GENERAL APPROACH TO MODELING OF NON-CONTACT SEALS AND THEIR EFFECT ON THE DYNAMICS OF A CENTRIFUGAL MACHINE ROTOR

Ukraine There is a constant demand for higher equipment parameters, such as pressure of a sealing medium and shaft rotation speed. However, as the parameters rise it becomes more difficult to ensure hermetization efficiency. More-over, sealing systems affect the overall operational safety of the equipment, especially vibratory. Non-contact seals are considered as hydrostatody-namic supports that can effectively damp rotor oscillations. Models of an impulse and a groove seals, models of rotor-seals system and rotor-auto-unloading system, model of a shaftless pump are studied to evaluate an effect of these sealing systems on oscillatory characteristics of rotor. Analytical dependencies for computation the dynamic characteristics of impulse seals, hydromechanical systems rotor-seals and rotor-auto-unloading, as well as shaftless pumps are obtained. These dependencies describe the ra-dial-angular vibrations of a centrifugal machine rotor in seals-supports. Equations for computation the amplitude-frequency characteristics are given. The directions of improving the оperational safety of critical pumping equipment by purposefully increasing the rigidity of non-contact seals that leads to higher rotor vibration stability have been on the study of models of non-contact seals and models of rotor-groove seals-auto unloading analytical dependencies are obtained that describe the effect of non-contact seals on the dynamics of a centrifugal It is shown that a purposeful choice of the design parameters of the seals can influence the vibration state of the rotor. In this case, the initially "flexible" in the dynamic sense rotor combined with correctly designed seals becomes "rigid". This is especially important for machines with high parameters. The studies carried out make it possible to determine the directions of increasing the vibration stability of centrifugal


Introduction
Centrifugal pumps and compressors are widely used throughout all industries. There is a steady tendency towards an increase in operating parameters of pumps: capacity, pressures and speeds, i.e. to the higher concentration of parameters in a single unit.
Thermal and nuclear power stations constantly require an increase in parameters of feed, main circulation and other pumps. This need stimulated a detailed study of hydrodynamic processes in non-contact seals and their influence on the vibration state of centrifugal machines rotors [1]. Designing turbopump units for high-power rocket engines for reusable spacecraft have further raised the researchers' interest in the dynamic characteristics of non-contact seals and rotor vibrations in seals [2].
The effect of the pumping medium is especially significant with large gradients of velocities and pressures. These conditions are typical for small gaps of non-contact seals, on which large pressure is throttled, and one of the surfaces belongs to a rotor that rotates and vibrates.
Therefore, when modelling groove seals, besides their designated purpose -to reduce volume losses, we need to ensure the required vibration characteristics of rotor, which is also a very important function.
Noncontact seals, on which a huge pressure is throttled, can play the role of static, and with the right design approaches, dynamic supports. This must be considered when designing critical power equipment in order to improve its environmental safety.

Literature Review
Current approaches for refinement of mathematical models of oscillatory systems according to experimental data are presented in the work [3]. The monograph [4] evaluates coefficients of mathematical models for oscillatory systems, which includes rotary systems for multistage centrifugal machines. The work [5] addresses the phenomena of rotor rotation stability loss at rolling bearing.
Modern approaches in the field of linear and non-linear rotor dynamics and their practical applications are presented in the paper [6]. The work [7] provides an assessment of segment bearing stiffness with the balancing procedure for flexible rotors of turbocharge units in the accelerating-balancing stand. Current methods for determination of active magnetic bearings stiffness and damping identification from frequency characteristics of control systems were introduced in [8]. Application of the finite element analysis for stiff-This work is licensed under a Creative Commons Attribution 4.0 International License.  Serhii S. Shevchenko, 2022 ness and critical speed calculation of a magnetic bearing-rotor system for electrical machines was described in the paper [9]. Article [10] provides stability and vibration analysis of a complex flexible rotor bearing system. A phenomenon of subharmonic resonance of a symmetric ball bearing-rotor system is investigated in the paper [11]. The paper [12] proposes models for investigation critical frequencies of the centrifugal compressor rotor that considers non-linear stiffness characteristics of bearings and seals.
As indicated in [13], energy of volumetric losses can be converted into net energy, if the groove seals are used simultaneously as hydrostatic bearings, that are able to have not only high radial rigidity but also to effectively damp the rotor fluctuations to the acceptable level even if there is a significant disbalance. This effect is especially considerable if there are existing steep velocity and pressure gradients, which are peculiar to close gaps of the groove seals, on which high pressure differentials are chocked and one of the surfaces belongs to rotor that both rotates and vibrates [14].
The dynamic characteristics of groove seals as intermediate supports have been studied in the paper [15]. However, the problems of rotor dynamics in groove seals are slightly neglected as to solve them it is necessary to account for the hydrodynamic characteristics of groove seals. And this is a separate problem in the hydrodynamics of three-dimensional unsteady viscous fluid flows in annular channels, whereof surfaces rotate and simultaneously perform radial-angular oscillations [16]. Since the problems of the rotor dynamics without groove seals have been mainly solved, this paper focuses more on the analysis of oscillatory processes caused by the hydrodynamic characteristics of seals. Figure 1 shows a model of the groove seal that is an annular throttle formed by inner cylinder (shaft) with a small taper angle A  and outer cylinder (sleeve) with a taper angle , and also perform radial and angular oscillations.

Problem statement
Thus, when developing groove seals, it is necessary to consider not only their direct purpose to reduce volumetric losses, but also their equally important function, which is to provide the necessary vibration characteristics of the rotor.
The paper has provided an assessment of the force characteristics for laminar and turbulent flow regimes taking into consideration local resistances and in view of flow swirl at the gap inlet [16].

Models of non-contact sealing systems Impulse Seal Model
Impulse seals (Fig. 2) with selfadjusting clearance have several undeniable advantages over conventional mechanical seals [2]. In impulse seals, as rotational speed rises, the face clearance increases, meaning that increase in energy losses due to friction is negligent. Therefore, impulse seals are especially effective in high-speed machines. There is negative feedback between the face clearance z (adjustable value) and the force F s (control response), which ensures self-regulation of the face clearance (Fig. 3).
The operation of impulse compaction is accompanied by Loading area А е + Object of regulationan axially movable ring Automatic regulator Fig. 3. The model of impulse seal as the automatic control system complex unsteady hydrodynamic processes, the mathematical description of which is complicated. Success in the analytical solution of the dynamic computation problem depends on the correct choice of a simplified computational model of compaction. The simplifications can only be justified by the results of experimental studies [17].
Groove Seals A simplified model of the rotor-groove seals system is shown in Fig. 4. Radial (x, y) and angular (ϑ x , ϑ y ) oscillations of the rotor are largely determined by hydrodynamic forces (F) and moments (M) arising in the sealing gaps (in annular throttles), and the very forces and moments depend on the nature of rotor movement. There is another feedback between geometric shape of the gap (average radial gap H and taper ϑ 2 ) and pressure in the gap p(z, φ): deformations of the sealing rings are determined by pressure distribution, and the latter is very sensitive to changes in size and shape of the gap.

Automatic balancing devices as sealing systems
Automatic balancing devices have many different designs [18], however there is a general principle: negative feedback is created between the balancing force and the axial position of the rotor, providing only small deviations of the axial position of the rotor from some predetermined position (Fig. 5).
The groove seals of the rotors have clearances of the same order as journal bearings. Therefore, the seal is a full-grip bearing, the bearing capacity of which is provided not only by the rotation of the eccentrically located shaft, but also by the significant pressure drop throttled on the seal. The hydrostatic component of the bearing capacity is predominant since it usually exceeds the bending stiffness of the shaft and the stiffness of radial journal bearings. Face groove seals of auto-unloading systems simultaneously serve as thrust hydrostatic bearings.
High sealing pressure results in significant losses of the sealed medium. To reduce them, the gaps are made as small as possible. Therefore, the balancing device additionally functions as a face seal with self-regulating leaks.

Shaftless pumps with seals -bearings
In high-pressure pumps, groove seals, in addition to their main purpose -to limit the crossflows between cavities with different pressures, can be used as rotor supports. Implicitly, these sealing functions have always been implemented in centrifugal pumps. There are recent designs where groove seals are also functioning as only supports (Fig. 6.) [19]. The advantages of shaftless pumps are possible only if there is no contact between the rotating impeller and the stationary casing. In turn, the non-contact mode of operation is determined by the hydrodynamic characteristics of groove seals and axial force auto-unloading system.

Systems frequency response
Oscillations of the rotor under the action of gyroscopic forces and moments arising in the seals are described by the equations [20]: The notations can be found in [20]. General view of the solution to these equations: , which after substitution in (1) leads to a system of algebraic equations for complex amplitudes Using standard programs one can immediately find a numerical solution to these equations. However, the traditional approach produces analytical expressions of amplitudes and phases, which allow to see how different forces and moments affect them.
After switching to dimensionless frequencies and introducing some notation, equations (2) take the form: Here 22 22 11 11 , are proper operators of the independent radial and angular oscillations respectively. Cross operators 21 21 12 12 , characterize the effect of angular oscillations on radial ones and the reverse effect, i.e., interconnection of these oscillations.
Substituting the values of the determinants into (3), we obtain the amplitudes and phases expressed in terms of external disturbances: For impulse seals, the amplitudes and phases are expressed by the formulas [2]: The corresponding notations can be found in [2]. The amplitude frequency characteristics can be used to estimate the dimensional values of the amplitudes of the forced axial oscillations of the ring at any rotation frequency.
Amplitude and phase frequency characteristics of the rotor-gap seals system for the corresponding external influences: The frequency characteristics of the rotor-balancing device system are calculated in a similar way: and of shaftless pumps:

Stability criteria
The stability is determined using the Routh-Hurwitz criterion for a system of 4th order which for the rotor-groove seals system is reduced to the form: Inequality (4) shows that the main destabilizing factor is the circulating force, characterized by the coefficient 5 а . Damping 21 а , gyroscopic force 4 а and shaft bending stiffness 0 и  stabilize the rotor. For impulse seal it is possible to determine the chamber volume admissible in stability: . ) )( )( 1 (  3   30  10  10  3  30  1 It can be seen from inequality (5) that the stability region of the seal expands due to a decrease in the volume of the chambers and in the coefficient of hydrostatic stiffness.
For balancing device The design of the chamber can be easily changed since its depth is an independent parameter. Therefore condition (7) can be used for pump design to ensure its stability.

Conclusions
Based on the study of models of non-contact seals and models of rotor-groove seals-auto unloading systems, analytical dependencies are obtained that describe the effect of non-contact seals on the dynamics of a centrifugal machine. It is shown that a purposeful choice of the design parameters of the seals can influence the vibration state of the rotor. In this case, the initially "flexible" in the dynamic sense rotor combined with correctly designed seals becomes "rigid". This is especially important for machines with high parameters. The studies carried out make it possible to determine the directions of increasing the vibration stability of critical power centrifugal machines.