A transition to clean hydrogen energy will not be possible until the issues related to its production, transportation, storage, etc., are adequately resolved. Currently, however, it is possible to use methane-hydrogen mixtures. Natural gas can be transported using a pipeline system with the required pressure being maintained by gas compression stations. This method, however, is affected by some problems too. Compressors emergency stops can be induced by vibrations because in some cases, mechanical methods are not able to reduce the vibration amplitude. As an example, it is known that a gas-dynamic flow effect in labyrinth seals can lead to increased vibrations. This paper presents the numerical simulation of rotor oscillations taking into account a gas-dynamic load. The influence of a transported mixture on the oscillatory process is investigated. Mixtures consisting of methane and hydrogen in various proportions and an air mixture are considered. The results are discussed for various operating pressures and include the rotor motion trajectories and oscillation frequency spectra obtained numerically. It is shown that the gas mixture composition has a significant effect on the oscillations and their occurrence. Hydrogen as a working fluid reduces the vibration amplitude. Operating a compressor with hydrogen leads to a decrease in the resonant frequency, bringing it closer to the operating one. However, the operating pressure at which maximum oscillations are observed depends slightly on the gas mixture composition.

Currently, issues related to the possibility of transition from fossil hydrocarbon fuels to hydrogen fuel are being widely studied. An undeniable advantage of hydrogen fuels is their high environmental friendliness, since the combustion product is water vapor. In addition, the combustion temperature and specific heat of combustion of hydrogen fuel exceed those of fossil hydrocarbon fuels [

An economic analysis [

A number of works [

The gas transmission system consists not only of pipelines and distribution units connecting production sites with consumers but also gas compressor stations that maintain the pressure of the transported gas, which decreases due to friction losses.

An emergency shutdown of a station can lead to losses. One of the reasons for emergency stops is the increased level of vibrations. Compressor rotor vibrations near bearing supports must not exceed up to 40 microns.

Vibration causes may include residual mechanical imbalances due to inaccurate parts manufacturing, asymmetrical elements, non-uniformity of work piece material, and assembly errors. To eliminate mechanical imbalances, balancing is used [

To reduce leaks between compressor stages, labyrinth seals are used, which are subject to high requirements for operational reliability [

Thus, in a number of cases, existing mechanical balancing methods fail to reduce the magnitude of vibrations to the required values. It is necessary to clarify existing methods and search for unaccounted factors that lead to a vibration occurrence. Such a factor may be taking into account gas dynamics when modeling rotor dynamics.

There is a linear model of gas-dynamic forces acting on a rotor, which is used in a rotor dynamics analysis [

Researchers perform many experiments in an attempt to identify the influence of various geometric factors, but using a linear model does not allow studying a process nature.

The work [

Despite a large number of studies on rotor vibrations caused by unsteady forces in labyrinth seals, there are practically no studies devoted to a compressor operation using hydrogen. Therefore, the work purpose is to assess a rotor vibrations possibility in a hydrogen compressor. To achieve this purpose, it is necessary to solve the following tasks:

1. Assess the influence of a gas mixture composition on a vibrations occurrence in labyrinth seals.

2. Determine the effect of hydrogen concentration in a mixture on vibrations amplitude.

3. Compare vibration frequencies at which a maximum vibration amplitude is observed.

Replacing natural gas with a methane-hydrogen mixture in gas pipelines may lead to a change in the level of vibration of compressor rotors and require additional assessment of the performance of existing compressor stations, so this issue requires research.

The numerical model is presented in _{0}. The outer ring is non-deformable. The gap at the initial time moment is filled with gas at given initial values of pressure and temperature. The calculations explicitly took into account a rotation and Earth’s gravity. At one of the ends of the rotor, an axial movements limitation was set to unambiguously define the model in space. A detailed model description is presented in [

The mathematical model is based on non-stationary equations of fluid dynamics and mechanics of deformable solids. Fluid motion equations includingmass, momentum and energy governing laws, are closed by the ideal gas state equation and the turbulence model [

Continuity equation:

Momentum equation:

The relationship for determining viscous stresses has the form:

Energy equation:

The equations described above are supplemented with constitutive relations of state for the density and enthalpy. For an ideal gas the following equations are valid:

The compressor rotor movement is described by differential equations in displacement within the framework of the linear elasticity theory. Limiting ourselves to the linear elasticity theory to describe the rotor movement, the smallness of possible deformations is assumed, which are determined through displacement gradients:

Since the elastic deformation process is associated with inertial forces acting on structural elements at the moment of rotation, and external influences caused by the influence of fluid mass transfer in the gas compressor compressor, forces called stresses arise on the surface of the solid body. The relationship between the resulting stresses and deformations is isotropic Hooke’s law [

The Lamé parameters are determined by the following equations:

The differential motion equations of a continuous medium, which is the compressor rotor, follow from the static equilibrium equations when taking into account volumetric inertia forces:

The equations of static equilibrium when taking into account volumetric inertia forces have the form:

In this case, the material density is a constant value, since the rotor material is assumed to be homogeneous, i.e., its physical and mechanical properties are the same at all points of the body. The mass forces density, in the general case, is a function of spatial coordinates and is formulated from physical considerations about the processes occurring inside the body:

However, this article does not imply the presence of internal mass sources, i.e., external forces are determined from the boundary conditions caused by external interaction.

Using the above relationships, it is possible to obtain differential motion equations of the rotor compressor in displacements:

Calculations were carried out for several types of mixtures presented, see

Mixture | Mixing ratio, % | ||
---|---|---|---|

Air | Methane | Hydrogen | |

1 | 100 | – | – |

2 | – | 100 | – |

3 | – | 95 | 5 |

4 | – | 90 | 10 |

5 | – | – | 100 |

To analyze the rotor operation stability, the point movements on the shaft axis in the disk section over time were considered (

Using mesh methods of computer modeling implies performing a study of the mesh convergence, which consists of searching for mesh model parameters for which changing a elements number does not affect a result obtained. Since a problem being solved is multidisciplinary, a mesh convergence study for a complete computational model requires large computational costs. Therefore, the analysis was performed separately for each of subdomains (mechanical and fluid).

The convergence analysis of a rotor mesh model was performed by varying the number of elements along a shaft length. This choice is due to a fact that a disk mounted on a shaft has significant rigidity due to its geometric dimensions and does not undergo significant deformation. A geometric shaft dimensions were chosen in such a way that a rotor rigidity is low and an aeroelastic effects manifestation is more pronounced. Therefore, a rotor experiences the greatest deformation as a shaft deflection result in the radial direction. A static shaft deflection under an gravity influence and first natural bending frequency were chosen as criteria for mesh convergence. The results of a mesh model convergence analysis are presented in

As can be seen on plots, changing elements number in a wide range along a rotor shaft length has little effect on the parameters under consideration. For all points considered, a deviation does not exceed 1%, therefore, for further calculations, an intermediate option was chosen from the considered ones-75 elements along a rotor shaft length. This choice reduces the need for computing resources and at a same time provides some margin in a solution accuracy obtained. The rotor mesh model was 7016 elements and 22,292 nodes.

Before constructing a mesh model for fluid calculations, a first layer thickness near a wall was estimated in order to obtain the dimensionless distance from a wall y +< 1 required by the selected SST turbulence model. A first layer thickness was 0.8 µm. According to the simulation results performed, a requirement for a dimensionless distance near a wall was met.

To analyze a mesh model convergence of a fluid region, a number of elements along the ring and along a gap thickness were changed. An elements number in an axial direction varied in proportion to an elements number in a circumferential direction. Since radial rotor vibrations are considered, to analyze a mesh model convergence, an inner wall of a fluid region adjacent to a rotor disk was made with eccentricity. The wall was shifted by the static rotor deflection value of 60 μm. Thus, on the inner wall, when a gas moved in a circumferential direction, a gas-dynamic force arose due to an uneven gap thickness. A gas-dynamic force magnitude was chosen as a convergence criterion. The mesh model convergence results are presented in

An element number in the circumferential direction was chosen to be 1500. With this value, a calculation series wasperformed with different element numbers along the gap thickness (

The resulting mesh models are shown in

The selected two-way coupling between fluid and structure (2FSI) model was verified using experimental data and computational experiment results obtained in a shock tube with an installed plate, which deforms when interacting with a shock wave [

At the beginning of the experiment, there are high and low-pressure areas in the pipe, separated by a thin membrane. At the end of a low-pressure area there is a deformable plate rigidly fixed to the base. When a threshold pressure value is exceeded, the membrane ruptures, and expansion wave, contact surface and incident shock begin to propagate in a pipe. When interacting with a wave, a plate begins to bend.

During the experiment, shadow photography of a flow and plate deformation was performed. From photographs, a movement dependence of a plate’s upper edge over time was obtained. A sensor located on a top pipe wall at a distance of 10 mm from a plate recorded pressure oscillations.

A similar numerical experiment was performed, which made it possible to perform a qualitative and quantitative comparison. Results obtained are presented in the form shadow images (

In general, one can note a good agreement a plate movement calculated using an accepted model and experimental data obtained in [

When analyzing rotor vibrations, a key reliability criterion is a low vibration displacement magnitude. Results obtained showed a correct prediction of a structure deformation under a gas flow influence, which allows a described model to be used for vibration simulation.

The resulting trajectories of motion of a point on the rotation axis in the disk section, obtained in a computational experiment, are presented in

It is clear from the figures that divergent vibrations are observed for air at 10 and 14 MPa. Moreover, at 14 MPa the vibration amplitudes are higher. At 5 and 20 MPa, steady motion is observed. At 14 MPa in 0.12 s the vibration amplitude exceeds 100 µm.

When considering methane, divergent rotor vibrations occur at a pressure of 14 MPa. The vibration increase rate is lower than for air. At time 0.12 s, the rotor oscillation amplitude exceeds 40 μm.

An increase in rotor oscillation amplitude occurs as a result of a fluid wedge interaction that occurs between a rotor and seal sleeve. Under identified conditions, a fluid wedge rotates along a circumference with a constant frequency and phase difference.

At the considered 5 and 20 MPa, a vibration is stable. At 10 MPa, oscillations with a constant amplitude of about 20 μm are observed. With a slight (5%–10%) admixture of hydrogen, a multiple decrease in the amplitude of oscillations is observed throughout the entire pressure range considered. Moreover, at a pressure of 14 MPa, the vibrations have a greater amplitude than at 5, 10, and 20 MPa.

When pumping pure hydrogen, the transient oscillation process changes qualitatively. The amplitude of the oscillations of the transient process is greater than that of mixtures No. 3 and 4. In this case, the nature of the oscillations is damped. The largest amplitude of oscillations of the transient process is observed at 5 MPa and decreases with increasing pressure.

According to

For further analysis, the results are summarized in the plots form of the rotor oscillation amplitudes over the initial pressure (

For mixture No. 1 (air) in

A three-dimensional numerical simulation of the compressor rotor vibrations was performed taking into account a gas-dynamic influence of a seal. The assumptions made in the model are described. An influence study of a mesh model on results obtained was carried out, and a convergence condition was achieved. The proposed numerical model was verified with experimental data, and agreement between results was obtained. A series of computational studies were carried out for various mixtures and operating pressures. For each modeled point, rotor movement trajectories were obtained, which were summarized in a plot form. From the results obtained, the following conclusions can be drawn:

1. The gas mixture composition has a significant influence on the occurrence of vibrations in labyrinth seals.

2. Hydrogen as a working fluid reduces the oscillatory amplitude.

3. Hydrogen as a working fluid increases a shaft deflection and reduces an oscillatory processes amplitude.

4. Resonance pressure weakly depends on the gas mixture composition.

None.

Fluid density

Velocity vector

Time

Pressure

Dynamic viscosity

Kronecker delta function

Total enthalpy

Fluid temperature

Thermal conductivity

Molar mass

Universal gas constant

Isobaric heat

Vector displacement

Lame parameters

Unit tensor

Young’s modulus

Poisson’s ratio

Material density

Mass force density

External forces density

The research was carried out with financial support from the Russian Ministry of Education and Science, Project FSNM-2023-0004 “Hydrogen Energy. Materials and Technology for Storage, Transportation and Use of Hydrogen and Hydrogen-Containing Mixtures”.

The authors confirm their contribution to the paper as follows: study conception and design: Vladimir Ya. Modorskii; data collection: Ivan E. Cherepanov; analysis and interpretation of results: Vladimir Ya. Modorskii, Ivan E. Cherepanov; draft manuscript preparation: Ivan E. Cherepanov. All authors reviewed the results and approved the final version of the manuscript.

Data on which this paper is based is available from the authors upon reasonable request.

The authors declare that they have no conflicts of interest to report regarding the present study.