on the lubrication of mechanical face … the lubrication of mechanical face seals proefschrift ter...

ON THE LUBRICATION

OF

MECHANICAL FACE SEALS

Harald Lubbinge

The research project was sponsored by Flowserve B.V.and was carried out at the University of Twente.

ISBN: 90-3651240-9

Printed by FEBO druk B.V., Enschede

Copyright c©1999 by H. Lubbinge, Enschede

ON THE LUBRICATION OF MECHANICAL FACE SEALS

PROEFSCHRIFT

ter verkrijging vande graad van doctor aan de Universiteit Twente,

op gezag van de rector magnificus,prof.dr. F.A. van Vught,

volgens besluit van het College voor Promotiesin het openbaar te verdedigen

op vrijdag 15 januari 1999 te 15.00 uur

door

Hans Lubbinge

geboren op 9 juni 1971

te Giethoorn

Dit proefschrift is goedgekeurd door:

Promotor: Prof.ir. A.W.J. de GeeAssistent–promotor: Dr.ir. D.J. Schipper

voor Tineke

ACKNOWLEDGEMENTS

This research is sponsored by Flowserve B.V., which is gratefully acknowl-edged.

I would like to thank the members of the tribology group, who created apleasant work environment during the last four years: Ton de Gee, JohanLigterink, Hans Moes, Wijtze ten Napel, Dik Schipper, Kees Venner, Laurensde Boer, Willy Kerver, Walter Lette, Erik de Vries, Jan Bos, Bernd Brogle,Rob Cuperus, Edwin Gelinck, Rudi ter Haar, Qiang Liu, Henk Metselaar,Elmer Mulder, Daniel van Odyck, Patrick Pirson, Matthijn de Rooij, JanWillem Sloetjes, Ronald van der Stegen, Harm Visscher, Andre Westenengand Ysbrand Wijnant.

Special thanks are deserved by:Ton de Gee, my promotor, for his valuable contribution to this thesis. EdwinGelinck, who was my roommate during the last 2 years, for the many usefuldiscussions and suggestions. From Flowserve, Jan Keijer, Seb Bakx, Jan vander Velden and Erik Roosch for the discussions and their support. The em-ployees of Flowserve in Dortmund for lapping and measuring seal faces. Gerritvan der Bult, Willie Olthof and Willie Kerver for making parts for the test rig.From the Philips laboratories, Bram Pepers and Cor Adema, who accuratelyprepared the seal faces for testing. Arie de Jong of the Netherlands Foundationfor Research in Astronomy for the interference microscope measurements. Theemployees of the IMC, who made some specific parts of the test rig. Gerben teRiet o.g. Scholten of the former AID, who did a great job with respect to theelectronics of the test rig. Laurens de Boer and Erik de Vries for their techni-cal assistance concerning the test rig. Ieke van Gaalen and Peter Wijlhuizen,who worked on my project for their MSc. degree and delivered a significantcontribution to my thesis. Marcel de Boer for his help with the design of thetest rig. The ladies of the secretariat, Debbie Vrieze, Annemarie Teunissenand Carolien Post for their administrative assistance. Katrina Emmett for herhelp concerning the English language. Lieselot IJsendoorn for her contributionto the design of the cover.

I especially thank my mentor Dik Schipper, for his stimulating discussions,

ii Acknowledgements

comments on this thesis and, moreover, his optimism and great support.Laurens and Erik are also thanked for their assistance as paranimf.Although he cannot read this, Bob is also thanked, because of his pleasantcompany during the weekends and the evenings at the university.I thank my parents for their encouragement and support.

Finally, I thank my girl-friend Tineke for being my best friend, for her loveand patience.

Harald Lubbinge

Enschede, January 1999

SAMENVATTING

Om de lekkage van een mechanische asafdichting te minimaliseren, als gevolgvan de steeds strenger wordende milieu eisen, dient de separatie zo klein mo-gelijk te zijn. Als gevolg hiervan zal zowel de wrijving (vermogens verlies) alsde slijtage (verkorte levensduur) toenemen.Er dient dus gezocht te worden naar een operationele conditie waarbij de slij-tage en de wrijving aanvaardbaar zijn, en de lekkage tot een minimum wordtgebracht. Wanneer gekeken wordt naar de Stribeck curve, waarin de wrijv-ingscoefficient wordt uitgezet tegen bijvoorbeeld de snelheid of een of andersmeringskental, zijn er drie smeringsregimes te onderscheiden. Dat zijn hetgrensgesmeerde regime, het gemengde smeringsregime en het hydrodynamischgesmeerde regime. Grensgesmeerd zou ideaal zijn voor een minimale lekkage,maar is echter niet geschikt met betrekking tot de wrijving en slijtage. Daar-entegen bestaat er onder in het gemengde smeringsregime, in het gebied vande overgang van hydrodynamisch naar gemengd gesmeerd, een situatie die welgeschikt is. Hier is namelijk de separatie klein, zodat de lekkage relatief laagis. Daarnaast zijn zowel de slijtage als de wrijvingscoefficient laag.In de literatuur bestaan er verschillende modellen die de filmdikte in een mech-anische asafdichting bepalen. Een nadeel van deze modellen is dat meestal uit-sluitend naar de hydrostatische druk component van de af te dichten vloeistofwordt gekeken, terwijl vaak, zo niet altijd, een hydrodynamische componentaanwezig is. De hydrostatische druk component wordt bepaald door de matevan coning die er zich op de afdichting bevindt. Het resultaat van een dergelijkmodel is dat het theoretisch voorspelde gedrag niet overeenkomt met de prak-tijk situatie.Vaak bevinden er zich op het contactoppervlak van een mechanische asafdicht-ing een tweetal golven (waviness) in omtreksrichting. Deze ontstaan gedurendehet voorbewerkingsproces, het vlakleppen van de afdichting. Maar ook tijdensbedrijf onstaan er tengevolge van slijtage, mechanische deformatie en thermis-che effecten, golven in omtreksrichting op het oppervlak. Dergelijke golvenmet amplitudes van enkele tienden van een micrometer, zijn voldoende omeen aanzienlijke hydrodynamische vloeistofdruk te genereren, met als resul-taat een grotere separatie en daarmee een hogere lekkage. Een gewenst effectvan een dergelijke golving is dat, mocht de hydrostatische component falen om

iv Samenvatting

een of andere reden, deze golving kan blijven zorgen voor de nodige smeringen vloeistofdruk in het contact. Een ander nadeel van de bestaande mod-ellen is dat uitsluitend wordt gekeken naar volle film condities, terwijl juist,onder gemengde smeringscondities, tevens gekeken dient te worden naar eencontactmodel.In dit proefschrift wordt daarom een model gepresenteerd waarmee een volledigeStribeck curve voor een mechanische asafdichting berekend kan worden, endaarmee het overgangsgebied van volle film naar gemengde smering als functievan de operationele condities. Dit model is gebaseerd op de combinatie van eencontact model met een filmvergelijking. In dit model, dat overigens isother-misch is, wordt rekenschap gehouden met onder andere de golving, coning,geometrie van de asafdichting, ruwheid, druk van de af te dichten vloeistof enbelasting. Uit literatuuronderzoek bleek dat een filmvergelijking voor mecha-nische asafdichtingen, die ook rekening houdt met hydrodynamische effecten,niet bestond en deze is daarom ontwikkeld en in dit proefschrift beschreven.Om gebruik te kunnen maken van het contactmodel, diende er een schattinggemaakt te worden van het nominale contactoppervlak. In dit proefschrift is,gebaseerd op numerieke berekeningen, een funktiefit gemaakt voor het nomi-nale contactoppervlak als funktie van de amplitude van de golving, de coninghoek, de elasticiteitsmodulus en de belasting.Tenslotte, om het model te verifieren, is er een testopstelling ontworpen engemaakt waarmee Stribeck curves aan mechanische asafdichtingen gemetenkunnen worden. Ook zijn er slijtagemetingen en belasting proeven uitgevo-erd. Slijtagemetingen om de veranderingen in de microgeometrie te kunnenanalyseren, en belastingproeven om de belastbaarheid voor de wrijvingsexper-imenten vast te stellen.Het wrijvingsmodel komt zeer goed overeen met de gemeten wrijvings curves.Het effect van de operationele conditions, zoals de geometrie (ruwheid, coningen golving), druk van de af te dichten vloeistof en de belasting, op de transitievan volle film smering naar gemengde smering is geanalyseerd. Afhankelijkvan de operationele condities, wordt de transitie van volle film smering naargemengde smering sterk bepaald door onder andere de coning hoek, de belast-ing en de ruwheid en in mindere mate door de amplitude van de golving, deruwheidsverdeling en de gereduceerde elasticiteits modulus.

ABSTRACT

In order to minimize leakage of a mechanical face seal, due to environmentalregulations, the separation between the faces should be as small as possible.As a consequence, an increase of friction (power loss) and wear (reducing lifetime) occurs. Hence, an operational condition is sought for which wear andfriction are acceptable, and, moreover, the leakage is minimized. Taking theStribeck curve into consideration, in which the coefficient of friction is plottedas a function of the velocity or some lubrication parameter, three lubricationregimes can be distinguished. These are the boundary lubrication regime, themixed lubrication regime and the hydrodynamic lubrication regime. Boundarylubrication would be the ideal regime regarding leakage, but it is not suitablewith regard to friction and wear. In the lower region of the mixed lubricationregime, however, i.e. the transition region from hydrodynamic to mixed lu-brication, a suitable operational situation exists. Here, the film thickness orseparation is relatively small, and, therefore, the leakage is low. In addition,wear as well as friction are low.In the literature, different models are described which calculate the film thick-ness in a mechanical face seal. Unfortunately, these models mostly only con-cern the hydrostatic fluid pressure, which is the result of the pressure of thefluid to be sealed, whereas often, if not always, a hydrodynamic component isalso present. The hydrostatic pressure is determined by the amount of coningpresent on a seal face. The result of such a model is that the theoreticallypredicted behaviour does not correspond with the practical situation.Often, a two-wave waviness exists on the circumference of a seal face. Thesewaves develop during the preprocessing, i.e. flat lapping of the face, but alsoduring seal operation when, as a result of wear, mechanical distortion andthermal effects, waves develop on the face circumference. Such waves withamplitudes of a few tenths of a micrometer, are enough to generate a consider-able hydrodynamic fluid pressure, resulting in a larger separation and, hence,a greater leakage. A desirable effect of waviness is that, when the hydrostaticcomponent fails for some reason, lubrication and interfacial fluid pressure ofthe faces is maintained. Another disadvantage of such models is that theyonly strictly apply in the full film lubrication regime; for the mixed lubricationregime a contact model must also be incorporated.

vi Abstract

Hence, in this thesis, a model is presented which is able to calculate a completeStribeck curve for a mechanical face seal and, as a consequence, the transitionfrom full film to mixed lubrication as a function of the operational conditions.This model is based on a combination of a contact model and a film thicknessequation. The model, which is isothermal, incorporates waviness, coning, facegeometry, roughness, pressure of the fluid to be sealed and load. From theliterature it appeared that no film thickness equation for mechanical face seals,which also accounts for hydrodynamic effects, exists and this is, therefore,developed and described in this thesis. In order to use the contact model, anominal contact area is required. In this thesis, based on numerical data, afunction fit is made for the nominal contact area as a function of wavinessamplitude, coning angle, modulus of elasticity and load.Finally, in order to verify the model, a test rig was designed and built. With thetest rig, Stribeck curves of mechanical face seals were measured. Furthermore,wear measurements and load carrying capacity tests were performed. Wearmeasurements were carried out to analyze the change in micro-geometry, loadcarrying capacity tests were performed to determine the maximum applicableload during friction experiments.The friction model agrees very well with the friction experiments performed.The effect of the operational conditions, i.e. geometry (roughness, coning andamplitude), pressure of the fluid to be sealed and load on the transition full-film lubrication to mixed lubrication is shown. It was found that, dependingon the operational conditions, the transition from hydrodynamic to mixedlubrication significantly depends on the coning angle, load, and roughness andto a lesser extent on the waviness amplitude, the height distribution and thereduced modulus of elasticity.

CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Mechanical face seals . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Objective of this research . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. Review on the lubrication of mechanical face seals . . . . . . . 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Principle of mechanical face seals . . . . . . . . . . . . . . . . . 8

2.2.1 Inside vs. outside pressurized seals . . . . . . . . . . . . 8

2.2.2 Balance ratio . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Hydrostatic lubrication . . . . . . . . . . . . . . . . . . . 9

2.2.3.1 Effect of seal radii on hydrostatic pressure dis-tribution – parallel flat faces . . . . . . . . . . . 9

2.2.3.2 Effect of coning on hydrostatic pressure distri-bution . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.4 Hydrodynamic lubrication . . . . . . . . . . . . . . . . . 17

2.2.4.1 Thermal wedge . . . . . . . . . . . . . . . . . . 18

2.2.4.2 Viscosity wedge . . . . . . . . . . . . . . . . . . 18

2.2.4.3 Microasperity lubrication . . . . . . . . . . . . 18

2.2.4.4 Asperity-asperity collisions . . . . . . . . . . . 19

2.2.4.5 Squeeze film . . . . . . . . . . . . . . . . . . . . 19

2.2.5 Force equilibrium . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . 20

viii Contents

3. Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Modelling friction . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Friction under full film lubricated conditions . . . . . . . 23

3.2.2 Friction under boundary lubricated conditions . . . . . . 24

3.2.3 Friction under mixed lubricated conditions . . . . . . . . 25

3.3 Contact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Nominal contact area . . . . . . . . . . . . . . . . . . . . 29

3.3.2 Function fit for the nominal contact area . . . . . . . . . 32

3.4 Full film model . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.3 Cavitation boundary conditions . . . . . . . . . . . . . . 39

3.4.4 Dimensionless variables . . . . . . . . . . . . . . . . . . . 39

3.4.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . 43

3.4.5.1 γ- and α-dependence . . . . . . . . . . . . . . . 43

3.4.5.2 Pf -dependence . . . . . . . . . . . . . . . . . . 44

3.4.5.3 ψ-dependence . . . . . . . . . . . . . . . . . . . 44

3.4.5.4 ρc-dependence . . . . . . . . . . . . . . . . . . . 45

3.4.6 Film thickness equation . . . . . . . . . . . . . . . . . . 52

3.4.6.1 Asymptotes . . . . . . . . . . . . . . . . . . . . 52

3.4.6.2 Film thickness for Pf = 0 . . . . . . . . . . . . 54

3.4.6.3 Film thickness for 0 < Pf ≤ 1 . . . . . . . . . . 54

3.4.6.4 Film thickness for 1 < Pf ≤ 1.75 . . . . . . . . 55

3.4.7 Friction under full film lubricated conditions . . . . . . . 60

3.4.8 Leakage under full film lubricated conditions . . . . . . . 60

3.5 Calculating Stribeck curves . . . . . . . . . . . . . . . . . . . . 64

3.5.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5.2 Preliminary model results . . . . . . . . . . . . . . . . . 66

3.5.2.1 Waviness amplitude, A . . . . . . . . . . . . . . 66

3.5.2.2 Coning angle, a . . . . . . . . . . . . . . . . . . 67

3.5.2.3 Roughness . . . . . . . . . . . . . . . . . . . . . 67

3.5.2.4 Non-Gaussian height distribution; χ2n-height dis-

tribution . . . . . . . . . . . . . . . . . . . . . 68

3.5.2.5 Axial load FN . . . . . . . . . . . . . . . . . . . 69

3.5.2.6 Viscosity . . . . . . . . . . . . . . . . . . . . . 70

3.5.2.7 Reduced modulus of elasticity . . . . . . . . . . 70

3.5.2.8 Hydrostatic fluid pressure . . . . . . . . . . . . 70

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Contents ix

4. The test rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 Design of the test rig . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2.1 Stationary part . . . . . . . . . . . . . . . . . . . . . . . 854.2.2 Rotating part . . . . . . . . . . . . . . . . . . . . . . . . 874.2.3 Data acquisition and control of the operational parameters 88

4.3 Validation of the test rig; preliminary results . . . . . . . . . . . 924.3.1 Friction curve measurements . . . . . . . . . . . . . . . . 924.3.2 Load carrying capacity tests . . . . . . . . . . . . . . . . 934.3.3 Wear rate measurements . . . . . . . . . . . . . . . . . . 95

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5. Verification of model with experimental results . . . . . . . . . 1015.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2 Experimental procedure and materials . . . . . . . . . . . . . . 101

5.2.1 Experimental procedure . . . . . . . . . . . . . . . . . . 1015.2.2 Material specifications . . . . . . . . . . . . . . . . . . . 102

5.3 Theoretical vs. experimental results . . . . . . . . . . . . . . . . 1035.3.1 Effect of micro-geometry on hydrodynamic pressure gen-

eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.3.2 Macroscopic features . . . . . . . . . . . . . . . . . . . . 103

5.3.2.1 Waviness amplitude, A, variations . . . . . . . 1035.3.2.2 Radial coning angle, a . . . . . . . . . . . . . . 1055.3.2.3 Influence of macroscopic features on the tran-

sition from full film to mixed lubrication . . . . 1065.3.3 Axial load variations . . . . . . . . . . . . . . . . . . . . 1075.3.4 Pressure of fluid to be sealed variations . . . . . . . . . . 1085.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6. Conclusions and recommendations . . . . . . . . . . . . . . . . . 1196.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Appendix 123

A. Analytical solution of the hydrostatic fluid pressure for flatand coned faces — Polar coordinates . . . . . . . . . . . . . . . 125A.1 Flat parallel faces . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.2 Flat faces with a convergent coning . . . . . . . . . . . . . . . . 126

B. Photo impression of the test rig . . . . . . . . . . . . . . . . . . 131

x Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

NOMENCLATURE

a coning angle [rad]ai contact radius of an individual asperity [m]A waviness amplitude [m]Ac real contact area [m2]Aci area of contact of a single asperity i [m2]Af sealing interface area [m2]Ah hydraulic loading area [m2]AH hydrodynamic contact area [m2]Anom nominal contact area [m2]

Anom dimensionless nominal contact area Anom =Anom

b2[–]

Aseal seal area [m2]b equivalent radius of contact [m]

Br balance ratio Br =AhAf

[–]

B radial seal width [m]c compliance [m]dd distance between ds and dh [m]ds mean plane of the summits heights [m]dh mean plane of the surface heights [m]Db balance diameter [m]Di inner face seal diameter [m]Dm mean face seal diameter [m]Do outer seal face diameter [m]E ′ reduced modulus of elasticity [Pa]Ei elasticity modulus of contacting surface i (i = 1, 2) [Pa]f coefficient of friction [–]fci coefficient of friction of a single asperity i [–]fc coefficient of friction in the boundary lubrication regime [–]FC load carried by the asperities [N]Ff friction force [N]

Ff dimensionless friction force Ff =FfB2pm

√pmηω

[–]

xii Nomenclature

Fi load of an individual asperity [N]FH load carried by the hydrodynamic component [N]FN axial load [N]Fs spring load [N]G duty parameter [–]h film thickness, separation [m]

H dimensionless film thickness H =h

B

√pmηω

[–]

h∗ separation [m]hmin minimum film thickness [m]

Hmin dimensionless minimum Hmin =hmin

B

√pmηω

[–]film thickness

k number of circumferential waves on a seal face [–]ks specific wear rate [mm3/N.m]K pressure gradient factor [–]p pressure [Pa]

P dimensionless pressure P =p

pm[–]

p mean pressure [Pa]

p dimensionless pressure p =p

pi − po[–]

pa atmospheric pressure [Pa]pC asperity pressure [Pa]pci pressure in a single asperity i [Pa]pcav cavitation pressure [Pa]pf sealed fluid pressure [Pa]

Pf dimensionless fluid pressure Pf =pfpm

[–]

ph Hertzian contact pressure [Pa]pH hydrodynamic pressure [Pa]pi pressure at inner seal face diameter [Pa]pinside mean pressure of an inside pressurized seal [Pa]pm mean pressure [Pa]po pressure at outer seal face diameter [Pa]poutside mean pressure of an outside pressurized seal [Pa]ps spring pressure [Pa]pT total pressure [Pa]qc liquid fraction [–]qm leakage [m3/s]

Qm dimensionless leakage Qm =qmη

B3pm

(√pmηω

)3

[–]

Nomenclature xiii

r radius [m]

r dimensionless radius r =r

ro[–]

rb balance radius [m]ri inner radius [m]rm mean radius [m]ro outer radius [m]Rx radius of curvature [m]s sliding distance [m]t time [s]U velocity [m/s]Useal velocity of seal face at mean radius rm [m/s]v velocity [m/s]vt transiton velocity from HL to ML [m/s]

vt dimensionless transition velocity vt =vtexp

vtcal[–]

from HL to MLvtcal calculated transition velocity from HL to ML [m/s]vtexp measured transition velocity from HL to ML [m/s]V volume [mm3]w compliance [m]x cartesian coordinate [m]X dimensionless coordinate [–]y cartesian coordinate [m]Y dimensionless coordinate [–]z cartesian coordinate [m]

Greek symbols

α dimensionless coning α = a

√pmηω

[–]

αnom dimensionless coning angle αnom =aRx

b[–]

β radius of asperities [m]

βnom dimensionless seal widthB

b[–]

γ dimensionless waviness γ =A

B

√pmηω

[–]

γ shear rate [s−1]γML Adaptation parameter for hydrodynamic component [–]

in mixed lubrication regime

xiv Nomenclature

δ distance [m]η dynamic viscosity [Pa·s]ηs density of asperities [1/m2]θ angular coordinate [rad]

λ dimensionless separation λ =h

σs[–]

ν Poisson’s ratio [–]

ξ curvature variable ξ =riro

[–]

ρ density [kg/m3]

ρc cavitation variable ρc =

∣∣∣∣pcav

pm

∣∣∣∣ [–]

σ standard deviation of the surface height distribution [m]σafter standard deviation of the surface height distribution [m]

after the experimentσini standard deviation of the surface height distribution [m]

before the experimentσs standard deviation of the height distribution of the summits [m]τci shear stress at the asperity i [Pa]τH hydrodynamic shear stress [Pa]φ distribution of the asperities [–]

ψ dimensionless seal face geometry ψ =roB

[–]

ω angular velocity [rad/s]

Abbreviations

BL Boundary LubricationHL Hydrodynamic LubricationML Mixed Lubrication

1. INTRODUCTION

1.1 Mechanical face seals

Mechanical face seals are used to seal a fluid at places where a rotating shaftenters an enclosure. Figure 1.1 shows schematically the configuration of a me-chanical face seal. The rotating seal is fixed to the shaft and rotates with it,whereas the stationary seal is mounted on the housing. The secondary seals(o-rings) prevent leakage between the rotating shaft and the rotating seal, andthe housing and the stationary seal, respectively. The rotating seal is flexiblymounted in order to accomodate angular misalignment and is pressed againstthe stationary seal by means of the fluid pressure and the spring. The primarysealing occurs at the sealing interface of both seal faces, where the rotating faceslides relative to the stationary face. For proper functioning of a mechanicalface seal, a fluid film is maintained between the faces. In the configuration ofFig. 1.1 the sealed fluid may also act as a lubricant. Applications of mechanical

Housing

Pressurized fluid

Rotating shaft

Stationary sealRotating sealSpring

Secondary seal

Sealing interface

Fig. 1.1: Mechanical face seal, schematically.

face seals are numerous. The most common example of application is in pumps

2 1. Introduction

for the chemical industry. Also propellor shafts in ships and submarines, com-pressors for air conditioners of cars and turbo jet engines and liquid propellantrocket motors in the aerospace industry require mechanical face seals.

Mechanical face seals have become the first choice for sealing rotating shaftsoperating under conditions of high fluid pressures and high speeds, at theexpense of soft-packed glands. The reason for this is lower leakage, less main-tenance and longer life. A disadvantage of face seals is that when they fail,they do so completely, whereas a soft-packed gland can continue, although lessefficiently.

1.2 Problem definition

Due to increasing technical and environmental requirements, operational con-ditions are becoming more severe. Face seals have to operate at higher pres-sures and higher speeds, so a sufficient fluid pressure in the sealing interface isvital if excessive wear, friction and temperature rise (frictional heating) are tobe avoided and a long seal life is to be ensured. However, a too thick fluid filmis unfavourable with regard to leakage, as this is proportional to the cube ofthe film thickness. Due to environmental demands, leakage must be minimizedby reduction of the separation between the faces. From the above it is clearthat the demands with regard to optimum sealing are contradictory. Ideally,a mechanical face seal should operate with a fluid film as thin as possible, toreduce the leakage and to restrict wear.

In Fig. 1.2 the coefficient of friction is schematically plotted as a function of alubrication parameter, which yields the generalized Stribeck curve (Schipper,1988). Figure 1.2 also shows the separation h. The lubrication parameteris defined in many ways in the literature. It contains, for instance, the vis-cosity, the velocity of the surfaces, the contact pressure and the roughnessof the surfaces, see Gelinck (1999). In this graph, three lubrication regimescan be distinguished, i.e. Hydrodynamic Lubrication(HL), Mixed Lubrication(ML) and Boundary Lubrication(BL). The different lubrication regimes areschematically represented in Fig. 1.3. The faces are hydrodynamically lubri-cated, when they are fully separated by a fluid film, due to pressure build-up,which is caused by rotation of the faces. The load is transmitted by the fluid.When the fluid pressure for some reason is not capable of fully separating themating seal faces, asperity contact will occur. Then, the load on the faces iscarried by both the fluid and the asperities. This type of interfacial contactis called mixed lubrication. When there is no fluid pressure build-up at all,the load is completely carried by the interacting asperities and this is calledboundary lubrication, i.e. a layer is present which protects the surface. Each

1.2 Problem definition 3

ab

BL ML HL

Lubrication parameter (log)

Coeffi

cien

tof

fric

tion

f

Sep

arat

ionh

Fig. 1.2: Generalized Stribeck curve; coefficient of friction and theseparation are schematically plotted as a function of a lubricationparameter.

lubrication mode is characterized by a typical friction behaviour. In the BL-regime shear takes place in the boundary layers or at the interface of bothlayers. When the boundary layers are damaged, direct contact between theasperities occurs, and shear takes places at this interface or in the weaker as-perities, which results in material transfer from one surface to the other. In thehydrodynamically lubricated regime, the faces are fully separated by a fluid,and all shear, as a result of motion of one of the faces, takes place in the fluid.In the mixed lubricated regime, shear in both the fluid and the boundary layertakes place. The transitions HL–ML and ML–BL are defined by the inter-sections obtained by extrapolating the curves representing the coefficients offriction of the HL regime and the BL regime, respectively, with the tangent ofthe ML regime.For mechanical face seals an optimum operational region would be around thetransition from hydrodynamic to mixed lubrication, indicated by position a.In this region a low coefficient of friction is accompanied by a low wear rate(hardly any interaction between the opposing surfaces is present) and a lowleakage, as the separation is rather small. Position b, where face seals mayoperate as well, will also show a low coefficient of friction and hardly any wearas the faces are fully separated by a fluid film. However, as shown in thegraph, position b is accompanied by a much larger separation and hence, alarge leakage.Several researchers performed friction measurements in order to establish thetransitions between the different lubrication regimes. Lebeck (1987) collected a

4 1. Introduction

Boundary lubrication

Mixed lubrication

Hydrodynamic lubrication

Fig. 1.3: Lubrication modes.

lot of these friction measurements and plotted them in a graph as a function ofa lubrication parameter, in this case called the “duty parameter” G (Fig. 1.4),which is frequently used. The duty parameter is defined by:

G =ηv∆r

FN, (1.1)

where η is the dynamic viscosity, v the velocity, ∆r the width of the seal faceand FN the axial load acting on the seal (Lubbinge et al., 1997). Figure 1.4shows that the friction is not characterized adequately. For example the G–value for the transition from hydrodynamic– to mixed lubrication differs byat least 2 powers of 10. The reasons for this are that a) the duty parameterG does not contain any surface roughness parameter and b) the load per unitwidth does not represent the real pressure between the seal faces (Lubbinge

1.3 Objective of this research 5

Coeffi

cien

tof

fric

tion

Duty parameter G

Fig. 1.4: Stribeck curves of mechanical face seals [from Lebeck (1987)].

et al., 1997). From this graph it is therefore clear that further investigation ofthe lubrication of mechanical face seals is required.

1.3 Objective of this research

In the previous sections it was pointed out that the lubrication of mechani-cal face seals becomes quite complicated due to the increasing technical andenvironmental demands. There are many factors that affect the interfacialfluid pressure and thus the transition between the lubrication regimes as madeclear by Fig. 1.4. Therefore, the objective of this research is to develop a modelwhich predicts the frictional behaviour of mechanical face seals as a functionof the operational conditions. The existing duty parameter is not adequate.When the lubrication mode under specific conditions can be predicted, it ispossible to optimize the seal configuration with respect to leakage, frictionand wear. In this thesis the model is restricted to the iso-thermal situation.Clearly, experimental friction data are required to verify the model. Thus, anew test rig was designed and built to measure the friction of mechanical faceseals.

6 1. Introduction

1.4 Overview

Chapter 2 presents an overview of existing knowledge on the lubrication ofmechanical face seals. In Chapter 3 the development of the theoretical modelis discussed. The model is based on a combination of a modified contact modeland a newly developed film thickness equation. Chapter 4 describes the testrig, which is used to collect experimental data to verify the model. In addition,also wear rate measurements and load carrying capacity tests were performedand analyzed. In Chapter 5, the experimental results are compared with thetheoretical results and, finally, in Chapter 6, conclusions and recommendationsfor further research are presented.

2. REVIEW ON THE LUBRICATIONOF MECHANICAL FACE SEALS

2.1 Introduction

Already a lot of research has been performed on the lubrication of mechanicalface seals, particularly with respect to the fluid pressure between the matingfaces. The pressure generating mechanisms can be divided into two maincategories, i.e. hydrostatic mechanisms and hydrodynamic mechanisms.The mechanism of hydrostatic pressure generation has been extensively inves-tigated by e.g. Doust and Parmar (1986), Young and Lebeck (1982), Lebeck(1991) and Etsion (1978a; 1978b; 1994), and it is in general well understood.A more difficult area is, however, hydrodynamic lubrication of mechanical faceseals, because when two flat parallel surfaces slide parallel to each other in thepresence of a liquid, there is, according to the Reynolds’ equation, no mecha-nism to generate pressure in the fluid. The Reynolds’ equation is derived fromthe Navier-Stokes equations (Reynolds, 1886). It describes mathematically fullfilm lubrication and, for surfaces that do not deform in the direction of flow,is given in polar coordinates by:

1

r

∂

∂θ

(ρh3∂p

∂θ

)+

∂

∂r

(rρh3∂p

∂r

)= 6ηrω

∂(ρh)

∂θ︸︷︷︸wedge term

+ 12ηr∂(ρh)

∂t︸︷︷︸squeeze film term

, (2.1)

where r and θ are the polar coordinates within the fluid, p is the local pressurewithin the fluid film, ρ and η are, respectively, the density and the dynamicviscosity of the lubricant, h is the film thickness, and ω the angular velocity ofthe rotating seal face. For liquids, the density variations are negligibly smalland, also because the fluid pressures are relatively low, the density ρ can beomitted entirely from Eq. (2.1).The physical interpretation of the two terms which describe hydrodynamicpressure generation is as follows:

1. Wedge term — Pressure building as a result of a narrowing gap in theflow direction of the fluid.

8 2. Review on the lubrication of mechanical face seals

2. Squeeze film term — Pressure building as a result of film thickness chang-ing with time.

When two flat seal faces slide parallel to each other, there is no convergingwedge and, under such conditions there is no hydrodynamic pressure genera-tion. If a constant load is present and both surfaces are flat, the squeeze termreduces to zero. Therefore, separation of the flat parallel sliding surfaces canonly be achieved by hydrostatic action.

In experiments, however, it has been shown that besides the hydrostatic pres-sure imposed by the sealed fluid pressure, hydrodynamic pressure often devel-ops, see e.g. Sneck (1969), Pape (1968), Stanghan-Batch and Iny (1973), Annoet al. (1968) and Lebeck (1991). The possible reasons for this are discussed inSection 2.2.4.

2.2 Principle of mechanical face seals

2.2.1 Inside vs. outside pressurized seals

Mechanical face seals can be mounted in two different ways in, for instance, apump or a sealed vessel:

1. Inside mounting. In this configuration the pressurized fluid is to be sealedon the outside of the seal, which is called an outside pressurized seal ,see Fig. 2.1. This is the most common arrangement.

2. Outside mounting. The pressurized fluid is on the inside of the seal asshown in Fig. 2.2 and the seal is called an inside pressurized seal .

Ah

Db

Af

pf

Do Di

pa

Dh

Fig. 2.1: Unbalanced outsidepressurized seal, internallymounted.

Db Ah

pa

Do Di

pf

Dh

Af

Fig. 2.2: Balanced insidepressurized seal, externallymounted.

2.2 Principle of mechanical face seals 9

2.2.2 Balance ratio

An important parameter, well known in the sealing industry, is the balanceratio, which for an outside pressurized seal is defined as:

Br =hydraulic loading area

sealing interface area=AhAf

=

(Do

2 −Db2)(

Do2 −Di

2) . (2.2)

For an inside pressurized seal the hydraulic loading area is given by 14π(Db

2 −Di2),

thus

Br =AhAf

=

(Db

2 −Di2)(

Do2 −Di

2) . (2.3)

The balance ratio controls the axial load, acting on the seal interface. WhenBr is greater than 1, the seal is called unbalanced , whereas a balanced seal hasa Br-value lower than 1. Seals operating at high pressures are mostly of thebalanced type, Br < 1, whereas many low-pressure seals operate at Br > 1,the unbalanced type.

2.2.3 Hydrostatic lubrication

Mechanical face seals always operate with a radial pressure gradient across theface; the pressurized fluid, pf , on the one side and the atmospheric pressure,pa, on the other side, see Figs. 2.1 and 2.2. The pressure distribution in the gapis determined by the shape of the sealing interface and therefore the averagepressure is strongly affected by this shape. The average hydrostatic pressurein the gap is expressed as K·pf , the K-factor (or the pressure gradient factor)times the fluid pressure to be sealed. The following sections discuss the effectof different seal face geometries on the hydrostatic pressure distribution.

2.2.3.1 Effect of seal radii on hydrostatic pressure distribution –parallel flat faces

As mechanical face seals are circularly shaped, the radial hydrostatic pressuredistribution is affected by the degree of curvature, expressed by the ratio ofthe inner radius and the outer radius, ri/ro.The hydrostatic pressure distribution across a seal face for the statically loaded,parallel face situation (Fig. 2.3), Eq. (2.1) reduces to:

∂

∂r

(r∂p

∂r

)= 0. (2.4)


Integrating this expression yields:

r∂p

∂r= C ⇒ ∂p = C

∂r

r⇒ p = C ln r +D. (2.5)

As shown in Fig. 2.3, the boundary conditions for Eq. (2.5) read:

p = pi at r = ri

p = po at r = ro.(2.6)

Solving the equation for the integration constants C and D by substituting

pi po

riro

Fig. 2.3: Boundary conditions in the sealing interface.

these boundary conditions gives the solution for the hydrostatic pressure acrossthe seal face:

p =pi ln

rro + po ln rir

ln riro

. (2.7)

By defining r =r

ro, p =

p

pi − poand ξ =

riro

, Eq. (2.7) can be written as:

p =ln r

ln ξ+

popi − po

. (2.8)

Figures 2.4 and 2.5 show the pressure distribution for an outside and an insidepressurized seal, respectively, for different values of ξ. For the sake of com-pleteness, the exact solution for the hydrostatic pressure across a rectangulargeometry (Fig. 2.6) is given below:

p =po − piB

y + pi. (2.9)


(�(d (��(��

(�x(�n

(�d

Dimensionless radius r

ξ =

ξ

1

10

Hydro

stat

icpre

ssurep

Fig. 2.4: Pressure distribu-tion for different values ofξ for an outside pressurizedseal.

(�(d (��(��

(�x(�n

(�d

Dimensionless radius r

ξ =

ξ

1

10

Hydro

stat

icpre

ssurep

Fig. 2.5: Pressure distribu-tion for different values ofξ for an inside pressurizedseal.

Here the pressure drops linearly from the higher pressure on the one side tothe lower pressure on the other side, which thus results in a mean hydrostaticpressure of p = (po + pi) /2.As a result of the seal radii, the average pressure across the sealing interfacediffers from a rectangular geometry, as shown in Table 2.1 and Fig. 2.7. Thedifference increases with decreasing value of ξ. Furthermore the followingapplies:

limξ→0

poutside → 1 and limξ→0

pinside → 0. (2.10)

In practice, however, the value of ξ is about 0.9, so with regard to the hydro-static pressure, the effect of seal radii is relatively small, see Table 2.1.As well as the seal radii, a much more important factor with regard to hydro-static pressure in the contact is the coning of the faces. This is discussed inthe next section.

2.2.3.2 Effect of coning on hydrostatic pressure distribution

Another important geometrical feature of the seal face, which affects the hy-drostatic pressure distribution between the faces, is the so-called coning orradial taper. Figure 2.8 shows the three possible gaps. The coning is conver-gent if the sealing gap narrows in the flow direction of the fluid, it is divergent


pi po

y = 0 y = B yx

z

Fig. 2.6: Rectangular geometry.

ξ pinside poutside

0.01 0.21 0.790.1 0.33 0.660.3 0.40 0.600.5 0.44 0.560.7 0.47 0.530.9 0.49 0.51

Table 2.1: Mean pressure in the contact for different ratios of the radii,where pinside and poutside relate to an inside pressurized seal and anoutside pressurized seal, respectively.

if the coning narrows in the opposite direction. When the gap is parallel inthe radial direction there is no coning. As for large values of ξ, the effect ofcurvature on the hydrostatic pressure is relatively small, the hydrostatic pres-sure distribution has been derived in cartesian coordinates in order to showthe effect of coning. In Appendix A the Reynolds’ equation is solved in polarcoordinates (ξ << 1), expressed in dimensionless form.

The Reynolds’ equation in cartesian coordinates reads:

∂

∂x

(h3

η

∂p

∂x

)+

∂

∂y

(h3

η

∂p

∂y

)= 6U

∂h

∂x. (2.11)


p

ξ

poutside

pinside

p = 0.5

0

0

1

1

Fig. 2.7: Mean hydrostatic pressure as a function of ξ.

After substituting the following dimensionless variables:

X =x

B

Y =y

B

P =p

pm

H =h

B

√BpmηU

Hmin =hmin

B

√BpmηU

α = a

√BpmηU

Pf =pfpm,

(2.12)

the Reynolds’ equation can be written in a dimensionless form as follows:

∂

∂X

(H3 ∂P

∂X

)+

∂

∂Y

(H3∂P

∂Y

)= 6

∂H

∂X. (2.13)

With coning only, the solution of the hydrostatic pressure distribution, Eq. (2.13),reduces to:

∂

∂Y

(H3∂P

∂Y

)= 0. (2.14)


hminhmin

∆h∆h

parallel gap converging gap diverging gap

hmin

Fig. 2.8: Three different gap geometries. In each situation the seal isoutside pressurized.

The film thickness equation for a seal face with only convergent coning reads:

H = Hmin + αY, (2.15)

whereas for divergent coning the film thickness equation becomes:

H = Hmin + α(1 − Y ). (2.16)

Solving Eq. (2.14) with convergent coning results in:

P (Y,Hmin, α, Pf ) = PfY(Hmin + α)2(2Hmin + αY )

(2Hmin + α)(Hmin + αY )2. (2.17)

The hydrostatic pressure distribution in a diverging gap reads:

P (Y,Hmin, α, Pf ) = −PfY Hmin2(αY − 2Hmin − 2α)

(2Hmin + α)(αY −Hmin − α)2. (2.18)

Besides the coning angle of the gap, the minimum film thickness is also im-portant for the shape of the pressure distribution, as shown by Eqs. (2.17)and (2.18). Furthermore, the local pressure P depends linearly on the fluidpressure Pf .The effect of different coning angles α on the pressure distribution is shown inFig. 2.9. Lines are plotted for four different values of α, viz. 0, 0.5, 1 and 5.The fluid pressure Pf and the minimum film thickness Hmin are taken to beconstant and set at 1.75 and 1, respectively. With increasing α, the curvatureincreases. In the case of convergent coning (solid lines), the resistance in thedirection of the flow increases with increasing α, resulting in a more convexcurve and therefore a higher hydrostatic mean pressure. A divergent coning


results in a more concave curve (dashed lines) with increasing α. The sameeffect is observed when the minimum film thickness Hmin is varied; a smallerHmin results in greater curvature. In Fig. 2.10 the pressure P is plotted as afunction of the seal face width Y for different values of Hmin, viz. 0.1, 0.5, 1and 5. The constants Pf and α are set at 1.75 and 1, respectively. The solidlines indicate convergent coning, whereas the dashed lines indicate divergentconing.

( d

(

d

1

Seal face width Y

P

α =

5

5

1

1

0.5

0.5

0

Fig. 2.9: Hydrostatic pressure distribution for different values of α,Pf = 1.75 and Hmin = 1. The solid lines indicate convergent coning,whereas the dashed lines reflect divergent coning. In the case ofα = 0, the gap is parallel (dotted line).

From the above it becomes clear that the mean pressure p is strongly influencedby Hmin when a certain degree of coning is present, as opposed to the parallelgap case, where p is independent of the film thickness hmin, see Eq. (2.7). For aconstantly convergent coning angle, p increases with decreasing hmin, whereas


( d

(

d

1

Seal face width Y

P

Hmin =

0.1

0.1

0.5

0.5

1

1

5

5

Fig. 2.10: Hydrostatic pressure distribution for different values ofHmin, Pf = 1.75 and α = 1. The solid lines indicate convergentconing, whereas the dashed lines reflect divergent coning.

for a constantly divergent coning angle p decreases with decreasing hmin. Thesame result is valid for the angle of coning; an increasing coning angle resultsin an increasing p in the convergent case, but results in a decreasing p inthe divergent case. It is clear that sealing with a diverging taper leads toan unstable situation. As the face seals operate in equilibrium with the fluidpressure (the balance ratio equals the K-factor), a small disturbance (e.g. as aresult of temperature or pressure variations), resulting in decreasing hmin, willlead to a collapse of the fluid film and severe mechanical contact of both sealfaces, accompanied by heavy wear and high temperatures.

On the other hand, a convergent coning gives a stable hydrostatic fluid film;disturbances are immediately compensated for by the increasing mean pres-sure when hmin decreases. However, an excessive convergence is undesirable,


because it opens up the gap, reduces the stiffness and increases the leakagerate, as shown theoretically by Cheng et al. (1968) and Lebeck (1991) and alsoexperimentally by Snapp and Sasdelli (1973) and Young and Lebeck (1982).Unfortunately during operation, as a result of thermal effects, pressure dis-tortions and wear, the taper is not constant (Young and Lebeck, 1982). Infact, it can vary from convergent to parallel and even to divergent. Interfacialdivergence can also occur as a consequence of misalignment, (Etsion, 1978a).When this happens, the hydrostatic pressure across the sealing interface is nolonger able to restore the seal interface to equilibrium with the surroundingfluid pressure. In that case, it would be beneficial if a hydrodynamic pressuregenerating mechanism were present, which could still separate the seal facesby a fluid film.

From the literature it appears that a lot of research has been performed on thesubject of possible mechanisms which might lead to hydrodynamic pressurebuilding. Among these mechanisms, circumferential waviness seems to be themost prominent one. The next section will discuss briefly the reviewed litera-ture concerning hydrodynamic pressure generating mechanisms in mechanicalface seals.

2.2.4 Hydrodynamic lubrication

In the past, many different theories have been developed which describe themechanisms causing pressure generation in the contact of theoretically flatparallel surfaces. An extensive literature survey and evaluation of the possiblemechanisms has been given by Lebeck (1987). Lebeck concludes that devia-tion from the parallel, like waviness and misalignment, is the strongest effectcausing hydrodynamic fluid pressure, but he does not exclude the possibilitythat there is another, as yet unknown mechanism present (Lebeck, 1991).Also Nau’s (1967) review on the possible sources of pressure build-up showsthat waviness and misalignment of the faces play an important role with regardto hydrodynamical operation.Stanghan-Batch (1971) demonstrated experimentally that hydrodynamic pres-sure developed as a result of a sinusoidally shaped two-humped surface profileon one face, produced by the lapping process. Furthermore, carbon faces spon-taneously developed waves during testing, as a result of wear. In the regionof diverging film thickness cavitation occurred, so a net hydrodynamic loadsupport remained. The same phenomena were observed by Pape (1968). Filmthickness fluctuations at twice shaft speed frequency were measured. Surfacetopography measurements revealed a two-cycle sinusoidally shaped wave inthe circumferential direction. Pape concluded therefore that macroroughnessor waviness was the only feasible source of the observed phenomena. Ruddy


et al. (1982) studied the mechanism of film generation in seals, in which bothfaces had a circumferential 2-wave waviness. The cyclical variation in filmthickness resulted in an axial movement of the face. Hence, also the squeeze-film term (see Eq. (2.1)) was taken into account. It was shown that lowamplitude circumferential waviness, combined with relative axial movement ofthe sealing faces, generated hydrodynamic load support.In the following sections, possible sources of hydrodynamic load support, otherthan waviness and misalignment, are briefly summarized.

2.2.4.1 Thermal wedge

Fogg (1946) studied hydrodynamic lift in a thrust bearing with nominallyparallel faces and explained the lift in terms of a thermal wedge effect. As aresult of a temperature gradient in the direction of the sliding motion, a densityvariation occurred, which led to the generation of hydrodynamic pressure.When lubricant enters the bearing, it heats up due to viscous friction, whichresults in a reduction of its density. Since continuity requires that the massflow rate must be constant, the volume flow rate has to increase, which is onlypossible if there is an increasingly negative pressure gradient. This requirementplus the boundary conditions, i.e. the pressure must become ambient at eachend of the slider, causes a load supporting pressure in the film. In his articleLebeck (1987) shows that only under conditions of high speeds and lubricationwith oil, can some load support develop, however, for liquids like water thethermal wedge is negligible. Therefore, the thermal wedge is not strong enoughto explain the load support under parallel sliding conditions, see also Cameron(1966). Dowson and Hudson (1963) and Neal (1963) concluded that, ratherthan a thermal wedge, thermal distortion acts as a source for the observedload support.

2.2.4.2 Viscosity wedge

Cameron (1966) analyzed the effect of varying viscosity caused by temperaturegradients across the film. However, Dowson and Hudson (1963) showed that,when considering parallel surfaces, this mechanism will decrease load supportrather than enhance it.

2.2.4.3 Microasperity lubrication

In this case an asperity on the surface acts like a step bearing in the fluid. Thepressure increases when the asperity is approached and decreases when theasperity is left behind. As the pressure tends to decrease below the cavitationpressure, the fluid starts to cavitate, and a net hydrodynamic load support


remains. Pape (1969) showed that microasperity lubrication does not appearto be strong enough to explain the observed hydrodynamic load support inflat parallel faces.

2.2.4.4 Asperity-asperity collisions

Pressure load support is supposed to develop when two asperities of the matingrough surfaces collide in the presence of a lubricant film. This mechanism hasbeen studied extensively by Fowles (1975). Although some load support isobserved in the thin film lubrication regime, it is not enough to explain theobserved hydrodynamic effects in mechanical face seals.

2.2.4.5 Squeeze film

When two faces oscillate in the axial direction, for instance due to vibrationsof the machine itself, fluid pressure can be developed, see Eq. (2.1). Cameron(1966) showed that when the medium to be sealed is compressible, a load iscarried. However, a large excitation is required, as for small movements theload curve is practically symmetrical, resulting in a zero net load when inte-grating over a full cycle. At small amplitudes or at low frequencies, the viscousforces dominate the compressible forces. Fluids are hardly compressible, so thenet pressure would be near to zero. A little load support is generated by cavi-tation of the fluid, as shown experimentally by Parkins and May-Miller (1984).Finally, due to inertia effects, fluid pressure can develop, see e.g. Kuhn andYates (1964) and Kauzlarich (1972). More recently, Lebeck (1987) showedthat in order to develop noticeable fluid pressure, a much higher excitationfrequency is needed than is likely for mechanical face seals.

2.2.5 Force equilibrium

In the previous sections the different pressure generating mechanisms are pre-sented, i.e. hydrostatic and hydrodynamic. The following relation applies, seeFig. 2.11:

pfπ(ro2 − rb

2) + Fs = Kpfπ(ro2 − ri

2) + prestπ(ro2 − ri

2). (2.19)

The left side of Eq. (2.19) represents the load on the face seal, which consists ofthe spring load Fs, plus the fluid pressure pf times the balance area π(ro

2−rb2).This load has to be supported by the mean hydrostatic pressure in the contactarea of the faces, which is defined as the pressure gradient factor K times thefluid pressure pf times the face area (ro

2 − ri2). When the mean hydrostatic

fluid pressure is not capable of supporting the load — i.e. when Kpf < Brpf +


ps — the rest of the load must be supported by prest which consists of materialcontact and/or hydrodynamic fluid pressure. Equation (2.19) can be solvedfor prest and then reads:

prest = pf (Br −K) +Fs

π(ro2 − ri2)= pf (Br −K) + ps, (2.20)

where ps is the spring pressure and the balance ratio Br is defined as inEq. (2.2).

Fs

pf

Face seal

pf prest

roro

rirb

+

Fig. 2.11: Equilibrium of forces acting on a mechanical face seal [afterLebeck (1991)].

2.3 Summary and conclusions

As stated in the introduction of this chapter, when two flat parallel faces slideagainst each other, according to the Reynolds’ equation there is no mechanismpresent which could generate any hydrodynamic fluid pressure. From the lubri-cation theory it is known that, at thin film lubrication, only a small variationin the film thickness in the direction of sliding is enough to generate a consid-erable hydrodynamic load support. In fact, if a mechanical face seal operatesat a film thickness of 1 µm, a variation of the order of 0.1 µm of the flatnesswould be sufficient. It was shown by e.g. Pape (1969) and Lebeck (1984) thatmany factors exist which can cause a variable film thickness. For example,during the lapping of the faces, often some waviness (Fig. 2.12) remains on thesurface as a result of non-axisymmetric loading of the face. Waviness may alsodevelop during the running-in period and the wear process afterwards. Duringoperation waviness may also develop as a result of thermal and mechanicaldistortions, induced by frictional heating and the pressure of the sealed fluid,respectively. So from the above it becomes likely that during operation the

2.3 Summary and conclusions 21

Fig. 2.12: Seal face with a circumferential two wave waviness (exag-gerated).

seals are not really flat and parallel, and that an accidental source for gen-erating fluid pressure will usually be present. In Chapter 3 hydrodynamiclubrication in mechanical face seals will be analysed further.

3. MATHEMATICAL MODEL

3.1 Introduction

In this chapter the development of the theoretical friction model is discussed.With this model the frictional behaviour of a mechanical face seal can be pre-dicted for the complete lubrication range, i.e. from hydrodynamic to boundarylubrication. The calculated coefficient of friction can be plotted in a Stribeckdiagram as a function of the operational conditions and interfacial shapes,i.e. waviness and coning. The model consists of a combination of a contactmodel and the full film lubrication model. For friction, the separation betweenthe opposing surfaces is important. In the full film regime the separation,in combination with the lubricant properties and the operational conditions,determines the shear rate and hence the level of friction. The same holds forthe boundary lubrication regime, where the separation determines the aver-age contact pressure at the interacting asperities and, hence, the height of theshear stress in these contacts.In Section 3.2 the friction model is discussed in general. It can subsequentlybe applied to mechanical face seals. The contact model is based on the workof Greenwood and Williamson (1966); it is discussed in Section 3.3. In orderto determine the separation under full film lubricated conditions, a film thick-ness equation is required. The full film model is described in Section 3.4. InSections 3.4.7 and 3.4.8 some design diagrams for full film friction and full filmleakage, respectively, are presented as a function of dimensionless variableswhich contain the operational and the geometrical parameters.

3.2 Modelling friction

The following sections discuss the friction in the contact of seal faces in thedifferent lubrication regimes.

3.2.1 Friction under full film lubricated conditions

In general, three components contribute to friction under full film lubricatedconditions, i.e. the rolling-or-squeezing component, the sliding component

24 3. Mathematical model

and the geometrical component. Because of the simple sliding situation inmechanical face seals, we are mainly concerned with the sliding component.The other two friction components are negligible; they only become of interestin the case of pure rolling, see Moes (1997).Sliding friction is caused by shear of the lubricant in the contact as a resultof relative movement of the mating faces. In general, the hydrodynamic shearstress can be written as a function of the shear rate γ as:

τH = f(γ), (3.1)

where:

γ ≡ ∂v

∂z. (3.2)

The shear stresses in the lubricant are relatively low. Therefore, the lubricantbehaviour in this thesis will be regarded as Newtonian, or:

τH = ηγ = ηUseal

h. (3.3)

Finally, the total friction force can be obtained by the following relation:

Ff =

∫∫AH

τH dAH =

∫∫AH

ηωr

hdAH , (3.4)

with AH the contact area of the hydrodynamic component and Useal = ωr.In order to determine the friction under full film lubricated conditions withEq.(3.4), the separation, h, is required. However, as mentioned in the intro-duction, a film thickness equation for mechanical face seals is not available. InSection 3.4 the development of such an equation is discussed.

3.2.2 Friction under boundary lubricated conditions

In the boundary lubrication regime the friction is determined by the shearstrength of the protective boundary layers at the surface. The fundamen-tal mechanisms, responsible for forming these layers, are physical adsorption,chemical adsorption and chemical reaction, see e.g. Godfrey (1968). The shearstrength of the boundary layer depends on the way the layer is formed.The friction force in the boundary lubrication regime is defined by:

Ff =N∑i=1

∫∫Aci

τci dAci , (3.5)

3.2 Modelling friction 25

with N the number of asperities in contact, Aci the area of contact of the singleasperity i and τci the shear stress at the asperity contact i. The coefficient offriction fci of a single asperity reads:

fci =τcipci, (3.6)

with pci the pressure in a single asperity. It is assumed that the ratio of τciand pci is constant, fc, for all asperity contacts (Briscoe et al., 1973):

N∑i=1

∫∫Aci

fci pci dAci = fcFC , (3.7)

where FC is the total load carried by all asperities. The value of fc must bedetermined from friction experiments, performed under boundary lubricatedconditions or obtained from experiments, as performed by Briscoe et al. (1973)and Georges et al. (1992), in which the shear stress, τc, of surface layers ismeasured as a function of the contact pressure, pC .

3.2.3 Friction under mixed lubricated conditions

In the previous sections it is shown how the friction force is defined in the hy-drodynamic lubrication regime and in the boundary lubrication regime. Now,by combining both friction components, the friction can be calculated in themixed lubrication regime.The friction force Ff is the sum of the friction force at the interacting asperitiesand the shear force of the hydrodynamic component (Eqs. (3.4) and (3.5)):

Ff =N∑i=1

∫∫Aci

τci dAci +

∫∫AH

τH dAH , (3.8)

with N the number of asperities in contact, Aci the area of contact of the singleasperity i, τci the shear stress at the asperity contact i, AH the contact areaof the hydrodynamic component and τH the shear stress of the hydrodynamiccomponent. The coefficient of friction is defined by:

f =FfFN

. (3.9)

By substituting the Eqs. (3.3) and (3.7) into Eq. (3.8), the following expressionfor the coefficient of friction can be derived:

f =FfFN

=

fcFC +∫∫AH

τH(γ) dAH

FN. (3.10)


In order to solve Eq. (3.10), the load carried by the asperities, FC , as well as theseparation, h, must be calculated. The value of h is needed in the calculationof γ (Eq. (3.3)).

In order to calculate FC the contact model of Greenwood and Williamson(1966) is used. This is discussed in Section 3.3. In Section 3.4 a film thicknessequation for mechanical face seals is presented. This equation is needed tocalculate the separation, h.

3.3 Contact model

As mentioned previously, the contact model is based on the contact model ofGreenwood and Williamson (1966). The G&W model has been shown to bequite accurate.

The G&W model was developed for the contact of two flat planes; one surface isassumed smooth, and the other surface rough, shown in Fig. 3.1. Greenwood

h∗

β smooth surface

centre line rough surface

Fig. 3.1: Schematic representation of contact between a rough surfaceand a smooth surface [after Greenwood and Williamson (1966)].

and Tripp (1970-71) extended the model to two nominally flat rough surfaces.Only elastic deformation of the asperities is considered. Furthermore, it isassumed that the asperities have spherical summits, all with equal and constantradius β. Their heights may vary randomly and the summits are uniformlydistributed over the rough surface with a density ηs, the number of summitsper unit area. The probability that the height of an asperity lies between s ands + ds above some reference plane is given by φ(s) ds. Two reference planescan be distinguished (Fig. 3.2):

1. The mean plane of the summit heights, ds.

2. The mean plane of the surface heights, dh.

3.3 Contact model 27

dd

dh

ds

Fig. 3.2: Schematic representation of the two reference planes, i.e. themean plane of the summit heights and the mean plane of the surfaceheights.

In Fig. 3.2 dd is defined as the distance between the mean plane of the summitheights, ds, and the mean plane of the surface heights, dh, which is character-ized by Whitehouse and Archard (1970) by:

dd = 0.82σ, (3.11)

where σ is the standard deviation of the surface height distribution (RMS).The deformation of an individual asperity is described by the Hertzian equa-tions. The contact radius ai, the area Aci , and the load Fi of an individualasperity contact can be expressed in terms of the compliance w as:

ai = β12w

12i Aci = πβwi Fi =

2

3E ′β

12w

32i (3.12)

where the reduced modulus of elasticity, E ′, is defined as:

2

E ′ =1 − ν1

2

E1

+1 − ν2

2

E2

. (3.13)

The compliance of the summit is the amount which the summit deflects in thedirection normal to the mean plane.If the separation between the smooth surface and the reference plane of thesurface heights of the rough surface is equal to h∗, contact will occur for eachasperity with an original height greater than h∗, see Fig. 3.1.Since w = s − h∗, the normalized expected total area of asperity contact isfound to be (Greenwood and Williamson, 1966):

Ac = πηsβσsAnom

∫ ∞

h∗σs

(s− h∗

σs

)φ(s)ds. (3.14)

with ηs the surface density of the asperities, Anom the nominal contact areaand σs the standard deviation of the height distribution of the summits.


In the same way the load, carried by the asperities, can be found and is givenby (Greenwood and Williamson, 1966):

FC =2

3ηsβσs

√σsβE ′AnomFj

(h∗

σs

)(3.15)

where:

j =3

2and Fj

(h∗

σs

)=

∫ ∞

h∗σs

(s− h∗

σs

)jφ(s)ds, (3.16)

with φ(s) the height distribution function. Most machined surfaces have aGaussian asperity height distribution, Fig. 3.3, which is given by:

φ(s) =1√2π

exp

(−s2

2

). (3.17)

Also a real height distribution of a 3D surface measurement can be imple-

Fig. 3.3: Measured surface asperity height distribution vs. Gaussiansurface asperity height distribution.

mented, see for instance Westeneng (1996). 3D discrete surface scans can be


produced using an advanced contactless interference microscope, described inmore detail by Lubbinge (1994). With the data from this device also otherparameters required, e.g. ηs, β, σs and σ can be calculated. Determination ofηs, β and σs is described by de Rooij (1998).Finally, in order to solve the load carried by asperity interaction, Anom isrequired. If both faces are nominally flat, Anom is simply defined by 1

4π(Do

2 −Di

2). However, if waviness and/or coning are present, which is common inmechanical seal faces, Anom is different. In the next section Anom is determinedas a function of the waviness amplitude, coning angle, the seal dimensions, thereduced modulus of elasticity and the load.

3.3.1 Nominal contact area

The contact of a seal face with coning and waviness can be considered as thecontact between a cylinder, with a length B, at an angle a with a smooth flatsurface, as shown in Fig. 3.4. The cylinder is represented by a parabolicallyshaped body. In this Figure, a is the coning angle, B the width of the seal

a

B

y

x

h

δ

Rx

B/2

Fig. 3.4: Parabolically shaped body, representing a cylinder, on a flatsurface.

face, δ the distance between the “cylinder” and the flat surface at half the sealwidth, B/2, and Rx the radius of curvature of the waviness. The expressionfor Rx is derived below.The waviness on a seal face is described by a cosine function:

h(x) = h0 + A

(1 + cos

(2πk

πDm

x+ π

))= h0 + A

(1 − cos

(2πk

πDm

x

)),

(3.18)


with h0 a constant, A the amplitude of the waviness, k the number of wavesand Dm the mean diameter of the seal face. In order to find the radius ofcurvature at h(x) = h0 for x = 0, minus cosine is taken. Applying a Taylorseries expansion results in:

h(x) = h0 +2Ak2x2

Dm2 . (3.19)

The expression for the gap between a cylinder and a flat surface, approximatedby a parabolic function, reads:

h(x) = h0 +x2

2Rx

. (3.20)

Finally, equating Eq. (3.19) with Eq. (3.20) gives an expression for Rx:

Rx =Dm

2

4Ak2. (3.21)

When the “cylinder” is pressed onto the flat surface, deformation occurs anda contact is formed, defined as Anom. The expression for the gap, h(x, y), is:

h(x, y) = −δ +x2

2Rx− ay +

2

πE ′

∫ ∞

−∞

∫ ∞

−∞

p(x′, y′) dx′ dy′√(x− x′)2 + (y − y′)2︸︷︷︸

elastic deformation

. (3.22)

The derivation of the expression for the elastic deformation can be found inDowson and Higginson (1966). Eq. (3.22) can be solved with the followingcomplementarity conditions:

h · p = 0 and h ≥ 0 and p ≥ 0. (3.23)

Next, by introducing suitable dimensionless groups and variables, the num-ber of parameters in Eq. (3.22) can be reduced significantly. Eq. (3.22) isnormalized using the Hertzian dry contact parameters (Hertz, 1881):

• b, the equivalent radius of the contact, defined as:

b =

(3FNRx

2E ′

)1/3

(3.24)

• c, the compliance, defined as:

c =b2

Rx

(3.25)


• ph, the contact pressure, defined as:

ph =3FN2πb2

(3.26)

Applying the following substitutions:

h = H · c p = P · ph (3.27)

δ = ∆ · c x′ = X ′ · b (3.28)

x = X · b y′ = Y ′ · b (3.29)

y = Y · b, (3.30)

yields the dimensionless form of Eq. (3.22):

H = −∆ +X2

2− aRx

bY +

2

π2

∫ ∞

−∞

∫ ∞

−∞

P (X ′, Y ′) dX ′ dY ′√(X −X ′)2 + (Y − Y ′)2

. (3.31)

The range of the spatial coordinate Y can be derived as follows:

−B2< y <

B

2=⇒ −B

2b< Y <

B

2b. (3.32)

From the dimensionless equations (Eq. (3.31) and (3.32)) it follows that theproblem is a two parameter problem, i.e. the nominal contact area Anom canbe described by two parameters:

αnom =aRx

b=

aRx

3

√3FNRx

2E ′

, (3.33)

βnom =B

b=

B

3

√3FNRx

2E ′

. (3.34)

The variable αnom incorporates the coning, whereas βnom is a measure for theradial seal width. Equation (3.31) is solved numerically, by using multigridtechniques, see e.g. Wijnant (1998). The dimensionless nominal contact area,Anom, is defined as:

Anom =Anom

b2. (3.35)

It is calculated by summing up the number of grid points where the pressureP > 0, and, subsequently, by multiplying this number by the square of thegrid distance (the grid points are equidistant).Figure 3.5 shows some results of the calculations. Based on these data, afit is made for the nominal contact area Anom, which is discussed in the nextsection.


3.3.2 Function fit for the nominal contact area

The fit for Anom can be described by two asymptotes, i.e. Anom as a function ofαnom and Anom as a function of βnom, respectively. Both asymptotes, as shownin Fig. 3.6, are fitted by the following equation:

y =((A · xB)E + (C · xD)E

)1/E, (3.36)

where x is substituted by αnom or βnom, respectively.Subsequently, the two fits are combined in the same way:

Anom =(y(αnom)A + y(βnom)A

)1/A, (3.37)

where A is the fitting parameter.The total fit reads:

Anom =

([((2.27 ∗ αnom

−0.4)−3 + (1.72 ∗ αnom−1)−3

)−1/3]−6

+[(

(3.2 ∗ βnom)−2 + (2.62 ∗ βnom0.5)−2

)−1/2]−6)−1/6

.

(3.38)

The fit is accurate within ±3%. Figure 3.6 shows the contact area, Anom, asa function of αnom for different values of βnom.


−0.5 0 0.5

−1.5

−1

−0.5

0

0.5

1

1.5

Y

X

(a) αnom = 0, βnom = 1 and Anom = 2.029

−0.5 0 0.5

−1.5

−1

−0.5

0

0.5

1

1.5

Y

X

(b) αnom = 0.5, βnom = 1 and Anom = 1.933


−0.5 0 0.5

−1.5

−1

−0.5

0

0.5

1

1.5

Y

X

(c) αnom = 1, βnom = 1 and Anom = 1.516

−0.5 0 0.5

−1.5

−1

−0.5

0

0.5

1

1.5

Y

X

(d) αnom = 5, βnom = 1 and Anom = 0.343

Fig. 3.5: Contact of a “cylinder” with a flat surface for βnom = 1and different values of αnom, with on the left-hand side the pressuredistribution and on the right-hand side the nominal contact area.Note the scaling of the axes.


d(Tx d(T; d(Tn d(T1 d(Td d(( d(d d(1

d(T1

d(Td

d((

d(d

αnom

Anom

0.01

0.1

0.5

1

2

4

10

βnom =

Fig. 3.6: Fit for the nominal contact area, Anom, as a function of αnom

for different values of βnom.


3.4 Full film model

In order to determine the separation, h, for mechanical face seals, a film thick-ness equation is required. A study of the literature showed that, as yet, nosuch equation is available. A numerical program, suitable for journal bearings(Moes, 1997), has been adapted for mechanical face seals. Plots are con-structed, based on the output data of this program. Finally, fits are generatedfrom these plots.In this section the procedure of the development of a film thickness equationfor mechanical face seals is discussed.

3.4.1 Assumptions

The following assumptions apply to the situation for which the film thicknessequation is developed;

• Steady state model. One face is flat, the other face has a coning and/ora sinusoidally shaped waviness with a constant amplitude, resulting innon-time-dependent solutions. The applied load and the fluid pressureare constant.

• The number of waves is always 2, as this is the most common situationin practice.

• The model is restricted to the iso-thermal situation.

• Seal face deformations due to pressure variations are neglected.

• The lubricant is Newtonian and considered incompressible. The viscosityof the lubricant does not depend on the fluid pressure.

• The flow is laminar (as in fluid seals).

3.4.2 Equations

The film thickness equation will be based on a modified Reynolds’ equationin polar coordinates, to which an extra parameter qc, the liquid fraction, hasbeen added (Eq. (3.39)). This is further explained in Section 3.4.3.

1

r

∂

∂θ

(qch

3∂p

∂θ

)+

∂

∂r

(rqch

3∂p

∂r

)= 6ηrω

∂(qch)

∂θ︸︷︷︸wedge term

+ 6ηrqch∂ω

∂θ︸︷︷︸stretch term

+ 12ηr∂(qch)

∂t︸︷︷︸squeeze film term

.

(3.39)

3.4 Full film model 37

In this equation the stretch term can be neglected as the angular velocitydoes not vary with a change of the θ-coordinate. Furthermore, as one face isassumed to be flat, the film thickness will not vary in time and, therefore, alsothe squeeze term can be omitted. Equation (3.39) now reduces to:

1

r

∂

∂θ

(qch

3∂p

∂θ

)+

∂

∂r

(rqch

3∂p

∂r

)= 6ηrω

∂(qch)

∂θ. (3.40)

For an outside pressurized face seal, the following boundary conditions apply,see Fig. 3.7:

p = pa if r = ri

p = pf if r = ro.(3.41)

Furthermore, for the pressure over the entire width of the seal face, the follow-ing applies:

p|θ=0 = p|θ=2π. (3.42)

The film thickness for an outside pressurized seal with a convergent coning(corresponding to an inside pressurized divergent coning) is defined as follows:

h = hmin + a(r − ri)︸︷︷︸taper

+A(1 + cos(kθ))︸︷︷︸waviness

, (3.43)

where hmin is the minimum film thickness, a the coning angle, A the amplitudeof the waviness and k the number of waves on the circumference, see Figs. 3.7and 3.8. The film thickness for an outside pressurized seal with a divergentconing reads:

h = hmin + a(ro − r) + A(1 + cos(kθ)). (3.44)

Equation (3.44) also applies to an inside pressurized seal with convergent con-ing.Furthermore, the equations for the mean fluid pressure pm in the contact, theleakage qm and the friction force Ff read, respectively:

pm =1

π (ro2 − ri2)

∫ 2π

0

∫ ro

ri

p(r, θ) r drdθ, (3.45)

qm = − rm12η

∫ 2π

0

h3∂p

∂r

∣∣∣∣r=rm

dθ, (3.46)


riro

B

a

pa pf

ω

Fig. 3.7: Boundary conditions.

ω

A

hmin

Fig. 3.8: Waviness amplitude.

Ff =1

rm

∫ 2π

0

∫ ro

ri

(qcηωr

h− h

2

∂p

r∂θ

)r2 drdθ. (3.47)

When waviness is present on the face, according to the Reynolds’ equation anegative pressure will occur in the diverging zones. As a fluid cannot with-stand negative pressures below the vapour pressure, the fluid starts to cavi-tate. Mostly, researchers solved this problem by applying the Half-Sommerfeldcondition and letting p = 0 in those regions where negative pressures are cal-culated, see e.g. Pape (1968) and Etsion (1980). However, by doing so, thecontinuity of flow across the cavitation boundary is not satisfied. The next sec-tion discusses the cavitation boundary conditions used in the present model.


3.4.3 Cavitation boundary conditions

A “free boundary” is introduced at the exit of the pressurized film, defined bythe extra conditions (Reynolds or Swift-Stieber conditions):

∂p

∂θ= 0 and

∂p

∂r= 0, (3.48)

and so the continuity condition is satisfied. This condition may, however, leadto problems in numerical calculations (Moes, 1997). Fortunately, as shownby Christopherson (1941), the same result is obtained when the conditionp < pcav is not permitted during numerical relaxation of the discrete Reynolds’equation.

As lubricant starvation due to cavitation cannot be simulated in this way,other conditions are required. Suppose that cavitation occurs and that a puredrag flow of a mixture of liquid and gas occurs in the entire cavity, thenhydrodynamic lubrication may be defined in terms of the complementaritycondition (Kostreva, 1984). Then, for a completely filled gap, the Reynolds’equation must be supplemented by the initial conditions:

qc = 1,∂qc∂t

(p− pcav) = 0,∂qc∂t

≤ 0 and p− pcav ≥ 0. (3.49)

Immediately after the gap is no longer completely filled with lubricant, rup-tured areas occur with qc < 1, and the following conditions apply:

(1 − qc)(p− pcav) = 0, 1 − qc ≥ 0 and p− pcav ≥ 0. (3.50)

The cavitated fraction (1 − qc) represents a cloud of evenly distributed tinycavitation bubbles, containing a mixture of vaporized liquid and exuded gases.Now, according to the conditions at the cavitation boundaries (Eq. (3.50)), thecontinuity of the flow will be maintained, and therefore the Reynolds cavitationboundary condition will be satisfied simultaneously.

3.4.4 Dimensionless variables

As shown in Section 3.4.2, the film thickness equation depends on a largenumber of parameters. By substituting the following parameters:

r = y, ∂r = ∂y, θ =x

ro, ∂θ =

∂x

roand ri = ro −B, (3.51)

the following equations result:


Reynolds’ equation:

∂

∂x

(qch

3 ∂p

∂x

)+

y

ro2∂

∂y

(qcyh

3 ∂p

∂y

)= 6ηy2 ω

ro

∂qch

∂x(3.52)

Film thickness:

h = hmin + a(y − r0 +B) + A

(1 + cos

(kx

ro

))(3.53)

Mean pressure:

pm =1

roπ (2roB − B2)

∫ 2πro

0

∫ ro

ro−Bp y dydx (3.54)

Leakage:

qm = − 1

12η

(1 − B

2ro

)∫ 2πro

0

h3 ∂p

∂y

∣∣∣∣r=rm

dx (3.55)

Friction force:

Ff =2

2ro −B

1

ro

∫ 2πro

0

∫ ro

ro−B

(qcηωy

h− hro

2

∂p

y∂x

)y2 dydx (3.56)

By defining a number of non-dimensional variables, the amount of compu-tational work required can be reduced. An algorithm, Optimum SimilarityAnalysis, developed by Moes (1992), is used to determine a set of dimension-less variables. Based on the former equations and variables, the followingdimensionless variables are obtained:

Variables:

X =x

B(3.57)

Y =y

B(3.58)

P =p

pm(3.59)

H =h

B

√pmηω

(3.60)

qc (3.61)


Input parameters:

Coning:

α = a

√pmηω

(3.62)

Waviness:

γ =A

B

√pmηω

(3.63)

Fluid pressure:

Pf =pfpm

(3.64)

Seal face geometry:

ψ =roB

(3.65)

Cavitation pressure:

ρc =

∣∣∣∣ pcpm∣∣∣∣ (3.66)

Number of waves:

k (3.67)

Output parameters:

Minimum film thickness:

Hmin =hmin

B

√pmηω

(3.68)

Leakage:

Qm =qmη

B3pm

(√pmηω

)3

(3.69)


Friction force:

Ff =FfB2pm

√pmηω

(3.70)

Rewriting Eqs. (3.52) to (3.56) in terms of these non-dimensional variablesyields:

Reynolds’ equation:

∂

∂X

(qcH

3 ∂P

∂X

)+Y

ψ2

∂

∂Y

(qcY H

3∂P

∂Y

)= 6

Y 2

ψ

∂qcH

∂X(3.71)

Film thickness:

H = Hmin + α(Y − ψ + 1) + γ

(1 + cos

(kX

ψ

))(3.72)

Mean Pressure:

1 =1

ψπ (2ψ − 1)

∫ 2πψ

0

∫ ψ

ψ−1

P Y dY dX (3.73)

Leakage:

Qm = − 1

12

(1 − 1

2ψ

)∫ 2πψ

0

H3∂P

∂Y

∣∣∣∣Y=ψ−0.5

dX (3.74)

Friction force:

Ff =2

2ψ − 1

1

ψ

∫ 2πψ

0

∫ ψ

ψ−1

(qcY

H− Hψ

2

∂P

Y ∂X

)Y 2 dY dX (3.75)

The Reynolds’ equation (Eq. (3.71)) cannot be solved analytically, as it con-tains a double partial derivative of P with respect to coordinates X and Y .Hence, a numerical solver was used. The numerical program was based onmultigrid techniques. For detailed information concerning the numerical as-pects, the reader is referred to Wijlhuizen (1997). In Appendix A the analyticalsolution, in non-dimensional variables, is given for flat parallel faces and forfaces with a radial taper.


3.4.5 Numerical results

In this section we discuss the results obtained with the numerical program. InFigs. 3.9 (a) to (d) some 3-dimensional pressure distributions of the numericalresults are presented. In the Figures the effect of coning and sealed fluidpressure is clearly shown. Cavitation occurs in the blue-coloured areas.In Table 3.1 the ranges of the dimensionless input parameters defined inEqs. (3.62) to (3.67), according to both theory and practice, are presented.

Table 3.1: Ranges of the non-dimensional parameters.

Non-diml. Theoretical Practicalparameter lower upper lower upper

γ −∞ ∞ 0 ≈ 100α 0 ∞ 0 ≈ 5ψ 1 ∞ 6 13Pf 0 2 0 1.75ρc 0 ∞ 0 0.5k 0 ∞ 2 2

The following sections concern the effect of these dimensionless parameters onthe film thickness. The results all relate to an outside pressurized seal, witha parallel gap or a convergent coning, i.e. a narrowing gap in the direction offlow of the fluid. In the numerical program, however, an arbitrary number ofwaves, k, can be chosen, the seal configuration may be inside pressurized andthe coning divergent.

3.4.5.1 γ- and α-dependence

Figure 3.10 shows plots of the minimum film thickness Hmin as a function ofthe waviness γ. In each graph, plots for different values of the coning α and fortwo different values of Pf and ψ are presented. When fluids with small fluidpressures have to be sealed, Pf = 0.5 in Fig. 3.10 (a), the shape of the curves isparabolic. When γ is increased, the waviness amplitude increases (supposingthe other parameters are kept constant), resulting in an increase of Hmin. Sowith an increase of γ, more hydrodynamic pressure is generated. At a certainvalue of γ an optimum is reached. When γ is further increased, Hmin starts todecrease, indicating that less hydrodynamic fluid pressure is generated. It isfound that a parabolically shaped curve is always generated when Pf rangesbetween 0 and 1.


In Fig. 3.10 (b) it is clearly shown that when the fluid to be sealed has a ratherlarge pressure (Pf = 1.5), the coning angle α is very important with regard tothe minimum film thickness Hmin (left side of graph). A larger coning angle αresults in a larger Hmin, as made clear in Section 2.2.3.2. When γ increases,Hmin decreases. An increasing waviness amplitude results in a larger mean filmthickness, which results in less hydrostatic fluid pressure. Furthermore, it isobserved that at a certain value for γ a point of inflection in the curve occurs,see e.g. α = 0.75 and α = 1, indicating that the seal starts to cavitate and willgenerate a net hydrodynamic fluid pressure. Without cavitation, there is nohydrodynamic fluid pressure and Hmin would collapse, as shown in Fig. 3.16.

3.4.5.2 Pf-dependence

The pressure of the fluid to be sealed also greatly influences the minimum filmthickness. The sealed fluid pressure mainly affects the interfacial hydrostaticfluid pressure; a larger fluid pressure will result in a higher hydrostatic inter-facial pressure, as shown by the left side in both graphs of Fig. 3.11. Whenhydrodynamic pressure starts to develop, the negative effect of the coning onthe hydrodynamic pressure generation is stronger than the positive effect ofconing on the development of hydrostatic fluid pressure, as a lower maximumHmin results for α > 0.

Furthermore, it is shown in both graphs that the higher Pf , the later the sealstarts to cavitate; the top of the curves is shifted to the right with a higher fluidpressure Pf . When a higher hydrostatic interfacial fluid pressure is present,it becomes harder to reach the cavitation pressure of the fluid. Therefore thehydrodynamic pressure generating mechanism must be stronger, i.e. a greaterwaviness amplitude is required.

3.4.5.3 ψ-dependence

Figure 3.12 shows Hmin for three different values of the seal face geometryparameter ψ. In (a) there is almost no hydrostatic component present, asα and Pf are small. On the left-hand side of the graph, no effect of ψ onthe hydrodynamic pressure is visible. On the right-hand side, a decrease ofψ results in a more rapid decrease of Hmin (the curvature increases), which isalso shown in (b). Here, greater values for α and Pf are taken, so a greaterhydrostatic pressure is developed (left-hand side of graph). With a decreasingψ, the curvature of the seal face increases, resulting in a higher hydrostaticpressure, and therefore a higher Hmin, see also Section 2.2.3.1.


3.4.5.4 ρc-dependence

The cavitation parameter ρc appears to have a very small effect on Hmin.Figure 3.13 (a) shows that a lower ρc value results in a somewhat higher Hmin.A lower ρc involves a lower cavitation pressure, resulting in a higher net hydro-dynamic pressure. The cavitation parameter ρc has, as expected, no effect inthe hydrostatic region (Fig. 3.13 (b)). As the effect of ρc on the film thicknessis very small, ρc is taken constant at a value of 0.097 (pcav = −97 kPa andpm = 1 MPa, both relative to ambient pressure).


(a) α = 0, γ = 0.5, ψ = 4, k = 2, ρc = 0.097, Pf = 0 andHmin = 0.23.

(b) α = 0.5, γ = 0.5, ψ = 4, k = 2, ρc = 0.097, Pf = 0 andHmin = 0.025.


(c) α = 0, γ = 0.5, ψ = 4, k = 2, ρc = 0.097, Pf = 1 andHmin = 0.32.

(d) α = 0.5, γ = 0.5, ψ = 4, k = 2, ρc = 0.097, Pf = 1 andHmin = 0.13.

Fig. 3.9: Pressure distribution in the contact of a mechanical face seal,for different values of the dimensionless parameters, which are de-rived in the previous section.


d(T1 d(Td d(( d(d d(1

(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

Hmin

γ

α =0

0.1

0.2

0.3

0.4

0.5

(a) Pf = 0.5, ψ = 9.5, k = 2 and ρc = 0.097.


d(T1

d(Td

d((

d(d

((�d

(�1

(�n

(�;

(�xd

1

x

((�d(�1(�n(�;(�x(��x

d

(��x

Hmin

γ

α =

α =

(b) Pf = 1.5, ψ = 6, k = 2 and ρc = 0.097.

Fig. 3.10: Minimum film thickness, Hmin, as a function of the waviness,γ, for different values of the coning angle, α.



(�(

(�d

(�1

(�n

(�;

(�x

(�E

(��

1.75

1.5

1.251

0.50

Pf =

Hmin

γ

(a) α = 0, ψ = 6, k = 2 and ρc = 0.097.


(�(

(�d

(�1

(�n

1.25

1

0.750.5

0

Pf =

Hmin

γ

(b) α = 0.3, ψ = 9.5, k = 2 and ρc = 0.097.

Fig. 3.11: Minimum film thickness, Hmin, as a function of the waviness,γ, for different values of the sealed fluid pressure, Pf .



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

13

6

9.5

ψ =

Hmin

γ

(a) α = 0.2, Pf = 0.75, k = 2 and ρc = 0.097.


d(Td

d((

d(d

13

13

6

6

9.5

9.5

ψ =

ψ =

Hmin

γ

(b) α = 0.5, Pf = 1.75, k = 2 and ρc = 0.097.

Fig. 3.12: Minimum film thickness Hmin as a function of the waviness,γ, for different values of the seal geometry, ψ.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(�nx

Hmin

γ

0

0.1

0.2

0.3

0.4

α =ρc = 0.0485ρc = 0.097

(a) Pf = 0.5, ψ = 9.5 and k = 2.

d(Tn d(T1 d(Td d(( d(d d(1

d(T1

d(Td

d((

d(d

Hmin

γ

1

0.5

α =

ρc = 0.097ρc = 0

(b) Pf = 1.75, ψ = 6 and k = 2.

Fig. 3.13: Minimum film thickness Hmin as a function of the waviness,γ, for different values of the cavitation parameter, ρc.


3.4.6 Film thickness equation

Based on the numerical results, three different film thickness equations will befitted for the conditions Pf = 0, 0 < Pf ≤ 1 and 1 < Pf ≤ 1.75, respectively.This is done because for these three Pf -values the shapes of the curves differconsiderably. Firstly, the asymptotes for respectively γ = 0, α = 0 and Pf = 0are derived.

3.4.6.1 Asymptotes

Film thickness for γ = 0

In Section 2.2.3.2 an equation for the pressure distribution is given in cartesiancoordinates for a seal face with no waviness and a convergent gap, Eq. (2.17).The mean pressure in the contact, Pm is obtained by integrating Eq. (2.17)from 0 to 1 and reads:

Pm = PfHmin + α

2Hmin + α(3.76)

The mean pressure in cartesian coordinates can also be defined as follows:

pm =1

LB

∫ B

0

∫ L

0

p dxdy. (3.77)

Rewriting this equation in a dimensionless form, by substituting the dimen-sionless variables of Eq. (2.12), yields:

1 =1

L

∫ 1

0

∫ L

0

PdXdY, (3.78)

so the mean pressure Pm is equal to 1. Equation (3.76) can now be solved forHmin and reads:

Hmin = −αPf − 1

Pf − 2. (3.79)

This is the minimum film thickness for ψ approaching infinity, in polar coor-dinates.Since, when ψ approaches infinity, the minimum film thickness has to approachEq. (3.79) for any value of α and Pf , Hmin can be written as follows:

Hmin = −α Pf − 1

Pf − 2g(ψ, Pf ), (3.80)


with:

g(ψ, Pf ) = 1 +1

g2(ψ, Pf), (3.81)

a function which approaches 1, when ψ approaches infinity. The functiong2(ψ, Pf ) approaches infinity when ψ approaches infinity. Based on the dataof the numerical program, the following film thickness equation for γ = 0 isfound:

Hmin|γ=0 = −αPf − 1

Pf − 2

(1 +

1

5.41 + 0.794ψ1.5 − Pf (3.09 + 0.405ψ1.5)

).

(3.82)

Film thickness for α = 0

The basic equation which describes the film thickness for α = 0 and for Pfranging from 0 to 1.75 is the so-called Asymmetric Double Sigmoidal equation,which is defined as:

Hmin =a

1 + exp(−(x− b+ c

2)/d) (1 − 1

1 + exp(−(x− b− c

2)/e)) . (3.83)

This equation is fitted on the numerical data, resulting in the following valuesfor the different coefficients:

a =1

3.20 − 0.433 exp(Pf )

b =

(1

2.95 − 5.46/√ψ

+ exp

(−6.02 + 22.6

lnψ

ψ

)Pf−(

−0.637 +11.1

ψ

)2

Pf2 + (exp(4.04 − 3.29 lnψ) − 0.1)Pf

3

)−1

c = exp(1.50 + 0.313

√Pf

)− 0.0163ψ2 + 0.539ψ − 2.64

d = 0.661 + 0.0297 exp

(Pf

0.478

)e = 0.967.


3.4.6.2 Film thickness for Pf = 0

In this case, the fluid to be sealed is pressureless, so the seal faces will operatein a purely hydrodynamic way. The film thickness equation for Pf = 0 reads:

Hmin =a

1 + exp(−(ln(γ) − b+ c

2)/d) (1 − 1

1 + exp(−(ln(γ) − b− c

2)/e)) ,(3.84)

where:

a = exp (1.39 − 2.58 exp(α))

b =1

0.499 + 35.0/ψ2 + (1/(1.03√ψ − 2.25))α2

c =√

20.6 − 30.3α− 0.0163ψ2 + 0.539ψ − 2.64

d =1

1.62 + 4.99α

e = exp(0.116 − 0.956

√α).

The maximum deviation that can occur between the fit and the numericalresults is 10%. Figure 3.14 shows some plots of the numerical results and thefitted data for ψ = 6 and ψ = 9.5, respectively.

3.4.6.3 Film thickness for 0 < Pf ≤ 1

The curve-fit is based on Eq. (3.84), the same equation as used for Pf = 0.The expressions for the coefficients read:

a = exp( (

1.20 − 0.538√Pf

)2

−(0.773 + 1.82 exp(−Pf)) exp(α)

)b =

1 − (0.671 + 0.297Pf2.5)2α

0.848 + 0.519Pf − 0.282Pf2 − 0.0654Pf

3

−1.68 + 0.235ψ − 0.00612ψ2

c = exp(1.51 + 0.313

√Pf − (2.44 − 0.670 exp(Pf))α

1.5)

−0.0163ψ2 + 0.539ψ − 2.64

d =1

1.42 − 0.341Pf2 + exp

(1.80 − 1.95Pf

2.5)α

e =1

2.90 − 1.86 exp(−α).


The maximum deviation that can occur between the fit and the numericalresults is 15%. Figure 3.15 shows examples of numerical data and fitted data.

3.4.6.4 Film thickness for 1 < Pf ≤ 1.75

As shown by Fig. 3.10 (b) the shape of the curve changes completely whenthe hydrostatic pressure becomes more relevant. The shape of these kinds ofcurves is well described by the following equation:

Hmin =(Hstat

3 +Hdyn3)1

3 , (3.85)

where Hstat is an expression for the hydrostatic region and Hdyn is defined byEq. (3.84).

Hydrostatic region

In order to eliminate the fluid pressure due to hydrodynamic effects, the nu-merical program has been adapted. It was assumed that the pressure in thecontact can drop below the vapour pressure, so that the hydrodynamicallygenerated negative pressure equals the hydrodynamically generated positivepressure. Figure 3.16 shows plots of Hmin with and without cavitation, i.e.with and without hydrodynamic effects. It is clearly shown that, when cavita-tion of the fluid occurs, a larger Hmin is maintained (the dashed line). Withoutcavitation, Hmin decreases rapidly with increasing γ (the solid line).

The expression for Hstat was found to be:

Hstat = Hmin|γ=0 exp

(− 1

Hmin|γ=0

γ

), (3.86)

where Hmin|γ=0 is given by Eq. (3.82).

Complete region

Unfortunately, the film thickness fit in the hydrodynamic region could not befound for ψ. Therefore, ψ is taken constant at a practical mean value of 9.5.For ψ = 9.5 the following components for Eq. (3.85) were derived:

Hmin|γ=0,ψ=9.5 = −αPf − 1

Pf − 2

(1 +

1

28.7 − 14.9Pf

). (3.87)


The coefficients for the expression of Hdyn, Eq. (3.84), were found to be:

a = exp

(1

0.823 + 0.585 ln(Pf )− 1 −√

−4.75 +6.37√Pf

exp(α)

)

b =

(−2.92 + 28.6 exp

( −Pf0.366

))α−

1

−1.08 + 0.0944Pf3

c = exp

(2.00 −

(0.794 − 3.59

ln(Pf)

Pf

)α1.5

)d =

11

0.451 + 0.287Pf2 + 0.530α

e =1

2.90 − 1.86 exp(−α).

The maximum deviation that can occur between the fit of Eq. (3.85) and thenumerical results is 20%. Figure 3.17 shows some curve fits for Pf = 1.5 andα = 0, 0.6 and 1, respectively.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

0

0.1

0.2

0.3

0.4

α =

Numerical dataFitted data

Hmin

γ

(a) Pf = 0, ψ = 6.


(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

0

0.1

0.2

0.3

0.4

α =


Hmin

γ

(b) Pf = 0, ψ = 9.5.

Fig. 3.14: Minimum film thickness as a function of the waviness, γ,for different values of the coning angle, α. Numerical data vs. fitteddata.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(�nx

0

0.1

0.2

0.3

0.4

0.5

α =


Hmin

γ

Fig. 3.15: Minimum film thickness as a function of the waviness, γ,for different values of the coning angle, α. Numerical data vs. fitteddata, for: Pf = 0.75, ψ = 13.


d(T1

d(Td

d((

d(d

1

0.5

0.1

α =

Hmin

γ

Without cavitationWith cavitation

Fig. 3.16: Minimum film thickness as a function of the waviness, γ, fordifferent values of the coning angle, α. With and without cavitation,for: Pf = 1.75, ψ = 6.



d(T1

d(Td

d((

d(d

10.6

0

α =


Hmin

γ

Fig. 3.17: Minimum film thickness as a function of the waviness, γ,for different values of the coning angle, α. Numerical data vs. fitteddata, for: Pf = 1.5, ψ = 9.5.


3.4.7 Friction under full film lubricated conditions

This section discusses briefly some numerical results for the friction force, Ff ,between the faces. Figure 3.18 shows Ff for three different values for ψ. Anincrease in ψ results in an increase in Ff , see also Eq. (3.75).Figure 3.19 shows the effect of α on Ff . In the hydrostatic region significantdifferences are found. In this region a larger coning angle will result in ahigher Hmin, and therefore a decrease in Ff . Also, Fig. 3.19 shows that, assoon as the seal starts to cavitate, there is hardly any further effect of α onFf . Besides that, Ff decreases rapidly with increasing γ. With an increase ofγ, the cavitation area increases too, thus the area where the gap is completelyfilled with fluid decreases. Since the viscosity of a liquid is of the order of 1000times larger than the viscosity of a gas, clearly, as the cavitation area increasesFf will decrease.

3.4.8 Leakage under full film lubricated conditions

As shown by Eq. (3.74), the leakage depends on the film thickness to the thirdpower. In the hydrostatic region the film thickness is determined by the coningangle and not by the waviness. A larger coning angle results in a larger filmthickness and, consequently, a larger leakage rate will occur, see Fig. 3.20.For larger γ the leakage increases, as now the mean film thickness also stronglyincreases. The geometrical parameter ψ has a relatively small effect on theleakage, see Fig. 3.21. An increase of ψ results in a larger seal circumferenceand therefore a slightly greater leakage. In Fig. 3.22 it is shown that, as wellas ψ, another phenomenon is responsible for the leakage, as the leakage forψ = 6 is higher than for ψ = 13 in the hydrostatic region, but lower in thehydrodynamic region. In Fig. 3.12 (b) it is shown that a lower ψ-value resultsin a higher Hmin in the hydrostatic region. Apparently, for Pf = 1.75 the effectof the increase of Hmin with a decreasing ψ is stronger than the decrease ofcircumference for decreasing ψ.Figure 3.23 shows the leakage for different values of Pf . As expected, anincrease in Pf results in a greater leakage.



d(d

d(1

d(n

13

9.5

6

ψ =

Ff

γ

Fig. 3.18: Friction force, Ff , as a function of the waviness, γ, for dif-ferent values of the seal geometry, ψ, for: Pf = 1.75, α = 0.5, k = 2and ρc = 0.097.


d(d

d(1

d(n

d(;

((�d(�1(�n(�;(�x

d

1

x

Ff

γ

α =

Fig. 3.19: Friction force, Ff , as a function of the waviness, γ, for dif-ferent values of the coning angle, α, for: Pf = 1.5, ψ = 9.5, k = 2and ρc = 0.097.



d(T1

d((

d(1

d(;

d(E

d(M

x

1

(�x

(�;

(�1

(�d

(

Qm

γ

α =

Fig. 3.20: Leakage as a function of the waviness, γ, for different valuesof the coning angle, α, for: Pf = 1.75, ψ = 9.5, k = 2 and ρc = 0.097.


d((

d(1

d(;

d(E

d(M

dn��xE

ψ =

Qm

γ

Fig. 3.21: Leakage as a function of the waviness, γ, for different valuesof the seal geometry, ψ, for: Pf = 1.5, α = 0.75 and ρc = 0.097.



d(1

d(;

d(E

d(M

dn��x

E

dn��xE

ψ =

ψ =

Qm

γ

Fig. 3.22: Leakage as a function of the waviness, γ, for different valuesof the seal geometry, ψ, for: Pf = 1.75, α = 0.5, k = 2 and ρc =0.097.


d(T1

d((

d(1

d(;

d(E

d(M

d��x

d�xd�1x

Pf =

Qm

γ

Fig. 3.23: Leakage as a function of the waviness, γ, for different valuesof the sealed fluid pressure, Pf , for: α = 0.1, ψ = 6, k = 2 andρc = 0.097.


3.5 Calculating Stribeck curves

In the preceding sections, the nominal contact area and the film thicknessequation for mechanical face seals were derived. The Stribeck curves can becalculated by combining the contact model and the film thickness equation.In Section 3.5.1 the procedure for calculating the Stribeck curve is discussed.In Section 3.5.2 results of the calculations are shown and the effects of thedifferent parameters, like waviness, coning and contact model parameters areanalyzed.

3.5.1 Procedure

The total axial load FN , which acts on the contact of the mating faces inthe mixed lubrication regime, is shared between the asperity contact and thegenerated hydrodynamic force, therefore:

FN = FC + FH , (3.88)

FC is the load carried by the interacting asperities, defined by Eq. (3.15), andFH the load carried by the hydrodynamic component.Dividing Eq. (3.88) by the nominal contact area Anom, which is determined inSection 3.3.1, gives an expression for the pressure:

pT = pH + pC . (3.89)

As shown in Fig. 3.24, in the mixed lubrication regime the total pressure, pT ,can be divided into the asperity pressure pC and the hydrodynamic pressure,pH , according to Johnson et al. (1972). Based on Fig. 3.24, the followingrelation may be defined for pT :

pT = γMLpH , (3.90)

with γML a constant equal or greater than unity. The dimensionless variablesin Section 3.4.4, which are used in the film thickness equations (3.84) and(3.85), have to be adapted according to Eq. (3.90):

α → α

√1

γML

ψ → ψ

γ → γ

√1

γML

Hmin → Hmin

√1

γML

(3.91)

Pf → Pf

As the friction is determined in the nominal contact area, the film thicknessequation, Eq. (3.72), simplifies to H = Hmin.

3.5 Calculating Stribeck curves 65

pT

pH

pC

Fig. 3.24: Pressure distribution in the mixed lubrication regime.

Finally by rewriting Eq. (3.90) as pH = pT/γML and substituting it into rela-tion (3.89), the following equation results:

pT/γML − pT + pC = 0. (3.92)

This equation can be solved iteratively, as there are 3 equations (Eq. (3.15),(3.84) or (3.85) and (3.88)) with 3 variables (PC , PH and h).

The coefficient of friction can be calculated, using Eq. (3.10). The mean shearstress due to hydrodynamic lubrication can be substituted by Eq. (3.3). AHis defined as Anom − Ac, with Ac defined by Eq. (3.14):

f =fcFC + ηUseal

hAH

FN. (3.93)

The dimensionless film thickness Hmin, which is obtained from Eq. (3.84) or(3.85), is converted to the dimensional film thickness h, which is equated withh∗ in Eq. (3.15). Furthermore, in the calculations the mean plane of the surfaceheights is used, rather than the mean plane of the summits, see Section 3.3.Therefore dd has to be subtracted from the separation in Eq. (3.15), whichbecomes:

FC =2

3ηsβσs

√σsβE ′AnomFj

(h∗ − ddσs

). (3.94)


3.5.2 Preliminary model results

When calculating the Stribeck curve, there are a number of parameters whichcan be varied. In the following sections the effect of the different parameterson the behaviour of the Stribeck curve are shown. When a specific param-eter is changed, the other model parameters are kept constant with valuesgiven in Table 3.2. Geometrical effects on the Stribeck curve are presented

Table 3.2: Operational conditions.

ηs β σs fc E ′ B Do FN Pf

[m−2] [m] [m] [–] [Pa] [m] [m] [N] [–]

×1010 ×10−5 ×10−8 ×1011 ×10−3 ×10−3

1.45 5.59 6.1 0.25 4.4 6 82 240 0

in Sections 3.5.2.1 and 3.5.2.2 (macro-geometry), whilst the micro-geometricaleffects on the Stribeck curve are given in Section 3.5.2.3. The effect of theoperational parameters, load and temperature, i.e. viscosity, on the Stribeckcurve are presented in Sections 3.5.2.5 and 3.5.2.6, respectively. The influenceof the material property, E ′, is given in Section 3.5.2.7. In Section 3.5.2.8 theeffect of the hydrostatic sealed fluid pressure is shown.

3.5.2.1 Waviness amplitude, A

By changing the waviness amplitude, the hydrodynamic pressure generationis affected. The coefficient of friction in the boundary lubrication regime, fcis taken as 0.25, which is typical for silicon carbide/silicon carbide seal facecombinations, see e.g. Summers-Smith (1988).The different values, taken for the waviness amplitude, A, are presented inTable 3.3. In Figure 3.25 the coefficient of friction, f , and the separation, λ,

Table 3.3: Different values chosen for the waviness amplitude, A. Thenumbers 1–13 correspond with the numbers in Fig. 3.25.

1 2 3 4 5 6 7 8 9 10 11 12 13

A [µm] 0.5 0.75 1 2 5 10 15 20 25 30 35 40 50

are plotted as a function of the velocity, where λ is defined as the ratio of the


film thickness, h, and the standard deviation of the height distribution of thesummits, σs. The numbers 1 to 13 in Fig. 3.25 correspond with the numbers inTable 3.3. It is shown that the waviness amplitude has a rather small effecton the transition from hydrodynamic lubrication (HL) to mixed lubrication(ML). Figure 3.25 (a) shows that the transition velocity, vt, acquires its lowestvalue for A = 2µm. For both lower and higher amplitudes the transitiontakes place at higher velocities. When the amplitude is further increased, seeFig. 3.25 (b), the transition from HL to ML shifts even further to the right.For high speeds the separation, λ, no longer increases, but remains constant,as shown for A = 0.5µm (number 1 in the Figure).

3.5.2.2 Coning angle, a

In Fig. 3.26 a number of Stribeck curves (solid lines) are presented for differentvalues of the coning angle, a. In Fig. 3.26 (a) the amplitude of the wavinessis taken to be 1µm, in Fig. 3.26 (b) the amplitude is taken to be 10µm. InTable 3.4 the different values for the coning angle are shown, the numbers 1 to9 correspond with the numbers in Fig. 3.26 (a) and (b). The coning angle has a

Table 3.4: Different values chosen for the coning angle, a. The numbers1–9 correspond with the numbers in Fig. 3.26.

1 2 3 4 5 6 7 8 9

a [rad] × 10−5 0 1 2.5 5 7.5 10 15 20 25

rather large effect on the transition from HL to ML, especially in Fig. 3.26 (a),where a smaller waviness amplitude of 1µm is chosen. A larger coning angle isunfavourable with regard to hydrodynamic pressure generation, and therefore,as expected, the transition HL–ML shifts to the right for higher coning angles.The separation, λ, becomes smaller for higher coning angles. As shown inFig. 3.25 (a), an amplitude of A = 10µm can lead to higher separations thanan amplitude of A = 1µm. As a consequence the transitions HL–ML will occurat lower speeds, as shown in Fig. 3.26 (b).

3.5.2.3 Roughness

ηsβσs:

From the literature it is known that the product of ηsβσs is about 0.05. InFig. 3.27 the effect of this product is presented. Three different values are


chosen, ηsβσs =0.03, 0.05 and 0.07, respectively. In this product, only the ηsand the β are varied, both with the same factor. The operational conditions,including σs, are kept constant (see Table 3.2). Furthermore, two differentvalues are chosen for the amplitude as well as for the coning. In Fig. 3.27 (a),A = 1µm and a = 0 rad and in Fig. 3.27 (b) A = 10µm and a = 1.5×10−4 rad.It is shown that the product of ηsβσs hardly has any effect on the shape andthe transitions of the Stribeck curve. In Fig. 3.27 (a) the transition moves alittle to the right for a higher value of ηsβσ. Besides that, the separation, λ,is a little greater for a larger value of ηsβσs in the boundary and the mixedlubrication regime, as for a greater value of ηsβ, when both are increased bythe same factor, the stiffness of the surface is greater. As a consequence, morehydrodynamic pressure has to be generated in order to obtain separation ofthe faces, which results in a small shift to the right for the HL–ML transition.In Fig. 3.27 (b) the effect is even smaller. The transition shifts very little tothe right with increasing values of ηsβσ.

σs:

The ratio of h to σs determines the lubrication regime. With an increasingvalue of λ the lubrication regime changes from boundary to mixed and frommixed to hydrodynamic lubrication. In this section the standard deviation ofthe height distribution of the summits, σs, is varied. The other parameters arekept constant, and are given in Table 3.2. The product of ηsβσ is kept constantat 0.05. When σs is multiplied by a factor x, ηs and β are both divided by afactor

√x.

In Fig. 3.28 two different combinations of the amplitude and the coning arechosen, in Fig. 3.28 (a) A = 1µm and a = 0, in Fig. 3.28 (b) A = 10µm anda = 1.5 × 10−4 rad. Six different values voor σs are chosen, which are given inFig. 3.28.As expected, for larger values of σs the transition from HL to ML shifts tothe right. A larger film thickness h, and with that more hydrodynamic fluidpressure has to develop before full separation of the faces occurs.

3.5.2.4 Non-Gaussian height distribution; χ2n-height distribution

In the friction model a Gaussian height distribution is assumed. However,when rough surfaces run in, the summits flatten, and a roughness distributionwith a negative skewness remains, see e.g. Lubbinge (1994). Figure 3.29 showsa profile measurement of a run-in seal face with flattened summits. An exam-ple of a non-Gaussian height distribution function is the so-called M -invertedχ2n distribution function, see Adler and Firman (1981). This distribution func-

tion is particularly suitable for studying the effect of negative skewness. The


probability density function of an M -inverted χ2n-distribution is defined as:

φχ2n(s) =

exp(−M−s

2N

)2

n2N

n2 Γ(n2

) (M − s)n2−1, (3.95)

with:

M =n√2n

and:

N =1√2n. (3.96)

Figure 3.30 shows the effect of n on the χ2n-height distribution. For large values

of n the Gaussian height distribution is approached, for smaller values of n theskewness becomes more negative and the kurtosis increases, see also de Rooij(1998).In Fig. 3.31 some Stribeck curves are calculated for different shapes of the χ2

n-height distribution, by varying n. The different values for n are indicated in theFigure. In the case of no coning, Fig. 3.31 (a), a lower value for n results in alower transition velocity vt. As for lower values of n the χ2

n-height distributionbecomes steeper (right tail of curve 1 in Fig. 3.30), sooner full separation ofthe faces occurs, resulting in lower vt. In Fig. 3.31 (b), with coning, the sametrend is found, however the effect of the different height distributions is small.As expected from Fig. 3.30, the Stribeck curve for n = 100 is the same asStribeck curve no. 3 in Fig. 3.25.

3.5.2.5 Axial load FN

In Fig. 3.32 the axial load FN is varied. In Table 3.5 the different values for FNare given. The numbers 1 to 5 correspond with the numbers in the figure. Twodifferent combinations of the amplitude and the coning angle are taken, withthe same values as in the previous sections. The other operational conditionsare given in Table 3.2.In both graphs (a) and (b) it is shown that for a greater load the Stribeckcurves shift to the right. It is clear that when a greater load is present, morehydrodynamic pressure has to be generated, in order to separate the faces.In Fig. 3.32 (b), where coning is present, the transitions take place at highervelocities compared to Fig. 3.32 (a), where a = 0; note the different scales.Furthermore, it is shown that there is no effect of the amplitude and theconing on the relative displacement of the HL–ML transition. In both graphs,the velocity at which the transitions occur, changes almost with the samefactor as the load, FN , does.


Table 3.5: Different values for the axial load FN . The numbers 1–5correspond with the numbers in Fig. 3.32.

1 2 3 4 5

FN [N] 100 500 1000 2000 5000

3.5.2.6 Viscosity

Three different values for the fluid viscosity, η, are taken, i.e. η = 1×10−3, 1×10−2 and 5 × 10−2 Pa·s. The product of η and the velocity v determines thehydrodynamic pressure generation, as shown by Fig. 3.33. An increase of theviscosity results in lowering of the velocity at which the transition from HL toML occurs. The effect of the viscosity is the same for both graphs, i.e. linearwith the viscosity.

3.5.2.7 Reduced modulus of elasticity

In Fig. 3.34 three Stribeck curves are presented for three different values of E ′.With an increase of E ′ the transition HL–ML moves to the right. When coningis present (Fig. 3.34 (b)), the effect of E ′ becomes smaller. For a stiffer materialthe separation in the ML regime is larger, and therefore a higher hydrodynamicpressure is required to enable separation of the faces. Furthermore, a lower E ′

shows a higher coefficient of friction in the HL regime, due to an increase of thecontact area. Practical values of E ′ for mechanical face seals range between2 × 1010 Pa for hard-soft seal face combinations and 4 × 1011 Pa for hard-hardseal face combinations.

3.5.2.8 Hydrostatic fluid pressure

In Section 3.4.6, three different film thickness equations have been derived forthree different fluid pressure regions, i.e. Pf = 0, 0 < Pf ≤ 1 and 1 < Pf ≤1.75. The previous sections all concern calculated Stribeck curves with Pf = 0.In the following Figures the effect of the sealed fluid pressure is demonstrated.

0 < Pf ≤ 1

In both graphs of Fig. 3.35, with waviness and coning values as in the for-mer sections, the transitions shift to the left with an increasing fluid pressure,Pf . When a sealed fluid pressure is present, the hydrostatic component be-comes active in the sealing interface. As a result, less hydrodynamic pressure

3.6 Summary 71

is required in order to separate the faces. In Fig. 3.35 (b) the HL–ML tran-sition takes place at higher velocities, compared to Fig. 3.35 (a), as coning isunfavourable for generating hydrodynamic fluid pressure.

1 < Pf ≤ 1.75

When the fluid pressure, Pf , is further increased, the transition from HL toML occurs at an even lower velocity, compare for instance Fig. 3.36 (a) withFig. 3.35 (a) (coning angle and waviness amplitude are the same). When, aswell as Pf , the coning angle is also increased, a = 1.5 × 10−4 rad in Fig. 3.36 (b),the effect of Pf becomes more significant. The numbers 1 to 7 in this graphcorrespond with the numbers in Table 3.6. When Pf is larger than 1.7, alreadyin the region where boundary lubrication is expected, the load is carried byboth the surface asperities and the hydrostatic fluid pressure. So, according tothe definition given in Section 1.2, mixed lubrication occurs. In the case wherePf = 1.75, the load is mainly carried by the hydrostatically pressurized fluid.At a velocity of about 0.07 m/s the transition to full film lubrication occurs,indicated by the small decrease of the coefficient of friction.

Table 3.6: Different values for the sealed fluid pressure, Pf . The num-bers 1–7 correspond with the numbers in Fig. 3.36 (b).

1 2 3 4 5 6 7

Pf [–] 1.25 1.5 1.7 1.735 1.737 1.74 1.75

3.6 Summary

In the present chapter a friction model was developed by which a Stribeckcurve under specific operational conditions for mechanical face seals can bepredicted.The friction model is a combination of an asperity contact model and a filmthickness equation. The asperity contact model was based on the work ofGreenwood and Williamson, a film thickness equation for mechanical face sealswas developed based on the full-film lubrication theory.With the film thickness equation the separation in a mechanical face seal canbe calculated as a function of the waviness, coning, seal geometry, load andsealed fluid pressure, which are combined in dimensionless parameters. In thisway, the amount of numerical calculations could be significantly reduced.


In Chapter 5 the theoretically obtained Stribeck curves will be verified withthe experimentally determined Stribeck curves.

3.6 Summary 73

d(Tx d(T; d(Tn d(T1 d(Td d(( d(d

(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1

;

E

M

d(

d1

d;

dE

dM

1(

1

2

3

456

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

4 → 3 → 2 → 5 → 1 → 6

λ=h/σ

s

(a) Amplitude, A, ranging from 0.5 − 10µm


(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1

;

E

M

d(

d1

d;

dE

dM

1(78910

111213

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

7 → 8 → 9 → 10 → 11 → 12 → 13

λ=h/σ

s

(b) Amplitude, A, ranging from 15 − 50µm

Fig. 3.25: Calculated Stribeck curves and λ curves for different valuesof the waviness amplitude, A.


d(T; d(Tn d(T1 d(Td d(( d(d d(1

(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1

;

E

M

d(

d1

d;

123

456

7

89

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

1 → 2 → 3 → 4 → 5 → 6 → 7 → 8 → 9

λ=h/σ

s

(a) Waviness amplitude = 1µm


(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1

;

E

M

d(

d1

d;

dE

dM

1(123456789

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s

1 → 2 → 3 → 4 → 5 → 6 → 7 → 8 → 9

(b) Waviness amplitude = 10µm

Fig. 3.26: Calculated Stribeck curves and λ curves for different valuesof the coning angle, a.

3.6 Summary 75

d(T; d(Tn d(T1 d(Td d(( d(d

(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1

;

E

M

d(

d1

d;

dE

dM

1(

0.03 → 0.05 → 0.07

0.070.05

0.03

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s

(a) Amplitude A = 1µm, coning angle a = 0 rad

d(T; d(Tn d(T1 d(Td d(( d(d d(1 d(n

(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1(

;(

E(

M(

d((

d1(

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s

(b) Amplitude A = 10µm, coning angle a = 1.5 × 10−4 rad

Fig. 3.27: Calculated Stribeck curves and λ curves for different valuesof the product ηsβσs. Other parameter values are given in Table 3.2.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1(

;(

E(

M(

d((

σs [m] × 10−8 1

1

22

4

4

6

68

8 10

10

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

x(

d((

dx(

1((

1x(

n((

nx(

σs [m] × 10−8

1 → 2 → 4 → 6 → 8 → 10

1

2

46810

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s


Fig. 3.28: Calculated Stribeck curves and λ curves for different valuesof the standard deviation of the height distribution of the summits,σs. Other parameter values are given in Table 3.2.

3.6 Summary 77

Fig. 3.29: Roughness profile measurement of a run-in seal face.

T; Tn T1 Td ( d 1 n ;

(�(

(�d

(�1

(�n

(�;

(�x

;

d

1

n

φ

s/σs

Fig. 3.30: A Gaussian height distribution (dashed line) vs. M -invertedχ2n distributions for different values of n. 1: n = 5, 2: n = 8, 3:n = 25, 4: n = 100.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

d

1

n

;

x

E

�

M

�

510152025

100

5 → 10 → 15 → 20 → 25 → 100n =

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

d(

1(

n(

510152025

100

5 → 10 → 15 → 20 → 25 → 100n =

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s


Fig. 3.31: Calculated Stribeck curves and λ curves for different shapesof the χ2

n-height distribution. Other parameter values are given inTable 3.2.

3.6 Summary 79


(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1

;

E

M

d(

d1

d;

1

1

2

2

3

3

4

4

5

5

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s


d(Tn d(T1 d(Td d(( d(d d(1 d(n d(;

(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1(

;(

E(

M(

d((

d1(

d;(

1 12 23 34 45 5

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s


Fig. 3.32: Calculated Stribeck curves and λ curves for different val-ues of the axial load FN , numbers referring to Table 3.5. Otheroperational conditions are given in Table 3.2.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1

;

E

M

d(

d1

d;

50

50

10

10

1

1

η [mPa·s] =

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1(

;(

E(

M(

d((

50

50

10

10

1

1

η [mPa·s] =

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s


Fig. 3.33: Calculated Stribeck curves and λ curves for three differ-ent values of the viscosity, η. Other parameter values are given inTable 3.2.

3.6 Summary 81


(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

1

;

E

M

d(

d1

d;

E′ [GPa] =

1 → 10 → 100

110

100

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

d(

1(

n(

;(

x(

E(

E′ [GPa] =

1 → 10 → 100

110

100Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s


Fig. 3.34: Calculated Stribeck curves and λ curves for three differentvalues of the reduced modulus of elasticity, E ′. Other parametervalues are given in Table 3.2.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

x

d(

dx

1(

1x

n(

nx

;(0.9

0.6

0.3

0.10.9 → 0.6 → 0.3 → 0.1

Pf =

Pf =

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

d(

1(

n(

;(

x(

E(

�(

0.9

0.6

0.3

0.1

0.9 → 0.6 → 0.3 → 0.1

Pf =

Pf =

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s


Fig. 3.35: Calculated Stribeck curves and λ curves as a function of Pfbetween 0 and 1.

3.6 Summary 83


(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

x

d(

dx

1(

1x

n(

1.75

1.51.25

1.75 → 1.5 → 1.25

Pf =

Pf =

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

(

d(

1(

n(

;(

x(

1

1

2 2334

4

5

5

6

6

7

7

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

λ=h/σ

s


Fig. 3.36: Calculated Stribeck curves and λ curves as a function of Pfbetween 1 and 1.75. Operational conditions are given in Table 3.2.

4. THE TEST RIG

4.1 Introduction

In order to verify the theoretical model, experimental data, i.e. Stribeck-likefriction curves, are required. In the literature experimental data are available,however, very often important information on seal design, fluid temperature,operational conditions, etc. is left out. Therefore, a test rig was built forcollecting experimental data. As well as the friction experiments, also wearand load carrying capacity tests were to be performed with the test rig. Thiswas in order to determine the change in geometry and the maximum load tobe applied during the friction tests.

The test rig was based on designs of the BHRA-group, see Nau (1989). How-ever a number of modifications were made to allow accurate measurements,see Lubbinge (1995) and Lubbinge et al. (1997). In Section 4.2 the design ofthe test rig is discussed. In order to validate the newly developed test rig, inSection 4.3 a number of preliminary tests are presented.

4.2 Design of the test rig

In Fig. 4.1 a schematic representation of the test rig is given. In Appendix Ba photographic impression of the test rig is shown. The centre of the test rigis indicated by the dashed circle. Here the stationary seal face is in contactwith the rotating seal face. The left-hand side of Fig. 4.1 is the stationarypart of the test rig (indicated with the dashed box) and contains the load unit.The right-hand side of this figure, the rotating part, contains the drive unit.In Section 4.2.1, the stationary part is discussed, in Section 4.2.2 the rotatingpart.

4.2.1 Stationary part

In the following, the main elements of the stationary part will be describedaccording to the numbering in Fig. 4.1. The stationary part consists of:

86 4. The test rig

Bellows (3)A bellows, mounted on support (2), was used to apply a closing force

between the seal faces. By using a bellows, the axial load can be con-trolled accurately. A pneumatic cylinder, which might have been appliedinstead of a bellows, would have suffered from static friction between thepiston and cylinder. Additionally, a pressure vessel was installed (notshown in the Figure) in order to eliminate pressure fluctuations from theair supply.

Bar (5)A bar between the bellows and the stationary seal construction was usedfor two reasons:

• Any misalignment of the bellows with regard to the support (6) willbe minimized.

• When the bar is removed, the complete stationary construction(nos. 6–14) can be moved towards the bellows, which enables chang-ing the seals in the housings (9 & 15).

Support unit (6, 10 to 14)On the support (6) the hybrid force transducer (7) is mounted. Thissupport is placed on a plate (14), which is connected to the slide (12) bymeans of spring blades (11). The slide (12) can slide over the bar guides(10) and can be fixed to the bar guides by means of a screw (13). Duringa test run the slide (12) is fixed to the bar guides. Small movements ofthe plate (14) with regard to the slide (12), for instance due to wear ofthe faces, are possible, as the spring blades (11) can bend.

Hybrid force transducer (7)A specially designed hybrid transducer, made by Hottinger BaldwinMesstechnik (HBM), is able to measure both the normal force, appliedto the seal combination, and the torque caused by the rotating seal face.The transducer, equipped with strain gauges, can measure a maximumclosing force of 5 kN and a maximum torque of 16 Nm. The strain gaugesare applied in such a way that there is almost no effect of the torque onthe signal of the axial force and almost no effect of the axial force on thetorque signal. Besides that, the transducer is mounted directly onto thestationary face housing, so parasitic forces, such as bearing friction, areavoided.

Stationary seal assembly (8 & 9)To significantly improve the alignment of the stationary seal face parallel

4.2 Design of the test rig 87

to the rotating seal face, a new construction has been made for thesuspension of the stationary seal housing (9). The stationary face isfixed in the tangential direction by two pin/notch connections. Figure 4.2shows an exploded view of the parts (8) and (9) of Fig. 4.1, such that theupper drawing is a top view and the lower drawing is a side view. At thetop and the bottom the stationary housing (9) is mounted on a frame(8) by means of crossed spring blades. In this way, the stationary sealhousing can rotate around the z -axis. Frame (8) (Fig. 4.2) is connected toframe (2), also by crossed spring blades (parts 4, 5 & 6). The stationaryseal housing (9), including the frame (8), can rotate around the y-axis.

All together, the stationary seal face can be aligned adequately parallelto the rotating face, whilst the construction is stiff in both the x -directionand the tangential direction (rotation around the x -axis), see Fig. 4.2.The stationary seal assembly is connected by (1) to the hybrid trans-ducer. The fluid to be sealed by the face seals is supplied by connectionson the housing (9) (not shown in Fig. 4.2). The fluid is added from theinside of the seals, so the seals in the test rig are of the “inside pressur-ized” type.

4.2.2 Rotating part

The different parts of the rotating assembly are listed below. The most im-portant aspect of the rotating part is that the rotating housing (15) is in linewith the stationary face.

Rotating housing (15)A simple housing, made of stainless steel, is used to mount the rotatingface. Also here, the face is tangentially fixed by two pins, that fit in thenotches on the back of the faces.

Bearing house (16)The bearing house, mounted on the ground plate (17), consists of twobearings. On the right-hand side an angular contact ball bearing ismounted, as this type of bearing is particularly suitable for the acco-modation of combined loads, i.e. radial and axial loads. So the axialload of the stationary face is completely carried by this bearing. On theleft-hand side a deep groove ball bearing is used, in order to lock up theshaft (20) in the radial direction.

Coupling (19)A torsion-stiff coupling is used, so there is no tolerance in the tangentialdirection between the shaft of the motor and shaft (18).

88 4. The test rig

Motor (20)The rotating face is driven by a DC-motor. This motor has a widevelocity range, i.e. from 3 rpm to 2000 rpm. An important advantage ofa DC-motor is that a high torque can be generated at low speeds. Thisis required for the boundary lubricated regime, as in this regime very lowspeeds are associated with a relatively high friction/torque.

4.2.3 Data acquisition and control of the operationalparameters

The test rig is controlled by a computer equipped with an AD-card (Analog-Digital data-acquisition card), which has 8 different analog input channels, 2analog output channels, 8 digital input and output channels and 1 counterinput for counting a TTL-compatible input signal. The following itemizationshows which data are acquired from the test rig. The digital input channelsare not used.

Analog input channels:

• Axial load, measured by the hybrid transducer.

• Friction force (torque), also measured by the hybrid transducer.

• Three channels for measuring the temperature in the gap of the matingfaces. The stationary carbon seals contain 3 equidistant holes close tothe contact surface, in which the thermocouples, type J, are placed.

Analog output channels:

• The speed of the rotating face is controlled by sending a specific voltageto the DC-motor.

• The bellows, which applies the axial force, is controlled by transmittinga specific voltage to the pressure regulator.

Digital output channels:

• Switching the motor on and off.

• Switching the pressure of the pressure regulator on and off, and, thus,the axial force on the seals.


Counter input channel:

• The counter input is used for counting the pulses of the pulse generator,which measures the velocity of the rotating seal. The pulse generatorproduces 500 pulses per revolution and is connected to the shaft (18)by means of a belt with a transmission ratio of 1:1. By counting thenumber of pulses per unit time, the velocity can be determined. Thisway accurate speed measurements can be performed, especially at lowvelocities.

90 4. The test rig

Fig. 4.1: Schematic representation of the test rig.


Fig. 4.2: Construction for the alignment of the stationary seal faceparallel to the rotating seal face.

92 4. The test rig

4.3 Validation of the test rig; preliminary

results

In order to validate the newly developed test rig, test experiments were per-formed. These experiments consisted of the measurement of friction curves,load carrying capacity tests and wear rate measurements. The reasons forthe load carrying capacity tests and the wear measurements are given in theintroduction. In these preliminary tests, carbon/SiC seal combinations wereused.

4.3.1 Friction curve measurements

Figures 4.3 and 4.4 show examples of measured friction curves (Stribeck curves).The coefficient of friction is plotted as a function of the velocity. The plots

d(T1 d(Td d(( d(d

(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

Velocity [m/s]

Coeffi

cien

tof

fric

tion

Fig. 4.3: Three Stribeck like friction curves of a carbon/silicon carbideseal combination, with FN = 400 N and η = 1 mPa·s.

show that excellently reproducible friction measurements can be performedwith the modified test rig. The three lubrication regimes can be clearly distin-guished. The friction measurements were performed at constant loads of 400 N

4.3 Validation of the test rig; preliminary results 93

d(T1 d(Td d(( d(d

(�((

(�(;

(�(M

(�d1

(�dE

(�1(

Velocity [m/s]

Coeffi

cien

tof

fric

tion

Fig. 4.4: Three Stribeck like friction curves of a carbon/silicon carbideseal combination, with FN = 475 N and η = 1 mPa·s.

and 475 N, respectively, by changing the velocity step-wise from 1.3 m/s downto 0.025 m/s and vice versa and using water as the fluid. Firstly, the curvemarked (•) was measured with decreasing velocity, next, the curve marked(H) was measured with increasing velocity and, finally, the curve marked (�)was measured, again with decreasing velocity. The experimental procedure isdescribed in Chapter 5.

4.3.2 Load carrying capacity tests

In addition to the friction experiments, load carrying capacity tests or pv–experiments were performed, in order to determine the maximum allowableaxial load on the seal combination. During these experiments the velocity, v,contact pressure, p, coefficient of friction and the temperature of the carbonface were measured. In order to measure the temperature, three equidistantholes were made in the stationary seal, close to the contacting face. The pv–experiments were performed by keeping the velocity constant and increasingthe contact pressure step-wise. The first tests were performed using water asa lubricant. In that case the seals did not fail at all, so no pv-values were

94 4. The test rig

obtained. It was therefore decided to use no lubricant. In Fig. 4.5 an exampleof such an experiment is shown, the velocity was 4 m/s and there was nolubricant supply. Here, when the contact pressure was increased above 7 MPa,the coefficient of friction showed a sudden strong increase from about 0.03 to0.2 and higher, while the temperature increased from approximately 70 ◦C to150 ◦C.

( d 1 n ;

d

1

n

;

x

E

�

M

(�((

(�(1

(�(;

(�(E

(�(M

(�d(

(�d1

(�d;

(�dE

(�dM

(�1(

(�11

;(

E(

M(

d((

d1(

d;(

dE(

Time [min.]

Mea

nco

nta

ctpre

ssure

[MPa]

Coeffi

cien

tof

fric

tion

Con

tact

tem

per

ature

[◦C

]

PressureCoefficient of frictionTemperature

Fig. 4.5: Load carrying capacity experiment with v = 4 m/s, no lubri-cant supply.

A number of such load carrying capacity experiments were performed at dif-ferent velocities. In Fig. 4.6 the results of these experiments are collected inone plot. On the left axis the failure pressure is given and on the right axisthe temperature at the moment of failure is given. In the Figure a line fora constant pv-value of 22 MPa·m/s was drawn (dotted line marked (◦)). Themeasured curve, marked (•), shows that in reality failure does not occur at aconstant pv-value. On the other hand, Fig. 4.6 shows that failure of the sealsoccurs at a practically constant face temperature of 75±5 ◦C. Most probably,seal failure occurs at a particular value of the seal face contact temperature,rather than at a fixed pv-value (which, when multiplied by the coefficient offriction, is a measure for the heat flux into the seal faces). A similar conclusion


was drawn by Honselaar and de Gee (1989) for polymeric bearing materialssliding against steel. These load carrying capacity tests showed that for thelubricated situation no failure took place, therefore, no effects of (local) failureon the Stribeck like friction curves have to be taken in consideration.

n ; x E � M � d(

(

d

1

n

;

x

E

�

M

(

d(

1(

n(

;(

x(

E(

�(

M(

�(

d((

velocity [m/s]

Fai

lure

pre

ssure

[MPa]

Fai

lco

nta

ctte

mper

ature

[◦C

]

Experimental failure pressure p [MPa]Theoretical failure pressure, pv = constantTemperature [◦C]

Fig. 4.6: Load carrying capacity experiments. Failure at a contacttemperature of between 70–80 ◦C.

4.3.3 Wear rate measurements

The last type of experiments performed on the test rig, was the measurementof wear of the seal faces. In these experiments different material combinationsof silicon carbide/carbon seals were tested.In order to determine the wear, three methods were tried out. The mostelegant method would have been a real time measurement, as during such atest the wear behaviour can be observed continuously.Schipper and Odi-Owei (1992) performed several wear tests on a pin-on-discdevice. They designed a wear measuring system which corrected for tempera-ture and swell effects of the materials. In their construction two displacementtransducers were used: one measured the distance change caused by wear and

96 4. The test rig

swell as well as temperature effects, whereas the other transducer measuredonly the distance change caused by temperature variations and swell. By sub-tracting the signals from each other, the actual wear displacement remained.A disadvantage of this method is that wear of the complete “tribo-system” ismeasured and not that of the pin and disc separately. In seal faces, based ona combination of a soft (carbon) and a hard (silicon carbide) seal, however,wear occurs nearly entirely on the soft carbon seal. Unfortunately, a similarconstruction was not possible on the author’s test rig. Therefore efforts weremade to measure as close as possible to the seal contact with only one trans-ducer to avoid temperature effects as much as possible. A capacitive probeand a laser probe were tried out, but both transducers failed. Small environ-mental temperature variations had a much larger effect on the signal of thetransducers than the actual wear. Summarizing, real time wear measurementson the test rig were not succesful.The second method considered was based on mass loss of the seals. By deter-mining the mass before and after a test, the volume loss ∆V can be calculated(if the density of the seal material is known). Before and after a wear testthe seals were dried in an oven at a temperature of 110 ◦C. However, it tookapproximately 600 hours (= 25 days) before a steady weight was reached. Fig-ure 4.7 shows the mass decrease, plotted against time. In view of this, themethod is hardly appropriate for measuring wear of seal faces.A third method turned out to be more suitable. On the carbon faces smallcircumferential steps were made, as shown in Fig. 4.8. The steps were madeon the inside circumference and on the outside circumference of the seal face.Before and after a wear test the step-height was measured without contactby means of an interference microscope, which is described in more detail byLubbinge (1994). Figure 4.9 shows an example of a step-height measurement.Where the reflection of the steps was too low, step-height measurements wereperformed using a form tester, i.e. a stylus-like device, which however doesnot mate a contactless measurement. Figure 4.10 shows an example of a mea-surement made using a form tester.As carbon faces may swell or shrink as a result of changing temperature and/orhumidity (Flitney and Nau, 1987), the step-height also depends on this swellingor shrinking. Therefore a second step is made (Fig. 4.8), which will not wear,because it is not in contact with the counter face. The height of this step isalso measured, and is used to correct for swell/shrink or temperature effects.The actual wear height can then be calculated using the following formula (seeFig. 4.8):

∆h = at1 − at2 × bt1bt2, (4.1)

where:


( 1(( ;(( E(( M((

d�1�(

d�1�x

d�n�(

d�n�x

d�;�(

d�;�x

d�x�(

d�x�x

Mas

s[g

]

Time [h]

Fig. 4.7: Mass loss of a carbon seal face in an oven at a temperatureof 110 ◦C.

∆h : decrease of the height caused by wear of the face [mm]at1 : height of the upper step before the wear testat2 : height of the upper step after the wear testbt1 : height of the lower step before the wear testbt2 : height of the lower step after the wear test

The specific wear rate, ks, is calculated according to the following equation:

ks =∆V

FN × s=Aseal∆h

FN × s

[mm3

N·m], (4.2)

where ∆V is the worn volume of the seal, FN the normal force, s the slidingdistance and Aseal the contacting seal area.Different silicon carbide/carbon seal combinations were tested. To avoid hy-drodynamic effects, which reduce the wear rate significantly, the experimentswere performed at very low speeds, i.e. in the boundary lubrication regime.The specific wear rates of all combinations were below ks = 3.5×10−7mm3/Nm,see Table 4.1, indicating that very little wear occurred. Wear experiments were

98 4. The test rig

s

{

A

Fig. 4.8: Circumferential steps (exaggerated). Dimensions of the stepsare about a = b ≈ 0.02 − 0.04 mm and c ≈ 0.5 mm.

also performed on a pin-on-disc device (Schipper, 1992). The results of theseexperiments agree well with the author’s results. The resin impregnated

Table 4.1: Specific wear rates for different seal material combinations.

Material ks (×10−7 mm3/Nm) ks (×10−7 mm3/Nm)

combination Test rig Pin-on-disc

RC – RBSiC 3.3 6.6

AC – RBSiC 2.2 1.3

RC – SSSiC 2.4 3.0

AC – SSSiC 1.9 0.7

carbon faces (RC) showed more wear than the antimony impregnated car-bon faces (AC) when run against reaction bonded silicon carbide (RBSiC) aswell as against self-sintered silicon carbide (SSSiC). Both carbon faces showedmore wear against reaction bonded silicon carbide face than when run againstself-sintered silicon carbide. More details about the wear measurements aredescribed by van Gaalen (1996).

4.4 Summary 99

Fig. 4.9: Example of a measurement of the steps with the interferencemicroscope.

After the experiments, it was found that waviness developed on the carbonface, which in these experiments (BL) did not affect the friction. However, forfriction curve experiments, this, by wear generated waviness might affect thefriction results. Hence, for the friction curve experiments a silicon carbide vs.silicon carbide seal combination will be used.

4.4 Summary

With the newly developed test rig it is possible to perform friction measure-ments, load carrying capacity tests and wear experiments in a reproducibleway. A great advantage of this test rig is that the transducer, which measuresboth the torque and the axial load, is mounted directly behind the seal hous-ing. Furthermore, a stiff alignment construction has been developed (Fig. 4.2),which inhibits misalignment of the faces. In this way, the real forces acting onthe seal combination are measured.The load carrying capacity tests showed that seals do not fail at a constantpv-value, but seem to fail at a constant seal face temperature. When there issufficient lubricant present, no seal failure occurs.

100 4. The test rig

Fig. 4.10: Example of a measurement of the steps with a form tester.

From the wear experiments it can be concluded that, provided that enoughlubricant is present to generate a boundary layer, hard-soft seal combinationshardly show any wear.Finally a number of friction experiments were performed in order to analyzewhether Stribeck-like measurements (see Section 1.2), can be performed withthe test rig. Is it shown that reproducible measurements can indeed be carriedout. In Chapter 5 the friction experiments and the friction model are discussedin detail.

5. VERIFICATION OF MODELWITH EXPERIMENTALRESULTS

5.1 Introduction

In Chapter 3 a model was described by which frictional behaviour can bepredicted for different conditions. In order to validate the theoretical results,experimental data are required. Therefore, a test rig was developed and built.It was described in Chapter 4. In the following section, the test procedureand the materials are discussed and, subsequently, in Section 5.3, the model isverified with experimental results (friction data).

5.2 Experimental procedure and materials

5.2.1 Experimental procedure

Before an actual friction measurement was started, the seal faces were cleanedwith acetone and both the micro-geometry (roughness) and the macro-geometry(waviness and coning) of the silicon carbide face and the carbon face were mea-sured. The roughness was measured on a 3D interference microscope. The sizeof the surface scan was 766µm×597µm. Waviness and coning were measuredon a Talyrond 200 precision spindle and on an interference microscope. AStribeck curve was measured under a constant normal load and at a constantwater supply temperature, by varying the velocity step-wise from 0.012 to 8m/s (3 – 2000 rpm) and back. By using this procedure the experiments wereconducted under reasonably iso-thermal conditions. At high velocities the co-efficient of friction is low and at low velocities the coefficient of friction is high.Therefore, the heat, generated in the contact, which is equal to f ·FN ·v, withf the coefficient of friction, FN the normal force and v the sliding velocity,will be reasonably constant during an experiment. All tests were conductedwith water at 20 oC as a lubricant. After each velocity change, the coefficientof friction was measured and recorded by a computer, as soon as a constant

102 5. Verification of model with experimental results

frictional behaviour was reached. After a friction experiment, the seal faceswere cleaned with acetone and the roughness and the macro-geometry weremeasured.

5.2.2 Material specifications

Both the rotating and the stationary seal faces were made of silicon carbide.In this way the original seal face macro-geometry was maintained. When usingcarbon faces, very often a two-wave waviness develops as a result of wear; seealso Section 2.2.4. The face seals were specially designed for this test rig. Inorder to minimize the effect of material deformation as a result of the actingpressures, both seals were made with a thickness of 31.7 mm. The essential sealdimensions for both the rotating and the stationary face are given in Fig. 5.1.

In Table 5.1 some typical physical and thermal properties of the silicon carbideface seal are given.

Table 5.1: Material properties.

Properties Tested seal

Type SSSiC

SiC [wt %] 100

Bending strength [MPa] 550

Compression strength [MPa] 3900

Elasticity modulus [GPa] 400

Hardness [GPa] 28

Density [kg/m3] 3100

Conductivity [W · m−1 · K−1] 126

In order to validate the theoretical model, i.e. the calculated Stribeck curves,seal faces with different waviness amplitudes and seal faces with different con-ing angles are required. A number of production methods were tried, amongwhich the most promising seemed to be elastic deformation of the face, fol-lowed by lapping the face flat and, finally, releasing the face. Unfortunately, asa result of this method, coning also developed, which was unwanted. Besidesthat, the coning in the circumferential direction varied from divergent to con-vergent and vice versa, as shown in Fig. 5.2. Finally, at the Philips Research

5.3 Theoretical vs. experimental results 103

Laboratories, an accurate two-wave waviness with no coning was applied to theseal faces, by means of a three-axes grinding device. In Fig. 5.3 a 3D surfacemeasurement of such a ground seal face is shown.Unfortunately, the grinding method appeared not to be so successful for apply-ing a coning to the seal face. One has to be aware that on an average diameterof 80 mm and a seal width of 13 mm, in a controlled way, a height differenceof a few µm had to be realized. Measurements showed that all the seals hadapproximately the same coning angle, whereas seal faces with different coningangles were required. Figure 5.4 shows a coning measurement. In this Figure,the contact with the counter face is indicated as “contact width”.The counter seal face, which will be the rotating face in the test rig, must beas flat as possible. At the Philips Research laboratories this was achieved bylapping. The flatness was within 0.1µm. In Fig. 5.5 an example of a flat sealface is shown.

5.3 Theoretical vs. experimental results

5.3.1 Effect of micro-geometry on hydrodynamicpressure generation

According to classical hydrodynamic theory, when both faces are flat and par-allel there is no mechanism present able to generate hydrodynamic pressure,see also Section 2.1. Therefore a number of experiments were performed with asilicon carbide face and a carbon face, both lapped as flat as possible. Fig. 5.6shows the friction–time characteristic of such an experiment. The coefficientof friction stays at a constant level, corresponding with that of the bound-ary lubrication regime, even though the velocity is as high as 1 m/s, i.e. highenough for hydrodynamic lubrication to occur. This plot shows that in a situa-tion without macroscopic geometry variations, like waviness, no hydrodynamicfluid pressure is generated, so micro-geometry alone cannot generate hydrody-namic fluid pressure. The initial CLA roughness of the carbon and siliconcarbide seals was 0.28 and 0.22µm, respectively. After the test, the roughnessof both faces had significantly decreased, i.e. to 0.08µm for the carbon faceand to 0.17µm for the silicon carbide face.

5.3.2 Macroscopic features

5.3.2.1 Waviness amplitude, A, variations

The effect of the waviness on the Stribeck curve was measured for 5 differentseal combinations. Both seals were made of self-sintered silicon carbide, the


rotating seal face flat to within 0.1µm, the stationary seal face with 2 wavesand no coning. In Table 5.2 the operational conditions are given for the 5 com-binations, where “s” is the stationary seal face, “r” the rotating seal face andσini and σafter the standard deviation of the surface height distribution beforeand after the experiment, respectively. In Table 5.1 the material propertiesof the seal faces are given. The standard deviation of the summits, σs, andthe radius of the summits, β, are the combined values of both faces after anexperiment. They are defined as:

σs =√σs12 + σs22, (5.1)

1

β=

1

β1

+1

β2

. (5.2)

We were concerned only with hydrodynamic effects, and therefore the mea-surements were performed without sealed fluid pressure, i.e. Pf = 0. In

Table 5.2: Seal face properties of seals with varying waviness and noconing.

Experimentalconditions 1 2 3 4 5

rotating/stationary r s r s r s r s r s

A [µm] 0 1.1 0 1.7 0 2.9 0 3.8 0 5.0

σini [nm] 17 287 11 400 33 467 7 457 4 454

σafter [nm] 29 140 12 112 32 115 6 110 10 57

ηs (×109) [m−2] 14.5 12.3 11 10.7 6.4

β [µm] 55.9 70.0 105 97.4 174.4

σs [nm] 62 58 43 48 45.0

FN [N] 240 240 240 240 240

vtexp [m/s] 0.15 0.12 0.10 0.11 0.11

Fig. 5.7 the measured Stribeck curves of experimental condition 1 (Table 5.2)are presented, together with the predicted Stribeck curve (dashed line), whichwas calculated on the basis of the model presented in Chapter 3 and the pa-rameters of Table 5.2. It can be seen that the measured curves are predictedquite accurately by the calculated Stribeck curve. Unfortunately, the speed


of the motor could not lowered any further, so the transition from mixed toboundary lubrication was not measured. In the model, the value of the frictioncoefficient for the boundary lubrication regime, fc, was taken to be 0.25, whichis a practical value for silicon carbide/silicon carbide contacts under boundarylubricated conditions, see e.g. Summers-Smith (1988). Another comparisonof an experimental Stribeck curve with a predicted curve is shown in Fig. 5.8.Experimental condition 5 in Table 5.2 was taken, and also here the measuredStribeck curves were very well predicted by the theoretical Stribeck curve.

As shown in Table 5.2, the roughness of the stationary seal face decreasedstrongly during the test. The roughness was measured at the top of the waves,where the contact of the faces occurs. No roughness change was observed inthe valleys of the seal faces with waviness.

5.3.2.2 Radial coning angle, a

As mentioned in Section 5.2.2, it was not possible to obtain seals with differentconing angles. The experiments all showed the same frictional behaviour.

Table 5.3: Seal face properties of experiments with constant coning.

rotating/stationary r s

Coning a (×10−4) [rad] 0 2.3

A [µm] 0 0.9

σini [nm] 26.4 232

σafter [nm] 18.6 23.9

ηs (×109) [m−2] 7.9

β [µm] 338

σs [nm] 16

FN [N] 240

vtexp [m/s] 0.1

In Figure 5.10, 4 measured Stribeck curves are shown, with operational condi-tions as given in Table 5.3. The material properties are presented in Table 5.1.From the theoretical results it was expected that, as a result of coning, thetransition from hydrodynamic lubrication to mixed lubrication would shift tothe right, see Fig. 3.26. However, the transition HL–ML occurred at a lowervelocity than that observed without coning; see Fig. 5.7.


The dashed Stribeck curve in Fig. 5.10 was calculated on the basis of thedata of Table 5.3. It can be seen that the measured Stribeck curves are notwell predicted by the calculated Stribeck curve. In fact, the measured curvesare located more than one order of magnitude in velocity to the left of thecalculated curve. After the experiment, 3D interference scans of the completesurface were made. The scans revealed that on the outside contact diameter ofthe wavy face with coning, the face was locally worn flat in the radial direction,with a width of about 2.4 mm, as indicated in Fig. 5.9.

Thus, another Stribeck curve was calculated, this time according to the situ-ation shown in Fig. 5.9. The seal width was taken as 2.4 mm, and the coningangle was set to a = 0 rad. This new situation is described by the dash-dottedcurve and it can be seen that this curve predicts the measured transition fromhydrodynamic to mixed lubrication rather well. Measurements with lower ve-locities could not be performed, due to the operational limitations of the testrig and because the frictional behaviour was unstable.

Hence, it can be concluded that the measured Stribeck curves were very wellpredicted by the calculated Stribeck curves. It appeared that friction mea-surements with coning, as described by the model, could not be performed, asthe coning wears partly away in a very short time and a radially flat-to-flatcontact between the faces remains.

In the next section the transition from hydrodynamic to mixed lubrication willbe analyzed.

5.3.2.3 Influence of macroscopic features on the transition fromfull film to mixed lubrication

As mentioned in Section 1.2, the transition from hydrodynamic to mixed lubri-cation would be the ideal operational regime for mechanical face seals. Here,a low coefficient of friction is associated with a relatively low leakage and lowwear. Therefore, it is fortunate that this regime can now be determined as afunction of the present operational conditions.

The transitions from full film lubrication to mixed lubrication, which are de-termined from the Stribeck curve according to the method described in Sec-tion 1.2, can be reflected in a so-called transition diagram. In Fig. 5.11, vt,defined as the experimentally determined transition, vtexp , divided by the cal-culated transition, vtcal , is plotted as a function of the waviness amplitude. Itis shown that the experimentally determined values of vt (vtexp) are in goodagreement with the calculated values (vtcal). In fact, the maximum differencebetween vtexp and vtcal amounts to not more than 10%.

According to Fig. 3.25, vt should be nearly independent of amplitude, A, forthe amplitude range given in Table 5.2. The differences between different vtexp


values are nearly entirely due to differences in some of the other parameters,notably σs.

5.3.3 Axial load variations

Three different values for the axial load, FN , were chosen, i.e. 240, 500 and750 N. In Table 5.4 the operational conditions are given. In each experiment,one seal was flat within 0.1µm and one seal had a waviness amplitude of 2.9µmand no coning. It was expected from the preliminary results in Section 3.5.2.5

Table 5.4: Seal face properties of experiments with different loads.

Experimentalconditions 1 2 3

rotating/stationary r s r s r s

A [µm] 0 2.8 0 2.8 0 2.8

σini [nm] 33 467 32 115 43 92

σafter [nm] 32 115 43 92 24 43

ηs (×109) [m−2] 11 7.8 6.9

β [µm] 105 181 168

σs [nm] 43 36 43

FN [N] 240 500 750

vtexp [m/s] 0.10 0.13 0.19

that, with an increase of the load, FN , the transition velocity, vt, would in-crease. From Table 5.4 it can be seen that indeed vt shifts to the right withincreasing FN . Furthermore, it is shown in Table 5.4 that in each experimentalcondition σ decreased.The Figs. 5.12 and 5.13 show the measured Stribeck curves and the calculatedStribeck curves with loads of 500 and 750 N, respectively. The calculated curvesare based on the data of Tables 5.1 and 5.4. The Stribeck of experimentalcondition 1 in Table 5.4 is the same as experimental condition 3 in Table 5.2(see also Fig. 5.11. Again, it is shown that the measured Stribeck curves arevery well predicted by the calculated Stribeck curves. In fact, the maximumdeviation of the predicted transition velocity is 10% (experimental conditionno.1). In case of FN = 750 N (Fig. 5.13), the measured Stribeck curve isperfectly predicted by the calculated Stribeck curve.


5.3.4 Pressure of fluid to be sealed variations

In order to verify the model, when there is pressure of the fluid to be sealed, pf ,present, a number of experiments were performed with different values for pf .In each experiment the axial load was set to FN = 500 N. The experimentalconditions are given in Table 5.5. As the type of sealing of the test rig is insidepressurized, the axial load has to be corrected for the pressure inside the sealingchamber. The load in opposite direction, which has to be distracted from FN ,equals pf × πri

2, the fluid pressure times the inside area of the seal. The fluidpressure was measured relative to the environment by a sensor, mounted insidethe sealing chamber.

Table 5.5: Seal face properties of experiments with different pressuresof the fluid to be sealed.

Experimentalconditions 1 2 3 4

rotating/stationary r s r s r s r s

A [µm] 0 1 0 1 0 1 0 1

a [rad] 0 0 0 0 0 0 0 0

σini [nm] 29 140 25 115 24 98 24 75

σafter [nm] 25 115 24 98 24 75 23 68

ηs (×109) [m−2] 11 12 14 14

β [µm] 77 89 100 102

σs [nm] 59 47 36 34

FN (nett) [N] 385 308 231 116

Pf [–] 0.11 0.23 0.43 1.23

pf (×105) [Pa] 0.3 0.5 0.7 1

vtexp [m/s] 0.7 0.14 0.08 –

In Fig. 5.14 the measured Stribeck curves and the predicted Stribeck curves,based on the data of Table 5.5, are shown. The transitions from hydrodynamicto mixed lubrication are well predicted by the model, however the slope of themeasured curves in the mixed lubrication regime is much smaller then the pre-dicted slopes. The coefficient of friction in the boundary lubrication regimecould not be measured, due to limitations of the test rig. The same behaviouris also predicted by the model, see Fig. 3.36, but occurs for higher values of


Pf . An explanation could be as indicated in Fig. 5.9. After the measurements,it was found that the contact between the faces mainly occurred at the out-side diameter. Thus, a perfectly flat contact exists at the outside diameter,while also some coning exists at the inside diameter, probably as a result ofmechanical deformation. This results in extra hydrostatic fluid pressure in thecontact, resulting in a lower coefficient of friction in the lubrication regimewere boundary lubrication is expected.

Table 5.6: Seal face properties of experiments with different pressuresof the fluid to be sealed and with coning.

Experimentalconditions 1 2 3

rotating/stationary r s r s r s

A [µm] 0 1 0 1 0 1

a (×10−4) [rad] 0 2.5 0 2.5 0 2.5

σini [nm] 17 97 – – – –

σafter [nm] – – – – 17 79

ηs (×109) [m−2] 6

β [µm] 165

σs [nm] 50

FN (nett) [N] 385 308 231

Pf [–] 0.11 0.23 0.43

pf (×105) [Pa] 0.3 0.5 0.7

vtexp [m/s] 0.13 0.03 0.03

Next to the experiments without coning, also some experiments were per-formed with coning. As, after the experiments the seal face exhibited a geom-etry as shown in Fig. 5.9, the Stribeck curves could not be calculated for thesetests. In Fig. 5.15 a number of friction curves are presented for different valuesof Pf . The experimental conditions are given in Table 5.6. The coning anglewas about 2.5 × 10−4 rad. It is shown that already for lower values of Pf thecoefficient of friction is lower in the mixed lubrication regime, compared withFig. 5.14. As a partially flat and partially coned face exists, there is next to asignificantly generated hydrodynamic pressure, also considerable hydrostaticpressure present due to the coned part.


5.3.5 Summary

In this chapter it is shown that the friction model developed for mechanicalface seals predicts the friction and, hence, the transition from full film tomixed lubrication rather well. When hydrostatic fluid pressure is present, thetransition from full film to mixed lubrication is well predicted, however, thecalculated slope of the friction curve in the mixed lubrication regime differsfrom the slope in the mixed lubrication regime of the measured friction curve.This is probably due to the geometry of the seal face which was present afterthe experiment (see Fig. 5.9). Unfortunately, this geometry is not included inthe friction model.


ÿþýü

��

ü�

(a) Rotating face.

ÿþýü

��

��

(b) Stationary face.

Fig. 5.1: Dimensions in mm of the tested face seals.


Fig. 5.2: Lapped waviness by means of elastic deformation of the seal,involving a varying coning angle.

Fig. 5.3: A two-wave waviness applied by means of grinding tech-niques.


1µm

Contact width

Fig. 5.4: Coning applied on the face seal by means of grinding tech-niques.

Fig. 5.5: Seal face, flat to within 0.1µm, obtained by lapping tech-niques.


( 1 ; E M d( d1 d;

(�((

(�(1

(�(;

(�(E

(�(M

(�d(

(�d1

(�d;

(�dE

(�dM

Time (h)

Coeffi

cien

tof

fric

tion

Fig. 5.6: Friction–time characteristics of a flat parallel seal face com-bination. FN = 400 N, v = 1 m/s, η = 1 mPa·s.


(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

Fig. 5.7: Measured Stribeck curves vs. a calculated Stribeck curve(dashed line) as a function of the velocity. Experimental conditionsaccording to no. 1 in Table 5.2.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

Fig. 5.8: Measured Stribeck curve vs. a calculated Stribeck curve(dashed line) as a function of the velocity. Experimental conditionsaccording to no. 5 in Table 5.2.

2.4 mm

Seal width

Fig. 5.9: Schematic representation of wear of a seal face with coning.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

Fig. 5.10: Measured Stribeck curves with coning vs. calculatedStribeck curves (dashed line and dash-dot line) as a function of thevelocity. Experimental conditions according to Table 5.3.

( d 1 n ; x E �

(�x

(�E

(��

(�M

(��

d�(

d�d

d�1

d�n

d�;

d�x

Waviness amplitude, A [µm]

vt

f

v [m/s]vt

Fig. 5.11: Transition from hydrodynamic to mixed lubrication as afunction of different waviness amplitudes.



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]



(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]




(�((

(�(x

(�d(

(�dx

(�1(

(�1x

(�n(

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

0.110.230.431.23

0.11

0.23

0.43

1.23

Pf =

Fig. 5.14: Measured Stribeck curves vs. calculated Stribeck curves(dashed line) as a function of the velocity for different values ofPf . Experimental conditions according to Table 5.5.

d(T1 d(Td d(( d(d

(�((

(�(x

(�d(

(�dx

Coeffi

cien

tof

fric

tion

,f

Velocity [m/s]

0.11

0.23

0.43

Pf =

Fig. 5.15: Measured Stribeck curves of a seal with coning as a functionof the velocity for different values of Pf .

6. CONCLUSIONS ANDRECOMMENDATIONS

In this thesis a model was presented by which the frictional behaviour of me-chanical face seals can be predicted as a function of the operational conditions.Also a test rig was designed and built, in order to verify the friction model.Based on the work presented in the previous chapters, a number of conclusionscan be drawn. In Section 6.2, recommendations for further research are given.

6.1 Conclusions

Friction model

• Film thickness equations for mechanical face seals have been developed.In order to reduce the number of calculations, the parameters weregrouped into dimensionless quantities. Three different film thicknessequations were required, depending on the pressure range of the fluid tobe sealed.

• In order to apply the contact model of Greenwood and Williamson, thenominal contact area was required. The nominal contact area between aflat seal face and a seal face with waviness and with or without coning wassimulated by pressing a parabolically shaped cylinder on a flat surface,in which the centre line of the cylinder made an angle to the flat surfaceequal to the coning angle. A contact area is formed as a result of elasticdeformation of both surfaces. Based on the numerical data, a functionfit for the nominal contact area was made as a function of the wavinessamplitude, coning angle, modulus of elasticity and load.

• By combining the contact model and the film thickness equation, Stribeckcurves can be calculated. With this friction model the effects of the dif-ferent parameters, like coning, waviness, fluid pressure, roughness, etc.on the transitions between the different lubrication regimes can be ana-lyzed.

120 6. Conclusions and recommendations

Test rig and experiments

• A test rig was designed and built with which accurate and reproduciblefriction measurements, wear measurements and load carrying capacitytests could be performed.

– To align the rotating face parallel to the stationary face a new align-ment construction was developed. The construction was based onelastic hinges, which resulted in a stiff construction in both the ro-tating and the axial directions, while maintaining a full alignment.

– A specially designed hybrid transducer was used for measuring theaxial load as well as the torque simultaneously. By mounting thetransducer directly behind the contacting seal faces, the real forcesacting in the contact were measured.

• Load carrying capacity tests were performed in order to determine themaximum applicable load. Measurements revealed that when lubricantwas present, no failure at all occurred. Without lubricant supply, sealfailure occurred. These tests showed that seals do not fail at a constantpv-value, as often assumed in the literature, but appear to fail at aconstant seal face temperature.

• Friction experiments were performed in order to determine the transitionfrom hydrodynamic to mixed lubrication. In this region an optimum sit-uation with regard to leakage, friction and wear exists for mechanicalface seals. The experiments showed that the friction curves and, con-sequently, the transition from hydrodynamic to mixed lubrication couldbe obtained in a reproducible way.

Model versus experiment

• The friction model was verified with experimental friction data, obtainedfrom the newly developed test rig. It was found that the model accuratelypredicts the friction in mechanical face seals.

• Friction experiments with coned seals showed a much lower transitionvelocity than the model predicted. Surface measurements of the conedseal revealed that the seal on the outside contact diameter was locallyworn flat. After adapting the model to the new situation, the modelshowed good agreement with the measured Stribeck curve.

6.2 Recommendations 121

Hydrodynamic to mixed lubrication

• It was shown that, depending on the operational conditions, the transi-tion from hydrodynamic to mixed lubrication significantly depends on:

– coning angle, a

– standard deviation of the height distribution of the summits, σs

– load, FN

– viscosity, η

– pressure of the fluid to be sealed, Pf

and to a lesser extent on:

– waviness amplitude, A

– the value of ηsβσs

– the height distribution, analyzed with a χ2n-height distribution func-

tion

– the reduced modulus of elasticity, E ′

• Depending on the seal geometry and pressure of the fluid to be sealed,the transition from hydrodynamic to mixed lubrication disappears.

6.2 Recommendations

• Models already exist which can calculate the shape of the seal face asa function of thermal and mechanical distortions. By combining theauthor’s friction model with such a model, the frictional behaviour of amechanical seal face can be predicted accurately.

• Due to frictional heating, two-phase lubrication may result. This effectcould be implemented in the model. In fact, the model already assumestwo-phase lubrication, as also cavitation is taken into account. By mak-ing the liquid fraction parameter, qc, also a function of the thermal effects,two-phase flow can be incorporated.

• In this thesis it is shown that there is a considerable number of parame-ters which affect the transition from hydrodynamic to mixed lubrication.A lubrication parameter is required, other than the commonly used dutyparameter, G, which incorporates these effects, and which will thus ad-equately determine this transition.

122 6. Conclusions and recommendations

APPENDIX

A. ANALYTICAL SOLUTION OFTHE HYDROSTATIC FLUIDPRESSURE FOR FLAT ANDCONED FACES — POLARCOORDINATES

In this Appendix, an analytical solution is derived for the hydrostatic fluidpressure, both for a flat parallel seal face and for a flat seal face with a coningangle. The equations are derived for the dimensionless variables, given inSection 3.4.4.

A.1 Flat parallel faces

In the case of flat parallel faces, Eq. (3.72) reduces to H = Hmin and Eq. (3.71)simplifies to:

∂

∂Y

(Y∂P

∂Y

)= 0. (A.1)

Solving this equation for an outside pressurized seal gives:

P (Y ) =Pf

ln

(ψ − 1

ψ

) ln

(ψ − 1

Y

). (A.2)

Equation (A.2) shows that the hydrostatic interfacial pressure for flat faces isonly controlled by the fluid pressure Pf and the seal face geometry ψ.The mean pressure in the gap is obtained by integrating Eq. (A.2) from ψ− 1to ψ and reads:

Pm(ψ, Pf ) = Pf

ψ ln

(ψ − 1

ψ

)+ 1

ln

(ψ − 1

ψ

) = 1. (A.3)

126 Flat faces with a convergent coning

Figure A.1 shows a plot of Pf as a function of ψ. When ψ approaches 1, Pfgoes to 1, as pm equals pf . As ψ approaches infinity, the curvature becomesnegligible, and a rectangular situation is approximated. Now Pf approaches2, as pm = pf/2.

( x d( dx 1(

d

1

ψ

Pf

Fig. A.1: Non-dimensional fluid pressure Pf as a function of seal facegeometry ψ for flat parallel faces.

A.2 Flat faces with a convergent coning

For an outside pressurized seal face with a convergent coning, the film thicknessequation reads:

H = Hmin + α(Y − ψ + 1). (A.4)

The part of the Reynolds’ equation (Eq. (3.71)) which has to be solved isreduced to:

∂

∂Y

(Y H3∂P

∂Y

)= 0. (A.5)

Solving this equation with the boundary conditions for a convergent gap:

P = 0 at Y = ψ − 1

P = Pf at Y = ψ,(A.6)

Flat faces with a convergent coning 127

will give the following pressure distribution:

P (Y, α, ψ,Hmin, Pf ) = Pf · 1αψ−α−Hmin

ln[

(ψ−1)(Hmin+αY−αψ+α)HminY

]− 1

2(3Hmin−αψ+α)

Hmin2 + 1

2(3Hmin+2αY−3αψ+3α)

(Hmin+αY−αψ+α)2

1(αψ−α−Hmin)

ln[

(Hmin+α)(ψ−1)ψHmin

]+ 1

2α

(−4Hmin2−5Hminα+2Hminαψ+α2ψ−α2)

(Hmin+α)2Hmin2

.

(A.7)

Integrating the equation above over Y from ψ− 1 to ψ gives an expression forthe mean pressure:

Pm(α, ψ,Hmin, Pf ) = Pf(Hmin+α)·[2Hmin

2ψ(Hmin+α)−α+αψ−Hmin

ln (ψ−1)(Hmin+α)Hminψ

+ (α2ψ−α2−3Hminα−2Hmin2)

2Hmin2(Hmin+α)2

−α+αψ−Hminln (ψ−1)(Hmin+α)

Hminψ+α(α2ψ−α2+2Hminαψ−5Hminα−4Hmin

2)

],

(A.8)

which must equal one.

( x d( dx 1( 1x

d

1 0

0.5

1

5

α =

Hmin = 1

Pf

ψ

Fig. A.2: The dimensionless fluid pressure, Pf as a function of thedimensionless seal geometry, ψ, for different values of α.

In Figs. A.2 and A.3 the fluid pressure, Pf , is plotted as a function of the sealgeometry variable, ψ, for different values of the coning, α. In Fig. A.2 Hmin istaken to be 1 and in Fig. A.3 Hmin is taken to be 10. In both Figures, with anincreasing value of α, the mean hydrostatic fluid pressure, pm, in the sealing


interface increases, which results in a lower value for Pf , defined as pf/pm.Furthermore, it is shown that for a smaller value of Hmin (Fig. A.2), the effectof a varying α is stronger than for a larger value of Hmin (Fig. A.3), see alsoSection 2.2.3.2. In the case of α = 0, a situation as described in the previoussection occurs.

( x d( dx 1( 1x

d

1 01

2.5

5

10

α =

Hmin = 10

Pf

ψ

Fig. A.3: The dimensionless fluid pressure, Pf as a function of thedimensionless seal geometry, ψ, for different values of α.

Finally, in Fig. A.4, Pf is plotted as a function of Hmin for different valuesof α. The seal geometry, ψ, is taken to have a mean value of 9.5. In thecase of no coning, α = 0, pm does not depend on the value of Hmin, resultingin a horizontal line. With an increasing value of α, pm increases, and, as aconsequence, Pf decreases. An increasing value of Hmin shows a higher valuefor Pf , as pm decreases.

Flat faces with a convergent coning 129

( x d( dx 1(

d

10

0.51

2.5

5

10

α =

ψ = 9.5

Pf

Hmin

Fig. A.4: The dimensionless fluid pressure, Pf as a function of thedimensionless minimum film thickness, Hmin, for different values ofα.

B. PHOTO IMPRESSION OF THETEST RIG

Fig. B.1: Complete test rig. On the left-hand side the stationary unit,on the right-hand side the rotating part.

Fig. B.2: Alignment construction, side view.

132 B. Photo impression of the test rig

Fig. B.3: Close-up of rotating seal and stationary seal.

Fig. B.4: Alignment construction, front view.

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INDEX

Aasperity

coefficient of friction, 25contact, 25contact area, 25density, 26elastic deformation, 26load, 25pressure, 25reference plane, 26shape, 26

asperity-asperity collisions, 19assumptions

contact model, 26full film model, 36

Bbalance area, 19balance ratio, 9, 20

balanced, 9unbalanced, 9

bellows, 86boundary layer, 24boundary lubrication, see lubrica-

tion

Ccalculating

Stribeck curve, see Stribeck curvecarbon, 98carbon seal

shrink, 96swell, 96

cavitation, 19, 38, 55, 60dimensionless parameter, 45

cleaningseal face, 101, 102

coefficient of friction, 3, 25, 65, 101asperity, 25boundary lubrication regime, 25,

66, 105measuring, 101micro-geometry, 103

compliance, 27coning, 11, 23, 29, 31, 101, 102

analytical solution, 42convergent, 11divergent, 11, 43effect, 43production methods, 103transition, 105

contactheat, 101material, 20

contact areaasperity, 25hydrodynamic, 65hydrodynamic lubrication, 25nominal, 29

dimensionless, 31fit, 31

contact model, 6, 23, 26curvature, 9, 30

radius, 30

Ddata acquisition, 88deformation

elastic, 30asperity, 26

140 Index

distributionasperity height, 28Gaussian, 28, 69M -inverted χ2

n, 68measured, 28standard deviation, 27

duty parameter, 4, 5

Eelastic deformation

asperity, 26elasticity

reduced modulus, see reducedmodulus of elasticity

equilibriumforce, 19

experimental procedure, 101experiments

frictionconing, 105lubricant, 102materials, 102micro-geometry, 103waviness amplitude, 103

iso-thermal, 101roughness, 103

Ffilm thickness, 24, 60, 67film thickness equation, 6, 23, 36,

520 < Pf ≤ 1, 541 < Pf ≤ 1.75, 55Pf = 0, 54α = 0, 53γ = 0, 52assumptions, 36convergent coning, 14diverging coning, 14fit, 36

force equilibrium, 19form tester, 96

friction, 3, 5, 88BL, 3boundary lubrication regime, 24cavitation, 60coning, 60effect of

coning, 105macro-geometry, 103micro-geometry, 103waviness amplitude, 103

experimental procedure, 93force, 37full film lubrication, 24, 60

design diagram, 60geometrical component, 23HL, 3measurements, 85, 92mixed lubrication regime, 25ML, 3rolling component, 23seal geometry, 60sliding component, 23squeezing component, 23static, 86Stribeck curve, see Stribeck curvewaviness, 60

full film lubrication, 7leakage, 60

full film model, 23, 36dimensionless equations, 42dimensionless variables, 40

GGreenwood & Williamson, 26–28

HHalf-Sommerfeld condition, 38height distribution, see distributionheight distribution function, 28Hertz, 30

contact area, 27contact load, 27

Index 141

contact parameters, 30contact radius, 27

hybrid transducer, 86hydrodynamic lubrication, see lubri-

cationhydrodynamic pressure, see pressurehydrostatic lubrication, see lubrica-

tionhydrostatic pressure, see pressure

Iinside pressurized seal, 8, 43, 87, 108

film thicknessconvergent coning, 37divergent coning, 37

interference microscope, 29, 96, 101,106

iso-thermal, 5, 101

KK-factor, 9, 16, 19kurtosis, 69

Lλ-variable, 66leakage, 3, 5, 37, 60, 106

coning, 60design diagram, 60waviness, 60

liquid fraction, 36, 39load

asperities, 28, 64hydrodynamic, 64spring, 19

load carrying capacity, 93experimental procedure, 93measurements, 85

results, 95lubricant behaviour

Newtonian, 24lubrication

boundary, 2, 71, 88

hydrodynamic, 2, 17, 39, 104contact area, 25

hydrostatic, 9microasperity, 18mixed, 2, 64, 71modes, 2, 23regimes, 2, 23

Mmacro-geometry, 101material

deformation, 102dimensions, 102silicon carbide

properties, 102specification, 102

mean planesummit heights, 26, 65surface heights, 26, 65

measuring procedure, 101micro-geometry, 101

coefficient of friction, 103microasperity lubrication, 18misalignment, 17, 86mixed lubrication, see lubricationmixed lubrication regime, see lubri-

cationmodel, 5

contact, 6, 23, 26assumptions, 26

full film, 23, 36assumptions, 36

Stribeck, 64mounting

inside, 8outside, 8

Nnominal contact area, 29, 64

dimensionless, 31fit, 31, 32

142 Index

OOptimum Similarity Analysis, 40outside pressurized seal, 8

boundary conditions, 37film thickness

convergent coning, 37divergent coning, 37

Ppin-on-disc device, 95, 98pneumatic cylinder, 86predicting

Stribeck curve, see Stribeck curvepressure

asperity, 25, 64distortions, 17fluctuations, 86fluid, 19, 37

effect, 44hydrodynamic, 17, 20, 64

macro-geometry, 103micro-geometry, 103

hydrostatic, 8, 19, 55cartesian coordinates, 10curvature, 9measuring, 108polar coordinates, 9

mixed lubrication, 64vapour, 38, 55

pressure distributionconing, 14minimum film thickness, 14

pressure gradient factor, see K-factorpressure vessel, 86pv, see load carrying capacity

Rradial taper, see coningreduced modulus of elasticity, 27,

29, 70reference plane

summit heights, 26

surface heights, 26Reynolds’ equation, 7

cartesian coordinates, 12cavitation, 39polar coordinates, 12, 36squeeze film term, 8, 18squeeze term, 37stretch term, 37wedge term, 7

roughness, 4, 101, 105

Sseal dimensions, 102seal face

cleaning, 101flat, 103grinding, 102lapping, 102, 103

seal geometryeffect, 44

seal width, 31separation, see film thicknessshear, 24

boundary layer, 24lubricant, 24rate, 24stress, 24, 25

silicon carbide, 98, 102properties, 102wear, 106

skewness, 69spring

load, 19squeeze film, 19standard deviation, 27, 107

summits, 67, 104Stribeck curve, 2, 92

calculating, 64effect ofσs, 106axial load, 69, 107

Index 143

coning, 67, 105ηsβσs, 67hydrostatic fluid pressure, 70,

108reduced modulus of elasticity,

70roughness, 67σs, 68viscosity, 70waviness amplitude, 66, 103,

106χ2n-height distribution, 68

experimental procedure, 101experimental validation, 101experiments

materials, 102friction, 23lubricant, 102mixed lubrication regime, 64predicting, 64prediction vs. experiment, 104procedure, 64roughness, 105summary, 71theoretical results, 66transition, 3

summitsradius, 104standard deviation, 67, 104

TTalyrond, 101taper, see coningTaylor series expansion, 30test rig

data acquisition, 88pulse generator, 89rotating part, 87stationary part, 85suspension, 87

thermal effects, 17

thermal wedge, 18transition, 5

definition, 3diagram, 106effect ofηsβσs, 68σs, 68, 106axial load, 107coning, 67, 105hydrostatic fluid pressure, 70,

108load, 69M -inverted χ2

n, 69reduced modulus of elasticity,

70viscosity, 70waviness amplitude, 67, 106

measuring, 3, 104, 106velocity, 67, 107

Vvelocity

angular, 37viscosity, 70viscosity wedge, 18

Wwaviness, 17, 23, 29, 101, 102

amplitude, 29effect, 43number of waves, 43production methods, 102transition, 106

wear, 3, 5, 16, 20, 102, 106measurements, 85, 95

methods, 95results, 98

specific wear rate, 97wedge

thermal, 18viscosity, 18

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