# Temperature Field Calculation of the Hybrid Heat Pipe Cooled Permanent Magnet Synchronous Motor for Electric Vehicles Based on Equivalent Thermal Network Method

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## Abstract

**:**

## 1. Introduction

## 2. Hybrid Heat-Pipe Cooling PMSM for EVs

#### 2.1. Motor Model

#### 2.2. The Equivalent Thermal Conductivity of Hybrid Heat Pipe

## 3. Thermal Network Analysis of Hybrid Heat-Pipe Cooling Motor

#### 3.1. Thermal Network Model Construction

- The motor’s cooling effect and temperature distribution are circumferentially symmetrical.;
- All the heat generated in the motor is regarded as all derived from the coolant, and the coolant temperature is regarded as the external node;
- Due to the heat pipe’s relatively large thermal conductivity, heat pipe’s additional heat path is regarded as heat conduction only through the heat pipe.
- All the losses of the rotor and the permanent magnet are transmitted from the heat pipe to the coolant in the hollow shaft.

#### 3.2. Thermal Resistance Calculation of Hybrid Heat-Pipe Cooling Structure

_{1}is the thermal conductivity of the contact part 1, λ

_{2}is the thermal conductivity of the contact part 2, r

_{1}is the inner meridian of the cylinder, r

_{2}is the interface radius of the two-cylinder parts, and r

_{3}is the outer diameter of the cylinder.

- The thermal resistance from the end winding to the annular fixed heat pipe is R
_{87}and R_{1011}. Taking R_{87}as an example:$${R}_{87}=\frac{{d}_{\mathrm{cudb}}}{2{\lambda}_{\mathrm{cu}}{S}_{87}}+\frac{1}{{\lambda}_{\mathrm{hp}}{S}_{87}}$$_{cudb}is the radial thickness of the end winding, and λ_{hp}is the static thermal conductivity of the heat pipe. - The thermal resistance of the rotor yoke through the rotating heat pipe to the cooling water inside the hollow shaft includes R
_{1828}, R_{1928}and R_{2028}. The thermal resistance R_{1828}is analyzed as an example:$${R}_{1828}=\frac{1}{2\pi {\lambda}_{\mathrm{stj}}l/3}\mathrm{ln}\frac{{r}_{\mathrm{re}}}{{r}_{\mathrm{m}}}+\frac{1}{{\alpha}_{\mathrm{q}}{S}_{1828}}$$_{q}is the convective heat transfer coefficient between the heat pipe and the cooling water, and S_{1828}is the equivalent heat transfer area between the rotating heat pipe and the cooling water. - The thermal resistance of the annular fixed heat pipe at the end of the winding and the rotating heat pipe at the rotor yoke can be obtained by the temperature difference between the evaporation section and the condensation section of the heat pipe obtained by the CFD simulation above, namely:$$R=\frac{{T}_{\mathrm{E}}-{T}_{\mathrm{c}}}{P}$$
_{e}is the steady-state temperature of the evaporation section of the heat pipe, T_{c}is the steady-state temperature of the condensation section of the heat pipe, P is the heat load of the evaporation section of the heat pipe, and R is the equivalent thermal resistance of the heat pipe.

#### 3.3. Thermal Resistance Calculation of Other Parts

#### 3.3.1. The Stator Yoke

- The thermal resistance from the stator yoke to the winding in the slot is R
_{18}, R_{29}and R_{310}. Considering the consistency of thermal resistance, R_{18}is taken as the analysis object:$${R}_{18}=\frac{{d}_{\mathrm{de}}/2}{{\lambda}_{\mathrm{stj}}{S}_{18}}+\frac{{d}_{\mathrm{jy}}}{{\lambda}_{\mathrm{jy}}{S}_{18}}+\frac{{d}_{\mathrm{cu}}/2}{{\lambda}_{\mathrm{cu}}{S}_{18}}$$_{stj}is the radial thermal conductivity of the silicon–steel sheet in the stator yoke, λ_{jy}is the thermal conductivity of insulation in the slot, λ_{cu}is the thermal conductivity of the winding’’s copper wire, d_{de}is the radial thickness of the stator yoke, d_{cu}is the radial thickness of the winding, S_{18}is the equivalent heat-transfer area between node 1 and node 8, and the same below. - Taking the thermal resistance from the stator yoke to shell R
_{133}from node 1 to node 33 as an example:$${R}_{133}=\frac{1}{2\pi {\lambda}_{\mathrm{stj}}\frac{l}{3}}\mathrm{ln}\frac{{r}_{\mathrm{dw}}}{{r}_{\mathrm{den}}}+\frac{1}{2\pi {\lambda}_{\mathrm{sh}}\frac{l}{3}}\mathrm{ln}\frac{{r}_{\mathrm{sh}}}{{r}_{\mathrm{dw}}}$$_{Sh}is the shell thermal conductivity, r_{den}is the inner diameter of the stator yoke, r_{dw}is the outer diameter of the stator yoke, and r_{sh}is the outer diameter of the motor shell. - The thermal resistance from the stator yoke to the potting material is R
_{131}and R_{132}, taking R_{131}as an example:$${R}_{131}=\frac{1}{2}{R}_{12}+\frac{1}{{\lambda}_{\mathrm{gf}}{S}_{131}}$$_{12}is the axial thermal resistance of stator yoke, λ_{gf}is the thermal conductivity of the potting adhesive, and S_{131}is the equivalent heat-transfer area between the stator yoke and potting material. - The thermal resistances from the stator yoke to stator teeth are R
_{14}, R_{25}and R_{36}. Taking R_{14}as an example:$${R}_{14}=\frac{({r}_{\mathrm{dw}}-{r}_{\mathrm{dn}})/2}{{\lambda}_{\mathrm{stj}}{S}_{14}}$$_{dn}is the inner diameter of the stator. - The axial thermal resistance of the stator yoke is R
_{12}and R_{23}. Taking R_{12}as an example:$${R}_{12}=\frac{l/3}{{\lambda}_{\mathrm{stz}}{S}_{12}}$$_{stz}is the axial thermal conductivity of the stator silicon–steel sheet, and l is the axial length of the stator.

#### 3.3.2. The Winding

- The thermal resistance from the winding to the stator yoke is, taking R
_{81}between node 8 and node 1 as an example:$${R}_{81}={R}_{18}$$ - The thermal resistance from the winding to the stator teeth includes R
_{84}, R_{95}, and R_{106}, taking R_{84}as an example:$${R}_{84}=\frac{{d}_{\mathrm{cu}}/2}{{\lambda}_{\mathrm{cu}}{S}_{84}}+\frac{{d}_{\mathrm{jy}}}{{\lambda}_{\mathrm{jy}}{S}_{84}}+\frac{{d}_{\mathrm{cu}}/2}{{\lambda}_{\mathrm{stj}}{S}_{84}}$$_{dc}is the radial thickness of stator teeth. - The axial thermal resistance of the winding includes R
_{89}and R_{910}, taking R_{89}as an example:$${R}_{89}=\frac{l/3}{{\lambda}_{\mathrm{cu}}{S}_{89}}$$

#### 3.3.3. The Stator Teeth

- The thermal resistance from the stator teeth to the stator yoke includes R
_{41}, R_{51}, and R_{63}. Taking R_{41}as an example for calculation:$${R}_{41}={R}_{14}$$ - The thermal resistance from the stator teeth to the winding includes R
_{48}, R_{59}, and R_{610}, taking R_{48}as an example:$${R}_{48}={R}_{84}$$ - The axial thermal resistance of stator teeth includes R
_{45}and R_{56}. Taking R_{45}as an example:$${R}_{45}=\frac{l/3}{{\lambda}_{\mathrm{stz}}{S}_{45}}$$ - The thermal resistance from stator teeth to air gap includes R
_{426}, R_{526}and R_{626}. Taking R_{426}as an example:$${R}_{426}=\frac{{h}_{\mathrm{st}}}{2{\lambda}_{\mathrm{stj}}{S}_{426}}+\frac{1}{{\alpha}_{\mathrm{cq}}{S}_{426}}$$_{cq}is the convective heat transfer coefficient between the stator teeth and the air-gap.

#### 3.3.4. The Rotor Boot

- The thermal resistance from the rotor boot to the air gap includes R
_{1226}, R_{1326}and R_{1426}. Taking R_{1226}as an example:$${R}_{1226}=\frac{1}{2\pi {\lambda}_{\mathrm{stj}}l/3}\mathrm{ln}\frac{{r}_{\mathrm{rw}}}{{r}_{\mathrm{re}}}+\frac{1}{{\alpha}_{\mathrm{rq}}{S}_{1226}}$$_{rw}is the outer diameter of the rotor, r_{re}is the outer diameter of the rotor yoke, and α_{cq}is the convective heat-transfer coefficient between the rotor boot and the air gap. - The thermal resistance from rotor boot to the permanent magnet includes R
_{1215}, R_{1316}and R_{1417}. The thermal resistance R_{1215}is analyzed as an example:$${R}_{1215}=\frac{{d}_{\mathrm{pm}}/2}{{\lambda}_{\mathrm{pm}}{S}_{1215}}+\frac{1}{2\pi {\lambda}_{\mathrm{stj}}l/3}\mathrm{ln}\frac{{r}_{\mathrm{rw}}}{{r}_{\mathrm{re}}}$$_{pm}is the thermal conductivity of the permanent magnet and d_{pm}is the thickness of the permanent magnet. - The thermal resistance from the rotor boot to the rotor yoke includes R
_{1218}, R_{1319}and R_{1420}. Taking R_{1218}as an example for analysis:$${R}_{1218}=\frac{1}{2\pi \frac{180}{360}{\lambda}_{\mathrm{stj}}l/3}\mathrm{ln}\frac{{r}_{\mathrm{rw}}}{{r}_{\mathrm{m}}}$$_{m}is the inner diameter of the rotor. - The axial thermal resistance inside the rotor boot includes R
_{1213}and R_{1314}. Taking R_{1213}as an example for analysis:$${R}_{1213}=\frac{l/3}{{\lambda}_{\mathrm{stz}}{S}_{1213}}$$

#### 3.3.5. The Permanent Magnet

- The thermal resistance from permanent magnet to rotor boot includes R
_{1512}, R_{1613}and R_{1714}. Taking R_{1512}as an example for analysis:$${R}_{1512}={R}_{1215}$$ - The thermal resistance from the permanent magnet to the rotor yoke includes R
_{1518}, R_{1619}and R_{1720}. The thermal resistance R_{1518}is analyzed as an example:$${R}_{1518}=\frac{{d}_{\mathrm{pm}}/2}{{\lambda}_{\mathrm{pm}}{S}_{1518}}+\frac{1}{2\pi {\lambda}_{\mathrm{stj}}l/3}\mathrm{ln}\frac{{r}_{\mathrm{rew}}}{{r}_{\mathrm{m}}}$$ - The axial thermal resistance of permanent magnet includes R
_{1516}and R_{1617}. Taking R_{1516}as an example for analysis:$${R}_{1516}=\frac{l/3}{{\lambda}_{\mathrm{pm}}{S}_{1516}}$$

#### 3.3.6. The Rotor Yoke

- The thermal resistance from the rotor yoke to the rotor boot includes R
_{1812}, R_{1913}and R_{2014}. The thermal resistance R_{1812}is analyzed as an example:$${R}_{1812}={R}_{1218}$$ - The thermal resistance from the rotor yoke to the permanent magnet includes R
_{1815}, R_{1916}and R_{2017}. The thermal resistance R_{1815}is analyzed as an example:$${R}_{1815}={R}_{1518}$$ - The axial thermal resistance of the rotor yoke includes R
_{1819}and R_{1920}. Taking R_{1819}as an example for analysis:$${R}_{1819}=\frac{l/3}{{\lambda}_{\mathrm{stz}}{S}_{1819}}$$

#### 3.4. Solution of Equivalent Thermal Network Model

## 4. The Temperature Field Analysis of Hybrid Heat-Pipe Cooling PMSM for EVs

#### 4.1. Hybrid Heat-Pipe Cooling PMSM for Evs

#### 4.2. Equivalent Thermal Conductivity Experiment of Heat Pipe

_{e}is the steady-state temperature of the evaporation section of the heat pipe, T

_{c}is the steady-state temperature of the condensation section of the heat pipe, P is the heat load of the evaporation section of the heat pipe, Δt represents the average temperature difference between the evaporation and condensation section, R is the equivalent thermal resistance of the heat pipe, k is the equivalent heat transfer coefficient of the heat pipe, and A is the equivalent contact area of the heat pipe.

#### 4.3. Equivalent Thermal Conductivity Experiment of Heat Pipe

## 5. Conclusions

- The equivalent thermal resistance of the rotating heat pipe decreases first, then increases, and finally decreases with the increase of the rotor rotation speed when the thermal load is constant. The thermal resistance is low at the rated speed of 4000 rpm, and the maximum equivalent thermal conductivity can reach 76,804.9W/m·k;
- By establishing an equivalent thermal network for the motor, the temperature of each node was analyzed and calculated. When comparing the calculation results with the CFD simulation results, it was found that the temperature error of each node was within 5%, which is within the acceptable range. In view of the error, the analysis in this paper was mainly due to the complex shape of the cooling structure of the rotating heat pipe and the cooling structure of the heat pipe at the end winding, and the existence of potting glue between the end winding and the fixed heat pipe, which affects the accuracy of the thermal resistance calculation, so there is a certain error in the calculation results;
- In this paper, by comparing the temperature field calculation method of hybrid heat-pipe cooling motor based on equivalent thermal network with the traditional CFD simulation time, it was found that the iteration time of the proposed calculation method is within 194 s, which greatly shortens the temperature field calculation time of the hybrid heat-pipe cooling PMSM for EVs.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Equivalent thermal conductivity experimental equipment: (

**a**) experimental platform; and (

**b**) drive and measuring equipment.

**Figure 8.**Temperature field under non-sinusoidal excitation: (

**a**) rated condition; (

**b**) maximum speed condition; and (

**c**) maximum torque condition.

Parameter | Date | Parameter | Date |
---|---|---|---|

Rated power | 90 kW | Pole-Slot number | 8p48s |

Rated torque | 215 Nm | Pitch | 5 |

Rated speed | 4000 r/min | Stator yoke diameter | 144 mm |

Peak Power | 160 kW | Rotor yoke diameter | 48 mm |

Maximum torque | 380 Nm | Air gap length | 0.5 mm |

Maximum speed | 12000 r/min | Polar arc coefficient | 0.64 |

DC Bus Voltage | 320 V | Insulation level | H |

Motor Speed (rpm) | Thermal Resistance (°C/W) | Thermal Conductivity (W/m·k) |
---|---|---|

0 | 0.018 | 23,560.3 |

600 | 0.023 | 18,723.8 |

4000 | 0.006 | 74,628.2 |

Property | Unit | Date |
---|---|---|

Thermal conductivity | W/m·k | 3.2–3.4 |

Breakdown voltage | kV/mm | ≥15 |

Working temperature | °C | −55–180 |

Modeling Area | Material | Thermal Conductivity (W/m·k) |
---|---|---|

Shell | Iron | 39.2 |

Stator yoke (radial/axial) | Silicon steel sheet | 38.7/3.7 |

Rotor (radial/axial) | Silicon steel sheet | 38.7/3.7 |

Winding | Copper | 385 |

Slot insulation | Insulating paper | 0.2 |

Permanent magnet | Nd-Fe-B | 9 |

Shaft | Steel | 42 |

Bearing | Steel | 42 |

Cavity | Air | 0.026 |

Node | Loss at Rated Condition (W) | Loss at Maximum Torque Condition (W) | Loss at Maximum Speed Condition (W) |
---|---|---|---|

1–3 | 254.26 | 23.1 | 2147.53 |

7–11 | 6749.9 | 15,835.8 | 6145.6 |

4–6 | 484.57 | 123 | 766.97 |

12–14 | 32.4 | 12.5 | 126.75 |

18–20 | 52.4 | 11.1 | 200.7 |

15–17 | 11.1 | 5.8 | 29.3 |

21–22 | 126.1 | 18.9 | 432 |

Motor Speed (rpm) | Temperature (°C) | Thermal Resistance (°C/W) | Thermal Conductivity (W/m·k) | |
---|---|---|---|---|

T_{c} | T_{e} | |||

0 | 109.78 | 113.37 | 0.018 | 25,601.6 |

600 | 16.95 | 21.62 | 0.023 | 20,036.1 |

1200 | 22.71 | 25.12 | 0.012 | 38,402.5 |

1800 | 24.96 | 28.45 | 0.017 | 27,107.6 |

2400 | 26.40 | 30.31 | 0.019 | 24,254.2 |

3000 | 30.41 | 33.62 | 0.016 | 28,801.8 |

3600 | 32.59 | 34.96 | 0.012 | 38,402.5 |

4000 | 35.14 | 36.40 | 0.006 | 76,804.9 |

Speed (rpm) | Equivalent Experiments (W/m·k) | CFD Calculations (W/m·k) |
---|---|---|

0 | 25,601.6 | 23,560.3 |

4000 | 76,804.9 | 74,628.2 |

Operating Conditions | Calculation Method | Winding | Stator | PM | Rotor |
---|---|---|---|---|---|

Rated condition | Thermal network | 89.34 °C | 44.32 °C | 35.12 °C | 33.98 °C |

CFD | 86.48 °C | 45.4 °C | 35.54 °C | 34.33 °C | |

Error | 3.2% | 2.4% | 1.2% | 1% | |

Maximum torque condition | Thermal network | 141.82 | 45.62 | 27.12 | 27.09 |

CFD | 138.77 | 46.55 | 27.8 | 27.62 | |

Error | 2.2% | 2% | 2.4% | 2% | |

Maximum speed condition | Thermal network | 87.62 | 49.27 | 37.11 | 33.98 |

CFD | 85.26 | 49.81 | 37.72 | 34.59 | |

Error | 2.7% | 1.1% | 1.6% | 1.8% |

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## Share and Cite

**MDPI and ACS Style**

Wang, H.; Zhang, C.; Guo, L.; Chen, W.; Zhang, Z.
Temperature Field Calculation of the Hybrid Heat Pipe Cooled Permanent Magnet Synchronous Motor for Electric Vehicles Based on Equivalent Thermal Network Method. *World Electr. Veh. J.* **2023**, *14*, 141.
https://doi.org/10.3390/wevj14060141

**AMA Style**

Wang H, Zhang C, Guo L, Chen W, Zhang Z.
Temperature Field Calculation of the Hybrid Heat Pipe Cooled Permanent Magnet Synchronous Motor for Electric Vehicles Based on Equivalent Thermal Network Method. *World Electric Vehicle Journal*. 2023; 14(6):141.
https://doi.org/10.3390/wevj14060141

**Chicago/Turabian Style**

Wang, Huimin, Chujie Zhang, Liyan Guo, Wei Chen, and Zhen Zhang.
2023. "Temperature Field Calculation of the Hybrid Heat Pipe Cooled Permanent Magnet Synchronous Motor for Electric Vehicles Based on Equivalent Thermal Network Method" *World Electric Vehicle Journal* 14, no. 6: 141.
https://doi.org/10.3390/wevj14060141