number was about 20. As a result of the yield stress, the fluid formed a cavern around the impeller (Figure 1). The cavern was successfully captured by PTV measurements (Figure 2).

Figure 1: Extension of the cavern formed by a Herschel-Bulkley fluid agitated at 400 rpm, visualized by injection of a dye.

Figure 2: Detail of the Lagrangian trajectories in the impeller discharge region obtained with PTV (left) and 2D average Eulerian velocity field obtained by post-processing (right).

References:

1. Bashiri, H., Bertrand, F., Chaouki, J. (2016), Chem Eng J, 297: 277–294.
2. Galletti, C., Brunazzi, E., Yianneskis, M., Paglianti, A. (2003), Chem Eng Sci, 58: 3859–3875.
3. Wernersson, E.S., Tragardh, C. (1998), Chem Eng J, 70: 37–45.
4. Alberini, F., Liu, L., Stitt, E.H., Simmons, M.J.H. (2017), Chem Eng Sci, 171: 189 – 203.
5. Metzner, A.B., Otto, R.E. (1957), AIChE J, 3: 3 – 10.