**2. Mathematical Model**

The classical Fourier heat equation can be written as:

$$\mathbf{q} = -\lambda \,\nabla \,\theta\_{\prime} \tag{1}$$

where **q**—heat flux vector, *ϑ* = *T* − *To* and *λ*—thermal conductivity.

When combined with the energy conservation principle, the heat equation can be used to calculate the local deformation:

$$-\nabla \mathbf{q}(t) = \rho \, \mathbb{C}\_p \, \frac{\partial \theta(t)}{\partial t},\tag{2}$$

with *ρ* being the density and *Cp* the specific heat capacity.

Many researchers have examined the application of differential calculus or integral calculus involving the use of derivatives of fractional orders. Fractional calculus is a natural extension of the notions of differentials and integrals used in classical differential calculus and integral calculus, respectively. Recently, there has been much interest in the use of fractional differential calculus to explain and model many physical phenomena. For example, fractional differential calculus has been used to design and/or model *PIλD<sup>μ</sup>* controllers, heat conduction [23–25], thermoelasticity [26], complex nonlinear systems [7], supercapacitors, electrical and mechanical systems [27–29], electrical filters, dielectric relaxation, diffusion, and viscoelasticity [30].

The time fractional heat conduction equation for the rotor can be written as [31]:

$$\frac{\partial^{\alpha}\theta}{\partial t^{\alpha}} = \kappa \,\Delta\theta \qquad 0 < \alpha \le 2 \,\,and \,\,\kappa = \frac{K}{\rho \,\, \mathcal{C}\_p}. \tag{3}$$

For the case considered here, it was assumed that *α* = *α*(*t*). The function *α*(*t*) was defined as a function of time (see Figure 1).

**Figure 1.** Fractional derivative order α vs. time.

The function describing the change in the coefficient *α*(*t*) can be written using the following formula:

$$a(t) = \begin{cases} 1 + \frac{1}{1 + 10^{-3(t - t\_0)}} & t \in (0, 2) \\ 1 & t > 2 \end{cases},\tag{4}$$

The transition to the time fractional heat conduction equation with the boundary conditions, was described, for example, in [32] as:

$$\frac{\partial^{\alpha}\theta}{\partial t^{\alpha}} = \kappa \left( \frac{1}{r} \frac{\partial \theta}{\partial r} + \frac{\partial^{2}\theta}{\partial r^{2}} + \frac{\partial^{2}\theta}{\partial z^{2}} \right), \text{ for } r\_{i} \le r \le r\_{o}; \ 0 \le z \le L; \ t > 0,\tag{5}$$

where *L* is the thickness of the sealing rings.

The Caputo derivative of the fractional order can be defined as described in [15]:

$$\frac{\partial^{\alpha}f(t)}{\partial t^{\alpha}} = \left\{ \begin{array}{c} \frac{1}{\Gamma(n-a)} \int\_{0}^{t} (t-\tau) & \frac{d^{n}f(\tau)}{d\tau^{n}} d\tau, \ n-1 < \alpha < n, \\\\ & \frac{d^{n}f(t)}{dt^{n}} \; \; \; a = n, \end{array} \right. \tag{6}$$

where Γ(*α*) is the gamma function, with the initial conditions being: *t* = 0 *ϑ* = 0, 0 < *α* ≤ 2, *t* = 0 *∂ϑ <sup>∂</sup><sup>t</sup>* = 0, 1 < *α* ≤ 2.

The heat conduction equation for the stator is thus rewritten as:

$$\frac{1}{r}\frac{\partial\theta}{\partial r} + \frac{\partial^2\theta}{\partial r^2} + \frac{\partial^2\theta}{\partial z^2} = 0,\text{ for }r\_i \le r \le r\_o;\ 0 \le z \le L,\tag{7}$$

The heat transfer model analyzed here also needs to take into consideration the heat flux generated in the gap, which can be described with a relationship based on the simplified energy equation. A similar method was used in [32]:

$$
\mu \left( \frac{\partial \nu\_{\phi}}{\partial z} \right)^{2} + \lambda^{f} \frac{\partial^{2} T^{f}}{\partial z^{2}} = 0 \tag{8}
$$

The velocity of the fluid particles *νϕ* is linearly variable. Ranging from zero on the stator surface to the value of (*ω r*) on the rotor surface, it can be described as:

$$\frac{\partial \nu\_{\phi}}{\partial z} = \frac{\omega r}{h} \tag{9}$$

The energy Equation (8) was solved taking into account relationship Equation (9) to calculate the distribution of temperature in the gap.

Considering sealing rings with unmodified face surfaces and neglecting other factors affecting their geometry, e.g., mechanical deformations, we can assume that the height of the gap is constant: *h* = *const*. The gap height considered by other researchers is that discussed, for example, in reference [33].

As dynamic viscosity is a temperature-dependent parameter, here it can be defined using the formula proposed by Li et al. [1]:

$$
\mu = \mu\_o \exp(-b \left(T\_m - T\_o\right)).\tag{10}
$$

The average temperature of the medium in the gap can be determined from:

$$T\_m = \frac{1}{h(r)} \int\_0^{h(r)} T^f dz. \tag{11}$$

Equation (10) is a function describing the distribution of dynamic viscosity *μ*(*r*) in the radial direction.

The seal performance is largely affected by the forces acting on the stator and the rotor. The closing force produced by the spring ensures the leak tightness of the device in the OFF mode and the contact of the stator and the rotor. However, when the device operates

(is ON), the opening force is generated by the pressure of the medium in the gap. The distribution of pressure can be described using the one-dimensional Reynolds equation:

$$\frac{d}{dr}\left(r\frac{\rho}{\mu}\frac{h^3}{dr}\frac{dp}{dr}\right) = 0,\tag{12}$$

where the boundary conditions are:

$$p(r)|\_{r=r\_i} = p\_{i\prime} \ p(r)|\_{r=r\_0} = p\_o \tag{13}$$

Equation (12) is the simplified Reynolds equation describing the changes in pressure of the incompressible medium in the gap only in the radial direction.
