Analysis of Unsteady Pressure Fluctuation in a Semi-Open Cutting Pump

: In order to understand the pressure ﬂuctuation characteristics of a semi-open cutting pump, the three-dimensional unsteady ﬂow ﬁelds were calculated. External and internal ﬂow characteristics of four schemes with di ﬀ erent relative angles between the rotary cutter and the impeller were studied. The pressure ﬂuctuations in the lower plate, the upper plate, the clearance between the rotary cutter and the ﬁxed cutter, the ﬁrst section in volute and nearby parts of the tongue were all analyzed, which are all the places that pressure distributions are greatly a ﬀ ected by the static and dynamic interaction, and at the same time, the force on the impeller was also analyzed. The results show that the ﬂuctuations at di ﬀ erent positions change periodically; the main frequency is blade frequency. The amplitude of pressure ﬂuctuation decreases from near the rotating part to far away, from near the tongue to far from the tongue. Due to the inﬂuence of both impeller and rotary cutter, the pressure ﬂuctuation on the lower plate is the largest. The pressure ﬂuctuation is a ﬀ ected by ﬂow rate, the larger the ﬂow rategreater the pressure ﬂuctuation. The radial and axial forces of the impeller change periodically with time, and the number of wave peaks and wave valleys is the same as the number of blades.


Introduction
Sewage pumps are widely used in municipal engineering, environmental protection engineering, power industrial engineering, sewage treatment station, mining, metallurgy, medical and health services and many other industries. Compared with traditional sewage pumps, the semi-open cutting pump is not easy to be blocked, which can be used to transport sewage, waste water, rainwater and urban domestic water containing solid particles and various fibers. As a kind of rotating machinery, the semi-open submersible cutting pump faces with vibration, noise, operation instability and other problems, these problems are related to the unsteady pressure fluctuations in the pump cavity, namely the axial and radial forces.
Adkins et al. [1] and Barrio et al. [2] studied the radial force distribution of a centrifugal pump rotor through theoretical analysis and numerical calculation. In order to improve the force on the impeller of single blade spiral centrifugal pump, Yuan et al. [3] used a numerical simulation method to calculate the internal flow field of the pump, and the characteristics of pressure pulsation at the outlet of spiral centrifugal pump and the radial force acting on the impeller and volute were obtained. González et al. [4] Barrio et al. [5] Solis et al. [6] studied the radial force characteristics of the impeller and the pressure fluctuation at the tongue under different tongue clearance conditions of a centrifugal pump. It was found that the changes of tongue clearance had a great influence on the radial force

Calculation Method and Grid Division
In order to obtain the pressure fluctuation characteristics and the distribution of radial force and axial force in the semi-open submersible cutting pump, the commercial computational fluid dynamics (CFD) software ANSYS CFX 15.0 (ANSYS Inc., Pittsburgh, PA, USA) was used to solve the internal Reynolds averaged equations of three dimensional flow fields. Based on the results of steady calculation, the unsteady internal flow fields of 0.6Qopt, 0.8Qopt, 1.0Qopt, 1.2Qopt and 1.4Qopt were calculated. The standard k-ε turbulence model was adopted in the simulation as the model is one of the most widely used models to study the flow field of sewage pumps [12][13][14]. The rotating parts include rotary cutter, impeller with back blades. The static domain includes inlet extension section, volute, and water ring in volute, lower plate, upper plate and outlet extension section, the whole computational domain is shown in Figure 3. The rotation speed is 1480 r/min; the medium is water and the temperature is 25 degrees centigrade. Frozen rotor is used for the interfaces of impeller and inlet extension section, impeller and upper plate, impeller and lower plate. The interface between the static and static fields is set to be none. The GGI (grid-grid interface) method with strong

Calculation Method and Grid Division
In order to obtain the pressure fluctuation characteristics and the distribution of radial force and axial force in the semi-open submersible cutting pump, the commercial computational fluid dynamics (CFD) software ANSYS CFX 15.0 (ANSYS Inc., Pittsburgh, PA, USA) was used to solve the internal Reynolds averaged equations of three dimensional flow fields. Based on the results of steady calculation, the unsteady internal flow fields of 0.6Qopt, 0.8Qopt, 1.0Qopt, 1.2Qopt and 1.4Qopt were calculated. The standard k-ε turbulence model was adopted in the simulation as the model is one of the most widely used models to study the flow field of sewage pumps [12][13][14]. The rotating parts include rotary cutter, impeller with back blades. The static domain includes inlet extension section, volute, and water ring in volute, lower plate, upper plate and outlet extension section, the whole computational domain is shown in Figure 3. The rotation speed is 1480 r/min; the medium is water and the temperature is 25 degrees centigrade. Frozen rotor is used for the interfaces of impeller and inlet extension section, impeller and upper plate, impeller and lower plate. The interface between the static and static fields is set to be none. The GGI (grid-grid interface) method with strong

Calculation Method and Grid Division
In order to obtain the pressure fluctuation characteristics and the distribution of radial force and axial force in the semi-open submersible cutting pump, the commercial computational fluid dynamics (CFD) software ANSYS CFX 15.0 (ANSYS Inc., Pittsburgh, PA, USA) was used to solve the internal Reynolds averaged equations of three dimensional flow fields. Based on the results of steady calculation, the unsteady internal flow fields of 0.6Q opt , 0.8Q opt , 1.0Q opt , 1.2Q opt and 1.4Q opt were calculated. The standard k-ε turbulence model was adopted in the simulation as the model is one of the most widely used models to study the flow field of sewage pumps [12][13][14]. The rotating parts include rotary cutter, impeller with back blades. The static domain includes inlet extension section, volute, and water ring in volute, lower plate, upper plate and outlet extension section, the whole computational domain is shown in Figure 3. The rotation speed is 1480 r/min; the medium is water and the temperature is 25 degrees centigrade. Frozen rotor is used for the interfaces of impeller and inlet extension section, impeller and upper plate, impeller and lower plate. The interface between the Energies 2020, 13, 3657 4 of 16 static and static fields is set to be none. The GGI (grid-grid interface) method with strong compatibility is chosen for the connection of the domain interface. The inlet boundary is set as total pressure of one standard atmospheric pressure, 1.01325 × 10 5 p a , the outlet boundary is set as mass flow rate, and walls are set to be non-slip. The roughness of the main flow parts such as rotary cutter and impeller is set to be 0.025 mm. In the solver, high resolution is selected for the numerical dispersion method of the convection term in the transport equation. The discretization of the turbulence equation is set as the second-order upwind, the maximum time step is 2000, and the convergence accuracy is 10 −5 .
Energies 2020, 13, x FOR PEER REVIEW 4 of 15 compatibility is chosen for the connection of the domain interface. The inlet boundary is set as total pressure of one standard atmospheric pressure, 1.01325 × 10 5 pa, the outlet boundary is set as mass flow rate, and walls are set to be non-slip. The roughness of the main flow parts such as rotary cutter and impeller is set to be 0.025 mm. In the solver, high resolution is selected for the numerical dispersion method of the convection term in the transport equation. The discretization of the turbulence equation is set as the second-order upwind, the maximum time step is 2000, and the convergence accuracy is 10 −5 .
rotary cutter upper plate impeller volute water ring back blade upper plate During unsteady simulation, calculation time step is set to be 3.37838 × 10 −4 s, which is the time for every three degrees of impeller rotation, and the total calculation time is set to be 0.405405405 s. That is to say, it takes 1200 steps to rotate the impeller for 10 circles. The dynamic and static interface is changed to transient rotor-stator. The convergence accuracy is set to 10 −5 too, and the maximum iteration is 20, so as to ensure the accuracy and time of calculation. In order to obtain more reliable unsteady calculation results, calculation data of the last four circles are selected for analysis.
To verify the mesh independence, when the number of grid nodes reaches 4.08 million, the head and efficiency tend to be stable, as shown in Figure 4, the number of grids and y+ value in each part are shown in Table 1.

Location of Pressure Pulsation Monitoring Point
Monitoring points are set up at the entrance of the rotary cutter, the clearance between the rotary cutter and the fixed cutter, the groove of the lower and upper plate, the cross-section of the volute, the impeller outlet and the impeller runner. The locations of the monitoring point are shown in Figure  5. Dynamic and static interaction is the main cause of the pressure fluctuation. The rotary cutter inlet, the rotary cutter and the fixed cutter are mainly within the influence of the rotary cutter. The lower During unsteady simulation, calculation time step is set to be 3.37838 × 10 −4 s, which is the time for every three degrees of impeller rotation, and the total calculation time is set to be 0.405405405 s. That is to say, it takes 1200 steps to rotate the impeller for 10 circles. The dynamic and static interface is changed to transient rotor-stator. The convergence accuracy is set to 10 −5 too, and the maximum iteration is 20, so as to ensure the accuracy and time of calculation. In order to obtain more reliable unsteady calculation results, calculation data of the last four circles are selected for analysis.
To verify the mesh independence, when the number of grid nodes reaches 4.08 million, the head and efficiency tend to be stable, as shown in Figure 4, the number of grids and y+ value in each part are shown in Table 1  ring unsteady simulation, calculation time step is set to be 3.37838 × 10 −4 s, which is t y three degrees of impeller rotation, and the total calculation time is set to be 0.4054 o say, it takes 1200 steps to rotate the impeller for 10 circles. The dynamic and static i ed to transient rotor-stator. The convergence accuracy is set to 10 −5 too, and the ma is 20, so as to ensure the accuracy and time of calculation. In order to obtain more y calculation results, calculation data of the last four circles are selected for analysis. verify the mesh independence, when the number of grid nodes reaches 4.08 million, t iency tend to be stable, as shown in Figure 4, the number of grids and y+ value in ea n in Table 1.

Location of Pressure Pulsation Monitoring Point
Monitoring points are set up at the entrance of the rotary cutter, the clearance between the rotary cutter and the fixed cutter, the groove of the lower and upper plate, the cross-section of the volute,   As shown in Figure 5a, from the bottom to the top, the monitoring point JK is at the rotary cutter inlet, Y is at the clearance between rotary cutter and fixed cutter, XP is at the groove of lower plate and SP is at the groove of upper plate. As shown in Figure 5b, monitoring points P1, P3, P5, P7, P8 are respectively set at Sections I, III, V, VII and VIII of the volute, G is at the tongue and CK is at the volute outlet.

Flow Characteristics for Different Relative Angles
As the core part of the submersible cutting pump, the impeller affects the head and efficiency of the pump. Due to the existence of the rotary cutter, the distribution of the flow field in the pump changes, which will affect the head and efficiency of the pump. The experimental pump is of semiopen impeller, the number of impeller blades is three, and the number of rotary cutter blades is three. The impeller and the rotary cutter are cast together. When the rotary cutter rotates 120 degrees, the relative position of the rotary cutter and the impeller coincides. Therefore, 120 degrees relative angle between the rotary cutter and the impeller is taken as the research object here.

External Characteristic Analysis
The performance curve of the pump at different relative angles under 1.0Qopt flow rate is shown in Figure 6. When the relative angle is 20 degrees, 40 degrees, 80 degrees and 100 degrees, it is found that, when the relative angle between the rotary cutter and the impeller is 20 degrees, the head is 13.8 m, and the error rate of the design head which is calculated by (13.8 − 13)/13 = 6.15% is the smallest, the efficiency of about 46.2% is relatively acceptable. In general, the efficiency difference of the pump with different angles is small when the relative angle is in the range of 20 to 110 degrees, and the internal flow characteristics are analyzed. Figure 7 is the schemes of rotary cutter and impeller under four different angles. As shown in Figure 5a, from the bottom to the top, the monitoring point JK is at the rotary cutter inlet, Y is at the clearance between rotary cutter and fixed cutter, XP is at the groove of lower plate and SP is at the groove of upper plate. As shown in Figure 5b, monitoring points P1, P3, P5, P7, P8 are respectively set at Sections I, III, V, VII and VIII of the volute, G is at the tongue and CK is at the volute outlet.

Flow Characteristics for Different Relative Angles
As the core part of the submersible cutting pump, the impeller affects the head and efficiency of the pump. Due to the existence of the rotary cutter, the distribution of the flow field in the pump changes, which will affect the head and efficiency of the pump. The experimental pump is of semi-open impeller, the number of impeller blades is three, and the number of rotary cutter blades is three. The impeller and the rotary cutter are cast together. When the rotary cutter rotates 120 degrees, the relative position of the rotary cutter and the impeller coincides. Therefore, 120 degrees relative angle between the rotary cutter and the impeller is taken as the research object here.

External Characteristic Analysis
The performance curve of the pump at different relative angles under 1.0Q opt flow rate is shown in Figure 6. When the relative angle is 20 degrees, 40 degrees, 80 degrees and 100 degrees, it is found that, when the relative angle between the rotary cutter and the impeller is 20 degrees, the head is 13.8 m, and the error rate of the design head which is calculated by (13.8 − 13)/13 = 6.15% is the smallest, the efficiency of about 46.2% is relatively acceptable. In general, the efficiency difference of the pump with different angles is small when the relative angle is in the range of 20 to 110 degrees, and the internal flow characteristics are analyzed. Figure 7 is the schemes of rotary cutter and impeller under four different angles. Analysis of Internal Flow Field he pressure distributions at different relative angles under 1.0Qopt flow rate are shown in Fi location of the comparison plane is the middle section of the volute, as shown the A-A inte in Figure 1. Although the pressure distributions at A-A plane is slightly different u ent schemes, the overall changing rule is similar. The pressure increases gradually from ler inlet to the volute outlet along the main flow direction. The maximum value appears in ion section of the volute, and the relative low-pressure area appears in the suction sides o ler near the blade inlet. With the increase in the relative angle between the rotary cutter an ler, the area of the relative low pressure gradually decreases. The relative high-pressure ar lute and the diffusion section will gradually increase. When the angle is 100 degrees, the h ure area has occurred at the tongue. On the same section, the pressure at the volute in usly smaller than that at the volute outer wall, and the difference becomes obvious with se in the relative angle between the rotary cutter and the impeller.

Analysis of Internal Flow Field
The pressure distributions at different relative angles under 1.0Qopt flow rate are shown in Figure  8. The location of the comparison plane is the middle section of the volute, as shown the A-A interface plane in Figure 1. Although the pressure distributions at A-A plane is slightly different under different schemes, the overall changing rule is similar. The pressure increases gradually from the impeller inlet to the volute outlet along the main flow direction. The maximum value appears in the diffusion section of the volute, and the relative low-pressure area appears in the suction sides of the impeller near the blade inlet. With the increase in the relative angle between the rotary cutter and the impeller, the area of the relative low pressure gradually decreases. The relative high-pressure area in the volute and the diffusion section will gradually increase. When the angle is 100 degrees, the highpressure area has occurred at the tongue. On the same section, the pressure at the volute inlet is obviously smaller than that at the volute outer wall, and the difference becomes obvious with the increase in the relative angle between the rotary cutter and the impeller.

Analysis of Internal Flow Field
The pressure distributions at different relative angles under 1.0Q opt flow rate are shown in Figure 8. The location of the comparison plane is the middle section of the volute, as shown the A-A interface plane in Figure 1. Although the pressure distributions at A-A plane is slightly different under different schemes, the overall changing rule is similar. The pressure increases gradually from the impeller inlet to the volute outlet along the main flow direction. The maximum value appears in the diffusion section of the volute, and the relative low-pressure area appears in the suction sides of the impeller near the blade inlet. With the increase in the relative angle between the rotary cutter and the impeller, the area of the relative low pressure gradually decreases. The relative high-pressure area in the volute and the diffusion section will gradually increase. When the angle is 100 degrees, the high-pressure area has occurred at the tongue. On the same section, the pressure at the volute inlet is obviously smaller than that at the volute outer wall, and the difference becomes obvious with the increase in the relative angle between the rotary cutter and the impeller.
Energies 2020, 13, 3657 7 of 16 impeller near the blade inlet. With the increase in the relative angle between the rotary cutter and the impeller, the area of the relative low pressure gradually decreases. The relative high-pressure area in the volute and the diffusion section will gradually increase. When the angle is 100 degrees, the highpressure area has occurred at the tongue. On the same section, the pressure at the volute inlet is obviously smaller than that at the volute outer wall, and the difference becomes obvious with the increase in the relative angle between the rotary cutter and the impeller.    The hydraulic loss depends on the velocity vector bandwidth. If the velocity vector width is smaller, the jet-wake structure is weaker, and the loss is smaller. It can be clearly seen from Figure 9 that when the relative angle of the rotary cutter and the impeller is 20 degrees, the velocity vector bandwidth is the smallest and the jet-wake structure is the weakest. With the increase in relative angle, the velocity vector bandwidth increases and the jet-wake structure becomes stronger. The scheme of relative angle between the rotary cutter and the impeller 20 degrees has been chosen for further research.

Time Aomain Analysis of Pressure Pulsation at Different Positions
In order to study the pressure fluctuation in the pump, dimensionless pressure pulsation coefficient Cp is defined as, where ∆ is the difference between the instantaneous pressure and the average pressure, pa; is fluid density, kg/m 3 ; u2 is the peripheral speed of the impeller outlet, m/s. Figure 10   The hydraulic loss depends on the velocity vector bandwidth. If the velocity vector width is smaller, the jet-wake structure is weaker, and the loss is smaller. It can be clearly seen from Figure 9 that when the relative angle of the rotary cutter and the impeller is 20 degrees, the velocity vector bandwidth is the smallest and the jet-wake structure is the weakest. With the increase in relative angle, the velocity vector bandwidth increases and the jet-wake structure becomes stronger. The scheme of relative angle between the rotary cutter and the impeller 20 degrees has been chosen for further research.

Time Aomain Analysis of Pressure Pulsation at Different Positions
In order to study the pressure fluctuation in the pump, dimensionless pressure pulsation coefficient C p is defined as, where ∆p is the difference between the instantaneous pressure and the average pressure, p a ; ρ is fluid density, kg/m 3 ; u 2 is the peripheral speed of the impeller outlet, m/s. Figure 10    It can be seen that the amplitude of pressure fluctuation at each monitoring point changes periodically, with three peaks and three valleys. This is the same as the number of impeller blades, but the amplitude at each monitoring point has a large difference under different flow rates. Under the condition of small flow rate, the amplitude is XP > SP > y > G > P1, under the 1.0Qopt flow rate, XP > Y> G > P1 > SP, and under the condition of large flow rate, XP > P1 > G > Y> SP. Overall, the maximum amplitude of pressure fluctuation appears at XP at different flow rates. This is because XP is between the rotary cutter and the impeller, which is affected by the double interference. In addition, there is an obvious time difference between wave peaks and wave valleys of each monitoring point, which is caused by the difference in relative positions of dynamic and static interactions.    It can be seen that the amplitude of pressure fluctuation at each monitoring point changes periodically, with three peaks and three valleys. This is the same as the number of impeller blades, but the amplitude at each monitoring point has a large difference under different flow rates. Under the condition of small flow rate, the amplitude is XP > SP > y > G > P1, under the 1.0Q opt flow rate, XP > Y> G > P1 > SP, and under the condition of large flow rate, XP > P1 > G > Y> SP. Overall, the maximum amplitude of pressure fluctuation appears at XP at different flow rates. This is because XP is between the rotary cutter and the impeller, which is affected by the double interference. In addition, there is an obvious time difference between wave peaks and wave valleys of each monitoring point, which is caused by the difference in relative positions of dynamic and static interactions.   It can be seen that the amplitude of pressure fluctuation at each monitoring point changes periodically, with three peaks and three valleys. This is the same as the number of impeller blades, but the amplitude at each monitoring point has a large difference under different flow rates. Under the condition of small flow rate, the amplitude is XP > SP > y > G > P1, under the 1.0Qopt flow rate, XP > Y> G > P1 > SP, and under the condition of large flow rate, XP > P1 > G > Y> SP. Overall, the maximum amplitude of pressure fluctuation appears at XP at different flow rates. This is because XP is between the rotary cutter and the impeller, which is affected by the double interference. In addition, there is an obvious time difference between wave peaks and wave valleys of each monitoring point, which is caused by the difference in relative positions of dynamic and static interactions.     Figure 13. It can be seen from the figure that the m cy at each monitoring point is 74.01 Hz. The pressure fluctuation amplitude at frequency he largest. In general, the pressure fluctuation frequency of each monitoring point is mai trated within 0~600 Hz, and there are obvious low peaks in 6fn, 9fn, 12fn, 15fn, 18fn, 21fn, 24fn, frequency increases, the peak value decreases until it nearly disappears. At XP point of plate, the main frequency amplitude is much larger than other monitoring points, and ring point Y at the clearance between the rotary cutter and the fixed cutter is the place w est number of frequency divisions.

Frequency Domain Analysis of Pressure Pulsation at Different Positions
The fast Fourier transform (FFT) is applied to the time domain signals of pressure fluctuation at each monitoring point, and the frequency domain of pressure fluctuation at the corresponding monitoring points is obtained. In this paper, when the impeller speed n is 1480 r/min, the shaft frequency f n is 24.67 Hz, the number of impeller blades and rotary blades is three. The frequency of the impeller blade and the frequency of the rotary cutter blade are 3f n , which is 74.01 Hz. Under 1.0Q opt flow rate, the frequency domain is shown in Figure 13. It can be seen from the figure that the main frequency at each monitoring point is 74.01 Hz. The pressure fluctuation amplitude at frequency of 3f n , is the largest. In general, the pressure fluctuation frequency of each monitoring point is mainly concentrated within 0~600 Hz, and there are obvious low peaks in 6f n , 9f n , 12f n , 15f n , 18f n , 21f n , 24f n , etc. As the frequency increases, the peak value decreases until it nearly disappears. At XP point of the lower plate, the main frequency amplitude is much larger than other monitoring points, and the monitoring point Y at the clearance between the rotary cutter and the fixed cutter is the place with the largest number of frequency divisions.

requency Domain Analysis of Pressure Pulsation at Different Positions
The fast Fourier transform (FFT) is applied to the time domain signals of pressure fluctuat monitoring point, and the frequency domain of pressure fluctuation at the correspo itoring points is obtained. In this paper, when the impeller speed n is 1480 r/min, the ency fn is 24.67 Hz, the number of impeller blades and rotary blades is three. The frequen mpeller blade and the frequency of the rotary cutter blade are 3fn, which is 74.01 Hz. Under 1 rate, the frequency domain is shown in Figure 13. It can be seen from the figure that the ency at each monitoring point is 74.01 Hz. The pressure fluctuation amplitude at frequen s the largest. In general, the pressure fluctuation frequency of each monitoring point is m entrated within 0~600 Hz, and there are obvious low peaks in 6fn, 9fn, 12fn, 15fn, 18fn, 21fn, 24 e frequency increases, the peak value decreases until it nearly disappears. At XP point r plate, the main frequency amplitude is much larger than other monitoring points, an itoring point Y at the clearance between the rotary cutter and the fixed cutter is the place argest number of frequency divisions.     Figure 15 shows the radial force distribution on the pump impeller in one rotation period under the flow rates of 0.6Qopt, 0.8Qopt, 1.0Qopt, 1.2Qopt, and 1.4Qopt. The radial force on the impeller changes with time as shown in Figure 15a with three peaks and three valleys, which are consistent with three impeller blades. Under 0.6Qopt flow rate, radial force on the impeller is the largest, with the peak value of 306 N. With the increase in the flow rate to 0.8Qopt, 1.0Qopt and 1.2Qopt, radial force on the impeller gradually decreases, and the peak value under each flow rates is 274 N, 162 N and 87.3 N, respectively. Because of the complex interference of the internal flow field to radial force, there are obvious secondary peaks waves of radial forces except the largest peak. Figure 15b is the polar diagram of radial force on the impeller changing with rotation angle, where a rotation period of 360 degrees is selected. It can be seen from the figure that radial force changes with rotation angle, which is a periodic change of irregular clover petals. There are secondary peaks at 30 degrees, 150 degrees, and 270 degrees, lower peaks at 115 degrees, 225 degrees, and 345 degrees, and secondary valleys at many places.   Figure 15 shows the radial force distribution on the pump impeller in one rotation period under the flow rates of 0.6Q opt , 0.8Q opt , 1.0Q opt , 1.2Q opt , and 1.4Q opt . The radial force on the impeller changes with time as shown in Figure 15a with three peaks and three valleys, which are consistent with three impeller blades. Under 0.6Q opt flow rate, radial force on the impeller is the largest, with the peak value of 306 N. With the increase in the flow rate to 0.8Q opt , 1.0Q opt and 1.2Q opt , radial force on the impeller gradually decreases, and the peak value under each flow rates is 274 N, 162 N and 87.3 N, respectively. Because of the complex interference of the internal flow field to radial force, there are obvious secondary peaks waves of radial forces except the largest peak. Figure 15b is the polar diagram of radial force on the impeller changing with rotation angle, where a rotation period of 360 degrees is selected. It can be seen from the figure that radial force changes with rotation angle, which is a periodic change of irregular clover petals. There are secondary peaks at 30 degrees, 150 degrees, and 270 degrees, lower peaks at 115 degrees, 225 degrees, and 345 degrees, and secondary valleys at many places.

Analysis of Impeller Force
obvious secondary peaks waves of radial forces except the largest peak. Figure 15b is the polar diagram of radial force on the impeller changing with rotation angle, where a rotation period of 360 degrees is selected. It can be seen from the figure that radial force changes with rotation angle, which is a periodic change of irregular clover petals. There are secondary peaks at 30 degrees, 150 degrees, and 270 degrees, lower peaks at 115 degrees, 225 degrees, and 345 degrees, and secondary valleys at many places.   Figure 16 is the radial force vector of the pump impeller under the flow rates of 0.6Q opt , 1.0Q opt and 1.4Q opt . The horizontal and vertical coordinates represent the components of radial force along X and Y axes, respectively. The number of sample points in each figure is 120 record points in one cycle. It can be seen from the figure that, at each flow rate, radial force on the impeller exhibits a triangular distribution, and its size and direction will change at any time. Under 0.6Q opt flow rate, radial force on the impeller is in the range of −280 to 300 N in X axis direction, and in the range of −250 to 300 N in Y axis direction. Under flow rate of 1.0Q opt , the radial force range of the impeller decreases, but the amplitude fluctuation increases; compared with small flow rate, the force distribution ranges of X axis and Y axis reduce to between −150 and 150 N, that is to say, the radial force decreases significantly by nearly half. Under flow rate of 1.4Q opt , the radial force on impeller is similar to an irregular clover petal, and the periodicity is more obvious than the former two flow rates. Although there are three larger radial force peaks, more points concentrate at the center circle point where a smaller radial force is distributed.
Energies 2020, 13, x FOR PEER REVIEW 11 of 15 and Y axes, respectively. The number of sample points in each figure is 120 record points in one cycle. It can be seen from the figure that, at each flow rate, radial force on the impeller exhibits a triangular distribution, and its size and direction will change at any time. Under 0.6Qopt flow rate, radial force on the impeller is in the range of −280 to 300 N in X axis direction, and in the range of −250 to 300 N in Y axis direction. Under flow rate of 1.0Qopt, the radial force range of the impeller decreases, but the amplitude fluctuation increases; compared with small flow rate, the force distribution ranges of X axis and Y axis reduce to between −150 and 150 N, that is to say, the radial force decreases significantly by nearly half. Under flow rate of 1.4Qopt, the radial force on impeller is similar to an irregular clover petal, and the periodicity is more obvious than the former two flow rates. Although there are three larger radial force peaks, more points concentrate at the center circle point where a smaller radial force is distributed.  Figure 17 shows the axial force on impeller under flow rates of 0.6Qopt, 0.8Qopt, 1.0Qopt, 1.2Qopt, and 1.4Qopt. Axial force on impeller changing with time is shown in Figure 16a, it changes periodically with time showing three peaks and three valleys. Compared with the radial force time distribution, the fluctuations are more regular and almost all sinusoidal curves have better stability. Under 0.6Qopt flow rate, the axial force on the impeller is the smallest, with a peak value of 70.2 N. As flow rate increases, axial force on the impeller gradually increases. The peak axial forces experienced by the impeller under 0. 8Qopt, 1.0Qopt, 1.2Qopt, and 1.4Qopt flow rates are 78.7 N, 98.3 N, 116 N, and 139 N, respectively. In order to prevent the impeller from bearing large axial force and to extend the service life of the pump, it is necessary to avoid the pump running under the condition of large flow rate as much as possible. It can be seen from Figure 17b Figure 17 shows the axial force on impeller under flow rates of 0.6Q opt , 0.8Q opt , 1.0Q opt , 1.2Q opt , and 1.4Q opt . Axial force on impeller changing with time is shown in Figure 16a, it changes periodically with time showing three peaks and three valleys. Compared with the radial force time distribution, the fluctuations are more regular and almost all sinusoidal curves have better stability. Under 0.6Q opt flow rate, the axial force on the impeller is the smallest, with a peak value of 70.2 N. As flow rate increases, axial force on the impeller gradually increases. The peak axial forces experienced by the impeller under 0.8Q opt , 1.0Q opt , 1.2Q opt , and 1.4Q opt flow rates are 78.7 N, 98.3 N, 116 N, and 139 N, respectively. In order to prevent the impeller from bearing large axial force and to extend the service life of the pump, it is necessary to avoid the pump running under the condition of large flow rate as much as possible. It can be seen from Figure 17b that, axial force changes with the rotation angle, showing a periodic change of an irregular clover petal. When the impeller rotates to 90 degrees, 210 degrees and 330 degrees, there are obvious peaks. When the impeller rotates to 45 degrees, 165 degrees and 285 degrees, there are obvious valleys, as is consistent with the time domain of Figure 17a.
with time showing three peaks and three valleys. Compared with the radial force time distribution, the fluctuations are more regular and almost all sinusoidal curves have better stability. Under 0.6Qopt flow rate, the axial force on the impeller is the smallest, with a peak value of 70.2 N. As flow rate increases, axial force on the impeller gradually increases. The peak axial forces experienced by the impeller under 0.8Qopt, 1.0Qopt, 1.2Qopt, and 1.4Qopt flow rates are 78.7 N, 98.3 N, 116 N, and 139 N, respectively. In order to prevent the impeller from bearing large axial force and to extend the service life of the pump, it is necessary to avoid the pump running under the condition of large flow rate as much as possible. It can be seen from Figure 17b

Experiment
The external characteristics and pressure fluctuation test of the model pump are carried out on the multi-functional pump model test stand, and the layout of the test device is shown in Figure 18. The main instruments include semi-open submersible cutting pump, turbine flow-meter, flow regulating valve, outlet pressure transmitter, power distribution box, data acquisition instrument and so on.

Experiment
The external characteristics and pressure fluctuation test of the model pump are carried out on the multi-functional pump model test stand, and the layout of the test device is shown in Figure 18. The main instruments include semi-open submersible cutting pump, turbine flow-meter, flow regulating valve, outlet pressure transmitter, power distribution box, data acquisition instrument and so on. The comparison of simulation curves and test curves of head and efficiency of the cutting pump under different flow rates is shown in Figure 19. The predicted results of numerical simulation are basically consistent with the test results. The simulation values of head and efficiency are larger than the test values, the head error is about 1.13 m, and the efficiency error is about 5%. Because of the selection of model, parameter, empirical formula, simplification of three-dimensional modeling and the technology of the pump in the process of simulation, there are some deviations between the simulation value and the test value. However, the trends are relatively consistent, so as to verify the reliability of this data calculation, suggesting that simulation can replace some tests for performance prediction. The comparison of simulation curves and test curves of head and efficiency of the cutting pump under different flow rates is shown in Figure 19. The predicted results of numerical simulation are basically consistent with the test results. The simulation values of head and efficiency are larger than the test values, the head error is about 1.13 m, and the efficiency error is about 5%. Because of the selection of model, parameter, empirical formula, simplification of three-dimensional modeling and the technology of the pump in the process of simulation, there are some deviations between the simulation value and the test value. However, the trends are relatively consistent, so as to verify the reliability of this data calculation, suggesting that simulation can replace some tests for performance prediction.
test values, the head error is about 1.13 m, and the efficiency error is about 5%. Because of the selection of model, parameter, empirical formula, simplification of three-dimensional modeling and the technology of the pump in the process of simulation, there are some deviations between the simulation value and the test value. However, the trends are relatively consistent, so as to verify the reliability of this data calculation, suggesting that simulation can replace some tests for performance prediction. During the pressure fluctuation test, the flow rate is adjusted by adjusting the valve at the outlet of the pipeline. Through the outlet pressure transmitter, pump outlet pressure is obtained, and the electrical measurement method is used to measure and calculate the pump shaft power. The submersible high-frequency pressure sensor with an accuracy class of 0.25 and the measurement range of 0 to 1 Mpa is used. The sampling frequency is 2960 Hz, and each sampling time lasts 20 s. The preliminary processing of data is completed by HSJ-2010 hydraulic machinery comprehensive tester (Huazhong University of science and technology, Wuhan, China). The main test instruments are shown in Figure 20.
The structure of the semi-open submersible cutting pump is compact and complex, so it is difficult for the sensor to insert deeply into the pump to measure the pressure distributions at upper and lower plates and other positions. Therefore, the pressure fluctuation sensors are installed at the volute section and outlet where the pressure fluctuations have a great influence on the sewage      the tested time domain of pressure fluctuations under flow rates o n be seen that periodic fluctuations exist at the first section, the f d the volute outlet, which is consistent with the simulation results s is 3 too, which is consistent with the number of blades. Under there are obvious secondary peaks at each monitoring point. A plitude of the pressure fluctuation at the monitoring point P1 furth than those at other locations, which is consistent with the simulati veforms of the fifth section, the eighth section and the volute outlet a ever, as the flow rate increases, due to the internal high frequency se field at cross sections, the waveform regularity at the volute outle e at other two positions, which is slightly different from the simula ditions, the manufacturing process of test pump and its test environ een the simulation value and test value. But the fluctuation tre whole.  It can be seen that periodic fluctuations exist at the first section, the fifth section, the eighth section and the volute outlet, which is consistent with the simulation results. The number of peaks and valleys is 3 too, which is consistent with the number of blades. Under the condition of small flow rate, there are obvious secondary peaks at each monitoring point. As the flow rate increases, the amplitude of the pressure fluctuation at the monitoring point P1 further increases and is more obvious than those at other locations, which is consistent with the simulation. Under small flow rate, the waveforms of the fifth section, the eighth section and the volute outlet are more regular and clearer. However, as the flow rate increases, due to the internal high frequency sensor disturbing the internal flow field at cross sections, the waveform regularity at the volute outlet is significantly higher than those at other two positions, which is slightly different from the simulation. The setting of boundary conditions, the manufacturing process of test pump and its test environment leads to a certain gap between the simulation value and test value. But the fluctuation trend is relatively consistent on the whole.    Figure 22 is the tested time domain of pressure fluctuations under flow rates of 0.6Qopt, 1.0Qopt and 1.4Qopt. It can be seen that periodic fluctuations exist at the first section, the fifth section, the eighth section and the volute outlet, which is consistent with the simulation results. The number of peaks and valleys is 3 too, which is consistent with the number of blades. Under the condition of small flow rate, there are obvious secondary peaks at each monitoring point. As the flow rate increases, the amplitude of the pressure fluctuation at the monitoring point P1 further increases and is more obvious than those at other locations, which is consistent with the simulation. Under small flow rate, the waveforms of the fifth section, the eighth section and the volute outlet are more regular and clearer. However, as the flow rate increases, due to the internal high frequency sensor disturbing the internal flow field at cross sections, the waveform regularity at the volute outlet is significantly higher than those at other two positions, which is slightly different from the simulation. The setting of boundary conditions, the manufacturing process of test pump and its test environment leads to a certain gap between the simulation value and test value. But the fluctuation trend is relatively consistent on the whole.  Figure 23 shows the frequency domain comparison of pressure fluctuation at each monitoring point under different flow rates. It can be seen from the figure that, although there are some differences of the amplitudes results from simulation and experiment, their fluctuation trends are more consistent. The maximum peak appears at the three times axis frequency, indicating that the main frequency is the blade frequency, and there are obvious fluctuations at the blade frequency. In  Figure 23 shows the frequency domain comparison of pressure fluctuation at each monitoring point under different flow rates. It can be seen from the figure that, although there are some differences of the amplitudes results from simulation and experiment, their fluctuation trends are more consistent.
The maximum peak appears at the three times axis frequency, indicating that the main frequency is the blade frequency, and there are obvious fluctuations at the blade frequency. In the low frequency area of each monitoring point under low flow rates, the frequency division components are more and their density is large. With the increase in the flow rate, the density and intensity of the frequency division of each monitoring point show a decreasing trend, and the simulation and test results are more consistent.
Energies 2020, 13, x FOR PEER REVIEW 14 of 15 the low frequency area of each monitoring point under low flow rates, the frequency division components are more and their density is large. With the increase in the flow rate, the density and intensity of the frequency division of each monitoring point show a decreasing trend, and the simulation and test results are more consistent.

Conclusions
(1) The pressure fluctuation at each flow component shows a periodic change: the amplitude attenuates from a position near the rotating part to a position away from the rotating part, and from a position near the tongue to a position away from the tongue, and the fluctuation increases as the flow rate increases. Under different flow rates, the pressure fluctuation amplitude of lower plate XP is the largest, because the position is between the rotary cutter and the impeller, which is affected by the double interference.
(2) The pressure fluctuation main frequency is the blade passing frequency. In the low frequency region with multiple integrals of blade frequency and axial frequency, wave amplitudes are obvious, and in the high frequency region, wave amplitudes are weak. At each flow rate, the main frequency amplitude at the monitoring point of lower plate XP is much larger than those at other monitoring points. However, there are many low-frequency signals at the monitoring point Y between the rotary cutter and the fixed cutter, and the frequency components of the pressure fluctuation are complicated.
(3) The radial force and axial force on the impeller change periodically with time, and the number of peaks and valleys is consistent with the number of blades in one turn. Under different flow rates, the distributions of radial force and axial force show a strong similarity. The axial force experienced by the impeller increases with the increase in flow rate, while the radial force experienced by the impeller at flow rate of 0.6Qopt is the largest.

Conflicts of Interest:
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.