# Performance Optimization of Slotted Blades for Low-Specific Speed Centrifugal Pumps

Authors: Qidi Ke; Lingfeng Tang
Journal: Advances in Civil Engineering
 (2023)

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## Abstract

Centrifugal pumps are widely used for the transport of fluids, but low-specific-speed centrifugal pumps widely have problems with serious backflow and low efficiency. In this paper, a low-specific-speed centrifugal pump with a specific speed of 55 is used as a research object. By combining numerical simulation and orthogonal experiment, the pressure distribution and velocity distribution of the flow channel are analyzed, the priority of each geometric factor for slits on pump performance is determined, the geometric parameter structure of the slotted blade is optimized by entropy weight on TOPSIS, the optimal impeller slit solution is obtained. The results show that with a balance of head and efficiency, the order of influence of the factors is: slit center widthb> diameter of slit D> shrinkage ratio of slit f> depth of slit h> deflection angle of slit β. The optimal combination of slit geometry parameters is: slit center width is 3 mm, diameter of the slit is 200 mm, shrinkage ratio of the slit is 0.5, depth of the slit is 6 mm, and the deflection angle of the slit is 20°. Through ANSYS FLUENT simulation and experiment of closed pump experiment system, confirmed that hydraulic performance is improved.


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## Body

## 1. Introduction

The low-specific-speed pump has the characteristics of a low flow rate and a high head and is widely used in agricultural irrigation and industrial water supply fields. To improve the performance of low-specific-speed pumps, scholars have done research in various areas [1–3]. De Donno and Thakkar et al. used a genetic algorithm and a multiobjective optimization algorithm to enhance the performance of the centrifugal pump, and the effect is significant [4, 5]. Al-Obaidi and Towsyfyan developed an efficient signal processing method based on envelope spectral analysis, which was used to monitor the vibration characteristics of centrifugal pumps, and after that, they investigated the effect of different operating conditions using different statistical features in a time domain analysis (TDA) and detected and diagnosed the cavitation in centrifugal pumps [6, 7]. Zhang et al. and Zhang et al. proposed and optimized the equations and parameters of 3D blade formation [8, 9]. Bai et al. built a vibration test stand to examine the vibration and stability of cantilever multistage centrifugal pumps at different flow rates [10]. Feng et al. studied the rotational stall characteristics inside the vane diffuser of a centrifugal pump based on ANSYS CFX [11]. Stel et al. conducted numerical and experimental studies on the motion of air bubbles in the impeller of a centrifugal pump and also used numerical particle tracking methods to evaluate the bubble motion [12]. Namazizadeh et al. studied the effect of adding separator blades and modifying their geometry to optimize the impeller of the centrifugal pump [13]. Liu et al. proposed some very excellent and practical optimization and prediction methods for multipump and mixed-flow pumps [14–16]. Li et al. combine the nondominated sorting genetic algorithm-II (NSGA-II) and a modified technique for order preference by similarity to an ideal solution (TOPSIS) based on the Shannon entropy and explore the optimal design of grooves in axial-flow pumps [17].The major problem with low-specific-speed centrifugal pumps is the backflow inside the flow channel, which reduces the hydraulic performance [18]. To block partial backflow, usually use the method of setting the splitter blade. The blade method was originally used in the aviation field, but it has not been well studied in centrifugal pumps. Currently, some scholars have studied the relationship between the shape of the slit vane and the performance of centrifugal pumps by using orthogonal tests, but they are not very comprehensive [19, 20]. Zhang et al. and Wei et al. investigated the effect of slot width on drainage performance [21, 22]. The study found that slot drainage will reduce the head, but it will increase its hydraulic efficiency. Zhang et al. aimed to improve the cavitation performance of centrifugal pumps and proposed a new type of impeller for centrifugal pumps with slots, and this worked well [23].This paper designs the typical low-specific-speed centrifugal pump with a specific speed of 55, analyzed the reasons for its inefficiency, investigated the effect of slit geometry parameters on the hydraulic performance and internal flow field, and performed a multiobjective optimization. This study provides a reference for improving the hydraulic performance of a low-specific-speed centrifugal pump.


## 2. Centrifugal Pump Structure Design

### 2.1. Determination of Structural Parameters

The pump studied in this paper is a low-specific-speed centrifugal pump, and the main design parameters are shown in Table1.Table 1
Main design parameters.
FlowQm3/h−1HeadHmSpeednr/min50251440Design the structure of the centrifugal pump according to the given parameters. Hydraulic design is the first step of the study, and the value of the structural parameters will directly affect the performance of the pump.(1)
Pump inlet diameter and inlet speed(1)Ds=ksQn3,whereDs—inlet diameter m; Q—inlet speed m3/h; the value of the coefficient ks ranges from 4∼5, and 4 is taken here. After calculation: Ds = 85.15 mm, rounding to 80 mm in accordance with the common caliber. Based on the inlet diameter and flow rate, the selected flow velocity Vs = 2.76 m/s.(2)
Pump outlet diameterThe formula is as follows:(2)Dd=kdQn3,whereDd—outlet diameter m; The value of the coefficient kd ranges from 3.5∼4.5, and 3.5 is taken here. So pump outlet diameter Dd = 74.5 mm(3)
Specific speed(3)ns=3.65nQH3/4,wherens—specific speed; H—head m. After calculation: ns=55.407(4)
Efficiency estimationThe hydraulic efficiency of the pump is calculated according to the following equation:(4)ηh≈1+0.0835lgQn3.After calculation:ηh=0.8604.The volumetric efficiency is calculated by(5)ηv≈11+0.68ns−2/3.After calculation:ηv=0.9553The mechanical efficiency of the disc friction loss is calculated by the following equation:(6)ηm=1−0.071ns/1007/6.After calculation:ηm=0.8606.The total efficiency of the pump is(7)η=ηmηvηh.After calculation:η=0.7073.


### 2.2. Impeller Size Determination

The structure form of centrifugal pump impeller is mainly semiopen type and closed type. The semiopen impeller has the advantage of easy cleaning and easy forming, therefore, the impeller used in this paper is a semiopen type. Used the velocity coefficient method for the design of the impeller structure.


#### 2.2.1. Impeller Inlet Diameter

This pump is not a through-shaft impeller; instead, it is a cantilever impeller without a hub. Impeller inlet diameter is calculated according to the following equation:(8)D1=k0k1Qn3,where coefficient k0 = 3.6. And the coefficient k1 is generally taken as 0.7∼1.0, this is determined by the specific rotation number, the lower ratio speed takes the larger value, so take 0.9 here. Finally, D1 = 72 mm.


#### 2.2.2. Impeller Outlet Diameter

Impeller outlet diameterD2 is calculated according to the following formula:(9)D2=kDQn3,kD=9.35kD2ns100−1/2.The selection ofkD2 is related to the form of the pump and the specific speed, so kD2 = 1.075 in this paper. After calculation: D2 = 291.2 mm. After rounding, take D2 = 295 mm.


#### 2.2.3. Impeller Outlet Width

Impeller outlet widthb2 is calculated according to the following formula:(10)b2=kbQn3,kb=0.64kb2ns1005/6,where kb2 = 1.34. After calculation: b2 = 11.2 mm.


#### 2.2.4. Other Data Selection

Blade linear selection of logarithmic spiral, blade inlet angleβ1 is 19°, blade outlet angle β2 is 33°, blade wrap angle φ is 123°, and blade thickness select 4 mm [24, 25].


#### 2.2.5. Blade Number

The number of blades is calculated according to the following equation:(11)Z=6.5D2+D1D2−D1sinβ1+β22.Take the values ofD1, D2, β1, and β2 into calculation and get Z = 4.732, so we take Z = 5.


### 2.3. Determination of Volute Size

The volute is one of the overflow components that transform energy. The shape of the volute has no significant effect on performance and select rectangle here. The main dimensions of the volute are designed as follows:(1)
Base circle diameterD3To make the flow channel smooth, usually taken as follows:(12)D3=1.03∼1.08D2.The coefficient for low-specific speed-pumps should be lower. So we takeD3 = 305 mm(2)
Volute inlet widthb3Volute inlet width is determined by considering the 8th cross-section; therefore we try to make the geometry of the 8th cross-section reasonable.(13)b3=b2+5∼10.After calculation:b3 = 20 mm(3)
Setting angle of the volute tongueφ0Based on factory experience, takeφ0 = 15° in this paper(4)
Area of each cross-sectionWe use the speed coefficient method for similar conversions as follows:(14)v3=k32gH,where v3−the average speed of each section; k3−Speed coefficient, we take 0.46 here.After calculation:v3 = 10.19 m/s.The flow through section 8 is nearly equal to the actual flow. Therefore, the area of the 8th section can be calculated by the following formula:(15)F8=Qv3.Other section areas are calculated at equal speeds in each section.(16)Fφ=φ360F8.The area of each section is obtained as follows Table2:Table 2
Section area table.
Section12345678Wrap angle (°)4590135180225270315360Area (mm2)170.4340.8511.1681.5851.91022.31192.61363


## 2.1. Determination of Structural Parameters

The pump studied in this paper is a low-specific-speed centrifugal pump, and the main design parameters are shown in Table1.Table 1
Main design parameters.
FlowQm3/h−1HeadHmSpeednr/min50251440Design the structure of the centrifugal pump according to the given parameters. Hydraulic design is the first step of the study, and the value of the structural parameters will directly affect the performance of the pump.(1)
Pump inlet diameter and inlet speed(1)Ds=ksQn3,whereDs—inlet diameter m; Q—inlet speed m3/h; the value of the coefficient ks ranges from 4∼5, and 4 is taken here. After calculation: Ds = 85.15 mm, rounding to 80 mm in accordance with the common caliber. Based on the inlet diameter and flow rate, the selected flow velocity Vs = 2.76 m/s.(2)
Pump outlet diameterThe formula is as follows:(2)Dd=kdQn3,whereDd—outlet diameter m; The value of the coefficient kd ranges from 3.5∼4.5, and 3.5 is taken here. So pump outlet diameter Dd = 74.5 mm(3)
Specific speed(3)ns=3.65nQH3/4,wherens—specific speed; H—head m. After calculation: ns=55.407(4)
Efficiency estimationThe hydraulic efficiency of the pump is calculated according to the following equation:(4)ηh≈1+0.0835lgQn3.After calculation:ηh=0.8604.The volumetric efficiency is calculated by(5)ηv≈11+0.68ns−2/3.After calculation:ηv=0.9553The mechanical efficiency of the disc friction loss is calculated by the following equation:(6)ηm=1−0.071ns/1007/6.After calculation:ηm=0.8606.The total efficiency of the pump is(7)η=ηmηvηh.After calculation:η=0.7073.


## 2.2. Impeller Size Determination

The structure form of centrifugal pump impeller is mainly semiopen type and closed type. The semiopen impeller has the advantage of easy cleaning and easy forming, therefore, the impeller used in this paper is a semiopen type. Used the velocity coefficient method for the design of the impeller structure.


### 2.2.1. Impeller Inlet Diameter

This pump is not a through-shaft impeller; instead, it is a cantilever impeller without a hub. Impeller inlet diameter is calculated according to the following equation:(8)D1=k0k1Qn3,where coefficient k0 = 3.6. And the coefficient k1 is generally taken as 0.7∼1.0, this is determined by the specific rotation number, the lower ratio speed takes the larger value, so take 0.9 here. Finally, D1 = 72 mm.


### 2.2.2. Impeller Outlet Diameter

Impeller outlet diameterD2 is calculated according to the following formula:(9)D2=kDQn3,kD=9.35kD2ns100−1/2.The selection ofkD2 is related to the form of the pump and the specific speed, so kD2 = 1.075 in this paper. After calculation: D2 = 291.2 mm. After rounding, take D2 = 295 mm.


### 2.2.3. Impeller Outlet Width

Impeller outlet widthb2 is calculated according to the following formula:(10)b2=kbQn3,kb=0.64kb2ns1005/6,where kb2 = 1.34. After calculation: b2 = 11.2 mm.


### 2.2.4. Other Data Selection

Blade linear selection of logarithmic spiral, blade inlet angleβ1 is 19°, blade outlet angle β2 is 33°, blade wrap angle φ is 123°, and blade thickness select 4 mm [24, 25].


### 2.2.5. Blade Number

The number of blades is calculated according to the following equation:(11)Z=6.5D2+D1D2−D1sinβ1+β22.Take the values ofD1, D2, β1, and β2 into calculation and get Z = 4.732, so we take Z = 5.


## 2.2.1. Impeller Inlet Diameter

This pump is not a through-shaft impeller; instead, it is a cantilever impeller without a hub. Impeller inlet diameter is calculated according to the following equation:(8)D1=k0k1Qn3,where coefficient k0 = 3.6. And the coefficient k1 is generally taken as 0.7∼1.0, this is determined by the specific rotation number, the lower ratio speed takes the larger value, so take 0.9 here. Finally, D1 = 72 mm.


## 2.2.2. Impeller Outlet Diameter

Impeller outlet diameterD2 is calculated according to the following formula:(9)D2=kDQn3,kD=9.35kD2ns100−1/2.The selection ofkD2 is related to the form of the pump and the specific speed, so kD2 = 1.075 in this paper. After calculation: D2 = 291.2 mm. After rounding, take D2 = 295 mm.


## 2.2.3. Impeller Outlet Width

Impeller outlet widthb2 is calculated according to the following formula:(10)b2=kbQn3,kb=0.64kb2ns1005/6,where kb2 = 1.34. After calculation: b2 = 11.2 mm.


## 2.2.4. Other Data Selection

Blade linear selection of logarithmic spiral, blade inlet angleβ1 is 19°, blade outlet angle β2 is 33°, blade wrap angle φ is 123°, and blade thickness select 4 mm [24, 25].


## 2.2.5. Blade Number

The number of blades is calculated according to the following equation:(11)Z=6.5D2+D1D2−D1sinβ1+β22.Take the values ofD1, D2, β1, and β2 into calculation and get Z = 4.732, so we take Z = 5.


## 2.3. Determination of Volute Size

The volute is one of the overflow components that transform energy. The shape of the volute has no significant effect on performance and select rectangle here. The main dimensions of the volute are designed as follows:(1)
Base circle diameterD3To make the flow channel smooth, usually taken as follows:(12)D3=1.03∼1.08D2.The coefficient for low-specific speed-pumps should be lower. So we takeD3 = 305 mm(2)
Volute inlet widthb3Volute inlet width is determined by considering the 8th cross-section; therefore we try to make the geometry of the 8th cross-section reasonable.(13)b3=b2+5∼10.After calculation:b3 = 20 mm(3)
Setting angle of the volute tongueφ0Based on factory experience, takeφ0 = 15° in this paper(4)
Area of each cross-sectionWe use the speed coefficient method for similar conversions as follows:(14)v3=k32gH,where v3−the average speed of each section; k3−Speed coefficient, we take 0.46 here.After calculation:v3 = 10.19 m/s.The flow through section 8 is nearly equal to the actual flow. Therefore, the area of the 8th section can be calculated by the following formula:(15)F8=Qv3.Other section areas are calculated at equal speeds in each section.(16)Fφ=φ360F8.The area of each section is obtained as follows Table2:Table 2
Section area table.
Section12345678Wrap angle (°)4590135180225270315360Area (mm2)170.4340.8511.1681.5851.91022.31192.61363


## 3. Numerical Simulation

### 3.1. Computational Model

Construct the water body parts of the suction section, impeller, and volute. To ensure adequate flows, reduce the influence of the inlet section on the water flow, so it is necessary to increase the length of the inlet section appropriately. The calculation model is shown in Figure1.Figure 1
Flow field diagram.


### 3.2. Determination of the Turbulence Model and Boundary Conditions

In CFD flow field calculation, the accuracy of the computational model has a decisive influence on the results of the flow field simulation. Commonly, thek−ε model and k−ω model have a wide range of applications. The RNGk−ε model is derived from a rigorous statistical technique, considered turbulent vortex formation, fully investigated the low Reynolds number flow viscosity calculation, and is more suitable for flow in the near wall area. This model is chosen in this paper [26, 27]. The specific expression is shown in the following equation:(17)∂ρk∂t+∂ρkui∂xi=∂∂xjαkμeff∂k∂xj+Pk−ρε,∂ρε∂t+∂ρεui∂xi=∂∂xjαεμeff∂ε∂xj+C1εεkPk−C2εε2kρ.In the formula(18)μeff=μ+μt,μt=ρCμk2ε,where Cμ=0.0845; αk=αε = 1.39; C1ε=1.42; C2ε=1.68.The mesh file displayed in FLUENT is shown in Figure2. Inlet boundary selection speed inlet: set the value to 2.76 m/s. Outlet boundary selection mass flow rate: the flow rate is set to 13.89 kg/s. The rotational speed is set to 1440 r/min. Set the impeller blades and back and forward cover plates to the rotating wall; the values are 0 rev/min. Set the shear condition of the static wall to a no-slip wall [28].Figure 2
Boundary condition setting diagram.


### 3.3. Mesh

In numerical simulations, there are often various factors that affect the final results [29, 30]. To ensure the accuracy of the numerical calculation, it needs to ensure high quality in the near-wall region to enable capture of the flow in the boundary layer, as in Figure 3. Uses y+ to denote the dimensionless distance from the center of mass of the grid nearest the wall to the wall, the defining equation is as follows:(19)y+=∆yuτv,where ∆y is the distance from the first layer mass center of the mesh to the wall, uτ is friction velocity, and v is kinematic viscosity.Figure 3
Boundary layer mesh diagram.In the numerical simulations, the calculatedy+ is shown in Figure 4. Basically around 10∼140. The requirement of the RNG k−ε model is 0∼300, meets the near-wall mesh quality requirements [31].Figure 4
y+ diagram of the impeller boundary.To ensure the mesh numbers have no effect on the calculation results, get 5 different numbers of meshes. The mesh irrelevance analysis is shown in Table3. As shown in the table, with the number of meshes increasing, the effect of quantity on the calculation results can be ignored. The finalized mesh number is 3391623. However, the calculation of the hydraulic efficiency requires the measurement of the moment M at the pump shaft, as shown in the following equation:(20)η=ρgQHMω.Table 3
Mesh independence check.
Program numberScheme 1Scheme 2Scheme 3Scheme 4Scheme 5Mesh number18104582554012339162347194236238245Head (m)25.02625.03525.47025.52725.467Hydraulic efficiency (%)73.9173.2672.5572.7372.66Total efficiency (%)61.4660.9360.3560.5060.41


### 3.4. Flow Field Analysis

Simulation of centrifugal pump rated conditions, the result is shown in Figure5. The simulated head value is 25.470 m, slightly higher than the expected target; this is because leakage losses are not considered.Figure 5
Diagram of head simulation. (a) Head curve diagram. (b) Head monitoring documentation diagram.
(a)(b)Figure6 shows the pressure nephogram of the centrifugal pump. It can be seen that the pressure in the flow path inside the impeller increases gradually along the flow direction, which is the positive pressure gradient. There is a significant pressure difference between the pressure face and the suction face. At an impeller of the same radius, the pressure on the pressure surface is higher than the suction surface. At the intersection of the impeller channel and volute base circle, small low-pressure areas appear; this is a direct result of diffusion at the channel outlet of the low-specific speed pump. The model has a good pressure distribution that is consistent with reality.Figure 6
Pressure nephogram.Figure7 shows the velocity streamlines of the flow field. The lower velocity region occupies most of the blade’s working surface in the figure, and backflow occurs throughout the middle channel, only away from the volute tongue can be slightly relieved. Backflow causes channel blockage and consumes large amounts of fluid kinetic energy, decreasing the performance of centrifugal pumps.Figure 7
Velocity streamlines of the flow field.The Reynolds number is usually used to determine the flow state of viscous fluids; it means the ratio of the inertial force to the viscous force. The ideal flow state inside the impeller is a steady laminar flow along the channel. But as the flow progresses to the trailing edge, the Reynolds number increases, the thickness of the laminar boundary layer gradually thickens, and flow disturbances begin to develop. When creating coordinates, set along the impeller from the inlet to the outlet direction as the positive direction of thex-axis, the vertical direction of the blade is set as the positive direction of the y-axis, assumed constant incompressible fluid, the N−S equation is(21)∂u∂x+∂v∂y=0,u∂u∂x+v∂u∂y=−1ρ∂p∂x+v∂2u∂x2+∂2u∂y2,u∂v∂x+v∂v∂y=−1ρ∂p∂y+v∂2v∂x2+∂2v∂y2.Ignoring smaller magnitudes, simplify equation (21) to(22a)∂u∂x+∂v∂y=0,(22b)u∂u∂x+v∂u∂y=−1ρ∂p∂x+v∂2u∂y2,(22c)0=∂p∂y.From equation (22c), we can get the following result: the pressure p of the fluid boundary layer does not change along the y direction. This indicates that the fluid disturbance is caused by a pressure change in the x-axis direction: during flow channel diffusion, there is a negative pressure gradient in the x-axis direction, that is, ∂p/∂x<0 in equation (22b). When the pressure is less than the resistance during flow and the flow direction changes, turbulence begins to develop.From a macro perspective, in the flow channel between two blades, the differential pressure between the back and working surface is high pressure and low speed at the working surface, meanwhile, the opposite is shown for the suction surface. This pressure gradient forces the fluid to flow towards the back of the blade, which will create secondary flows and increase flow losses. With integrated consideration of boundary layer control methods and low-specific-speed centrifugal pump internal flow conditions, this paper uses slit blades to improve the performance of centrifugal pumps.


## 3.1. Computational Model

Construct the water body parts of the suction section, impeller, and volute. To ensure adequate flows, reduce the influence of the inlet section on the water flow, so it is necessary to increase the length of the inlet section appropriately. The calculation model is shown in Figure1.Figure 1
Flow field diagram.


## 3.2. Determination of the Turbulence Model and Boundary Conditions

In CFD flow field calculation, the accuracy of the computational model has a decisive influence on the results of the flow field simulation. Commonly, thek−ε model and k−ω model have a wide range of applications. The RNGk−ε model is derived from a rigorous statistical technique, considered turbulent vortex formation, fully investigated the low Reynolds number flow viscosity calculation, and is more suitable for flow in the near wall area. This model is chosen in this paper [26, 27]. The specific expression is shown in the following equation:(17)∂ρk∂t+∂ρkui∂xi=∂∂xjαkμeff∂k∂xj+Pk−ρε,∂ρε∂t+∂ρεui∂xi=∂∂xjαεμeff∂ε∂xj+C1εεkPk−C2εε2kρ.In the formula(18)μeff=μ+μt,μt=ρCμk2ε,where Cμ=0.0845; αk=αε = 1.39; C1ε=1.42; C2ε=1.68.The mesh file displayed in FLUENT is shown in Figure2. Inlet boundary selection speed inlet: set the value to 2.76 m/s. Outlet boundary selection mass flow rate: the flow rate is set to 13.89 kg/s. The rotational speed is set to 1440 r/min. Set the impeller blades and back and forward cover plates to the rotating wall; the values are 0 rev/min. Set the shear condition of the static wall to a no-slip wall [28].Figure 2
Boundary condition setting diagram.


## 3.3. Mesh

In numerical simulations, there are often various factors that affect the final results [29, 30]. To ensure the accuracy of the numerical calculation, it needs to ensure high quality in the near-wall region to enable capture of the flow in the boundary layer, as in Figure 3. Uses y+ to denote the dimensionless distance from the center of mass of the grid nearest the wall to the wall, the defining equation is as follows:(19)y+=∆yuτv,where ∆y is the distance from the first layer mass center of the mesh to the wall, uτ is friction velocity, and v is kinematic viscosity.Figure 3
Boundary layer mesh diagram.In the numerical simulations, the calculatedy+ is shown in Figure 4. Basically around 10∼140. The requirement of the RNG k−ε model is 0∼300, meets the near-wall mesh quality requirements [31].Figure 4
y+ diagram of the impeller boundary.To ensure the mesh numbers have no effect on the calculation results, get 5 different numbers of meshes. The mesh irrelevance analysis is shown in Table3. As shown in the table, with the number of meshes increasing, the effect of quantity on the calculation results can be ignored. The finalized mesh number is 3391623. However, the calculation of the hydraulic efficiency requires the measurement of the moment M at the pump shaft, as shown in the following equation:(20)η=ρgQHMω.Table 3
Mesh independence check.
Program numberScheme 1Scheme 2Scheme 3Scheme 4Scheme 5Mesh number18104582554012339162347194236238245Head (m)25.02625.03525.47025.52725.467Hydraulic efficiency (%)73.9173.2672.5572.7372.66Total efficiency (%)61.4660.9360.3560.5060.41


## 3.4. Flow Field Analysis

Simulation of centrifugal pump rated conditions, the result is shown in Figure5. The simulated head value is 25.470 m, slightly higher than the expected target; this is because leakage losses are not considered.Figure 5
Diagram of head simulation. (a) Head curve diagram. (b) Head monitoring documentation diagram.
(a)(b)Figure6 shows the pressure nephogram of the centrifugal pump. It can be seen that the pressure in the flow path inside the impeller increases gradually along the flow direction, which is the positive pressure gradient. There is a significant pressure difference between the pressure face and the suction face. At an impeller of the same radius, the pressure on the pressure surface is higher than the suction surface. At the intersection of the impeller channel and volute base circle, small low-pressure areas appear; this is a direct result of diffusion at the channel outlet of the low-specific speed pump. The model has a good pressure distribution that is consistent with reality.Figure 6
Pressure nephogram.Figure7 shows the velocity streamlines of the flow field. The lower velocity region occupies most of the blade’s working surface in the figure, and backflow occurs throughout the middle channel, only away from the volute tongue can be slightly relieved. Backflow causes channel blockage and consumes large amounts of fluid kinetic energy, decreasing the performance of centrifugal pumps.Figure 7
Velocity streamlines of the flow field.The Reynolds number is usually used to determine the flow state of viscous fluids; it means the ratio of the inertial force to the viscous force. The ideal flow state inside the impeller is a steady laminar flow along the channel. But as the flow progresses to the trailing edge, the Reynolds number increases, the thickness of the laminar boundary layer gradually thickens, and flow disturbances begin to develop. When creating coordinates, set along the impeller from the inlet to the outlet direction as the positive direction of thex-axis, the vertical direction of the blade is set as the positive direction of the y-axis, assumed constant incompressible fluid, the N−S equation is(21)∂u∂x+∂v∂y=0,u∂u∂x+v∂u∂y=−1ρ∂p∂x+v∂2u∂x2+∂2u∂y2,u∂v∂x+v∂v∂y=−1ρ∂p∂y+v∂2v∂x2+∂2v∂y2.Ignoring smaller magnitudes, simplify equation (21) to(22a)∂u∂x+∂v∂y=0,(22b)u∂u∂x+v∂u∂y=−1ρ∂p∂x+v∂2u∂y2,(22c)0=∂p∂y.From equation (22c), we can get the following result: the pressure p of the fluid boundary layer does not change along the y direction. This indicates that the fluid disturbance is caused by a pressure change in the x-axis direction: during flow channel diffusion, there is a negative pressure gradient in the x-axis direction, that is, ∂p/∂x<0 in equation (22b). When the pressure is less than the resistance during flow and the flow direction changes, turbulence begins to develop.From a macro perspective, in the flow channel between two blades, the differential pressure between the back and working surface is high pressure and low speed at the working surface, meanwhile, the opposite is shown for the suction surface. This pressure gradient forces the fluid to flow towards the back of the blade, which will create secondary flows and increase flow losses. With integrated consideration of boundary layer control methods and low-specific-speed centrifugal pump internal flow conditions, this paper uses slit blades to improve the performance of centrifugal pumps.


## 4. Experimental Design and Results Analysis

### 4.1. Orthogonal Experimental Design

There are many geometric parameters of the slotted blade, and with a balance between the number of experiments and the mastery of experimental rules, an orthogonal experiment of 5 factors and 4 levels for analysis was designed. According to the characteristics of the flow channel and the structure of the blade, decided to select the factors as the diameter of slitD, slit center width b, deflection angle of slit β, depth of slit h, shrinkage ratio of slit f, and denoted by A, B, C, D, E in order. Where the shrinkage ratio f is defined as the ratio of the width of the slit at the pressure surface to the width of the slit at the suction surface. The parameters are defined in Figure 8.Figure 8
Slit geometry parameters.In order for the slit to act directly on the low-velocity backflow area and effectively improve the performance: Slits will be set in areas of severe backflow, and 4 slit positions will be evenly selected. Meanwhile, to investigate the effect of different slit shapes, other factors were also selected at 4 levels based on blade size. Based on the selected factors and levels, selectL1645 orthogonal table. The specific values are shown in Table 4.Table 4
Factor and level table.
LevelFactorDiameter of slitDmmSlit center widthbmmDeflection angle of slitβ (°)Depth of slithmmShrinkage ratio of slitf113622020.5216835040.832002.3−2061.242322.7−5081.5Part of the slit shape is shown in Figure9. The figure shows the fluid field with the presence of slits.Figure 9
Slit watershed shape diagram: (a) Group 1; (b) Group 7; (c) Group 9; (d) Group 15.
(a)(b)(c)(d)Table5 shows the calculation results of 16 orthogonal tests at the rated flow rate.Table 5
Orthogonal experiment data table.
NumberABCDETotal efficiencyη%HeadHm113622020.560.8225.490213635040.860.2925.33731362.3−2061.259.5425.47541362.7−5081.559.0625.517516825061.560.4025.340616832081.259.8725.46471682.3−5020.860.5425.41581682.7−2040.560.4025.39692002−2080.860.2825.682102003−5060.560.7425.598112002.32041.560.7825.483122002.75021.260.3925.347132322−5041.258.4924.872142323−2021.560.3025.422152322.35080.559.8525.120162322.72060.860.3025.859


### 4.2. Process and Analysis of Experimental Results

#### 4.2.1. Direct Analysis

Figure10 shows the results of 16 groups of orthogonal tests, the specific parameters are shown in Table 5.Figure 10
Orthogonal test results figure.Visual analysis by Figure10: the effect on pump performance is very significant after opening the blade slits, and the factors influence and constrain each other. The shape of the slit with the highest head is the 16th group, at 25.859 m, up 0.389 m over the original model. Group 11 is the most efficient, 60.78%, up 0.43% over the original model. Additionally, the group 13 model head reduction was 0.598 m, and the efficiency dropped by 1.86%. It can be seen that the unreasonable shape of the slit can seriously damage the performance of the centrifugal pump.The flow field of these centrifugal pumps is shown in Figure11. As seen by this, a proper slit shape (e.g., group 11, group 16) will significantly compress the backflow area of the flow channel, reduce the internal consumption of fluid kinetic energy, and increase the head and efficiency of centrifugal pumps. An unreasonable slit shape (group 13) will cause fluid of the pressure surface through the slit to neither improve the fluid exit velocity triangle nor destroys the backflow structure at the end of the flow channel. Instead, the backflow is locally enhanced by the blowing effect, which causes degradation of centrifugal pump performance.Figure 11
Flow field diagram of each model.


#### 4.2.2. Extremum Difference Analysis

In order to investigate the influence degree of each factor on the performance of the flow field, do an extreme difference analysis for head and efficiency, respectively, the results are shown in Tables6 and 7.Table 6
Extreme difference analysis of head.
IndexFactorABCDEK1101.819101.384102.296101.674101.604K2101.615101.821101.144101.088102.293K3102.11101.493101.975102.272101.158K4101.273102.119101.402101.783101.762K1¯25.4547525.34625.57425.418525.401K2¯25.4037525.4552525.28625.27225.57325K3¯25.527525.3732525.4937525.56825.2895K4¯25.3182525.5297525.350525.4457525.4405Optimal level34132R0.209250.183750.2880.2960.28375Order45213Table 7
Extreme difference analysis of efficiency.
IndexFactorABCDEK1239.71239.99241.77242.05241.81K2241.21241.2240.93239.96241.41K3242.19240.71240.52240.98240.98K4238.94240.15238.83239.06240.54K1¯59.927559.997560.442560.512560.4525K2¯60.302560.360.232559.9960.3525K3¯60.547560.177560.1360.24560.245K4¯59.73560.037559.707559.76560.135Optimal level32111R0.81250.30250.7350.74750.3175Order15324R is the extreme difference value in the table, the larger the value, indicates that the greater the influence of the factor on the index.For the head, changes in factor C (deflection angle of slitβ) and factor D (depth of slit h) have a greater effect on the head, and factor B (slit center width b) has a small effect on the head by changing. The order of impact from largest to smallest is D>C>E>A>B. The optimal combination of the head is A3B4C1D3E2.For efficiency: factor A (diameter of slitD) has a large effect on the variation of efficiency, while factor B (slit center width b) and factor E (shrinkage ratio of slit f) have a small effect on the efficiency. The order of impact from largest to smallest is A>D>C>E>B. The optimal combination for efficiency is A3B2C1D1E1.


## 4.1. Orthogonal Experimental Design

There are many geometric parameters of the slotted blade, and with a balance between the number of experiments and the mastery of experimental rules, an orthogonal experiment of 5 factors and 4 levels for analysis was designed. According to the characteristics of the flow channel and the structure of the blade, decided to select the factors as the diameter of slitD, slit center width b, deflection angle of slit β, depth of slit h, shrinkage ratio of slit f, and denoted by A, B, C, D, E in order. Where the shrinkage ratio f is defined as the ratio of the width of the slit at the pressure surface to the width of the slit at the suction surface. The parameters are defined in Figure 8.Figure 8
Slit geometry parameters.In order for the slit to act directly on the low-velocity backflow area and effectively improve the performance: Slits will be set in areas of severe backflow, and 4 slit positions will be evenly selected. Meanwhile, to investigate the effect of different slit shapes, other factors were also selected at 4 levels based on blade size. Based on the selected factors and levels, selectL1645 orthogonal table. The specific values are shown in Table 4.Table 4
Factor and level table.
LevelFactorDiameter of slitDmmSlit center widthbmmDeflection angle of slitβ (°)Depth of slithmmShrinkage ratio of slitf113622020.5216835040.832002.3−2061.242322.7−5081.5Part of the slit shape is shown in Figure9. The figure shows the fluid field with the presence of slits.Figure 9
Slit watershed shape diagram: (a) Group 1; (b) Group 7; (c) Group 9; (d) Group 15.
(a)(b)(c)(d)Table5 shows the calculation results of 16 orthogonal tests at the rated flow rate.Table 5
Orthogonal experiment data table.
NumberABCDETotal efficiencyη%HeadHm113622020.560.8225.490213635040.860.2925.33731362.3−2061.259.5425.47541362.7−5081.559.0625.517516825061.560.4025.340616832081.259.8725.46471682.3−5020.860.5425.41581682.7−2040.560.4025.39692002−2080.860.2825.682102003−5060.560.7425.598112002.32041.560.7825.483122002.75021.260.3925.347132322−5041.258.4924.872142323−2021.560.3025.422152322.35080.559.8525.120162322.72060.860.3025.859


## 4.2. Process and Analysis of Experimental Results

### 4.2.1. Direct Analysis

Figure10 shows the results of 16 groups of orthogonal tests, the specific parameters are shown in Table 5.Figure 10
Orthogonal test results figure.Visual analysis by Figure10: the effect on pump performance is very significant after opening the blade slits, and the factors influence and constrain each other. The shape of the slit with the highest head is the 16th group, at 25.859 m, up 0.389 m over the original model. Group 11 is the most efficient, 60.78%, up 0.43% over the original model. Additionally, the group 13 model head reduction was 0.598 m, and the efficiency dropped by 1.86%. It can be seen that the unreasonable shape of the slit can seriously damage the performance of the centrifugal pump.The flow field of these centrifugal pumps is shown in Figure11. As seen by this, a proper slit shape (e.g., group 11, group 16) will significantly compress the backflow area of the flow channel, reduce the internal consumption of fluid kinetic energy, and increase the head and efficiency of centrifugal pumps. An unreasonable slit shape (group 13) will cause fluid of the pressure surface through the slit to neither improve the fluid exit velocity triangle nor destroys the backflow structure at the end of the flow channel. Instead, the backflow is locally enhanced by the blowing effect, which causes degradation of centrifugal pump performance.Figure 11
Flow field diagram of each model.


### 4.2.2. Extremum Difference Analysis

In order to investigate the influence degree of each factor on the performance of the flow field, do an extreme difference analysis for head and efficiency, respectively, the results are shown in Tables6 and 7.Table 6
Extreme difference analysis of head.
IndexFactorABCDEK1101.819101.384102.296101.674101.604K2101.615101.821101.144101.088102.293K3102.11101.493101.975102.272101.158K4101.273102.119101.402101.783101.762K1¯25.4547525.34625.57425.418525.401K2¯25.4037525.4552525.28625.27225.57325K3¯25.527525.3732525.4937525.56825.2895K4¯25.3182525.5297525.350525.4457525.4405Optimal level34132R0.209250.183750.2880.2960.28375Order45213Table 7
Extreme difference analysis of efficiency.
IndexFactorABCDEK1239.71239.99241.77242.05241.81K2241.21241.2240.93239.96241.41K3242.19240.71240.52240.98240.98K4238.94240.15238.83239.06240.54K1¯59.927559.997560.442560.512560.4525K2¯60.302560.360.232559.9960.3525K3¯60.547560.177560.1360.24560.245K4¯59.73560.037559.707559.76560.135Optimal level32111R0.81250.30250.7350.74750.3175Order15324R is the extreme difference value in the table, the larger the value, indicates that the greater the influence of the factor on the index.For the head, changes in factor C (deflection angle of slitβ) and factor D (depth of slit h) have a greater effect on the head, and factor B (slit center width b) has a small effect on the head by changing. The order of impact from largest to smallest is D>C>E>A>B. The optimal combination of the head is A3B4C1D3E2.For efficiency: factor A (diameter of slitD) has a large effect on the variation of efficiency, while factor B (slit center width b) and factor E (shrinkage ratio of slit f) have a small effect on the efficiency. The order of impact from largest to smallest is A>D>C>E>B. The optimal combination for efficiency is A3B2C1D1E1.


## 4.2.1. Direct Analysis

Figure10 shows the results of 16 groups of orthogonal tests, the specific parameters are shown in Table 5.Figure 10
Orthogonal test results figure.Visual analysis by Figure10: the effect on pump performance is very significant after opening the blade slits, and the factors influence and constrain each other. The shape of the slit with the highest head is the 16th group, at 25.859 m, up 0.389 m over the original model. Group 11 is the most efficient, 60.78%, up 0.43% over the original model. Additionally, the group 13 model head reduction was 0.598 m, and the efficiency dropped by 1.86%. It can be seen that the unreasonable shape of the slit can seriously damage the performance of the centrifugal pump.The flow field of these centrifugal pumps is shown in Figure11. As seen by this, a proper slit shape (e.g., group 11, group 16) will significantly compress the backflow area of the flow channel, reduce the internal consumption of fluid kinetic energy, and increase the head and efficiency of centrifugal pumps. An unreasonable slit shape (group 13) will cause fluid of the pressure surface through the slit to neither improve the fluid exit velocity triangle nor destroys the backflow structure at the end of the flow channel. Instead, the backflow is locally enhanced by the blowing effect, which causes degradation of centrifugal pump performance.Figure 11
Flow field diagram of each model.


## 4.2.2. Extremum Difference Analysis

In order to investigate the influence degree of each factor on the performance of the flow field, do an extreme difference analysis for head and efficiency, respectively, the results are shown in Tables6 and 7.Table 6
Extreme difference analysis of head.
IndexFactorABCDEK1101.819101.384102.296101.674101.604K2101.615101.821101.144101.088102.293K3102.11101.493101.975102.272101.158K4101.273102.119101.402101.783101.762K1¯25.4547525.34625.57425.418525.401K2¯25.4037525.4552525.28625.27225.57325K3¯25.527525.3732525.4937525.56825.2895K4¯25.3182525.5297525.350525.4457525.4405Optimal level34132R0.209250.183750.2880.2960.28375Order45213Table 7
Extreme difference analysis of efficiency.
IndexFactorABCDEK1239.71239.99241.77242.05241.81K2241.21241.2240.93239.96241.41K3242.19240.71240.52240.98240.98K4238.94240.15238.83239.06240.54K1¯59.927559.997560.442560.512560.4525K2¯60.302560.360.232559.9960.3525K3¯60.547560.177560.1360.24560.245K4¯59.73560.037559.707559.76560.135Optimal level32111R0.81250.30250.7350.74750.3175Order15324R is the extreme difference value in the table, the larger the value, indicates that the greater the influence of the factor on the index.For the head, changes in factor C (deflection angle of slitβ) and factor D (depth of slit h) have a greater effect on the head, and factor B (slit center width b) has a small effect on the head by changing. The order of impact from largest to smallest is D>C>E>A>B. The optimal combination of the head is A3B4C1D3E2.For efficiency: factor A (diameter of slitD) has a large effect on the variation of efficiency, while factor B (slit center width b) and factor E (shrinkage ratio of slit f) have a small effect on the efficiency. The order of impact from largest to smallest is A>D>C>E>B. The optimal combination for efficiency is A3B2C1D1E1.


## 5. Multiobjective Optimization Based on Entropy Weight TOPSIS

### 5.1. Analysis of Entropy Weight TOPSIS

Direct extreme difference analysis for a single target only. To fully judge the performance indicators of centrifugal pumps, we need to transform multiobjective problems into single-objective problems through empowerment to facilitate a comprehensive evaluation of pump performance. The entropy weighting method determines the weights by the entropy value of each index. The weight coefficients are calculated as follows:(1)
Constructs the head and efficiency from the orthogonal test results as a decision matrix ofXm×n as shown in the following equation:(23)Xm×n=x11x12…x1nx21x22…x2n…………xm1xm2…xmn,wherem—Number of experiments; n—Number of test indexes.(2)
To facilitate comparison of results, regularization is performed according to the following equation:(24)λij=xij∑i=1mxij2,whereλij—Regularization results of the j indicators in the number i test.(3)
Calculate the weight of the sum of test values for thei th value under the j th indicator.(25)pij=λij∑i=1mλij.(4)
Calculate the entropy value of each index.(26)ej=−k∑i=1mpijlnpij,wherek—1/lnm.(5)
Calculate the information entropy redundancy of each index.(27)dj=1−ej.(6)
Calculate the weighting coefficients of each index.(28)ωj=dj∑j=1ndj×100%.Bringing the results of the orthogonal tests into equations (26) to (28) in turn, the weights of efficiency and head were obtained as 60.08% and 39.92%. Through the TOPSIS method, we ranked the 16 sets of experimental results according to their proximity to the idealized target to evaluate their relative merits. The calculation steps are as follows:(1)
We create a weighting matrix as follows:(29)Z=zij=z11⋯z1n⋮⋱⋮zm1⋯zmn,wherei=1,2,3,⋯,m; j=1,2,,⋯,n; andzij=λij×ωj.(2)
To find the optimal and inferior solutions we use the following equations:(30)zij+=maxm,nz1+,z2+,⋯,zm+,zij−=minm,nz1−,z2−,⋯,zm−.(3)
To calculate the distance between the evaluation object and the optimal/inferior solutions we use the following equation:(31)Di+=∑jzij−zj+2,Di−=∑jzij−zj−2.(4)
To calculate the relative proximity of the evaluation object to the optimal state we use the following equation:(32)Ci=Di−Di++Di−.Finally, ranking according to the value ofCi. The larger the Ci is the closer the evaluation object to the optimal state. The evaluation results are shown in Table 8.Table 8
TOPSIS evaluation results.
Number of testsPositive ideal solution distanceDi+Negative ideal solution distanceDi−Relative proximityCiRank10.1471.4210.906220.3811.0970.7421130.7840.6750.4631441.0660.4280.2871550.3261.1630.781760.5920.8620.5931270.2441.2510.837480.3131.1660.789590.3321.1230.7729100.1151.3820.9231110.1521.3970.9023120.3291.1570.7788131.4540016140.3581.1090.75610150.6530.8230.55813160.3121.1570.7876Obtained from Table8, with the balance of head and efficiency, the shape of the slit in the 10th group is optimal: its efficiency rises by 0.39% compared to the original model, the head rises by 12.8cm, and both indicators have improved significantly.


### 5.2. Multiobjective Optimization

In order to get the optimal level that takes into account head and efficiency, we can perform an extreme variance analysis on the weighted index. The results are shown in Table9.Table 9
Weighted index extreme deviation table.
IndexFactorABCDEK1¯46.1659846.1646246.5229946.4029846.45994K2¯46.3709246.3899846.2818646.1305746.46862K3¯46.5675246.2836446.3032146.5019445.88673K4¯45.9958346.2620145.9921946.0647646.28496Optimal level32131With a compromise between efficiency and head, the optimal combination is:A3B2C1D3E1; that is, the diameter of the slit D is 200 mm, the slit center width b is 3 mm, the deflection angle of the slit β is −20°, depth of the slit h is 6 mm, and the shrinkage ratio of the slit f is 0.5. Performed hydraulic performance simulations on the optimized slotted impeller geometry parameters, its efficiency rises by 0.32% compared to the original model, the head rises by 0.113m, proven to have some optimization effect. Its flow channel pressure nephogram is shown in Figure 12, the velocity streamline diagram is shown in Figure 13.Figure 12
Optimized pressure nephogram.Figure 13
Optimized velocity streamlines.By optimizing the structural parameters of low-specific-speed centrifugal pumps, we compare the optimization results and obtain Table10.Table 10
Comparison of optimization results.
IndexRankParameter combinationsEfficiency (%)HeadmOriginal model——60.3525.470Optimal efficiencyA > D > C > E > BA3B2C1D1E160.8525.410Optimal headD > C > E > A > BA3B5C1D3E260.3525.870MultiobjectiveB > A > E > D > CA3B2C1D3E160.6725.583


### 5.3. Experimental Verification

To verify the reasonableness and reliability of the slotted blade, conduct physical experiments on the model. 3D printed the overcurrent parts used, as shown in Figure14.Figure 14
Overcurrent parts figure (a) impeller and (b) volute.
(a)(b)Assemble the closed pump performance test bench to completion, as shown in Figure15.Figure 15
Closed pump performance test bench diagram: (a) front view and (b) lateral view.
(a)(b)Underrated conditions (1440r/min; 50 m3/h−1), we conducted several hydraulic performance tests on the original model blade and the optimized one, and the obtained results were taken as the mean value. The original model head is 24.13 m, after optimization, the head is 24.58 m. After measuring the moment M of the pump shaft by the parameters measurement instrument. The total efficiency of the test system is calculated using equation (32). The original model efficiency was 55.33%, and the efficiency after optimization was 57.82%. There was a good performance enhancement effect.(33)η=ρgQH1000UI.


## 5.1. Analysis of Entropy Weight TOPSIS

Direct extreme difference analysis for a single target only. To fully judge the performance indicators of centrifugal pumps, we need to transform multiobjective problems into single-objective problems through empowerment to facilitate a comprehensive evaluation of pump performance. The entropy weighting method determines the weights by the entropy value of each index. The weight coefficients are calculated as follows:(1)
Constructs the head and efficiency from the orthogonal test results as a decision matrix ofXm×n as shown in the following equation:(23)Xm×n=x11x12…x1nx21x22…x2n…………xm1xm2…xmn,wherem—Number of experiments; n—Number of test indexes.(2)
To facilitate comparison of results, regularization is performed according to the following equation:(24)λij=xij∑i=1mxij2,whereλij—Regularization results of the j indicators in the number i test.(3)
Calculate the weight of the sum of test values for thei th value under the j th indicator.(25)pij=λij∑i=1mλij.(4)
Calculate the entropy value of each index.(26)ej=−k∑i=1mpijlnpij,wherek—1/lnm.(5)
Calculate the information entropy redundancy of each index.(27)dj=1−ej.(6)
Calculate the weighting coefficients of each index.(28)ωj=dj∑j=1ndj×100%.Bringing the results of the orthogonal tests into equations (26) to (28) in turn, the weights of efficiency and head were obtained as 60.08% and 39.92%. Through the TOPSIS method, we ranked the 16 sets of experimental results according to their proximity to the idealized target to evaluate their relative merits. The calculation steps are as follows:(1)
We create a weighting matrix as follows:(29)Z=zij=z11⋯z1n⋮⋱⋮zm1⋯zmn,wherei=1,2,3,⋯,m; j=1,2,,⋯,n; andzij=λij×ωj.(2)
To find the optimal and inferior solutions we use the following equations:(30)zij+=maxm,nz1+,z2+,⋯,zm+,zij−=minm,nz1−,z2−,⋯,zm−.(3)
To calculate the distance between the evaluation object and the optimal/inferior solutions we use the following equation:(31)Di+=∑jzij−zj+2,Di−=∑jzij−zj−2.(4)
To calculate the relative proximity of the evaluation object to the optimal state we use the following equation:(32)Ci=Di−Di++Di−.Finally, ranking according to the value ofCi. The larger the Ci is the closer the evaluation object to the optimal state. The evaluation results are shown in Table 8.Table 8
TOPSIS evaluation results.
Number of testsPositive ideal solution distanceDi+Negative ideal solution distanceDi−Relative proximityCiRank10.1471.4210.906220.3811.0970.7421130.7840.6750.4631441.0660.4280.2871550.3261.1630.781760.5920.8620.5931270.2441.2510.837480.3131.1660.789590.3321.1230.7729100.1151.3820.9231110.1521.3970.9023120.3291.1570.7788131.4540016140.3581.1090.75610150.6530.8230.55813160.3121.1570.7876Obtained from Table8, with the balance of head and efficiency, the shape of the slit in the 10th group is optimal: its efficiency rises by 0.39% compared to the original model, the head rises by 12.8cm, and both indicators have improved significantly.


## 5.2. Multiobjective Optimization

In order to get the optimal level that takes into account head and efficiency, we can perform an extreme variance analysis on the weighted index. The results are shown in Table9.Table 9
Weighted index extreme deviation table.
IndexFactorABCDEK1¯46.1659846.1646246.5229946.4029846.45994K2¯46.3709246.3899846.2818646.1305746.46862K3¯46.5675246.2836446.3032146.5019445.88673K4¯45.9958346.2620145.9921946.0647646.28496Optimal level32131With a compromise between efficiency and head, the optimal combination is:A3B2C1D3E1; that is, the diameter of the slit D is 200 mm, the slit center width b is 3 mm, the deflection angle of the slit β is −20°, depth of the slit h is 6 mm, and the shrinkage ratio of the slit f is 0.5. Performed hydraulic performance simulations on the optimized slotted impeller geometry parameters, its efficiency rises by 0.32% compared to the original model, the head rises by 0.113m, proven to have some optimization effect. Its flow channel pressure nephogram is shown in Figure 12, the velocity streamline diagram is shown in Figure 13.Figure 12
Optimized pressure nephogram.Figure 13
Optimized velocity streamlines.By optimizing the structural parameters of low-specific-speed centrifugal pumps, we compare the optimization results and obtain Table10.Table 10
Comparison of optimization results.
IndexRankParameter combinationsEfficiency (%)HeadmOriginal model——60.3525.470Optimal efficiencyA > D > C > E > BA3B2C1D1E160.8525.410Optimal headD > C > E > A > BA3B5C1D3E260.3525.870MultiobjectiveB > A > E > D > CA3B2C1D3E160.6725.583


## 5.3. Experimental Verification

To verify the reasonableness and reliability of the slotted blade, conduct physical experiments on the model. 3D printed the overcurrent parts used, as shown in Figure14.Figure 14
Overcurrent parts figure (a) impeller and (b) volute.
(a)(b)Assemble the closed pump performance test bench to completion, as shown in Figure15.Figure 15
Closed pump performance test bench diagram: (a) front view and (b) lateral view.
(a)(b)Underrated conditions (1440r/min; 50 m3/h−1), we conducted several hydraulic performance tests on the original model blade and the optimized one, and the obtained results were taken as the mean value. The original model head is 24.13 m, after optimization, the head is 24.58 m. After measuring the moment M of the pump shaft by the parameters measurement instrument. The total efficiency of the test system is calculated using equation (32). The original model efficiency was 55.33%, and the efficiency after optimization was 57.82%. There was a good performance enhancement effect.(33)η=ρgQH1000UI.


## 6. Conclusions and Foresights

### 6.1. Conclusions

(1)
Simulated the flow field of a low-specific-speed centrifugal pump and analyzed the reasons for the inefficiency. Low-specific-speed pump flow channel outlet diffusion is serious, and there is a pressure gradient between the back of the blade and the working surface, which generates extremely large backflow areas, causes blockage of flow channels, and consumes large amounts of fluid kinetic energy.(2)
Designed theL1645 orthogonal table, and conducted orthogonal experiments on the impeller slit geometry parameters. The optimal combination of geometric parameters for the head is A3B4C1D3E2, that is, the diameter of slit D is 200 mm, slit center width b is 2.7 mm, the deflection angle of slit β is 20°, depth of slit h is 6 mm, shrinkage ratio of slit f is 0.8. The ranking of impacts from largest to smallest is the depth of slit h> deflection angle of slit β> shrinkage ratio of slit f> diameter of slit D> slit center width bThe optimal combination of efficiency is:A3B2C1D1E1, that is, the diameter of slit D is 200 mm, slit center width b is 3 mm, the deflection angle of slit β is 20°, depth of slit h is 2 mm, shrinkage ratio of slit f is 0.5. The importance is the diameter of slit D> depth of slit h> deflection angle of slit β> shrinkage ratio of slit f> slit center width b.The combination of slit geometry parameters that take into account both head and efficiency isA3B2C1D3E1, that is, the diameter of slit D is 200 mm, slit center width b is 3 mm, the deflection angle of slit β is 20°, depth of slit h is 6 mm, shrinkage ratio of slit f is 0.5. The order of effects is: slit center width b> diameter of slit D> shrinkage ratio of slit f> depth of slit h> deflection angle of slit β.(3)
Performed simulation of the optimized model by FLUENT, the obtained head and efficiency are 25.583m and 60.67%, respectively, which is an improvement over the original model. Verified through experiments that the hydraulic performance has indeed improved. It is concluded that the entropy power TOPSIS method is useful for the optimization of structural parameters of centrifugal pumps.


### 6.2. Foresights

Improvements in centrifugal pump performance mean progress for the entire drainage and water supply industries. Based on the above conclusions, further research may be conducted on the following aspects:(1)
In this paper, the flow field state at the cross-section of the slit is observed. We could select multiple cross-sections to describe the flow field of the slit blade in more detail.(2)
In this paper, four representative levels of each factor were selected. Researchers could do this by numerical twinning, etc., to obtain a more accurate hydraulic model.


## 6.1. Conclusions

(1)
Simulated the flow field of a low-specific-speed centrifugal pump and analyzed the reasons for the inefficiency. Low-specific-speed pump flow channel outlet diffusion is serious, and there is a pressure gradient between the back of the blade and the working surface, which generates extremely large backflow areas, causes blockage of flow channels, and consumes large amounts of fluid kinetic energy.(2)
Designed theL1645 orthogonal table, and conducted orthogonal experiments on the impeller slit geometry parameters. The optimal combination of geometric parameters for the head is A3B4C1D3E2, that is, the diameter of slit D is 200 mm, slit center width b is 2.7 mm, the deflection angle of slit β is 20°, depth of slit h is 6 mm, shrinkage ratio of slit f is 0.8. The ranking of impacts from largest to smallest is the depth of slit h> deflection angle of slit β> shrinkage ratio of slit f> diameter of slit D> slit center width bThe optimal combination of efficiency is:A3B2C1D1E1, that is, the diameter of slit D is 200 mm, slit center width b is 3 mm, the deflection angle of slit β is 20°, depth of slit h is 2 mm, shrinkage ratio of slit f is 0.5. The importance is the diameter of slit D> depth of slit h> deflection angle of slit β> shrinkage ratio of slit f> slit center width b.The combination of slit geometry parameters that take into account both head and efficiency isA3B2C1D3E1, that is, the diameter of slit D is 200 mm, slit center width b is 3 mm, the deflection angle of slit β is 20°, depth of slit h is 6 mm, shrinkage ratio of slit f is 0.5. The order of effects is: slit center width b> diameter of slit D> shrinkage ratio of slit f> depth of slit h> deflection angle of slit β.(3)
Performed simulation of the optimized model by FLUENT, the obtained head and efficiency are 25.583m and 60.67%, respectively, which is an improvement over the original model. Verified through experiments that the hydraulic performance has indeed improved. It is concluded that the entropy power TOPSIS method is useful for the optimization of structural parameters of centrifugal pumps.


## 6.2. Foresights

Improvements in centrifugal pump performance mean progress for the entire drainage and water supply industries. Based on the above conclusions, further research may be conducted on the following aspects:(1)
In this paper, the flow field state at the cross-section of the slit is observed. We could select multiple cross-sections to describe the flow field of the slit blade in more detail.(2)
In this paper, four representative levels of each factor were selected. Researchers could do this by numerical twinning, etc., to obtain a more accurate hydraulic model.



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*Source: 9612947-2023-01-05.xml*