Direct Stiffness Coefficients Research Articles

Dynamic Performance of Liquid Smooth Annular Seal Operating in the Transition Regime

Abstract This paper presents the experimental leakage and rotordynamic performance for a liquid smooth annular seal operating in the transition regime. The test conditions include pressure differentials up to 64 bars with 1∼2 bar increments for 6 rotor speeds (2.5, 3.8, 5, 7.5, 8.8, and 10 krpm), as well as nonrotating rotor case under zero preswirl condition. The rotordynamic coefficients for all the test conditions are obtained by pseudo-random excitation of the seal at multiple subsynchronous frequencies. By considering the transition Reynolds number (1000 < Re < 3000) and the Taylor Number (Ta) versus Axial Reynolds Number (Rez), the variations in the direct stiffness coefficients (K) can used as an indicator of the flow regime transition boundaries. The direct stiffness K resulting from the Lomakin and hydrodynamic effects significantly drops until Rez reaches ∼1500. For higher Rez, K increases mainly due to hydrodynamic effects. When K drops, the cross-coupled stiffness k, the direct damping C and the cross-coupled virtual mass m increase while the cross-coupled damping c and virtual mass M decrease. None of predictions based on either laminar or turbulent flow show the variations in rotordynamic coefficients obtained from experimental results. The leakage is not highly influenced by rotor speeds for low speed cases crossing laminar boundary as ΔP increases, however, results for higher speeds in the superlaminar region show reduced leakage rates as rotor speed increases.

Evaluation of the flow state and static performance of smooth annular liquid seals

The static fluid-induced force and stiffness coefficient of the smooth annular seal directly affect the rotor system stability. In this paper, a computational fluid dynamics method is applied to investigate the flow characteristics of a smooth annular seal for various eccentricities, discharge/supply pressures and rotational speeds under different flow conditions (laminar, transition, and turbulent flow). The influence factors and formation mechanism of the static instability in the smooth annular liquid seal are analyzed. Results show that laminar flow dominates the flow state at a rotational speed of ω= 2000 rpm. As the rotational speeds increase, the transition flow (2000-7000 rpm) gradually transits to the turbulent regime (ω> 7000 rpm). The direct static stiffness decreases first and then increases from laminar to transition flow state, and the viscosity effect is the dominant factor. For transition and turbulent flow with high eccentricities (ε= 80 %), the dominant viscous effect and inertial effect lead to the negative radial force and negative direct static stiffness coefficients. The smooth annular liquid seal shows best performance in the laminar flow and worst performance in the turbulent flow.

CFD-Predicted Rotordynamic Characteristics for High-Temperature Water Liquid Seal Considering Tooth Deformation

With the development of high-temperature centrifugal pump, the temperature of the medium in the pump must be higher than the normal water temperature. It is particularly important to study the rotordynamic characteristics of the seal at high temperature due to it being the core component of the rotor system. This paper takes the high temperature water liquid seal as a research object to study its rotordynamic characteristics based on the fluid-solid-thermal coupling, the deformation of seal teeth under thermal and dynamic loads was calculated. Based on the test rig, the leakage flow rate and drag power loss of water liquid seal at 20 °C, 50 °C, and 86 °C temperatures were tested and compared with the CFD (Computational Fluid Dynamics) calculation. Meanwhile, the DEFINE-CG-MOTION and DEFINE-PROFILE control macro were used to establish the rotor whirling equation, the frequency-independent rotordynamic coefficients (K, k, C, c) and frequency-dependent rotordynamic coefficients (Keff,Ceff) were evaluated by transient CFD method. This analysis was done at three different pressure drops (2.08, 4.12, and 8.25 bar) and three rotational speeds (2000, 4000, and 6000 r/min). The results show that with the increase of water temperature, both the leakage flow rate and drag power loss decrease, indicating the 86 °C water seal has a better sealing capacity. From the rotordynamic perspective, with the increase of water temperature, the direct stiffness coefficient decreases, and the effective stiffness coefficient Keff for 20 °C water seal possesses a better stiffness capability than the other two temperature seals. The effective damping coefficient Ceff for 20 °C water seal is larger than the other two temperature seals, which means it is more stable for the rotor system.

Thermal Management and Dynamic Performance of a Cooler Design for the Supercritical CO2 Turbine Shaft Cooling

This paper presents the results of a numerical study on a supercritical CO2 (SCO2) turboshaft cooling device by considering heat transfer and hydrodynamics. A micro-cooling device with a radius clearance of 50 micron and a nozzle diameter of 4 mm was designed and used to investigate the heat transfer characteristics of a micro-spacing impinging and gas film dynamics. Sixteen nozzles (N = 16) are equally spaced around the shaft in single or double rows. Investigations include journal speed- and eccentricity-dependent forward and cross-coupled force coefficients, and the effects of nozzle layouts and mass flow rate on the heat transfer efficiency. Analysis of the correlation coefficient shows that the gas density in the radial clearance is the dominant factor affecting the convective heat transfer performance, while the fluid velocity is a secondary factor. And the cooling efficiency (mass flow utilization rate) at low cooling pressure (ps < 0.7 MPa) is significantly greater than that at high cooling pressure (ps > 0.7 MP). In addition, considering the structure alone, a dual-row cooler exhibits a higher average Nusselt number, also registers a higher mass flow rate at the same pressure. Once the shaft is heated only one end, the difference in effectiveness between single- and dual-row cooling is not significant, so coolers with a single-row configuration should be preferred. Then, experimental values for the temperature of the heated rotor are provided under specific cooling airflow conditions. Dynamic analysis results show that the force coefficient of the single-row configuration is more dependent on the journal rotation speed and eccentricity ratio, and exhibits a negative direct stiffness coefficient at higher inlet pressure and journal rotation speed. Moreover, cross-coupled terms (stiffness coefficient) generally have a more explicit variation tendency than direct terms, and are more sensitive to changes in shaft speed and eccentricity. Small clearance cooling is a relatively complex technology aimed to improve heat dissipation efficiency in gas cooling devices while minimizing the effect of hydrodynamic pressure. Comparing the gas force coefficients of different journal speeds reveals a drastic increase in the effect of hydrodynamic pressure when the journal is eccentric. The cooler may be considered for operation with compliant support (such as bump foil) to generate additional damping and appropriately compensate for the eccentricity of the rotor.

Rotordynamic analysis and leakage performance study of a hole diaphragm labyrinth seal using the CFD method

Improving rotor stability and reducing leakage at various working parts of gas turbines is an important topic that experts and scholars have always pursued. In this article, a new type of hole-diaphragm-labyrinth seal (HDLS) structure is obtained by introducing an additional damping source. The HDLS structure is built by the traditional diaphragm labyrinth seal (DLS) with a single hole added to the diaphragm. The simulation under different working conditions of three different seal structures is carried out by ANSYS Fluent. Subsequently, the tangential force under different whirl frequencies was obtained. Meanwhile, the empirical parameters in Childs’ model of the dynamic performance of rotor-seal structures were obtained. The results show that the HDLS achieved a 71.4% improvement in the direct stiffness coefficient and a 71.8% improvement in the direct damping coefficient of the traditional labyrinth seal. The leakages of HDLS at different rotation speeds were lower than those of DLS by 1.53% and LS by 3.64%. The HDLS can be comprehensively considered the most promising seals among LS, DLS, and HDLS not only by stability but also by mass flow leakage in the rotor system.

Dynamic Characteristics of High-speed Water-Lubricated Spiral Groove Thrust Bearing Based on Turbulent Cavitating Flow Lubrication Model

The water-lubricated bearings are usually the state of turbulent cavitating flow under high-speed conditions. And the distribution of cavitation bubbles and the interface effect between the two phases have not been included in previous studies on high-speed water-lubricated bearings. In order to study the influence of interface effect and cavitation bubble distribution on the dynamic characteristics of high-speed water-lubricated spiral groove thrust bearings (SGTB).A turbulent cavitating flow lubrication model based on two-phase fluid and population balance equation of bubbles was established in this paper. Stiffness and the damping coefficients of the SGTB were calculated using the perturbation pressure equations. An experimental apparatus was developed to verify the theoretical model. Simulating and experimental results show that the small-sized bubbles tend to generate in the turbulent cavitating flow when at a high rotary speed, and the bubbles mainly locate at the edges of the spiral groove. The simulating results also show that the direct stiffness coefficients are increased due to cavitation effect, and cross stiffness coefficients and damping coefficients are hardly affected by the cavitation effect. Turbulent effect on the dynamic characteristics of SGTB is much stronger than the cavitating effect

Effect of slip boundary condition and non-newtonian rheology of lubricants on the dynamic characteristics of finite hydrodynamic journal bearing

In the present study, the influence of various slip zone locations on the dynamic stability of finite hydrodynamic journal bearing lubricated with non-Newtonian and Newtonian lubricants has been investigated. Linearized equation of motion with free vibration of rigid rotor has been used to find the optimum location of the slip region with maximum stability margin limit. It has been observed that bearing with interface of slip and no-slip region near the upstream side of minimum film-thickness location is effective in improving the direct and cross stiffness coefficient, critical mass parameter, and critical whirling speed. The magnitude of dynamic performance parameters with slip effect is highly dependent on the rheology of lubricant. Shear-thinning lubricants combined with slip boundary condition shows higher dynamic stability as compared to the Newtonian lubricants under the conventional boundary condition. For all considered rheology of lubricants, the dynamic stability of bearing with slip effect is improving by increasing the eccentricity ratio.

Analytical Study of Non-Recessed Conical Hole Entry Hybrid/Hydrostatic Journal Bearing with Constant Flow Valve Restrictor

The present work studies the analysis of a non recessed hole entry conical hybrid/hydrostatic journal bearing adjusted for constant flow valve (CFV) restriction. The paper provides effectiveness between the conical bearings with hole entry operating in hybrid and hydrostatic mode. The Reynolds formulae, for the flow of fluid through the mating surfaces of a conical journal and bearing, are numerically worked out in both the modes considering the finite element analysis (FEA) and the necessary boundary preconditions. Holes in double row are marked on conical bearing circumference to accommodate the CFV restrictors, the angular distance between two holes are 30o apart from the apex. Qualitative features of the conical journal bearing system with hole entry have been elaborated to analyze bearing performance for radial load variation Wr = 0.25-2. Numerical results obtained from the present study indicate that load carrying capacity of conical bearing, operating in hydrostatic mode, is enhanced by the maximum pressure, direct fluid film damping and direct film stiffness coefficients vis-a-vis corresponding hybrid mode. 

Thermo-Elasto-Hydrodynamic Characteristics Analysis of Journal Microbearing Lubricated with Rarefied Gas.

Temperature rise and elastic deformation are unavoidable issues occurring in high-speed gas microbearings due to the dominant small-scale fluid dynamics in rarefied gas flow applications. In this paper, thermo-elasto-aerodynamic analysis requires simultaneously solving the modified Reynolds equation, modified energy equation, temperature–viscosity relationship and the elasticity equations for predicting the lubrication characteristics of microbearings. A thermo-elasto-aerodynamic lubrication is systematically investigated by using the partial derivative method, finite difference formulation and the finite element approach. The results indicate that, compared with rigid microbearing which has a constant viscosity gas lubricant, the temperature effect increases the load capacity, friction coefficient and stiffness coefficients, and it decreases the attitude angle and damping coefficients of the microbearing. The flexibility of the bearing pad also leads to the increase in load capacity and direct stiffness coefficients, while it remains to further decrease the direct damping coefficients on the basis of thermo-aerodynamic performance. The present study is conducive to accurately analyze the microscopic flow properties in a microbearing-rotor system.

Dynamic characteristics of compound textured journal bearing

An investigation of the compound texture effect on the dynamic characteristics of the journal bearing film is conducted. In this work, eight dynamic coefficients of the compound textured journal bearing and critical speed of the rotor supported by textured bearings are obtained and compared. Meanwhile, the elastic deformation effect of the bearing and rotor is obtained using the continuous convolution fast Fourier transform (CC-FFT) method. It is found that the reasonably arranged compound texture brings out an obvious increment in the direct stiffness coefficients and damping coefficients compared with the simple one, which results in the high critical speed of the bearing-rotor system. The above changes are close to the texture distribution, second-layered texture length, and width-length ratios of the compound texture. Moreover, there exists a critical compound texture depth to improve the critical speed of the bearing-rotor system.

Thermohydrodynamic Model of Cavitating Flow and Dynamic Characteristic Calculation for High-Speed Water-Lubricated Pump-Out Spiral Groove Bearing

A complete thermohydrodynamic lubrication model for pump-out spiral groove thrust bearings (SGTB) with cavitating flow is established based on two-phase flow and transport theory. Various effects (cavitating flow interface effect, convective heat transfer effect, heat conduction effect, inertia effect, breakage and coalescence of bubbles) of water lubricant at high speed are included when modeling. The bubble size distributions, bubble number density of the cavitating flow, and the dynamic coefficients of the SGTB are predicted using this model. The probability distribution of the bubble size, bubble number density, and axial film stiffness coefficient agree with the experimental values. The results show that the number of small-sized bubbles is much larger than the number of large-sized bubbles in cavitating flow. When the SGTB has a specified spiral angle, the direct stiffness coefficients increase and the cross-stiffness coefficients decrease due to the cavitation effect, but the influence of the cavitation effect on the damping coefficient is not obvious.

The Static Instability Characteristics of Labyrinth Seals: Experiments and Computational Fluid Dynamics Verification

Abstract The paper proposed an experimental measurement to identify the fluid-induced force and static stiffness coefficients. A three-dimensional fluid model of the labyrinth seals was also established. And the static characteristics under different eccentricities and inlet pressure were studied. Theoretical results show a good agreement with the experiment results. The labyrinth seal with different eccentricity ratios produces a de-centered fluid-induced force and a negative direct static stiffness, which cause the rotor deviating from the geometric center of the stator. Both the fluid-induced force and negative direct static stiffness coefficient with a value of −13.5 kN/m to −72 kN/m increase with increasing eccentricity and inlet pressure. Particularly, the direct static stiffness coefficient shows a relatively significant increase with a high eccentricity ratio (∼>40%). The static instability is mainly due to the fact that the fluid velocity in the small clearance increases more rapidly along the leakage path. The inertial force is increased significantly and the pressure decreases. This results in a greater pressure distribution in the larger clearance. The fluid force and direct static stiffness coefficient that tend to push the rotor away from the stator center are produced, and eventually lead to the static instability of the labyrinth seal.

Research on Static and Dynamic Characteristics of Gas-lubricated Microbearings for Microfluidic Devices

Aerodynamic journal microbearings with microtribological phenomena significantly influence the operating stability of microfluidic devices. The modified Reynolds equations including different rarefaction models are derived and are solved by the partial derivative method and relaxation iteration algorithm. The effects of Knudsen number and bearing parameters on the static and dynamic characteristics of microbearings are investigated in detail. The results show that the rarefaction effect plays a crucial role in the ultra-thin gas film lubrication. The maximum film pressure of Fukui–Kaneko (FK) model is lowest and the result in Boltzmann model is largest in small Knudsen number regions. As the Knudsen number increases further, the curve of FK model is coincident with that of the Boltzmann model in the transition regime. The direct stiffness coefficients of Boltzmann model increase with the increase of eccentricity ratio and aspect ratio, whereas the effect of Knudsen number on the damping coefficients yields a relatively complicated trend.

Experimental Rotordynamic Force Coefficients for a Diffusion Bonded Compliant Hybrid Journal Gas Bearing Utilizing Fluid-Filled Hermetic Dampers

Abstract Dynamic force coefficients are presented from experimental results of a radial gas bearing with hermetically sealed squeeze film dampers (HSFDs) in the bearing support. HSFDs are a relatively new technology aimed to increase damping levels in gas bearings while sustaining an oil-free bearing sump. Past HSFD designs proved bulky and contained many components making it difficult to employ in size-limited environments such as jet engines, while the diffusion bonded bearing discussed in this paper provides a compact integral design. Details of the design are found in a companion paper by Ertas (Ertas, B. H., 2019, “Compliant Hybrid Gas Bearing Using Integral Hermetically-Sealed Squeeze Film Dampers,” ASME Paper No. GT2018-76312). Test results for a 3 in. (76.2 mm) diameter bearing using a test rig providing static loads up to 80 lbs (356 N), controlled-dynamic orbital motion, and speeds up to 27 krpm are shown. Results include frequency- and speed-dependent direct and cross-coupled rotordynamic force coefficients. Dynamic testing showed little dependence on rotor speed or static load and exhibited frequency dependency at lower excitation frequencies. Cross-coupled terms are generally an order of magnitude lower than direct terms. Results show the direct stiffness coefficients increasing with frequency, while direct damping decays radically with frequency. Comparison of the overall gas bearing coefficients with the companion paper (Ertas, B. H., 2019, “Compliant Hybrid Gas Bearing Using Integral Hermetically-Sealed Squeeze Film Dampers,” ASME Paper No. GT2018-76312), showing bearing support coefficients, reveals a drastic reduction in damping when engaging the gas film. The results also indicate that the bearing can withstand vibration levels representative of a large rotor system critical speed at lower excitation frequencies.

Static instability of the smooth annular seals with choked/unchoked flow

Abstract A computational fluid dynamics (CFD) method is applied to investigate the flow characteristics of a smooth annular seal for various eccentricities(e), length-diameter ratios (L/D) and taper coefficients(ƒ) under choked and unchoked flow conditions. For the unchoked flow condition, the viscous effect is the dominant factor for the negative static direct stiffness coefficient K of the experimental seal (L/D = 1.0) with high eccentricities. For the choked flow, the dominant viscous and inertial effect produce the negative K even at low eccentricities. Increasing L/D leads to a negative K eventually. The K of the divergent and convergent annular seal is negative and positive, respectively. The viscous effect in the minimum clearance of the tapered seal results in a negative K.

Dynamic and stability performance improvement of the hydrodynamic bearing by using triangular-shaped textures

In the present study, the effect of triangular shape textured on the bearing dynamic and stability performance has been investigated. The triangular-shaped texture having variation in their depth size, number of textures, and location has been used in the study to find the stiffness, damping, and stability parameters and compared with un-textured bearing. The pressure and fluid-film thickness in the lubricant flow domain having characteristics of Newtonian and isothermal and which is governed with Reynold's equation have been computed by discretizing the domain into four-nodded quadrilateral isoparametric by using finite element method. Four different cases of texture distribution on the bearing surface have been studied. The study has been carried out considering the bearing operation only under average eccentricity ratios of 0.6. From obtained results, it is found that the value of direct stiffness coefficient and threshold speed is found maximum at lower texture depth and the direct damping coefficient is found maximum at higher value of texture depth corresponding to different texture distribution on the bearing surface. The optimum triangular-shaped textured parameters have been also finalized to get maximum dynamic performance and stability, which may be expected to be valuable for bearing designers. For the purpose of a better insight into the stability aspect of the optimal textured journal bearing, the journal center trajectories are also drawn and compared with un-textured bearing.

Lubrication Characteristics of Water-Lubricated Rubber Bearings With Partial Wear

Abstract Water-lubricated rubber bearings are widely used in the propulsion shafting of military craft and ships. These bearings may wear down after a period of service, and consequently, their lubrication characteristics will change, affecting the operation of the shaft. Hydrodynamic lubrication characteristics of water-lubricated rubber bearings with partial wear at the bottom are studied by finite difference method. The steady-state characteristics of water-lubricated rubber bearings with wear and elastic deformation of rubber liner considered are solved with successive over-relaxation iteration method, and the dynamic characteristics are further calculated with finite perturbation method. The results show that water film thickness and distribution of water film pressure is significantly changed by wear. With the same eccentricity ratio, the maximum water film pressure, load capacity, attitude angle, and friction force are reduced by wear, but the friction coefficient is increased by wear. Under the same load, the minimum water film thickness and the maximum water film pressure are slightly affected by wear, the eccentricity ratio increases with the increase of wear, and the attitude angle and the friction coefficient decrease with the increase of wear. For large eccentricity ratios, the direct stiffness coefficients and direct damping coefficients are decreased by wear. The maximum allowable wear depth is approximately 1/6 of the bearing clearance.

Finite element analysis of multirecess hybrid spherical journal bearing system

The present work deals with finite element method analysis of a multirecess hybrid spherical journal bearing system. The governing equations have been discretized using Galerkin’s technique and are solved simultaneously using a suitable iterative technique. The effect of span angle on the static and dynamic behavior of a hybrid spherical journal bearing compensated with membrane restrictor is investigated in the present work. Numerical results indicate that larger values of span angle provide enhanced value of minimum fluid-film thickness [Formula: see text], reduced lubricant flow requirement [Formula: see text], and higher value of frictional torque [Formula: see text]. Further, the results have been compared with a correspondingly similar capillary-compensated bearing. The comparison of numerically results demonstrates that the value of direct fluid-film stiffness coefficient [Formula: see text] could be 45.90% higher than that of correspondingly similar capillary-compensated bearing. The numerical results presented in this work may be useful as design guidelines for a recessed hybrid spherical journal bearing.

Study on dynamic characteristics for high speed water-lubricated spiral groove thrust bearing considering cavitating effect

Abstract The purpose of this paper was to study dynamic characteristics for high-speed water-lubricated spiral groove thrust bearing considering cavitating effect. A lubrication model was established with multiple gas-liquid interface effects, the evolution of bubble volume caused by breakage and coalescence of bubbles was quantitatively described using the population balance equation. Dynamic characteristics of the spiral groove thrust bearing (SGTB) were predicted. An experimental rig was developed to measure the dynamic coefficients of the SGTB. The result shows that, when spiral angle is in small range, direct stiffness coefficients of the SGTB with the cavitating effect are larger than those without cavitating effect. The cavitating effect has less influence on the damping coefficients than the stiffness coefficients.

Non-Axisymmetric Flows and Rotordynamic Forces in an Eccentric Shrouded Centrifugal Compressor—Part 1: Measurement

Abstract Centrifugal compressors can suffer from rotordynamic instability. While individual components (e.g., seals, shrouds) have been previously investigated, an integrated experimental or analytical study at the compressor system level is scarce. For the first time, non-axisymmetric pressure distributions in a statically eccentric shrouded centrifugal compressor with eye-labyrinth seals have been measured for various eccentricities. From the pressure measurements, direct and cross-coupled stiffness coefficients have been determined. Thus, the contributions of the pressure perturbations in the shroud cavity and labyrinth seals have been simultaneously investigated. The cross-coupled stiffness coefficients in the shroud and labyrinth seals are both positive and one order of magnitude larger than the direct stiffness coefficients. Furthermore, in the tested compressor, contrary to the common assumption, the cross-coupled stiffness in the shroud is 2.5 times larger than that in the labyrinth seals. Thus, not only eye-labyrinth seals but also shrouds need to be considered in rotordynamic analysis.