Multi-Parameter Optimization Design of Axial-Flow Pump Based on Orthogonal Method

Abstract

The axial-flow pump is a widely used piece of general machinery which consumes large amounts of energy. In this study, an axial-flow pump with the specific speed of 536 is firstly designed and experimentally measured; then, the orthogonal method is employed to conduct the energy performance optimization. Five optimization parameters, including hub control point, hub stagger angle, shroud stagger angle, camber angle and centroid position were set with four levels. Sixteen individual pumps were designed according to the orthogonal method; then, a numerical simulation was implemented to obtain the energy performance and flow pattern. Results showed that the shroud stagger angle has the maximum influence on the pump head and efficiency, and the hub stagger angle and camber angle are also very important. At a design point of flow rate 70 kg/s, the efficiency of the optimal pump is 86.29%, which improved by 2.05% in comparison with the baseline pump. The pressure gradient of the optimal pump from blade inlet to outlet becomes more fluent than that of baseline pump, and the low-velocity region of the optimal pump at the blade head shrinks, compared to that of the baseline pump.

1. Introduction

Axial-flow pumps are widely used in agricultural irrigation, the aerospace industry, marine power, hydraulic engineering and other fields, due to their characteristics of high flow, low head and wide operation range [1]. Axial-flow pumps consume large amounts of energy, especially in works on a low design level and off design conditions; therefore,, it is of great benefit to optimize the hydraulic performance of the axial-flow pump for energy conservation.

Many investigations have been carried out to improve the hydraulic performance of the axial-flow pump. Cao et al. [2,3,4,5], proposed and developed a design method of a mixed-flow pump, based on fourth-order polynomial velocity moment distribution function and studied the influence of several parameters related to a polynomial on blade shape and pump energy performance. Zangeneh et al. [6,7,8,9] focused on the parameters of blade loading distribution and studied the correlation between load distribution, pump efficiency and cavitation performance. Tan et al. [10,11] proposed an impeller design method based on direct and inverse problem iteration, which simultaneously solves the continuity and motion equations of the fluid and shapes the blade geometry.

Chakraborty et al. [12] calculated the internal flow of a centrifugal pump and pointed out that the efficiency increased with the number of blades. Tan et al. [13] found that the flow of the maximum swap pump is fluent in passage, and the head and efficiency are also higher than other pumps. Bin et al. [14] studied the effects of different blade numbers on impellers and guide vanes on the performance of a pump, and suggested that the number of guide vane blades should be large than that of impellers, which converts kinetic energy into pressure energy and eliminates circulation. Zhao et al. [15] found that the appropriate blade placement angle could reduce the vortex in flow passage and increase efficiency. Shi et al. [16] studied six different guide-vane sweeping schemes and suggested that the guide vane of forward sweep 16° had a better hydraulic performance. Liu et al. [17,18] studied the influence of different blade tip clearance on pump performance and found that, with the increase in tip clearance size, the head and efficiency of the pump decreased. The above researchers focus on the single parameter, while the influence of multi-parameters is complex and mutual. Therefore, multi-parameter optimization is critical to further improve the pump performance.

With the development of computer technology, multi-parameter optimization becomes possible, which is widely used in pump blade design. Liu et al. [19] optimized pumps by employing the backpropagation (BP) neural network and radial basis function (RBF) neural network. Both BP and RBF neural networks can predict the data points in the design space; the maximum relative errors were 0.571% of efficiency and 0.411% of head, respectively. Zhang et al. [20] used the RBF neural network and Latin hypercube sampling (LHS) to set up a network and the final efficiency was improved by 3.87%. Zeng Cheng [21] combined the RBF neural network and NSGA-Ⅱ to establish the mapping relationship and search for the best design parameters automatically, and found that it was necessary to consider the importance of various indicators and select parameters in the design process. The corresponding surface method [22,23,24], simulated annealing algorithm [25], particle swarm optimization [26] and other intelligent algorithms were also used to solve multi-parameter optimization with satisfying results. The above experiments usually establish a response relationship between parameters and performance, which should calculate the performance of many pump samples and consume a large amount of computational resources.

The orthogonal method was employed to study the influence of multi-parameters with less calculation resources. On the basis of the orthogonal method, researchers can quickly determine test samples after selecting the factors and levels. Xu et al. [27] established a five-factor and four-level orthogonal table and pointed out that the trailing blade angle and blade leading edge position have the maximum impact on efficiency and cavitation performance. By choosing appropriate parameters, the pump efficiency increases 3.09% and pump cavitation decreases 1.45 m. Liu et al. [28] chose five pump factors; a hub leading angle; a shroud leading angle; a shroud trailing angle; and control coefficients at the hub and the shroud, effectively improving the pump pressure rise by 12.8 kPa at the design flow rate. Fan et al. [29] optimized a mixed-flow fan through the orthogonal method and greatly reduced the flow separation and tip leakage flow. Yuan et al. [30] took the weighted average efficiency around design flow as the optimization objective, and the efficiency of the optimal pump as 4.68% higher than that of baseline pump. Zheng et al. [31] designed an orthogonal experiment with four factors and three levels, and studied the influence of blade number, foil, hub ratio and distance between impeller and guide vane on performance and stability of the axial-flow pump. The results found that the hub ratio had the greatest impact. Many researchers [32,33,34,35,36] have designed impellers with excellent hydraulic performance by using the orthogonal optimization method, which proves the efficiency of the orthogonal method.

In this work, the baseline axial-flow pump is first designed and experimentally measured. Five parameters, (including hub control point, hub foil stagger angle, shroud foil stagger angle, camber angle and hub foil centroid position) and four levels are chosen to establish the orthogonal table; then, sixteen orthogonal tests with corresponding parameters are calculated. Finally, the optimal pump is determined after the orthogonal analysis. The performance of the optimal pump is better than the baseline one, especially at a low flow rate and around the design point. The inner flow between the baseline pump and the optimal pump at different flow rates is analyzed.

2. Experimental Platform

The baseline axial-flow pump consisted of a six-blade impeller and a seven-blade guide vane. The impeller was designed by our laboratory, with a diameter of 180 mm. The rated flow rate Q_d and rated head H_d of the pump were 0.07 m³/s and 3.6 m, respectively, and the rotational speed was 1450 r/min. The nominal specific speed of the pump was 536.

The test platform of the axial-flow pump was a closed-loop rig located at Tsinghua University, as shown in Figure 1. In the test rig, an electromagnetic flow meter with a measuring range of 0–635.85 m³/h and precision of 0.5% was used to measure the flow rate. Two pressure sensors with a measuring range of 0–0.6 MPa and precision of 0.25% were used to measure the pressure at the inlet and outlet of the tested axial-flow pump. A torque meter with a measuring range of 0–100 N·m and precision of 0.2% was used to measure the rotational speed and the torque. All measurement of synchronous data acquisition was completed through a data acquisition card.

The pump head H and efficiency η can be calculated as follows:

$$H=\frac{P_{out}-P_{in}}{\rho g}$$

(1)

$$\eta =\frac{\rho gQH}{2\pi nM}$$

(2)

where P_out is the pressure at pump outlet; P_in is the pressure at pump inlet; ρ is the density of water; g is the acceleration of gravity; Q is the flow rate; n is the rotation speed; and M is the torque.

3. Computational Domain and Numerical Method

3.1. Computational Domain

The computational domain consisted of inlet pipe, impeller, guide vane and outlet pipe. The length of the inlet pipe was twice that of the pipe diameter, and the length of the outlet pipe was three times that of the pipe diameter. Figure 2 shows the computational domain of the axial pump.

3.2. Computational Mesh

The overall computational domain of the axial-flow pump was divided by the structured mesh. The inlet pipe domain and outlet pipe domain were divided by the H-grid structured mesh. The impeller domain and guide vane domain were divided by the hexahedral structured mesh. The meshes near the wall were refined to capture the detailed flow structure. Figure 3a shows the impeller mesh, which includes the hub and six blades, and Figure 3b shows the whole mesh of the guide vane, which includes the hub and seven vanes.

3.3. Mesh Independence Test

The mesh number has an influence on the numerical calculation results. Too few meshes will reduce the calculation accuracy, while too many meshes will consume additional computational resources. Therefore, it is necessary to conduct the mesh independence test first. The mesh numbers of the inlet pipe, guide vane and outlet pipe were 154,440, 664,818 and 254,100, respectively, and they remained the same in the mesh independence test. Five groups of impeller mesh, ranging from 5 × 10⁵ to 1.5 × 10⁶, were set to conduct the test.

Table 1. Independence test of mesh number.

Number	Mesh1	Mesh2	Mesh3	Mesh4	Mesh5
Inlet pipe	154,440	154,440	154,440	154,440	154,440
Impeller	502,355	725,580	995,450	1,268,745	1,501,830
Guide vane	664,818	664,818	664,818	664,818	664,818
Outlet pipe	254,100	254,100	254,100	254,100	254,100
Total mesh	1,575,713	1,798,938	2,068,808	2,342,103	2,575,188
Relative head	1	0.995797	0.992838	0.990733	0.990468
Relative efficiency	1	0.996977	0.993128	0.992751	0.991628

lists the numerical results of the relative head and relative efficiency of the pump at a design point of 70 kg/s. When the impeller mesh elements exceeded 1.2 × 10⁶, the relative head and relative efficiency changed little. Finally, the total mesh number of 1268745 for the axial-flow pump was chosen in the following calculation.

3.4. Numerical Method

The software that conducts numerical calculation for the axial-flow pump is ANSYS CFX 2020 R1. The shear stress transport model (SST k-ω), which introduces the influence of shear stress propagation on turbulent viscosity, was set as the turbulence model. The impeller domain was set as the rotation domain, and the rest of the parts were set as the static domains. Frozen rotor technology was adopted at the dynamic static interface. a no slip wall condition was adopted at all solid interfaces. The boundary conditions were total pressure at inlet and mass flow rate at outlet. The convergence condition was set as 10⁻⁵. The calculation process of the external characteristic curve was completed by gradually reducing the mass flow rate.

The hydraulic efficiency η_h can be calculated by numerical simulation, while the volume efficiency η_v and mechanical efficiency η_m are determined by their empirical functions. Therefore, the total efficiency η was calculated by η = η_h × η_v × η_m. Figure 4 shows the comparison of pump head and efficiency between experimental measurement and numerical simulation. The experimental and numerical results agree well, especially at the designed flow rate, which validates the accuracy of the numerical method.

4. Orthogonal Optimization

4.1. Parameters Setting

The impeller is the core component of the axial-flow pump for energy conversion. The three-dimensional shape of the blade directly affects the hydraulic performance. In this work, the meridian passage, hydrofoil shape and centroid stacking line are chosen as the design parameters to conduct the optimization.

The shape of the meridian flow passage is very important for the flow pattern of the impeller. Figure 5 shows the meridian flow passage of the axial-flow pump. The hub profile is determined by the quartic Bézier curve with five control points. The coordinate Z of hub control point 3 is selected as the first optimization parameter.

The foil can determine the lift and drag, and further affects the hydraulic performance, of the axial-flow pump. NACA65 foil was selected in this work, and the blade’s three-dimensional (3D) shape can be controlled by the stagger angle (γ) and camber angle (φ). Figure 6 shows the foil shape, stagger angle and camber angle. Usually, the stagger angle at the hub and shroud are different, which mainly affects the blade twist in the radial direction. The stagger angles at the hub and shroud are selected as the second and third optimization parameters. The camber angle is selected as the fourth parameter.

Figure 7 shows the foil stacking line. The position of the stack line affects the work area of the blade. The Z coordinate of the hub foil centroid is used to determine the whole position of the line from which the foil is stacking to form a 3D blade. The foil centroid position is selected as the fifth parameter.

4.2. Orthogonal Table

The orthogonal method is a method used to arrange and analyze multi-parameter optimization, which can greatly improve optimization efficiency. Three elements of the orthogonal method are the orthogonal parameter, parameter level and orthogonal table. The orthogonal parameters are determined in the above paragraph, and the parameter level can be determined by our experience. The baseline pump was designed and tested in our laboratory, and single parameter analysis and optimization has been conducted in a larger range of values. The value range of the hub control point is from 0.2 to 0.8, due to the geometrical limitation. The value range of the centroid position is from 66 mm to 69 mm, due to the guide vane position.

Table 2. Orthogonal parameter and level.

Level	Hub Control Point (A)	Hub Stagger Angle/° (B)	Shroud Stagger Angle/° (C)	Camber Angle/° (D)	Centroid Position/mm (E)
1	0.2	51	15	7.5	66
2	0.4	53	18	8.3	67
3	0.6	55	21	9.1	68
4	0.8	57	24	9.9	69

shows the orthogonal parameter and level of the axial-flow pump in this work.

Finally, an orthogonal table (L₁₆) with five parameters’ factors and four levels is established. There are sixteen individuals in this orthogonal table, which evenly distribute in the range of the overall 1024 (4⁵) cases.

Table 3. Orthogonal table.

Individual No.	Factor
	A	B/°	C/°	D/°	E/mm
1	0.2	51	15	7.5	66
2	0.2	53	18	8.3	67
3	0.2	55	21	9.1	68
4	0.2	57	24	9.9	69
5	0.4	51	18	9.1	69
6	0.4	53	15	9.9	68
7	0.4	55	24	7.5	67
8	0.4	57	21	8.3	66
9	0.6	51	21	9.9	67
10	0.6	53	24	9.1	66
11	0.6	55	15	8.3	69
12	0.6	57	18	7.5	68
13	0.8	51	24	8.3	68
14	0.8	53	21	7.5	69
15	0.8	55	18	9.9	66
16	0.8	57	15	9.1	67

shows the details of individuals in this orthogonal table.

5. Results and Discussion

5.1. Orthogonal Analysis

Sixteen tested impellers with different parameter levels were firstly designed; then, the numerical simulation was conducted to obtain the pump head and efficiency.

Table 4. Axial-flow pump performance.

Individual No.	Head/m	Efficiency/%
1	3.23	84.13
2	3.94	85.28
3	4.37	80.96
4	4.45	75.66
5	3.92	85.78
6	3.64	85.80
7	4.09	74.04
8	4.38	79.96
9	4.2	81.78
10	4.37	76.23
11	3.74	86.06
12	4.15	83.02
13	4.37	77.66
14	4.25	81.27
15	4.26	84.38
16	3.98	85.01

shows the simulated pump head and efficiency for different individuals at the design point. The pump head is in a range of 3.23 m to 4.45 m, and the pump efficiency is in a range of 74.04% to 86.06%.

In the orthogonal method, to evaluate the influence of every specific parameter, the average value $\overline{K_i}$ of head is defined as follows:

$$\overline{K_i}=\frac{1}{N}{\displaystyle \sum _{j=1}^N{H}_j}$$

(3)

where i is the level of the parameter; N is the individual number of the corresponding parameter at level I; and H_j is the head of tested pumps with level i. The average value of efficiency is defined as the same formula.

Range R describes the influence of weight on each parameter, and can be defined as follows:

$$R=\max (\overline{K_i})-\min (\overline{K_i}),i=1,2,3,4$$

(4)

Table 5. Range analysis for pump head.

Head/m	Factor
	A	B	C	D	E
$\overline{K_1}$	3.997	3.93	3.647	3.93	4.06
$\overline{K_2}$	4.008	4.05	4.067	4.107	4.053
$\overline{K_3}$	4.115	4.115	4.3	4.16	4.133
$\overline{K_4}$	4.215	4.24	4.32	4.137	4.09
R	0.218	0.31	0.673	0.23	0.08

shows the range analysis of the pump head for five parameters. According to the results of the range analysis, the pump head increases as the A, B and C levels increase, while it has the maximum value at D3 and E3.

Table 6. Range analysis for pump efficiency.

Efficiency/%	Factor
	A	B	C	D	E
$\overline{K_1}$	81.51	82.34	85.25	80.61	81.17
$\overline{K_2}$	81.39	82.14	84.61	82.24	81.53
$\overline{K_3}$	81.77	81.36	80.99	82.00	81.86
$\overline{K_4}$	82.08	80.91	75.9	81.91	82.19
R	0.69	1.43	9.35	1.63	1.02

shows the range analysis of pump efficiency for five parameters. The parameter A has little influence on the pump efficiency. According to the results of range analysis, the pump efficiency decreases as the B and C levels increase, while it has the maximum value at A4, D2 and E4.

Therefore, the influence order on the pump head can be concluded as follows: C > B > D > A > E, while the influence order on pump efficiency can be concluded as follows: C > D > B > E > A. From the range analysis, it can be found that the parameter C, the shroud stagger angle, has the maximum influence on the pump head and efficiency. Parameter B and parameter D, corresponding to hub stagger angle and camber angle, respectively, are also very important.

This range analysis shows that, as the hub control point value increases, the area of meridian passage also increases, which makes the flow pattern more even. The camber angle mainly influences the three-dimensional geometry of the blade. With the increase in the camber angle, the twist of the blade is more obvious, which strengthens the control ability on fluid and improves the pump’s energy performance. The foil centroid position mainly influences the blade position and the interval between the impeller and vane. The proper foil centroid position can balance the stator–rotor interaction effect and flow pattern in a vaneless zone, and then improve the pump’s energy performance.

5.2. Orthogonal Optimization

With comprehensive consideration of the design requirements and optimization, the levels of five parameters are chosen as A3, B3, C1, D3 and E4.

Table 7. Levels of five parameters for baseline pump and optimal pump.

Pump	Levels of Five Parameters
	A	B/°	C/°	D/°	E/mm
Baseline pump	0.4	45	20	6.7	67
Optimal pump	0.6	55	15	9.1	69

shows the levels of the five parameters for the baseline pump and optimal pump.

Figure 8a shows the foil stagger angle of the baseline pump and optimal pump from hub to shroud. The foil stagger angle of the optimal pump near the hub is larger than that of the baseline pump, although it is lower than that of the baseline pump near the shroud. Figure 8b shows the comparison of blade geometry between the baseline pump and the optimal pump. The blade geometry of the optimal pump is more twisted than that of the baseline pump near the blade rear and tip.

Figure 9 shows the comparison of head and efficiency between the baseline pump and the optimal pump at different flow rates. The efficiency of the optimal pump is higher than that of the baseline pump at all flow rates, especially at a low flow rate. At a design point of flow rate 70 kg/s, the efficiencies of the baseline pump and optimal pump are 84.24% and 86.29%, respectively, improving by 2.05%.

To further investigate the energy loss in different components of the axial-flow pump,

Table 8. Energy performance of baseline pump and optimal pump.

Pump	Q (kg/s)	H (m)	η (%)	Inlet Pipe Loss (%)	Impeller Loss (%)	Guide Vane Loss (%)	Outlet Pipe Loss (%)
Baseline pump	70	3.67	84.24	0.26	8.90	3.20	3.39
Optimal pump	70	3.80	86.29	0.26	7.49	2.99	2.97

shows the energy loss details of baseline pump and optimal pump.

For the inlet pipe, the energy loss is the same, due to the same geometry and the same inlet condition. For the impeller, the energy loss of the optimal pump reduces by 1.41%, which validates the effectiveness of the orthogonal optimization. For the downstream components, the energy losses of optimal pump in the guide vane and outlet pipe reduce by 0.21% and 0.42%, respectively.

5.3. Flow Pattern Analysis

To further investigate the optimal mechanism, the flow pattern in the pump should be revealed. Figure 10 shows the blade load at different span sections for the baseline pump and optimal pump. Results show that the blade load is mainly concentrated at the front and middle part of the blade. At 10% blade height, the blade load of the optimal pump is similar to the baseline pump, because the blade geometry is nearly the same at the hub and obviously different near the shroud. The blade load in this section is relatively low, which corresponds to being of weak workability on fluid. The blade load of the optimal pump is higher than that of the baseline pump, which validates the effectiveness of the orthogonal optimization method. At 50% blade height, the blade load of the optimal pump near the blade head is lower than that of baseline pump, while on the middle and rear part of blade it is higher than that of the baseline pump. The pressure on the pressure side is nearly the same for both the baseline pump and the optimal pump, and the pressure of the optimal pump on suction side is obviously lower than that of baseline pump. At 90% blade height span, the blade load and pressure distributions between the two pumps are similar to that at 50% blade height.

Figure 11 shows the pressure distribution at blade suction surface for the baseline pump and the optimal pump. The results show that the pressure gradually increases from the blade inlet to outlet for both the pressure and suction surfaces. For the baseline suction surface, a low-pressure area appears near the blade head, while this local region greatly reduces for the optimal suction surface, which will be beneficial for the cavitation performance. The reason may be that the blade angle for the baseline pump is not matched. Meanwhile, the pressure in this location greatly increases for the optimal suction surface. The stagger angle of the shroud for the optimal pump is smaller than that of the baseline pump, and the flow impact on the blade head is weak. The pressure gradient near the blade rear for the optimal pump is more even than that for the baseline pump, which makes the work on fluid more stable and improves the energy performance.

Figure 12 shows the velocity distribution in the impeller at 90% blade height for the baseline pump and the optimal pump under different flow rates. The results show that as the flow rate increases, the velocity increases for the baseline pump and the optimal pump. Under the low flow rate of 50 kg/s, there is an obvious low-velocity region in the flow passage, which will block the flow passage and reduce the energy performance of the pump. Under the design flow rate of 70 kg/s, the velocities in the flow passage for the baseline pump and the optimal pump are both fluent. There is a very small high-velocity region at the blade head of the baseline pump, due to the unsuitable blade angle at the blade leading edge. Under the large flow rate of 90 kg/s, the velocity distributions are relatively even for the baseline pump and the optimal pump.

6. Conclusions

In this work, an axial-flow pump is optimized by the orthogonal method, based on experimental measurement and numerical simulation. Five optimization parameters including hub control point, hub stagger angle, shroud stagger angle, camber angle and centroid position are chosen( named as factor A, B, C, D and E in the orthogonal table). Four levels for the optimization parameters are determined by experience. Conclusions can be drawn as follows:

(1)

According to the range analysis of orthogonal method, the influence level of optimization parameters on the pump head is sorted by C > B > D > A > E, and the influence level on pump efficiency is sorted by C > D > B > E > A;

(2)

The energy performance of the optimal pump improves in comparison of the baseline pump. At a design point of flow rate 70 kg/s, the efficiencies of the baseline pump and the optimal pump are 84.24% and 86.29%, respectively, improving by 2.05%;

(3)

The flow pattern in the optimal pump greatly improves. The pressure gradient of the optimal pump from blade inlet to outlet becomes more fluent than that of the baseline pump. The low-velocity region of the optimal pump at the blade head shrinks compared to that of the baseline pump.