A Numerical Analysis of the Effect of Impeller Rounding on Centrifugal Pump as Turbine

: Pump as turbine (PAT) is one of the micro-hydro system components. It can be used alone as a generator in remote areas without power supply, and in hydraulic networks instead of pressure-reducing valves. However, its hydraulic optimization still remains an open research problem. One of the optimization techniques is the rounding of the sharp edges at the blade periphery. Existing studies are mostly based on prototype experiments to obtain the optimization effect. In order to more intuitively analyze the inﬂuence of this structural optimization on the internal ﬂow of PAT, this paper uses the CFD method to study the inﬂuence of the leading edge of the centrifugal pump blade and the ﬁllet of the impeller on the turbine performance. By simulating the PAT performance at different ﬂow rates and speeds, the internal hydraulic performance changes caused by the inverted circular blade are analyzed. Simulation results show that, under various operating conditions, the impeller inverted circle improves the efﬁciency of PAT to different degrees. At the speed of 1500 rpm, the efﬁciency is most obviously improved, which can reach 8.09%. Internal ﬂow results show that the efﬁciency increases along with the decrease in impeller inlet resistance and the ﬂow separation region in the impeller. This paper provides an effective method for studying the PAT hydraulic optimization problem.


Introduction
Along with the increasing energy demand, it is urgent to develop renewable and environment-friendly energy. Since the oil crisis of the 1970s, small hydropower has gained great attraction. Compared to the huge hydraulic turbines, pumps are easy to manufacture and maintain, and easily available all over the world. Experiments have shown that, at relatively low power outputs [1], pumps with high technological standards can compete with conventional turbines in terms of maximum efficiency in reverse operation [2].
An experimental study of district heating in Sweden shows that pump as turbine (PAT) can recover energy and supply power, thus reducing operating costs and providing temporary heating as a pump in case of failure of the main pump [3,4]. In another study, PAT was applied to the regional heating system in Poland for experimental study. The experimental results show that venue operating system (VOS) needs to be introduced for stable energy recovery. In addition, by using PAT instead of the pressure-reducing valve, the noise and vibration of the pipe network can be reduced and the damage of the pipe can be prevented or reduced [5,6].
The hydraulic losses in PAT are due to the long flow passage and unmatched fluid flow across the turbine, as it is not designed to run in reverse flow [7]. Modifications to optimize the performance were proposed to enhance the hydraulic characteristics and consequently increase the efficiency [8]. The proposed modifications include trimming the impeller blades, adjusting the blade number, adding splitter blades, installing guide

Structure of Impeller Rounding
The experimental evaluation of mass transfer distributions and oil flow visualization in long and short 90 • elbows were conducted in a water tunnel [22]. It consists of a tank, a pump, and a settling chamber. Recently, in the study of PAT performance optimization, the effect of impeller rounding has been paid attention by many researchers. The authors of [23] compare three rounding schemes: (1) taking two times the gap width as the rounding radius; (2) taking one half of the blade thickness as the rounding radius; (3) on the basis of scheme (1), using small radius rounding for the back outer edge of the blade. Considering that the hydraulic efficiency is greatly improved and the head drop is small, the rounding method is adopted in this paper. In this study, impeller rounding can be divided into two modifications of impeller, namely, blade rounding and shroud rounding, and the fillet radius is one half of the blade thickness. The extent of the rounding of the peripheral edges of the impeller is shown in Figure 1, which comprises of 'bullet shaped' rounding of the blade edges in the front view and the shrouds in the side view. In order to understand the modification of blade rounding more intuitively, Figure 2 shows a schematic diagram of the changes before and after blade rounding.
edges of impellers on PAT performance through computation simulation modeling. This enables internal hydraulic analysis that can determine the hydraulic changes causing the variations in PAT performance, which provides more comprehensive evidence for the study of theoretical models and experimental methods.

Structure of Impeller Rounding
The experimental evaluation of mass transfer distributions and oil flow visualization in long and short 90° elbows were conducted in a water tunnel [22]. It consists of a tank, a pump, and a settling chamber. Recently, in the study of PAT performance optimization, the effect of impeller rounding has been paid attention by many researchers. The authors of [23] compare three rounding schemes: (1) taking two times the gap width as the rounding radius; (2) taking one half of the blade thickness as the rounding radius; (3) on the basis of scheme (1), using small radius rounding for the back outer edge of the blade. Considering that the hydraulic efficiency is greatly improved and the head drop is small, the rounding method is adopted in this paper. In this study, impeller rounding can be divided into two modifications of impeller, namely, blade rounding and shroud rounding, and the fillet radius is one half of the blade thickness. The extent of the rounding of the peripheral edges of the impeller is shown in Figure 1, which comprises of 'bullet shaped' rounding of the blade edges in the front view and the shrouds in the side view. In order to understand the modification of blade rounding more intuitively, Figure 2 shows a schematic diagram of the changes before and after blade rounding.  In order to study the influence of impeller rounding on the flow region, the variable to be considered is the radial velocity [24]. The flow rate can be represented by the radial flow velocity component at the inlet impeller area, as shown in Equation (1): where 1 r C is the radial velocity at the impeller inlet, and 1 A is the flow area of the impeller inlet.
As seen from Equation (1), radial flow velocity depends on the flow rate and the flow area under consideration. It can be seen from Figure 1 that the impeller rounding increases the flow area at the impeller inlet. The change to the radial flow velocity component causes a change to the inlet vortex angle as well. However, as seen from the velocity triangles in edges of impellers on PAT performance through computation simulation modeling. This enables internal hydraulic analysis that can determine the hydraulic changes causing the variations in PAT performance, which provides more comprehensive evidence for the study of theoretical models and experimental methods.

Structure of Impeller Rounding
The experimental evaluation of mass transfer distributions and oil flow visualization in long and short 90° elbows were conducted in a water tunnel [22]. It consists of a tank, a pump, and a settling chamber. Recently, in the study of PAT performance optimization, the effect of impeller rounding has been paid attention by many researchers. The authors of [23] compare three rounding schemes: (1) taking two times the gap width as the rounding radius; (2) taking one half of the blade thickness as the rounding radius; (3) on the basis of scheme (1), using small radius rounding for the back outer edge of the blade. Considering that the hydraulic efficiency is greatly improved and the head drop is small, the rounding method is adopted in this paper. In this study, impeller rounding can be divided into two modifications of impeller, namely, blade rounding and shroud rounding, and the fillet radius is one half of the blade thickness. The extent of the rounding of the peripheral edges of the impeller is shown in Figure 1, which comprises of 'bullet shaped' rounding of the blade edges in the front view and the shrouds in the side view. In order to understand the modification of blade rounding more intuitively, Figure 2 shows a schematic diagram of the changes before and after blade rounding.  In order to study the influence of impeller rounding on the flow region, the variable to be considered is the radial velocity [24]. The flow rate can be represented by the radial flow velocity component at the inlet impeller area, as shown in Equation (1): where 1 r C is the radial velocity at the impeller inlet, and 1 A is the flow area of the impeller inlet.
As seen from Equation (1), radial flow velocity depends on the flow rate and the flow area under consideration. It can be seen from Figure 1 that the impeller rounding increases the flow area at the impeller inlet. The change to the radial flow velocity component causes a change to the inlet vortex angle as well. However, as seen from the velocity triangles in In order to study the influence of impeller rounding on the flow region, the variable to be considered is the radial velocity [24]. The flow rate can be represented by the radial flow velocity component at the inlet impeller area, as shown in Equation (1): where C r1 is the radial velocity at the impeller inlet, and A 1 is the flow area of the impeller inlet. As seen from Equation (1), radial flow velocity depends on the flow rate and the flow area under consideration. It can be seen from Figure 1 that the impeller rounding increases the flow area at the impeller inlet. The change to the radial flow velocity component causes a change to the inlet vortex angle as well. However, as seen from the velocity triangles in Figure 3, the change to radial flow velocity could affect the relative flow entry (β1) at the inlet, which could, in turn, have an influence on the hydraulics in zone I and zone II.  Figure 3, the change to radial flow velocity could affect the relative flow entry (β1) at the inlet, which could, in turn, have an influence on the hydraulics in zone I and zone II.

Governing Equation
The governing Navier-Stokes equations are based on the conservation laws for physical properties of fluids, which include mass, energy, and momentum. The standard k-ε turbulence model is selected to combine robustness with reasonable accuracy in turbomachinery analysis [24]. The mathematical equations applied to the momentum and energy can be expressed by the following equations.
Momentum equation: Momentum equation with rotational forces:

Oil Flow Visualization
At present, the standard k-ε model is the most basic of the double equations model, the most widely used being the eddy viscosity model. The equations are as follows: Turbulence kinetic energy k: Dissipation rate ε: where μ is the turbulent viscosity, μ = 2 ; is the turbulent kinetic energy, = ( + ) ; is the vicious stress, = 1.0; is the turbulence stress, = 1.3;

Governing Equation
The governing Navier-Stokes equations are based on the conservation laws for physical properties of fluids, which include mass, energy, and momentum. The standard k-ε turbulence model is selected to combine robustness with reasonable accuracy in turbomachinery analysis [24]. The mathematical equations applied to the momentum and energy can be expressed by the following equations.
Momentum equation: Momentum equation with rotational forces:

Oil Flow Visualization
At present, the standard k-ε model is the most basic of the double equations model, the most widely used being the eddy viscosity model. The equations are as follows: Turbulence kinetic energy k: Dissipation rate ε: where µ t is the turbulent viscosity, µ t = ρC µ k 2 ε ; G k is the turbulent kinetic energy,

Original Modeling and Model Modification of the PAT
The model used in the numerical analysis of this paper is a horizontal single-stage centrifugal pump [25], as shown in Figure 3, where (a) runs in pump mode and (b) runs in PAT mode. The centrifugal pump uses the high-speed centrifugal rotation force to throw Processes 2021, 9, 1673 5 of 14 the water, which increases the water flow energy and flows out through the pump outlet. On the other hand, PAT utilizes the reverse rotation of the pump as a turbine for power generation.
The structure of the impeller is shown in Figure 4, which is composed of the front shroud, back shroud, and the blade. The design specifications in the pump mode are given in Table 1.

Original Modeling and Model Modification of the PAT
The model used in the numerical analysis of this paper is a horizontal single-stage centrifugal pump [25], as shown in Figure 3, where (a) runs in pump mode and (b) runs in PAT mode. The centrifugal pump uses the high-speed centrifugal rotation force to throw the water, which increases the water flow energy and flows out through the pump outlet. On the other hand, PAT utilizes the reverse rotation of the pump as a turbine for power generation.
The structure of the impeller is shown in Figure 4, which is composed of the front shroud, back shroud, and the blade. The design specifications in the pump mode are given in Table 1.  The modified part is to round off the edge of the PAT impeller. The rounding of the impeller can be divided into two parts, namely, the blade rounding and the shroud rounding. The radius of rounding is half of the thickness of the blade and shroud, respectively. The modification to the impeller is shown in Figure 5.  The modified part is to round off the edge of the PAT impeller. The rounding of the impeller can be divided into two parts, namely, the blade rounding and the shroud rounding. The radius of rounding is half of the thickness of the blade and shroud, respectively. The modification to the impeller is shown in Figure 5.

Flow Region and Grid Generation
In FLUENT 18.2, the flow region of the PAT required for the simulation is extracted. As shown in Figure 6, the fluid was split into five component parts; these were volute, impeller, casing, inlet pipe, and outlet pipe. This separation allowed each mesh to be generated individually and tailored to the flow requirements in that particular component, as shown in Figure 7. To get a relatively stable inlet and outlet flow, four times the pipe diameter had been extended in the PAT inlet and outlet section. The number of grids is about 2.15 million.

Flow Region and Grid Generation
In FLUENT 18.2, the flow region of the PAT required for the simulation is extracted. As shown in Figure 6, the fluid was split into five component parts; these were volute, impeller, casing, inlet pipe, and outlet pipe. This separation allowed each mesh to be generated individually and tailored to the flow requirements in that particular component, as shown in Figure 7. To get a relatively stable inlet and outlet flow, four times the pipe diameter had been extended in the PAT inlet and outlet section. The number of grids is about 2.15 million.

Flow Region and Grid Generation
In FLUENT 18.2, the flow region of the PAT required for the simulation is extracted. As shown in Figure 6, the fluid was split into five component parts; these were volute, impeller, casing, inlet pipe, and outlet pipe. This separation allowed each mesh to be generated individually and tailored to the flow requirements in that particular component, as shown in Figure 7. To get a relatively stable inlet and outlet flow, four times the pipe diameter had been extended in the PAT inlet and outlet section. The number of grids is about 2.15 million.

Setting of Boundary Conditions for Numerical Analysis
The boundary conditions setting in FLUENT 18.2 is shown in Table 2. The numerical analysis uses steady-state simulation and sets the gravity in the reverse direction of y. For the turbulence model, the standard k-ε model is selected. The material of the fluid is set as water with normal temperature, and the surface roughness of all walls in the flow region is set to 50 micrometers. The boundary conditions of the inlet and outlet are the mass flow inlet and the pressure outlet, and the corresponding simulation results are obtained by the change of the mass flow; finally, the performance curve can be drawn.
Three fixed speeds of 1000 rpm, 1250 rpm, and 1500 rpm are selected for numerical simulation [7]. The reason for this choice is that, when used as a generator in the actual inlet and the pressure outlet, and the corresponding simulation results are obtained by the change of the mass flow; finally, the performance curve can be drawn. Three fixed speeds of 1000 rpm, 1250 rpm, and 1500 rpm are selected for numerical simulation [7]. The reason for this choice is that, when used as a generator in the actual situation, it is always matched with a four-pole induction generator, and its rated speed is 1500 rpm. However, due to the influence of part-load conditions in actual situations, the rated speed may not be reached. Therefore, in order to test the effect of impeller rounding more comprehensively, three representative rotation speeds in the wide speed range of 1000 rpm to 1500 rpm were selected for testing.

Description of Performance Evaluation Parameters
When evaluating the performance of PAT, the three main parameters are water head, shaft power, and efficiency, among which efficiency is the most important parameter.
Herein, the head is indicated by H with the unit of meter. The head of the pump represents the effective energy provided by the pump to the liquid per unit weight, while the head of the PAT is the opposite. The head of the PAT represents the effective energy provided by the unit weight of water to the PAT. As shown in Equation (1), when the density and the acceleration of gravity are constant, the pressure difference between the inlet and outlet of the PAT obtained in FLUENT 18.2 is required when calculating the head [26].
The shaft power of the PAT represents the power output from the axial rotation of the impeller when the water flowing through the PAT drives the impeller to rotate. The shaft power is indicated by p shaft with the unit of pascal. It can be calculated by multiplying the torque value in the simulation result and the angular velocity of the impeller rotation.
Efficiency is the most important parameter. PAT efficiency is the ratio of shaft power to total input power; it is indicated by η [27]. As shown in Equation (7), when the flow rate is constant, it can be calculated by substituting the shaft power and the head.

Draw Performance Curves by Changing the Flow Rate
In this study, three representative speeds are selected for the performance test. Under these three rated speeds, the head, shaft power, and efficiency are obtained by changing the inlet mass flow, and then the efficiency curve is drawn. The calculated values of head, shaft power, and efficiency under various working conditions have been recorded in Tables 3-5. According to the expression of Reynolds number, the Reynolds number under various conditions is calculated. The efficiency curve is shown in Figure 8. In the form of fixed Reynolds number, the effects on the efficiency of the two models before and after rounding are compared. Firstly, the Reynolds number of the model before rounding is calculated, and the Reynolds number of the rounding model is fixed to be the same as the original model, as shown in Equation (8), derived from Re = vd υ and v = Q/ πd 2 4 , then, the efficiency and the results before and after rounding are compared.
In Figure 8, the light curve represents the original PAT and the dark curve represents the rounding PAT. Comparing the two curves at the rated speed of 1000 rpm, the figure shows that PAT reaches the best efficiency point (BEP) when the Reynolds number is about 580. Compared with the original PAT, the efficiency of the PAT after impeller rounding has been improved in the entire flow rate range, with an increase of 0.32% to 1.34%. Although the efficiency has been improved under each working condition, the improvement is not obvious. Comparing the two curves at the rated speed of 1250 rpm, the BEP is reached when the Reynolds number is about 720. Compared with 1000 rpm, the Reynolds number needed to reach the optimal efficiency point is increased. Under low flow conditions, the increase in efficiency is small and the minimum increase is only 0.39%. At a higher flow rate, the efficiency can be increased by 2.26%.

Comparison of Internal Flow Visualization Results
Visualization of internal flow is one of the advantages of CFD simulation. In order to understand the flow situation of water in PAT clearly, the flow situations of the internal flow region before and after the impeller rounding were observed and compared in CFD post-processing. First, a cross-section is provided in the front-view direction of the PAT, and the cross-section is located on the middle surface of the entire PAT. Then, the pressure contour, speed contour, and speed vector on the cross-section are inserted. The internal flow visualization results of PAT at the best efficiency points (BEP) at 1000 rpm, 1250 rpm, and 1500 rpm are selected for comparison. Figures 9-11 are pressure contour, speed contour, and speed vector diagrams at 1000 rpm, respectively. It can be observed from the pressure contour in Figure 9 that the rounded impeller PAT has a small pressure gradient at the inlet of the impeller. In Figure  10, the low flow velocity region inside the modified PAT has been reduced, and the velocity distribution inside the impeller has improved. It can be seen in Figure 11 (velocity It can be seen from the figure that, compared with the results of the previous two rated speed tests, the influence of impeller roundness is the most obvious at 1500 rpm. When the Reynolds number is about 850, the optimal efficiency point is achieved. In the flow range, PAT efficiency can be improved from 0.44% to 8.09%.

Comparison of Internal Flow Visualization Results
Visualization of internal flow is one of the advantages of CFD simulation. In order to understand the flow situation of water in PAT clearly, the flow situations of the internal flow region before and after the impeller rounding were observed and compared in CFD post-processing. First, a cross-section is provided in the front-view direction of the PAT, and the cross-section is located on the middle surface of the entire PAT. Then, the pressure contour, speed contour, and speed vector on the cross-section are inserted. The internal flow visualization results of PAT at the best efficiency points (BEP) at 1000 rpm, 1250 rpm, and 1500 rpm are selected for comparison. Figures 9-11 are pressure contour, speed contour, and speed vector diagrams at 1000 rpm, respectively. It can be observed from the pressure contour in Figure 9 that the rounded impeller PAT has a small pressure gradient at the inlet of the impeller. In Figure 10, the low flow velocity region inside the modified PAT has been reduced, and the velocity distribution inside the impeller has improved. It can be seen in Figure 11 (velocity vector) that the overall flow separation region and secondary flow zones have been reduced to some extent.

Comparison of Internal Flow Visualization Results
Visualization of internal flow is one of the advantages of CFD simulation. In order to understand the flow situation of water in PAT clearly, the flow situations of the internal flow region before and after the impeller rounding were observed and compared in CFD post-processing. First, a cross-section is provided in the front-view direction of the PAT, and the cross-section is located on the middle surface of the entire PAT. Then, the pressure contour, speed contour, and speed vector on the cross-section are inserted. The internal flow visualization results of PAT at the best efficiency points (BEP) at 1000 rpm, 1250 rpm, and 1500 rpm are selected for comparison. Figures 9-11 are pressure contour, speed contour, and speed vector diagrams at 1000 rpm, respectively. It can be observed from the pressure contour in Figure 9 that the rounded impeller PAT has a small pressure gradient at the inlet of the impeller. In Figure  10, the low flow velocity region inside the modified PAT has been reduced, and the velocity distribution inside the impeller has improved. It can be seen in Figure 11 (velocity vector) that the overall flow separation region and secondary flow zones have been reduced to some extent.  The visualization results at 1250 rpm are shown in Figures 12-14. It can be seen from the pressure distribution in Figure 12 that the flow obstruction at the entrance of the rounded impeller is reduced. This leads to a pressure reduction in the whole inlet pipe, which reduces the head. The low-speed region in Figure 13 is similar to 1000 rpm, and it is reduced to some extent. The low-velocity zones in Figure 13 are reduced to some extent, which is similar to the 1000 rpm. The flow separation and secondary flow zones in Figure  14 are also reduced. The visualization results at 1250 rpm are shown in Figures 12-14. It can be seen from the pressure distribution in Figure 12 that the flow obstruction at the entrance of the rounded impeller is reduced. This leads to a pressure reduction in the whole inlet pipe, which reduces the head. The low-speed region in Figure 13 is similar to 1000 rpm, and it is reduced to some extent. The low-velocity zones in Figure 13 are reduced to some extent, which is similar to the 1000 rpm. The flow separation and secondary flow zones in Figure  14 are also reduced. The visualization results at 1250 rpm are shown in Figures 12-14. It can be seen from the pressure distribution in Figure 12 that the flow obstruction at the entrance of the rounded impeller is reduced. This leads to a pressure reduction in the whole inlet pipe, which reduces the head. The low-speed region in Figure 13 is similar to 1000 rpm, and it is reduced to some extent. The low-velocity zones in Figure 13 are reduced to some extent, which is similar to the 1000 rpm. The flow separation and secondary flow zones in Figure 14 are also reduced. The visualization results at 1250 rpm are shown in Figures 12-14. It can be seen from the pressure distribution in Figure 12 that the flow obstruction at the entrance of the rounded impeller is reduced. This leads to a pressure reduction in the whole inlet pipe, which reduces the head. The low-speed region in Figure 13 is similar to 1000 rpm, and it is reduced to some extent. The low-velocity zones in Figure 13 are reduced to some extent, which is similar to the 1000 rpm. The flow separation and secondary flow zones in Figure  14 are also reduced.  Similar to the pressure distribution at 1250 rpm, when the rated speed is 1500 rpm, the flow obstruction at the inlet of the impeller is also reduced after modification. As shown in Figure 15, the pressure in the whole inlet pipe becomes lower. It can be observed in Figure 16 that the velocity distribution between each blade is almost improved. Compared with the original model, the velocity distribution after impeller rounding is more uniform. Figure 17 shows that the flow separation regions behind the blade are relatively large at 1500 rpm because of the faster rotating speed, and this situation has been significantly reduced after modification. Similar to the pressure distribution at 1250 rpm, when the rated speed is 1500 rpm, the flow obstruction at the inlet of the impeller is also reduced after modification. As shown in Figure 15, the pressure in the whole inlet pipe becomes lower. It can be observed in Figure 16 that the velocity distribution between each blade is almost improved. Compared with the original model, the velocity distribution after impeller rounding is more uniform. Figure 17 shows that the flow separation regions behind the blade are relatively large at 1500 rpm because of the faster rotating speed, and this situation has been significantly reduced after modification. Similar to the pressure distribution at 1250 rpm, when the rated speed is 1500 rpm, the flow obstruction at the inlet of the impeller is also reduced after modification. As shown in Figure 15, the pressure in the whole inlet pipe becomes lower. It can be observed in Figure 16 that the velocity distribution between each blade is almost improved. Compared with the original model, the velocity distribution after impeller rounding is more uniform. Figure 17 shows that the flow separation regions behind the blade are relatively large at 1500 rpm because of the faster rotating speed, and this situation has been significantly reduced after modification.
the flow obstruction at the inlet of the impeller is also reduced after modification. As shown in Figure 15, the pressure in the whole inlet pipe becomes lower. It can be observed in Figure 16 that the velocity distribution between each blade is almost improved. Compared with the original model, the velocity distribution after impeller rounding is more uniform. Figure 17 shows that the flow separation regions behind the blade are relatively large at 1500 rpm because of the faster rotating speed, and this situation has been significantly reduced after modification.   Table 6 compares the results of numerical analysis of three different rotating speeds. It can be seen that, as the rotational speed increases, the flow required to achieve best efficiency point (BEP) also increases. At the rotating speed of 1500 rpm, the efficiency improvement range is the greatest. It can be increased by 0.44%~0.89% in the whole flow range, and the highest efficiency can reach 76.18%. The reason for the higher efficiency increase at higher speeds is that the flow obstruction at the inlet of the original model impeller is also higher, so the improvement is more obvious after the modification.   Table 6 compares the results of numerical analysis of three different rotating speeds. It can be seen that, as the rotational speed increases, the flow required to achieve best efficiency point (BEP) also increases. At the rotating speed of 1500 rpm, the efficiency improvement range is the greatest. It can be increased by 0.44%~0.89% in the whole flow range, and the highest efficiency can reach 76.18%. The reason for the higher efficiency increase at higher speeds is that the flow obstruction at the inlet of the original model impeller is also higher, so the improvement is more obvious after the modification.   Table 6 compares the results of numerical analysis of three different rotating speeds. It can be seen that, as the rotational speed increases, the flow required to achieve best efficiency point (BEP) also increases. At the rotating speed of 1500 rpm, the efficiency improvement range is the greatest. It can be increased by 0.44~0.89% in the whole flow range, and the highest efficiency can reach 76.18%. The reason for the higher efficiency increase at higher speeds is that the flow obstruction at the inlet of the original model impeller is also higher, so the improvement is more obvious after the modification.

Conclusions
In this study, the effect of the optimization method 'Impeller rounding' on PAT was simulated by CFD numerical analysis. Simulation results show that impeller rounding has improved the efficiency of PAT, and, through the observation of the internal flow region, the reason for the performance improvement has been analyzed. First, compared with the original model, the efficiency of the rounded impeller PAT is improved under all test conditions. In addition, the increase is more obvious at higher rotating speeds. This shows that impeller rounding is very beneficial for the common PAT with small diameter and high rotating speed.
The reason for the increase in efficiency is that, in terms of shaft power, the head is reduced in almost all operating conditions. In addition, when both the shaft power and the head are reduced, the head is reduced more, which leads to an increase in efficiency. The reduced flow obstruction at the entrance of the modified impeller is the main reason for the head drop, which shows that water passes better when passing through the rounded impeller. It can be observed from the visualization results that the pressure distribution inside the rounded impeller is more uniform, and the wakes and secondary flow are also reduced. CFD and flow visualization results have important reference significance for the study of the effect of impeller rounding on PATs.