## 5 Results and Discussion
### Flow Analysis in Design and Off-Design Conditions  
The flow characteristics related to vortical flow inside the blade passage of the microbubble pump were analyzed by three-dimensional numerical simulation in two off-design conditions (\(\O\) = 0.31 and 0.47) and in the design condition (\(\O\) = 0.39), as well. Figure 11 shows the streamlines inside the microbubble pump colored by velocity at the design operating condition. The flow comes through the inlet duct and is accelerated by the rotation of the impeller. The flow into the side of the 62 impeller blades is observed to be relatively fast at the blade tip due to the vortical flow generated inside each blade, as shown in Figure 12. The velocity imparted by the blade tip is mixed and discharged in the buffer tank after passing through the impeller. It is noted that the behavior of the streamlines inside the blades is similar regardless of the flow conditions. Figure 10: Relative uncertainty in measured variables: (**a**) flowrate; (**b**) outlet pressure. Figure 13 shows the turbulent kinetic energy and the tangential velocity vectors on the normal planes near the inlet region. The three planes are in the vicinity of the impeller after passing through the inlet duct at the location indicated by the dashed line in Figure 11a. As shown in the figure, it can be seen that a large vortical flow is observed inside each impeller, which is the concave part of the 'S-shape' impeller. Turbulent kinetic energy has a higher value as the flowrate is relatively small compared to the design flow condition. It is noted that violent mixing between the internal flow inside the impeller and the channel flow increases turbulent kinetic energy. In the design flow condition, the highest magnitudes of the recirculating flow were observed as compared to the other two flow conditions. The highest recirculating flow observed in the design flow condition helps to achieve a better momentum exchange, thus increasing the overall efficiency of pump. The relatively high turbulent kinetic energy in lower-flow conditions corresponds to high relative uncertainty, as shown in Figure 10a. Figure 11: Streamlines inside microbubble pump colored by the velocity at the design flow condition: (**a**) positions; (**b**) streamlines. Figure 12: Streamlines colored by velocity near the inlet in the design and off-design conditions. (**a**) \(\oslash\) = 0.31; (**b**) \(\oslash\) = 0.39; (**c**) \(\oslash\) = 0.47. Figure 14 shows velocity and tangential velocity vectors on the mid-width plane of the pump impeller for the design and off-design operating conditions. As the flowrate increases, relatively high concentrated flow velocity distributions are observed in the direction of impeller rotation. It is noted that a relatively vital mixing flow at lower flow conditions is distributed inside the buffer tank due to high fluctuation. In the space between the pump casing and the tip of the impeller, the combined flow with the rotational and the normal direction of the impeller is observed. As [PERSON] et al. [7] reported, the angle between incoming fluid direction and blade is of key importance for the overall performance of the pump impeller in terms of lower vortex generation. The figure also demonstrates this statement at three different flow conditions. Due to the optimal design of the impeller with 'S-shape' blades, velocity vectors observed near the impeller are mostly in line with impeller blades. Figure 15 shows the tangential velocity vectors on the mid-width plane of the pump impeller shown in Figure 11a for the design and off-design operating conditions. As shown in Figure 11a, the three measuring positions correspond to the pump inlet (\(\theta\) = 0\({}^{\circ}\)), middle (\(\theta\) = 120\({}^{\circ}\)), and outlet (\(\theta\) = 240\({}^{\circ}\)). The small circles in the figure represent the center of the recirculating flow.