Heat Transfer Analysis for Non-Contacting Mechanical Face Seals Using the Variable-Order Derivative Approach

Slawomir Blasiak

doi:10.3390/en14175512

Abstract

This article presents a variable-order derivative (VOD) time fractional model for describing heat transfer in the rotor or stator in non-contacting mechanical face seals. Most theoretical studies so far have been based on the classical equation of heat transfer. Recently, constant-order derivative (COD) time fractional models have also been used. The VOD time fractional model considered here is able to provide adequate information on the heat transfer phenomena occurring in non-contacting face seals, especially during the startup. The model was solved analytically, but the characteristic features of the model were determined through numerical simulations. The equation of heat transfer in this model was analyzed as a function of time. The phenomena observed in the seal include the conduction of heat from the fluid film in the gap to the rotor and the stator, followed by convection to the fluid surrounding them. In the calculations, it is assumed that the working medium is water. The major objective of the study was to compare the results of the classical equation of heat transfer with the results of the equations involving the use of the fractional-order derivative. The order of the derivative was assumed to be a function of time. The mathematical analysis based on the fractional differential equation is suitable to develop more detailed mathematical models describing physical phenomena.

Highlights

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
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
One of the shortcomings that most models have to overcome is that, mathematically, velocity is an instantaneous velocity defined at a point. It can be seen from the literature that differential calculus is used in the heat transfer theory

Summary

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. 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. One of the shortcomings that most models have to overcome is that, mathematically, velocity is an instantaneous velocity defined at a point. It can be seen from the literature that differential calculus is used in the heat transfer theory

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Results

Conclusion