Pipe flow properties

Figure 4.7a shows a velocity map in the z-axis direction of pipe flow in the Couette cell with stationary inner cylinder, at Q = 7.2 cm3 min−1

. The dark part in the middle of the image corresponds to the inner cylinder. An average over 50 cross-section lines of this velocity map shows the typical Poiseuille velocity profile (Figure 4.7b). Due to the shape of the Couette cell, the maximum velocities on the profile are found closer to the inner than the outer cylinder32. The inner surface of the outer cylinder being larger than the surface of the inner cylinder, higher friction is experienced on the outer side. The right-hand part of the velocity

profile is slightly higher than the left hand-side. This symmetry break, probably caused by inhomogeneous distribution at the inlet or a gap irregularity, has to be considered when studying the VFR flow.

Figure 4.7. (a) Velocity map in the z direction of pipe flow at Q =7.2 cm3 min−1

. (b) Average velocity profile in the radial direction.

Figure 4.8 shows a plot of average (Uax) and maximum (Umax) velocities for different flow rates. As expected, there is a linear relation between the average velocity and the pump flow rate. Due to the Couette cell shape, the measured pipe flow profile is lying between Poiseuille profile in a tube and Poiseuille profile between two parallel planes32. For low radii ratio, _η, the Couette cell looks more like a tube where typically Uax = 1/2 Umax, while for high η values the TC cell gap is similar to the gap between two parallel planes where Uax = 2/3 Umax. So, the ratio Uax / Umax is characteristic of each VFR device. In these results, Uax / Umax is found to be constant at 0.64 ± 1 (Note that 1/2 < 0.64 < 2/3).

Figure 4.8. Plot showing Umax and Uax velocities of pipe flow in the Couette cell against the applied flow rate.

Uax measurements were also used to calculate the Reynolds numbers for this range of flow rates. Re was found to vary between 1.7 and 5.3. Also, since the average axial velocity is the same for pipe and VFR flows at the same Q, knowing Uax allows calculating the velocity drift for the VFR.

4.3.3 Velocity drift

Figure 4.9 shows the evolution of the velocity drift, Vd = UTV / Uax, with flow and rotation rate. Vd was found to vary between 1.02 and 1.21. Vd > 1 indicates that the vortices move faster than the axial flow rate. Vd ~ 1, where the vortices travel at approximately the same velocity as the axial flow, is considered as an important condition for good plug-flow properties of a VFR. If the vortices move at a different speed than the average velocity, inter- vortex mixing is expected to occur, compromising the plug-flow properties.

Figure 4.9 Plots of the velocity drift against flow (a) and rotation (b) rate.

For Q > 4 cm3 min−1

, Vd does not seem significantly influenced by the flow rate (Figure 4.9a). In fact, as both UTV and Uax were found to have a linear dependence on the flow rate, Vd = UTV / Uax is expected to be constant at this range of parameters. Vd is found to be limited so as to the characterisation of flow properties for increasing flow rates. An increase in rotation rate causes a Vd decrease for Q > 4 cm3 min−1. The lower Vd values, where the most plug-like flow properties are expected, are found for ω = 2.5 Hz. The only exception is found for Q = 4 cm3 min−1

and _{ω = 2.5 Hz, where a higher vortex pair velocity is observed, leading to a V}d increase. Notice that despite Vd > 1 for all our experiments, images obtained from dye injection experiments (Figure 4.3) give the impression of plug-flow, indicating that the vortices are being pushed as a stack. The dye experiments might indeed provide misleading

information about the properties of the reactor. MR velocity imaging was performed to get a better understanding of the VFR system.

4.3.4 MR Velocity imaging of travelling vortices

The travelling vortex period values, TTV, measured in the dye experiments (Figure 4.5 and Figure 4.6) were used to time the PGSE velocity imaging sequence. Series of 2 q-slice MR experiments were performed, where the triggering time of the pulse sequence varied from TTV – 500ms to TTV + 500ms, with 20 ms steps. The experimental time for each image acquired was of 18 to 40 minutes, depending on TTV. Figure 4.10 shows intensity maps obtained at different loop time periods.

Figure 4.10. Intensity maps of VFR flow at ω = 1 Hz and Q = 7.2 cm3 min−1, obtained at different PGSE loop times. The loop that eliminated artifacts had a period of 8.16 s (= TTV).

When the loop period of the sequence is different from TTV ± 20ms, several phase artifacts are observed, with high signal observed in the region corresponding to the inner cylinder and low signal regions in the gap area. This behaviour was observed for all direction experiments on this VFR. As the PGSE loop period is approaching the estimated TTV value, less artifacts appear on the intensity maps. For each one of the imaging series, only one measurement gives

an intensity map with no apparant artifacts (no signal outside the gap region and no signal loss inside the gap region). The loop time producing this intensity map corresponds to TTV. Similar experiments were conducted on a second VFR, giving the same results (Appendix A). The data acquired during this measurement were post-treated to produce a velocity map of the measured flow.

Figure 4.11 shows axial (b), radial (c) and azimuthal (d) MR velocity maps at ω = 1 Hz and Q = 7.2 cm3 min−1, in comparison with a dye visualisation picture at the same regime (a). The vortex pair wavelength measured on the velocity maps is in agreement with the dye experiment results (λ ∼ 85 mm). These velocity maps present also several similar features with the TVF velocity maps at ω = 1 Hz (cf. 3.3.1). In fact, the addition of pipe flow does not seem to have a strong effect on the velocity field. The velocity map in the direction of the translating vortices, Uz, is more affected, since pipe flow does not have radial and azimuthal components at this range. Figure 4.10b shows a slalom-like, high-velocity region, suggesting the presence of by-pass flow. The higher velocities in the axial direction are found in the parts of the vortices situated close to the inner cylinder. This can be explained by the fact that both TVF and pipe flow exhibit enhanced velocities in this region. The radial direction velocity map shows the secondary flow jets towards the outer cylinder.

Figure 4.11. (a) Digital photo of translating vortices in the VFR system at _{ω = 1 Hz and Q =} 7.2 cm3 min–1. (b)–(d) MR velocity maps of VFR flow in the axial, Uz (b), radial, Uy (c) and azimuthal, Ux (d), directions.

of the secondary flow is probably due to the higher axial velocities found close to the inner cylinder. As with TVF, velocities in-between vortex pairs in the azimuthal direction (Figure 4.10d) are an order of magnitude higher compared to those inside the vortex pairs, so they are not noticeably deformed. These strong inter-vortex flows, in combination with the periodic outwards flow structures in the radial direction, create the sharp interfaces observed in the dye-injection experiments. Despite the fact that the vortices move faster than the average axial velocity (Vd ~ 1.2) and the observed high-velocity slalom-like region in the axial direction velocity maps, these interfaces give the misleading impression of plug-flow, where vortices appear as non inter-mixing translating reactors.

To compare with other studies of travelling vortex flow in a VFR, several approaches can be used. A contour plot obtained using the axial direction map (Figure 4.12a) allows better comparison with contour plots found in theoretical studies of the travelling vortex flow, where streamlines are found to slalom around the vortex cores33. A 2D velocity vector map, produced using the MR velocity maps (Figure 4.12b), facilitates comparison with particle imaging velocimetry measurements12, typically used to study this flow. Moreover, a 2D map can give a better visualisation of the vortical structures within a VFR. In this map, the MR velocity maps (Uz and Uy) were used to define the magnitude of each vector in the axial and radial directions. In the resulting 2D velocity vector map, the centre of the successive vortices can be easily identified, as well as the slight distortions caused by the secondary inflows and outflows in the radial direction.

Figure 4.12. (a) Contour representation of the left part of the axial direction velocity map (Uz) and (b) 2D velocity vector map obtained by combining the axial and radial direction velocity maps.

MR experiments using 8 q-slices were also performed to obtain more accurate measurements. The experimental time for each image to be acquired with the pulse sequence timed to the flow period was of 2 - 3 hours. The main challenge in performing these experiments was to maintain the vortex translation stable over the course of the 8 q-slice experiments. As mentioned before, the precision required for the data synchronisation to eliminate motion artifacts was of the order of 20 ms, and small axial flow rate variations (of the order of 0.05 cm3 min−1

) tended to desynchronise the pulse sequence timing, resulting in motion artifacts. Figure 4.13 shows a velocity map of VFR flow in the axial direction obtained with an 8 q-

slice PGSE experiment. Artifacts can be noticed outside the gap (right-hand side) but also in the inter-vortex regions where opposed direction velocities are found.

Figure 4.13. MR velocity map of VFR flow (at ω = 1 Hz and Q = 7.2 cm3 min-1) in the axial direction obtained with an 8 q-slice PGSE experiment.

Despite the vortical shape being clearly identifiable and the wavelength in agreement with the 2 q-slice experiments, the overall velocities are found to be much lower, probably due to disparities in the successive slices. Hence, 2 q-slice experiments were used to study the relative effects of pipe flow and TVF in the resulting VFR flow.