**1. Introduction**

Over the last few years, there have been many studies using fractional-order differential and integral operators to generalize classical differential and integral calculus with the aim of further understanding the nature of complex systems. Currently, attempts are being made to apply fractional calculus to solve various physical, mechanical, biological, or chemical problems. While classical integer-order operators are dependent only on the local behavior of the function, fractional-order operators accumulate all the information about the function. Another fundamental feature of fractional derivatives is that they are defined along a segment, not at a point, as is the case with classical derivatives. This feature ensures a more accurate and effective analysis of different phenomena. One of the shortcomings that most models have to overcome is that, mathematically, velocity is an instantaneous velocity defined at a point. Thus, it can be seen from the literature that differential calculus is used in the heat transfer theory.

It has been found, for example, that variable thermal conductivity is a key physical property of materials, especially when it is dependent on temperature. Variable thermal conductivity is of significance in a wide range of applications, including modern physics and mechanical engineering. It is taken into consideration primarily to determine the effect of temperature on the performance of machine elements. Special attention is paid to this property when rapid changes in temperature occur, as they may affect the operation of mechanical assemblies and subassemblies, and consequently the whole machine. The relationship between variable thermal conductivity and fractional differential calculus

**Citation:** Blasiak, S. Heat Transfer Analysis for Non-Contacting Mechanical Face Seals Using the Variable-Order Derivative Approach. *Energies* **2021**, *14*, 5512. https:// doi.org/10.3390/en14175512

Academic Editor: Chi-Ming Lai

Received: 20 July 2021 Accepted: 31 August 2021 Published: 3 September 2021

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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

has been investigated extensively for problems related to fluid mechanics [1,2], viscous elasticity [3], relaxation [4], thermal elasticity and conductivity [5,6], control theory [7,8], and many others [9–11]. Over the years, researchers have extended the classical theory of elasticity and introduced the theory of thermoelasticity.

The mechanical properties, especially the relaxation, of additively manufactured materials were analyzed, for example, in [4]. A mathematical model was developed using fractional differential calculus to further fit the relaxation curves with the experimental data.

Warbhe et al. [5] used the quasi-static approach to the fractional-order theory of thermoelasticity to solve a two-dimensional problem for a thin circular plate with zero temperature across the lower surface and constant and evenly distributed temperature across the insulated upper surface. The integral transform technique was employed to solve the physical problem, while the displacement potential function was applied to calculate thermal stresses. Povstenko and Kyrylych [6] studied the interface between two solids. They considered generalized boundary conditions of a non-ideal thermal contact to solve the heat conduction equation of the fractional order as a function of time using the Caputo derivative.

The extensive monograph by Kaczorek [7] presented the theoretical and practical aspects of the application of fractional calculus to fractional discrete-time linear systems.

The research by Nowacki, described for instance in [12–14], was a breakthrough in this field. His findings are still valid and thus widely cited, as they can be applied to solve a large number of theoretical and practical engineering problems.

Knowledge of all theories available in this area is crucial to understand the deformations of elastic materials. Their thermal and mechanical behavior needs to be taken into account whenever thermoelastic deformations result in leaks, as is the case with noncontacting face seals, where the gap height may range from one to several micrometers.

Fractional differential calculus has proved well-suited to predict many physical phenomena associated with elastic media, e.g., thermal conductivity, heat transfer, and viscoelasticity. The classical differential equations describing physical phenomena are modified using the variable-order derivative (VOD) time fractional approach. Recently, there has been much research focusing on extending the applications of fractional differential calculus, e.g., [15–18]. These studies provide an insight into many important aspects associated with heat conduction. In [15], for example, Povstenko presented fundamental solutions to the time-fractional advection diffusion equation for two cases: the plane and the half-plane. He also considered Cauchy problems, inverse source problems, and Dirichlet problems. The solutions were expressed in terms of Bessel integral functions combined with Mittag–Leffler functions. In his other articles [16,17], Povstenko discussed solutions for heat transfer in composite media, in which he employed the time-fractional heat conduction equation with the Caputo derivative of fractional order to describe heat transfer in both constituent materials (0 < α ≤ 2 and 0 < β ≤ 2, respectively). The problem was solved under ideal contact conditions. This suggests that the temperature of the two materials and the heat fluxes were the same.

Liu et al. [19] proposed an approach to solving the time fractional nonlinear heat conduction equation. Similar solutions were proposed by Koca and Lotfy [20,21].

Another issue is an inverse problem in heat transfer. Liu and Feng [22], for instance, solved it using the fractional differential equation for a time-dependent derivative. The solution to the 2D time-fractional inverse problem of diffusion was based on an improved kernel technique. Maximum a posteriori estimation was required. Convergence was achieved using the regularization term and deriving a priori probability.

The primary aim of this article was to show how the results obtained by solving the classical equation of heat transfer differ from those obtained for the same physical phenomenon by employing the fractional differential equation. The study involved using the derivative order as a time-dependent function. The results from the two approaches were compared graphically. The analytical solutions of the main physical quantities were developed using the Marchi–Zgrablich transform, the finite Fourier cosine transform, and, finally, the Laplace transform.
