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

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 an 𝐿₂₇(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.

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 Figure 1. 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 𝑄_𝑑=10 m³/h, total head 𝐻_𝑑=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 Table 1 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.

Definition 1. The following is the matrix of the experimental investigation index. The study order is 𝐾_𝑖𝑗=𝑘_𝑖𝑗. The matrix is established on the conditions that 𝑙 factors exist in the experiment, each factor has 𝑚 levels, 𝑘_𝑖𝑗 is the average index of factor 𝐴_𝑖 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 𝐾_𝑖𝑗=1/𝑘_𝑖𝑗.

Definition 2. The matrix of the factor layer: 𝑇_𝑖=1/∑^𝑚_𝑗=1𝐾_𝑖𝑗 to build (2) matrix.

Definition 3. The matrix of the level layer: the range of 𝐴_𝑖 in the orthographic experiment is set as 𝑠_𝑖. The study order is 𝑆_𝑖=𝑠_𝑖/∑^𝑙_𝑖=1𝑠_𝑖. Equation (3) matrix is then established.

Definition 4. The weight matrix that may affect the test index:𝜔^𝑇=𝑀𝑇𝑆. 𝐾𝑀=¹¹𝐾00⋯0₁₂𝐾00⋯0⋯⋯⋯⋯⋯_1𝑚00⋯00𝐾₂₁0⋯00𝐾₂₂0⋯0⋯⋯⋯⋯⋯0𝐾_2𝑚⋯⋯⋯⋯⋯⋯000⋯𝐾_𝑙1000⋯𝐾_𝑙2⋯⋯⋯⋯⋯000⋯𝐾_𝑙𝑚𝑇,𝑇=¹0000𝑇₂00⋯⋯⋯⋯000𝑇_𝑙𝑆,𝑆=¹𝑆₂⋯𝑆_𝑙𝜔_𝑇,=𝜔₁,𝜔₂,…,𝜔_𝑚. (1)(2)(3)(4)
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 head 𝐻_𝑡 is 𝐻_𝑡=𝑢₂𝑣_𝑢2−𝑢₁𝑣_𝑢1𝑔=𝑢₂𝑔𝑢₂ℎ₀−𝑄_𝑡𝐹₂tan𝛽₂, (5) where 𝑢₁ (m/s) is peripheral speed at the inlet; 𝑣_𝑢1 is the peripheral velocity component of the blade inlet; 𝑣_𝑢1=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; 𝑄_𝑡(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: 𝑃^′=𝜌𝑔𝐻_𝑡𝑄_𝑡=𝜌𝑢₂²𝑄_𝑡ℎ₀−𝑄_𝑡𝑢₂𝐹₂tan𝛽₂. (6)

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 𝑃_𝑚, additional shaft power, as it enters the guiding blade channel. Hence, shaft power 𝑃=𝑃^′+𝑃_𝑚 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 outlet 𝑏₂, 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 𝑑_2min, 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, 𝑛_𝑠=3.65𝑛𝑄𝐻^3/4, (7) 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.

Table 3 

Orthogonal test schemes.

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]: 𝜕𝜌+𝜕𝜕𝑡𝜕𝑥_𝑖𝜌𝑢_𝑖𝜕=0,𝜌𝑢_𝑖+𝜕𝜕𝑡𝜌𝑢_𝑖𝑢_𝑗𝜕𝑥_𝑗=−𝜕𝑝𝜕𝑥_𝑖+𝜕𝑒_𝑖𝑗𝜕𝑥_𝑗. (8)

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 group 𝑘−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).

Table 5 

Shaft power at design flow contrast of different stages (kW).

Table 6 

Efficiency at design flow contrast of different stages (%).

Tables 4 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 𝑄_𝑑 was regarded as an example. The calculated results are shown in Table 7.

Table 7 

Results of different mesh sizes.

Table 7 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 Figure 4; 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 at 1.5𝑄_𝑑 was selected as the monitoring value for overload judging. An electric pump works safely at the range of 0.7𝑄_𝑑 to 1.3𝑄_𝑑. If the shaft power at 1.5𝑄_𝑑 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 1.5𝑄_𝑑, and efficiency 𝜂 of stage two under the rated condition are shown in Table 8.

Table 8 

Simulation statistics.

5.1. Visual Analysis

Orthographic experimental data were analyzed. With the first column of rated point head index 𝐻 as an example, 𝑘_1𝐴=()×(9)32.17+32.81+30.25+38.08+34.97+35.22+35.94+36.78+38.56₋₁𝑘=34.977,_2𝐴=()×(9)34.88+35.04+35.64+35.80+36.89+35.04+37.12+34.50+35.72₋₁𝑘=35.627,_3𝐴=()×(9)38.01+35.42+35.36+34.16+33.78+34.24+35.89+37.45+35.30₋₁=35.513. (9) Where 𝑘_1𝐴, 𝑘_2𝐴 and 𝑘_3𝐴 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 𝑘_1𝐴, 𝑘_2𝐴,  and 𝑘_3𝐴 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 1.5𝑄_𝑑 were obtained. 𝑘₁, 𝑘₂, 𝑘₃,  and 𝑠 of each factor are listed in Table 9.

Table 9 

Range analysis of orthogonal test data.

Visual analysis indicates that the order of the importance of the index for the head is 𝐶𝐽𝐵𝐷𝐸𝐴𝐻𝐹𝐺𝐼, 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, 𝐾_𝑖𝑗=𝑘_𝑖𝑗, 𝑇_𝑖=1/∑^𝑚_𝑗=1𝐾_𝑖𝑗, 𝑆_𝑖=𝑠_𝑖/∑^𝑙_𝑖=1𝑠_𝑖, and 𝜔=𝑀⋅𝑇⋅𝑆.

In general, higher efficiency is also considered good. Hence, 𝐾_𝑖𝑗=𝑘_𝑖𝑗, 𝑇_𝑖=1/∑^𝑚_𝑗=1𝐾_𝑖𝑗, 𝑆_𝑖=𝑠_𝑖/∑^𝑙_𝑖=1𝑠_𝑖. Furthermore, low shaft power is desirable. Thus, 𝐾_𝑖𝑗=1/𝑘_𝑖𝑗, 𝑇_𝑖=1/∑^𝑚_𝑗=1𝐾_𝑖𝑗, and 𝑆_𝑖=𝑠_𝑖/∑^𝑙_𝑖=1𝑠_𝑖. The general weight matrix in (14) is obtained after substituting the above calculations into (4) as follows: 𝑀₁=𝑇34.97735.62735.51334.40135.35536.362⋯⋯⋯⋯35.41235.43435.27135.08736.53534.495₁,=⋯𝑆0.009420.00942₁,=𝜔0.06200.18700.24420.13530.07110.02110.015870.053420.015590.1945₁,=𝑀₁⋅𝑇₁⋅𝑆₁, (10)(11)(12)(13)𝜔𝜔=¹+𝜔₂+𝜔₃3=++×(3)0.02040.02080.02070.06060.06230.06410.07790.08250.08380.04420.0460.04520.0240.02370.02350.0070.0070.0070.00530.00530.00530.01780.0180.01770.00520.00520.00520.06430.06690.06320.01150.01150.01150.03510.0350.03470.04360.04330.0430.06040.06030.05930.0060.0060.0060.01150.01150.01140.03230.03270.03260.01880.01890.01880.00390.00390.00390.11070.11290.1090.01510.01460.01460.0350.03360.03210.05730.0520.04980.02380.02240.02250.00970.00970.00990.00530.00540.00540.00180.00180.00180.00480.00480.00480.00870.00870.00880.00660.00660.0067⁻¹==0.015667𝐴0.0156330.01560.0435670.0436330.0436230.05960.0592670.0588670.04280.04290.0423330.0132330.0131320.0131330.0079330.0079670.0079330.0131330.0132670.0132330.01380.01390.0137670.0059330.0059320.0059670.0605330.0621330.059633¹𝐴₂𝐴₃𝐵₁𝐵₂𝐵₃𝐶₁𝐶₂𝐶₃𝐷₁𝐷₂𝐷₃𝐸₁𝐸₂𝐸₃𝐹₁𝐹₂𝐹₃𝐺₁𝐺₂𝐺₃𝐻₁𝐻₂𝐻₃𝐼₁𝐼₂𝐼₃𝐽₁𝐽₂𝐽₃. (14)

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 is 𝐴₁𝐵₂𝐶₂𝐷₂𝐸₁𝐹₂𝐺₂𝐻₂𝐼₃𝐽₂, 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 𝑑_2min=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.

(a) Head flow curve

(b) Efficiency flow curve

(c) Shaft power flow curve

Figure 5 

Contrast diagram between the simulation and experiment.

Comparison 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 vane 𝛽₃ 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 is 𝛽₂𝛽₃𝑏₂𝑧𝑆₂𝑑_2minΔ𝛽₁𝛼₂𝑑₁𝛾₃, and the geometric parameters of optimal plan are as follows: 𝑑_2min=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 𝛽₃𝑧𝛽₂𝑏₂𝑑₁Δ𝛽₁𝑑_2min𝛼₂𝑆₂𝛾₃, and the geometric parameters of optimal plan are as follows: 𝑑_2min=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 𝛽₂𝑏₂𝑧𝑑_2min𝑆₂𝛾₃𝛽₃𝛼₂Δ𝛽₁𝑑₁, and the geometric parameters of optimal plan are as follows: 𝑑_2min=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, 𝑑_2min=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.

Acknowledgments

This work was financially supported by the State Key Program of National Natural Science of China (Grant no. 51239005) and the National Science & Technology Pillar Program (Grant no. 2011BAF14B04) of China. This research was also sponsored by the Research and Innovation Project for College Graduates of Jiangsu Province (CXZZ12_0679).