# Influence of Blade Profiles on Plastic Centrifugal Pump Performance

Authors: Hui Zhang; Lingfeng Tang; Yongtao Zhao
Journal: Advances in Materials Science and Engineering
 (2020)

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## Abstract

To study the influence of blade profiles of the plastic centrifugal pump on pump performance, the impeller blade profiles were designed and drawn by the single arc method, double arc method, logarithmic spiral method, and B-spline curve method, respectively, with the known structural parameters.The shape and size of four profiles were drawn, and two-dimensional models and three-dimensional models of four impellers and volute were completed, respectively. The impeller models were printed by 3D printing technology, and the performance experiments of the plastic centrifugal pump were carried out. The numerical simulation of the internal flow field was performed. From the contours of the velocity and pressure, the vapor volume fraction distribution, and fluid-structure interaction analysis of impellers, the impeller drawn by the logarithmic spiral method was better than others. The optimization of the logarithmic spiral method was completed. The impeller inlet and outlet diameters (D1 and D2) and impeller inlet and outlet installation angles (β1 and β2) were taken as control variables, and the total power loss and the minimum NPSHr of the pump were taken as the objective functions. The optimization results were that D1 = 58 mm and D2 = 162 mm and β1 = 17° and β2 = 31°. The hydraulic efficiency was increased by 1.68%.


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## Body

## 1. Introduction

As a type of general machinery, centrifugal pumps are widely used such as in mechanical engineering, aerospace, and petrochemical industries [1–3]. The flow components of plastic centrifugal pumps are made of engineering plastic, with less manufacturing costs and better corrosion resistance. Plastic centrifugal pumps are suitable for most corrosive environments such as acid, alkali, and salt and can be applied to the transportation of corrosive materials in chemical and petroleum industries. However, it is well known that inevitable cavitation in centrifugal pumps will affect the pump performance and cause noise and vibrations [4–10].The impeller is the core part of the centrifugal pump, and the impeller blade profile plays a vital role in the fluid flow which will directly impact the performance of centrifugal pumps. At present, there have been many studies on the influence of the impeller parameters on the flow characteristics of centrifugal pumps [11–20]. Tao et al. [21] used the numerical method to study how the blade thickness influenced the flow characteristics in the impeller passages and volute casing. They concluded that the pressure pulsations increased at the leading edges but decreased at the trailing edges according to the increase in the blade thickness. However, their work was only helpful for the design of ceramic pumps. Tao et al. [22] studied the influence of blade leading-edge shape on the cavitation and concluded it delayed the inception cavitation and critical cavitation. Peng et al. [23] analyzed various blade outlet angles without changing other impeller parameters and concluded that if the blade outlet angle was increased, the flow separation and the axial vortex along the blade working surface would be more serious. However, their work was aimed at low specific speed submersible pumps. Yu-Qin and Ze-Wen [24] studied how different blade numbers influenced the flow-induced noise of the centrifugal pump based on the idea of acoustic-vibration coupling. They found that the 6-blade model pump was the optimal value. Han et al. [25] presented an impeller and volute shape optimization design method that improved the hydraulic performance of the pump. Zhang et al. and Wang et al. [26, 27] explored the effects of slotted blades on the performance of centrifugal pumps. Li et al. [28] studied the influence of the blade angle of a low specific speed plastic centrifugal pump on the pump performance. They applied the one-way fluid-structure interaction but did not take the impact of structural deformation on performance into consideration.The blade profile directly affects the bending of the blade and has an important impact on the fluid flow in the impeller channel. Studying the impact of the blade profile on the performance of the centrifugal pumps is beneficial to improve the performance and efficiency of the centrifugal pumps. Hu et al. [29] designed an iterative method of double arc blade profile for drawing cylindrical blade profile which overcame the shortcomings of blade profile drawing methods. El-Gazzar and Hawash [30] used the splitting technique to change the shape of the impeller and concluded that the impeller blade split caused the uneven mass distribution around the impeller, which led to the imbalance of the impeller. Ming et al. [31] proposed a new variable-angle spiral equation that was complementary to the traditional variable-angle spiral equation by changing the blade placement angle function.In existing research studies, the studies on blade profile are mainly focused on metal pumps and a single profile, while only a few pieces of research are on the performance of plastic centrifugal pumps. Moreover, there is no regularity between the various blade profiles and the performance parameters of centrifugal pumps. In this paper, four design methods of blade profiles, which are the single arc method, double arc method, logarithmic spiral method, and B-spline curve method, are proposed and the influence of four kinds of blade profile on plastic centrifugal pump performance is studied. The flow field simulation of pumps with various impellers is analyzed in detail.


## 2. Design Methods of Blade Profiles

### 2.1. Single Arc Method

Figure1 shows the design principle of the single arc method. In Figure 1, D1 and D2 are impeller inlet and outlet diameters, respectively, β1 and β2 are the impeller inlet and outlet installation angles, respectively, and arc AB is the blade profile.Figure 1
Blade profile of single arc method.Take a random point (pointA) on the entrance circle D1. Take point A as the fixed point. Draw a line with a β1 angle and a length of ρ in the negative direction of the x-axis. Take point E as the endpoint. Then, take point E as the center and ρ as the radius to make the arc intersect the inlet and outlet circles at A and B. Therefore, the arc AB is the blade profile of the single arc method.The radius of curvature of the single arc:(1)ρ=R22−R122R2cosβ2−R1cosβ1,where ρ = EA—radius of curvature of the single arc (mm) and R1, R2—impeller inlet and outlet radii (mm).


### 2.2. Double Arc Method

The principle of the double arc method is to find the radius of rotation according to the drawing method to determine the first arc and then calculate the value of the radius of rotation to determine the second arc according to the formula.In Figure2, D1 and D2 are impeller inlet and outlet diameters and β1 and β2 are the impeller inlet and outlet installation angles. Take D1 as the diameter to make a circle and divide the circle into equal parts according to the number of blades Z. The equal points are A1, A2, A3, A4, A5, and A6. Take the diameter δ = D1sin β1 as the concentric circle with the diameter D1. Make tangent lines A1E1, A2E2, A3E3, A4E4, A5E5, and A6E6 from each equal point to the circle of diameter δ. Take E1 as the center of the circle (∠E1A1O = β1) and E1A1 as the radius to make an arc, which intersects the auxiliary circle (point F); then, the arc FA1 is the first profile of the blade. The second arc needs to meet three conditions which determine the position of point P. Three conditions are that (1) F, E1, and M points are collinear; (2) ∠OPM = β2; and (3) PM = MF = ρ.Figure 2
Blade profile of double arc method.The value ofρ is(2)ρ=12R22−Rf2R2cosβ2−Rfcosβf,where ρ—radius of rotation of the second arc (mm); Rf—radius of rotation of the point F (mm); and βf—blade angle of point F (°).After the position of pointP is determined, the center point M can be determined and the second arc can be determined according to ρ.


### 2.3. Logarithmic Spiral Method

After determining the impeller inlet and outlet installation angles and the impeller inlet and out diameters, the blade wrap angle has a larger value range. Therefore, the advantage of the logarithmic spiral method in the shape of the cylindrical impeller’s blade is also more prominent. Design and draw the impeller blade profile according to the equivariant angle spiral, and its design principle of the profile is shown in Figure3.Figure 3
Blade profile of logarithmic spiral method.In Figure3, β is impeller installation angle (°); β1 and β2 are impeller inlet and outlet installation angles (°); D1 and D2 are the impeller inlet and outlet diameters (mm); and θ is the variation of wrap angle ψ (°).Whenβ1 ≠ β2, the change law of β is(3)β=β2−β1•θφ+β1.Profile equation:(4)r=r1cosβ1cosβ2−β1/φθ+β1Ψ/β2−β1.Wrap angle:(5)φ=β2−β1lnr2/r1lncosβ1/cosβ2.Substituteβ1 = 20, β2 = 30, D1 = 58 mm, and D2 = 162 mm into the program written by MATLAB software, and the resulting blade profile is shown in Figure 4(a).Figure 4
Blade profiles drawn by MATLAB software. (a) Logarithmic spiral method. (b) B-spline curve method.
(a)(b)


### 2.4. B-Spline Curve Method

Use 5-point (or 4-point) Bezier curve to fit the studied impeller profile, which eliminates the inconvenience of curve adjustment caused by traditional cubic polynomial fitting and makes the design of blade profile simpler, smoother, and easier to control. Determine the coordinates of the control points first. According to the Bezier equation,n = 4,(6)Pt=1−t4P0+4t1−t3P1+6t21−t2P2+4t31−tP3+t4P4.The coordinate of any pointP (xp, yp) on the Bezier curve is(7)xp=1−t4x0+4t1−t3x1+6t21−t2x2+4t31−tx3+t4x4,yp=1−t4y0+4t1−t3y1+6t21−t2y2+4t31−ty3+t4y4.Figure5 shows the design principle of the B-spline curve method. In Figure 5, the impeller inlet and outlet installation angles β1 and β2 and the impeller inlet and outlet diameters D1 and D2 are the known parameters.Figure 5
Blade profile of B-spline curve method.Substituteβ1 = 20, β2 = 30, D1 = 58 mm, D2 = 162 mm, and ψ = 88.3 into the calculation and obtain P0 (0, 29), P1 (−26.32, 38.58), P2 (60.87, 12.77), P3 (−66.79, 25.49), and P4(−81, 2.83). From the obtained coordinates of P0, P1, P2, P3, and P4, the designed blade profile can be drawn by MATLAB software, as shown in Figure 4(b).


## 2.1. Single Arc Method

Figure1 shows the design principle of the single arc method. In Figure 1, D1 and D2 are impeller inlet and outlet diameters, respectively, β1 and β2 are the impeller inlet and outlet installation angles, respectively, and arc AB is the blade profile.Figure 1
Blade profile of single arc method.Take a random point (pointA) on the entrance circle D1. Take point A as the fixed point. Draw a line with a β1 angle and a length of ρ in the negative direction of the x-axis. Take point E as the endpoint. Then, take point E as the center and ρ as the radius to make the arc intersect the inlet and outlet circles at A and B. Therefore, the arc AB is the blade profile of the single arc method.The radius of curvature of the single arc:(1)ρ=R22−R122R2cosβ2−R1cosβ1,where ρ = EA—radius of curvature of the single arc (mm) and R1, R2—impeller inlet and outlet radii (mm).


## 2.2. Double Arc Method

The principle of the double arc method is to find the radius of rotation according to the drawing method to determine the first arc and then calculate the value of the radius of rotation to determine the second arc according to the formula.In Figure2, D1 and D2 are impeller inlet and outlet diameters and β1 and β2 are the impeller inlet and outlet installation angles. Take D1 as the diameter to make a circle and divide the circle into equal parts according to the number of blades Z. The equal points are A1, A2, A3, A4, A5, and A6. Take the diameter δ = D1sin β1 as the concentric circle with the diameter D1. Make tangent lines A1E1, A2E2, A3E3, A4E4, A5E5, and A6E6 from each equal point to the circle of diameter δ. Take E1 as the center of the circle (∠E1A1O = β1) and E1A1 as the radius to make an arc, which intersects the auxiliary circle (point F); then, the arc FA1 is the first profile of the blade. The second arc needs to meet three conditions which determine the position of point P. Three conditions are that (1) F, E1, and M points are collinear; (2) ∠OPM = β2; and (3) PM = MF = ρ.Figure 2
Blade profile of double arc method.The value ofρ is(2)ρ=12R22−Rf2R2cosβ2−Rfcosβf,where ρ—radius of rotation of the second arc (mm); Rf—radius of rotation of the point F (mm); and βf—blade angle of point F (°).After the position of pointP is determined, the center point M can be determined and the second arc can be determined according to ρ.


## 2.3. Logarithmic Spiral Method

After determining the impeller inlet and outlet installation angles and the impeller inlet and out diameters, the blade wrap angle has a larger value range. Therefore, the advantage of the logarithmic spiral method in the shape of the cylindrical impeller’s blade is also more prominent. Design and draw the impeller blade profile according to the equivariant angle spiral, and its design principle of the profile is shown in Figure3.Figure 3
Blade profile of logarithmic spiral method.In Figure3, β is impeller installation angle (°); β1 and β2 are impeller inlet and outlet installation angles (°); D1 and D2 are the impeller inlet and outlet diameters (mm); and θ is the variation of wrap angle ψ (°).Whenβ1 ≠ β2, the change law of β is(3)β=β2−β1•θφ+β1.Profile equation:(4)r=r1cosβ1cosβ2−β1/φθ+β1Ψ/β2−β1.Wrap angle:(5)φ=β2−β1lnr2/r1lncosβ1/cosβ2.Substituteβ1 = 20, β2 = 30, D1 = 58 mm, and D2 = 162 mm into the program written by MATLAB software, and the resulting blade profile is shown in Figure 4(a).Figure 4
Blade profiles drawn by MATLAB software. (a) Logarithmic spiral method. (b) B-spline curve method.
(a)(b)


## 2.4. B-Spline Curve Method

Use 5-point (or 4-point) Bezier curve to fit the studied impeller profile, which eliminates the inconvenience of curve adjustment caused by traditional cubic polynomial fitting and makes the design of blade profile simpler, smoother, and easier to control. Determine the coordinates of the control points first. According to the Bezier equation,n = 4,(6)Pt=1−t4P0+4t1−t3P1+6t21−t2P2+4t31−tP3+t4P4.The coordinate of any pointP (xp, yp) on the Bezier curve is(7)xp=1−t4x0+4t1−t3x1+6t21−t2x2+4t31−tx3+t4x4,yp=1−t4y0+4t1−t3y1+6t21−t2y2+4t31−ty3+t4y4.Figure5 shows the design principle of the B-spline curve method. In Figure 5, the impeller inlet and outlet installation angles β1 and β2 and the impeller inlet and outlet diameters D1 and D2 are the known parameters.Figure 5
Blade profile of B-spline curve method.Substituteβ1 = 20, β2 = 30, D1 = 58 mm, D2 = 162 mm, and ψ = 88.3 into the calculation and obtain P0 (0, 29), P1 (−26.32, 38.58), P2 (60.87, 12.77), P3 (−66.79, 25.49), and P4(−81, 2.83). From the obtained coordinates of P0, P1, P2, P3, and P4, the designed blade profile can be drawn by MATLAB software, as shown in Figure 4(b).


## 3. Numerical Simulation Method

### 3.1. Computation Model

Take the plastic centrifugal pump designed by a certain unit as an example, and its main design parameters are shown in Table1. According to the known structural parameters of a plastic centrifugal pump, the impeller blade profiles were designed and drawn by the single arc method, double arc method, logarithmic spiral method, and B-spline curve method, respectively.Table 1
Design parameters of plastic centrifugal pump.
Performance parametersGeometrical dimensionQ (m3/h)20Impeller inlet diameter (mm)58H (m)30Impeller outlet diameter (mm)162n (r/min)2900Impeller inlet width (mm)23ns60.871Impeller outlet width (mm)10Pt (kW)5.5Impeller inlet installation angle (°)20NPSHa (m)4Impeller outlet installation angle (°)30Volute width (mm)33Base circle diameter of volute (mm)175Figure6 shows a two-dimensional diagram of the impeller. According to the parameters of the impeller in Figure 6, the UG software is used to design the three-dimensional model of the impeller. The three-dimensional models of the impeller with four profile design methods are shown in Figure 7.Figure 6
Two-dimensional diagram of the impeller.Figure 7
Models of impellers drawn by four profile design methods. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)Figure8 shows a two-dimensional diagram of the volute. According to the parameters of the volute in Figure 8, the UG software is used to design the three-dimensional model of the volute. The three-dimensional model of the volute is shown in Figure 9.Figure 8
Two-dimensional diagram of the volute.Figure 9
Model of volute.


### 3.2. Meshing

According to the above completed three-dimensional models, ANSYS-ICEM software is employed to utilized to mesh the computational domains. When we mesh the models, the actual grid cannot reach the ideal shape. If the grid deforms or deformation exceeds a certain limit, the accuracy of the calculation results will change accordingly. Therefore, in the initial division of the grid, it is necessary to use appropriate measures to control or measure the quality of the grid and try to achieve the best grid. The final determination of the number of grid cells in each part of the computational domains is shown in Table2.Table 2
The number of grid cells.
Computational domainsInlet extensionImpellerVoluteOutlet extensionThe number of grid cells5811023975263932661502055The final meshing is shown in Figure10.Figure 10
Meshing.


### 3.3. Computational Setup

Numerical calculations were performed in FLUENT software. The turbulence model was a k-epsilon (equation (2)) turbulence model. The computational impeller domain was set to a rotational domain. All other computational domains were set to static.The boundary conditions were set to velocity inlet and outflow. The relevant setting parameters were set by default. The reference pressure was set to standard atmospheric pressure. The wall surface was placed under a non-slip boundary condition, and a standard wall surface function was applied. The calculation method was SIMPLEC. The convergence accuracy was set to 10−4.


## 3.1. Computation Model

Take the plastic centrifugal pump designed by a certain unit as an example, and its main design parameters are shown in Table1. According to the known structural parameters of a plastic centrifugal pump, the impeller blade profiles were designed and drawn by the single arc method, double arc method, logarithmic spiral method, and B-spline curve method, respectively.Table 1
Design parameters of plastic centrifugal pump.
Performance parametersGeometrical dimensionQ (m3/h)20Impeller inlet diameter (mm)58H (m)30Impeller outlet diameter (mm)162n (r/min)2900Impeller inlet width (mm)23ns60.871Impeller outlet width (mm)10Pt (kW)5.5Impeller inlet installation angle (°)20NPSHa (m)4Impeller outlet installation angle (°)30Volute width (mm)33Base circle diameter of volute (mm)175Figure6 shows a two-dimensional diagram of the impeller. According to the parameters of the impeller in Figure 6, the UG software is used to design the three-dimensional model of the impeller. The three-dimensional models of the impeller with four profile design methods are shown in Figure 7.Figure 6
Two-dimensional diagram of the impeller.Figure 7
Models of impellers drawn by four profile design methods. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)Figure8 shows a two-dimensional diagram of the volute. According to the parameters of the volute in Figure 8, the UG software is used to design the three-dimensional model of the volute. The three-dimensional model of the volute is shown in Figure 9.Figure 8
Two-dimensional diagram of the volute.Figure 9
Model of volute.


## 3.2. Meshing

According to the above completed three-dimensional models, ANSYS-ICEM software is employed to utilized to mesh the computational domains. When we mesh the models, the actual grid cannot reach the ideal shape. If the grid deforms or deformation exceeds a certain limit, the accuracy of the calculation results will change accordingly. Therefore, in the initial division of the grid, it is necessary to use appropriate measures to control or measure the quality of the grid and try to achieve the best grid. The final determination of the number of grid cells in each part of the computational domains is shown in Table2.Table 2
The number of grid cells.
Computational domainsInlet extensionImpellerVoluteOutlet extensionThe number of grid cells5811023975263932661502055The final meshing is shown in Figure10.Figure 10
Meshing.


## 3.3. Computational Setup

Numerical calculations were performed in FLUENT software. The turbulence model was a k-epsilon (equation (2)) turbulence model. The computational impeller domain was set to a rotational domain. All other computational domains were set to static.The boundary conditions were set to velocity inlet and outflow. The relevant setting parameters were set by default. The reference pressure was set to standard atmospheric pressure. The wall surface was placed under a non-slip boundary condition, and a standard wall surface function was applied. The calculation method was SIMPLEC. The convergence accuracy was set to 10−4.


## 4. Test Verification

### 4.1. 3D Printing of Plastic Centrifugal Pump Impellers

UTR9000 material was used for 3D printing of impellers, which was an ABS-like stereo light modeling resin with accurate and durable characteristics. The durability of components made of UTR9000 resin was more than 6.5 months. According to the three-dimensional models of the impellers drawn by four profile design methods (Figure6), the impeller models printed by 3D printing technology are shown in Figure 11.Figure 11
3D printing of impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)


### 4.2. Performance Tests of Plastic Centrifugal Pump

Figure12 shows the pump performance test bench, which mainly included a motor and a flow control valve.Figure 12
Pump performance test bench.The pump test system (electrical measurement method) V8.97 software on the data acquisition platform was used to collect data at each operating point by adjusting the flow control valve. The pump performance experiments were performed on the impellers of the four types of profiles. The parameter settings before the experiment are shown in Table3, and the collected data are shown in Tables 4–7.Table 3
Parameter settings of pump performance experiment.
Product numberFluid density (kg/m3)Fluid viscosity (CST)Meter factor (f/1)Inlet diameter (m)Outlet diameter (m)50UHB15-32WaterWater79.890.050.032Table 4
Performance experiment of single arc method.
Measured valuen = 2900 (r/min)Inlet pressureOutlet pressureQnPinQHPinη(kPa)(kPa)(m3/h)(r/min)(kW)(m3/h)(m)(kW)(%)1−70.00170.0020.952998.605.8520.2625.725.2926.832−80.00220.0019.933001.445.8319.2631.195.2631.113−70.00240.0018.923002.215.8118.2831.945.2430.374−60.00260.0016.353001.815.5615.8032.475.0127.875−50.00280.0014.863001.815.3714.3633.214.8426.826−40.00300.0012.933002.065.1212.4933.904.6224.997−30.00310.0010.913002.364.9610.5433.664.4721.628−20.00330.008.283001.634.688.0034.394.2217.769−20.00340.006.333001.544.536.1235.214.0914.3610−10.00350.002.443001.344.262.3635.063.845.8611−10.00360.001.493001.164.221.4436.003.813.7112−10.00360.000.213000.814.150.2035.993.750.53130.00380.000.003000.904.030.0036.943.640.00Table 5
Performance experiment of double arc method.
Measured valuen = 2900 (r/min)Inlet pressureOutlet pressureQnPinQHPinη(kPa)(kPa)(m3/h)(r/min)(kW)(m3/h)(m)(kW)(%)1−80.00190.0021.193000.595.4720.4828.604.9432.312−80.00220.0019.982998.585.3319.3231.264.8234.133−70.00240.0018.092997.925.2117.5031.894.7232.234−70.00250.0016.892998.155.1816.3432.644.6930.985−50.00280.0014.252998.205.0113.7833.204.5327.506−40.00290.0012.632998.184.8212.2233.004.3625.177−30.00310.0010.852997.944.6010.5033.764.1623.188−20.00330.008.562998.144.448.2834.504.0219.369−20.00340.006.702998.484.326.4835.313.9115.9510−10.00350.004.832998.584.204.6735.203.8011.7911−10.00360.002.062998.723.991.9936.063.615.4212−10.00370.000.242999.023.820.2336.993.450.6813−10.00380.000.002999.523.710.0037.933.350.00Table 6
Performance experiment of logarithmic spiral method.
Measured valuen = 2900 (r/min)Inlet pressureOutlet pressureQnPinQHPinη(kPa)(kPa)(m3/h)(r/min)(kW)(m3/h)(m)(kW)(%)1−70.00180.0021.332998.605.5820.6326.755.0529.792−80.00190.0021.082998.075.3620.3928.624.8532.773−80.00210.0020.243001.455.2119.5630.304.7034.354−80.00230.0018.983002.295.2018.3331.954.6934.055−60.00250.0016.503001.645.0115.9431.554.5233.326−50.00280.0014.653001.624.8014.1533.184.3329.557−30.00310.0011.303001.654.4810.9233.724.0424.828−30.00320.001.543001.624.4210.1834.593.9924.089−20.00330.008.463001.774.208.1734.403.7920.2310−20.00340.007.353001.724.117.1035.273.7118.4111−20.00340.005.933001.503.985.7335.193.5915.3012−10.00350.002.633001.253.742.5435.063.377.1913−10.00370.000.003001.343.430.0036.933.090.00Table 7
Performance experiment of B-spline curve method.
Measured valuen = 2900 (r/min)Inlet pressureOutlet pressureQnPinQHPinη(kPa)(kPa)(m3/h)(r/min)(kW)(m3/h)(m)(kW)(%)1−80.00180.0021.272998.155.4020.5727.704.8931.772−80.00220.0020.053001.805.3119.3731.214.7934.393−80.00240.0017.693001.605.1317.0932.694.6332.904−80.00250.0016.653001.515.0316.0933.484.5432.345−60.00280.0014.913001.364.8114.4134.174.3430.916−40.00300.0012.753001.134.6012.3233.904.1527.417−30.00310.0011.083000.934.4410.7133.714.0124.548−30.00330.008.293000.634.268.0135.373.8520.079−20.00340.007.063000.614.176.8235.283.7617.4210−20.00350.005.813000.564.055.6236.163.6615.1311−10.00350.004.753000.453.924.5935.163.5412.4212−10.00360.003.293000.553.823.1836.053.459.0513−10.00370.002.143000.563.722.0736.973.366.20


### 4.3. Data Analysis

Figures13 and 14 show the comparison of head and efficiency in performance experiments of four types of impellers. From the shut-off point to the maximum flow condition during the test, 13 flow condition points were selected to completely reflect the external characteristic variation curve. It can be seen that the efficiency of the logarithmic spiral method was better than that of the other three methods, and the efficiency of the single arc method was the worst.Figure 13
Q-H curves of four profiles.Figure 14
Q-η curves four profiles.


## 4.1. 3D Printing of Plastic Centrifugal Pump Impellers

UTR9000 material was used for 3D printing of impellers, which was an ABS-like stereo light modeling resin with accurate and durable characteristics. The durability of components made of UTR9000 resin was more than 6.5 months. According to the three-dimensional models of the impellers drawn by four profile design methods (Figure6), the impeller models printed by 3D printing technology are shown in Figure 11.Figure 11
3D printing of impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)


## 4.2. Performance Tests of Plastic Centrifugal Pump

Figure12 shows the pump performance test bench, which mainly included a motor and a flow control valve.Figure 12
Pump performance test bench.The pump test system (electrical measurement method) V8.97 software on the data acquisition platform was used to collect data at each operating point by adjusting the flow control valve. The pump performance experiments were performed on the impellers of the four types of profiles. The parameter settings before the experiment are shown in Table3, and the collected data are shown in Tables 4–7.Table 3
Parameter settings of pump performance experiment.
Product numberFluid density (kg/m3)Fluid viscosity (CST)Meter factor (f/1)Inlet diameter (m)Outlet diameter (m)50UHB15-32WaterWater79.890.050.032Table 4
Performance experiment of single arc method.
Measured valuen = 2900 (r/min)Inlet pressureOutlet pressureQnPinQHPinη(kPa)(kPa)(m3/h)(r/min)(kW)(m3/h)(m)(kW)(%)1−70.00170.0020.952998.605.8520.2625.725.2926.832−80.00220.0019.933001.445.8319.2631.195.2631.113−70.00240.0018.923002.215.8118.2831.945.2430.374−60.00260.0016.353001.815.5615.8032.475.0127.875−50.00280.0014.863001.815.3714.3633.214.8426.826−40.00300.0012.933002.065.1212.4933.904.6224.997−30.00310.0010.913002.364.9610.5433.664.4721.628−20.00330.008.283001.634.688.0034.394.2217.769−20.00340.006.333001.544.536.1235.214.0914.3610−10.00350.002.443001.344.262.3635.063.845.8611−10.00360.001.493001.164.221.4436.003.813.7112−10.00360.000.213000.814.150.2035.993.750.53130.00380.000.003000.904.030.0036.943.640.00Table 5
Performance experiment of double arc method.
Measured valuen = 2900 (r/min)Inlet pressureOutlet pressureQnPinQHPinη(kPa)(kPa)(m3/h)(r/min)(kW)(m3/h)(m)(kW)(%)1−80.00190.0021.193000.595.4720.4828.604.9432.312−80.00220.0019.982998.585.3319.3231.264.8234.133−70.00240.0018.092997.925.2117.5031.894.7232.234−70.00250.0016.892998.155.1816.3432.644.6930.985−50.00280.0014.252998.205.0113.7833.204.5327.506−40.00290.0012.632998.184.8212.2233.004.3625.177−30.00310.0010.852997.944.6010.5033.764.1623.188−20.00330.008.562998.144.448.2834.504.0219.369−20.00340.006.702998.484.326.4835.313.9115.9510−10.00350.004.832998.584.204.6735.203.8011.7911−10.00360.002.062998.723.991.9936.063.615.4212−10.00370.000.242999.023.820.2336.993.450.6813−10.00380.000.002999.523.710.0037.933.350.00Table 6
Performance experiment of logarithmic spiral method.
Measured valuen = 2900 (r/min)Inlet pressureOutlet pressureQnPinQHPinη(kPa)(kPa)(m3/h)(r/min)(kW)(m3/h)(m)(kW)(%)1−70.00180.0021.332998.605.5820.6326.755.0529.792−80.00190.0021.082998.075.3620.3928.624.8532.773−80.00210.0020.243001.455.2119.5630.304.7034.354−80.00230.0018.983002.295.2018.3331.954.6934.055−60.00250.0016.503001.645.0115.9431.554.5233.326−50.00280.0014.653001.624.8014.1533.184.3329.557−30.00310.0011.303001.654.4810.9233.724.0424.828−30.00320.001.543001.624.4210.1834.593.9924.089−20.00330.008.463001.774.208.1734.403.7920.2310−20.00340.007.353001.724.117.1035.273.7118.4111−20.00340.005.933001.503.985.7335.193.5915.3012−10.00350.002.633001.253.742.5435.063.377.1913−10.00370.000.003001.343.430.0036.933.090.00Table 7
Performance experiment of B-spline curve method.
Measured valuen = 2900 (r/min)Inlet pressureOutlet pressureQnPinQHPinη(kPa)(kPa)(m3/h)(r/min)(kW)(m3/h)(m)(kW)(%)1−80.00180.0021.272998.155.4020.5727.704.8931.772−80.00220.0020.053001.805.3119.3731.214.7934.393−80.00240.0017.693001.605.1317.0932.694.6332.904−80.00250.0016.653001.515.0316.0933.484.5432.345−60.00280.0014.913001.364.8114.4134.174.3430.916−40.00300.0012.753001.134.6012.3233.904.1527.417−30.00310.0011.083000.934.4410.7133.714.0124.548−30.00330.008.293000.634.268.0135.373.8520.079−20.00340.007.063000.614.176.8235.283.7617.4210−20.00350.005.813000.564.055.6236.163.6615.1311−10.00350.004.753000.453.924.5935.163.5412.4212−10.00360.003.293000.553.823.1836.053.459.0513−10.00370.002.143000.563.722.0736.973.366.20


## 4.3. Data Analysis

Figures13 and 14 show the comparison of head and efficiency in performance experiments of four types of impellers. From the shut-off point to the maximum flow condition during the test, 13 flow condition points were selected to completely reflect the external characteristic variation curve. It can be seen that the efficiency of the logarithmic spiral method was better than that of the other three methods, and the efficiency of the single arc method was the worst.Figure 13
Q-H curves of four profiles.Figure 14
Q-η curves four profiles.


## 5. Analysis of Numerical Simulation Results

### 5.1. Velocity in the Flow Field

Figure15 shows the contours of the velocity of the four impellers, respectively. As shown in Figure 15, when the other parameters were consistent, the impellers drawn by four profile design methods had insufficient fluid flow, and the internal flow fields in the impellers drawn by the single arc method and B-spline curve method had obvious vortex. The internal flow fields in the impellers drawn by the double arc method and logarithmic spiral method were significantly better than those drawn by the B-spline curve method and single arc method.Figure 15
Contours of the velocity of the four impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)


### 5.2. Pressure in the Flow Field

Figure16 shows the contours of the pressure of the four impellers, respectively. As shown in Figure 16, when the other parameters were consistent, the internal pressure trends of the impellers drawn by four design methods were correct, and the stratification was obvious. There was negative pressure near the impeller inlet in the flow fields of impellers drawn by the single arc method and B-spline method, while negative pressure did not occur in the other two flow fields of impellers. The maximum pressure values corresponding to the four methods were 437563 Pa, 441628 Pa, 438368 Pa, and 435518 Pa. The pressure fields in the impellers drawn by the double arc method and logarithmic spiral method were significantly better.Figure 16
Contours of the pressure of the four impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)


### 5.3. Cavitation in the Flow Field

Figure17 shows the vapor volume fraction distribution in four impellers when NPSHr was 1 m. As shown in Figure 17, when NPSHr was 1 m, the bubbles almost filled the entire impellers. The vapor volume fractions of the impellers drawn by the double arc method and logarithmic spiral method were 0.9455 and 0.9449, which were significantly lower than those in the other two impellers. The impeller drawn by the single arc method had the largest value.Figure 17
Vapor volume fraction distribution (NPSHr = 1 m). (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)Figure18 shows the vapor volume fraction distribution in four impellers when NPSHr was 2 m. As shown in Figure 18, when NPSHr was 2 m, bubbles filled one third of the entire impellers. The vapor volume fractions of the impellers drawn by the double arc method and logarithmic spiral method were 0.861 and 0.8174, which were significantly lower than those in the other two impellers. The impeller drawn by the B-spline curve had the largest value.Figure 18
Vapor volume fraction distribution (NPSHr = 2 m). (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)In summary, when NPSHr = 1 m and 2 m, the impeller drawn by the logarithmic spiral method is significantly better than the other three impellers, which means that the logarithmic spiral method is the best design method.


### 5.4. Analysis of Efficiency

According to the above conclusion, CFD-Post software was used to calculate the performance parameters of the four profiles, as shown in Table8. From Table 8, it can be concluded that the impeller of the logarithmic spiral method was the optimal impeller under the design condition.Table 8
Performance parameters comparison of four methods.
MethodSingle arcDouble arcLogarithmic spiralB-spline curveH (m)35.443535.465335.54135.3835P (kW)3.504853.36623.300893.51129η (%)76.9981.5381.9876.72


### 5.5. Analysis of Blade Profiles Based on Fluid-Structure Interaction

The purpose of fluid-structure interaction was to find how the flow field influenced the impeller, which was mainly reflected in the deformation of the impeller caused by the fluid, and the force on the impeller. ANSYS-CFX software was used to complete the fluid-structure interaction analysis.Figure19 shows the total deformation of the impellers drawn by four profile design methods under the same working condition. As a whole, the total deformation distribution of the four impellers was symmetrical and stratified obviously. The deformation of the impellers increased continuously with the increase of the radius, the maximum deformation occurred at the edge of the impellers, and the minimum deformation occurred at the inlet of the impellers.Figure 19
Total deformation of the impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)The maximum total deformation of the impellers drawn by four profile design methods was 0.61503 mm, 0.58614 mm, 0.57442 mm, and 0.61609 mm, respectively. The following can be concluded for total deformation: B-spline curve method > single arc method > double arc method > logarithmic spiral method.Figure20 shows the equivalent stress of the impellers drawn by four profile design methods under the same working conditions. As a whole, the impeller stress was mainly concentrated on the blade root, where the equivalent stress was significantly greater than that in other areas. Therefore, compared to other positions, the blade root near the impeller inlet was more prone to stress concentration.Figure 20
Equivalent stress of the impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)The maximum equivalent stress of the impellers drawn by four profiles was 41.357 MPa, 41.43 MPa, 40.909 MPa, and 44.891 MPa, respectively. The following can be concluded for equivalent stress: B-spline curve method > double arc method > single arc method > logarithmic spiral method.


## 5.1. Velocity in the Flow Field

Figure15 shows the contours of the velocity of the four impellers, respectively. As shown in Figure 15, when the other parameters were consistent, the impellers drawn by four profile design methods had insufficient fluid flow, and the internal flow fields in the impellers drawn by the single arc method and B-spline curve method had obvious vortex. The internal flow fields in the impellers drawn by the double arc method and logarithmic spiral method were significantly better than those drawn by the B-spline curve method and single arc method.Figure 15
Contours of the velocity of the four impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)


## 5.2. Pressure in the Flow Field

Figure16 shows the contours of the pressure of the four impellers, respectively. As shown in Figure 16, when the other parameters were consistent, the internal pressure trends of the impellers drawn by four design methods were correct, and the stratification was obvious. There was negative pressure near the impeller inlet in the flow fields of impellers drawn by the single arc method and B-spline method, while negative pressure did not occur in the other two flow fields of impellers. The maximum pressure values corresponding to the four methods were 437563 Pa, 441628 Pa, 438368 Pa, and 435518 Pa. The pressure fields in the impellers drawn by the double arc method and logarithmic spiral method were significantly better.Figure 16
Contours of the pressure of the four impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)


## 5.3. Cavitation in the Flow Field

Figure17 shows the vapor volume fraction distribution in four impellers when NPSHr was 1 m. As shown in Figure 17, when NPSHr was 1 m, the bubbles almost filled the entire impellers. The vapor volume fractions of the impellers drawn by the double arc method and logarithmic spiral method were 0.9455 and 0.9449, which were significantly lower than those in the other two impellers. The impeller drawn by the single arc method had the largest value.Figure 17
Vapor volume fraction distribution (NPSHr = 1 m). (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)Figure18 shows the vapor volume fraction distribution in four impellers when NPSHr was 2 m. As shown in Figure 18, when NPSHr was 2 m, bubbles filled one third of the entire impellers. The vapor volume fractions of the impellers drawn by the double arc method and logarithmic spiral method were 0.861 and 0.8174, which were significantly lower than those in the other two impellers. The impeller drawn by the B-spline curve had the largest value.Figure 18
Vapor volume fraction distribution (NPSHr = 2 m). (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)In summary, when NPSHr = 1 m and 2 m, the impeller drawn by the logarithmic spiral method is significantly better than the other three impellers, which means that the logarithmic spiral method is the best design method.


## 5.4. Analysis of Efficiency

According to the above conclusion, CFD-Post software was used to calculate the performance parameters of the four profiles, as shown in Table8. From Table 8, it can be concluded that the impeller of the logarithmic spiral method was the optimal impeller under the design condition.Table 8
Performance parameters comparison of four methods.
MethodSingle arcDouble arcLogarithmic spiralB-spline curveH (m)35.443535.465335.54135.3835P (kW)3.504853.36623.300893.51129η (%)76.9981.5381.9876.72


## 5.5. Analysis of Blade Profiles Based on Fluid-Structure Interaction

The purpose of fluid-structure interaction was to find how the flow field influenced the impeller, which was mainly reflected in the deformation of the impeller caused by the fluid, and the force on the impeller. ANSYS-CFX software was used to complete the fluid-structure interaction analysis.Figure19 shows the total deformation of the impellers drawn by four profile design methods under the same working condition. As a whole, the total deformation distribution of the four impellers was symmetrical and stratified obviously. The deformation of the impellers increased continuously with the increase of the radius, the maximum deformation occurred at the edge of the impellers, and the minimum deformation occurred at the inlet of the impellers.Figure 19
Total deformation of the impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)The maximum total deformation of the impellers drawn by four profile design methods was 0.61503 mm, 0.58614 mm, 0.57442 mm, and 0.61609 mm, respectively. The following can be concluded for total deformation: B-spline curve method > single arc method > double arc method > logarithmic spiral method.Figure20 shows the equivalent stress of the impellers drawn by four profile design methods under the same working conditions. As a whole, the impeller stress was mainly concentrated on the blade root, where the equivalent stress was significantly greater than that in other areas. Therefore, compared to other positions, the blade root near the impeller inlet was more prone to stress concentration.Figure 20
Equivalent stress of the impellers. (a) Single arc. (b) Double arc. (c) Logarithmic spiral. (d) B-spline curve.
(a)(b)(c)(d)The maximum equivalent stress of the impellers drawn by four profiles was 41.357 MPa, 41.43 MPa, 40.909 MPa, and 44.891 MPa, respectively. The following can be concluded for equivalent stress: B-spline curve method > double arc method > single arc method > logarithmic spiral method.


## 6. Optimization of the Logarithmic Spiral Method

### 6.1. Objective Function

Based on the above analysis, it is shown that the logarithmic spiral method has the highest efficiency. To maximize the efficiency, the impeller inlet and outlet diameters (D1 and D2) and impeller inlet and outlet installation angles (β1 and β2) were taken as control variables and the total power loss and the minimum NPSHr of the pump were taken as the objective functions to optimize the impeller structure of the logarithmic spiral method.


#### 6.1.1. Minimum NPSHr

The basic equation of NPSHr:(8)NPSHr=v022g+λω022g.Assuming that there is no prerotation at impeller inlet,(9)vuo=0,ω02=v02+u02,v0=k24QD12−dh2πηv,u0=k1πnDj60,where k1 = 0.876; k2 = 0.91; Dj—impeller inlet diameter (m); dh—impeller hub diameter (m); ηv=Q/Qt; λ—blade inlet pressure drop coefficient, λ = 0.15–0.3; v0—the absolute velocity of the slightly forward part of the blade inlet (m/s); and ω0—the relative velocity of the slightly forward part of the blade inlet (°/s).Then,(10)NPSHr=12g1+λk2242Q2D12−dhπ2ηv2+λk1π2n2Dj2602×2g.Therefore, the minimum objective function of NPSHr is min [f1(x) = NPSHr].


#### 6.1.2. Minimum Power Loss

The centrifugal pump losses mainly include volume, mechanical, and hydraulic losses. The volume loss accounts for a small proportion of the total loss, so the volume loss can be ignored. The optimization model is established with the mechanical loss and hydraulic loss.(11)P=Pm1+Pm2,where P—total loss power (kW); Pm1—hydraulic loss power (kW); and Pm2—mechanical loss power (kW).(12)Pm1=t1ρQv222,Pm2=0.35t2×10−2ρω3R25,where t1—velocity energy loss coefficient in the pressurized water chamber, t1 = 0.15∼0.25; t2—dimensionless disc friction coefficient, t2 = 0.8∼1.0; v2—the absolute velocity of impeller outlet (m/s); ω—the rotational angular velocity of the shaft (°/s); and R2—impeller radius (m).Assuming that there is no prerotation at impeller inlet,u1 = 0. According to Stodola’s equation,(13)Ht=u2gσu2−Vm2tanβ2,u2=nπR230,σ=1−π2sinβ2,Vm2=2Qtb2ψ2R2π.Then,(14)Ht=1gnπR23021−πsinβ2Z−nQt60b2ψ2tanβ2,where u1, u2—the circumferential velocity of inlet and outlet (m/s); vu1, vu2—the circumferential absolute velocity of inlet and outlet (m/s); Ht—theoretical head of the pump (m); n—rotational speed (rad/s); ψ2—crowding coefficient of impeller outlet, ψ2 = 0.842; and σ—Stodola’s slip coefficient.Impeller outlet speed triangle:(15)v22=vm22+vu22,vm2=2QtπR2b2ψ2,vu2=gHtu2.Then,(16)P=1−πsinβ2/ZgHt+nQt/60b2ψ2tanβ2•t1ρQ2g2Ht2+nQt60b2ψ22+14×602•0.35×10−2t2ρnπ260gHtnπ31+nQt60gHtb2ψ23/2.Therefore, the minimum objective function ofP is min[f2 (x) = P].


#### 6.1.3. Unified Objective Function

The weighting coefficient transformation method is used to deal with the two objective functions. According to the actual situation, the power loss of the plastic centrifugal pump has a greater impact on the total efficiency of the pump than NPSHr. Therefore, it is assumed that NPSHr accounts for 30% and the power loss accounts for 70% for optimization and the unified objective function is min [f(x) = 0.3P+0.7NPSHr].


### 6.2. Constraint Conditions

The constraint ranges of design variables were determined by combining the statistical formula of the velocity coefficient method with the actual excellent model pump. Then, we increased the constraint ranges appropriately.Constraint formulas:(17)2Qn3<D1<4Qn3,9.35ns1001/2Qn3<D2<10.5ns1001/2Qn3,10°<β1<30°,20°<β2<40°.


### 6.3. Multiobjective Function Optimization

According to the creation of a multiobjective optimization program, the four parameters of impeller inlet and outlet diameters and impeller inlet and outlet installation angles were optimized. The final optimization results were that the impeller inlet and outlet diameters were 57.778 mm and 161.57 mm and the impeller inlet and outlet installation angles were 16.836° and 30.68°. The optimization results were rounded and compared with those before optimization, and the design variables are shown in Table9.Table 9
Comparison of optimization design variables.
ParametersD1 (mm)D2 (mm)β1 (°)β2 (°)Before optimization581622030After optimization581621731


### 6.4. Simulation Comparison and Verification

The optimized parameters were used to remodel the impeller. Under the same working conditions, the internal flow field simulation was performed. The results after analysis are shown in Figure21. It can be seen that after optimization, the flow in the impeller was more sufficient. The efficiency was 83.66% after calculation, which was 1.68% higher than before. Therefore, the performance of the optimized impeller had been improved.Figure 21
Optimization results.
(a)(b)


## 6.1. Objective Function

Based on the above analysis, it is shown that the logarithmic spiral method has the highest efficiency. To maximize the efficiency, the impeller inlet and outlet diameters (D1 and D2) and impeller inlet and outlet installation angles (β1 and β2) were taken as control variables and the total power loss and the minimum NPSHr of the pump were taken as the objective functions to optimize the impeller structure of the logarithmic spiral method.


### 6.1.1. Minimum NPSHr

The basic equation of NPSHr:(8)NPSHr=v022g+λω022g.Assuming that there is no prerotation at impeller inlet,(9)vuo=0,ω02=v02+u02,v0=k24QD12−dh2πηv,u0=k1πnDj60,where k1 = 0.876; k2 = 0.91; Dj—impeller inlet diameter (m); dh—impeller hub diameter (m); ηv=Q/Qt; λ—blade inlet pressure drop coefficient, λ = 0.15–0.3; v0—the absolute velocity of the slightly forward part of the blade inlet (m/s); and ω0—the relative velocity of the slightly forward part of the blade inlet (°/s).Then,(10)NPSHr=12g1+λk2242Q2D12−dhπ2ηv2+λk1π2n2Dj2602×2g.Therefore, the minimum objective function of NPSHr is min [f1(x) = NPSHr].


### 6.1.2. Minimum Power Loss

The centrifugal pump losses mainly include volume, mechanical, and hydraulic losses. The volume loss accounts for a small proportion of the total loss, so the volume loss can be ignored. The optimization model is established with the mechanical loss and hydraulic loss.(11)P=Pm1+Pm2,where P—total loss power (kW); Pm1—hydraulic loss power (kW); and Pm2—mechanical loss power (kW).(12)Pm1=t1ρQv222,Pm2=0.35t2×10−2ρω3R25,where t1—velocity energy loss coefficient in the pressurized water chamber, t1 = 0.15∼0.25; t2—dimensionless disc friction coefficient, t2 = 0.8∼1.0; v2—the absolute velocity of impeller outlet (m/s); ω—the rotational angular velocity of the shaft (°/s); and R2—impeller radius (m).Assuming that there is no prerotation at impeller inlet,u1 = 0. According to Stodola’s equation,(13)Ht=u2gσu2−Vm2tanβ2,u2=nπR230,σ=1−π2sinβ2,Vm2=2Qtb2ψ2R2π.Then,(14)Ht=1gnπR23021−πsinβ2Z−nQt60b2ψ2tanβ2,where u1, u2—the circumferential velocity of inlet and outlet (m/s); vu1, vu2—the circumferential absolute velocity of inlet and outlet (m/s); Ht—theoretical head of the pump (m); n—rotational speed (rad/s); ψ2—crowding coefficient of impeller outlet, ψ2 = 0.842; and σ—Stodola’s slip coefficient.Impeller outlet speed triangle:(15)v22=vm22+vu22,vm2=2QtπR2b2ψ2,vu2=gHtu2.Then,(16)P=1−πsinβ2/ZgHt+nQt/60b2ψ2tanβ2•t1ρQ2g2Ht2+nQt60b2ψ22+14×602•0.35×10−2t2ρnπ260gHtnπ31+nQt60gHtb2ψ23/2.Therefore, the minimum objective function ofP is min[f2 (x) = P].


### 6.1.3. Unified Objective Function

The weighting coefficient transformation method is used to deal with the two objective functions. According to the actual situation, the power loss of the plastic centrifugal pump has a greater impact on the total efficiency of the pump than NPSHr. Therefore, it is assumed that NPSHr accounts for 30% and the power loss accounts for 70% for optimization and the unified objective function is min [f(x) = 0.3P+0.7NPSHr].


## 6.1.1. Minimum NPSHr

The basic equation of NPSHr:(8)NPSHr=v022g+λω022g.Assuming that there is no prerotation at impeller inlet,(9)vuo=0,ω02=v02+u02,v0=k24QD12−dh2πηv,u0=k1πnDj60,where k1 = 0.876; k2 = 0.91; Dj—impeller inlet diameter (m); dh—impeller hub diameter (m); ηv=Q/Qt; λ—blade inlet pressure drop coefficient, λ = 0.15–0.3; v0—the absolute velocity of the slightly forward part of the blade inlet (m/s); and ω0—the relative velocity of the slightly forward part of the blade inlet (°/s).Then,(10)NPSHr=12g1+λk2242Q2D12−dhπ2ηv2+λk1π2n2Dj2602×2g.Therefore, the minimum objective function of NPSHr is min [f1(x) = NPSHr].


## 6.1.2. Minimum Power Loss

The centrifugal pump losses mainly include volume, mechanical, and hydraulic losses. The volume loss accounts for a small proportion of the total loss, so the volume loss can be ignored. The optimization model is established with the mechanical loss and hydraulic loss.(11)P=Pm1+Pm2,where P—total loss power (kW); Pm1—hydraulic loss power (kW); and Pm2—mechanical loss power (kW).(12)Pm1=t1ρQv222,Pm2=0.35t2×10−2ρω3R25,where t1—velocity energy loss coefficient in the pressurized water chamber, t1 = 0.15∼0.25; t2—dimensionless disc friction coefficient, t2 = 0.8∼1.0; v2—the absolute velocity of impeller outlet (m/s); ω—the rotational angular velocity of the shaft (°/s); and R2—impeller radius (m).Assuming that there is no prerotation at impeller inlet,u1 = 0. According to Stodola’s equation,(13)Ht=u2gσu2−Vm2tanβ2,u2=nπR230,σ=1−π2sinβ2,Vm2=2Qtb2ψ2R2π.Then,(14)Ht=1gnπR23021−πsinβ2Z−nQt60b2ψ2tanβ2,where u1, u2—the circumferential velocity of inlet and outlet (m/s); vu1, vu2—the circumferential absolute velocity of inlet and outlet (m/s); Ht—theoretical head of the pump (m); n—rotational speed (rad/s); ψ2—crowding coefficient of impeller outlet, ψ2 = 0.842; and σ—Stodola’s slip coefficient.Impeller outlet speed triangle:(15)v22=vm22+vu22,vm2=2QtπR2b2ψ2,vu2=gHtu2.Then,(16)P=1−πsinβ2/ZgHt+nQt/60b2ψ2tanβ2•t1ρQ2g2Ht2+nQt60b2ψ22+14×602•0.35×10−2t2ρnπ260gHtnπ31+nQt60gHtb2ψ23/2.Therefore, the minimum objective function ofP is min[f2 (x) = P].


## 6.1.3. Unified Objective Function

The weighting coefficient transformation method is used to deal with the two objective functions. According to the actual situation, the power loss of the plastic centrifugal pump has a greater impact on the total efficiency of the pump than NPSHr. Therefore, it is assumed that NPSHr accounts for 30% and the power loss accounts for 70% for optimization and the unified objective function is min [f(x) = 0.3P+0.7NPSHr].


## 6.2. Constraint Conditions

The constraint ranges of design variables were determined by combining the statistical formula of the velocity coefficient method with the actual excellent model pump. Then, we increased the constraint ranges appropriately.Constraint formulas:(17)2Qn3<D1<4Qn3,9.35ns1001/2Qn3<D2<10.5ns1001/2Qn3,10°<β1<30°,20°<β2<40°.


## 6.3. Multiobjective Function Optimization

According to the creation of a multiobjective optimization program, the four parameters of impeller inlet and outlet diameters and impeller inlet and outlet installation angles were optimized. The final optimization results were that the impeller inlet and outlet diameters were 57.778 mm and 161.57 mm and the impeller inlet and outlet installation angles were 16.836° and 30.68°. The optimization results were rounded and compared with those before optimization, and the design variables are shown in Table9.Table 9
Comparison of optimization design variables.
ParametersD1 (mm)D2 (mm)β1 (°)β2 (°)Before optimization581622030After optimization581621731


## 6.4. Simulation Comparison and Verification

The optimized parameters were used to remodel the impeller. Under the same working conditions, the internal flow field simulation was performed. The results after analysis are shown in Figure21. It can be seen that after optimization, the flow in the impeller was more sufficient. The efficiency was 83.66% after calculation, which was 1.68% higher than before. Therefore, the performance of the optimized impeller had been improved.Figure 21
Optimization results.
(a)(b)


## 7. Conclusions

(1)
According to the basic parameters of the centrifugal pump and the four different design methods of impeller profile, the shapes and sizes of several profiles were determined. Three-dimensional models were drawn by UG software. ANSYS-ICEM software was utilized to mesh the computational domains.(2)
According to the three-dimensional models drawn by four profile design methods, the impeller models were printed by 3D printing technology. The performance experiments of the plastic centrifugal pumps were carried out. The experimental results showed that the logarithm spiral method was the best and the single arc method was the worst.(3)
The analysis of numerical simulation was performed by FLUENT software. From the contours of velocity and pressure, the flow field in the impeller drawn by the logarithmic spiral method was significantly better. The maximum pressure values corresponding to the four methods were 437563 Pa, 441628 Pa, 438368 Pa, and 435518 Pa. From the vapor volume fraction distribution, when NPSHr = 1 m and 2 m, the impeller drawn by the logarithmic spiral method was significantly better than the other three impellers. According to the calculation of performance parameters, it can be concluded that the impeller drawn by the logarithmic spiral method was the optimal impeller under the design condition.(4)
ANSYS-CFX software was used to complete the fluid-structure interaction analysis. The maximum total deformation of the impellers drawn by the four profile design methods was 0.61503 mm, 0.58614 mm, 0.57442 mm, and 0.61609 mm, respectively. Deformation: B-spline curve method > single arc method > double arc method > logarithmic spiral method. The maximum equivalent stress of the impellers drawn by the four profile design methods was 41.357 MPa, 41.43 MPa, 40.909 MPa, and 44.891 MPa, respectively. Equivalent stress: B-spline curve method > double arc method > single arc method > logarithmic spiral method.(5)
Based on the analysis, the impeller drawn by the logarithmic spiral method had the highest efficiency. To maximize the efficiency, the impeller inlet and outlet diameters (D1 and D2) and impeller inlet and outlet installation angles (β1 and β2) were taken as control variables and the total power loss and the minimum NPSHr of the pump were taken as the objective functions. The final optimization results were that the impeller inlet and outlet diameters were 58 mm and 162 mm and the impeller inlet and outlet installation angles were 17°and 31°. The hydraulic efficiency was increased by 1.68% after calculation, which indicated that the impeller structure had been improved.



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*Source: 6665520-2020-12-22.xml*