**3. Boundary Conditions**

The schematic diagram in Figure 2 illustrates the non-contacting face seal with the flexibly mounted stator (1) and the rotor (2) connected to the shaft (6) of the turbo machine. The gap between the stator and the rotor, with a height ranging from one micrometer to several or even more than ten micrometers, is filled with a sealing medium, e.g., water. The distribution of temperature on the stator and rotor surfaces is dependent mainly on the heat flux in the gap.

**Figure 2.** Non-contacting face seal made up of the flexibly mounted stator (1), the rotor (2), the spring (3), the housing (4), the O-ring (5), and the shaft (6).

The boundary conditions are represented graphically in Figure 3.

**Figure 3.** Boundary conditions for the heat transfer in the non-contacting face seal. (a) Heat flux, (b) convection, and (c) insulated surfaces.

In this analysis, it is assumed that the inner and bottom surfaces of the rotor (*ri*) and the inner and top surfaces of the stator are not in contact with the surroundings; thus, heat transfer for these surfaces takes the general form: *∂ϑ <sup>∂</sup><sup>n</sup>* <sup>=</sup> *∂θ <sup>∂</sup><sup>n</sup>* = 0. In a real system, these surfaces are in contact with other elements of the seal characterized by different physical properties. Under certain conditions, heat transfer taking place between these elements can be assumed to be negligible. Another reason to introduce the boundary conditions ( *∂ϑ <sup>∂</sup><sup>n</sup>* = 0) for the surfaces is the considerable simplification of the calculations for the analytically solved model.

As the gap is limited by the ring faces, the boundary conditions for the rotor and the stator are as follows:

$$
\lambda \frac{\partial \theta}{\partial z} \bigg|\_{z=L} = \lambda^f \frac{\partial \theta^f}{\partial z} \bigg|\_{z=0} \text{ and } \theta = \theta^f \text{ \AA} \tag{14}
$$

$$
\lambda \left. \frac{\partial \theta^f}{\partial z} \right|\_{z=h} = \lambda \left. \frac{\partial \theta}{\partial z} \right|\_{z=0} \text{ and } \theta^f = \theta \text{, respectively} \tag{15}
$$

For the outer surface of the rotor (*ro*), heat transfer is assumed to be by convection, and it can be expressed by:

$$-\lambda \left. \frac{\partial \theta}{\partial r} \right|\_{r=r\_0} = a^{\int} \left. \theta \right|\_{r=r\_0} \text{ for the rotor} \tag{16}$$

$$-\lambda \left. \frac{\partial \theta}{\partial r} \right|\_{r=r\_0} = \alpha^{\ell} \left. \theta \right|\_{r=r\_0 \prime} \text{ for the stator.} \tag{17}$$

where *α<sup>f</sup>* is the convection coefficient.

All the above boundary conditions are necessary to solve the system of three differential equations.
