# Multiobjective Optimization of Low-Specific-Speed Multistage Pumps by Using Matrix Analysis and CFD Method

Authors: Qiaorui Si; Shouqi Yuan; Jianping Yuan; Chuan Wang; Weigang Lu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The implementation of energy-saving and emission-reduction techniques has become a worldwide consensus. Thus, special attention should be provided to the field of pump optimization. With the objective of focusing on multiobjective optimization problems in low-specific-speed pumps, 10 parameters were carefully selected in this study for anL27(310) orthogonal experiment. The parameters include the outlet width of the impeller blade, blade number, and inlet setting angle of the guide vane. The numerical calculation appropriate for forecasting the performance of multistage pumps, such as the head, efficiency, and shaft power, was analyzed. Results were obtained after calculating the two-stage flow field of the pump through computational fluid dynamics (CFD) methods. A matrix method was proposed to optimize the results of the orthographic experiment. The optimal plan was selected according to the weight of each factor. Calculated results indicate that the inlet setting angle of the guide vane influences efficiency significantly and that the outlet angle of blades has an effect on the head and shaft power. A prototype was produced with the optimal plan for testing. The efficiency rating of the prototype reached 58.61%; maximum shaft power was within the design requirements, which verifies that the proposed method is feasible for pump optimization.


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## Body

## 1. Introduction

Pumps are widely utilized in various fields of the national economy [1]. Where there is water, there are pumps. Low-specific-speed centrifugal pumps are commonly employed in sewage treatment and well pumping because of their small flow and high head. They consume large amounts of energy and have a potential for energy saving. Low efficiency and overload tendency in long-term large-flow operations are the two obstacles in designing such pumps; the impeller diameter of low-specific-speed centrifugal pumps is relatively large, and the flow channel is long and narrow, resulting in significant disc friction and hydraulic loss. Thus, the efficiency of low-specific-speed centrifugal pumps is relatively low; the shaft power curves of such pumps increase sharply. The greater flow method is usually adopted in the design of low-specific-speed pumps to enhance flow efficiency. Thus, the ratio of macroshaft power to designed power becomes much higher than that of common centrifugal pumps. A motor whose export pipeline has no valve burns out easily [2, 3]. Pumps are usually designed in multistages to increase the amount of pressure because of limitations imposed by costs or diameter, such as the case of well pumps. Increasing the single-stage head may decrease the number of stages to save costs, energy, and materials. High efficiency in maintaining the head and low power are the two goals of multistage pump designs. Thus, the study on multitarget optimal pump designs is important in implementing energy saving and emission reduction.Mathematical methods, such as neural networks, orthographic experiments, genetic algorithms, and grey theory, are often adopted in multitarget optimization design [4, 5]. An orthographic experiment design can optimize the experimental conditions to achieve the target with fewer experiments [6, 7]. It has been widely adopted in pump design. Shouqi et al. [8] employed an orthogonal tableL8(27)to thoroughly study the effect of impeller geometric parameters and the throat area on pump performance. He proposed a practical framework for a centrifugal pump without overload and its design method. Wang et al. [9] focused on the effect of main impeller geometric parameters on deep-geometric performance via orthographic experiments. Shen et al. [10] designed nine models of complex impeller centrifugal pumps with orthographic experiments and discovered the effect order of geometric parameters on pump performance. Zhou et al. [11] identified the key factors that affect the performance of guiding blades by studying conduit guiding blades and by conducting orthographic experiments and designing high-performance guiding blades. The orthogonal experiment method is therefore a multivariate and multilevel optimization method suitable for pump design. The studies mentioned above examined the influence of parameters on the impeller or guiding blades separately; however, the factors were not enough to completely reflect the effect of geometric parameters on the experiment. Furthermore, the multiobjective test was usually transformed into a single-objective test in the experimental analysis. Plans were then comprehensively selected through single-index analyses, including intuitive analysis and variance analysis. These methods neglect the significance and differences of indexes.QS10-68, a typical multistage electric submersible pump (ESP), was regarded as an example in the present study. A method that combines orthogonal experimentation and numerical simulation was applied for optimization. A simple matrix method was also introduced to calculate the effect of factors on each index and to directly decide the factor order according to weight, which would perfectly solve the issue of selecting an optimal plan in an orthographic experiment design. The effect of 10 geometric parameters of the impeller and guiding vane on the multistage head, efficiency, and shaft power was also studied. Lastly, a set of optimum geometric parameters was obtained based on weight calculation. A prototype that employs the optimal plan was then tested for verification.


## 2. Pump Model

The structure of the multistage ESP is displayed in Figure1. The figure shows a multistage centrifugal pump operating in a vertical position [12]. The pump shaft is connected to the protector by a mechanical coupling at the bottom of the pump. Fluids from the well enter the pump through an intake screen and are lifted by the pump stages. Produced liquids, after being subjected to great centrifugal forces caused by the high rotational speed of the impeller, lose their kinetic energy in the diffuser. Kinetic energy is converted to pressure energy in the diffuser. The design parameters for this pump are as follows: flow rate Qd=10 m3/h, total head Hd=68 m, speed n=2850 r/min, four stages, and the efficiency of the necessary electric motor is 4 kW.Figure 1
Model structure chart.


## 3. Matrix Analysis Model and Experimental Scheme

In a multiobjective optimization problem, engineers adopt an orthographic experimental method as a solvent. The orthographic experimental method is a scientific method utilized to arrange and analyze multiple factor experiments by means of a table based on orthogonal principles. Through scientific arrangement and analysis of the result, a study can discover the ideal production conditions and techniques. However, problems, including large calculated amounts and confirming the weights unreasonably, exist in the multitarget orthogonal test method. A matrix analysis model was presented to solve this problem. In this model, a three-layer structure and the layer structure matrix of the orthogonal test are established first. The weight matrix of the test index is then calculated by multiplying the matrix of each layer. The weighs of the factors and levels that affect the tests results are calculated. Finally, the optimal plan and the importance order of the factors that influence the test index values are determined according to the weights.


### 3.1. Matrix Analysis Model

An orthographic experiment was designed (ignoring the interaction for the first). The three-layer structural model based on data structure was established. The first layer in Table1 is the investigation index, the second layer is the factor layer, and the last one is the level layer. The data of each layer determine the definition of the matrix as follows.Table 1
Data structure of the multi-target orthogonal test.
First level
Experimental index
Second level
FactorA1
FactorA2
FactorAl
Third level
A
11
…
A
1
m
-
1
A
1
m
A
21
…
A
2
m
-
1
A
2
m
A
l
1
…
A
l
m
-
1
A
l
mDefinition 1.
The following is the matrix of the experimental investigation index. The study order isKij=kij. The matrix is established on the conditions that l factors exist in the experiment, each factor has m levels, kij is the average index of factor Ai at j level, and the investigation index is good when it is high. A low investigation index is considered good when the study establishes a matrix supposing Kij=1/kij.Definition 2.
The matrix of the factor layer:Ti=1/∑j=1mKij to build (2) matrix.Definition 3.
The matrix of the level layer: the range ofAi in the orthographic experiment is set as si. The study order is Si=si/∑i=1lsi. Equation (3) matrix is then established.Definition 4.
The weight matrix that may affect the test index:ωT=MTS. 									(1)M=[K1100⋯0K1200⋯0⋯⋯⋯⋯⋯K1m00⋯00K210⋯00K220⋯0⋯⋯⋯⋯⋯0K2m⋯⋯⋯⋯⋯⋯000⋯Kl1000⋯Kl2⋯⋯⋯⋯⋯000⋯Klm],(2)T=[T10000T200⋯⋯⋯⋯000Tl],(3)S=[S1S2⋯Sl],(4)ωT=[ω1,ω2,…,ωm].
In the matrix above mentioned,ω1=K11T1S1, K11T1 refers to the ratio of A1’s first level index to the sum of all A1’s level indexes. S1 is the ratio ofA1’s range to the sum of all ranges. The product of these two data reflects the effect of the first level of A1 on the index and range of A1. The other factors and levels are identical. The weight of each factor and level was obtained after calculation. The optimal plan and the factor order in the index were established based on the weight value.


### 3.2. Experimental Scheme

According to Euler equations, theoretical headHt is
(5)Ht=u2vu2-u1vu1g=u2g(u2h0-QtF2tanβ2),
where u1 (m/s) is peripheral speed at the inlet; vu1 is the peripheral velocity component of the blade inlet; vu1=0 in the straight cone suction chamber; u2 (m/s) refers to the circular velocity at the impeller outlet; g(m2/s) is acceleration because of gravity; h0=1-(π/z) is the stodala slip coefficient; z represents blade numbers; Qt(m3/h) is the theoretical flow; F2=πD2b2ψ2 is the cross-section area of the impeller outer; ψ2 is the blade expelling coefficient;b2(m)is the width of the blade outlet; β2 is the blade outlet angle. P′, the hydraulic power input, can be deduced by the following equation:
(6)P′=ρgHtQt=ρu22Qt(h0-Qtu2F2tanβ2).In (6), ρ (kg/m3) refers to water density. The equation shows that both theoretical head and hydraulic input power are determined by the impeller. However, rolling flow causes Pm, additional shaft power, as it enters the guiding blade channel. Hence, shaft power P=P′+Pm and pump efficiency η=P′/P. Pump shaft power and efficiency are determined by both the impeller and the guide vane in hydraulics.The following are the main geometric parameters of the model: width of the blade outletb2, suction diameter of impeller d1, inclination of the back cover board α2, inlet attack angle Δβ1, inlet angle in axial plane γ3, blade outlet angle β2, blade numberz, outer diameter of the back cover d2min, blade thickness S2, inlet angle of the guide vane β3, outlet angle of the guide vane α3, import width of the guide vane b3, blade number of the guide vane z1, axial length of the guide vane γ3, inlet diameter of the guide vane D2, and outlet diameter of the guide vane D1. α3 is equal to 90° to eliminate the rotation component. The last five factors were determined by the basic geometric parameters of the impeller. The first 10 geometric parameters were selected as testing factors.According to the specific speed equation,(7)ns=3.65nQH3/4,
where Q refers to the flow and H refers to the head of the pump. The design-specific speed of the pump is low, namely, 65.6, because it is lower than 80. Relative parameters can be obtained with the velocity-coefficient method under structure demand; these factors are shown in Table 2. Three levels were selected in each factor. The experimental scheme, shown in Table 3, is based on the L27(310) orthogonal table. In the 27 group tests, different numerals appear at the same frequency in any column, and any two-digit number appears equivalent in any row. After deciding the experimental plan, tests were implemented in numerical order without any arbitrary changes.Table 2
Factor level table.
Level
Factor
A
B
C
D
E
F
G
H
I
J
1
122
8
10
6
2
88
34
5
0
15
2
124
9
15
7
3
89
36
10
5
20
3
126
10
20
8
4
90
38
15
10
25
A—d2min/mm, B—b2/mm, C—β2/(°), D—z, E—S2/mm, F—α2/(°), G—d1/mm, H—Δβ1/(°), I—γ3/(°), J—β3/(°).Table 3
Orthogonal test schemes.
Test number
A/mm
B/mm
C/(°)
D/
E/mm
F/(°)
G/mm
H/(°)
I/(°)
J/(°)
1
122
8
10
6
2
88
34
5
0
15
2
122
8
10
6
3
89
36
10
5
20
3
122
8
10
6
4
90
38
15
10
25
4
122
9
15
7
2
88
34
10
5
20
5
122
9
15
7
3
89
36
15
10
25
6
122
9
15
7
4
90
38
5
0
15
7
122
10
20
8
2
88
34
15
10
25
8
122
10
20
8
3
89
36
5
0
15
9
122
10
20
8
4
90
38
10
5
20
10
124
8
15
8
2
89
38
5
5
25
11
124
8
15
8
3
90
34
10
10
15
12
124
8
15
8
4
88
36
15
0
20
13
124
9
20
6
2
89
38
10
10
15
14
124
9
20
6
3
90
34
15
0
20
15
124
9
20
6
4
88
36
5
5
25
16
124
10
10
7
2
89
38
15
0
20
17
124
10
10
7
3
90
34
5
5
25
18
124
10
10
7
4
88
36
10
10
15
19
126
8
20
7
2
90
36
5
10
20
20
126
8
20
7
3
88
38
10
0
25
21
126
8
20
7
4
89
34
15
5
15
22
126
9
10
8
2
90
36
10
0
25
23
126
9
10
8
3
88
38
15
5
15
24
126
9
10
8
4
89
34
5
10
20
25
126
10
15
6
2
90
36
15
5
15
26
126
10
15
6
3
88
38
5
10
20
27
126
10
15
6
4
89
34
10
0
25


## 3.1. Matrix Analysis Model

An orthographic experiment was designed (ignoring the interaction for the first). The three-layer structural model based on data structure was established. The first layer in Table1 is the investigation index, the second layer is the factor layer, and the last one is the level layer. The data of each layer determine the definition of the matrix as follows.Table 1
Data structure of the multi-target orthogonal test.
First level
Experimental index
Second level
FactorA1
FactorA2
FactorAl
Third level
A
11
…
A
1
m
-
1
A
1
m
A
21
…
A
2
m
-
1
A
2
m
A
l
1
…
A
l
m
-
1
A
l
mDefinition 1.
The following is the matrix of the experimental investigation index. The study order isKij=kij. The matrix is established on the conditions that l factors exist in the experiment, each factor has m levels, kij is the average index of factor Ai at j level, and the investigation index is good when it is high. A low investigation index is considered good when the study establishes a matrix supposing Kij=1/kij.Definition 2.
The matrix of the factor layer:Ti=1/∑j=1mKij to build (2) matrix.Definition 3.
The matrix of the level layer: the range ofAi in the orthographic experiment is set as si. The study order is Si=si/∑i=1lsi. Equation (3) matrix is then established.Definition 4.
The weight matrix that may affect the test index:ωT=MTS. 									(1)M=[K1100⋯0K1200⋯0⋯⋯⋯⋯⋯K1m00⋯00K210⋯00K220⋯0⋯⋯⋯⋯⋯0K2m⋯⋯⋯⋯⋯⋯000⋯Kl1000⋯Kl2⋯⋯⋯⋯⋯000⋯Klm],(2)T=[T10000T200⋯⋯⋯⋯000Tl],(3)S=[S1S2⋯Sl],(4)ωT=[ω1,ω2,…,ωm].
In the matrix above mentioned,ω1=K11T1S1, K11T1 refers to the ratio of A1’s first level index to the sum of all A1’s level indexes. S1 is the ratio ofA1’s range to the sum of all ranges. The product of these two data reflects the effect of the first level of A1 on the index and range of A1. The other factors and levels are identical. The weight of each factor and level was obtained after calculation. The optimal plan and the factor order in the index were established based on the weight value.


## 3.2. Experimental Scheme

According to Euler equations, theoretical headHt is
(5)Ht=u2vu2-u1vu1g=u2g(u2h0-QtF2tanβ2),
where u1 (m/s) is peripheral speed at the inlet; vu1 is the peripheral velocity component of the blade inlet; vu1=0 in the straight cone suction chamber; u2 (m/s) refers to the circular velocity at the impeller outlet; g(m2/s) is acceleration because of gravity; h0=1-(π/z) is the stodala slip coefficient; z represents blade numbers; Qt(m3/h) is the theoretical flow; F2=πD2b2ψ2 is the cross-section area of the impeller outer; ψ2 is the blade expelling coefficient;b2(m)is the width of the blade outlet; β2 is the blade outlet angle. P′, the hydraulic power input, can be deduced by the following equation:
(6)P′=ρgHtQt=ρu22Qt(h0-Qtu2F2tanβ2).In (6), ρ (kg/m3) refers to water density. The equation shows that both theoretical head and hydraulic input power are determined by the impeller. However, rolling flow causes Pm, additional shaft power, as it enters the guiding blade channel. Hence, shaft power P=P′+Pm and pump efficiency η=P′/P. Pump shaft power and efficiency are determined by both the impeller and the guide vane in hydraulics.The following are the main geometric parameters of the model: width of the blade outletb2, suction diameter of impeller d1, inclination of the back cover board α2, inlet attack angle Δβ1, inlet angle in axial plane γ3, blade outlet angle β2, blade numberz, outer diameter of the back cover d2min, blade thickness S2, inlet angle of the guide vane β3, outlet angle of the guide vane α3, import width of the guide vane b3, blade number of the guide vane z1, axial length of the guide vane γ3, inlet diameter of the guide vane D2, and outlet diameter of the guide vane D1. α3 is equal to 90° to eliminate the rotation component. The last five factors were determined by the basic geometric parameters of the impeller. The first 10 geometric parameters were selected as testing factors.According to the specific speed equation,(7)ns=3.65nQH3/4,
where Q refers to the flow and H refers to the head of the pump. The design-specific speed of the pump is low, namely, 65.6, because it is lower than 80. Relative parameters can be obtained with the velocity-coefficient method under structure demand; these factors are shown in Table 2. Three levels were selected in each factor. The experimental scheme, shown in Table 3, is based on the L27(310) orthogonal table. In the 27 group tests, different numerals appear at the same frequency in any column, and any two-digit number appears equivalent in any row. After deciding the experimental plan, tests were implemented in numerical order without any arbitrary changes.Table 2
Factor level table.
Level
Factor
A
B
C
D
E
F
G
H
I
J
1
122
8
10
6
2
88
34
5
0
15
2
124
9
15
7
3
89
36
10
5
20
3
126
10
20
8
4
90
38
15
10
25
A—d2min/mm, B—b2/mm, C—β2/(°), D—z, E—S2/mm, F—α2/(°), G—d1/mm, H—Δβ1/(°), I—γ3/(°), J—β3/(°).Table 3
Orthogonal test schemes.
Test number
A/mm
B/mm
C/(°)
D/
E/mm
F/(°)
G/mm
H/(°)
I/(°)
J/(°)
1
122
8
10
6
2
88
34
5
0
15
2
122
8
10
6
3
89
36
10
5
20
3
122
8
10
6
4
90
38
15
10
25
4
122
9
15
7
2
88
34
10
5
20
5
122
9
15
7
3
89
36
15
10
25
6
122
9
15
7
4
90
38
5
0
15
7
122
10
20
8
2
88
34
15
10
25
8
122
10
20
8
3
89
36
5
0
15
9
122
10
20
8
4
90
38
10
5
20
10
124
8
15
8
2
89
38
5
5
25
11
124
8
15
8
3
90
34
10
10
15
12
124
8
15
8
4
88
36
15
0
20
13
124
9
20
6
2
89
38
10
10
15
14
124
9
20
6
3
90
34
15
0
20
15
124
9
20
6
4
88
36
5
5
25
16
124
10
10
7
2
89
38
15
0
20
17
124
10
10
7
3
90
34
5
5
25
18
124
10
10
7
4
88
36
10
10
15
19
126
8
20
7
2
90
36
5
10
20
20
126
8
20
7
3
88
38
10
0
25
21
126
8
20
7
4
89
34
15
5
15
22
126
9
10
8
2
90
36
10
0
25
23
126
9
10
8
3
88
38
15
5
15
24
126
9
10
8
4
89
34
5
10
20
25
126
10
15
6
2
90
36
15
5
15
26
126
10
15
6
3
88
38
5
10
20
27
126
10
15
6
4
89
34
10
0
25


## 4. Numerical Simulation

In traditional orthogonal experiments, research procedures involve manufacturing prototypes, performance tests, and results analyses. However, creating prototypes from the 27 groups of impellers and guide vanes would be a waste of time and money; moreover, a large number of prototype tests would inevitably cause significant manufacturing and test errors. The prediction of pump performance has become possible in engineering applications owing to the rapid development of computational fluid dynamics (CFD) [13]. Reasonable CFD calculations reflect the actual internal flow of the pump and accurately predict the pump head performance, efficiency, shaft power, and so forth at specific conditions [14, 15]. Therefore, the use of CFD technology is more feasible than the use of a prototype in pump design optimization and in the establishment of a preliminary forecast of pump performance.


### 4.1. Governing Equations and Boundary Conditions

The flow field information of pumps can be described by Navier-Stoke equations. A numerical simulation is performed to solve the following governing equations [16]:
(8)∂ρ∂t+∂∂xi(ρui)=0,∂(ρui)∂t+∂(ρuiuj)∂xj=-∂p∂xi+∂eij∂xj.Fluent 6.2 was utilized in this study. The 3D unsteady flow of centrifugal pumps was calculated based on Reynolds-averaged equations that resemble the standard renormalization groupk-ε turbulence model. Velocity, turbulent kinetic energy, and eddy viscosity coefficients were provided as a first-order upwind scheme. The velocity at the inlet and free flow at the outlet were selected with boundary conditions. A solid wall was set as the no-slip condition, and a smooth wall condition was employed as the near-wall function. The convergence precision was set to 10−5.


### 4.2. Computational Domain

The flow domain of the submersible pump is composed of the inlet, a multistage distorting impeller, a multistage space diffuser, and an outlet. Its flow pattern is more complex than that of a single-stage pump. A rotational flow exists at the impeller inlet, but the first stage and flow in the channel are similar. Demands on computer performance would be made if all stages are considered in the calculation. Therefore, selecting only the appropriate stages is important. Four kinds of computational domains in different stages are simulated for scheme1. The relevant results are presented in Tables 4, 5, and 6.Table 4
Head at design flow contrast of different stages (m).
Stage
1st head
2nd head
3rd head
4th head
Total head
1
16.93
16.56
2
16.89
16.53
32.17
3
16.87
16.62
16.77
48.87
4
16.79
16.60
16.92
16.84
64.50Table 5
Shaft power at design flow contrast of different stages (kW).
Stage
1st Shaft power
2nd Shaft power
3rd Shaft power
4th Shaft power
Total shaft power
1
0.853
0.853
2
0.855
0.856
1.711
3
0.857
0.854
0.851
2.562
4
0.851
0.854
0.858
0.861
3.424Table 6
Efficiency at design flow contrast of different stages (%).
Stage
1st efficiency
2nd efficiency
3rd efficiency
4th efficiency
Total efficiency
1
59.25
58.38
2
59.34
61.77
60.78
3
59.47
61.73
61.79
60.81
4
59.41
61.82
61.93
61.87
60.85Tables4 and 5 indicate that the head and shaft power of each stage in the pump are very similar that the two-stage values can be regarded as the corresponding values of the pump. Table 6 shows that only a few differences exist in terms of efficiency; however, many differences were noted between the first stage and other stages after the second. No changes were observed after stage two. In summary, it is appropriate to select two full-flow models to analyze pump performance. Similarly, the use of the head and shaft power of two stages as well as the efficiency of stage two to predict performance is also feasible. The computational domain composed of the inlet, the distorting impeller of two stages, the space diffusers, and the outlet is shown in Figure 2. The first pump water chamber is not shown in the figure.Figure 2
Schematic of the computational domain.


### 4.3. Model Meshing

The quantity and quality of the grid are two important factors that affect computation accuracy and duration. The unstructured tetrahedral mesh provided by GAMBIT, the professional software for grid generation, was utilized on the entire basin. The mesh has strong adaptability. Encryption and nonequidistance were processed in the near-wall region. Five different mesh sizes were selected to determine the appropriate number of grids and to conduct a grid-independent analysis. Scheme1at Qd was regarded as an example. The calculated results are shown in Table 7.Table 7
Results of different mesh sizes.
Parameter
Mesh size/mm
1.6
1.8
2.0
2.2
2.4
Grids/104
184
129
91
69.4
53.4
Head/m
32.58
32.33
32.17
32.03
31.87
Efficiencyη/%
61.82
61.81
61.77
61.46
60.53
Shaft PowerP/kW
1.716
1.706
1.711
1.694
1.686Table7 shows that the difference between pump head does not exceed 0.5%, efficiency does not exceed 0.06%, and shaft power does not exceed 0.3% when the mesh size is smaller than 2.0 mm. y+, which was utilized to examine the closest node to the surface, is not greater than 150. Thus, the criteria of standard renormalization group k-ε turbulence model calculation were met. Therefore, numerical simulation grid size of 2.0 mm was selected after considering computation accuracy and duration. The grid view of computational domain is shown in Figure 3.Figure 3
Grid view of computational domain.One of the convergence plots is shown in Figure4; this plot can be used to justify the ultimate choice of grid for the analysis. According to Figure 4, all the convergence precisions reached 10−5, which satisfied the requirement of the calculation.Figure 4
Convergence plots of the CFD calculation.


## 4.1. Governing Equations and Boundary Conditions

The flow field information of pumps can be described by Navier-Stoke equations. A numerical simulation is performed to solve the following governing equations [16]:
(8)∂ρ∂t+∂∂xi(ρui)=0,∂(ρui)∂t+∂(ρuiuj)∂xj=-∂p∂xi+∂eij∂xj.Fluent 6.2 was utilized in this study. The 3D unsteady flow of centrifugal pumps was calculated based on Reynolds-averaged equations that resemble the standard renormalization groupk-ε turbulence model. Velocity, turbulent kinetic energy, and eddy viscosity coefficients were provided as a first-order upwind scheme. The velocity at the inlet and free flow at the outlet were selected with boundary conditions. A solid wall was set as the no-slip condition, and a smooth wall condition was employed as the near-wall function. The convergence precision was set to 10−5.


## 4.2. Computational Domain

The flow domain of the submersible pump is composed of the inlet, a multistage distorting impeller, a multistage space diffuser, and an outlet. Its flow pattern is more complex than that of a single-stage pump. A rotational flow exists at the impeller inlet, but the first stage and flow in the channel are similar. Demands on computer performance would be made if all stages are considered in the calculation. Therefore, selecting only the appropriate stages is important. Four kinds of computational domains in different stages are simulated for scheme1. The relevant results are presented in Tables 4, 5, and 6.Table 4
Head at design flow contrast of different stages (m).
Stage
1st head
2nd head
3rd head
4th head
Total head
1
16.93
16.56
2
16.89
16.53
32.17
3
16.87
16.62
16.77
48.87
4
16.79
16.60
16.92
16.84
64.50Table 5
Shaft power at design flow contrast of different stages (kW).
Stage
1st Shaft power
2nd Shaft power
3rd Shaft power
4th Shaft power
Total shaft power
1
0.853
0.853
2
0.855
0.856
1.711
3
0.857
0.854
0.851
2.562
4
0.851
0.854
0.858
0.861
3.424Table 6
Efficiency at design flow contrast of different stages (%).
Stage
1st efficiency
2nd efficiency
3rd efficiency
4th efficiency
Total efficiency
1
59.25
58.38
2
59.34
61.77
60.78
3
59.47
61.73
61.79
60.81
4
59.41
61.82
61.93
61.87
60.85Tables4 and 5 indicate that the head and shaft power of each stage in the pump are very similar that the two-stage values can be regarded as the corresponding values of the pump. Table 6 shows that only a few differences exist in terms of efficiency; however, many differences were noted between the first stage and other stages after the second. No changes were observed after stage two. In summary, it is appropriate to select two full-flow models to analyze pump performance. Similarly, the use of the head and shaft power of two stages as well as the efficiency of stage two to predict performance is also feasible. The computational domain composed of the inlet, the distorting impeller of two stages, the space diffusers, and the outlet is shown in Figure 2. The first pump water chamber is not shown in the figure.Figure 2
Schematic of the computational domain.


## 4.3. Model Meshing

The quantity and quality of the grid are two important factors that affect computation accuracy and duration. The unstructured tetrahedral mesh provided by GAMBIT, the professional software for grid generation, was utilized on the entire basin. The mesh has strong adaptability. Encryption and nonequidistance were processed in the near-wall region. Five different mesh sizes were selected to determine the appropriate number of grids and to conduct a grid-independent analysis. Scheme1at Qd was regarded as an example. The calculated results are shown in Table 7.Table 7
Results of different mesh sizes.
Parameter
Mesh size/mm
1.6
1.8
2.0
2.2
2.4
Grids/104
184
129
91
69.4
53.4
Head/m
32.58
32.33
32.17
32.03
31.87
Efficiencyη/%
61.82
61.81
61.77
61.46
60.53
Shaft PowerP/kW
1.716
1.706
1.711
1.694
1.686Table7 shows that the difference between pump head does not exceed 0.5%, efficiency does not exceed 0.06%, and shaft power does not exceed 0.3% when the mesh size is smaller than 2.0 mm. y+, which was utilized to examine the closest node to the surface, is not greater than 150. Thus, the criteria of standard renormalization group k-ε turbulence model calculation were met. Therefore, numerical simulation grid size of 2.0 mm was selected after considering computation accuracy and duration. The grid view of computational domain is shown in Figure 3.Figure 3
Grid view of computational domain.One of the convergence plots is shown in Figure4; this plot can be used to justify the ultimate choice of grid for the analysis. According to Figure 4, all the convergence precisions reached 10−5, which satisfied the requirement of the calculation.Figure 4
Convergence plots of the CFD calculation.


## 5. Results Analysis

Given that the shaft power of low-specific-speed centrifugal pumps increases rapidly with the enlargement of flow, the shaft power at1.5Qd was selected as the monitoring value for overload judging. An electric pump works safely at the range of 0.7Qd to 1.3Qd. If the shaft power at 1.5Qd is less than the motor power, then the pump will not experience an overload. A total of 27 numerical simulations were performed. The results of two-stage head H under the rated condition, shaft power P at 1.5Qd, and efficiency η of stage two under the rated condition are shown in Table 8.Table 8
Simulation statistics.
Test number
H/m
η/%
P/kW
1
32.17
61.77
1.711
2
32.81
63.41
1.669
3
30.25
60.45
1.561
4
38.08
62.08
2.108
5
34.97
61.05
1.983
6
35.22
60.84
2.01
7
35.94
57.41
2.17
8
36.78
59.11
2.212
9
38.56
61.02
2.242
10
34.88
59.12
2.054
11
35.04
60.10
2.006
12
35.64
61.58
1.987
13
35.80
61.08
2.064
14
36.89
61.21
2.097
15
35.04
60.12
2.064
16
37.12
62.32
2.029
17
34.50
59.28
1.989
18
35.72
61.13
2.005
19
38.01
61.73
2.151
20
35.42
60.03
2.108
21
35.36
60.21
2.092
22
34.16
59.62
1.924
23
33.78
60.64
1.978
24
34.24
60.90
1.831
25
35.89
60.52
2.080
26
37.45
61.88
2.110
27
35.30
59.59
2.069
H (m) is two-stage head under Q; η (%) is the efficiency of stage two under Q; P (kW) is two-stage shaft power at 1.5Qd.


### 5.1. Visual Analysis

Orthographic experimental data were analyzed. With the first column of rated point head indexH as an example,
(9)k1A=(32.17+32.81+30.25+38.08+34.97+35.22+35.94+36.78+38.56)×(9)-1=34.977,k2A=(34.88+35.04+35.64+35.80+36.89+35.04+37.12+34.50+35.72)×(9)-1=35.627,k3A=(38.01+35.42+35.36+34.16+33.78+34.24+35.89+37.45+35.30)×(9)-1=35.513.
Where k1A, k2A and k3A in the equations stand for the average efficiency values at the rated point of factor A.s, the range, is introduced to reflect the effect of various factors on the target value. A large range signifies an important factor whose change greatly affects the objective. If the range is small, then the factor has minimal influence and is usually not important.swas obtained by the subtraction of the minimum from the maximum among k1A, k2A,  and k3A in each column. The calculation methods for the other nine columns were similar to the first one. The efficiency under the rated condition and shaft power at 1.5Qd were obtained. k1, k2, k3,  and s of each factor are listed in Table 9.Table 9
Range analysis of orthogonal test data.
Index
A
B
C
D
E
F
G
H
I
J
H/m
k
1
34.977
34.401
33.862
34.625
35.784
35.473
35.281
35.366
35.412
35.087
k
2
35.627
35.355
35.831
36.045
35.295
35.252
35.447
35.656
35.434
36.535
k
3
35.513
36.362
36.424
35.447
35.038
35.393
35.389
35.096
35.271
34.495
s
0.650
1.961
2.562
1.420
0.746
0.221
0.167
0.560
0.163
2.040
η/%
k
1
60.795
60.935
61.059
61.116
60.630
60.739
60.285
60.530
60.678
60.600
k
2
60.661
60.838
60.754
60.965
60.747
60.755
60.920
60.897
60.713
61.795
k
3
60.571
60.254
60.214
59.946
60.650
60.532
60.822
60.601
60.636
59.632
s
0.224
0.681
0.844
1.170
0.117
0.223
0.635
0.367
0.077
2.163
P/kW
k
1
1.963
1.927
1.855
1.936
2.033
2.027
2.009
2.015
2.017
2.018
k
2
2.033
2.007
2.045
2.053
2.017
2.000
2.008
2.022
2.031
2.025
k
3
2.038
2.101
2.134
2.045
1.985
2.007
2.017
1.998
1.987
1.992
s
0.075
0.174
0.279
0.116
0.048
0.027
0.009
0.024
0.043
0.033Visual analysis indicates that the order of the importance of the index for the head isCJBDEAHFGI, and its optimal plan is A2B3C3B2E1F1G2H2I2J2. The order of importance of the index for efficiency is JDCBGHAFEI, and its optimal plan is A1B1C1D1E2F2G2H2I2J2. Similarly, the order of importance for shaft power isCBDAEIJFHG, and its optimal plan is A1B1C1D1E3F2G2H3I3J3. The integrated balance method even drawing was utilized to select the optimal program, resulting in large amounts of calculation and difficulty in establishing choices for multitarget optimization. If the matrix method introduced in this study is selected, an optimal plan can be immediately established because only three investigation indexes are calculated. The optimal program is determined by the weight value.


### 5.2. Weight Matrix Method

Weight matrix analysis is necessary to calculate the weight matrix of the investigation index. The weight calculation of the head was regarded as an example; high values were considered good. Thus,Kij=kij, Ti=1/∑j=1mKij, Si=si/∑i=1lsi, and ω=M·T·S.In general, higher efficiency is also considered good. Hence,Kij=kij, Ti=1/∑j=1mKij, Si=si/∑i=1lsi. Furthermore, low shaft power is desirable. Thus, Kij=1/kij, Ti=1/∑j=1mKij, and Si=si/∑i=1lsi. The general weight matrix in (14) is obtained after substituting the above calculations into (4) as follows:
(10)M1=(34.97735.62735.51334.40135.35536.362⋯⋯⋯⋯35.41235.43435.27135.08736.53534.495),(11)T1=(0.00942⋯0.00942),(12)S1=(0.06200.18700.24420.13530.07110.02110.015870.053420.015590.1945),(13)ω1=M1·T1·S1,(14)ω=(ω1+ω2+ω3)3=([0.02040.02080.02070.06060.06230.06410.07790.08250.08380.04420.0460.04520.0240.02370.02350.0070.0070.0070.00530.00530.00530.01780.0180.01770.00520.00520.00520.06430.06690.0632]+[0.01150.01150.01150.03510.0350.03470.04360.04330.0430.06040.06030.05930.0060.0060.0060.01150.01150.01140.03230.03270.03260.01880.01890.01880.00390.00390.00390.11070.11290.109]+[0.01510.01460.01460.0350.03360.03210.05730.0520.04980.02380.02240.02250.00970.00970.00990.00530.00540.00540.00180.00180.00180.00480.00480.00480.00870.00870.00880.00660.00660.0067])×(3)-1=[0.0156670.0156330.01560.0435670.0436330.0436230.05960.0592670.0588670.04280.04290.0423330.0132330.0131320.0131330.0079330.0079670.0079330.0131330.0132670.0132330.01380.01390.0137670.0059330.0059320.0059670.0605330.0621330.059633]=[A1A2A3B1B2B3C1C2C3D1D2D3E1E2E3F1F2F3G1G2G3H1H2H3I1I2I3J1J2J3].The calculations provide the weight of the result of each factor with three levels. Hence, the optimal plan for orthographic experiments is determined quickly. The optimal plan isA1B2C2D2E1F2G2H2I3J2, and the importance order of the index is JDCGHEABFI.


## 5.1. Visual Analysis

Orthographic experimental data were analyzed. With the first column of rated point head indexH as an example,
(9)k1A=(32.17+32.81+30.25+38.08+34.97+35.22+35.94+36.78+38.56)×(9)-1=34.977,k2A=(34.88+35.04+35.64+35.80+36.89+35.04+37.12+34.50+35.72)×(9)-1=35.627,k3A=(38.01+35.42+35.36+34.16+33.78+34.24+35.89+37.45+35.30)×(9)-1=35.513.
Where k1A, k2A and k3A in the equations stand for the average efficiency values at the rated point of factor A.s, the range, is introduced to reflect the effect of various factors on the target value. A large range signifies an important factor whose change greatly affects the objective. If the range is small, then the factor has minimal influence and is usually not important.swas obtained by the subtraction of the minimum from the maximum among k1A, k2A,  and k3A in each column. The calculation methods for the other nine columns were similar to the first one. The efficiency under the rated condition and shaft power at 1.5Qd were obtained. k1, k2, k3,  and s of each factor are listed in Table 9.Table 9
Range analysis of orthogonal test data.
Index
A
B
C
D
E
F
G
H
I
J
H/m
k
1
34.977
34.401
33.862
34.625
35.784
35.473
35.281
35.366
35.412
35.087
k
2
35.627
35.355
35.831
36.045
35.295
35.252
35.447
35.656
35.434
36.535
k
3
35.513
36.362
36.424
35.447
35.038
35.393
35.389
35.096
35.271
34.495
s
0.650
1.961
2.562
1.420
0.746
0.221
0.167
0.560
0.163
2.040
η/%
k
1
60.795
60.935
61.059
61.116
60.630
60.739
60.285
60.530
60.678
60.600
k
2
60.661
60.838
60.754
60.965
60.747
60.755
60.920
60.897
60.713
61.795
k
3
60.571
60.254
60.214
59.946
60.650
60.532
60.822
60.601
60.636
59.632
s
0.224
0.681
0.844
1.170
0.117
0.223
0.635
0.367
0.077
2.163
P/kW
k
1
1.963
1.927
1.855
1.936
2.033
2.027
2.009
2.015
2.017
2.018
k
2
2.033
2.007
2.045
2.053
2.017
2.000
2.008
2.022
2.031
2.025
k
3
2.038
2.101
2.134
2.045
1.985
2.007
2.017
1.998
1.987
1.992
s
0.075
0.174
0.279
0.116
0.048
0.027
0.009
0.024
0.043
0.033Visual analysis indicates that the order of the importance of the index for the head isCJBDEAHFGI, and its optimal plan is A2B3C3B2E1F1G2H2I2J2. The order of importance of the index for efficiency is JDCBGHAFEI, and its optimal plan is A1B1C1D1E2F2G2H2I2J2. Similarly, the order of importance for shaft power isCBDAEIJFHG, and its optimal plan is A1B1C1D1E3F2G2H3I3J3. The integrated balance method even drawing was utilized to select the optimal program, resulting in large amounts of calculation and difficulty in establishing choices for multitarget optimization. If the matrix method introduced in this study is selected, an optimal plan can be immediately established because only three investigation indexes are calculated. The optimal program is determined by the weight value.


## 5.2. Weight Matrix Method

Weight matrix analysis is necessary to calculate the weight matrix of the investigation index. The weight calculation of the head was regarded as an example; high values were considered good. Thus,Kij=kij, Ti=1/∑j=1mKij, Si=si/∑i=1lsi, and ω=M·T·S.In general, higher efficiency is also considered good. Hence,Kij=kij, Ti=1/∑j=1mKij, Si=si/∑i=1lsi. Furthermore, low shaft power is desirable. Thus, Kij=1/kij, Ti=1/∑j=1mKij, and Si=si/∑i=1lsi. The general weight matrix in (14) is obtained after substituting the above calculations into (4) as follows:
(10)M1=(34.97735.62735.51334.40135.35536.362⋯⋯⋯⋯35.41235.43435.27135.08736.53534.495),(11)T1=(0.00942⋯0.00942),(12)S1=(0.06200.18700.24420.13530.07110.02110.015870.053420.015590.1945),(13)ω1=M1·T1·S1,(14)ω=(ω1+ω2+ω3)3=([0.02040.02080.02070.06060.06230.06410.07790.08250.08380.04420.0460.04520.0240.02370.02350.0070.0070.0070.00530.00530.00530.01780.0180.01770.00520.00520.00520.06430.06690.0632]+[0.01150.01150.01150.03510.0350.03470.04360.04330.0430.06040.06030.05930.0060.0060.0060.01150.01150.01140.03230.03270.03260.01880.01890.01880.00390.00390.00390.11070.11290.109]+[0.01510.01460.01460.0350.03360.03210.05730.0520.04980.02380.02240.02250.00970.00970.00990.00530.00540.00540.00180.00180.00180.00480.00480.00480.00870.00870.00880.00660.00660.0067])×(3)-1=[0.0156670.0156330.01560.0435670.0436330.0436230.05960.0592670.0588670.04280.04290.0423330.0132330.0131320.0131330.0079330.0079670.0079330.0131330.0132670.0132330.01380.01390.0137670.0059330.0059320.0059670.0605330.0621330.059633]=[A1A2A3B1B2B3C1C2C3D1D2D3E1E2E3F1F2F3G1G2G3H1H2H3I1I2I3J1J2J3].The calculations provide the weight of the result of each factor with three levels. Hence, the optimal plan for orthographic experiments is determined quickly. The optimal plan isA1B2C2D2E1F2G2H2I3J2, and the importance order of the index is JDCGHEABFI.


## 6. Experimental Verification

The optimal model,J2D2C2G2H2E1A1B2F2I3, was prototyped for the experiment to test the optimized programs. The geometric parameters of the hydraulic model were as follows: width of the blade outlet b2=9mm, inclination of back cover board α2=89°, inlet attack angel Δβ1=10°, inlet angle in axial plane γ3=10°, blade outlet angle β2=15°, blades number z=7, outer diameter of back cover d2min=122mm, blade thickness S2=2mm, inlet angle of guide vane β3=20°, and axial length of the guide vane γ3=10°. The test was performed in accordance with the national standards GB/T 12785-2002 of China. Figure 5 presents the results of the experiment and the entire flow field simulation.Contrast diagram between the simulation and experiment.
(a)
Head flow curve
(b)
Efficiency flow curve
(c)
Shaft power flow curveComparison of the simulation and experimental results indicates that the simulated head and efficiency are slightly higher than those of the experiment; however, the simulated power is slightly lower. Error analysis indicates the following:(1)the numerical simulation did not consider the loss of leakage at the oral ring and between stages;(2)small differences exist because of limitations imposed by casting accuracy, casting errors, casting model (particularly the impeller), and simulated model; these differences can cause disparity between experimental and simulated values. However, the error is less than 5%, and both experimental and simulated values exhibit similar trends when the flow changes. Hence, the simulation methods employed in this study can meet the demand of practical usage in programs for design optimization.The efficiency of QS10-68/4 based on the national standard of China is 51%. After optimization, the pump’s rated head is 68.9 m and rated efficiency is 58.61%, which is approximately 7% higher than the national standard. The maximum shaft power is 3.83 kW. The performance of nonoverload also meets the demand, which is 4 kW. All of the above data indicate that optimization was successful.


## 7. Conclusions

The influence of 10 factors and three levels on the head, efficiency, and shaft power was analyzed by conducting 27 orthographic experiments through numerical simulation. Weight matrix analysis was then performed to determine the optimal model. The principal conclusions drawn from this research are as follows.(1)
The simulation results indicate that the inlet setting angle of the guide vaneβ3 significantly affects the efficiency and that the outlet angle of blades β2 has a significant effect on the head and shaft power.(2)
The importance order of the index for the head isβ2β3b2zS2d2minΔβ1α2d1γ3, and the geometric parameters of optimal plan are as follows: d2min=124mm, b2=10mm, β2=20°, z=7, S2=2mm, α2=88°, d1=36mm, Δβ1=10°, γ3=5°, and β3=20°. The importance order of the index for efficiency is β3zβ2b2d1Δβ1d2minα2S2γ3, and the geometric parameters of optimal plan are as follows: d2min=122mm, b2=8mm, β2=10°, z=6, S2=3mm, α2=89°, d1=36mm, Δβ1=10°, γ3=5°, and β3=20°. For shaft power, the importance order of the index is β2b2zd2minS2γ3β3α2Δβ1d1, and the geometric parameters of optimal plan are as follows: d2min=122mm, b2=8mm, β2=10°, z=6, S2=4mm, α2=89°, d1=36mm, Δβ1=15°, γ3=10°, and β3=25°.(3)
By performing weight matrix analysis, optimal model was obtained based on the influence of factors and levels on the head, efficiency, and shaft power. The geometric parameters of the hydraulic model were as follows:b2=9mm, α2=89°, Δβ1=10°, γ3=10°, β2=15°, z=7, d2min=122mm, S2=2mm, β3=20°, and γ3=10°. The performance of the prototype in the test indicates that efficiency at the rated point is 7% higher than the national standard of China. This result verifies the potential use of the orthographic design combined matrix analysis with numerical simulation in pump optimization.(4)
The changes in the head, efficiency, and shaft power determined from the simulation are in accordance with the test results. The error between numerical simulation and test result is less than 5% in the rated flow, which further verifies the possibility of forecasting the performance of multistage pumps through numerical simulation.



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*Source: 136195-2013-12-10.xml*