We prepared two thermal models of the HMB. The first was two dimensional and the second was three dimensional. Femm software was used for the preparation and calculation of the temperature within the 2D modelling [2], while Opera 3D Tempo package was applied in the 3D model [3].

According to Fourier’s law for heat conduction, the heat flux density is proportional to the gradient of temperature ∇*T*, [2]:

$$\begin{array}{}\overrightarrow{q}=-\kappa \mathrm{\nabla}T\end{array}$$(1)

where *κ* is the thermal conductivity. Heat flux density must comply with Gauss law, which states that the heat flux which is coming out any closed volume equals to the heat which is generated within that volume [2]:

$$\begin{array}{}\mathrm{\nabla}\cdot \overrightarrow{p}=p\end{array}$$(2)

where *p* is the power density generated by Joule’s heat.

Inserting equation (1) into (2), we obtain Poisson’s equation, with the function *T* which is the temperature distribution being investigated:

$$\begin{array}{}\mathrm{\nabla}\cdot (\kappa \mathrm{\nabla}T)=-p\end{array}$$(3)

Thermal calculations are difficult to execute, mostly due to difficulties in ascertaining the material property data in the analysed area and in determining the boundary conditions. The HMB actuator is a device that consists of few components. Stator and rotor are made from isotropic silicon steel M530-50A, where the silicon content equals to 1.43%. The producer of the silicon steel does not provide information about the thermal conductivity of the stack of sheets. According to [3], the thermal conductivity of silicon steel can vary from 19 W/(m⋅K) to 55 W/(m⋅K) for different silicon contents (from 5% to 0%).

The thermal conductivity of silicon steel with 1.5% silicon content equals to 36 W/(m⋅K). Additionally, the thermal conductivity of the stator and rotor is an anisotropic quantity, and its axial component is a complex function of the clamping pressure, stacking factor, lamination thickness and lamination surface finish [4]. Unfortunately, there is still not enough information about the value of the thermal conductivity component across the laminated stack. In our calculation model, we assumed that the thermal conductivity of the stator and rotor are isotropic quantities and equal to 33 W/(m⋅K).

The next confusing issue is the determination of thermal conductivity in the winding volume. The winding is a complex object for thermal analysis because it is a heterogeneous structure, that consists of copper wires, wire insulation, slot insulation, impregnation substance and air. The thermal conductivity of the non-impregnated winding is equal to 0.1 W/(m⋅K). For the impregnated one, this value is almost double and eguals 0.18 W/(m⋅K) [4]. In the calculation models, we assumed the equivalent thermal conductivity of the windings to be equal to 0.1 W/(m⋅K). The housing of the HMB is made of aluminium alloy PA6, with a thermal conductivity equal to 164 W/(m⋅K) at a temperature of 293.16K [2]. This value was used in the numerical model. The HMB is surrounded by air, with thermal conductivity 26.19 mW/(m⋅K) at a temperature of 300 K. Thermal parameters of the calculation model domains are presented in .

Table 1 Thermal parameters of calculation model domains

For the mathematical modelling, we made the following assumptions:

descriptionthe gap between stator and rotor has thermal properties of air. Heat transfer between the stator and rotor exists only due to thermal conduction,

the heat is generated only by the windings in the result of Joule’s heat,

the losses generated in the cores of stator and rotor are omitted.

Such assumptions are acceptable, because the HMB is powered from a direct current source and the magnetic field changes mostly in the rotor during its rotation. Additionally, the tests of the heating process were carried out in all windings powered by a direct current source. Thus, the value of the generated power density in the HMB is given by the equation:

$$\begin{array}{}{\displaystyle p=\frac{R{I}^{2}}{{V}_{winding}}}\end{array}$$(4)

where *R* is the winding resistance, *I* is the current intensity flowing through the winding and *V*_{winding} is the winding volume. The power density is identical for all three windings. Additionally, we assumed a constant value of the winding resistance in the operating temperature range.

Two linearized boundary conditions have been used in the calculation models. The first is Dirichlet’s boundary condition on the outer edges of calculation area, which defines constant ambient temperature. The second is Robin’s convective boundary condition, which describes the convection phenomenon on the HMB outer surfaces [5]:

$$\begin{array}{}{\displaystyle \kappa \mathrm{\nabla}T\cdot \overrightarrow{n}+{h}_{combined}(T-{T}_{ambient})=0}\end{array}$$(5)

where *n⃗* denotes the normal vector to the surface and the combined convection coefficient on the surface, which represents heat transfer by convection and radiation. The average value of the *h*_{combined} is described by the equation [6]:

$$\begin{array}{}{\displaystyle {h}_{combined}={h}_{conv}+{h}_{rad}}\end{array}$$(6)

where *h*_{conv} denotes the heat transfer coefficient by convection and *h*_{rad} is the heat transfer coefficient by radiation. Both heat transfer coefficients were combined into one, because the Opera 3D Tempo software package does not allow the inclusion of heat transfer by radiation. Similarly, the Femm software only allows the setting of one type of heat transfer.

The convection heat transfer coefficient *h*_{conv} was calculated separately for each side of the housing and windings based on empirical correlations for the average Nusselt number for natural convection over the surface [2]. The values of *h*_{conv} were calculated for ambient temperature *T*_{ambient} equal to 294 K. The radiation heat transfer coefficient *h*_{rad} is described [5] by the equation:

$$\begin{array}{}{\displaystyle {h}_{rad}=4\sigma \u03f5{\left(\frac{T+{T}_{ambient}}{2}\right)}^{3}}\end{array}$$(7)

where *σ* is the Stefan–Boltzmann constant and *ε* is the emissivity of the surface. Heat transfer by radiation of the aluminium housing was calculated for the emissivity *ε* = 0.1, while heat transfer by radiation from the copper windings was calculated for the emissivity *ε* = 0.2 [2]. Calculation of the heat transfer coefficient is a difficult task because coefficients are nonlinear functions of temperature. In order to reduce the complexity of solving the model, the combined convection coefficients were approximated by a linear function. Figure 2 shows the values of the combined convection coefficient calculated within a temperature range from 290 K to 400 K for different surfaces of the model.

Figure 2 Values of the combined transfer coeflcients

In the 2D model, the convective boundary condition was fixed on the outer edges of the HMB construction, while Dirichlet’s boundary condition has been set at the distance of 40 cm from the HMB. In the 3D model the combined convection coefficient has been assumed on the front surface of the housing and windings in addition to the 2D model boundary conditions.

The 2D model contains 120980 elements and computation time is equal to 15 s. The 3D model contains 1724755 elements and computation time is equal to 157 s. Both calculations were carried out using a computer with two Intel Xeon E5620 2.4 Ghz processors and 32GB RAM.

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