School of Industrial and Information Engineering
Master of Science in Mechanical Engineering
Internal Combustion Engines and Turbomachinery
DESIGN AND ANALYSIS OF A 3D PRINTED COOLED
PAD FOR TILTING PAD JOURNAL BEARINGS
Supervisor: Ing. Steven CHATTERTON
Graduation thesis of:
Luigi BAIO 874933
Alessandro GRANATA 876578
Acknowledgments
Ringraziamenti
Vorremmo ringraziare innanzitutto il Professor e nostro relatore Steven Chatterton insieme al Professor Paolo Pennacchi per i preziosi consigli, il supporto, la pazienza, il grande aiuto per la fase sperimentale e infine, l’incredibile opportunità di questo bellissimo progetto. Un grosso ringraziamento va anche a Guglielmo, nostro compagno tesista, il cui aiuto nella fase sperimentale è stato fondamentale. Vogliamo anche ringraziare Alexandra E. per il suo impegno nella divulgazione scientifica e per averci trasmesso la sua conoscenza.
Vogliamo ringraziare anche tutti i nostri Amici e Compagni di dipartimento con i quali abbiamo condiviso le fatiche, le gioie e i dolori dei progetti di tesi. In ordine sparso, Ila, Fra, Giulio e Ale, il Biondo e Frigio: la vostra compagnia ha alleviato e reso piacevoli molti momenti di svolgimento e di stesura di questa tesi. Il dipartimento rimarrà un luogo nel nostro cuore soprattutto grazie a voi.
Al termine di questi lunghi anni di università voglio ringraziare tutti coloro che mi hanno fatto vedere la bellezza della vita, condividendo insieme le fatiche degli studi (Enrico, Fabio, Francesco, Federico, Ciro, Abdullah, Nicola, Elena, Chantal), le risate davanti ad una buona birra e un piatto di nachos (Federico, Nicolò, Guglielmo, Riccardo) e le avventure ruolate al Libra (Nicolò, Fabrizio, Sergio, Luca, Andrea, Riccardo). Grazie a chi mi ha fatto innamorare di un percorso di vita bellissimo (Cesare, Eleonora), a chi l’ha condiviso e lo sta condividendo con me (Giacomo, Federica, Niccolò, Simone, Daniele, Isabel e tutti gli altri) e a chi per altre strade mi aiuta a tenere lo sguardo aperto (Agnese, Martina). Grazie al mio compagno d’avventura Alessandro, capace di sopportarmi e sempre pronto al sorriso. Grazie alla mia famiglia, che non ha mai smesso di essermi vicina e di appoggiarmi in tutte le mie scelte, soprattutto in quelle più difficili da prendere e da condividere. E infine grazie a tutti gli altri che non ho nominato, ma che hanno lasciato la loro impronta passando.
Senza tutte queste vite incrociate non sarei la persona che sono oggi.
Abraham Lincoln
Il mio primo ringraziamento va Gigi, per la sua infinta pazienza dimostrata soprattutto in questo ultimo periodo, per le sue incredibili capacità di ingegnere, e anche per quelle più artistiche, che col tempo ho avuto il piacere di apprezzare e imparare: hai avuto un grande coraggio a buttarti in questo progetto con me.
Vorrei ringraziare Ahmed, Fonta e Ale perché la distanza e il tempo passato non sembra aver cambiato mai le cose tra di noi: in quel giorno di quasi 6 anni fa, mentre aspettavamo una professoressa che non si sarebbe fatta vedere, mai avrei immaginato di farmi strada in questo lungo percorso con la forza di Quadri e Parry.
Un abbraccio ai Fantastici 3, Ste, Tommy e Marco: perché non finiscano mai le scorte di birra, le serate, le grigliate, le partite ai Coloni, i tuffi e i film con voi. Ma soprattutto le scorte di birra. Grazie a voi per esserci, sempre. Grazie di essere i migliori amici che mai
avrei immaginato di poter avere.
Un grosso grazie anche a tutti gli altri miei amici, al gruppo super unito del liceo e ai miei compagni di basket, con cui ho condiviso troppi bei momenti da poter essere elencati. Grazie ai miei genitori, per i sacrifici e il supporto, per avermi cresciuto così come sono: nell’eterna battaglia su chi sia il più fortunato tra di noi, col tempo mi sono accorto di dover ringraziarvi io per quanta fortuna ho avuto ad essere vostro figlio. Grazie a mio fratello Andrea, per essere sempre stato un modello irraggiungibile e un esempio inimitabile per me. Per avermi cresciuto, sopportato e supportato sempre in tutto. Grazie a Mari, per essere un concentrato di buone azioni da cui ho sempre cercato di prendere spunto, fin da quando ero un bambino delle elementari. E grazie anche al piccolo Gege, che ha l’innata capacità da ormai tre anni, di farmi tornare sempre il sorriso.
Grazie a te Greta infine, per le infinite cose che mi hai insegnato e per essere sempre pronta ad aiutarmi e spronarmi fin da quel capodanno del 2016. Grazie per i momenti insieme, per quelli belli e quelli difficili. Grazie per i posti che mi hai fatto vedere e quelli che mi farai vedere. Qualcuno mi ha insegnato che la vita è fatta di scelte: spero, insieme a te, di prendere sempre quelle giuste.
Summary
Acknowledgments ... 3 Summary ... 6 List of Figures ... 8 List of Tables ... 13 Abstract ... 14 Chapter 1 Introduction ... 15Chapter 2 State of the Art ... 17
2.1 Tilting pad journal bearings ... 17
2.2 Oil supply and pad cooling ... 20
2.3 Additive manufacturing ... 29
Chapter 3 Geometry and Manufacturing of the Pad ... 34
3.1 FEM preliminary static analysis ... 39
3.2 Manufactured pads ... 44
Chapter 4 Simulation Model ... 50
4.1 Main THD model ... 50
4.1.1 Dynamic coefficients ... 56
4.2 3D Thermal model ... 58
4.2.1 Thermal mixing model ... 62
4.3 Model of the cooled pad ... 69
4.3.1 Oil model ... 74
4.3.2 Convergence analysis ... 76
4.3.3 Details of Matlab-Fluent integration ... 76
Chapter 5 Simulation’s Results ... 78
5.1 Ansys model validation ... 78
5.2 Cases choice ... 80
5.3 Effect of cross section shape ... 90
5.4 Effect of the inlet temperature of the cooling circuit ... 91
5.5 Effect of cooling flow rate for water fluid ... 92
5.8 Effect of number of cooled pads ... 98
Chapter 6 Thermal FEM Static Analysis ... 101
Chapter 7 Test Rig and Experimental Tests ... 104
7.1 Test rig ... 104
7.2 Test procedure ... 109
7.3 Validation of fluid pressure ... 112
Chapter 8 Conclusion and Future Developments ... 118
Bibliography ... 120
List of Figures
Figure 2.1 - LBP bearing with two pivoted pad in the bottom half [2] ... 18
Figure 2.2 - LOP bearing with rocker-type pads [2] ... 18
Figure 2.3 - Damaged pad [2] ... 20
Figure 2.4 - Oil nozzles in TPJB [2] ... 21
Figure 2.5 - LEG and oil jacking pump pocket on a LOP bottom pad [2] ... 22
Figure 2.6 – Bearing with heat-pipe grooves [4] ... 22
Figure 2.7 - Cross section of a bearing shell with cooling channels [5] ... 23
Figure 2.8 - Profile of an axial-thrust bearing pad with cooling channels [5] ... 23
Figure 2.9 - JP4930290 (2007) [6] ... 24
Figure 2.10 – (a) JP09032848 (1997) [7], (b) JP58106616U (1983) [8] ... 24
Figure 2.11 - Sector shaped pad with embedded pipe [9] ... 25
Figure 2.12 – Numerical simulation results at different depths: top surface (a) and cooling circuit depth (b) [9] ... 26
Figure 2.13 - (a) 3-D solid model and (b) sketch for thrust bearing with novel design with characteristic dimensions [10] ... 26
Figure 2.14 – Numerical simulation results [10] ... 27
Figure 2.15 - Image of 2D finite element models for pipe array calculations [12] ... 29
Figure 2.16 - Laser powder bed fusion machine [14] ... 30
Figure 2.17 - AMAZE “millipipe” CAD and AM piece [12] ... 31
Figure 2.18 - Pure copper concept heat exchanger [13] ... 31
Figure 2.19 – (a) Pin-fin arrays[12], (b) Wavy fins plate [16] ... 32
Figure 2.20 – Flow Guiding Unit (FGU) [17] ... 32
Figure 3.1 – Investigated bearing section ... 34
Figure 3.2 – Pad with thermal probe DN100 ... 35
Figure 3.3 – 3D CAD model of “Circular” pad ... 36
Figure 3.4 – 3D CAD model of “6 Squares” pad ... 36
Figure 3.5 – Section drawing of “Circular” ... 37
Figure 3.6 – Section drawing of “6 Squares” ... 37
Figure 3.9 – Static analysis model geometry ... 40
Figure 3.10 – FEM mesh for the static analysis ... 40
Figure 3.11 - Pressure field applied in the FEM model ... 41
Figure 3.12 - Total deformation (Circular, max load 8 MPa) ... 42
Figure 3.13 - Equivalent Von-Mises stresses (Circular, max load 8 MPa)... 43
Figure 3.14... 44
Figure 3.15... 45
Figure 3.16 - View of the cut supports: This is the bottom surface during the deposition process ... 45
Figure 3.17 - Comparison with traditional-machined pad (AM one on the left, traditional one on the right) ... 45
Figure 3.18 – (a) AM pad surface, (b) Circular section hole view ... 46
Figure 3.19 – (a) Thermal probe hole, (b) 6 squares hole view ... 46
Figure 3.20 – 6 squares, detail of the rounded internal edge of the hole ... 46
Figure 3.21 – Finished pads overview ... 47
Figure 3.22 - Front (a) and back (b) views of finished pad ... 48
Figure 3.23 - Side view of the pads ... 48
Figure 3.24 - Microscope images of finished pads ... 49
Figure 4.1 -Generalized film between two surfaces [23] ... 51
Figure 4.2 - Geometry and coordinates for a single pad [18] ... 52
Figure 4.3 - PDE mesh grid ... 53
Figure 4.4 - PDE pressure results ... 54
Figure 4.5 - Pressure results on the nodal points ... 54
Figure 4.6 - Flowchart of the THD model used ... 55
Figure 4.7 - Mesh of the exploded model ... 58
Figure 4.8 - Bodies and meshes of the thermal model ... 59
Figure 4.9 – Matlab-Fluent model interactions ... 61
Figure 4.10 - Oil inlet groove [26] ... 62
Figure 4.13 – Mixing model mass balance ... 66
Figure 4.14 - Mixing model energy balance ... 67
Figure 4.15 - ANSYS Workbench Fluent module and software relations map... 69
Figure 4.16 – Meshes of Pad [6 Squares Model (6SQ)] (a), Coating [6SQ] (b), Circular fluid path (c), 6 squares fluid path (d), Temperature probe [6SQ] (e) ... 71
Figure 4.17 – Boundary conditions used in the Fluent model ... 73
Figure 4.18 - Density for VG46 oil ... 75
Figure 4.19 - Dynamic viscosity for Oil VG46 ... 75
Figure 4.20 - Residuals convergence for case S5 ... 76
Figure 5.1 -NoFluent - Solid ... 79
Figure 5.2 – TPJB scheme ... 80
Figure 5.3 - Section surface used for subfigure (e) of thermal results ... 83
Figure 5.4 – Case Solid, temperature results [°C] ... 84
Figure 5.5 – Case C1, temperature results [°C] ... 85
Figure 5.6 – Case S8, temperature results [°C] ... 86
Figure 5.7 - Pressure distribution among the pads [MPa] ... 87
Figure 5.8 - Polar plots for case S3 of pressure (a) and oil-film thickness (b) ... 87
Figure 5.9 - Oil-film thickness over pad #1 [µm] ... 88
Figure 5.10 - Temperature difference [°C], C1 – S1 ... 90
Figure 5.11 - Temperature difference [°C], S1 – S2 ... 91
Figure 5.12 - Temperature difference [°C], S5 – S8 ... 93
Figure 5.13 - Temperature [°C] VS inlet velocity, by using water as cooling fluid ... 94
Figure 5.14 - Flow direction difference between case S3 (a) and S4 (b) ... 95
Figure 5.15 - Temperature difference [°C], S3 – S4 ... 96
Figure 5.16 - Temperature difference [°C], S3 – S8 ... 97
Figure 5.17 - Cases S3 (a), S33(b) and S35(c): the highlighted pads are the ones cooled .. 98
Figure 5.18 - Temperature difference [°C], S3 – S33 ... 99
Figure 5.19 - Temperature difference [°C], S33 – S35 ... 100
Figure 6.1 - ANSYS Workbench simulation map ... 101
Figure 6.2 - Deformed and undeformed structure comparison in the simulation ... 102
Figure 7.1 – Test apparatus’s scheme ... 105
Figure 7.2 – Actuators and load cells’ scheme ... 105
Figure 7.3 - Picture of the test rig ... 106
Figure 7.4 - Pump, tank and heat exchanger of the cooling circuit ... 107
Figure 7.5 – Thermal and pressure probes of the cooling circuit ... 108
Figure 7.6 – Cooling circuit: connection between the probes and the pad ... 108
Figure 7.7 – Cooled pad installed with the others traditional pads inside the case of the TPJB ... 109
Figure 7.8 – picture of the shaft with the frontal bearing equipped with a cooled pad and two proximity probes ... 110
Figure 7.9 - Time history of pressure test, Circular case ... 112
Figure 7.10 - "6 Squares" case, experimental data ... 113
Figure 7.11 - "Circular" case, experimental data ... 114
Figure 7.12 - ΔT1cooling, time averaged values ... 115
Figure 7.13 - Shaft centre position, 6 Squares... 116
Figure 7.14 - Shaft centre position, Circular ... 117
Figure 8.1 -6 square pad printed in copper (Courtesy of BEAM-IT S.p.A.) ... 118
Figure 9.1 - Case S1, temperature results [°C] ... 122
Figure 9.2 - Case S2, temperature results [°C] ... 123
Figure 9.3 - Case S3, temperature results [°C] ... 124
Figure 9.4 - Case S4, temperature results [°C] ... 125
Figure 9.5 - Case S9, temperature results [°C] ... 126
Figure 9.6 – Case S33, temperature results [°C] ... 127
Figure 9.7 – Case S33, temperature results, section at half-pad [°C] ... 128
Figure 9.8 – Case S33, temperature results [°C] ... 129
Figure 9.9 - Case S35, temperature results [°C] ... 130
Figure 9.10 - Case S35, temperature results [°C] ... 131
Figure 9.11 - Case S35, temperature results [°C] ... 132
Figure 9.14 – Case S83, temperature results [°C] ... 135 Figure 9.15 - Case S5, temperature results [°C] ... 136 Figure 9.16 - Case S7, temperature results [°C] ... 137
List of Tables
Table 2.1 - Cross sections [11] ... 28
Table 2.2 - Mechanical properties of SS316L parts fabricated by AM compared to those of their traditionally processed counterparts [14] ... 33
Table 3.1 – Bearing data and settings ... 34
Table 3.2 – White metal properties [19] ... 41
Table 4.1 - 3D thermal model BCs ... 60
Table 4.2 - Number of elements (#E) and nodes (#N) of each body ... 72
Table 4.3 - Boundary conditions used in Fluent model ... 73
Table 4.4 - VG-46 oil properties as polynomial coefficients (T [K]) ... 74
Table 5.1 – Cases definition ... 81
Table 5.2 - Cases Results: temperature, pressure and thickness in the oil film ... 81
Abstract
Hydrodynamic journal bearings are essential power transmission elements which are carrying increasingly high loads because of the growing volume specific power in numerous machines. Industrial rotating machineries with high loads, such as centrifugal compressors, steam turbines, pumps and motors, employ journal bearings for supporting rotor. The load applied is balanced by the pressure field generated in the oil wedge between the two relatively moving surfaces, due to the hydrodynamic lubrication phenomenon. The working principle of these bearings is based on the convergent shape of the film between the two surfaces, that produces a not uniform pressure field.
The aim of the thesis is to develop an innovative way to reduce the oil-film temperature: the investigated solution consists in a cooling circuit located directly inside the pad. This feature is obtained with additive manufacturing, that nowadays is suitable to obtain a functional prototype. Taking advantage of this potential technology, the cooling circuit is characterised by a suitable path to cover uniformly the pad surface and by a multi-channel configuration that allows a high exchanging area/volume ratio.
A numerical model for investigating the influence of the different variables of the problem has been developed. The thermo-hydro-dynamic model of the bearing is coupled with a CFD model of the cooled pad that has been made to consider the specific geometry and the fluid-dynamic nature of the heat exchange. Finally, the numerical results have been validated by experiments, by means of a test-rig designed for common TPJB and specifically adapted for this work.
Chapter 1
Introduction
Fluid film bearings are mechanical components in which two relatively moving surfaces are divided by an oil film. The applied load is balanced by the pressure field generated in the oil wedge, due to the hydrodynamic lubrication phenomenon.
The working principle of these bearings is based on the convergent shape of the film between the two moving surfaces, that produces a not uniform pressure field.
During the years different types of bearings have been developed, to satisfy the specific needs.
Among the family of journal bearings in which the load acts in the radial direction, Tilting Pad Journal Bearing (TPJB) is the bearing type widely used in rotating machines, such as turbomachinery. In TPJB the pad can tilt freely allowing an adaption of the oil-film geometry respect to a sleeve bearing: while for sleeve bearing the problem of the shaft instability at high rotational speeds can’t be avoided, it is well known that the TPJB configuration has higher stability, function of stiffness and damping coefficients, according to lubrication theory (Lund, 1964) [1]. The behavior of hydrodynamic bearings is strongly influenced by the temperature distribution in the bearing, i.e. given by the heat generated by shear stresses in the oil-film. The high dependence of the oil viscosity on the temperature affects the pressure distribution in the oil-film. Eventually, the coating materials, such as Babbitt metals, usually have a limited maximum operating temperature. Therefore, prediction of temperature distribution in the bearing or at least of the maximum oil-film temperature is mandatory during the design phase and controlling this temperature becomes a critical task to achieve higher performances in terms of loads and speeds.
The aim of the thesis is to develop an innovative way to reduce the oil-film temperature: hydrodynamic bearings nowadays are cooled mainly by the injection of cold oil (that is the same oil used for the lubrication) in particular pockets or in the space between two consecutive pads through nozzles, that can have different shapes. Also, some patents have been developed during the years on new methodologies to cool the bearing and the pads, but never achieved an industrial stage. After a bibliographical investigation, the best solution results in a cooling pipe located directly inside the pad. This feature is hard, if not impossible, to be obtained with traditional manufacturing techniques, but the results achieved nowadays by the additive manufacturing technology allow to obtain a functional prototype. Taking advantage of this technology potential, the cooling circuit is characterised by a suitable path (in order to cover uniformly the pad surface) and a multi-channels configuration (a
into account, with the aim of obtaining a prototype ready for an industrial application. A structural analysis is needed to investigate the new design under the pressure load and the thermal load.
Then a parametrical numerical study has been made to investigate the influence of the different boundary conditions applicable to the problem: coolant inlet flow rate, inlet cooling temperature, coolant type and circuit geometry are all parameters that have an influence on the result. The thermo-hydro-dynamic model used for the bearing is based on partial differential equations (PDEs) solutions. In this model the fluid-dynamic component is approached with the Reynolds equation, that is the classical simplification of the Navier-Stokes equation applied to the TPJB problem. The TPJB is modelled as 11 objects (the shaft, 5 oil-films, 5 pads) linked by the convergence of their thermal boundary conditions. A coupling of the main code with a CFD model of the cooled pad is necessary to consider the specific geometry and the fluid-dynamic nature of the heat exchange.
The numerical model has been validated subsequentially by a series of experiments performed on a test-rig. The test-rig was designed for common TPJB and has been equipped with a cooling plant specifically designed for this work.
Chapter 2
State of the Art
This chapter will be focused on the technology of the TPJB, by investigating solutions aimed at cooling the pads. Then the additive manufacturing technology is examined as the suitable one for the pad manufacturing.
2.1 Tilting pad journal bearings
The first bearing classification criterion is the direction of loading: we talk about “journal bearing” if the load is acting perpendicularly to the shaft axis and “thrust bearing” if the load is acting in the direction of the shaft axis.
Journal bearings are classified themselves on the type of sliding surfaces: if the bearing has fixed sliding surfaces, then it is called “sleeve bearing”, otherwise the bearing can have several pivoted pads, which can tilt freely and it is defined as “tilting pad journal bearing” (TPJB).
TPJBs may be classified on many different characteristics: a good investigation on the different designs developed in the years was made by Pennacchi in 2016 in “Introduction of advanced technologies for steam turbine bearings” [2].
The classifications are based on the characteristic of the pads:
Position: they can be located all around the shaft or only in the lower half of the holding.
Relative dimensions: they can be equal or asymmetric: the most used disposition is the symmetric one, which has higher stability, but in specific cases are used also asymmetrical pads such as in big turbine used in the nuclear energy field.
Load position: the difference for symmetrical pads is between Load Between Pads (LBP, Figure 2.1) and Load On Pad (LOP, Figure 2.2) configurations for load acting vertically downwards (represented in the figures by the blue arrows). The LBP configuration allow the support over a larger area, increasing the damping as Lai et al. explained in 2014 [3] with a theoretical study upon the load direction influence. From their study it results that the LOP loading capacity is approximately 24% lower than the LBP one.
Pivot: it can be ideal (Rocker type pad) or obtained machining the surface or with a harder insert (pivoted type pad). In the rocker pad pivot the contact acts on a straight line between the holding and the convex surface of the pad. The pivot function is to allow the free tilting: this involves that all the load, theoretically, is concentrated on it.
Figure 2.1 - LBP bearing with two pivoted pad in the bottom half [2]
Figure 2.2 - LOP bearing with rocker-type pads [2]
Each component of the bearing is made with a specific material as a function of their needed characteristics.
The holding and the pad’s base are made generally with structural steel, i.e. S355J0, which have high strength and suitable machining and welding properties. Sometimes the pads have
insert it is composed of a low alloy martensitic steel, i.e. 100Cr6, that, due to the concentrated load, it is chosen for is high hardness, his wear resistance and his suitability to cold deformation machining.
Because of the possible contact with the shaft, pads are coated with a material that must have a low friction coefficient and a hardness lower than the shaft’s one.
This is due to two main reasons:
At start up and in case of contact the pad acts as sacrificial element respect to the shaft, using a softer material the damaging of the shaft is avoided and the maintenance costs strongly decrease.
At the start up if there isn’t an oil film between rotor and pad the low friction coefficient reduces strongly the breakaway torque necessary.
The standard material used for coating is white metal, called also Babbitt as eponym from his inventor Isaac Babbitt, that create the first of this alloy category in 1839. Normally are used the tin-based ones. The main distinction is based on the lead percentage, which has serious environmental consequences.
The characteristics that made these alloys a standard are excellent anti-seizure and corrosion resistance, high conformability (the ability of bearing to conform to the shaft alignment variation), embeddability (the ability to permit foreign particles to become embedded in the coating, preventing damages such as scratching) and galling resistance. The bad mechanical properties of Babbitt are low resistance to fatigue and creep and low melting temperature. The wanted low hardness implies a Young modulus that’s about a fourth of the steel’s one. For those reasons the average load allowed is usually lower than 2-3 MPa, to avoid damages such as wiping (Figure 2.3).
Figure 2.3 - Damaged pad [2]
Nowadays, the power-generation applications prefer bearing coatings made with different types of white metals, namely ECKA TEGO V738 for heavy duty and more recently ECKA Tegostar, which have become an industrial standard de facto, because of their better performances in terms of resistance to creep. Tegostar has also the remarkable advantage that the alloy is without lead, cadmium, nickel, and arsenic.
Alternatives to metallic coatings are polymeric materials, such as polytetrafluoroethylene (PTFE) and polyether ether ketone (PEEK), and ceramic materials. These alternatives to metallic coatings are characterized by a low friction coefficient with steel and an “environment-friendly” composition. The coating of bearing components is made in special heated centrifuges.
The drawbacks of TPJBs are the hot oil carry over, the higher costs, a difficult determination of clearances and the risk of flutter of unloaded pads such as the higher ones. Anyway, the high overcome of advantages on the disadvantages makes the TPJBs the most common choice in the turbines field.
2.2 Oil supply and pad cooling
The viscosity of the lubricant is a key parameter for the performances of the bearings (load capacity, film thickness, friction coefficient, power loss and dynamic coefficients) as much as it is the oil-feed of the bearing.
The simplest lubrication method is the so called “flooded lubrication”, where the oil is directly injected in the vanes between the components of the bearing and the sealings. The
efficiency due to the turbulent mixing in the pockets (for the sleeve bearings) or in the gaps between the pads (for tilting pad bearings).
Th alternative solution suitable for tilting pad bearings is the so called “direct lubrication”, characterized by a lower operating temperature reachable, up to 25°C less than the flooded lubrication, and a significant decrease of oil flow needed for equal temperature effect. It can be obtained with injectors located between the pads such as nozzles and spray bars (Figure 2.4) or with a machined leading edge groove (LEG) on each pad, with oil fed by an orifice (Figure 2.5).
Often big machines are equipped with oil jacking pumps, whose function is to provide hydrostatic support to the shaft when the sliding velocity in the bearing is not sufficient to produce full film hydrodynamic lubrication (i.e. at the start-up). In these cases, the oil is fed at high pressure to special pockets machined in the journal bearings, by creating a complete film able to support the shaft, even at zero sliding speed. Figure 2.5 shows the pocket in a LOP TPJB.
In TPJBs, the pockets are usually in the radial correspondence of the pivot, due to basic considerations of static equilibrium.
Figure 2.5 - LEG and oil jacking pump pocket on a LOP bottom pad [2]
In order to obtain an internal cooling of the pads innovative designs were developed in the years.
In 1999 Chen et al. [4] developed an isothermal journal bearing by employing heat-pipe cooling technology for improved thermal and tribological performances. They designed and constructed a bearing with several heat-pipe grooves which used methanol as working fluid (Figure 2.6). The experimental results indicated that the heat pipe uniformly distributed the frictional heat along the entire circumference of the bearing. As a result, the bearing became a nearly isothermal element, moreover it could work at much higher thermal loads.
In 2000 Becker registered an EPO patent (EP1002965) [5] for cooling of both journal and thrust bearings. In Figure 2.7 it is shown a representation of a journal bearing in which an open cooling circuit is implemented: parallel to the sliding surface (11) in the bearing shell (3) two type of channels are realized. One in the axial direction (31) and one in circumferential (33). By means of a circulating flow inside the channels (e.g. lubricant oil), the sliding surface and the bearing body can be cooled.
Figure 2.7 - Cross section of a bearing shell with cooling channels [5]
In Figure 2.8 it is represented the scheme of a thrust bearing cooling system: a pathway (7) under the pad helps to form the fluid flow that enters in the channels in direction 39. There are two different directions for flow paths: one tangential to the radial direction (41) and one in the radial direction (35). The two sets of channels lie on different planes. The cooling channels are formed by cylindrical drilling.
In Figure 2.9 it is represented the scheme of Japan Patent n. JP4930290 (2007). Two circular holes form the cooling passages (7) of the tilting pad: they penetrate the pad transversely, from the outer diameter and the leading edge side (10d) to the inner diameter and the trailing edge side (10e). The lubricant enters in the bearing through different oil supply holes (4) positioned in the shell (9). Then the oil acts both as a film sustaining the load and as a cooler through the pad channels (7). Its motion is induced naturally by the rotation of the shaft (11).
Figure 2.9 - JP4930290 (2007) [6]
Some other solutions are described in Figure 2.10. Also in these configurations the pad cooling is permitted by the same lubricant of the film.
(a) (b)
cooling. They analyzed the thermal effect on a thrust bearing pad: they incorporate a suitable cooling circuit, that follows an M-shaped path (Figure 2.11). The water circulation is allowed by an external pump and the channel lies just beneath the Babbitt coating of the pad.
Figure 2.11 - Sector shaped pad with embedded pipe [9]
As a first step to study the effect of water cooling arrangement, they performed an analysis on the heat transfer for the bearing pads without the cooling. This was followed by introducing water pipe network in the bearing solid and the estimation of heat carried away by the water: their assumption was that all the heat flow from the wall entering the fluid is equal to the heat carried away by it. The analysis of heat transfer to water-pipe network involved proper estimation of heat transfer parameters such as Reynolds number (they performed both laminar and turbulent analysis) and Nusselt number (i.e. the ratio of convective to conductive heat transfer across a boundary): this was achieved by dividing the pipe length into several segments. The methodology begins with the determination of the temperature profiles along the depth of the pad. For this purpose, it was essential to determine the temperature of the top surface of the pad. Once obtained this temperature, three dimensional conduction equation was solved and appropriate heat losses (runner, end flow, side leakage, vertical and bottom sides of the pad, groove) were obtained. Also, the variation of viscosity was taken into account using a standard ASTM curve.
After their numerical simulation, they obtained that the temperature profile of the pad is strongly influenced by the cooling circuit and that the overall temperature reduced significantly as compared to traditional cooling systems. This aspect was much stronger at the depth of the cooling circuit. Therefore, along with the thickness of the pad, the thermal
effect varied from top surface to below the pipeline: The results, for two different sections obtained with a flow velocity of 1 𝑚/𝑠, are shown in Figure 2.12.
(a) (b)
Figure 2.12 – Numerical simulation results at different depths: top surface (a) and cooling circuit depth (b) [9]
In 2016 [10] they improved the investigation with a novel cooling arrangement (Figure 2.13) to better study the influence on heat transfer in thrust bearing. The new configuration for the pipeline is formed by a bifurcation in the most heated zone, near the trailing edge.
Figure 2.13 - (a) 3-D solid model and (b) sketch for thrust bearing with novel design with characteristic dimensions [10]
The simulations were performed with different flow velocity of coolant, ranging from 0.5m/s to 2.0m/s. The same model and the same procedure of [9] were used: the pipeline is divided
method (FDM) was used to solve the partial differential and Gauss-Seidel iterative method for iteration purpose.
They observed a significant amount of heat is removed from the bearing with the increase of flow velocity of coolant in the embedded duct within the pad, thus increasing the maximum pressure in the film wedge and therefore the load carrying capacity of the bearing (Figure 2.14).
Figure 2.14 – Numerical simulation results [10]
In the thesis work of Navatta (2016) [11] many different cooling circuit designs have been investigated by simulations. The pad heating has been modeled by considering a fixed heat flux applied on the active (upper) surface of the pad for all the simulations, with a heat flux density given by a triangular distribution. With this simplification, the interaction between the cooled pad and the lubricating fluid in the active surface has been neglected. The different profiles investigated at same conditions of pathlength, inlet flow temperature and velocity are shown in Table 2.1.
Table 2.1 - Cross sections [11]
The results show how sections with multiple holes (i.e. G2, G3) have better performances due to a better distribution and a larger heat exchange surface. The same consideration is confirmed by Hancock et al. (2018) [12] (Figure 2.15): the finite element models of an array of between 1 and 6 cooling pipes take as constant conditions the uniform heat flux applied,
results of this analysis show a reduction in peak wall temperature and stress.
Figure 2.15 - Image of 2D finite element models for pipe array calculations [12]
2.3 Additive manufacturing
The development of an internally cooled pad requires the use of non-conventional technologies: in particular, the most suitable one is the additive manufacturing (AM). AM is the official industry standard name of all applications of the technology defined as the process of joining materials to make objects from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing methodologies. AM first emerged as a novel technology which application was limited to design and model functional prototypes. With manufactured components spanning the different commercial, industrial, medical and thermal domains, AM is revolutionizing fabrication concepts and capabilities and creating an immense range of applications that were once bound by conventional and traditional manufacturing techniques, according to Singer et al. (2017) [13]. The state of the art of this continuously evolving technology is well described by DebRoy et al. (2018) [14] in the paper “Additive Manufacturing of Metallic Components – Process, Structure and Properties”. The AM processes are characterized by their production times, maximum size of the component that can be fabricated, ability to produce intricate parts and the product qualities such as defects and dimensional accuracy. The production time of the powder-based AM processes is high due to the limitations of powder feeding rate, scanning speed and low layer thickness. In contrast, filler wires allow relatively higher mass flow (deposition) rate in wire-based processes. As a result, the powder-wire-based processes are considered suitable for relatively smaller parts and the wire-based processes are considered suitable for fabrication of large-size components, typically heavier than 10 kg.
The process suitable for producing an internal cooled pad is called laser powder bed fusion: the dedicated software begins with a solid or surface CAD model, orienting it within a build
and the specific machine configuration. The part forms by spreading thin layers of powder and fusing pass-by-pass, layer upon layer, within an inert chamber, incrementally lowering the Z-axis after each layer. Fusion occurs by a raster motion of the laser heat source using galvanometer-driven mirrors, resulting in melting and solidification of overlapping melt tracks (Figure 2.16).
Figure 2.16 - Laser powder bed fusion machine [14]
Commonly used alloys in AM are aluminum based, maraging steel, stainless steel, titanium based, cobalt chrome, nickel super alloys and precious metals. In particular, in the “energy, oil and gas” field the most use is the stainless steel. For this project, the focus was in particular on the characteristics of the SS316L: the mechanical characteristics of the final part are strictly dependent from the settings of the laser beam and the feedstock, as can be seen in Table 2.2. With a properly set machine performances can be achieved in terms of strength comparable to the ones of the traditional casted pieces (a study concerning the influence of orientation during the printing of SS316 was performed by Tolosa et al. (2010) [15]).
Another material interesting for thermal applications is copper: until few years ago, AM of copper and its alloys was considered as one of the major limitations of metal AM, due to his high thermal conductivity and ductility. In our study copper alloys are not suitable for their poor mechanical properties, but the actual development in heat exchange field is well presented by Singer et al. (2017) [13] and Hancock et al. (2018) [12] (Figure 2.17 and Figure 2.18).
Figure 2.17 - AMAZE “millipipe” CAD and AM piece [12]
Figure 2.18 - Pure copper concept heat exchanger [13]
Other heat exchanger geometries allowable by AM are pin-fin arrays (Figure 2.18 (a), [12]), plate-fin with wavy fins (Figure 2.18 (b), [16]) and fluid flow guiding elements based on freeform surfaces (Figure 2.19, [17]).
(a) (b) Figure 2.19 – (a) Pin-fin arrays[12], (b) Wavy fins plate [16]
Table 2.2 - Mechanical properties of SS316L parts fabricated by AM compared to those of their traditionally processed counterparts [14]
Chapter 3
Geometry and Manufacturing of the Pad
The bearing data are described in Table 3.1 whereas the cross-section of the bearing is showed in Figure 3.1 and the detail of the pad in Figure 3.2. The LOP configuration was chosen to allow a reasonable test with only one 3D-printed pad placed in the direction of the load. The oil-feed nozzles located between each pad act also as blockage to keep them in the chosen configuration.
Bearing data and settings
Shaft radius 49.94 mm
Housing internal radius 66.00 mm
Pads Number 5
Pads Configuration LOP
Pivot Type Rocker
Nominal Load 5 kN
Rotational Speed 3000 rpm
Table 3.1 – Bearing data and settings
The first step of the thesis was the development of the 3D CAD (Computer Aided Design) files with the Autodesk software Inventor®.
In the thesis the following 3 configurations will be considered:
Solid Pad (Solid Case): is the reference case, without the cooling circuit. It’s similar to the real pad obtained with standard milling manufacturing technology (Figure 3.2). Circular section circuit (Circular Case): the cooling circuit has a circular
cross-sectional area, such as a common pipe (Figure 3.3 and Figure 3.5).
Squared section circuit (6 Squares Case): the cross-sectional area of the cooling circuit is in the shape of six squares disposed in a 2x3 grid, taken from [11] (Figure 3.4 and Figure 3.6).
The pad is cooled by an M-shaped path circuit, in order to cover uniformly the pad surface taking into account the presence of the thermal probe and the connection holes.
Figure 3.3 – 3D CAD model of “Circular” pad
Figure 3.4 – 3D CAD model of “6 Squares” pad
As evidenced in [12], the multi-channels design allows a better cooling due to the increase of the exchanging surface.
The path line was designed to cover homogeneously all the space available and is kept at a midlevel of the semi-finished steel part (the pad without coating). The two cases have the same connection holes to the external cooling circuit, with standard G 1/8 threaded holes (for oil-feed design simplicity), resulting into a cross-sectional area of 50 mm2.
Figure 3.5 – Section drawing of “Circular”
Figure 3.6 – Section drawing of “6 Squares”
During the design phase the common rules for AM design have been also considered, such as avoiding printing cantilevered parts with angles respect to the horizontal lower than 45° degrees, interspaces between parts lower than 0.3 mm or “vertical” holes with a radius higher than 10mm. Alternative designs with rhomboidal-section chambers (Figure 3.7) have been developed in order to avoid the presence of any supports, but they involve a detrimental effect on the fluid dynamic point-of-view. Moreover, in the CAD models two vents were implemented in proximity of the path curves, considering the potential residual powder trapped in the cooling circuit, to allow a better inspection for removing it.
Figure 3.7 – “6 Squares” pad. Version optimized for AM
In order to understand which design optimal and which printing problems may arise BEAM-IT S.p.A, a company specialized in additive manufacturing, did a preliminary evaluation about the manufacturing design of the pad (Figure 3.8). In order to avoid possible deformation of the path during the manufacturing, the company developed the printing process with an oblique positioning of the pad. The problem is furthermore prevented by the use of internal supports in the channels. The figure also shows the base support (in colour blue) that will have to be removed with a subsequent process.
Figure 3.8 – CAD “6squares” model with AM support preview (Courtesy of BEAM-IT S.p.A.)
3.1 FEM preliminary static analysis
A preliminary investigation on the mechanical deformation of the designed geometry was made with ANSYS®. The model of the coated pad is clamped to a holding with the same radius of the real one: due to the small difference in the curvature of the two components the software identify a constant contact region with a no-negligible width (Figure 3.9 and Figure 3.10).
Figure 3.9 – Static analysis model geometry
Figure 3.10 – FEM mesh for the static analysis
Two different materials have been considered in the analysis: Structural steel for the base part of the pad
Tin-based white metal for the antifriction layer
The load was given by a typical pressure distribution: the pressure field above the pad was modeled with a centered paraboloid with the a mean value of 4 𝑀𝑃𝑎 and a maximum value of 8 𝑀𝑃𝑎 (Figure 3.11), given as “.csv” file. This approximation fits with a good accuracy the models found in bibliography [18].
White metal Grade-1 Tin-based
Density 7272 kg/m³
Thermal Expansion Coefficient 2.1e-05 1/K
Young’s Modulus 53 GPa
Poisson’s Ratio 0.35
Tensile Yield Strength 30 MPa
Compressive Ultimate Strength 89 MPa
Specific Heat 230 J/ (kg K)
Table 3.2 – White metal properties [19]
Figure 3.11 - Pressure field applied in the FEM model
The deformation due to the pressure load is small (below 10 𝜇𝑚) and comparable with the oil film thickness. The equivalent Von-Mises stresses predicted are acceptable (well below the Yield stress limit).
Figure 3.12 and Figure 3.13 show the results for a paraboloidal pressure. Moreover, it is shown the fictitious maximum stress due to the clamp constrain (Figure 3.13 (c)).
(a)
(b)
(a)
(b)
3.2 Manufactured pads
The CAD files delivered to BEAM-IT S.p.A. take into account an oversize of 0.5 mm on each surface excepting the top one, to allow the clamping of the pad during the coating procedure and the following finishing of the piece.
The following figures (Figure 3.14 to Figure 3.20) are pictures of the half-processed pieces produced by BEAM-IT S.p.A. . In the pictures the AM process signatures can be recognized as the deposition layers and the base supports cut off (Figure 3.16). It’s also presented a comparison with traditional manufactured pad (Figure 3.17).
Figure 3.15
Figure 3.16 - View of the cut supports: This is the bottom surface during the deposition process
With the use of a digital microscope the following pictures of details were taken (from Figure 3.18 to Figure 3.20).
(a) (b)
Figure 3.18 – (a) AM pad surface, (b) Circular section hole view
(a) (b)
Figure 3.19 – (a) Thermal probe hole, (b) 6 squares hole view
The classic procedure for the application of Babbitt metal involves the use of a centrifuge in which the liquid coating metal is “spread” over a solid tube. The tube is then lathed and cut to obtain the final pads.
The coating and the refinement of the printed half-processed pieces were executed by EUROBEARINGS S.R.L., which adapted their machines with specific tools for single-pads managing and coating.
In particular they had to face the problem of the coating of the SS316 steel: in fact, with the traditional procedure the Babbitt metal bond with a stainless steel isn’t strong enough to guarantee the resistance of the pad. The problem was solved by the company with a surface treatment done with chemical products to deeply clean and enhance the bounding between the two materials.
Then the white metal was applied all over the pad and removed with the removal of the stock allowance. In the following images (Figure 3.21 to Figure 3.24) the finished pieces are shown: the details (Figure 3.24) of threaded holes (a,b), surface processing (c) and defects due to the difficulties into the Babbitt bounding (d) are taken with a digital microscope.
(a) (b)
Figure 3.22 - Front (a) and back (b) views of finished pad
Figure 3.23 - Side view of the pads
The defects visible in Figure 3.23 are due to the difficulties faced by the company in the re-centering of the pieces after the coating: a future development of the manufacturing design could involve the additive manufacturing of a cylinder that includes the five pads. The object could be then processed in the traditional way, with the coating in the centrifuge and the sub sequential cutting into the 5 pads.
(a) (b)
(c) (d)
Chapter 4
Simulation Model
In this chapter the model used for the numerical simulation will be described. The main model for TPJB developed with Matlab is explained, with a focus on the 3D thermal model. Also the topic of the thermal mixing model has been explored, exposing different solutions adopted in literature. Then is explained the use of ANSYS Fluent for the cooled pad and the coupling between the two softwares.
4.1 Main THD model
The behavior of hydrodynamic bearings is strongly influenced by the temperature distribution in the bearing, i.e. given by the heat generated by shear stresses in the oil-film. The high dependence of the oil viscosity on the temperature affects the pressure distribution in the oil-film. Eventually, the coating materials, such as Babbitt metals, usually have a limited maximum operating temperature. Therefore, prediction of temperature distribution in the bearing or at least of the maximum oil-film temperature is mandatory during the design phase and they have been widely investigated in the literature. For example, a thermo-hydrodynamic modeling (THD) with temperature-viscosity variation through the thickness of the lubricating film was developed by Rohde and Oh (1975) [20], where both conduction and convection through lubricating film have been also provided.
In THD models, the pressure distribution in the oil-film can be obtained by integrating the generalized Reynolds equation, presented by Reynolds (1886) [21] that can be written in cylindrical coordinates under steady state and incompressible fluid hypothesises according to Vohr (1981) [22]: 1 𝑟 𝜕 𝜕𝜗 ℎ 𝜇 𝜕𝑝 𝜕𝜗 + 𝜕 𝜕𝑟 𝑟ℎ 𝜇 𝜕𝑝 𝜕𝑟 = 6𝜔𝑟 𝜕ℎ 𝜕𝜗 (4.1)
flow, invariant pressure in the film thickness direction, a negligible shaft curvature effect and negligible fluid inertia following Hori [23], it becomes:
𝜕 𝜕𝑥 𝜌ℎ 𝜇 𝜕𝑝 𝜕𝑥 + 𝜕 𝜕𝑧 𝜌ℎ 𝜇 𝜕𝑝 𝜕𝑧 = 6 (𝑈 − 𝑈 ) 𝜕 𝜕𝑥(𝜌ℎ) + 𝜌ℎ 𝜕 𝜕𝑥(𝑈 + 𝑈 ) + 2𝜌(𝑉 − 𝑉 ) (4.2)
where h is the oil-film thickness, p is the pressure in the fluid film, 𝜇 is the lubricant dynamic viscosity, z is the axial direction, x is the tangential direction, and 𝜌 is the density of the oil. The velocity vector components of the shaft and the pads are described by 𝑈 , 𝑉 , and 𝑈 , 𝑉 , respectively, represented in Figure 4.1.
Figure 4.1 -Generalized film between two surfaces [23]
In particular the viscosity 𝜇 and the density 𝜌 of the lubricating fluid have been assumed to be a function only of the temperature and they can be expressed as:
𝜇 = 𝜇( ° )
𝑇 𝑇( ° )
(4.3)
𝜌 = 𝜌( ° ) 1 + 𝛼 𝑇( ° )− 𝑇 (4.4)
the generalized Reynolds equation and the energy equation. If a steady state, laminar flow, incompressible and Newtonian fluid is assumed, the energy equation can be expressed as:
𝜌𝑐 𝑢𝜕𝑇 𝜕𝑥+ 𝑣 𝜕𝑇 𝜕𝑦+ 𝑤 𝜕𝑇 𝜕𝑧 = 𝑘 𝜕 𝑇 𝜕𝑥 + 𝜕 𝑇 𝜕𝑦 + 𝜕 𝑇 𝜕𝑧 + 𝜇 𝜕𝑢 𝜕𝑦 + 𝜕𝑤 𝜕𝑦 (4.5)
where 𝑐 and 𝑘 are the heat capacity and the conductivity of the lubricant, respectively. The first term of Eq. (4.5) represents the thermal energy transfer due to convective phenomenon, whereas the other terms in the right side of the equation are respectively the transfer of thermal energy by conduction and the dissipation ratio of mechanical energy due to oil viscosity. More recently with the growth of computational power thermo-elastic-hydro-dynamic (TEHD) models, that include the pads deformation due to pressure and temperature distributions, have been implemented widely. The more complex models are fundamental in the bearing design phase because the deformed pads strongly influence the oil-film wedge, that is linked to temperature and pressure by the Reynolds equation.
The THD model used is similar to the one developed and described by Dang et al. (2016) [18]: the model does not compute the pad deformations, but it considers the flexibility of the pivot, based on the displacement of the pivot position along the radial direction 𝜂 (Figure 4.2).The bearing modeled is a five-pad rocker-type TPJB and the pad tilts by angle 𝜃 about the line contact P.
Figure 4.2 - Geometry and coordinates for a single pad [18]
The hydrodynamic model is based on the well-known Reynolds equation (4.2). A finite difference code had been developed by Chatterton et al. [24] for the integration of the Reynolds equation. For instance, the pressure 𝑝, at node (i, j) of the mesh grid is given by
a combination of pressures of the nearest nodes (Eq.(4.7)):
𝑝, = 𝑎 + 𝑎 𝑝 , + 𝑎 𝑝 , + 𝑎 𝑝, + 𝑎 𝑝, (4.7)
The Partial Differential Equation Toolbox implemented in Matlab can be used to solve the Reynolds equation: instead of using a Finite Difference Method the PDE Toolbox makes a Finite Element Analysis on a triangular mesh. From the code point-of-view, the drawback of using this toolbox is the necessity of interpolate the results on an orthogonal grid to employ the data.
Figure 4.3 - PDE mesh grid
In Figure 4.4 and Figure 4.5 are plotted the first iteration results of the pressure field achieved with the PDE solution, respectively on the PDE grid and on the nodal grid.
-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 x [m] -0.03 -0.02 -0.01 0 0.01 0.02 0.03 y [m ]
Figure 4.4 - PDE pressure results
convergence of the pressure distribution in each pad; convergence of the temperature distribution in each pad; equilibrium of the forces on each pad;
equilibrium of the forces on the shaft for the given static load.
The results of the numerical simulations have been obtained by using a code based on Matlab®: the main program flowchart is shown in Figure 4.6.
4.1.1 Dynamic coefficients
The dynamic behaviour of the rotating machine is influenced by the forces that take place in the oil-film and act on the shaft. Those forces can be linearized in order to model the dynamic characteristics of the mechanical system: the horizontal and vertical hydrodynamic oil film forces acting on the shaft can be written as the sum of the static value at equilibrium and perturbation contributions:
𝑓, = 𝑓, , + 𝛥𝑓, = ∑𝑓, , + ∑𝛥𝑓,
𝑓, = 𝑓, , + 𝛥𝑓 , = ∑𝑓, , + ∑𝛥𝑓 , (4.8)
The oil-film forces on the shaft are obtained by the sum of the oil-film forces Δf of each k-th pad. The perturbation contribution forces can be linearized around the equilibrium position by means of the full dynamic coefficients:
⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡Δ𝑓𝑥,𝑜𝑖𝑙 𝑘 Δ𝑓𝑦,𝑜𝑖𝑙𝑘 Δ𝑓𝜃,𝑜𝑖𝑙𝑘 Δ𝑓𝜂,𝑜𝑖𝑙𝑘 ⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎤ = − ⎣ ⎢ ⎢ ⎢ ⎢ ⎡𝑘𝑥𝑥𝑘 𝑘𝑥𝑦𝑘 𝑘𝑥𝜃𝑘 𝑘𝑥𝜂𝑘 𝑘𝑦𝑥𝑘 𝑘𝑦𝑦𝑘 𝑘𝑦𝜃𝑘 𝑘𝑦𝜂𝑘 𝑘𝜃𝑥𝑘 𝑘𝜃𝑦𝑘 𝑘𝜃𝜃𝑘 𝑘𝜃𝜂𝑘 𝑘𝜂𝑥𝑘 𝑘𝜂𝑦𝑘 𝑘𝜂𝜃𝑘 𝑘𝜂𝜂𝑘 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ Δ𝑥 Δ𝑦 Δ𝜃 Δ𝜂 − ⎣ ⎢ ⎢ ⎢ ⎢ ⎡𝑐𝑥𝑥𝑘 𝑐𝑥𝑦𝑘 𝑐𝑥𝜃𝑘 𝑐𝑥𝜂𝑘 𝑐𝑦𝑥𝑘 𝑐𝑦𝑦𝑘 𝑐𝑦𝜃𝑘 𝑐𝑦𝜂𝑘 𝑐𝜃𝑥𝑘 𝑐 𝜃𝑦 𝑘 𝑐 𝜃𝜃 𝑘 𝑐 𝜃𝜂 𝑘 𝑐𝜂𝑥𝑘 𝑐 𝜂𝑦 𝑘 𝑐 𝜂𝜃 𝑘 𝑐 𝜂𝜂 𝑘 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎡Δ𝑥Δ𝑦̇̇ Δ𝜃̇ Δ𝜂̇ ⎦⎥ ⎥ ⎤ (4.9)
Where Δ𝑓, and Δ𝑓 , are respectively the moment of the oil-film forces with respect to the pivot and the resultant of the oil-film forces along the radial direction of the pivot position.
By considering a harmonic excitation with a forcing frequency 𝜔, the motion of the system for each degree of freedom 𝑞 will be:
Δ𝑞(𝑡) = Δ𝑄(𝜔)𝑒 Δ𝑄̇(𝜔) = 𝑖𝜔Δ𝑄(𝜔) (4.10) For each pad, the amplitude of oil-film forces in frequency domain can be expressed by
⎣ ⎢ ⎢ ⎡Δ𝐹 , Δ𝐹 , Δ𝐹 , Δ𝐹 , ⎦ ⎥ ⎥ ⎤ = − ⎣ ⎢ ⎢ ⎢ ⎡𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 ⎦⎥ ⎥ ⎥ ⎤ 𝑘 Δ𝑋 Δ𝑌 ΔΘ Δ𝑁 (4.11)
⎣ ⎢ ⎢ ⎢ ⎡ 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 𝑍 ⎦⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎡ 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 ⎦⎥ ⎥ ⎥ ⎤ + 𝑖𝜔 𝑐𝑐 𝑐𝑐 𝑐𝑐 𝑐𝑐 𝑐 𝑐 𝑐 𝑐 [𝑍] = [𝐾] + 𝑖𝜔[𝐶] (4.12)
4.2 3D Thermal model
As said before the 3D thermal model is based on the one developed by Chatterton et al. in 2018 [24].
The distribution of the temperature in the entire bearing is obtained by means of a three- dimensional thermal model that includes a portion of the shaft (S), the oil-films (O) and the pads (P). The energy equation for each oil-film is given by Eq.(4.5), valid for laminar flow: in case of turbulent flow, all the terms in Eq.(4.5) can be seen as the sum of an average value and a fluctuation term.
Temperature distributions in the pads and shaft at steady state are governed by the following equations:
−𝛻(𝑘 𝛻𝑇) = 0
−𝛻(𝑘 𝛻𝑇) = 0 (4.13)
where 𝑘 and 𝑘 are the thermal conductivity coefficients of the bearing and shaft, respectively. Equations (4.5) and (4.13) have been solved by means of the finite element approach using a structured mesh for the oil-films and unstructured meshes for the pads and shaft, as shown in Figure 4.7 and Figure 4.8.
The boundary conditions (BC) applied to the faces of the bodies of the model are listed in Table 4.1.
Region BC # Face# Description BC Type
k-th OIL FILM
1 O1 Oil Inlet Temperature @ oil inlet T
2 O2 Oil Outlet Convection with oil @ T supply
3 O3 Lateral Convection with oil @ T supply
4 O4 Lateral Convection with oil @ T supply
5 O5 Bearing interface Heat flux from k-th pad
6 O6 Shaft interface T from shaft
k-th PAD
7 P1 External surface Convection with oil @ T supply 8 P2 Oil interface T from k-th oil part
9 P3 Lateral Convection with oil @ T supply
10 P4 Lateral Convection with oil @ T supply
11 P5 Pad Inlet Convection with oil @ T supply
12 P6 Pad Outlet Convection with oil @ T supply
SHAFT
13 S1 Cylindrical Convection with air @ room T
14 S2 Oil Interface Heat flux from oil parts
15 S3 Cylindrical Convection with air @ room T
16 S4 Lateral Temperature @ T supply
17 S5 Lateral Temperature @ T supply
18 S6 Lateral Convection with oil @ T supply
19 S7 Lateral Convection with oil @ T supply
Table 4.1 - 3D thermal model BCs
The bodies of the model are connected at interfaces, where the temperature and the heat flux of two adjacent bodies must be the same. By considering the k-th oil-film body, the conditions are (see Figure 4.9):
𝑇 , = 𝑇 ,
𝑄̇ , = 𝑄̇ ,
𝑇 , = 𝑇 ,
𝑄̇ , = 𝑄̇ ,
been solved with ANSYS Fluent ® software, to compute the heat transfer of the fluid path with a CFD simulation. The Matlab-Fluent interaction is well described in Figure 4.9: the THD model computed in Matlab gives a temperature distribution to be applied on the top surface of the Babbitt coating in Fluent, which computes the heat transfer in the pad and gives back to Matlab the total heat flux from the top surface.
Figure 4.9 – Matlab-Fluent model interactions
By considering the rotation of the shaft, a constant temperature along the circumferential direction of the shaft can be assumed.
The solution of the thermal model is obtained by iterating the solution of each subsystem until the convergence criteria, based on the relative iteration error of interface conditions, is reached. A smooth function between consecutive iterations was adopted on the boundary conditions at interfaces to reduce the oscillations of the numerical solution.
4.2.1 Thermal mixing model
The temperature of lubricant flowing into the leading edge plays an important role in the bearing dynamic behaviour. In literature the thermal mixing in the groove between two consequential pads is modelled with many different approaches, looking for a trade-off between physical accuracy and computational time. In [24] the inlet temperature 𝑇 , of
each oil-film part was simply evaluated by considering a weighted average of the supply oil at the cold supply temperature 𝑇 and the outlet hot oil from the previous oil-film part at temperature 𝑇 , (where “k” denotes the pad number) by means of the coefficient
“m”:
𝑇 , = 𝑚 ∗ 𝑇 , + (1 − 𝑚) ∗ 𝑇 (4.15)
Equation (4.15) comes from an energy balance between the hot oil coming from the previous pad, the fresh oil injected in the groove and the warm oil entering the next pad neglecting the mass flow rate of oil leaked and the different values for density and specific heat. In Suh and Palazzolo (2015) [26] the schematic diagram used for the mixing temperature calculation is shown in Figure 4.10. Also this model does not consider the amount of oil that is not carried over and it considers constant density and specific heat.
Figure 4.10 - Oil inlet groove [26]
The amount of the downstream flow “𝑄 ” is the sum of the upstream and supply flows. For the calculation of the inlet temperature “𝑇 ”, mixing temperature theory is adopted: conventional mixing temperature formula is applied in Eq. (4.16). In the case of 𝑄 − 𝑄 ≤ 0, there should be a compensation term to prevent any problems. Moreover, if 𝑄 − 𝑄 = 0, all recirculated fluid goes into the leading edge resulting in a physically wrong case. A mixing coefficient “𝜂” is introduced to prevent these cases and the modified
factor. 𝑇 =𝑄 ∗ 𝑇 + 𝑄 − 𝑄 ∗ 𝑇 𝑄 (4.16) 𝑇 (𝑧) = ⎩ ⎪ ⎨ ⎪ ⎧𝑄 ∗ 𝑇 + 𝑄 − 𝑄 ∗ 𝑇 𝑄 𝑖𝑓 𝜂𝑄 > 𝑄 𝜂 ∗ 𝑄 ∗ 𝑇 + 𝑄 − 𝜂 ∗ 𝑄 ∗ 𝑇 𝑄 𝑖𝑓 𝜂𝑄 ≤ 𝑄 (4.17)
An alternative approach is presented by Stachowiak and Batchelor (2000) [27]: their study is focused on a thrust bearing but the same procedure can be applied on a TPJB. According to them, a certain amount of space is required between the pads for the hot lubricant discharged from one pad to be replaced by cool lubricant before entering in the following pad. In practice, the replacement of lubricant is never perfect, and a phenomenon known as hot oil carry over is almost inevitable. This phenomenon is illustrated schematically in Figure 4.11. It was found from boundary layer theory that the lubricant inlet temperature can be calculated from Eq.(4.18):
𝑇 = 𝑇 (1 − 𝑚)
(1 − 0.5 ∗ 𝑚)+
(0.5 ∗ 𝑇 ∗ 𝑚)
(1 − 0.5 ∗ 𝑚) (4.18)
The hot oil carry over coefficient is function of sliding speed and space between adjacent: in [27] is reported that the coefficient goes for a small gap width of 5 𝑚𝑚 between pads, from a value of 0.8 at 20 𝑚/𝑠 to 0.7 at 40 𝑚/𝑠. For a large gap width of 50 𝑚𝑚, the coefficient varies between 0.55 and 0.5 for respectively 20 𝑚/𝑠 and 40 𝑚/𝑠. The minimum value of hot oil carry over coefficient occurs at approximately 40 𝑚/𝑠 with a sharp rise beyond this speed.
The last approach presented from the literature is developed by Rindi et al. (2015) [28]: in their work they proposed an innovative strategy for the simulation of thermal-hydraulic systems based on the decoupling between momentum (resistive elements) and mass/energy balances (capacitive elements). Following this approach, the lubricant supply model, for the calculation of the supply and leakage flows of the system, it is composed by two components: the sump and the duct models.
In the first one, the most important for the interests on this work, the sump represents the cavity between two consecutive pads where supply lubricant flows mix (Figure 4.12). Thus, this sub model evaluates the balance of these flow rates. Its inputs are the inlet and the outlet flow rates “𝑄 ” and “𝑄 ” calculated previously in the oil film model, the supply flow rate 𝑄 sent from the supply plant to the bearing, and the leakage flow rate 𝑄 , i.e., the flow rate that does not go over the pad but leaves the bearing through the seals.
Figure 4.12 – Scheme of duct and sump models [28]
These two flow rates are calculated using the duct sub-model, which represents, as a lumped parameters system, the ducts that reach the sump coming from the supply; it allows the
model allows to calculate the sump pressure, that is needed as a boundary condition for the trailing and leading edges in the oil film model.
The model developed by Rindi et al. [28] is expensive from the point of view of the computational cost and not feasible to be implemented in the code used for this project. Although, it introduces the calculation of the oil mass flow rate lost (or leaked).
4.2.1.1 Novel mixing model
The mixing model has been improved, in particular on this topic, with a better modelled energy balance in the groove between two pads.
As it can be seen in Figure 4.13, the model is similar to the one in Figure 4.12. The Matlab code, before entering in the thermal model, computes all the mass flow rates on the pads, in particular:
the mass flow rate of the coolant carried over the pad (𝑚̇ ).
the mass flow rate of the coolant exiting the pad from the trailing edge (𝑚̇ ,). the mass flow rate exiting from the lateral sides of the pad (𝑚̇ ).
the supply amount of coolant (𝑚̇ = 𝑚̇ ⁄𝑁 ).
The 𝑚̇ dependence from the circumferential position of the groove in the bearing is neglected. The supply mass flow rate is known and imposed. The unknowns of the problem are 𝑇 and 𝑚̇ , where 𝑇 is the temperature of the coolant at the leading edge of the pad and 𝑚̇ is the mass flow rate that is not carried over the pad, as Stachowiak and Batchelor [27] theorized: the replacement of lubricant is never perfect and the phenomenon called hot oil carry over takes place.
Figure 4.13 – Mixing model mass balance
The mass balance equations are:
𝑚̇ + 𝑚̇ = 𝑚̇ + 𝑚̇ (4.19)
𝑚̇ = 𝑚̇ + 𝑚̇ (4.20)
respectively for the k-th groove (4.19) and for the k-th pad (4.20).
An equilibrium on the mass balance to verify that the total mass flow rate leaked from the bearing is equal to the total mass flow supplied:
𝑚̇ = 𝑚̇ + 𝑚̇ = 𝑚̇ (4.21)
The energy equation for the groove is:
𝑄̇ + 𝑄̇ = 𝑄̇ + 𝑄̇ (4.22)
where the heat exchange contribution from the pads’ sides is neglected (i.e. the groove is considered as an adiabatic system).
𝑚̇𝑐 (𝑇 − 𝑇 ) + 𝑚̇𝑐 (𝑇 − 𝑇 )
= 𝑚̇𝑐 (𝑇 − 𝑇 ) + 𝑚̇𝑐 (𝑇 − 𝑇 )
(4.23)
From Equation (4.23) and neglecting the c variation is obtained
𝑚̇𝑐 𝑇 + 𝑚̇𝑐 𝑇 = 𝑚̇𝑐 𝑇 + 𝑚̇𝑐 𝑇 (4.24)
Figure 4.14 - Mixing model energy balance
Another unknown is the coolant leaking temperature 𝑇 , in the last term of the equation. It
is modelled as:
𝑇 = (1 − 𝛽)𝑇 + 𝛽 𝑇 (4.25)
where β is the mixing coefficient that takes into account the degree of mix of the fresh coolant in the groove and it is equal to:
when 𝑚̇ ≫ 𝑚̇ , 𝛽 is equal to 1, so 𝑇 = 𝑇 , it means that the amount of fresh oil injected in the groove is much more than the amount of oil entering the following pad such that the temperature of the leaking oil is not affected by the hot oil exiting from the pad and remains constant and equal to the temperature of the fresh oil supplied.
When 𝛽 = 0, 𝑚̇ and 𝑚̇ are comparable, it is like considering a perfect mixing to the equilibrium, in fact 𝑇 is equal to 𝑇 , i.e. in equilibrium with the oil temperature entering the following pad.
If the fresh oil flow rate is very low, it means that 𝑚̇ ≫ 𝑚̇ , this leads to a negative 𝛽. In this case the coefficient is imposed to be equal to zero.
From Equation (4.23) isolating the unknown 𝑇 :
𝑇 = 𝑚̇𝑐 𝑚̇𝑐 𝑇 + 𝑚̇𝑐 𝑚̇𝑐 𝑇 − 𝑚̇𝑐 𝑚̇𝑐 𝑇 (4.27) 𝑇 = 𝐴 ∗ 𝑇 + 𝐵 ∗ 𝑇 − 𝐶 ∗ 𝑇 (4.28) Where: 𝐴 = 𝑚̇𝑐 𝑚̇𝑐 𝐵 = ṁc ṁc 𝐶 = 𝑚̇𝑐 𝑚̇𝑐
Introducing the coefficient 𝛽:
𝑇 = 𝐴 ∗ 𝑇 + 𝐵 ∗ 𝑇 − 𝐶 ∗ (1 − 𝛽)𝑇 + 𝛽 𝑇 (4.29)
Finally:
𝑇 , =
𝐴 ∗ 𝑇 + (𝐵 − 𝛽𝐶) ∗ 𝑇
