## 1 Introduction  
A microbubble pump (MBP) with a regenerative impeller is widely used in various fields, such as water treatment, medical industry, biogas refining, and fisheries [1, 2]. The pump has the advantage of continuous micro-sized bubble generation. The regenerative impeller is composed of disk-shaped vanes and generates high pressure by rotating the impeller [3]. Microbubbles are produced by the high-pressure mixing of water and air supplied by the pump inlet due to the strong swirling flow due to the rotating of the impeller and the vanes of the impeller blade [4, 5]. A microbubble pump is used as a core part of water purification systems in public facilities, such as sewage treatment plants, and is often operated not only in the design flow condition but also in off-designed flow conditions, depending on the inflow rate. According to the shape of the impeller blades, regenerative impellers are classified into two types: open and side channel impellers. Open channel impellers are widely used in industrial fields due to relatively low manufacturing costs. Microbubble pumps with an open channel impeller have a disadvantage in that the durability is low in high pressure operating conditions because only one side of the blade is fixed to the impeller hub, forming a cantilever structure. In other words, in conventional open channel impellers, a pair of swirling flows are generated between rotating blades, as in the momentum exchange theory [6], resulting in high pressure between the individual blades and large loads on blade surfaces. Using experiments and numerical analysis, many researchers have investigated the effect of the regenerative impeller's geometric parameters on pump performance. [PERSON] et al. [7] experimentally examined the effects of blade angle on the performance of an open-channel type pump impeller. They stated that a 30\({}^{\circ}\) chevron angle produces the best results in terms of efficiency and pressure increase. The difference between the tangential velocity of the incoming fluid and the rotational speed of the impeller was lowest in the case of a 30\({}^{\circ}\) chevron angle, which improved the momentum exchange between the flow through the blade space and the flow in the side channel. [PERSON] et al. [8] selected the blade wrapping angle as a design variable to analyze its effect on pump efficiency. They stated that a small vortex is formed at the blade tip due to the interaction between the incoming and internal flows formed near the inside wall of the pump casing. The vortex propagates from the inlet to the outlet, resulting in head loss. The vortex reduces the fluid tangential velocity, which dissipates the fluid energy needed to overcome the fluid friction and turbulence, thereby increasing pressure loss. [PERSON] et al. [9] optimized impeller blades to enhance pump performance using two geometric design parameters: hub height and upper blade height. They showed that the vortex intensity at the pump outlet was reduced by optimizing the impeller blades. [PERSON] et al. [10] also performed a similar study that considered three different design variables: the ratio of blade height to the impeller diameter, the blade width to the impeller diameter, and the blade angle. Throughout blade optimization, a higher-velocity region formed near the pump inlet, and the outlet could be suppressed, increasing pump efficiency. [PERSON] and [PERSON] [11] optimized the blade angle and number of blades for microbubble pump impellers for a building fuel cell system. Overall efficiency and pressure were chosen as objective functions to evaluate the new design's performance, and the response surface method (RSM) was then adopted as the design methodology. They not only presented empirical values of the objective functions but also analyzed the internal flow conditions using pressure, velocity, and vorticity contours, as well as velocity streamlines and vectors at the design flow condition.