**6. Results and Discussion**

The simulations were performed to determine the influence of the factor *α* on the distribution of temperature in the seal (Figure 4).

**Figure 4.** Distribution of temperature in the cross-section of the non-contacting face seal at different values of α. (**a**) for *α* = 1, (**b**) for *α*(*t*) = var.

Figure 4 illustrates the distributions of temperature in the rotor over time, ranging from 0.1 s to 1.5 s. When the classical heat equation is used (Figure 4a), the temperature of the rotor increases gradually because of the heat flux generated in the gap. The greatest difference between the diagrams is observed for the time range 0.1–1.25 s. With the assumption that *α*(*t*), Figure 1 suggests that if values higher than *t*<sup>0</sup> = 1 *s* are used, the order of the derivative tends toward unity.

The heat flux provided by the sealing ring face at a given moment of time for *t* = 0.05 (s) causes a local increase in temperature of 2.1 ◦C, which may result in thermal deformations of the sealing ring in contact with the gap. In the case of the fractional differential equation (for *t* = 0.05 (s)), the equation is hyperbolic in nature, as illustrated in Figure 5b.

**Figure 5.** (**a**) Distribution of temperature in the seal for *t* = 0.05 (s) and *α* = 1.998, (**b**) temperature vs. the rotor thickness for *r* = 0.0545 (m).

The solution based on the variable-order derivative (VOD) time fractional model provides a better correlation between the predicted data and the observed results than the classical approach.
