Sommerfeld Number Research Articles

Influence of Atmospheric Plasma Spraying Parameters on Porosity Formation in Coatings Manufactured from 45S5 Bioglass® powder

Atmospheric plasma spraying parameters were studied to understand their influence on the splat stacking and the porosity formation in coatings made from commercial Schott 45S5 Bioglass® powder onto AISI 304 L stainless steel substrate. The spraying parameters were established from numerical simulation carried out by the Jets&Poudres software. The plasma spraying was performed by using an Oerlikon Metco F4 MB gun with a 6-mm-internal-diameter nozzle. Three Ar/H2 mixtures were used as plasma-forming gas (95/5, 88/12 and 84/16 vol.%), and the current intensity was varied from 450 to 650 A and the spray distance from 60 to 100 mm. The cooling of the samples during spraying was controlled by air jet pressure. The x-ray diffraction and the scanning electron microscopy were used to determine phases and structural characteristics of coatings, respectively. The numerical simulation results for each set of spray parameters have shown suitable Sommerfeld number for proper splat stacking. However, Na2O and P2O5 evaporation occurred experimentally during spraying of 45S5 Bioglass® promoted the porosity formation into the coating structure. Low porosity was obtained for 45S5 Bioglass coatings manufactured under the appropriate relationship between plasma-forming gas mixture ratio, current intensity, spray distance and cooling air jet pressure to achieve thermal energy input enough to overcome the glass transition temperature and to avoid the evaporation of volatile oxides.

Investigation of cracked rotor analytically and numerically

The aim of this paper is analyzing the cracked Jeffcott with an offset disk rotor system. Mathematical modeling of a Jeffcott rotor with a crack while the depths of crack changes allows analyzing the dynamic behavior of the rotor. The effect of changing speeds has been investigated on the dynamic parameters of journal bearings (stiffness and damping) and then on Sommerfeld number. The relation of the dynamic response with the static response of rotor with cracks (dynamic load factor DLF) and with the ratio of depth crack to radius rotor has been found. The results have obtained analytically and numerically by using MATLAB and ANSYS, respectively. The results show when crack’s depths increase the DLF increases and reach to maximum when a/R = 0.8.

Effect of journal cylindricity error with saddle or drum distribution on the performance of hydrodynamic journal bearing systems

The present study investigates the effect of cylindricity error on the performance of hydrodynamic journal bearing systems. Two types of cylindricity errors of the journal, namely the drum shape distribution cylindricity error (DCE) and the saddle shape distribution cylindricity error (SCE), are considered. The Legendre–Fourier model is used to characterize the profile of each journal. Based on the nonlinear analysis method, the dynamic characteristics and stability of hydrodynamic journal bearing systems are analyzed. The results indicate that cylindricity error affects the system stability, and the effect is related to the type and level of the error. DCE used with a certain range of operating speed and load is not harmful and is even beneficial to the stability of hydrodynamic journal bearing systems; conversely, SCE decreases the system stability. When the Sommerfeld number is between 0.02 and 0.05, cylindricity errors have a minimal effect on the system stability. Additionally, the results indicate that the effect of cylindricity error on the bearing system is more significant than that of roundness error.

Scuffing analysis of roller-shoe mechanism after an aggressive test

The roller-shoe assembly is a known and efficient mechanism used for movement transmission in order to obtain mechanical energy. This mechanism is found in high pressure pumps and its role is to change the rotating motion in a translation one. During the running time, some wear types can be identified such as scuffing, corrosion, material removal. Scuffing is the main type of wear appearing in this mechanism. The aim of our paper is to analyse the behaviour of these two components (roller and shoe) over an aggravating test. The main defect is low clearance between components. The roller and shoe assembly is mounted in a diesel fuel pump for test. The test chosen for this experiment is Low Sommerfeld Number. It was stopped after 10% from time set. After visual analysis, it was observed the roller-shoe scuffing. The shoe was analysed and it was observed that the diamond like carbon (DLC) layer is non-conform. In the same pump, the affected components were replaced with new components to be convinced that the wear was caused by low clearance between them correlated to non-conform DLC layer. The test results confirm the reason of scuffing.

Lubrication effects on magmatic mush dynamics

Silicic magma bodies are formed by repeated injections of mobile magma and reside as a crystal-rich mush. Numerical studies of open-system events have revealed the complexity of mixing and rheological behavior. This is associated with the dilation of the crystal network and the possible occurrence of a lubricated regime. Lubrication forces are hydrodynamic interactions occurring when neighboring crystals have relative motion. The effect of such dissipative forces has not yet been explored in the case of magmatic mush. Here, we investigate the effects of lubrication on mush dynamics and on magma transport. First, we propose scaling relationships to assess the relative importance of the forces controlling the motion of one crystal within a mush by adding lubrication terms into the Basset-Boussinesq-Oseen equation that describes crystal motion in a viscous melt. We then investigate lubrication effects at the macroscopic scale with computational fluid dynamics with discrete element modeling (CFD-DEM) simulations that include these forces. We explore two cases: crystal mush sedimentation and the injection of a crystal-free magma inside a mush. We perform all simulations twice, with and without lubrication forces, and compare the results. At the grain scale, we show that three dimensionless numbers and the crystal content can describe the competition between viscous drag, buoyancy, and lubrication. Two of these numbers (Stokes and Froude numbers) have been previously employed in the context of dilute suspensions. The third is a new form of the Sommerfeld number that measures the importance of lubrication. At the macroscopic scale, simulation pairs (with and without lubrication forces) exhibit very similar behavior when in steady state. The duration of the transient regime preceding steady state, however, is increased when lubrication forces are included. Lubrication causes an apparent bulk strain hardening followed by softening at the initiation of the mush motion. Our results show that lubrication opposes dilation and the initiation of motion within the magmatic mush during this transient phase. Our results highlight the control that the crystal network exerts on magma transport and provide a novel way to evaluate when lubrication matters.

Analysis on the steady state performance of a multi pad externally adjustable fluid film bearing

PurposeThis study aims to investigate the performance characteristics of an externally adjustable bearing with multiple pads in steady state conditions. The proposed adjustable bearing geometry can effectively control the hydrodynamic operation in bearing clearances by adjusting the pads in radial and tilt directions. These pad adjustments have a significant role in improving the bearing characteristics such as load capacity, attitude angle, side leakage, friction variable and Sommerfeld number, which will be analysed in this paper.Design/methodology/approachThe adjustable bearing is designed with circumferentially spaced four bearing pads subjected to similar radial and tilt adjustments. Tilt angles are applied along the leading edges of bearing pads. A modified film thickness equation is used to incorporate the pad adjustments and accurately predict the variation in film profile. Finite difference approximation is adopted to solve the Reynolds equation and discretize the fluid film domain.FindingsFor negative radial and tilt adjustments, higher hydrodynamic pressures are generated in bearing clearances, which increases the bearing load capacity at different eccentricity ratios. From comparative analysis for different pad adjustments, superior bearing performance is observed for bearing pads under negative radial and negative tilt adjustments.Originality/valueThis research presents a detailed theoretical approach to analyse the performance capability of a four pad adjustable bearing geometry, which is not available in literatures. Improved bearing performances with negative pad adjustments can attract bearing designers to implement the proposed adjustability-bearing concept in rotating machineries.

Mechanical behaviour of fluid-lubricated faults

Fluids are pervasive in fault zones cutting the Earth's crust; however, the effect of fluid viscosity on fault mechanics is mainly conjectured by theoretical models. We present friction experiments performed on both dry and fluid-permeated silicate and carbonate bearing-rocks, at normal effective stresses up to 20 MPa, with a slip-rate ranging between 10 μm/s and 1 m/s. Four different fluid viscosities were tested. We show that both static and dynamic friction coefficients decrease with viscosity and that dynamic friction depends on the dimensionless Sommerfeld number (S) as predicted by the elastohydrodynamic-lubrication theory (EHD).Under favourable conditions (depending on the fluid viscosity (η), co-seismic slip-rate (V), fault geometry (L/H02) and earthquake nucleation depth (∝σeff)), EHD might be an effective weakening mechanism during natural and induced earthquakes. However, at seismic slip-rate, the slip weakening distance (Dc) increases markedly for a range of fluid viscosities expected in the Earth, potentially favouring slow-slip rather than rupture propagation for small to moderate earthquakes.

Necessary parameter setting of the finite‐difference method for determining selected rotor‐dynamics characteristics

AbstractThe aim of the study is to analyse the adequacy of the finite difference‐method (FDM) for determining the stability boundary of a rigid rotor mounted on short hydrodynamic plain journal bearings through the solution of the Reynolds partial differential equation (PDE). The Reynolds equation, obtained by coupling the equation of motion with the continuity equation, governs the pressure distribution in a lubricant film. The quasi stationary form of this equation requires normalization prior to solving, for the numerical error of the FDM to be minimized. Except for numerical benefits, the dimensionless form of the PDE is universally applicable to a bride spectrum of tribological problems, including hydrodynamic plain journal bearings. The present study first examines the adequacy of solver settings, including the computational grid, through comparison of the corresponding numerical results with an analytical solution obtained by means of the Short Bearing Theory (SBT). Two characteristic tribological quantities are selected as the basis for comparison: the modified Sommerfeld number and the bearing attitude angle. Further, dimensionless stiffness and damping tensors, valid for any short hydrodynamic plain journal bearing, are determined from the numerically obtained distribution of the dimensionless hydrodynamic pressure. These two dimensionless tensors allow for the subsequent estimation of the stability boundary, which marks the dimensionless speed at which a given rigid rotor becomes dynamically unstable. The results of the current study demonstrate sensitivity of the FDM results to the numerical step size. The corresponding numerical error is verified using the SBT and presented as function of the dimensionless (relative) journal eccentricity together with rotor start‐up curves. As a major finding, a dimensionless range of the step size is identified for the error to remain below 5%.

Steady-State Analysis of Journal Bearing Including Inertia Effect

Classical theory of hydrodynamic journal bearing ignores the effect of fluid inertia, due to its negligible contribution as compared to viscous forces. However, there are some situations like high density / low viscosity fluids, sudden change in cross-section area, super laminar flow etc., where inertia effect is dominant. Inertia effect is noticeable for moderate range of velocity where modified Reynolds number is in between 0 to 1.4. In the present work, steady-state analysis of hydrodynamic journal bearing including the inertia effect is performed. In order to account inertia effect, modified Reynolds equation has been derived using perturbation technique. Zeroth order and first order equations are descritised using finite difference method and are solved by Gauss-Seidel iterative scheme. Effect of inertia on various parameters i.e. load, Sommerfeld number, attitude angle, friction parameter, end flow, has been studied. It is observed that the effect of inertia is prominent on end flow and there is marginal effect on load and attitude angle.

Stability analysis of fluid film bearings under laminar and turbulent regimes

Stability of short journal bearings is investigated in bith the linear and nonlinear regimes under laminar and turbulent operating conditions with consideration of drag force effects. Exerted forces on fluid film bearings comprises of two different components. (1) Pressure force components in radial and tangential directions, and (2) Drag force (shear/friction force) components in radial and tangential directions. In most existing literature, related to rotor bearing system supported on journal bearing stability analysis, the effect of fluid film drag force has been neglected. Utilizing Ng-Pan-Elrod model and the modified Reynold’s equation, detailed calculation of the turbulent pressure distribution is carried out. Local stability of periodic solutions was studied using Hopf bifurcation theory. The shaft stiffness and inclusion of drag force and turbulent effects were found to play an important role in stability regions and bifurcation profiles of a flexible rotor bearing system.It was found that for shafts supported on short journal bearings, consideration of drag force effect could lead to expansion of un-stable region at high Sommerfeld numbers (low shaft static loads). Dynamic coefficients of short length journal bearing were analytically calculated under laminar and turbulent regimes based on Ng–Pan–Elrod model. Linear stability charts of a flexible rotor supported on laminar and turbulent journal bearings are found by calculating the threshold speed of instability associated to the start of instable oil whirl phenomenon. Local journal trajectories of the rotor-bearing system were found at different operating conditions solely based on the calculated dynamic coefficients in laminar and turbulent flow. Results show discrepancy between laminar and turbulent models, when shaft stiffness is great than 10, for the entire range of Sommerfeld number by increasing the Reynolds number in turbulent models.

Numerical study of a hydrodynamic journal bearing with herringbone grooves for oil leakage reduction

Hydrodynamic journal bearings and herringbone-grooved journal bearings were numerically analyzed using finite volume-based computational fluid dynamics software. Navier–Stokes equations were solved to describe pressure and velocity distributions in three dimensions. Values of the Sommerfeld number were calculated as a benchmark case for a hydrodynamic journal bearing. The predicted values of the Sommerfeld number are comparable to theoretical results from engineering tribology literature. For a hydrodynamic journal bearing, it is found that there is a positive correlation between the maximum pressure and the eccentricity ratio. For a hydrodynamic herringbone-grooved journal bearing, the peak pressures are found to occur at the tips of the grooves. It is further found that a herringbone-grooved design can reduce average axial velocity at the outlet by 84%, compared to a hydrodynamic journal bearing without eccentricity. This design could also address the oil leakage issue occurring often in hydrodynamic bearings. These findings may be useful for oil leakage reduction in a hydrodynamic herringbone-grooved journal bearing design.

Numerical and experimental investigation of texture shape and position in the macroscopic contact

Abstract In this work, the influence of operating conditions on the shape parameters of surface texture is investigated by means of both numerical and experimental investigations. The analysed texture consists of micro-dimples obtained through laser surface texturing on a pin-on-disc configuration. From the numerical point of view, particular attention is paid to the faithful representation of the 2D surface of the experimental set-up and to modelling cavitation phenomena through a mass conserving algorithm. As results, the dimple depth shows a higher relevance than diameter in determining the optimal texture shape (both in terms of friction reduction and load carrying capacity). Moreover, the dimple depth, corresponding to the minimal friction, is found to scale with the square root of the Sommerfeld number in agreement with the experimental results. Finally, it is found that a numerical approach with the present hydrodynamic model cannot account for friction reduction obtained experimentally with different orientation of the texture.

Influence of Rotor Suspension Anisotropy on Oil Film Instability

A crucial problem of turbomachinery is the oil film instability on increasing the angular speed, which is correlated with the asymmetry of the bearing stiffness matrix and resembles the hysteretic instability somehow. As a beneficial effect is exerted on the latter by the anisotropy of the support stiffness, some favorable effects have been recently found by the author also for the former, whence a systematic analysis has been undertaken. The instability thresholds may be detected by the usual conventional methods, but a detailed analysis may be carried out by closed-form procedures in the hypothesis of symmetry of the rotor-shaft-support system, which condition approaches the real working of turbomachines quite often. Altogether, the results point out an improvement of the rotor stability for low Sommerfeld numbers by softening and locking the support stiffness in the vertical and horizontal directions, respectively. Nonetheless, the partial support release on one plane implies lower instability thresholds for large Sommerfeld numbers, but this drawback may be obviated by a sort of “two-mode” stiffness management, with some vertical flexibility for heavy loads and full blocking for light loads. Otherwise, it is possible to combine the anisotropic supports with journal bearing types that offer favorable stability behavior in the range of large Sommerfeld numbers. Basing on approximate but realistic models, the present analysis elucidates the changes of the rotor-shaft unstable trend on varying the external stiffness of the supports and gives tools for a rapid calculation of the expected instability thresholds.

Mejoramiento de la adherencia en recubrimientos de Hidroxiapatita elaborados mediante proyección térmica oxiacetilénica, a partir de resultados de simulación numérica

Los resultados de la simulación numérica realizada con el software Jets et Poudres fueron utilizados para depositar recubrimientos de hidroxiapatita mediante proyección térmica oxiacetilénica, con el fin de aumentar su cohesión y adhesión sobre sustratos de Ti6Al4V. Se determinó el efecto de una llama neutra, una oxidante y una súper-oxidante, así como de la distancia de proyección (7, 9,5 y 12 cm) entre el sustrato y la antorcha, sobre el número adimensional de K-Sommerfeld (K). Dicho número adimensional es influenciado por la distancia de proyección y no por las llamas evaluadas. Para recubrimientos fabricados a una distancia de 12 cm el número de Sommerfeld varía entre 34 y 37, lo que les confiere una estructura más homogénea que la de aquellos depositados a 7 y 9,5 cm en los que existe un alto contenido de partículas parcialmente fundidas (K=0), o con salpicaduras (K>75), que reducen la adhesión de los recubrimientos.

A Confrontation of Lattice Boltzmann, Finite Difference and Taguchi Experimental Design Results for Optimizing Plasma Spraying Operating Conditions Toward Deposit Requirements

The aim of the present work is the confrontation of three numerical techniques results to optimize the operating conditions of thermal plasma spraying process. The Lattice Boltzmann method (LBM) is used to scrutinize dispersion effects of injection parameters on droplet impact characteristics when impacting substrate. The validation of the developed model shows good agreement with former findings. The results of spraying Zirconia particles give the values Kmin=88.2, Kmax=367.4, Kmean=273.8 and a standard deviation of 48.0 for the Sommerfeld number. The Taguchi experimental design study is conducted for five operating parameters of two levels. The ensuing retained factors combination give Kmean=258.9. To assess drawn conclusions, confirmation test was performed using the Jets&Poudres software. The results show that the prior way is to coat and particles of dp< 40.3 µm have evaporated, particles 40.3 = dp = 49 µm are fully molten and all particles of dp = 71.9 µm arrive fully solid.

Anomalous Nernst and Righi-Leduc Effects in Mn_{3}Sn: Berry Curvature and Entropy Flow.

We present a study of electric, thermal and thermoelectric response in noncollinear antiferromagnet Mn_{3}Sn, which hosts a large anomalous Hall effect (AHE). Berry curvature generates off-diagonal thermal (Righi-Leduc) and thermoelectric (Nernst) signals, which are detectable at room temperature and invertible with a small magnetic field. The thermal and electrical Hall conductivities respect the Wiedemann-Franz law, implying that the transverse currents induced by the Berry curvature are carried by Fermi surface quasiparticles. In contrast to conventional ferromagnets, the anomalous Lorenz number remains close to the Sommerfeld number over the whole temperature range of study, excluding any contribution by inelastic scattering and pointing to the Berry curvature as the unique source of AHE. The anomalous off-diagonal thermo-electric and Hall conductivities are strongly temperature dependent and their ratio is close to k_{B}/e.

On the kinematics and dynamics of crystal‐rich systems

AbstractPartially molten rocks, often called a mush, are examples of a hydrogranular mixture where the dynamics are controlled by both fluid and crystal‐crystal interactions. An obstacle to progress in understanding high‐temperature hydrogranular systems has been the lack of adequate levels of description of microphysical processes. Here we rationalize the hydrogranular kinematic and dynamic states by applying the concept of particle (crystal) force chains. We exemplify this with discrete‐element computational fluid dynamic simulations of the intrusion of a basaltic melt into an olivine‐basalt mush, where crystal‐scale force chains, crystal transport, and melt mixing are resolved. To describe the microscale kinematics of the system, we introduce the coordination number and the fabric tensors of particle contacts and forces. We quantify the changing contact and force fabric anisotropy, coaxiality, and the connectedness of the mush, under dynamic conditions. To describe the dynamics, particle and fluid characteristic response times are derived. These are used to define local and bulk Stokes numbers, and viscous and inertia numbers, which quantify the multiphase coupling under crystal‐rich conditions. We employ the Sommerfeld number, which describes the importance of crystal‐melt lubrication, with a viscous number to illustrate the dynamic regimes of crystal‐rich magmas. We show that the notion of mechanical “lock up” is not uniquely identified with a particular crystal volume fraction and that distinct mechanical behaviors can emerge simultaneously within a crystal‐rich system. We also posit that this framework describes magmatic fabrics and processes which “unlock” a crystal mush prior to eruption or mixing.

Influence of Hydrodynamic Journal Bearings With Multiple Slip Zones on Rotordynamic Behavior

This paper mainly reports stability investigations of rotors supported on fluid film journal bearings possessing multilocational slip-no-slip zones at the bush–film interface. The coupled solution of governing equations (Reynolds equation, energy equation, heat diffusion equation, lubricant rheological relation, and thermal boundary conditions) has been used to find pressure distributions in the lubricating film followed by evaluation of bearing coefficients. These coefficients have been used to determine stability limit speed (SLS) of the system and its robustness for both short (nearly inflexible) and long (flexible) rotors. Numerical simulations show that the pattern of pressure distribution with multiple slip-no-slip zones is similar to that obtained for multilobe bearings, resulting in substantial improvement of rotor–bearing stability irrespective of eccentricity ratio. A reduction in friction force (up to Sommerfeld number 1.8) and an increase in SLS and robustness compared to conventional bearings are observed when used with short rotors. Typically, up to six pairs of slip-no-slip zones improve SLS of the rotor–shaft system and robustness for short rotors, although more pairs deteriorate both. However, for long rotors, where dynamic rotor forces also act, these bearings provide marginal improvement in stability and robustness only for a small range of slip length.

Linear stability analysis of finite length journal bearings in laminar and turbulent regimes

Dynamic coefficients of a finite length journal bearing are numerically calculated under laminar and turbulent regimes based on Ng–Pan–Elrod and Constantinescu models. Linear stability charts of a flexible rotor supported on laminar and turbulent journal bearings are found by calculating the threshold speed of instability associated to the start of instable oil whirl phenomenon. Local journal trajectories of the rotor-bearing system were found at different operating conditions solely based on the calculated dynamic coefficients in laminar and turbulent flow. Results show no difference between laminar and turbulent models at low loading while significant change of the size of the stable region was observed by increasing the Reynolds number in turbulent models. Stable margins based on the laminar flow at relatively low Sommerfeld numbers [Formula: see text] were shown to fall inside the unstable region and hence rendering the laminar stability curves obsolete at high Reynolds numbers. Ng-Pan turbulent model was found to be generally more conservative and hence is recommended for rotor-bearing design.

Impact of Clearance Contact on the Performance of Hydrodynamic Journal Bearing System

The paper presents the analytical study of the impact of clearance contact on hydrodynamic journal bearing system’s static performance. The governing model of hydrodynamic lubrication i.e. the non-dimensional Reynolds equation is solved with using Garlerkin’s FEM approach along with appropriate boundary conditions. A cavitation zone, which is unknown priori is established using Reynolds boundary condition through iteratively. A clearance parameter is described and its influence on fluid-film pressure and static performance parameters of hydrodynamic journal bearing is studied for a commonly used range of sommerfeld number. The static performance parameters include maximum pressure, load carrying capacity, attitude angle, and coefficient of fluid-film friction etc. The results of the study are useful to the bearing designer from the view point of static performance parameters as the effect of sommerfeld number has been used for commonly used clearance parameter.