Multi-objective optimization of intake parameters based on response surface methodology

The open intake is an important hydraulic building in pumping stations. This paper adopts the simple coupled level-set and volume of fluid and bifurcation model for numerical simulation. Based on the response surface methodology (RSM), multi-objective optimization of the structural parameters of the open intake is carried out. Considering the air-entrained vortex and irreversible energy loss of the intake pool, the response surface optimization model of the intake is established. The results verify the accuracy of this model. It is shown that the influence of floor clearance and back-wall clearance on the objective function is significant; After gradient optimization calculation, the optimal parameter combination is C = 0.4172D and B = 0.7208D. This optimization scheme suppressed the air-entrained vortex and other adverse flow patterns. And these results provide relevant theoretical references for the future design of intake.


Introduction
The pump station plays an important role in agricultural irrigation, flood drainage, municipal water supply and trans regional water transfer.Therefore, it is important to improve the performance of pump stations.However, during the operation of the pump station, the vortex in the intake significantly affects the pump performance, and the most influential one is the air-entrained vortex.The air will be sucked into the pump by the air-entrained vortex, reducing the efficiency of the pump and enhancing cavitation, so that the safe and stable operation of the pump station cannot be ensured.For these reasons, it is necessary to study the air-entrained vortex.
According to previous studies, the intake parameters can significantly affect the vortex and flow pattern of the intake [1][2][3].At present, many scholars have studied the influence of the floor clearance and back-wall clearance on the flow pattern of the pump station inlet flow field by using the single factor comparison method, and given the corresponding optimization value or value range.However, these optimizations do not concentrate on the air-entrained vortex.
The key to optimizing design lies in selecting appropriate optimization objectives.According to our previous research, the relative air-entrainment rate in the bell mouth suction pipe can measure the degree of the air-entrained vortex and can be used as the objective parameter for optimization [4,5].There are many adverse flow patterns in the intake, and the air-entrained vortex is only one of them.Due to the fact that the relative air-entrainment rate can only represent the strength of the air-entrained vortex, it cannot fully reflect the performance of the intake.
In order to better characterize the inflow performance of the intake, this paper uses entropy production to characterize the irreversible energy loss in the intake.The term 'entropy' is a thermodynamic term, which has a basic meaning for various processes occurring in fluid flow, The unreasonable design of key parameters in the intake leads to the occurrence of airentrained vortex in the open intake, which affects the operation of the pump station and can lead to cavitation and erosion in severe cases.At present, the research on the optimization objective of air-entrained vortex is seldom reported.In addition, the design of the intake has not considered the self-generated irreversible energy loss.In this paper, the response surface methodology (RSM) is used to optimize the parameters of the intake, combined with the multi-objective optimization method of entropy production and relative air-entrainment rate.The distribution of entropy production in the intake, and the parameters of the intake after multi-objective optimization are given.The air-entrained vortex significantly suppressed and reduced other adverse flow patterns by optimization.

RSM and numerical simulation description
In order to comprehensively study the influence of floor clearance and back-wall clearance on the objective function, combined with the research results of ANSI/HI9.8[20] and relevant scholars [1][2][3], this paper expands the range of floor clearance (C) and back-wall clearance (B), taking C = (0.25~0.45)D and B = (0.65~0.85)D. To ensure the accuracy of the fitting, take 5 levels with a moderate spacing between the two factors in their respective value ranges.Table 1 shows the value levels of floor clearance and back-wall clearance.At the suction of the pipe, a bell mouth with a diameter of D = 0.15 m was designed.The intake width is 2D and the length is about 10D.The distance between the centre of pipe and both sides of the intake is equal, both of which are 1D.As shown in figure 1 (a).
Table 1 The levels of two-factors response surface optimization design.Intake model.This paper uses OpenFOAM 2.2.2 for numerical simulation.The turbulence model is BM [13][14][15], and the interface tracking method is S-CLSVOF [17,16].The initial condition and boundary conditions of numerical simulation, as well as the iteration schemes and interpolation schemes are the same as references [4,5].

Relative air-entrainment rate
The relative air-entrainment rate is an indicator of the strength of the air-entrained vortex.The calculation method of the relative air-entrainment rate is proposed by Möller et al. [19].The calculation method is as follows: According to our previous research, the relative air-entrainment rate will change periodically after 5 s.[11,12] The average relative air-entrainment rate (  ) is averaged over t = 5 s~14 s.In order to characterize the volume of entrained air in the inlet pipe, the air volume in the straight section of the inlet pipe at each time steps are calculated and averaged over time (t = 5 s ~ 14 s).

Entropy production
The entropy production rate s  can clearly describe the irreversible energy loss generated in fluid.In this paper, the calculation equations were proposed by Kock [21].According to Reynolds timeaveraged Navier-Stokes equations, each transient quantity is decomposed into time-averaged movement and the fluctuating terms. Where, is the direct entropy production rate, , pro TD s  is turbulent entropy production rate Kock et al [22] and Walsh et al [23] proposed a calculation method based on turbulence model for , pro TD s  , as follows: In addition, the high velocity gradient and pressure gradient existing on the blade surface of the rotating fluid machine cause a strong wall effect in the flow field, resulting in an irreversible flow loss that cannot be ignored.Scholars also considered the entropy production caused by the wall effect [10].The entropy production rate in the tip area from the wall to the first layer of the grid can be solved by equation ( 4) where,   indicates the shear stress in the wall; v  indicates the velocity of the first grid near the wall.Therefore, the total entropy production can be obtained by the volume integral of the direct entropy production rate and turbulent entropy production rate and the surface integral of the wall entropy production rate.
To facilitate the establishment of the weighting function, the total entropy production is dimensionless: The

Objective function
As the air-entrainment is the main factor considered for this case, after comprehensive consideration and based on the author's experience, the weight coefficient is chosen as 5:1.In this way, the objective function is as follows:

Two-factors CFD calculation results
According to the above method, perform a full factor combination of the two factors in table 1.And the calculation results of 25 schemes are shown in table 2.  Based on the data in table 2, the response surface optimization design method is used to carry out multiple regression fitting analysis, and the polynomial response surface regression equation of Y with floor clearance and back-wall clearance is established.This paper fit the response surface model with a quadratic polynomial, as shown in equation.(8).From figure 2, it can be seen that the floor clearance and back-wall clearance have a significant impact on the objective function.From the overall trend of RSM (as shown in figure 3), with the increase of floor clearance and the decrease of back-wall clearance Y gradually reduce, so reduce the back-wall clearance and increase the floor clearance, which has a certain effect on Y.As shown in figure 3 (a), when the back-wall clearance is large, Y rapidly decreases with the increase of the floor clearance.Therefore, the floor clearance should not be less than 0.3D.When the back-wall clearance is small, the amplitude of change is relatively gentle.From figure 3 (b), it can be observed that under different floor clearance, the relationship between the objective function and the back-wall clearance is consistent.The tendency shows a slow increase and then a rapid increase after B/D = 0.8.This indicates that the value of the back-wall clearance should be below 0.8D.
By using the gradient optimization method to calculate the optimal value of the RSM, it can be obtained that the objective function is optimal at C = 0.4172D and B = 0.7208D, with an optimal value of 0.006761.Numerical simulation was conducted on the optimization results, and the calculation result was Y = 0.007176, with an RSM prediction error of 4.23%.

Air-entrained vortex discussion
In section 4.1, we found that when the higher floor clearance or larger back wall clearance, the larger objective function value (as shown in figure 3), which indicated that these schemes have poor flow patterns.Therefore, these options can be excluded in following research.
The method of combining l  and Q-criterion is used to identify vortices in the intake, as shown in figure 4. The red color represents the iso-surface of l  =0.95, and the blue color represents the iso- surface of Q = 1500.Comparing the vortices of different schemes, it is not difficult to find that the optimal scheme predicted by RSM has fewer vortices.This is due to the reduced intake of air (red isosurface) in the optimal scheme.This indicates that the combination of C = 0.4172D and B = 0.7208D have significantly improved the air-entrained vortex of the intake.

Entropy production discussion
From figure 4, it can be observed that there are many subsurface vortices near the bell mouth, and it is one-sided to measure the flow pattern of the intake through the air-entrained vortex.Therefore, this article adopted entropy production to analysis other adverse flow pattern.Before this, it is necessary to analysis the distribution of entropy production.
Figure 5 shows the Q and , pro D s  distributions on the cross-section through which the vortex core passes.It is easy to see from figure 5 that the entropy production is lower at the vortex core, while the entropy production is higher in the area around the vortex.This is due to the stable flow at the vortex core and low turbulent kinetic energy.The rotation of the vortex drives the surrounding fluid, causing it to rotate around the vortex, resulting in an increase in the turbulent kinetic energy of this part of the fluid.
Figure 6 shows the distribution of local entropy production on a cross-section with x = 0 mm.It is not difficult to find that entropy production is mainly concentrated near the bell mouth and on the wall.This is due to the concentration of vortices near the bell mouth, resulting in poorer flow pattern and higher irreversible losses.The high entropy production on the wall is due to the presence of large wall shear stress on the wall.In the optimization scheme (figure 6(e)), the distribution of entropy production is uniform, and the value of local entropy production is relatively low.From figures 6 (a), (b) and (d), there is a high local entropy production concentration region, which is due to the strong sub-surface vortex near the bell mouth.

Conclusion
This paper adopts CFD for numerical simulation and uses RSM method to optimize the design of intake parameters for air-entrained vortex and entropy production.The following conclusions are drawn: 1.The established optimization method of pump station intake parameters comprehensively considers the air-entrained vortex and irreversible energy loss of the intake, and determines the average relative air-entrainment rate (  ) in the calculation domain and dimensionless entropy production (ep), the objective function Y which can characterize the air-entrained vortex and irreversible energy loss in the intake is established.
2. An entropy production analysis was conducted in the intake, and it was found that the entropy production at the vortex core was low, while the entropy production around the vortex was high.In addition, entropy production is mainly concentrated near the bell mouth and at the wall.
3. After 2-factor optimization design analysis and gradient optimization method calculation, the optimal parameter combination for the vertically arranged bell mouth suction pipe in this article is C = 0.4172D and B = 0.7208D.

Figure 1
Figure1Intake model.This paper uses OpenFOAM 2.2.2 for numerical simulation.The turbulence model is BM[13][14][15], and the interface tracking method is S-CLSVOF[17,16].The initial condition and boundary conditions of numerical simulation, as well as the iteration schemes and interpolation schemes are the same as references[4,5].
of Regression Models, where R 2 = 0.8919, F = 31.36,P = 1.501×10-7 .The value of P less than 0.01 indicates excellent significance of the response surface model, while R 2 = 0.8919 indicates that more than 89% of the response values can be explained by this model.Therefore, the obtained equation can accurately reflect the impact of floor clearance and back-wall clearance on the objective function.

Figure 2 Figure 3
Figure 2 Response surface of Y.

Figure 5
Figure 5 Q and , pro D s  at different time (scheme 4).

Table 2
Two-factors CFD results and objective function values.

Table 3
Comparison of RSM prediction results with CFD results.