The influence of physical properties of materials used for slide rings on the process of heat transfer in the non-contacting face seals

The paper presents the results of analytical solution of the model of heat transfer for noncontacting face seals. Comparative analyses were performed for various physical properties of materials used for slide rings. A mathematical model includes a series of differential equations of partial derivatives with generally used boundary conditions, i.e. the Reynold’s equation, energy equation and heat transfer equations, which describe the heat transfer in sealing rings with surrounding medium. Heat transfer equation is written in the Cartesian coordinate system and solved using the Green's functions method. Theoretical studies made it possible to draw a number of practical conclusions on the phenomena of heat transfer in the node seal. The presented model will allow more accurate identification of the heat transfer mechanism in the node seal. The results will help to select appropriate materials for sealing rings, depending on operating conditions of non-contacting face seals.

The studies published in the above mentioned works are especially important due to the wide range of temperatures for contactless face seals. Manufacturers of the contactless face seals declare working temperature ranges of those components to be between -20 and 220 °C for gas dynamics seals, and between -54 and 650 °C for contactless face seals co-functioning with other media [20] and under extremely different operating parameters [14]. Such a wide range of working temperatures creates a series of problems that researchers have to deal with. Because the assumed height of the radial gap varies by several micrometres, there might occur a rapid evaporation of the working medium in the fluid film. This determines how research is undertaken and conducted. It also determines how to develop improved mathematical models describing the phenomenon of two-phase radial gap flow [29], and how to conduct research on those phenomena under laboratory conditions [41]. By carrying out more specific literature analysis in terms of heat transfer of the contactless face seals, for both the fluid film and cofunctioning rings, it is appropriate to mention the works that present complex thermodynamic (THD) and thermoelastohydrodynamic (TEHD) models, such as the works of: [8] and [13] as well as the works characterized by numerical solutions of the proposed models of heat transfer and thermoelastic deformations of the ring seals [12].
The development of new technologies allows for preparation of materials with physical properties that enable dissipation of huge amounts of heat flux into the environment. It is especially important in cases, where contactless face seals are used in devices that pump hazardous media, such as chemically aggressive or explosive substances whose leakage could lead to contamination of the natural environment or other accidents.
diagram of the FMR-type (Flexible Mounted Stator) contactless seal is shown in Fig. 1. The primary condition that determines proper functioning of the contactless face seals is the maintenance of gap that keeps the co-functioning rings apart by few micrometres. That condition can be met only when balance of forces acting on the rings system is maintained. At this point one should distinguish two main types of forces acting on the susceptibly positioned rotor, i.e. the closing force, which comes mainly from the compression spring, and the opening force generated in the fluid film. The opening force is dependent on the shape of the gap and on the pressure in the fluid film that results from it. The mathematical model of the contactless face seals is commonly known and has been presented in the previous works of the author. At this point, the only cited elements are the primary formulas that govern the processes taking place in the fluid film and ring seals.
Reynolds' single-dimensional (in the radial direction) equation describes pressure layout in the fluid film: Temperature layout in the fluid film is determined based on a simplified energy equation: A linear change in the fluid flow v along the height of the fluid film (Couette's Flow) was adopted. Change in the fluid speed by the coordinate x is noted as: The equality of temperatures-T r (y) = T f (y) as well as heat flux: between the main ring and the fluid film was adopted for the S1 surface. On the S2 surface -it is assumed similarly that the temperatures of the fluid film and the stationary ring are equal -T f (y) = T s (y) for x = h, and that the temperature on contact between the fluid film and the stator reaches the maximum value: The mathematical model formulated in this way has been solved analytically using the Green's Function Method.

Analytical Solution
The first stage of the analytical solution consisted in determining the temperature layouts in the ring seal by outlining the general functions satisfying the Laplace's Equation (5) for both main rings. The solution included the above boundary conditions and employed a coordinate system to simplify analytical calculations.
Further part of the study will include analytical solution for the stator. Calculations for the rotor are characterized with full analogy, and hence final formulas will be quoted. After placing the adopted functions in the differential equation (5), the outcome was: After the transformation, two independent equations were determined: General solutions of the (7) and (8) The c n,m coefficient has been determined similarly, as in the work of [56].
The temperature layout in the cross-section of the stator has been noted as: The Green's Function for the G s (x, y | x ' , y ' ) is presented by the following dependency: Using the substitution:  (19) In the end, the dependency describing the temperature layout in the stator takes the following form: By using the dependencies set out in the work of [56], the Green's Function for the rotor takes the following form: The further part of the study involved an analysis regarding how physical properties of materials used for manufacturing of the ring seals influence the heat transfer in the fluid film-main rings system.

Results and Discussion
Numerical calculations of the heat transfer in mechanical seals have been conducted with the assumption that the stator is made of resin-impregnated carbon, whereas the rotor, whose properties are compiled in Table 1, is made of the most commonly used materials in manufacturing ring seals. The parameters from Table 2 were adopted for the purpose of calculations. In Table 2, the working medium is water with temperature of 20 °C on the process side -T o . It has been assumed that the external radius and the internal radius have the length of 45 mm and 40 mm respectively, for both co-functioning ring seals. A radial gap with the height of 1 ȝm was used in the calculations.  Table 1).  The analysis of the results shown in Fig. 4 was followed by a determination of the smallest differences in temperature layouts of materials with similar physical properties. This concerns chemically bound silicon carbide and sintered silicon carbide. The thermal resistance of the whole studied system increases when the heat transfer coefficient decreases.

Conclusion
The main task of the contactless face seals is to keep tightness, regardless of external conditions. There is a close relation between the geometry of the radial gap and the occurrence of a leakage. The major factors connecting the above mentioned parameters are any deformations caused by uneven temperature layout in the ring seals. A development of more precise mathematical models and calculation apparatuses, in the course of the layout design process, will allow to develop the temperature of the rings-fluid film system. That will allow researchers and constructors to estimate the temperature range and select proper materials for the manufacture of ring seals. This knowledge, in turn, will lead to the achievement of high reliability and a long operational life of the developed structures.