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Journal bearing optimization and analysis using streamline upwind Petrov-Galerkin finite element method Zengeya, Miles 2009

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Journal Bearing Optimization And Analysis Using Streamline Upwind Petrov-Galerkin Finite Element Method  by  Miles Zengeya B.S.M.E University of Arkansas, U.S.A., 1988 M.Sc., Loughborough University of Tech., England, 1992   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering)   THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) May 2009 © Miles Nyamana Zengeya, 2009   ii  Abstract A three-dimensional finite element thermo-hydrodynamic lubrication model that couples the Reynolds and energy equations is developed. The model uses the streamline upwind Petrov-Galerkin (SUPG) method. Model results indicate that the peak temperature location in slider bearing is on the mid-plane well as when pressure boundary conditions are altered in such a way that the inlet/outlet pressure is higher than the side pressure. The adiabatic temperature profiles of an infinite and square sliders are compared. The wider slider shows a higher peak temperature. Side flow plays a major role in determining the value and position of the peak temperature. Model results also indicate peak side flow at a width-to-length ratio of 2. A method of optimizing leakage, the Flow Gradient Method, is proposed. The SUPG finite element method shows rapid convergence for slider and plain journal bearings and requires no special treatment for backflow in slider bearings or special boundary conditions for heat transfer in the rupture zone of journal bearings. A template for modeling thermo-hydrodynamic lubrication in journal bearings is presented. The model is validated using experimental and analytical data in the literature. Maximum deviation from measured temperatures is shown to be within 40 per cent. The model needs no special treatment of boundary conditions in the rupture zone and shows rapid and robust convergence which makes it quite suitable for use in design optimization models and in obtaining closed relations for critical parameters in the design of journal and slider bearings. Empirically derived simulation models for temperature increase; leakage; and power loss are proposed and validated using the developed finite element model and experimental results from literature. Predictions of temperature increase, leakage, and power loss are  iii  better than those obtained for available relations in the literature. The derived simulation models include five important design variables namely the radial clearance, length to diameter ration, fluid viscosity, supply pressure and groove position. The derived model is used to minimize a multi-objective function using weight/scaling factors and Pareto optimal fronts. The latter method is recommended as preferable, and Pareto diagrams are presented for common bearing speeds. Including the groove location in the optimization model is shown to have a significant effect on the results. The lower bound of groove location appears to result in preferred power loss/side leakage values. Significant power loss savings may be realized with appropriate groove location.                iv  TABLE OF CONTENTS   Abstract…………...………………………………………………………………….……ii Table of Contents…...………………………………………………………….…………iv List of Tables………..………………………………………………………..………....viii List of Figures…………………………………………………………..………..……......x List of Symbols, Abbreviations and Parameters…...………………………….…….…...xv Acknowledgements…………………………………..……………………….…..……..xix  Chapter 1 Introduction……………………………..………………………..…….……1 1.1   Thermal Effects in Slider and Journal Bearings…..………………….….….…….1             1.1.1    Historical Background……………………..………………….…..………1             1.1.2    Thermal Effects and Boundary Conditions……..……….……….……….2 1.2 Literature Review………………………………………..………………………..3  1.2.1 Slider Bearings and Thermo-hydrodynamic Lubrication…...…..………...3  1.2.2 Journal Bearing THD Modeling………………………………..…………3 1.2.3 Convection-Dominated Flows and Upwinding………………..…….........5 1.2.4 Design Optimization…………..……………………………..…………....6  1.2.5 Journal Bearings and Optimization Models………………..…….….……7 1.3 Motivation For The Research……………………………………………………11 1.4 Structure of Research Work…………………………..…………..…….………..12  Chapter 2 Three-Dimensional SUPG FE Formulation………………...…….……....14 2.1 Introduction……………………………………….……………………………...14 2.2 Governing Equations……………………………………………...………….….14  v   2.2.1 Pressure Computations……………………….………………………….14  2.2.2 Energy and Viscosity Variation……………….…………………………18 2.3 Finite Element Formulation with Streamline Upwind Petrov-Galerkin…...….....24 2.4 Modeling Procedure…………………………………………….……….………30 2.5 Chapter summary………………………………………………..………….……34  Chapter 3 Application of (SUPG) FE to Slider Bearings............................................35 3.1 Introduction………….…………………………………………...……….……...35 3.2 Lebeck Slider Bearing Data....……………………………………..….…....……35 3.2.1 Model Convergence Performance...…….…………………....…..….…...36 3.2.2 Predicted Pressure Values……………………...………..…..…..…....….38 3.2.3 Thermal Profile……………………...…………………..…….….…...…40 3.2.4 Leakage Flow Pattern – Flow Gradient Method………...…............…….46 3.3 Chapter Summary……………..………………………………..….…..…...……51  Chapter 4 Streamline Upwind Petrov-Galerkin FE and Journal Bearings…...…... 52 4.1 Introduction…………………………….………………………………………...52 4.2 Model Formulation……………………..…………………………..…….….…..52 4.3 Simulation Results………………………….…………………...………….…....55 4.3.1 Mesh Density, Convergence and Performance………..…….…………...55 4.3.2 Pressure and Pressure Gradient Predictions…………..…….…....………59 4.3.3 Pressure Gradient and Effect on Dissipation…………..…...……….…...60 4.3.4 Dissipation Source Terms…………………...………..…………….…....63 4.3.5 Thermal Profiles……………………………...…………...……….……..66  vi  4.3.5.1 Standard Error and Goodness of Fit……………………………………...68 4.3.6 Pressure Simulations of Costa’s Data………………........………………77 4.3.7 Flow Predictions…………...…………..……………...……………...….79 4.4 Chapter Summary and Conclusions…………………….……..…………..….…79  Chapter 5 Formulation of Empirically Derived Model and Simulation Results......81 5.1 Introduction…………………………………...……………………………….....81 5.2 Governing Equations………………….…………………………………......…..86 5.2.1 Temperature Increase………………………………………………….....86 5.2.2 Side Leakage…………………………………………………………......88 5.2.3 Power Loss Determination…………………………………………...…..89 5.2.4 Pressure Computations…………………………………………….……..90 5.2.5 Formulation of the Optimization Problem…………………………...…..91 5.2.6 Implementation in Matlab…………………………………………...…...92 5.3 Validation of Proposed Equations……………………………………...….…….94  5.3.1 Effect of Supply Pressure ps………………………………...………...…94  5.3.2 Temperature Increase ΔT(X)……………………………………………..96  5.3.3 Leakage Flow QL Predictions…………………………….…………….102  5.3.4 Variation of C on Performance Parameters ΔT and QL….……………..104  5.3.5 Temperature Increase ΔT, Side Leakage QL as Function of Diameter....107  5.3.6 Simulation Results with ΔT (f1) and QL (f2) as Objective Functions…...108 5.4 Power Loss PL (F1) and Leakage QL (F2) as Objective Functions…………...…115  5.4.1 Power Loss Predictions from Model……………………………...….....115  5.4.2 Determination of Weighting Parameters and Scaling Factors…...…..…116  vii   5.4.3 Pareto Optimal Front Method………………………….…..……….….117  5.5 Extended Optimization Model……………………………………..…….…….125  5.5.1 Leakage Flow Equation and Validation……………………………...…125  5.5.2 Power Loss Equation and Validation……………………………….…..131  5.5.3 Optimization Simulations……………………………………..………..134 5.6 Interpretation of Optimization Results………………………………..………..140  5.6.1 Pareto Optimal Fronts with X܂ ൌ ሺܥ, ߣ, ߤ, ݌௦ሻ and XT ൌ ሺܥ, ߣ, ߤ, ݌௦, ߠሻ 140  5.6.2 Uncertainty in the Results………………...…………………………….141 5.7 Chapter Summary………………………………..…………………....….….…141  Chapter 6 Summary, Conclusions and Future Work…………….…..…….….…..143 6.1 Summary and Conclusions………………………….…………..….….….…...143 6.2 Recommendations for Future Work………………………….…….…..……..146  Bibliography……………………………………………………..……….…….……..144 Appendices……………………………………………………….…….…..….….…..157 Appendix A  Velocity fields u, v and w………………..…………………….157 Appendix B      Summary of Martin’s Procedure.………..……….……………159 Appendix C      Optimization Procedure………………..………………………160 Appendix D     Tables….……………………………….………………………166 Appendix E     Verification Examples……………….…………………………174  viii  List of Tables  Table 3.1  Input values for the model……………………………...……….……….36 Table 3.2 Mesh density and effect on prediction………………..…….…………....36 Table 3.3  Convergence criteria…..……………………………..………….…….…37 Table 3.4  Isothermal peak pressure and load-carrying capacity, k=0.4..…….........38 Table 3.5  THD peak pressure and load capacity comparisons………….…….…....40 Table 3.6  Dimensionless flow rate as a function of L/B……………..…..….…......47  Table 4.1 Convergence of Peak Pressure Predictions…………….………..……….56 Table 4.2 Computation Times for one Iteration…………….……………..………..57 Table 4.3 Model predictions of analytical and Dowson’s data….……….…….…...59 Table 4.4  Effect of pressure gradients ∂p/∂x, and ∂p/∂z on temperature rise ΔT…...59 Table 4.5 Standard Error and r2 correlation for Temperature Predictions……..…...69 Table 4.6 Predicted versus measured temperatures, Costa’s data……….…..……..75 Table 4.4 Predicted versus measured temperatures as function of supply press...…76  Table 5.1 Computation Times for Optimization Model Time Stepping…….…..….84 Table 5.2  Values a, b, and c computation of maximum temperature Tmax….....…...87 Table 5.3 Standard error and square of the Pearson correlation……………..……..95 Table 5.4  Local Reynolds number from a sample of the data…………...….….…..97 Table 5.5  Summary of ΔT(X) and QL data comparisons………………..…….…...104 Table 5.6  Typical Pareto Table for Bearing at 4000 rpm, W=10kN…….…...…..125 Table 5.7 Comparison of variance between model and Dowson’s predictions…...130 Table 5.8 Variation of Performance Parameters PL and QL………………….…..140  ix   Table D.1   Input data from Dowson and Song bearings………………….….......…166 Table D.2   Input values for the bearing tested by Boncompain et al. ……….....….169 Table D.3   Input values for Costa’s bearing………………………………...……...170 Table D.4  Input values for Ferron’s bearing………………………….……...……171 Table D.5  Additional input data to run optimization simulations…….…..…….…172 Table D.6 Input data for Dowson and Claro bearings………………….…..……...173  Table E1 Minimum and minimum values of the solution………………...………177 Table E2  Comparison of SUPG and Galerkin predictions………….…...………..179 Table E3 Prediction comparison, analytical vs SUPG FE………………...…...…182 Table E4 Input parameters and predictions……………………………...….…….183 Table E5 Input values for Mitsui bearing……………………………...……...…..184 Table E6 Maximum and average error for Mitsui’s data……………..………….185 Table E7 Input parameters for journal tested.…………………………..….…….186 Table E8 Error on thermal predictions, SUG vs THD model…………...………..189 Table E9 Performance parater predictions for Pierre’s data……..…………....…189         x  List of Figures  Figure 1.1 Flow chart of a simple genetic algorithm……………………..……....…10 Figure 1.2 Thesis roadmap…………………………………………….…….………13  Figure 2.1a Slider bearing………………………….……………………….………...16 Figure 2.1b Journal bearing and notation………………………………….….…....…16 Figure 2.2  Inlet temperature variation for Dowson’s bearing data……….…......…..21 Figure 2.3  Three dimensional slider bearing geometry……………………...……....25 Figure 2.4a Solution domain Ω showing boundary Γ1 and unit normal vector n…….26 Figure 2.4b Transformation of domain Ω……………………………………....….....26 Figure 2.5  Flowchart for thermo-hydrodynamic lubrication model…………......…..33  Figure 3.1  Convergence of load capacity solution…………………………………..37 Figure 3.2 Non-dimensional isothermal pressure predictions……….……....………39 Figure 3.3a  Normalized temperatures for 2-D SUPG model………….…..….………42 Figure 3.3b Temperature predictions for 3-D SUPG model………………..…......….42 Figure 3.4  Contour map of thermal field, L/B=10….…………………...…………..43 Figure 3.5  Thermal field with L/B=0.5……...………………...………..…..….…....44 Figure 3.6a  Model isotherms for Lebeck’s data……………………………...….........45 Figure 3.6b Comparison of temperature profiles from SUPG and Kumar………..….45 Figure 3.7 Dimensionless side leakage LQ as a function of L/B, k=0.4………......…48 Figure 3.8 Flow Gradient, k=0.4…………………………………...………….….....48 Figure 3.9 Side leakage as a function of L/B, k=0.4……………….……..……...….49 Figure 3.10  Load capacity as a function of k……………………………..…….….....50  xi   Figure 4.1a  Schematic representation of journal bearing……..……………………....53 Figure 4.1b Journal bearing unwrapped…………………………...………………….53 Figure 4.1c Isometric view of finite element mesh………………...…………………54 Figure 4.2  Cross section of the film region……………………………………...…..54 Figure 4.3 Convergence analysis on Dowson’s data…………………….…..…...….56 Figure 4.4 Peak Pressure Prediction for Dowson’s exp. Value of 35MPa…………..57 Figure 4.5 Model, experimental, and analytical predictions, Dowson’s data……….58 Figure 4.6 Convergence analysis on pressure gradient dp/dz……………………….61 Figure 4.7a  Dimensionless pressure predictions at bearing mid-plane, model.............62 Figure 4.7b Dimensionless pressure predictions at bearing mid-plane, Bonc............. 62 Figure 4.7c Pressure map showing the rupture and reformation interfaces………..…62 Figure 4.8a  Pressure gradient ∂p/∂x variation along the bearing z axis…………..…..63 Figure 4.8b Pressure gradients ∂p/∂x, ∂p/∂z variation along the bearing x axis……...63 Figure 4.9a  Velocity gradient ∂u/∂y as a function of X …………………………….65 Figure 4.9b Magnified view at the peak gradients……………..…….………….…....65 Figure 4.10 Velocity gradient ∂w/∂y as a function of X ……………………..…..….66 Figure 4.11a  Thermal profile, Boncompain’s data………………..…….……………..71 Figure 4.11b Figure 7 from Boncompain et al. ………………………………..…...…71 Figure 4.12a,b Thermal profile in the inlet, cavity and bearing  for Costa’s data….…...72 Figure 4.13  Bush thermal profile……………………………………………...……...73 Figure 4.14  Thermal predictions at the bush/oil interface – Costa’s data………........74 Figure 4.15  Predicted versus measured peak temperatures………………..…………76  xii  Figure 4.16  Pressure prediction as a function of groove location θ………….…........78 Figure 4.17  Flow predictions: model, measured data and Martin’s formula…............79  Figure 5.1 Flowchart for optimization model……………………………..…..…….85 Figure 5.2  Model versus Khonsari’s predictions – Costa’s data…………..….....….95 Figure 5.3  ΔT(X) as function of speed Ns for Dowson’s bearing…………...………98 Figure 5.4  ΔT(X) as a function of speed Ns for Ferron’s bearing…………..….........99 Figure 5.5  ΔT(X) as function of speed Ns, Costa’s bearing…………………..….…100 Figure 5.6a ΔT(X) as function of clearance ratio C, Dowson data…………..…...…101 Figure 5.6b ΔT(X) as function of diameter D, Dowson data………………..…..…..101 Figure 5.7  Side leakage predictions for Dowson’s bearing………………....….......102 Figure 5.8  Side leakage predictions – Ferron’s bearing…………………..………..103 Figure 5.9  Side leakage predictions – Costa’s bearing…………………..….……..103 Figure 5.10  Side leakage as a function of clearance C, speed 25 to 201 rps…....…..105 Figure 5.11  ΔT(X) versus clearance ratio C, Dowson’s bearing, Ns=201 rps…….…106 Figure 5.12  Side leakage as a function of clearance and three different speeds….....106 Figure 5.13  ΔT(X) as a function of diameter D……………………………….….....107 Figure 5.14  QL as a function of diameter D, Dowson’s bearing, Ns=25 rps…...…...108 Figure 5.15a  Objective function from Song et al. ……………………………..…....111 Figure 5.15b f(X) predictions from SQP and hybrid algorithm………………..….…111 Figure 5.16a  Clearance ratio C as a function of speed for Song’s model……..….....112 Figure 5.16b Clearance ratio C as a function of speed, current model…………....…112 Figure 5.17a   Length to diameter ratio λ as function of speed for Song’s model…....113  xiii  Figure 5.17b Length to diameter ratio λ as function of speed, current models…….....113 Figure 5.18a  Viscosity variation, Song’s model……………………………………...114 Figure 5.18b Viscosity variation versus speed, hybrid and SQP………….…..…...…114 Figure 5.19  Power loss prediction – Dowson’s data……………………...…......…..115 Figure 5.20  Power loss predictions – Costa’s data………………………...…...……116 Figure 5.21  f(X) as function of α1/α2, scaling parameters: β1=1, β2=105 …..…........118 Figure 5.22  Model predictions with α1/α2=1/5, β1=1/400, β2=105 ………....……...119 Figure 5.23  Model predictions with α1/α2=1/5, β1=1/35, β2=105 ……………….....120 Figure 5.24  Model predictions with α1/α2=1/5, β1=1, β2=105 ……………...…...…121 Figure 5.25  Pareto optimal front for speeds of (a)1500 rpm and (b) 2000 rpm…......123 Figure 5.26  Pareto optimal front for (a) 3000 rpm and (b) 4000 rpm…...…….....….124 Figure 5.27a Leakage flow prediction and data as function of supply pressure……...128 Figure 5.27b Data from Costa et al. ……………………………………………….....128 Figure 5.28 Leakage flow as function of supply pressure, Costa………...……….…129 Figure 5.29 Non-dimensional leakage flow as function of eccentricity ratio…….…129 Figure 5.30 Leakage flow as function of speed, ps0= and 170 kPa, Dowson…….…130 Figure 5.31 Non-dimensional leakage flow as function of eccentricity ratio….……131 Figure 5.32 Torque predictions as function of groove position, Costa……..….……133 Figure 5.33 Torque predictions as function of groove non-dim pressure……….…..133 Figure 5.34 Variation of torque as function of speed, Costa…………...…..…..……134 Figure 5.35 Variation of XT as a function of speed, W=10kN…..……………..……135 Figure 5.36 Variation of XT as a function of speed, W=700N…..……………..……138 Figure 5.37a,b Pareto optimal front for speed of 1500 rpm, W=10kN….………..…….139  xiv   Figure D1 Thermocouple location for pressure and temp. measurements…….…...168  Figure E1 Region of the skew…………………………………………………..….176 Figure E2 Mesh for convection-diffusion problem………………………….…….177 Figure E3 Three-dimensional plots…………………………………………..…….178 Figure E4 SUPG and Galerkin predictions as a function of speed………….……..180 Figure E5 Slider geometry ……………………………………………...….……...181 Figure E6 SUPG predictions versus measured temperatures, Mitsui………….…..185 Figure E7 Experimental and THD temperature predictions as function of angle....187 Figure E8 Comparison of SUPG and Pierre’s THD model………………………..188               xv  List of Symbols, Abbreviations and Parameters  B  length of slider (x-dimension)  Bib, Bis  Biot number for bush, shaft cf, cb, cs  specific heats of lubricant, bush and shaft, [J/kg.K] C  radial clearance, [m] D  journal diameter, [m] πD  Bearing circumferential length [m] e    eccentricity, [m] Ec  Eckert number Ec = U2/(cfΔT) F  objective function;  Ffr  frictional force, [N] f  coefficient of friction;  fR   reaction load on bearing, [N] h  film thickness, [m];  hp  thickness of the pad Hb,s  convective heat transfer coefficients for bush, shaft, [W/m2.k] IP  point of inflection where gradient is zero k  ratio of slider inlet (hi) to outlet (ho) dimensions (k=hi/ho) kf, kb, ks, kg  thermal conductivities fluid, bush (pad), shaft (runner) and gas, [W/m.K] L  width of bearing perpendicular to the direction of motion (z-dimension), [m] n    outward normal vector N  mesh density (number of elements in the mesh) Ni  the ith basis (shape) function Nu  Nusselt number, Nu=(HbR/kb) Nv  number of volume (brick) elements Ns  journal speed, rev/s  xvi  Pe  Pecklet number, Pe = ρocpωC2/kf PL  Power loss, [W] p    film pressure, [N/m2];  op ambient pressure [Pa]; ps  supply pressure [Pa] Pr  Prandtl number, Pr = cpμ/kf Qs leakage flow rate [m3/s];  Qv  the volume flow rate [m3/s] R1, R2, R3 bush inner, shaft, and bush outer radii respectively, [m] (see figure 2.1b) r2 the square of the Pearson product moment correlation coefficient T   lubricant temperature, [K]; Ta, Ti  ambient, inlet temperature, [K] Ts, Tb, Tr  shaft, bush and runner temperature, [K] Tfr   frictional torque [N.m] U  shaft surface velocity, [m/s] u    velocity vector u, v, w   fluid velocity component in x, y, z-axis respectively, [m/s] X  vector of design variables W  load capacity of the bearing, [N] x, y, z coordinates in the circumferential, radial, and axial bearing directions z1  parameter for computing attitude angle ϕ α   thermal diffusivity of the lubricant (α=kf/(ρcf)) α1, α2  Weigthing parameters, Optimization Model β   temperature-viscosity coefficient [1/K] β1,β2  Scaling parameters, Optimization Model ε   eccentricity ratio, ε = e/C εr   target error magnitude  xvii  ζ  streamline upwind parameter μ   fluid viscosity [Pa.s] δ   lubricant thermal expansivity [1/K], κ1, κ2  temperature rise parameters λ, λs   length to diameter ratio for journal (L/D) and slider (L/B) bearings respectively ρ   density of lubricant [kg/m3] σd standard deviation of predictions θ   circumferential coordinate of angle from oil groove position (Fig. 1), [radians] ϕ   attitude angle (Fig. 1), [radians] ϕcav  angle from oil groove to the cavitation boundary/interface, [radians] Ω     volume domain, discretized for finite element; Γ   boundary to domain Ω  Subscript a     indicates ambient conditions i      indicates conditions at the oil inlet o      indicates conditions at the outlet of slider (trailing edge)  Superscript  ‘¯’ indicates a dimensionless parameter (note: ‘+’ is used for normalized temperature iT TT =+ (in °K) which is a ratio to show the magnitude of temperature increase. Also μ+ as defined in non-dimensional parameters below) e denotes elemental values s  denotes a shape function for surface element (finite element formulation)  xviii  se denotes number of integration points, surface element ve denotes number of integration points, volume element   Non-dimensional parameters (those in brackets are specific to sliders). Note that symbols are defined above U uu = ; U v C Rv = ; U ww = ; D xx π= ;     h yy =   ; L zz = ; C hh =  iT TT =+ ; i s s T TT = ;   )( iTTT −= β ;  iTββ = ;  iTδδ = iρ ρρ =+ ;  ( )11 −−= +Tδρ ; [ ]T−= expμ ; effμ μμ = ; iμ μμ =+ if i cr Tk U EP 2μ= ; sNR pCp 2 2 μ= ;  ( UB ph p i o μ 2 = ) (for sliders) b b ib k RHB 3= ; s s is k LHB = ( ⎟⎠ ⎞⎜⎝ ⎛= B h UB QQ o2 ); ( 2UB hTT a or r μ= );  ( ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛= 2 3 o i h BU WW μ ) For the bush and pad (see fig. 2.1b): ݔҧ௕ ൌ ௫್ గ஽ ,   ݕത௕ ൌ ௬್ೞିோభ ோయିோభ ,     ݖҧ௕ ൌ ௭ ௅                             ݔҧ௣ ൌ ௫೛ ஻ ,      ݕത௣ ൌ ௬೛ ௛೛ ,       ݖҧ௣ ൌ ௭ ௅  For the shaft: ݕത௦ ൌ ௬ೞ ோమ  where ys is measured from the origin as indicated in figure 2.1b. Other non-dimensional values presented equation 2.4 and 2.7.   xix  Acknowledgements  This thesis would not have been possible without the assistance, support and encouragement from those who interact with me on a regular basis.  Top on that list is my supervisor Professor Mohamed S. Gadala for his academic and financial support. His guidance and encouragement have been crucial in finishing this work.  Special mention goes to Dr Guus Segal whose sharp numerical modeling abilities were invaluable in implementing the model in the finite element method. His proof reading of the journal papers is much appreciated. Sabrina Crispo, a co-worker at Koerner’s Pub who helped in proof reading the papers.  I also would like to thank Professor Michel Fillon, Université de Poitiers, France for kindly agreeing to include figures from some of the papers he contributed.  Finally I am grateful to my family whose moral support at crucial times made a difference in this work. And yes dad, you made such a difference in my life. You continue to inspire me even when times are hard and quitting seems the only option.     Miles Zengeya    1 Chapter 1 Introduction and Literature Review  1.1  Thermal Effects in Slider and Journal Bearings Slider and journal bearings are widely used in applications such as mechanical seals, collar thrust bearings, machine tool ways, automotive engine crankshaft and connecting- rods, steam turbines in power generating stations, railway wheels, piston rings, and many others. They have high reliability, good load-carrying capacity, excellent stability and durability and low coefficient of friction. Relative motion between two surfaces leads to not only an increase in temperature due to friction and viscous energy dissipation but thermal gradients that can develop on the surfaces. Almost all the frictional energy is dissipated in the form of heat. A thorough understanding of thermal effects in bearings is important in order to predict bearing performance. Investigation of the lubricant temperature fields leads to a better understanding of the load-carrying capacity of bearings and how complete loss of bearing clearance known as seizure can be avoided [1].  1.1.1 Historical Background Historically hydrodynamic lubrication theory was developed without consideration of thermal effects [2]. Isothermal models were used and adjusted for thermal effects and these models were adequate for relatively slow machinery where viscous energy was not very high. However, with modern machinery that runs at high speeds and loads thermal effects become significant. Not only must the non-linear effects associated with material property changes be addressed, the system thermal effects which couple the fluid film  2 with the overall mechanical system and its environment must be taken into account. The thermo-hydrodynamic (THD) problem is not easy to deal with considering the difficulty in extracting from the plethora of variables and boundary conditions those few parameters which embody the essentials of the process. Another problem is the difficulty in manipulating the complex set of governing differential equations – the Reynolds and energy equations, and other auxiliary material property expressions as an interconnected and coupled system. Lastly, the problem must be formulated such that solutions obtained are applicable to as many bearing configurations as possible.  1.1.2   Thermal Effects and Boundary Conditions Temperature variations are due to several causes, one of which is the heat generated by viscous dissipation. Temperature increase due to this phenomenon leads to a change in fluid properties, particularly its viscosity which lowers which in turn leads to lower load- carrying capacity of the film; this might cause the two surfaces to come into contact and eventually cause bearing seizure. A properly designed bearing will operate in the range where fluid property variations are in a range that will not cause such severe effects. The second mechanism is heat transfer to and from the fluid film which depends on material properties of the shaft and bush as well as the fixtures that keep the two in place. The shaft might be operating in high temperature environments that may result in heat transfer to the film, or the bush may be surrounded by hot ambient conditions that influence heat transfer at the interface. A third mechanism that affects the thermal map is the inlet mixing conditions. The hot recycled oil mixes with a fresh stream at the supply groove. Many mixing models appear in the literature with varying degrees of prediction accuracy [3,4,5,6].  3 Thermal aspects in fluid film bearings are important due to [2]: a) Their effect on bearing performance as mentioned above. This includes film thickness h, power loss PL, and side leakage rates QL. b) Magnitude and location of the maximum temperature. c) Effects of thermal gradients on the geometry of the fluid film (thermal distortion). d) Heat flow to and from the components of the system. Although their importance may not be the same, consideration of the above aspects plays some part in most thermo-hydrodynamic (THD) analysis. Besides determining performance and possible seizure due to degradation of the film, thermal gradients can also cause failure of the bearing material by instigating cracking. It is generally believed that proper design of journal and slider bearings is by appropriate consideration of thermal effects. Thermal effects, however, are not the only focus of the work carried out here. The bearing is viewed as a system and optimization of variables that designers have control over will also be investigated. Literature review of the state of thermo- hydrodynamic lubrication follows.  1.2  Literature Review 1.2.1 Slider Bearings and Thermo-hydrodynamic Lubrication Many studies on slider bearings appear in the literature [1,2,4,7,8,9,10,13,14]. Recent examples include the work of Rodkiewicz [7], Schumack [8] and Kumar et al. [9,10]. Schumack derived a spectral element scheme for thermo-hydrodynamic (THD) lubrication whereas Kumar et al. [9] used two dimensional finite elements to numerically analyze the load-carrying capacity of slider bearings under thermal influence by solving the coupled nonlinear partial differential equations governing fluid film temperature field  4 with either isothermal or adiabatic boundary conditions. At high speeds convection dominates the flow resulting in numerical oscillations due to the convective terms in the energy equation. Kumar et al. [9] used a finite element numerical scheme with streamline upwind Petrov-Galerkin (SUPG) as proposed by Brooks and Hughes [11] and Grygiel [12] to formulate a two dimensional version. Zengeya and Gadala [13] used a control volume finite difference (CVFD) scheme that recognizes location of the nodes with respect to fluid flow and applies appropriate weighting to solve the 2D thermo- hydrodynamic problem. The models mentioned above mainly focus on the two dimensional solution of the coupled equations. This works well for predicting the behavior of infinitely wide bearings. One important aspect of the work to be considered is modeling bearings in three dimensions and investigating the effect of aspect ratios L/B and L/D on leakage flow. Both finite and infinite bearings would be analyzed. Although other workers used three- dimensional models for slider bearings [14,15,16,17,18], the emphasis here would be on formulating the thermo-hydrodynamic problem in a way that applies to both sliders and plain journal bearings with little modification and to use the new template in deriving optimization equations for journal bearings.  1.2.2 Journal Bearing THD Modeling The field of application of journal bearings is large. Crankshafts and connecting-rods in automotive engines are two common examples; they are also used extensively in steam turbines and railway wheels. Properly installed and maintained, journal bearings have essentially infinite life. However, a thorough investigation of the lubricant temperature fields leads to a better understanding of the load-carrying capacity of the bearing and to  5 minimization of seizure. Higher temperatures affect bearing performance and reduce the margin against seizure. There are numerous studies on thermal effects in journal bearings. Khonsari [1] gave a good review of much of the work up to 1992. The work edited by Pinkus [2] is also valuable as a reference on this important area. Numerical solution of the thermal problem in journal bearings can be broadly classified into the finite volume (FV), finite difference (FD) and finite element (FE) methods. Significant contributions in the work done using finite volume is presented by Paranjpe and Goenka [19], Paranjpe and Han [20,21]. Stachowiak and Batchelor [22] present a numerical CVFD model while Booker and Huebner [23], Goenka [24], Labouff and Booker [25], Bayada et al. [26], and Kucinschi et al. [27] use finite elements. An important consideration in journal bearing modeling is cavitation, and some of the most significant contributions in this area were made by Elrod [28], Kumar and Booker [29,30], and more recently by Optasanu and Bonneau [31].  1.2.3 Convection-Dominated Flows and Upwinding The lubricant flow problem is convection dominated, resulting in numerical oscillations or ‘wiggles’ if the symmetrically weighted Galerkin method is used. Some form of upwinding whereby nodes upstream of the flow direction are assigned a higher weighting than downstream ones is required. Gero and Ettles [16] solve the problem using a backward difference Galerkin scheme. Colynuck and Medley [32] use a control volume finite difference (FD) scheme that uses compass directions (east, west, north, south) to enforce flow directionality into the controlling FD equations.  Kumar et al. [9] use a finite element numerical scheme with streamline upwind Petrov-Galerkin (SUPG) scheme for two dimensional THD problems.  6 It should be emphasized that there should be a way to define upstream and downstream positions in such algorithms. A method to account for possible backflow at the inlet must also be included.  1.2.4 Design Optimization Tied to the thermal performance of journal bearings is the choice of variables a designer may control. A designer may choose the type of oil and its initial viscosity for both slider and journal bearings. Other parameters that are determined at the early stages of the design process include the choice of materials for the two surfaces, general bearing dimensions, the speed at which the bearing is expected to run, load and oil supply pressure.  Choosing these variables influences a second group of variables, usually referred to as performance parameters group,  that the designer has no direct control over, and these may include: the coefficient of friction f, temperature rise ΔT, volume flow rate Q (which in turn affects leakage flow (QL), and the minimum film thickness hmin. Certain limitations on the performance parameters must be imposed by the designer to ensure satisfactory performance. One can surmise the design of journal and slider bearings as choosing the first group of variables such that limits on the second group are not exceeded [33]. The second part of the thesis develops an optimization scheme to help designers select the best possible set of design variables for a particular application. In order to properly carry out the optimization procedure, certain relations of key parameters must be established. In the work of the second part of the thesis such relations for leakage rate and power loss are proposed, tested and verified. A brief review of literature on journal bearing optimization follows.  7  1.2.5 Journal Bearings and Optimization Multi-objective optimization of bearings can be viewed as a minimization of temperature increase and side leakage as proposed by Hashimoto [34], Hashimoto and Matsumoto [35] and used extensively by Song et al. [36], and Yang et al. [37] for high-speed, short journal bearings (0.2<λ<0.6). Hashimoto uses weighting and scaling factors to combine the two objective functions into one multi-objective function. However this has the disadvantage that the designer must have prior knowledge about the relative importance of each objective. Hirani and Suh [38] propose minimizing the leakage and power loss since these two are ‘independent axioms’ [39]. Hirani [40,41] utilizes an alternative approach to multi-objective optimization, namely Pareto optimal fronts. This approach uses a posterior articulation of the weights in that the designer initially generates a number of non-inferior (a set of equally efficient) solutions from which a final decision is made on any one solution. These two methods can be formally defined as [42] • Prior articulation of preferences: involves combining individual objective functions using pre-decided weight factors into a single utility function [34,35,38,43] and solving problem as a single objective optimization problem. Weight factors express relative importance of objective function in the overall utility measure. • Posteriori articulation of preferences: involves the optimizer generating a number of non-inferior (a set of equally efficient) solutions from which a final decision is made on any one solution. This approach is often referred to as Pareto optimal approach. As a set of many trade-off solutions are already available with their pros and cons, Pareto optimal approach helps high level qualitative decisions.  8  Many optimization solution techniques are available depending on the nature of the objective function [42]. A popular classical solution technique for multi-objective functions with nonlinear constraints is the sequential quadratic programming (SQP). The method uses a quasi-Newton updating procedure and approximates the Hessian matrix of the optimization problem by an identity matrix. Various techniques may be used to improve the calculation of the search direction and to control the step size. Also, various modifications may be used to improve the identity matrix assumption of the Hessian matrix. Based on the work of Powell [44] the method mimics Newton’s method for constrained optimization whereby at each major iteration an approximation is made of the Hessian which is based on knowledge from previous steps. The method requires an initial guess for the design variables. Genetic and evolutionary algorithms are the current state of the art solution techniques for optimization problems with discontinuous objective functions or those with many local minima which make SQP unsuitable [44,45]. The term evolutionary suggests the natural process associated with biological evolution, particularly the Darwinian rule of selection of the fittest individuals in a population. Genetic algorithms (GA) were first proposed by Holland [46] and extended further by DeJong [47] and Goldberg [45]. They are an efficient search technique which apply the rules of natural genetics to explore a given search space [48]. They are robust, adaptive and well behaving for problems with combination of complex, discontinuous and discrete functions. Genetic algorithms maintain a population of encoded solutions, and guide the population towards the optimum solution [47]. Thus they search the space of possible individuals and seek to find the best fitness strings. Rather than start from a single point within the search space  9 as in SQP, GA’s start with an initial set of random solutions of population. The solutions are represented by strings (chromosomes) which are coded as a series of zeros and ones or a vector solution. GA’s are non-deterministic and do not require the evaluation of objective function differentials, hence their classification as zero-order optimization methods [49] which require only function evaluation. Figure 1.1 shows a simple flowchart for a genetic algorithm [50]. The process starts with the selection of a randomly chosen population and evaluation of fitness values for the individuals. An iterative process follows until satisfaction of the termination criteria. Within the reproduction phase, each individual’s fitness is evaluated, genetic operators that include crossover and mutation are applied to produce the next generation. The new individuals replace the parent generation and re-evaluation of the fitness of new individuals performed. A new set of chromosomes are produced at each generation using information from the previous generation. Related to the two main solution techniques is a group of ‘hybrid schemes’ that combine the benefits of both SQP and GA to solve optimization problems. The genetic algorithm is used to identify the area of the global minimum and SQP finds the exact location of the minimum [51]. Hybrid schemes are popular for solving journal bearing problems where the objective function has local peaks and valleys or is discontinuous and use of SQP alone may lead to erroneous solutions.      10                       Figure 1.1 Flow chart of a simple genetic algorithm [50].     Start Input: Design variable coding Initial population Objective Function SE ED IN G  Generation 1 Perform selection Parent I Parent II Perform crossover Perform mutation Perform other genetic operators Until temporary population is full R EP R O D U C TI O N  Updating existing generation EvaluationEV A LU A TI O N  Termination criteria satisfied Yes Write final result End No Generation = Generation + 1  11 1.3 Motivation For The Research The THD models described in section 1.2.1-1.2.3 are generally specifically formulated for slider or plain journal bearings. This approach obviously works well for dedicated computations in slider or journal bearings. The need for a general thermo-hydrodynamic formulation applicable to both slider and plain journal bearings with minimum modification gives motivation for the work. A holistic approach is adopted whereby the theoretical equations are formulated, validated and used in analyzing bearing phenomenon. Encompassed in this approach is optimization of design variables that engineers can control. A formal statement of the research objectives: • to formulate the three-dimensional finite element THD solution in a form applicable to both slider and journal bearings, is robust, and converges rapidly to be used in validating optimization equations developed in subsequent sections (chapter 5). Key elements of the model include wide applicability, to be used for slider as well as journal bearings, ease of applying boundary conditions, and ease of implementation in batch runs for optimization procedures. • The model should have minimal requirements for boundary conditions that are easy to implement. No need for complex boundary conditions to account for dissipation and thermal fields in the rupture zone as explained in section 4.3 where pressure gradients are analyzed in detail. • Is based on the streamline upwind Petrov-Galerkin finite element formulation for its accuracy in convection-dominated flows and its ability to account for backflow in the groove region.  12 • The idea of a ‘thermo-hydrodynamic bearing template’ is proposed as a direct outcome of the general formulation proposed. The template is easy to implement and gives reasonable predictions for bearing operating characteristics as shown in subsequent chapters. • The template is implemented such that groove location θ can be changed which requires generating a new mesh. This novel dynamic aspect of generating output proved critical in validating the optimization model in the second part of the work. • Empirically derived bearing optimization equations are proposed as a major contribution of this work. The equations are derived from first principles and validated with the aid of the developed template.  1.4 Structure of Research Work In the first part of the thesis, a robust thermo-hydrodynamic (THD) solution scheme applicable to both slider and plain journal bearing is proposed. The scheme, developed in the finite element method, has to be robust and exhibit rapid convergence characteristics due to the iterative nature of the optimization procedures proposed in the second part. Application of the scheme to slider bearing follows with analysis and discussion on certain aspects of obtained results.  Chapter 4 shows how the scheme can be applied to plain journal bearings with minor adjustments. Application and discussion of results obtained for journal bearings are presented. The second major section of the work focuses on using the developed finite element scheme to formulate optimization model for slider and journal bearings. New relations for leakage rate and power loss are proposed, tested and verified. Data on journal  13 bearings from the literature is used to simulate and validate the optimization equations. Two main optimization solution methods are compared and recommendations on the preferred method made in the final section. Figure 1.2 summarizes the flow of the work carried out.              Figure 1.2 Thesis roadmap         Develop robust THD scheme suitable for slider and plain journal bearings Validate model with slider bearing data from literature Extend THD model to plain journal bearings and validate with data from literature Use developed THD scheme to develop and validate an Optimization procedure  for plain journal bearings Use developed Optimization Scheme to make Design Recommendations via Pareto Charts Report on optimization results & conclusions  14 Chapter 2 Three-Dimensional SUPG FE Formulation  2.1 Introduction The previous chapter explored literature in thermo-hydrodynamic lubrication and optimization. This chapter develops the three-dimensional model that extends current models. A general scheme that can be used for both slider and journal bearings is developed. Governing equations for hydrodynamic lubrication and energy dissipation are initially presented before formulation of the streamline upwind Petrov-Galerkin finite element method. The formulation represents the first contribution in this work [52,53].  2.2 Governing Equations 2.2.1 Pressure Computations The Reynolds equation describes pressure variations in thin film flow where the film thickness h is very small compared to the size of the bearing. One can show by dimensional analysis that the continuity and Navier-Stokes equations describing fluid flow in this case may be approximated by the Reynolds equation for incompressible flow. Assumptions made in deriving Reynolds equation are: 1. Pressure is invariant across the fluid film (in the y direction, see figure 2.1) 2. Inertial forces are negligible 3. The flow is laminar 4. The lubricant behaves as a Newtonian fluid and there is no slip at the boundaries. The Reynolds equation takes the form  15  ( ) ( ) 0)212 21 3 =−++⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −+−∇−∇ ob ppkdt dhUUhph (fγμ       (2.1) where ∇ denotes the divergence, h and dh/dt are the film thickness and rate of change of the film thickness, μ the fluid viscosity (Pa.s), u the velocity vector of the runner and pad (figure 2.1a), p and po the gauge and ambient pressure respectively, fb body forces, and γ and k are constants. Figure 2.1 illustrates a slider and journal bearing with notations. Dowson [54] derived a generalized form of the Reynolds equation which allows variations in fluid properties along and across the film. Dowson further simplified the momentum equation by an order of magnitude analysis and neglecting body forces ( )122 0 1 1 0 1 222 vvx hU F FU F FhU xz pF zx pF x −+∂ ∂−⎥⎦ ⎤⎢⎣ ⎡ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −∂ ∂=⎭⎬ ⎫ ⎩⎨ ⎧ ∂ ∂ ∂ ∂+⎭⎬ ⎫ ⎩⎨ ⎧ ∂ ∂ ∂ ∂  (2.2) where x, y, and z coordinates in the flow direction, across the film, and axial direction respectively (x, y, z in the equations refers to (xs,ys,zs) on fig 2.1a and b as the origins). Note all reference to), v1 and v2 are the runner and pad velocities in the transverse y- direction and ∫= h dyF 00 1μ ,   ∫= h dyyF 01 μ   ∫ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −= h dy F FyyF 0 0 1 2 μ   (2.3) Slider Bearings: Applying the following dimensionless group for slider bearings (the bar on top of each variable indicates dimensionless quantities as defined in the nomenclature) 1U uu = ; 1U v hh Bv oi − = ; 1U ww = ; B xx = h yy =  ; L zz = ; oi hh hh −= ; ( ) 1 2 BU hhp p oiμ −=    (2.4)  16 and assuming U2 (stationary pad) and (v1-v2=0), the dimensionless form of the steady- state generalized Reynolds equation for slider bearings is [55] ⎥⎦ ⎤⎢⎣ ⎡ ∂ ∂=⎭⎬ ⎫ ⎩⎨ ⎧ ∂ ∂ ∂ ∂+⎭⎬ ⎫ ⎩⎨ ⎧ ∂ ∂ ∂ ∂ 0 1 2 3 22 3 4 F Fh xz pFh zx pFh x sλ    (2.5) where B L s =λ  is the aspect ratio for sliders and ∫= 100 1 ydF μ ;  ∫= 1 01 ydyF μ ;  ∫ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −= 1 0 0 1 2 ydF FyyF μ    (2.6)       (a)         (b) Figure 2.1(a) Slider bearing (b) Journal bearing and notation Stationary pad Runner ho h B U1 u(x,y) v(x,y) v1 U2=0 hi v2  zs  xs  ys  xp  yp zp ϕ e R1 R2 Oil supply hole h W ybxb θ y x z ω R3 Oil groove fys fxs xs ys  17 and pressure boundary condition spp =  at 0=x ; 0=p at 0=z ; and 0=∂∂ zp  at 2/1±=z     (2.7)  Journal Bearings: Dimensionless parameters for journal bearings are  R xx π2= ; h yy = ; L zz = U uu = ; U v C Rv = ; U ww = ; C hh = ; si NR PCP 2 2 μ=   (2.8) These are substituted into the generalized Reynolds equation to give [55]  ⎥⎦ ⎤⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −∂ ∂=⎭⎬ ⎫ ⎩⎨ ⎧ ∂ ∂ ∂ ∂+⎭⎬ ⎫ ⎩⎨ ⎧ ∂ ∂ ∂ ∂ 0 1 2 3 22 3 11 F Fh xz pFh zx pFh x λ     (2.9) where λ is the aspect ratio L/D and 0F , 1F  and 2F  are defined in equation 2.6, In addition to pressure boundary conditions equation 2.7, the Swifter-Stieber pressure boundary condition 0=∂∂ xp  and 0=p  at cavxx =   applies at the rupture zone. Similarities in equations 2.5 and 2.8 for slider and journal bearings are clearly evident. Only minor adjustments are required to specify the length in the flow direction as x=2πR in journal bearings. The flow boundary condition on surface S1 (mid-plane) is zero which can be expressed mathematically as [56]     0nu =⋅−∂ ∂ 212 3 h x ph iμ          (2.10) where xi takes the value of i=1,2 and corresponds to the coordinates, u is the velocity vector, and n is unit normal vector. It must be emphasized, however, that some of the gradient terms ߲݌ҧ ߲ݔҧ௜⁄  are not zero at the mid-plane as noted in section 4.3.2.    18 2.2.2 Energy and Viscosity Variation The three dimensional steady state energy equation is used to account for variations in film temperature.    ( ) sff fpTTkTc +∇⋅=∇∇−∇⋅ uu ρδρ    (2.11) where the first term is convective, the second determines conduction, u is the velocity vector, kf the fluid thermal conductivity, cf the fluid heat capacity, ρ the fluid density, δ the compressibility, and fs represents source terms. Simplifying assumptions made are [2] 1. Conduction in the sliding (x) direction is negligible 2. Thermal conductivity and specific heat are constant, and 3. Velocity gradients in all but the transverse (y) direction are negligible. The form of the equation used in the finite element formulation is [2] ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂+∂ ∂+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+∂ ∂+∂ ∂ 22 2 2 y w y u z pw x puT y Tk z Tw y Tv x Tuc ff μρδρ   (2.12)  in which the energy is expressed per unit volume. Dowson [54] and Ezzat and Rohde [57] show how to compute the velocity profiles u, v and w from the momentum and continuity equations (reproduced in Appendix A). (x, y, z in the equations refers to (xs,ys,zs) on fig 2.1a and b as the origins). Two methods of non-dimensionalising equation 2.12 are presented. The dimensionless parameter group the Pecklet (Pe), Prandtl (Pr), and Eckert (Ec) numbers present one method of making the energy equation non-dimensional if c Tc UE = ; Lk Uhc P f of e 2ρ= ; if i cr Tk U EP 2μ=              (2.13)  19 where the subscript i indicates an input’s value at the inlet. These are substituted into equation 2.12, with the resulting equation ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+∂ ∂+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+∂ ∂+∂ ∂ ++++ y pv x pu P EP y T Pz Tw y Tv x Tu e cr e 2 21ρ  ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂+⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂+ 22 z w z u P EP e cr μ       (2.14a)  where T+=T/Ti indicates the extent of temperature increase from inlet conditions. This formulation is applicable to both slider and journal bearing with little modification. However Jang et al. [55] derive a closed form of the energy equation 2 212 2 22 2 1 11 ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+∂ ∂=∂ ∂ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −+∂ ∂ y u hy T hy T xd hdyuv hx Tu μκκ κ   (2.14b) where two key parameters κ1 and κ2 referred to as temperature rise parameters appear and defined for slider and journal bearings as 2 2 1 ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −= oif i hh B Bk Uβαμκ , f i k U βμκ 22 =  2 2 1 ⎟⎠ ⎞⎜⎝ ⎛= C R Rk U f i βαμκ , f i k U βμκ 22 =  (2.14c) All terms can be determined from inlet conditions. The first temperature rise parameter κ1 is associated with viscous dissipation while κ2 relates oil properties and velocity. Slider Bearings:  The heat flux at the oil film/pad interface is assumed equal  01 == ∂ ∂=∂ ∂ pb y p p p y f T kTk yy      (2.15a) while heat conduction in the pad is determined from    02 2 2 2 =∂ ∂+∂ ∂ p p p p z T y T      (2.15b)   20    where kf and kp are the fluid and pad thermal conductivities, ݔҧ௣, ݕത௣ and ݖҧ௣ the non- dimensional coordinates of the pad as defined in the nomenclature, and ( )ipp TTT −= β  is the non-dimensional pad temperature, and β the temperature-viscosity coefficient. At the pad/ambient interface ⎟⎠ ⎞⎜⎝ ⎛ −=∂ ∂ = = oypip yp TTB y T p p 1 1             (2.16a) where തܶ௣, തܶ௢ are the pad surface and ambient temperatures respectively, and Bip, the biot number for the pad. At the film/runner interface heat flux is assumed equal  00 == ∂ ∂=∂ ∂ ss ys s s y f TkTk yy      (2.16b)  and the temperature determined from      0 2 2 2 2 =∂ ∂+∂ ∂ s s s s z T y T       (2.16c)  Journal Bearings: The energy equation as presented applies to journal bearings with modifications to account for heat transfer in the rupture zone. The rupture zone presents challenges as it comprises regions of alternating fluid and gas forming finger-like striations. Boncompain et al. [15] use two different energy equations in the rupture zone, one for the fluid and another for the gas sections. Critical to any modeling attempt is the identification of the rupture and reformation interfaces. It is expected that the shear rate decreases whereas the heat convection increases in the rupture zone while the lubricant is mixed with gas thus decreasing viscous dissipation significantly. Another approach to account for thermal characteristics of lubricant in this zone is to use the ‘effective length’ [15,58] which is defined as L’(x)=[hcav/h(x)]L and in dimensionless form as L  21    ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ ∫ ⎥⎦ ⎤⎢⎣ ⎡ ⎥⎦ ⎤⎢⎣ ⎡ = 2/1 0 1 0 2/1 0 1 0 ,, ,, zdydzyxuxh zdydzyxuxh xL cav     for 1<< xxcav  (2.17) where h(xcav) is the film thickness at the rupture interface and the other symbols have their usual meanings. Effective values of fluid conductivity 'fk , specific heat ' fc or viscosity 'μ  are determined using a linear relationship of the form     Ψ’=ΨG+(ΨF-ΨG) ( )xL     (2.18) where Ψ represents the property and the subscript denote gas (G) or fluid (F). Fluid film properties are assumed constant in the continuous sections in the transverse (y) direction at each position along the bearing length. The heat flux at the oil film/bush interface is  01 == ∂ ∂=∂ ∂ bs yb b b y f y TkTk y               (2.19a) where kf and kb are the fluid and bush thermal conductivities and ( )ibb TTT −= β  the non- dimensional bush temperature. Heat conduction in the bush is  0 2 2 2 2 =∂ ∂+∂ ∂ b b b b z T y T                                                          (2.19b)  Convection applies at the outer surface of the bush and shaft. At the shaft end [15,75]    ⎟⎠ ⎞⎜⎝ ⎛ −−=∂ ∂ ±=±= ozsis s s zs s TTB k H z T s s 2/1 2/1     (2.20) where Hs is the convective heat transfer coefficient at the shaft/ambient interface, Bis the Biot number for the shaft, and oT  the dimensionless ambient temperature. At the bush ends convection carries heat away from the shaft ends [15,75]  22    ( )ozbib b b z b TTB k H z T −−=∂ ∂ ±=±= 2/12/1     (2.21) Inlet temperature conditions present some challenges as the fresh oil mixes with circulating oil in the bearing. The oil mixing model was verified inversely in that the oil temperature at the supply hole was assumed to be Ti and the model determines how the inlet conditions vary. A parabolic mixing model shown in Figure 2.2 for Dowson’s [59] data is determined, similar to the findings of other researchers [5,6].  Figure 2.2 Inlet temperature variation for Dowson’s [59] bearing data.  Computation of the inlet temperature was done following recommendations by Costa et al. [6] whereby the ratio of the oil groove width Lg to bearing width L determines the particular method used. If Lg/L>0.5 the energy conservation principle is applied at the oil groove whereas if Lg/L≤0.5 the energy conservation principle is applied to the section immediately downstream of the supply groove, over the full bearing width. Film pressure in the groove region for the case where oil groove is aligned with the load line as indicated in figure 2.1b was shown to be equal to the supply pressure. Simplifications made by Costa et al. [6] were used: T = -5E+08z2 + 6358.5z + 312.4 R2 = 0.9983 36.00 37.00 38.00 39.00 40.00 41.00 0.0E+00 2.0E-05 4.0E-05 6.0E-05 8.0E-05 Te m p (C ) y-axis (m) Temperature variation across inlet film Temp  23    Tag=Ts+Cms(Tin-Ts)   for Lg/L>0.5    Tag=Tin    for Lg/L≤0.5    Tig=Tag    for all Lg/L  (2.22) where Tag is the temperature of the oil coming out of the groove in the axial direction, Tig is the temperature of the oil in the bearing clearance on the groove region, and Cms is an empirical mixing coefficient between zero and one (0.5 was used as recommended by Costa et al.). If film reformation was detected upstream of the groove,  ( )[ ] agmsin sagmssrvrvupup i QCQ TQCQQTQT T + −−++= 1  for a/b>0.5  sub sssubup i QQ QTQT T + +=     for a/b≤0.5  (2.23) If the film reformation was over the groove region or downstream, the oil inlet temperature was determined by  ( )[ ] agmsin sagmssrvrvupup i QCQ TQCQQTQT T + −−++= 1  for a/b>0.5  sre ssreup i QQ QTQT T + +=    for a/b≤0.5  (2.24) In equations 2.23 and 2.24 Tup is the average lubricant temperature of the re-circulating oil flow, Trv is the average lubricant temperature of the reverse flow, Qs is the oil supplied to the bearing, Qre is the re-circulating flow coming from the rupture region, Qup is the oil flow calculated upstream of the groove for the groove axial extent, Qub is the oil flow upstream of the groove calculated for the full bearing width, Qin the flow rate at the inlet section, and Qrv is the reverse flow.  24 Viscosity and temperature equations form the final couple of equations in the set. They are related to temperature variations by     ( )[ ]ii TT −−= βμμ exp     (2.25a)     ( )[ ]io TT −−= δρρ 1         (2.25b) where β is the temperature-viscosity coefficient and μi the inlet viscosity and δ the temperature-density coefficient. The viscosity equation couples the Reynolds and energy equations and can be expressed in dimensionless form by introducing the dimensionless temperature ( )iTTT −= β  which, when substituted into 2.25 gives     [ ]T−= expμ       (2.26a)     ( )11 −−= Tδρ       (2.26b) Equation 2.26a is based on the effective viscosity concept [55] (as defined in the List of Symbols) while 3.26b eminates from equation 2.14a i e cr T P EP δδ =  (very small). The viscosity-temperature relationship has a limited range of applicability - temperature variations on the order of a few tens of degrees – thus will not be applied to peak temperature predictions well in excess of 100°C.  The last two sections present literature review on methods of solving the thermo- hydrodynamic problem in journal bearings. Critical in this endeavor is proper specification of boundary conditions. Although not always easy to do, specification of boundary conditions at the groove/oil inlet is critical.  2.3 Finite Element Formulation with Streamline Upwind Petrov-Galerkin The slider in  figure 2.3 is  shown for  reference (only half the bearing  is  used  due  to  25 symmetry). Mid-plane surface, S1 is shown hatched. Numerical solution of the pressure problem is based on the variational/weak formulation. Assume the domain of interest (shown in figure 2.4) is bounded in R3 with a piecewise boundary Γ and let n( zyx ,, ) denote the spatial coordinates of the outward unit vector of a generic point on Ω . Let a portion of the boundary be Γ1 on which pressure computations are performed. The pressure p is the solution of the minimization problem [60]      ( ) ( ) Γ⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂+∂ ∂−⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂+⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂=ΠΠ ∫ Γ d z p x ph z p x php 1 223 p 2 1 6  with ,pmin μ   (2.27) As mentioned earlier the Galerkin weighted residual formulation is not reliable in convective dominated flows as it leads to spurious node-to-node oscillations in the solution. Some form of upwinding is needed, and the streamline upwind Petrov-Galerkin (SUPG) shows good stability in initial and boundary value problems and can handle backflow at the inlet [61]. The energy equation is multiplied by a test function w~  and integrated over the domain Ω  to get the weak formulation. 0 1 ~ 22 2 2 =Ω ⎥⎥ ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎢ ⎣ ⎡ ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂− ⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂+∂ ∂−⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂−⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+∂ ∂+∂ ∂ ∫Ω d y w y u P EP z pw x pu P EP y T Pz Tw y Tv x Tu w e cr e cr e μ ρ   (2.29) The second term in equation 2.29, ( )( ) Ω∂∂∫ Ω dyTPw e 221~  is integrated by parts to reduce the derivatives by one order. The weight function w~  consists of two parts: the classical Galerkin weighting function w and a discontinuous part ζ  (the streamline upwind parameter) which is added so that the weighting function w~  becomes ζ+= ww~ .  The  26 finite element weak formulation is solved by approximating the solution by a finite number of basis functions and approximating the weighting function w by the same set.         Figure 2.3 Three dimensional slider bearing geometry. Surface S1 is the mid-plane, S2 and S5 are the inlet and outlet surfaces.     (a)               (b) Figure 2.4(a) Solution domain Ω showing boundary Γ1 and unit normal vector n. (b) Domain Ω transformation (only a small part of the transformation is shown). L/2 B U hi ho Outlet Inlet S1 A B  R2  R3 Ω Γ1 n( zyx ,, ) Γ  R1 Pressure calculated on bottom surface Projection of nodal pressure values on Γ1 into volume domain Ω S5 S3 hp  yp  zp  xp  zs  xs  ys  27   This results in the standard finite element interpolation or shape functions Ni which doesn’t yet account for the extra part in the weighting function. The ζ-part of the basis function is defined at the element level as ζe [62]  u u 2 i e Nx ∇⋅Δ= ξζ      (2.30) where Δx is the size of the element in the direction of flow, ξ is a parameter defining the type of upwinding (in the classical upwind scheme ξ=1), ∇Ni is the gradient of the interpolation (shape) functions, and ||u|| is the square root of the inner product of u, (||u||=(uiui)1/2). In Appendix E a comparison of thermal prediction results from Galerkin and SUPG is explored. In the finite element method approach the lubrication problem is discretized into a finite number of unknowns. This is achieved by dividing the solution domain Ω into hexahedral (brick) elements and expressing the field variables (pressure, velocity and temperature) in terms of approximating trial functions (Ni) within each element. The volumetric domain Ω is divided into Nve linear hexahedral elements. The field variables (velocity and temperature) are expressed in terms of approximating trial or shape functions (Ni) within each element whereas the boundary surface Γ1 is discretized into Ns bilinear rectangular elements such that [56]   ڂ Ωഥ௘ ൌ Ω ேೡ೐ ௘ୀଵ ,   ځ Ωഥୣ ൌ Ԅଵ ேೡ೐ ௘ୀଵ     (2.31a)   ڂ Γതଵ௘ ൌ Γଵ ேೞ ௘ୀଵ ,  ځ Γതଵୣ ൌ Ԅଶ ேೞ ௘ୀଵ    (2.31b) where φ1 represents the volumetric solution field and φ2 the surface solution vector, and ڂ and ځ represent the union and intersect of the elements. The discretized field variables are given by [52,53]  28 ∑ = ≈ ne e N k k v k uNu 1 ,  ∑ = ≈ ne ke N k k vvNv 1 ,  ∑ = ≈ ne ke N k k TTNT 1 , and ∑ = ≈ se e N k k p k pNp 1    (2.32) where Nne=8 for the brick elements and Nse=4 for the bilinear rectangular elements and ܰ௩, ்ܰand ܰ௣ are the shape functions for velocity, temperature and pressure respectively. These elemental discretizations are introduced into equation 2.27 for the pressure [52,53]      ∫ ∑∑∑Γ === =Γ⎪⎪⎭ ⎪⎪⎬ ⎫ ⎪⎪⎩ ⎪⎪⎨ ⎧ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ∂ ∂− ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ∂ ∂+⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ∂ ∂ e se j se j se j dNp x hNp y Np x h N j p j e N j p j e N j p j e 1 0 12 1 1 2 1 2 1 3 μ   (2.33) Equation 2.33 is minimized with respect to the elemental field variable ejp  and on further simplification becomes [52,53]        ∫∑∫ Γ= Γ =∂∂−⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ∂ ∂+∂ ∂ ∂ ∂ e se e xd x N hp z N z N x N x Nh pie j N j p i p j p i p j 11 0 6 1 3 μ              (2.34) The energy equation can be discretized in a similar manner to yield [52,53,63] ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ + ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −=⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ∂ ∂− ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ∂ ∂ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +∂ ∂ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +∂ ∂ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −− ∑∑∑∑ ∑∑ ∑∫ ∑∫ ∑∑∑∑ ==== == =Ω = Ω ==== nene k nene k ne k ne k ne e ve e ne k ne k ne k ne N k e k v k N k v k e N k e k v k N k v k e N k v k e N k v k e N k e k v k e cr i e j v ji e N j v j N k v k e v j N k v k e v j N k v k e N k e k v ki pN zd dNwpN xd dNu NwNu y TN P EPwT y N y W P z N Nw y N Nv x N NuTNw 1111 2 1 2 1 1 1 1111 exp~1 11~ β                                                                                                        (2.35) where Nne is the number of nodes for each elemental, vkN is the shape function of the k th volume element, wi and iw~  are the Galerkin and SUPG weighting functions respectively and equation 2.26 has been substituted for ρ  and μ  while the subscript e represents a variable at the elemental level. One important observation about the discretization in  29 equation 2.35 is that the gradient pressure vectors xp ∂∂ / and zp ∂∂ / on boundary Γ1 have to be extended into the volume domain Ω in order to compute the velocity components u and v (figure 2.4b). The governing equations are used to develop the model using the finite element program Sepran [62], an FE program developed at Delft University. The program is structured into ‘standard problems’ that mainly encompass solution to second- order elliptic and parabolic differential equations. One advantage the program has is the ease with which user subroutines are implemented in the Fortran programming language. An example is the computation of compressibility terms in equation 2.35, velocity components u, v, and w, effective viscosity μeff and density ρ. Performance parameters are the load capacity, frictional drag force, and side flow. The non-dimensional load carrying capacity and frictional force for a slider are [9]    ∫ ∫ ⎥⎥⎦⎤⎢⎢⎣⎡ ⎥⎦⎤⎢⎣⎡∂∂−⎭⎬⎫⎩⎨⎧ ∂∂∂∂+⎭⎬⎫⎩⎨⎧ ∂∂∂∂ 1 0 1 0 0 1 2 3 22 3 4 zdxd F Fh xz pFh zx pFh x sλ              (2.36a)      ܨ௦ ൌ ׬ ଵ ிതబ ଵ ଴ డ௨ഥ డ௬ത ݀ݔҧ                                              (2.36b) where velocity component ݑത ൌ ݑതଶ െ ݑതଵ is determined in Appendix A and డ௨ഥ డ௬ത  is the differential of equation A_2 with respect to y. Shaft frictional torque Tfr in journal bearings is determined by integrating shear stresses (߬ҧ ൌ ௛ ഥ ଶ ቀడ௣ҧ డ௫ҧ ቁ ൅ ሺݑതଶ െ ݑതଵሻ/ܨത௢ሻat the shaft and bush interfaces [54]     ௙ܶ௥ ൌ ׭ ቂ ௛ഥ ଶ ቀడ௣ҧ డ௫ҧ ቁ ൅ ሺ௨ഥమି௨ഥభሻ ிത೚ ሻቃଵ଴ ݀ݔҧ݀ݖҧ             (2.37) where velocity components ݑതଶ and ݑതଵ are presented in Appendix A. The first term is due to film pressure and second due to velocity-induced shearing. Load capacity for journal  30 bearings is presented in section 2.4. Equations 2.36 and 2.37 were solved using user subroutine which performs the integrations using current values of viscosity μ . The flow rate  has a pressure and velocity-induced component [64]      ݍതሶ ൌሶ ׬ ݑത݀ݕതଵ଴      (2.38) where ݑത is determined in appendix A. Side flow in slider bearings can also be determined by assuming conservation of mass flow and finding the difference between the inflow and outflow, oiL QQQ −= .  2.4 Modeling Procedure The equations presented in sections 2.2 and 2.3 form the core of the bearing template developed. A flow chart of the implementation is shown in figure 2.5. Current values of viscosity μ  are used to compute the pressure field from Reynolds equations (2.5 and 2.8) with the boundary conditions equations 2.7 and 2.9. A conjugate gradient iteration method is used. The pressure is used in the next step where the Reynolds equation is solved with the Swift-Stieber boundary condition using an over-relaxation technique with constraint p≥0. Two important forces on the bearing are the load capacity as determined from the Reynolds equation Wd and the actual load W. A force balance must exist between the applied load and load capacity. The procedure used to achieve this is a. Assume an initial value for eccentricity e, compute eccentricity ratio (ε=e/c) (normal range 0.5≤ε≤0.9) as well as attitude angle ϕ and compute the film thickness h(θ) from (see figure 2.1b) h(θ)=c(1+εcos(θ-ϕ))                          (2.39a)  31 where c=(R1-R2) is the radial clearance, θ and ϕ the angle from the load line and attitude angle respectively. The minimum film thickness is located at θ=π+ϕ    hmin=c(1-ε)      (2.39b) b. Solve the Reynolds equation for pressure and load-carrying capacity Wd. c.  For this capacity compute the reaction load fR=(fx,fz)T and attitude angle from    ( )∫Ω Ω−= dpf x ϕcos                 (2.40a)    ( )∫Ω Ω−= dpf z ϕsin      (2.40b)   and  fR ൌ ඥ ௫݂ଶ ൅ ௬݂ଶ      (2.40c) and attitude angle ϕ which is ⎟⎠ ⎞⎜⎝ ⎛= − z y f f1tanϕ     (2.40d) Sommerfeld number can be computed from  2 ⎟⎠ ⎞⎜⎝ ⎛= C R W DLN S seff μ                                        (2.40e)  d. If fR<W increase the value of eccentricity ratio ε, if fR>W decrease its value. e. Compute the attitude angle from equation 2.40d. If ϕ<ϕa (where ϕa is value from step a) decrease value of ϕ (in step a), or if ϕ>ϕa increase value of ϕ (in step a). f. Go back to step a and repeat procedure. g. When convergence is obtained on both eccentricity ratio and attitude angle, the flowchart in figure 2.5 continues at stage 4 by computing velocity components u, v, w.  32 h. modify the attitude angle until the direction of the calculated load from 5c is equal to the imposed load direction. i. Once equilibrium is achieved the flowchart in figure 2.5 continues at stage 3 by computing pressure gradients dp/dx and dp/dz followed by velocity components u, v, w. The equations for the velocity fields are numerically integrated in a user subroutine using the trapezoidal rule. Velocity component w is determined from the continuity equation A-7 (Appendix A). Once the velocity field is determined, the energy equation is solved for temperature, which is used to compute new density and viscosity fields (note that density variation is so small it may be ignored). Computation of the thermal field in the rupture zone is achieved by considering effective length equation 2.17 and the linear relationship in 2.18 for other physical properties (thermal conductivity and heat capacity). New values of effective viscosity are determined with the thermal field and compared to previous values. The effective viscosity is the average viscosity value over the film nodes determined using equation 2.26a and dividing by the number of nodes. If this value is more than a predetermined value εr (set to μത≤0.01 or if the number of iterations iter exceeds a specified limit Lm (set to 100) in case of slow/no convergence) the iteration repeats (outer loop in figure 2.5) otherwise the program proceeds to computing bearing operating parameters ഥܹௗ, തܳ௅, ௅ܲ.      1−−≤ iir μμε      (2.42) Jang and Khonsari [55] recommend an empirical equation for determination of ε that may be used to verify equation 2.40 (note that this is used for verification purposes only and used at the end of the process).  33   ( )S DL ln20986.0 / 49737.036667.0 −+−=ε  for 0.5 ≤ L/D ≤ 1.5  (2.41) Periodical boundary conditions are applied to surfaces S3 and S5 in figure 2.3. Similar conditions are set for the bush and shaft ends. A typical run using data from Dowson et al. [59] is given below a. Initial guesses: ε=0.5, ϕ=0.9 radians. b. Initial results from solving Reynolds: horizontal component of load fx:  2.81838E+03   vertical component of load  fy:   4.94780E+03   modulus of load    fR:                 5.69421E+03   attitude angle:                           1.05301E+00  c. W=11 000N, therefore W>fR increase ε to 0.64. Second iteration gave the results   horizontal component of load   7.05888E+03   vertical component of load       8.51415E+03   modulus of load             1.10598E+04   attitude angle             8.78576E-01  Third iteration with ε=0.63:  horizontal component of load   7.24169E+03    vertical component of load       8.28417E+03    modulus of load           1.10032E+04    attitude angle           8.52443E-01  The modulus of load or fR=11 003 ≈ W; the iterations can be stopped. The flowchart continues at stage 4 in figure 2.5.   2.5 Chapter summary The streamline upwind Petrov-Galerkin finite element method is formulated for three- dimensional thermo-hydrodynamic lubrication problems. Governing equations and boundary conditions in the critical areas of the groove and rupture zone are presented. The model is used to simulate slider bearings thermal behavior in chapter 3 and journal bearings in chapter 4.   34                                          Figure 2.5 Flowchart for thermo-hydrodynamic lubrication model.   Start. Input parameters Ti, μa,, n, k,ρ, D, C Find gradients ∂p/∂x, ∂p/∂y Compute the velocity components u, v, w using Navier-Stokes and continuity equations Use current value of μ  to compute p using constrained minimization and over-relaxation. Determine ε, h, ϕ (eqns. 2.39-2.40) at force balance equilibrium Compute new pressure field pnew with updated viscosity , frictional torque Tr, power loss PL,, QL for current values of viscosity Is |µnew-µold|≤εr Or is Iter>Lm Yes No Compute output parameters rT , LQ and solve for conduction in pad/bush  End Compute new viscosity and density fields Change value of θ? No Yes Solve energy equation iteratively for temperature field T in film. Use conduction eqn to determine temperature in pad/bush  35 Chapter 3 Application of SUPG FE to Slider Bearings  3.1 Introduction The streamline upwind Petrov-Galerkin (SUPG) finite element method formulated in chapter 2 is chosen for its robust nature and rapid convergence in convection dominated flows. The method is applied to slider bearings analysis and design in this chapter. Slider bearings are widely used in applications such as mechanical seals, collar thrust bearings, machine tool guides, and piston rings. They have good load-carrying capacity, excellent stability and durability. A tilted pad slider bearing consists of two surfaces as shown in figure 2.1a separated by a lubricant film. The direction of the runner motion and inclination of the planes are such that a converging oil film is formed between the two surfaces and the physical wedge pressure-generating mechanism is developed in the lubricant film. This enables the bearing to support a load. The flow is viscous, leading to heat generation, dissipation and increase in film temperature.  3.2 Lebeck Slider Bearing Data  The test case presented by Lebeck [65] and used by Kumar [9] in a two-dimensional model with input values in Table 3.1 is used. Mesh density sensitivity analysis was performed with the thermo-hydrodynamic SUPG model; results appear in Table 3.2. The maximum number of pressure and temperature iterations for convergence is also indicated. An analytical load capacity is included and used as a benchmark. A 30x30x10 (x-y-z) mesh is finally chosen. The length (x) and height (y) values are similar to those used by Kumar [9] while an axial value of ten elements is a good compromise in  36 computation time and accuracy. The mesh size was chosen after trials with other densities – a compromise had to be made on mesh density versus computing time.  Table 3.1 Input values for model [65]. Input Parameter Value Sliding Speed, U 20 m/s Leading edge height, hi 5x10-5 m Bearing Length L 0.1 m Pad thickness hp 0.1m Lubricant heat capacity cf 1926 J/kg K Lubricant conductivity kf 0.132 W/m K Pad conductivity kb 60 W/m K Runner conductivity ks 50 W/mK Inlet temperature Ti 310 K Inlet lubricant density ρi 897.1 kg/m3 Inlet viscosity μi 0.0174 Pa.s Viscosity-temp coeff. β 0.035 /K Density-temperature coeff. δ 0.0012 /K Ratio k (ho/hi) 0.4 Inlet pressure pi (gauge) 0.0   Table 3.2 Mesh sensitivity analysis k Mesh size (x,y,z) Load Capacity Diff. (from analytical model) % 0.4 5x5x5 0.095 12.74 10x10x10 0.106 2.94 20x20x10 0.108 0.66 30x30x10 0.109 0.22 40x40x10 0.109 0.07 50x50x10 0.109 0 Analytical: 0.109 Max. no. of pressure iterations 6 Temperature iterations 10   3.2.1 Model Convergence Performance The iterative procedure in figure 2.5 showed convergence for load capacity computations of less than ten iterations for L/B=10 and less than five for L/B=1 as shown in figure 3.1.  37 Table 3.3 has the limiting values for the outer loop in figure 2.5 and an indication of the convergence values. The power and advantage of SUPG is evident in the finite bearing case where, at L/B=1 convergence takes place after only three iterations. A similar pattern is evident in journal bearing simulations.   Figure 3.1 Convergence of load capacity solution.    Table 3.3 Convergence criteria Maximum number of iterations – outer loop 15 Percentage variation – 1st to 2nd iteration 57.9 Percentage variation – 5th to 6th iteration 4.9 Pressure iteration: max no. 6 Temperature iterations: 10  Load Capacity as Function of Number of Iterations 0.00E+00 2.00E+05 4.00E+05 6.00E+05 8.00E+05 1.00E+06 1.20E+06 1.40E+06 1 2 3 4 5 6 7 8 9 10 Number of iterations Lo ad  C ap ac ity , N /m 2 L/B=10 L/B=5 L/B=3 L/B=2 L/B=1  38  3.2.2 Predicted Pressure Values The effect of the width to length ratio L/B on predicted pressures generally indicates whether a bearing is infinite or finite. Initially adiabatic conditions on upper and lower surfaces and an inlet temperature Ti=To are assumed. The 3-D SUPG model formulated in section 2.5 is used to predict peak pressure and load capacity. Results appear in table 3.4 and figure 3.2. Values of peak pressure for L/B ratios of ∞ (2-D model), 20, 10, and 5 are coincident as expected [66]. Percent difference in table 3.4 indicates model predictions for the infinitely wide (2-D) and 3-D sliders as a function of L/B ratio for both peak pressure and load capacity. Both fall dramatically at L/B values less than 2. What is significant here is the sudden drop in load carrying capacity from L/B of 2 to 1 (square slider). A 3-D model should be used for L/B less than 3.  Table 3.4 Isothermal peak pressure and load capacity, k=0.4  L/B ratio  Peak Pressure ( p )  Difference (%) Load capacity W Difference to 2D model (%) 2D Model 0.2564 - 1.572 - 20 0.2564 0.00 1.572 0 10 0.2564 0.00 1.572 0 5 0.2562 0.09 1.571 0.06 2 0.2378 7.3 1.453 7.5 1 0.1669 34.9 0.9996 36.4 0.5 0.0806 68.6 0.4437 71.8   39 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 x/L SUPG Predictions as function of L/B ratio p_2D p_20:1 p_10:1 p_5:1 p_2:1 p_1:1 p_0.5:1 Figure 3.2 Non-dimensional isothermal pressure predictions as a function of L/B. The first four lines (L/B=∞, 20, 10, 5) coincide.   Arnell [66] suggests sliders are considered finite when L/B<2 which corresponds to a peak pressure difference of 7%. Of more significance is the thermo-hydrodynamic effect on peak pressure predictions as shown in Table 3.5 which indicates the difference between isothermal and THD values for several L/B ratios. Consideration of the thermo- hydrodynamic effect must be made at the bearing speed tested (20 m/s).           40 Table 3.5 THD peak pressure and load capacity comparisons L/B ratio Peak Pressure Isothermal ( p ) Peak Pressure THD ( p ) Difference Load Capacity, Isothermal W Load Capacity, THD W Difference 2D Model 0.2564 0.144 0.112 1.572 0.950 0.622 20 0.2564 0.144 0.112 1.572 0.895 0.677 10 0.2564 0.144 0.112 1.572 0.439 1.13 5 0.2562 0.144 0.112 1.571 0.208 1.36 2 0.2378 0.135 0.103 1.453 0.068 1.39 1 0.1669 0.096 0.071 0.9996 0.022 0.978 0.5 0.0806 0.045 0.036 0.4437 0.005 0.439   3.2.3 Thermal Profile Thermal fields in the film from the two-dimensional [9] and SUPG models for L/B=10 are compared in figure 3.3. The isotherms follow a similar pattern in both for this aspect ratio. The scale of the y axis representing the film thickness is magnified 1000 times since it is much smaller than the other two dimensions (length L and width B). Adiabatic conditions and an inlet temperature Ti=310°K and L/B=10 are used similar to Kumar’s model for comparison. Peak temperature prediction for the two dimensional model is 75.4°C (normalized temperature T+=1.124) at the upper corner of the trailing edge while that for the three dimensional model is 74.7°C (T+=1.121) at the same location. These results assume adiabatic conditions. Figure 3.4 is a 3D model of the film showing the predicted thermal field. Maximum temperatures occur at the trailing edge on the mid-plane surface (S1 in figure 2.3, 0,1,1 === ZYX ) at the film/pad interface. Peak temperature reduces to 66.3°C for L/B=0.5 indicating the increased effect of heat carried out through leakage flow (figure 3.5). Peak temperature is at the mid-plane surface. Table 3.6 shows that the fraction of leakage flow increases from 7 to 50% as L/B reduces from 10 to 0.5.  41 More industrially applicable conditions, such as those used by Kumar [9]: 0.1=ip , )372(2.1 KTi =+ , )434(4.1 KTp =+ and )310(0.1 KTs =+  are used to run the model. These normalized temperatures are relative to the inlet temperature in °K, all other variables are in Table 3.1). Inflow boundary conditions were checked and updated as well as using a finer grid mesh N=30x30x30 with improved results as shown in figure 3.6a. A clearer picture emerges when predictions are compared to Kumar’s results by digitizing and superimposing these onto figure 3.6a. Figure 3.6b shows a couple of isotherms. What is evident from this is that although relatively close, it is difficult to reproduce Kumar’s results exactly. Isotherms generally follow similar patterns in both cases although model predictions appear to be shifted slightly upwards especially at the lower regions.  42  Figure 3.3(a) Normalized temperatures for 2-D and (b) 3-D SUPG models showing general patterns of the isotherms.     (a) (b)  43  Figure 3.4 Contour map of the thermal field, L/B=10 (hi=50μm). The arrow shows flow direction.          Y + 0     0 .2    0 .4    0 .6    0 .8     1   44   Figure 3.5 Thermal field with L/B=0.5 (hi=50μm).       0     0 .2    0 .4    0 .6    0 .8     1           Y +        45  (a)  (b) Figure 3.6(a) Model isotherms for Lebeck’s data [65] (b) Comparison of temperature profiles from SUPG model (solid lines) and Kumar [9] (dashed lines).   46 3.2.4 Leakage Flow Pattern – Flow Gradient Method The SUPG model is used to predict flow patterns as a function of L/B in more detail. Table 3.6 shows the average side flow as a function of L/B, also shown in figure 3.7 for 0.1<k<0.9 and different L/B ratios. Flow is determined at k intervals of 0.1 and averaged out. Boxed numbers represent the ratio of side leakage to inflow as a percentage. It increases significantly for L/B ratios less than two. The values presented in table 3.6 and figure 3.7 are for average dimensionless flow rates for the range 0.1<k<0.9 and represents the general behavior of slider bearings as a function of aspect ratio L/B. Figure 3.8 follows from the general observations evident from figure 3.7 and is applicable to slider bearings in general. The gradient in figure 3.8 is determined manually at each point on figure 3.7. A differencing technique is used to compute the change in tilt ratio d(L/B) and flow ݀ሺܳሻതതതത. The gradient gives an indication of how the flow is changing as a function of length to width ratio L/B. The minimum point IP represents, for a given tilt angle k, rate of change of leakage flow as a function of L/B ratio is zero. This point is at L/B=2 in figure 3.8. Lower values of the L/B give more leakage flow but far less load-carrying capacity whereas higher L/B values give more load capacity but less leakage flow and consequently higher operating temperatures. Thus Ip gives a good compromise between load capacity and leakage flow for optimum operating temperature. As pointed out earlier higher leakage flows are beneficial in lowering peak operating temperatures. The gradient of side flow ∂QL/∂(L/B) is high from zero to L/B=1, decreases rapidly between L/B=1 and L/B=2 to almost zero, gradually increases from two to ten, then becomes nearly constant above ten. It decreases rapidly from zero to one. It can be argued that the influence of side flow is most significant between zero and two. Even though load capacities are relatively low in this region, the beneficial effects of lower  47 operating temperatures are realized. Another interesting observation is that even at L/B=60 the percentage of side flow to inlet flow is 4.9 although the slider is in the infinite range. Figure 3.8 illustrates the Flow Gradient Method of determining optimum leakage flow by computing the gradient Qd / ( )BLd /  for 0.1<k<0.9. The graph bottoms out at a ratio of two, increases up to a ratio of ten and remains nearly constant thereafter. To the best of our knowledge this work represents the first definition of the Flow Gradient Method.  Table 3.6 Dimensionless flow rate as a function of L/B1  L/B  vQ   oQ   LQ  Ratio of side leakage to inflow, % 60 46.4 44.1 2.3 4.9 40 30.9 29.4 1.6 5.1 20 15.5 14.7 0.87 5.6 10 7.86 7.29 0.57 7.2 5 4.05 3.60 0.46 11.2 2 1.79 1.37 0.42 23.3 1 1.02 0.63 0.39 38.0 0.5 0.57 0.29 0.28 49.7 0.25 0.30 0.13 0.17 55.4 ܳ௩, ܳ௢, and ܳ௅ are the inflow, outflow and side leakage flow respectively   F 0. 0. 1. 1. 2. 2. 3 igure 3.7 Di numbe 0 5 0 5 0 5 0 55.4 49.7 8.0 23.3 11.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 mensionless rs represent Fig 10 Dimens 7.2 2 4 6 Flow G  side leakag the ratio of ure 3.8 Flow 20 Leng ionless leak 5.6 8 10 12 1 radient a 48 e LQ as a fu side leakage    gradient,  30 th to width rati age flow as 4 16 18 20 L/B s Functio nction of L/  to inflow a 0.1<k<0.9. 40 o L/B function of L 5.1 22 24 26 2 n of L/B, B, 0.1<k<0 s a percenta 50 /B, 01<k<0. 8 30 32 34  0.1<k<0 dQ/d(L/B .9. Boxed ge. 60 9 - 4.9 36 38 40 .9 )    49 In Figure 3.9 the non-dimensional side leakage pattern from point A to B in figure 2.3a is shown as a function of L/B for k=0.4. Numbers in boxes indicate the width to length ratio L/B. Side leakage increases from L/B=20 to a peak at L/B=1 and L/B=2 then declines as L/B decreases. This is an important observation, as it implies the bearing is optimized in terms of side leakage in the range L/B=1-2.   Figure 3.9 Side leakage as a function of L/B, k=0.4.    Although the fact that finite sliders run at significantly lower temperatures than higher width ones has been long established [67] the effect on the point of the most intense viscous dissipation has not been widely reported. Peak temperature increase lowers lubricant viscosity which leads to lower load carrying capacity. Slider geometry can be Side leakage profiles 0.00 0.05 0.10 0.15 0.20 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N on -d im en si on al  le ak ag e flo w  L/B=20 L/B=10 L/B=1 L/B=0.5 L/B=2 L/B=3 L/B=20 10 0.5 1  2 3  50 adjusted or new geometries can be tried in order to reduce the effect of higher temperatures on load carrying capacity.   The effect of tilt ratio k on peak temperatures and load carrying capacity are shown in figure 3.10 for L/B of 20, 10, 5, and 1. All variables not specified are in Table 3.1. Boxed numbers show peak temperature predictions at each k value. A high tilt angle (low k value) results in high peak temperatures. The highest load capacity occurs at k=0.5 in both cases. Even though high values of k have a lower peak temperature, load capacity is less. A tilt ratio of 0.5 seems ideal for slider design, although slightly higher values (k=0.6-0.7) are preferred for their lower peak temperatures. The high peak temperatures at L/B=10 and 20 and tilt ratio k=0.1 suggest high risk of metal to metal contact for these values.  Figure 3.10 Load capacity as a function of k. Non-dimensional load capacity is as defined in nomenclature.   0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N on D im  L oa d C ap ac ity  Parameter k Load Capacity as a Function of k L/B=10 L/B=1 L/B=5 L/B=20 64 C69 C 49 C 82 C 75 C 69 C 64 C 61 C 58 C 56 C 58 C 87 C 61 C 77 C 71 C 66 C 62 C 59 C 57C 56 C 75 C 82 C 75 C 69 C 64 C 61 C 58 C 56 C 55 C 56 C 83 C  51 3.3 Chapter Summary The three dimensional SUPG thermo-hydrodynamic model developed is used to analyze slider bearing characteristics at different width to length and tilt ratios. Pressure predictions for the non-isothermal case are significantly less than the isothermal case as expected.  Model predictions show square sliders running at lower temperatures than wider ones. This is due to the higher side leakage; however their load carrying capacity is significantly less. The significance of side leakage on peak temperature is evident. The model also shows that side leakage peaks in the region L/B=1-2. The Flow Gradient Method to predict optimum flow conditions is proposed. Even though high values of the tilt ratio have a lower peak temperature, the load capacity is less and an tilt ratio of 0.5- 0.7 seems ideal for slider design. Lower peak temperatures are predicted if inlet pressure is raised from ambient to 0.1=ip . Ways to do this should be implemented on slider bearing if higher load capacities are required.            52 Chapter 4 Streamline Upwind Petrov-Galerkin FE and Journal Bearings  4.1 Introduction The streamline upwind Petrov-Galerkin finite element for thermo-hydrodynamic modeling is used for the analysis and design of journal bearings. Theoretical basis of the model applicable to plain journal bearings with requirements to account for cavitation effects was presented in chapter 2. A novel THD template for journal bearings in Cartesian coordinates that shows good convergence is developed.  4.2 Model Formulation Figure 4.1(a) shows the classical two-dimensional journal geometry while figure 4.1(b) illustrates the model development whereby the bearing is unwrapped into regions that correspond to the bush, fluid and shaft. The oil groove presents a good reference point whereas attitude angle φ (fig 4.1(a)) depends on bearing load and speed among other factors. The computational region for pressure is mapped onto a rectangle, the x-z plane while the y axis represents the film thickness h as a function of angle θ from groove position. The minimum film thickness hmin is at an angle ϕ  from the groove position as illustrated in figure 4.1a. Figure 4.2 shows the fluid region where film thickness h is a function of the x-coordinate. The cutting plane splits the oil groove in half. Periodic boundary conditions identify surfaces S3 and S5 as ‘mating’, representing the circular nature of the template.  53 The film thickness h is determined from equation 2.39. The film thickness h(θ) from equation 2.39 can be transformed to h(x) from the relation h(x)= h(θ)/(πD).            Figure 4.1(a) Schematic representation of journal bearing.   4.1(b) Journal bearing unwrapped. The cross-sectional areas of the two regions are equal  Bush Fluid region (exaggerated) Shaft ϕ e R1 R2 Oil supply hole h W yb xb θ y x z ω R3 Oil groove fys fxs xs ys  54  4.1(c) Isometric view of finite element mesh of the model                Figure 4.2 Cross section of the film region. Note that the origin is on line C3 and surfaces S3 and S5 are defined on C4 and C3 respectively.    Z Y X Journal (lower surface) bush Oil grooves Mid-plane surface S1 Side surface S3 Side surface S5 C1 End surface S4 C2 ps bc’s C3 C4 x  55 Film rupture and reformation effects have to be considered in journal bearing computations. Cavitation algorithms fall into two main categories, the Swift-Stieber (Reynolds) boundary condition which states that ∂p/∂ϕ=0 at ϕ=ϕcav, where ϕcav is the rupture interface, and the mass conserving algorithms by Elrod [28] and Kumar and Booker [29,30] among others. The Swift-Stieber condition has gained wide acceptance for steady state stationary loaded journal bearings [55] even though it does not work as well at the reformation boundary, will be used.  4.3 Simulation Results 4.3.1 Mesh Density, Convergence and Performance The experimental data from Dowson [59] represents one of the landmark work in plain journal bearings and is often used to validate many journal bearing models. Initial model validation was performed on this data.  Input parameters and thermocouple setup information is given in table D.1 and section D1.1. The same problem is solved analytically by Khonsari [68] using the Boyd and Raimondi tables and Yang and Khonsari’s [55] proposed empirical equations. No data on the bush dimensions and geometry of the oil supply groove thus a convective heat transfer coefficient for the bush/ambient interface of 100W/m2.K is used and found to give results close to experimental. The published peak pressure of 3.5 MPa obtained by Dowson [59] is used in convergence analysis. Table 4.1 shows the effect of varying grid size from N=10x10 to N=80x80 for both linear and quadratic elements. The sequence N=10x10, 20x20 …is used since it is recommended in numerical analysis to halve the step size in a convergence study. As expected quadratic elements converge faster although they are  56 computationally more expensive in terms of time. Figure 4.4 shows the convergence performance of the model while figure 4.5 presents the predictions for each mesh compared to the experimental value of 3.5 MPA.  Table 4.1 Convergence of Peak Pressure Predictions from Experimental Value N Pressure Prediction Exp. Value Linear Elements: Devation = (Pexp-Ppred)/Ppred Quadratic Elements: Devation = (Pexp-Ppred)/Ppred   Linear Elements Quadrilateral Elements 10x10 2.405E+06 2.747E+06 3.50E+06 0.3128 0.2152 20x20 3.074E+06 3.194E+06   0.1217 0.0876 40x40 3.418E+06 3.453E+06   0.0235 0.0134 80x80 3.468E+06 3.476E+06   0.0093 0.0067     Figure 4.3 Convergence analysis on Dowson’s data  0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 10 20 30 40 50 60 70 80 90 Pe rc en t D ev ia tio n No. of Elements N Percentage Divation vs No. of Elements N Linear Quad  57   Figure 4.4 Peak pressure prediction for Dowson’s experimental value of 3.5 MPa    Quadratic element mesh achieves a deviation of only 1.3% with a 40x40 mesh size while an 80x80 mesh is required for linear ones. Computation times as a function of mesh density for linear elements are shown in table 4.2 (using a Pentium IV 3.25 GHz 1 GB RAM). These are for a single iteration in figure 2.5. Considering that each cycle required at least three iterations for convergence and that the process is repeated at Ns=2 rev/s intervals in the optimization model, computation time can be considerable. A target deviation of 1% was used which coincides with a mesh of N=80x80 elements (0.9%).  Table 4.2 Computation Times for One Iteration   Computation Times (Sec) N Mesh Computing Total 10x10 0.0313 0.0156 0.0469 20x20 0.0781 0.0469 0.125 40x40 0.2812 0.4219 0.7031 80x80 1.0156 1.9219 2.9375  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Pr es su re , M pa Pressure Predictions vs No. of Elements N Pressure@N=80x80 Pressure@N=10x10 Pressure@N=20x20 Pressure@N=40x40  58 Model performance in predicting peak temperatures for Dowson’s data is shown figure 4.5 (experimental value is 47°C). In order to compare predicted temperatures to experimental outlet drain temperatures and Khonsari’s [68] analytical computations, nodal temperatures are summed and divided by the number of nodes. Khonsari performs six iterations in his computations as shown in figure 4.5 which also indicates Dowson’s experimental value and the proposed SUPG model. The rapid convergence of the model is evident; only three iterations produce stability in predictions. The average temperature prediction of 48.4˚C from the model is 1.4˚C higher than experimental values; the peak temperature prediction of 51.2˚C compares favorably with 51.6˚C from Dowson. The peak temperature is located at the mid-plane of the journal. Table D.1 (Appendix) has input parameters for the bearing while table 4.3 shows some of the performance parameter predictions (input parameters not available from Dowson’s data were estimated). These results are preliminary; more detailed analysis follow.  Figure 4.5 Model, experimental, and analytical predictions, Dowson’s data. 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 1 2 3 4 5 6 7 Te m pe ra tu re  (C ) No. of Iterations SUPGThermal Predictions, Analytical Method, and Experimental Temperature T_measured T_model T_analyt Exp. temperature Oil inlet  59 Table 4.3 Model predictions, analytical [68] and Dowson’s [59] experimental data.   Analytical SUPG Exp. Attitude angle ϕ° 51.0 51.0 57.0 Peak Pressure (Mpa) 3.60 3.46 3.50 Side Leakage cm3/s 24.2 26.1 26.2 Friction Force, N 67.4 65.7 64.4 Shaft Torque, N.m 3.42 3.41 3.39    4.3.2 Pressure and Pressure Gradient Predictions The experimental data from Boncompain et al. [15] and Costa et al. [69] are used to perform further validation of the model. The top surface is stationary (bush) and the bottom is the shaft in figure 4.1(b); figure 4.1(c) shows the mesh model. Input parameters for the bearings appear in Table D.2 and D.3 respectively. The effect of pressure gradients ∂p/∂x and ∂p/∂z on the energy equation 2.12 is through the compressibility term (second term on right hand side). This term is retained here even though most researchers ignore it due to its relatively small contribution [1,2,55,74]. The behavior of the two gradients near and in the rupture zone form a critical part of the discussion to follow, hence the need to retain them. Table 4.4 shows the effect of including the compressibility terms on temperature predictions.  Table 4.4 Effect of pressure gradients ∂p/∂x and ∂p/∂y on temperature rise ΔT Data Ti, °C ΔT, With °C ΔT Without, °C Difference, °C Dowson 36.8 11.39 11.48 0.09 Boncompain 40 11.77 12.13 0.36 Costa 35 19.07 19.19 0.13   60 Model convergence performed on pressure gradient dp/dz for the experimental data from Boncompain et al. [15] is presented in figure 4.6. The bearing analyzed has aspect ratio L/B=0.8 which is short and pressure gradient dp/dz falls rapidly from the bearing mid- plane. Lower values of the gradients at തܺ ൎ 0.5 are due to the increased effect of film pressure in the narrowing (converging) part of the bearing. Pressure predictions from the proposed model and the rupture/reformation interfaces are shown in figure 4.7 for Boncompain’s [15] bearing. The rupture zone is in the range 00 332227 << θ  which is in good agreement with Boncompain’s results (figure 4.7b). Figure 4.8a presents pressure gradient ∂p/∂x as a function of Z at several locations around the bearing. Boxed numbers indicate position along the X coordinate. Figure 4.8b compares ∂p/∂x and ∂p/∂z along the x axis and reveals that the magnitude of ∂p/∂x is much greater than ∂p/∂z, which is zero except for the rupture zone which consists of finger-like fluid/gas regions. It appears ∂p/∂x has a contribution on the compressibility term of the energy equation since it is significantly larger than ∂p/∂z although this contribution is negligible as indicated in Table 4.3.  4.3.3   Pressure Gradients and Effect on Dissipation Of more significance is the position of the rupture interface and how it affects the gradients. The boxed numbers on the ordinate of figures 4.8a and 4.8b show the position along the x axis for Boncompain’s [15] input data (Table D2 in appendix). The magnitude of ∂p/∂x increases along the x direction, peaks at 5.0=X , changes sign with a minimum at 592.0=X which is located just before the rupture interface and becomes zero at the reformation interface. Noting that the gradients ∂p/∂x and ∂p/∂y are multiplied  61 by the density-temperature coefficient in equation 2.12 and w is several orders of magnitude less than u [2], the contribution of these gradients to dissipation is minimal (positive values increase dissipation). Thus investigation of the dissipation phenomenon from a mathematical point of view gives an insight as to why experimental temperatures peak just after the rupture interface and decrease thereafter.    Figure 4.6. Convergence analysis on pressure gradient dp/dz.     ‐7 ‐6 ‐5 ‐4 ‐3 ‐2 ‐1 0 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 dp /d z,  M N /m 3 Model Convergence for gradient dp/dz vs No of Elements N N=10 N=20 N=30 N=50  62 (a)  (b)   (c) Figure 4.7(a) Model pressure predictions at bearing mid-plane, model vs (b) Boncompain’s result [15]. (c) Pressure map showing the rupture and reformation interfaces. Model Pressure Prediction, MPa 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 40 80 120 160 200 240 280 320 360 Angle from oil hole, degrees Pr es su re , M Pa THD pressure Cavitation zone Rupture interface Reformation interface  63  (a)   (b) Figure 4.8(a) Pressure gradient ∂p/∂x variation along the bearing x coordinate, Costa et al. [69]. (b) Pressure gradients ∂p/∂x, ∂p/∂z variation along the x axis.    4.3.4    Dissipation Source Terms The dissipation source terms (last two terms in equation 2.12) in the energy equation are evaluated for their contribution to the thermal profile. Figure 4.9 shows the values of the -250 -200 -150 -100 -50 0 50 100 150 0.00 0.10 0.20 0.30 0.40 0.50 dp /d x,  M N/ m 3 Pressure gradient dp/dx 1. dpdx@X=0.11 2. dpdx@X=0.21 3. dpdx@X=0.46 4. dpdx@X=0.5 5. dpdx@X=0.53 6. dpdx@X=0.54 7. dpdx@X=0.55 8. dpdx@X=0.61 9. dpdx@X=0.65 10. dpdx@X=0.92 11. dpdx@X=0.99 12. dpdx@X=0.63 1, 2,10,11 8 12 7 9 6 5 3 4 ‐250 ‐200 ‐150 ‐100 ‐50 0 50 100 150 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 G ra di en ts  d p/ dx  a nd  d p/ dz , M N /m 3 Pressure Gradient Variations dp/dx dp/dz  64 gradient ∂u/∂y (normalized by dividing by the peak value) as a function of x . Peak values occur at the rupture interface 1 which is just before peak temperatures. This is an important observation as it correctly indicates the region of maximum viscous dissipation without specifying thermal boundary conditions at this interface as done by Boncompain [15] and other researchers [6]. Velocity gradient ∂w/∂y is shown in figure 4.10 (normalized by dividing by peak ∂u/∂y and multiplying by 100). An order of magnitude analysis reveals this gradient varies between -0.38 percent to +0.18 percent of ∂u/∂y making its contribution small and hence justifies ignoring the term [2]. The peak value occurs at 98.0=X  which is close to the oil groove. Discontinuities (figure 4.10) appear to be caused by the groove/cavity interface and mixing zone phenomenon.     65  (a)  (b) Figure 4.9(a) Velocity gradient ∂u/∂y as a function of X . (b) Magnified view at the peak gradients point.  66    Figure 4.10 Velocity gradient ∂w/∂y as a function of X .   4.3.5   Thermal Profiles One benefit of the model is the detailed thermal profile of the film. Figure 4.11(a) shows the lower surface is in contact with the shaft and the top with the bush. The bottom surface is assumed to be at a constant temperature whereas the bush contact surface varies significantly. As mentioned above temperature measurements are normally taken on the bush or shaft; knowledge of the fluid profile is sketchy at best. The isotherms in  67 the axial (z) direction are mainly horizontal indicating no thermal gradients confirming what other researchers noted [2,67]. Their density depends on the number of transverse divisions considered (ten in the model). The variation in temperature across the film thickness is apparent due to the mixing of the fresh oil with circulating oil. Comparison of proposed model predictions (figure 4.11(a)) and figure 4.11(b) (figure 7, adopted from Boncompain et al. [15]) at the minimum film thickness position (indicated by the centerline) and position )36(1.0 0≈x  in figure 4.11(a) shows similarities. The circular peak just after hmin in the figure is also very similar to the circular peak just after hmin in figure 4.11(b). A peak temperature of 53.5°C occurs at an angle of 232° from the oil supply groove (position (0.65,0,0.12)). The geometry of the oil supply hole and cavity are not given in [15] which makes setting boundary conditions difficult. Conservation of mass flow at the inlet was used to determine oil temperature after mixing. Experimental data from Costa et al. [69] and Kucinschi et al. [27] are analyzed next. Input parameters for the bearings are shown in Table D.3. Temperature profiles from the proposed model are presented in Figure 4.12(a) and (b), including the inlet hole and cavity as a function of inlet boundary conditions (note that the x-axis compressed due to large aspect ratio between this axis and the z-axis). The proposed model predicts Costa’s data well. Note that Costa et al. [69] present data and no model in their paper. In figure 4.12(a) the supply hole and cavity are assumed to be at Ti whereas in 4.12(b) it is assumed to be at Ti. The latter assumption results in higher cavity temperatures. A peak temperature of 62.7°C at position 70.0≈x  (252°) is in agreement with 62°C (figure 5 in Costa [69]) although the position is slightly lagging. The cavity thermal profile depends on boundary conditions imposed on the supply hole and cavity. In figure 4.12(a) both the  68 supply hole and cavity are assumed to be at temperature Ti whereas in figure 4.12(b) the entrance to the supply hole is at Ti. Clearly a mixing phenomenon is evident in the latter case which appears closer to reality. The type of boundary condition used plays an important role in model predictions. Kosasih and Tieu [5] assumed that most parts of supply region are filled with re-circulating oil and most of the injected cold oil does not immediately enter the oil film, which tends to support the case in figure 4.10(b). The influence of the inlet oil at a lower temperature Ti on the cavity oil dynamics is also clearly evident. Figure 4.13(a) shows the bush upper thermal profile while in figure 4.13(b) the bush lower surface is exposed to show how the temperature varies along the axial direction (note that both coordinates y, z are reported in real figures and not in non-dimensional form). Coordinate z is the actual bearing half width dimension and y is magnified 4000 times since it is very small compared to the other two). Model results of the bush temperatures are in agreement with Costa’s et al. [69] experimental results. The peak temperature of 62.7°C occurs at position 70.0≈x . As expected the outer edge ( 04.0=z ) has lower surface temperatures than the mid-plane due to convection to the ambient and leakage flow (the top isotherm of 51.3°C does not get to the outer surface).  4.5.3.1 Standard Error and Goodness of Fit To validate the model further, Costa’s experimental temperature data at the mid-plane of the bush are compared to model predictions in figure 4.14 and Table 4.3. Standard error σe is used to estimate the error  ߪ௘ ൌ ට ∑ ∑ ሺ௬೔ೞሻమ ೙ ೔సబ ೘ ೞసభ ൫௡೤ିଵ൯൫௡೤൯                   (4.3)  69 where n is the number of points in each series, i the point number in series s, yis the data value of series s and the ith point, and ny the total number of values in all series. The standard error of a method of measurement or estimation is the estimated standard deviation of the error in that method. Specifically, it estimates the standard deviation of the difference between the measured or estimated values and the true values. Notice that the true value of the standard deviation is usually unknown and the use of the term standard error carries with it the idea that an estimate of this unknown quantity is being used. Looking at figure 4.14 model predictions at θ=-30° has the most points outside the standard error estimate of both the data and predictions (9 points) whereas all points at θ=0° are within and 2 points are outside at θ=+30°. Table 4.5 summarizes predictions from the three positions (θ=-30°,0,+30) including the square of the Pearson correlation coefficient r2. A possible explanation for the high error for θ=-30° is that this groove location might be in the cavitation zone and so applied boundary conditions are not applicable.  Table 4.5 Standard Error and r2 correlation for Temperature Predictions θ=-30 θ=0 θ=+30 Standard Error 3.18 2.12 2.66 r2 0.796 0.939 0.921 Note: r2 represents the square of the Pearson product moment correlation coefficient through data points.   As explained in section 2.4.2 the inlet film temperature Ti can be determined using conservation of energy in the groove  ௜ܶ ൌ ொೝ ொ೔೙ ௥ܶ ൅ ቀ1 െ ொೝ ொ೔೙ ቁ ௦ܶ  where Ts, Tr are the supply and re-circulating oil temperatures respectively and Qr, Qin the re-circulating and oil inflow. Maximum model prediction deviation from experimental values is 0.09 which  70 occurs in the region after oil mixing prior to the minimum film thickness at groove location θ=-30°. A probable cause of the higher error here is that the groove is probably in the rupture zone. This indicates a need for a more accurate modeling of the mixing phenomenon for this case. Peak temperature predictions are within 0.03 in each case. It has to be emphasized here that unlike other models in the literature [20,27], the model as presented is dynamic in that groove location can be changed and results may be easily obtained. Appendix E explores predictions from other models and shows the superior performance of SUPG in steady state THD lubrication. Peak experimental temperatures as a function of supply pressure for a 2 kN load and 50 rev/s speed are shown in figure 4.15. Boxed values indicate experimental temperatures while standard error bars are also included. Deviation of model predictions is small as shown in Table 4.7. Experimental values differ from model prediction by a maximum of 1.6 percent.    71  (a)  (b) Figure 4.11 (a) Thermal profile from Boncompain et al. [15] data. (b) Figure 7 from Boncompain. The geometry of the oil supply hole and cavity are not given which makes setting boundary conditions difficult.  72  (a)  (b) Figure 4.12(a and b) Thermal profile in the inlet, cavity and bearing, Costa.  0   0. 2   0. 4    0 .6    0 .8    1          Y +               Y +    0 0 2 0 4 0 6 0 8 1  73  (a)    (b)  Figure 4.13 Bush thermal profile, (a) top and (b) bottom view   74  (a)  (b)  (c) Figure 4.14 Thermal predictions at the bush/oil interface – Costa’s data. Input parameters given in Table D3. 30 35 40 45 50 55 60 65 0 30 60 90 120 150 180 210 240 270 300 330 360 Te m pe ra tu re , C Angle, degrees Model Prediction vs Exp. temperatures, θ=-30 SUPG Exp. data 30 35 40 45 50 55 60 65 70 0 30 60 90 120 150 180 210 240 270 300 330 360 Te m pe ra tu re , C Angle, degrees Model Prediction vs Exp. temperatures, θ=0 SUPG Exp. data 30 35 40 45 50 55 60 65 0 30 60 90 120 150 180 210 240 270 300 330 360 Te m pe ra tu re , C Angle, degrees Model Prediction vs Exp. temperatures, θ=+30 Exp. data SUPG  75  Table 4.6 Predicted versus experimental temperatures, Costa’s data.     Exp. Values [69] Model Predictions Difference %    Angle (Degrees) T(-30) Fillon T(0) Fillon T(+30) Fillon θ =-30 θ =0 θ =+30 Diff. % (-30) Diff. % (0) Diff. % (+30) 0.0000 0 35.0 35.0 35.0 35.0 35.0 35.0 0.0 0.0 0.0 0.0080 9 35.0 35.0 35.0 35.0 35.0 35.0 0.0 0.0 0.0 0.0109 13 35.5 35.5 35.5 36.4 36.3 36.3 2.5 2.2 2.2 0.0393 45 39.0 39.0 37.0 42.6 42.9 41.1 8.5 9.1 10.0 0.0654 75 41.8 41.0 38.0 46.0 45.5 42.2 9.1 9.9 10.0 0.0916 105 42.0 42.0 39.0 46.7 46.7 43.3 10.0 10.0 9.9 0.1178 135 43.0 44.0 44.0 47.8 48.9 48.8 10.0 10.0 9.8 0.1309 150 44.0 46.0 49.0 48.9 51.1 53.9 10.0 10.0 9.1 0.1440 165 45.0 47.5 50.0 50.0 52.8 54.6 10.0 10.0 8.4 0.1571 180 46.5 49.0 52.0 51.7 54.4 55.4 10.0 9.9 6.1 0.1702 195 47.0 51.0 54.0 52.3 55.3 56.2 10.0 7.8 3.9 0.1833 210 49.0 54.0 58.0 54.5 56.2 57.1 10.0 3.9 1.6 0.1963 225 53.0 58.0 60.0 57.1 57.2 58.1 7.2 1.4 3.3 0.2182 250 55.0 60.0 60.5 57.8 57.9 58.7 4.8 3.6 3.1 0.2313 265 58.0 61.0 60.0 57.7 57.7 58.6 0.5 5.7 2.4 0.2443 280 59.0 62.0 58.0 56.9 56.9 57.8 3.7 9.0 0.3 0.2662 305 58.0 58.0 56.0 53.2 53.2 53.9 9.0 9.0 3.9 0.3033 348 46.0 46.0 40.0 42.0 41.8 39.8 9.5 10.0 0.5 0.3107 356 35.0 35.0 35.0 35.0 35.0 35.0 0.0 0.0 0.0 0.3142 360 35.0 35.0 35.0 35.0 35.0 35.0 0.0 0.0 0.0 Input Parameters Bearing load, kN 8.0 Bearing speed, Ns rev/s 50.0 Supply pressure, Ps, kPa 300.0 Inlet temperature, C 35.0 Inlet viscosity, Pa.s 0.0 All other input parameters same as Table D3    x  76  Figure 4.15 Predicted versus experimental peak temperatures as function of oil supply pressure, Costa et al. [69]. The y-axis starts at 60.5°C hence the apparent large difference between prediction and experimental data.    Table 4.7 Predicted versus experimental temperatures as function of supply pressure.  Supply Pressure, kPa Model Predictions, °C (Angle) Exp. Values, °C Difference % 50 63.5 (300°) 64.4 1.40 70 63.2 (298°) 63.0 0.32 140 62 (293°) 63.2 1.90 200 61.4 (288°) 62.0 0.97 300 60.7 (279°) 61.2 0.82 Load, kN 2 Groove location θ  (+30) Bearing Speed, Ns rev/s 50 Oil supply temperature, C 35    59.5 60 60.5 61 61.5 62 62.5 63 63.5 64 64.5 65 0 50 100 150 200 250 300 350 Pe ak  T em pe ra tu re , C Supply Pressure, kPa Temperature as function of supply pressure T_model Data 64.4 63.4 63.9 63.2 62.5  77 4.3.6   Pressure Simulations of Costa’s Data Pressure predictions presented some challenges, the main one being how to take the effect of supply pressured into account. It appears from Costa et al. [69] that supply pressure is insignificant at groove locations (theta=-30 and 0 degrees). However, an initial pressure of 200 kPa is detected at groove location theta=+30. Figure 4.16 presents model predictions versus experimental pressures. Standard error bars show predictions to be within the ‘confidence’ range.            78  (a)  (b)  (c) Figure 4.16 Pressure prediction as a function of groove location, Costa et al. [69]. Applied load: 6 kN, speed: 50 rev/s.   0 200 400 600 800 1000 1200 1400 1600 1800 0 30 60 90 120 150 180 210 240 270 300 330 360 P re ss ur e,  k P a Angle, degrees Pressure Predictions vs Exp. data, θ=-30 P_Model Exp. data 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 30 60 90 120 150 180 210 240 270 300 330 360 Pr es su re , k Pa Angle, Degrees Pressure Model Prediction Vs Measured, Theta=0 P_model P_measured 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 30 60 90 120 150 180 210 240 270 300 330 360 Pr es su re , k Pa Angle, degrees Pressure Prediction vs Model, Theta=+30 P_exp P model  79 4.3.7 Flow Predictions Flow predictions are compared to experimental data [69] at a bearing load of 8 kN, 50 rev/s speed, and an inlet angle of -30˚ in figure 4.17. An analytical prediction from Martin’s formula [70] (see Appendix B) is also included for comparison. The model predictions are within 2.8 percent of measured values while Martin’s formula gives an accuracy of 11.2 percent. Standard error bars are included for both models. More detailed analysis of flow predictions is presented in chapter 5.   Figure 4.17 Flow predictions: model, experimental data and Martin’s formula [70].     4.4 Chapter Summary and Conclusions An effective and systematic procedure to model the thermo-hydrodynamic phenomenon is presented. The cylindrical coordinates are transformed to Cartesian ones with the resultant ‘bearing template’ used to analyze plain journal bearings. This transformation 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0 30 60 90 120 150 180 210 240 270 300 330 Fl ow , L /m in Supply Pressure, kPa Flow Predictions at W=8kN, Theta=-30 Martin,W=8kN,Theta=-30 Measured Flow_model  80 makes bearing analysis easier as shown by the rapid convergence of the model. On average only two iterations are required to produce stability in thermal predictions. This difference between model and experimental data is consistently less than 10 percent. No special treatment of the rupture interface is required in this model due to the high stability of the streamline upwind Petrov-Galerkin method. The velocity gradient ∂u/∂y is maximum at the rupture interface while ∂w/∂y, which is significantly less, has its lowest values after the reformation interface (no side flow in this region). The model also shows the high influence boundary conditions at the oil inlet/groove have on the results. More investigations need to be carried out to determine the most realistic boundary conditions in the mixing zone.  Improvement on the mixing model by implementing conservation of energy in the groove led to improved predictions shown in Appendix E. Pressure simulations are more difficult due to a lack of knowledge about the effect of supply pressure on inlet boundary conditions. Peak pressure predictions are however, reasonably good. Flow predictions from the model show a maximum deviation of 2.8 per cent as compared to predictions from Martin’s formula. The model is thus a significant improvement on methods currently available. Further verification of SUPG appears in Appendix E where the method is shown to be effective in convection-diffusion problems such as the one analyzed by Kelly et al. [71]. Improvements to the inlet boundary condition modeling by implementing groove mixing using conservation of energy resulted in improved temperature predictions for Mitsui’s [72] and Pierre’s et al.[73] data (Appendix E). Prediction accuracy of the SUPG model  81 can be improved significantly with better groove mixing computations which are complex due to groove size and geometry.                    82 Chapter 5 Formulation of Empirically Derived Model and Simulation Results  5.1   Introduction The SUPG finite element model developed in the previous chapters is used to validate optimization equations that are introduced here to complete the integrated design process for journal bearings. Empirical equations for temperature increase ΔT, leakage flow QL and power loss PL are formulated from first principles. Coefficients for these equations are determined by simulating measured bearing data in the literature. For a given set of design variables the FE model is run at speeds from zero to 200 rev/s and the three parameters are determined at each speed. Figure 5.1 gives details of how the optimization model is structured. The main focus in this work is to use the developed THD model to validate a proposed optimization scheme for medium range (length to diameter ratio 0.4<λ<1.5) medium speed (0<Ns<200 rev/s) plain journal bearings. Journal bearing design and optimization involves choosing a set of design variables that include but are not limited to radial clearance C, length to diameter ratio λ, viscosity μ, oil supply pressure ps for optimum performance and groove location θ. The goal is to find a set of design variables that minimizes the objective function. The objective function may include temperature increase ΔT, side leakage QL, and power loss PL.  Derivation of equations for temperature rise ΔT, side leakage QL, power loss PL, and constraint equations for maximum pressure pmax, whirl onset speed ωcr, and oil supply pressure ps will be considered. It should be emphasized that selecting design variables may be influenced by additional factors such  83 as journal strength/stiffness requirements for minimum misalignment. Thus the minimum journal size could also be determined by consideration of shaft deflection and strength which is not considered here. A formal definition of the optimization model and its formulation is presented in chapter 5. Multi-objective function equations with linear and non-linear constraints form the basis of the optimization models developed. Two solution techniques, namely the method of assigning weighting/scaling factors and using Pareto optimal fronts will be explored. The targeted range of aspect (L/D) ratios is 0.4<λ<1.5 and a speed range (0<Ns<200 rev/s). Although Hirani and Suh [54] propose an optimization model for this range of bearings, their model requires complex numerical integration for power loss PL and use of mass conserving finite difference equations to solve for angle θ  around the bearing circumference. Their model also involves reference to tables (table 1 in reference 38) which makes their model even more cumbersome to implement as a flow prediction program. Optimization equations are derived from first principles and empirical validation of coefficients is achieved using the bearing template developed in the previous chapter as well as proving these equations using data from literature. Convergence and simulation times for both meshing and computations was addressed in section 4.3. As noted each iteration took about 2.9 seconds and at least three iterations were required for convergence for a total of 8.7 seconds. For optimization computations which required obtaining operating characteristics at Ns=2 rev/s intervals, a typical cycle time for Costa’s data (Tbale D3 in appendices) is analyzed in table 5.1.    84 Table 5.1 Computation Times for Optimization Model Time Stepping Task Time (Seconds) Meshing 4.1 Time for convergence at each Ns 26.6 Total time for 0<Ns<200 rev/s  2510.5 Time required for SQP algorithm 5.0 Time for seven attempts with SQP 35.0 Time required for hybrid algorithm 15.0  The total time of 2510 seconds translates to approximately 42 minutes for the iterations. The computer specifications are Pentium IV clock speed of 3.25 GHz and 1 GB RAM. The genetic algorithm generally required more time than SQP, however the problem with SQP is the result depends on the initial guess. If the initial guess is in a region with a local minimum the algorithm gives erroneous results. On average seven attempts were made to get reasonable results with SQP. The SUPG finite element method predicts temperature increase from given input parameters and further validates the coefficients in equation 5.7. Table 5.2 presents the coefficients.          85                    Figure 5.1 Flowchart for Optimization Model      Start with initial speed Ns Run SUPG FE model to compute ΔT1, QL1, and PL1 Predict ΔT2, QL2, and PL2 using empirically derived model Are predictions (ΔT2, QL2, PL2 ) within specified tolerances to (ΔT1, QL1, PL1) and data in literature? No Readjust coefficients for empirical model Yes Empirical model validated at current speed Ns Increase speed Ns at 2 rev/s intervals up to 200 rev/s Determine optimized design variables using Hybrid Technique End  86 5.2 Governing Equations 5.2.1   Temperature Increase The journal bearing shown in figure 4.1(a) is used for reference. The method proposed follows Jang and Khonsari’s approach [55] whereby an initial estimate of the Sommerfeld number S is made 2 ⎟⎠ ⎞⎜⎝ ⎛= C R W DLN S seff μ     (5.1) where the oil viscosity at inlet conditions is taken as the effective viscosity μeff. Eccentricity ratio ε is determined to be a function of Sommerfeld number and length to diameter ratio λ, ε=ε(S,λ) using the approximation ( )Sln20986.049737.036667.0 −+−= λε   0.5 ≤ λ ≤ 1.5  (5.2) A closed form of the energy equation can be derived by assuming adiabatic conditions at the oil/bush interface ( 0=∂∂ yT  at 0=y ) and ∫ =∂∂ π2 0 0xd y T   and  shaftTT =   at 1=y   (5.3) while isothermal conditions apply at the shaft/oil interface ( shaftTT =  at 1=y ), the so- called ISOADI (isothermal shaft, adiabatic interface) condition resulting in the dimensionless energy equation [74], as presented in equation 2.14b. Two key parameters κ1 and κ2 that appear in equation 2.14b are referred to as temperature rise parameters. The first temperature rise parameter κ1 is associated with viscous dissipation while κ2 relates oil properties and velocity. The energy equation is solved for temperature increase ΔT(X); it is evident ΔT(X) is a function of eccentricity ratio, κ1 and κ2  87                                                           ΔT(X) = f(ε,κ1,κ2) (5.4) The dimensionless temperature equation proposed by Jang [55] forms the basis for temperature increase computations                                                     ⎥⎦ ⎤⎢⎣ ⎡ ++= 5.0 2 5.0 1 max exp κκ cbaT  (5.5) where constants a, b, and c are empirically determined from simulating bearing temperature data in the literature [59,71,75].  Table 5.2 Values a, b, and c computation of maximum temperature Tmax. Note that the valid range of temperature rise parameters is: 0.001≤κ1≤0.5 and 0.01≤κ2≤5   [ ]5.025.01max //exp κκ cbaT ++= Eccentricity ratio Constants  a b c 0.1<ε<0.3 1.58105 -0.14250 -1.04050 0.3≤ε≤0.7 1.5917 -0.13962 -1.24999 0.7<ε≤0.9 1.63973 -0.10627 -1.45216  These constants are functions of the eccentricity ratio. Temperature predictions with the constants in table 5.2 are compared to the finite element and other models in section 5.4 and shown to give accurate results. The maximum temperature can be determined from the relation ( )iTTT −= βmax  once maxT  is known. Maximum shaft temperature is determined in a similar manner  1 5.0 2 5.0 1 − ⎥⎦ ⎤⎢⎣ ⎡ ++= κκ cbaTshaft            (5.6) Effective viscosity μeff is updated using the shaft temperature Tshaft  and  equation 2.26. The Sommerfeld number and eccentricity ratio are updated using the new viscosity  88 values; maximum temperature and shaft temperature are re-evaluated using the updated values. The iteration process is terminated when convergence occurs to a predetermined value.  5.2.2     Side Leakage A key difference with other models is in the empirical relation for the leakage flow LQ . Khonsari’s model [55] utilizes LsL QCDLNQ 2 π=  while Hashimoto [34] uses επ 2 4 CDNQ sL =  for short bearings. Consideration of where the hydrodynamic film starts along the bearing circumference is important [68], however the curve fitted empirical model developed here appears to give good results. The non-dimensional flow LQ  is expressed as a function of ε and λ as proposed by Martin [70] and used by Jang and Khonsari [55] ( )[ ]ελ ln05.1500.020.0exp 5.1 +−=LQ       0.5≤λ≤1.5;  0.1≤ε≤0.9 (5.7) where the coefficients were determined empirically by Gadala and Zengeya [76] who simulated pressure-fed bearings in the literature and performing curve-fitting to the data given that LQ =f(ε,λ). Model predictions are compared to those from Khonsari’s model in simulating Costa’s data in section 4.4. Predictions from Khonsari’s model are not as accurate since they don’t specifically consider supply pressure ps. The proposed model was also validated using measured leakage data from literature [59,69,75] and compared with predictions from Song [36], Jang and Khonsari [55], and the finite element model.   89 5.2.3   Power Loss Determination Power loss forms an important consideration in journal bearing design. Frictional losses in machines result in wasted energy and the generation of heat which affects the life of materials including the lubricant. A significant portion of total fluid film bearing losses in high-speed turbomachinery may be consumed simply in overcoming frictional losses. At surface speeds below approximately 51 m/s, these losses are generally less than 10 percent of total power input to the bearing [77]. At higher velocities, however, they rapidly with surface speed and can reach 25 to 50 percent of the total bearing loss in large turbine-generators and related machinery. Power loss is determined from ∫∫ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂⋅ ∂+= dzURd R Ph h U P effL θθ μ 2     (5.8) where the effective viscosity is determined at the current iteration. Hirani and Suh [38] use a mass conserving finite difference scheme to solve for the power loss. The drawback in this approach is that it involves much more computational effort than proposed. The approach in emanates from the established fact the term  ( ) fCR  is a function of λ and ε. Power loss is determined from    ⎟⎟⎠ ⎞⎜⎜⎝ ⎛ −+= ελ 97838.4 73622.29994.1expf C R   (5.9a)            Ffr=fW      (5. 9b)   PL=Ffr(2πRNs)      (5.9c) where equation 5.9a is determined empirically following Jang and Khonsari’s  approach [55]. The frictional force Ffr is determined once f is known, and W the load on the bearing.   90 5.2.4 Pressure Computations Pressure prediction for the optimization equations are implemented by integrating the Reynolds equation to get a non-dimensional form [55] 2 ⎟⎠ ⎞⎜⎝ ⎛−= C R N pp P si s μ      (5.10) with the empirical formula proposed by Jang and Khonsari [55] used to evaluate non- dimensional pressure P ⎥⎦ ⎤⎢⎣ ⎡ +−= 3max 77721.439491.025062.2exp ελP    (5.11) The onset of whirl is important due to the instabilities it creates in journal bearings. However, for speeds considered here (Ns<200 rev/s) is was determined not to be a major problem even for the upper range. Hashimoto [34] gives a brief account on the effects of whirl especially for high-speed, short bearings. Whirl velocity is determined from the Routh-Hurwitz stability criterion and found to be a function of eccentricity ratio ε. An empirical expression is presented ( ){ }873.2318.199.6exp0584.0 07.2 +−= εεω C g cr   (5.12) where ωcr is the critical whirl speed and g the gravitation acceleration. 5.2.5 Formulation of the Optimization Problem The first part of the work investigated minimization of temperature increase ΔT(X) and side leakage QL by combining the two objective functions F1 and F2 into a multi- objective function F(X) using weighting and scaling parameters. This works well if the relative importance of each functional is known apriori. Hashimoto [34,35] uses this method in analyzing high-speed, short journal bearings. The concept of axiomatic design  91 [39] helps determine the independence of objective functions, and as explained by Hirani [40,41] ΔT (F1) and QL (F2) are not independent of each other as ΔT is a function of W and QL which means F1=func(W, F2). However objective functions W (F1) and QL (F2) are making them the preferred choice. In addition to the method of weighting functions Pareto optimal fronts at several common speeds are produced as an alternative optimization technique that requires no weighting or scaling parameters. Pareto optimal fronts are sets of equally efficient or non-inferior alternative design vectors. The design vector X is considered non-inferior if the following conditions are fulfilled [41] Wi≤Wj and Qi<Qj or     (5.13a) Wi<Wj and Qi≤Qj     (5.13b) Four design variables are chosen; namely the radial clearance ratio C, length to diameter ratio λ, oil viscosity μ and supply pressure ps. The design variable set can be written in a vector X, ( )sT pCX ,,, μλ= . Supply pressure is specifically included as emphasis is on pressure-fed bearings and variation of this variable will be closely monitored in the results. State variables vary with the given operating conditions of the bearing such as the load W, rotational speed Ns and ambient temperature To. These include the eccentricity ratio ε, film pressure p, film temperature T (K), side leakage QL (m3/s), and whirl onset speed ωcr. The given choice of design variables affects the state variables.  It should be emphasized that selection of design variables may be influenced by additional factors such as journal strength/stiffness requirements for minimum misalignment. Thus the minimum journal size could also be determined by shaft deflection which is not considered here.  92 The multi-objective optimization problem can be formally stated as [42] Find X to minimize F(X) subject to the constraints gi(X) ≤ 0 where F(X) = α1β1PL + α2β2QL      (5.14a) and gi(X) ≤ 0,         (5.14b) with  g1 = ha – C{1 - ε(X)},  g2 = ΔT(X) – ΔTa, g3 = ω - ωcr,   g4 = pmax – pa non-linear constraints and           Cmin ≤ C ≤ Cmax,    λmin ≤ λ ≤ λmax, μmin ≤ μ ≤ μmax,   ps, min ≤  ps ≤  ps, max (5.14c) are lower and upper bounds linear constraints for C, λ, μ and ps  respectively. Constants α1 and α2 are scaling while β1 and β2 fulfill a weighting role in equation 5.14a. Note that these are not necessary if the problem is solved using Pareto optimal fronts.  5.2.6 Implementation in Matlab The program structure follows the steps given in Appendix C and is implemented in Matlab [51,78] with the input data in table D.5. The hybrid scheme invokes the genetic algorithm (GA) for rapid convergence to the global minimum region and SQP once this region is identified for accurate determination of the minimum point. Equation 5.14(a) is the fitness function and design variable set ( )sT pCX ,,, μλ=  comprises the set of genes. Highlights of the GA include a population of 100 individuals and a double vector population type, an initial penalty of 10 and a penalty factor of 100 for the non-linear constraints. The penalty factor increases the penalty parameter when the problem is not solved to required accuracy and constraints are not satisfied. A rank-based fitness scaling is used. Rank scales the raw scores based on the rank of each individual, rather than its  93 score. The rank of an individual is its position in the sorted scores. The rank of the fittest individual is 1, the next fittest is 2 and so on. Rank fitness scaling removes the effect of the spread of the raw scores. The selection function used is stochastic. Stochastic uniform lays out a line in which each parent corresponds to a section of the line of length proportional to its expectation. The algorithm moves along the line in steps of equal size, one step for each parent. At each step, the algorithm allocates a parent from the section it lands on. The first step is a uniform random number less than the step size. However as Hirani and Suh [38] point out it is important to maintain diversity in non-inferior solutions by including a degree of sharing. Reproduction options determine how the next generation is created. Elite individuals are guaranteed survival to the next generation; the elite count was set to 2. Crossover fraction, which specifies the fraction of the next generation other than elite individuals that are produced by crossover, was set at 0.8. The remaining individuals, other than elite individuals, in the next generation are produced by mutation. Mutation functions make small random changes in individuals in the population, which provide genetic diversity and enable the GA to search a broader space. The mutation method used is Gaussian which adds a random number to each vector entry of an individual. This random number is taken from a Gaussian distribution centered on zero. The variance of this distribution can be controlled with two parameters. The Scale parameter determines the variance at the first generation. The Shrink parameter controls how variance shrinks as generations go by. If the Shrink parameter is 0, the variance is constant. If the Shrink parameter is 1,  94 the variance shrinks to 0 linearly as the last generation is reached. Both scale and shrink parameters are set at 1. Crossover combines two individuals, or parents, to form a new individual, or child, for the next generation. The scattered option, which creates a random binary vector and then selects the genes where the vector is a 1 from the first parent, and the genes where the vector is a 0 from the second parent, and combines the genes to form the child, is chosen. Migration is the movement of individuals between subpopulations if population size is greater than one. Every so often, the best individuals from one subpopulation replace the worst individuals in another subpopulation. Migration was controlled by using the forward option where migration takes place toward the last subpopulation, that is the nth subpopulation migrates into the (n+1)th subpopulation. A migration fraction of 0.2 is used, and this controls how individuals move between subpopulations. A migration interval of 20 is set, and this controls number of generations before migration takes place. The Matlab code is included in Appendix C.  5.3 Validation of Proposed Equations 5.3.1  Effect of Supply Pressure ps The effect of supply pressure ps on leakage flow with model predictions on Costa’s data [69] is shown in figure 5.2. Jang and Khonsari’s [55] predictions are also included for comparisons. Although Khonsari does not specifically include supply pressure in the simple expressions of Table 4, reference 55, predicted values in figure 5.2 for this model are not constant. This may be due to an indirect effect of supply pressure on eccentricity ratio ε and the way coefficients of LQ  are determined in the (ref. 55). Standard error and the square of the Pearson product moment correlation coefficient appear in table 5.3. It is  95 interesting to note that r2 for Khonsari’s model is better (0.999) than the model (0.998) although it is evident from the graph that the ‘goodness of fit’ is poor.  Figure 5.2 Model versus Khonsari’s predictions – Costa’s data.   Table 5.3 Standard error and square of the Pearson correlation     Model Khonsari 2 kN STD Error 8.87E-07 2.88E-06 r2 0.986 0.857   5.3.2 Temperature Increase ΔT(X) The equations were validated by comparing predictions to measured data from literature. Three bearings with input data in Tables D.1 (Dowson et al. [59]), D.3 (Costa et al. [69]) and D.4 (Ferron et al. [75]) are used for the analysis. Temperature increase ΔT(X) as a function of speed for Dowson’s bearing is shown in figure 5.3(a) and 5.3(b), figures 000E+0 10E-6 20E-6 30E-6 40E-6 50E-6 60E-6 70E-6 0 50 100 150 200 250 300 350 Q L, m 3 /s Supply Pressure, kPa Leakage Flow Data: Costa's Bearing QLModel, 8kN QLKhonsari, 8kN Measured, 8kN QLModel@2kN QLKhonsari@2kN Measured@2kN  96 5.4(a) and 5.4(b) for Ferron’s data (ΔT is 18 °C at 66.7 rev/s), while figures 5.5(a) and (b) show simulations for Costa’s data. The dashed lines indicate measured data points and journal speed on all the figures. The first figure in each case has all four models (FE, proposed model, Khonsari, and Song); the second shows a smaller speed range for better clarity of measured points/values. Temperature increase ΔT(X) is determined as the leakage oil temperature while model predictions represent the average of all nodal points in the film mesh. In all cases in the figures, temperature increase is assumed to be the measured temperature of leakage oil. Standard error bars indicated on 5.3b, 5.4b and5.5b show model predictions in the 95% confidence level. In all cases the proposed model predictions are more accurate as the formulations are based on both theoretical and empirical validation for the range of bearings under consideration. The equations were derived from first principles using ISOADI and adjusted empirically using Jang and Khonsari [55] as a guide. Also included is Hashimoto/Song model [34,36] for short, high speed bearing configurations. It is clear from figures 5.3(a) and 5.4(a) that predictions from Hashimoto are inaccurate for finite and long bearings under consideration. The model is, however, included for completeness in the discussion of journal bearing optimization. A different perspective to the results is shown in Figure 5.6 where temperature increase is shown as a function of diameter and radial clearance for Dowson’s data. Measured values are indicated on the graphs. An important consideration is establishing whether the range of bearing performance is laminar and below the speed at which shaft whirl starts. This is generally determined by  97 computing the Reynolds number (Re=ρCU/μ). A sampling of data points was used where the variable set XT=(C,λ,μ,ps) for some of data is shown in Table 5.4.  Table 5.4 Local Reynolds number from a sample of the data XT in (μm,-,Pa.s,kPa) Bearing Diameter, m Re (60,0.4,0.03,300) 0.191 2871 (50,0.4,0.03,300) 0.191 3402 (80,0.4,0.03,300) 0.191 2284 (40,0.6,0.03,300) 0.128 1511  It was determined from analysis that the range of Reynolds numbers determined were in the laminar range although the second value in Table 5.4 is high. Onset of whirl occurred at Ns>195 rev/s for this particular set and higher speeds should be avoided. For Dowson’s bearing, a Reynolds number of 820 is determined at Ns=200 rev/s for XT=(40,0.75,0.03,300) which is well below both whirl onset and turbulent flow. Costa and Ferron bearings also operate below turbulent flow.       98  (a)  (b) Figure 5.3 ΔT(X) as function of speed Ns for Dowson’s bearing. 0 100 200 300 400 500 600 700 800 10 30 50 70 90 110 130 150 170 190 210 ΔT,  C Speed, rev/s Delta T Comparisons, Dowson's Bearing DT_model DT_FE DT_Khonsari DT_Song/Hashimoto 0 20 40 60 80 100 120 140 160 180 200 220 5 10 15 20 25 30 35 40 45 50 55 ΔT,  C Speed, rev/s Model, FE, Khonsari and Song/Hashimoto ΔT Comparisons, Dowson's Bearing Exp. Point DT_FE DT_model DT_Khonsari DT_Song/Hashimoto ΔTmesured at 25rps = 14.7 C  99  (a)  (b) Figure 5.4 ΔT(X) as a function of speed Ns for Ferron’s bearing.  0 50 100 150 200 250 300 350 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 ΔT,  C Speed, rev/s Model, FE, Khonsari, Song/Hashimoto Temp Dif f  Comparisons: Ferron Brg DT_model DT_FE DT_Khonsari DT_Song/Hashimoto ΔTmeasure, ns=67rps=18C 0 5 10 15 20 25 30 35 40 45 50 55 60 10 20 30 40 50 60 70 ΔT,  C Speed, rev/s Model, FE, Khonsari and Song/Hashimoto ΔT Comparisons. Ferron Brg Exp. Point DT_model DT_FE DT_Khonsari DT_Song/Hashimoto ΔTmeasured, ns=67rps=18  100  (a)  (b) Figure 5.5 ΔT(X), QL as function of speed Ns, Costa. 0 50 100 150 200 250 300 350 400 450 500 550 600 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 ΔT , C Speed, rev/s Delta T Comparisons, Costa DTModel DT_FE DT_Khonsari DT_Song/Hashimoto 000E+0 10E-6 20E-6 30E-6 40E-6 50E-6 60E-6 70E-6 80E-6 90E-6 100E-6 110E-6 120E-6 130E-6 140E-6 150E-6 160E-6 170E-6 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 Q L, m 3 /s Speed, rev/s Leakage flow comparisons, Costa Exp. Point QL_Model QL_Khonsari QL_Song/Hashimoto QL_FE Measured Leakage Data QL ns=33.3 = 45.8x10-6 m3/s QL ns=50 = 54.2x10-6 m3/s QL ns=66.7 = 62x10-6 m3/s  101   (a)  (b) Figure 5.6 (a) ΔT(X) as function of clearance ratio C, Dowson data, and (b) as function of diameter D  0 5 10 15 20 25 30 40E-6 60E-6 80E-6 100E-6 120E-6 140E-6 ΔT , C Clearance, C (m) Delta T Comparisons, Ns=25 rev/s, Dowson's data Exp. Point FE Model Khonsari Measured Data Clearance C = 63.8 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 ΔT , C Diameter D, (m) Delta T vs Diameter for Dowson's Brg, C=60μm, L=0.0765m Exp. Point FE Model Khonsari Song/Hashimoto ΔT Data (Celsius) Measured = 13.7 Model = 13 Khonsari = 15.8 Song = 39.2  102 5.3.3 Leakage Flow QL Predictions Leakage results were also analyzed for the examples above and presented in Figures 5.7, 5.8 and 5.9. The finite element predictions are closest to the measured values at 25 rev/sec speed. Figure 5.8 presents leakage flow data from Ferron’s bearing (measured at 66.7 rev/s); similar results are evident.  Standard error bars are included for the model prediction as well as a select few (to reduce clutter). Model predictions are good in all cases. Hashimoto’s model is significantly off for the range of bearings tested since it is derived for high-speed, short bearings. Another shortcoming with this model ( επ 2 4 CDNQ sL = ) is that it does not specify supply pressure ps explicitly.  Figure 5.7 Side leakage predictions for Dowson’s bearing. 000E+0 10E-6 20E-6 30E-6 40E-6 50E-6 60E-6 70E-6 80E-6 90E-6 100E-6 110E-6 120E-6 130E-6 140E-6 150E-6 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 Q L, m 3 /s Speed, rev/s Leakage Flow Rate Predictions: Dowson Bearing Exp. Points Q_FE Q_Khonsari Q_Song/Hashimoto Q_model Ps = 280 kPa QL measuredns=25 = 26.2x10-6 m3/s QL measuredns=33.3 = 30.1x10-6 m3/s Song/Hashimoto  103  Figure 5.8 Side leakage predictions – Ferron’s bearing.  Figure 5.9 Side leakage predictions – Costa’s bearing.   000E+0 20E-6 40E-6 60E-6 80E-6 100E-6 120E-6 140E-6 160E-6 180E-6 200E-6 220E-6 240E-6 260E-6 280E-6 300E-6 320E-6 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 Q L, m 3 /s Speed, rev/s Leakage Comparisons: Ferron Brg Exp. Point QL_FE QL_Khonsari QL_Song/Hashimoto QL_Model QL exp. = 130.6x10-6 m3/s 000E+0 10E-6 20E-6 30E-6 40E-6 50E-6 60E-6 70E-6 80E-6 90E-6 100E-6 110E-6 120E-6 130E-6 140E-6 150E-6 160E-6 170E-6 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 Q L, m 3 /s Speed, rev/s Leakage flow comparisons, Costa Exp. Point QL_Model QL_Khonsari QL_Song/Hashimoto QL_FE Measured Leakage Data QL ns=33.3 = 45.8x10-6 m3/s QL ns=50 = 54.2x10-6 m3/s QL ns=66.7 = 62x10-6 m3/s  104 A summary of the data is presented in Table 5.5. The figures in brackets show percentage difference to measured values.  The proposed model gives consistently better results than Khonsari and Song models.  Table 5.5 Summary of ΔT(X) and QL data comparisons. Note that all leakage flow data are 610−× m3/s, ΔT values are in °C and based on (Tmax-Ti).  ΔT(X) Finite Element Model Khonsari Song Measured  Dowson  14.2  (-3.4) 16 (8.8) 19.1 (29.9) 70.7 (380) 14.7 Ferron   16.9  (6) 19.6 (8.8) 21.8 (21) 42.2 (133)  18 Costa  13.3 (2.3) 18.1 (9.5) 23.1 (0.4) 16.0 (23) 22.2 (11) 27.6 (20) 20.0 (54) 28.0 (40) 35.0 (52) 66 (400) 122 (500) 190 (700) 13 20 23  QL  Dowson 26.2 - 22.5 (14) 17.5 (-33) 1.05 (-95.9)  26.2 Ferron 130 (0.46) 134.9 (3.4) 106.1 (-18.8) 5.47 (-96)  130.6 Costa  43.4 (5.2) 58.9 (8.7) 71.7 (15.6) 41.4 (9.6) 56.9 (5) 69.7 (12.4) 32.6 44.6 54.7 1.55 1.61 1.67 45.8 54.2 62     5.3.4 Variation of C on Performance Parameters ΔT and QL The model is run to obtain temperature increase ΔT(X) as a function of clearance ratio C in the range 40≤C≤150 μm for Dowson’s bearing. Figure 5.10 shows values at Ns=201 rev/s. The model, FE, and Khonsari predictions follow a similar pattern while Song’s model predictions are well off. Results for side leakage as a function of C for all four models at a speed of 25, 101 and 201 rev/s are shown in figure 5.11. It appears that side leakage is linear at lower speeds and changes to a power function at higher speeds as shown in figure 5.12. A least squares curve fitting was performed in MS Excel with the results QL ∝ C   Ns < 50 rev/s  105 QL ∝ C1.5   50 ≤ Ns ≤ 120 rev/s QL ∝ C1.3   Ns > 120 rev/s   (5.15) Included in figure 5.12 are r2 values for each trendline. It has to be mentioned that the bearings analyzed have a diameter close to 100 millimeters. A wider range of sizes needs to be tested to give the results wider applicability.   Figure 5.10 Side leakage as a function of clearance C, speed 25 to 201 rev/s. 000E+0 50E-6 100E-6 150E-6 200E-6 250E-6 300E-6 350E-6 400E-6 40E-6 60E-6 80E-6 100E-6 120E-6 140E-6 Le ak ag e,  m 3 /s Clearance ratio C, (m) Leakage Flow at two speeds, Dowson FE:ns=25 Model:ns=25 Khonsri:ns=25 Song:ns=25 FE:ns=201 Model:ns=201 Khonsari:ns=201 Song:ns=201  106  Figure 5.11 ΔT(X) versus clearance ratio C, Dowson’s bearing, Ns=201 rev/s.    Figure 5.12 Side leakage as a function of clearance at three different speeds. 0 200 400 600 800 1000 1200 40E-6 60E-6 80E-6 100E-6 120E-6 140E-6 160E-6 ΔT , C Clearance, C (m) Delta T Comparisons for Dowson's bearing, Ns=201 DTFE:ns=201 DTModel:ns=201 DTKhonsari DTSong/Hashimoto R² = 0.993 R² = 0.999 R² = 0.998 000.0E+0 50.0E-6 100.0E-6 150.0E-6 200.0E-6 250.0E-6 300.0E-6 350.0E-6 400.0E-6 40.0E-6 60.0E-6 80.0E-6 100.0E-6 120.0E-6 140.0E-6 160.0E-6 Si de  L ea ka ge  Q L,  m 3 /s Radial Clearance C, (m) QL v C, λ=0.75, μi=0.03 Pa.s, Dowson "n_s=25" n_s=101 n_s=200 Linear ("n_s=25") Power (n_s=101) Power (n_s=200) QL α C QL α C1.5 QL α C1.3  107 5.3.5 Temperature Increase ΔT, Side Leakage QL as Function of Diameter The effect of varying diameter D while holding the length L constant for Dowson’s bearing is analyzed. Figure 5.13 shows the temperature increase ΔT(X) while figure 5.14 shows leakage variation. Figure 5.13 is plotted by holding the length of Dowson’s bearing constant and varying the diameter. Song’s model is included for completeness of the discussion although it is clearly unsuitable for the range of bearings tested. Their model is based on the friction force which is calculated by considering the local Reynolds number (Re=ρCU/μ). There appears to be no method of updating the viscosity for use in computation of the Reynolds number in their model. The other models show steady rise with increasing diameter although the gradient ∂ΔT/∂D increases in the region 0.1<D<1.25 after which it decreases.    Figure 5.13 ΔT(X) as a function of diameter D. 0 50 100 150 200 250 300 350 400 450 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 ΔT , C Diameter D, (m) Delta T vs D, Ns=101rps, Dowson FE@101 Model@101 Khonsari@101 Song/Hashi@101  108   Figure 5.14 QL as a function of diameter D, Dowson’s bearing, Ns=25 rev/s. The relation is linear as expected at this speed.   5.3.6 Simulation Results with ΔT (f1) and QL (f2) as Objective Functions The set of equations developed are used to minimize a multi-objective function of temperature increase ΔT(X) (F1) and side leakage QL (F2) using the design variable set ( )μλ,,CX T = , (same set used by Hashimoto [34] and Song et al. [36] ). Hashimoto’s model is for high speed (up to 240 rev/s) and short bearings (0.2<λ<0.6) whereas the model developed here is for medium speed (up to 200 rev/s) and medium range (0.4<λ<1.0). The two objective functions are combined using scaling and weighting factors to form one multi-objective function, F(X)=α1β1ΔT+α2β2QL which is minimized subject to linear and non-linear constraints given in equation 5.14(b) and (c). Additional 0.00E+00 1.00E-05 2.00E-05 3.00E-05 4.00E-05 5.00E-05 6.00E-05 7.00E-05 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Le ak ag e Q L, m 3 /s Diameter D, (m) Leakage Flow Vs Diameter, Dowson's Data, C=80 mm Model Khonsari FE Song/Hashimoto  109 input parameters for Dowson’s bearing to run the optimization model are given in table D.5. Results will be compared to those obtained by Song et al. for high-speed, short bearings where a similar objective function was used (included for comparison only as input parameters are different). Results from both the classical (SQP) and a hybrid method are compared and presented in figures 5.15, 5.16, 5.17 and 5.18. At least three runs with many sub-iterations were performed for each method. Figure 5.15 compares results of the objective function from the two methods at load 10 kN. The hybrid algorithm needed at most two runs to produce a consistent result. SPQ however required at least four runs while adjusting the initial guess to get reasonable predictions. Comparison with Song’s results show a similar pattern to the objective function although Song’s values are lower due to different input parameters and range (see table D.5). Clearance ratio predictions from the model (figure 5.16(b)) show a rapid initial increase and leveling off at speeds above 80 rev/s. This is different from Song’s model prediction (figure 5.16(a)), which shows an initial decline, leveling off, and gradual increase at speeds greater than 120 rev/s. Actual values of C in Song’s model are much lower than model predictions. Length to diameter λ predictions are shown in figure 5.17(b) for the model and 5.17(a) for Song’s model. Both show a decline to the lower bound of λ, the model showing a faster descend than Song’s model. Thus a similar pattern is exhibited by both models. Finally viscosity variation μ for both models show predictions in the lower bound values as shown in figure 5.18 (a) and (b). In summary, the following points are evident: • Objective function predictions F(X) show a smooth increase for both models  110 •  Clearance ratio requirements for wider bearings appear to be higher than shorter bearings • Length to diameter ratio λ decreases to the lower bound in both cases, declining faster for wider bearings. • Viscosity remains at the lower bound in both cases. Obviously viscosity values should be as low as possible to minimize viscous dissipation. The next section extends the design variable set to four ( ( )sT pCX ,,, μλ= ) by including supply pressure variation; the optimization work primarily focuses on pressure fed bearings. The multi-objective function is also reformulated by considering minimization of the power loss PL and leakage QL ;  these two are axiomatically independent [39].         111  (a)  Figure 5.15 (a) Objective function from Song et al. [36], and (b) F(X) predictions from SQP and hybrid algorithm, Wd=10kN for both cases.  2 4 6 8 10 12 14 16 18 50 100 150 200 250 O bj ec tiv e  Fu nc iti on  f( X) Speed, rev/s Objective Function: SQP vs GA SQP GeneticAlg Objective Function f(X): SQP vs Hybrid 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 Speed, rps f(X ) f(X): Hybrid SQP: 7 Attempts W = 10 kN  112  (a)   (b) Figure 5.16 (a) Clearance ratio C as a function of speed for Song’s model (b) Model predictions, Wd=20 kN  40 50 60 70 80 90 100 50 100 150 200 250 Ra di al  C le ar an ce  C , μm Speed, rev/s Radial Clearance, μm SQP GA Radial Clearance C : SQP vs Hybrid 000E+0 50E-6 100E-6 150E-6 200E-6 250E-6 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 Speed, rps R ad ia l C le ar an ce  C , m C: SQP C: Hybrid W = 10 kN  113  (a)   Figure 5.17 (a) Length to diameter ratio λ as function of speed, Song’s model, and (b) Length to diameter ratio λ as function of speed for SQP and hybrid algorithm for developed model.  0.10 0.15 0.20 0.25 0.30 50 100 150 200 250 Le ng th  to  D ia m et er  λ Speed, rev/s Length to Diameter Variation λ SQP GA Length to Diameter Ratio λ Comparisons 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 Speed, rps λ 1:SQP: 7 attemps 2: Hybrid W = 10 kN  114  (a)   (b)  Figure 5.18(a) Viscosity variation, Song/Hashimoto’s model [36] (b) Viscosity variation versus speed, hybrid and SQP.    0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 50 100 150 200 250 Vi sc os ity  μ,  Pa .s Speed, rev/s Viscosity Variation μ SQP GA Viscosity μ Comparisons 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0.03 0.033 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 Speed, rps Vi so co si ty  μ,  (P a. s) 2: Hybrid SQP: 7 Attempts W = 10 kN  115 5.4 Power Loss PL (F1) and Leakage QL (F2) as Objective Functions 5.4.1 Power Loss Predictions from Model Power loss comparisons for the proposed model, Jang and Khonsari’s [55], finite element, and Hashimoto/Song’s models [34,36] is shown in figure 5.19 for Dowson’s bearing [59]. Five measured data points and standard error bars are included; these show the consistently better prediction from the model. Figure 5.20 presents simulation results for Costa’s data [69]. Both figures show the relative accuracy of the proposed model for the range of bearings tested (0.4<λ<1.5 and 0<Ns<200 rev/s). Model predictions are well within the 95% confidence level. Another advantage is the ease with which the model can be implemented. Equation set 5.9 computes the coefficient of friction f, friction force F, and power loss PL.  Figure 5.19 Power loss prediction – Dowson’s data.  0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 P ow er  L os s,  W Speed, rev/s Power Loss Comparisons - Dowson's Data Exp. Points FE Model Khonsari Song/Hashimoto Exp. Data Speed, rps Power Loss, W 11.7             153.1 13.3             189.2 16.7             280.8 20.8             406.4 25                532  116  Figure 5.20 Power loss predictions – Costa’s data.   5.4.2 Determination of weighting parameters and scaling factors The method of weighting/scaling factors is used to determine optimum values for the design variables. Input data shown in Tables D.1 and D.5 is used. Values of leakage QL were weighted appropriately by using a suitable value for β2 while the scaling factors were selected after comparing the absolute values of the objective function F(X) as a function of the ratio α1/α2. Three values of the ratio were tried, namely 5, 1, and 1/5. It is assumed the objective functions f1 and f2 are of equal importance and this means the magnitude of the two should be equal or nearly so. The effect of weighting factor β2 of 105 is to transform leakage values, which are in the order 10-5 m3/s to the same order of magnitude as power loss. Values of f1 and f2 are adjusted to be of equal magnitude by 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 8500 9000 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 Po w er  L os s,  W Speed, rev/s Power Loss Comparisons, Costa Bearing, 70 kPa Exp. Point FE Model Khonsari Song/Hashimoto Measured Data PowerLossns=33rps=612 W PowerLossns=50rps=1125 W PowerLossns=67rps=1663 W  117 selecting α1/α2=1/5, β1=1/400, β2=105 as shown in figure 5.22a. It is important that values of the objective functions be similar if they are of equal importance (somewhat like ‘comparing oranges with oranges’). It must be emphasized that the ultimate choice of the weighting parameters depends on the relative importance of each objective function f1 and f2 [34].  These graphs show that power loss f1 increases more rapidly than leakage f2 thus having a greater influence on the objective function value F(X). Depending on the relative importance of minimizing the power loss or leakage, figure 5.21 gives an indication on what route to take. An assumption is made here that the goal is to obtain the lowest possible value of the objective function and use α1/α2=1/5. The effect of scaling factors β1 and β2 with α1/α2=1/5 is shown in figures 5.22, 5.23 and 5.24. Also included are the values of F1, F2 and the design variable sets (C,λ,μ,ps). One observation is increasing clearance ratio reduces power loss but increases side leakage significantly. Figure 5.21 shows results with β1=1/400 and β2=105 which has the effect reducing power loss contribution to F(X). The net result is minimizing side leakage has more weight and clearance ratio is at the minimum value in 5.21c while length to diameter ratio λ is at the upper limit of 1. Supply pressure predictions show rapid increase with increasing speed to the upper limit at a speed of 40 rev/s. Viscosity is virtually at the minimum allowable in all cases.  The trends change significantly however, when β1 is increased to 1/35 in figure 5.22, making minimization of power loss (F1) more important (figure 5.22a). Supply pressure increases rapidly to the upper limit as before, however higher clearance ratio values are predicted. Length to diameter values shift lower while viscosity remains at the lower values. Significant changes occur with β1=1 (figure 5.23) as supply pressure increases less rapidly, viscosity remains at the  118 lower limit, clearance ratio is initially high and reduces, while length to diameter ratio λ reduces further to values in the region of 0.65 at higher speeds.   Figure 5.21 f(X) as function of α1/α2, scaling parameters: β1=1, β2=105       Objective Function Values for Different Scaling Factors 0 500 1000 1500 2000 2500 3000 0 20 40 60 80 100 120 140 160 180 200 Speed, rps O bj . F un ct io n f(x ) α1/α2=5 α1/α2=1 α1/α2=1/5  119  (a)   (b)   (c)   (d)    (e)  Figure 5.22 Model predictions with α1/α2=1/5, β1=1/400, β2=105   Objective function F(X), F 1 , F 2  for β 1 =1/400 0 5 10 15 20 25 30 0 20 40 60 80 100 120 140 160 180 200 Speed, rps O bj ec tiv e fu nc tio n f (X) f_1 f_2 000E+0 50E+3 100E+3 150E+3 200E+3 250E+3 300E+3 350E+3 400E+3 0 20 40 60 80 100 120 140 160 180 200 p s , k Pa Speed, rps Supply pressure ps variation p_s Variation of clearance ratio C 30E-6 32E-6 34E-6 36E-6 38E-6 40E-6 42E-6 0 20 40 60 80 100 120 140 160 180 200 Speed, rps R ad ia l C le ar an ce  C , m C Viscosity variation μ 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0 20 40 60 80 100 120 140 160 180 200 Speed, rps Vi sc os ity  μ,  (P a. s) μ Length to diameter ratio λ 0.9 1 1.1 0 20 40 60 80 100 120 140 160 180 200 Speed, rps λ λ  120  (a)  (b) (c) (d)  (e)  Figure 5.23 Model predictions with α1/α2=1/5, β1=1/35, β2=105 Objective function F(X) , F 1 and F 2 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 160 180 200 Speed, rps O bj ec tiv e fu nc tio n f f(X) f_1 f_2 Variation of supply pressure p s 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 120 140 160 180 200 Speed, rps Su pp ly  p re ss . p s , k Pa p_s Variation of clearance ratio C 39E-6 40E-6 41E-6 42E-6 43E-6 44E-6 45E-6 46E-6 47E-6 48E-6 0 20 40 60 80 100 120 140 160 180 200 Speed, rps R ad ia l C le ar an ce  C , m C Variation of inlet viscosity μ 0 0.002 0.004 0.006 0.008 0.01 0.012 0 20 40 60 80 100 120 140 160 180 200 Speed, rps Vi sc os ity  μ,  (P a. s) μ Variation of λ: Hybrid 0.8 0.85 0.9 0.95 1 1.05 0 20 40 60 80 100 120 140 160 180 200 Speed, rps λ λ  121  (a)   (b) (c)  (d)  (e)  Figure 5.24 Model predictions with α1/α2=1/5, β1=1, β2=105   Objective functions F(X), F 1, F 2 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 20 40 60 80 100 120 140 160 180 200 Speed, rps O bj . f un ct io n f (X) f_1 f_2 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 120 140 160 180 200 p s , k P a Speed, rps Variation of supply pressure ps p_s 4.0E-05 4.5E-05 5.0E-05 5.5E-05 6.0E-05 6.5E-05 7.0E-05 7.5E-05 8.0E-05 8.5E-05 0 20 40 60 80 100 120 140 160 180 200 C , m Speed, rps Variation of clearance ratio C C Variation of inlet viscosity μ 0 0.002 0.004 0.006 0.008 0.01 0.012 0 20 40 60 80 100 120 140 160 180 200 Speed, rps Vi sc os ity  μ,  (P a. s) μ Length to diameter ratio λ 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 180 200 Speed, rps λ λ  122 5.4.3 Pareto Optimal Front Method The method of Pareto diagrams eliminates the need for scaling and weighting factors [40,42]. Pareto fronts for a load of W=10 kN are developed. The two objective functions are plotted on the axes and a series of non-inferior points plotted. These charts can be very effective in helping designers choose design variable sets depending on the relative importance they place on each objective function. Pareto optimal fronts at common speeds for these bearings (1500 rev/min (25 rev/s), 2000 (33.3), 3000 (50) and 4000 (66.7)) are shown in figures 5.25 and 5.26. Included in each chart is a boxed value for the recommended operating point. The design variable set for this point is also shown. The power loss range is small in these plots because the Pareto fronts are developed by changing the supply pressure and determine its effect on both objective functions. The Pareto optimal fronts all satisfy the condition in equation 5.12. The Pareto table at 4000 rpm is shown in Table 5.6.    123  (a)   (b)  Figure 5.25 Pareto optimal front for speeds of (a)1500 rpm and (b) 2000 rpm.  Pareto Optimal Front, N s =25 rps 25.70E-6 25.80E-6 25.90E-6 26.00E-6 26.10E-6 26.20E-6 26.30E-6 26.40E-6 26.50E-6 26.60E-6 26.70E-6 418.5 419 419.5 420 420.5 Power Loss, W Le ak ag e m 3 / s {6.94E-5,0.734,0.01,75E3}  Pareto Optimal Front, Ns=33.3 rps 30.60E-6 30.80E-6 31.00E-6 31.20E-6 31.40E-6 31.60E-6 31.80E-6 583.6 584.1 584.6 Power Loss, W Le ak ag e,  m 3 / s {65.2E-6,0.697,0.01,107E3}  124  (a)   (b)  Figure 5.26 Pareto optimal front for (a) 3000 rpm and (b) 4000 rpm. Power loss in Watts.      Pareto Optimal Front, N=50 rps 41.80E-6 41.90E-6 42.00E-6 42.10E-6 42.20E-6 42.30E-6 42.40E-6 931.7 931.85 932 932.15 932.3 932.45 Power Loss, W Le ak ag e,  m 3 /s {6.11E-5,0.664,0.01,184E3}  Pareto Optimal Front, Ns=66.7 rps 53.20E-6 53.30E-6 53.40E-6 53.50E-6 53.60E-6 53.70E-6 53.80E-6 53.90E-6 54.00E-6 54.10E-6 54.20E-6 1257.3 1257.45 1257.6 1257.75 1257.9 1258.05 Power Loss, W Le ak ag e,  m 3 /s {59.9E-6,0.652,0.01,250E3}  125 Table 5.6 Typical Pareto Table for Bearing at 4000 rpm, W=10kN  Power Loss PL, Watts Leakage QL (m3/s) Clearance C (μm) Length to Diameter ratio λ Supply pressure ps 1257.354 54.138E-6 60.050 0.65197 140.0E+3 1257.354 54.051E-6 60.051 0.65205 150.0E+3 1257.381 53.950E-6 60.031 0.65201 160.0E+3 1257.419 53.801E-6 60.002 0.65203 179.1E+3 1257.420 53.796E-6 60.002 0.65203 180.0E+3 1257.470 53.670E-6 59.965 0.65200 200.0E+3 1257.540 53.546E-6 59.913 0.65195 225.0E+3 1257.560 53.516E-6 59.898 0.65192 235.0E+3 1257.607 53.464E-6 59.864 0.65188 250.0E+3 1257.652 53.421E-6 59.830 0.65182 260.0E+3 1257.687 53.394E-6 59.805 0.65178 270.0E+3 1257.721 53.371E-6 59.780 0.65173 279.5E+3 1257.776 53.339E-6 59.740 0.65166 293.9E+3 1257.815 53.323E-6 59.711 0.65162 305.0E+3 1257.877 53.298E-6 59.666 0.65151 320.0E+3 1257.916 53.288E-6 59.638 0.65146 330.0E+3    5.5  Extended Optimization Model 5.5.1 Leakage Flow Equation and Validation The flow equation presented by Zengeya and Gadala [76] and detailed in section 5.3 assumes flow is a function of aspect ratio L/D and eccentricity ratio ε, (QL=f(λ,ε)) only; it does not explicitly consider the effect of supply pressure and oil groove geometry on leakage flow rate. The method proposed by Martin [70] considers leakage flow as the sum of a hydrodynamic pressure-induced and velocity-induced components. The drawback of Martin’s work is in the many considerations of bearing inlet conditions (oil hole/oil groove, film thickness at inlet port hg, and parameter fg which is a function of the oil groove or hole dimensions). This justifies the need for formulating a simplified leakage flow model with less complex considerations while maintaining accurate predictions. The  126 work presented by Zengeya and Gadala [76] is used as a base and expanded here to take the effect of groove geometry Lg/L, groove location θ, groove length to diameter ratio Lg/D into account, i.e., ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛= L L D L pFQ ggsL ,,,,, θλε . Based on extensive numerical simulations using the streamline upwind Petrov-Galerkin finite element method, the following equation for leakage flow rate is proposed:  ( )( ) ( )[ ]ελθ ln05.15.02.0exp15.13.0117.016.11 5.1 +−⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +++= L L D L pQ ggsL     (5.16) where sp , the non-dimensional supply pressure 2 2 ⎟⎠ ⎞⎜⎝ ⎛= R C N pp s s s πμ is determined using the fluid pressure ps, the fluid viscosity μ, the journal speed Ns (rev/s), and the radial clearance C. Other parameters in equation 5.16 include the length to diameter ratio λ = L/D, the groove length Lg, and the groove location θ.  It should be noted that θ can be positive or negative as shown in figure 4.1a; it varies between ±π/2. The exponential part of the equation is adopted from the work of reference 76.  Depending on the nature of the bearing set-up, appropriate terms in equation 5.16 may be ignored. Justification for the choice/form of sp and θ  (linear) in equations 5.16 follows. Figures 5.27 to 5.31 show the linear relationship between leakage flow and supply pressure, groove location and eccentricity ratio for some bearing data. Figure 5.27a illustrates the effect of supply pressure on flow rate for Claro’s data [79] at eccentricity ε=0.7, speed Ns=8.6 rev/s and supply pressure variation from 8 to 60 kPa. Maximum error of 17 per cent from experimental data is better than 25 per cent reported in Claro’s work (error=(predicted-experimental)/experimental value)x100). Standard error bars also  127 reveal two data points outside the 95% confidence level. Figure 5.27b shows model predictions on Costa’s data [69] at speed Ns=50 rev/s, eccentricity ε=0.5 and supply pressure values ranging from 70 to 300 kPa. Maximum error is 19 per cent with two points outside the 95% confidence level. Figure 5.28 presents flow as a function of speed at two supply pressures for Dowson’s data [80,81]. Predictions from the model show a maximum error of 14 per cent while Dowson’s is 21. Both models fall outside the 95% confidence level for the experimental data at ps=170kPa. Figure 5.29 shows the effect of both supply pressure and eccentricity ratio on leakage flow for Dowson [80] whose model predictions are included for comparison. Both models are accurate at zero pressure; however the model developed here shows better prediction of the data as indicated by determining the variance of each in table 5.7. The goodness of fit, however, is poor at ݌ҧ௦ ൌ 1.0. The accuracy of the model at two supply pressures is tested in figure 5.30 while figure 5.31 presents non-dimensional leakage flow as a function of eccentricity ratio from Boncompain [15] and Ferron [75]. Two other predictions are included for comparison. Error bars in figure 31a indicate all models fail to accurately predict higher eccentricity values.         128  (a)  (b) Figure 5.27 (a) Leakage flow prediction and data as a function of supply pressure for Claro’s data [79]. Groove at +90° to the load line. Other bearing parameters as in Table D2. (b) Data from Costa et al. [69]. Groove location: +30°  0 0.1 0.2 0.3 0.4 0.5 0.6 0 10 20 30 40 50 60 70 N on -d im . le ak ag e Supply pressure, kPa Measured vs Predicted Leakage, Claro et al., θ=+90o Measured Model LQ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.05 0.1 0.15 0.2 N on -d im . le ak ag e Non-dim supply presure Predicted vs Measure values, Costa ( =0.03,0.04,0.09,0.13,0.19 kPa, θ=+30) Model Measured LQ sp sp  129  Figure 5.28 Leakage flow as function of supply pressure, Costa [69]. Ns=50 rpm, ε=0.5.    Figure 5.29 Non-dimensional leakage flow as a function of eccentricity ratio for Boncompain [15] and Ferron [75]. 0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05 2.50E-05 0 5 10 15 20 Le ak ag e flo w  Q L, m 3 /s Speed, rev/s Predicted, Measured, and Dowson Leakage Data Measured:ps=170kPa Q_LDowson:ps=170kPa QMeasured:ps=0 Q_LModel:ps=0 Q_LDowson:ps=0 Q_LModel:ps=170kPa Q_LKhonsari:ps=0 Q_LKhonsari:ps=170kPa 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 No n- di m en si on al flo w  Eccentricity ratio ε Non-dimensional Flow: Predicted vs Measured Values Measured:ps=0 Qmodel:ps=0 Measured@ps=0.5 Qmodel:ps=0.5 Mesured@ps=1 Qmodel:ps=1 Dowson:ps=0 Dowson:ps=0.5 Dowson:ps=1  130 Table 5.7 Comparison of variance between model and Dowson’s predictions.    Variance   Model Dowson 0 0.0022 0.0015 0.5 0.0065 0.0157 1.0 0.0121 0.0176   Figure 5.30 Leakage flow as a function of speed ps=0 and 170 kPa, Dowson’s [80].            Leakage Flow Vs Speed, Dowson/Taylor 0.0E+00 1.0E-05 2.0E-05 3.0E-05 4.0E-05 5.0E-05 6.0E-05 0 20 40 60 80 100 120 140 160 180 200 Speed, rps Le ak ag e flo w , m 3 /s ps170 ps=0 Model predictions (measured data in brackets) ps=0: QL,ns=13.3=5.4x10-6 (5.2x10-6) m3/s ps=170kPa: QL,ns=13.3=22.1x10-6(21x10-6) m3/s sp  131  (a)   (b)   (c)    (d) Figure 5.31 Non-dimensional leakage flow LQ  as a function of eccentricity ratio for Boncompain [15] and 75 [75].   5.5.2 Power Loss Equation and Validation Power loss determination follows a similar procedure presented in section 5.4. The coefficient of friction parameter (R/C)f in reference 76 is expanded to include effects of 0 0.2 0.4 0.6 0.8 1 1.2 0.30 0.40 0.50 0.60 0.70 0.80 0.90 N on -d im en si on al  fl ow Eccentricity ε Leakage Flow vs Eccentricty, 1500 rpm Measured Martin Khonsari Model LQ 0.0 0.2 0.4 0.6 0.8 1.0 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 N o n- d im en si o na l f lo w Eccentricity ε Leakage Flow vs Eccentricity, 2000 rpm model khonsari Martin Measured LQ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.30 0.40 0.50 0.60 0.70 0.80 N on -d im en si on al  f lo w Eccentricity ε Leakage Flow vs Eccentricity, 3000 rpm model Exp. Data khonsari martin LQ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.20 0.30 0.40 0.50 0.60 0.70 N on -d im en si on al  fl ow Eccentricity ε Leakage Flow vs Eccentricity, 4000 rpm model khonsari martin Exp. data LQ  132 supply pressure and groove position θ  and with the aid of extensive numerical simulations and curve-fitting techniques:    ( )( ) ⎥⎦ ⎤⎢⎣ ⎡ ⎟⎟⎠ ⎞⎜⎜⎝ ⎛ −+++−= ελθθ 97838.4 73622.29994.1exp6.0105.025.01 2 spfC R     (5.17) where f is the coefficient of friction and ε   is the eccentricity ratio, ε = e/C.  The above equation introduces the effects of supply pressure sp  and groove position θ in an explicit way. It will become evident that the groove position plays a significant role in calculating the friction and the subsequent torque and power values from validation simulations. The effect of supply pressure, although more significant at low eccentricity ratios [80], is included in both equations 5.16 and 5.17. The quadratic nature of coefficient of friction as a function of groove location in equation 5.17 is evident in figure 5.32 for Costa’s data speed Ns=50 rev/s, eccentricity ε=0.21 and 0.5 (load W is 2 and 8 kN respectively). The magnitude of each coefficient is determined empirically and averaged over the bearing data. Maximum error is 9.4 per cent. Variation of torque as a function the non-dimensional supply pressure is illustrated in figure 5.33, Costa [69]. Supply pressure varies from 50 kPa to 300 kPa (non-dimensional values from 0.04 – 0.23) at two different bearing loads (2 kN, ε=0.2 and 8 kN, ε=0.5) which is in the light to medium bearing load range [80]. Maximum error is 14 per cent. Figure 5.34 confirms the accuracy of the model in predicting Costa’s data at three speed values (Ns=33.3, 50 and 66.7 rev/s, θ=+30°) and eccentricity of 0.5. Values in brackets are experimental.     133  Figure 5.32 Torque predictions as a function of groove position, Costa [69]. Supply pressure of 70 kPa.   Figure 5.33 Torque predictions as a function of non-dimensional supply pressure, Costa et al. [69]. 2.00 2.50 3.00 3.50 4.00 4.50 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 To rq ue , N .m Groove position, radians Predicted Vs Measured Torque: ps=70kPa TvTheta:Model_W=2kN Measured@2kN TvTheta:Mode_W=8kN Measured@8kN Data Measured at 2kN: 2.7 N.m Predicted at 2kN: 2.9 N.m (9.4%) Measured at 8kN: 3.7 N.m Predicted at 8kN: 4.0 N.m (6.8%) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 0 0.05 0.1 0.15 0.2 0.25 To rq ue , N .m Non-dimensional supply pressure Predicted vs measured torque values, θ=-30o Model@2kN Measured@2kN Model@8kN Measured@8kN  134      Figure 5.34 Variation of torque as function of speed, Costa [69]. Three data points are shown.   5.5.3 Optimization Simulations The program structure follows the steps in Appendix C and is implemented in Matlab [51] with input data in tables D5 (Dowson) and D6. The two objective functions are chosen to be: F1 the non-dimensional friction parameter ((R/C)f) and F2 the leakage flow (QL). A weighting factor β2 of 1x105 is used to make the two objective functions of the same order of magnitude while α1 and α2 are both set to 1. Figure 5.35 shows the variation of design variables ( )θμλ ,,,,XT spC=  as a function of speed. Optimum radial clearance increases with speed to a peak at approximately 30 rev/s before  decreasing  and  leveling off close to 60 μm. It is interesting to compare this value 0.00 1.00 2.00 3.00 4.00 5.00 6.00 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 To rq ue , N .m Speed, rev/s Torque T as function of speed, θ=+30o, ps=70kPa Exp. Data T Measured data (Predictions in brackets): TNs=33.3=3.1 N.m (3.08) TNs=50= 3.6 N.m (3.66) TNs=66.7=4.05 N.m (4.02)  135 (a) (b) (c)  (d)  (e) Figure 5.35 Variation of XT as a function of speed, W=10kN. Model predictions with α1/α2=1, β1=1, β2=105  Optimum radial clearance C 0.0E+00 2.0E-05 4.0E-05 6.0E-05 8.0E-05 1.0E-04 1.2E-04 1.4E-04 0 20 40 60 80 100 120 140 160 180 200 Speed, rps R ad ia l c le ar an ce  C C Optimum L/D λ ratios 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 180 200 Speed, rpm L/ D  ra tio  λ λ Optimum viscosity μ 0 0.005 0.01 0.015 0 20 40 60 80 100 120 140 160 180 200 Speed, rps V is ci si ty , P a. s μ Optimum supply pressure p_s 0 10 20 30 40 50 60 0 20 40 60 80 100 120 140 160 180 200 Speed, rps Su pp ly  p re ss ur e,  k Pa p_s Optimum groove position θ -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0 20 40 60 80 100 120 140 160 180 200 Speed, rps G ro ov e po si tio n θ θ  136 with 138.8µm used by Dowson [80]. The predicted value is obtained with groove position θ=-π/2. Their data shows bearing speeds up to 16.7 rev/s. Another interesting observation is evident when figure 5.35a is compared to figure 11c in reference 76 where the groove is positioned at the load line. The predicted peak clearance of 120 μm at 15 rev/s in figure 5.35a is significantly higher than a peak of 79 μm at 10 rev/s in figure 11c, reference 76. Corresponding C values at each speed are generally higher in figure 5.35a, suggesting optimum groove location allows for higher optimum clearance ratios. Length to diameter ratio values (figure 5.35b) decrease from a high of 0.94 and levels off at 0.67. This pattern is similar for both bearings simulated and is also comparable to figure 11c, reference 76. Optimum viscosity is at the lower bound as expected. This is because lower viscosity lowers power loss as well as increases leakage for the same set of bearing input variables. Similar findings are reported by other researchers [34,36]. Supply pressure is predicted to be at the lower bound for all speeds. This is significantly different from predictions in figure 11b, reference 76 where it increases from the lower to upper bounds. Groove location appears to have a significant effect on supply pressure. Lower values can be used when optimum location is used in journal bearing design. Groove location prediction is interesting in that the model indicates optimum placing of the groove at the lower bound (-π/2), figure 5.35e. Although this position has the least leakage flow, it has significant power loss minimization. Obviously no consideration of whether this position is in the rupture zone and what effect that it has on mechanical stresses is taken. For convenience most journal bearings are designed with groove  137 location in line with the load. The other potential beneficial position, although not shown in the figures, is at the upper bound (+π/2) which gives slightly higher F(X) values. Figure 5.36 shows variation of the design variables as a function of speed for a load of W=700N.  The results are similar to those in figure 5.35 although peak C value of 107 μm occurs at higher speed (25 rev/s). Length to diameter ratio levels off at a lower value of 0.45 in figure 5.36b compared to 0.67 in figure 5.35b. Pareto charts present an alternative method of showing optimum design points. A typical pareto optimal front for load W=10kN and speed 50 rev/s is shown in figure 5.37a. In figure 5.37b the load is 700N and speed 25 rev/s. Two finite element predictions with the boxed XT values are included, showing the accuracy of the optimization model. The graph shows significant power loss variation. Pareto charts present a set of non- dominated solutions, thus each point on the front has its own merits. A wide range of these charts can be produced at small speed increments. Designers can then make use of these charts in design decisions. An interesting observation is in the power loss values predicted in figure 5.37a (740W) compared to figure 13a (932W), reference 76 for similar input values. Significant power loss savings can be realized with proper groove location.        138  (a) (b)  (c) (d)  (e)  Figure 5.36 Variation of XT as a function of speed, load W=700N. Optim um  radial clearance C as  function of speed 0.0E+00 2.0E-05 4.0E-05 6.0E-05 8.0E-05 1.0E-04 1.2E-04 0 20 40 60 80 100 120 140 160 180 200 Speed, rps R ad ia l c le ar an ce  C , m C Variation of L/D ratio λ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 20 40 60 80 100 120 140 160 180 200 Speed, rps λ λ  Optimum viscosity μ  as function of speed 0 0.002 0.004 0.006 0.008 0.01 0.012 0 20 40 60 80 100 120 140 160 180 200 Speed, rps Vi sc os ity , P a. s μ Optim um  supply pressure ps 0 10 20 30 40 50 60 0 20 40 60 80 100 120 140 160 180 200 Speed, rps S up pl y pr es su re . k P a ps Optimum groove location θ -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0 20 40 60 80 100 120 140 160 180 200 Speed, rps G ro ov e lo ca tio n,  ra di an s θ  139  (a)   (b)  Figure 5.37(a) Pareto optimal front for speeds of 1500 rpm, W=10kN; (b) Pareto optimal front, W=700N Pareto front, W= 10kN, N s =50 rev/s 3.00E-05 3.10E-05 3.20E-05 3.30E-05 3.40E-05 3.50E-05 3.60E-05 3.70E-05 710 720 730 740 750 760 770 780 Power Loss, W Le ak ag e flo w , m 3 /s Pfront FE 63.3e-6,0.818,01,50e3,-1.571 68.1e-6,0.699,01,50e3,-1.571 Pareto Chart, Speed=1500 rpm, Load=700N 0.0E+00 2.0E-06 4.0E-06 6.0E-06 8.0E-06 1.0E-05 1.2E-05 35 45 55 65 75 85 Power Loss, W Le ak ag e Fl ow , m 3 / s Ns=1500rpm FE 1.16e-4,0.728,01,50e3,-1.571 4.72e-5,0.47,01,50e3,-1.571  140 5.6 Interpretation of Optimization Results A brief overview of the optimization results is appropriate here. Results from the four variable model are analyzed in more detail and compared to five variable ones.  5.6.1 Pareto Optimal Fronts with X܂ ൌ ሺܥ, ߣ, ߤ, ݌௦ሻ and X܂ ൌ ሺܥ, ߣ, ߤ, ݌௦, ߠሻ The Pareto charts in figure 5.25 show a variation of power PL and leakage flow QL of less than one and four percent respectively. The power loss variation is small, and in engineering terms might be interpreted to mean other factors have more influence in determining bearing performance. Noting that for this set the load is in line with the oil groove, it is clear this position limits the extent to which power loss and side leakage can be influenced. The situation is however significantly different when groove location θ is included in the design set. As shown figure 5.37a power loss variation is 8% while leakage flow varies 23% which gives a significant influence on the two performance parameters. Table 5.7 summarizes the results from the two sets. Further analysis of the results appear in reference 76.  Table 5.8 Variation of Performance Parameters PL and QL  Characteristic Variation % X܂ ൌ ሺܥ, ߣ, ߤ, ݌௦ሻ Power Loss 0.2  Leakage flow 1 X܂ ൌ ሺܥ, ߣ, ߤ, ݌௦, ߠሻ Power Loss 8   Leakage flow 23      141 5.6.2 Uncertainty in the Results The optimization model is based on fundamental engineering principles of lubrication flow and temperature increase as proposed by the respected research teams Jang and Khonsari [55] and Khonsari et al. [74]. There is no reason to doubt the validity of the temperature rise parameters in reference 76 or the form of the flow equation in reference 76. What might be questioned is the accuracy of the data used. Dowson’s data [59] is regarded as a benchmark in lubrication circles while Costa et al. [69] present a wide variety of performance data which is also well taken in the industry. Unfortunately both sets of researchers do not present any error analysis with their data. The finite element simulations presented in earlier chapters (and Appendix E) were shown to be accurate in predicting experimental data from Boncompain [15], Ferron [75], Costa [69] and others [72,73,79,80,81]. Prediction results were compared to available models and shown to be at least as good or better [73].  5.7 Chapter Summary An Empirically Derived Optimization Model (EDOM) is proposed for length to diameter ratio 0.4≤λ≤1.0. Both minimization of the temperature increase ΔT and leakage QL as well as power loss PL and leakage QL are analyzed. The objective function minimizing power loss PL and side flow QL is preferable as it minimizes two independent functions. The proposed approach can be used by design engineers who need simple and effective models to help them choose optimum design variables using readily available optimization programs such as Matlab’s Optimization Toolbox. Such an approach presents advantages over more complex methods such as Hirani’s model in that no complex integration techniques are required.  142 Model predictions for temperature increase, leakage flow, and power loss are accurate in comparison to models in the literature. The equations are empirically derived and make the need for complex integrations unnecessary. The method of weighting functions/scaling parameters as well as Pareto optimal fronts are used. The benefits of each method are assessed. Pareto optimal diagrams eliminate the need to determine the weights and scaling required in the first method and should be used more often in journal bearing design. The model is extended to include groove position and supply pressure computations explicitly by introducing terms width to diameter ratio (wg/D) and groove length to bearing length ratio (Lg/L).  Maximum error of 15% on leakage is observed from the proposed model.  The model is comprehensive yet simple to implement, and depending on the nature of the problem some terms in the equations may be ignored.  Power loss and torque predictions are within 10% of experimental data.  These results show the effectiveness of the model and the ease of use in bearing optimization problems. Model results show that groove location has a significant effect on the optimization of the bearing. The model predicts groove location should be at the lower bound (-π/2).  Higher values of clearance ratio C may be used when optimum groove position is chosen. Significant power loss savings may be realized with appropriate location of oil groove.       143 Chapter 6 Summary, Conclusions and Future Work  6.1 Summary and Conclusions Journal bearing design and optimization requires accurate and reliable prediction of thermal fields due to viscous dissipation. The effect of temperature increase on load carrying capacity of the bearing can be dramatic due to a decrease in the viscosity of the film. Several thermo-hydrodynamic (THD) models in the literature focus on formulating a robust, rapid converging THD model that can be used in optimization where many iterations at varying speeds are required. This requirement presents one of the main challenges for researchers and is the focus of this work. A robust and effective FEA procedure is proposed and a novel and simple model for journal bearing is introduced. Applications of the developed models to the analysis and optimization of practical cases are discussed. Main outcomes from the work are summarized in the following. 1. Rapid Thermal Predictions: Streamline Upwind Petrov-Galerkin FE Method The first goal was to formulate a THD model in a robust, efficient and as general as possible form. The streamline upwind Petrov-Galerkin finite element (SUPG-FE) method for three dimensional THD lubrication modeling is developed using Cartesian coordinates. It is shown to be effective in slider as well as journal bearings with some minor modifications. In summary the streamline upwind Petrov-Galerkin FE method can be effectively used to model journal bearings using Cartesian coordinates. The model can be refined, especially the boundary conditions in the groove area, to give more accurate  144 results. The same model can also be used for slider bearings with less implementation requirements. A more detailed model procedure is given in sections 2.4 and 3.3. 2. Slider Bearing Thermal Predictions Some important observations on slider bearings are noted. An aspect ratio between 2 and 1 is ideal in terms of leakage flow and minimum peak temperatures. A unique way of identifying this region is illustrated where the differential of the flow Q∂ as a function of aspect ratio )/( BL Q ∂ ∂  reduces rapidly to minimum before increasing again and becoming constant at BL /  ratios higher than 10. The above findings along with the flow gradient method (FGM), uniquely defined in this work, helps optimize slider bearing tilt ratio for minimum temperature rise. 3. Development of Journal Bearing Template A template for modeling THD lubrication in journal bearings is shown to be effective in modeling thermal effects, leakage flow, and cavitation effects. The model as presented has no special consideration of thermal boundary conditions at the rupture interface. Although the approach proposed by Boncompain [15] is used in the rupture zone, focus on the dynamics of velocity gradients ∂v/∂y and ∂w/∂y near the rupture and reformation interfaces are critically analyzed and appear to explain the rapid decline in temperature. Thermal predictions are within ten percent of measured values. The model is robust and converges in two or three iterations making it ideal for iterative applications such as in validating optimization equations developed in the second part of the research. 4. Empirically Derived Optimization Model (EDOM) A novel set of optimization equations derived from first principles is proposed. The set of equations require no complex integrations and can be easily implemented using standard  145 optimization tools such as Matlab. Optimization equations for temperature increase ΔT, power loss PL, and leakage flow QL are developed; constants and other values in the equations are determined empirically for the range of bearings under consideration. The set of equations are well-designed and easy to use while giving accurate results. Comparisons with models in literature show that the equations consistently give better predictions. The approach used by Jang and Khonsari [55] whereby temperature rise parameters are derived and empirically verified is expanded on and used to formulate temperature increase ΔT(X) model. Such an approach is used for the first time in journal bearing optimization. It presents an alternative to Hirani and Suh’s model [38] which requires a finite difference mass conserving algorithm to determine temperature increase and integration techniques to determine  leakage flow and power loss. Once empirical values in the temperature increase ΔT(X), leakage QL, and power loss PL are determined, the optimization model can be used for the range of bearings under consideration to find optimum values of the design variables (X). This work presents the developed equations as an optimization model. 5. Optimization using Weight/Scaling Parameters and Pareto Optimal Fronts The developed set of equations are used to minimize a multiobjective function optimization model using the weight/scaling factors and Pareto optimal fronts and results are compared to Hashimoto’s [34]. It is clear Hashimoto’s model is for short bearings with L/D ratios up to 0.5 and is not accurate for medium range bearings (0.4<L/D<1.0) under consideration here. In the Pareto optimal front approach, posterior articulation of the weights is used whereby the designer initially generates a number of non-inferior (a set of equally efficient)  146 solutions from which a final decision is made on any one solution. Advantages and disadvantages of each method are given, however Pareto charts appear to be preferable as they give a set of non-inferior, non-dominated solutions and final choice depends on the merits of a particular design.  6.2 Recommendations for Future Work 1. Journal Bearing Template The developed SUPG FE model, although effective in predicting thermal effects, should be extended to thermo-elastohydrodynamic (TEHD) lubrication where expansion of the shaft and bush are taken into account. This will give a better prediction of the load carrying capacity of the bearing. Prediction of the clearance ratio C could be adjusted accordingly. The journal bearing template developed here could be tested on transient phenomenon such as the squeeze effect and thermal predictions during start-up and shut down. This is an important area as it is also linked to the study of bearing time to seizure and how this can be avoided. 2. Journal Bearing System Optimum Design The model presented here focuses on individual journal bearings. A possible expansion of the work might involve looking at the ‘system’ of a shaft and two journal bearings at the end. Such a task would the overcome the qualifying statement in section 4.3.5 that other factors may influence the choice of design variable set (X) such as the shaft stiffness and misalignment problems that may be encountered.    147 3. Expansion of the Empirically Derived Optimization Model The optimization model developed here works well for medium range journal bearings. An extension of the model could be developed for short and long range bearings. This would be very useful to design engineers who use bearings in the two ranges and also make the EDOM applicable to the full range of aspect ratios.                     148 Bibliography  1. Khonsari, M.M., Wang, S.H., “Note on transient thermo-hydrodynamic   lubrication effects in lubricating films”, STLE Tribol. Trans., 35,   pp. 177-183, 1992. 2. Pinkus, O., Thermal Aspects of fluid film tribology, ASME Press, New York, 1990. 3. Ettles, C.M.M., Cameron, A., “Considerations of oil flow across a bearing groove”, Trans. ASME J. Lub. 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Claro, J.C.P., and Miranda, A.A.S., “Analysis of hydrodynamic journal bearings considering lubricant supply conditions”, IMechE Part C:  J. Mech. Eng. Sci., 207, 93-101, 1993. 80. Dowson, D., Taylor, C.M., Miranda, A.A.S., “The prediction of liquid film journal bearing performance with a consideration of lubricant film reformation Part 1: theoretical results”, Proc. IMechE J. Eng. Trib., 199(C2), 95-102, 1985. 81. Dowson, D., Taylor, C.M., Miranda, A.A.S. “The prediction of liquid film journal bearing performance with a consideration of lubricant film reformation Part 2: experimental results”, Proc. IMechE J. Eng. Trib., 199(C2), 103-111, 1985.  157 APPENDICES APPENDIX A A.1   Velocity fields u, v, and w The equilibrium condition for forces acting in the x-direction gives      0=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ∂ ∂−∂ ∂ y u yx p μ      (A-1) where u is the velocity of the lubricant in the direction of sliding. The boundary conditions are u=0 at y=h and u=U at y=0. For the one-dimension bearing considered here the value of the lubricant flow is constant along the bearing assuming no side leakage; this fact is used to calculate v. Algebraically this means that the lubricant velocity across the film thickness is constant, that is ( ) 0 0 =∂∂ ∫hudyx . Equation A-1 can be integrated in the y-direction to get v, the component of u in that direction .     ∫ ∫ ++= y y UdyAdyydxdpyu 0 0 2)( μμ      (A-2) where A is given by  ( ) ∫ ∫ −⎟⎟⎠⎞⎜⎜⎝⎛−= h h dy Udyydxdp A 0 2 0 μ μ     (A-3) and U the slider velocity in m/s. Non-dimensional value becomes ݑതଶ ൌ ݑଶ/ܷଶ. Similar equilibrium condition for forces acting in the y-direction is      0=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ∂ ∂−∂ ∂ y w yz p μ     (A-4) with boundary conditions w=0 at y=0 and w=0 at y=h. Equation A-4 is integrated with respect to y to give  158    ( ) ( ) ( )( ) ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ −= ∫ ∫ ∫∫y h hy dy dyydy dyy dz dpw 0 0 00 μ μμμ    (A-5) Since pressure is constant through the film thickness, 0=∂∂ yp  and the continuity of flow requires that      0=∂ ∂+∂ ∂+∂ ∂ z w y v x u     (A-6) The continuity equation is more conveniently solved by differentiating with respect to y giving                  0 2 2 =⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂+∂ ∂ ∂ ∂+∂ ∂ z w x u yy v     (A-7) with boundary conditions u=w=0 at y=h and u=U, w=0  at y=0               159 APPENDIX B The term (1+εcosϕ)3 in equation 5.12, referred to as (hg/C)3 in Martin’s model, relates to the case where the oil groove is line with the load. The same term applies for an inlet hole of diameter DH in line with the load but fg in this case becomes 75.1 4.0675.0 ⎟⎠ ⎞⎜⎝ ⎛ += L Df Hg      0.1<DH<0.25   (B.1) and '1' , S p S mtotL QQQ −=          (B.2) where S’=0.6.    Another important case is when the oil groove is at the maximum film thickness position hmax where (hg/C)3 becomes ( )3 3 1 ε+=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ C hg          (B.3) equation B1 for fg is valid, and total leakage becomes m g mtotL L L QQ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛=,    0.3<Lg/L<0.8     (B.4) where 27.0 27.0 ⎟⎠ ⎞⎜⎝ ⎛= p L Q Qm  . The proposed model here uses equation 10 to determine the flow due to film pressure QL while Qp is evaluated using equation 13 with the appropriate value of (hg/C)3.      The power value to eccentricity ratio in equation B4 applies when the hydrodynamic film starts at hmax and L/D<1.0, 0.3≤λ≤0.9.    160 APPENDIX C Optimization Procedure     The procedure presented here will help designers implement the optimization model proposed. This can be easily done in Matlab.  A1. Sequential Quadratic Programming (SQP) This requires three main components namely the main program, objective function, and non-linear constraint m-files. The call to the optimization subroutine ‘fmincon’ is done in the main program while computations in part B are done in the objective function m-file and part C computes the non-linear constraints. Define global variables that will be passed between the three m-files (ΔT, ε, Ns). The three m-files are attached at the end of the appendix. While care was taken in preparing these files, no guarantee is given that the codes are error free. A brief description of the main parts follows. Create a folder and copy the three files. Run the main program which will call the other m-files.   Part A: Main Program  -Define upper and lower limits for design variable ( )sT pCX ,,, μλ=  and provide these as lb and ub in the call to ‘fmincon’  -Provide program with initial guess for design variable x0 -Set options and call ‘fmincon’ (this calls both Part B and C)       161 Part B: Objective Function  -Set all the problem parameters here (since the hybrid technique will use same objective function) For Ns from Ns1 to Ns max Step 1: Compute journal surface velocity U             2: Use μI to compute Sommerfeld number S         3: Find eccentricity ratio ε         4: Compute the temperatures Tmax and Tshaft and temperature increase ΔT         5: Update the viscosity μ         6: Go to step 2 and re-compute S until convergence.         7: Compute the side leakage QL         8: Compute the objective function F(X) given the constraints gi(X)         9: Record F(X) 10. Loop and update Ns  Part C: Non-linear Constraint Computation  -Define minimum allowable hmin, ΔTa,  ωcr ,and pa  -Compute pmax, ω, and ωcr  -Define non-linear constraints g1, g2, g3, and g4   C.2 Matlab M-files Main Program  function [X,FVAL,REASON,OUTPUT,POPULATION,SCORES] =  dowprob1 %%   This is an auto generated M file to do optimization with the Genetic Algorithm and %    Direct Search Toolbox. Use GAOPTIMSET for default GA options structure.  %%Fitness function fitnessFunction = @dowtaylorobj; %%Number of Variables  162 nvars = 5 ; %Linear inequality constraints Aineq = []; Bineq = []; %Linear equality constraints Aeq = []; Beq = []; %Bounds LB = [4e-005             0.4            0.01           50000         - 1.5708 ]; UB = [0.0003                1             0.03           350000 1.5708 ]; %Nonlinear constraints nonlconFunction = []; %Start with default options options = gaoptimset; %%Modify some parameters options = gaoptimset(options,'PopInitRange' ,[-2.5 ; 2.5 ]); options = gaoptimset(options,'PopulationSize' ,100); options = gaoptimset(options,'MutationFcn' ,{ @mutationgaussian 1  1 }); options = gaoptimset(options,'Display' ,'off'); options = gaoptimset(options,'PlotFcns' ,{ @gaplotbestf @gaplotbestindiv }); options = gaoptimset(options,'HybridFcn' ,{ @fminsearch [] }); %%Run GA [X,FVAL,REASON,OUTPUT,POPULATION,SCORES] = ga(fitnessFunction,nvars,Aineq,Bineq,Aeq,Beq,LB,UB,nonlconFunction,opti ons);  Objective Function  function [ret,f1,f2] = objective(x) global ns rho c_p D W e_0 deltaT Q_tot rfric1 p_max p_s f1 f2 a1 b1 c1 omega1 omega_cr tht_mn; global tht_mx p_smin p_smax;      n_s = 3;                 % Journal speed, 40-240 rps     n_smx =50;     rho = 860;               % Density of fluid, kg/m3     c_p = 2.0e3;             % Specific heat of lubricant     mu_i = 0.0358;           % Inlet viscosity of oil, Pa.s     T_i1 = 313;              % Inlet oil temperature, K     T_i = T_i1 - 273;        % Inlet temp in C     k_f = 0.13;              % Thermal conductivity of lubricant, W/(m K)     alpha = 0.756e-7;        % Thermal diffusivity of lubricant, m^2/s     beta = 0.0414;           % Temperature-viscosity coefficient, K^(- 1)     L1 = 0.08;               % Bearing length     L_g1 = 0.07;             % Length of groove     L_Lg1 = L1/L_g1;           % Ratio of groove/bearing length     w_g = 12e-3;           % Width of groove     D = 0.1;              % Journal diameter     W = 20e3;                 % Load on bearing, N     C_mn = 40.0e-6;          % Radial clearance  163     C_mx = 300.0e-6;     lambda_mn = 0.20;        % L/D ratio     lambda_mx = 1.0;     mu_mn = 1.0e-3;          % Viscosity     mu_mx = 4.0e-2;     p_smin = 50e3;           % Supply pressure     p_smax = 350e3;      beta1 = 1;               % Scaling factor     beta2 = 1e5;     alpha1 = 1;              % Weighting factor     alpha2 =alpha1;  % Nonlinear objective function %x0 = [50e-6 0.6 2e-2];      for ns = n_s:n_smx,       % Define the variables       C=x(1);       La=x(2);       mu=x(3);       p_s=x(4);       theta = x(5);       L = La*D; L_g = L_g1; R_C = D/(2*C);       ps_bar = p_s*(1/R_C)^2/(mu*2*pi*ns);       U = ns*2*pi*D/2; Re = rho*C*U/mu;       S = ns*mu*D*L*(D/(2*C))^2/W;       e_0 = -0.36667+0.49737/sqrt(La)-0.20986*log(S);       if (e_0 > 1),           e_0 = 0.6121;  %         return;       end;        % Compute side leakage Q_Leak       Q_leak1 = exp(0.20-0.5*La^1.5+1.05*log(e_0));       Q_bar4 = (1+1.6*ps_bar)*(1+0.17*theta)*(1+0.3*w_g/D)*(1.15*L_g/L)*Q_leak1;       Q_leak4 = Q_bar4*pi*ns*D*C*L;       Q_leak = Q_leak1*pi*ns*D*C*L;       % find leakage due to supply pressure       z = 4*(1 + La*(1 - 1.25*e_0^0.18));       attit = atan((pi/(z*e_0))*sqrt(1-e_0^2));            f_g = (1.25-0.25*(L_g/L))/(3*(L/L_g-1)^(0.33333))+...                 (w_g/D)/(3*La*(1-L_g/L));           h_g = (1 + e_0*cos(attit))^3;       Q_p =f_g*h_g*p_s*C^3/mu_i; % Leakage due to pressure       Qm = Q_leak + Q_p - 0.3*sqrt(Q_leak*Q_p); % Datum flow rate       S_prime = 0.7*(L_g/L)^0.7 + 0.4;       Q_tot = Qm^S_prime*Q_p^(1-S_prime);         % Total leakage        %Compute temperature increase       kappa1 = alpha*mu_i*beta*U*(D/(2*C))^2/(k_f*D/2);       kappa2 = U*sqrt(mu_i*beta/k_f);       % Determine parameters a, b, and c %      if (e_0 > 0.1) && (e_0 < 0.3), %        a1 = 1.58105;  b1 = -0.14250;  c1 = -1.04050; %      end;       if (e_0 >= 0.1) && (e_0 <= 0.7),  164         a1 = 1.5917;   b1 = -0.13962;  c1 = -1.24999;       end;       if (e_0 > 0.7) && (e_0 <= 0.9),         a1 = 1.63973;  b1 = -0.10627;  c1 = -1.45216;       end;        % Compute T_max       T_max1 = exp(a1 + b1/kappa1^0.5 + c1/kappa2^0.5);       T_max = (T_max1/beta + T_i) + 273;       deltaT = T_max - T_i1;        % Compute Power loss       rfrict = exp(1.9994+2.73622/sqrt(La)-4.97838*sqrt(e_0));       rfric1 = (1- 0.25*theta^2+0.23*abs(theta))*(1+0.6*ps_bar)*(rfrict);       torqueM = rfrict/R_C*W*D/2;       torq2 = rfric1/R_C*W*D/2;       P_Loss = torqueM*2*pi*ns;       Ploss2 = torq2*2*pi*ns;       % Recompute mu       x(3) = mu_i*exp(-beta*deltaT);        % Objective function       f1 = alpha1*beta1*Ploss2;        % Friction parameter obj function       f2 =  alpha2*beta2*Q_leak4;       ret = f1 + f2;      end  Non-linear Constraints M-file  function [c, ceq] = nonlin_cons(x) global ns rho c_p D W e_0 deltaT Q_tot rfric1 p_max p_s f1 f2 a1 b1 c1 omega1 omega_cr tht_mn; global tht_mx p_smin p_smax;      h_a = 5e-6;            % Allowable minimum film thickness     p_a = 30e6;             % Allowable maximum pressure, Pa     deltaT_a = 70;          % Allowable temperature rise, K  p_max1 = exp(2.25062-0.39491/x(2)+4.77721*e_0^3); p_max = p_max1*x(3)*(D/2)^2*ns/x(1)^2 + p_s; omega1 = 2*pi*ns; omega_cr = sqrt(9.81/x(1))*(0.0584*exp(6.99*e_0^2.07)-1.318*e_0+2.873);  g7 = h_a -x(1)*(1-e_0); g8 = deltaT - deltaT_a; %g9 = omega1 - omega_cr; g10 = p_max - p_a;  c = [g7 g8 g10]; % nonlinear equality constraints returned as vector ceq  ceq = []; % there are no nonlinear equality                 % constraints in   165 C3. Hybrid Scheme The genetic algorithm toolbox with the Hybrid option was used. The fitness function is the objective function defined in section C1 and non-linear constraints are the same as in C1. Other settings are as shown in Table D5.                     166 APPENDIX D Tables Table D.1  Input data from Dowson [59] and Song [36] bearings (empty cells mean unspecified values).  Input Parameter Dowson Song Bearing Diameter D, m 0.102  0.1 Bearing width W, m 0.07625 - Bush outer diameter, m 0.2 - Journal Speed (Ns=25rev/s), U 8.01 m/s - Load, W 11000  10000 L/D 0.76 - Radial clearance ratio R/C 800 - Groove width 4.76 x 10-3 m - Groove length 0.067 m - Specific heat cf, cb, cs 2000, 380, 490 4190 Conductivities kf, kb, ks 0.13, 50, 65 W/m K - Convective heat transfer coefficient H1, W/m2.K 100 - Film/bush heat transfer coefficient H2, W/m2.K 200 - Inlet temperature Ti 308 K - Inlet lubricant density ρi 867 kg/m3 - Inlet viscosity μi 0.03 Pa.s - Viscosity-temperature coeff. β 0.0414 /K - Density-temperature coeff. δ 0.0012 /K - Thermal diffusivity (αt=ρcp/kf) 0.756 x 10-7 m2/s - Supply pressure ps (gauge) 0.276 x 106 Pa -  D1.1 Location of thermocouples Figure D1 shows the location of thermocouples (TC) on the bush and shaft in Dowson’s experimental setup [59]. The temperature of the lubricant at inlet was recorded by two iron-constantan thermocouples fitted in a radial inlet hole located within the bush. One of the thermocouples was located near the outer radius of the bush while the other was  167 within 1.59 mm of the entry to the oil groove. Four outlet temperature thermocouples were fitted in special oil scoops located at the ends of the bush to collect lubricant flow leaving the bearing. For bush temperature measurements, the thermocouples were mounted in groups of four on bronze plugs which fitted closely into 34 radial holes in the bush. The hot junction of each TC was located on the centerline of each plug and spaced at distances of 1.59, 17.5, 33.3, and 49.2 mm from the dead end of the radial hole adjacent to the oil film. The bronze plugs were located in four planes at the axial locations shown in figure D1a. Shaft temperatures were measured using twelve TC’s mounted in plugs similar to the bush setup. The TC plugs were tightly fitted into radial holes in the shaft with their outer ends flush with the shaft surface. The TC’s were located in four planes disposed about the edge of the bearing as shown in figure D1b to enable temperature gradients to be determined. Wires from the thermocouples ran along a hole drilled in the shaft to a slip- ring assembly that was maintained at a constant temperature by air cooling at the end of the shaft. The shaft surface temperature was recorded by means of a specially constructed thin film platinum resistance thermometer mounted on a glass rod located in a radial hole in the shaft at a distance of 16 mm from the central plane of the bearing. The exposed surface of the glass plug was contoured to conform with the shaft surface. The leads fro the platinum film were carried down the internal bore of the shaft to the slip ring assembly.  Figure D1. Thermocouple location for 168   pressure an  d temperature measurem  ents [59]  169   Table D.2  Input values for the bearing tested by Boncompain et al. [15]  Input Parameter Value Bearing Radius R, m 0.050 Bearing width W, m 0.08 Bush outer radius R2, m 0.100 Journal Speed (Ns=33.3rev/s), U 10.5 m/s Eccentricity, ε 0.8 Radial clearance at 20°C, C, m-3 145µm Groove angle 18° Groove length 0.070 m Groove Location 0° Specific heats cf, cb, cs 2000, 380, 490 Conductivities kf, kb, ks, kg, W/m K 0.13, 250, 50, 0.025 Convective heat transfer coefficient , bush/ambient H1, W/m2.K  100 Convective heat transfer coefficient , bush/oil and shaft/oil interfaces H2, W/m2.K  500 Inlet temperature Ti, K 313 Ambient Temperature To, K 318 Inlet lubricant density ρi 860 kg/m3 Inlet viscosity μi 0.028 Pa.s Viscosity-temperature coeff. β 0.0414 /K Density-temperature coeff. δ 0.0012 /K Supply pressure ps (gauge) 0.070 x 106 Pa           170  Table D.3 Input values for Costa’s bearing [69]  Input Parameter Value Bearing Diameter D, m 0.100 Bearing width W, m 0.08 Bush outer diameter, m 0.2 Journal Speed (Ns=50rev/s), U 15.7 m/s Load, W 8000 N L/D 0.75 Radial clearance C, m-6 93.5 Groove width 16 x 10-3 m Groove length 0.070 m Groove Location 0° Specific heats cf, cb, cs 2000, 380, 490 Conductivities kf, kb, ks 0.13, 50, 65 W/m K Convective heat transfer coefficient H, W/m2.K 100 Inlet temperature Ti 308 K Inlet lubricant density ρa 867 kg/m3 Inlet viscosity μi 0.0358 Pa.s Viscosity-temperature coeff. β 0.0414 /K Density-temperature coeff. δ 0.0012 /K Supply pressure ps (gauge) 0.300 x 106 Pa                      171   Table D.4 Input values for Ferron’s [75] bearing.  Input Parameter Ferron Bearing Radius R, m 0.05 Bearing width L, m 0.08 Bush outer radius R2(assumed), m 0.1 Radial clearance at 20°C, C, m-6 145 Groove angle, ° - Groove length, m-3 70 (assumed) Groove Location, ° 0 Specific heats cf, cb, cs 2000, 380, 490 Conductivities kf, kb, ks, kg, W/m K 0.13, 250, 50, 0.025 Convective heat transfer coefficient , bush/ambient H1, W/m2.K  80 Convective heat transfer coefficient , bush/oil and shaft/oil interfaces H2, W/m2.K  250 Inlet temperature Ti, K 313 Ambient Temperature To, K 313 Inlet lubricant density ρI, kg/m3 860 Inlet viscosity μi, Pa.s 0.0277 Viscosity-temperature coeff. β, /°K 0.0414 Supply pressure ps (gauge), kPa 70            172 Table D.5 Additional input data to run optimization simulations.  Parameter Dowson Song Minimum radial clearance Cmin=40μm Cmin=40μm Maximum radial clearance Cmax=300μm Cmax=300μm Minimum length to diameter ratio λ λmin=0.4 λmin=0.2 Maximum length to diameter ratio λ λmax=1.0 λmax=0.6 Minimum lubricant viscosity, Pa.s μmin=0.01  μmin=0.001 Maximum lubricant viscosity, Pa.s μmax=0.03  μmax=0.01 Minimum allowable film thickness hmin=10 μm hmin=10 μm Maximum allowable film pressure pmax=10 MPa pmax=10 MPa Allowable temperature rise ΔTa=70 °C ΔTa=70 °C Scaling (normalizing) factor β1=1, β2=105 β1=1, β2=105 Weighting factor α1=1, α2=α1/5 α1=1, α2=α1/5 Load, kN 10, 20 10, 20 Population Size (Hybrid Technique) 100 20 Population Type Double Vector Initial Search Range -2.5 to 2.5 Fitness Scaling Rank Selection Function Stochastic Uniform Reproduction: Cross-over Fraction 0.8 Mutation Function Gaussian Cross-over Function Scattered Migration Direction Forward       173 Table D6 Input data for Dowson [80] and Claro [79] bearings Input Parameter Dowson et al. Claro Bearing diameter D, m 0.0635 0.05 Bearing width L, m 0.0635 0.05 Radial clearance at 20°C, C, μm 138.8  125 Groove length Lg, m 0.026 0.04 Groove width wg, m 12x10-3 10x10-3 Groove Location, ° 0 +90° Specific heats cf, cb, cs 2000, (840), (490) (2000, 380, 490) Conductivities kf, kb, ks, kg, W/m K 0.13, (1.05), 50, 0.025 0.13, 250, 50, 0.025 Inlet temperature Ti, K 313 313 Ambient Temperature To, K (298) (298) Inlet lubricant density ρI, kg/m3 867.5  860 Inlet viscosity μi, Pa.s 0.0224 0.075 Viscosity-temperature coeff. β, /°K (0.0414)  (0.0414) Supply pressure ps (gauge), kPa 0,170  8-60 Load W, N 700 824             174 APPENDIX E Verification Examples E1 Introduction This section presents some further validation examples of the streamline upwind Petrov- Galerkin finite element method developed. The classical method is compared to the Galerkin and SUPG methods, namely the classical upwind scheme defined in section 2.5, the doubly asymptotic, and discontinuity capturing methods. Brief definitions of each follow. E1.1 Upwinding Methods If in the energy equation 2.11 the convective part ߩ ௙ܿu · ׏ܶ dominates the diffusive part ׏൫݇௙׏ܶ൯, an improvement of the accuracy may be possible by applying the upwinding method. It should be noted that upwinding does not always improve the accuracy and moreover is more computationally expensive in building the matrices. The SUPG method used in this work is based on the method proposed by Brooks and Hughes [11]. Essentially the method introduces an extra term to the standard Galerkin equation     ׬ ሺܦܶ െ ݂ሻݓ෥݀Ω௘      (E.1) where e is the element, DT is the differential equation (energy equation in this case), f the source term, and ݓ෥  the weighting function that contains two parts, the Galerkin weighting function and ζ defined in equation 2.30. The parameter defining the type of upwinding ξ in equation 2.30 defines the type of upwinding. For the definition of ζ we introduce η=uTΛu where Λ is the matrix with elements Λij and ν ൌ ԡuԡ∆௫ଶη . The type of upwinding becomes  175 1. Classical SUPG upwinding:  ζ=1 which is used in all the results in thesis. 2. Doubly asymptotic approximation:  ξ ൌ ቊ ν ଷ        െ 3≤ν≤3 ݏ݅݃݊ሺνሻ          |ν|>3 3. Discontinuity capturing type 1 from Hughes et al. [11] where a pseudo velocity uצൌ u·׏c ԡ௖ԡ૛  where c is the degree of freedom (solution to be computed) is defined and the upwind basis function ζ=(τ1u+τ2uצ)· ׏ ௜ܰ where ∇Ni is the divergence of the basis functions and ߬ଵ ൌ ∆௫క ଶԡܝԡ  and ߬ଶ ൌ ∆௫క ଶฮܝצฮ . For this case ξ is chosen according to the doubly asymptotic approximation.    (E.2) More upwinding schemes appear in the work of Brooks and Hughes [11], however only these three will be analyzed and results compared to the Galerkin method. The first example analyses a convection-diffusion problem while the second compares SUPG and Galerkin on lubrication data.  E.1.2 Convection Skew to the Mesh E.1.2.1 Problem Statement Consider a convection-diffusion problem defined by the general equation     డ௖ డ௧ ൅u· డ௖ డ௫೔ െ ܦ డ మ௖ డ௫೔ మ ൌ ௦݂    (E.3) where D is the diffusivity and c is the concentration of a molecular species. We assume steady state conditions (∂c/∂t=0) and no source term (fs=0). We consider the flow in the square shown in figure E1 with diffusivity D=10-6 with unidirectional and constant flow ԡuԡ ൌ 1 (convection dominates the flow) . Boundary conditions: Lower boundary (y=0): Dirichlet (c=1)  176 Left boundary  (x=0):  Dirichlet (c=1 for y≤0.2 and c=0 for y>0.2) Natural boundary condition, డ௖ డ௫೔ ൌ 0 imposed on all other boundaries Flow Conditions: The flow is unidirectional and constant (ԡuԡ ൌ 1) The angle of the flow α  is an input variable chosen to be 45°.        Figure E1. Region for skew convection   Exact Solution: The exact solution is equal to one in the region starting with lower boundary condition c=1 and following a straight line with the angle of flow. Obviously the value of c lies between o and 1 (0≤c≤1)  E.1.2.2 Solution Procedure The diffusivity D is much less than flow velocity u thus the problem is convection- dominated. The behavior of the standard Galerkin and the three upwind methods is compared. Linear triangular elements are used. Figure E2 shows the mesh. 0.2 c=1 c=0 c=1 x y 1 1 α 0 0  177  Figure E2. Mesh for convection-diffusion problem   E.1.2.4   Simulation Results Appearance of wiggles in the solution is evident in the minimum and maximum values. Table E1 compares results from the four methods.  Table E1. Minimum and maximum values of the solution Method Minimum Maximum Galerkin -1.83421E-04 1.33327 SUPG first order -3.78362E-02 1.17226 SUPG double asymptotic -3.78362E-02 1.17226 SUPG discontinuity capturing -8.77571E-04 1.05377  The Galerkin method has the highest over-prediction in Table E1. Wiggles are most apparent with the method as show in figure E3a. The discontinuity capturing SUPG method (figure E3d) exhibits the least amount of wiggles although it is the most expensive in terms of computing time and effort.    178  (a)  (b)  (c)  (d)  Figure E3. Three-dimensional plots of solutions from (a) Galerkin, (b) first order SUPG, (c) double asymptotic SUPG, and (d) discontinuity capturing SUPG   E.1.3 SUPG/Galerkin Comparison on Journal Bearings Experimental data from Costa et al. [69] and Dowson [59] are simulated using the three methods. An estimate of the error is determined from     ܧݎݎ݋ݎ ൌ ்೛ೝ೐೏ష ೐்ೣ೛ ೐்ೣ೛                  (E.4)  179 where Tpred and Texp are the temperature predicted by the model and experimental temperatures respectively at each radial position on the bush. Table E2 and figure E4 show the results of the simulation.  Table E2 Comparison of SUPG and Galerkin predictions N , rpm T max ,Pred T max ,Exp. Error T max , Pred. T max , Exp. Error T max T max ,Exp. Error T max , Pred. T max ,Exp. Error T max ,Pred T max ,Exp. Error T max , Pred. T max , Exp. Error Galerkin 41.0 0.078 51.6 0.199 56.6 0.098 59.0 0.110 67.9 0.152 76.6 0.126 SUPG 1 38.6 0.016 45.4 0.057 51.3 0.004 52.9 0.003 59.2 0.005 65.3 0.040 SUPG 2 38.9 0.024 45.5 0.058 52.2 0.014 52.9 0.003 59.5 0.010 65.8 0.032 38 500 53.1 59.051.5 1500 3000 4000 43 68.0 1000 2000   Results indicate a steady increase in error for Galerkin method. The classic SUPG and double asymptotic methods give progressive better results with increasing speed. Although the latter method requires more computational effort in terms of time and speed, it appears there is no benefit in using it except for the highest speed of 4000 rev/min considered here. There is no doubt, however, of the benefit of better thermal predictions of SUPG especially for higher speed simulations. Considering that bearing speeds of 18000 rev/min are not uncommon, SUPG should be the method of choice in such cases. This, to the best of knowledge, represents the first such analysis in lubrication problems. More validation simulations of data in the literature follow in subsequent sections.  180  Figure E4 SUPG and Galerkin predictions as a function of speed  E2 Slider bearing simulation This example is from Arnell [66] (example 5.4.3, page 136) and analyzes three slider bearings using the infinitely long approximations (ILA) for maximum pressure pm, flow rate Qi, load capacity Wd, and frictional power loss PL. This example is from Arnell [66] (example 5.4.3, page 136) and analyzes three slider bearings using the infinitely long approximations (ILA) for maximum pressure pm, flow rate Qi, load capacity Wd, and frictional power loss PL. E2.1 Problem Statement A thrust bearing of length L=0.2 m and width B=0.05 m operates with a minimum film thickness of 0.0001m and sliding velocity U=3 m/s, effective viscosity is µ=0.04 Pa.s, 30 35 40 45 50 55 60 65 70 75 80 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Te m pe ra tu re , C Speed, rpm Performance of SUPG versus Galerkin Experimental Galerkin Classic SUPG SUPG: Doubly Assymptotic  181 and three tilt ratios K=(h2/h1-1) of 1.19, 2.0, and 0.50. The tilt ratio is defined for the three cases by keeping h1 constant at 0.0001 m and varying h2. Operating characteristics maximum pressure pm, flow rate Qi, load capacity Wd, and power loss PL are to be determined. Figure E5 shows the setup.        Figure E5 Slider geometry.  E.2.2 Solution Procedure Input data and results from ILA and SUPG FE are compared in table E3. The ILA expressions for maximum pressure pm, flow rate Qi, load capacity Wd, and power loss PL are [66]     ݌௠ ൌ ቀ ఓ೐೑೑௎௅ ௛మ మ ቁ ቀ ଷ௄ ଶሺ௄ାଵሻሺ௄ାଶሻ ቁ                                     (E5) ܳ௅ ൌ ሺܮ݄ଵܷሻ ቀ ௄ାଵ ሺ௄ାଶሻ ቁ                                             (E6) ௗܹ ൌ ൜ߤ௘௙௙ܷܮ ቀ ஻ ௛భ ቁ ଶ ൠ ቂ ଺ ௄మ ቄ݈݊ሺܭ ൅ 1ሻ െ ଶ௄ ௄ାଶ ቅቃ                            (E7) ௥ܶ ൌ ቄ ఓ೐೑೑௎௅ ௛భ ቅ ቂସ ௄ ቄ݈݊ሺܭ ൅ 1ሻ െ ଺ ௄ାଶ ቅቃ                            (E8)       ∆ܶ ൌ ೝ்௎ ொಽఘ஼೛                               (E9) h2 Stationary pad Runner h1 h B  y  x z U  182 Deviation values εr are defined as the difference between SUPG FE prediction and ILA divided by the SUPG figure.  Table E3 Prediction comparison, analytical vs SUPG FE U , m/s μeff (Pa.s) h 1 (m) k p max (Pa) εr Q L , m 3 εr W d          N εr P L , W εr Δ T, °C εr 150000 4.50E-05 887.5 25.1 0.34 ILA 147879 4.5E-05 742.3 24.1 0.32 SUPG 153293 4.12E-05 961.5 27.2 0.4 ILA 150914 4.12E-05 830.4 26.3 0.39 SUPG 120000 3.60E-05 787.0 30.4 0.34 ILA 118201 3.65E-05 650.4 29.8 -0.32 SUPG 0.00010.04 0.33 0.46 -0.014 -0.016 -0.015 0.001 0.012 -0.196 -0.158 -0.2100.67 Performance Parameters 3 Bearing Input -0.043 -0.031 -0.019 -0.043 -0.032 2.043 0.010  The values of εr on pressure, flow rate, and power loss are small (between 1 and 3%) which is very good. L/B ratio for this bearing is 4 which is not a long bearing, so ILA over-predicts the performance parameters.  E3 Journal Bearing Simulation E3.1 Problem Statement Arnell [66] (page 169): A hydrodynamic journal bearing of length 160 mm operates with a shaft 200 mm diameter which rotates at 1200 rev/min. The effective viscosity is assumed to be 40 cP (centipoises) and diametrical clearance is 0.18 mm. Determine the operating characteristics of the bearing at an eccentricity ratio of 0.7. E3.2 Solution Procedure  The problem is analyzed using SUPG FE and results compared to the analytical method used in Arnell. The method [68] uses tabulated numerical solutions for performance characteristics flow rate Qi, frictional power loss PL, and maximum pressure pm. Table E4 shows input parameters and predictions from the two methods.  183 E3.3 Simulation Results The deviation between the two methods as defined in equation E4 above is small (εr<2.5%) for maximum pressure, flow rate and power loss. Temperature increase shows a deviation of 5.8%. The deviation between the two methods as defined in equation E4 above is small (εr<2.5%) for maximum pressure, flow rate and power loss. Temperature increase shows a deviation of 5.8%.  Table E4 Input parameters and predictions. μ eff , Pa.s 0.04 p max Pa Q L m 3 /s P L Δ T k f , W/m.K 0.13 2.64E+07 1.38E-04 8988 31.40 ILA C,  m 9.00E-05 2.64E+07 1.40E-04 9193 29.68 SUPG U, rev/s 20 0.0002 0.0143 0.0223 -0.058 ε r L , m 0.16 D, m 0.2 C p , J/kg.K 1880 ε 0.7 Performance ParametersBearing Input   E4 Bearing data from Mitsui Mitsui’s work [72] represents a landmark in journal bearing thermal predictions. The work presents bearing thermal performance as a function of speed Ns, clearance ratio C, type of lubricant, load capacity. Three different oils (transformer, no. 90 and no. 140) with different absolute viscosities are used. Experimental data and model predictions in figure E6 show the accuracy of the proposed model. Mitsui’s predictions on no. 90 and 140 oil are included for comparison. SUPG appears to give reasonable results in all cases. Of significance is the improved boundary conditions imposed in the groove/mixing zone. Conservation of mass flow (equation 2.24) is imposed, resulting in better predictions than  184 in the results of section 4.3 and figure 4.12. Model predictions are still least accurate in converging region of the bearing (0<θ<150°) similar to the pattern in figure 4.12. It appears Mitsui’s model under-predicts the same region.   Table E5. Input values for Mitsui bearing Input Parameter Mitsui [72] Bearing Radius R, m 0.05 Bearing width L, m 0.07 Bush outer radius R2(assumed), m 0.1 Radial clearance at 20°C, C, m-6 78.5 Groove angle, ° 10 Groove length, m-3 60 Groove Location, ° 0 Specific heats cf, cb, cs 1940, 380, 490 Conductivities kf, kb, ks, kg, W/m K 0.13, 250, 50, 0.025 Convective heat transfer coefficient , bush/ambient H1, W/m2.K  80 Convective heat transfer coefficient , bush/oil and shaft/oil interfaces H2, W/m2.K  250 Inlet temperature Ti, K 313 Ambient Temperature To, K 300 Inlet lubricant density ρI, kg/m3 865 Inlet viscosities μi, Pa.s 0.0736 (transf. oil) 0.0192 (No. 90 oil) 0.0469 (No. 140 oil) Viscosity-temperature coeff. β, /°K 0.0414 Supply pressure ps (gauge), kPa 98       185   Figure E6. SUPG predictions versus experimental temperatures, Mitusi [72]. Included is Mitsui’s prediction for no.90 oil.    A summary of the maximum and average errors for SUPG and Mitsui’s model are shown in table E6. SUPG predictions are significantly better in each case.  Table E6 Maximum and average error for Mitsui’s data. Error Summary   Transformer oil No. 90 oil No. 140 oil   SUPG Mitsui SUPG Mitsui SUPG Mitsui Max Error 0.017 0.039 0.014 0.030 0.015 0.028 Ave. error 0.008 0.017 0.026 0.064 0.025 0.073   40 42 44 46 48 50 52 54 56 58 60 0 30 60 90 120 150 180 210 240 270 300 330 360 Te m pe ra tu re , C Angle, degrees Predictions vs  Exp. Temperatures Mitsui:No.140 No.140 No.90 Transformer SUPG:Transf SUPG:No.90 SUPG: No.140 Mitsui:No.90  186 E5. SUPG performance in predicting Pierre’s data The data and THD model presented by Pierre and Fillon [73] is analyzed and compared to SUPG predictions. The researchers present thermal and performance parameters for bearing input data presented in table E7.  Table E7 Input parameters for journal tested [73] Input Parameter [unit] Value Diameter D, [m] 0.1 Bearing length L, [m] 0.08 Radial clearance C, [µm] 123 Rotational speed Ns, [rev/min] 3000 Oil supply temperature Ti, [°C] 35 and 50 Bearing load Wd, [N] 9000 Dynamic viscosity µì40°C at 40 °C (10-3 Pa.s) 30 Dynamic viscosity µì100°C at 100 °C (10-3 Pa.s) 4.05 Lubricant density ρ, [kg/m3] 860 Lubricant specific heat Cp, [J/kg.K] 2000   Experimental and predicted temperatures for the two speeds are shown in figure E7 (figure 2 in reference 73). Their THD model appears to be less accurate in the convergent and divergent zones.   187  Figure E7. Experimental and THD temperature predictions as a function of angle from inlet groove [73].  The two cases in figure E7 are analyzed using SUPG and presented in figure E8. Included on figure E8(a) is Pierre’s prediction for reference. Table E7 gives a summary of the error magnitudes from the two methods. These show SUPG to give better results in both cases. Both models are off on the convergent region while SUPG gives more accurate predictions in the divergent zone (240<θ<360). Table E8 gives a summary of performance parameter predictions for Pierre’s data. L/D ratio is 1, input oil temperature Ti is 40°C (other input parameters as shown in table E6). The bearing characteristic number S. The error, as determined as before (equation E3) is small, showing the accuracy of SUPG.   35 40 45 50 55 60 65 70 0 60 120 180 240 300 360 Te m pe ra tu re , C Angle from inlet groove, deg Pierre's Model T35Exp. T50Exp. Pierre@35C Pierre@50C  188  (a)  (b) Figure E8. Comparison of SUPG and Pierre’s THD model [73]. 35 40 45 50 55 60 65 0 30 60 90 120 150 180 210 240 270 300 330 360 Te m pe ra tu re , C Angle from groove pos., degrees SUPG vs Pierre Predictions, Ti=35 C Exp. Data SUPG Pierre 40 45 50 55 60 65 70 0 30 60 90 120 150 180 210 240 270 300 330 360 Te m pe ra tu re , C Angle from inlet groove, degrees SUPG vs Pierre Predictions, Ti=50 C Exp. data SUPG Pierre  189 Table E8. Error on thermal predictions, SUPG vs Pierre’s THD model Error Summary   T35 T50   SUPG Pierre SUPG Pierre Max Error 0.047 0.052 0.058 0.076 Ave. error 0.105 0.128 0.029 0.037   Table E9. Performance parameter predictions for Pierre’s data S=0.227 Exp. Data SUPG Error PL 820 819.5 0.0006 pmax, MPa 2.3 2.26 0.0174 Tmax, C 58 59.5 0.0250 QL, m3/s 4.50E-05 4.60E-05 0.0222 

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