Multiobjective Combination Optimization of an Impeller and Diﬀuser in a Reversible Axial-Flow Pump Based on a Two-Layer Artiﬁcial Neural Network

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Article Multiobjective Combination Optimization of an Impeller and Diﬀuser in a Reversible Axial-Flow Pump Based on a Two-Layer Artiﬁcial Neural Network

Fan Meng 1, Yanjun Li 1,*, Shouqi Yuan 1, Wenjie Wang 1 , Yunhao Zheng 1 and Majeed Koranteng Osman 1,2

1 Research Center of Fluid Machinery Engineering and Technology, Jiangsu University, Zhenjiang 212013, China; [email protected] (F.M.); [email protected] (S.Y.); [email protected] (W.W.); [email protected] (Y.Z.); [email protected] (M.K.O.) 2 Mechanical Engineering Department, Wa Polytechnic, Wa, Upper West, Ghana * Correspondence: [email protected]

Received: 8 January 2020; Accepted: 4 March 2020; Published: 7 March 2020

Abstract: This study proposed a kind of optimization design for a reversible axial-flow pump based on an ordinary one-way pump. Three-dimensional (3D) Reynolds-averaged Navier–Stokes (RANS) equations was used to predict the pump performance, and the optimized design was validated by an external characteristic test. Six main geometry parameters of an impeller and diffuser based on an orthogonal experiment were set as design variables. The efficiency and head under forward and reverse design conditions were set as the optimization objective. Based on 120 groups of sample designs obtained from Latin hypercube sampling (LHS), a two-layer artificial neural network (ANN) was used to build a non-linear function with high accuracy between the design variables and optimization objective. The optimized design was obtained from 300 groups of Pareto-optimal solutions using the non-dominated based genetic algorithm (NSGA) for multiobjective optimization. After optimization, there was a slight decrease in the forward pump efficiency and head. The reverse pump efficiency and head on the other hand was largely improved and the high efficiency range was also widened.

Keywords: reversible axial-ﬂow pump; computational ﬂuid dynamics; surrogate model; multiobjective optimization; two-layer ANN

1. Introduction Axial-flow pump with reversible impeller can meet bidirectional pumping needs by varying the direction of the rotation of the impeller, and this has widely been used in flood control and drainage [1–3]. Due to the special blade airfoil, the flow characteristic in the reversible axial-flow pump is different from that of the conventional axial-flow pump. Thus, providing a reliable and efficient optimization design method especially for a reversible axial-flow pump is very necessary. The key point of an optimization design is accurate prediction for pump performance. In recent years, the application of computational fluid dynamics (CFD) in the research about pressure fluctuation [4,5], cavitation performance [6,7] and hydraulic losses [8,9] in the different types of pump has proved numerical simulation to be an efficient method for predicting pump performance. To improve pump performance, researchers have applied the design of experiment (DOE) based on CFD for pump optimization to reduce experiment cost and save time. Long et al. [10] applied the orthogonal test to optimize the diffuser in the reactor coolant pump to improve pump performance. The optimized design

Processes 2020, 8, 309; doi:10.3390/pr8030309 www.mdpi.com/journal/processes Processes 2020, 8, 309 2 of 22 was validated by an experiment. Liu et al. [11] used the optimal Latin hypercube sampling (LHS) method in the multicondition optimization of a mixed-flow pump and the optimization objective was chosen as weighted average efficiency at three flow rates. However, optimized design obtained from DOE is the optimal solution within the discrete design domain. Combination of the approximation model and intelligent algorithm can get the optimal solution within the continuous design domain. An approximation model is therefore being to construct a function between design variables and optimization objective. This function can then be solved by an optimization algorithm to obtain optimal optimized solutions. Pei et al. [12] therefore combined LHS, the artificial neural network (ANN) and modified particle swarm optimization (PSO) to obtain higher centrifugal pump efficiency at three flowrates. Miao et al. [13] applied the combination of neural networks and modified PSO algorithms to improve the pump efficiency and cavitation performance. Shim et al. [14,15] completed the multiobjective optimization based on approximation model and non-dominated based genetic algorithm (NSGA) to improve stability, efficiency and cavitation performance for different types of centrifugal pump. Wang et al. [16,17] used different surrogate models to optimize the impeller and diffuser of a centrifugal pump based on CFD. However, there is not much optimization design on a reversible axial-flow pump. The reversible axial blade pump generally has two-way impeller airfoils, which can be grouped into an arc, S-shape and polynomial curve. The S-shaped impeller can obtain similar pump performance under the forward and reverse condition. In some actual engineering, the pump operation time under the forward condition is much longer than that under the reverse condition. To obtain high pump efficiency under the forward condition and also to ensure that the pump meets a basic standard under the reverse condition, the bidirectional impeller airfoils are chosen as an asymmetric arc or polynomial curve. Ma [18,19] investigated the pressure fluctuation in a reversible axial-flow pump with an S-shaped impeller, and compared the inner flow in an S-shaped and arc bidirectional impeller. It provided some useful suggestions for hydraulic design of a reversible axial-flow pump. Jung [20] combined an orthogonal experimental design, response surface model (RSM) and feasible direction method to optimize the arc two-way impeller with a symmetric chord airfoil. In this study, the one-way axial-flow pump was selected as the basis of combination optimization design for a two-way impeller and diffuser. The optimization objective was to improve the reverse pump performance and maintain forward pump performance at a certain level. The 3D Reynolds-averaged Navier–Stokes (RANS) equation was solved to predict pump performance and six geometry parameters of the impeller and diffuser were set as the design variables based on an orthogonal experimental design. The relationship between design variables and optimization objective was constructed with 120 groups of sample points using LHS and two-layer ANN. Finally, the optimized design was obtained based on 300 groups of Pareto-optimal solutions by using NSGA, and the external characteristic experiment of optimized design was performed to verify the optimization results.

2. Numerical Method

2.1. Computational Model and Mesh The original design is a one-way pump, which consists of a straight pipe, a bidirectional impeller, a diffuser and an elbow as shown in Figure1. The optimized pump is a reversible axial-flow pump consisting of a straight pipe, a two-way impeller, a diffuser and an elbow. The main design parameters of the original design are listed in Table1, and the specific speed can be calculated by Equation (1).

3.65NQ0.5 ns = (1) H0.75 where ns is the speciﬁc speed, N is rotational speed, Q is the volume ﬂow rate and H is the head. Processes 2020, 8, 309 3 of 24

the other components was obtained by preprocessing software of computer aided engineering (CAE), ICEM CFD, which is a module in the commercial software ANASYS. This software can generate a hexahedral mesh semi-automatically and generate a tetrahedral mesh automatically. Figure 3 shows

the grid independence analysis of the original design for the impeller and diffuser. In this figure, 휼1 stands for pump efficiency under a forward design flow rate, 휼2 stands for pump efficiency under a reverse design flow rate. Figure 3a,b shows the tendency of 휼1 and 휼2 with a grid number of the impeller increasing. The maximum relative deviation was smaller than 0.08% when the grid number 6 was greater than 1.25 × 10 . Figure 3c,d shows the tendency of 휼1 and 휼2 with a grid number of the diffuser increasing. The maximum relative deviation was smaller than 0.16% when the grid number was greater than 1.47 × 106. Hence, the number of nodes of the inlet section, impeller, diffuser and outlet section were finally chosen as approximately 0.74 × 106, 1.25 × 106, 1.47 × 106 and 0.95 × 106 respectively. Y+ is a dimensionless number and it can define the distance from the wall to the first node. The average Y+ of any component can be obtained by calculating the mean Y+ values from each Processes 2020node, 8, 309on all walls. The average Y+ of the straight pipe without the inlet section, impeller, diffuser and 3 of 22 elbow pipe without the outlet section were 21.8, 24.3, 13.52 and 45.4 respectively.

Impeller

Elbow

Diffuser

Straight pipe

FigureFigure 1. 3D 1. 3D model model of of the theoriginal original design. design. Table 1. Main parameters of the original design. Table 1. Main parameters of the original design.

DesignDesign Parameters Parameters 3 Rated flowRated rate flow (m rate3/s) (m /s) 0.40.4 Rotational speed speed (r/min) (r/min) 1450 1450 Head on rated flow rate Head on rated flow rate (m) 3.873.87 Specific Specific speed speed 1213.1 1213.1 (m) GeometryGeometry Parameters Parameters ImpellerImpeller blade numberblade number 33 DiDiffuserffuser blade number number 5 5 Impeller diameter (mm) 300 OutletOutlet diameter diameter of of di ffdiffuseruser (mm) 312.8 Impeller diameter (mm) 300 312.8 Optimization Parameters(mm) Optimization Parameters Forward design flow rate (m3/s) 0.36 Efficiency on forward design flow rate (%) 78.0

Reverse design flow rate (m3/s) 0.3 Efficiency on reverse design flow rate (%) 51.91

The straight pipe and elbow were set as an inflow pipe and outflow pipe respectively, under the forward operation. In the reverse operation, the straight pipe and elbow were set as the outlet pipe and inlet pipe respectively. Figure2 shows the mesh of the entire computational domain. The hexahedral mesh of the impeller and diffuser was generated by TurboGrid. The hexahedral mesh of the other components was obtained by preprocessing software of computer aided engineering (CAE), ICEM CFD, which is a module in the commercial software ANASYS. This software can generate a hexahedral mesh semi-automatically and generate a tetrahedral mesh automatically. Figure3 shows the grid independence analysis of the original design for the impeller and diffuser. In this figure, η1 stands for pump efficiency under a forward design flow rate, η2 stands for pump efficiency under a reverse design flow rate. Figure3a,b shows the tendency of η1 and η2 with a grid number of the impeller increasing. The maximum relative deviation was smaller than 0.08% when the grid number was greater than 1.25 106. Figure3c,d shows the tendency of η and η with a grid number of the diffuser increasing. × 1 2 The maximum relative deviation was smaller than 0.16% when the grid number was greater than 1.47 106. Hence, the number of nodes of the inlet section, impeller, diffuser and outlet section were × finally chosen as approximately 0.74 106, 1.25 106, 1.47 106 and 0.95 106 respectively. Y+ is a × × × × dimensionless number and it can define the distance from the wall to the first node. The average Y+ Processes 2020, 8, 309 4 of 24 Processes 2020, 8, 309 4 of 22 Forward design flow Efficiency on forward design Processes 2020, 8, 309 0.36 78.0 4 of 24 rate (m3/s) flow rate (%) of any componentRFeverseorward design design can flow beflow obtained rate by calculatingEfficiency theEfficiency mean on on Y forward+ reversevalues design design from each node on all walls. 0.360.3 758.01.91 The average Y+rateof(m (m the3/s)3/s) straight pipe without the inlet section,flowflow impeller, rate rate (%) (%) di ffuser and elbow pipe without Reverse design flow rate Efficiency on reverse design the outlet section were 21.8, 24.3, 13.520 and.3 45.4 respectively. 51.91 (m3/s) Diffuser flow rate (%)

Diffuser

Impeller

Impeller Straight pipe Straight pipe

Elbow pipe Elbow pipe Figure 2. Mesh of the original design. Figure 2. Mesh of the original design. Figure 2. Mesh of the original design.

(a) (b)

Figure 3. Grid independence of the original design for the (a) impeller under a forward design (c) (d) flowrate, (b) impeller under a reverse design flow rate, (c) diffuser under a forward design flowrate and (d) diffuser under a reverse design flow rate.

2.2. Numerical Setup The k-ω formulation has good performance near the wall treatment for low Reynolds number computations, so it was applied widely in the numerical simulation [21] in the pump. In addition, Processes 2020, 8, 309 5 of 22 the k-ω based Shear-Stress-Transport (SST) model with an automatic wall treatment can present a smooth transition from a low-Reynolds number form to a wall function formulation, which shows less sensitivity to Y+ [22]. So, it was used to solve a steady 3D RANS equation in this study. The governing equation was defined as: ∂vj = 0 (2) ∂xi ∂ ρv v ∂(ρvi) i j ∂p ∂ ∂vi + = + (µ ρv0iv0 j) + ρ fi (3) ∂t ∂xj −∂xi ∂xj ∂xj − where v is the time-average speed and p is the time-average pressure. ρ is the fluid density. µ stands for the dynamic viscosity and ρv0iv0 j represents the Reynolds stress. ρ fi is the source item. The reference pressure is 1 atm. The inlet boundary condition was set as the ‘Mass Flow Rate’ under a forward operation. The outlet boundary condition was set as ‘Static Pressure’ and the outlet static pressure was initially specified as 0. In the reverse operation, the inlet boundary condition was also set as the ‘Mass Flow Rate’. However, the outlet boundary condition was set as the ‘Opening’ due to flow recirculation and the opening pressure was initially specified as 0. The interface condition between the impeller and stator was set as a ‘Frozen rotor’ (the relative position of the impeller and stator across the interface was fixed through calculation). The interface condition between stators was set as ‘None’. The wall boundary condition was considered as smooth with no slip. The convergence criteria and advection scheme were set as 1 104 and 1st-order upwind respectively. The numerical × calculation was to be stopped when it achieved convergence criteria or periodic stability (few calculated results shown periodic oscillations, but the amplitude of efficiency is less than 0.5%.).

3. Optimization Procedure The optimization procedure for the improvement of pump efficiency and the head is shown in Figure4. Firstly, the shape of the impeller blade and di ffuser blade was controlled by the airfoil section. The main geometric parameters of the airfoil section were set as design variables and described in detail in Section 3.2. In order to reduce the design variables of the diffuser to improve the optimization speed, the orthogonal experimental design was applied to obtain the effect of main geometric parameters on the optimization objective. Thirdly, LHS was used to construct 120 groups of sample designs in design space. The CFD results of the optimization objective were obtained by CFX automatically executed by MATLAB as shown in Figure5. A two–layer ANN was designed to construct the non-linear function between the design variables and optimization objective. Finally, the 300 groups of Pareto-optimal solutions were obtained by NSGA. Optimized design was based on one of these solutions and verified byProcesses experiment 2020, 8 data., 309 6 of 24

Orthogonal test of Combination optimization diffuser of diffuser and impeller

Orthogonal test Choose design design LHS NSGA variables of impeller and diffuser Reduce design variables Database of of diffuser Database Optimized design sample designs Sample points

Determine optimization Verify with Two-layer ANN objectives Range analysis experiment data

Figure 4. The optimization procedure of the reversible axial-ﬂow pump. Figure 4. The optimization procedure of the reversible axial-flow pump.

Figure 5. Optimization flow chart in the ANSYS Workbench.

3.1. Optimization Functions

According to engineering requirements, the forward design flow rate was chosen as 0.36 m3/s and the reverse design flow rate was chosen as 0.3 m3/s. The optimization objectives can be described as follow:

Minimum: 풁(풙) = {풁1(푥), 풁2(푥), 풁3(푥)} (4)

where 풙 is a vector of design variables. 풁 stands for a vector of three optimization objectives 풁1, 풁2 and 풁3. In this study, three optimization objectives need be minimized to improve hydraulic

performance to meet the engineering requirement. The first optimization objective 푍1 is the negative hydraulic efficiency under the forward design flow rate. The second optimization objective 푍2 is the negative hydraulic efficiency under the reverse design flow rate. The third optimization objective 푍3 is the negative sum of the head under a forward design flow rate and head under a reverse design flow rate. Three objective functions were defined as:

푍1 = −휂1 (5)

푍2 = −휂2 (6)

Processes 2020, 8, 309 6 of 24

Orthogonal test of Combination optimization diffuser of diffuser and impeller

Orthogonal test Choose design design LHS NSGA variables of impeller and diffuser Reduce design variables Database of of diffuser Database Optimized design sample designs Sample points

Determine optimization Verify with Two-layer ANN objectives Range analysis experiment data

Processes 2020, 8, 309 6 of 22 Figure 4. The optimization procedure of the reversible axial-flow pump.

Figure 5. Optimization flow chart in the ANSYS Workbench. Figure 5. Optimization flow chart in the ANSYS Workbench. 3.1. Optimization Functions 3.1. Optimization Functions 3 AccordingAccording to engineering to engineering requirements, requirements, the the forward forward design design flow flow rate rate was was chosenchosen as 0.36 m 3/s/s and the reverseand design the reverse flow ratedesign was flow chosen rate was as chosen 0.3 m3 /ass. 0.3 The m3 optimization/s. The optimization objectives objectives can can be describedbe described as follow: as follow: Minimum : Z(x) = Z1(x), Z2(x), Z3(x) (4) Minimum: 풁(풙) = {풁1(푥), 풁2(푥), 풁3(푥)} (4) where wherex is a vector풙 is a vector of design of design variables. variables.Z 풁stands stands for for a avector vector of three of three optimization optimization objectives objectives 풁1, 풁2 Z1, Z2 and Z3and. 풁3. In thisIn study, this three study, optimization three optimization objectives objectives need be need minimized be minimized to improve to improve hydraulic hydraulic performance performance to meet the engineering requirement. The first optimization objective 푍1 is the negative to meet the engineering requirement. The first optimization objective Z1 is the negative hydraulic hydraulic efficiency under the forward design flow rate. The second optimization objective 푍2 is the efficiency under the forward design flow rate. The second optimization objective Z2 is the negative negative hydraulic efficiency under the reverse design flow rate. The third optimization objective 푍3 hydraulicis the e ffinegativeciency sum under of the the head reverse under designa forward flow design rate. flow The rate third and head optimization under a reverse objective designZ 3 is the negativeflow sum rate. of Three thehead objecti underve functions a forward were defined design as: flow rate and head under a reverse design flow rate. Three objective functions were defined as: 푍1 = −휂1 (5)

Z푍2 ==−휂η2 (6) (5) 1 − 1

Z = η (6) 2 − 2 Z = H = (H + H ) (7) 3 − 3 − 1 2 where η1 and H1 are hydraulic efficiency and a head under the forward design flow rate. η2 and H2 are hydraulic efficiency and the head under the reverse design flow rate. H3 is the sum of H1 and H2. The hydraulic efficiency η and head H can be calculated by the equations as follow:

(Pout Pin)Q η = − (8) TN

Pout P H = − in (9) ρg where Pout is the total pressure in the outlet section, Pin is the total pressure in the inlet section. Q is the volume ﬂow rate. T is the torque on the pump shaft during pump operation. N is the forward rotation speed of the pump shaft. ρ is the ﬂuid density and g is the gravitational acceleration.

3.2. Geometry Variables Figure6a shows the impeller shape was defined by five airfoil sections ( I-I, II-II, III-III, IV-IV and V-V) obtained by radial equidistant division. The main geometry parameters of every airfoil section are the length of the chord line L, blade angle (angle between the chord line and horizontal line) θ, deformation of the mean line (midpoint connection of the local geometric thickness of the airfoil section Processes 2020, 8, 309 7 of 22 along chord line) and thickness distribution along the chord line, as shown in Figure6b. In this study, every airfoil section was designed based on the NACA 65 series. So, the deformation of the mean line in every section could calculate the form lift coefficient σ, which was obtained from the camber angle Φ. The definition of Φ was shown in Figure6c and the deformation function of the mean line was described in detail as follows [23]: 2π Φ σ = tan (10) ln(2) 4 Φ = λ ε (11) " − # yc σ x x x x = 1 c ln 1 c + c ln c (12) L −4π − L − L Li L where σProcessesis the 2020 theoretical, 8, 309 lift coefficient of the airfoil, Φ is the camber angle. λ and ε are the8 of inlet 24 angle and outlet angle of the skeleton line. yc is the ordinate of the airfoil profile (direction perpendicular to chord line). xc is the abscissa of the airfoil profile (direction parallel to the chord line).

Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ

Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ (a) λ

Equivalent circular skeleton line Tangent to the Mean line skeleton line

Chord Φ θ line

Tangent to the skeleton line ε (b) (c)

FigureFigure 6. The 6. deﬁnition The definition of theof the main main geometry geometry variables of of the the impeller: impeller: (a) definition (a) deﬁnition of the airfoil of the airfoil section, (b) main geometry variables of the airfoil and (c) the rest of the geometry variables. section, (b) main geometry variables of the airfoil and (c) the rest of the geometry variables. Figure 7a shows the diffuser shape was defined by five airfoil sections (I-I, II-II, III-III, IV-IV and In this hydraulic design, the thickness distribution and blade angle θ were predetermined, so the V-V) obtained by radial equidistant division. Figure 7b shows that every airfoil section was defined shape ofby the the airfoilinlet angle section 훼, outlet was angle controlled 훽, blade by height the length휁 and wrap of the angle chord 휑. In line thisL study,and camber the outletΦ angle. In order to improve훽 theand optimizationblade height 휁 speed,were predetermined the design variables based on previous of the impeller engineering was research. only chosen Only the from wrap the chord length andangle camber 휑 and inlet angle angle in 훼 section in sectionI-I Iand-I and section section V-VV-V were: L1, setL5 ,asΦ geometry1 and Φ 5variables.. The L Theand 휑Φ andin the rest sections훼 werein the determined rest of the sections by the were equation calculated as as follow: follow: ((휑L5 − L휑1) 휑휍 = 5 1 × 𝜍 + 휑1 (15) Lκ = −4 κ + L1 (13) 4 × (훼5 − 훼1) 훼휍 =(Φ5 Φ1)× 𝜍 + 훼1 (16) Φκ = −4 κ + Φ (14) 4 × 1 where, 𝜍 is the section number of the diffuser. 휑5 is the wrap angle in section V-V and 휑1 is the wrap angle in section I-I. 훼5 is the inlet angle in section V-V and 훼1 is the inlet angle in section I-I.

Processes 2020, 8, 309 8 of 22

where, κ is the section number of the impeller. Φ1 and Φ5 are the camber angle in section I-I and section V-V. L1 and L5 are the length of the chord line in section I-I and section V-V. Figure7a shows the di ffuser shape was defined by five airfoil sections (I-I, II-II, III-III, IV-IV and V-V) obtained by radial equidistant division. Figure7b shows that every airfoil section was defined by the inlet angle α, outlet angle β, blade height ζ and wrap angle ϕ. In this study, the outlet angle β and blade height ζ were predetermined based on previous engineering research. Only the wrap angle ϕ and inlet angle α in section I-I and section V-V were set as geometry variables. The ϕ and α in the rest of the sections were calculated as follow:

(ϕ5 ϕ1) ϕς = − ς + ϕ (15) 4 × 1

(α5 α1) ας = − ς + α (16) 4 × 1 where, ς is the section number of the diﬀuser. ϕ5 is the wrap angle in section V-V and ϕ1 is the wrap angle in sectionProcesses 2020I-I., 8α, 5309is the inlet angle in section V-V and α1 is the inlet angle in section9 of I-I24 .

Outfolw Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ

Ⅰ Ⅱ Ⅲ Ⅳ Inflow Ⅴ (a)

Tangent to the mean line β

Mean line Rotation center (Bezier Ⅰ ζ curves) Ⅱ Ⅲ Ⅳ Ⅰ Ⅱ Ⅴ Ⅲ Ⅳ Ⅴ α Trailing edge Tangent to Leading edge the mean line In view of diffuser outlet (b)

Figure 7. FigureThe definition 7. The definition of the of mainthe main geometry geometry variablesvariables of ofthe the diffuser: diffuser: (a) definition (a) definition of the airfoil of the airfoil section andsection (b) main and (b geometry) main geometry variables variables of of the the airfoil. airfoil.

3.3. Orthogonal3.3. Orthogonal Test Design Test Design of Di ffofuser Diffuser According to Section 3.1, the impeller shape was controlled by 퐿 , 퐿 , 훷 and 훷 , and the According to Section 3.1, the impeller shape was controlled by L 1, L 5, Φ1 and Φ5 , and the diffuser diffuser shape was controlled by 휑1, 휑5, 훼1 and 훼5. The focus was to1 optimize5 1 the impeller5 and shape wasdiffuser controlled at the same by ϕ time1, ϕ in5, thisα1 study.and α However,5. The focusif all these was eight to optimizegeometry variables the impeller were set andas diffuser design variables, the computing resource consumption would increase and optimization speed would decline. So, the number of design variables had to be reduced. Considering the importance of impellers for hydraulic performance, the total number of design variables should be reduced by reducing geometry variables of the diffuser. Orthogonal test design [24] is a kind of economical and fast design method, which was used for a multi factor with a different level. Some representative designs with evenly dispersed and comparable were selected. Then the range analysis based on these representative designs was obtained to evaluate the qualitative contribution of each geometry variables on the optimization

Processes 2020, 8, 309 9 of 22 at the same time in this study. However, if all these eight geometry variables were set as design variables, the computing resource consumption would increase and optimization speed would decline. So, the number of design variables had to be reduced. Considering the importance of impellers for hydraulic performance, the total number of design variables should be reduced by reducing geometry variables of the diffuser. Orthogonal test design [24] is a kind of economical and fast design method, which was used for a multi factor with a different level. Some representative designs with evenly dispersed and comparable were selected. Then the range analysis based on these representative designs was obtained to evaluate the qualitative contribution of each geometry variables on the optimization objective. Finally, the geometry variables that were influential on the optimization objective were selected as design variables.

3.3.1. Orthogonal Test Scheme

Table2 shows that ϕ1, ϕ5, α1 and α5 were set as factor A, B, C and D respectively. Each of these factors was considered at ﬁve levels in the orthogonal test. Table3 shows the orthogonal test scheme, and the correspondence between the serial number of the factor and geometric parameters. Three optimization objectives under the orthogonal test scheme were obtained by numerical simulation.

Table 2. Level of the orthogonal factor.

Factor Level A B C D α5 α1 ϕ5 ϕ1 Level 1 70 85 26 4.5 Level 2 65 80 30 6.5 Level 3 60 75 34 8.5 Level 4 55 70 38 10.5 Level 5 50 65 42 12.5

Table 3. Orthogonal test scheme.

Test Factor Corresponding Parameter Optimization Objective

Number ABCD α5 α1 ϕ5 ϕ1 η1/(%) η2/(%) H3/(m)

1 A1 B1 C1 D1 70 85 26 4.5 73.62 54.04 7.30 2 A1 B2 C2 D2 70 80 30 6.5 75.43 53.30 7.21 3 A1 B3 C3 D3 70 75 34 8.5 76.86 52.30 7.12 4 A1 B4 C4 D4 70 70 38 10.5 77.85 51.12 6.99 5 A1 B5 C5 D5 70 65 42 12.5 78.90 49.69 6.88 6 A2 B1 C2 D3 65 85 30 8.5 75.98 52.52 7.17 7 A2 B2 C3 D4 65 80 34 10.5 77.37 51.52 7.06 8 A2 B3 C4 D5 65 75 38 12.5 78.79 50.31 6.98 9 A2 B4 C5 D1 65 70 42 4.5 77.75 51.23 6.99 10 A2 B5 C1 D2 65 65 26 6.5 77.32 51.96 7.03 11 A3 B1 C3 D5 60 85 34 12.5 77.74 50.73 7.00 12 A3 B2 C4 D1 60 80 38 4.5 76.43 51.75 7.03 13 A3 B3 C5 D2 60 75 42 6.5 78.22 50.53 6.94 14 A3 B4 C1 D3 60 70 26 8.5 77.92 51.34 7.01 15 A3 B5 C2 D4 60 65 30 10.5 79.38 49.85 6.92 16 A4 B1 C4 D2 55 85 38 6.5 76.96 50.95 6.98 17 A4 B2 C5 D3 55 80 42 8.5 78.45 49.66 6.87 18 A4 B3 C1 D4 55 75 26 10.5 78.54 50.45 6.96 19 A4 B4 C2 D5 55 70 30 12.5 79.82 48.90 6.85 20 A4 B5 C3 D1 55 65 34 4.5 77.64 49.81 6.79 21 A5 B1 C5 D4 50 85 42 10.5 78.77 48.56 6.80 22 A5 B2 C1 D5 50 80 26 12.5 79.06 49.67 6.91 23 A5 B3 C2 D1 50 75 30 4.5 77.19 50.54 6.87 24 A5 B4 C3 D2 50 70 34 6.5 78.47 49.07 6.76 25 A5 B5 C4 D3 50 65 38 8.5 80.34 47.07 6.69 Processes 2020, 8, 309 10 of 22

3.3.2. Range Analysis

Ki, Ki and R were set as the evaluation index of the range analysis. Ki is the sum value of each level for a certain factor. Ki is the average value of Ki, which can reﬂect the optimal levels of a certain factor for η1, η2 and H3. R stands for the maximum range that the optimization objective changes as the factor changes. The larger the range size, the greater the qualitative contribution of the factor to the optimization objective. These evaluation indexes can be calculated as follows:

X5 Ki = Yij (17) j=1

5 1 X K = Y (18) i 5 ij j=1 R = max K , K , K , K , K min K , K , K , K , K (19) 1 2 3 4 5 − 1 2 3 4 5 where Yij stands for the optimization objective of the i factor in level j. Ki is the sum value of each level for a certain factor. Ki is the average value of Ki. Table4 shows the range analysis of the pump e fficiency under a forward design condition. The qualitative contribution of the factor to η1 according to R was arranged from high to low: D, A, B and C. Table5 shows the range analysis of the pump e fficiency under the reverse design condition. The qualitative contribution of the factor to η2 according to R was arranged from high to low: A, B, D and C. Table6 shows the range analysis of the sum of the head under the forward design condition and the head under the reverse design condition. The R was so little from the range analysis of H3. So, the qualitative contribution of the factor to H3 was not obvious. In summary, factor D (ϕ1) and A (α5) had the most important influence on η1 and η2 based on R. So, ϕ1 and α5 were set as design variables in the next combination optimization of the impeller and diffuser.

Table 4. Range analysis of η1

Test Indices D A B C

K1 382.63 382.66 383.08 386.45 K2 386.41 387.21 386.73 387.81 K3 389.54 389.70 389.60 388.08 K4 391.90 391.41 391.81 390.37 K5 394.32 393.83 393.59 392.10 K1 76.53 76.53 76.62 77.29 K2 77.28 77.44 77.35 77.56 K3 77.91 77.94 77.92 77.62 K4 78.38 78.28 78.36 78.07 K5 78.86 78.77 78.72 78.42 R 2.34 2.23 2.10 1.13

Table 5. Range analysis of η2

Test Indices A B D C

K1 260.45 256.80 257.38 257.45 K2 257.55 255.90 255.81 255.10 K3 254.20 254.14 252.88 253.44 K4 249.76 251.65 251.50 251.21 K5 244.91 248.38 249.30 249.68 K1 52.09 51.36 51.48 51.49 K2 51.51 51.18 51.16 51.02 K3 50.84 50.83 50.58 50.69 K4 49.95 50.33 50.30 50.24 K5 48.98 49.68 49.86 49.94 R 3.11 1.68 1.62 1.55 Processes 2020, 8, 309 11 of 22

Table 6. Range analysis of H3

Test Indices A B C D

K1 35.51 35.25 35.22 34.98 K2 35.24 35.09 35.03 34.93 K3 34.90 34.87 34.72 34.85 K4 34.45 34.60 34.67 34.73 K5 34.03 34.31 34.49 34.62 K1 7.10 7.05 7.04 7.00 K2 7.05 7.02 7.01 6.99 K3 6.98 6.97 6.94 6.97 K4 6.89 6.92 6.93 6.95 K5 6.81 6.86 6.90 6.92 R 0.30 0.19 0.14 0.07

3.4. Latin Hypercube Sampling LHS [25] is a method of approximately random sampling. This method can divide the sampling unit into diﬀerent layers and randomly sample from diﬀerent parts based on a certain ratio. In this study LHS was applied to obtain design points to construct the surrogate model. A lack of enough design points can result in losing some regions of space design since large sample properties of simulations are required during LHS. So, the LHS was used to generate 120 groups of design points to ensure a full display of the design space. Among these sample points, 90 groups of design points were selected as training data and 30 groups of design points were used as test data.

3.5. Multilayer Artificial Neural Networks The approximation model can build the function between the design variables and optimization objective based on the design points obtained from DOE. The optimization objective can be predicted by this function. Common approximation models include RSM [26], Kriging Model [27], radial basis function (RBF) [28] and ANN [29]. ANN is a kind of computing system inspired by the biological neural networks. It can be divided into several layers and the function between inputs and outputs can be constructed through layer-by-layer mapping. As shown in Figure8, the first layer is called the input layer and the last layer is called the output layer. Layers between the input layer and output layer are collectively called the hidden layers. The mapping relationship between layers is called the transfer function. The multilayer ANN with more than one hidden layer can build a strong non-linear function between inputs and outputs. The kind of ANN can be divided into a feedforward neural network and cascade-forward neural network [30]. In this study, the multilayer cascade-forward ANN was applied as shown in Figure6. The quantity of neurons in the first and second hidden layers is 6 and 6 respectively. In addition, the transfer function between the input layer and hidden layer and that between the hidden layer and output layer are the tansig function and purlin function respectively, which was defined as follows:   Xm Xn 2 2  1 1 1 2 y = purlin( w tan sig w u + b ) + b (20) j ×  i,j i j  j=1 i=1

2 tan sig(x) = 1 (21) 2x 1 + e− − purlin(x) = ax + d (22) where, superscript 1 and 2 stands for the coefficients from the first and second hidden layer. w and b mean the weight and biases for every neuron respectively. a and d is the constant coefficient. Processes 2020, 8, 309 13 of 24

purlin(x) = 푎x + 푑 (22)

Processeswhere,2020 superscript, 8, 309 1 and 2 stands for the coefficients from the first and second hidden layer. 12푤 ofand 22 푏 mean the weight and biases for every neuron respectively. 푎 and 푑 is the constant coefficient.

Input Layer Hidden Layer Output Layer

......

Figure 8. The structure of the cascade-forward multilayer neural network. Figure 8. The structure of the cascade-forward multilayer neural network. In order to prove that the cascade-forward two-layer ANN built function with high accuracy betweenIn order the design to prove variables that the and cascade optimization-forward objectives, two-layer theANN prediction built function accuracy with of high optimization accuracy objectivebetween comparedthe design withvariables other and common optimization approximation objectives, models the prediction is shown accuracy in Table 7of. optimization Compared withobjective other compared common approximatewith other common models, approxi a two-layermation ANN models model is shown can construct in Table the7. Compared function withwith highestother common accuracy approximate between design models, variables a two and-layer optimization ANN model objectives. can construct The the R-squared function was with set highest as the evaluationaccuracy between basis of the design prediction variables accuracy and andoptimization it is defined objectives. as follows: The R-squared was set as the evaluation basis of the prediction accuracy and it is defined as follows: Xn 푛 2 SSE = τi(yi yî) (23) − 2 푆푆퐸 =i=∑1 𝜏푖(푦푖 − 푦̂푖) (23) 푖=1 n X푛 2 SST = τi yi y (24) 푆푆푇 = ∑ 𝜏 (푦 −− 푦̅i )2 i=1 푖 푖 푖 (24) 푖=1 P 2 (y yˆ) 2 R2 = 1 ∑(푦−− 푦̂) (25) 푅2 = 1 −P 2 (25) − ∑((y푦 −y푦)̅)2 − wherewhereSSE 푆푆퐸, SST, SSTand andR2 mean푅2 mean the sumthe ofsum squares of squares due to duethe error, to the total error, sum total of squares sum of and squares coefficient and ofcoefficient determination, of determination respectively., respectively.y, yˆ and y stand 푦, for푦̂ and the outputs푦̅ stand offor the the sample outputs points, of the predicted sample outputs points, bypredicted approximation outputs modelsby approximation and average models outputs and for average sample outputs points. for sample points.

TableTable 7.7. TheThe accuracyaccuracy ofof optimizationoptimization objectiveobjective predictedpredicted byby didifferentﬀerent approximation approximation models. models.

ퟐ2 ퟐ2 ퟐ 2 ퟐ 2 Approximation Model Model 푹R ofofη 1휼/%ퟏ/% 푹R of η휼2/ퟐ%/% 푹R ofof H푯1ퟏ/m/m 푹R ofof H푯2ퟐ//mm Two-layerTwo-layer ANN ANN Model Model 0.9820.982 0.990 0.993 0.993 0.996 0.996 RSMRSM 0.9130.913 0.962 0.970 0.970 0.994 0.994 KrigingKriging Model Model 0.7920.792 0.953 0.938 0.938 0.948 0.948 RBF Model 0.905 0.983 0.972 0.993 RBF Model 0.905 0.983 0.972 0.993

3.6. NSGA for Multiobjective Optimization The non-dominated sorting genetic algorithm with the elitism algorithm (NSGA-II) was invoked from the genetic algorithm for multiobjective optimization. The advantage of this optimization algorithm is that the elitism sorting algorithm can reduce computational complexity. NSGA-II [31] is

Processes 2020, 8, 309 14 of 24 one of the popular multiobjective evolutionary algorithms. In this study, the objective function values were calculated by using two-layer ANN models. The size of the population and total number of the generation are 300 and 100, respectively. So, the NSGA-II was used to solve these approximate models. There were 300 Pareto-optimal solutions for the three optimization objectives as shown in Figure 9. The Pareto-optimal solution is a set of solutions that are not inferior to all other feasible solutions. Generally it refers to improving the objective functions of a multiobjective problem while degrading some other objectives values. This is also referred to as non-dominated [32]. The optimized solutions were selected according to design requirements. In order to illustrate the relationship between the optimization objectives in detail, the Pareto-optimal solutions in 2D space was shown in Figure 10.

Figure 10a shows the trade-off among Pareto-optimal solutions in the 휂1– 퐻3 space, where 휂1 increased with 퐻3 decreasing. Additionally, Figure 10b shows a trade-off among Pareto-optimal solutions in 휂 –휂 space, where 휂 increased with 휂 decreasing. However, Figure 10c shows a Processes 2020, 8, 3091 2 1 2 13 of 22 weak relationship among Pareto-optimal solutions in 휂2– 퐻3 space. 휂1, 휂2 and 퐻3 cannot be improved at the same time, but the sum of 휂1 and 휂2 can be improved after optimizing the pump 3.6.and NSGA maintain foring Multiobjective a certain Optimizationlift. One representative Pareto-optimal solution was chosen based on Equations (26) and (27). The 휂 , 휂 and 퐻 predicted by NSGA-II were 78.13%, 61.01% and 8.07 m. The non-dominated sorting1 genetic2 algorithm3 with the elitism algorithm (NSGA-II) was invoked This representative Pareto-optimal solution was considered as the finally optimization design after a from the genetic algorithm for multiobjective optimization. The advantage of this optimization minor adjustment of the design variables based on experience. algorithm is that the elitism sorting algorithm can reduce computational complexity. NSGA-II [31] is one of the popular multiobjective evolutionaryMaximum: algorithms. 퐻3 In this study, the objective function values(26) were calculated by using two-layer ANN models. The size of the population and total number of the 휂1 > 78 Constraint: { II (27) generation are 300 and 100, respectively. So, the NSGA-휂2 >was61 used to solve these approximate models. There were 300 Pareto-optimal solutions for the three optimization objectives as shown in Figure9.

Figure 9. Pareto-optimal solutions in 3D function-space. Figure 9. Pareto-optimal solutions in 3D function-space. The Pareto-optimal solution is a set of solutions that are not inferior to all other feasible solutions. Generally it refers to improving the objective functions of a multiobjective problem while degrading some other objectives values. This is also referred to as non-dominated [32]. The optimized solutions were selected according to design requirements. In order to illustrate the relationship between the optimization objectives in detail, the Pareto-optimal solutions in 2D space was shown in Figure 10. Figure 10a shows the trade-off among Pareto-optimal solutions in the η1–H3 space, where η1 increased with H3 decreasing. Additionally, Figure 10b shows a trade-off among Pareto-optimal solutions in η1–η2 space, where η1 increased with η2 decreasing. However, Figure 10c shows a weak relationship among Pareto-optimal solutions in η2–H3 space. η1, η2 and H3 cannot be improved at the same time, but the sum of η1 and η2 can be improved after optimizing the pump and maintaining a certain lift. One representative Pareto-optimal solution was chosen based on Equations (26) and (27). The η1, η2 and H3 predicted by NSGA-II were 78.13%, 61.01% and 8.07 m. This representative Pareto-optimal solution was considered as the finally optimization design after a minor adjustment of the design variables based on experience. Maximum : H3 (26) ( η > 78 Constraint : 1 (27) η2 > 61 Processes 2020, 8, 309 14 of 22 Processes 2020, 8, 309 15 of 24

(a) (b)

(c) Figure 10. Pareto-optimal solutions in 2D function-space: (a) 휂 – 퐻 , (b) 휂 – 휂 ⁡and (c) 휂 – 퐻 Figure 10. Pareto-optimal solutions in 2D function-space: (a) η1–H3,, (b) η11–η22and (c) η2–2H3.3

4.4. Results and Analysis

4.1.4.1. Comparison of Pump Performance FigureFigure 1111 showsshows thethe comparisoncomparison ofof pumppump eefficiencyfficiency andand headhead betweenbetween thethe originaloriginal designdesign andand optimizedoptimized design. design. Under Under the the forward forward condition, condition, the tendency the tendency of performance of performance curves for curves the optimized for the designoptimized was design similar was with similar that for with the original that for design, the original and the design, high e ffiandciency the high zone wasefficiency obtained zone under was 3 Qobtained= [0.36, under0.4] (m 푄/s). = The [0.36, pump 0.4] performance (m3/s). The underpump the performance forward condition under the and forward reverse condition condition could and not be improved at the same time. In order to take η2 and H2 into consideration during the optimization reverse condition could not be improved at the same time. In order to take 휂2 and 퐻2 into design,consideration forward during pump the performance optimization under design, all flowrates forward slightly pump dropped performance after optimization. under all flowrates The η1 and H1 of the optimized design were 77.61% and 4.21 m. However, under the reverse condition, slightly dropped after optimization. The 휂1 and 퐻1 of the optimized design were 77.61% and 4.21 them. However, improvement under of pump the reverse efficiency condition, and the the pump improvement head were soof obviouspump efficiency and the highand ethefficiency pump range head was also widened. The η2 and H2 of the optimized design were 61.58% and 3.8 m. were so obvious and the high efficiency range was also widened. The 휂2 and 퐻2 of the optimized design were 61.58% and 3.8 m.

Processes 2020,, 8,, 309309 1615 of of 2224

Processes 2020, 8, 309 16 of 24

Original η Optimized η Original η Optimized η OriginalOriginal H η OptimizedOptimized Hη OriginalOriginal η H OptimizedOptimized η H 85.00 Original H Optimized5.50 H 65.00 Original H Optimized5.50 H 85.00 5.50 65.00 5.50 5.00 60.00 80.00 4.50 5.00 60.00 80.00 4.50 4.50 55.00

75.00 4.50 55.00 3.50

/m /m

75.00 /% 3.50

/% 4.00 50.00

η η

/m /%

70.00/% 4.00 50.00 2.50

η η H 70.00 3.50 H 45.00 2.50 3.50 45.00 65.00 1.50 65.00 3.00 40.00 1.50 3.00 40.00 60.00 2.50 35.00 0.50 60.000.30 0.33 0.36 0.39 0.42 2.50 35.00 0.23 0.26 0.29 0.32 0.350.50 0.30 0.33 0.36 0.39 0.42 0.23 0.26 0.29 0.32 0.35 Q /(m3/s) Q /(m3/s) Q /(m3/s) Q /(m3/s)

(a()a ) (b(b) )

FigureFigure 11. 11. Comparison Comparison of of pump pump performanceperformance between the the original original and and optimized optimized design: design: (a )( aunder) under thethe forward forward condition condition and and (( b(b)) ) underunder under thethe the reversereversereverse condition.condition. 4.2. Analysis of Inner Flow 4.2.4.2. Analysis Analysis of ofInner Inner Flow Flow Figure 12 shows the turbo surface of the impeller with diﬀerent span, and the span was deﬁned as FigFigureure 12 12 shows shows the the turbo turbo surface surface ofof thethe impellerimpeller withwith differentdifferent s pan,span, and and the the span span was was defined defined Equationas asEquation Equation (28). (28). (28). Span = (r r )/(rt r ) (28) − h − h SpanSpan == (푟 − 푟ℎ))//((푟푟푡푡−−푟푟ℎℎ)) (28()28 ) where, r is the calculated ring radius, rh is the hub radius and rt is the radius of the impeller rim. 푟 푟 where,where, r is r isthe the calculated calculated ring ring radius, radius, 푟ℎℎ isis thethe hub radius andand 푟푡푡 isis the the radius radius of of the the impeller impeller rim. rim. Span==00..55 SpanSpan==00.9.9

Span=0.1 Span=0.1 Impeller Impellerblade blade

FigureFigure 12. 12. TheThe turbo surface with with different diﬀerent s spans.pans. Figure 12. The turbo surface with different spans. FigureFigure 13 13 shows shows the the velocityvelocity axial distribution distribution in in the the expansion expansion diagram diagram of the of turbo the turbo surface surface of ofthe theFig impeller impellerure 13 shows with with s pan spanthe velocity= =0.5,0.5, under axial under the distribution the forward forward design in designthe flowexpansion ﬂow rate. rate. In diagram the In expansion the of expansionthe turbo diagram, surface diagram, the of thevertical verticalimpeller axis axis with w wasas sparallelpan parallel = 0.5, to to the under the impeller impeller the forward rotation rotation designaxis axis and and flow the the rate. horizontal horizontal In the axis expansion axis was was parallel parallel diagram, to tothe the circumferentialverticalcircumferential axis was direction directionparallel of toof the the the impeller. impeller. impeller TheThe rotation Ps and axis Ss Ss stands standsand the for for horizontalthe the pressure pressure axis side side was and and parallelsuction suction side. to side. the IncircumferentialIn both both the the original original direction and and optimized ofoptimized the impeller. design,design, The thethe Ps high and velocity Ss stands was was for obse observedtherved pressure near near theside the suction and suction suction side side and side. and theIn theboth low low velocitythe velocityoriginal near and near the optimized leadgingthe leadging edge design, due edge the to due the high presence to velocity the presence of was the stagnation obse of rvedthe stagnationnear point. the After suction point. optimization, side After and the velocity low velocity increased near at the pressureleadging side edge and due appeared to the to presence be declining of the near stagnation the suction point. side. After

Processes 2020, 8, 309 17 of 24

optimization, the velocity increased at the pressure side and appeared to be declining near the suction Processes 2020, 8, 309 16 of 22 side.

Low velocity near leading edge

-1 Vz /(m·s )

Original design

Optimized design

Figure 13. Comparison of the velocity axial distribution in the impeller between the original design Figure 13. Comparison of the velocity axial distribution in the impeller between the original design and optimized design under the forward design ﬂow rate (span = 0.5). and optimized design under the forward design flow rate (span = 0.5).

FigureFig 14urea 14a shows shows the the axial axial velocity velocity distribution distribution in in the the impeller impeller outlet outlet under under the theforward forward design design flowflow rate. rate. The The trend trend of theof the velocity velocity curve curve inin thethe optimized design design was was similar similar with with that thatof the of original the original design.design. The The low low axial axial velocity velocity could could be be seen seen nearnear s spanpan == 0 0and and span span = 1.0= 1.0due due to the to viscous the viscous effects eff ects causedcaused by the by frictionthe friction between between the the fluid fluid and and wall.wall. In In the the middle middle interval, interval, the the axial axial velocity velocity increased increased slowlyslowly with with increasing increasing span. span. In In addition, addition, whenwhen s spanpan << 0.6,0.6, the the axial axial velocity velocity in the in theoriginal original design design was smallerwas smaller than th thatan that of theof the optimized optimized design, design, butbut when s spanpan > >0.6,0.6, the the axial axial velocity velocity in the in original the original designdesign was was larger larger than than that that in in the the optimized optimized design.design. Fig Figureure 14b 14 bshows shows the the circulation circulation distribution distribution in in the impeller outlet. The circulation was defined as Equation (29). The tendency of the velocity curve the impeller outlet. The circulation was defined as Equation (29). The tendency of the velocity curve in in the optimized design was also similar with that in the original design. The circulation rose slowly the optimized design was also similar with that in the original design. The circulation rose slowly with with span increasing firstly and then it increased rapidly with span increasing. Furthermore, the spancirculation increasing in firstly the optimized and then design it increased was smaller rapidly than with that span of the increasing. original design Furthermore,, which would the circulationcause in thethe optimized pump head design to decline was after smaller optimization. than that of the original design, which would cause the pump head to decline after optimization. Z 퐶 = ∫ 풗 푑풍 (29) Cuu = v푢udl (29) 퐿 L where, 푣푢 is the absolute circumferential velocity and 푑푙 is the unit length of the ring. where, vuFigisure the 15 absolute shows circumferentialthe axial velocity velocitydistribution and indl theis theexpansion unit length diagram of the of ring.the turbo surface Figurewith span 15 =shows 0.5 under the the axial forward velocity design distribution flow rate. inAs theshown expansion in the figure, diagram both ofin the the original turbo surface design with spanand= 0.5 optimized underthe design, forward the difference design flow of the rate. flow Asstructure shown near in theeach figure, diffuser both blade in was the so original obvious. design and optimizedThe high velocity design, could the di beff erence found near of the some flow diffuser structure blades near and each the di obviousffuser blade low velocity was so was obvious. The highobserved velocity near couldthe rest be diffuser found nearblades, some which diff coulduser bladesresult in and hydraulic the obvious losses. low velocity was observed near the rest diffuser blades, which could result in hydraulic losses.

Processes 2020, 8, 309 18 of 24

Optimized design Original design Optimized design Original design Processes 2020, 8, 309 17 of 22 Processes 2020, 8, 309 18 of 24 8.00 12.20

6.00 Optimized design Original design Optimized design Original design 8.60

8.00 /s) 12.20 2

4.00

/(m/s)

/(m z

6.00 u V

C 8.605.00 /s) /s)

2.00 2

4.00

/(m/s)

/(m

u V 0.00 C 5.001.40 2.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Span normalized Span normalized 0.00 1.40 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Span(a) normalized Span (normalizedb)

Figure 14. Comparison in the impeller outlet between the original and optimized design for the (a) (a) (b) velocity axial distribution and (b) circulation distribution (under the forward design flow rate). FigureFigure 14. 14.Comparison Comparison in in thethe impellerimpeller outlet outlet between between the theoriginal original and optimized and optimized design designfor the (a for) the (a) velocityvelocity axial axial distributiondistribution and and (b ()b circulation) circulation distribution distribution (under (under the forward the forward design design flow rate). ﬂow rate).

-1 Vz /(m·s ) -1 Vz /(m·s )

OriginalOriginal designdesign

Optimized design Optimized design Figure 15. Comparison of the velocity axial distribution in the diffuser between the original and Figure 15. Comparison of the velocity axial distribution in the diffuser between the original and Figureoptimized 15. Comparison design under of thethe forward velocity design axial flow distribution rate (span = in 0.5). the diffuser between the original and optimized design under the forward design flow rate (span = 0.5). optimized design under the forward design flow rate (span = 0.5). Figure 16 is the distribution of the axial velocity in the expansion diagram of the turbo surface withFigure span 16 =is 0.5 the under distribution the reverse of design the axial flow velocity rate. These in thetwo expansion designs show diagram a similar of flow the turbostructure surface Figure 16 is the distribution of the axial velocity in the expansion diagram of the turbo surface withwith span weak= 0.5 velocity under thenear reverse the leading design edge flow, which rate. was These due to two the designs presence show of the a stagnation similar flow point structure of with span = 0.5 under the reverse design flow rate. These two designs show a similar flow structure withthe weak leading velocity edge, near as well the as leading the high edge, velocity which near was the due suction to the side. presence In the optimized of the stagnation design, the point area of the with weak velocity near the leading edge, which was due to the presence of the stagnation point of leadingof the edge, high as velocity well as near the highthe suction velocity side near bec theame suction larger, and side. the In velocity the optimized near the design, pressure the side area of the leading edge, as well as the high velocity near the suction side. In the optimized design, the area the high velocity near the suction side became larger, and the velocity near the pressure side became of the high velocity near the suction side became larger, and the velocity near the pressure side lower. The velocity gradient near the pressure side in the original design was larger than that in the optimized design.

Processes 2020, 8, 309 19 of 24

became lower. The velocity gradient near the pressure side in the original design was larger than that in the optimized design. Figure 17 shows the comparison of the velocity in the impeller outlet between the original and optimized design. As shown in Figure 17a, the original impeller had a non-uniform axial velocity distribution, and it first increased, then decreased with increasing span in the middle interval. The Processesaxial velocity 2020, 8, 309of the optimized design remained almost stable in the middle interval, which meant19 of 24 that the impeller outflow was much more stable. As shown in Figure 17b, the tendency of the beccirculationame lower. curve The forvelocity the optimized gradient near design the waspressure similar side with in the that original for the design original was design.larger than The that incirculation the optimized initially design. rose slowly and later became rapid with increasing span. The optimized impeller outletFig howeverure 17 shows had higher the comparison circulation whenof the svelocitypan > 0.25 in andthe impellerit could result outlet in between a higher the pump original head and Processesoptimizedcompared2020, 8 ,withdesign. 309 the Asoriginal shown design. in Fig ure 17a, the original impeller had a non-uniform axial 18velocity of 22 distribution, and it first increased, then decreased with increasing span in the middle interval. The axial velocity of the optimized design remained almost stable in the middle interval, which meant Low velocity near that the impellerleading outflow edge was much more stable. As shown in Figure 17b, the tendency of the circulation curve for the optimized design was similar with that for the original design. The -1 circulation initially rose slowly and later becameVz /( m·srapid) with increasing span. The optimized impeller outlet however had higher circulation when span > 0.25 and it could result in a higher pump head compared with the original design.

Low velocityPs near leading edge Ss -1 Vz /(m·s )

Original design

Optimized design

Original design Figure 16. Comparison of the velocity distribution in the impeller between the original design and optimizedFigure 16. design Comparison under the of reversethe velocity design distribution flow rate in (span the impeller= 0.5). between the original design and optimized design under the reverse design flow rate (span = 0.5). Figure 17 shows the comparison of the velocity in the impeller outlet between the original and optimized design. As shown in Figure 17a, the original impeller had a non-uniform axial velocity Optimized design Original design Optimized design Original design distribution, and it first increased, then decreased with increasing span in the middle interval. The axial velocity of6.30 the optimized design remained almost stable14.50 in the middle interval, which meant that the impeller outflow was much more stable.Optimized As shown design in Figure 17b, the tendency of the circulation

curve for the4.20 optimized design was similar with that for10.00 the original design. The circulation initially /s) /s)

rose slowlyFigure and 16. laterComparison became of rapid the velocity with increasing distribution span. 2 in the The impeller optimized between impeller the original outlet howeverdesign and had /(m/s) higher circulation when span > 0.25 and it could result/(m in a higher pump head compared with the

optimizedz design under the reverse design flow rate (span = 0.5).

u V original design.2.10 C 5.50

Optimized design Original design Optimized design Original design 0.00 1.00 6.30 0.00 0.25 0.50 0.75 1.00 14.500.00 0.25 0.50 0.75 1.00 Span normalized Span normalized

4.20 10.00

(a) /s) (b) 2

/(m/s)

/(m

u V

2.10 C 5.50

0.00 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Span normalized Span normalized

(a) (b)

Figure 17. Comparison in the impeller outlet between the original and optimized design for the (a) axial velocity distribution and (b) circumferential velocity distribution (under the reverse design flow rate). Processes 2020, 8, 309 20 of 24

4.3. Validation of the Optimization Results In order to prove the validity of the optimization design and accuracy of the numerical simulation, the external characteristic test for the optimized design was carried out. Figure 18 shows the vertical bench test of the axial-flow pump and the flow rate was regulated by the gate valve. The main technical parameters of the measurement equipment were shown in Table 8. Due to the Processes 2020structure, 8, 309 of the test bench, the rotation speed of 1350 r/min was used for the external characteristic 19 of 22 test. Then the experiment data was transformed by the equivalent efficiency formula to compare with the numerical simulation as shown in Figure 19. The performance curves from the numerical 4.3. Validationsimulation of the had Optimization good agreement Results with experiment data, and the deviation of η1 and η2 were about 1.23% and 3.56%. In addition, the main design parameters of this optimized reversible axial-flow In orderpump to is prove shown the in validityTable 9. The of theresults optimization shows that the design reverse and pump accuracy performance of the was numerical improved simulation, the externalactually, characteristic at the same test time for of maintaining the optimized the forward design pump was performance. carried out. Figure 18 shows the vertical bench test of the axial-flow pumpTable 8. andMain thetechnical flow parameters rate was of regulatedmeasurement by equipment. the gate valve. The main technical parameters of the measurement equipment were shown in Table8. Due to the structure of the test bench, Measurement Instrument Measurement Measurement Equipment Name the rotation speedItems of 1350 r/min was used for the externalModel characteristicRange test. ThenAccuracy the experiment data was transformed by the equivalentIntelligent effi ciency formula to compare with the numerical simulation as shown in FigureFlow 19 rate. The performanceelectromagnetic curves fromOPTIFLUX2000F the numerical 0– simulation1800 m3/h had good<±0.2% agreement with flowmeter experiment data, and the deviation of η1 and η2 were about 1.23% and 3.56%. In addition, the main Intelligent differential Head EJA 0–10 m <±0.1% design parameters of this optimizedpressure transmitter reversible axial-flow pump is shown in Table9. The results shows that the reverseTorque pump performanceIntelligent was torque improved actually, at the same time of maintaining the forward JCL1 0–200 N·m <±0.1% pump performance.Rotation speed speed sensor

Processes 2020, 8, 309Figure Figure 18. The 18. The experiment experiment bench of of the the axial axial-ﬂow-flow pump. pump. 21 of 24

EXP η CFD η EXP H CFD H 90.00 8.00

70.00 6.00 /m

/% 50.00 4.00

η H H 30.00 2.00

10.00 0.00 0.16 0.21 0.26 0.31 0.36 0.41 0.46 Q /(m3/s)

(a) CFD η EXP η EXP H CFD H 70.00 8.00

6.00

50.00 /m

/% 4.00

η H 30.00 2.00

10.00 0.00 0.18 0.22 0.26 0.30 0.34 0.38 Q /(m3/s)

(b)

Figure 19. The comparison of pump performance between the simulated data and experiment data Figure 19. The comparison of pump performance between the simulated data and experiment data under the (a) forward operation and (b) reverse operation. under the (a) forward operation and (b) reverse operation. Table 9. The main design parameters of the optimized reversible axial-flow pump.

Design Parameters Value Forward rated flowrate (m3/s) 0.368 Efficiency under forward rated condition (%) 78.73 Head under forward rated condition (m) 3.72 Reverse rated flowrate (m3/s) 0.298 Efficiency under reverse rated condition (%) 63.85 Head under reverse rated condition (m) 3.51 Rotation speed (rev/min) 1450

5. Conclusions In this paper, a multiobjective optimization was performed to improve the total pump performance under the forward and reverse design condition. Some conclusions could be drawn.

(1) α5 and 휑1 had the most important effect on the pump performance, according to the orthogonal experimental design. 퐿1, 퐿5, 휃1, 휃5, 훼5 and 휑1 were set as the final design variables for the combination optimization of the impeller and diffuser. (2) Compared with other approximate models, the two-layer ANN could construct the functions with highest accuracy between the design variables and optimization objectives, based on 120 groups of sample points generated from LHS. (3) 300 Pareto-optimal solutions were obtained by NSGA and one representative Pareto-optimal

solution was selected as the basis of the optimized design. The 휂1 and 퐻1 of the optimized

Processes 2020, 8, 309 20 of 22

Table 8. Main technical parameters of measurement equipment.

Measurement Measurement Measurement Equipment Name Instrument Model Items Range Accuracy Intelligent electromagnetic Flow rate OPTIFLUX2000F 0–1800 m3/h < 0.2% ﬂowmeter ± Intelligent diﬀerential Head EJA 0–10 m < 0.1% pressure transmitter ± Torque Intelligent torque speed JCL1 0–200 N m < 0.1% Rotation speed sensor · ±

Table 9. The main design parameters of the optimized reversible axial-ﬂow pump.

Design Parameters Value Forward rated flowrate (m3/s) 0.368 Efficiency under forward rated condition (%) 78.73 Head under forward rated condition (m) 3.72 Reverse rated flowrate (m3/s) 0.298 Efficiency under reverse rated condition (%) 63.85 Head under reverse rated condition (m) 3.51 Rotation speed (rev/min) 1450

5. Conclusions In this paper, a multiobjective optimization was performed to improve the total pump performance under the forward and reverse design condition. Some conclusions could be drawn.

(1) α5 and ϕ1 had the most important effect on the pump performance, according to the orthogonal experimental design. L1, L5, θ1, θ5, α5 and ϕ1 were set as the final design variables for the combination optimization of the impeller and diffuser. (2) Compared with other approximate models, the two-layer ANN could construct the functions with highest accuracy between the design variables and optimization objectives, based on 120 groups of sample points generated from LHS. (3) 300 Pareto-optimal solutions were obtained by NSGA and one representative Pareto-optimal solution was selected as the basis of the optimized design. The η1 and H1 of the optimized design decreased by 0.50% and 10.06%, respectively, compared with the original design. The axial velocity gradient near the suction side of the optimized diffuser was larger than that of the original diffuser. The circulation of the impeller outlet had decreased after optimization. However, the η2 and H2 of the optimized design increased by 18.62% and 60.40%, compared with the original design. In addition, the curve slope of the axial velocity and circulation in the impeller outlet of the optimized design decreased and increased after optimization. (4) The external characteristic test for the optimized design was completed to prove the validity of this optimization method. In addition, the test highest efficiency values under the forward condition and reverse condition were 78.73% and 63.85% respectively, which could meet engineering needs.

The design method in this paper could be used as a reliable and high time-efficient methodology to design the reversible axial-flow pump based on a one-way axial-flow pump.

Author Contributions: Conceptualization, F.M.; Data curation, F.M.; Methodology, F.M.; Project administration, Y.L., S.Y. and W.W.; Supervision, Y.L.; Validation, F.M.; Writing—original draft, F.M.; Writing—review & editing, Y.Z. and M.K.O. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Key Research and Development Plan (Grant No.2018YFB0606103), Science and Technology Plan of Wuhan (Grant No.2018060403011350). Primary Research and Development Plan of Jiangsu Province (Grant No. BE2019009-1). Natural Science Foundation of Jiangsu University (Grant No. BK20190851), China Postdoctoral Science Foundation funded project (Grant No. 2019M651736). Acknowledgments: The author sincerely thanks the 4Cpump researcher group for its help. Conﬂicts of Interest: The authors declare no conﬂict of interest. Processes 2020, 8, 309 21 of 22

Nomenclature

Q (m3/s) Flow rate H1 (m) Head under forward design condition H2 (m) Head under reverse design condition H3 (m) Sum of H1 and H2 η1 (%) Efficiency under forward design condition η2 (%) Efficiency under forward design condition N (r/min) Rotating speed θ (◦) Blade angle of impeller L (mm) Chord length of impeller ∅ (◦) Included angle between camber line and chord σ (◦) theoretical lift coefficient λ (◦) Outlet angle of skeleton line ε (◦) Inlet angle of skeleton line α (◦) Inlet angle of diffuser β (◦) Outlet angle of diffuser ϕ (◦) Wrap angle of diffuser 2 Cu (m /s) Circulation Vz (m/s) Axial velocity Abbreviations 3D Three-Dimensional RANS Reynolds-Averaged Navier–Stokes LHS Latin Hypercube Sampling ANN Artificial Neural Network NSGA Non-domination based Genetic Algorithm CFD Computational Fluid Dynamics DOE Design of Experiment RSM Response Surface Model EXP Experiment

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