The article deals with the issues related to the mutual influence of vibrations and gas dynamic processes in the labyrinth seals (LS) of gas transmittal unit compressors. The mutual influence of vibrational gas dynamic processes in LS and vibrations of the rotor is studied. Within the framework of the unified algorithm, a solution is obtained for an unsteady aeroelastic one-dimensional gas flow problem in a deformable LS. A new factor (the rotor diameter in the LS region), which affects the pulsation magnitude of the gas dynamic force in the LS, is revealed. Changing the diameter of the rotor, you can reduce vibration. In this case, it is possible to reduce the designated clearances in the LS and to reduce the leakage.
High-Performance Computing / L.N. Butymova, V.Ya. Modorskii oscillations of displacements, velocities, pressures and gas-dynamic force acting on the rotor in the LS area are recorded at the control points. The displacement of these oscillations (φ U) may take place with respect to two parameters: the gas-dynamic forces acting on the rotor in the LS area and the rotor displacements. With different phases (φ U) convergent, divergent and steady oscillatory processes can be observed.
The mathematical description of the gas-elastic process in this formulation includes the following relationships:
(
г EVxг г E P г k 1 E Vxг 2 2 x («Sticking») Vxк 0 ,
Vxк 0 , xx Pгр The boundary condition for the piston motion:
Vxк Vxг
Vleft.pisto n V0 sin(t ) ; Vright.piston V0 sin(t ) where V0 – amplitude of the piston oscillations, – the piston oscillation frequency.
(
For solving gas dynamic tacks was used method of large particles. Using the same method for solving gas dynamic and stress-strain state tasks provide unity mesh for gas dynamic region and stress-strain state tasks. For this used unified system of differential equations to ensurecoupled solving for elastic tasks and gas dynamic tasks. Thus we used method of large particles for calculations.
The main idea of method’s large particles consisted in splitting into physical processes of the initial non-stationary system of Euler equations which written in the form’s conservation laws. The space is modeled by particle system which coincides with cell’s Euler grid in the moment. If stationary solving is, we get it in process stabilized solving.So all process solving composed multiple repetitions of time steps.
Each computational cycle is divided into seven stages. The first three stages are designed to solve the gas dynamic tasks. The next four stages are designed to evaluate the parameters dynamic stress-strain state of the structure.
In a unidirectional aeroelastic formulation, the solution using equations (
When investigating the dependence of pressure fluctuations in the LS gas-dynamic cavity on the LS geometric characteristics the rotor diameters in the LS area variedand were equal to 65, 130, 195 or 260 mm. The working body of the gas -dynamic 3 cavity is air, adiabatic exponent equals 1.4, density ρ= 1,29 kg / m .
With an increase in the shaft diameter from 65mm to 260mm, the pressure oscillations in the gas-dynamic cavity are noted, which have a time periodic character (Fig. 2). The pressure amplitude is 101368.8 Pa with D = 65mm, with D = 130 mm the pressure amplitude is 148792.2 Pa, and with D = 195mm the pressure amplitude is 107577.7 Pa, with D = 260mm the pressure amplitude is 101732.1Pa.
High-Performance Computing / L.N. Butymova, V.Ya. Modorskii at D = 130mm the temperature amplitude is 323.6043K, and at D = 195mm the temperature amplitude is 279.0668K, at D = 260mm the temperature amplitude is 274.4728K
Thus, when designing the LS it is necessary to take into account the geometric dimensions of the rotor and the gap between the rotor and the LS in order to reduce possible vibrations. 5.2. Analysis of the influence of kinematic parameters
The dependence of pressure oscillations in the LS gas dynamic cavity on the kinematic parameters of propagation speed of gas oscillations in the circumferential direction varied and equaled 3.5, 7.0 or 10.5 m/s, the working fluid of the gas-dynamic cavity is air, the adiabatic index is 1.4, ρ = 1, 29 kg/m3.
With speed increase from 3.5 m/s to 10.5 m/s, the pressure oscillations of the gas-dynamic cavity are noted, which have a periodic character (Fig. 4). The pressure amplitude is 100918.3 Pa at V = 3.5 m/s. At V = 7.0 m/s, the pressure amplitude is 103,000 Pa, and at V = 10.5 m/s the pressure amplitude is 104419Pa.
When studying the temperature dependence of the gas dynamic cavity in the LS on kinematic parameters, the propagation speed of the gas oscillations in the circumferential direction varied and assumed values of 3.5, 7.0 or 10.5 m/s, the working fluid of the gas-dynamic cavity is air, adiabatic index is 1.4, air density ρ = 1, 29 kg/m3.
High-Performance Computing / L.N. Butymova, V.Ya. Modorskii
With speed increase from 3.5 m/s to 10.5 m/s oscillations in the temperature of the gas-dynamic cavity are observed, which have a periodic character (Fig. 5). The temperature amplitude is 271.9472K at V = 3.5 m/s. At V = 7.0 m/s the temperature amplitude is 275.1519 K, and at V = 10.5 m/s the temperature amplitude is 279.8211K.
3 10.5
When studying the dependence of pressure oscillations in the LS gas-dynamic cavity on the working fluid characteristics the adiabatic index varied and was 1.1, 1.25 or 1.4 at density ρ = 1,29kg /m3.
With an increase in the adiabatic index from 1.1 to 1.4, the pressure oscillations of the gas-dynamic cavity are observed, which have a periodic character (Fig. 6). The pressure amplitude is 102880.5 Pa for k = 1.1. For k = 1.25, the pressure amplitude is 102816.3 Pa, and for k = 1.4 the pressure amplitude is 1003003Pa.
When studying the dependence of temperature oscillations in the LS gas-dynamic cavity on the working fluidcharacteristics the adiabatic index varied and was 1.1, 1.25 or 1.4, with density ρ = 1,29 kg/m3.
With an increase in the adiabatic indexfrom 1.1 to 1.4, the temperature oscillations of the gas-dynamic cavity are observed, which have a periodic character (Fig. 7). The temperature amplitude is 271.8515 K for k = 1.1. For k = 1.25 the temperature amplitude is 274.0991K, and for k = 1.4 the temperature amplitude is 275.1212K.
In the bi-directional aeroelastic formulation the LS calculation scheme was generated (Fig. 8) and a solution was obtained
using equations (
When studying the dependence of pressure oscillations in the LS gas-dynamic cavity on physico-mechanical characteristics of the LS the material density was set ρ = 7800 kg/m3,Poisson ratioμ = 0.35, the elasticity modulus ranged within 50, 100, 150, 200 GPa. With an increase in elasticity modulus from 50 GPa to 200 GPa, the amplitude of the periodic pressure oscillations of the gas dynamic cavity decreases by 5 times. The pressure amplitude is 1 MPa at E = 50 GPa. At E = 100 GPa the pressure amplitude is 0.5 MPa, and at E = 150 GPa the pressure amplitude is 0.3 MPa, at E = 200 GPa the pressure amplitude is 0.2 MPa (Fig. 9). Fig. 9 shows the dependence of pressure oscillations in the LS gas-dynamic gap on time for LS various physicomechanical characteristics.
When studying the dependence of displacements in the LS structure on physico-mechanical characteristics the density of the LS material was setρ = 7800 kg/m3, poisson ratio μ = 0.35, adiabatic index k = 1.4, air density ρ = 1.29 kg/m3,P0 = 0.1MPa, the calculated ratio of cells in the structure to the total number of cells calculated FL = 0.96, the elastic modulus varied within 50, 100, 150 and 200 GPa. Near the gas-dynamic gap with an increase in E from 50 GPa to 200 GPa oscillations of displacements in the LS structure are observed. The oscillations are periodic in nature and stable in time. The displacement amplitude is 1 × 10-2 microns at E = 50GPa. When E = 100GPa the displacement amplitude is 5 × 10 -3 m, and when E = 150 GPa the displacement amplitude is 3 × 10 -3 m, for E = 200 GPa the displacement amplitude is 2.3 × 10 -3 m (Fig.10). Figure 11 shows the relationship of displacements in the LS structure versus time for LS different physico-mechanical characteristics. 1. With an increase in the compression wave velocity arising from the approach of the rotor to the surface of the LS under vibrations from 3.5 m/s to 10.5 m/s the amplitude of the gas dynamic force increases from 23.6Н to 45.5Н. The frequency does not change and is equal to 400 Hz. At a natural rotor frequency of 808 Hz, one can expect that with a minimum value of the gas flow velocity in the circumferential direction, weak vibrations may appear. As the speed increases, one can expect an increase in the LS vibrations.
2. With the shaft diameter increase from 65mm to 260mm, the maximum amplitude of the gas dynamic force of 82.7N is observed at a diameter equal to 130 mm, the minimum amplitude of the gas dynamic force is observed at a diameter of 65mm from 7.7H, the frequency is 400Hz. The maximum frequency of the gas dynamic force is 770 Hz with a diameter of 65 mm. The minimum frequency of the gas-dynamic force is 406 Hz with the diameter equal to 260 mm. It can be seen that as the rotor diameter increases, the nominal values of the gas dynamic force increase. This is due to the increase in the rotor area at a constant nominal pressure. In this case the maximum amplitudes of gas-dynamic forces are observed when the oscillation frequency f P of the rotor is equal to the first natural frequency of the gas-dynamic pressure fluctuations of the circumferential cavity in the gap. Oscillation amplitude of gas-dynamic forces arelower at the rotor oscillation frequency fP equal to the second natural frequency of the gas-dynamic pressure oscillations of the circumferential cavity in the gap.Even lower are the amplitudes of gas-dynamic force oscillations at the rotor oscillation frequency fP equal to the fourth natural frequency of the circumferential pressure oscillations of the gas-dynamic cavity in the gap.The oscillation amplitudeof the gas-dynamic forces was also low at the natural frequency of the gas-dynamic cavity nonmultiple for the rotor frequency.
3. With an increase in the adiabatic index k from 1.1 to 1.4 the gas-dynamic force oscillation amplitude increases from 29.63N to 35.15N, and the oscillation frequency of gas-dynamic forces decreases from 392Hz to 388Hz. Thus, we note a weak influence of the working fluid characteristics on the LS vibrations. 4 200 0.2
The study was performed with a grant from the Russian Science Foundation (project №14-19-00877).
High-Performance Computing / L.N. Butymova, V.Ya. Modorskii 4. With increase in elastic modulus from 50GPa to 200GPa the pressure oscillation amplitude decreases from 0.97MPa to 0.19MPa, pressure oscillation frequency rises from 134kHz to 256kHz. Analysis of the influence of physical and mechanical characteristics on the LS vibrations demonstrated that with a hard material the strain rate is lower than with a soft material. The displacements in a softer material under given loads are 1 × 10 -2 microns. In a harder material the displacement amplitude is much lower.
5. By changing the rotor diameter, it is possible to reduce vibration. Thus, it is possible to reduce the gaps in the LS and reduce leakage.