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Modelling power transfer in electron beam heating of cylinders Tripp, David William 1994

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MODELLIN( POWER TRANSFER IN ELECTRON BEAM HEATING OFCYLINDERSbyDAVID WILLIAM TRIPPB.A.Sc., The University of British Columbia, 1984M.A.Sc., The University of British Columbia, 1987A THESIS SUBMIFI’ED TN PARTIAL FULFILMENTOF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of Metals and Materials EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1994© DAVID WILLIAM TRIPP, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of fkl.5? ZX(i 1’s\The University of British ColumbiaVancouver, CanadaDate P/L- /3( (/DE-6 (2J88)AbstractThe electron beam remelting process is used extensively for the refining andrecycling of titanium and its alloys. The success of the process relies on its ability toprovide a thermal environment capable of removing impurities and deleterious particleswhile allowing control of chemical composition and solidification. These aspects of theprocess hinge on the accurate control of power input to the melt stock. Over the years,the effects of various parameters (such as chamber pressure) on the power delivery to themelt stock have largely been ignored.In this work, a series of laboratory scale experiments using instrumentedcylinders was conducted. In parallel, a finite element model of the electron beam heatingprocess was developed to analyze the experimental results.The experimental results show that the temperature regime within a targetcylinder is affected by variations in chamber pressure. The magnitude of the temperaturechanges measured as a result of pressure changes was on the order of two to three timesthe intrinsic error in the thermocouples. Thus these perceived temperature changes wereclose to the limit of our ability to measure them. The experimental results are selfconsistent in that a pressure variation produced a similar trend at each thermocouplelocation.Analysis with the model has shown that the effect of pressure is to alter the powerdistribution within the beam and not the efficiency of power transfer. The model can bemade to reproduce both qualitatively and quantitatively the measured temperatureresponse by varying the beam spreading parameter with chamber pressure. Such a claimcannot be made when varying the power transfer efficiency with chamber pressure.By fitting the model to the experimental thermocouple results it has been shownthat that the beam power disthbution is adequately represented by a Gaussian or normaldistribution. Additional analysis has led to empirical relationships for the effect of— 11 —111chamber pressure on beam focusing characteristics under the conditions used in thelaboratory.-The indirect link between chamber pressure and beam power distribution isreinforced using careful and self-consistent experiments, a mathematical model that canreproduce both quantitatively and qualitatively the results of the experiments and thephysics of beam - gas particle interactions.Table of ContentsAbstract iiTable of Contents ivList of Tables viiList of Figures ixList of Symbols xvAcknowledgments xviiChapter 1. : Introduction 11.1. Commercial Hearth Remelting Furnaces 21.2. Raw Materials 21.3. Defects in Titanium 31.3.1. High Density Inclusions 31.3.2. Interstitial Rich Inclusions 41.4. Consolidation Processes 51.4.1. The Vacuum Arc Remelting Process 51.4.2. The Electron Beam Cold Hearth Remelting Process 61.4.3. Electron Beam Cold Hearth Remelting Process Advantages 71.4.4. Electron Beam Cold Hearth Refining Process Disadvantages 81.5. The Thermal Regime During EBCHR 9Chapter 2. : Literature Review 102.1. Electron Beam Generation and Heating 102.1.1. Beam Generation, Propagation and Control 102.1.2. Energy Losses in Electron Bombardment 122.1.3. Other Sources of Beam Power Loss 152.1.4. Measurement of Beam Parameters 162.1.5. Summary 172.2. Electron Beam Welding 172.2.1. Process Description 182.2.2. Pressure Effects 182.2.3. Mathematical Models 212.2.4. Summary 222.3. Electron Beam Remelting 232.3.1. Ingot Models 232.3.2. Hearth Models 252.3.3. Other Electron Beam Heat Transfer Models 252.3.4. Summary 26Chapter 3.: Scope and Objectives 273.1. Scope of the Research Programme 273.2. Objectives of the Research Programme 28Chapter 4. : Experimental Procedures and Results 294.1. Laboratory Scale Experimental Procedures 29- iv -V4.1.1. Equipment.294.1.2. Tantalum Experiments 304.1.3. Titanium Experiments 354.1.4. Experimental Difficulties 414.2. Industrial Scale Experiments 434.2.1. Equipment 434.2.2. Industrial Results 44Chapter 5. : Mathematical Model 525.1. Formulation 525.1.1. Discretization of the Spatial Derivatives 535.1.2. Solution 545.1.3. Element Type 545.1.4. Numerical Integration 565.1.5. Treatment of Temperature Dependent Materials Properties 565.1.6. Phase Changes 575.2. Basic Verification of the Computer Code 585.3. Inverse Heat Transfer Calculations 625.4. Summary 65Chapter 6. Application of the Finite Element Model To Electron Beam Heatingand Melting 666.1. Boundary Conditions 666.1.1.Domain 666.1.2. Side Boundary Condition 666.1.3. Bottom Boundary Condition 666.1.4. Symmetry Conditions 676.1.5. Top Surface Boundary Condition 676.1.6. Three Dimensional Model 696.1.7. Initial Conditions 716.1.8. Fluid Flow 716.2. Sensitivity Analysis 726.3. Results 936.3.1. Analysis of Tantalum Experiments 936.3.2. Analysis of Titanium Experiments 1006.3.3. Summary 1126.4. Discussion 1136.4.1. The Effect of Pressure on Power Delivery 1136.4.2. Chamber Gas Composition 1166.4.3. The Heat Balance 1176.4.4. Error Assessment 1216.5. Summary 123Chapter 7. Industrial Results 1247.1. Methodology 1247.2. Results 1267.3. Discussion 1317.4. Summary 137viChapter 8. : Conclusions 1388.1. Conclusions 1388.2. Recommendations for Future Work 142References 144Appendix 1. : Thermophysical Properties of Materials Used in the Models 150A 1.1 Titanium 150A1.2 Tantalum 153Appendix 2.: Finite Element Meshes 155A2.l. Titanium Mesh 155A2.2. Tantalum Mesh 155List of TablesTable 1: Thermocouple Positions in Tantalum Experiments Conducted at UBC 32Table 2 : Thermocouple Positions in Titanium Experiments Conducted at UBC 36Table 3 : Location of the Thermocouples Used in the Axel Johnson Experiments 44Table 4 : Typical Process Conditions for Axel Johnson Experiments 46Table 5 : Time of Each of the MR Scan Sets 48Table 6: Range of Parameters Studied in the Sensitivity Analysis of the ThermalResponse of the Titanium Block to Changes in Process Parameters 72Table 7 : The Sensitivity of Temperatures Within the Titanium Cylinder toChanges in the Conditions Shown in Table 6 73Table 8 : The Sensitivity of Temperatures Within the Tantalum Cylinder toChanges in the Conditions Shown in Table 6 84Table 9: Summary of Differences Between the Temperatures Predicted for theThree Dimensional Model and the Shifted Axisymmetrical Modelat the Thermocouple Locations Used in the Tantalum Experiments 88Table 10 : The Temperature within the Tantalum Cylinder at Locations Removedfrom the Axis of the Beam Pattern 91Table 11: The Relative Sensitivity of the Temperatures within the TantalumCylinder to Variations in Net Applied Power and Beam SpreadingFactor for Various Offsets of the Beam Pattern from the CylinderAxis 92Table 12 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 10 95Table 13 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 9 97Table 14: Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 11 97Table 15 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 4. (Pressure =0.02 Pa) 104Table 16 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 5. (Pressure =0.l6Pa) 106Table 17 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 6. (Pressure =0.33 Pa) 108- vii -viiiTable 18 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 8. (Pressure =0.04 Pa) 108Table 19 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 7. (VariablePressure) 112Table 20: Summary of Beam Spreading Factors and Power Transfer Efficienciesas a Function of Pressure for All of the Titanium Experiments 113Table 21: Summary of the Results of Fitting the Calculated Power Distributionsto Equation (48) 127Table 22: Summary of Input Power, Power Delivered to the Cylinder and PowerEfficiency for the Axel Johnson Experiments 128Table 23 : Heat Balances at the Time of Each MR Scan During the IndustrialExperiments 136List of FiguresFigure 1: Schematic Diagram of the Vacuum Arc Remelting Process 5Figure 2: Schematic Diagram of the Cold Hearth Refining Process 6Figure 3 : Schematic Diagram of a High Power Electron Gun Used in MeltingApplications 11Figure 4 : The Various Interactions Between a Beam and a Solid Target 13Figure 5 : The Number of Backscattered Electrons as a Function of frradiatedMaterial for a Normal Electron Beam 13Figure 6: The Effect of Beam Angle on the Number of Backscattered Electrons 14Figure 7 : The Energy of Backscattered Electrons as a Function of frradiatedMaterial 14Figure 8 : The Power Losses Associated with Backscattered Electrons as aFunction of Irradiated Material 15Figure 9 : The Effect of Pressure on Penetration in Electron Beam Welds 19Figure 10: Measured Current Density Distribution in the Focal Spot at VariousWorking Pressures in a Welding Chamber (UB = 60 kV, ‘B =90mA, zB = 525 mm.) 20Figure 11: Relative Power Input, Absorbed Power, Depth of Weld in NiCr-Steeland Unscattered Part of EB Current as a Function of Pressure forUB=75kV and Power of 5.1 kW at Optimal Focus 21Figure 12: Power Absorbed by the Workpiece as a Function of Pressure for a 8.7kW Beam at 100 KV 21Figure 13: Photograph of the 30 kW Electron Beam Melting Facility at UBC 29Figure 14: Schematic Diagram of the System Used For Pressure Control in theLaboratory Scale Experiments 30Figure 15: Schematic Diagram of Tantalum Cylinder Used in the Experiments 31Figure 16: Photograph of the Tantalum Cylinder Before Heating Showing theLocation in the Furnace and the Location of the ThermocoupleAssemblies 32Figure 17 : Results of Tantalum Experiment Designated Test 9 34Figure 18 : Results of Tantalum Experiment Designated Test 11 35Figure 19: Schematic Diagram of the Titanium Cylinder Showing the Locationsof the Thermocouples 36Figure 20: Results of Titanium Experiment Designated Test 4 37Figure 21: Appearance of the Titanium Cylinder at the Conclusion of Test 4 37- ix -xFigure 22: Tracing of the Extent of the Liquid Pool as Taken at the ConclusionofTest4 38Figure 23 : Results of Titanium Experiment Designated Test 7 39Figure 24: Appearance of the Titanium Cylinder at the Conclusion of Test 7 40Figure 25 : Tracing of the Extent of the Liquid Pool as Taken at the ConclusionofTest7 40Figure 26: Photomacrograph of the Through Diameter Cross Section of theTitanium Cylinder Showing the Pool Developed During Test 8 41Figure 27: Schematic Diagram of Titanium Cylinder Used for Experiments atAJNI 43Figure 28 : The Power Distribution Used in the Axel Johnson Experiments 44Figure 29: Photograph of the Titanium Cylinder Used in the IndustrialExperiments Taken Before the Experiments Occurred 45Figure 30 : Applied Power, Chamber Pressure and Temperature Response of theTitanium Cylinder Used in The AJMI Experiments, Segment A,First Heat Up Cycle 46Figure 31: Applied Power, Chamber Pressure and Temperature Response of theTitanium Cylinder Used in The AJMI Experiments, Segment C,Second Heat Up Cycle 47Figure 32 : Pool Cross Section Obtained by Doping the Axel Johnson Cylinderwith Copper at 18:30 September 02, 1992 50Figure 33 : Temperature Distribution on the Surface of the AJMI Cylinder at18:32 - September 02, 1992, Obtained from the NIR Camera 50Figure 34: Photograph of the Cylinder Used in the Industrial Experiments TakenAfter the Experiments 51Figure 35 : Eight-node, Isoparametric, Curve Sided Element in Both x-y Spaceand u-v (transformed) Space 55Figure 36 : Twenty-node, Isoparametric, Curve Sided Element in Both x-y-zSpace and u-v-w (transformed) Space 55Figure 37 : Comparison Between the Finite Element Solution and the AnalyticalSolution for an Infinite Cylinder Losing Heat by Convection forVarious Initial Temperatures (k = 0.2 W cm1 Cp = 0.5 J g4oC..l p = 4.54 g cnv3) 60Figure 38 : Comparison Between the Analytical Solution For a Cylinder with aConstant Heat Flux Applied to One End and the Finite ElementSolution, (k = 0.2 W cm1 °C1,Cp = 0.5 J g1 C1, p = 4.54 gcm3,Qo =25 W cm2) 61xiFigure 39: Schematic Diagram of the Target Cylinder Showing the BoundaryConditions 70Figure 40: Sensitivity of the Thermal Response of the Titanium Cylinder toChanges in the Contact Heat Transfer Coefficient 74Figure 41: Sensitivity of the Thermal Response of the Titanium Cylinder toChanges in Net Applied Power 75Figure 42: Variation of the Liquid Pool Dimensions and Cylinder SurfaceTemperature Distribution with Changes in Net Applied Power forthe Titanium Cylinder 76Figure 43 : Sensitivity of the Thermocouple Response of the Titanium Cylinderto Changes in the Beam Pattern Radius 77Figure 44: Variation of the Liquid Pool Dimensions and Cylinder SurfaceTemperature Distribution with Changes in the Beam PatternRadius 78Figure 45: Sensitivity of the Thermal Response of the Titanium Cylinder toChanges in the Beam Spreading Parameter 79Figure 46: Variation of the Liquid Pool Dimensions and Cylinder SurfaceTemperature Distribution with Changes in the Beam SpreadingParameter 80Figure 47 : Sensitivity of the Thermal Response of the Titanium Cylinder toDecreasing the Thermal Conductivity of the Liquid 81Figure 48 : Variation of the Liquid Pool Dimensions and Cylinder SurfaceTemperature Distribution with Changes in the Liquid ThermalConductivity Multiplier 82Figure 49: Sensitivity of the Thermal Response of the Tantalum Cylinder toChanges in the Bottom Surface Contact Heat Transfer Coefficient 85Figure 50: Sensitivity of the Thermal Response of the Tantalum Cylinder toChanges in the Net Applied Power 86Figure 51: Sensitivity of the Thermal Response of the Tantalum Cylinder toChanges in the Beam Spreading Parameter 87Figure 52: Sensitivity of the Thermal Response of the Tantalum Cylinder toChanges in the Beam Pattern Radius 87Figure 53 : The Sensitivity of the Thermal Response of the Tantalum Cylinder toChanges in the Center of the Beam Pattern with Respect to theAxis of the Cylinder 88Figure 54: Calculated Temperature Distribution For an Offset Beam Pattern AxisUsing the 3-Dimensional Model (dashed lines) Overlaid with theCalculated Result Using the Axisymmetric Model (solid lines)Offset by the Same Amount 89xiiFigure 55 : The Temperature as a Function of Radius for the Tantalum Cylinderfor Different Beam Spreading Parameters 90Figure 56: The Temperature as a Function of Radius for the Tantalum CylinderFor Different Applied Powers 90Figure 57 : The Thermal Response of the Tantalum (left) and Titanium (right)Cylinders to Changes in Power (top) and Beam SpreadingParameter (bottom) 93Figure 58 : Model Predictions and Experimental Results for Test 10 94Figure 59 : Model Predictions and Experimental Results for Test 9 96Figure 60: Model Predictions and Experimental Results for Test 11 96Figure 61: The Response of the Three Thermocouples to Changes in Pressure inTantalum Test 11 98Figure 62: Model and Experimental Results for Power Efficiency Variation withPressure in Tantalum Heating Test 11 99Figure 63 : Model and Experimental Results for Beam Spread Factor Variationwith Pressure in Tantalum Heating Test 11 99Figure 64: Experiment and Model Results for Test 4 (1=0.95, cYb = 17.5mm) 101Figure 65 : The Model Predictions and Thermocouple Response for TitaniumTest 4 After Power Off 102Figure 66: Predicted Temperature Distribution for Test 4 at 2403 seconds AfterPower On 103Figure 67 : Experimental and Model Results for Test 5 = 0.95, cYb = 15.mm) 105Figure 68 : Predicted Temperature Distribution for Test 5 at 2300. seconds AfterPower On 105Figure 69 : Experimental and Model Results for Test 6 (n = 1.02, 0b = 17.5mm) 107Figure 70: Predicted Temperature Distribution for Test 6 at 2310. seconds AfterPower On 107Figure 71: Experimental and Model Results for Test 8 (T = 0.99, Gb = 19.mm) 109Figure 72 : Predicted Temperature Distribution for Test 8 at 2025. seconds AfterPower On 109Figure 73 : Photomacrograph of the Through Diameter Cross Section of theTitanium Cylinder Showing the Apparent Liquidus (solid line) andthe Predicted Liquidus (dashed line) 110xliiFigure 74 : Experimental and Model Results for Test 7 Using a Simplified PowerSchedule (r10—0.95-0.99, Gb:= 17. -21. mm) 111Figure 75 : Beam Spreading Factor as a Function of Chamber Pressure for AllTantanlum Experiments 114Figure 76: Beam Spreading Factor as a Function of Chamber Pressure for AllTitanium Experiments 115Figure 77 : Power Efficiency as a Fuñtion of Pressure for All the TitaniumExperiments 116Figure 78 : The Heat Removed from the Various Surfaces of the TantalumCylinder as a Function of Time as Calculated for Test 11 118Figure 79: Schematic Heat Flow Diagram of the Tantalum Cylinder During Test11 as Calculated at 3000 seconds from Power On 118Figure 80: The Heat Removed from the Various Surfaces of the TitaniumCylinder as a Function of Time as Calculated for Test 4 119Figure 81: Schematic Heat Flow Diagram of the Titanium Cylinder During Test4 as Calculated at 2300 seconds from Power On 120Figure 82: The Heat Removed from the Various Surfaces of the TitaniumCylinder as a Function of Time as Calculated for Test 7 120Figure 83: Schematic Heat Flow Diagram of the Titanium Cylinder During Test7 as Calculated at 3000 seconds from Power On 121Figure 84: The Calculated Power Distribution for Segment A of the AJMIExperiments Using the Measured Temperature Distribution Acrossthe Entire Cylinder, the Liquid Region Only and The CombinedPower Distribution 125Figure 85 : The Calculated Power Distributions For Each of Segments A, C, GandH 126Figure 86 : The Spreading Parameter as a Function of Pressure for the IndustrialExperiments 127Figure 87 : Model Results and Measured Temperatures for Segment A Using theCalculated Power Distribution 128Figure 88 : Model Results and Measured Temperatures for Segment C Using theCalculated Power Distribution 129Figure 89 : Model Results and Measured Temperatures for Segment G Using theCalculated Power Distribution 130Figure 90 : Predicted Temperature Response of the Cylinder Using the CalculatedPower Disthbution for Segment A with = 0.6 133Figure 91: Measured and Calculated Temperature Distributions for Segment A at134 minutes 133xivFigure 92 : Calculated Power Distributions and MR Camera Scans for SegmentsA,C,GandH 135Figure 93 : The Calculated Beam Power Distribution as a Function of the LiquidThermal Conductivity Multiplier for Segment A 137Figure 94 : The Thermal Conductivity of Titanium as a Function of Temperature 150Figure 95 : The Heat Capacity of Titanium as a Function of Temperature 151Figure 96 : The Enthalpy Function for Titanium 152Figure 97 : The Thermal Conductivity of Tantalum as a Function of Temperature 153Figure 98 : The Heat Capacity of Tantalum as a Function of Temperature 154Figure 99 : The Emissivity of Tantalum as a Function of Temperature 155Figure 100 : The Mesh Used in the Finite Element Modelling of the LaboratoryScale Titanium Cylinder 157Figure 101 : The Mesh Used in the Finite Element Modelling of the ExperimentsConducted at Axel Johnson Metals 158Figure 102 The Axisymettric Mesh Used in the Finite Element Modelling of theTantalum Laboratory Experiments 159List of Symbolsr- radiusRmax - maximum value for the radius- in cylindrical co-ordinates (the angle)z - the axial distanceZmax - maximum value for the axial co-ordinatek- thermal conductivityasb - Stefan-Bolzman Constant = 5.67 x 10-12 W cm-2K- emissivityT,- ambient temperature- heat transfer coefficient for the contact region on the bottom surface of thetargett -timeT- temperatureth- mass flux of a material due to evaporation (g cm-2)R- gas constantM - molecular weightp° - equilibrium vapour pressure of a material‘11evap - heat of vapourization- beam pattern parameter, the distance between the center of a circular beampattern and the center of the beam spot- beam spot parameter, a measure of the spreading of the beam spot,spreading factor- beam power efficiency, the ratio between dial power and power delivered tothe targetC - constant used in the specification of a beam power distribution- in a non-centered beam pattern the distance between the center of the targetand the center of the beam patternx, y, z - co-ordinates in a Cartesian co-ordinate systemT0 - the initial temperature- Heat Capacity per unit mass of target materialp - target density- xv-xviAH25,x - the change in enthalpy per unit mass of material relative to 25 °C- thermal diffusivity = k/pCUb - beam accelerating voltage- beam currentE- energy of an electrone - the charge on an electrond - beam diameter and- the distance between the beam generation system and the workpiece inelectron beam welding- eddy thermal conductivity and eddy viscosity, these parameters are used inthe so called k-epsilon approach to modelling turbulent fluid flow.AcknowledgmentsNo work of this nature takes place in a vacuum. Many people have contributed inmany ways to the production of this thesis. I would first like to acknowledge thecontributions of Axel Johnson Metals Inc. and Cabot Corporation for the materials usedin the experiments. I would also like to thank the people at Axel Johnson and GeneralElectric Aircraft Engines for allowing me to use the results of experiments theyconducted. Specifically I would like to thank Dr. Janine Borofka of Axel Johnson andDr. Gordon Hunter of General Electric Aircraft Engines (at the time) for their help.Without the help of Cohn Edie the laboratory experiments might never have taken place.I owe Cohn my thanks.My thanks to Alec Mitchell for the opportunity to publish papers and attendnumerous conferences while the work was in progress and my heartfelt thanks to SteveCockcroft who finally made me understand that this publication must come first. I wouldalso like to thank the National Sciences and Engineering Research Council for theirfinancial support.I would like to acknowledge the support of three families. The first is theextended family that comprises all the people in the Department of Metals and MaterialsEngineering at UBC. You’ve all helped! Secondly, my brothers, sister and parents havehelped me immensely over the years it took to produce this document and long before.Finally, I want to acknowledge my immediate family: my daughter, Natasha, and mywife, Sue. Words don’t exist to express how much they mean to me. All I can do is givethem my thanks and my love.- xvii -Chapter 1.: Introduction“On July 19, 1989, United Airlines Flight 232 was on a regularlyscheduled transit from Denver, Colorado to Chicago, Illinois, with 285passengers and]] crew members on board. While in level flight at 37,000feet and Mach 0.83 there was a loud report, followed by vibration and/orshuddering of the aircraft, sensed by the flight crew. The center engine(position number 2) had incurred an inflight separation of the stage 1 fandisk with subsequent damage to the aircraft, resulting in the depletion ofhydraulicfluidfrom all three systems powering theflight controls.The aircraft flewfor 44 minutes after the disk separation, with thecrew experiencing difficulty in controlling the aircraft due to the loss ofhydraulic power to the flight controls. Flight 232 crashed during landingat Sioux Gateway Airport, Sioux City, Iowa. Fatalities included 110 passengers and oneflight attendant. “[45]The demise of United Airlines Flight 232, as described above, was attributed toan interstitial rich defect in the titanium alloy employed in the first stage compressor ofengine number 2. This titanium failure arose from material double Vacuum ArcRemelted (VAR) in the early 1970’s. The Titanium Rotating Components Review Team(struck to investigate the UA232 engine failure) concluded that VAR technology may beapproaching state-of-the-art minimum limitations in defect size and defect rate and thatcold hearth remelting for ingots promises a substantial reduction in disk failures frommetallurgical defects[45].In the time period preceding the crash of United Airlines 232, at least one jet engine manufacturer in the United States was actively involved in the development ofhearth remelting technology to improve the cleanliness of rotating grade titanium for jetengines. Both the Electron Beam Cold Hearth Remelting (EBCHR) process and thePlasma Arc Remelting (PAR or PAM) processes have been examined with an aim todecreasing the defect rate in titanium disks. Since the UA232 crash, the Federal AviationAdministration has specified that all titanium in rotating parts in jet engines should behearth remelted.-1-21.1. Commercial Hearth Remelting FurnacesIn North America there are currently four electron beam hearth remelting furnaces producing titanium in significant quantity. Two of these furnaces are operated byAxel Johnson in Morgantown, Pennsylvania. One of the furnaces at Axel Johnson is a 2.0MW installation using 4- 600 kW von Ardenne electron guns. The second furnace is a3.3 MW installation that uses 3 - 600 kW and 2 - 750 kW von Ardenne electronguns[78]. The two furnaces give Axel Johnson a total production capacity of some 7,000tonnes per year[291. The other two furnaces are operated by Viking Metallurgical inVerdi, Nevada. One of the Viking furnaces is a 1.2 MW installation and the other is a 2.4MW furnace using 2- 1.2 MW von Ardenne guns.In addition to these titanium melting furnaces, a number of smaller installationsexist in North America which are used primarily for the production and refining ofsuperalloy materials for foundry applications.In addition to the EB furnaces, at least two plasma hearth remelting furnaces arecoming on line for the production of titanium for aerospace applications. These furnacesare owned by Wyman-Gordan and Teledyne Aflvac respectively[72j.Despite a substantial industrial base and growing market for hearth remelted titanium (both in the aerospace industry and elsewhere), there are many aspects of the process that are not well understood and have not been well quantified.1.2. Raw MaterialsAll virgin titanium material is manufactured in a batch process via either theKroll process or the Hunter process. Both processes involve the reaction of titaniumtetrachioride with a reducing metal to produce a sponge. In the Kroll process magnesiumis used and in the Hunter process the reductant is sodium. The titanium sponge is thenmelted using one or more of the primary consolidation processes (VAR, EBCHR, PAR)to produce ingot material. Virgin titanium sponge can contain volatile elements which3are released during melting. The release of these elements during melting can causetransient pressure fluctuations during the consolidation processes.The jet engine manufacturing industry generates a large amount of high qualitytitanium scrap due primarily to the high buy-to-fly ratios (on the order of 10:1) arising inthe production of engine components. Titanium scrap generally contains contaminatedmaterial resulting from manufacturing processes such as pieces of tool bits and variousother exogenous materials. The economics of the titanium industry demand that thisscrap be recycled.In order to make titanium scrap suitable for remelting into rotating grade ingot,machine turnings and other scrap pieces are degreased, washed and passed over magneticseparating units to minimize the incidence of magnetic tool bit chips in the scrap material. If the material is to be further processed using Vacuum Arc Remelting (VAR), thenthe scrap is extensively X-rayed and any batch containing an X-ray indication of a highdensity particle is rejected[47].1.3. Defects in TitaniumPrimary melting defects in titanium can be grouped into two broad categoriesHigh Density Inclusions (HDI’s) or Interstitial Rich Inclusions (IRIs). Interstitial RichInclusions are also known in the industry as hard-ct defects, Type I defects or Low Density Inclusions (LDI’s).1.3.1. High Density InclusionsHigh Density Inclusions in titanium result from two sources : high density material contained in recycled scrap (such as a tungsten carbide tool bit chip) and high densitymaterial introduced as a result of alloy additions.To ensure that a high density inclusion does not exist in a titanium ingot, thescrap preparation process must be extremely rigorous and the consolidation process mustensure their removal from the liquid metal prior to solidification. The consolidation proc4ess used must therefore provide a means of removing the particle by sedimentation in theliquid and subsequent separation from the solidifying ingot or ensure that a defectcausing particle spends an adequate time at a high enough temperature during theconsolidation process to dissolve in the liquid. In general, the HDI causing particles havehigh melting temperatures and require significant periods of time to dissolve completelyin liquid titanium.1.3.2. Interstitial Rich InclusionsInterstitial rich inclusions, which contain high concentrations of strong alphastabilizers in titanium (C, 0, N), can result from a number of sources[71]. These particlesvary widely in composition depending on their source and are stable at elevatedtemperatures. Once incorporated into titanium, or a titanium alloy, the particles form ahard region in a relatively soft matrix and act as fracture nucleation sites. Since the interstitially enriched region is coherent with the surrounding matrix material, it is difficultto locate by non-destructive means such as ultrasonic inspection and only detectable if itis very large, there is a void associated with the defect or it intersects the surface of thecomponent and is discovered when the near-finished component is tested metallographically.The density of IRI’s varies with composition as well. The bulk density of pure,monolithic titanium nitride indicates that such a particle would sink in liquid titanium[1].In real material, the concentration of the alpha stabilizers and the source of the defectmay reduce the density such that the particle floats, or in the worst case, has a neutralbuoyancy[41j.An IRI particle can be removed from liquid titanium by flotation, dissolution orsedimentation. If the particle is of neutral density then it must be dissolved. Should theparticle be less dense than the liquid, the particle can float and be dissociated on contactwith the electron beam, plasma flame or electric arc depending on the consolidation5process used. A IRI denser than the liquid can be removed by dissolution orsedimentation. In any case the only way to guarantee the removal of IRI causing particlesis to ensure that the process conditions are conducive to dissolution during the period inwhich they are exposed to liquid metal.1.4. Consolidation ProcessesOnly two methods of conversion of both sponge and scrap into ingot form areused commercially today Vacuum Arc Remelting and Cold Hearth Remelting usingeither electron beams or gas plasma as the heat source.1.4.1. The Vacuum Arc Remelting ProcessA schematic diagram of the VAR process is shown in Figure 1. In the VARprocess for consolidation of rotating grade titanium, sponge and scrap are first compacted. These compacts are then welded together to form a primary electrode or “stick”.The stick is then melted in a VAR furnace toproduce an ingot of slightly larger diameter.This ingot is then inverted and remeltedusing a larger ingot crucible to form a largeringot. The process is repeated once morebefore the ingot is ready for use. Thisprocess of melting the input titanium threetimes is known as “triple-VAR” and has beenthe industry standard for rotating gradetitanium since the early 1970’s. Prior to theuse of “triple-VAR”, double vacuum arcremelting was considered adequate.The configuration of the electrodeand ingot in the VAR process provides aFigure 1: Schematic Diagram of theVacuum Arc RemeltingProcess.Water CooledCopper CrucibleRemeltElectrode6“short circuit” route between electrode and ingot. It is possible for a defect causingparticle to enter the liquid pooi and proceed to the solidifying interface without beingexposed to liquid titanium for a period of time sufficient to dissolve the particle. Thepresence of this short circuit route provides the possibility of a defect causing particlesurviving triple melting.1.4.2. The Electron Beam Cold Hearth Remelting ProcessA schematic diagram of the cold hearth remelting process is shown in Figure 2. Inthis process particulate or consolidated feed is input to the furnace where either an elecEleciron Guns orPlasma Torches \. tron beam or plasma jet melts theM’ria1 material. The liquid metal is thenallowed to flow over a water-cooled copper hearth for a periodof time and subsequently into awater-cooled copper cruciblewhere an ingot is withdrawn.The removal of a defectFigure 2: Schematic Diagram of the Cold Hearthcausing particle using a coldRefining Process.hearth remelting process is, inprinciple, fairly straight forward. During the refining stage of the hearth remeltingprocess, HDI’s and high density WI’s sink to the bottom of the liquid pool and are thentrapped by the skull and ultimately dissolved. These defect causing particles are thendissolved over an extended period of time. A well designed and properly operated hearthremelting furnace will give low density IRI’s sufficient time in the refining heath to floatto the surface and be dissociated by the power source. Neutral density particles must havesufficient time at a sufficiently high temperature to dissolve completely.71.4.3. Electron Beam Cold Hearth Remelting Process AdvantagesThe advantages of electron beam hearth remelting result from the presence of ahigh vacuum, a relatively large amount of liquid metal at a reasonably high temperature,and the ability to decouple the melting process from the refining process.The presence of a high vacuum (0.13 Pa or I Im of Hg) allows various volatileimpurities (such as Mn and Mg in titanium) to evaporate, thus refining the liquid material. Since the metal is maintained as a liquid over the length of the refining portion ofthe hearth, a significant quantity of liquid metal is available for gravimetric separation ofHDI’s and high density IRI’s. This liquid volume also provides a medium for the dissolution of IRI causing particles of neutral or lower density and HDI’s.In comparison to EBCHR, the VAR process lacks the ability to decouple themelting process from the refining and solidification process. In VAR, if the operatorwishes a higher heat input, the only option is to increase the melting current which inturn increases the melt rate. An increase in the melt rate will use up more power in themelting of the material reducing or eliminating the desired effect of increasing the heatinput to the process (i.e. the thermal regime in the ingot remains unaffected). For a givenalloy system, the VAR process operates over narrow ranges of melt rate and ingotdiameter to reduce the occurrence of macrosegregation and surface defects. These qualityconsiderations allow for only small adjustments to the heat input to the process. In coldhearth renielting, the heat source is decoupled from the melt stock. Thus if a highersuperheat in the refining portion of the hearth is desired, the operator of the furnaceadjusts the beam control systems to deposit more power in the refining zone. This greatercontrol of the heat source in hearth remelting is the most significant advantage overVacuum Arc Remelting.The ability to control the power source in hearth remelting also allows a varietyof different shapes to be cast. Squares, slabs, rounds and hollow ingots have all beensuccessfully manufactured{3 1,44]. The decoupling of the power source from the8feedstock also reduces the amount of macrosegregation due to melt rate instabilities[5O,improves the surface quality of the finished ingot[44] and reduces the amount of productlost due to the formation of shrinkage cavities during final solidification[511.1.4.4. Electron Beam Cold Hearth Refining Process DisadvantagesApplication of electron beam technology in melting and refining processes is verycomplex requiring a broad knowledge of physics, vacuum systems, electronics andmetallurgical processes. This requires a very high level of operator competence. Inaddition, the various systems that make up an electron beam melting installation--transformers, guns, computer controls, vacuum equipment-- are costly making theprocess extremely capital intensive.One of the advantages of electron beam cold hearth remelting is also its largestdetractor. The high vacuum level that is responsible for the removal of volatile impuritiesalso results in the removal of desired alloying elements from titanium. This isparticularly important in the melting of titanium alloys such as Ti6A14V and Titanium 17that contain large portions of such elements as aluminum and chromium that have highvapour pressures at the processing temperatures[39,53,821. Thus chemistry control whenmelting these alloys becomes very difficult. Another problem associated with alloyremelting is the lack of chemical homogeneity that can result from poor mixing withinthe liquid pooi during ingot solidification.Current practice for the production of titanium alloy ingots is to melt the materialin an electron beam cold hearth furnace while trying to attain chemistries that are withinthe alloy specification. At the conclusion of EBCHR, the ingot is melted in a VAR furnace to provide better ingot structure for further forging and chemical homogeneity. Inaddition, the ingot chemistry can be altered by adding alloying elements that mayevaporate during the EBCHR step.9Obviously the VAR step is an added expense and it would be economically desirable to eliminate it. Extensive work has been undertaken by the US Government, enginemanufacturers and hearth melters to move towards a “hearth-melt only” or “in-spec”hearth melting process[72].1.5. The Thermal Regime During EBCHRThe success of the cold hearth refining processes depends largely on accurate andconsistent control of the thermal fields found within the melting hearth, refining hearthand casting area. In these areas the ability to dissolve and/or remove defect causingparticles as well as the ability to control chemistry is critically dependent on the controlof the thermal regime. Variations in the temperature field in operating electron beamfurnaces have been observed when pressure flucutations occur.While crucial to the success of the cold hearth refining processes, the heat transferprocesses that occur in EBCHR have received very little attention over recent years.Mathematical models by various researchers have focused on such things as fluid flowwithin the solidifying ingot, the solidification process itself and the steady state thermalprofile in the hearth region of the furnace. While these aspects of the process are important in the quantifying of the thermal fields generated during melting, refining andcasting, the process is also influenced strongly by parameters (such as residual chamberpressure) that affect the delivery of power by the electron beam to the various areas ofthe furnace. Unfortunately, this important aspect of the process has been largely ignoredprior to this work. Fundamental studies of power delivery in electron beam hearthremelting would provide a clearer understanding of the process leading to improvementsin product quality. Such studies would also enable the further development of betterpredictive models to characterize other factors affecting refining, melting and solidification in the EBCHR process.Chapter 2.: Literature ReviewThe heat transfer processes which occur in electron beam remelting are criticallydependent on a number of factors. Firstly, the power delivery to the meltstock isdependent on beam generation in an electron gun and the propagation of the beamthrough the melt chamber to the target. Once the beam strikes its target, the deposition ofenergy within the material is also important. Finally, the transport of the thermal energythrough the meltstock to the water cooled copper hearth and crucible both by convectiveand conductive mechanisms will also influence the resultant thermal regime.2.1. Electron Beam Generation and HeatingElectron Beam Remelting furnaces require a number of systems to operateproperly. Beam generation and control systems are responsible for the production of ahigh energy particle beam and the control of that beam on the work piece - in this casethe melting and refining hearths and the solidification area. In order for an electron beamto be generated and propagated to the hearth a high vacuum is required. This requirementimplies integrated beam generation/control and vacuum systems.2.1.1. Beam Generation, Propagation and ControlThe generation and control of electron beams are based on the physics of highenergy particles and electron optics and although many different types of electron gunsare manufactured ranging in power from the microwatt to the megawatt levels, thefundamentals of beam generation and control remain the same.A schematic diagram of a high power electron beam gun used for melting isshown in Figure 3. The first stage of beam generation is the formation of electrons. Formelting operations a tungsten electrode emitter is usually indirectly heated. At highvacuum -- usually around 1O Pa (lOs 1m Hg) in melting applications -- the tungstenemitter produces electrons. A high voltage (20- 50 kV for melting) is then appliedbetween the hot emitter or cathode and a positive electrode or anode. The resulting- 10 -11electrostatic fieldcauses the electrons to___—cabtc tonnctionbe pulled away fromthe cathode at extremely high speed witha high kinetic energy.The choice ofheating method, cathode size and shape, aswell as electrode configuration depends onthe application, the operating parameters ofthe gun, conditions inthe melt chamber anddesired cathode life.[5jModem meltingelectron guns can operate in two modesFigure 3: Schematic Diagram of a High Power Electron GunUsed in Melting Applications, space charge limited ortemperature controlled.In space charge mode, electrons are produced faster than they are accelerated away fromthe cathode resulting in an electron cloud in front of the cathode. This results in a stablebeam which is independent of the temperature of the cathode. Electron guns operating inspace charge mode are easier to control. In temperature controlled operation, as soon asan electron is emitted from the cathode it is accelerated towards the work piece. In thismode the rate at which electrons are generated is controlled by the temperature of the12cathode (or the rate of generation of electrons). Electron emission is a very sensitivefunction of temperature and as such the control of beam power during temperature controlled operation is very difficult.In addition to the two required electrodes (anode and cathode) a beam generationsystem may contain electrodes for focusing and shaping of the electron beam. Selectionand use of these extra electrodes will affect the main beam parameters such as currentdensity, aperture and size and shape of the beam spot.Once generated and focused, the beam can be manipulated using a series ofmagnetic lenses. These lenses control such parameters as location of the beam spot andcan influence current density, size and shape of the beam spot. In general the beamfocusing and deflection systems are computer controlled in most industrial meltingapplications.The details of beam generation and control for axial guns used in melting andother types of guns such as ring, line and transverse guns can be found in theliterature[4,69,74j.2.1.2. Energy Losses in Electron BombardmentEnergy losses in electron beams result from various beam interactions with thetarget material as shown in Figure 4. The energy losses due to x-ray generation,thermionic emissions and secondary electrons amount to less than 0.5% of the totalincident beam energy. Backscattered electrons can account for 10-40% of the incidentenergy depending on the material being melted and the accelerating voltage.[69]13BackscatteredSecondary 4 ElectronsElectrons\0000000 jF I I I F ThermionicElectron Beam Emissions...................... )‘ Interaction zone.Figure 4: The Various Interactions Between an Electron Beam and aSolid Target.The number of backscattered electrons is related to the atomic number of theirradiated material, the angle of incidence of the electron beam and the acceleratingvoltage as shown in Figures 5 and 6. The energy distribution of these backscatteredelectrons can also be determined experimentally and this is shown in Figure 7. Doing anBackscatterBeam80.0 100.0Figure 5: The Number of Backscattered Electrons as a Function ofIrradiated Material for a Normal Electron Beam.0.0 20.0 40.0 60.0Atomic Number14appropriate integration of the energy distribution results in the power losses due tobackscattered electrons (see Figure 8).BackscaUer::o.co 15Xn 3JXO 45fl 6JXO 75.co coAngIe fimNmil (Ies)Figure 6: The Effect of Beam Angle on the Number of Backscattered Electrons. 0.7‘back 0.6eUU 0.5‘back(pb) 1.0EeUbFigure 7: The Energy of Backscattered Electrons as aFunction of Irradiated Material.0.4 0.6 0.8150.40.3Backscatteredp 0.2Beam0.10.0Figure 8: The Power Losses Associated with BackscatteredElectrons as a Function of Irradiated Material.The application of these relationships to electron beam remelting is somewhatcomplicated by the presence of liquid material. Although the backscattering of electronsis independent of the physical state of the irradiated material, the surface of the targetmaterial during the melting operation is rarely planar especially in the region of the beamspot. During melting it has been observed that the liquid surface deforms, forming adepression at the beam spot[841.2.1.3. Other Sources of Beam Power LossBeam losses are also affected by the path the beam must take between the cathodeand the work piece. An optimally designed and manufactured beam generation systemreduces power losses at the anode to 0.1%. There may be additional losses in the beamguidance system (i.e. the magnetic lenses, deflection system and possibly the diaphragmorifices). These losses in the deflection system are caused by beam spreading due to theinherent space charge of the beam and collisions with gas atoms and molecules as thebeam passes through the components of the deflection and guidance systems.0 10 20 30 40 50 60 70 80 90Atomic Number16Stephan[76] indicates that at pressures lower than 0.013 Pa (10 torr) and withprecise adjustment of the lens current, the losses in the guidance system can be reducedto 1% or less. At higher pressures (0.13 Pa) losses in the beam guiding tube can increaseto 5 or 10%.Losses between the deflection system and the target are largely a function of thechamber pressure. Beam divergence due to the space charge[61 and scatter due tocollisions with the remaining gas atoms and metal vapour are all dependent on thechamber pressure. At lower pressures the beam divergence increases. At pressuresbetween iO Pa and 10 Pa space charge divergence will be so pronounced that anominal 60 kW beam will not melt tantalum[76].Bas, et al.[101, measured the space charge divergence using a 2 kW beam whichpassed from a pressure of Pa through a 5 mm diameter diaphragm into a 10-2 Paregion. It was noticed that the beam did not continue to diverge in the higher pressureregion but was focused. This divergence was accompanied by a decrease in the currentflow of the beam. It would appear that as the electrons strike the residual gas atoms, ionsform which counteract the space charge of the beam. These electrons also lose energy inthe process. As the chamber pressure rises further, the beam begins to diverge throughscatter and the two mechanisms reduce the beam intensity to the point where the beambecomes incapable of delivering adequate power to the target.Stephan also suggests that same phenomenon should occur at the surface of themelt due to metal atoms boiling off the surface.2.1.4. Measurement of Beam ParametersThe direct measurement of various beam parameters (such as focal spot diameteror beam current distribution), as a function of conditions in the beam generation systemand in the propagation region, is very difficult. The beam current and the current density-- and ultimately the beam power distribution -- can be measured using a thin, slitted17plate protecting a Faraday cup[20,33] to measure the current as the beam moved acrossthe slits. Other researchers[66,67] have used rotating probes to measure the beam currentdistribution in low power beams.Sze[77] used yttha coated graphite sheets to observe the beam trajectory andbeam waist of a line gun for various accelerating voltages and beam currents. In addition,the beam current density was measured using a Faraday collector having a narrow slitbetween water cooled tungsten jaws. The beam was translated across the slit by varyingthe magnetic field. Current distributions for accelerating voltages ranging from 0 - 10 kVwith gun currents from 0.38 to 1.5 A were measured. In addition to the Faraday collectormethod, a pinhole camera was used to form an image of the x-rays produced by the beamincident on the target surface. Measurement of the x-ray intensity then led to a measureof the beam current density on the target surface.2.1.5. SummaryIn general, power distribution of an electron beam on a target is dependent on alarge number of interdependent parameters. The mode of operation of the gun, thecathode heating method, the anode - cathode distance, gun vacuum, chamber vacuum,focusing current, beam focus location and beam deflection system all play a crucial rolein defining the current distribution (and thus the power distribution) in the beam spot.2.2. Electron Beam WeldingThe electron beam welding process is closely related to the electron beam hearthremelting process in that high power electron guns are used to melt metals in a controlledway. Thus the literature on the effects of various parameters on the power delivery inelectron beam welding can provide insight to the similar processes in electron beamremelting.182.2.1. Process DescriptionElectron Beam Welding (EBW) is a process of fusion joining in which energy isdeposited along the depth of a cavity as opposed to being transferred from the surface byconvection and conduction as in arc welding processes. This cavity is produced when avery high energy electron beam impinges on the surface of the weld material. Since theenergy of the beam is deposited in a very shallow layer 1 jim thick), the weld materialis vapourized. The reactive effect of the evaporating metal forces the surrounding liquidaside exposing unmelted material. The weld pool very quickly reaches a quasi-steadystate which moves through the material as the electron beam is translated across theworkpiece. [27]As with electron beam melting a great deal of work has been done to try tounderstand the thermal regimes and fluid flow regimes that arise from using electronbeams in welding.The electron beam welding process differs from the electron beam meltingprocess in the following ways:• the guns used employ a much higher accelerating voltage (150 kV istypical) when compared to electron beam melting guns (typically50 kV),• the beam currents are usually small (100 mA),• the workpiece may be extremely close to the electron gun,• the beam spot is highly focused to enable precision welding and• working pressures range from 0.013 Pa (10 torr) to atmospheric inelectron beam welding.2.2.2. Pressure EffectsThe effect of pressure on the penetration of electron beam welds has beenexamined on an empirical basis by a number of researchers[13,48,75]. The results ofsome of these investigations are shown in Figure 9. Since the penetration of an electronbeam weld depends on parameters like the distance to the workpiece and the acceleratingvoltage of the gun it is difficult to make any fundamental conclusions about the effect of19pressure on EB power density in electronbeam remelting based on research inelectron beam welding machines.Other fundamental studies of theeffect of pressure on beam powerdistribution have been done for electronbeam welding guns. Dumonte[20} used hisFaraday cage device to measure thecurrent distribution and thus the powerdensity distribution as a function ofFigure 9: The Effect of Pressure onPenetration in Electron Beam pressure at both 30 and 60 kV and 90Welds.mAmps. In this study the pressure wasvaried from 1.3 x iO Pa to 14.3 Pa (10.2- 107.3 im Hg). The results obtained from this study indicate that, at 60 kV and withworking distances less than 525 mm, welds obtained between 1.3 x iO and 6.5 Pa arealmost identical. Schiller[69] presents experimental work on the current density over apressure range of 10-2 to 15 Pa (0.08 - 112.5 im Hg) shown in Figure 10. This workindicates that with an accelerating voltage of 60 kV and a 500 mm beam path, scatteringhas very little effect on the electron beam up until about 4 Pa (30. im Hg) In the pressurerange 5-15 Pa (37.5 - 112.5 pm Hg) there is a significant effect on the currentdistribution within the focal spot.I20Figure 10: Measured Current Density Distribution in theFocal Spot at Various Working Pressures in aWelding Chamber (UB = 60 kV, ‘B = 90 mA,ZB = 525 mm.)Schiller[68] has done experiments measuring the thermal efficiency of powertransfer for high accelerating voltage electron beam guns at various pressures and underdifferent focusing conditions. The results of his experiments can be seen in Figures 11and 12. During his experiments, Schiller found:• The incident energy and energy transmitted to the workpiece decreasemonotonically at pressures up to 1300 Pa (10 torr).• With adequate power, working distance and focusing current, theefficiency of energy transfer, r= Q /Qmcjd , remains at about 0.98even at higher pressures.• The energy transmitted to the workpiece is larger than the expectedvalues based on theory of elastic scattering. This implies a portion ofthe scattered electrons also reaches the workpiece.• Energy transmitted to the workpiece at some pressures higher than0.013 Pa (10 torr) was increased over the low pressure region. Thisindicates some self-focusing effects.• The distribution of the power density in the central zone is a normalGaussian distribution.Schiller suggests that the distribution could be represented by the followingequation:.4I, — 4Pti—0.6—t —0.2 Dl 0.4 mm Ep(r)=pje’ +p2e_a2) (1)21Figure 11: Relative Power Input, Absorbed Power, Depth of Weld in NiCr-Steel andUnscattered Part of EB Current as a Function of Pressure for UB=75kV andPower of 5.1 kW at Optimal Focus.Figure 12: Power Absorbed by the Workpiece as a Function of Pressure for a 8.7 kWBeam at 100 kV.where Pi P2 and (Y2 are functions of pressure and a1 is a constant and r is the radius. Inthis formulation, P1 would tend to zero at high pressure and P2 would be zero at lowpressures.(p)b*.•0NpIIp2.2.3. Mathematical ModelsMathematical models of welding processes are generally models in which a heatsource of some description is moved across a semi-infinite solid. The heat source may bea single point or line[65], a two-dimensional source[56,62,83] or a three dimensional22source[28}. The application of a specific model to a specific process is thus dependent onthe description of the heat source.-The effects of beam convergence angle, focal spot size and distance from thefocal spot to the workpiece have been analyzed using a planar heat source model[83]. Inthis investigation the energy flux was assumed to be a Gaussian distribution in anyhorizontal cross section of the beam. Using the mathematical model, the authors wereable to conclude that a complete specification of beam convergence angle, energydistribution at the focal spot and the location of the focal spot relative to the workpiecesurface was required to determine the effects of beam focusing characteristics in welding.In addition, the depth of penetration and diameter of the cavity are primarily controlledby the energy distribution at the workpiece surface. Finally the model shows that aprecise determination of the energy flux at the workpiece (which is a function of focalspot location relative to the surface, energy distribution at the focal spot and convergenceangle) is required to reduce experimental error. The investigation did not address thespecific effects of pressure on the energy distribution at the focal spot of the electronbeam.In the three-dimensional analysis of heat flow in welds, the energy may beassumed to be distributed in a volume described by a double ellipsoid[281. In this casethe energy distribution is Gaussian. The authors of this model claim that it can be appliedto all sorts of welding processes (arc, electron beam or laser) provided the parameters ofthe ellipsoid and power distribution are chosen correctly. The model predictions werecompared to thermal histories and weld pool shapes of actual welds and were found topredict the experiments reasonably well.2.2.4. SummaryThe effect of work chamber pressure in electron beam welding is reasonably wellunderstood in a qualitative sense. Mathematical models have been used to emphasize that23the determination and specification of the energy distribution at the beam spot is crucialto the quantitative understanding of the process.2.3. Electron Beam RemeltingThe fundamentals of the delivery of energy to the workpiece in electron beammelting have not been studied in as much detail as that of the electron beam weldingprocess. Most of the attention in electron beam melting has been on the development ofmathematical models to determine the nature of the heat flow once power has beendelivered to the melt stock.2.3.1. Ingot ModelsEarly work on the thermal regime in the consumable ingot remeltingprocesses[16,22,23,54,63] led to the development of unsteady-state heat transfermodels[7,25]. These models incorporated the unsteady state nature of the heat flowprocess by allowing an ingot to grow in the water cooled crucible. The models includedtemperature dependent thermal physical properties, radiation and conduction boundaryconditions and a fixed temperature condition on the ingot surface. Fluid flow within theliquid was characterized using an artificially increased thermal conductivity in the liquidmaterial and the latent heat of solidification was released using an increased heat capacityin the so-called “mushy” zone. The models have been verified against ingots produced inthe VAR process and good agreement between predicted and measured pooi profiles hasbeen obtained[7,25,52]. The major deficiency in these ingot models is the inability topredict the thermal regime from only melt rate information. Since the models require afixed top surface temperature distribution on the ingot surface, this distribution must beinferred from process conditions such as melt rate, melting current and melting voltage.As there is no strong theoretical relationship between these parameters and the fixedsurface temperature distribution, the models are empirical at least in this respect.24The fundamental physical processes taking place on the surface of an electronbeam remelted ingot are much easier to characterize than those in VAR. Modificationshave been made to earlier models to reflect the energy transfer processes actuallyoccurring at the surface of an electron beam remelted ingot[12,l 1,55]. Beam input powerwas distributed on the surface and radiation allowed to occur. The calculated toptemperature distribution was fitted to a measured one using a series of adjustableparameters including a beam loss factor and an additional thennal conductivitymultiplying term to account for the surface tension driven flows on the surface of theingot. The model results were compared to both an experimentally determined surfacetemperature distribution and an ingot pooi profile. Model and experimental results werefound to agree quite favourably.Other models have examined the coupled heat transfer fluid flow problem in theVAR process[40,85]. One model was used to determine and predict the effects of meltingcurrent on the macrosegregation in a Uranium 6w/o Niobium alloy being vacuum arcremelted[85j. This model uses a radiation heat balance and a simplified model of therernelting electrode to determine the top temperature distribution, then imposes thisdistribution on the ingot surface. Another approach is to couple the heat and fluid flowequations when the shape of the molten pool is known apriori[40]. Such a model iscapable of predicting fluid flow velocities and thermal transport number during ingotmelting. The requirement for prior knowledge as to the nature of the liquid pool shape isa serious detriment to this type of model, however.Other work on the electron beam remelting of Uranium 6 w/o Niobium[70] hasshown that inertially driven recirculation interacts with a counter rotating buoyancydriven recirculation. Both of these flows heavily influence the shape of the liquid pool.The ic-s model of turbulent flow has also been applied to the electron beamremelting process.[73]. The model is a coupled fluid flow, heat transfer treatment of theEBR process. Unlike earlier models this model calculates the location of the liquid/solid25interface and also takes into consideration the formation of a “mushy” zone during thesolidification process. The model also predicts pool profiles that are reasonably close tothose seen in practice. This approach predicts a pool shape that is closer to the actualshape than does the earlier work which treats fluid flow by increasing the thermalconductivity.2.3.2. Hearth ModelsMathematical models of the electron beam cold hearth remelting furnace havebeen developed[3,12,64,80,841. Initial work was done using the finite differencetechnique, temperature variant thermophysical properties, thermal conductivityenhancement to simulate fluid flow and a time averaged power distribution on thesurface of the hearth[801. This model was verified using some independent observationsof hearth furnaces in operation, some x-ray photographs of the hearth at Axel Johnsonand some rudimentary experiments on small pieces of titanium heated with an electronbeam. Other researchers have developed a coupled heat transfer, fluid flow model of thehearth region of the electron beam furnace[12j. This model includes energy transport dueto the turbulent nature of the fluid flow and makes some time. averaged assumptionsabout the power distribution to the ingot surface. Unfortunately, the authors do not haveaccess to operating parameters and industrial data to validate their model in any way. Atthis time, the applicability of the model is restricted to qualitative observations about thehearth remelting process. In other work, the evaporation of aluminum from a hearth hasbeen examined using a finite element model[841. In this model, a deforming mesh hasbeen employed to track both the liquid/solid interface and the liquid/vapour interface.2.3.3. Other Electron Beam Heat Transfer ModelsA number of heat transfer studies have been done in order to understand thedynamics of the thermal regime underneath the beam spot. These models have been usedprimarily to evaluate the effect of beam speed or scanning frequency on the evaporation26rate in electron beam processing. Nakamura[58] developed a heat transfer model topredict the evaporation rate of aluminum from titanium 6A1 4V alloy during electronbeam processing. This model was a 2-dimensional unsteady state treatment in which amoving heat source was used to calculate the thermal behaviour on the surface of a liquidbeing heated with an electron beam. The model was successful in showing that for highscanning rates (> 10 Hz), the time averaging technique for approximating the powerdistribution in either a hearth or ingot model is quite good.In a more fundamental study, the Navier-Stokes equations and the unsteady stateheat transfer equation in two dimensions were used to construct a mathematical model ofthe region just below the beam spot[42]. The pressure balance which results from theintense evaporation of material just under the beam spot -was also considered.Unfortunately, the computer model developed was unable to converge to a steady statesolution. The authors postulated that there may exist some type of chaotic flow in thethird dimension which would not allow the solution to converge. It is also distinctlypossible that the physical processes which are occurring under the beam impingementspot are such that the shape of the liquid pool is chaotic but stable.2.3.4. SummaryIn summary, the heat transfer processes which occur in electron beam melting arenot well understood. Initial research into the questions of heat flow, ingot solidificationand fluid flow in the electron beam remelting process have led to some insights into thefundamentals of the process. The state of mathematical modelling of the electron beamremelting process requires that the boundary conditions which are important bequantified exactly. This leads to the conclusion that the nature of power transfer to thesurface of an EB hearth or an EB ingot be investigated in some detail.Chapter 3.: Scope and Objectives3.1. Scope of the Research Programme-The major thrust of this research is to determine the effect of pressure on thepower transfer to material being melted using one or more moving electron beams. Oncethis relationship has been determined and supported by experimental and qualitativeevidence, the effect of chamber pressure on the hearth remelting and ingot solidificationprocesses can be determined.Due to the difficulties in measuring the power density distributiondirectly[20,68,77] it was necessary to evaluate this parameter using indirect methods.These methods include the extensive use of laboratory and industrial scale experimentalmelting trials (described in Chapter 4). In the experiments, thermocouples and the outputof a scanning infrared pyrometer are used to determine the thermal regime within a blockof material being electron beam remelted. In order to interpret the results of theexperimental work a heat transfer model capable of calculating the temperaturedistribution within the electron beam remelted material was developed. The formulationof the heat transfer model is described in Chapter 5.Following Cockcroft[171, finite element techniques have been used for both threedimensional and two dimensional heat transfer models of the electron beam meltingprocess. The finite element technique has been employed due to the ability to make useof an already developed solving engine and data visualization software library. Althoughthe experiments can be analyzed using a two dimensional analysis, the finite element programs have been developed into three dimensions as well. This capability allows forgreater flexibility in the analysis of the experimental data should the need arise. Finally,the finite element analysis lends itself inverse heat flux calculations from the calculatedtemperature distributions. Since the aim of the research is to determine the power density- 27 -28or heat flux distribution on the surface of the irradiated material, this capability is rathersignificant.Once formulated and developed, the mathematical model was validated using aseries of analytical solutions to simplified problems. In addition to comparisons to analytical solutions, the results of laboratory heating trials were used to validate the modeland investigate the electron beam heating and melting process.Once validated the results of the industrial experiments could be interpreted. Themodel was run using a fixed temperature boundary condition on the top surface (the temperatures at steady state having been measured using scanning infrared pyrometry). Thethermal regime thus generated was used to calculate a steady state heat flux that wouldproduce the equivalent surface temperature distribution. This heat flux distribution wasthen correlated to the various parameters used to generate the thermal data. The results ofthese calculations are discussed in Chapter 7.3.2. Objectives of the Research ProgrammeThe objectives of the research program can be summarized as follows:[1] To formulate, develop and verify a mathematical model capable ofcalculating the thermal field in a volume of material irradiated by anelectron beam during heating.[2] To calculate the temperature distribution of a volume of materialbeing heated in an electron beam furnace and to use these calculationsand the results of experiments on the laboratory and industrial level toassess the effect of process parameters on power delivery to thetarget.[3] To apply the mathematical model to the electron beam heating of avolume under different chamber pressures to determine therelationship between chamber pressure and power transfer in electronbeam melting.Chapter 4. : Experimental Procedures and ResultsIn order to investigate the effect of various process and beam generationparameters on the power delivered to the workpiece in electron beam remelting, anumber of experimental tests were conducted. These experiments ranged from lowpower, small scale laboratory experiments to high power, large scale trials in anoperating industrial hearth furnace at Axel Johnson Metals Inc. In this chapter, thevarious experimental configurations and procedures will be described and the results ofeach of the different experiments will be shown.4.1. Laboratory Scale Experimental ProceduresLaboratory scale experiments were conducted on two targets: a titanium cylinderand a tantalum cylinder.4.1.1. EquipmentThe laboratory scale experiments were all conducted in the Electron BeamFigure 13 : Photograph of the 30 kW Electron Beam Melting Facility atUBC.- 29 -30Furnace at UBC. This furnace is shown in Figure 13. The power supply wasmanufactured by North Hill Enterprises and is rated at 37.5 kW. The EB gun is a vonArdenne EH-30/20 with a nominal power rating of 30 kW at 20 kV. The gun can operatein either current limited mode or space charge limited mode depending on the gunsettings. The gun is controlled manually by operating dials which vary the filamentcurrent, the exciter voltage and the accelerating voltage. The position of the electronbeam pattern on the target is controlled using a static magnetic field which is manuallyset. Superimposed on the static field is a dynamic field which defmes a beam pattern.The electron beam pattern is controlled using an IBM personal computer containing twoChromatograph QuaTech WSB- 10 waveform generators.ValveRegulator ,.7’ These generators are capable of outputtingChamber synchronized waveforms which drive the xVacuumNeedle and y deflection coils independently. BeamValveArgon pattern amplitude, frequency and shape areCylindercontrolled using software developed at UBCduring the course of the experimental work.In some laboratory experimentsFigure 14: Schematic Diagram of the pressure was maintained by using aSystem Used For PressureControl in the Laboratory controlled flow of argon into the furnaceScale Experiments.chamber. The flow was controlled using anargon regulator, a gas chromatograph valve and the needle valve used to vent the furnace.A schematic of this arrangement is shown in Figure Tantalum ExperimentsA cylinder of tantalum measuring 152.4 mm in diameter by 101.6 mm in lengthwas obtained on loan from Cabot Corporation in Boyertown, Pennsylvania. Three holes2.38 mm in diameter and 76.2 mm long were drilled into the tantalum cylinder at 25.431mm, 50.8 mm and 76.2 mm from one surface of the cylinder (see Figure 15). Aphotograph of the tantalum cylinder before heating is shown in Figure 16. Threads weremachined at the end of each hole to accommodate a modified SWAG-LOC’ fitting.tungsten 3% rhenium- tungsten 26% rhenium (Type D) thermocouples were constructedand the wires fitted through 2.38 mm double bore alumina tubing. These thermocoupleswere placed in the two holes closest to the target surface. A single chromel alumel (TypeK) thermocouple was constructed and placed in the third hole. The fittings were thenused to hold the thermocouples firmly in place and provide good contact at the centerlineof the tantalum block. From the end of the solid alumina tube the bare wires of the TypeD thermocouples were threaded through small sections of double bore alumina tube toprovide protection from the heat and metal vapour and to allow the wires to be bent andattached to the thermocouple junctions.F— 152.4 mm The thermocoupleswere connected to a waterTC325.4 mm cooled copper junction inside25.4 mm the furnace. The temperatureTC2 1 101.6 mm25.4 mm at this junction was moniTCIX25.4 mm tored using the temperatureof the cooling water exitingFigure 15: Schematic Diagram of Tantalum Cylinder the furnace. Wires wereUsed in the Experiments.taken from the junction tovacuum feed-throughs inside a water cooled umbilical pipe. Thermocouple leads werethen attached to 2 Kippen-Zonen 2 pen chart recorders.Pressure was measured using an Edwards APG-L-NW16 pressure gauge head.The output of the gauge head was measured using one of the Kippen-Zonen chartrecorders.32Power was measured using the output dials on the EB 30/20 gun control panelmanufactured by North Hill Enterprises. The meters were checked periodically and thevalues of filament current, filament voltage, exciter current, exciter voltage, beam currentand beam voltage recorded by hand. The experimental error associated with the beamcurrent measurement is ± 0.05 Amps and with the beam voltage, ± 500 V.Figure 16: Photograph of the Tantalum Cylinder Before Heating Showing the Locationin the Furnace and the Location of the Thermocouple Assemblies.Distance Below Distance from ThermocoupleSurface Centerline Type(mm) (mm)25.4 0.0 W/WRh50.8 0.0 W/WRh76.2 0.0 Type KTypically each experiment proceeded as follows:• the chart recorders and timer were synchronized to one of the twocharts,Table 1: Thermocouple Positions in Tantalum Experiments Conducted at UBC.33• the filament was warmed up, the exciter voltage turned on and set atthe operating value-- this generally caused a low power beam (<0.5kW) to operate,• the beam accelerating voltage was increased until a visible beam spotwas present,• the beam was then centered on the tantalum block visually,• the beam pattern (a circle 5O.8 mm in diameter) was initiated,• and the beam power increased to the desired level.This procedure took 1.5 to 2 minutes to complete. During this time the powerlevel was constantly monitored.The tantalum block was heated three times, the results of which are shown inFigures 17 and 18. In the first two experiments (Figure 17 is typical), no attempt wasmade to control the pressure to a specific value and the pressure was easily maintainedbelow 0.15 Pa (1.13 jim Hg). In the third experiment (Figure 18), the pressure was raisedperiodically after allowing the thermocouples to reach a reasonably constant temperaturebefore altering the pressure. Since Test 11 was expected to last much longer than theprevious tantalum tests, the dial power was reduced to 75% of that used in the constantpressure experiments.34- 15- 12-90-I;c.-3-o00I-.1250100075050025000.60 250 500 750 1000Time (seconds)Figure 17: Results of Tantalum Experiment Designated Test 9.35i500 1.0 —1- 0.9-0.8 -8- 0.71000&o-0.6 -6-IzE-0.4,. -4500- 0.3-0.2 -2- 0.10 —- o.o - 00 1000 2000 3000 4000Time (seconds)Figure 18: Results of Tantalum Experiment Designated Test Titanium ExperimentsFor the titanium experiments a cylinder 203.2 mm in diameter by 161.9 mm inheight was used. Thermocouples were located at or near the centerline at distances of20.1 mm, 45.5 mm, 75.2 mm, 100.3 mm and 125.7 mm below the surface. In each setthe top two thermocouples were W3%Rh / W26%Rh while the remaining thermocoupleswere Type K (chromel-alumel). The location of the thermocouples is summarized inTable 2 and shown in Figure 19. The thermocouples were held in place using a systemsimilar to that used in the tantalum experiments.EL I11 4.- A•:..IIIiIIa.,• a• ‘.• a TC1>A TC2AA7]at-•Aa• TC- A 0 DialPowera__________Prswe36I1,2O.lmmc 25.4mm29.7mm25.1 mm15.4mm203.2mm161.9mmFigure 19: Schematic Diagram of the Titanium Cylinder Showing the Locations of theThennocouples.Each experiment was conducted using a procedure identical to that of thetantalum experiments. At the conclusion of the experiment, the cylinder was allowed tocool and a transparent film placed over the target surface. The pool boundary was thentraced onto the transparent film.Table 2: Thermocouple Positions in Titanium Experiments Conducted at UBC.Distance Distance ThermoBelow from coupleSurface Centerline Type(mm) (mm)20.1 2.38 W/WRh45.5 0.0 WIWRh75.2 0.0 Type K100.3 0.0 Type K125.7 0.0 Type KFour experiments were conducted without modifying the pressure within thefurnace during the course of the experiment. Typical results of the constant pressureexperiments (Tests 4, 5, 6 and 8) are shown in Figures 20 to 22. The pressure wasmaintained at a constant value during each of these experiments and ranged from 0.02 Pain Test 4 to 0.33 Pa in Test 6.37:1500 -1.0aD0a-0.9Qaa[T I >1250 1aDalia f Tf JLH.,_i000 [I aj0 a!-0.6a%-••a. VVVVVVV,,11 -- 0.5S•750 VTV AAAAAAAAAA4aAA,VVTAA.1 -0.4 -•.,V. -500 - aA TC2 - 0.3•• V A•V AV A. V.V250- • A•• 1t4-0.2a -a V AA Da1 Power• V A• -0.1• V A • Pressure0 I I I I —0.00 500 1000 1500 2000 2500 3000Time (seconds)Figure 20: Results of Titanium Experiment Designated Test 4.10-8-7-6-5-4-3-2-1-0CCD-IFigure 21: Appearance of the Titanium Cylinder at the Conclusion of Test 4.38Figure 22: Tracing of the Extent of the Liquid Pool as Taken at the Conclusion of Test 4.In Test 7, the pressure was allowed to vary during the course of the experiment.At each pressure regime, the pressure was held constant over a period of about 5 minutesto allow the top thermocouple to attain steady state. The experiment was done on a verywet day and the chamber had been open earlier in the day (approximately 4 hours earlier)for a 20 minute period for the post-mortem on Test 6. When power was initially turnedup (0.7 kW) the pressure sensor measured a pressure spike. This was attributed to therelease of adsorbed water on surfaces on the inside of the furnace.The run proceeded without complication after the initial pressure spike. As seenin Figure 23, the pressure was increased over a two orders of magnitude range atapproximately 1600 seconds from power on. This two orders of magnitude correspondsto 0.4 V on the pressure sensor output at the low end of the sensitivity range. No arcingor glow discharge was visible in the chamber at this pressure level. On subsequentpressure increases a glow discharge in the chamber began to appear. This discharge was abright blue by the time the pressure had approached about 1.07 Pa.At around 3700 seconds and 1.07 Pa pressure the pressure became unstablemaking several peaks, the maximum of which was about 3.33 Pa. During this time themolten pool appeared to be moving across the face of the titanium block. At roughly394200 seconds from power on, the beam generation system failed and the experiment wasended. Pressure spikes on the same order of magnitude as experienced while the beamwas on were seen at the end of the experiment as well.After the experiment was concluded and the block allowed to cool down over atwo day period, transparent plastic film was placed over the surface of the block and thelocation of the liquid pooi traced out. As Figure 25 shows, the pooi appears to havemoved on at least four occasions from the original starting point. Pool diameter isdifficult to measure given that the evidence of the initial pool site has been eradicated.During Test 8, there were no pressure instabilities within the chamber similar tothose experienced at the end of Test 7. Such instabilities would have indicated a vacuumsystem malfunction. At the conclusion of Test 8, a through diameter section was takenand the cylinder macroetched to show the location of the liquid pooi. The macrograph isshown in Figure 26.4.0-10-900aa Dial Power:: of aO UHTht0 •jA 0750 - TC4• , rcza • -5a ‘ AAA—•VA••v A• A50(3- V AA•V A• A •___-2• V250 - A I PressureA • 0.5.• I • I -1I—I--oo 1000 2000 3000Time (seconds)a a aa a a a —°TCS a3. 23: Results of Titanium Experiment Designated Test 740Figure24:AppearanceoftheTitaniumCylinderattheConclusionofTest7.Figure25:TracingoftheExtentoftheLiquidPoolasTakenattheConclusionofTest7.41Figure 26: Photomacrograph of the Through Diameter Cross Section of the TitaniumCylinder Showing the Pool Developed During Test Experimental DifficultiesDuring the course of the laboratory experimental work, a number of problemswere encountered. In all of the experiments, the startup sequence of generating a beam,centering it, adjusting the beam pattern and then centering the pattern had to be donevisually due to equipment limitations. This resulted in some variation of the beam patternsize and location from experiment to experiment.During the individual experiments a variety of difficulties arose. The tantalumblock was placed in the furnace and the thermocouples installed on a very wet day. Theprocedure itself took on the order of five hours to accomplish. This allowed a largeamount of water to adsorb on the inside surface of the chamber.During the first tantalum trial (Test 9), a blue glow was observed on the interiorback wall of the furnace. A significant instability in the ultimate vacuum level prior to10mm42heating was also seen. After the initial experiment was concluded and the tantalum blockwas allowed to cool, the furnace was opened and a substance-- which appeared to bediffusion pump oil -- was removed from the furnace wall.The chamber was then closed and evacuated for a period of several days duringwhich hot water was passed through the chamber walls to encourage desorption of water.At this point, the next experiment (Test 10) was conducted. The ultimate vacuumattained prior to Test 10 was much superior to that attained during Test 9. As thetantalum cylinder was heated it radiated to the furnace walls causing them to heat up.This heating caused any remaining water to quit the surface of the chamber wall and betaken into the pumping system. During this experiment the pressure also varied significantly but was under much better control than during Test 9.Test 11 was hampered by a power interruption due to a failure in the gun at 33.7minutes, the beam was recovered and turned back on within 60 seconds.During the last two experiments (Test 10 and Test 11), some difficulties wereexperienced with thermocouple #3 (closest to the target surface). In both experiments, thethermocouple output dropped significantly just after the power was increased to thedesired level. After a period of time (400 seconds in Test 10 and 300 seconds in Test 11)it appeared that the output of the thermocouple had attained levels consistent with theother two thermocouples. This aberration was attributed to differential expansion in thealumina tubing containing the thermocouple wires that caused the wires to twist and shortout between the fitting and the water cooled copper junction. After a period of time thetube reached an equilibrium and the wires were then untwisted removing the short.Other than the pressure spike at the onset of experiment 7 and the glow dischargeand beam pattern movement observed at the end of the same experiment, the titaniumexperiments were carried out without additional problems.434.2. Industrial Scale Experiments4.2.1. EquipmentThe experiments were conducted at Axel Johnson Metals Inc. in Morgantown,Pennsylvania. They were carried out by personnel from Axel Johnson and GeneralElectric Aircraft Engines.A cylindrical stub of commercial purity titanium was placed inside the ingotcasting mould of the older furnace at AJMI (this furnace has 4 600 kW von Ardenne EBguns for a total rated capacity of 2.0 MW). The stub was 500 mm in diameter by 482.6mm in length.Horizontal holes 9.5 mm in diameter were drilled in the cylinder. Tantalumsheathed tungsten/tungsten rhenium thermocouples were inserted in the holes and held inH place with compression fittings (see500.0mm Figure 27 and Table 3). As well as2SAmm recording the temperatures from the762=§ .. embedded thermocouples, a near101.6=infra-red scanning pyrometer was101.6 482.6 mmused to measure the temperature atselected points as a function of time.The MR camera was also used toperiodically scan the top surface ofFigure 27: Schematic Diagram of Titanium the cylinder to obtain the surfaceCylinder Used for Experiments atAJMI. temperature distribution.44Distance Distance DistanceBelow From FromSurface Centerline Centerline(mm) (mm) (mm)25.4 0. (#1) N/A101.6 0. (#2) 224.7 (#5)203.2 0. (#3) 224.7 (#6)304.8 0. (#4) 224.7 (#7)Power was applied to the surface ofthe cylinder using the power distributionshown schematically in Figure 28. Thepower distribution consisted of a series of 8spots in a circle about the center of thetarget. The beam spots described an annulusTC’s of inside diameter 76.2 mm and an outsidediameter of 175.3 mm. The beam patternwas generated using Axel Johnson’s beamsweep system. During the experiments oneof the beam spots (#5) was accidentally leftflatter than the others.Over a two day period the cylinder was subjected to a series of heat up and cooldown cycles at power levels ranging from 11 kW to 89 kW. Three different chamberpressures were examined - 0.073 Pa, 0.36 Pa and 4.4 Pa. These conditions aresummarized in Table Industrial ResultsDuring the experiments one of the thermocouples attached to the surface of thecylinder became unresponsive. Temperature at the six other thermocouple positions,power, chamber pressure and the output of the NIR camera were recorded at 10 secondTable 3 Location of the Thermocouples Used in the Axel Johnson ExperimentsCentrllne TCsFigure 28: The Power Distribution Usedin the Axel JohnsonExperiments.45intervals. The raw data was filtered by examining temperature and pressure at eachminute and time averaging both the NIR camera output and the applied power level overone minute intervals. In order to break the problem down to a manageable size, theexperiments were split into eight parts (arbitrarily separated into data files provided byAJMI). Power was applied to the target during five of the eight data segments. Typicalexperimental results for each of these five segments of interest are shown in Figures 30and 31.‘. .-..... .Figure 29: Photograph of the Titanium Cylinder Used in the Industrial ExperimentsTaken Before the Experiments Occurred.46Table 4 : Typical Process Conditions for Axel Johnson ExperimentsSegment Date/Time Chamber Pressure Power Focus Current(Pa) (kW) (Amp)A 01-Sep-92; 15:39 - 19:54 0.035 43.13 0.182B 01-Sep-92; 19:54 - 20:30 CoolingC 01-Sep-92; 20:30 - 00:43 0.240 I 43.07 0.185D 02-Sep-92; 00:43 - 04:58 CoolingE 02-Sep-92; 04:58 - 09:3 1 CoolingF 02-Sep-92; 09:3 1 - 15:45 0.03 - 0.08 33- 50 0.097 - 0.184G 02-Sep-92; 15:45 - 17:30 0.046 42.44 0.140H 02-Sep-92; 17:59 - 21:03 4.270 89.00 0.2051500-1000a. i.iiuii. — 50.......a....IIII..• A1•11••• A A2.....I.I • A,I I • A, -40•_____I I ° A1..III 0 A1IV I.I •—.—.—.—.—. p1:• p2:. S 0ALALAAALAAAAAA•A.J\LA- 20AL- 0.20- 0.15- 0.10a’0.055000100 150 2000-0.000 50Time (minutes)Figure 30: Applied Power, Chamber Pressure and Temperature Response of theTitanium Cylinder Used in The ATMI Experiments, Segment A, First Heat UpCycle.47‘I)1.0250 300 350 400 450Time (minutes)150011000500040300C20100Time (minutes)Figure 31: Applied Power, Chamber Pressure and Temperature Response of theTitanium Cylinder Used in The AJMI Experiments, Segment C, Second HeatUp Cycle.5002.01.50.5250 300 350 400 4500.050048At selected times during the course of a heat up cycle, dip thermocouplemeasurements were taken, and at other times, the MR camera was scanned in both the xand y directions to measure the top surface temperature distribution on the ingot. Thetime of each scan set and the segment in which it was taken is shown in Table 5.Table 5 : Time of Each of the NIR Scan Sets.Scan Name Time Data Segment2[2inX,linYj 17:39SepOl A3[2inX,2inYJ 18:32SepOl A4[2inX,2inYl 22:22SepOl C5[2inX,2inY] 22:37SepOl C6 [2 in X, 2 in YJ 23:02 Sep 01 C6E[2inX 2inY] 23:31 Sep01 Cat edge of pooi3R[6inX,6inY] 10:3OSepO2 F8 [2 in X, 2 in Yl 15:12 Sep 02 F9[2inX,2inYJ 15:54Sep02 F40A [2 in X, 2 in Yl 18:59 Sep02 GNumbers in brackets indicate the number of scans done in each direction.The output from the NIR camera required calibration in two ways. Firstly, thevoltage output of the camera was calibrated for temperature. This procedure was done atAxel Johnson using the melting point of CP titanium as the calibration point. A constantvalue for the emissivity of liquid titanium was assumed. After calibration, the output ofthe MR camera were converted to temperature using the modified Plank’s equation:T = [_273.15+in1+BAet (2)(v-C)where T is the temperature in degrees Celsius, V is the output of the MR camera in volts,(et) = 0.004230, C = 0, A = 1064330 at f8, and 587129 at f16 and B = 14974.2 at f8 and4915841.2 at f16. The aperture was set to f8 until part way through segment H at whichtime it was changed to f16.In addition to calibrating the MR camera for temperature, the scan was alsocalibrated for x and y distances. Axel Johnson supplied video tape output from the NIRcamera. The image taken on September 02, 1992 at 18:30 just after the doping of theliquid pool measures 177.8 mm in diameter on a television monitor. This corresponds tothe extent of the liquid pooi or 273.1 mm taken from the cross-section of the liquid pooias shown in Figure 32. To calibrate an optical scan using this information, the scan waslocated on the videotape. The periphery of the liquid pooi was assumed to correspond toan inflection point on the trace of the NIR camera output. The difference in voltageoutput of the distance location corresponds to a specific distance on the video imagewhich can be correlated to an actual distance using the calibration to the liquid pool. Forexample in scan 3YA, the voltage difference between the inflection points marking theedges of the liquid pooi is 2.0 V. The video image at the exact time of scan 3YA was127.0 mm in diameter. Thus the full scale distance (corresponding to 5.0 V) is calculatedto be 487.7 mm. From this information, and the output of the MR camera, a scan oftemperature as a function of distance can be obtained. Such a scan is shown in Figure 33.The MR camera has been accurate to within 3% in past measurements[26j. Thecamera designer believes the camera to be accurate to within 25 °C when the material isliquid but feels the camera is likely not accurate below the solidus temperature[371.50Maximum Depth31.73mmCylinder not level in furnaceGrid Area25.4 mmx25.4 mmFigure 32: Pool Cross Section Obtained by Doping the Axel Johnson Cylinder withCopper at 18:30 September 02, 1992.1900..uI•.18001700 T=1666°C1600-C-)1500 -a 1400-mill’a”1300 -1200-a aa- I‘a.0 50 100 150 200 250Radial Co-ordinate (mm)Figure 33: Temperature Distribution on the Surface of the ATMI Cylinder at 18:32-September 02, 1992, Obtained from the MR Camera.51Figure 34: Photograph of the Cylinder Used in the Industrial Experiments Taken Afterthe Experiments.Chapter 5.: Mathematical ModelTo interpret the results of the various experiments, it was necessary to develop amathematical model that could calculate the thermal regime in the target cylinder as afunction of time under various conditions.The mathematical model developed here makes use of the general finite elementengine developed by Cockcroft[17]. This approach was used in order to eliminate anyredundancy in the development of appropriate solving and time stepping techniques.While the “finite-element engine” is the same as that developed by Cockcroft, significantdevelopment work has been done to ensure that the model is representative of theelectron beam melting/heating process.5.1. FormulationThe governing differential equation for heat transfer in three dimensions is:( aT ( a’ ai a’ dT (3)with the boundary conditions of the formT = c1(x,y,z,t) on surface Si, fort> 0 (4)andaTk —ni +k +k —n +q(x,y,z,t)+ h(x,y,z,t)T = 0 (5ax ay azon surface S2. for t> 0with appropriate initial conditions;T=T0(x,y,z)indomain2,t=O (6)where S1 and S2 represent surfaces of an arbitrary domain 12 on which the boundary andinitial conditions are applied. The terms n, and n represent the directional cosines ofthe normal to the surface in question. For an isotropic material, the thermal conductivityis independent of direction and the directional subscripts on the thermal conductivity, k,can be dropped. Equation (3) is solved by applying a finite-element discretization of the- 52 -53spatial derivatives. The non-linear system of ordinary differential equations which resultsfrom this approach is then solved using a step-by-step recursion technique.5.1.1. Discretization of the Spatial DerivativesWithin a typical element the temperature may be assumed to obey the followingrelationship:Te(x,y,z)= N(x,y,z)T(t) (7)where, N1 are the nodal interpolation functions and n is the total number of nodes perelement. The interpolation functions (or shape functions) are dependent on the number ofnodes in each element and are usually simple polynomials of some degree depending onthe nature of the approximation. Application of the Galerkin criterion[18,35,86] andapplying equation (7) to the governing equation, equation (3), gives a system ofequations at the elemental levelCeI+KeT+r=o (8)dtwhere= ,f BTkBJ dV $S.h1NINJ dS2 (9)C JNPCpNjd’1 (10)= $ QN1 dV IS2 Nq1 dS2L. NJhTb dS2 (11)In equation (8), the K matrix is the so-called stiffness matrix or temperatureinfluence matrix, which is dependent on the B matrix (equations (12) and (13)) and thethermal conductivity matrix, k, defined in equation (14).B”=VTN (12)B=VN (13)k 0 0k= 0 k 0 (14)0 0 k154The force vector, f, equation (11), includes those terms which drive the systemsuch as internal sources of heat (Q), applied heat transfer coefficient boundary conditions(h) and external sources or sinks of heat (q).5.1.2. SolutionTo solve the problem for the entire volume, the system of equations for eachelement is assembled into a global system of equations of a form similar to that ofequation (8). Since the equations that make up the global system are non-linear they mustbe solved by a time-stepping procedure similar to the finite difference techniques. TheCockcroft engine uses a three-point recurrence or Dupont scheme, namely:K(3T1++i) + C(T1÷, —T1) (15)4where T represents the temperature at the ith time step.Equation (15) requires the temperature to be known at two successive time steps.This means that a two-point recurrence scheme must be employed initially to start theprocess. For this model the Crank-Nicolson technique has been used for the first timestep.K(T. +T. ) C(T. —T. )2+‘ =f (16)Equation (15) can be manipulated to:AT1=B (17)which can be explicitly solved. The matrix, A, is dimensioned n x n, where n is thenumber of nodes in the entire system. The matrix is generally banded and symmetricabout the diagonal. An in-core profile solver was employed for the solution of the systemof equations.5.1.3. Element TypeBased on a review of the element types available, Cockcroft chose the curvesided, isoparametric, quadratic temperature elements. Eight node (Figure 35) and twenty554 7 3 node (Figure 36) versions wereused in the two- and three8 4’ 6 8 -dimensional cases respectively.I . u For the purposes of this2investigation, the isoparametric1 5 2 5 formulation gives the ability toFigure 35: Eight-node, Isoparametric, Curve Sided represent elements of arbitraryElement in Both x-y Space and u-v(transformed) Space. shape (i.e. cylindrical shapes)with good accuracy and with areasonable amount of computational overhead. The ability of these types of elements torepresent any curved surface or volume arrives from the isoparametric formulation. Inthis formulation, the Cartesian co-ordinate system is transformed into a rectilinear coordinate system using the same interpolation functions as those used to approximate thetemperature across the finite element. The co-ordinate transformation is valid only at thelocal level. The penalty for using this type of element is a slightly increasedcomputational cost as equations (9) through (11) must be integrated numerically.7 18 6VFigure 36: Twenty-node, Isoparametric, Curve Sided Element in Both x-y-z Space andu-v-w (transformed) Space.2565.1.4. Numerical IntegrationThere are various procedures for numerical integration over the domain of anelement. Gauss-Quadrature or Legendre-Gauss methods are best suited to the finiteelement techniques since the number of function evaluations is kept to a minimum. Theexpressions for numerical integration in two- and three dimensions are as follows:f(u,v)dudv=W1f(u,v3) (18)j=I i=1ff’ f’ f(u, v, w) du dv dw WWJW1f(u1,v , Wk) (19)k=1 j=1 i=1where W, W and Wk are the weight factors at locations i, j and k, respectively, and m isthe number of integration (gauss) points within the domain of the element. For mostapplications involving quadratic temperature elements, 2 x 2 (two-dimensional) or 2 x 2x 2 (three-dimensional) Gauss points are adequate for integration.5.1.5. Treatment of Temperature Dependent Materials PropertiesThe problem of solving the heat transfer equation for a block of materialundergoing electron beam remelting involves a wide range of temperatures (25 °C -2000 °C) over which the material may go through one or more phase transformations(i.e. solid — liquid). In order to generate a solution that represents reality, the modelmust include temperature dependent thermophysical properties. The standard techniquefor handling this in the finite element method is to evaluate the various thermophysicalproperties at the gauss points during the generation of the elemental equations (equation.(8) - (16)). This approach introduces another non-linearity into a system of equationswhich is already non-linear.The non-linearity of the finite element equations represents a concern for theaccuracy of the solution. Generally, non-linearities are handled using some sort ofiteration scheme (e.g. Newton-Raphson method) over an individual time step until thesolution between successive iterations differs by less than a specified tolerance. Due to57the numerically intensive nature of the problem, this approach was deemedcomputationally too expensive.Instead of an iterative scheme, Cockcroft chose to develop code which woulddynamically alter the time step based on the results of the previous two time steps. In thisapproach, the time step is set such that the maximum temperature change between twosuccessive time steps is less than some user specified value. Thus for rapid heating orcooling problems, a small value for this maximum temperature change results in thethermophysical properties being updated over very small temperature changes. Inaddition to ensuring the accuracy of the model, this method makes the computer codemuch more efficient as the physical system approaches steady state. As the temperatureschange slowly, the time step is chosen to be significantly large to allow faster timestepping over long simulation times. To greater improve the resolution of the temperaturedependent properties of the material, Cockcroft chose to use a three point Gauss-Quadrature integration scheme.5.1.6. Phase ChangesTitanium (as an example) undergoes at least two complete phase changes in thetemperature ranges typical in this study. This makes it necessary for the model to accountfor the consumption (or release) of heat during these phase transformations. There arenumerous methods for accounting for the consumption of latent heat of transformation ina mathematical model. In situations where the latent heat of transformation does notdominate the process there are mainly two methods 1) the distributed source methodand 2) the heat capacity method. In the distributed source method, the latent heat oftransformation is included as a distributed heat source, Q, see equation (11). This methodis subject to instability problems because the distributed heat source term arises in thedefinition of the forcing vector, f. In his engine, Cockcroft chose to use the heat capacitymethod. In this approach, the heat capacity of the material is artifiàially raised to account58for the latent heat as in equation (20). As the heat capacity arises in the definition of thecapacitance matrix, C, (see equation (10)) an increase in this component represents anincrease in the damping of the system and this is numerically more stable. The effectiveheat capacity through a phase change is defined asc = +; (20)transfonnaticiwhere Alltransformarion is the latent heat of the transformation and Tansfor,tion is thetemperature range over which the transformation takes place.When using the heat capacity method, care must be taken to guard against thepossibility of one or more integration points skipping the peak in the heat capacity curvecausing the heat of transformation to be missed. There are various computationalmethods for avoiding this situation. In his FEM engine, Cockcroft chose to useLemmons method, an enthalpy technique, where the heat capacity over thetransformation range is calculated based on the enthalpy function and temperaturegradients. The resulting expressions for the effective heat capacity per unit volume in twoand three dimensions are:Ceff _1(dH/th)2+(/dy)2 (21)‘) ijdT/dx)2+(dT/ y1(H/ +(dH/dy)2+(dH/dz)2 (22)p (dT/dx)+(dT/dyH z5.2. Basic Verification of the Computer CodeCockcroft verified his computer code quite extensively using analytical solutionsfor various specific conditions. This verification included one-dimensional heatconduction with a heat-transfer coefficient boundary condition and one-dimensional heatconduction with a phase change. As the computer code is unchanged, with respect to howthe boundary conditions associated with such a problem are handled, reverification wasdeemed unnecessary.59Cockcroft restricted his verification procedures to rectangular co-ordinates,however. Thus he did not examine the specific case of a two-dimensional axi-symmetricproblem.The differential equation for heat flow in a cylinder of a material whoseproperties are invariant with respect to temperature and there are not gradients in thetheta direction is:2T lT 2T 1 T—+——+-—=——- (23).ar2 rar az2 atTo check the finite element engine against an analytical solution for the cylindrical case,the problem of convective heat flow from the surface of an infinite cylinder wasconsidered. The problem is stated mathematically as:=h(T—TjT(r) = T0, t=O (24).-oz=O — aZ z=Z—The analytical solution of the problem described by Equations (23) and (24) has beenobtained in the literature[15] and is given by Equation (25).T2 (25)T0,(2+(RH))0()whereHzkand the eigenvalues I3ri’ n = 1, 2, 3, ..., are the roots of= J0(13).where and J1 are the Bessel functions of the zeroth and first order respectively.The solution to Equation (25) was obtained using programs developed by othersat UBC[32] for conditions which are similar to those found in the electron beam heatingof titanium. Thermophysical properties were set to the average for titanium over the60()0E2r=98.25 mmI I I I I I I0 20 40 60 80 1.00Time (seconds)Figure 37: Comparison Between the Finite Element Solution and the Analytical Solutionfor an Infinite Cylinder Losing Heat by Convection for Various InitialTemperatures (k = 0.2 W cm1 °C’, Cp = 0.5 J g1 O4, p = 4.54 g cm-3)Since the electron beam remelting process involves a heat flux boundarycondition, an analytical solution for a constant heat flux applied to one plane of a slab hasbeen obtained[15]. In the case when all but one end of a cylinder are perfectly insulated,the solution is also valid for conduction down the length of the cylinder. The boundaryand initial conditions for the problem are defined as follows:temperature range expected in the electron beam heating and were independent oftemperature to allow for the analytical solution. The heat transfer coefficient (h) waschosen to give a heat flux equivalent to that expected for radiation at the initialtemperature. A comparison between the finite element solution and the analyticalsolution is shown in Figure 37.r=98.25mm900 -Solid Lines are Analytical SolutionSymbols are the FEM SolutionR = 101.6mm61=0rRaT1.lz=o=O=Q0aZIZ...LT(z,r) = 0, t=0The solution to the problem defined by Equations 23 and 26 is given asT= Q0t + Q0L f 3z2 —L2 2 (—l) , nirze L cos—pCL k 6L2 jt2 n2 L(26)(27).For verification, thermophysical properties were set to averages for titanium (asdiscussed earlier). The heat flux (Q0)was set such that the total heat input to the cylinderwas 8 kW (approximately the power input in the titanium experiments). Figure 38 showsthe comparison between the analytical solution and the finite element model.150000—1000250000 500 1000 1500Time (seconds)Figure 38: Comparison Between the Analytical Solution For a Cylinder with a ConstantHeat Flux Applied to One End and the Finite Element Solution, (k = 0.2 Wcm oC4 C = 0.5 J g1 0l, p = 4.54 g cm3,Q0 =25 W cm2).625.3. Inverse Heat Transfer CalculationsNumerical techniques can also be used to determine the heat flux distribution onthe basis of temperatures measured with embedded thermocouples. In one technique[38],a heat flux distribution is assumed, and a model used to predict temperatures at thethermocouple locations. Using a least squares formulation, changes in the assumed heatflux distribution are calculated to better predict the measured temperatures. This methodcan be used with either a Finite Difference model or a FEM formulation. The techniquerequires at least one thermocouple for each point on the heat flux distribution to becalculated.Given certain information -- namely the top surface temperature distribution inthe material being heated at steady state -- the current FEM formulation can be used tocalculate the heat flux on the top surface. In the industrial experiments conducted at AxelJohnson Metals, a scanning near-infrared pyrometer was used to measure the top surfacetemperature distribution at conditions approaching steady state. Thus it is possible tocalculate the heat flux required to produce this distribution using the followingformulation.In cartesian coordinates, the finite element 2D transient heat flow equation withapproriate boundary conditions is written asr etti aT oN. aT oN.—“ dxdy+![ x ax ay dy(28).f[k.z+k_iiy]N1ds_j1pcpNZdxdy =0Using the finite element method in approximation to the variation in temperatureover the domain of the element the equation becomes_ff[k[j{Te}ç+ k[jfT1]dxdY (29)+fqN1ds (30)63+ hjN]TC}N1ds— fhTN1ds (31)_11pcLNj—dxdy (32)Where terms (30) and (31) represent the contributions of the heat flux boundarycondition and the convective/radiative boundary condition respectively.At steady state the capacitance term (Equation (32)) is zero and we can then writethe equations in matrix form (making the substitutions which are required when using thefinite element method) arriving at{Kj{r}_{çJ +[K]{Te }— f:} = 0 (33).where f: } and [f: } are the temperature independent components of terms 30 and 31.The heat flux profile that we wish to obtain from the temperature distribution isthe global vector fq }. Rearranging Equation (33) gives{c}=[Ke+KjTe}_{f:} (34).The stiffness matrix K is known or can be calculated from known quantitiessuch as the thermal conductivity provided that the elemental temperatures are alsoknown. The heat transfer coefficient matrix K can also be calculated provided theelemental temperatures are known as can {f}. Therefore on an elemental level the valueof {ç} can be determined.Equation (30) can be rewritten using standard finite element method techniquesand Gauss quadrature integration as{ç}=wq1fNJds (35).For cylindrical coordinates Equation 35 becomes[f}= +J (36)64for the particular surface of interest (the equation is dependent on the integration locationand direction as indicated by the subscript “k” on the terms of the Jacobian used tocalculate the value of ds).For any given element lying on a boundary for which a heat flux boundary isrequired, there will only be three entries into {ç }. We can thus expand the force vectorinto the following formN1_______(37)f2 = 27tw1iq N2 (4j +i=1 N3For the jth element this equation isf = 2it(w1rqjN(gj1 +j2 ) +w2rN(4J1+ J2 ) +w3rqN(4J1+ (38).Similar expansion in the other entries into the force vector yields a system ofthree equations such thatw1rN1J w1rN2J w1rjN3J q,f2 = 2it w2rN1J w2rN2J w2rN3J q,f3 w3rN1J w3rN2J w3rN3J q,or{ç} = 2t[S]{q, } (39).where [SI is the coefficient matrix representing the coefficients of integration for theforce vector and ? = .JJ + J.Inverting the matrix [S] then gives the values of the heat flux at the Gauss points.Using the shape functions to describe the values of q at the Gauss points as a function ofthe heat flux at the nodes leads to{q } = [Njfq } (40)where the rows of the matrix [N] are the row vectors of the shape functions evaluated ateach Gauss point.65The solution to equation (40) gives the values of the heat flux at the nodes foreach element. These values can then be combined to produce the heat flux distribution.Since the elements used in the program are only “zerot’-order continuous they are notContinuous Ofl the first derivatives and the values at the corner nodes do not necessarilyagree between elements. The heat flux distribution was smoothed out by averaging thevarious contributions to each node from each associated element. This technique isanalogous to aproaches used in finite element stress analysis.5.4. SummaryThe computation engine developed by Cockcroft has been modified to reflect theconditions found in electron beam heating. The mathematical model has been verifiedagainst analytical solutions which reflect the conditions likely to be found in the electronbeam heating of titanium in the laboratory scale experiments. The comparison betweenthe mathematical model and the analytical solutions indicates that the formulation of themodel is numerically correct. An inverse heat transfer calculation method which can beused provided the surface temperature distribution is known has also been developed.Chapter 6.: Application of the Finite Element Model To Electron Beam Heatingand MeltingFor the finite element model of the electron beam heating or melting problemdescribed in Chapter 5 to be completed, boundary conditions, initial conditions andmaterials properties must also be specified.6.1. Boundary Conditions6.1.1. DomainCylindrical co-ordinates have been used in all of the specifications of themathematical model. Thus the physical domain can be described as:0.0 r RmaxO.O92ir (41).0.0 z Z6.1.2. Side Boundary ConditionAll electron beam processing takes place in a vacuum. The measured vacuum forall experiments conducted at UBC did not exceed 1.33 Pa (10 1.im Hg) and at AxelJohnson, the vacuum did not rise higher than 4.67 Pa (35 j.tm Hg). At these low pressuresand the high temperatures present during electron beam heating, the effects of convectiveheat transfer can be ignored. Assuming a small gray body in a large gray volume and aconstant furnace wall temperature (T,j, the boundary condition on the cylindrical surfaceof the targets can be written as—k= —T) (42)arwhere Gsb is the Stefan-Boltzman constant, and 6 is the emissivity of the target material.6.1.3. Bottom Boundary ConditionOn the bottom surface of the target, the thermal situation is somewhat less clear.In the experimental furnace at UBC, the targets were rested on a water cooled copperplate. At AJMI, the target was placed on a water cooled copper dove-tail puller used for- 66 -67ingot removal during the remelting process. In both situations, there was some contactbetween the uneyen surface of the target and the copper surface. In these areas, heat istransferred by conduction to the cooling water within the support. In the areas wherethere is no contact, radiation heat transfer is assumed to take place. This combination ofheat transfer mechanisms has been assumed to be described adequately by a boundarycondition of the form=hC(TBS _Tj+YSb8(TBS —T) (43)aZwhere is the heat transfer coefficient representing the heat transfer over the pointcontacts on the bottom surface.The value of the contact heat transfer coefficient (hconmct) has been estimated byBallantyne in VAR remelting as 0.084 W cm2s ocl[7]. In modelling the EBCHRprocess, Tripp[80j used a value of 0.067 W cm-2 s’ °C’ to account for the poor contactbetween the electron beam skull mould and the skull itself. These values are inreasonable agreement with those used by others[16,22].6.1.4. Symmetry ConditionsIn the axi-symmetric model (two dimensional case), it is assumed that there areno temperature gradients in the theta direction allowing for=0 (44).6.1.5. Top Surface Boundary ConditionThe characterization of the top surface boundary condition is the primaryobjective of this research. In general, there are as many as three heat transfer processesoperating at this surface. These are 1) heat input due to the impingement of the beam, 2)heat loss due to radiation to the furnace and 3) heat loss due to evaporation of materialfrom the surface of the liquid (should a liquid be formed). Mathematically, we can writethe top surface boundary condition as68_kj= cTthE(T —T)— evap (r,t)+ beam (r,t) (45).where evap and qiam describe the heat fluxes related to the evaporation of material andthe power delivered by the beam respectively.The Langmuir equation[46] gives the evaporation rate of a liquid in a nearvacuum as(46)%JRMXTwhere p is the equilibrium vapour pressure of the material (x) at temperature T (K), Mis the atomic weight and R is the gas constant. The equation assumes that the evaporativeprocess is rate limited by the actual evaporation step, not by mass transfer in either thegas or the liquid. In the case of a pure material (such as commercial purity titanium), therate of evaporation cannot be controlled by liquid mass transfer. The equation as writtenassumes that any vapour that leaves the surface does not return. This assumption is goodat pressures less than 13.33 Pa (100 urn) and equation (47) is therefore valid in electronbeam remelting. Thus, the heat flux related to the evaporation of material isqevap (r,t) = thxt1evap,x (47)where AHevap is the heat required to evaporate one mass unit of material at thetemperature T.In both the experiments conducted at UBC and AJMT, the beam spot was movedover the surface of the target material. Since the beam spot is a moving area source, themodel would have to be three-dimensional and quite complex to handle the temperaturefluctuations which would result. Nakamura[58] has shown that provided the beam spot ismoving at a sufficiently high speed in a repetitive pattern (Nakamura quotes 94.2 cm s4as sufficient) then the power distribution can be reasonably approximated by a timeaveraged power distribution on the surface of the target. In general if the heat signatureof the beam spot is visible on the surface then the beam is moving too slowly to time69average the power on the top surface. In all of the experiments reported in this thesis, theheat signature was not visible permitting the use of the time average approximation.A review of the literature on modelling electron beam welding and thecharacterization of electron beams[20,28,33,66,67,83] suggests that the powerdistribution within the beam spot can best be represented by a Gaussian or normaldistribution. Other models of energy and mass transport in the electron beam hearth havealso used a Gaussian distribution for power within the beam spot[12,64,84j. Thus, whenthe beam spot is being moved rapidly in a circle the power distribution can berepresented by a double normal distribution. Mathematically, the power distribution wasrepresented asQ.—r)2 Q.+r)2p(r)=C(e 202 +e 22 ) (48);where ?. is the distance between the center of the beam spot and the center of the beampattern and cY is a measure of the amount of spreading in the beam spot itself. The valueof C was chosen such that the total power to the target (given by the area integral ofequation (48) over the surface of the target) was equal to the power efficiency (r.)multiplied by the power reading on the gun control panel for each experiment. Thus threeparameters (, a and r) are sufficient to describe the power distribution when acircular beam pattern is centered on a cylindrical target, the axi-symmetric case.6.1.6. Three Dimensional ModelThe three dimensional model is handled in a somewhat different way than theaxi-symmetric model. In the 3-D formulation, a Cartesian co-ofdinate system is used.This facilitates the use of the finite element engine previously described. All boundaryconditions described also apply in the 3-D case. In practice the three-dimensional modelhas been used to determine the effects of power distributions which are not centered on acylindrical target. In all cases in which the 3-D version has been used, the powerdistribution has its own axis of symmetry. Provided this axis crosses the center of the70circular cross section of the target, the resulting thermal profile will have two-foldsymmetry. Thus the boundary conditions for the 3-D version of the model can be writtenas (see also Figure 39):-kan lx2iy2R-kai0-kaz Iz=Z= cythe(T —T)= h(T—Tj+athE(T —T)= cy,e(T — T) +q (r, t)—q (r, t)=0.ay(49).In this instance the axis of symmetry of the half cylinder lies along on the x axis.kaT — (TBS — T)÷—az— aSbe(TBS — T)Figure 39: Schematic Diagram of the Target Cylinder Showing the BoundaryConditions.=—p(r)azlevap += aSh E(T —T)=0zt= aSh E(T —T.)71When the beam pattern is not centered on the target an additional parameter (%)is required. This parameter is defined as the distance between the center of the beampattern and the center of the target. Using this definition, the power distribution used inthe three dimensional model was.—r)2 ().+r)2p(r—q)=C(e 202 +e 202 ) (50).6.1.7. Initial ConditionsMost of the tests conducted at UBC on either the tantalum or titanium cylinderswere started from an initial temperature reasonably close to the temperature of thecooling water. Between experiments the targets were left for a period of time to allow theblocks to cool to a uniform temperature. This was generally confirmed by ensuring thatall thermocouples indicated the same temperature prior to starting the next experiment.Mathematically, the initial condition isT(r,z,O)=T0 (51).In the AJMI experiments, the target was much larger (and therefore had a greaterthermal mass) than those used in the UBC experiments. This fact coupled with the cost oftime on a production furnace, resulted in the AIMI target not being completely cooledand thermally equilibrated between heating cycles. This required the use of the model topredict the starting temperature distribution for all but the first heat up cycle.6.1.8. Fluid FlowHeat transfer in a liquid is enhanced significantly by the bulk movement of theliquid. Since the model does not calculate the fluid velocities and the associated increasein heat transfer, this effect must be accounted for in another way. The easiest method ofaccounting for the increased heat transfer is to artificially increase the thermalconductivity of the liquid by a liquid thermal conductivity multiplying factor (LTCMF).This method has been used effectively in modelling VAR ingots[7] and the electronbeam hearth[80]. In modelling the energy and mass transfer behaviour of liquid titanium72under the influence of both a stationary and moving electron beam, Powell[64J used amultiplier of 30 for both the turbulent viscosity and the turbulent conductivity. For liquidtitanium a multiplier of 20 corresponds to an average velocity of 1.6 cm s1 in alldirections. Velocities of this magnitude are consistent with observations of theexperiments conducted under this programme.6.2. Sensitivity AnalysisPrior to modelling the heating experiments (Tests 4 through 11), a sensitivityanalysis was done to determine the effects of various parameters on the thermal regime inthe targets. The parameters chosen for the sensitivity analysis were those parameterswhich are not well understood (i.e. the bottom surface contact heat transfer coefficient,and the spreading factor, ab) and those which represent possible process controlparameters (i.e. net applied power and beam impingement point, ?). Since, in thetitanium experiments liquid metal formed, it was also necessary to examine the effects ofcertain assumptions with respect to the enhanced heat flow due to bulk movement of theliquid. Thus the effects of the magnitude of the thermal conductivity multiplying factorand the variation of the LTCMF over the transition range on the thermal regime(including the size and shape of the liquid pool) were also examined. The range ofparameters examined is shown in Table 6. The sensitivity to the changes in the variousTable 6: Range of Parameters Studied in the Sensitivity Analysis of the ThermalResponse of the Titanium Block to Changes in Process Parameters.Description Parameters VariedBase = 0.02092W cm2Ol, Net Applied Power = 8. kW, ). = 15 mm,Gb = 15 mm, LTCMF = 20., p =0.Base + = 0.0279 and 0.0 1396 W cm2 °C1 (also for tantalum)power Base + Net Applied Power = 6. and 10. kW (also for tantalum)? Base + ?. = 10 and 20 mm (also for tantalum)(Yb Base + Gb = 10 and 20 mm (also for tantalum)LTCMF Base + LTCMF =1. and 5.Base + = 15 mm. (for tantalum only)73parameters were evaluated at the locations of the thermocouples used in the experiments.The results of the sensitivity analysis for the titanium cylinder used in theexperiments are shown in Figures 40 through 48 and shown numerically in Table 7.Table 7 : The Sensitivity of Temperatures Within the Titanium Cylinder to Changes inthe Conditions Shown in Table 6.Desc. A z=20 mm z=45.5 mm z = 75.2 mm z = 125.7 mm(%) °C (A%) °C (A%) °C (A%) °C (A%)Base 1684.70 1225.60 873.82 523.6333 1680.07 -0.27 1210.64 -1.22 852.04 -2.49 487.58 -6.88-33 1684.00 -0.04 1244.90 1.57 900.20 3.02 569.82 8.82Power 25 1701.80 1.02 1483.90 21.08 1077.10 23.26 654.21 24.94-25 1437.20 -14.69 963.34 -21.40 674.44 -22.82 399.76 -23.662 33 1677.00 -0.46 1199.00 -2.17 863.26 -1.21 520.17 -0.66-33 1688.00 0.20 1257.60 2.61 883.42 1.10 526.19 0.49ab 33 1673.90 -0.64 1188.20 -3.05 858.72 -1.73 518.56 -0.97-33 1690.20 0.33 1268.90 3.53 888.64 1.70 527.99 0.83LTCMF -75 1682.90 -0.11 1173.20 -4.28 835.78 -4.35 501.40 -4.25-95 1420.70 -15.67 993.81 -18.91 712.03 -18.52 429.21 -18.03Note: All temperatures obtained at 3000 seconds from power on for the titaniumsensitivity analysis. A% = (TTBe) .100TBe74I I I I I I I I I I Iz=20.Omm below the surface1500 - -z=45.5 mm below the surface —(_)0.-I- -(1)1000- -z=712 mm below the surface__—-_— ._-—. • \\\‘\0) \H—z=125.7 mm below the surface500/Z0 1000 2000 3000Time (seconds)h = 0.01396 W cm2 °C’h=0.02092Wcm2°Ch=O.02790Wcni°4Figure 40: Sensitivity of the Thermal Response of the Titanium Cylinder to Changes inthe Contact Heat Transfer Coefficient.75C-)C)4-CeIC-)C.)aCeC)F-0 1000 2000 3000Time (seconds)Power = 6. kW- Power=&kWPower=lOkW0 1000 2000 3000Time (seconds)Figure 41: Sensitivity of the ThermalNet Applied Power.Response of the Titanium Cylinder to Changes in76C-)001-I-.00EC)C)0C,)= 1666 °C20 30Radial Co-ordinate (mm)160—‘ 1500I.1401301200200018001600140012001000800600100Figure 42: Variation of the Liquid Pool Dimensions and Cylinder Surface TemperatureDistribution with Changes in Net Applied Power for the Titanium Cylinder.0 20 40 60 80Radial Co-ordinate (mm)77I I I I I I I I I—.———---.—-—-.— — —— z=20.Omm below the surface1500 - _——, —7 —/ —/7 ./ / z=45.5 mm below the surfaceI / /I /I ‘ // /c_) ,/o / ——I —— —1000-2 75.2 mm below the surface—7—.I, —Ii2 z- •zHI /f1/ z=l 25.7 mm below the surface0 1000 2000 3000Time (seconds)= 10. mm=15.mm20. mmFigure 43: Sensitivity of the Thermocouple Response of the Titanium Cylinder toChanges in the Beam Pattern Radius.78= 1666 °Cl ‘160--/150--140- - - -15. mm.= 10. mmI 3( III II I0 10 20 30 40 50Radial Co-ordinate (mm)2000 --1800 --2=15.mm800 -- =20mm64)() I I I I I I I0 20 40 60 80 100Radial Co-ordinate (mm)Figure 44: Variation of the Liquid Pool Dimensions and Cylinder Surface TemperatureDistribution with Changes in the Beam Pattern Radius.79I I I I I I I I j— ———.—zmm below the surface1500- --I — —I —7 // ,/ / z=45.5 mm below the surfaceI’ ‘I /I / /————I ./ --I!I /_———.1000 - 1/1/ ——75.2 mm below the surfaceIi - —.-.-‘ ——I/ ,.—C—’ / , .—///Ij—/ /j z=2i7 mm below the surface500 / ‘/IIII/iI!’Ic -,,4.1 cI I I I0 1000 2000 3000Time (seconds)ab_IO.mmab_l5.mmb—2O.mmFigure 45: Sensitivity of the Thermal Reponse of the Titanium Cylinder to Changes inthe Beam Spreading Parameter.80= 1666 °CI,,160f‘150 - //- /I-.C / /o-- /U --- -,(Yb= 20. mm -- - //140 - - - -- - - — - ---Cr =15. mmbGb— 10. mm13c I I0 10 20 30 40 50Radial Co-ordinate (mm)2000 -1800 -800 -- clb=2O.mm600 Iii unit I0 20 40 60 80 100Radial Co-ordinate (mm)Figure 46: Variation of the Liquid Pool Dimensions and Cylinder Surface TemperatureDistribution with Changes in the Beam Spreading Parameter.C.)0)SI1500C-)0a 1000c’SC.2500Figure 47: Sensitivity of the Thermal Response of the Titanium Cylinder to Decreasingthe Thermal Conductivity of the Liquid.810 1000 2000 3000Time (seconds)LTCMF=J.- LTCMF=5.LTCMF=20.0 1000 2000 3000Time (seconds)82T —1666°Contour1Iz150- / -0-, /LTCMF =5. -.145- —- .—‘ -/LTCMF = 20.140 -___•----135 I I0 10 20 30 40Radial Co-ordinate (mm)2000-1800 --1600 - -I—1400--1200 --1—-S S5—1000-____—,------------LTCMF=1.LTCMF=5.800 -- LTCMF=20.-600 I I I I I0 20 40 60 80 100Radial Co-ordinate (mm)Figure 48: Variation of the Liquid Pool Dimensions and Cylinder Surface TemperatureDistribution with Changes in the Liquid Thermal Conductivity Multiplier.83The results of the analysis indicate that the temperature response at the topthermocouple location is most sensitive to changes in the net applied power. Thethermocouple response is also affected by changes in liquid thermal conductivity, beamspreading factor, beam pattern radius and the contact heat transfer coefficient indecreasing order of significance.In comparison to the other parameters, the large changes in thermal response thatresult when the liquid thermal conductivity is altered are not unexpected given the size ofthe changes used in this analysis. In the sensitivity analysis, the liquid thermalconductivity multiplier is altered by a factor of 20 whereas the other parameters arevaried ± 25 - 33%. A LTCMF of 1 is equivalent to a stagnant liquid and represents nofluid flow within the melt. During the experiments, evidence of some fluid flow wasobserved indicating that a multiplier greater than 1 should be chosen. In the sensitivityanalysis, it was necessary to vary the liquid thermal conductivity over the range of 1 to20 to show the effect of liquid movement on the thermal field. Figure 47 shows that thereis a much smaller change in the thermal response when the multiplier is decreased from20 to 5 (-4.3% at 45.5 mm below the surface) than when there is no liquid movement(-19.0% at the same location). The pooi shape and dimensions change quite significantlywhen a multiplier of 1 is used (see Figure 48) but there are only small changes in pooidepth and pool radius when the multiplier is decreased from 20 to 5.Changing the contact heat transfer coefficient has very little effect on thetemperature at the top thermocouple position (< 0.3%) (see Figure 40) but affects theresponse at the bottom thermocouple position more strongly (± 7%) than all of the otherfactors except beam power and effective liquid thermal conductivity.Figure 41 shows that all the temperatures in the cylinder are very sensitive tochanges in the net applied power consistently at all thermocouple positions (± 23-24% --Table 7). This is in contrast to the sensitivity of the temperatures to changes in the beampower distribution parameters. The temperatures at the thermocouple positions furthest84away from the surface of the cylinder are relatively unaffected (± 0.75%) by changes inthe beam power distribution parameters (see Figures 43 and 45). Near the surfacechanges in the beam power distribution parameters produce 2-3% changes in thetemperature response.The effect of target material has been examined by repeating the titaniumsensitivity analyses for tantalum using the base case and the variations of power,impingement point and beam spreading factor. The results of the analysis are shown inFigures 49 to 53 and summarized in Table 8. The temperatures in the tantalum cylinderare more sensitive to variations in the contact heat transfer coefficient (see Figure 49) --± 8% at 76.2 mm below the surface in tantalum, ± 3% at 75.2 mm below the surface intitanium. This is likely due to the higher thermal conductivity of tantalum compared totitanium. The sensitivity of the thermal response to variations in the power distributionparameters (net applied power, beam spreading parameter and impingement point) isTable 8 : The Sensitivity of Temperatures Within the Tantalum Cylinder to Changes inthe Conditions Shown in Table 6.Desc. , z=Omm z=25.4mm z=50.8mm z=76.2mm(%) °C (z%) °C (A%) (A%) (A%)Base 1869.40 1345.80 1123.20 991.78hj-n 33 1827.70 -2.23 1294.90 -3.78 1063.40 -5.32 921.02 -7.13-33 1915.60 2.47 1403.20 4.27 1191.40 6.07 1073.00 8.19Power 25 2129.86 13.93 1530.93 13.76 1275.58 13.57 1124.63 13.40-25 1519.10 -18.74 1097.20 -18.47 917.09 -18.35 810.33 -18.302 33 1693.10 -9.43 1318.90 -2.00 1117.80 -0.48 990.52 -0.13-33 2066.20 10.53 1369.30 1.75 1125.30 0.19 990.27 -0.15b 33 1746.50 -6.57 1317.40 -2.11 1118.30 -0.44 987.06 -0.48-33 1969.70 5.37 1374.00 2.10 1123.90 0.06 992.30 0.05pmm 10 1836.44 -1.76 1332.27 -1.01 1116.89 -0.56 987.42 -0.4420 1725.90 -7.68 1298.40 -3.52 1103.70 -1.74 979.82 -1.21Note: All temperatures obtained at 1000 seconds from power on for the tantalumsensitivity analysis. Percent difference is (T Tiase).100.85qualitatively similar to that in titanium. The temperature response is uniformly sensitiveto changes in net applied power (± 13-18%), and ranges from ± 2% at the topthermocouple location to almost no variation at the bottom thermocouple location as aresult of variations in the other two parameters-- see Figures 51 and 52. In general, thesensitivity to changes in the beam power parameters is less in tantalum than it is intitanium (i.e. at 76 mm below the surface, a 33% increase in beam spreading parametercauses a 1.7% variation in the titanium cylinder and an 0.06% variation in the tantalumcylinder).Figure 54 shows the temperature calculated using the three dimensional modeland an off-axis beam power distribution (q=lO mm) overlaid on the temperature distribution calculated using the axi-symmetric model. Table 9 summarizes the results for a10 mm offset of the beam pattern. The Figure and Table clearly show that for a variationin asymmetry of 10 mm, the temperatures at the centerline of the cylinder can be wellrepresented by the axi-symmetric model at a radius equal to the value of the axis offset.20001500U0a)z1000H50000 200 400 600 800 1000 1200Time (seconds)Figure 49: Sensitivity of the Thermal Response of the Tantalum Cylinder to Changes inthe Bottom Surface Contact Heat Transfer Coefficient.86C)zUI-3E200015001000500020000 200 400 600 800Time (seconds)Power= 10kW- Power=8kWPower=6kW1000 12001500100050000 200 400 600Time (seconds)Figure 50: Sensitivity of the Thermal Response of the Tantalum Cylinder to Changes inthe Net Applied Power.800 1000 12001200Time (seconds)Figure 52: Sensitivity of the Thermal Response of the Tantalum Cylinder to Changes inthe Beam Pattern Radius.87200 400 600 800 100020001500C)0ISC)F—50000 1200Time (seconds)Figure 51: Sensitivity of the Thermal Response of the Tantalum Cylinder to Changes inthe Beam Spreading Parameter.2000150000C)I100050000 200 400 600 800 10008815000010005000Time (seconds)1200Figure 53: The Sensitivity of the Thermal Response of the Tantalum Cylinder toChanges in the Center of the Beam Pattern with Respect to the Axis of theCylinder.Table 9: Summary of Differences Between the Temperatures Predicted for the ThreeDimensional Model and the Shifted Axisymmetrical Model at the ThermocoupleLocations Used in the Tantalum Experiments.Location 3D Model Shifted Axisymmetric DifferenceP = 10 mm Model (%)(°C) (°C)Surface 1836.4 1838.5-0.1125.4 mm below surface 1332.3 1335.6-0.2550.8 mm below surface 1116.89 1120.43-0.3276.2 mm below surface 987.4 990.8-0.34Note: Percentage differences calculated using (T30T11)T0 200 400 600 800 1000Axisyrnmetzic ModelShifted3 DzmensionalModelFigure 54: Calculated Temperature Distribution For an Offset Beam Pattern Axis Usingthe 3-Dimensional Model (dashed lines) Overlaid with the Calculated ResultUsing the Axisymmetric Model (solid lines) Offset by the Same Amount.Figures 55 and 56 show the calculated temperature as a function of radius atdifferent thermocouple locations within the tantalum cylinder for different power levelsand beam spreading parameters.Given that the temperature distribution that results from an off-axis beam patterncan be estimated by offsetting the temperature distribution calculated using the axisymmetric model (as shown in Figure 54), the sensitivity of the absolute temperature atthe thermocouple locations to an off-axis temperature distribution can be estimated (seeTable 10). At the thermocouple 25.4 mm from the surface of the tantalum cylinder, a 10mm offset of the beam pattern results in less than 1% variation in the predictedtemperature. For a 20 mm offset this variation increases to 3%. The relative sensitivity89120100180E6000L)< 40200-75 -50 -25 0 25 50 75Radial Coordinate (mm)90of the temperature to variations in net applied power and beam spreading parameterwhen the beam pattern is off-axis can also be estimated.120011501100Radial Coordinate (mm)Figure 55: The Temperature as a Function of Radius for the Tantalum Cylinder ForDifferent Beam Spreading Parameters.UI1600140012001000Radial Coordinate (mm)Figure 56: The Temperature as a Function of Radius for the Tantalum Cylinder ForDifferent Applied Powers.140013501300U12500 10 20 30 40 50 60 700 10 20 30 40 50 60 7091Table 10: The Temperature within the Tantalum Cylinder at Locations Removed fromthe Axis of the Beam Pattern.z = 76.2 mm z=50.Smm z = 25.4 mm- z 0 mmr=0 r=1O r=20 r=0 r=10 r=20 r=O r=10 r=20 r=0 r=10 r=20Base, P = 8 kW991.8 0.10% 0.36% 1123.3 0.25% 0.94% 1345.8 0.76% 2.83% 1869.4 1.65% 7.25%°b lSmm0b = 10mm 987.1 0.11% 0.42% 1123.9 0.31% 1.16% 1374.0 1.00% 3.73% 1969.7 -0.93% 5.42%ab=2Omm 992.3 0.08% 0.31% 1118.3 0.20% 0.75% 1317.4 0.57% 2.16% 1746.5 1.61% 6.10%P=6kW 810.3 0.08% 0.32% 915.1 0.24% 0.86% 1097.2 0.73% 2.72% 1519.1 1.65% 7.18%P=lOkW 1127.3 0.11% 0.41% 1278.8 0.27% 1.01% 1536.9 0.79% 2.97% 2146.9 1.66% 7.35%Note: Temperatures at r = 0 mm obtained from the tantalum sensitivity analysis at 1000 seconds afterpower on in °C. Other entries are the percentage difference between the temperature at r=0 mmand the temperature at r (10,20) mm using the relationship, = C”° — T1).100.TTable 11 shows the sensitivity of the temperatures at the surface and the threethermocouple locations within the tantalum cylinder to variations in beam spreadingparameter and net applied power when the beam pattern is not coaxial with the cylinder.The sensitivity of the temperatures to a 25% change in power ranges between 18.3% and18.7% and is essentially independent of radius. For a 33% variation in beam spreadingparameter, the temperature varies by 5-8% at the surface and by less than 0.5% at thethermocouple location furthest removed from the surface. The sensitivity is essentiallyindependent of radial location at the bottom two thermocouples and slightly more so atthe top thermocouple location (2.1% at r=0, 1.9% at r=lOmm).Figure 57 shows the temperature response of the titanium and tantalum cylinderssubject to variations in beam power and beam spreading parameter. The titaniumtemperature has been predicted at a location 45.5 mm below the surface and for tantalumat 25.4 mm below the surface. Refering to Figure 57, at any given time and thermocouplelocation, the range of temperatures predicted for a 25 % change in net applied power islarger than that for a 33% change in the beam spreading parameter. Moreover, this rangeincreases as time increases. This behaviour indicates that modifying the powerdistribution parameters (namely the beam spreading factor) has a smaller role to play in92the determination of the final temperature attained at any one thermocouple location thendoes the net applied power. For example, in the tantalum disk, a thermocouple at thecenterline 25.4 mm below the surface will attain a steady state temperature which differsby about 50 °C over the range of beam spreading parameters examined (10-20 mm). Onthe other hand, the steady state temperature attained will vary by about 450 °C when thepower is raised from 6 kW to 10 kW.Table 11: The Relative Sensitivity of the Temperatures within the Tantalum Cylinder toVariations in Net Applied Power and Beam Spreading Factor for Various Offsets of theBeam Pattern from the Cylinder Axis.z76.2mm z=50.8mm z=25.4mm z=Ommr=O r=1O r=20 r=0 r=10 r=20 r=0 r=10 r=20 r=0 r=10 r=20Base P=SkW991.8 990.8 988.2 1123.2 1120.4 1112.7 1345.8 1335.8 1307.7 1869.4 1838.5 1733.8a, = 15mma), 10mm -0.48% -0.49% -0.54% 0.06% 0.00% -0.16% 2.10% 1.85% 1.15% 5.37% 8.14% 7.45%a),2Omm 0.05% 0.07% 0.11% -0.44% -0.39% -0.25% -2.11% -1.93% -1.43% -6.57% -6.53% -5.42%P=6kW -18.3% -18.3% -18.3% -18.5% -18.5% -18.5% -18.5% -18.5% -18.4% -18.7% -18.7% -18.7%P 10 kW 117% 13.7% 13.6% 13.9% 13.8% 13.8% 14.2% 14.2% 14.0% 14.9% 14.8% 14.7%Note: Temperatures for the base case obtained from the tantalum sensitivity analysis at 1000 secondsafter power on in °C. Other entries are the percentage difference between the base casetemperature and the temperature at the position and conditions indicated using the relationship,(T—T )%= .100.T),936.3. Results6.3.1. Analysis of Tantalum ExperimentsThe axi-symmetric model was used to predict the thermal regime which occurredduring the experiments reported in Chapter 4. The tantalum experiments were modelledfirst to eliminate the added complications of metal evaporation and liquid metal flowfrom the modelling process. Since the conditions surrounding the tantalum experiment,designated Test 10, were felt to be the most controlled, this experiment was used as theTime (seconds) Time (seconds)0 500 1000 0 1000 2000 3000Time (seconds) Time (seconds)Figure 57: The Thermal Response of the Tantalum (left) and Titanium (right) Cylindersto Changes in Power (top) and Beam Spreading Parameter (bottom).94foundation for further work. The four basic parameters (Gb, Tw and hcot) wereestimated using literature, observations and earlier results. The results of the calculationswere then compared to the experimental data. The values of the various parameterswhere then modified until the results shown in Figure 58 were obtained. The values ofthe contact heat transfer coefficient, the beam spreading parameter and the beam spotlocation were kept constant over the entire run. As can be seen easily from the Figure,the fit to the experimental data is quite good. The top most thermocouple does show asignificant contact problem between 100 and 450 seconds.1500U02 1000Figure 58: Model Predictions and Experimental Results for Test 10.Table 12 shows the measured beam power and the power required to generate thepredictions of Figure 58. The power efficiencies used by the model are in reasonableagreement with those shown in Figure 8. The model power efficiency (0.73) is withinexperimental error of the literature value (0.68).1.000.750.500.250.00I50000- 10.07.5C-5.0- 2.5- 0.0500 1000Time (seconds)95Table 12 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 10.Time Beam Power Error in Model Error inBeam PowerPower(kW) (kW) (kW)0 0.25 0.28 0.18 .73 .8020 0.35 0.38 0.26 .73 .7830 0.90 0.50 0.66 .73 .4060 1.95 0.73 1.42 .73 .2770 3.25 0.78 2.37 .73 .1790 4.20 0.85 3.06 .73 .15100 12.25 1.23 8.94 .73 .07232 12.42 1.25 9.06 .73 .07284 12.60 1.25 9.19 .73 .07577 12.60 1.25 9.19 .73 .07697 12.42 1.25 9.06 .73 .07937 12.42 1.25 9.06 .73 .07Experimental errors in the model power and power efficiency were calculatedusing the standard methods of calculating experimental errors. As discussed earlier, thebeam current is known to within ± 0.05 Amp and the accelerating voltage to within ± 500V. The error in the beam power (expressed as a percentage) is then the sum of the errorsin the measurement of the beam voltage and current (also expressed as a percentage).Finally, the error in the power efficiency (expressed as a percentage) is the same as theexperimental error in the measured power (since the model power is known exactly). Forexample, in Test 10 at 30 seconds, the measured beam current was 0.10 A and the beamvoltage 9000 V. The error in the beam current is 50% (0.05/0.10) and the error in thebeam voltage 5.56% (500/9000). Thus the total error in beam power is 55.56% or0.50 kW. This is also the error in the beam power efficiency which is 0.40. As theexperiment progresses the error on the beam power efficiency decreases because the errorin beam power input decreases as the relative error in beam current and acceleratingvoltage decrease.The model results for Tests 9 and 11 are shown in Figures 59 and 60. Theseresults were obtained using the same model parameters as the ones that generated the961.000.750.500.25Test 10 results except that the power factor was adjusted slightly at certain times to moreaccurately predict the experimental data. Tables 13 and 14 show the measured power,and the power used in the model for the additional tests.10.015007.5—1000C)o 0500- 2.50-0.00 500 1000 1500Time (seconds)Figure 59: Model Predictions and Experimental Results for Test 9.1500-71250-61000-750500-22.500 -00 500 1000 1500 2000 2500 3000 3500Time (seconds)Figure 60: Model Predictions and Experimental Results for Test,,C0.4E-0.20.0400097Table 13 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 9.Time Beam Error in Model Error inPower Beam PowerPower() (kW) (kW) (kW)10 0.80 0.90 0.64 0.80 .7174 12.96 1.40 0.64 0.05 .01786 12.96 1.40 8.80 0.68 .07966 12.96 1.40 8.80 0.68 .071086 12.60 1.40 8.80 0.70 .081206 12.60 1.40 8.80 0.70 .08Table 14: Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 11.Time Beam Error in Model Error inPower Beam PowerPower(kW) (kW) (kW)0 0.95 0.53 0.76 0.80 0.4438 0.95 0.53 0.76 0.80 0.4450 3.90 0.80 4.00 1.03 0.21295 8.00 1.05 6.50 0.81 0.11611 8.00 1.05 7.10 0.89 0.12731 7.84 1.05 7.10 0.91 0.12851 7.84 1.05 6.78 0.86 0.121091 7.84 1.05 6.51 0.83 0.111256 7.84 1.05 6.22 0.79 0.111597 7.89 1.05 6.26 0.79 0.111855 7.89 1.05 6.26 0.79 0.112021 0.00 0.00 0.00 0.00 0.002080 7.94 1.06 6.29 0.79 0.112095 7.94 1.06 6.29 0.79 0.112259 7.99 1.06 6.34 0.79 0.112497 8.15 1.07 6.34 0.78 0.102721 8.20 1.07 6.45 0.79 0.103656 8.20 1.07 6.45 0.79 0.10Tantalum Test 11 was conducted while deliberately varying the pressure in thework chamber. As can be seen in Figure 60, there is little or no effect of changing thepressure on the thermocouple response until 2950 seconds into the experiment at whichtime the temperature can be seen to decrease slightly as the pressure rises from 0.43 Pa98(3.25 ).Lm Hg) to 0.67 Pa (5 .tm Hg). Similarly, in a subsequent change from 0.57 Pa(4.25 Im Hg) to 0.93 Pa (7 tm Hg), at 3150 seconds, the temperature also decreased.The region of Figure 60 which shows these results is expanded in Figure 61.-1.01300ciJ tJ D 0 0 0 0 kxj o 0 a a a 0 0 0 0- 0.8A’ B’ b°1200- —- ...:- -- 0604F2 1100- 1 BI •———>00 ocUm) 0 0 00 00 cjyo 0 o4o 00.21000 ----I I I- U.U2775 2900 3025 3150 3275 3400Time (seconds)Figure 61: The Response of the Three Thermocouples to Changes in Pressure inTantalum Test 11.During the experiments the beam pattern radius did not change. The only possibleexplanations for the variation in temperature with pressure are that pressure affected thepower transfer efficiency or beam spreading parameter. The model was used to fit thethermal response of the top thermocouple using changes in the net power (and thus thepower efficiency) and the beam spreading parameter. Figure 62 shows the results of themodel fit using variations in net power and Figure 63 shows similar results usingvariations in beam spreading parameter.99•10.0j-’..Pressure (lOPa) —a.-1300- I-5’ 1200 L P_-__! ii -‘.-__..J1——-1I, *I — IH11100 -- I ‘.. -F- I----I...- l-250 0 OTflD 0 03)0 0 0 GJfl) 0 o ccI 0 O) 010002525 2650 2775 2900 3025 3150 3275 3400 3525Time (seconds)Figure 62: Model and Experimental Results for Power Efficiency Variation withPressure in Tantalum Heating Test 11.—1.01300- A B!) C’ -1200DXjja-0.8I I Pressure 0.6-041100-/ B -o 0 0 02X) 0 C) 0 0 0 0 0 Q) 0 0 0 910 01000 .----------------—-------------------------------------—-r-• ,,, I .12775 2900 3025 3150 3275 3400Time (seconds)Figure 63: Model and Experimental Results for Beam Spread Factor Variation withPressure in Tantalum Heating Test 11.100Since the top thermocouple was used to fit the model to the experiments in bothcases, Figures 62 and 63 show good agreement between the response at the topthermocouple location and the model prediction. In contrast, there are notable differencesat the bottom two thermocouples in that when the beam power efficiency is varied withpressure (Figure 62) the predicted temperatures change quite substantially. Thesechanges are not reflected in the measured thermocouple response. When the beamspreading factor is varied with pressure (Figure 63) the predicted temperatures at thebottom two thermocouple locations do not change appreciably and are much closer to theactual measured temperatures. This difference in behaviour is also indicated in thesensitivity analysis for tantalum where it was demonstrated that the temperatures at allthe thermocouple locations are sensitive to variations in power (see Figure 50) but onlythe temperature at the thermocouple closest to the surface is appreciably sensitive tochanges in the beam spreading parameter (see Figure 51). These results indicate that thepower transfer efficiency is relatively independent of pressure while the size of the beamspot is affected significantly by changes in pressure.6.3.2. Analysis of Titanium ExperimentsUsing the results of the sensitivity analysis and observations, the variousparameters were changed to fit the experimental results to the results of the model. Toreduce the impact of pressure variation, the tests in which the pressure remained constantwere used to determine the validity of the model. The results of the model fit to theexperimental data for the experiment, designated Test 4, is shown in Figure 64. The fit ofthe model to the experimental results is fairly good in all areas with a few exceptions.Over the first 1800 seconds, the model under predicts the temperature at the topthermocouple position. During the experiment, the cooling water to the furnace chamberwas inadvertently left off. Thus the furnace wall was not maintained at a constanttemperature. The mathematical model assumes a constant ambient temperature and is101unable to accurately predict temperatures when the temperature of the furnace wall varieswith time as in this experiment. At about 1400 seconds into the run, the outside of thefurnace was quite hot to the touch and the cooling water to the furnace chamber wasturned on. After this point the fit to the top thermocouple becomes much better.In the initial stages of the heat up cycle the temperatures measured by the bottomthree thermocouples (which are insulated chromel - alumel thermocouples) do notcorrespond well to the model predictions. Once a reasonably high temperature is attainedthe fit of the model to the thermocouple response at these three locations improves agreat deal. This behaviour is typical of poor contact at the thermocouple tip.150012501000C-)0750500250876CV00T3TCITC2TC3TC4TC554Cc•r’c.30210 500 1000 1500 2000 2500 3000Time (seconds)0Figure 64: Experimental and Model Results for Test 4 (r,,=0.95, cYb=l7.5mm).Once the effects of poor thermocouple contact and the lack of cooling water tothe furnace walls have dissipated the model fit to the experimental results is quite gooduntil the power is shut off. The time period following power off is shown in Figure 65.102The model predicts the response of the top two thermocouples reasonably well for aperiod of about 300 seconds after power off. About 200 seconds after power off the topthermocouple enters a thermal arrest associated with the -cz transition at about 882 °C.In the model this arrest stops distinctly at 2760 seconds while the measurements show agradual decrease in the temperature starting about 80 seconds later. As the coolingcontinues, the model under predicts the temperatures at the five thermocouple locations.This phenomenon is not reproduced in the tantalum experiments (Figures 58, 59 and 60).The timing of the deviation from the model and the temperature at which it occurssuggests that the deviation is a result of the way in which the (3-a transition is modelled.The transition itself is solid state and thus reasonably sluggish. The release of heat duringthe transition is time dependent. The model assumes that the transition occurs like achange of state and is dependent only on the temperature. This may explain the modeldeviation from the experimental results.liii.—8T1250-6D TCI00 -5 TC2V TC3TC.1O00 \\ ° TC5o N.vv - 2V-.--..750 --AA A--_A A — — — —— 1—03, 0, I I I I I2300 2400 2500 2600 2700 2800 2900 3000Time (seconds)Figure 65: The Model Predictions and Thermocouple Response for Titanium Test 4After Power Off.V_•103The results of fitting the model to the experimental data for Test 4 indicate thatthere is essentially no transfer of heat by contact at the bottom surface of the cylinder.Thus a value of 0.00 W cm-2 °C1 was used in modelling the titanium experiments. Thevalue used for the tantalum experiments was 0.02092 W cm2 °C1. The value for thetitanium experiments is not unreasonable since the surface was rough and indented whencompared to the machined tantalum surface. The shape of the initial thermal transients onthe bottom three thermocouples indicates that the thermal conductivity multiplier shouldbe set to 20.In the case of Test 4, the predicted pool diameter is 55 mm (see Figure 66) andthe actual measured diameter is 63.75 mm (as taken from the transparent film) which isin reasonable agreement. Since determining the pool depth after each experiment wouldhave required the destruction of the titanium block the pool depth and shape wereunattainable for all of the tests except Test 8.,66150£ 1001.0C)5000 20 40 60 80 100Radial Co-ordinate (mm)Figure 66: Predicted Temperature Distribution for Test 4 at 2403 seconds After PowerOn.104Table 15 Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 4. (Pressure 0.02 Pa)Time Beam Error in Model Error inPower Beam PowerPower) (kW) (kW) (kW)0 7.20 0.99 0.528 0.07 0.0146 7.20 0.99 6.33 0.88 0.12110 7.20 0.99 6.84 0.95 0.13303 7.20 0.99 6.84 0.95 0.13403 7.44 1.015 7.07 0.95 0.13703 7.44 1.015 7.07 0.95 0.13803 7.49 1.02 7.12 0.95 0.131103 7.49 1.02 7.12 0.95 0.131203 7.58 1.03 7.20 0.95 0.131303 7.63 1.035 7.25 0.95 0.131603 7.68 1.04 7.25 0.94 0.131703 7.63 1.035 7.26 0.95 0.131903 7.68 1.04 7.25 0.94 0.132003 7.63 1.035 7.26 0.95 0.132203 7.68 1.04 7.25 0.94 0.132303 7.68 1.04 7.25 0.94 0.132403 7.68 1.04 7.26 0.95 0.13The same procedure was used to evaluate the fit to the other experimental tests.For further calculations, the value of the contact heat transfer coefficient was set to0.00 W cm2 °C1 and the liquid thermal conductivity multiplier set to 20. The experimental results of Test 5 are shown in Figure 67. As in Test 4, the model does not fit verywell in the initial stages of the experiment especially at the lower thermocouples. Themodel does appear to fit reasonably well after power shutdown. During Test 5, variationsin filament current and bombardment voltage in the gun at about 720 seconds after poweron caused a significant power drop. This variation in the beam generating systemapparently caused a change in the beam characteristics as well. In order to match thetemperature variation in the cylinder after the power drop it was necessary to change thebeam spreading parameter as indicated in Figure 67.6’iooo0z7505001500125081057-Tci5 TC2TC3TC4° TXDS62500Time (seconds)Radial Co-ordinate (mm)Figure 68: Predicted Temperature Distribution for Test 5 at 2300.On.100seconds After Power106Table 16: Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 5. (Pressure 0.16 Pa)Time Beam Error in Model Error inPower Beam PowerPower) (kW) (kW) (kW)30 0.85 0.56 0.24 0.28 0.1640 6.00 0.16 0.24 0.04 0.0195 6.93 0.14 5.70 0.82 0.12295 7.11 0.14 6.58 0.93 0.13495 7.11 0.14 6.75 0.95 0.14695 7.16 0.14 6.75 0.94 0.13895 6.08 0.16 7.30 1.20 0.20935 7.68 0.14 7.30 0.95 0.131135 7.84 0.13 7.30 0.93 0.121335 7.84 0.13 7.45 0.95 0.132205 7.84 0.13 7.45 0.95 0.13The predicted pool diameter using the conditions shown in Figure 67 is 70. mmwhile the measured pooi diameter is 80 mm. which is a reasonable agreement given thedifficulties in determining the position of the liquid pooi from the top surface of thecylinder.The model results for Test 6 are shown in Figure 69. As with the previous results,the model does not agree well with the experimental results during the initial stages ofheating and after power shutoff. Once the heating pattern has become established, the fitto the experimental results is quite good. The predicted pool diameter is 69. mm whilethe actual pool was measured at 72.0 mm.107----1;TCiT2TC3TC4TCS1ooo‘2)f260 100Radj Co-ordinate (mm)Figure 70: Predicted Temperature Distbut0for Test 6 at 2310. seconds After Poweron.108Table 17 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 6. (Pressure = 0.33 Pa)Time Beam Error in Model Error inPower Beam PowerPower() (kW) (kW) (kW)31 4.05 0.83 0.21 0.05 0.01111 4.05 0.83 4.25 1.05 0.21191 6.75 0.98 7.35 1.09 0.16391 6.84 0.99 7.35 1.07 0.15591 6.89 0.99 7.35 1.07 0.15791 6.78 0.99 7.35 1.08 0.16991 6.82 1.00 7.35 1.08 0.161191 6.51 0.99 7.35 1.13 0.171332 6.98 1.00 6.94 0.99 0.141532 7.13 1.01 6.94 0.97 0.141732 6.98 1.00 6.94 0.99 0.142310 6.98 1.00 6.94 0.99 0.14The results of Test 8 are shown in Figure 71. The model predicts the pooldiameter for Test 8 to be 70. mm and was measured at 75 mm. Figure 73 shows a crosssection of the titanium block taken after Test 8. The predicted pool shape and diameter isshown superimposed on the photograph.Table 18 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 8. (Pressure = 0.04 Pa)Time Beam Error in Model Error inPower Beam PowerPower(sec) (kW) (kW) (kW)157 7.68 1.3 7.60 0.99 0.17357 7.68 1.3 7.60 0.99 0.17957 7.68 1.3 7.60 0.99 0.171157 7.36 1.3 7.60 1.03 0.181357 7.36 1.3 7.60 1.03 0.181557 7.78 1.31 7.39 0.95 0.161957 7.73 1.31 7.34 0.95 0.162025 7.73 1.31 7.34 0.95 0.16100C0UCFigure 72:109T0 TCjTC2V TC3TC4TCsFigure100Radial Coordjnate (mm)on.Predjct Temperature Distrjbutjo0for Test 8 at 2025 secofld After Power110Figure 73: Photomacrograph of the Through Diameter Cross Section of the TitaniumCylinder Showing the Apparent Liquidus (solid line) and the PredictedLiquidus (dashed line).During Test 7 the pressure was varied and the temperature in the cylinder allowedto equilibrate at each pressure level. Due to extensive variations in measured powerlevels during the experiment, the effects of one specific variable on the temperatureresponse were difficult to isolate. Accordingly, the power schedule was simplified andthe model fitted to the measured thermal response using modifications to the beamspreading parameter. The first low pressure portion was used to determine the radius ofthe beam pattern. The results of fitting the model to the measured temperatures can beseen in Figure 74.Apparent Liquidus10 mm Predicted Liquidus11110-4.015009 T1—81250- 3.0 EZZZ7 _I’sTC1TC2‘1000 6 V TC3TC4S 5 -2.0 0 TC5750-0. CD4—500 3- 1.0225010 -0.00 1000 2000 3000 4000 5000Time (seconds)Figure 74: Experimental and Model Results for Test 7 Using a Simplified PowerSchedule (r = 0.95- 0.99,‘b=17 - 21. mm).Since the beam pattern traveled across the surface of the cylinder at the end ofTest 7, it was difficult to get a measured pool diameter, thus the predicted pool diameterwas not obtained (see Figure 25).Since a simplified power schedule was used to fit the experimental results of Test7, a greater variation in the power efficiency would be expected. This is shown in Table19 where the power efficiency varies from 0.91 ± 0.13 to 1.01 ± 0:14.112Table 19 : Measured Power, Power Used in the Model Calculations, the PowerEfficiency Factor and Associated Errors for Test 7. (Variable Pressure)Time Beam Error in Model Error inPower Beam PowerPower(sec) (kW) (kW) (kW)31 0.70 0.40 7.56 0.09 0.0592 6.90 0.98 7.560 0.9i.. 0.13292 7.58 1.03 7.560 1.00 0.14492 7.58 1.03 7.560 1.00 0.14792 7.63 1.04 7.560 1.01 0.14992 7.16 1.02 7.560 0.95 0.131192 7.16 1.02 7.560 0.95 0.131315 7.00 1.02 7.245 0.97 0.141515 6.84 1.01 7.245 0.94 0.141722 6.88 1.02 7.245 0.95 0.142021 7.20 1.03 7.245 0.99 0.142261 7.20 1.03 7.245 0.99 0.142501 7.16 1.02 7.245 0.99 0.142661 7.16 1.02 7.245 0.99 0.142861 7.20 1.03 7.245 0.99 0.143291 7.20 1.03 7.245 0.99 SummaryThe calculated beam spreading parameters and average power efficiencies for allof the titanium experiments are summarized in Table 20. For Test 7, the simplified powerdistribution has been used to calculate the power efficiency (see Table 19). The powerefficiency reported for the constant pressure tests is a time average of the powerefficiencies reported in Tables 15, 16, 17 and 18.The mathematical model has been used to predict the thermal response of the twocylinders under the experimental conditions. The analyses of three tantalum experimentsand five titanium experiments have been shown. The comparison between the measuredand predicted temperatures shows quite conclusively that the power distribution withinthe beam spot is adequately represented by Gaussian distribution. Using a combination ofsensitivity analysis and model fitting it has been demonstrated that the power transfer113efficiency during electron beam melting and heating is largely independent of residualchamber pressure and that the power distribution within the beam spot is significantlyaffected by the chamber pressure.Table 20: Summary of Beam Spreading Factors and Power Efficiencies as a Function ofPressure for All of the Titanium Experiments.Experiment Pressure Beam Spread (ab) Power Efficiency (iw)See Figures (64,67,69,71,74) (Pa) (mm)Test 7 : A - B 0.004 20 0.96Test 4 0.02 17.5 0.95Test 8 0.04 19 0.99Test7:B-C 0.12 17 0.95Test 5 0.16 15 0.95Test 7 : C - D 0.26 17 0.99Test 6 0.33 17.5 1.02Test 7 : D - E 0.54 19 0.99Test 7 : E - F 0.76 21 0.996.4. Discussion6.4.1. The Effect of Pressure on Power DeliveryUsing the results of the laboratory experiments it is possible to develop empiricalrelationships between pressure and beam spread for the electron beam heating oftantalum and titanium in an electron beam furnace under the conditions of the laboratoryscale experiments.Observations during the experiments suggest that the beam spot does not dropbelow 6.35 mm in diameter. This value was adopted as a minimum. To determine therelationship between pressure and beam spreading factor for tantalum, the beamspreading factor was used to fit the temperature response over segment A-A’ on Figure63. A value of beam spread (ab) of 10 mm was found to reproduce the temperatureresponse quite well. Over segment B-B’ (also on Figure 63), the beam spread factorwhich best fitted the temperature response was found to be 20 mm. Since changes intemperature response were not observed below approximately 0.37 Pa (2.8 pm Hg), this114pressure was used to fix the lower end of the functional relationship between pressureand beam spreading factor.Using these parameters the quadratic relationship given in Equation 52 isobtained. The function is also show graphically in Figure 75.=1 6.4mm p <0.37 Pa (52).b l2. 64—33.265p + 44.276p2 0. 37 Pa p 0. 93 Pa1917. 1513Cl)1150.1 0.2 0.3 0.4 0.5 0.6Pressure (Pa)21C T9T10Til07a0.7 0.8 0.9 1Figure 75: Beam Spreading Factor as a Function of Chamber Pressure for All TantanlumExperiments.To determine a similar functional relationship between pressure and the beamspread parameter in titanium a pressure - beam spread pair was obtained for eachpressure regime in Test 7 and in each of the other constant pressure experiments. Theseresults are summarized in Table 20 and plotted in Figure 76.115Figure 76: Beam Spreading Factor as a Function of Chamber Pressure for All TitaniumExperiments.The best fit to the data presented in Table 20 was obtained by splitting the datainto two pressure ranges and fitting quadratics to the data. The resulting empirical equatishown as the solid line in Figure 76 and is given by:f121.4209p —40.5255p+19.5l32 p 0.2 (53)— 1 0.6921p—7.7660p+14.6840 O.2pO.8In heating tantalum, the beam was observed to defocus as pressure was increasedbeyond a lower limit (0.37 Pa). In contrast, in melting titanium, under the conditions ofthese experiments, the beam is initially diffuse at low pressure, then focuses and finallydiffuses as the pressure increases. This focusing behaviour has been attributed to positiveions created by beam gas interactions compensating for some of the negative intrinsicspace charge of the beam[69]. This ability of an inert gas to aid in focusing an electronbeam is used to advantage in high power (> 100 kW) gun design when argon is injectedinto the beam deflection region with the intent of creating positively charged ions tocounteract the space charge of the beam. Defocusing of the electron beam has beenattributed to excessive electron scattering as the beam interacts with too many gas2221120E181716150 T40 T5T6o T7• T8[]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Pressure (Pa)116particles. The focusing/defocusing effect has also been observed in an operatingindustrial hearth fumace[30].1.021.01T41.000 °T5V0.990.98 T60.97°T70.96T80.950.94 I0 0.2 0.4 0.6 0.8Pressure (Pa)Figure 77: Power Efficiency as a Function of Pressure for All the Titanium Experiments.While it is possible that there is a slight variation in power transfer efficiency inthe melting of titanium under the conditions of this programme, it is difficult to state anyconclusions due to the experimental error in the determination of the input power.6.4.2. Chamber Gas CompositionTarget material has a significant effect on the chamber gas composition. Thevapour pressure of tantalum expected at the surface of the cylinder is very small ( 1 x10 Pa) in comparison to the imposed argon pressure (0.37 - 0.93 Pa). On the otherhand, at the temperatures expected at the surface of the titanium cylinder, the titaniumvapour pressure is 1.85 Pa. Due to its high reactivity with the common residual gasses(namely oxygen and nitrogen), heated titanium is used as a getter in the vacuum industry.Therefore in the titanium experiments where liquid titanium is produced, any nitrogen oroxygen present in the chamber prior to melting would be gettered. No gettering wouldoccur when heating tantalum. Thus the atmosphere within the chamber in tantalum117heating is made up of non-condensible gaseous elements (Ar, 02 and N2) while theatmosphere during titanium melting is essentially argon and titanium vapour.The gauge used in the experiments is capable of measuring only the pressure ofthe non-condensible species. The tantalum experiments provide a baseline for the effectsof the residual gas species on an electron beam independent of the presence of metalvapour. The experimental results show that the beam spreading factor during titaniummelting is higher than that for the tantalum experiments at any given measured pressure.This result could be attributed to the presence of metal vapour in the gas phase above thetarget cylinder during titanium melting. As the partial pressure of titanium vapour isestimated to be a minimum of twice that of the measured pressure of the noncondensibles there is a greater number of gaseous atoms available to interact with thebeam.6.4.3. The Heat BalanceTo gain insight into the heat transfer processes occurring in the electron beamheating of tantalum and titanium, an energy balance has been done to show the relativecontributions of the various heat transfer processes. The total heat removed from each ofthe surfaces of the target tantalum cylinder as a function of time is shown in Figure 78.The Figure shows that at the conclusion of the experiment the cylinder had obtained asteady state. In addition, most of the heat supplied to the target is removed through thebottom of the cylinder. This is also shown in the schematic heat flow diagram of Figure79. In tantalum heating at steady state under the conditions of Test 11, 59% of the netapplied heat is removed through the bottom surface. Figure 78 also shows how the heatbalance adjusts to the changes in beam spreading factor which occur as a result of thepressure changes during Test 11. At the time of the pressure changes (= 3175 secondsafter power on), the heat removed from the top surface due to radiation decreases slightlycorresponding to decreased surface temperatures as a result of a defocused beam.1187000-J‘—1____I_______________ Input Heat6000 -5000 - ,7 ToI/ Bourn/ Right/ Top4000— /3000 //2000 I’//.—-—• __•—•—•—.—-—-—ZZ—z:\/—-——.—-———-woo I--.-—1’ ‘.I_______________0 I 11111110 1000 2000 3000Time (seconds)Figure 78: The Heat Removed from the Various Surfaces of the Tantalum Cylinder as aFunction of Time as Calculated for Test 11.8200 W13 1.2%1188W 6250WNc19% I 100% I \J Beam Losses1950W____________________31.2%Tantalum CylinderTest 113000 seconds D’358 WAccumulation 0.3 % 21.7%= 0.76U3689 W59%Figure 79: Schematic Heat Flow Diagram of the Tantalum Cylinder During Test 11 asCalculated at 3000 seconds from Power On.119.CH0F-7000500Similar calculations were performed for some representative titaniumexperiments. Figures 80 to 83 show the results of the calculations. In the titaniumexperiments, evaporation accounts for less that 3% of the total heat balance. In addition,the heat flow through the bottom of the cylinder is only about 7% of the total heat inputto the cylinder. This is likely due to the absence of contact heat transfer at the bottomsurface. The lower thermal conductivity of titanium and the longer cylinder used in thetitanium experiments will also account for some of the decreased heat transfer throughthe bottom surface.600050004000300020004000C1-Il20010010000 00Figure 80: The Heat Removed from the Various Surfaces of the Titanium Cylinder as aFunction of Time as Calculated for Test 4.500 1000 1500 2000 2500 3000Time (seconds)1207680 WRadiation 105.8%Evaporation108W 7260W1.5% 100% Beam Losses__________4/Titanum CylinderTest 4___2300 seconds \, 2103 WAccumulation 23 % / 29%Th=0•945U388W3.1%Figure 81: Schematic Heat Flow Diagram of the Titanium Cylinder During Test 4 asCalculated at 2300 seconds from Power On.8000— _— —7000 - Net Applied Powet - -S/To(1 -- 400 .,-Evum — —6000 -—-——Right — ,‘ _——Evap — -H 5000-_-__-300— ‘IC 4000-z‘I__-2003000 - - ./ — .-.t_-——---__i._________.— —— I‘ 2000 - - ,- - -. _----———--—--— ‘v___ ---—5) _.--• r— ,---,-__--1000-0 I I 00 500 1000 1500 2000 2500 3000Time (seconds)Figure 82: The Heat Removed from the Various Surfaces of the Titanium Cylinder as aFunction of Time as Calculated for Test 7.1217200 WRadiation 99.4%Evaporation144W 7245W24% 100% Beam LossesTitanum CylinderTest 7_____3000 seconds 2388 WAccumulation 15.7 % / °“°1.006U472 W6.5%Figure 83: Schematic Heat Flow Diagram of the Titanium Cylinder During Test 7 asCalculated at 3000 seconds from Power On.6.4.4. Error AssessmentDuring the experiments and the subsequent modelling work, differences betweenthe experimental results and the predicted temperatures can result from a number ofsources both in the experimental technique and apparatus and in the modelling procedure.As previously mentioned in Chapter 4., at the start of each experiment, the beamwas initiated and centered manually then the beam pattern was initiated and centered.During this setup process a low power beam (<0.5 kW) irradiated the target material.While the startup phase was occurring, the power level was monitored constantly. It wasdifficult to determine the location of the low power beam during startup which couldhave affected the temperature response during the initial phase of each experiment. Sinceheating in each experiment was continued until the top thermoãouple had approachedsteady state it is unlikely that the small amount of energy deposited outside of the beampattern significantly affected the measured temperatures during the later portions of theexperiment.122The model and experimental results differ quite substantially during the initialheat up portion of each experiment especially at the lower thermocouples (see forexample Figure 58). In all experiments this disagreement disappears by the time thetemperature has reached 400 or 500 °C. This behaviour is typical of a contact problem atthe thermocouple tips.In addition to the contact problems at the thermocouple tips, there is inherenterror in the thermocouple measurements. The standard error of the Type D (tungsten 3%rhenium//tungsten 26% rhenium) is ± 1% or 15 °C at 1275 °C. The standard error of theType K thermocouples (chromel II alumel) is ± 2.2% or 20 °C at 750 °C[59J.Manual alignment of the beam pattern and cylinder axes was difficult. If the beampattern was off-center in the experiment, then there would be some error in assuming thatthe beam pattern was on-axis in the model. Since it has been shown in the sensitivityanalysis that the temperature distribution generated by an off-axis beam pattern can bewell approximated by shifting an on-axis temperature distribution, the magnitude of thiserror can be estimated by examining the variation of temperature with radius. Figure 55shows the radial temperature distribution at each thermocouple position for differentbeam spreading parameters. Figure 56 gives the same information using different valuesof the net power. Table 10 shows the temperatures at the cylinder axis for an on-axisbeam pattern and compares the temperatures at positions removed from the axis of thecylinder for different beam spreading parameters and net applied powers. Table 10 showsthat there is less that a 1% error in the beam power distribution being off-axis by 10 mmand an error of less than 3% in a 20 mm displacement of the beam pattern at thethermocouple positions used in the experiments. Thus the error in aligning the beampattern by 10 mm or less is of the same magnitude as the intrinsic error in thethermocouple measurements.The magnitude of the effect of pressure changes on the beam spreading parameter(and subsequently on the measured target temperatures) are reasonably small (less than12340 °C in the tantalum experiments and between 30 and 60 °C in the titaniumexperiments). These changes are at or near the limit of the accuracy of the temperaturemeasurements (i.e. ± 15 °C at 1275 °C). However, the results are self-consistent in thatthey are reproduced accurately in the model and at each thermocouple location.6.5. SummaryA fundamentals based mathematical model has been tised to determine thethermal response of two cylinders of different materials under conditions of electronbeam heating. This model has been verified using the experimental results conducted in alaboratory scale electron beam furnace.The experiments and model have been used to extract the effect of pressure on thetemperature response during electron beam heating or melting. The model andexperiments are self consistent in a qualitative sense in that the effect of a change in thechamber pressure is observed in the output of all of the thermocouples and the responsecan be readily reproduced using the mathematical model.The model and experimental data show that when heating tantalum the beampower distribution is adequately represented by a Gaussian power distribution which isfocused at pressures less than 0.37 Pa (1 .tm Hg) and defocuses as the pressure isincreased. In melting titanium, the beam focuses and defocuses as the pressure isincreased. These focusing and defocusing effects are likely associated with interactionsbetween the electron beam and gases within the furnace chamber (either residual or fromvapourization of the target).Chapter 7. : Industrial Results7.1. MethodologyUnlike the laboratory scale experiments, the industrial experiments providetemperature data on the top surface of the liquid pool. This information can be used tocalculate the spatially dependent heat flux needed to bring about the measuredtemperature distribution and thus provide a means of verifying some of the assumptionsmade with respect to the form of the power distribution employed in the model. Theformulation of the finite element based two dimensional inverse heat conduction methodemployed in this analysis is presented in detail in Chapter 5. In summary the methodmakes use of the finite element formulation to separate the contributions of radiationconvection and conduction on a steady state temperature distribution. Once thesecontributions are determined as a function of spatial variables, then the balance must be aresult of an applied power distribution. A stand-alone utility was developed to performthese calculations. The method assumes that the cylinder is at steady state and requiresthat the top surface temperature distribution be known.The procedure for calculating the power distribution which produced a toptemperature distribution is as follows:• the measured top temperature distribution is used as a fixedtemperature boundary condition on the top surface of the cylinder,• the model is run to steady state,• the temperature distribution which results is then input to the programfor power distribution determination described in Chapter 5.In the industrial experiments, three of the segments approached steady state:namely segments A, C and G (see Figures 31 and 32). The NIR scans which are closestto representing the steady state temperature field for these segments are Scan 3 (segmentA), Scan 6 (segment C) and Scan 9 (segment G) (see also Table 5). Although a scan wasnot taken during the time when the cylinder was approaching steady state during segment-124-1251 100I50H, Scan 40A was used to calculate the power distribution during the high pressureregime used in this segment.Owing to inaccuracies in the NIR pyrometer at temperatures below theliquidus[37], the calculations were performed twice. In the first calculation, thetemperature distribution was fixed across the entire cylinder. In the second, thetemperature was fixed in the liquid region only. Both approaches generated powerdistributions which are somewhat suspect (see Figure 84). In the first approach, the totalcalculated input power exceeded the actual input power during the experiment. In thesecond case, a secondary heat flux peak was observed in the region of the periphery ofthe predicted liquid pooi. This peak is likely a numerical artifact associated with a15000 50 100 150 200Radial Coordinate (mm)250Figure 84: The Calculated Power Distribution for Segment A of the AJMI ExperimentsUsing the Measured Temperature Distribution Across the Entire Cylinder(Line), the Liquid Region Only (Circles) and The Combined PowerDistribution (Squares).126discontinuity in the first derivative of the thermal conductivity function used or thechange in boundary condition type at that point. The final calculated power distributionwas determined by fitting the results of the heat flux calculations to equations of the formof Equation 68. The fitted distribution provides a reasonable combination of the twocalculated heat flux distributions.(+r)2 ().—r)2pd(r) = C(e 2 + e 2 (48)7.2. ResultsThe calculated power distributions for each of segments A, C, G and H are shownin Figure 85. Table 21 shows the parameters obtained in fitting Equation 35 to the eachof the calculated power distributions. Figure 86 shows the results in graphical form.Since it was only possible to obtain four points from these experiments, it is difficult to300250200C-)150gioc500250///IIII/ I’\I, ‘‘I,I,Segment A - Power DensitySegmentC-PowerDensitySegmentG-Power DensitySegrnentH -Power Density0 50 100 150 200Radial Coordinate (mm)Figure 85: The Calculated Power Distributions For Each of Segments A, C, G and H.127obtain any empirical correlation between pressure and beam spreading parameter. Thebeam spreading pressure function was also difficult to obtain because the beam wasfocused using the focusing current adjustment prior to each heat up section. It isindicated that at low pressures (0.04 - 0.36 Pa) there is an effect of pressure on the beamspreading parameter and thus the temperature within the cylinder.Table 21: Summary of the Results of Fitting the Calculated Power Distributions toEquation (48).Segment Pressure a C Focus(Pa) (mm) (mm) Current(Amp)A 0.07 36.46 46.24 119.37 0.182C 0.36 21.26 56.17 178.07 0.185G 0.04 29.33 47.72 118.35 0.140H 4.40 24.00 64.47 268.45 0.2050 1 2 3 4 5Chamber Pressure (Pa)Figure 86: The Spreading Parameter as a Function of Pressure for the IndustrialExperiments.The total power delivered to the cylinder can then be calculated by integratingEquation (35) over the surface of the cylinder. Table 22 shows the results of thesecalculations and the respective power efficiencies.III4035302520151050U128Table 22: Summary of Input Power, Power Delivered to the Cylinder and PowerEfficiency for the Axel Johnson Experiments.Segment Pressure Dial Power Delivered Power Power Efficiency(Pa) (kW) (kW)A 0.07 43.2 33.5 0.77C 0.36 43.2 31.6 0.77G 0.04 42.5 26.8 0.63H 4.40 89.0 65.2 0.73Using the power distribution, the calculated power efficiency and the AxelJohnson power schedule, each experiment was simulated using the mathematical model.The comparison between the model results and the experimental results for segments A,C and G are shown in Figures 87, 88 and 89.1500L)0I100050000Time (minutes)Figure 87: Model Results and Measured Temperatures for Segment A Using theCalculated Power Distribution.The agreement between the mathematical model and the experimental results forthe first 200 minutes (as shown in Figure 87) is poor. The model consistently over50 100 150 200129predicts the temperatures at each of the four centerline thermocouple positions. Thecalculated temperatures at the top centerline thermocouple (T1) show the best agreementwith the measured temperatures. This agreement does improve after 80 minutes whenthe measured temperature indicates the presence of liquid titanium. At 134 minutes frompower on (the time of the near infrared camera scan used to determine the powerdistribution) the predicted and measured temperatures differ by as much as 165 °C at thethermocouple 101.6 mm (4”) below the surface of the cylinder. The model does predictreasonably well the onset and temperature of the thermal arrest associated with the 13 to otransition (882 °C) at 142 minutes. By the end of the segment (207 minutes), thepredicted and measured temperatures disagree by 80 °C.1500U0010005000Figure 88: Model Results and Measured Temperatures for Segment C Using theCalculated Power Distribution.250 300 350 400 450Time (minutes)130The results of the simulation of Segment C also show poor agreement betweenpredicted and measured temperatures. The temperatures at all the thermocouples on thecenterline differ from the calculated ones by as much as 170 °C at 400 minutes (the timeof the scan used to calculate the power distribution). In addition the predicted thermalarrests at 310 minutes and 440 minutes are not seen in the experimental data.The experimental results for Segment G clearly indicate that the data fromthermocouple one is in error after about 1360 minutes. The agreement between theexperimental data and the calculated response 101.6 mm below the surface is the best ofthe three experiments (within 80 °C at 1410 minutes).15oo/ 00 1 0 A1I 0 AI i 3I 0 A1ooo- I a ___--- -I A A— A A0— AF— 1I - AJ / AI /0I ,/500-_._-0000 0 0 0 OQI I I I I I1300 1350 1400 1450Time (minutes)Figure 89: Model Results and Measured Temperatures for Segment G Using theCalculated Power Distribution.1317.3. DiscussionOther than beam accelerating voltage, power level and size of the cylinder, theindustrial experiments do not differ substantially from the those conducted in thelaboratory. Thus the poor agreement between the measured and calculated response inthe industrial experiments are puzzling in light of the results of the laboratory scaleexperiments reported in Chapter 6. This indicates that the industrial experiments wereconducted differently from the laboratory experiments (which produced greater error inthe measured temperatures), the method of calculating the power distribution is flawed orthat the either boundary conditions or fluid flow parameters are different in the largerscale experiments.The thermocouples used in the industrial experiments were quite different fromthose used in the laboratory experiments. The thermocouples were tantalum sheathed,inserted into the holes and held in place using compression fittings. Poor contact betweenthe thermocouple tip and the cylinder could have resulted in erroneous data. The additionof the tantalum sheathing also provides for some thermal resistance producing additionalerrors. There is some evidence to suggest that this has occurred. The response forthermocouple 1, Segment A (Figure 87) is over predicted by the model during the initial110 minutes. At approximately 110 minutes into the experiment, the difference betweenthe measured temperature and the predicted temperature decreases - with no apparentexternal influence. This shift coincides with measured temperatures near or above theliquidus temperature, at that location, indicating the presence of liquid metal and adecrease in the thermal contact resistance. All of the thermocouples show a thermal lagafter the power is initiated. This behaviour is similar to that at the lower thermocouplepositions in the laboratory experiments which is indicative of poor thermocouple contact.The problems of acquiring an accurate thermocouple measurement in an electronbeam furnace are amplified in an industrial setting. The distances over which the signalmust be transported are larger making the signal more susceptible to noise.132During the experiments, thermocouple #1 was in contact with liquid titanium atan elevated temperature for a significant period of time. During this time it is possiblethat the thermocouple was poisoned. This would result in erroneous data at the topthermocouple position. If the data from thermocouple #1 is treated as unreliable(especially in the later stages of the experiments) and the model is fitted to thermocouple#2, a power efficiency of approximately 0.6 is required. Figure 96 shows the predictedtemperature response at the thermocouple locations given a power efficiency of 0.6 forSegment A. Although the agreement between the measured and calculated response atthe lower thermocouple positions is improved, the differences at 134 minutes are still onthe order of 100 °C (see Figure 90). The model also predicts a smaller liquid pool thanwas observed on the videotape of the experiments (177.8 mm diameter observed, 130mm predicted) and a surface temperature profile 50 °C below that measured by theMR camera in the liquid region (see Figure 91). In addition, using a power transferefficiency of 0.6, the model does not predict that the thermocouple 25.4 mm below thesurface would attain the temperatures necessary to poison the thermocouple.1330000T1500- 00000000— T,J I --------------Ti0 A,A,/0 0 A,/0010100°- f/ ° - - IIIa ‘0I a5 /oIA.-.0000—0 /....I ‘ .—.— 0v.., ——Ia / 00 ——/ A <>o00 .—/A——fAA •.o00 ——A .— I0 -. 50 100 150 200Time (minutes)Figure 90: Predicted Temperature Response of the Cylinder Using the Calculated PowerDistribution for Segment A with = 0.6.1800•..‘ .-.‘1600—1400 -\\ \1200- \. - - - -0.EE 1000800-_________Calculated 134.38 mmNlRScanl34minutes600 - ----- Calculated TSmf 11=0.6I I0 50 100 150 200 250Radial Coordinate (mm)Figure 91: Measured and Calculated Temperature Distributions for Segment A at 134minutes.134The cylinder was instrumented using 9.5 mm holes to position the thermocouplesat the centerline. For the thermocouple closest to the heat source (thermocouple #1) thisis unlikely to present a source of error in the temperature measurement. For thethermocouples below the top position, the thermocouples closer to the surface and theirassociated holes represent a possible interruption to the heat flow path which may resultin decreased temperatures being measured at the other positions. This effect may alsohave existed in the laboratory experiments but the holes were almost one-quarter of thesize of the holes used in the industrial experiments.The calculation of the power distribution depends critically on the measurementof the top temperature distribution. Figure 92 shows the calculated power distributionsand the temperature distributions that produced them. It is quite clear that small changesin the measured temperature distribution bring about significant changes in the calculatedpower distribution. For example a difference in 40 °C in the centerline surfacetemperatures of Segments A and G results in a difference of 65 W cm2 at the centerline.By corollary, it is also true that significant changes in the power distribution will bringabout only small changes in the top temperature distribution. Since the MR camera isbelieved to be accurate to within 3% (or 60 K at 1973 K- 60 °C at 1700 °C), the error inthe calculated power distributions is expected to be at least of the order of 3%. Due to thesensitivity of the power distribution to surface temperature, this error is likely to begreater.The calculation of the power distribution also depends on the cylinder havingattained steady state. At the time of the scans used to determine the power distribution,the cylinder has not yet reached steady state (see Figures 87, 88 and 89) as indicated byincreasing temperatures in the cylinder interior. The heat balance at the time of each MRscan can be determined using the model results. The results of these calculations (shownin Table 23) show that the cylinder did not attain steady state during any of theexperiments. From the temperature response of the top thermocouple in each of theU0zI2t)F0135experiments, it appears that the near surface region has reached a quasi-steady state.Since the surface and near surface temperatures will determine the applied power (in themethod used here) it is possible that the errors associated with the cylinder not being atsteady state are small when compared with the errors in the actual measurement of thetemperature and the calibration of the NIR camera.300250-,I.\r, -LUJ 011501005000 50 100 150 200 250Radial Co-ordinate (mm)Figure 92: Calculated Power Distributions and NIR Camera Scans for Segments A, C, Gand H.136Table 23 Heat Balances at the Time of Each NTR Scan During the IndustrialExperiments.Surface Segment A Segment C Segment G Segment HTime (mm) 134 400 410 1597Bottom (W) 369.5 610.4 496.9 575.1Edge (W) 5420.7 6068.0 5127.3 6884.7Evap(W) 1549.7 2343.8 1402.5 6631.4Top (W) 16697.5 17233.8 14607.5 26746.0Total Output 24037.4 26256.0 21634.2 40837.2Applied Power 31580.0 34570.0 28115.9 66241.1Diff 7542.6 8314.0 6481.7 25403.9% Close 24 % 24 % 23 % 38 %The power distribution calculation is also dependent on the conditions of fluidflow within the liquid pool or on the specification of the liquid thermal conductivityfunction. A proper determination of the thermal conductivity function could be carriedout by comparing the predicted pool dimensions to the actual pool dimensions. Due tothe lack of reliable data about the power distribution used at the end of the experiments,it is quite difficult to compare the predicted pool profile and the actual one. A coupledheat and fluid flow model may be able to correctly predict the transport of heat by bulkmotion and consequently remove the need for a thermal conductivity function but wouldalso require a priori knowledge of the power distribution. Figure 93 shows the effect ofdifferent thermal conductivity multiplying factors on the calculated power distribution.As the LTCMF is increased from 1. to 20., the calculated impingement point movestowards the axis and the calculated beam spreading factor decreases. Since the model wasable to predict the thermal response of the thermocouples in the laboratory experimentsusing the thermal conductivity function discussed in Section 6.2, this thermalconductivity function was used for the determination of the power distribution. Unlessthe fluid flow conditions in the larger cylinder are significantly different than those in thelab scale cylinder, this thermal conductivity function should provide accurate results.137150‘l0015000 50 100 150 200 250Radial Coordinate (mm)Figure 93: The Calculated Beam Power Distribution as a Function of the Liquid ThermalConductivity Multiplier for Segment A.7.4. SummaryThe Constraints of using a production facility for experiments combined withother difficulties associated with doing experiments in an operating electron beamfurnace make it difficult to reach conclusions about the effect of pressure on powerdelivery with the same degree of certainty as the laboratory experiments. Nevertheless,the results of the experiments, although unable to provide a correlation between pressureand beam spreading parameter under the conditions used, show quite well that pressurewill significantly affect the temperature regime in the target cylinder.Chapter 8.: Conclusions8.1. ConclusionsA series of laboratory experiments was conducted to elucidate the effect ofchamber pressure on power transfer in electron beam melting. Cylinders of titanium andtantalum have been instrumented with embedded thermocouples in order to measure andrecord the temperature at several locations within material being heated by an electronbeam while varying the chamber pressure. The tantalum was used to examine the effectin the absence of a molten pooi and metal vapour. In parallel, a mathematical model ofthe electron beam heating process has been developed and used to analyze these experimental results.The experimental results clearly show that the temperature regime within a targetcylinder is significantly affected by variations in the chamber pressure. In the tantalumexperiments the temperature at the top thermocouple position, located 25.4 mm belowthe top surface, was observed to change by 40 °C in association with a pressure changefrom 0.37 to 0.93 Pa. In titanium, a pressure change from 0.004- 0.76 Pa was observedto produce temperature changes of up to 60 °C at the thermocouple 20.1 mm from belowthe top surface. The magnitude of the observed temperature changes was on the order oftwo to three times the intrinsic error in the thermocouples themselves. Thus, arguably theobserved variations in temperature were very near the limit of our ability to measurethem. However, the experimental results were all self-consistent in that a pressure variation produced a similar trend in the temperature response at each thermocouple location. Furthermore, while these temperature changes are small in comparison to the actualtemperatures attained in electron beam melting, they are significant in that they couldhave an impact on the ability of the electron beam remelting process to removedeleterious particles.Subsequent analysis of the experiments with the model has revealed that theeffect of pressure is to alter predominately the power distribution and not the beam- 138 -139transfer efficiency. The predictions of the model can be made to reproduce quantitativelythe behaviour of the top thermocouple (used to fit the model) and qualitatively the behavior of the lower thermocouples in a given cylinder by varying the power distribution parameters as a function of chamber pressure. In contrast, when fitting the response of thetop thermocouple by varying the power transfer efficiency with chamber pressure, themodel does not qualitatively reproduce the response at the other thermocouple locations.In the tantalum experiments, it was found that the beam focused below 0.37 Paand defocusecl as the pressure was increased to 0.93 Pa. In the titanium experiments, thebeam first defocused, then focused and defocused again as the pressure was increasedfrom 0.004 Pa to 0.76 Pa.VThe results of a series of analyses in which the model was fitted to the experimental thermocouple results show that the beam power distribution is adequately representedby a Gaussian or normal distribution. Further analysis has led to the development ofempirical relationships for the effect of pressure on the beam focusing characteristics(namely the standard deviation in the normal distribution). Under the conditions used inthe laboratory experiments the relationship between pressure and beam spreadingparameter in tantalum heating is given by1 6.4mm pczo.37Pa (52),b l2.64—33.265p+44.276p 0.37Pa p 0.93Paand for titanium melting:J12L4209p—40.5255p+19.5132 p0.2Pa (53)b.0.6921p—7.7660p+14.6840 0.2Pap0.8PaIt should be noted that these empirical relationships are only applicable to the experimental scale furnace conditions of beam accelerating voltage, beam-target distance, pressurerange and target material. An important conclusion of the mathematical analysis is that afundamentally based model is able to reproduce the behaviour observed in thethermocouple measurements.140The observed focusing effect is consistent with the literature[68] and has beenattributed to positively charged gaseous ions (created by beam interactions with thechamber gas) compensating for some of the intrinsic negative space charge of theelectron beam. As the pressure is increased, more electrons are scattered when interactingwith the chamber gas and the beam is defocused. At any given pressure, the beam is lessfocused when striking a titanium target than it is for a tantalum one. At the temperaturesexpected at the surface of the target cylinders, the vapour pressure of titanium is nearlytwice that of the pressure of non-condensable gases while the vapour pressure oftantalum is almost negligible. Thus the difference in the effect of pressure on focusingbetween the two materials indicates an increase in the impediments to the propagation ofthe electron beam (i.e. more gas atoms) in the case of titanium and not a difference in thefundamental scattering process as a result of the presence of liquid metal.In summary, careful and self-consistent experiments, a mathematical model thatcan reproduce, both quantitatively and qualitatively, the results of the experiments andsome elementary physics with respect to the interactions between gas atoms and an electron beam provide a solid, although indirect, link between chamber pressure and thebeam power distribution.From a mathematical modelling standpoint, the work demonstrates that, for smallcylinders being heated by electron beam in the laboratory scale furnace, the enhancedheat transfer due to bulk motion of the liquid can be adequately represented by anenhance thermal conductivity in the liquid.A similar program of experiments was carried out in an industrial scale furnace atAxel Johnson Metals. A titanium cylinder was instrumented with thermocouples andsubjected to heating by an electron beam. Given the constraints on the industrial furnace,the experiments were not ideal and subsequent analysis with the model was difficult tocarry out. Nevertheless, the experimental results (including the data from a near infra-redcamera), the mathematical model and inverse heat flux calculations were used to attempt141to elucidate the relationship between chamber pressure and the beam power distributionon an industrial scale. Limited data, poor agreement between the model and the measuredtemperatures and some inconsistency in the focusing current made conclusions regardingthe link between chamber pressure and beam power distribution difficult. Theinvestigation did indicate that the beam focusing characteristics in the industrial furnaceare affected by variations in chamber pressure.The results of the laboratory experimental work show reasonably well the effectof pressure on power transfer in electron beam melting. In attempting to transfer thelaboratory experimental results to industrial practice, a number of important differencesbetween a typical industrial furnace and the laboratory furnace should be noted.In the laboratory furnace the accelerating voltage did not exceed 20 kV. In theindustrial experiments the accelerating voltage was 30 kV while typical industrial operation is at 40 kV or higher. The accelerating voltage will play a major role in determiningthe ability of the electron beam to penetrate the gas atmosphere above the target. In addition, the laboratory furnace is much smaller than a typical industrial furnace implyingthat the distance between the beam exit point and the workpiece is larger in industrialmelting. The larger the distance over which the beam must be propagated, the more collisions there will be between the beam and the gas atoms in the chamber increasing thelikelihood of scattering. In addition, high power melting guns usually use some form ofinert gas self-focusing that may affect the scattering characteristics of an electron beam inthe industrial context.The laboratory and industrial experiments were restricted to pure (or commercially pure) materials. Although CP titanium accounts for the bulk of the material meltedby electron beam commercially, other alloys are also melted. During electron beammelting of the workhorse titanium alloy (Ti6AI4V), for example, the metal vapour phasewill differ considerably from that found in either the industrial or laboratory experimentsdiscussed in this thesis.142The fluid flow conditions during electron beam hearth melting are also muchmore complex than the fluid flow conditions encountered in these experiments. Even inthe ingot solidification stage, which is reasonably approximated by the conditions in theexperiments, there are additional fluid flows generated by the introduction of hot metalinto the crucible at the pour lip. In the refining and melting regions of the hearth furnace,the fluid flows are much more complex.These factors of beam accelerating voltage, alloy element evaporation, furnacesize and differences in fluid flow regimes combine to suggest that the results of theseexperiments are not readily applicable in a quantitative way to the commercial electronbeam melting of titanium and its alloys. However, the methods and principles used in thisthesis for the determination of the link between chamber pressure and power transfer arereadily applicable to industrial scale experiments.8.2. Recommendations for Future WorkThe industrial experiments discussed in this thesis did not provide sufficientinformation to determine the relationship between chamber pressure, beam power distribution and ultimately temperature variation in an operating industrial furnace. Toimprove our ability to predict the power transfer as a function of operating parameters inan industrial hearth furnace a program of additional experiments and further analysisshould be conducted.Additional industrial experiments should attempt to remove the dependence onthe MR camera and the target attaining steady state. By using the inverse heat transfercalculation method used by Imwinkelreid[38j, these goals would be met. Since themethod uses thermocouple measurement to determine the input heat flux, the NTRcamera would not be required to determine the power distribution. In addition, themethod does not require that the target be at steady state to complete the analysis. Thisfeature alone would significantly reduce the impact of the experiments on the production143furnace. To use the method successfully, thin (100-150 mm) disks 500 mm in diameterof the appropriate target material would be instrumented with a large number ofthermocouples (15-20) located at different radial locations on the bottom surface of thetarget.The laboratory work clearly identifies target material and chamber pressure asimportant variables in the delivery of power to the surface of the target. On an industrialscale, the beam accelating voltage and the beam focus current may also significantlyaffect the power transfer charteristics of an electron beam. Any additional experimentsshould be done to determine the effects of these variables on power transfer in electronbeam heating.In all instances, industrial scale experiments should talce place within the rangesassociated with normal industrial practice (i.e. pressures between 0.13 Pa and 4.00 Pa).With these considerations, experiments with tantalum, commercial purity titanium,Ti6A14V and Titanium-17 would include all of the materials of interest on an industrialbasis (using tantalum would describe the baseline for the absence of metallic vapour).Since the power distribution could be determined after reasonably short time, alloyelement depletion need not be a factor.References[1] “CRC Handbook of Chemistry and Physics”, 63t Ed., 1982-1983.[2] “Thermal Properties of Titanium Alloys” Defense Materials Information Center,Battelle Memorial Institute, Columbus, Ohio.[3] Ablitzer, D, M. Allibert, C. Avare, J.P. Bellot, Y. Bienvenu, Y. Fautrelle, P.Gillon, E. Hess, A. Jardy and A. Pasturel: “Electron Beam Remelting andRefining - State of the Art 1992”, R. Bakish, ed., Bakish Materials Corporation,Englewood, NJ, pp. 85 - 94.[4] Amboss, K. : “Electron and Ion Beam Technology”, Proc. Conf., 1974, TheElectrochemical Society, Princeton, NJ., pp. 482 - 517.[5) Apelian, D. and C.H. Entrekin Jr. : International Metals Reviews, No. 2, 1986,pp. 77-89.[6] Ardenne, M. v. : Tables of Electron Physics, Vol. 1, VEB, Deutcher Verlag derWissenschaften, Berlin, 1956, p. 126.[7] Ballantyne, A.S. : Ph.D. Thesis, University of British Columbia, 1979.[8] Barin, I and 0. Knacke: “Thermochemical Properties of Inorganic Substances”,Springer, Berlin & Verl., Stahleisen, Dusseldorf, 1973 and 1977.[9] Barin, I.: “Thermochemical Data of the Pure Substances”,Verlagsgesellschaft.mbH, Weindheim (FRG), 1989.[10] Bas. E.B. and G. Cremosnik: “First International Conference on Electron and IonBeam Science and Technology”, Proc. Conf., 1964, J. Wiley and Sons, Inc. NewYork, New York.[11] Bellot, J.P. et. al. : Thermal Modelling of Solidification and Cooling of anElectron Beam Titanium Ingot, 7th World Conference on Titanium, San Diego,1992, TMS-AIME, Warren, Ohio, In Press.[12] Bellot, J.P., E. Floris, A. Jardy and D. Ablitzer: Numerical Simulation of theEBCHR Process, “Electron Beam Remelting and Refining- State of the Art1993”, To be published.[13] Brandon, G.W. : The Influence of Pressure Variations on the Penetration ofElectron Beam Welds, Pamphlet, Society of Manufacturing Engineers TechnicalPaper No. AD75-975, Society of Manufacturing Engineers, Dearborn Michigan,1975.[14] Carpenter, L.G. and F.R. Reavell : Nature, 163, 1949, p. 527.- 144 -145[151 Carslaw, H.S. and J.C. Jaeger: “The Conduction of Heat in Solids”, OxfordUniversity Press, 1959. p. 112.[161 Carvajal, L. Fernando and G.E. Geiger: Met. Trans., Vol. 2, Aug. 1971, pp. 2087- 2092.[171 Cockcroft, S.L. : Ph.D. Thesis, University of British Columbia, September 1990.[181 Cook, R.D. : “Concepts and Application of Finite Element Analysis”, 2nd ed.,John Wiley and Sons, New York, New York, 1974.[191 Crawley, A.F. : International Metals Reviews, Vol. 19, 1974, Rev. 180, pp. 32-48.[201 Dumonte, P. and G. Sayegh: “Sixth International Conference on Electron and IonBeam Science and Technology”, Proc. Conf., 1964, J. Wiley and Sons, Inc. NewYork, New York. p. 346 - 365.[211 Edwards, J.W., H.L. Johnston and W.E. Ditmars : J. Amer. Chem. Soc., 75,1953, pp. 2467-2470.[22] Eisen, W.B. and A. Campagna : Met. Trans., Vol. 1, April 1970, pp. 849 - 856.[231 Elliot, J.F. and A. Maulvault : Elec. Furnace Proc., 1970, vol. 28, p. 13.[24] Entrekin, C.H. : “Electron Beam Remelting and Refining - State of the Art 1985Pt. I”, R. Bakish, ed., Bakish Materials Corporation, Englewood, NJ, pp. 40 - 47.[25] Falk, L., A. Jardy, D. Ablitzer and P. Paillère: “Mathematical Modelling ofMaterials Processing Operations”, Proc. Conf., Palm Springs, 1987, J. Szekely,et.al. eds, TMS-AIME, Warren, Ohio. pp. 1041 - 1054.[26] GEAE Technical Memorandum, TM91-160.[27] Giedt, W.H. : “Trends in Welding Research in the United States”, S.A. David,ed., American Society for Metals, Metals Park, Ohio, 1981, p. 109 - 131.[28] Goldak, John, A. Chakravarti and M. Bibby: Met Trans B, Vol. 15B, June 1984,p. 299 - 305.[29] Harker, H. R. : “Electron Beam Remelting and Refining - State of the Art 1989 -Part Two”, R. Bakish, ed., Bakish Materials Corporation, Englewood, NJ, pp. 185- 186.[30] Harker, H.R. : Private Communication.146[31] Harker, H.R. and C.H. Entrekin International Conference on Titanium Productsand Applications, Proc. Conf., 1987, Titanium Development Assoc., Dayton,Ohio, pp. 939 - 947.[32] Hernandez-Morales, B. : “Transient Heat Conduction in 2-D AxisymmetricProblems”, Internal Communication, October 1990.[33] Hicken, G.H., W.H. Giedt and A.E. Bentley Welding Journal ResearchSupplement, March 1991, Vol. 70, pp. 69s - 75s.[34] Ho, C.Y., R.W. Powell, P.E. Liley, : J. Phys. Chem. Ref. Data, Vol.3, Suppi. 1,“Thermal Conductivity of the Elements”, 1974.[35] Huebner, K.H.: “The Finite Element Method for Engineering”, John Wiley andSons, New York, New York, 1975.[36] Hultgen, R, P.D. Desai, D.T. Hawkins, M.Gleiser, K.K. Kelley and D.D. Wagner“Selected Values of Thermodynamic Properties of the Elements”, ASM, MetalsPark, Ohio, 1973[37] Hunter, G.B. : Private Communication.[38] Imwinkelreid, T. : Ph.D. Thesis, 1993, EPFL, Lausanne, Switzerland.[39] Isawa, T., H. Nakamura and K. Murakami : ISIJ International, Vol. 32 (1992),No. 5, pp. 607 - 615.[40] Jardy, A. and D. Ablitzer: “Mathematical Modelling of Materials ProcessingOperations”, Proc. Conf., Palm Springs, 1987, J. Szekely, et.al. eds, TMS-AIME,Warren, Ohio. pp. 1027 - 1040.[411 Jarrett, R.N. : “Electron Beam Remelting and Refining- State of the Art 1986”,R. Bakish, ed., Bakish Materials Corporation, Englewood, NJ, pp. 332 - 346.[42] Kheshgi, H.S. and P.M. Gresho: “Electron Beam Remelting and Refining- Stateof the Art 1986”, R. Bakish, ed., Bakish Materials Corporation, Englewood, NJ,pp. 68 - 79.[43] Kiemens, P.G., Journal of Applied Physics, Vol. 47, No. 5, May 1976, p. 2165 -2174.[44] Knecht, J.A. II and C.H. Entrekin: “Electron Beam Remelting and Refining-State of the Art 1987”, R. Bakish, ed., Bakish Materials Corporation, Englewood,NJ,pp.46-59.[45] Koenig, R.J et. al.: “Titanium Rotating Components Review Team Report”,United States of America, Federal Aviation Administration, Aircraft CertificationService, Engine and Propellor Directorate, December 14, 1990.147[46] Kubaschewski, O.A. and C.B. Alcock: “Metallurgical Thermochemistry”,Permagon Press, New York, New York, 1979.[47] Laughlin, J.P. : International Conference on Titanium Products and Applications,Proc. Conf., 1987, Titanium Development Assoc., Dayton, Ohio, pp. 879-883.[48] Meier, J.W. : “Electron and Ion Beam Science and Technology - Proceedings ofthe 2nd International Conference”, ed. R. Bakish, AIME, New York, p. 505 - 533.[49] Menzies R.G, C.E. Shamblen, D.G. Backman and G.B. Hunter: Process Designand Control for Aircraft Engine Materials, “Electron Beam Remelting andRefining - State of the Art 1993”, To be published.[50] Mitchell, A and D.W. Tripp : International Conference on Titanium Products andApplications, Proc. Conf., 1987, Titanium Development Assoc., Dayton, Ohio,pp. 1011 - 1019.[51] Mitchell, A and D.W. Tripp : Proc. Vac. Metl. Soc. Conf., Pittsburgh 1989, edsL. Lherbier, G.K. Bhat, pp. 126 - 135.[52] Mitchell, A. and D.W. Tripp: “Sixth World Conference on Titanium”, P.Lacombe, R.Tricot, G. Béranger, eds., Société Française de Métallurgie, LesEditions de Physique, Les Ulis, France. pp. 603 - 608.[53] Mitchell, A. and K. Takagi: “Electron Beam Remelting and Refining - State ofthe Art 1984”, R. Bakish, ed., Bakish Materials Corporation, Englewood, NJ, p.89.[54] Mitchell, A. and S. Joshi : Met. Trans., Vol. 4, March 1973, pp. 631 - 642.[55] Mitchell, A., D.W. Tripp, S.L. Cockcroft and C.H. Entrekin : “Electron BeamRemelting and Refining- State of the Art 1991”, R. Bakish, ed., Bakish MaterialsCorporation, Englewood, NJ, pp. 248 - 260.[56] Miyazaki, T. and W.H. Geidt: Int. J. Heat and Mass Trans., Vol. 25, 1982, pp.807-8 14.[57] Nakamura, H : M.A.Sc. Thesis, University of British Columbia, 1989.[58] Nakamura, H. and Mitchell, A : ISIJ International, Vol. 32 (1992), No. 5, pp. 583- 592.[591 OMEGA Engineering: “The Temperature Handbook”, Vol. 28, pp. Z-4l.[601 Ozisik, M.N.: “Heat Conduction”, John Wiley and Sons, New York, New York,1980.[61] Pam, V.M. : Sbornik Nauch Rabot Inst. metallofiz, 20, 1964, pp. 130-132.148[62] Pavelic, V., R. Tanbakuchi, O.A. Uyehara and P.S. Myers : Welding JournalResearch Supplement, 1969, Vol. 48, pp. 295s-305s.[63] Pocklington, D.N. and B. Patrick : Met. Trans. B, Vol. lOB, Sept. 1979, pp. 359366.[64) Powell, A., B. Damkroger, R. Fisher, J. VanDenAvyle and J. Szekely : Modellingof Melting by a Stationary Electron Beam, “Electron Beam Remelting andRefining- State of the Art 1993”, To be published.[65] Rosenthal, D. : Trans. ASME, 1946, Vol. 68, pp. 849-865.[66] Rykalin, N., A. Uglov, I. Zuev and A. Koroka: “Laser and Electron BeamMaterial Processing Handbook”, MW Publishers, Moscow, U.S.S.R., 1988.[67] Sanderson, A. : Br. Weld J., 1968, Vol. 15, pp. 509-23.[68] Schiller, S. U. Heisig and G. Jaesch: “Electron and Ion Beam Science andTechnology- Proceedings of the 4th International Conference”, ed. R. Bakish,The ElectroChemical Society, New York, New York, p. 115 - 135.[69] Schiller, S., U. Heisig and S. Panzer: “Electron Beam Tecnology”, 1982, Wileyand Sons, New York, New York.[70] Schunk, P.R., R.W. Fisher and F.J. Zanner: Proceedings of the 1991 VacuumMetallurg Conference on The Melting and Processing of Specialty Materials,Proc. Conf., 1991, Iron and Steel Society, Warrendale, PA. p. 41 - 43.[71] Shamblen, C.E., and G.B. Hunter: “Electron Beam Remelting and Refining-State of the Art 1989 - Part I”, R. Bakish, ed., Bakish Materials Corporation,Englewood, NJ, pp. 50- 62.[72] Shamblen, C.E. : “Electron Beam Remelting and Refining- State of the Art1990”, R. Bakish, ed., Bakish Materials Corporation, Englewood, NJ, pp. 49 - 54.[73) Shyy, W., Y. Pang, G.B. Hunter, D.Y. Wei and M.-H. Chen : Tnt. J. Heat MassTransfer., Vol. 35, 1992, No. 5, pp. 1229-1245.[74] Smith, H.R.: “Electron Beam Processing”, 1972, Universal TechnologyCorporation, Dayton, Ohio, pp. lal - 1a66.[75] Solomon, J.L. and H.A. James: “Electron and Ion Beam Science and Technology- Proceedings of the 2nd International Conference”, ed. R. Bakish, AIME, NewYork, p. 553 - 574.[76] Stephan, H.: “Electron Beam Metallurgical Processing Seminar”, February,1971, Universal Technology Corp., Dayton, Ohio., pp lal - lalOO.149[77] Sze, J.S. and S.A. Self: “Electron Beam Remelting and Refining- State of theArt 1989”, R. Bakish, ed., Bakish Materials Corporation, Englewood, NJ, pp. 21 -35.[78] Tilmont, S.M and H. Harker: “Electron Beam Remelting and Refining- State ofthe Art 1990”, R. Bakish, ed., Bakish Materials Corporation, Englewood, NJ, pp.135 - 145.[79] TPRC Data Series: “Thermophysical Properties of Matter”, Volume 1,IFI/Plenum Data Co., New York, New York, January, 1981.[80] Tripp, D.W. : M.A.Sc. Thesis, University of British Columbia, 1987.[81] Ul’yinov, R.A. and S.F. Korfun : Phys. Metas Metallogr., 1964, 17, No. 4, pp. 24-30.[821 Watanabe, S, K. Suzuke and K. Nishikawa: ISIJ International, Vol. 32 (1992),No.5,pp.625-629.[83] Wei, P.S. and Y.T. Chow : Met. Trans. B, Vol. 23B, February 1992, p. 81 - 90.[84] Westerberg, K.W., M.A. McClelland and B. Finlayson : Numerical Simulation ofMaterial and Energy Flow in an E-Beam Melt Furnace, “Electron BeamRemelting and Refining- State of the Art 1993”, To be published.[85] Zanner, F. and L.A. Bertram: “Computational and Experimental Analysis of aU-6w/oNb Vacuum Consumable Arc Remelted Ingot”, Sandia NationalLaboratories, SAND8O-1 156, National Technical Information Service.[86] Zienkiewicz, O.C. : “The Finite Element Method”, 3rd ed., McGraw-Hill, 1977.Appendix 1. : Thermophysical Properties of Materials Used in the ModelsThe mathematical model requires thermal conductivity (k) and heat capacity (Cr)as functions of temperature. For materials undergoing a phase change, an enthalpyfunction is also required. Determination of the enthalpy function requires thedetermination of heats of transformation. For the most part, the thermophysicalproperties of titanium and tantalum are reasonably well established and available in theliterature.A1.1 TitaniumThe thermal conductivity of titanium at various temperatures has been obtainedfrom the literature[34,79]. These values have been used to fit a polynomial to describetitanium thermal conductivity as a function of temperature. Even though titanium goesthrough a phase change (room temperature stable HCP cx Ti to BCC Ti) at 882 °C[9,21,81] the thermal conductivity can be represented by one polynomial, namelyk = O.218l57187—1.441O8x10T+2.58465x1OT (54)—1.32338x10’°T +2.58346x10’This relationship is shown graphically in Figure 94.0.3c-) II I I • I0 200 400 600 800 1000 1200 1400 1600 1800Temperature (C)Figure 94: The Thermal Conductivity of Titanium as a Function of Temperature.-150-151Fluid flow in the liqud metal pool produced during titanium heating results inenhanced heat transfer within the liquid pooi. This enhanced heat transfer has beenmodelled by increasing the thermal conductivity in the liquid pool by a multiplyingfactor. This particular approach has been used successfully in modelling the VARprocess[7]The heat capacity of titanium is reported in the literature[8,9,46] asC(x)=O.5188+2.2O3x1OTJgnf o1 OT882°CC()=O.4593+1.656x1OT 882T1666°C (55),C(1)=O.72O7 T>1666°Cand is shown graphically in Figure 95.0.750.70.65Vc 0.6c.)-..0.550.5 I I I I I I0 200 400 600 800 1000 1200 1400 1600 1800Temperature (C)Figure 95: The Heat Capacity of Titanium as a Function of TemperatureTo be consistent with the fixed mesh required in the finite element method anaverage density of 4.54 g cm3[l] was used.As discussed in Chapter 5., the enthalpy method is used for latent heat evolutionassociated with a phase change. In this method, an enthalpy function is used to calculatea modified heat capacity for those elements that have a node above or within a phasetransition range. Once through a phase transition range, the enthalpy technique is152bypassed to reduce the computational load. Titanium goes through two transitions in thetemperature range of interest, the x- transition at 882 °C and the solid to liquidtransition at 1666 °C. Since there is no transition gap in a pure material, an artificiallysmall transformation range was imposed on the system. Thus the melting range of CPtitanium was assumed to be from 1664 °C to 1668 °C and the ct-f transition was assumedto take place from 880 °C to 884 °C. The latent heat of the solid-solid transition isreported as 87.68 Jg1[9,36] and the latent heat of fusion as 307.85 J g’[9,36,46,61}.Using the heat capacity function (Equation (55)), heats of transition and the associatedtemperature ranges, the following enthalpy function for titanium can be determined (thefunction is shown graphically in Figure 9.1.101x10T2÷0.5188T—13.039 Jg1 0°C T 880°C= 22.580T—19341.629 880°CT884°C= 8.299x10T +0.4593T+148.2312 884°C T1664°C (56)= 77.6906T—128134.87 1664°CT 1668°C= O.7207T+250.8985 T>1668°C1800160014001200‘100080060040020000 200 400 600 800 1000 1200 1400 1600 1800 2000Temperature (C)Figure 96: The Enthalpy Function for Titanium.The vapour pressure of titanium has been reported in the literature[46] as153111.74—O.661og(T+273)PTI(l)—’33.10 T+273 (Pa) (T in °C) (57).The emissivity of titanium has been reported as 0.4 and is relatively independentof temperature[2].Finally the heat of vapourization of titanium has been reported in theIiterature[8,9,14]. An average value of 8873. J mold has been used. Due to uncertainiesin the value used for the heat of vapourization no attempt has been made to adjust thequantity for changes in vapourization temperature.A1.2 TantalumThe thermal conductivity of tantalum as a function of temperature was obtainedfrom the literature[34] and this data fitted to obtain a polynomial for use in the model.The resulting thermal conductivity function iskTa = 0.573787+3.5l05420xl0T+6.8296xl092 (58)—2. 73308x10’ T3The data and the polynomial fit are shown in Figure 97.0.6700.6600.650, 0.6400.6300.6200.610c0.6000.5900.5800.5703000Figure 97: The Thermal Conductivity of Tantalum as a Function of Temperature0 500 1000 1500 2000 2500Temperature (C)The heat capacity of tantalum is reported in the literature{46] as154C, = 0.153770.—1.20238x10(T+273)1040 520 (59).+1.08677x1O(T+ 273)2— 2(T+273)Additional heat capacity data[341, the best fit to the data and the relationship describedby equation (59) are shown in Figure 98.55.00a50.00 a45.0O0 Data aAlcock40.0035.0030.0025.00B20.00 I I I I-200 300 800 1300 1800 2300 2800 3300Temperature (C)Figure 98: The Heat Capacity of Tantalum as a Function of TemperatureEquation (59) describes the data points in the temperature range of interest quite well andwas used in the calculations.Since tantalum does not go through any phase transformations in the temperaturerange expected in the experiments it is unnecessary to determine either the vapourpressure or the enthalpy function.An average value of 16.654 g cm3 was used for the density of tantalum.The emissivity of tantalum is a very strong function of temperature and surfaceconditions. Data for emissivity at a variety of temperatures was obtained from theliterature[341 and a polynomial fitted to the data. The emissivity of tantalum is given bythe following functionBTa = 0.06666+8.449xl05 (60).155and is shown in Figure 99.0.3500.3000.250-.• 0.200E0.1500.1000.0502500 3000 35000 500 1000 1500 2000Temperature (C)Figure 99: The Emissivity of Tantalum as a Function of TemperatureAppendix 2.: Finite Element MeshesA2.1. Titanium MeshThe mesh used for the titanium modelling is shown in Figure 100. The mesh ischaracterized by a fine mesh spacing over the 40 mm closest to the centerline of thecylinder and over the 20 mm closest to the surface of the cylinder. This is the region ofthe which was expected to have the largest temperature gradients. A total of 1201 nodesin 374 elements were used.The mesh used in modelling the Axel Johnson experiments is shown in Figure101. A total of 2121 nodes in 672 elements were used.A2.2. Tantalum MeshThe mesh used for the tantalum axi-symmetric modelling is shown in Figure 102.The tantalum mesh is similar to the titanium mesh in that it is more dense in the regionbelow the beam impingement point and near the top surface of the cylinder. A total of933 nodes in 288 elements were used.For the three dimensional problem used in the sensitivity analysis a total of 8469nodes in 1728 elements were used.- 156 -RadialCoordinate(mm)Figure100:TheMeshUsedintheFiniteElement ModellingoftheLaboratoryScaleTitaniumCylinder.150-I 100-4 0 C C-) • -4 >< <50 0oç-o-•0•0-G-0-0-0--G--0--0----0--—o——0——0——0——0—-G*-0--0-G-0-00-0----0--0---0——0-—--0——0--O-0--0-G0-0--0--0--o-Ø-o-,---—o—-—0——0-——0-——0-——0—0—0--00--00-0-0--0--0--0--0-,—-0——0-——0——0——0-——0-——0—-0—0--0-G0-0-0-0--0--0--0-*-0-—-O——0——0-——0——0———0——0——0-—G—0--0-G-0-00-0--0--0--0--0-—0——0.——0——0———0.-——0——0-——0.—G—0-0-0-00-0-0--0--0--0-—0——0——0——0——0——0——0——0—-0—0--00-0-0-0-0-0--0-0x0-—0——0——0—O—0--O-0-0-0-0--0--0--0--0-—0-——0——0-——0———0———-0——0-——0----—-0---—0—G—0--00-0-0-0-0--0+0--0--0-—0-——0-——0-——0-——0--——0———0-——0--———0——0-—0‘—0--G0-0-0-0-0--0--0-X.0-X0-—0——0——0-——0—--X_0_—0-——0--——--—‘‘————a--—0-0-00-0-0-0--0--0--0--0--0-—0---—0-——0-——0-——0--——0——0-——0-——0-——0—-00-0--0-0.-0-—0——0--—0——0---—0-——0---—0——0——0——0---0-0-0-0-0--0--0--0--0--0--0--0-—0-——0-——0-——0———0-——0-——0-——0———0——0-—0—0-*0-0--0--0--0--0--0--0--0—0——0-——0——0—--—0——0----—0———0——0---—0-—0-0-0-0--0--0--0--0--0--0--0--0—0——0——0-——0--——0-——0——0-——0-——0k—-0--0--0--0--0--0--0--0--0--0--0—0-——0----—0--——0——0-——0-——0———0——0-——0——‘-0--0--0--0--c’-----0--0--O--—n——0-——0-——n——O——0.——n--——0-——0----—O-—-IIIItrII020406080100IIIIII4-. Lit-Ii0-CDCDC.,CD-tCDtrlzCDrnCDCDCDz0-f-p CDrn>CDj.CD—p C’, 0C-)L0AxialCoordinate(mm)L’) 000000n00IjIIIIIII0000°F°0f -—0---———0-—--——0---——0————0—-—-0—-0——0-0—0--—0-——0--—0——0---0--0--0-—0-——0-——0——0--—0----0--0--0-C—0—----——0—----——0---——0-C0C—0----—0--——0---0-C——0-——0———0—-0--0-:—0-——0---——0——0——0-—-0--0--0-—o--——————0-————0——--—0-———0—-0--0-—o---———-—--——o—-————0———0————0-——-0—-0-—0-C—0——0-——0-—-0--0--0-C—0--——----——-—0—--—---—0--——0-—--—0-———0---0--0--—0———0———0--—--0--0--0-—0———0---——0--—0-——0—-0--0--0-C—O----------——0---------——0-—-—--—0-—--——0—-—-——0—-0——0——0———0-——0——0-——0----0---0-CC-—0-——0-——0——0--—0-----0--0--0-C—0-——0--——0-——0---—0—-0--0--0-CCC—0-——0--——0——0--—0-—-0--0--0-CC—0—--—0-——0---—0—-0--0-CC-C—0-——0-——0---—0—-0--0-CC)C—0——--—--———0-——-————0——-——0-——--—0——0—CC)C—0-—---——0--—--———0---——0---—,—0-———0——0---0--CC0C—0--——---—0------——0————0———0—-—-0——0--0--C0C—0—----——0------——0-———0-———0-——-0——0--0--CC0C—0——-———0------——0-———0--—--—0-------0-—0--0—C0C—0-—---———0--------—0-—-——0--——0--—0—CC0C—0--------—0————G—---—0-—0--0—CC0C8crRadialCoordinate(mm)Figure102:TheAxisymettriCMeshUsedintheFiniteElementModellingoftheTantalumLaboratoryExperiments.100 80 60C 020 0—0——0---0---0------0-—0——0——0--—0——0-——0——0——0—-0---*--<---0—0--—0——0——0---—0——0——0——0—-0--0--0--0-----0-—0——0——0--—0--—0——0---——6—-0--0--0--0-----0-—0——0——0——0——0——0——0-——0------0--0--0-0--0--0--—0-——0—--—0—.-.--0—-—0—-—0——0———0-—-0--0--o—X-o.—0--0--0--0-—0--—-—0-——0-——0-——0--——0—-——0-———-0--—-0--0--0—--0--0-—0--0--0-—0———0-——0-——0-———0-——0-——.—0-———0——-0--0---0—--0-—0--0--0--0-—0--——0-——0———0———0—-——0--——0-———0-—---0--0--0---O--0-—C---0--0--—0———0———0———‘0--———0-——0-—,—0---—0—-0--0--0---0--0-—0——0——0——0——0--—0--—0--—0—-0--0--0---0--0-—0——0--—6---—0--—0——0--———t:——:--——EEEEEEEEEEEEEE—0--—0—-0--0--0--0---0-—0——0——0——0--—0--—0---—0——0---0--0--0---0--0--0--0-—0——0——0——0---—0--—0———n---------------—•---—a--—n——n---—————J-C-----O-—0-—0---0-III020406080


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