Searching stochastic method of the submersible electric pump flow part optimal shape

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Introduction

Modern methods for calculating and designing dynamic pump flow parts with high energy efficiency indicators involve solving the problem of three-dimensional flow analysis of a viscous liquid in the computational domain of the object under study [1]. The efficiency of the pumping unit is largely determined by the shape of its impeller and guide vanes. Achieving optimal efficiency requires numerical optimization of the flow part of the hydraulic machine [2].

Various optimization approaches are used to identify the best geometric parameters for the flow parts. A comprehensive optimization approach [3] utilized the LP-tau method to enhance the hydraulic performance of the flow part of the sealed pump. The best approach for optimizing an axial pump was a combination of stochastic and gradient optimization methods. In the first approximation, a stochastic algorithm seeks the global extremum, followed by using a directional method to identify a local extremum [4]. Metamodels are increasingly used to minimize computational costs and time. These are developed based on direct calculation results from a compiled experiment matrix [5]. Well-developed approximation functions (metamodels) can accurately predict the extrema of target functions [6].

The aim of the work is to improve the hydraulic efficiency of pumps using a stochastic method, specifically the Latin hypercube sampling approach, to find the optimal shape of the flow part. This approach allows us to identify the global extremum of the target function.

The object of the study is a vertical submersible drainage electric pump with a speed coefficient ns = 205. Figure 1 shows a digital model of the pump that contains an inlet pipe — 1, n impeller — 2, guide vanes — 3, the annular area between the motor and pump housings — 4, and a discharge pipe — 5.

Method of numerical calculations

The numerical calculation method is based on solving the direct hydrodynamic problem of three-dimensional viscous liquid flow in the studied computational domain. Flow modeling in the flow section was performed using the finite volume method with the ANSYS CFX CFD package.

To validate the calculation method, a mathematical model of the initial flow part version of the object under study was calculated. The computational domain in its full scale and without periodicity conditions includes all elements of the flow section as shown in Fig. 1. The mesh is tetrahedral with 11 million cells. The boundary conditions applied include full pressure at the suction point and mass flow at the discharge. A k-epsilon turbulence model was used [7]. The walls are set as no-slip boundaries, considering equivalent roughness [8]. The flow parameters between the rotating and stationary domains are averaged around a circle, with Stage as the interface type. The calculations were carried out under stationary conditions.

 

Fig. 1. The digital model (a) and the computational region (b) of the submersible pump.

Рис. 1. Цифровая модель (a) и расчётная область (b) погружного электронасоса.

 

Seeking for the optimal shape of the flow part

In this work, the goal is to improve the energy indicators of the research object by identifying the global maximum of the target function, specifically the hydraulic efficiency of the pump in its nominal operating mode. In this regard, a stochastic method is used to find the optimal shape of the flow part within specified boundaries.

To conserve computational and time resources during optimization calculations, a simplified geometry of the computational domain with periodicity conditions is used.

In this work, the goal is to improve the energy indicators of the research object by identifying the global maximum of the target function, specifically the hydraulic efficiency of the pump in its nominal operating mode. In this regard, a stochastic method is used to find the optimal shape of the flow part within specified boundaries.

To conserve computational and time resources during optimization calculations, a simplified geometry of the computational domain with periodicity conditions is used.

The meridian projection of the parametric model of the computational domain is described by the inlet boundary of the suction pipe, the outlet boundary of the discharge pipe, the sleeve, and peripheral contours of the flow part, as well as the inlet and outlet edges of the impeller blades and guide vanes.

The impeller blade system is built along three flow lines: the sleeve, the periphery, and the middle line. The changes in blade angles and thicknesses are defined by a spline along three reference points: at the input and output edges, as well as in the middle part, which determines the coverage angle of the blade.

The cylindrical blade of the guide vanes is described by two flow lines: the sleeve and the periphery. The blade angle changes are linear, defined by two reference points at the input and output edges. Since the vanes are installed on the engine using studs passing through the blade, the thickness change is determined by design limitations.

The parametric model of the computational domain includes nine variable geometric parameters, namely sleeve and peripheral angles of the impeller blade at the inlet and middle parts of the current line (β_i1sl, β_i1per, β_i1-2sl, β_i1-2per), input and output angles of the guide vane blade (β_i3, β_i4), channel width of the guide vanes (b₃), channel width of the annular area (b), and channel width between the electric motor and discharge pipe (B).

Preliminary numerical calculations of the object under study indicate that the main hydraulic losses are concentrated in the discharge element. Therefore, the variable parameters in this flow section were determined across a wide range of values.

The values of these geometric parameters are listed in Table 1, with their graphical representation shown in Fig. 2.

 

Table 1. Values of geometric variables

Таблица 1. Значения варьируемых геометрических параметров

№	Parameter	Name	Min	Max
1	βi1sl	Installation angle of the impeller blade on the sleeve at the suction	16°	20°
2	βi1per	Installation angle of the impeller blade on the periphery at the suction	15°	19°
3	βi-2sl	Installation angle of the impeller blade on the sleeve in the middle part	25°	33°
4	βi1-2per	Installation angle of the impeller blade on the periphery in the middle part	23°	31°
5	βi3	Installation angle of the guide vanes at the inlet	11°	15°
6	βi4	Installation angle of the guide vanes at the outlet	30°	90°
7	b3 / D2	Channel width of guide vanes channel width	0,179	0,214
8	b / D2	Width of the annular area channel	0,089	0,268
9	B / D2	Channel width at the turn beyond the annular area	0,125	0,375

 

Fig. 2. Geometric variables of the computational region.

Рис. 2. Варьируемые геометрические параметры расчётной области.

 

In the chosen search method, the Latin hypercube sampling method (LHS method) calculated points are randomly distributed across an orthogonal mesh of the parameter space, ensuring two points share the same parameter values.

This stochastic LHS method provides an uniform distribution of geometric parameters within the search space.

Fig. 3 illustrates how channel width parameters of the annular area beyond (b / D₂ and B / D₂) are distributed in the search area.

 

Fig. 3. Distribution of the b / D2 and B / D2 variables in the search space.

Рис. 3. Распределение варьируемых параметров b / D2 и B / D2 в области поиска.

 

A table consisting of 1000 calculated points, each with different combinations of variable geometric parameters, has been created in the selected range of input parameter values. This table served as the basis for direct hydrodynamic calculations performed with the target function aimed at maximizing the hydraulic efficiency value of the pump.

The optimization calculations identified the most effective shape for the flow part. Subsequently, numerical calculations were conducted on this optimal design, using a mathematical model corresponding to that of the initial version tested earlier.

Results

Table 2 presents the input geometric parameters and the target function for both the initial and optimal versions of the flow parts, Fig. 4 providing a visualization of the flow structure.

 

Table 2. Values of initial parameters and the objective function of the flow section

Таблица 2. Значения входных параметров и целевой функции проточных частей

№	Parameter	Initial flow part	Optimal flow part
1	βi1sl	15,7°	19,4°
2	βi1per	14,9°	15,2°
3	βi1-2sl	28,5°	31,8°
4	βi1-2per	20,1°	26,9°
5	βi3	15,2°	13,3°
6	βi4	90,0°	67,9°
7	b3 / D2	0,230	0,181
8	b / D2	0,177	0,262
9	B / D2	0,266	0,302
10	КПДг	68,6%	75,6%

 

Fig. 4. Velocity fields in the initial (a) and the optimal (b) flow sections.

Рис. 4. Поля скоростей в исходной (a) и оптимальной (b) проточных частях.

 

The data in Table 2 reveal that the optimal flow part features increased channel width in the annular region (b / D₂ = 0.262) and between the electric motor and the discharge pipe (B / D₂ = 0.302). These adjustments result in decreased flow speeds and lower hydraulic losses in the discharge device, as shown in Fig. 4.

Conclusions

1. The stochastic Latin hypercube sampling method has proven effective in identifying the global maximum of the pump’s hydraulic efficiency. This result can serve as an initial approximation for locating a local maximum near the identified global extremum of the target function in the study.
2. Using the Latin hypercube sampling method, the hydraulic efficiency of the pump was increased by 7% compared to the initial flow part of the object of study.
3. The optimization results showed that an increase in the target function, specifically the hydraulic efficiency of the pump, was largely achieved by widening the channels between the motor and the pump housings. This modification led to reduced flow speeds and lower hydraulic losses in the sta-tor elements of the flow part of the pump under study.

Additional information

Authors’ contribution. D.A. Gorbatov ― literature review on the topic of the article, conducting computational research, writing the manuscript text; A.A. Zharkovskii ― scientific supervision, editing of the manuscript text, approval of the final version; A.V. Adrianov ― expert evaluation, agreement of the final version. Authors confirm the compliance of their authorship with the ICMJE international criteria. All authors contributed substantially to the conceptualization, research and preparation of the article, and read and approved the final version before publication.

Competing interests. The authors declare that they have no competing interests.

Funding source. This study was not supported by any external sources of funding.

Дополнительная информация

Вклад авторов. Д.А. Горбатов — обзор литературы по теме статьи, проведение расчётных исследований, написание текста рукописи; А.А. Жарковский ― научное руководство, редактирование текста рукописи, утверждение финальной версии; А.В. Адрианов ― экспертная оценка, согласование финальной версии. Авторы подтверждают соответствие своего авторства международным критериям ICMJE. Все авторы внесли существенный вклад в разработку концепции, проведение исследования и подготовку статьи, прочли и одобрили финальную версию перед публикацией.

Конфликт интересов. Авторы декларируют отсутствие явных и потенциальных конфликтов интересов, связанных с публикацией настоящей статьи.

Источник финансирования. Авторы заявляют об отсутствии внешнего финансирования при проведении исследования.

About the authors

Daniil A. Gorbatov

Peter the Great St. Petersburg Polytechnic University; SULAK

Author for correspondence.
Email: Da.Gorbatov@yandex.ru
ORCID iD: 0000-0002-3172-3346
SPIN-code: 5727-2661

Postgraduate of the Higher School of Power Engineering of the Institute of Energy, Leading Designer of the Main Division

Russian Federation, Saint Petersburg; Saint Petersburg

Alexander A. Zharkovskii

Peter the Great St. Petersburg Polytechnic University

Email: azharkovsky@gmail.com
ORCID iD: 0000-0002-3044-8768
SPIN-code: 3637-7853

Dr. Sci. (Engineering), Professor of the Higher School of Power Engineering of the Institute of Energy

Russian Federation, Saint Petersburg

Artemiy V. Adrianov

SULAK

Email: mr_a@inbox.ru
ORCID iD: 0000-0003-4853-0804
SPIN-code: 8117-4795

CEO

Russian Federation, Saint Petersburg

References

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