Thermal Analysis of High-Power Water-Cooled Permanent Magnet Coupling Based on Rotational Centrifugal Fluid–Structure Coupling Field Inversion

Abstract

An efficient and reliable heat dissipation system is essential for the safe and stable operation of high-power water-cooled couplers. However, thermal analysis methods accounting for the centrifugal effects on coolant flow remain limited. This paper presents a high-accuracy equivalent thermal network model (ETNM) for analyzing the temperature distribution in water-cooled permanent magnet couplers (WPMCs), based on fluid–structure interaction and rotational centrifugal flow-field inversion. First, the ETNM is established based on key assumptions. Subsequently, an eddy current loss calculation method based on permanent magnet mapping is proposed to accurately determine the heat source distribution. The convective heat transfer coefficient of the coolant is then precisely derived by inverting the flow field obtained from fluid–structure coupling simulations under rotational centrifugal conditions. Finally, the model is applied for temperature analysis, and its accuracy is verified through both finite element simulations and experimental tests. The calculated results show errors of only 3.2% compared to numerical simulation and 5.6% compared to experimental data, indicating strong agreement of the proposed thermal analysis method. The accuracy of copper conductor (CC) temperature prediction is improved by 32.73%, and that of permanent magnet (PM) prediction by 33.33%. Furthermore, this method enables accurate estimation of individual component temperatures, effectively preventing operational failures such as PM demagnetization, CC softening, and severe vibrations caused by overheating.

1. Introduction

The permanent magnet coupler (PMC) utilizes a magnetic field to transmit torque from a motor to a load by adjusting the air gap between the CC and the permanent magnet (PM). It offers advantages such as high transmission efficiency, relaxed alignment tolerances, and absence of harmonic interference [1,2], making it particularly suitable for high-power applications (above the MW level). Torque transfer in a PMC relies on slip between the CC and the PM. However, due to the resistance of the CC, the eddy currents generated by this slip cause power loss, leading to a temperature rise in the equipment. For PMs, magnetic properties decay exponentially with increasing temperature, with demagnetization occurring above 180 °C. For CCs, conductivity decreases at high temperatures, and softening under centrifugal force can degrade dynamic balance performance. These factors collectively reduce transmission capacity and may lead to operational failures. Therefore, effective thermal management for PMCs is crucial.

Research on PMCs has progressed, with Refs. [3,4,5,6,7,8] addressing electromagnetic behavior, heat dissipation, and mechanical structure. A thermophysical coupling methodology was implemented in Ref. [9] for the thermal design analysis of air-cooled permanent magnet eddy current couplings. Subsequently, Ref. [10] proposed a novel analytical model for a composite axial-radial permanent magnet eddy current coupler, conducting a detailed investigation into magnetic field distribution, torque-slip characteristics, the effects of structural parameters on performance, and introducing corrections for 3D effects and thermal drift. The thermal environments of low-power PMCs (under tens of kilowatts) were examined in Refs. [11,12,13] through orthogonal experiments, bidirectional electromagnetic-thermal coupling field analysis, and finite element simulations, where air cooling was the dominant strategy. However, high-power units generate considerable heat, necessitating water cooling for effective heat extraction. A significant knowledge gap exists regarding water-cooled heat dissipation for PMCs. Under high-power conditions, the temperature rise is more severe, highlighting the critical role of an efficient and reliable water-cooled system for ensuring stable performance. The characteristic high rotational speeds further complicate the thermal assessment of components in contact with the circulating coolant, impeding accurate temperature prediction.

Relevant studies on the temperature analysis of water-cooled permanent magnet couplers are scarce. Therefore, methodologies can be adapted from the field of permanent magnet motors. Currently, three main cooling channel models designed for high-power water-cooled outer-rotor permanent magnet synchronous motors are described in Ref. [14]: axial Z-shaped, rectangular-channel circumferential spiral, and circular-channel circumferential spiral. The optimal solution was obtained using a coupled electromagnetic-fluid-thermal field method. Literature [15,16] optimizes the spiral water jacket based on a fluid-thermal coupling model to improve cooling capacity. However, the cooling systems of high-power motors primarily involve wrapping the motor stator with a cooling water jacket. The coolant flow channels mainly include circumferential “Z”-shaped, circumferential spiral, circumferential semi-spiral, and axial “Z”-shaped types in Refs. [17,18,19,20]. A key distinction is that the cooling flow channel in a water-cooled motor is stationary, lacking rotational centrifugal motion. Traditional thermodynamic simulations or computational models can calculate the temperature field by specifying conditions such as coolant flow rate or velocity. Nevertheless, these methods cannot solve for the temperature field when considering heat removal by coolant flow under the influence of rotational centrifugal forces within the cooling channel.

This paper presents a high-accuracy thermal network model for water-cooled permanent magnet couplers (WPMCs), incorporating a novel rotational centrifugal flow-field inversion technique. The core contributions are twofold. First, an ETNM is established for the coupler under water-cooling conditions. Second, and most critically, the convective heat transfer coefficient (h) for the coolant is accurately determined by inverting the flow field derived from a Multiple Reference Frame (MRF) simulation that accounts for centrifugal effects. Unlike conventional CFD-derived h values used in prior ETNMs—which typically assume stationary or simplified flow—our method explicitly captures the strong centrifugal forces and complex flow patterns inherent to rotating cooling channels. Concurrently, an equivalent eddy current loss calculation method based on permanent magnet mapping is proposed to accurately define the heat source, significantly enhancing the overall accuracy of the thermal model.

The remainder of this paper is structured as follows. Section 2 introduces the geometric model of the WPMC and analyzes its water-cooling process and flow path. Section 3 details the ETNM, including node division and the calculation of key parameters. Section 4 describes the inversion process for determining the coolant flow rate and the convective heat transfer coefficient. Section 5 provides numerical simulation and experimental verification of the model. Finally, on-site validation is presented, and conclusions are drawn.

2. Geometric Model of WPMC

The water-cooled permanent magnet coupler is primarily constructed from four key components: the main coupling body, an outer casing, a speed regulation unit, and an oil–water heat exchange system, as detailed in Figure 1a. The central assembly incorporates the copper rotor, the permanent magnet rotor, an air gap adjustment mechanism, and the shafts. This study employs a symmetrically designed double-disc arrangement, as depicted in Figure 1b. The copper rotor integrates copper conductors, iron yokes, spoiler plates, and fin plates. Cooling channels are integrated into the conductor’s back plate. The permanent magnet rotor is fabricated from permanent magnets, aluminum yokes, and protective covers. In the operational setup, the motor is coupled to the copper rotor, while the load is linked to the permanent magnet rotor. When the motor drives the copper rotor, relative motion occurs through the magnetic field produced by the permanent magnets. This interaction induces eddy currents within the copper, establishing a counter-torque that resists the motion of the copper conductor. According to Newton’s third law, an equal and opposite force is applied to the permanent magnet rotor, causing it to rotate and thereby transmitting torque from the motor to the load.

Based on the geometric model, the cooling process of the WPMC is found to be concentrated mainly in the copper rotor. The composition of the copper rotor is shown in Figure 2. The cooling method involves the coolant carrying away heat from the heat source (the CC) under the rotating centrifugal action of the iron yoke (which contains the cooling channel). The coolant flow path is highly unpredictable, complicating the heat dissipation process.

The cooling water path is shown in Figure 3. Driven by a pump, the coolant enters the system inlet ① and flows through the inlet pipe ② to the inlet assembly ③. Under pressure, it enters the diffuser plate ⑤ within the copper rotor component ④. Rotation of the diffuser plate imparts centrifugal force to the water, directing it into the uniformly distributed inlets ⑥ on the conductor backplate. Under centrifugal rotation, the water accelerates through the cooling channels ⑦, where it exchanges heat with the CC heat source ⑧. It then exits through outlets ⑨ onto the fin plate ⑩. The fin turbulence directs the heated water into the spoon tube ⑪, which returns it via the outlet assembly ⑫ to the outlet pipe ⑬, completing the circuit.

Consequently, the high-speed rotation of the copper rotor and its open coolant inlets/outlets introduce significant complexity, a challenge largely unaddressed in existing thermal network models for PM machines and couplers. Within the cooling channels, the water experiences intense centrifugal forces, leading to considerable unpredictability in its flow behavior and distribution. This complexity, coupled with the intricate interplay between the coolant, CC, and iron yoke, has impeded the establishment of a reliable equivalent model for their fluid-solid coupled heat transfer. Critically, conventional thermal-network methods typically employ a CFD-derived convective heat transfer coefficient ($h$) obtained from stationary or simplified rotating flow conditions; these values fail to accurately capture the strong rotational centrifugal effects present in our WPMC system. Furthermore, the inability to precisely define the coolant’s boundary conditions at the entry and exit points under rotation has complicated accurate thermal calculation for such water-cooling systems.

To overcome these limitations, this paper introduces a “rotational centrifugal flow-field inversion” method. This novel approach integrates a fluid–structure interaction model based on a Multiple Reference Frame (MRF) to explicitly simulate the centrifugal flow field. From this simulation, a precise equivalent convection coefficient is inversely derived for the ETNM. This constitutes a key departure from and advancement over prior ETNMs. A comparison of thermal analysis methods for permanent magnet machines and couplers is presented in

Table 1. Comparison of Thermal Analysis Methods for PM Machines and Couplers.

Method	Cooling Type	Flow Treatment	Thermal Nodes	Accuracy	Application
Conventional CFD	Water/Air	Static Flow Assumption	/	Boundary-Sensitive	Static/Low-Speed Devices
Traditional ETNM	Air Cooling	Empirical Formulas	<15 nodes	Error > 8%	Low-Power PMCs
Our Method	Rotational Water Cooling	MRF-Based Centrifugal Flow Inversion	22 Nodes	Error ≤ 5.6%	High-Power WPMCs
Method	Cooling Type	Flow Treatment	Thermal Nodes	Accuracy	Application

.

To solve the temperature field for water cooling, the mathematical model of the PMC body is constructed, encompassing the permanent magnet rotor and copper rotor, which involve magnetic and thermal fields, respectively. These are coupled and analyzed together. Figure 4 provides the geometric shape, decomposition diagram, and geometric parameters of the studied PMC single disc, where the number of PM pole pairs is 10. The air gap $l_g$ is adjustable, typically between 3 and 8 mm. In the permanent magnet aluminum yoke, the sector-shaped permanent magnets follow an N/S alternating sequence.

Table 2. Detailed parameters of permanent magnetic coupler.

Name	Meaning	Value
$l_{i1}$	Thickness of the iron yoke (CC side)	22 mm
$l_{i2}$	Thickness of the iron yoke (PM side)	10 mm
$l_{cs}$	Thickness of the CC	8 mm
$l_{\mathrm{g}}$	Thickness of the air gap	3–34 mm can be changed
$l_h$	Thickness of the PM holder	34 mm
$l_p$	Thickness of the PM	33 mm
$r_1$	Inside radius of the CC	220 mm
$r_2$	Outside radius of the CC	400 mm
$r_{p1}$	Inside radius of the PM	250 mm
$r_{p2}$	Outside radius of the PM	360 mm
$r_a$	Average radius of the PM	355 mm
$r_c$	Inner diameter of the cooling channel	232 mm
$l_{c1}$	Cooling channel width	150 mm
$l_{c2}$	Cooling channel depth	4 mm
$l_{c3}$	Cooling channel length	10 mm
$l_{c4}$	Cooling channel spacing	22 mm
$H_p$	Coercive force of the PM	−900 KA/m
${\sigma }_{cs}$	Conductivity of the CC	5.8 × 107 S/m(20 °C)

presents the main parameters of the studied PMC, derived from practical engineering applications. The equipment prototype is shown in Figure 5.

3. ETNM

Figure 6 schematically depicts the configuration of the proposed equivalent thermal network method, showing the arrangement of thermal nodes and the overall model. Due to the system’s rotational symmetry, Figure 7 presents a simplified view of three permanent magnets from their average radius ($r=r_a$). This modeling framework positions thermal nodes across various components, such as the ambient environment, iron yoke, CC, air gap, aluminum yoke for the permanent magnets, the magnets themselves, and another iron yoke section. The network comprises a total of twenty-four nodes (0 to 24). In the diagram, these nodes are color-coded in red and black, with the red ones specifically indicating the locations of power loss, which serves as the heat source. It is important to note that this analysis accounts for conductive and convective heat transfer mechanisms, while the effects of radiation are considered negligible.

Solving the thermal network model requires calculating two key elements: power loss and the various thermal resistances. Among these, the equivalent convective heat transfer due to the cooling water is the most crucial factor in heat removal and requires precise calculation.

3.1. Power Loss Calculation

The CC of the permanent magnetic coupler is arranged parallel to the permanent magnet, with an air gap ranging from a few millimeters to tens of millimeters between them. Both components undergo relative rotational motion around the same axis. As shown in Figure 8, when the motor drives the CC to rotate, it cuts through the magnetic field lines of a periodically changing field with alternating N and S poles. Consequently, vortex-like and unevenly distributed eddy currents are induced in the CC on the side facing the permanent magnet. Due to the skin effect, these currents are confined to a thin layer on the CC surface adjacent to the magnetic poles. Since the permanent magnets embedded in the aluminum yoke are uniformly distributed with alternating N and S poles, the CC is exposed to a periodically changing magnetic field during rotation, making precise calculation difficult. Therefore, this paper proposes a heat loss calculation method based on permanent magnet mapping and equivalent small eddy current loops to determine the eddy current loss power.

The analysis focuses on a pair of N and S magnetic pole units. Based on magnetic circuit analysis, the following assumptions are made: (1) Due to the separation of adjacent permanent magnets by an aluminum yoke with high magnetic reluctance, coupled with the small gap between the CC and the aluminum yoke of the permanent magnet, magnetic flux leakage is relatively small and is neglected; (2) The resistivity of the CC and the relative magnetic permeability of the magnetic material are less affected by temperature during normal operation and are considered constant; (3) Magnetic circuit saturation is not considered.

Given the identical dimensions, materials, and regular arrangement of the permanent magnets within the aluminum yoke, the eddy currents generated by each pole on the CC are equivalently represented as a circular area of diameter $d$ ($A=\pi (d/2{)}^2$) for calculation ease. The direction of the eddy current circulation is illustrated in Figure 8. The area directly opposite a particular permanent magnet is denoted as $I$, $A_m$ representing the area of the CC directly facing the PM. $B$ represents the magnetic induction intensity within the air gap, which is the vector sum of the eddy current magnetic field and the permanent magnet’s magnetic field. Its magnetic flux is:

$${\varphi }_m=B{A}_m$$

(1)

The equivalent small circular loops of eddy currents generated by $n$ permanent magnets in the aluminum yoke of the permanent magnet interact with each other, resulting in a uniformly distributed eddy current loop on the entire CC.

The permanent magnet has alternating N and S poles, with adjacent N and S poles considered as a pair of magnetic poles. When a CC rotates from the position I area shown in the diagram, passing through area II and area III, respectively, the process of magnetic flux change is: ${\varphi }_m\to 0\to {\varphi }_m\to 0\to {\varphi }_m$, that is, from maximum to zero and then to negative maximum, repeating in a cycle, following an approximate cosine pattern. Its magnetic flux is:

$$\varphi =B{A}_m\cos \omega t$$

(2)

where, $\omega$ is the angular velocity of the magnetic field, $\omega =2\pi p\Delta n/60$, $P$ is the number of magnetic pole pairs, $\Delta n$ is the slip, and $t$ is time.

According to the law of electromagnetic induction, the induced electromotive force generated by an alternating magnetic field is:

$$\epsilon =d\varphi /dt=B{A}_m\sin \omega t$$

(3)

Due to the skin effect as shown in Refs. [21,22,23,24], many eddy current loops with a radius of $r$, a width of $dr$, and a skin depth of $\Delta h$ are formed on the side of the CC near the permanent magnet.

The resistance of the vortex ring is:

$$dR=\rho {\displaystyle \frac{2\pi r}{\Delta hdr}}$$

(4)

where $\rho$ is the resistivity of the CC, and $r$ is the radius of the water-cooled of the permanent magnet.

Skin depth:

$$\Delta h=\sqrt{{\displaystyle \frac{2\rho }{\omega {\mu }_0}}}$$

(5)

The induced eddy current on the CC is:

$$di={\displaystyle \frac{\epsilon }{dR}}={\displaystyle \frac{B\omega \Delta h\sin \omega t}{2\rho }}rdr$$

(6)

The instantaneous power of the vortex is:

$$dp=\epsilon di={\displaystyle \frac{\pi B^2{\omega }^2\Delta h\sin {\omega }^{2}t}{2\rho }}{r}^{3}dr$$

(7)

By integrating, we can obtain the eddy current loss power in the eddy current region corresponding to a permanent magnet as follows:

$$P_0={\displaystyle {\int }_0^{d/2}dp}={\displaystyle {\int }_0^{d/2}\frac{\pi B^2{\omega }^2\Delta h\sin {\omega }^{2}t}{2\rho }{r}^3 dr=\frac{\pi d^4{B}^2{\omega }^2\Delta h{\sin }^2\omega t}{128\rho }}$$

(8)

The number of circular areas with equivalent eddy current area on the CC is $n$:

$$n={\displaystyle \frac{{360}^{\circ }}{2\arcsin (d/2R_1)}}$$

(9)

So the total eddy current loss power on the CC is:

$$P_{loss}=n{P}_0={\displaystyle \frac{{n{A}_m}^2{B}^2{\omega }^2\Delta h\sin {\omega }^{2}t}{16\pi \rho }}$$

(10)

Sensitivity Analysis and Model Validation

To evaluate the robustness of the eddy current loss model under its stated assumptions, a comprehensive sensitivity analysis was conducted. The impact of variations in four key parameters—CC conductivity, permanent magnet (PM) coercivity, air gap length, and slip—on the calculated loss was quantified. The sensitivity analysis yielded the following results:

CC Conductivity vs. Temperature: Accounting for the temperature-dependent decrease in copper conductivity, a reduction of approximately 25% at the maximum operating temperature of 130 °C (compared to 20 °C) was calculated. This property variation leads to a corresponding increase in eddy current loss of about 12%.

PM Coercivity vs. Temperature: The analysis of Nd-Fe-B PM properties shows that within the operational temperature range (up to 110 °C), the temperature induced reduction in coercivity results in a negligible decrease (<2%) in the air gap magnetic flux density. The consequent effect on the total eddy current loss is less than 3%.

Air gap Range (3–34 mm): The air gap length is a dominant parameter. Increasing the air gap from its minimum (3 mm) to maximum (34 mm) value causes a significant attenuation in air gap flux density, resulting in a drastic reduction of over 80% in the eddy current loss. This confirms that air gap adjustment serves as the primary mechanism for torque and loss control. Slip: The model confirms the theoretical quadratic relationship between power loss and slip ($P\infty s^2$) at lower slip values. A slight deviation from this relationship is observed at higher slips (>10%), attributable to a reduction in magnetic field strength from increased reaction fields. The key findings of this sensitivity analysis are summarized in

Table 3. Sensitive period analysis results.

Parameter	Variation Range	Impact on Eddy Current Loss	Key Reason
CC Conductivity	20 °C to 130 °C	+12%	Resistivity increases with temperature.
PM Coercivity	20 °C to 110 °C	−<3%	Slight decrease in air gap flux density.
Air gap Length	3 mm to 34 mm	−>80%	Significant decrease in air gap flux density.

.

3.2. Thermal Resistance Calculation

(1) Heat conduction: The areas not in contact with air belong to heat conduction, which includes: the interior of the iron yoke (R_1~16, R_1~24, R_8~9, R_8~17, R_23~24), the interior of CS (R_3~14 and R_3~22), between the iron yoke and CS (R_22~23); the interior of the permanent magnet aluminum yoke (R_5~12, R_5~20, R_10~11, R_1l~12,R_18~19, and R_{19 ~20}); between the permanent magnet and the permanent magnet aluminum yoke (R_5~6, R_6~11, R_6~19, R_7~10, and R_7~18); the interior of the permanent magnet (R_6~7); between the permanent magnet and the iron yoke (R_7~8); and between the permanent magnet aluminum yoke and the iron yoke (R_9~10 and R_17~18). The thermal conduction resistance can be obtained according to Ref. [20]

$$R={\displaystyle \frac{L}{kA}}$$

(11)

where $L$ (m) represents the transmission path length, and $k$ (W/(m·°C)) denotes the thermal conductivity of the material.

$A$ (m²) represents the area of the transmission path.

Table 4. Thermal conductivity of materials.

Material	Symbol	Value (W/(m·°C))
Steel	${\kappa }_{st}$	36
Copper	${\kappa }_{cop}$	390
Aluminum	${\kappa }_{al}$	237
Nd-Fe-B	${\kappa }_{nd}$	9
Air	${\kappa }_a$	0.026

here displays the thermal conductivity of the materials.

The values for all conductive thermal resistances are provided in Appendix A.

(2) Modeling Conduction/Convection Heat Transfer: The challenging task of defining the flow regime within the air gap leads to a simplification for computational expediency: the air in this region can be modeled as a stationary solid medium. This approach allows the convective heat transfer mechanism to be represented by assigning an equivalent thermal conductivity to the air gap. Specifically, for the air gap associated with node 3, its equivalent thermal conductivity ($h_{air}$) is determined using an established empirical formula.

$$\left\{\begin{array}{l}{\upsilon }_{air}=\pi (n_{in}-n_{out})(r_2+r_{p2})/60\\ {\mathrm{Re}}_{air}={\upsilon }_{air}{l}_g/\gamma \\ h_{air}=0.0019{(l_g/r_2)}^{-2.9084}{{\mathrm{Re}}_{air}}^{0.4614\ln [3.33361(l_g/r_2)]}\end{array}\right.$$

(12)

Here, $v_{air}$ denotes the airflow velocity within the air gap, ${\mathrm{Re}}_{air}$ is the corresponding Reynolds number, and $\gamma$ represents the air’s kinematic viscosity.

Subsequently, using the relationships defined in Equations (11) and (12), the values for thermal resistances R_3~4, R_4~5, R_4~13, R_4~21, R_13~14, R_12~13, R_20~21, and R_21~22 can be sourced from Appendix B.

This empirical formulation for equivalent thermal conductivity is adapted from the work of Ref. [21], where it was successfully applied in thermal-network models of rotating electrical machines. The correlation is valid for rotational Reynolds numbers typically within the range of ${10}^3$ to ${10}^5$, which covers the operational regime of the WPMC under investigation

(3) Convective Heat Transfer: Within the model presented in Figure 6, convective heat exchange takes place at nodes interfacing with the ambient (node 0) and the cooling passages. For the permanent magnet assembly under investigation, the rotation of the conductor and its aluminum yoke generates a forced flow. The associated convective thermal resistance for this scenario is determined based on the methodology outlined in reference [22,23].

$$R={\displaystyle \frac{L}{hA}}$$

(13)

The formula uses h (W/(m·C)) for the convective heat transfer coefficient and A(m²) for the area through which heat is transferred.

The coefficient related to the environmental condition, which corresponds to node 0 and is denoted as $h_{am}$, is evaluated with the following relationship, where $v_{am}$ is the fluid velocity in the environment.

At the nodes representing the cooling channels, the convective heat transfer coefficient ($h_{am}$) is a critical fixed parameter. Its accurate determination requires knowledge of the coolant flow rate, necessitating finite element method (FEM) simulations. In the WPMC’s liquid-cooling system, heat rejection occurs predominantly through forced convection driven by the coolant flow, making it the pivotal heat dissipation mechanism. The accuracy of the final temperature field calculation depends critically on the precision of this convection coefficient, a parameter dominantly influenced by the coolant flow velocity. Therefore, accurate computation of the flow rate is a prerequisite for correct analysis.

A significant challenge arises because the coolant is propelled into the flow passages by the centrifugal force generated by the rotor’s spoiler. This creates a complex flow scenario with no directly applicable empirical correlation. Consequently, this work employs a systematic finite element numerical approach to solve for the coolant’s flow field and velocity. The convective thermal resistances derived from this analysis are provided in Appendix C.

To validate the equivalent thermal conductivity approach used for the air gap, a comparative analysis was conducted against an established rotational Nusselt number correlation for annular gaps with inner cylinder rotation as shown in Ref. [24]. The comparison focused on temperatures of nodes adjacent to the air gap and within the permanent magnets. The maximum temperature deviation observed for the permanent magnets was less than 2.5%. This demonstrates that the equivalent conductivity approximation is sufficiently accurate for this thermal model.

3.3. Discussion on Model Assumptions

The proposed ETNM involves assumptions that warrant discussion. Firstly, radiation heat transfer is neglected. An estimation based on the Stefan-Boltzmann law, considering surface temperatures up to 110 °C and typical emissivity for copper and aluminum, indicates that the radiative contribution constitutes less than 3% of the total heat dissipation, justifying its exclusion.

Secondly, potential minor coolant leakage or overflow effects are not considered. A sensitivity analysis suggests that a localized flow reduction of up to 5% could lead to a localized temperature rise of 4–6 °C at specific hotspots. However, such an effect is confined to very small regions and does not significantly alter the global temperature distribution. The dominant heat transfer mechanisms are therefore adequately captured.

4. Calculation of Cooling Water Flow Rate and Temperature Rise

4.1. Calculation of Cooling Water Flow Rate

This study focuses on an 800 kW WPMC. Given the bilateral symmetry of the coupler structure, a physical model is established based on the left half for analysis, as illustrated in Figure 9a. The structures used in the model include CCs and iron yokes with straight flow channels. The physical model is meshed, as shown in Figure 9b, with a mesh count of 2.32 million. To simplify the model, the following assumptions are made: (1) Minor leakage of cooling water between components is neglected. (2) Friction between cooling water and the turbulence device is neglected. (3) Overflow of cooling water when it flows back to the scoop tube is neglected.

The coolant employed is water. Its flow and heat transfer are governed by the conservation laws of mass, momentum, and energy. To analyze the internal flow dynamics, this work numerically solves the governing equations for both the fluid flow and thermal fields.

The physical model for the CFD simulation (Figure 9) includes solid domains (the iron yoke, copper conductor, permanent magnet holder, protective covers) and the fluid domain (cooling water). The iron yoke, copper conductor, and the cooling water domain are assigned to a rotating zone with a rotational speed of 1500 rpm around the Z-axis. The mesh for the copper conductor and flow regions was refined using hexahedral elements. Fluid-solid interfaces were defined as coupled wall pairs. The simulation employed a Conjugate Heat Transfer (CHT) model and the Multiple Reference Frame (MRF) approach. The cooling channel inlet was specified as a pressure-inlet with a temperature of 313.15 K, while the outlet was set as an outflow condition.

(1) Boundary conditions

The simulation employs a fluid–structure interaction model and an MRF multi-reference frame model [25,26,27,28]. The setup involved: assigning material properties; setting the rotating domain speed to the rated speed; designating the flow channel inlet as a velocity inlet (with velocity equal to the pump injection speed at the spoiler); setting the inlet temperature; and configuring the outlet as a fully developed free outflow. The cooling water flows turbulently under centrifugal force. The RNG $k-\epsilon$ turbulence model, which performs well for simulating rotational flows, was selected. The turbulence kinetic energy and dissipation rate equations are solved as follows:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial (\rho k{u}_i)}{\partial x_i}}=-{\displaystyle \frac{\partial }{\partial x_j}}\left[(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}){\displaystyle \frac{\partial k}{\partial x_j}}\right]+G_k+G_b-\rho \epsilon -Y_M$$

(14)

$${\displaystyle \frac{\partial \left(\rho \epsilon \right)}{\partial t}}+{\displaystyle \frac{\partial (\rho \epsilon u_i)}{\partial x_i}}=-{\displaystyle \frac{\partial }{\partial x_j}}\left[(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}){\displaystyle \frac{\partial k}{\partial x_j}}\right]+C_{1\epsilon }{\displaystyle \frac{k}{\epsilon }}(G_k+C_{3\epsilon }{G}_b)-C_{2\epsilon }\rho {\displaystyle \frac{{\epsilon }^2}{k}}$$

(15)

In the above two equations, $G_k$ represents the term induced by the average velocity gradient for the generation of turbulent kinetic energy;

$$G_k={\mu }_t\left\{2\left[{\left({\displaystyle \frac{\partial u}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial v}{\partial y}}\right)}^2+{\left({\displaystyle \frac{\partial w}{\partial z}}\right)}^2\right]+{\left({\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial z}}\right)}^2+{\left({\displaystyle \frac{\partial w}{\partial x}}+{\displaystyle \frac{\partial u}{\partial z}}\right)}^2+{\left({\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{\partial v}{\partial z}}\right)}^2\right\}$$

(16)

$u_i$,$u_j$ represents the fluid velocity, and $G_b$ denotes the buoyancy induced turbulence kinetic energy generation term; $Y_M$ denotes the contribution value of fluctuating expansion in compressible turbulence, and ${\sigma }_k$,${\sigma }_{\epsilon }$, respectively, are the Prandtl numbers corresponding to turbulence kinetic energy and dissipation rate, while $C_{1\epsilon }$, $C_{2\epsilon }$ and $C_{3\epsilon }$ are empirical constants.

The pressure velocity coupling method employs the SIMPLE algorithm, the pressure difference method adopts the standard pressure difference scheme, and the scheme is chosen as the second order upwind scheme.

(2) CFD Model Validation and Sensitivity Analysis

A mesh independence study was conducted. The Grid Convergence Index (GCI) for the outlet mass flow rate between medium and fine meshes was below 1.5%, indicating mesh-independent results. The medium-density mesh was adopted. The selection of the RNG $k-\epsilon$ turbulence model was justified by comparing its predictions with those of the Realizable $k-\epsilon$ and $k-\omega$ SST models. The discrepancy in the predicted average channel velocity between the RNG $k-\epsilon$ and $k-\omega$ SST models was found to be less than 2.5%, supporting the use of the RNG $k-\epsilon$ model for its computational efficiency in this specific rotational flow context. The uncertainty in the derived convective heat transfer coefficient was estimated at ±4.5%. The CFD model was experimentally validated. The simulated mass flow rate was compared against the volumetric flow rate measured by an electromagnetic flow meter during a 70 kW test. The deviation was less than 3.8%, supporting the model’s accuracy.

(3) Analysis of cooling water flow rate

Figure 10 depicts the distribution of cooling water velocity vectors in the cooling flow channel. The contour plot reveals a maximum flow velocity of 67 m/s just before the outlet. The enlarged vector diagram shows the coolant enters from the inlet, accelerates due to centrifugal force, and exits at approximately 45 m/s. The flow reaches its peak velocity before the outlet, but kinetic energy is dissipated through a right-angle bend after the inlet, reducing the overall flow velocity. By analyzing the fluid flow field, the average flow velocity in the channel is determined to be 46.8 m/s. This velocity is substituted into empirical correlations to precisely calculate the forced convection heat transfer coefficient of the cooling water.

4.2. Temperature Rise Calculation

Within the ETNM framework, the temperature increase at each node is determined by solving the governing equation:

$$[G][T]=[P]$$

(17)

Here, $[P]$ denotes the nodal power loss vector, $[G]$ is the global thermal conductance matrix, and $[T]$ represents the vector of temperature rises. The matrix $[G]$ is defined by the thermal resistances between nodes.

$$[G]=\left[\begin{array}{cccc}{\displaystyle \sum _{i=1}^n\frac{1}{R_{1~i}}}& {\displaystyle -\frac{1}{R_{1~2}}}& \cdots & -{\displaystyle \frac{1}{R_{1~n}}}\\ -{\displaystyle \frac{1}{R_{2~1}}}& {\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{R_{2~i}}}& \cdots & {\displaystyle -\frac{1}{R_{2~n}}}\\ ⋮& ⋮& \ddots & ⋮\\ -{\displaystyle \frac{1}{R_{n~1}}}& -{\displaystyle \frac{1}{R_{n~2}}}& \cdots & {\displaystyle \sum _{i=1}^n\frac{1}{R_{n~i}}}\end{array}\right]$$

For a model with n total nodes, the entries are defined by the conductance $R_{i~j}$ between connected nodes $i$ and $j$. It should be noted that the reciprocal resistance $1/R_{i~j}$ is typically set to zero for non-adjacent nodes, as no direct thermal exchange occurs between them, and the relationship is symmetric $R_{i~j}=R_{j~i}$.

To improve accuracy by incorporating the influence of ambient temperature, Equation (17) is modified as follows:

$$[T]=\left\{[P]-T_0[G_0]\right\}{[G]}^{-1}$$

(18)

In this corrected form, $T_0$ is the local ambient temperature, aligned with the laboratory conditions during testing, and $[G]=[-1/R_{1~0},-1/R_{2~0},\dots ,-1/R_{n~0}]$. For the 800 kW WPMC under investigation, the average thermal loss in the CC is derived from Equation (10). Furthermore, the final temperature for a component is reported as the mean value of all nodes within its corresponding region. For instance, the CC’s temperature is given by averaging the results from nodes 3, 14, and 22.

A dual approach, incorporating both numerical simulation and experimental measurement, was employed to corroborate the accuracy of the calculation results.

5. Numerical Simulation and Experimental Verification

5.1. Numerical Simulation Verification

Based on the fluid–structure interaction model from Section 4.1, the CC heat source and heat dissipation parameters were incorporated. The heat source was applied as a volumetric average. Simulations computed temperature contours for heat loss levels from 20 kW to 70 kW; the 70 kW case is examined in detail. Figure 11 shows the resulting temperature distribution contours for the CC and the iron yoke. The contours reveal a progressive temperature increase along the radial direction, reaching a maximum near the CC’s outer edge at approximately 130 °C (Figure 11a). Regions adjacent to coolant passages show lower temperatures, while areas between channels are comparatively higher. This pattern occurs because the centrifugally driven coolant absorbs heat from the CC, its temperature rising along the path and diminishing its cooling efficacy in outer radial regions.

Figure 11b illustrates the temperature distribution on the iron yoke, with a maximum of approximately 110 °C. The blue regions at the cooling water inlet show the lowest temperatures. Along with the flow channels, the temperature increases progressively (green zones), consistent with convective heat exchange characteristics.

5.2. Experimental Verification

5.2.1. Construction and Application of Experimental Platform

The experimental setup comprises an 800 kW WPMC, an infrared thermometer, torque and speed sensors, an 800 kW explosion-proof drive motor, a load motor of similar rating, a central control system, and a closed-loop water cooling circuit. The WPMC incorporates 36 cooling channels at each end. Each channel has a 10 mm diameter for entry/exit and a 4 mm thickness. Supporting components include oil–water heat exchangers, pressure/temperature indicators, an electromagnetic flow meter, circulating pumps, and reservoir tanks. A schematic is provided in Figure 12.

Operation begins when the drive motor is powered, rotating the input-side rotor. This induces relative motion inside the coupling, generating a magnetic field. Interaction with the eddy current field drives the loading motor. Output power is modulated by an electric actuator adjusting the air gap. System supervision is managed via a computer and controller. Torque/speed sensors are mounted on input and output shafts. Temperature indicators are fitted to the coolant inlet/outlet piping. The flow meter is at the pump discharge. Internal temperatures at critical locations (CC, aluminum yoke) are obtained via embedded sensors for comparison with the ETNM. As the ETNM represents a cross-section about 10 mm beneath the surface, the temperature difference is considered negligible for error estimation.

Experimental Procedure: To validate the simulated thermal field, measurements were taken at six uniformly distributed points (Figure 13). Thermal sensors recorded steady-state temperatures at these locations on the copper disc and permanent magnet regions. A correlation analysis was performed between measured and numerically computed results across heat losses from 20 kW to 70 kW. To ensure reliability, the system was maintained at each heat loss level for over one hour until steady state was confirmed. The entire experimental campaign was repeated more than three times on separate days.

5.2.2. Experimental Process

In the operational characteristics of permanent magnetic couplers, there exists a proportional relationship between slip and loss, that is ${\displaystyle \frac{n_{in}-n_{out}}{n_{in}}}\times p_t=p_{loss}$. By controlling the ratio of input to output rotational speeds, the output of loss can be controlled, thereby achieving thermal loss power loading.

Furthermore, to provide experimental validation, the analytically derived heat loss from Equation (10) was cross validated against the measured power loss. The measured loss $P_{loss}$ was determined from torque ($T$) and rotational speed ($\omega$) data acquired on the input and output shafts, calculated as $P_{loss}=T_{in}{\omega }_{in}-T_{out}{\omega }_{out}$. The results show excellent agreement, with a maximum deviation under 5%, validating the eddy current loss model.

Experimental Process: The electric actuator was adjusted to set the coupler at the maximum air gap (34 mm). The motor started, driving the PMC housing to its rated speed of 1480 rpm. The loading motor remained stationary. The actuator opening was gradually reduced to minimize the air gap, increasing transmitted torque. Upon reaching a stable output speed of 1405 rpm (slip near 5%), a heat loss of 20 kW per disc was achieved. The system ran until temperatures stabilized, then stopped. After cooling to room temperature, the process was repeated for heat losses of 30, 40, 50, 60, and 70 kW. At each level, the system ran to the thermal equilibrium. The water-cooling system’s outlet temperature was recorded throughout.

5.3. Verification of Result Analysis

The experimental, simulated, and calculated temperatures at six measurement points on the CC and the permanent magnet disc are compared for heat losses ranging from 20 kW to 70 kW, as shown in Figure 14. The calculated results are consistent with the trends of both the simulated and experimental data. The highest temperature occurs at position 6, while the lowest is at position 1. The absence of a cooling channel at position 6 results in a significant temperature rise. In contrast, position 1 corresponds to the cooling water inlet, where direct contact with the low temperature coolant provides optimal cooling and the minimum temperature.

A comparison between the CC and the permanent magnet disc shows that the CC, as the heat source, operates at a higher temperature. This difference arises because the air in the gap between them acts as a thermal insulator due to its low conductivity, which helps protect the permanent magnet. However, excessive heat loss can cause the Nd-Fe-B permanent magnet temperature to exceed 180 °C, leading to demagnetization and a complete loss of transmission performance. At a heat loss of 70 kW, the permanent magnet temperature reaches 110 °C, where the transmission performance has already degraded significantly—the transmitted torque measured by the torque sensor drops to approximately 70% of the rated value.

A comparison between experimentally measured temperatures of the CC and permanent magnet disks against both thermal network model computations and numerical simulations reveal a strong agreement. Position 6 reaches the highest temperature because it is located far from the cooling channels and situated at the outer edge of the copper conductor. This region not only experiences a relatively high eddy current loss density but also has a long heat dissipation path, resulting in the least effective cooling.

To CC, the deviation between the thermal network model and the experimental data remains within 3.2%, with a maximum temperature difference of 4.2 °C. Similarly, To PM, the numerical simulation results show a discrepancy within 5.6% from experimental values, peaking at 6.1 °C. These results confirm that the computational accuracy of the proposed model is acceptable. Further analysis of various heat loss conditions and positional temperature variations indicates that the average calculation error for the CC is 3.7 °C, while the average simulation error for the PM is 5.5 °C. For permanent magnets, the corresponding average errors are 3.2 °C and 4.8 °C, respectively. When compared to conventional numerical simulation approaches, the enhanced model demonstrates significant accuracy improvements of 32.73% for copper conductors and 33.33% for permanent magnets.

To validate model stability and parameter sensitivity, analysis investigated the influence of heat loss levels, number of cooling channels, and coolant flow velocity on the maximum system temperature. The error between model-predicted and simulated temperatures is less than 6%, demonstrating good robustness. Relevant results are in Figure 15.

Figure 15a shows the maximum temperature (Point 6) gradually decreases as the number of cooling channels increases, primarily due to higher coolant flow rate enhancing cooling capacity. However, beyond approximately 40 channels, the reduction effect diminishes. Figure 15b indicates the maximum temperature also decreases with increasing coolant flow velocity, as higher velocity increases heat removal. Therefore, the model can also be used for cooling performance optimization.

6. On-Site Experimental Verification

In the operational deployment of a high-power (900 kW) WPMC within the speed regulation and drive systems of large fans at a Datang Group power plant (Figure 16), the fully enclosed design restricts direct temperature monitoring to the coolant inlet and outlet. This prevents accurate determination of internal component temperatures (CC, PMs). The methodology presented enables effective estimation of these temperatures. By implementing real-time monitoring of the coolant outlet temperature and activating protective protocols (alarms, mechanical disengagement) when excessive temperatures are detected, the control system can prevent irreversible damage like demagnetization.

Figure 17 presents on-site monitored outlet coolant temperature data. As thermal load increases from 20 kW to 70 kW, the stabilized outlet temperature rises progressively from 37 °C to 56.5 °C. This correlation enables reliable estimation of internal component temperatures based on outlet readings. To justify using outlet temperature as a proxy and define a protection threshold, correlations between stabilized outlet temperature and maximum internal CC/PM temperatures are established in Figure 18.

The data shows a strong, consistent relationship. The figure demonstrates that an outlet water temperature of 60 °C corresponds to a CC temperature exceeding 130 °C and a PM temperature above 110 °C. At this point, PM magnetic properties have essentially vanished, and the severely overheated CC is susceptible to deformation, dynamic imbalance, and potential shaft failure. This condition signifies the WPMC is nearing thermal saturation due to overload, resulting in excessive slip, mandating immediate load adjustment or shutdown. Thus, the methodology provides a vital, data-driven foundation for predictive protection control.

7. Conclusions

This paper proposes a high-accuracy ETNM for analyzing the temperature distribution in water-cooled permanent magnet couplers (WPMCs), based on fluid–structure coupling and rotational centrifugal flow-field inversion. The method involves three key steps: (1) establishing an ETNM under water-cooling conditions; (2) calculating eddy current losses using a permanent-magnet mapping method; and (3) inverting the convective heat transfer coefficient via MRF-based centrifugal flow simulation. This integrated approach enables accurate temperature field prediction. Finite element simulations and experimental verification were conducted, yielding a maximum calculation error of 3.2% and a maximum simulation error of 5.6%. On average, the calculation accuracy for CCs was improved by 32.73%, and for permanent magnets by 33.33%. The strategy was applied to an actual temperature protection control system and validated on-site. This method enables internal temperature estimation based on external measurements, enabling protective disconnection of the load transmission mechanism under high-temperature conditions to ensure safe operation. In summary, this study establishes a necessary foundation for the design and reliable operation of WPMCs in high-temperature settings.