Influence of feeders on operating characteristics of the impulse seals

The paper presents an refined calculation of impulse face seals with a self-regulated gap. Unlike the existing methods of calculation, wherein, for the sake of simplicity, influence of the feeder conductance is neglected, this article describes conductance effect on the operating characteristics of the seal. Static calculation method of a face impulse seal of the high-speed pump is presented.


Design and working principle of the seal
The simplest design of a single-stage impulse seal (Figure 1) differs from the mechanical face seal only by the fact that closed chambers 2 are located on the end face of the axially movable ring 1, and several radial feeding channels 5 are made on the rotating support ring 6, and these channels are opened towards the sealing hole [1]. Sealing medium under sealing pressure 1 p is injected through these channels into the chambers during the short time intervals  The shorter is the time between injections, the lower is the depth of pressure drop min 2 p in the chambers, the higher is the averaged pressure 2 p in the chambers the greater is the force s F opening the end joint.
Pressure force   z F s opening the face joint depends on the pressure   z р 2 and, consequently, on the gap size. As the gap size is reduced, pressure increases, and its balance with the external force  Thus, there is a negative feedback between the face gap z (adjustable value) and the force s F (regulating effect), which provides self-regulation of the face gap ( Figure 4).  Seal performance is based on the developing of high-frequency pressure impulses in the discharge chambers, so it is called the impulse seal. Impulse frequency of the total axial force of pressure in the number of chambers) is large in comparison with rotation speed. Because of large damping in the micron face gap and inertia of the axially movable ring, the ring hardly reacts to such high-frequency impulses. For these reasons, highfrequency forced oscillations of the ring in the static calculation are not taken into account.
Static characteristic is found from the equation of axial equilibrium of an axially movable ring. In the equilibrium position, pressure force s F , opening the face gap, balances external pressure force e F and force of elastic elements . Averaged pressure 2 p in the chambers over a period between successive injections determines the force s F . For simplicity, a linear pressure change along the face gap radius is assumed ( Figure 5).

Calculation of the averaged pressure in the chambers
To estimate pressure in the chambers, it is necessary to consider a radial flow of viscous compressible fluid in a flat channel, which looks like a sector with a central angle с  and a radial size    through a liquid-filled chamber, steplike supplies pressure 1 р to it. As a result, pressure rises to the maximum value 1 р , compressing liquid in the chamber. After a feeder leaves the sector с  , liquid volume compressed in the chamber flows out and pressure decreases to the initial minimum value. The expansion process takes place over a period of time c t  Т . After that, compression takes place again, and the process repeats ( Figure 2, (a)). During compression, difference in the liquid volume inflowing into the feeder and inner gap and outflowing   с dt Q 3 from the outer one, is compensated by the liquid, which fills the volume dV  released in the chamber because of previous liquid compression: In the process of expansion, volume of the outflowing liquid   р dt Q 3 is greater than the volume of the inflowing one-  р dt Q 1 by the amount dV with the opposite sign: Equations (1) and (2) of the volume balance differ only by flow rate i Q through the feeder and the initial condition: compression begins with the minimum pressure, and expansion with the maximum one.
As defined by [5], the modulus of volume elasticity of a liquid is defined -dV dp , i.e. it is approximately equal to 1% of the chamber volume. Therefore, there is no need to expect a large effect from compressibility on the seal performance. The change of volume ΔV during compression and expansion differs only in sign.
Adding to term the equations (1) and (2), we obtain an equation for the entire period Т The sums of incremental components of time and pressure cover the entire period Т between the next injections, therefore, one can indicate: Differentials of independent variables are constant quantities: A pressure differential, accurate to a second-order quantity, can be presented as following: Taking into account the obtained relations, an equation (4) after dividing by dt represents an approximate, time-independent equation of flow rate balance: For a laminar flow, the flow rate is linearly dependent on pressure drops: where 2 рaveraged value of pressure in the chambers. Conductance values of the face throttles for laminar flows are proportional to the cube of the gap z [3,7] and are expressed by the equations: from which the averaged pressure in the chambers can be found, depending on the dimensionless face gap: Presents the resulting expression to the dimensionless form by introducing the following description:

Regulating effect and hydrostatic stiffness
Regulating effect is a total axial force of pressure influencing on the contact surface of the axially movable ring, which depends on the face gap. Using a linear pressure diagrams shown in Figure 5, we calculate the pressure force s F on the contact surface, opening the face gap (regulatory impact), pressure force e F which presses a ring to the support disk (external load) and the force k F of the elastic elements (control input): As it can be seen from expressions (12)  , taking into account the fact that the face gap value is negligibly small in comparison with the preliminary deformation of the elastic elements   z , after substitution of the forces (12) and (14) takes the form: where k is a reduced coefficient of axial stiffness of the elastic elements. It is necessary to divide this equation term-by-term by п Аp and pass on to dimensionless forces, introducing the following notations: As a result, we obtain a dimensionless regulatory impact effect: For further transformations, one can use the obvious equality   In this case, axial equilibrium equation (16) of an axially movable ring is: is also determined by the equation (11)   are rarely successful, since conductance of the face gap is determined by the micron-sized gap, and conductance of feeders -by the diameter of a channel -0.4 ... 1.0 mm. Many different designs of throttles have been developed for bearing supports with gas lubrication [8,9]. Such throttles can be also used for impulse seals. Coefficient of hydrostatic stiffness (19) is less than zero, and it provides stability of the equilibrium position. Hydrostatic stiffness makes it possible to estimate minimum value 0  of the own frequency of axially movable ring vibrations and, if necessary, to escape possible resonance. In the equilibrium position:

Static and discharge characteristics
Static characteristic is a dependence of the steady-state gap value on external disturbances level. It is determined from the joint solution of equations (18) and (11): From this equality one can find: and conductance (4) were determined for a sector with a central angle с  ( Figure 6). For the entire gap, the angle с  in equations (4)

Conclusion
The paper deals with a face impulse seal of a high-speed centrifugal pump, which has several advantages over traditional mechanical face seals. The extended analysis described above made it possible to estimate influence of feeder conductance on the static characteristics of the impulse seal. In particular, the steady-state value of a dimensionless face gap under equal external forces is proportional to the cubic root of the relative conductance . From the derived characteristics it can be seen, that in the set pressure range of the sealed liquid, face gap value slightly differs from the base value, and so optimal conditions of the seal operation are guaranteed.
The main problem of impulse seals is the development of structures and calculation of feeding channels with high hydraulic resistance (with low conductance). Impulse seal performance is significantly affected by the force of preliminary compression of elastic elements. Increasing of this force limits working area and increases static error, so it is necessary to monitor the load value and uniformity of its distribution along the circumference.