**8. Conclusions**

This article has presented a mathematical model to determine the heat transfer phenomena occurring in the non-contacting face seal used in a turbo machine. The problem was solved analytically by applying three integral transforms. The resulting relationships described the changes in temperature in the face seal cross-section. The boundary conditions were taken into consideration. The classical Fourier law was extended by applying the time fractional derivative where *α* = *α*(*t*). This model can be used to describe unstable heat transfer conditions or thermal shock.

Even small local changes in temperature of an order of 2–10 ◦C can cause thermoelastic deformations of the stator and the rotor. As a result, there is a change in the gap geometry. During the first moments of the startup, unstable conditions can be observed because the seal operates under dry friction conditions; thus, the heat flux generated between the rotor and the stator is much greater than during the non-contact operation of the seal assumed in this model. The deformations may contribute to a greater leak.

It should be mentioned that the mathematical analysis based on the fractional differential equation is suitable to develop more accurate models to analyze similar physical phenomena.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The author declares no conflict of interest.
