# Multiobjective Optimization Design and Experimental Investigation on the Axial Flow Pump with Orthogonal Test Approach

Authors: Yuquan Zhang; Yanhe Xu; Yuan Zheng; E. Fernandez-Rodriguez; Aoran Sun; Chunxia Yang; Jue Wang
Journal: Complexity
 (2019)

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## Abstract

A multiobjective optimization technique based on the computational fluid dynamics (CFD) simulations and the orthogonal test is proposed to reduce the pressure pulsation in this paper. Three levels of four well-known performance factorsL9 (34) were considered in the orthogonal test scheme: the number of blades, the blade setting angle, the hub ratio, and the distance between the blade and the guide vane. The evaluation indexes corresponded to the head, efficiency, shaft power, and pressure pulsation, respectively. An optimal configuration A2B1C2D3 was obtained by comprehensive frequency analysis method, after intuitive and range analysis. In comparison with the nonoptimized model, the new design’s head and efficiency increased by 17.8% and 4.26%, whilst the shaft power and the pressure pulsation coefficient reduced by 1.22% and 11%, respectively. Experiments conducted on the optimized pump were consistent with the CFD model. Six different rotational speed conditions in the optimal operating points were numerically calculated in order to explore the internal hydraulic characteristics of the optimized axial flow pump. It is verified that the comprehensive frequency analysis method based on the orthogonal test approach is effective for the multiobjective optimization of the axial flow pump.


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## Body

## 1. Introduction

Pump is a kind of widely used machinery with universal applications which accounts for almost 22% of the world’s energy consumed by electric motors [1–4]. Axial flow pumps, as the main pump type applied in low-lift and large-flow pumping stations, have attracted increasing attention in field irrigation and water diversion projects [5–7]. The internal flow of an axial flow pump is a three-dimensional complex flow field producing pressure pulsations and thus affecting the stability of the device [8, 9]. Therefore, it is of great significance to improve the pump efficiency and stability in order to meet targets of worldwide energy saving and security operation.Although several approaches exist for the purpose of multiobjective optimization design of the pumps and impellers, programs using computational fluid dynamics (CFD) continues to be the most popular method [10–12]. These programs, when run by high-performance computers, can forecast not only the functioning of the pump but also the flow field upstream and downstream of the pump [13–15]. For instance, Zhang et al. [16] validated a CFD model, used for optimizing the submerged propeller operated in an oxidation ditch. The results showed that the CFD model is able to characterize the flow pattern and energy consumption in the full-scale oxidation ditch. In a similar way, Shojaeefard et al. [17] simulated the 3D flow in the pump along with the volute by the CFD method, under different cases of passage width and outlet angle of the impeller. The pump performance was shown to improve with the modification of the original geometry because of the reduction of losses in the passage and outlet of the impeller. Asuaje et al. [18] conducted a 3D flow simulation by CFD tools based on the impeller performance. Spence and Amaral-Teixeira [19] presented a CFD approach for the analysis of the geometrical variations on the performance characteristics of a pump. Geometric recommendations were put forward to reducing vibration and increasing component life through the reductions in pressure pulsations. Fleder and Böhle [20] investigated the effect of blade number on the flow characteristics and performance of different pumps by CFD simulations and experiments. It is shown that the number of blades has a significant influence on the circulation velocity and flow angles between blades and side channel. Yang et al. [21] numerically researched the effect of adjustable guide vane on the pump performance with seven different adjustable angles. The results indicated that the hydraulic efficiency of the pump could be improved considerably by adjusting the guide vane angle so as to decrease the flow separation. Hou et al. [22] selected the geometrical parameter of blade setting angle to design a centrifugal pump. It was found that the optimal pump has an improved efficiency. A study on the axial flow pump performance improvement was presented by Park et al. [23], where the effect of hub tip ratio was investigated to design the guide vanes, and the performance of the pump was enhanced compared with the original model. Liu et al. [24] studied the centrifugal pump performance by extending the axial distances from 280 mm to 460 mm using experiments and CFD simulation. The results showed that the axial distance has a remarkable impact on the pressure fluctuations. Bonaiuti et al. [25] described the parametric design of a mixed-flow pump. CFD analyses were applied to assess the hydrodynamic performance of the different impeller and diffuser geometry configurations. A good agreement was found between experimental results and numerical simulations. It is indicated that the parametric study could provide a design guideline so as to find the optimal compromise in the process of pump design.The orthogonal test is a useful and statistical method using the representative parameter configuration to achieve a fast optimization process [26–29]. It was developed by Byrne [30] and plays an important role in the design of efficient pumps. Xu et al. [31] introduced an orthogonal method for designing a high-efficient, low-cost centrifugal pump. An orthogonal table designation L16 was established, where 16 individuals of blades were generated using 5 parameters. A centrifugal pump was suggested, having a better efficiency and cavitation performance than the original one. In Zhou et al.’s study [32], a pump was designed using 16 impellers and 5 main impeller geometric parameters; the pump efficiency and head of the optimal pump had a significant improvement compared with the original pump. Wang et al. [33] established a function relationship between three factors of impeller and the efficiency through the quadratic regression orthogonal test. The three factors included the impeller blade outlet width, the impeller blade outlet stagger angle, and the impeller outlet slope. The results demonstrated that orthogonal test is an effective method to improve the efficiency of stainless steel stamping multistage pump. Shi et al. [34] designed 16 schemes concerning the different flow conditions of the passages with orthogonal test to investigate the hydraulic performance of an axial flow pump device. The results showed that an optimized scheme of the inlet and outlet passages could be obtained by the approach of orthogonal test.It can be seen from the above literature review that CFD is a reliable and convenient approach to optimize the pump efficiency, and orthogonal test is a credible and effective method for pump optimization. Currently, lots of efforts have been made by the researchers on the prediction of pump’s overall performance, and majority of these studies mainly focused on the pump efficiency and power as the optimization index rather than on the pressure pulsation and stability of the pump. In fact, the pressure pulsation can lead to dangerous consequences with regard to pump reliability which may ultimately lead to device failure. Pressure pulsation has not been considered as the optimization parameter in the method of orthogonal test in previous studies, and no studies have reported the comprehensive frequency analysis method for the axial flow pump optimization. Therefore, it is paramount to improve the stability of the axial flow pump considering the flow rate, head, and shaft power to meet the design requirements.In the present paper, we propose a geometrical optimization technique to reduce the pressure pulsation and improve the overall performance of an axial flow pump, based on the CFD method with the implementation of the orthogonal test approach. Four factors influencing pump functioning were taken into account, namely, the number of blades, the blade setting angle, the hub ratio, and the distance between blade and guide vane. Parameters were analyzed in terms of the flow rate, head, efficiency, shaft power, and pressure pulsation. Finally, four design parameters, along with an optimal pump, were determined by means of the comprehensive frequency analysis method. Experiments were conducted to verify the method, and six different rotational speed conditions in the optimal operating points were numerically calculated.


## 2. Orthogonal Test Theory

### 2.1. Method of the Orthogonal Test

The traditional optimization method is mainly used to transform a multi-index into a single-index model analysis. The purpose of the method is to test the influence of a particular factor over the whole outcome, with the view of obtaining optimal configuration in terms of the performance levels [35]. However, enormous workload occurs, when the comprehensive test consists of many horizontal combinations. Overall, the traditional comprehensive test method is considered laborious and inefficient [36]. By contrast, the orthogonal test method uses an orthogonal table to analyze and arrange the multifactor test [37]. In this method, only some representative factors from the whole level combinations are selected, referred to as a partial test. This procedure allows a formal solution to be found, thus requiring less workload than the comprehensive test method.The basic procedure of the orthogonal test is indicated in Figure1. It is made up of the design of the scheme and the analysis of the test results.Figure 1
Basic design procedure of the orthogonal test.In this paper, the influence and weight of four factors were taken to analyze the pump performance. These include the number of blades (factor A), blade setting angle (factor B), hub ratio (factor C), and distance between blade and guide vane (factor D). The number of test schemes was 9, and the level of combination per factor was 3, and thusLn(ab) = L9(34). Table 1 details the parameters of the axial flow pump.Table 1
Orthogonal tableL9 (34).
Test scheme
Test factor
Factors and level combinations
A
B
C
D
1
1
1
1
1
A1B1C1D1
2
1
2
2
2
A1B2C2D2
3
1
3
3
3
A1B3C3D3
4
2
1
2
3
A2B1C2D3
5
2
2
3
1
A2B2C3D1
6
2
3
1
2
A2B3C1D2
7
3
1
3
2
A3B1C3D2
8
3
2
1
3
A3B2C1D3
9
3
3
2
1
A3B3C2D1A range analysis method was adopted to clarify the significance of the factors on the pump performance. The factorki is the sum of the test indexes corresponding to i levers under a certain factor, and its average score is named as Ki, whilst Rang is the range between each factor and reflects the difference between the maximal and minimal value of each factor’s level. The greater the value of Rang, the greater the effect on the test index [34].


### 2.2. Determination of the Orthogonal Test Scheme

The numerical setup of Table2 bears a close resemblance to Table 1, by using L9(34). The pump has a radial variation of blade angle (Φ) as depicted in Figure 2. The relative radius (r/R) is defined as the radius of the section (r) to the blade radius (R).Table 2
Orthogonal table of the axial flow pump.
Orthogonal test scheme
Four test factors
Number of blades (A)
Blade setting angle (B)
Hub ratio (C)
Distance between blade and guide vane (D) (mm)
1
3
B1
0.45
175
2
3
B2
0.47
210
3
3
B3
0.49
263
4
4
B1
0.47
263
5
4
B2
0.49
175
6
4
B3
0.45
210
7
5
B1
0.49
210
8
5
B2
0.45
263
9
5
B3
0.47
175Figure 2
Blade seating angle of each section.


## 2.1. Method of the Orthogonal Test

The traditional optimization method is mainly used to transform a multi-index into a single-index model analysis. The purpose of the method is to test the influence of a particular factor over the whole outcome, with the view of obtaining optimal configuration in terms of the performance levels [35]. However, enormous workload occurs, when the comprehensive test consists of many horizontal combinations. Overall, the traditional comprehensive test method is considered laborious and inefficient [36]. By contrast, the orthogonal test method uses an orthogonal table to analyze and arrange the multifactor test [37]. In this method, only some representative factors from the whole level combinations are selected, referred to as a partial test. This procedure allows a formal solution to be found, thus requiring less workload than the comprehensive test method.The basic procedure of the orthogonal test is indicated in Figure1. It is made up of the design of the scheme and the analysis of the test results.Figure 1
Basic design procedure of the orthogonal test.In this paper, the influence and weight of four factors were taken to analyze the pump performance. These include the number of blades (factor A), blade setting angle (factor B), hub ratio (factor C), and distance between blade and guide vane (factor D). The number of test schemes was 9, and the level of combination per factor was 3, and thusLn(ab) = L9(34). Table 1 details the parameters of the axial flow pump.Table 1
Orthogonal tableL9 (34).
Test scheme
Test factor
Factors and level combinations
A
B
C
D
1
1
1
1
1
A1B1C1D1
2
1
2
2
2
A1B2C2D2
3
1
3
3
3
A1B3C3D3
4
2
1
2
3
A2B1C2D3
5
2
2
3
1
A2B2C3D1
6
2
3
1
2
A2B3C1D2
7
3
1
3
2
A3B1C3D2
8
3
2
1
3
A3B2C1D3
9
3
3
2
1
A3B3C2D1A range analysis method was adopted to clarify the significance of the factors on the pump performance. The factorki is the sum of the test indexes corresponding to i levers under a certain factor, and its average score is named as Ki, whilst Rang is the range between each factor and reflects the difference between the maximal and minimal value of each factor’s level. The greater the value of Rang, the greater the effect on the test index [34].


## 2.2. Determination of the Orthogonal Test Scheme

The numerical setup of Table2 bears a close resemblance to Table 1, by using L9(34). The pump has a radial variation of blade angle (Φ) as depicted in Figure 2. The relative radius (r/R) is defined as the radius of the section (r) to the blade radius (R).Table 2
Orthogonal table of the axial flow pump.
Orthogonal test scheme
Four test factors
Number of blades (A)
Blade setting angle (B)
Hub ratio (C)
Distance between blade and guide vane (D) (mm)
1
3
B1
0.45
175
2
3
B2
0.47
210
3
3
B3
0.49
263
4
4
B1
0.47
263
5
4
B2
0.49
175
6
4
B3
0.45
210
7
5
B1
0.49
210
8
5
B2
0.45
263
9
5
B3
0.47
175Figure 2
Blade seating angle of each section.


## 3. Numerical Simulation

### 3.1. Simulation Methodology

The flows for the axial flow pump are highly turbulent, and the most common tool to model them is through the solution of the Reynolds-averaged Navier–Stokes (RANS) equations [1]. The given approach saves computational resource significantly, compared with other methods such as large eddy simulations [38]. RANS equations use the Reynolds decomposition of velocity and pressure as follows:(1)continuity:∂ui¯∂xi=0,(2)momentum:ρ∂ui¯∂t+ρ∂ui¯uj¯∂xj=ρf¯i+∂∂xj−p¯δij+2μS¯ij−ρui′uj′¯,where ui is velocity, xi is the spatial coordinate, p is the pressure, ρ is the fluid density, μ is the dynamic viscosity, f¯i is the time mean external force tensor, and S¯ij=0.5∂ui¯/∂xj+∂uj¯/∂xi is the mean rate of the strain tensor. The Dirac function δij = 1 for i=j and equals zero otherwise. The Reynolds stresses ui′uj′¯ are solved by using a turbulence model to closure the RANS equations. The k−ω SST (Shear Stress Transport) model, proposed by Menter [39], combines the best of both standard k−ω and k−ε turbulence models. SST model performs better in applications where adverse pressure gradients occur, so it is adopted in this study [40].


### 3.2. Boundary Condition and Mesh Refinement Study

The original axial flow pump of Jiangdu No. 1 pump station has been operating for many years, and as it is known, the practical operating mode always deviates from the design conditions. The computational domain includes the elbow-type inlet passage section, the blades, the guide vane, the siphon outlet flow section, and the outlet bend, as shown in Figure3. The basic parameters are as follows: design flow rate Q = 298.6 L/s, head H = 6.8 m, rotor’s rotational speed n = 1461 r/min, and number of blades Z = 4. An improved design provided a design flow of Q = 328.5 L/s and a head of H = 7.8 m.Figure 3
Computational domain of the pump.ICEM CFD software was used to obtain the mesh of the computing domain [14]. The unstructured grids were adopted due to its strength in the adaptability of the complicated and irregular geometry. In the domain of the blades and guide vanes, unstructured mesh with prisms was used to approximate the geometry of runner and guide vane. The y+, the dimensionless wall distance of the grid cells adjacent to the pump surface, was within 30 to 200, a range sufficiently enough for capturing turbulent flow near the surfaces.Ansys CFX was used to simulate the whole flow passage of the nine test schemes [16]. In order to transfer the data between the interfaces, the dynamic-static interface was set as Frozen Rotor Interface in the steady calculation and as Transient-Frozen Rotor/Stator interface in the unsteady calculation. The loss caused by small gaps between the pump body and the impeller was ignored. Mass flow and free outlet boundary condition were used to describe the inlet and outlet flow, respectively.For the unsteady simulation, the blades rotated three degrees at each time step. Periodic behavior was obtained by eight complete rotations. Four sets of pressure pulsation monitoring points were set up around the blades and guide vanes to obtain the pressure pulsation information inside the axial flow pump as shown in Figure4. The results of the last two cycles in the simulation were taken as the data for pressure pulsation analysis. Pressure pulsation coefficient (CP) was introduced to describe the pressure pulsation characteristics:(3)Cp=Pi−PavcPavc,where Cp is the pressure pulsation coefficient, Pi refers to the pressure at the monitoring point, and Pavc is the averaged static pressure within the rotation period.Figure 4
Four sets of pressure fluctuation monitoring points.Table3 shows the three mesh densities with coarse, medium, and fine levels whilst Table 4 illustrates the variation of the evaluation indexes. It is known that cases with a higher mesh density usually obtain better simulation results. However, the difference between fine and medium meshes is less than 2% which is negligible. Therefore, medium meshes were used considering both the computation accuracy and required resources.Table 3
Mesh size for the calculation model (million).
Mesh density
Number of elements (million)
Head (m)
Shaft power (kW)
Efficiency
Coarse
1.51
6.23
15.67
68.52
Medium
3.39
6.42
14.92
72.12
Fine
4.26
6.50
14.82
72.26Table 4
Variation of the evaluation indexes.
Mesh density
Number of elements (million)
Head (m)
Shaft power (kW)
Efficiency
Coarse
1.51
4.2%
5.7%
5.2%
Medium
3.39
1.2%
0.7%
0.2%
Fine
4.26
—
—
—


## 3.1. Simulation Methodology

The flows for the axial flow pump are highly turbulent, and the most common tool to model them is through the solution of the Reynolds-averaged Navier–Stokes (RANS) equations [1]. The given approach saves computational resource significantly, compared with other methods such as large eddy simulations [38]. RANS equations use the Reynolds decomposition of velocity and pressure as follows:(1)continuity:∂ui¯∂xi=0,(2)momentum:ρ∂ui¯∂t+ρ∂ui¯uj¯∂xj=ρf¯i+∂∂xj−p¯δij+2μS¯ij−ρui′uj′¯,where ui is velocity, xi is the spatial coordinate, p is the pressure, ρ is the fluid density, μ is the dynamic viscosity, f¯i is the time mean external force tensor, and S¯ij=0.5∂ui¯/∂xj+∂uj¯/∂xi is the mean rate of the strain tensor. The Dirac function δij = 1 for i=j and equals zero otherwise. The Reynolds stresses ui′uj′¯ are solved by using a turbulence model to closure the RANS equations. The k−ω SST (Shear Stress Transport) model, proposed by Menter [39], combines the best of both standard k−ω and k−ε turbulence models. SST model performs better in applications where adverse pressure gradients occur, so it is adopted in this study [40].


## 3.2. Boundary Condition and Mesh Refinement Study

The original axial flow pump of Jiangdu No. 1 pump station has been operating for many years, and as it is known, the practical operating mode always deviates from the design conditions. The computational domain includes the elbow-type inlet passage section, the blades, the guide vane, the siphon outlet flow section, and the outlet bend, as shown in Figure3. The basic parameters are as follows: design flow rate Q = 298.6 L/s, head H = 6.8 m, rotor’s rotational speed n = 1461 r/min, and number of blades Z = 4. An improved design provided a design flow of Q = 328.5 L/s and a head of H = 7.8 m.Figure 3
Computational domain of the pump.ICEM CFD software was used to obtain the mesh of the computing domain [14]. The unstructured grids were adopted due to its strength in the adaptability of the complicated and irregular geometry. In the domain of the blades and guide vanes, unstructured mesh with prisms was used to approximate the geometry of runner and guide vane. The y+, the dimensionless wall distance of the grid cells adjacent to the pump surface, was within 30 to 200, a range sufficiently enough for capturing turbulent flow near the surfaces.Ansys CFX was used to simulate the whole flow passage of the nine test schemes [16]. In order to transfer the data between the interfaces, the dynamic-static interface was set as Frozen Rotor Interface in the steady calculation and as Transient-Frozen Rotor/Stator interface in the unsteady calculation. The loss caused by small gaps between the pump body and the impeller was ignored. Mass flow and free outlet boundary condition were used to describe the inlet and outlet flow, respectively.For the unsteady simulation, the blades rotated three degrees at each time step. Periodic behavior was obtained by eight complete rotations. Four sets of pressure pulsation monitoring points were set up around the blades and guide vanes to obtain the pressure pulsation information inside the axial flow pump as shown in Figure4. The results of the last two cycles in the simulation were taken as the data for pressure pulsation analysis. Pressure pulsation coefficient (CP) was introduced to describe the pressure pulsation characteristics:(3)Cp=Pi−PavcPavc,where Cp is the pressure pulsation coefficient, Pi refers to the pressure at the monitoring point, and Pavc is the averaged static pressure within the rotation period.Figure 4
Four sets of pressure fluctuation monitoring points.Table3 shows the three mesh densities with coarse, medium, and fine levels whilst Table 4 illustrates the variation of the evaluation indexes. It is known that cases with a higher mesh density usually obtain better simulation results. However, the difference between fine and medium meshes is less than 2% which is negligible. Therefore, medium meshes were used considering both the computation accuracy and required resources.Table 3
Mesh size for the calculation model (million).
Mesh density
Number of elements (million)
Head (m)
Shaft power (kW)
Efficiency
Coarse
1.51
6.23
15.67
68.52
Medium
3.39
6.42
14.92
72.12
Fine
4.26
6.50
14.82
72.26Table 4
Variation of the evaluation indexes.
Mesh density
Number of elements (million)
Head (m)
Shaft power (kW)
Efficiency
Coarse
1.51
4.2%
5.7%
5.2%
Medium
3.39
1.2%
0.7%
0.2%
Fine
4.26
—
—
—


## 4. Results and Discussion

The results of the nine test schemes simulated by CFD with the designed flow rate are shown in Table5.Table 5
Results of the nine test schemes.
Test schemes
Evaluation indexes
Head,H (m)
Shaft power,P (kW)
Efficiency,η (%)
Pressure pulsation coefficient,Cp
1
6.42
14.92
72.12
1.19
2
6.58
16.64
78.47
1.16
3
5.59
13.09
72.66
1.11
4
8.01
20.29
78.59
0.31
5
6.70
15.67
72.56
0.80
6
6.90
16.74
75.28
0.74
7
7.48
17.58
72.94
0.53
8
7.72
18.47
74.25
0.99
9
7.70
19.76
79.65
0.34Each data was analyzed by the intuitively single-index method, followed by the range analysis. Finally, the comprehensive frequency analysis method was used to obtain the optimized levels.


### 4.1. Intuitive and Range Analysis

Flow rate is a working parameter belonging to the specified index which is different from the evaluation indexes of efficiency and pressure pulsation. It is our goal to increase the efficiency and reduce the pressure pulsation. The head belongs to the criterion index, and thus, only the schemes with head meeting the criterion could be available in the optimization process. Based on the results shown in Table5, the head of No. 1, No. 2, No. 3, No. 5, No. 6, and No. 7 test schemes did not meet the requirement at the condition of the designed flow rate, and thus, they were identified as nonoptimal models.The optimal test scheme, considering the index of head, was No. 4 (A2B1C2D3), compared with the index of efficiency’s No. 9 (A3B3C2D1), shaft power’s No. 8 (A3B2C1D3), and pressure pulsation coefficient’s No. 4 (A2B1C2D3).As mentioned in the above part of orthogonal test theory, range analysis is applied to clarify the significance levels of four influencing factors on the evaluation indexes.ki is calculated by the total of the test indices corresponding to i levers whilst Ki indicates the average value. Rang is subtracted by the maximal and minimal value of Ki (Tables 6 and 7).Table 6
Range analysis of the results on head and efficiency.
Range analysis
Head,H (m)
Shaft power,P (kW)
A
B
C
D
A
B
C
D
k
1
18.59
21.91
21.04
20.82
44.65
52.79
50.13
50.35
k
2
21.61
21.00
22.29
20.96
52.70
50.78
56.69
50.96
k
3
22.90
20.19
19.77
21.32
55.81
49.59
46.34
51.85
K
1
6.20
7.30
7.01
6.94
14.88
17.60
16.71
16.78
K
2
7.20
7.00
7.43
6.99
17.57
16.93
18.90
16.99
K
3
7.63
6.73
6.59
7.11
18.60
16.53
15.45
17.28
R
ang
1.43
0.57
0.84
0.17
3.72
1.07
3.45
0.50Table 7
Range analysis of the results on shaft power and pressure pulsation coefficient.
Range analysis
Efficiency,η (%)
Pressure pulsation coefficient,Cp
A
B
C
D
A
B
C
D
k
1
223.25
223.65
221.65
224.33
3.46
2.03
2.92
2.33
k
2
226.43
225.28
236.71
226.69
1.85
2.95
1.81
2.43
k
3
226.84
227.59
218.16
225.50
1.86
2.19
2.44
2.41
K
1
74.42
74.55
73.88
74.78
1.15
1.02
0.97
0.78
K
2
75.48
75.09
78.90
75.56
0.616
1.48
0.60
0.81
K
3
75.61
75.86
72.72
75.17
0.62
1.10
0.81
0.80
R
ang
1.19
1.31
6.18
0.78
0.534
0.46
0.37
0.03To verify the degree of factor influence on the four evaluation indexes, a visual comparison was made between the factor level and evaluation indexes. Such cases are depicted in Figures5(a)∼5(d).Figure 5
Relationship between factor lever and four evaluation indexes. (a) Head. (b) Shaft power. (c) Efficiency. (d) Pressure pulsation coefficient.
(a)
(b)
(c)
(d)As observed, an increase of blade numbers leads to higher head, and the efficiency reaches a peak at C2. As the shaft power is in direct proportion to the flow rate and head, the shaft power grows with increased blade numbers, as indicated in Figure 5(c). The number of blades (factor A) has a significant effect on the cascade solidity but has a little effect on the other structures of the pump. As a result, the load on each blade will be reduced with more blades, thus improving the flow pattern around the runner and thus increasing the shaft power [41].The circulation of velocityΓ is calculated by the formula: Γ = 2πrVu, where Vu is the circular velocity. In order to obtain the same head for each section of the blade, the circulation of velocity of the blade has to be the same in different radial direction, and the radius r, inversed with the circular velocity Vu. Therefore, the blade setting angle (factor B) near the hub is larger than that of the outer edge of the blade, distorting seriously the blade. Based on Figures 5(a)–5(d), a factor level changes from B3 to B1, resulting in a significant increase of the head and shaft power and slight reduction of the efficiency (2%). By reducing the blade seating angle of each section, the distortion degree of blade lowers, improving the power output.If the factor level of the hub ratio (factor C) changes from 0.45 to 0.49, the head, efficiency, and shaft power increase and decrease to low levels, peaking at C2. The smaller the hub ratio, the lesser the hydraulic friction losses and the greater the flow around the pump. However, the product will increase the distortion of the blades, disorder the flow pattern, and lower the efficiency of the pump. The hub ratio has an important influence on the flow pattern around the blades, and thus, it is an important geometric parameter in the hydraulic design and optimization of the axial flow pump.The distance between the blade and guide vane (factor D) was determined according to the formulaS=0.1∼0.15 Dh, where Dh is the diameter of the blades [17]. As we can see from Figures 5(a)–5(d), factor D has lesser influence on the evaluation indexes, when compared with factors A, B, and C. With the increase of the distance between the blade and guide vane, the head and shaft power increase monotonously and the efficiency increases but then decreases at higher levels.Through the above range analysis, the optimized test schemes can be obtained as follows: test scheme with optimum index of head was A3B1C2D3, of efficiency was A3B3C2D2, of shaft power was A1B3C3D1, and of pressure pulsation was A2B1C2D1. On the other hand, from the calculated result of range R, the impact from largest to smallest on head was A > C > B > D, as opposed to efficiency’s C > B > A > D, shaft power’s A > C > B > D, and pressure pulsation’s A > B > C > D.


### 4.2. Comprehensive Frequency Analysis

The four evaluation indexes are of equal importance. Table8 summarizes the results of the three levels corresponding to four factors, based on the test schemes obtained by the intuitive analysis and the test schemes obtained by the range analysis.Table 8
Comprehensive frequency analysis of three levels corresponding to four factors.
Factor level
A1
A2
A3
B1
B2
B3
C1
C2
C3
D1
D2
D3
Frequency
1/4
3/8
3/8
1/2
0
1/2
0
3/4
1/4
3/8
1/8
1/2Table8 shows A2 and A3 occur with the highest frequency (3/8). For factor B, the values are B1 and B3 (1/2) and for factor C, C2 (3/4), whilst for factor D, D3 (1/2). Consequently, the preliminary optimum test scheme becomes A2/A3B1/B3C2D3.The pump’s specific speed is estimated by equation (4), where ns is the specific speed of the pump. Since from Section 4.1, an increase of the number of blades corresponded to both lower specific speeds and the anticavitation performance of the pump. Therefore, a factor level of A2 with low number of blades becomes suitable for the model:(4)ns=3.65nQH3/4.Similarly, a factor lever of B1 with a smaller blade seating angle was adopted, on the basis of the serious distortion applied to the blades and the higher requirements for material and production technology of the blades.From the above comprehensive frequency analysis, the optimum test scheme was determined as A2B1C2D3, using a four-bladed pump with a blade seating angle of B1, hub ratio of 0.47, and distance between the blade and guide vane of 263 mm.


### 4.3. Performance of the Optimized Pump

The optimized test scheme A2B1C2D3, corresponding to scheme 4 under the design flow of Q = 328.5 L/s, was further verified in this section.The first main difference between the two models concerns with the streamlines around the blades and guide vanes at the half of the blade height. As observed in Figure6, the flow is more unevenly distributed in the original than in the optimized pump, leading to large hydraulic loss. Furthermore, two large-scale vortices are formed on the back of the original’s guide vane, compared with almost none in the optimized pump. The second difference relates to the high pressure observed near the hub of the original’s blade; this almost disappears on the optimized blade surface as shown in Figure 7.Figure 6
Distribution of streamline around the blade and guide vane at 0.5 times blade height. (a) Original model. (b) Optimized model.
(a)
(b)Figure 7
Distribution of pressure contours on the blade surface. (a) Original model. (b) Optimized model.
(a)
(b)With regard to the pressure pulsation,Cp increases gradually from hub to the blade tip with a cyclical trend, and the maximum coefficient is three times of the hub’s as indicated in Figure 8(a). From Figure 8(b), the flow at the outlet of the blades is affected by the rotor-stator interaction of the blades and guide vanes. The peak frequency of the pressure is seven times larger than the peak frequency of the guide vanes, and this is consistent with the number of the guide vanes. As expected, the amplitude of pressure pulsation increases from the hub to the tip, reaching a relatively higher value at a frequency of four times the rotational’s.Figure 8
Pressure pulsation in axial flow pump under design flow rate. (a) Time domain. (b) Frequency domain.
(a)
(b)In summary, the optimized axial flow pump is capable of meeting the requirements of the flow rate and head. Under the new design flow rate, the pump efficiency increases from 74.33% to 78.59% while the shaft power decreases from 20.54 kW to 20.29 kW. Besides, the absolute value of the pressure pulsation coefficient reduces by 11%. It is found that the comprehensive frequency analysis is an available method for the multiobjective optimization.


### 4.4. Comparison between Experiments and Simulation

To verify the method, tests were carried out in the optimized model at the multifunction test bench of Hohai University. Figure9 shows the three-dimensional schematic diagram of the hydraulic machinery multifunction test bench. The test bench is a vertical closed circulation system, including tail water tank, pressure water tank, dynamometer motor, and torque meter. Tests were carried out by using the equal head method, where the value of the prototype is the same with the model’s. Through the analysis of test error, the total error of this test was ±0.27%, satisfying the standard test regulations.Figure 9
Hydraulic multifunction test bench. (1) Tail water tank; (2) press water tank; (3) dynamometer motor; (4) pump model; (5) torque meter; (6) electromagnetic flow meter; (7) filling pump; (8) water supply pump; (9) solenoid valve.The physical model of the axial flow pump device is shown in Figure10. The blades were manufactured with brass material by numerical control machines. To ensure the inner walls of the connecting part are smooth, the inlet and outlet passages were welded with steel plates. The model scale of the pump device is 1 : 5.844.Figure 10
The device of the whole pump. (a) Blades. (b) Runner chamber. (c) Inlet passage section. (d) Pump device section.
(a)
(b)
(c)
(d)In order to eliminate the free gases and air bubbles in the system, the pump was operated for more than 30 minutes at rated conditions, prior to the recordings. The pump performance at a certain blade setting angle was investigated under the condition of no cavitation. The measuring points were reasonably distributed on the whole performance curve. The data of the flow rate, head, rotate speed, and shaft power were simultaneously obtained and processed, when the parameters remained stable for a period of time.Figure11 shows the comparison of the external characteristics between the numerical simulation and the experimental results. The head and the efficiency are slightly higher in the simulations than in the experimental measurements, when the flow rate is smaller than the designed discharge. The maximum error of head and efficiency is 4.6% and 3.2%, respectively. The small error could be due to the friction losses between the bearings and the seal ring, as well as due to the manual adjustment of the blade’s pitch angle. The head and efficiency of the numerical simulation are consistent with the variation of the test results, and the maximum error is not more than 5%. It shows that the numerical computation model and the method adopted in this paper have a good performance in predicting the external characteristics of the axial flow pump.Figure 11
Comparison of model pump test data with numerical simulation data. (a) Head curve. (b) Efficiency curve.
(a)
(b)A comprehensive characteristic curve of the axial flow pump model was drawn based on the experimental measurements. As shown in Figure12, the optimized pump model had a wide range of high-efficiency areas. The maximum efficiency of the model device is 78%, with a corresponding head and blade placement angle of 7.52 m and 0°, respectively. It is obvious that the pump with a blade placement angle of 0° is able to meet the requirement under the design point.Figure 12
Comprehensive characteristic curve.


### 4.5. Pump Performance under Variable Rotational Speed

Six different rotational speed conditions (n, 0.9n, 0.8n, 0.7n, 0.6n, and 0.5n) in the optimal operating points were studied to explore the internal hydraulic characteristics of the optimized axial flow pump.


#### 4.5.1. Hydraulic Performance of the Flow Passage

In an attempt to analyze the influence of rotational speed on the hydraulic loss in the passage of inlet, guide vane, and outlet, the hydraulic loss coefficientKp is introduced to expound the ratio of hydraulic loss between these three sections and the head of the pump device. The coefficient is defined as follows:(5)Kp=ΔhH×100%,where Δh is the hydraulic loss and δ indicates the ratio of hydraulic loss between different rotational speeds and the rated speed at 1461 r/min. Table 9 shows the hydraulic loss of the passages at variable rotational speed, along with the pump efficiency.Table 9
Hydraulic loss of the passages.
Rotational speed (r/min)
Passage of inlet
Passage of guide vane
Passage of outlet
Pump efficiency,Η (%)
K
p (%)
δ
K
p (%)
δ
K
p (%)
δ
1461 (n)
1.2257
1
11.7564
1
2.3165
1
93.58
1315 (0.9n)
1.2351
1.0077
11.8588
1.0087
2.3297
1.0057
93.52
1169 (0.8n)
1.2478
1.0180
12.0042
1.0211
2.3486
1.0139
93.44
1023 (0.7n)
1.2527
1.0220
12.1594
1.0343
2.3539
1.0161
93.34
877 (0.6n)
1.2803
1.0445
12.3238
1.0483
2.3887
1.0312
93.19
731 (0.5n)
1.3086
1.0676
12.6327
1.0745
2.4161
1.0430
93.09As seen from Table9, at rated rotational speed, the hydraulic loss ratio of the inlet passage, guide vane passage, and outlet passage is the smallest, whilst the hydraulic efficiency of the blades is the highest at 93.58%. At different rotational speeds, the hydraulic loss ratio Kp increases with the decrease of rotational speed. Evidently, the pump efficiency decreases with lower rotational speeds since speed influences the velocity circulation and the hydraulic loss of the inlet passage, leading to flow deviation from the ideal condition and thus higher impact losses.The hydraulic loss ratio of the inlet section is the smallest, whilst the guide vane section’s is the largest among the hydraulic loss ratios of each pump section. The change of rotational speed has the greatest influence on the guide vane section, and the corresponding hydraulic loss of 0.5n is 1.0745 times of rated speed’s. By contrast, the outlet channel has the smallest hydraulic loss, about 1.043 times the rated speed’s. These differences can be justified in part by the fixed rear guide vane used in the hydraulic design of the axial flow pump. Under the design condition, the flow angle of the runner outlet is equal to the inlet placement angle of the guide vane; therefore, the impact loss on the inlet of the guide vane is the smallest.


#### 4.5.2. Hydraulic Performance of the Blade Inlet and Outlet Section

Figure13(a) depicts the differences in axial velocity of the blades’ inlet section with respect to six rotational speeds. As can be seen, the axial velocity decreases from the tip to the hub and this variation trend with rotational speeds is basically similar. The change of axial velocity with respect to the blade section is largest at the hub and tip, and minimal, at the middle part of the blade. The velocity circulation (Figure 13(b)) increases along the radial direction, according to r∗=r−Rh/R−r, where r is the radius of a point at the blade section; Rh, the radius of the hub; and R, the radius of the blade. The velocity circulation increases smoothly when r∗ varies from 0.2 to 0.85, but rises significantly when r∗ varies from 0.85 to 1. This correlation can be attributed to the small and large blade setting angle at the hub and at the tip, respectively, resulting in a larger flow circulation through the outer edge of the blade, thus improving the anticavitation performance of the pump.Figure 13
Velocity of the blade inlet section. (a) Axial velocity. (b) Velocity circulation.
(a)
(b)Figure14 compares the distributions of axial velocity and velocity circulation at the blade outlet section under the six rotational speed conditions. A monotonous upward trend from hub to tip and low to high speed is observed with both velocities. Compared to the velocity circulation at the blade inlet section, the increase of velocity circulation at the blade outlet section is significantly due to the blade rotation and in direct proportion to the rotational speed.Figure 14
Velocity of the blade outlet section. (a) Axial velocity. (b) Velocity circulation.
(a)
(b)


## 4.1. Intuitive and Range Analysis

Flow rate is a working parameter belonging to the specified index which is different from the evaluation indexes of efficiency and pressure pulsation. It is our goal to increase the efficiency and reduce the pressure pulsation. The head belongs to the criterion index, and thus, only the schemes with head meeting the criterion could be available in the optimization process. Based on the results shown in Table5, the head of No. 1, No. 2, No. 3, No. 5, No. 6, and No. 7 test schemes did not meet the requirement at the condition of the designed flow rate, and thus, they were identified as nonoptimal models.The optimal test scheme, considering the index of head, was No. 4 (A2B1C2D3), compared with the index of efficiency’s No. 9 (A3B3C2D1), shaft power’s No. 8 (A3B2C1D3), and pressure pulsation coefficient’s No. 4 (A2B1C2D3).As mentioned in the above part of orthogonal test theory, range analysis is applied to clarify the significance levels of four influencing factors on the evaluation indexes.ki is calculated by the total of the test indices corresponding to i levers whilst Ki indicates the average value. Rang is subtracted by the maximal and minimal value of Ki (Tables 6 and 7).Table 6
Range analysis of the results on head and efficiency.
Range analysis
Head,H (m)
Shaft power,P (kW)
A
B
C
D
A
B
C
D
k
1
18.59
21.91
21.04
20.82
44.65
52.79
50.13
50.35
k
2
21.61
21.00
22.29
20.96
52.70
50.78
56.69
50.96
k
3
22.90
20.19
19.77
21.32
55.81
49.59
46.34
51.85
K
1
6.20
7.30
7.01
6.94
14.88
17.60
16.71
16.78
K
2
7.20
7.00
7.43
6.99
17.57
16.93
18.90
16.99
K
3
7.63
6.73
6.59
7.11
18.60
16.53
15.45
17.28
R
ang
1.43
0.57
0.84
0.17
3.72
1.07
3.45
0.50Table 7
Range analysis of the results on shaft power and pressure pulsation coefficient.
Range analysis
Efficiency,η (%)
Pressure pulsation coefficient,Cp
A
B
C
D
A
B
C
D
k
1
223.25
223.65
221.65
224.33
3.46
2.03
2.92
2.33
k
2
226.43
225.28
236.71
226.69
1.85
2.95
1.81
2.43
k
3
226.84
227.59
218.16
225.50
1.86
2.19
2.44
2.41
K
1
74.42
74.55
73.88
74.78
1.15
1.02
0.97
0.78
K
2
75.48
75.09
78.90
75.56
0.616
1.48
0.60
0.81
K
3
75.61
75.86
72.72
75.17
0.62
1.10
0.81
0.80
R
ang
1.19
1.31
6.18
0.78
0.534
0.46
0.37
0.03To verify the degree of factor influence on the four evaluation indexes, a visual comparison was made between the factor level and evaluation indexes. Such cases are depicted in Figures5(a)∼5(d).Figure 5
Relationship between factor lever and four evaluation indexes. (a) Head. (b) Shaft power. (c) Efficiency. (d) Pressure pulsation coefficient.
(a)
(b)
(c)
(d)As observed, an increase of blade numbers leads to higher head, and the efficiency reaches a peak at C2. As the shaft power is in direct proportion to the flow rate and head, the shaft power grows with increased blade numbers, as indicated in Figure 5(c). The number of blades (factor A) has a significant effect on the cascade solidity but has a little effect on the other structures of the pump. As a result, the load on each blade will be reduced with more blades, thus improving the flow pattern around the runner and thus increasing the shaft power [41].The circulation of velocityΓ is calculated by the formula: Γ = 2πrVu, where Vu is the circular velocity. In order to obtain the same head for each section of the blade, the circulation of velocity of the blade has to be the same in different radial direction, and the radius r, inversed with the circular velocity Vu. Therefore, the blade setting angle (factor B) near the hub is larger than that of the outer edge of the blade, distorting seriously the blade. Based on Figures 5(a)–5(d), a factor level changes from B3 to B1, resulting in a significant increase of the head and shaft power and slight reduction of the efficiency (2%). By reducing the blade seating angle of each section, the distortion degree of blade lowers, improving the power output.If the factor level of the hub ratio (factor C) changes from 0.45 to 0.49, the head, efficiency, and shaft power increase and decrease to low levels, peaking at C2. The smaller the hub ratio, the lesser the hydraulic friction losses and the greater the flow around the pump. However, the product will increase the distortion of the blades, disorder the flow pattern, and lower the efficiency of the pump. The hub ratio has an important influence on the flow pattern around the blades, and thus, it is an important geometric parameter in the hydraulic design and optimization of the axial flow pump.The distance between the blade and guide vane (factor D) was determined according to the formulaS=0.1∼0.15 Dh, where Dh is the diameter of the blades [17]. As we can see from Figures 5(a)–5(d), factor D has lesser influence on the evaluation indexes, when compared with factors A, B, and C. With the increase of the distance between the blade and guide vane, the head and shaft power increase monotonously and the efficiency increases but then decreases at higher levels.Through the above range analysis, the optimized test schemes can be obtained as follows: test scheme with optimum index of head was A3B1C2D3, of efficiency was A3B3C2D2, of shaft power was A1B3C3D1, and of pressure pulsation was A2B1C2D1. On the other hand, from the calculated result of range R, the impact from largest to smallest on head was A > C > B > D, as opposed to efficiency’s C > B > A > D, shaft power’s A > C > B > D, and pressure pulsation’s A > B > C > D.


## 4.2. Comprehensive Frequency Analysis

The four evaluation indexes are of equal importance. Table8 summarizes the results of the three levels corresponding to four factors, based on the test schemes obtained by the intuitive analysis and the test schemes obtained by the range analysis.Table 8
Comprehensive frequency analysis of three levels corresponding to four factors.
Factor level
A1
A2
A3
B1
B2
B3
C1
C2
C3
D1
D2
D3
Frequency
1/4
3/8
3/8
1/2
0
1/2
0
3/4
1/4
3/8
1/8
1/2Table8 shows A2 and A3 occur with the highest frequency (3/8). For factor B, the values are B1 and B3 (1/2) and for factor C, C2 (3/4), whilst for factor D, D3 (1/2). Consequently, the preliminary optimum test scheme becomes A2/A3B1/B3C2D3.The pump’s specific speed is estimated by equation (4), where ns is the specific speed of the pump. Since from Section 4.1, an increase of the number of blades corresponded to both lower specific speeds and the anticavitation performance of the pump. Therefore, a factor level of A2 with low number of blades becomes suitable for the model:(4)ns=3.65nQH3/4.Similarly, a factor lever of B1 with a smaller blade seating angle was adopted, on the basis of the serious distortion applied to the blades and the higher requirements for material and production technology of the blades.From the above comprehensive frequency analysis, the optimum test scheme was determined as A2B1C2D3, using a four-bladed pump with a blade seating angle of B1, hub ratio of 0.47, and distance between the blade and guide vane of 263 mm.


## 4.3. Performance of the Optimized Pump

The optimized test scheme A2B1C2D3, corresponding to scheme 4 under the design flow of Q = 328.5 L/s, was further verified in this section.The first main difference between the two models concerns with the streamlines around the blades and guide vanes at the half of the blade height. As observed in Figure6, the flow is more unevenly distributed in the original than in the optimized pump, leading to large hydraulic loss. Furthermore, two large-scale vortices are formed on the back of the original’s guide vane, compared with almost none in the optimized pump. The second difference relates to the high pressure observed near the hub of the original’s blade; this almost disappears on the optimized blade surface as shown in Figure 7.Figure 6
Distribution of streamline around the blade and guide vane at 0.5 times blade height. (a) Original model. (b) Optimized model.
(a)
(b)Figure 7
Distribution of pressure contours on the blade surface. (a) Original model. (b) Optimized model.
(a)
(b)With regard to the pressure pulsation,Cp increases gradually from hub to the blade tip with a cyclical trend, and the maximum coefficient is three times of the hub’s as indicated in Figure 8(a). From Figure 8(b), the flow at the outlet of the blades is affected by the rotor-stator interaction of the blades and guide vanes. The peak frequency of the pressure is seven times larger than the peak frequency of the guide vanes, and this is consistent with the number of the guide vanes. As expected, the amplitude of pressure pulsation increases from the hub to the tip, reaching a relatively higher value at a frequency of four times the rotational’s.Figure 8
Pressure pulsation in axial flow pump under design flow rate. (a) Time domain. (b) Frequency domain.
(a)
(b)In summary, the optimized axial flow pump is capable of meeting the requirements of the flow rate and head. Under the new design flow rate, the pump efficiency increases from 74.33% to 78.59% while the shaft power decreases from 20.54 kW to 20.29 kW. Besides, the absolute value of the pressure pulsation coefficient reduces by 11%. It is found that the comprehensive frequency analysis is an available method for the multiobjective optimization.


## 4.4. Comparison between Experiments and Simulation

To verify the method, tests were carried out in the optimized model at the multifunction test bench of Hohai University. Figure9 shows the three-dimensional schematic diagram of the hydraulic machinery multifunction test bench. The test bench is a vertical closed circulation system, including tail water tank, pressure water tank, dynamometer motor, and torque meter. Tests were carried out by using the equal head method, where the value of the prototype is the same with the model’s. Through the analysis of test error, the total error of this test was ±0.27%, satisfying the standard test regulations.Figure 9
Hydraulic multifunction test bench. (1) Tail water tank; (2) press water tank; (3) dynamometer motor; (4) pump model; (5) torque meter; (6) electromagnetic flow meter; (7) filling pump; (8) water supply pump; (9) solenoid valve.The physical model of the axial flow pump device is shown in Figure10. The blades were manufactured with brass material by numerical control machines. To ensure the inner walls of the connecting part are smooth, the inlet and outlet passages were welded with steel plates. The model scale of the pump device is 1 : 5.844.Figure 10
The device of the whole pump. (a) Blades. (b) Runner chamber. (c) Inlet passage section. (d) Pump device section.
(a)
(b)
(c)
(d)In order to eliminate the free gases and air bubbles in the system, the pump was operated for more than 30 minutes at rated conditions, prior to the recordings. The pump performance at a certain blade setting angle was investigated under the condition of no cavitation. The measuring points were reasonably distributed on the whole performance curve. The data of the flow rate, head, rotate speed, and shaft power were simultaneously obtained and processed, when the parameters remained stable for a period of time.Figure11 shows the comparison of the external characteristics between the numerical simulation and the experimental results. The head and the efficiency are slightly higher in the simulations than in the experimental measurements, when the flow rate is smaller than the designed discharge. The maximum error of head and efficiency is 4.6% and 3.2%, respectively. The small error could be due to the friction losses between the bearings and the seal ring, as well as due to the manual adjustment of the blade’s pitch angle. The head and efficiency of the numerical simulation are consistent with the variation of the test results, and the maximum error is not more than 5%. It shows that the numerical computation model and the method adopted in this paper have a good performance in predicting the external characteristics of the axial flow pump.Figure 11
Comparison of model pump test data with numerical simulation data. (a) Head curve. (b) Efficiency curve.
(a)
(b)A comprehensive characteristic curve of the axial flow pump model was drawn based on the experimental measurements. As shown in Figure12, the optimized pump model had a wide range of high-efficiency areas. The maximum efficiency of the model device is 78%, with a corresponding head and blade placement angle of 7.52 m and 0°, respectively. It is obvious that the pump with a blade placement angle of 0° is able to meet the requirement under the design point.Figure 12
Comprehensive characteristic curve.


## 4.5. Pump Performance under Variable Rotational Speed

Six different rotational speed conditions (n, 0.9n, 0.8n, 0.7n, 0.6n, and 0.5n) in the optimal operating points were studied to explore the internal hydraulic characteristics of the optimized axial flow pump.


### 4.5.1. Hydraulic Performance of the Flow Passage

In an attempt to analyze the influence of rotational speed on the hydraulic loss in the passage of inlet, guide vane, and outlet, the hydraulic loss coefficientKp is introduced to expound the ratio of hydraulic loss between these three sections and the head of the pump device. The coefficient is defined as follows:(5)Kp=ΔhH×100%,where Δh is the hydraulic loss and δ indicates the ratio of hydraulic loss between different rotational speeds and the rated speed at 1461 r/min. Table 9 shows the hydraulic loss of the passages at variable rotational speed, along with the pump efficiency.Table 9
Hydraulic loss of the passages.
Rotational speed (r/min)
Passage of inlet
Passage of guide vane
Passage of outlet
Pump efficiency,Η (%)
K
p (%)
δ
K
p (%)
δ
K
p (%)
δ
1461 (n)
1.2257
1
11.7564
1
2.3165
1
93.58
1315 (0.9n)
1.2351
1.0077
11.8588
1.0087
2.3297
1.0057
93.52
1169 (0.8n)
1.2478
1.0180
12.0042
1.0211
2.3486
1.0139
93.44
1023 (0.7n)
1.2527
1.0220
12.1594
1.0343
2.3539
1.0161
93.34
877 (0.6n)
1.2803
1.0445
12.3238
1.0483
2.3887
1.0312
93.19
731 (0.5n)
1.3086
1.0676
12.6327
1.0745
2.4161
1.0430
93.09As seen from Table9, at rated rotational speed, the hydraulic loss ratio of the inlet passage, guide vane passage, and outlet passage is the smallest, whilst the hydraulic efficiency of the blades is the highest at 93.58%. At different rotational speeds, the hydraulic loss ratio Kp increases with the decrease of rotational speed. Evidently, the pump efficiency decreases with lower rotational speeds since speed influences the velocity circulation and the hydraulic loss of the inlet passage, leading to flow deviation from the ideal condition and thus higher impact losses.The hydraulic loss ratio of the inlet section is the smallest, whilst the guide vane section’s is the largest among the hydraulic loss ratios of each pump section. The change of rotational speed has the greatest influence on the guide vane section, and the corresponding hydraulic loss of 0.5n is 1.0745 times of rated speed’s. By contrast, the outlet channel has the smallest hydraulic loss, about 1.043 times the rated speed’s. These differences can be justified in part by the fixed rear guide vane used in the hydraulic design of the axial flow pump. Under the design condition, the flow angle of the runner outlet is equal to the inlet placement angle of the guide vane; therefore, the impact loss on the inlet of the guide vane is the smallest.


### 4.5.2. Hydraulic Performance of the Blade Inlet and Outlet Section

Figure13(a) depicts the differences in axial velocity of the blades’ inlet section with respect to six rotational speeds. As can be seen, the axial velocity decreases from the tip to the hub and this variation trend with rotational speeds is basically similar. The change of axial velocity with respect to the blade section is largest at the hub and tip, and minimal, at the middle part of the blade. The velocity circulation (Figure 13(b)) increases along the radial direction, according to r∗=r−Rh/R−r, where r is the radius of a point at the blade section; Rh, the radius of the hub; and R, the radius of the blade. The velocity circulation increases smoothly when r∗ varies from 0.2 to 0.85, but rises significantly when r∗ varies from 0.85 to 1. This correlation can be attributed to the small and large blade setting angle at the hub and at the tip, respectively, resulting in a larger flow circulation through the outer edge of the blade, thus improving the anticavitation performance of the pump.Figure 13
Velocity of the blade inlet section. (a) Axial velocity. (b) Velocity circulation.
(a)
(b)Figure14 compares the distributions of axial velocity and velocity circulation at the blade outlet section under the six rotational speed conditions. A monotonous upward trend from hub to tip and low to high speed is observed with both velocities. Compared to the velocity circulation at the blade inlet section, the increase of velocity circulation at the blade outlet section is significantly due to the blade rotation and in direct proportion to the rotational speed.Figure 14
Velocity of the blade outlet section. (a) Axial velocity. (b) Velocity circulation.
(a)
(b)


## 4.5.1. Hydraulic Performance of the Flow Passage

In an attempt to analyze the influence of rotational speed on the hydraulic loss in the passage of inlet, guide vane, and outlet, the hydraulic loss coefficientKp is introduced to expound the ratio of hydraulic loss between these three sections and the head of the pump device. The coefficient is defined as follows:(5)Kp=ΔhH×100%,where Δh is the hydraulic loss and δ indicates the ratio of hydraulic loss between different rotational speeds and the rated speed at 1461 r/min. Table 9 shows the hydraulic loss of the passages at variable rotational speed, along with the pump efficiency.Table 9
Hydraulic loss of the passages.
Rotational speed (r/min)
Passage of inlet
Passage of guide vane
Passage of outlet
Pump efficiency,Η (%)
K
p (%)
δ
K
p (%)
δ
K
p (%)
δ
1461 (n)
1.2257
1
11.7564
1
2.3165
1
93.58
1315 (0.9n)
1.2351
1.0077
11.8588
1.0087
2.3297
1.0057
93.52
1169 (0.8n)
1.2478
1.0180
12.0042
1.0211
2.3486
1.0139
93.44
1023 (0.7n)
1.2527
1.0220
12.1594
1.0343
2.3539
1.0161
93.34
877 (0.6n)
1.2803
1.0445
12.3238
1.0483
2.3887
1.0312
93.19
731 (0.5n)
1.3086
1.0676
12.6327
1.0745
2.4161
1.0430
93.09As seen from Table9, at rated rotational speed, the hydraulic loss ratio of the inlet passage, guide vane passage, and outlet passage is the smallest, whilst the hydraulic efficiency of the blades is the highest at 93.58%. At different rotational speeds, the hydraulic loss ratio Kp increases with the decrease of rotational speed. Evidently, the pump efficiency decreases with lower rotational speeds since speed influences the velocity circulation and the hydraulic loss of the inlet passage, leading to flow deviation from the ideal condition and thus higher impact losses.The hydraulic loss ratio of the inlet section is the smallest, whilst the guide vane section’s is the largest among the hydraulic loss ratios of each pump section. The change of rotational speed has the greatest influence on the guide vane section, and the corresponding hydraulic loss of 0.5n is 1.0745 times of rated speed’s. By contrast, the outlet channel has the smallest hydraulic loss, about 1.043 times the rated speed’s. These differences can be justified in part by the fixed rear guide vane used in the hydraulic design of the axial flow pump. Under the design condition, the flow angle of the runner outlet is equal to the inlet placement angle of the guide vane; therefore, the impact loss on the inlet of the guide vane is the smallest.


## 4.5.2. Hydraulic Performance of the Blade Inlet and Outlet Section

Figure13(a) depicts the differences in axial velocity of the blades’ inlet section with respect to six rotational speeds. As can be seen, the axial velocity decreases from the tip to the hub and this variation trend with rotational speeds is basically similar. The change of axial velocity with respect to the blade section is largest at the hub and tip, and minimal, at the middle part of the blade. The velocity circulation (Figure 13(b)) increases along the radial direction, according to r∗=r−Rh/R−r, where r is the radius of a point at the blade section; Rh, the radius of the hub; and R, the radius of the blade. The velocity circulation increases smoothly when r∗ varies from 0.2 to 0.85, but rises significantly when r∗ varies from 0.85 to 1. This correlation can be attributed to the small and large blade setting angle at the hub and at the tip, respectively, resulting in a larger flow circulation through the outer edge of the blade, thus improving the anticavitation performance of the pump.Figure 13
Velocity of the blade inlet section. (a) Axial velocity. (b) Velocity circulation.
(a)
(b)Figure14 compares the distributions of axial velocity and velocity circulation at the blade outlet section under the six rotational speed conditions. A monotonous upward trend from hub to tip and low to high speed is observed with both velocities. Compared to the velocity circulation at the blade inlet section, the increase of velocity circulation at the blade outlet section is significantly due to the blade rotation and in direct proportion to the rotational speed.Figure 14
Velocity of the blade outlet section. (a) Axial velocity. (b) Velocity circulation.
(a)
(b)


## 5. Conclusion

The aim of the study was to identify the influence of four interrelated pump factors, with regard to pump’s stability and efficiency. To obtain this, a multiobjective optimization design method based on the orthogonal test was adopted. The results of range analysis showed the distance between the guide vane and the blade (factor D) and the hub ratio (factor C) were the least and most influential indexes, respectively. An optimal configuration A2B1C2D3 was obtained by the comprehensive frequency analysis method, after intuitive and range analysis.In order to understand how design affected efficiency, a comparison of the flow around the blades and guide vanes was made between the two models. In general, the flow in the two sections was more laminar in the optimized pump than in the original pump, leading to less hydraulic loss and higher efficiency. Furthermore, the high pressure built near the hub of the original pump almost disappears in the optimized design. Under the design point, the optimized axial flow pump was able to meet the requirements of the flow rate and head. In comparison with the nonoptimized model, the new design’s pump efficiency increased by 4.26%, whilst the shaft power and pressure coefficient reduced by 1.22% and 11%, respectively.The simulations of the pump’s head and efficiency were found consistent with experimental results, and the error was less than 5%. The results further prove the validity of the presented methods.In general, the decrease in rotational speed caused lager hydraulic loss ratio and smaller blade efficiencies. The hydraulic loss ratio of the blade inlet section was the smallest, and that of the guide vane was the largest among the hydraulic loss ratios of each pump section. The change of rotational speed had the greatest influence on the guide vane section. Compared to the velocity circulation at the blade inlet section, the increase of velocity circulation at the blade outlet section was significantly due to the blade rotation, and this was in direct proportion to the rotational speed.



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*Source: 1467565-2019-12-05.xml*