
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12613-8
10.1016/j.heliyon.2024.e36582
e36582
Research Article
Research on Multiphyics bidirectional coupling for small-diameter high-speed submersible permanent magnet synchronous motor
Tan Liping a
Wang Yucong a
Wu Yuyin a
Wang Teng a
Cui Junguo b19040005@s.upc.edu.cn
a⁎
Wang Hongyan wanghy@qust.edu.cn
b⁎⁎
a National Engineering Research Center of Ocean Geophysical Prospecting and Exploration Equipment, China University of Petroleum (East China), Qingdao, 266580, China
b College of Electromechanical Engineering, Qingdao University of Science & Technology, Qingdao, 266061, China
⁎ Corresponding author. b19040005@s.upc.edu.cn
⁎⁎ Corresponding author. wanghy@qust.edu.cn
30 8 2024
15 9 2024
30 8 2024
10 17 e3658225 6 2024
23 7 2024
19 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
The small-diameter high-speed submersible permanent magnet synchronous motor (SHS-PMSM) is essential equipment for rodless oil and gas extraction in slimhole wells and high-water content oil wells. The SHS-PMSM typically operates for extended periods of time underground in high temperatures. Because of its compact size, the heat is difficult to dissipate, which increases the risk of motor overheating and damage. In order to accurately predict temperature, the method of magnetic-heat-flow multiphysics bidirectional coupling is studied in this paper. A SHS-PMSM with an outer diameter of ø89mm is taken as the object, and its copper loss, friction loss and convective heat transfer coefficient are studied by analytical derivation. The relationship between them and temperature are expressed by functions which can be compiled into User-Defined Functions (UDFs) as variable during the calculation process of finite volume method. Both coupling calculations and experiments are conducted. The temperature calculated by magnetic-heat-flow bidirectional connection is higher than that produced by the conventional method and more in line with experimental results after the results of both simulations and experiments are carried out and compared. The accuracy of the magnetic-heat-flow bidirectional coupling method is verified and the design basis of temperature for SHS-PMSM can be provided.

Keywords

Slimhole wells
Small-diameter high-speed submersible PMSM
Magnetic-heat-flow coupling
Wingding loss
==== Body
pmc1 Introduction

With the continuous development of oil and gas exploration and development towards deepwater and unconventional oil and gas fields, the geological conditions and environment faced by oil and gas extraction are becoming increasingly complex [1,2]. With the benefits of high mechanical drilling speed, short drilling cycle, low production cost, and high safety and dependability, slimhole wells have emerged as a significant trend in the advancement of oil and gas extraction in the future [[3], [4], [5]]. At the same time, a large number of oil wells worldwide have an increasing water content, and higher displacement submersible pumps are needed. Compared with progressing cavity pumps, centrifugal pumps are increasingly used in oil and gas field development due to their advantages of high speed and high displacement [6,7].

In the oil production system of an electric submersible centrifugal pump for slimhole wells, the small-diameter high-speed submersible permanent magnet synchronous motor (SHS-PMSM) is the power source of the entire oil production system, and its performance directly affects the operating cost and efficiency of the whole system. Compared to conventional three-phase asynchronous motors, it has the advantages of stable operation, high efficiency, small size, and high power density [[8], [9], [10]]. However, SHS-PMSM usually works for a long time in high-temperature environments at depths of several kilometers underground, which is not conducive to heat dissipation [11,12]. At the same time, due to the limitation of the radial size of the casing in the slimhole wells, the outer diameter of the motor housing is small, the internal space of the motor is narrow, and the structure is compact, which is more unfavorable for heat dissipation. In addition, the high-speed rotation of the rotor inside the motor can lead to increased losses such as oil friction and core loss [13,14], which in turn increase the heat production of the motor. Based on the above, SHS-PMSM are prone to high temperature rise. However, excessive temperature can cause serious demagnetization of the permanent magnet [15,16], resulting in a significant decrease in motor efficiency and increased torque fluctuations, which can easily lead to motor overcurrent damage. In summary, the design requirements for SHS-PMSM are relatively high. Therefore, studying the heat source and factors affecting heat dissipation of SHS-PMSM and accurately predicting their temperature rise is of great significance for the development of electric submersible centrifugal pump oil production.

Currently, there are several methods used to calculate the temperature field: equivalent thermal resistance network method, finite difference method, finite element method, and finite volume method [[17], [18], [19]]. The equivalent thermal resistance network method correlates the temperature field element loss, temperature, and thermal resistance with the circuit element current, voltage, and resistance through the principle of similarity [20]. The average temperature of each component is determined by applying Newton's heat dissipation theorem and Fourier's rule of heat conduction [21]. This method is simple and fast, but the calculation accuracy is often low. The finite difference method discretizes the temperature solution domain, replacing the continuous solution domain with finite network nodes. Each node's partial differential equations are converted into difference equations, and each node's temperature is determined numerically. This method is simple to calculate but difficult to solve for temperature fields with complex boundary conditions [22]. The finite element method is based on the variational principle and discretizes the interpolation function of the mesh element into a problem of solving the extremum of multivariate functions. This method solves the domain with high computational accuracy under given boundary conditions, but for complex solving domains, the computation is relatively large. The finite volume method is based on the conservation of physical quantities, discretizes the interpolation function of the mesh elements, divides the temperature field solution domain into multiple element volumes, and integrates each volume to obtain a discrete equation system. This method has high accuracy, just like the finite element method. When solving a solution domain with complex boundary conditions, the mesh element of the solution domain can be adjusted to ensure accuracy and reduce computation. This method is suitable for solving temperature fields with complex boundary conditions [23].

In recent years, a large number of studies have been conducted on the temperature field solution of permanent magnet motors. Boglietti [24] computed the convective heat transfer coefficient using the fluid equation, examined the variables influencing the temperature field using the finite element method, and calculated the temperature distribution of small and medium-sized induction motors using the equivalent thermal resistance network method. Gilson et al. [25] established a temperature field model of a permanent magnet synchronous motor using the finite element method and analyzed the influence of the arrangement direction and structural size of the shell surface heat dissipation ribs on the temperature field. Nair et al. [26], Park et al. [27], Bhattacharya, and Bhattacharya [28] conducted temperature field analysis by finite element method and equivalent thermal resistance network method, demonstrating that the unidirectional coupling between magnetic field and temperature field (referred to as magnetic -thermal) can effectively predict the temperature distribution of conventional motors. Miao et al. [29] compared the temperature calculation results of the BLDCM equivalent thermal resistance network model and the finite element model, and analyzed the influencing factors of the temperature field based on the finite element model, providing a basis for dynamic thermal characteristic analysis. Moon et al. [30] established a coupled heat transfer model of TEFCM flow field and temperature field (referred to as flow-heat) using the finite volume method. The unidirectional coupling relationship between the two was considered, and the effect of fluid velocity on the motor temperature field was studied. Hruska K et al. [31] conducted a magnetic-heat unidirectional coupling analysis of an induction motor using the finite element method and proposed an equivalent method of replacing the air thermal conductivity with the average convective heat transfer coefficient in the air gap. Sun et al. [32] used the finite element method of magnetic-heat-mechanical coupling to analyze the performance of dual-mechanical-port machine, and verified the accuracy of this method by experiments. Shanel [33] calculated the fluid-solid convective heat transfer coefficient by field coupling of the finite volume method, avoiding the calculation error of the empirical formula. Zhang [34] established the fluid-solid heat transfer model of air-cooled induction motor based on the finite volume method, and used the magnetic-heat-flow unidirectional coupling method to study the heat dissipation performance of the motor. Meyghani et al. [35] compared the temperature calculation results under the equivalent thermal resistance network method and the finite element method, and the results showed that the calculation accuracy of the finite element method was higher than that of the equivalent thermal resistance network method. Daesuk Joo et al. proposed a three-dimensional finite element analysis method to solve the transient temperature distribution of permanent magnet motors and verified the effectiveness of this method through experiments [36]. In Refs. [37,38], the thermal network methods are used to estimate the temperature rise of the permanent magnet motor, and the influence of different speeds and loads on temperature rise is analyzed. In Ref. [39], adopts the finite element method and numerical method to calculate the temperature field distribution of the permanent magnet motor, and the magnetic-heat coupling method is used to calculate the temperature distribution. In Ref. [40], an extension of a FE-based EM/thermal coupled modeling method for electrical machines is proposed to provide a more accurate prediction of the machine's performance limits. In this paper, a linear relationship between the machine loss components and the temperature has been identified, but the friction loss and heat transfer coefficient are taken as constants at a certain temperature.

It is evident from the literature mentioned above that establishing a motor temperature field model, figuring out the magnetic-heat unidirectional coupling relationship, and figuring out the flow-heat unidirectional coupling relationship are the three keys to the temperature field. The magnetic-heat unidirectional coupling relationship is that the motor loss is loaded into the corresponding temperature field calculation area in the form of an average heat source. And the unidirectional fluid-heat coupling relationship is to assign the convection heat transfer coefficient to the fluid-solid interface as a constant and change the motor temperature by adjusting the fluid velocity. Through unidirectional coupling calculation, the temperature field distribution of a conventional motor can be calculated more accurately. However, considering that the working environment is different from that of conventional motors, the parameters affecting the temperature field of SHS-PMSM are mostly variable parameters, so the unidirectional coupling calculation method cannot be directly applied to the temperature field analysis of SHS-PMSM. At present, there are many applications of bidirectional coupling calculation methods [[41], [42], [43]] but there are still shortcomings. Wang et al. [130] and Chen [131] considered the magnetic-heat bidirectional coupling relationship in the finite element analysis, but the former only gave the calculation results and did not explain how to realize the bidirectional coupling; while the latter described how to realize the bidirectional coupling calculation, but did not consider the motor oil viscosity change when calculating the oil friction loss. Guo et al. [44] only gave the results when using the limited volume method, lacking the description of the bidirectional coupling calculation process and the setting of temperature boundary conditions. Kim and Lee [45] considered the bidirectional coupling relationship between flow and heat in the equivalent thermal resistance network method, and improved the calculation accuracy by constantly updating the thermal resistance related to the air conductivity coefficient. The results were compared and verified with the finite element method and the finite volume method respectively, and the error was within 5 %. However, in the verification process, they did not explain how to realize the coupling relationship, and the loss was regarded as a fixed value. Compared with only considering the bidirectional coupling between the two physical fields, there are very few studies that comprehensively consider the magnetic-heat bidirectional coupling and the flow-heat bidirectional coupling. Traxler samek et al. [46] coupled the cooling fluid network, loss calculation network and temperature field analysis network together, but the coupling model is too complex, so it can only calculate each part of the motor separately, which has a large error with the calculation results of the overall analysis of the motor. In Ref. [47], a multi-physics two-way coupled technique including thermal, fluid, and electromagnetic physics has been devised by the authors. In this paper the influence of temperature on copper loss has been studied and the loss is set as a variable when calculating the temperature of a submersible PMSM. However, the convective heat transfer coefficient of the housing external surface is considered as a constant and the effects of fluid velocity and temperature on the convective heat transfer coefficient are not taken into account.

To sum up, most of the current research on the accurate calculation method of PMSM temperature rise has a defect of insufficient considerations. Further more, there is no research on the temperature prediction of SHS-PMSM. Therefore, a more comprehensive temperature prediction method for SHS-PMSM needs to be deeply studied.

In this study, research on the temperature rise calculation method based on the finite volume method is conducted. A SHS-PMSM with an outer diameter of ø89mm is taken as an example, and the magnetic-heat-flow bidirectional coupling method of temperature rise calculation that takes into account heat sources and convective heat transfer coefficient changes is put forward. Firstly, based on the electromagnetic parameters and geometric topology of the SHS-PMSM, the coupling mechanism between fluid and solid is analyzed. The theoretical research is conducted on heat source and convective heat transfer coefficient, revealing the laws of copper loss and oil friction loss changing with temperature, as well as the laws of convective heat transfer coefficient changing with temperature and flow velocity. The temperature rise calculation method of magnetic-heat-flow bidirectional coupling that takes into account heat source and convective heat transfer coefficient changes is proposed. Secondly, taking the SHS-PMSM as an example, the changing heat source and convective heat transfer coefficient are compiled into a UDF that Fluent can recognize by C++, and unidirectional and bidirectional coupling calculations are carried out separately. Then, the prototype of SHS-PMSM and a centrifugal pump are used to build experimental system, and temperature measurement experiments are carried out under different working conditions. Finally, the experimental results are compared with those of the magnetic-heat-flow bidirectional coupling method, which verified the accuracy of the proposed magnetic-heat-flow bidirectional coupling method and provided a theoretical basis for the design of SHS-PMSM.

2 Method

The temperature rise of a motor is mainly determined by its own heat generation during operation and the heat exchange with the external environment. The heat source is the heat generated by the internal current, magnetic field, and inner friction during operation, and the convective heat transfer coefficient characterizes the heat transfer between the motor itself and the external fluid. The heat source of the motor itself mainly includes copper loss, core loss, and oil friction loss. where copper loss and oil friction loss are greatly affected by temperature. The convective heat transfer coefficient is greatly affected by the flow velocity and temperature of the fluid passing through the housing surface of the motor. This section conducts research on the variation law of heat source and convective heat transfer coefficient, and proposes a bidirectional coupling calculation method of magnetic-heat-flow that takes into account the variable heat source and convective heat transfer coefficient based on the finite volume method.

2.1 Multiphyics bidirectional coupling method

The finite volume method is utilised to compute the temperature field of SHS-PMSM based on ANSYS Fluent. In the calculation process, the heat source is loaded into the corresponding calculation domain in the form of volume heat generation rate, which is a three-dimensional volume domain composed of grids. The volumetric heat generation rate and convective heat transfer coefficient during loading can be either constant or variable. The SHS-PMSM internal structure, boundary conditions and calculation domain are very complex, and the coupling calculation process includes two parts: magnetic-heat coupling and heat-flow coupling. The traditional unidirectional coupling finite element calculation process is shown in Fig. 1. The loss is loaded into the corresponding calculation domain of the temperature field in the form of heat generation rate in the magnetic-heat unidirectional coupling. In the heat flow unidirectional coupling, the convection heat transfer coefficient is loaded into the corresponding calculation domain in the form of a fixed value, without considering the influence of temperature on it. This calculation method does not consider the influence of temperature on loss and convective heat transfer coefficient, and the calculation results are inaccurate. In some bidirectional coupling calculation methods, only the magnetic-heat coupling is considered, while the flow-heat coupling is ignored, which leads to a lack of accuracy in the calculation results.Fig. 1 Traditional temperature field calculation process.

Fig. 1

In the magnetic-heat-flow bidirectional coupling, the heat source and convective heat transfer coefficient are loaded as variables, and more accurate temperature field calculation results can be obtained. As shown in Fig. 2, in the process of magnetic-heat coupling calculation, in order to realize variable loading of copper loss and oil friction loss, equations (7), (9) are written as UDF (user defined functions) recognized by Fluent through the DEFINE_SOURCE macro in the c++ compilation environment. Through the DEFINE_ADJUST parameter transfer macro, the temperature after each iteration calculation can be transferred to formula (7) and formula (9), and the heat source term and boundary conditions at this temperature can be recalculated and loaded into the corresponding parts again. In the process of heat-flow coupling calculation, in order to realize variable loading of convective heat transfer coefficient, equations (15), (16), (17) are written as UDFs as well. Through the DEFINE_ADJUST parameter transfer macro, the temperature after each iteration can be transferred to formula (15) or (16) or (17), and the convective heat transfer coefficient at this temperature can be recalculated, and it can be reloaded to the housing outface as one of the boundary conditions.Each calculation's heat source and convective heat transfer coefficient are determined by using the temperature determined in the previous iteration. The process is repeated until the difference between the results of the calculations for the two adjacent temperature fields is less than the predetermined amount, at which point the calculation converges and comes to an end.Fig. 2 Magnetic-heat -flow bidirectional coupling calculation process.

Fig. 2

Based on the above analysis, the research for this new method mainly involves copper loss, oil friction loss and convective heat transfer coefficient. Taking a small diameter motor with an outer diameter of ø 89 mm as an example, the loss and convective heat transfer coefficient are analyzed respectively.

The SHS-PMSM has 4 poles and 18 slots, with a rated power of 60 kW, a rated speed of 3000 rpm, and a rated torque of 191 N m. The outer diameter of the motor housing is ø 89 mm and the rotor structure adopts a permanent magnet built-in type. The cooling method of the motor is oil cooling, and the lubricating oil flows into the air gap between the stator and rotor through the hollow oil hole inside the shaft as the motor rotates, thereby achieving cooling of the winding. Its radial cross-section is shown in Fig. 3.Fig. 3 Radial cross-sectional view of SHS-PMSM.

Fig. 3

The design parameters of the motor are shown in Table 1.Table 1 Part of the design parameters of SHS-PMSM.

Table 1Parameters	Value	
Rated power/kW	60	
Rated speed/rpm	3000	
Rated torque/N·m	191	
Stator out diameter/mm	79	
Stator iner diameter/mm	50	
Core length/mm	5200	
Air gap length/mm	0.6	
Magnet thickness/mm	3.7	
Magnet width/mm	25.4	

2.2 Copper loss

When the motor is running, the current flows through the winding copper and produces copper loss. The copper loss caused by three-phase alternating current can be calculated using Formula (1):(1) PCu=mI2R+m∑k=2∞Ik2Rk

where PCu denotes copper loss; m denotes number of phases;I denotes effective value of stator current; R为average resistance;Ik denotes effective current value of k-th harmonic;Rk denotes stator winding effective resistance of k-th harmonic.

The resistance calculation formula of winding material is as follows:(2) R=ρ·lA

in Formula (2), ρ denotesresistivity (Ωm); denotes length of winding (m); A denotes cross sectional area of winding conductor (m2).

It is well known that the resistivity of the winding material changes with the temperature [[48], [49], [50]]. When the temperature increases, the resistivity increases. It can be seen from Formula (2) that different resistivities will produce different resistance values. Formula (1) shows that when the current is the same, different resistances lead to different copper losses. Therefore, during the operation of the motor, the copper consumption is not a fixed value but increases with the increase of temperature. Therefore, in the coupling calculation of fluid and solid, the method of copper loss variable loading (bidirectional loading) will get more accurate results than the traditional fixed loading. In order to achieve variable loading of copper loss, it is necessary to study the law of copper loss changing with temperature.

The length and cross-sectional area of the winding are fixed values. When the motor is running, the temperature near the winding changes. With the change of temperature, the resistivity of the winding will change. The relationship between resistivity and temperature is:(3) ρT=ρ20[1+α(T−20)]

in formula (3),T is temperature (°C); ρ20 is resistivity at 20 °C (Ω.m); α is temperature coefficient (°C−1).

At 20 °C, the resistivity of copper is 1.678 × 10−8 Ω.m, and the temperature coefficient of resistance is 0.00393 °C-1. The resistivity of winding will increase linearly with the increase of temperature.

Ignoring the influence of high-order harmonics, the calculation formula of motor copper loss is:(4) PCu=mI2R

in formula (4),m number of phases,I is current of winding (A),R is winding resistance(Ω).

When the design of the motor is completed, the effective values of its phase number and phase current are determined values, and the copper loss is affected by the winding resistance value. Equations (5), (6) can be obtained from the comprehensive equations (2), (3), (4).(5) R=ρ·lA=ρ0[1+α(T−T0)]·lA

(6) PCu=mI2R=mI2ρ0[1+α(T−T0)]·lA

in formula (6), T0 is reference temperature of winding(°C),ρ0 is resistivity of copper at reference temperature (Ω.m).

According to equations (5), (6), when the temperature changes, the copper loss will change, and the higher the temperature, the greater the copper loss. The above formulas are theoretical. To obtain a more accurate relationship, Maxwell is adopted to conduct finite element calculations of the electromagnetic field. Based on the parameterized calculation results, the variation curves of copper loss at different temperatures are obtained.

The above is theoretical calculation formulas, in order to obtain a more accurate relationship, the finite element calculations of electromagnetic field is carried out by Maxwell. According to the parametric calculation results, the variation curves of copper loss at different temperatures are obtained. The copper loss at different temperature is calculated and the polynomial function of copper loss and temperature is obtained as follows:(7) PCu=4.4643×10−6T2+6.7612T+1585.2759

2.3 Friction loss

The internal cavities of the motor are filled with motor oil (lubrication oil). During the working process, the rotational oil friction loss of the rotor is mainly divided into rotor oil friction loss and stator oil friction loss. In the oil friction loss of high-speed permanent magnet synchronous motors, the stator oil friction loss can be ignored compared to the oil friction loss between the rotor and the motor oil. For slim submersible motors, the oil friction loss of the rotor is [51]:(8) Poil=2πμR23ω2δL

in formula (8), μ is dynamic viscosity of motor oil (μ/Pa·s); R2 is outer radius of rotor (m); ω is the rotational angular velocity of the rotor (rad/s); δ is the length of airgap (m); L is the axial length of the airgap (m).

The dynamic viscosity of motor oil varies with the temperature of the motor, and has an impact on the oil friction loss. Table 2 shows different dynamic viscosity of motor oil at different temperatures.Table 2 Dynamic viscosity of motor oil at different temperatures.

Table 2Temperature
T/°C	Dynamic viscosity
μ/(Pa·s)	Temperature
T/°C	Dynamic viscosity
μ/(Pa·s)	
4	0.4186	70	0.0146	
10	0.3001	80	0.0106	
20	0.1430	90	0.0083	
30	0.0712	100	0.0076	
40	0.0374	110	0.0073	
50	0.0279	120	0.0071	
60	0.0205			

Substitute the above data into formula (8) to obtain the curve of rotor oil friction loss with temperature.

Fig. 4 shows that the oil friction loss is inversely proportional to the motor temperature and decreases with increasing temperature. The expression for the variation of oil friction loss with temperature between 0 °C and 120 °C is as formula (9).(9) Poil=−1.058×10−7T6+3.368×10−5T5−3.067×10−3T4−5.567×10−2T3+23.895T2−1319T+2.463×104

Fig. 4 Curve of oil friction loss changing with temperature.

Fig. 4

The generation and transfer of heat inside a motor are very complex, and there are three types of heat transfer methods: convective heat transfer, heat conduction, and thermal radiation [52,53]. Due to the relatively small heat exchange rate of thermal radiation compared to the other two methods, the effect of thermal radiation is ignored. The PMSM motor is a closed type, and heat is transferred from the generating part to both inward and outward during operation. The heat transferred inward to the rotor is carried away by the motor oil circulating inside the shaft, and the heat transferred outward to the casing is carried away by the crude oil between the housing and the well casing. By exchanging heat with internal motor oil and external crude oil, the internal temperature of the motor reaches stability and is maintained within a reasonable range to ensure normal operation of the motor.

2.4 Convective heat transfer coefficient

The convective heat transfer coefficient reflects the heat transfer capacity and speed between the housing and the external crude oil. The larger the value, the faster the convective heat transfer. The factors that affect the value of convective heat transfer coefficient include flow velocity, fluid physical properties, and the geometric shape of the heat transfer surface, and fluid physical properties refer to fluid density, dynamic viscosity, thermal conductivity, and specific pressure heat capacity [54].

When the displacement of the pump is different, the flow rate of crude oil between motor housing and well casing is different, and the convective heat transfer coefficient between the motor housing and the fluid will also be different, which will affect the temperature rise of the motor.

The crude oil surrounding the motor housing is in a flowing state, and the Reynolds number needs to be calculated first to solve the convective heat transfer coefficient and determine the flow state of the crude oil. The Reynolds number Re outside the housing can be expressed as follows:(10) Re=v1D1ρ1μ1′

in formula (10),v1 is the flow rate of external crude oil (2.0 m/s,2.5 m/s,3.0 m/s);D1 is equivalent cooling diameter (D1 = 89 mm);r1 is fluide density; μ1′ is dynamic viscosity.

When the crude oil flowing around the housing has a Re ≤ 2320, the flow state can be regarded as laminar flow. The Nusselt number (Nu) of convective heat transfer coefficient on the outer surface of the housing is calculated by the following formula:(11) Nu=1.86(Re·P1r·D1L)0.33(μ1μ1′)0.14

in which, P1r is Prandtl number, L is effective length of motor, μ1 is dynamic viscosity of crude oil at ambient temperature.

When the crude oil flowing around the housing has a 2320<Re < 10000, the flow state can be regarded as laminar flow. The Nusselt number (Nu) of convective heat transfer coefficient on the outer surface of the housing is calculated by the following formula:(12) Nu=0.116(Re0.67−125)P1r0.33[1+(D1L)0.67](μ1μ1′)0.14

When the crude oil flowing around the housing has a Re ≥ 10000, the flow state can be regarded as laminar flow. The Nusselt number (Nu) of convective heat transfer coefficient on the outer surface of the housing is calculated by the following formula:(13) Nu=0.027Re0.8·P1r0.33(μ1μ1′)0.11

The convective heat transfer coefficient of the motor housing outface can be expressed as:(14) α=λNuD1

in which, λ is thermal conductivity.

The convective heat transfer coefficient on the outer surface of the housing under different temperatures and flow velocities were calculated, as shown in Table 3.Table 3 The convective heat transfer coefficient of the housing outer surface.

Table 3Temperature	Convective heat transfer coefficient α(W/(m2·K))	
v = 2.0 m/s	v = 2.5 m/s	v = 3.0 m/s	
40 °C	157.95	188.67	218.18	
50 °C	176.27	210.56	243.49	
60 °C	191.2	228.39	264.12	
70 °C	201.51	240.71	278.36	
80 °C	209.06	249.72	288.78	
90 °C	213.06	254.5	294.31	
100 °C	219.38	262.04	303.03	
110 °C	224.26	267.88	309.78	
120 °C	227.96	272.29	314.89	
130 °C	229.86	274.56	317.51	

Table 4 Material and thermal conductivity of each components.

Table 4Components	Material	Thermal conductivity(W/m·°C)	
Stator winding	Copper	397	
Housing	45#	46.47	
Sator core	50WW310	35	
Winding insulation	Polyimide film	0.16	
Magnet	SmCo30	9	
Shaft	40Cr	30.98	
Motor oil	Insulating oil	0.12	

The relationship between the convective heat transfer coefficient and temperature at different flow rates are obtained through the least squares fitting method.

When v = 2.0 m/s,the function is:(15) α2.0=−1.668×10−6T4+6.714×10−4T3−0.1043T2+7.862T−28.76

When v = 2.5 m/s,the function is:(16) α2.5=−1.995×10−6T4+8.031×10−4T3−0.1247T2+9.399T−34.5

When v = 3.0 m/s,the function is:(17) α3.0=−2.308×10−6T4+9.292×10−4T3−0.1442T2+10.9T−39.97

3 Simulations

3.1 Coupling calculations

In this study, ANSYS fluent are adopted to carry out coupling calculation. According to the structural parameters of the motor mentioned above, a three-dimensional finite element model is established and meshed. In order to facilitate meshing and improve the calculation speed, the axial length is taken as 100 mm, and the outer diameter of the motor housing is 89 mm. The calculation model is shown in Fig. 5. The grids are all octahedrons with a maximum twist of 0.65 and about 2.3 million elements.Fig. 5 Temperature field calculation model.

Fig. 5

The finite element calculation in Fluent needs to set material physical parameters (such as thermal conductivity), heat source, boundary conditions, etc. The material and thermal conductivity of each part of the SHS-PMSM are set according to Table 2.

In addition to the copper loss and oil friction loss mentioned above, the stator core will produce core loss, which is usually calculated by Bertotti model [54]. Fig. 6 shows the calculation results of core loss under rated working conditions of the SHS-PMSM by Maxwell.Fig. 6 Results of core loss.

Fig. 6

The changing magnetic field generated by the interaction between the permanent magnet and three-phase alternating current makes the silicon steel and permanent magnet produce hysteresis loss and eddy current loss. As the loss value of this part is small, the influence on the temperature field can be ignored. For mechanical loss and stray loss, 1 %∼1.5 % of motor power is usually taken according to engineering experience [55,56].

The settings of different heat sources are shown in Table 5.Table 5 Heat sources.

Table 5Loss	Unidirectional	Bidirectional	
Copper loss	4.945 kW	4.4643×10−6T2+6.7612T+1585.2759	
Oil friction loss	2.943 kW	−4.446×10−5T6+3.8×10−3T5−0.2002T4+6.126T3−83.36T2−448.4T+2.24×104	
Core loss	0.713 kW	0.713 kW	
mechanical and stray loss	0.7 kW	0.7 kW	

The convective heat transfer coefficient at different flow rates is shown in Table 6. It is assumed that the ambient temperature is 80 °C, and the housing surface temperature is 100 °C.Table 6 Convective heat transfer coefficient of motor housing outface at different flow rates.

Table 6Flow velocity	Unidirectional	Bidirectional	
2.0 m/s	209.06 W/(m2K)	−1.668×10−6T4+6.714×10−4T3−0.1043T2+7.862T−28.76	
2.5 m/s	249.72 W/(m2K)	−1.995×10−6T4+8.031×10−4T3−0.1247T2+9.399T−34.5	
3.0 m/s	288.78 W/(m2K)	−2.308×10−6T4+9.292×10−4T3−0.1442T2+10.9T−39.97	

3.2 Results and analysis

In order to compare the calculation results of the two coupling methods, unidirectional coupling and bidirectional coupling calculations are carried out under different flow velocities. The ambient temperature is set at 80 °C, the copper loss, oil friction loss and convective heat transfer coefficient are set as Table 5, Table 6, and the other parameters are set as Table 4.

When the crude oil flow rate on the outer surface of the housing is 2.0 m/s, the calculation results of copper loss, oil friction loss and convective heat transfer coefficient loded as constant and variable values are shown in Fig. 7, Fig. 8, respectively.Fig. 7 Temperature distribution under constant loading (unidirectional coupling) at flow rate of 2.0 m/s.

Fig. 7

Fig. 8 Temperature distribution under variable loading (bidirectional coupling) at flow rate of 2.0 m/s.

Fig. 8

It can be seen from Fig. 8, Fig. 9 that the highest temperature is located in the stator winding, followed by the winding insulation and stator core, the rotor and casing temperatures are low, and the shaft and electric oil temperatures are the lowest. The winding insulation is not easy to dissipate heat due to its low thermal conductivity, so its temperature is high. Because the motor carries out convection heat exchange with external crude oil through the housing, the temperature of housing is low. The maximum winding temperature calculated by constant value loading is 101.2 °C, the stator core is 85.07 °C, and the housing is 84.89 °C; And the maximum temperatures calculated by variable loading for the three components are 113.4 °C, 87.37 °C and 87.08 °C, respectively.Fig. 9 Average temperature of each component at different flow rates.

Fig. 9

In order to analyze the calculation results of the two methods under different flow rates, the calculations are carried out respectively for the case of crude oil flow rate of 2.5 m/s and 3.0 m/s, and the results are compared with the case of 2.0m/s Fig. 9 (a) and 9 (b) are the calculation results of constant value loading and variable loading under three flow rates respectively.

It can be seen from Fig. 9 (a) and 9 (b) that in the two calculation methods, the average temperature of each component decreases with the increase of flow rate. When the flow rate increases, the convective heat transfer coefficient on the shell surface increases, and the heat transfer between the motor and the external fluid increases, so the temperature of the motor decreases. When the flow rate increases from 2.0 m/s to 3.0 m/s, the average temperature of stator core decreases from 83.3 °C to 83.03 °C by unidirectional coupling calculation. Similarly, the stator core average temperature decreases from 113.25 °C to 112.75 °C through bidirectional coupling calculation. The calculation results under different flow rates and different calculation methods show that among the three components, the winding has the highest temperature, the stator core has the lowest temperature, and the housing temperature is slightly higher than the stator core. For example, when the flow rate is 2.5 m/s, the variable loading calculation results show that the average winding temperature is 112.95 °C, the stator core temperature is 84.62 °C, and the average housing temperature is 85.77 °C. On the one hand, the stator core carries out convection heat transfer with external crude oil through the housing; on the one hand, the heat exchange is carried out through the motor oil in the air gap, so the temperature of the stator core is slightly lower than that of the casing.

It can be seen from Fig. 10(a), (b) and (c) that the calculation results of variable loading are higher than those of fixed value loading. Among them, when the flow rate is 3.0 m/s, the winding temperature of variable load method is about 12 °C higher than that of constant value loading, and the temperature of stator core and housing is about 1.5 °C higher respectively.Fig. 10 Average temperature under unidirectional and bidirectional coupling method at different flow rates.

Fig. 10

According to Fig. 10(d), compared with other components, the winding has the most obvious increase (about 12 °C) at different flow rates, while comparing the variable loading temperature with the constant loading temperature. Because the copper loss is loaded by UDF written in C++, when the variable is loaded, and the rule that the copper loss increases with the increase of temperature can be considered, the winding temperature rise is higher than that of constant value loading method. Under different flow rates, the temperature obtained by variable loading method is higher than that of constant loading method, and is about 1.5 °C for housing and stator core. The average temperature of the husing decreases the fastest with the increase of flow rate, from 86.15 °C to 85.43 °C under variable loading, and from 84.23 °C to 83.74 °C under constant loading.

4 Experiments

In order to further verify the feasibility of the magnetic-heat-flow bidirectional coupling proposed in this paper, a SHS-PMSM prototype is developed, and the experimental equipment is built based on the experimental well to carry out the downhole temperature rise experiments.

4.1 Experimental system

The prototype of SHS-PMSM is manufactured, as shown in Fig. 11. The main processes are rotor processing, silicon steel lamination, winding threading and stator assembly and rotor assembly. In the winding threading process, the temperature sensor probe is embedded in the stator, as shown in Fig. 12.Fig. 11 SHS-PMSM prototype manufacturing.

Fig. 11

Fig. 12 Embedded temperature sensor.

Fig. 12

As shown in Fig. 13, the experimental system is composed of a downhole SHS-PMSM prototype, protector, multistage centrifugal pump, downhole sensor, experimental well, real-time monitoring system, control cabinet, frenquency transformer, high pressure circulating manifold, etc.Fig. 13 Experimental system.

Fig. 13

As shown in Fig. 14, the underground real-time monitoring system includes underground sensors, transformers, ground choke, ground control box HMI, etc (see Fig. 13). The downhole sensor unit is installed at the end of the SHS-PMSM through the reducer, which is responsible for measuring parameters such as winding temperature, discharge pressure, flow, etc., and transmitting the parameters to the ground control box (HMI) in the form of electrical signals.Fig. 14 Underground real-time monitoring system.

Fig. 14

As shown in Fig. 15, the ground control system includes the SHS-PMSM control cabinet and the centrifugal pump circulating pipeline console desk. The speed, power, current and other parameters of the SHS-PMSM are monitored through the motor control panel; The head, displacement and other parameters of the centrifugal pump are monitored through the circulating pipeline console desk.Fig. 15 Ground control system.

Fig. 15

The SHS-PMSM, downhole sensor unit, ground control system, multi-stage centrifugal pump, protector and other components are assembled to form an electric submersible pump unit, which is lowered to the experimental well. The test well is 60 m deep, with water as the medium in the well, high-pressure circulation pipelines on the well, and a complete set of submersible electric pump units underground. The speed and current of HSPMSM are set through the SHS-PMSM control cabinet, and the head and displacement of the centrifugal pump are set through the circulation pipeline console desk.

Experiments at different flow velocities under rated operating conditions are conducted. The power and speed of the motor are set to their rated values, and the head and displacement of the centrifugal pump are adjusted to test the temperature rise of the SHS-PMSM at different flow velocities. The experimental settings under diffrent flow velocities are shown in Table 7. Each experiment is conducted at an ambient temperature of 10 °C, and run continuously for about 250 min. During the experiment, the underground real-time monitoring system collected winding temperature data.Table 7 Experimental settings under diffrent flow velocities.

Table 7No.	Speed (rpm)	Rated power (kW)	Head (m)	Displacement (m³/d)	Flow velocities(m/s)	
1	3000	60	3244	80	2.0	
2	3000	60	2678	100	2.5	
3	3000	60	2301	120	3.0	

4.2 Comparison of results

Considering that the experimental well medium is water, which is different from the crude oil medium in actual production conditions underground, in order to control a single variable and achieve comparative analysis between experimental results and simulation results. Simulation calculations are carried out under different flow velocities with water as the medium, through the magnetic-heat-flow bidirectional coupling method proposed in this study. Then comparative analysis is conducted between the simulation results and the experimental results.

Experiments at different velocities are conducted and the temperature data is recorded automaticly by the underground real-time monitoring system.

The experiments results are shown in Fig. 16, Fig. 17, Fig. 18 are the HMI of real-time monitoring system at 2.0 m/s, 2.5 m/s and 3.0 m/s respectively and as shown in the three figures, the temperature is 37.20 °C,36.96 °C and 36.57 °C. Fig. 19, Fig. 20 show the trend of winding temperature changes throughout the entire experiment, and approximately 120 min after the start of the experiment, the winding temperature reaches stability. Fig. 20 displays the comparison of experimental results after the winding temperature reaches stability under three different flow rates. From the experimental results, it can be seen that when the flow rate is 2.0 m/s, the winding temperature is the highest (about 37.15 °C); When the flow rate is 3.0 m/s, the winding temperature is the lowest (about 36.54 °C).Fig. 16 HMI of at flow velocity 2.0 m/s.

Fig. 16

Fig. 17 HMI of at flow velocity 2.5 m/s.

Fig. 17

Fig. 18 HMI of at flow velocity 3.0 m/s.

Fig. 18

Fig. 19 Winding temperature rise curve at 2.5 m/s.

Fig. 19

Fig. 20 Winding temperature rise curve.

Fig. 20

The results of calculations under different flow velocities with water as the medium, through the magnetic-heat-flow bidirectional coupling method are shown in Fig. 21, Fig. 22, Fig. 23.Fig. 21 Temperature distribution by magnetic-heat-flow bidirectional coupling at displacement 80m³/d(Flow velocities 2.0 m/s).

Fig. 21

Fig. 22 Temperature distribution by magnetic-heat-flow bidirectional coupling at displacement 100m³/d (Flow velocities 2.5 m/s).

Fig. 22

Fig. 23 Temperature distribution by magnetic-heat-flow bidirectional coupling at displacement 120m³/d (Flow velocities 3.0 m/s).

Fig. 23

The temperature rise distribution of the motor in the simulated experimental well environment is shown in Fig. 22, Fig. 23, Fig. 24 show the globle temperature rise distribution of the motor, and Fig. 22, Fig. 23, Fig. 24 shows the winding temperature distribution.Fig. 24 Experimental results and calculation results of winding.

Fig. 24

In order to verify the accuracy of the temperature rise calculation method proposed in this article, the experimental results are compared with that of calculations by both variable and constant loading, as shown in Fig. 24 and Table 8.Table 8 Errors in simulation calculation results and experimental results.

Table 8Flow velocities	Constant	Variable	
Deviation	Percentage	Deviation	Percentage	
2.0 m/s	13.59 °C	36.53 %	1.44 °C	3.87 %	
2.5 m/s	13.33 °C	36.07 %	1.28 °C	3.46 %	
3.0 m/s	12.88 °C	35.22 %	0.93 °C	2.54 %	

According to Fig. 24, when the flow velocities is 2.0 m/s, the average temperature of the winding obtained through constant value loading and variable loading is 23.61 °C and 35.76 °C, respectively and the experimental winding temperature is 37.20 °C; When the flow rate is 2.5 m/s, the winding temperatures are 23.63 °C, 35.68 °C, and 36.96 °C, respectively; When the flow rate is 3.0 m/s, it is 23.69 °C, 35.64 °C, and 36.57 °C, respectively. The traditional constant value loading calculation method has a much lower temperature than the experimental results, with errors greater than 10 °C.

According to Table 8, when the flow velocities are 2.0 m/s, 2.5 m/s, and 3 m/s, the calculated temperature under constant loading method differs significantly from the experimental temperature, with error percentages greater than 30 %. The temperature calculated by variable loading differs from the experimental temperature by 1.44 °C, 1.28 °C, and 0.93 °C, respectively, with error percentages of 3.87 %, 3.46 %, and 2.54 %. At three different flow velocities, the error percentage of the temperature obtained by the variable loading method is around 3 %, which is relatively small. This verifies the effectiveness of the proposed magnetic-heat-flow bidirectional coupling method in predicting the temperature rise of HSPMSM.

5 Conclusions

Based on the finite volume method, this paper conducts research on the calculation method of temperature rise for submersible permanent magnet synchronous motors, and proposes magnetic-heat-flow bidirectional coupling method that takes into account changes of heat source and convective heat transfer coefficient. Furthermore, a 60 kW SHS-PMSM is taken as an example, and temperature calculations and experiments are conducted respectively. According to the comparative analysis, the following conclusions are drawn:

Copper loss and oil friction loss are greatly affected by temperature, and as the temperature increases, the loss increases. Therefore, the theoretical analysis shows that the temperature rise calculation results of constant loading method are smaller than those of variable loading. The simulation results show that the overall temperature of the motor calculated by the magnetic-heat-flow bidirectional coupling is higher than that of obtained by the traditional constant loading method, and is closer to experimental results. The feasibility of the magnetic-heat-flow bidirectional coupling method has been verified.

The convective heat transfer coefficient on the housing's outer surface is largely dependent on flow velocity; an increase in flow velocity causes an increase in the Reynolds number, which in turn causes an increase in the convective heat transfer coefficient. Since the convective heat transfer coefficient causes a decrease in the motor's temperature rise, increasing flow velocity can be used to reduce the temperature rise.

Funding

This research was funded by the 10.13039/501100002855 Ministry of Science and Technology of the People's Republic of China (2021YFB3401400 , 2021YFB3401402 ).

Data availability statement

Data will be made available on request.

CRediT authorship contribution statement

Liping Tan: Writing – original draft, Software, Data curation. Yucong Wang: Software, Resources, Formal analysis. Yuyin Wu: Software, Resources, Project administration. Teng Wang: Writing – original draft, Methodology, Investigation, Conceptualization. Junguo Cui: Validation, Methodology, Funding acquisition. Hongyan Wang: Writing – review & editing, Resources, Funding acquisition.

Declaration of competing interest

We declare that we have no competing interests.

Acknowledgments

The authors would like to thank the editor and the reviewers for their helpful comments and suggestions.
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