The centrifugal pump studied in this paper is a low specific rpm centrifugal pump, and other main parameters are as follows Table 1.

In Eq. (1), the

Ds -Pump inlet diameter (m);

Q -Pump flow rate (m³/h);

Vs -Pump inlet flow rate (m/s);

Take the pump inlet flow rate Vs = 2.73 m/s [1], then the pump inlet diameter Ds = 80 mm.(2)

Pump outlet diameter and outlet speed

The outlet diameter of the plastic centrifugal pump is calculated according to the following formula:(2) Dd=(0.7−1)Ds

In Eq. (2), the Dd -Pump outlet diameter (mm).

The contact factor is 0.9, so the pump outlet diameter Dd = 68 mm.

The pump outlet velocity is calculated as follows:(3) Vd=4QπDd2

In Eq. (3), the Vd -Pump outlet flow rate m/s.

The calculation gives Vd = 3.826 m/s.(3)

Specific speed(4) ns=3.65nQH34

In Eq. (4), the ns -Ratio speed. H -Head (m).

After calculation, we get: ns = 61.78.(4)

Hydraulic efficiency of the pump

The hydraulic efficiency of the pump is calculated according to Eq. (5) as follows:(5) ηh≈1+0.083511gQn3

Bringing in the data is calculated as ηh = 0.852.(5)

Volume efficiency of the pump

The volumetric efficiency of the plastic pump is calculated according to Eq. (6)(6) ηv≈11+0.68ns−23

Bringing in the data is calculated as ηv = 0.96.(6)

Mechanical efficiency of the pump

The mechanical efficiency of the disc friction loss is calculated by Eq. (7)(7) ηm≈1−0.07(ns100)76

Bringing in the data is calculated as ηm = 0.88.(7)

Total efficiency of the pump

The total efficiency of the plastic pump is calculated according to Eq. (8)(8) η=ηhηvηm

Bringing in the data is calculated as η = 0.72.

Plastic centrifugal pump head calculation formula.(9) H=Pout−Pinρg+Δz

In Eq. (9), the

Pout -Total pressure at the volute outlet, Pa. Pin -Total impeller inlet pressure, Pa. Δz -Difference in height between pump inlet and outlet, m. Plastic centrifugal pump shaft power calculation formula.(10) P=M∗ω where M is the torque, N⋅m , and ω is the angular velocity, rad/s.

The structure form of centrifugal pump impeller is mainly semi-open type and closed type, among which semi-open type impeller has the advantages of high strength and easy molding. Therefore, the impeller used in this paper is semi-open type. The structure of the impeller is shown in Fig. 1.

This paper applies the velocity coefficient method to design and calculate the impeller structural parameters.(1)

Impeller inlet diameter

Impeller inlet equivalent diameter is calculated according to Eq. (11)(11) D0=k0Qn3

In Eq. (12), the k0 -Factor, take k0 = 4.5^[1].

Substituting into the above equation gives D0 = 78 mm.

For impeller inlet diameter calculated by Eq. (12).(12) Dj=D02+dh2

In Eq. (12), the dh -Wheel diameter (mm);

dh is valid only for structural design, for hydraulic design, dh = 0; then Dj=D0  = 78 mm, take Dj = 80 mm.

The blade inlet diameter is estimated as follows:(13) D1=k1Dj

In Eq. (13), the k1 -Coefficient, generally taken from k1 = 0.7 to 1.0.

The coefficient k1 , which is determined by the specific speed, is taken as a large value for low specific speed and a small value for high specific speed. According to this paper, the specific speed is taken as k1 = 0.85, then D1 = 68 mm.(2)

Impeller inlet width

The inlet width of the impeller blade is determined by Eq. (14)(14) b1=D14k1ξ1

In Eq. (14), the ξ1 -Ratio of impeller inlet speed to inlet diameter.

Generally take ξ1 = 0.9∼1.0, and substitute D1=115mm,k1 = 0.85 into the Eq. (14) to get b1 = 20 mm.(3)

Impeller outlet diameter

The blade inlet diameter is first preliminarily estimated(15) D2=kDQn3 (16) kD=9.35kD2(ns100)−12

In Eq. (15), the kD - D2 Correction coefficient.

kD is related to the type and specific speed of the pump, this paper uses a single-stage pump, and the specific speed ns = 61.78, so choose kD2 = 1.038. The calculation results are D2 = 210 mm.(4)

Impeller outlet width

The blade exit width is calculated by the following two equations:(17) b2=kbQn3

(18) kb=0.78kb2(ns100)56

In Eq. (17), the kb−b2 Correction coefficient.

kb is related to the type and specific speed of the pump, this paper uses a single-stage pump, and the specific speed ns  = 61.78, so kb2  = 1.236 is chosen. Substitute into the calculation to get b2 = 11 mm.(5)

Number of blades

The number of blades is calculated by Eq. (19)(19) Z=6.5D2+D1D2−D1sinb1+b22

The values of D1 , D2 , b1 and b2 are substituted into the calculation; when the impeller diameter is the same, increasing the number of blades will increase the head, but not change its flow rate value, and the integrated given head H = 50 m, so take Z = 6.(6)

Impeller inlet and outlet placement angle

In this paper, the blade inlet settling angle β₁ is 19°, the outlet settling angle is 33°, and the wrap angle φ is 123° [17].(7)

Blade thickness design

In the design of plastic impeller, the reduction of thickness makes the flow channel wider, the slip coefficient is reduced so that the pump shaft power is increased, but the thickness can not be unlimited small, otherwise because of the lack of strength makes the impeller deformation is too large to work; the same thickness will make the flow channel narrower, the hydraulic loss is increased, so the thickness of the blade in this paper is 2 mm.

The overflow parts of the pump include the impeller and the volute, the volute is also one of the overflow parts to convert energy, the main dimensions of the volute are designed as follows:(1)

Diameter of base circle D3

The diameter of the base circle is too large and too small, the performance of the pump will have an impact, here the base circle is represented by D3 , usually taken

(20) D3=(1.03∼1.08)D2

High than the speed and size of the smaller pump coefficient to take the larger value, and vice versa to take the smaller value, substitute the value to get D3 = 220 mm.(2)

Inlet width of volute b3

The width of the worm housing inlet b3 is determined taking into account the 8th section, to give the 8th section a reasonable geometry, usually similar to round and square.(21) b3=b2+0.05D2

The calculation gives b3 = 22 mm.(3)

Angle of volute septum placement

The diaphragm placement angle is expressed by φ0 , and according to [1], φ0 = 22° in this paper.(4)

Area of each cross section

Similar conversions using the velocity factor method(22) v3=k32gH

In Eq. (22), the v3− Average velocity of the volute section. H− The single-stage head of the pump. k3− The value of the velocity coefficient, k3 , was found to be 0.34 by according to [1].

The calculation gives v3 = 10.65 m/s.

The flow rate through the 8th section and the actual flow rate of the plastic centrifugal pump are approximately equal, so the area of the 8th section can be calculated by the following formula:(23) F8=Qv3

The rest of the cross-sectional area is calculated by equalizing the area of each cross-section(24) Fφ=φ360F8

The area of each section was obtained as shown in Table 2 below.

The inducer structure designed in this paper adopts the simpler conical variable pitch form, and its structural form is shown in Figs. 2 and 3.

Theoretically speaking, the number of blades of the inducer to take 1 is the most ideal, in the conditions of the inducer blades on the liquid flow of the smallest crowding effect. But a single blade of the inducer pitch increases, the axial length increases, the hydraulic loss will increase, making it more difficult to manufacture, so the number of blades Zi is generally(25) Zi=2∼3 (2)

Inlet flow coefficient and leaf tip diameter

The inlet flow coefficient Φind has an important influence on the shaft power and cavitation performance of the high-speed pump, and the inlet flow coefficient is determined in the case of a fixed flow rate and speed, and the tip diameter of the inducer is also determined. Φind With smaller values, the cavitation performance of the inducer is relatively good. The literature [18] concluded on the basis of experiments that the best cavitation performance of the inducer is obtained when Φind = 0.0645. However, Φind takes a smaller value. The impeller tip diameter has to be increased, which is not conducive to the increase of pump shaft power. Taking into account the cavitation performance and shaft power, the value of Φind can be taken as [19].(26) Φind=0.06∼0.11

The inducer tip diameter Dt is taken as(27) Dt=42∼70mm (3)

Leaf grid consistency and leaf pitch

The inducer grid consistency is equal to the ratio of the blade spread length and pitch, which has a certain degree of influence on the cavitation performance of the inducer, Chebanowski [20] derived the relationship between the cavitation coefficient of the inducer and the grid consistency through experiments, at the number of blades Zi = 2∼4, if τ≥3 , the axial length of the inducer increases, the energy loss increases, so the range of values of τ should be(28) τ=2.0∼3.0

The blade pitch is given by the equation(29) t=πDtZi

The calculation can be obtained.(4)

Inlet punch angle αind and blade mounting angle βind

The inlet flow coefficient ϕind is determined, also determine the inlet liquid flow angle of the inducer βi , and the inlet blade installation angle of the inducer βind , then the inlet impulse angle αind and the liquid flow angle βi and, and the guide range S=πDttgβind , so determine the ϕind on the determination of S. Theoretically, αind take a smaller value can make the inducer to achieve higher cavitation performance, but the inducer must produce the pressure head that can make the centrifugal wheel work without cavitation, so αind can not be too small. According to the design practice and experience, it is desirable to(30) αind=3∘∼5∘≈arctan⁡Φind (31) βi1=arctan⁡Φind+αind=7∘∼10∘

For the variable pitch inducer, the inlet liquid flow impulse angle αind can be taken as zero or a smaller value, i.e., the inlet blade installation angle βi is equal to or slightly greater than the inlet liquid flow angle(32) βi1=arctan⁡Φind+0∘∼2∘

The exit blade installation angle of the variable pitch inducer βi2 is large to ensure that the exit head generated by the inducer can meet the energy requirements of the centrifugal wheel inlet. Under the design conditions, the centrifugal wheel is generally designed with a positive impulse angle and its blade inlet installation angle βi1=16∘∼22∘ . Therefore, the outlet blade installation angle of the variable pitch inducer βi2 can be taken as(33) βi2=12∘∼20∘

The blade processing of variable pitch inducer is more troublesome, and its processing method is different from the impeller blade processing method, so its angle change law cannot use the equal arc arrangement law or logarithmic spiral law.(5)

Import/export hub ratio

In order to make the inducer have better cavitation performance, the inlet hub ratio Rd1 should be taken as a smaller value, which is generally desirable(34) Rd1=0.15∼0.28

The exit hub ratio Rd2 should be taken as the larger value.(35) Rd2=0.45∼0.65 (6)

Leading edge wrap angle and leaf tip wrap angle

The inducers have the best cavitation performance less when the leading edge wrap angle is θ1=180∘ [20]. In practice, the cavitation performance of the inducer at θ1=90∘∼150∘ is close to that at θ1=180∘ , and the shorter axial length is favorable for machining. Considering all together, it should be taken(36) θ1=90∘∼150∘

The head and axial length of the leaf tip wrap angle of the equal-pitch inducer are calculated by Eqs. (37) and (38), respectively(37) θ2=360tτsin⁡βi1s (38) L=θ1+θ2360S

According to the above calculation formula, the specific design parameters of the inducer are as follows in Table 3.

The fluid dynamics control equations include the continuity equation, the momentum equation and the energy equation, and the flow inside the impeller machinery is subject to these three equations [21], and the energy conservation equation is not considered due to the relatively small amount of heat exchange involved in the calculations of this paper.(1)

Conservation of mass equation

The mass conservation equation, also known as the continuity equation, is a change in the total amount of some conserved quantity in any region, equal to the amount entering or leaving from the boundary; the conserved quantity cannot increase or decrease, but can only be transferred from one location to another.

(39) ∂ρ∂t+∂(ρu)∂x+∂(ρv)∂y+∂(ρw)∂z=0 where, ρ is the fluid medium density, t is the time quantity, and u, v, and w are the components of the velocity vector in the x, y, and z directions, respectively.(2)

Conservation of momentum equation

The conservation of momentum equation is a fundamental law that must be satisfied by any flow system, and is actually Newton’s second law, which is defined as: the rate of change of momentum in any control micro-element produced by a change in time is equal to the sum of external forces acting on the micro-element.

(40) ∂(ρu)∂t+u∂u∂x+v∂u∂y+w∂w∂z=−∂p∂x+∂τxx∂x+∂τyx∂y+∂τzx∂z+Fx

(41) ∂(ρv)∂t+u∂v∂x+v∂v∂y+w∂v∂z=−∂p∂x+∂τxy∂x+∂τyy∂y+∂τzy∂z+Fy

(42) ∂(ρw)∂t+u∂w∂x+v∂w∂y+w∂w∂z=−∂p∂x+∂τxz∂x+∂τyz∂y+∂τzz∂z+Fz

where τxx , τyy and τzz are the components of the viscous force τ in the x, y and z directions; F_x, F_y and F_z are the volume forces acting on the micro-element in the x, y and z directions. In this paper, the values of F_x, F_y and F_z are 0; p is the pressure on the micro-element.

If the working medium is a Newton fluid, then there is

(43) τxx=2μ∂u∂x+λdiv(u);τxy=τyx=μ(∂u∂y+∂v∂x)

(44) τyy=2μ∂v∂y+λdiv(u);τxz=τzx=μ(∂u∂z+∂w∂x)

(45) τzz=2μ∂w∂z+λdiv(u);τyz=τzy=μ(∂v∂z+∂w∂y)

where, μ is the dynamic viscosity, λ is the second viscosity, according to experience can generally take λ = 2/3, the above formula can be substituted to get.

(46) ∂(ρu)∂t+div(ρuu)=div((μ+μT)gradu)−∂p∂x+Su

(47) ∂(ρv)∂t+div(ρvu)=div((μ+μT)gradv)−∂p∂y+Sv

(48) ∂(ρw)∂t+div(ρwu)=div((μ+μT)gradu)−∂p∂z+Sw

where grad() = ∂ ()/ ∂ x + ∂ ()/ ∂ y + ∂ ()/ ∂ z and the symbols S_u, S_v and S_w are the generalized source terms of the momentum conservation equation, S_u = F_x + S_x, S_v = F_y + S_y, S_w = F_z + S_z, and S_x = S_y = S_z = 0 for incompressible flow.

The shear stress transport k-ω model (SST k-ω model for short), proposed by Menter, combines the respective advantages of the standard k-model and the standard k-ε model, with the addition of the transverse dissipative derivative term and cross-diffusion in the ω equation in the SST k-ω model compared to the two base models. This feature makes it more applicable to a wider range of applications and has higher accuracy and confidence. The model equations are.

(49) ∂(ρk)∂(t)+∂(ρujk)∂xi=∂∂xi[(μ+μtσk3)∂k∂xi]+pk−β′ρkω

(50) ∂(ρω)∂(t)+∂(ρuiω)∂xi=∂∂xj[(μ+μtσk3)∂ω∂xj]+a3ωkpk−β3ρω2+(1−F1)2ρσω21ω∂k∂xj∂ω∂xj

where pk is the turbulent kinetic energy production rate, whose expression is

(51) pk=μt2Sij∂ui∂xj

(52) Sij=12(∂ui∂xj+∂uj∂xi)

Turbulent viscosity coefficient

(53) μt=ρkω1max[1α∗,SF2a1ω]

where

(54) S=2SijSij

S_ij is the spin rate.

(55) F2=tanh⁡(Φ22)

(56) Φ2=max[2k0.09ωy,500μρy2ω]

where, y is the distance to the other face.

The SST k-ω model has been substantially optimized on the standard k-ω model. In order to make the final calculation results more accurate, the model further addresses the shortcomings of the large error near the wall in the calculation.

To ensure the accuracy of the numerical calculation and to ensure that the near-wall region can have a sufficient number of nodes for capturing the flow within the boundary layer, y⁺ is usually used. y⁺ denotes the distance of the nearest grid point to the wall surface and is a dimensionless parameter defined by(57) y+=Δyρμτwρ where, Δy is the distance from the node to the wall; τw is the wall shear stress: μ is the dynamic viscosity of the fluid.

The y⁺ is the basis for choosing the turbulence model, and the turbulence model is chosen differently for different y⁺ magnitudes. In the numerical simulation of this paper, the calculated y⁺ value is close to 30, and the turbulence model is chosen as the SST k-ω model, which satisfies its quality requirements [22].

In this paper, ANSYS ICEM CFD software was selected to mesh the model. The mesh delineation is shown as follows (Figs. 4–7).

The number of grids will limit the speed and accuracy of the numerical simulation. The smaller the number of grids and the coarser the mesh, the worse the simulation accuracy; the larger the number of grids and the finer the mesh, the longer the simulation time. In order to improve the simulation speed and study the shaft power without affecting the simulation accuracy, it is necessary to verify the grid irrelevance. The centrifugal pump with the same working condition and structure is selected, and the number of grids is changed to simulate the flow field of the whole machine, and the optimal number of grids is derived. The grid irrelevance validation is shown in the following table.

From the data in Table 4, it can be seen that the shaft power of the plastic centrifugal pump keeps a stable trend when the number of grids is above 1,169,818, indicating that when the number of grids reaches 1,169,818, its influence on the simulation results is small and negligible. In order to ensure the simulation speed and save computer resources, the mesh number is controlled to be about 1,169,818 million when the centrifugal pump simulation is carried out.

After the grid irrelevance verification, the number of grid nodes, the number of grids and the quality of grids are determined. The number of nodes in this simulation is 287,970, and there will be about 0.35% floating in different structural models; the number of grids is 397,207, and there is about 0.67%∼1% quantity difference in different models; 75% of the grid quality is 0.96, 22% of the grid quality is 0.63, and 3% of the grid quality is 0.27, and the grid quality is good.

Inlet boundary conditions: The inlet conditions of the flow are generally set as velocity inlet (for non-pressurizable flow), mass inlet (for pressurizable flow) or pressure inlet (for pressurizable and non-pressurizable flow). In this paper, we choose the pressure inlet boundary condition, and the size of the pressure value is chosen as 1 atm.

Outlet boundary condition: In this paper, the mass flow outlet boundary condition is selected to control the flow rate of the pump. According to the performance parameters of the plastic centrifugal pump, the size of mass flow rate is selected as 69.45 kg/s.

Object surface boundary conditions: The object surface includes the suction surface and pressure surface of the inducer and impeller blade, the wall surface of the axial gap, the front and rear cover of the impeller, the wall surface of the volute and the wall surface of the extension section, etc. Among them, the object surfaces related to the inducer and impeller do rotational motion with the inducer and impeller relative to the absolute coordinate system, and the rest of the object surfaces are relatively stationary. Because the turbulent flow evolves into laminar flow in the near-wall area, it is necessary to use the wall function method for the near-wall area to do the corresponding treatment. Generally, no-slip boundary conditions are used in the solution, and the relative velocity on the wall surface W = 0.

The boundary conditions in CFD calculations also include symmetric boundary conditions and periodic boundary conditions, etc. In this paper, the numerical calculation domain is the flow field of the whole centrifugal pump, and there is no periodicity problem or symmetry problem, so it is not considered.

With the above settings, the structure tree model in Fig. 8 and the fluid domain model in Fig. 9 can be obtained.

In this paper, the finite volume method is chosen to construct the discrete equations, and in this process the construction of the discrete format needs to be involved. The discrete format is the way in which the physical quantities at the interface of the control body and the nodes of their corresponding derivatives are interpolated. The commonly used discrete formats and their specific formats are described in the literature [23].

In this paper, considering the centrifugal pump flow field as a complex three-dimensional distorted fluid region, in order to ensure the unity of solution compatibility, convergence and stability in the numerical solution process. Therefore, the absolute stable windward format is chosen to solve the equations. First, the first-order windward format is used to solve the corresponding flow domain, and after reaching the convergence condition, the second-order windward format is used to continue the solution until convergence on the basis of the original solution results in order to improve the accuracy of the calculation.

After the CFX-Pre is defined, the solution is performed using the stability solver. In this paper, a time step of 3° of impeller rotation is chosen for the non-constant numerical calculation. According to the speed of centrifugal pump 2700 r/min, the time step of each step is calculated as 0.006 s (i.e., impeller rotation 3°), and the total calculation time is 11.52 s (i.e., impeller rotation 16 revolutions), the maximum number of iteration steps for the non-constant calculation is 1000, and the convergence criterion is the average value RMS with the value 1.E-5, and the SIMPLEC algorithm is used to calculate, and after the curve After convergence, pressure nephograms of the plastic centrifugal pump impeller and inducer impeller are obtained as follows Figs. 10–12.

Fig. 11 shows the pressure distribution diagram of the impeller of the plastic centrifugal pump. Analysis of Fig. 11 shows that the pressure value from the impeller inlet to the impeller outlet is gradually increasing, indicating that the impeller rotates during the process of doing work on water, which makes the water pressure gradually increase from the impeller inlet to the impeller outlet, and the water then reacts to the impeller, so that the impeller pressure appears to increase circumferentially.

Fig. 12 shows the pressure distribution of the impeller of the inducer. Analysis of Fig. 12 can be obtained, from the figure can be seen, the inducer impeller blade surface static pressure from the inlet to the outlet showing an increasing trend, blade pressure surface pressure is higher than the blade suction surface pressure, the lowest pressure area are in the blade suction surface inlet position.

The orthogonal test is a common experimental design scheme for solving multi-factor tests through the appropriate design of experimental protocols with orthogonal tables specified in mathematical statistics. The basic steps are shown below:(1)

Clear test objectives, determine the test indicators

In the orthogonal test design of the selected parameters, the test indexes should be clarified and the quality evaluation indexes should be determined.(2)

Selection of test factors and determination of their test levels

In the orthogonal test, the factors are shown in English A, B, etc., instead. When determining the test factors, the priority is given to the factors that have a greater influence on the test evaluation index. In order to obtain higher experimental efficiency, the number of factor level values is usually chosen from 2 to 4, and the interval of level values should be reasonable.

In this test, there are many factors affecting the head and shaft power, according to the relevant theory and requirements, selected blade thickness, lobe degree, blade flange angle, blade hub angle and hub length as the five factors of the orthogonal test, The selected factors and their level values are shown in Table 5, for the convenience of describing the results of the extreme difference analysis, they are denoted by A, B, C, D, E blade thickness S, cascade solidity t, blade rim angle β1, blade hub angle β2, hub length L, respectively.

It is a method to find out the curve by using mathematical formulae or computer-aided methods when the data of X and Y are known. In essence, it is a data processing method that approximates the discrete point data according to the analytical expressions, and approximates the relationship between the discrete point data by continuous smooth curves, and finds the relationship between the variables from these discrete point data, thus reflecting the change pattern of the discrete point data [24].

(1)

Set weights

In this paper, five factors are integrated into one by using the way of setting weights, and by referring to relevant information and considering the actual situation in the process of optimizing the structural parameters of the inducer, the weights of each factor are set as follows:(58) ω=(ω1,ω2,ω3,ω4,ω5)=(0.4,0.3,0.1,0.1,0.1) (2)

Numerical calculation

Using the weight values to weight the 16 sets of structural parameter combinations, the obtained is the X among the least squares method, whose values are 74.9, 75.48, 74.87, 76.84, 75.37, 76.48, 75.24, 76.6, 75.84, 75.5, 75.58, 78.37, 76.68, 75.97, 76.54, and 77.7. The evaluation indexes in this paper are shaft power and shaft power, so the Y values in the least squares fit are the values of head and shaft power for the 16 groups in the orthogonal test table, which can be obtained by checking Table 5.(3)

Curve fitting

In order to make the curve fitting more accurate, the weighted obtained data were imported into Matlab and the discrete points were connected, as in Fig. 13.

Fig. 13 shows the data distribution obtained after automatic import using Matlab software. Next, the software was used to simulate the curve for this set of discrete data, and through several attempts and optimization, the final fitted curve is shown in Fig. 14.

As shown in Fig. 14, the imported 16 sets of data are basically distributed on the fitted curve, and some points deviate from the fitted curve, because curve fitting itself is a process of infinite approximation, and it is impossible to guarantee that all the discrete points are on the fitted curve, but it is an important evaluation index to ensure that as many as possible are on the curve, so the fitted curve meets the requirements from the overall point of view. The analytic equation of the curve after fitting by the software is a 4th order Fourier polynomial. And Fig. 15 is a schematic diagram of fish school optimization, which mainly shows the activity state of fish school.