Generalized Reynolds Equation Research Articles

Thermo-Fluid–Structural Coupled Analysis of a Mechanical Seal in Extended Loss of AC Power of a Reactor Coolant Pump

We proposed a numerical method to investigate the thermo-fluid–structural coupled characteristics of a mechanical seal of a reactor coolant pump (RCP), especially during extended loss of AC power (ELAP) operation. We developed a finite element program for the general Reynolds equation, including the turbulence effect to calculate the pressure, opening force, and leakage rate of fluid lubricant and the two-dimensional energy equation to calculate the temperature distribution of the fluid lubricant. We verified the accuracy of the developed program by comparing the simulated temperature distribution and leakage rate of this study with those of previous research. Heat conduction and elastic deformation due to pressure and temperature changes at the seal structure were analyzed using an ANSYS program. The results showed that temperature more significantly affected the elastic deformation of the seal structure near clearance than pressure both under normal and ELAP operating conditions. High temperature and pressure of the coolant under ELAP operating conditions deform the seal structure, resulting in a much smaller clearance of the fluid film than normal operating condition. However, even with a small clearance under ELAP operation, the leakage rate slightly increases due to the high internal pressure of the coolant. This research will contribute to the development of robust mechanical seals for RCPs by accurately predicting the characteristics of mechanical seals, especially when the RCP is operating under ELAP.

Lubrication subjected to effects of electric and magnetic fields: recent research progress and a generalized MEMT-field Reynolds equation

Electric and magnetic fields have been used in various ways to enhance the performance of lubrication systems. The presence of these fields can significantly change the properties of lubricants. The rapid adoption of electric vehicles (EVs) has presented new lubrication-related challenges due to the presence of electric current. There is an urgent need for an in-depth study of lubrication systems subjected to such fields. This paper highlights recent research works on several key areas of lubrication involving electric or magnetic fields, which are:1) electric double layer in lubrication, 2) electrorheological fluids, 3) magnetorheological fluids, 4) ferrofluids, and 5) typical fluids used in the current EVs and typical surface failures of bearing components in EVs. Commonly used lubricants in each area are reviewed; lubrication mechanisms and related mathematical models are summarized; methods for and results from numerical analyses and experimental explorations are discussed; and common features of lubrications in different fields are explored. Based on the current research progress in these fields and the classic generalized Reynolds equation, a generalized mechanical-electro-magnetic-thermal-field (MEMT-field) Reynolds equation is proposed to describe the aforementioned lubrication scenarios and the effects of coupled mechanical, electric, magnetic, and thermal fields, which can be solved with a numerical iteration method.

Generalized Reynolds equation for microscale lubrication between eccentric circular cylinders based on kinetic theory

A microscale lubrication flow of a gas between eccentric circular cylinders is studied on the basis of kinetic theory. The dimensionless curvature, defined by the mean clearance divided by the radius of the inner cylinder, is small, and the rotation speed of the inner cylinder is also small. The Knudsen number, defined by the mean free path divided by the mean clearance, is arbitrary. The Boltzmann equation is studied analytically using the slowly varying approximation following the method proposed in the author's previous study (Doi,Phys. Rev. Fluids, vol. 7, 2022, 034201). A macroscopic lubrication equation, which is a microscale generalization of the Reynolds lubrication equation, is derived. To assess this, a direct numerical analysis of the Boltzmann equation in a bipolar coordinate system is conducted using the Bhatnagar–Gross–Krook–Welander kinetic equation. It is demonstrated that the solution of the derived lubrication equation approximates that of the Boltzmann equation over a wide range of the eccentricity and the whole range of the Knudsen number. It is also demonstrated that another lubrication equation derived by a formal application of the slowly varying approximation produces a non-negligible error of the order of the square root of the dimensionless curvature for large Knudsen numbers.

Study on the static characteristics of high-speed water-lubricated journal bearings considering thermal and cavitation effects

The influence of thermal and cavitation effects on the static characteristics of high-speed water-lubricated journal bearings are investigated in this study. Considering the transfer of momentum and energy on the gas-liquid interface, the generalized Reynolds equation and energy equation are established. Based on the theoretical model, the bearing static characteristics are calculated and analyzed. According to the calculation results, the cavitation effect reduces bearing maximum temperature and increases bearing load-carrying capacity. The simulation results of friction power loss are overestimated under isothermal conditions. The leakage flow rate was decreased as a result of the cavitation effect.

Vibration attenuation characteristics of squeeze film dampers in spiral bevel gear systems

A central drive bevel gear system of aero-engines extracts power from a high-pressure rotor. The high speed of the high-pressure rotor aggravates the gear system's vibration. This work adopts squeeze film dampers (SFD) as damping devices to suppress excessive vibration of the spiral bevel gear rotor system. This paper establishes the general Reynolds equation of SFD and calculates the nonlinear oil-film force of SFD in real-time based on the finite element method. Besides, the finite element method and Timoshenko beam element are employed to model the flexible gear shaft. Then, the dynamic model of the spiral bevel gear system supported by SFDs is established, including the nonlinear oil-film force of SFD and time-varying meshing stiffness of the gear system. The dynamic response of the spiral bevel gear system is analyzed theoretically and experimentally to study the vibration attenuation characteristics of SFD. The findings indicate that SFD can effectively suppress the vibration amplitude of the system at high speeds. The theoretical analysis and experimental amplitude are within an order of magnitude, and the changing trend is consistent.

Study of Dynamic Characteristics of Water-Lubricated Journal Bearings Using Thermohydrodynamic Cavitating Lubrication Model

Cavitation is prone to occur in high-speed water-lubricated journal bearings. However, the effect of cavitation on the dynamic characteristics of high-speed water-lubricated journal bearings has not been studied thoroughly. This study focuses on this topic. A thermohydrodynamic (THD) cavitating lubrication model was established for the high-speed water-lubricated journal bearings, including the generalized Reynolds equation, the energy equation, and the force balance equation for bubbles. An equilibrium distribution was introduced to describe the cavitation effect, and parameters were obtained by fitting the measured volume distribution of bubbles. The dynamic characteristics of the water-lubricated journal bearings were calculated, considering the influence of thermal and cavitation effects. An experimental setup was developed to validate the proposed model. The simulated results show that two secondary temperature peaks appear in the divergence zone of the bearing due to the cavitation effect. The cavitation effect should be considered when establishing the model for high-speed water-lubricated journal bearings, especially if it is under a large eccentricity ratio.

Theoretical and Experimental Comparisons for Rotordynamic Coefficients of a Multiscratched Tilting Pad Journal Bearing

Abstract This paper presents a thermo-elasto-hydrodynamics model (TEHD) based on a mass conservative generalized Reynolds equation, coupled with three-dimensional (3D) energy equation for the oil film, and a full 3D thermo-mechanical model for the pads. The model uses a regularization technique to consider film thickness discontinuities created by scratches. The numerical results are compared to the experimental results obtained from a rocker-backed, five tilting pads journal bearing (TPJB) with single and multiple scratches. The numerical model achieved good agreement, especially with respect to pad temperatures and dynamic coefficients. Discrepancies were observed at low speed but only on the stiffness in the load direction. The numerical model shows a significant drop in the pressure field at the location of the scratches, and a significant pressure rise between two consecutive scratches accompanied by a decrease in film thickness.

Theoretical and experimental investigation on local turbulence effect on mixed-lubrication journal bearing during speeding up

Due to the relatively low load-carrying capacity, low-viscosity lubricated bearings can easily operate in a mixed-lubrication regime. Furthermore, low lubricant viscosity may cause local turbulence in the area with relatively thick film. To analyze the influence of local turbulence on the transient characteristics of mixed-lubrication bearings, a simulation approach for unsteady mixed-lubrication bearings considering local turbulence effects is proposed. Transient journal center locations are solved via journal dynamic equations and numerical integration. The ratio of the film thickness to the roughness determines whether each node of the bearing is in the mixed-lubrication area or the pure hydrodynamic lubrication area. The liquid film in the mixed-lubrication area is analyzed by a transient average Reynolds equation to account for the surface roughness effect. A liquid film in the hydrodynamic lubrication area is analyzed by a transient generalized Reynolds equation to consider local turbulence. The transient average Reynolds equation and transient generalized Reynolds equation are derived in a form convenient for coupling with the governing equations of the journal center. The proposed model is experimentally validated. The influence of local turbulence on unsteady mixed-lubrication bearings is analyzed. The results show that local turbulence and asperity contact can coexist in low-viscosity lubricated bearings, even if the journal speed is relatively low. In the mixed-lubrication regime, local turbulence increases the minimum film thickness and decreases the friction coefficient as well as the journal speed at which the mixed-lubrication transitions to hydrodynamic lubrication.

A modified viscosity approach for shear thinning lubricants

Lubrication is essential to minimize wear and friction between contacting surfaces in relative motion. Oil based lubricants are often enhanced via polymer additives to minimize self-degradation due to the shear thinning effect. Therefore, an accurate estimate of the load carrying capacity of the thin lubricating film requires careful modeling of shear thinning. Available models such as the generalized Reynolds equation (GR) and the approximate shear distribution have drawbacks such as large computational time and poor accuracy, respectively. In this work, we present a new approach, i.e., the modified viscosity (MV) model, based on calculating the strain rate only in one point along the vertical direction. We investigate, for both MV and GR, the load, the maximum pressure, and the computational time for (i) sliding (non-cavitating) contacts, (ii) cavitating, and (iii) squeezing contacts. We observe that the computational time is reduced (i) considerably for non-cavitating sliding and rolling contacts and (ii) by several orders of magnitudes for cavitating and squeezing contacts. Furthermore, the accuracy of MV is comparable with the GR model within an appreciable range of bearing numbers. Finally, for each type of boundary motion, we have determined the optimal vertical location to calculate the shear strain rate for MV; while this optimal value is close to half the height of the contact for sliding configurations, for rolling dominated and squeezing contacts it is around one quarter (or three quarter) of their height. We finally provide an analysis to a priori estimate the optimal location of the strain rate.

An integrated finite volume framework for thermal elasto-hydrodynamic lubrication

Recent advances in computational methods and resources have fuelled numerical studies to explore novel techniques for solving thermal elasto-hydrodynamic lubrication (TEHL) problems. Commercial codes such as those that feature the Computational Fluid Dynamics (CFD) approach have gained much attention in the field of tribology due to the ability to tackle the limitations imposed by Reynolds equation assumptions and aptness in modelling complex geometries. However, the computational cost incurred by CFD software is significantly high due to the governing equations involved when solving the most general problem. The present contribution proposes an integrated finite-volume framework to solve TEHL problems to predict the lubrication performance of non-conformal contacts. Fluid flow effects are considered through the generalized Reynolds equation with the p−θ Elrod–Adams mass-conserving cavitation model to ensure the mass flow-rate conservation throughout the lubricated domain. The thermal behaviour of the lubricant film is predicted via the solution of the conservation of energy equation with appropriate boundary conditions for the fluid–solid interfaces. Novel partitioned iterative solution methods are integrated into the framework to handle the strong nonlinearities and improve the convergence of the coupled system of equations. Thermal and isothermal cases are presented and compared to results generated by existing CFD solvers and other experimental data to validate the proposed framework.

Monte Carlo Simulations of Shear‐Thinning Flow in Geological Fractures

AbstractThe hydraulic behavior of fractured rocks under shear‐thinning flow is a challenging topic of interest in several fields, related either to environmental remediation or to natural resources recovery. The compound effect of fluid rheology and medium heterogeneity strongly affects flow and transport in fractured geological formations. Here, a stochastic analysis is conducted via Monte Carlo simulations to investigate the flow of shear‐thinning fluids in fractures subjected to both natural and forced flow, considering different fracture dimensions, for a spatial correlation of the fracture that is an intrinsic parameter of the geological formation, independent of the fracture size. Considering the lubrication approximation, a generalized Reynolds equation for shear‐thinning fluids is solved using an ad hoc, finite volume‐based, numerical scheme. The influence of the rheology and aperture field heterogeneity on ensemble statistics of the velocity components and magnitude, as well as apparent fracture‐scale transmissivity, is quantified over 103 fracture realizations. The probability density functions (PDFs) and relative confidence intervals, obtained by averaging over the statistics, are analyzed to characterize the apparent transmissivity transition from Newtonian to shear‐thinning regime. The autocorrelation functions of velocity components are computed to understand the impact of rheology on spatial correlations the flow. Velocity components exhibit narrow PDFs with nearly exponential decay. More elevated pressure gradients emphasize the shear‐thinning behavior, inducing a more pronounced flow localization, under otherwise identical conditions. This translates at the scale of the fracture into a larger apparent transmissivity as compared to the same configuration with Newtonian rheology, by orders of magnitude.

Turbulence Effects in Tilting Pad Journal Bearings: A Review

This paper reviews the current knowledge on turbulence effects in tilting pad journal bearings. Turbulence is becoming increasingly important in the design of hydrodynamic bearings due to the trend to increase power density in turbomachines and consequently the operating speeds of the hydrodynamic bearings. Turbulence has a series of effects on the bearing performance which may be beneficial or detrimental, depending on the operating conditions. The main turbulence models are recalled and a historical overview on the evolution of numerical simulations of turbulent flow in tilting pad journal bearings is presented. The two broad simulation strategies used are the generalized Reynolds equation and computational fluid dynamics. The main experimental works are then reviewed, and a unified comparison of these works is provided. Novel results on the critical Reynolds number in a tilting pad journal bearing are reported. Much emphasis is given on the experimental evidence for laminar-to-turbulent transition. The evidence used in the literature is reviewed and its reliability is discussed. Lastly, some future research directions are suggested.

A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions

Inspired by the lubrication framework, in this paper we consider a micropolar fluid flow through a rough thin domain, whose thickness is considered as the small parameter ε while the roughness at the bottom is defined by a periodical function with period of order εℓ and amplitude εδ, with δ> ℓ >1. Assuming nonzero boundary conditions on the rough bottom and by means of a version of the unfolding method, we identify a critical case δ = 3/2ℓ − 1/2 and obtain three macroscopic models coupling the effects of the rough bottom and the nonzero boundary conditions. In every case we provide the corresponding micropolar Reynolds equation. We apply these results to carry out a numerical study of a model of squeeze-film bearing lubricated with a micropolar fluid. Our simulations reveal the impact of the roughness coupled with the nonzero boundary conditions on the performance of the bearing, and suggest that the introduction of a rough geometry may contribute to enhancing the mechanical properties of the device.

Active Fluid-Film Bearing With Deep Q-Network Agent-Based Control System

Abstract Despite the fact that the hydrodynamic lubrication is a self-controlled process, we designed control systems based on proportional integral (PI) controller, adaptive PI controller, and Deep Q network (DQN)-agent to minimize the friction torque in a conical fluid-film bearing. The bearing construction allows the shaft axial displacement and, as the result, the regulation of the bearing average clearance due to the controlled supply pressure. The main challenge is that the friction torque minimization may lead to the loss of load-carrying capacity and the contact in the shaft-bearing tribocouple. The other challenge is that random events may have an influence on the hydrodynamic lubrication parameters, therefore, changing the load-carrying capacity and the friction torque. So, the proposed control systems were designed and tested under the conditions of limited lateral shaft displacements and the action of a random external force. The tests were performed using simulation models of a controlled rotating machine in matlab software. The rotating machine simulation model includes modules of the rigid shaft, the coupling with linear axial reaction, and the conical bearing. The bearing module is based on the numerical solution of the generalized Reynolds equation and its nonlinear approximation with fully connected neural networks. The obtained results demonstrated that the application of an adaptive PI controller or a DQN agent allows decreasing friction torque in a bearing under the conditions of a random external force. The goal of a DQN agent is self-learning in contrast to an adaptive PI controller that needs to be tuned.

Thermal-Elastohydrodynamic Contacts Lubricated With Oil/Refrigerant Solutions: A New Cavitation Modeling Approach Based on Refrigerant Solubility

Abstract A first cavitation modeling with thermal effects for oil/refrigerant solutions lubricated elastohydrodynamic point contacts is reported in this work. The solubility of the oil/refrigerant system is introduced into the generalized Reynolds equation coupled with the elasticity equation and the energy conservation equation. The numerical results show a very good agreement with the published experimental results concerning film thickness prediction. Moreover, the present model describes the cavitation region on a physical basis. A discussion with other cavitation models from the literature is proposed. It puts into light the necessity of taking into account the solubility of the refrigerant into oil for such problems. Compared to pure oil, oil/refrigerant solutions can potentially reduce the amount of liquid oil for the next contact due to its higher cavitation intensity.

Reducing Rotor Vibrations in Active Conical Fluid Film Bearings with Controllable Gap

Despite the fact that the hydrodynamic lubrication is a self-controlled process, the rotor dynamics and energy efficiency in fluid film bearing are often the subject to be improved. We have designed control systems with adaptive PI and DQN-agent based controllers to minimize the rotor oscillations amplitude in a conical fluid film bearing. The design of the bearing allows its axial displacement and thus adjustment of its average clearance. The tests were performed using a simulation model in MATLAB software. The simulation model includes modules of a rigid shaft, a conical bearing, and a control system. The bearing module is based on numerical solution of the generalized Reynolds equation and its nonlinear approximation with fully connected neural networks. The results obtained demonstrate that both the adaptive PI controller and the DQN-based controller reduce the rotor vibrations even when imbalance in the system grows. However, the DQN-based approach provides some additional advantages in the controller designing process as well as in the system performance.

Analysis of underwater explosion shock on ship shaft stern bearing lubrication characteristics under different bearing working conditions

Warship always face the threat of underwater explosion shock, which may affect the ship stern tube bearing lubrication performance when the propeller shaft is bending. This paper builds the coupling model of shaft longitudinal vibration and bearing lubrication with consideration of shaft axial shock and misalignment. The performance of stern bearing under the influence of explosion shock was analyzed by solving the generalized Reynolds equation, shaft longitudinal vibration equation and explosion shock equation. The result shows that: the effect of underwater explosion shock on bearing lubrication performance is transient, and the effect will be more obvious when the bearing works at close quarter, larger misaligned angle, eccentricity and rotation speed. Moreover, the edge effect which caused by shaft bending will increase with the decrease of explosion distance and the increase of eccentricity.

An investigation into the thermo-elasto-hydrodynamic effect of notched mechanical seals

A 3D thermo-elasto-hydrodynamic model is developed to analyze the sealing performance of a notched mechanical seal applied in the reactor coolant pump. In the model, the generalized Reynolds equation, the energy equation coupled with notch heat balance equation, the heat conduction equations, and the deformation equations of the sealing rings are iteratively solved by the finite element method. The film pressure and temperature distribution are obtained, and the deformation of the sealing rings is revealed to study the mechanism of the notched mechanical seals. A parameterized study is conducted to analyze the sealing performance under different operating conditions. As a comparison, the sealing performance of non-notched seals is also studied. The results show that the hydrostatic effect is dominant in the load-carrying capacity of the fluid film due to the radial mechanical and thermal deformations. The notch can cool the fluid film and influence the thermal deformation of seal rings. The sealing performance is sensitive to the pressure difference, ambient temperature, and rotational speed. It is suggested to set the notches on the softer sealing rings to acquire the greater hydrodynamic effect. Compared with the non-notched, the notched end face holds a better lubrication performance, especially under lower rotational speed.

Theoretical analysis and experiment on gas film stiffness with slip flow in a spiral-grooved dry gas seal

Purpose This study aims to study the gas film stiffness of the spiral groove dry gas seal. Design/methodology/approach The present study represents the first attempt to calculate gas film stiffness in consideration of the slipping effect by using the new test technology for dry gas seals. First, a theoretical model of modified generalized Reynolds equation is derived with slipping effect of a micro gap for spiral groove gas seal. Second, the test technology examines micro-scale gas film vibration and stationary ring vibration to determine gas film stiffness by establishing a dynamic test system. Findings An optimum value of the spiral angle and groove depth for improved gas film stiffness is clearly seen: the spiral angle is 1.34 rad (76.8º) and the groove depth is 1 × 10–5 m. Moreover, it can be observed that optimal structural parameters can obtain higher gas film stiffness in the experiment. The average error between experiment and theory is less than 20%. Originality/value The present study represents the first attempt to calculate gas film stiffness in consideration of the slipping effect by using the new test technology for dry gas seals.

Theoretical Study of influence of MHD in an infinitely long rough Porous Journal Bearing

Analytical study of porous rough journal bearing with magneto hydrodynamic fluid in the presence of longitudinal and transverse roughness is investigated. Surface roughness effect is studied by using Christensen stochastic theory. The generalized Reynolds equations for both types of roughness patterns are derived. Reynolds boundary conditions are used to study the characteristics of infinitely long journal bearing. Cavitation angle is obtained by using bisection method. The expressions for pressure, load carrying capacity and attitude angle are obtained. The results are shown in the graph for longitudinal roughness pattern and concluded that, load carrying capacity increases for the increasing values of Hartmann number.