Heat Pipe Bending Effect on Cooling Effectiveness in Electrical Machines

Heat pipes (HPs) are being extensively explored in motor cooling scenarios for enhanced cooling capacity. HPs are commonly bent to adapt to the compact structure of electrical machines, whereas the bending effect on motor cooling effectiveness still requires further investigation. This article analytically and experimentally studies the effect of the bending process, including bending angle and bending radius, on HP thermal properties. Cooling density is defined and derived to analyze the tradeoff between HP thermal performance and additional volume due to HP installation. Practical guidelines for feasible HP bending solutions under different space constraints are provided to achieve a higher cooling density. Finally, the HP bending effect on motor cooling effectiveness under various cooling methods is quantitatively evaluated through a validated thermal model based on a stator-winding assembly. The bending process can degrade the HP equivalent thermal conductivity by up to 76%, thus leading to a temperature difference of 4.8 K under liquid cooling conditions.

systems [1], [2], [3]. In this case, the efficient cooling technique is a significant enabler for improving the performance metrics beyond the current state of the art. Whilst, the implemented temperature reduction is directly beneficial for active material saving and reliability promotion [4], [5].
Based on the phase change principle, the heat pipe (HP) presents outstanding thermal properties, i.e., very high thermal conductivity at a small volume and lightweight. Therefore, HPs are being widely applied in motor cooling systems with remarkable cooling performance validated through prototyping and testing [6], [7], [8], [9], [10]. During the design and implementation of motor cooling systems, HPs are often bent to adapt to the compact structure of high-power density motors. Fig. 1 illustrates some typical bent HP applications in electrical machines [11], [12], [13]. In these cases, the main heat source of the motor is the winding module, while the coolant is generally located externally, e.g., ambient air or liquid jacket in the housing. There is a long and curved cooling path between the heat source and the heat sink, which limits heat transfer and dissipation. Thereby, bent HPs are commonly adopted to achieve a shorter cooling path and lower thermal resistance.
However, studies in the field of thermodynamics and mechanics have confirmed that the bending process degrades the HP thermal performance, thus reducing cooling benefits [14], [15], Fig. 2. Destruction of the HP internal structure during the bending process. [16], [17]. The main reason lies in that the wick structure is torn away from the copper shell by machining and the internal channel is deformed, which blocks the vapor flow and backflow of the working medium, as presented in Fig. 2. Most of the existing studies in electrical machines only focus on the structural design and implementation of the motor cooling structures based on HPs, while ignoring the negative impact of the HP bending process on its own thermal performance and also motor cooling effectiveness. Hence, few publications carry out quantitative studies on this topic and practical guidelines for the promotion of bent HPs in electrical machines are necessary, but still lacking for engineers.
This study aims to fill the above knowledge gaps and the main contributions of this paper are summarized as follows: 1) The analytical model for bent HP is derived and validated by experimental results. The negative impact of the bending process, including the bending angle and the bending radius, on HP thermal properties is analytically and experimentally studied. 2) Cooling density is defined and derived to characterize the tradeoff between HP thermal conductivity and additional volume, coupled with practical guidelines provided to achieve a higher cooling density under different space constraints.
3) The bending effect on motor cooling effectiveness is quantitatively evaluated by a validated 3-D thermal model with common cooling methods. The remainder of this article is organized as follows. Section II briefly introduces the HP working principle and derives the analytical model of bent HP. In Section III, experimental investigation is conducted to quantify the bending effect on HP thermal performance and validate the above model. Section IV defines and derives the cooling density, together with discussions on feasible HP bending solutions. In Section V, a 3-D thermal model is developed and validated to evaluate the HP bending effect on motor cooling effectiveness. Finally, a conclusion is drawn in Section VI.

II. THERMAL MODEL OF HP
The copper-water HP with sintered wick structure is commonly the preferred choice in motor cooling scenarios [18], [19]. Therefore, it is adopted in this study and its detailed parameters are listed in Table I. A copper-water HP with sintered wick structure commonly consists of a hollow sealed copper shell, a sintered wick structure, and an internal vapor space, together with a small amount of working medium contained in the chamber, as shown in Fig. 3. The HP is based on the phase transition principle, with working medium from the liquid to vapor state and vice versa. The operating medium evaporates into vapor after absorbing heat in the evaporator section. The resulting vapor is driven by high pressure to flow through the adiabatic section into the condenser section, where it condenses into liquid again and releases a large amount of heat. The condensed liquid is pumped back to the evaporator section through the capillary force generated by the wick structure to complete a whole phase change cycle.
Based on the working principle of phase transition and HP structure, the heat transfer in the HP can be divided into the following three parts: (i) heat conduction through the copper shell; (ii) heat conduction through the wick structure; and (iii) phase change heat transfer in the vapor space. Fig. 3 presents a simplified thermal model of an HP, where thermal resistances in the model correspond to the above heat transfer process. It is assumed that there is a uniform temperature distribution in the evaporator and condenser sections.
The thermal model of the HP is established as follows, and all related symbols are defined in Table II [20], [21]. The conductive thermal resistances in copper shell are composed of radial (R se and R sc ) and axial (R s ) heat conduction, which are given by: Similarly, the conductive thermal resistances in wick structure include radial (R we and R wc ) and axial (R w ) heat conduction, The wick structure is a composite of sintered copper powder and working medium, which can be represented by a homogeneous material. The Chaudary and Bhandari model based on parallel method (λ pa ) and series method (λ se ) is adopted to estimate the thermal conductivity of such a homogeneous material [22], which is given by where the weighting factor n is usually between 0.8 and 0.9, and the value in this study is determined to be 0.8. Additionally, the thermal resistances in the vapor space can be divided into phase change in the evaporator and condenser sections (R phe and R phc ), and vapor flow in the vapor chamber (R v ), formalized as [20]: The HP bending process could narrow the internal vapor space and thus hinder the vapor flow. Two critical factors during the bending process are considered to be closely related to the HP thermal performance, i.e., bending angle α and bending radius r, which are described in Fig. 4. When the vapor flow through the bending section, there is an additional vapor pressure drop generated by hydraulic loss, which can be empirically evaluated by [23]: Therefore, the thermal resistance of vapor flow can be transformed into the following form: Since the thermal resistances of axial heat transfer in the copper shell (R s ) and wick structure (R w ) are much greater than that in parallel, thereby they are ignored in this study. Moreover, the thermal resistances of the phase change (R phe and R phc ) are far less than the conductive thermal resistances in copper shell and wick structure, so they can also be ignored. Finally, the HP thermal resistance can be simplified into the sum of thermal resistances of the following three parts: (i) the radial heat conduction in copper shell R cons ; (ii) the radial heat conduction in wick structure R conw ; and (iii) vapor flow in the vapor space R v . Additionally, the HP equivalent thermal conductivity can be also calculated as follows:

A. Experimental Setup
To validate the above HP numerical model, the HP thermal properties at various bending angles and bending radii are experimentally measured in this Section. A dedicated experimental setup based on the steady-state heat flow method is established for thermal tests, as shown in Fig. 5. The prepared specimen to be tested is affixed to the test bench with a clamp-on bench vise. The specimen is powered by an adjustable DC power supply and the input power is measured by a power meter (YOKO-GAWA WT333E). To provide a uniform cooling condition, an electric fan with large fan blades is employed and placed directly opposite the specimen. The wind velocity measured by an anemometer (OMEGA HHF144) is constant at 1.5 m/s. Moreover, the temperature variation and distribution along the specimen are monitored by thermal loggers (Pico TC-08), and then recorded in a laptop for further data processing and analysis.

B. PTCHE-Based Specimen Preparation
The positive temperature coefficient heating element (PTCHE), also known as self-regulating heater, is adopted as the heat source for the experimental specimen due to its advantages of the rapid warm-up, accurate temperature maintenance at fixed input voltage, absence of thermostats and overtemperature protection capability [24], as presented in Fig. 6(a). Compared to copper windings, the introduction of the PTCHE facilitates the specimen preparation and thermal test from the following three aspects: (i) small volume and timesaving for the preparation of large numbers of specimens, (ii) more precise affixation of thermocouples, thus beneficial for accurate measurement of temperature distribution along the specimen, (iii) a shorter duration before thermal equilibrium,   Table III. Fig. 6(b) sketches the structure of the PTCHE-based specimen. The PTCHE and aluminum fin are attached to both ends of the HP as the heat source and the heat sink, respectively. The length of the fin is chosen to match the PTCHE, as the HP presents favorable thermal performance when the length of the evaporator section approaches the condenser section [25]. To increase the contact area and enhance the heat conduction, a type of 92% micronized diamond powder added thermal compound (IC diamond) with a high equivalent thermal conductivity of 2000 W/m·K is applied on the contact interfaces between the PTCHE, fin and HP. Additionally, thermally conductive silica gel is used as an adhesive at both ends of the contact interface to bond them together for a more reliable connection. Finally, the entire specimen is wrapped with multi-layer thermal insulation material (Nitrile rubber) except for the fin to minimize heat losses. To obtain a detailed temperature profile along the specimen, a total of nine K-type thermocouples are affixed uniformly to different sections of the specimen by cyanoacrylate adhesive, where E 1 , E 2 and E 3 are located at the evaporator section, C 1 , C 2 and C 3 are attached to the condenser section, and F 1 , F 2 and F 3 are employed at the fin surface, respectively. Thermocouples  HP to ensure accurate temperature measurement, as detailed in Fig. 6(c).
To fully evaluate the bending effect on HP thermal performance, a set of PTCHE-based specimens is prepared based on the above processing method, where HPs are bent at bending angles of 0°, 30°, 60°, 90°, 120°, 150°, 180°and bending radii of 12.6 mm, 15.3 mm, 18.3 mm, 21.5 mm, 24.2 mm. The selection of bending radius depends on the fabrication process that can be provided by the supplier. The prepared specimens are presented in Fig. 7.

C. Data Processing
The HP thermal performance is characterized by its thermal resistance R HP , given by: where P is the heat load fed into the specimen, T e and T c are average temperatures of the evaporator and condenser sections. Consequently, the HP equivalent thermal conductivity can be finally calculated by: where S is the HP cross-sectional area and l is the HP equivalent length, i.e., the center-to-center distance between the evaporator and condenser sections. Additionally, the convective heat transfer coefficient of the fin can be calculated as: where A is the overall surface area of the fin, and T f is the temperature difference between the fin and ambient air, given by: where T f is the average temperature of the fin surface and T 0 is the ambient temperature.

D. Results and Analyses
During the experiments, the temperature data is sampled and logged once a second. When the temperature variation within 5 minutes is smaller than 0.2 K, thermal equilibrium is considered to be reached and the sensors data are recorded.
To distinctly demonstrate the bending effect on HP thermal properties, the variation rate V is defined to characterize the change of HP equivalent thermal conductivity in percentage, given by: where λ max and λ min are the maximum and minimum value of the HP equivalent thermal conductivity at each bending angle or bending radius. Fig. 8 illustrates the test results of HP equivalent thermal conductivities at different bending angles and bending radii. It can be found that the HP thermal performance deteriorates significantly with the increase of the bending angle, i.e., a slight decrease by 18.9% within 90°while a sharp degradation by 75.2% above 90°, as illustrated in Fig. 8(a). A greater impact of bending angle can be observed clearly with smaller bending radius, where the variation rate V declines from 75.2% (r = 12.6 mm) to 53.9% (r = 24.2 mm). As for the bending radius influence, the HP thermal performance gradually improves with increasing bending radius, i.e., smaller effect within 90°(V is less than 10%) while more significant influence over 90°(46% at 180°).
Based on the above results and analyses, the following brief conclusions can be drawn.
1) When the bending angle exceeds 90°, the bending effect becomes greater, including both bending angle and radius. 2) The HP thermal performance is more affected by the bending angle than the bending radius.
3) The influence of bending angle increases slightly with the decrease of bending radius, while the effect of bending radius enhances significantly with the increase of bending angle. The HP equivalent thermal conductivities at various bending angles and bending radii obtained by the experiment are further  compared with analytical results for validation, as presented in Fig. 9. The maximum deviation between them is within 15%, which verifies the analytical model in Section II.
Furthermore, the heat load measured by the power meter is 27.5 W and the total surface area of the fin is 0.0285 m 2 , coupled with an average temperature difference of 35 K between the fin and ambient air. Thereby the convective heat transfer coefficient of the fin at a wind speed of 1.5 m/s can be calculated according to (11):

IV. TRADEOFF BETWEEN HP THERMAL PERFORMANCE AND ADDITIONAL VOLUME
In motor cooling systems, HPs are generally bent to adapt to the curved cooling path and reduce additional volume, which refers to the extra space occupied by HP incorporation. On the other hand, the bending process has been proven to be detrimental to the HP thermal performance and even motor cooling effectiveness. Therefore, a tradeoff study between the HP thermal characteristics and additional volume is required to fully maximize the HP thermal benefits within a compact structure.

A. Additional Volume
The additional volume in an electrical machine consists of axial and radial sections, which can be illustrated in the sectional view of a motor equipped with bent HPs in Fig. 10. To simplify the derivation, the following assumptions are initially made in this study: (i) only the effect of bending angle is considered and bending radius is ignored; (ii) HPs are mounted evenly along the circumference of the motor surface; (iii) HPs are bent at the middle position so that the length of the HP insertion is the same as the exposed length, which results in favorable thermal characteristics [25].
For bending angle less than 90°, the additional volume is comprised of radial and axial parts, given by: = (L+L cos α)× π(R+L sin α) 2 −πR 2 +L cos α×πR 2 = 2πRL 2 sin α + πL 3 sin 2 α + 2πRL 2 sin α cos α + πL 3 sin 2 α cos α + πLR 2 cos α (15) where α is the HP bending angle, R is the motor radius and L is the length of HP insertion and exposure. As for bending angle above 90°, only radial additional volume is generated and calculated as: In summary, the additional volume V a can be expressed as:

B. Cooling Density
Generally, the HP is expected to demonstrate favorable thermal performance at a comparatively small additional volume. However, the additional volume of an electrical machine with bent HPs is contradictory to HP thermal benefits. For example, when the HP is bent at 180°, the minimum additional volume is achieved whilst the HP thermal property is also the poorest. Therefore, a new indicator, cooling density K V , is defined in this article to characterize the tradeoff between the HP thermal characteristics and the additional volume, which is expressed as the HP equivalent thermal conductivity per unit of resulted extra space, given by: It is considered to obtain a perfect tradeoff between HP performance and motor additional volume, when cooling density K V reaches its maximum. To facilitate derivation and calculation, the HP equivalent thermal conductivities obtained by the experiment are further fitted into a quadratic polynomial by polynomial regression, with a coefficient of determination of 0.993, which indicates a high accuracy. The expression is given by: As mentioned above, only bending angle α is considered when deriving the additional volume, so the above equation can be converted into the following form by substituting r = 12.6 mm: Finally, the cooling density K V can be expressed as: To obtain a favorable tradeoff between HP thermal benefits and motor additional volume, the maximum value of cooling density K V needs to be figured out. For bending angle below 90°, the maximal value of K V depends on the motor geometry: (i) when R (1 + √ 2)L, i.e., a motor with smaller length-toradius ratio, the maximal value appears at 90°; (ii) when R < (1 + √ 2), i.e., a motor with larger length-to-radius ratio, the maximal value is obtained at 0°. Regarding bending angle over 90°, K V reaches its maximal value at 180°.  Combined with space constraints for motor installation, the practical guidelines for feasible HP bending angle can be finally summarized as follows based on above derivations, as illustrated in Fig. 11.
1) For cases with limited radial mounting space, unbent or slightly bent HPs are well suited for this situation, where the external cylinder of the motor is not occupied and HPs provide favorable thermal performance. 2) When faced with axial mounting space limitations, the HP bending angle can be determined between 90°and 180°a ccording to the thermal load and cooling requirements. The right angle is ideal for high thermal load cases due to better HP thermal properties, while 180°is preferred for low thermal load scenarios to achieve a smaller additional volume. 3) Regarding space limitations in both axial and radial directions, a bending angle of 180°can be adopted for a minimum extra space or other structural implementation can be also considered based on the tradeoff between the HP performance and motor volume.

A. Stator-Winding Assembly
To fully evaluate the impact of HP bending process on motor cooling effectiveness, a stator-winding assembly is developed for numerical and experimental study, as detailed in Fig. 12. The copper wires are wound on a three-slot laminated stator module with slot liner used for electric insulation. An HP with a bending angle of 90°and a bending radius of 12.6 mm is inserted into the clearance between different winding layers. The same high-performance thermal compound (IC diamond) as the PTCHE-based specimen is used to fill the tiny gaps between windings and the HP to enhance heat conduction and electric insulation properties. Moreover, the fin arrangement is ensured to be identical to the PTCHE-based specimen, including material, geometry and installation method.
B. Thermal Modeling 1) Physical Model: A 3-D FEA thermal model is developed based on the established stator-winding assembly. To accurately quantify the temperature distribution across the thermal model, the winding module is divided into active part (in-slot winding) and end winding. The heat is transferred mainly in the axial direction (along the z-axis) in the in-slot winding, while along the y-axis in the end winding. The thermal properties of the materials involved in the thermal model are listed in Table IV. 2) Heat Source and Boundary Conditions: The heat source and cooling condition for the thermal model are the same as the aforementioned PTCHE-based specimen. The winding module is assigned as the heat source with an identical thermal load of 27.5 W. To merely evaluate the HP bending effect on the motor cooling effectiveness, it is assumed that the heat generated is only dissipated through the cooling path including the HP and the fin. Hence, the fin surface is defined as convective boundary conditions with a convective heat transfer coefficient of 27.6 W/m 2 ·K obtained in Section III, and the remaining surfaces are specified as adiabatic boundary conditions, as presented in Fig. 13.

C. Numerical Results
Fig. 14 describes the steady-state temperature distribution across the thermal model, particularly in the in-slot winding and end winding region. A higher temperature is observed in the end winding region with the hotspot temperature reaching 88.6°C. Compared to the end winding, the temperature of the in-slot winding is significantly lower due to the shorter cooling path to the HP. Moreover, favorable temperature uniformity can be observed in the HP, i.e., a very small temperature difference across the HP, which indicates remarkable thermal performance.

1) Specimen Preparation:
To experimentally verify the thermal model, an experimental specimen is further processed based on the stator-winding assembly developed in Part A, as detailed in Fig. 15. For a clear picture of the winding temperature profile, a total of six numbered thermocouples are affixed to the center of the winding section, where 3 and 4 are located at in-slot windings  and the others at end windings. According to the thermal model, the entire surface of the stator-winding specimen is wrapped with multi-layer thermal insulation material (Nitrile rubber) except for the fin to minimize the heat loss. Moreover, the fin arrays of the same material and geometry are attached to the HP.

2) Comparison of Simulation and Experimental Results:
The validation test is conducted on the established platform in Fig. 5 under the same heat load (27.5 W) and cooling conditions (27.6 W/m 2 ·K) as the thermal model. The comparison between calculated and measured winding temperatures is reported in Fig. 16, where the thermocouple numbers correspond to the numbering in Fig. 15(a). It can be observed that the experimental results agree well with FEA results with acceptable deviations within 4%, which confirms a good accuracy of the thermal model. The main reason for comparatively large deviations in the in-slot winding section lies in the rough thermocouple locations due to the narrow operating space in the slot.

E. Comparative Analysis Under Various Cooling Methods
Common cooling methods for electrical machines can be classified into natural convection, forced convection and liquid cooling. Their typical parameters are listed in Table V, including convective heat transfer coefficient intervals and corresponding achievable current density range [26].
The upper limits of the above ranges and HP equivalent thermal conductivities at different bending angles and bending radii are fed into the thermal model to evaluate the boundary of the bending effect on motor cooling effectiveness. The HP cooling performance is characterized by the average value of the steady-state temperature of the end winding which has been validated with higher accuracy. For a clearer presentation of the results, the end-winding temperature of the thermal model with unbent HP is assumed to be the base value. Hence, the temperature difference T between the thermal model with bent HP and the reference value is defined as follows to describe the negative effect on motor cooling performance, given by: where T bent and T unbent are end-winding temperatures of the thermal model with bent and unbent HPs. Fig. 17 reports the results of T under common cooling methods. An uptrend in temperature difference T is observed with enhanced thermal load and cooling conditions, i.e., T rises from 0.8 K under natural convection to 4.8 K under liquid cooling conditions, which indicates a strengthened HP bending effect on motor cooling effectiveness. It can be inferred that the HP bending effect requires less concern for electrical machines with lower operating current densities under the passive cooling method. However, particular attention should be paid to the impact on cooling effectiveness in high power density motors under enhanced cooling conditions. In this case, a reasonable HP bending angle can be determined according to the guidelines given in Section IV for favorable cooling effectiveness and compact structural realization.

VI. CONCLUSION
Although HPs are being widely applied in motor cooling systems, the bending effect on motor cooling effectiveness has received less attention to date. In this article, the effect of the bending process, including the bending angle and the bending radius, on HP thermal properties has been analytically and experimentally studied. Results have shown that the HP thermal performance can be weakened by up to 76% due to the bending process, where the effect of bending angle is more significant, especially over 90°. Moreover, the cooling density has been introduced and derived to characterize the tradeoff between the HP thermal performance and additional volume. The practical guidelines for a higher cooling density under different space constraints have been provided. Unbent or slightly bent HPs are ideal for motors with radial space constraints, while HPs bent at an obtuse angle are suitable for limited axial space cases. Finally, a 3-D thermal model based on a stator-winding assembly has been developed and validated to evaluate the HP bending effect on motor cooling effectiveness. The temperature difference due to HP bending effect increases with enhanced thermal load and cooling conditions, reaching 4.8 K under liquid cooling conditions. The obtained research findings could serve as practical guidelines for electrical machines with bent HPs to enhance cooling capacity and reduce additional volume.