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Mathematical modelling of heat transfer in the vacuum investment casting of superalloy IN718 Dominik, Barbara Eva 1993

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MATHEMATICAL MODELLING OF HEAT TRANSFERIN THE VACUUM INVESTMENT CASTINGOF SUPERALLOY 1N718byBARBARA EVA DOMINIKB.A.Sc., The University of British Columbia, 1986A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department of Metals and Materials Engineering)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1993© Barbara Eva Dominik, 1993In 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.(Signature)_____________________________Department of 4The University of British ColumbiaVancouver, CanadaDate_______________DE-6 (2/88)AbstractHeat transfer in vacuum investment casting of a nickel-based superalloy, 1N718, wasstudied using a finite-element-based solidification heat transfer code. For the externalradiation boundary, general 2- and 3-dimensional viewfactor calculation codes based on a raytracing approach were developed and verified. Heat transfer at the mould-metal interfacemay occur by contact conduction between the metal and the mould, and by radiation acrossthe interface gap areas. A simple, time-dependent model was developed to simulate thedecreasing contact conduction as solidification progresses. Temperature measurements weremade on casting moulds in a series of experiments done in collaboration with Deloro StelliteInc. of Belleville, Ontario. The model was applied to the experimental castingconfigurations. The model results were most influenced by the value of the mould thermalconductivity, the interface contact function and the radiation environment surrounding themould. The mould thermal conductivity which resulted in the best fit to the data ranged from0.9 to 1.1 W/m-deg C for the cylindrical castings and 0.8 - 0.9 W/m-deg C for the finnedcastings. The interface contact conduction function decreased from 1400 W/m2-deg C attime t = 0, to a value of 0 at t 700 seconds and t 200 seconds for the cylindrical andfinned castings respectively. The model was used to simulate casting conditions for a tensiletest bar which had been analyzed experimentally by Deloro Stellite Inc. Although the 2-dimensional model used gave results that were in qualitative agreement with the experimentsin terms of predicting effects on microporosity and secondary dendrite arm spacing, a 3-dimensional model altered the solidification pattern and time scale for solidification by anorder of magnitude. A 2-dimensional approximation, although requiring less model inputtime and computational time, may thus be misleading and result in incorrect conclusionsbeing drawn. The model developed in this work provides a strong tool which can be used inconjunction with experiments to develop relationships between heat flow and themicrostructural development of investment castings.11Table of ContentsPageAbstract iiTable of Contents iiiList of Tables vList of Figures viNomenclature xAcknowledgments xivCHAPTER 1 Introduction 11.1 Background of the investment casting process 21.2 Steps in the investment casting process 3CHAPTER 2 Literature review 5CHAPTER 3 Mathematical formulation of boundary conditions 183.1.1 Farfield radiation 183.1.2 Gap interface radiation 193.1.3 Enclosure radiation 233.1.4 Viewfactor calculation 273.2 Finite element implementation of boundary conditions 29CHAPTER 4 Verification of computer code 394.1 Viewfactor calculation verification 394.2 Radiation boundary condition verification 39CHAPTER 5 Experiments 465.1 Experimental procedures 465.2 Discussion of experimental results 505.2.1 Cylindrical castings 515.2.2 Finned castings 53CHAPTER 6 Sensitivity analysis and analysis of the casting process 716.1 Sensitivity analysis - cylindrical configuration 716.1.1 Interface heat transfer assumed contact time 726.1.2 Metal initial temperature 726.1.3 Mould thermal conductivity 736.1.4 Metal latent heat of solidification 746.1.5 Interface contact initial heat transfer coefficient 746.1.6 Fibrepaper shielding 746.1.7 Variables having minor effects 756.1.8 Analysis of heat flow resistances 766.2 Sensitivity analysis - finned configuration 796.2.1 Temperature variation with radial position 796.2.2 Temperature variation with differing faces 806.2.3 Effect of fibrepaper shielding 806.3 Comparison of experiments with model results 816.3.1 Cylindrical castings 816.3.2 Finned castings 83CHAPTER 7 Industrial application of casting model to testbar 105111Table of Contents (cont’d)Page7.1 Background 1057.2 Results of analysis - 2-dimensional testbar model 1057.3 Results of analysis - 3-dimensional testbar model 1077.4 Theoretical considerations in casting quality analysis 1087.4.1 Cavity porosity 1087.4.2 Microporosity 1097.4.3 Secondary dendrite arm spacing 1107.5 Correlation of model predictions with casting quality 111CHAPTER 8 Summary and recommendations 1208.1 Summary and conclusions 1208.2 Recommendations for further work 123Bibliography 124Appendix A Analytical solutions for verification of computer code 127ivList of TablesPageTable 4.1 Mean percent error and standard deviation for 41comparison of Monte Carlo viewfactorresults with analytically calculatedvalues (n=20)Table 5.1 Location of thermocouples in cylindrical castings 55Table 5.11 Location of thermocouples in finned castings 56Table 5.111 Vacuum furnace pressures during melting and cooling 57Table 5.IV Cylindrical mould initial cooling rates after removal 58from preheat furnaceTable 6.1 Sensitivity analysis parameters 85Table 6.11 Constant material properties 86Table 6.111 Effect of parameter variation on total solidification 86timeTable 6.IV Values of parameters used in fitting cylindrical 87experimental dataTable 6.V Constant values used in analysis of finned 87experimental dataTable 6.VI Values of parameters used in fitting finned 87experimental cataTable 7.1 Parameters used in testbar model analysis 114Table 7.11 Effect of model casting parameters on local 114solidification time for testbar castingsVList of FiguresPageFigure 1.1 Steps in the investment casting process 4Figure 3.1 Schematic of interface contact resistance model 36Figure 3.2 Nomenclature for radiative exchange between two 36surfacesFigure 3.3 Irradiation and radiosity for a grey surface 37Figure 3.4 Nomenclature for emission direction of a ray emitted 37by a surfaceFigure 3.5 Heat balance on the boundary of a conducting medium 38Figure 4.1 Configurations for viewfactor code verification 41Figure 4.2 Comparison of Monte Carlo viewfactor calculation 42with analytical resultsFigure 4.3 Configurations for radiation boundary condition 42verification of THERMAL codeFigure 4.4a Cooling of a 2D semi-infmite slab radiating to farfield 43environment at OKFigure 4.4b Cooling of a 3D semi-infinite slab radiating to farfield 43environment at OKFigure 4.4c Cooling of a 2D semi-infinite slab radiating to 44enclosure at OKFigure 4.4d Cooling of a 3D semi-infinite slab radiating to 44enclosure at OKFigure 4.4e Cooling of a 2D semi-infinite slab radiating across 45gap interface to solid at OKFigure 4.4f Cooling of a 3D semi-infinite slab radiating across 45gap interface to solid at OKFigure 5.la Dimension drawing - cylindrical test casting 59Figure 5. lb Dimension drawing - finned test casting 59Figure 52 Balzers VSGO5O Vacuum Induction Furnace (Deloro 60Stellite mc)Figure 5.3 Schematic of thermocouple installation in ceramic 60mouldFigure 5 .4a Typical cylindrical mould instrumented for testing 61Figure 5.4b Typical finned mould instrumented for testing 61Figure 5.4c Cylindrical mould in furnace after casting, showing 62manifold shieldingFigure 5.4d Finned mould in furnace after casting, showing 62manifold shieldingFigure 5.5 Thermocouple location in finned castings (Ref. Table 635.11)Figure 5.6 Pt-Ptl3Rh equivalent thermocouple circuit 63Figure 5.7 Experimental data: casting C5 64viList of Figures (cont’dFigure 5.8Figure 5.9Figure 5.10Figure 5.11Figure 5.12Figure 5.13Figure 5.14Figure 5.15Figure 5.16Figure 5.17Figure 5.18Figure 5.19Figure 6.laFigure 6.lbFigure 6.2Figure 6.3Figure 6.4Figure 6.5Figure 6.6Figure 6.7Figure 6.8Figure 6.9Figure 6.10Figure 6.11Figure 6.12Figure 6.13Figure 6.14Figure 6.15Figure 6.16Figure 6.17Figure 6.18Page646565666667676868696970888889898989909090909191919192Experimental data: casting C6Experimental data: casting C7Experimental data: casting C8Experimental data: casting C9Experimental data: casting ClOExperimental data: casting FlExperimental data: casting F4Experimental data: casting F5Experimental data: casting F6Experimental data: casting F7Experimental data: casting F8Experimental data: casting F92D cylindrical finite element model - unshielded2D cylindrical finite element model - shieldedSensitivity analysis - effect of mould thermalconductivitySensitivity analysis - effect of mould heat capacitySensitivity analysis - effect of mould initialtemperatureSensitivity analysis - effect of metal initialtemperatureSensitivity analysis - effect of mould emissivitySensitivity analysis - effect of metal emissivitySensitivity analysis - effect of ambient temj,eratureSensitivity analysis - effect of mould thicknessSensitivity analysis - effect of gap contact initial areafractionSensitivity analysis - effect of gap contact timeSensitivity analysis - effect of metal latent heat ofsolidificationSensitivity analysis - effect of fiberpaper shielding2D cylindrical model - circumferential effect offibrepaper shielding on temperatureprofilesComparison of model results with experimental data -casting C5Comparison of model results with experimental data -casting C6Comparison of model results with experimental data -casting C7Comparison of model results with experimental data -casting C893939494viiList of Figures (cont’d)PageFigure 6.19 Comparison of model results with experimental data- 95casting C9Figure 6.20 Comparison of model results with experimental data- 95casting Cl 0Figure 6.21 a 2D finned fmite element model- unshielded 96Figure 6.2 lb 2D finned finite element model- shielded 96Figure 6.22 Sensitivity analysis- variation of temperature with 97radial position: Face AFigure 6.23 Sensitivity analysis- variation of temperature with 97radial position: Face BFigure 6.24 Sensitivity analysis- variation of temperature with 97radial position: Face CFigure 6.25 Temperature difference between 90 deg and 135 deg 98face at same radial location: Position aFigure 6.26 Temperature difference between 90 deg and 135 deg 98face at same radial location: Position bFigure 6.27 Temperature difference between 90 deg and 135 deg 98face at same radial location: Position cFigure 6.28 Temperature difference between 90 deg and 135 deg 98face at same radial location: Position dFigure 6.29 Sensitivity analysis (shielded model) - variation of 99temperature with radial position: FaceAFigure 6.30 Sensitivity analysis (shielded model) - variation of 99temperature with radial position: FaceBFigure 6.31 Sensitivity analysis (shielded model) - variation of 99temperature with radial position: FaceCFigure &32 Temperature difference between 90 deg and 135 deg 100face at same radial location (shieldedmodel): Position aFigure 6.33 Temperature difference between 90 deg and 135 deg 100face at same radial location (shieldedmodel): Position bFigure 6.34 Temperature difference between 90 deg and 135 deg 100face at same radial location (shieldedmodel): Position cFigure 6.35 Temperature difference between 90 deg and 135 deg 100face at same radial location (shieldedmodel): Position dVII’List of Figures (cont’d)Figure 6.36 Variation in temperature with radial location on Face 101A: shielded model vs. unshieldedmodelFigure 6.37 Comparison of model results with experimental data: 101casting FlFigure 6.38 Comparison of model results with experimental data: 102casting F4Figure 6.39 Comparison of model results with experimental data: 102casting F5Figure 6.40 Comparison of model results with experimental data: 103casting F6Figure 6.41 Comparison of model results with experimental data: 103casting F7Figure 6.42 Comparison of model results with experimental data: 104casting F8Figure 6.43 Comparison of model results with experimental data: 104casting F9Figure 7.1 2D testbar - finite element mesh 115Figure 7.2 2D model results- base case; time 800 seconds 115Figure 7.3 2D model results - high initial mould temperature; 115time 900 secondsFigure 7.4 2D model results - high initial metal temperature; time 115700 secondsFigure 7.5 2D model results - thick mould; time 1000 seconds 116Figure 7.6 2D model results - mould wrap on upper half; time 1161300 secondsFigure 7.7 2D model results - mould wrap on sides and top; time 1162000 secondsFigure 7.8 2D model results - feeding to test section removed; 116time 400 secondsFigure 79 3D testbar - geometric representation 117Figure 7.10 3D model results - base case; time 25 seconds 118Figure 7.11 3D model results - base case; time 50 seconds 119ixNomenclatureChapter 2A, B, C Constants obtained from fitting data to exponential model of interface heattransferAca Contact area at gap interfaceAtotal Total interface area, = Aca +Void area at gap interfacec Roughness wavelength of rougher surface of contact interfaceD Thermal conductivity of airE Roughness of metal-mould interfaceF Dimensionless constant in local air gap conduction heat transfer modelViewfactor from integration point (x,y,z) on differential area dA1 toisothermal planar surface segment A2Viewfactor from elemental area dA to elemental area dAH Hardness of softer surface of contact interfaceh Overall gap interface heat transfer coefficienth Air gap conduction component of interface heat transfer coefficienthe/i Air gap local conduction heat transfer coefficienthe/rn Overall equivalent heat transfer coefficienthcornb Combined contact and void conduction interface heat transfer coefficienthcond Air gap conduction component of interface heat transfer coefficienth0t Surface contact conduction component of interface heat transfer coefficienthr Void radiation component of interface heat transfer coefficienthrad Void radiation component of interface heat transfer coefficienth Surface contact conduction component of interface heat transfer coefficientht Overall gap interface heat transfer coefficientka Thermal conductivity of gas in interface void spacesICC Thermal conductivity of castingkg Thermal conductivity of gas in interface void spaceskm - Mean thermal conductivity of materials contacting on interfacekm Thermal conductivity of mould1 Width of gap at contact interfaceHeat transfer gap interface normal vectorP Contact pressure at interfaceqcond Conduction heat flux at interfacegrad Radiation heat flux at interfaceRca Interface contact conduction resistanceF Vector from casting geometric center to casting-mould interfaceAr Air gap width (time- and space-dependant)distance between surface area elements dA1 and dA1R Interface void conduction resistanceT1, T2 Temperatures of surfaces on either side of a contact interfaceAverage temperatuer over volume of solidified metalxNomenclature (cont’d)Solidus temperatureV1x,Vly,vlz Direction cosines of A2 surface normalWg Total width of gap at contact interfaceWmg Width of gap formed due to solidification shrinkageWrm Initial gap width at contact interfacec Linear thermal expansion coefficient of metali’ Emissivities of surfaces on either side of a contact interface9, Oj Angle bewtween ru and surface normalsStefan Boltzman constant (5.669x108W/m2-K4)Chapter 3ao, al...an Coefficients of Polynomial describing gap interface conduction heat transfercoefficient as a function of timeA1,A2 Areas of isothermal surfaces exchanging heat by radiationAc/Atotal Contact area fraction in gap interfaceAi/Atotai Void area fraction in gap interfaceCp Specific heat of a materialdA1,dA2 Differential areas in radiation exchange calculationdq i -dA2 Radiation energy flux from dA1 to dA2Emissive power of a blackbodyFAB Viewfactor of surface A to B; the fraction of energy leaving surface Awhich strikes surface BG Radiosity of a surface; total radiation flux leaving the surfacehgap Gap interface conduction heat transfer coefficienthr Radiation heat transfer coefficienthrad Gap interface radiation heat transfer coefficient11,, 1 Energy emitted by surface element dA1 per unit time per unit projected areaper unit solid angle subtended by surface element dA21b Emissive intensity of a blackbodyi (2k) Spectral intensityJ Irradiation of a surface; total radiation flux striking a surfaceka, kb Thermal conductivities of equally rough materials on either side of a gapinterfacek1 Thermal conductivity of gas in void spaces of gap interfacekm Thermal conductivity of mould materialn Number of isothermal surfaces in an enclosure (Viewfactor calculation)n Number of nodes in an element (Finite element method)N Shape function for node i in Finite element Galerkin weighted residualmethodPQ,f3) Probability density function of total radiation energyq Internal heat source in heat conduction equationxiNomenclature (cont’d)qa Rate of radiation absorption per unit area of a surfaceqA i -A2 Total radiation energy flux from A1 to A2Constant heat flux to Finite element boundaryqe Rate of radiation emission per unit area of a surfaceqh Convective heat flux to fininte element boundaryqr Net incident radiation flux per unit area of radiating surfaceqr Radiative heat flux to fmite element boundaryr/cond conductance matrix contribution of radiation heat flux boundary condition tofinite element formulationqr/load Load vectyor contribution of radiation heat flux boundary condition to finiteelement formulationr Distance between differential areas dA1 and dA2R13 Cumulative distribution function relating the ray emission direction variable bto a random number uniformly distributed between 0 and 1R9 Cumulative distribution function relating the ray emission direction variable qto a random number uniformly distributed between 0 and 1S Surface domain of elementApproximation of temperature in a conducting mediumT Temperature of a surfaceT1, T2 Temperatures of surfaces on either side of a contact interfaceT Average surface temperature of an element surface based on the corner nodaltemperaturesT1 Temperature at node i in Finite element methodT1 Ambient or surrounding temperatureV Volume domain of elementWg Total interface gap widthW1 Weighting function for node 1, N1 in weighted residual methodcx Absorptivity of a surface13 Elevation angle of radiation emissionEmissivity of a surfacei Emissjvjties of surfaces on either side of a contact interface, (‘ 13) Directional spectral emissivityO Circumferential angle of radiation emission01, 02 Angles between ray connecting areas dA1 and dA2, and surface normals atdA1 and dA2p Density of a material (heat conduction equation)p Reflectivity of a surfacea Stefan Boltzman Constant (5.669x108W/m2-K4)Solid anglexiiNomenclature (cont’d)Chapter 4Cp Specific heat of a mediumH Boundary heat transfer coefficientk Thermal conductivity of a mediumm Exponent of temperature at boundary1r Number of rays traced per surfaceNumber of surfaces in enclosure for viewfactor calculationt Computational run timeTamb Ambient temperatureTenc Temperature of enclosure walls surrounding the mediumTopposing Temperature of opposing face at gap interfacev temperaturev0 Initial temperaturev Surface temperaturep density of a mediumChapter 7G Thermal gradientk Constant relating secondary dendrite arm spacing to local solidification time1 Channel (dendrite) lengthnrrR2 Total channel area for n channels of radius RCritical pressure below which bubbles nucleate from gases dissolved in meltP1 Pressure at dendrite rootPressure in the bulk meltAl? Pressure drop across dendrite arraySDAS Secondary dendrite arm spacingt Tortuosity factorLocal solidification timeATf Freezing range of solidifying metalV Liquidus isotherm velocity/XVfreezing Volume change associated with freezingxliiAcknowledgementsI would like to thank my advisers, Dr. Alec Mitchell and Dr. Steve Cockcroft for theirsupport and guidance, and Mr. Bob Dawson of Deloro Stellite Inc. for providing theopportunity to do experiments in an industrial setting. I would also like to acknowledge theNatural Sciences and Engineering Research Council for providing funding for this project.I am indebted to a host of people at UBC and at Deloro Stellite mc; in particular, AlSchmaltz for his technical help, and his unfailing good spirits during our plant trials; SergeMilaire for helping out with things electronic; Brock Barlow of Deloro Stellite for operatingthe furnace for us and for putting up with the engineers; Dave Tripp, the computer Guru; andSteve Cockcroft for his advice, discussions, patience and support. Many thanks.Finally, my thanks to my family for their faith, Sheldon Green and Rob Harley for theirfriendship, and especially my husband, Steve, for his example as an engineer, and for theencouragement which he provided when it was needed the most.xivChapter 1 IntroductionThe investment casting process for the production of complex components has manyadvantages over other more conventional casting methods, as well as over manufacturingroutes such as machining and assembly. It is a well-established near-net shape technologyallowing versatility of design, close tolerances and good surface quality, requiring little or noadditional machining. For these reasons, the process has become widely used in theaerospace industry, where a further advantage of design for weight savings can often berealized. Casting in vacuum extends the range of materials which may be successfully cast toinclude reactive materials such as titanium and alloys containing reactive elements such assuperalloys.Central to casting design is the ability to meet quality criteria in terms of porosity,defects and microstructure in order to obtain desired mechanical properties such as strength,ductility and fatigue life, which limit the use of a part in service. Traditionally, theexperience of the foundryman has been combined with extensive testing of finished parts toestablish the casting design and parameters. This method, however, can be costly, as severaldesign iterations are often necessary for intricate or thin-walled parts, where adequate feedingof molten metal may be difficult to achieve, resulting in non-fill or shrinkage porosity. Thus,considerable effort has gone into the development of computer-based casting design toolswhich will enable the prediction and location of possible macroscopic defects and correctionof the design before valuable resources in materials and shop time have been used. As well,the long term objective of such computer modelling tools is to predict microstructuralfeatures such as the location of microsegregation, and grain size, which ultimately determinethe mechanical behaviour of the cast part.The ability of computer-based models to predict correctly the flow of heat in castingprocesses hinges on the quantitative characterization of heat transfer in a casting from the1time molten metal is poured into the mold through solidification and cooling to ambienttemperature. Numerical formulations describe the phenomena of conduction through thecontinuous media (metal, ceramic mold) and heat transfer at the boundaries (mold externalsurface and mold-metal interface). Thus it is essential to have access to the relevantthermophysical properties at elevated temperatures and relationships describing the variousboundary conditions. Heat transfer at the boundaries is by convection, gap conduction andradiation. In vacuum casting , the primary mechanism of heat transfer at the boundaries is byradiation. This work concentrates on the development of boundary conditions for the case ofvacuum investment casting of 1N7 18, a nickel-based superalloy. The work has proceeded intwo stages: first, modifications have been made to a finite element-based mathematicalmodel of heat flow, developed in the Department of Metals and Materials Engineering at theUniversity of British Columbia [1], to include farfield radiation, enclosure radiation andinterface gap radiation in two and three dimensions. Second, the modified mathematicalmodel has been verified against industrial measurements made at Deloro Stellite Inc.,Belleville Ontario.1.1 Background of the investment casting processInvestment casting, also known as precision casting or the lost wax process, is one ofthe oldest known foundry techniques, used as far back as 1766 BC (Shang dynasty, China)[21. Other early examples of investment castings are attributed to ancient Columbian andAztec cultures, where it was primarily used in the production of intricate jewelry. Theprocess was discovered in Europe during the Italian Renaissance for casting large statues, andwas first used commercially in 1867 for dental fillings. It came to industrial prominenceduring the second world war, with the huge demand for aircraft parts. The near-net shapetechnology offered by investment casting allowed costly machining and assembly steps to bebypassed. In the past four decades investment casting has become a widely-used, well2established manufacturing technology. It is used extensively not only in aircraft andaerospace, but computers, electronic equipment, food processing machinery, gas turbines,machine tools, automotive applications, medical and dental uses, weapons systems, and manyother applications [2].The advantages of investment casting include excellent dimensional control, theability to cast thin sections and good surface finish with little or no fmishing required.Moreover, through the use of sophisticated coring, component geometry is virtuallyunlimited, allowing internal passages, undercuts, and features which cannot be obtained byother casting methods.1.2 Steps in the investment casting processA flow diagram of the investment casting process is shown in Figure 1.1. First, apattern (positive) is made to the dimensions of the finished part. From this, a wax injectionmold is made, and replicas of the part produced from wax. Depending on component sizeand geometry, several wax parts may be assembled onto a ‘tree’, with the feeding channels,pour cup, gating and risers. The wax assembly is dipped, or invested, in ceramic slurry andcoated with a face coat, consisting of very fine refractory flour or sand. The face coatmaterial is chosen to give the desired surface finish and to be as chemically unreactive withthe metal being poured as possible. It may also contain grain nucleating agents. The coat isallowed to dry, and a second face coat layer applied. Subsequent layers, or backup layers,are then added to give strength to the shell. The backup layers usually contain coarser solids(coarse sand or stucco). Five to eight backup layers may typically be applied. The fullyinvested assembly is next dewaxed, leaving a hollow ceramic shell, which is fired to increaseits strength. This step may also be the preheat for pouring of the metal. The shell is removedfrom the preheat furnace, placed under the melting crucible pour spout (in vacuum, if thematerial requires it) and the metal is poured into the shell. The casting is allowed to cool and3the ceramic is knocked off. Further treatment may include surface cleaning and finishingoperations such as drilling of holes or facing of surfaces.a: Produce wax paftem b: Assemble mold tree c: Dip in ceramic slurryd: Coat with stucco e: Dewax, tire, preheat f: Pour metalg: Break off moldFigure 1.1Steps in the Investment Casting ProcessUIi: Finished parth: Remove gating, clean4Chapter 2 Literature ReviewA large quantity of material has been published on the mathematical modelling ofsolidification processes, the majority of which has been devoted to primary castingprocesses such as ingot casting and continuous casting. No attempt will be made toreview this work. Rather, this review will focus on the modelling of boundary conditionsat the mould-metal interface and at the mould surface in vacuum investment casting. Inthe vacuum investment casting process heat flows between the casting and the mould byconduction and/or radiation and from the exterior of the shell by radiation. The complexgeometry of a typical shell requires that radiation exchange between different parts of theshell, where temperatures are changing spatially and with time, be accounted for. Theinterface heat transfer is also dependant on location and time due to the formation of agap between the mould and the solid shell as the metal solidifies.The literature was reviewed for details on previously applied techniques inmodelling radiation boundary conditions, viewfactor calculation and metal-mouldinterface heat transfer for use in the current work.Hamar {3J used a finite volume method to model microporosity formation ininvestment castings. The model included a mould filling algorithm, latent heatevolutionand feeding of the mould due to density differences in the liquid and solid phases. Intheir model the ceramic shell was encased in a cylinder of foundry sand. This techniqueallowed a simple farfield radiation boundary condition to be used; the approach is nontypical to the investment casting process, however.Duffy et. al. [4] applied the finite element method to the analysis of single crystalturbine blades. In this production method, the radiation boundary condition is changingwith time as the blade is withdrawn from the furnace. Several blades are typically cast ina cluster arrangement. Thus a mould may exchange heat with the furnace hot and cold5zones, radiation baffles, copper chill and with other moulds. This results in a modellingproblem similar to that of investment casting. The authors used a commercially availablefinite element package (MARC, MARC Analysis Inc.). This package did not incorporateradiation boundary conditions, which were added by the authors. View factors for theradiation exchange were calculated from an integration point on an element to a planarsurface segment. Contour integration was used to obtain the view factors. Both viewfactors and average surface temperatures were updated at the beginning of each time step.The view factor from an integration point (x,y,z) on differential area dA1 , to anisothermal planar surface segment is calculated as1 (z2—zl)dy2—(y2—yl)dz2 f (x2—xl)dz2—(z2—zl)&2 £ (y2—y1)d2—(x2—X1)dy2Fdl—2— Vlxy +V1y +vlzy21[ C2 C2 C22.1where vix, Vly and Viz are the direction cosines of the A2 surface normal. Thismethod does not account for partial views (shading of one surface by another), i.e. eitherone surface sees another surface entirely, or not at all. This limitation requires that thesubdivision of surface segments be carefully considered beforehand, and may require afiner surface mesh than the temperature field requires, increasing the computational sizeof the problem.Huang and Berry [5] dealt with solidification heat transfer in aluminuminvestment castings. They investigated the effects of metal superheat, mould preheat,casting thickness and mould thickness on solidification time, both experimentally andcomputationally. A finite difference scheme was used in the numerical model. Themould-metal interface heat transfer is discussed in some detail. Details of the boundaryconditions implementation are described in a Masters thesis by Huang.6Liu et. al. [6] discuss a finite difference model for unidirectional solidification ofsingle crystal high temperature alloys. Enclosure radiation with a coarse view factor gridis incorporated. The enclosure is divided in the horizontal plane into angles subtendingother moulds in the enclosure, and the furnace wall. Then in the wall zone, the verticalrange of exchange with the hot zone, baffle, cold zone and chill plate are determined. Inthe range of exchange with other moulds, the castings are divided into five exchangezones in the vertical direction. This approach, while it attempts to account for varyingenclosure conditions, is applied only to simple, non-convex (no self-irradiation) moulds,and lacks generality for application to complex investment casting shapes.Desbiolles et. al. [7] discuss the simulation of single crystal turbine bladesolidification using a finite element formulation and including enclosure radiation withshadowing effects. For 3-dimensional geometries view factors are calculated by therelation for elemental area dA1 to elemental area dA1:cos0icosOjFy= 2 dAj 2.2ltrywhere ru is the distance between surface area elements dA1 and dA1 and O and 0 arethe angles between and the surface normals. Implicit in this method is theassumption that, over the areas involved, cosOj, cos0j and ru are constant. This isapproximately true when rU is sufficiently large, or dA1 and dA1 are sufficiently small.For 2-dimensional, axisymmetric geometries, Desbiolles et. al. use a different approach.For each emitting facet 1, an external hemisphere is divided into a number of segments sothat the view factor from i to each segment is approximately equal. Each segment isassociated with a ‘shooting direction’, that is a direction from surface i to the segment.Each shooting direction is traced to its intersection with another facet j in the enclosure.7A final technique for calculation of radiation viewfactors, and the one which waschosen for use in this work, is a Monte Carlo ray tracing approach. This technique wasadopted for its ease of implementation and integration with the fmite element code andPATRAN preprocessor. It is generally applicable for all complex geometries, easilyincorporating shading effects, and requiring no numerical integration. The level ofaccuracy for a given run time is good, and computation time for large, complex problemsis comparable to the other numerical methods described above.The Monte Carlo method as it is used in engineering and the sciences today wasfirst formulated by Von Neumann and Ulam at Los Alamos National Labs in the 1940’sto investigate neutron transport phenomena [8]. The first applications to radiation heattransfer problems were done by Howell and Perimutter in 1964 [9,10]. They used themethod to determine heat transfer between grey walls through non-isothermal, absorbing-emitting media, a problem which, due to the gas attenuation, rapidly becomes extremelycomplex using integration or zoning methods. Corlett [11] and Toor and Viskanta [12]were also pioneers in the use of Monte Carlo for radiation heat transfer, investigatingradiation exchange between surfaces separated by non-participating media.Whereas the previous researchers calculated the total energy exchange betweenelements of their systems, several studies have used the Monte Carlo method primarily tocalculate exchange areas, or view factors, as in the present work. Modest [13] examined3-dimensional configurations, and curved surfaces, and allowed for arbitrary emission,absorption and reflection characteristics, as well as radiation through openings. Maitbyand Bums [14] allowed for 3-dimensional enclosures filled with a non-participatingmedium. They approximated curved surfaces by planar elements, and allowed for nonzero transmittance through the enclosure surfaces. Hoff and Janni [15] calculatedradiation shape factors and included intersection criteria for planar, cylindrical andspherical surfaces. Ikushima et. al. [16] compared computation time and accuracybetween area integration and Monte Carlo simulation to obtain the radiation view factors.8Plehiers and Froment [17] calculated exchange factors for the gas and furnace in steamreforming chemical process models.It is important to recognize that initially, in vacuum investment casting, heattransfer across the mould-metal interface may occur by more than one mechanism. On amicroscopic level, the surface of the mould is rough. When the metal flows into themould, there will be contact between the mould and the metal only at the asperities of themould surface. At a microscopic level, this contact will depend on the relative roughnessof the mould surface and on the degree to which the melt wets the surface of the mould.The result of this phenomenon is that, at least initially, several mechanisms are active inthe interface heat transfer. These are (1) conduction through solid contact between themetal and mould, (2) conduction in the gap between the metal and mould through theintervening gas (if present), and (3) radiation across the gap. A fourth possibility is heattransfer by convection in a gas across the gap. Initially, the solid contact conductioncomponent of heat transfer may be large. As the metal solidifies, it contracts away fromthe mould surface, reducing the effective area over which solid contact conductionoccurs, and radiation or gas conduction become more significant. Modelling thiscondition rigourously is a very complex undertaking, as the gap width is dependent on thegeometry of the part, as well as the solidification. If solidification shrinkage is. restricteddue to geometric constraints, the conduction component will be active longer than inunconstrained regions.Ho and Pehlke [18] investigated the first three mechanisms by solving an inverseheat conduction problem from thermocouple data using a permanent mould or chill ratherthan a sand mould. Interfacial gap size over time was measured to derive air gapcoefficients across the gas-filled gap. The overall gap heat transfer coefficient ht ishi=hs+hc+hr 2.39whereh3 = surface contact conduction coefficientair gap conduction coefficienthr = radiation coefficientThe air gap conduction coefficient, h is proposed to depend on a mean conductivity ofthe contacting materials, km, the roughness wavelength of the rougher surface, c, thecontact pressure F, and the hardness of the softer surface, H.h!-(FH) 2.4The air gap conduction coefficient ish =g 2.5Clwhere kg is the conductivity of the gas and 1 is the gap width.The radiation coefficient is given as(T2+T2)(T+T)26hr_ (i 1I —+—-—1‘.E1 6210where T1 and T2 are the interface temperatures of the metal and mould, and and E2are the interface emissivities of the metal and mould. The authors reference severalworks on the effect of the contact pressure on the mould-metal heat transfer coefficient.Experimental and numerical (finite difference) results for the change in heat transfercoefficient with gap length are given.Huang and Berry [5] use two models of interface heat transfer: the Ho- Pehlke(1984) model and the Kanetkar (1987) exponential model. The first model uses a heattransfer coefficient consisting of three terms:h = + hCOfld + hmd 2.7where h0t is the conduction component due to solid contact, hcond is conductionacross the interface gap, and hrad is radiation across the interface gap. They assumethat no solid contact conduction occurs, i.e. h0t= 0. The gap conduction termhCOfld=!c 2.8where Ka is the conductivity of the gas (air) and Wg is the gap width. The gap widthconsists of two components:= + 2.9where Wrm is the initial gap width, determined in the paper from average particle sizemeasurements on the mould to be approximately 0.2 mm, and Wmg is the ‘macrogap’which forms due to shrinkage during solidification. An overall gap width of 0.2- 0.4 mmis used in the work. This model, due to the assumption of negligible solid contact11conduction, is not applicable to a vacuum casting situation, where gap conduction isnegligible, and solid contact conduction may be significant at initial times.The second model of Kanetkar uses an equivalent heat transfer coefficientdescribed by a constant (steady state) plus and exponential (transient) term:hcimA+Be_Ct 2.10The constants A, B and C are adjusted to fit experimental data. The original work byKanetkar et. al. was not available at the time of writing, and no details are given in [5] asto the assumptions and validity of the model.Nishida et. al. [19] conducted experiments to determine the dominant mechanismof heat transfer in the formation of the air gap. The relative movement of the casting andmould were measured, and temperatures recorded at several locations in aluminum alloycastings. They concluded that air gap conduction dominated over radiation heat transfer,and that, as the air gap became larger, convection effects became significant. Theseconclusions are not applicable in the case of vacuum investment casting, whereconduction through the gas phase is negligible.Huang Ct. al. [20] and Huang [21] address the importance of estimation of the gapwidth in the mould-metal interface to calculate the mould-metal heat transfer coefficient.A ‘free thermal contraction’ method is used to describe the variation and distribution of airgap width, dependant on casting geometry, and thus enable estimation of the interfaceheat transfer coefficient. This method estimates the thermal contraction at an interfacelocation as:=c(F.ñ)(u_i) 2.1112whereAr = air gap widthF = vector from casting geometric center to casting-mould interfaceñ interface normal vectora = metal linear thermal expansion coefficientu = solidus temperature= average temperature over solidified metalGeometric restraints to contraction from the mould are considered as, for example, wherethe casting rests on the mould due to gravity or where contraction is resisted by aninternal core. The authors account for this by initially computing an unconstraineddisplacement, then adjusting the displacement at the constrained location and all locationscontracting toward the constrained location by the unconstrained amount. Finally, the airlocal gap conduction heat transfer coefficient is given by:D2.12E+FArwhereD thermal conductivity of airE = roughness of metal-mould interfaceF = dimensionless constant accounting for other effects such as mould dilatationThis model tries to account for geometry and time dependance of the air gap conductioncomponent. However, it is not applicable in vacuum situations, where the air gapconduction is negligible relative to the solid contact conduction and radiation heat13transfer components. No mention is made of solid contact conduction component in thiswork.Liu et. al. [6] discuss mould-metal interface heat transfer accounting for a solidconduction at contact points, conduction across the gap and radiation across the gap.They assume a contact area, Aca, and a void area, A such that Aca + A = Atotal.They also assume a gap width Wg. The contact conduction component of resistance toheat flow is thenWI 1 WIR =—-I I+—-I I 2.13ca 2 KcAca) 2 KmAca)The gap conduction resistance isR=W12.14gvwhereWg gap widthK = thermal conductivity of castingKm = thermal conductivity of mouldKg thermal conductivity of gas (air)Aca = area of solid contactA = void areaThe total resistance is141 114ca(2KmKc)AK 2.15Rtotai IJ’ Km + K VThe combined conduction heat transfer coefficient is thush i(2K111K+ A K 217comb j47 Atotal K, + K Atotaland the total heat transfer coefficient ishiotat = hCOfl,b + hrad 2.18It should be noted, however, that the two components and hrad are based ondifferent surface areas:qcond = hcombAtoialAT2.19grad hradAt4TThus, the direct addition of the heat transfer coefficients is not strictly correct. Theradiation heat transfer coefficient must be corrected by the factor Av/Atotai:grad = hrad AioiaiAT 2.20Atotalqrad = h,JdA,OlOlAT 2.2115had= 2.22Aioiaikotai = hcomb + h 2.23This correction is not applied in the paper by Liu et. a!. The parameters in this method,Wg and A, must be determined as functions of time and location. No details as to howthese are determined in the paper are given.The thermal properties of investment casting ceramic shells are not welldocumented in the literature. These properties depend strongly on the shell compositionand structure. A layer of the shell contains a flour or stucco and the silica bindercomponent. Each layer may have a different proportion of stucco to binder, as well asdifferent stucco grain size. Further, the shell is not completely dense, but contains voidswhich may account for up to 40% volume fraction of the shell. At any temperature, theshell properties are a function of composition and processing. Properties may also be afunction of temperature. At higher temperatures (>500 deg C) radiation across the voidspaces increases the apparent conductivity of the shell. Some researchers have furtherpostulated a radiation transmission mechanism through the fused silica at temperaturesgreater than 1000 deg C, as the silica becomes partially transparent. Some work has beendone on the theoretical calculation of conductivity of shell systems (for example, [28] -[32]). For this work, however, experimental values from the literature were used.Heames and Geiger [26] experimentally measured conductivities and diffusivitiesfor various shell systems. Values of thermal conductivities ranging from approximately0.550 to 0.750 W/m-deg C were reported for shells of primarily silica materials. Slightlylower values (0.450 to 0.60 W/m-deg C) were reported for shells containing zircon flour.In this work, conductivities do not appear to have consistent temperature dependencies.Thermal diffusivities of around 0.002 cm/sec are reported.Huang et. al. [27] performed experiments to determine shell conductivity as afunction of particle size, binder composition, and temperature, and simulated the effect of16porosity. Lower porosity volume fraction resulted in a higher thermal conductivity. Forvarious shell systems, thermal conductivities ranging from 0.75 to 1.5 W/m-deg C werereported. A linear variation of conductivity with temperature was indicated by theexperimental results over a temperature range of 30 to 750 deg C.Hamar [3] used a thermal conductivity increasing from 0.55 W/m-deg C at 100deg C to 0.86 W/m-deg C at 1000 deg C, and a thermal diffusivity of approximately0.003 cm2/sec.17Chapter 3 Mathematical Formulation of Boundary ConditionsThe solidification heat conduction code THERMAL, developed in the Department ofMetals and Materials Engineering at the University of British Columbia [1], was modified toincorporate the boundary conditions necessary to model the vacuum investment castingprocess.In the vacuum investment casting process, heat transfer from the molten pool to thesurrounding furnace takes place by the following five mechanisms: i) conduction andconvection in the liquid pool, ii) conduction through the solid shell, iii) conduction and/orradiation across the gap between the mould and casting, iv) conduction through the mould,and v) radiation from the mould exterior to the surroundings. The following sections detailthe implementation of radiative boundary conditions for radiation to far-field surroundings,radiation to an enclosure, and radiation and conduction across a narrow gap interface.3.1.1 Farfield RadiationA surface at temperature T, exchanging heat with its environment at temperature Tfemits radiation at a rate ofq=saT4 3.1where q is the rate of energy emission per unit area of the surface, a is the StefanBoltzman constant with a value of 5.669x108Wm2K4and is the total emittance of thesurface. The surface absorbs radiation from the environment at a rate ofq=cLa7 3.218where q is the rate of energy absorption per unit area of the surface, a is the absorptivityof the surface (fraction of incident radiation absorbed) and aTjn/ is the blackbody emissivepower of the environment. The net incident radiation per unit area is the difference betweenincoming and outgoing radiation flux:3.3If Kirchoffs law is assumed to be valid, that is, a = E, then3.4The radiation heat flux is non-linear in the dependant variable, T. A linearization isperformed as follows. It is desireable to express qr asq=h(2flf—T) 3.5Comparing equations 3.4 and 3.5 results in:hr==Ea(f+T2)(flf+T) 3.6(if-T)3.1.2 Gap interface radiationThe metal-mould interface heat transfer is complex. Referring to Figure 3.1 a, twosurfaces in contact will, in general, not be in perfect contact due to the fact that the surfaces19are not perfectly smooth, but rough, and actually contact only at projections and asperities onthe surfaces. Referring to Figure 3. ib, where it is assumed that both surfaces are equallyrough , an overall heat transfer coefficient, accounting for conduction across the contactpoints and across the gaps may be expressed as [22]:h i(A 2KaKbAvK 37gap tA1011 Ka + Kb Atotal f)wherehgap = interface heat transfer coefficient (W/m2-deg C)Wg = total interface gap width (m)= total contact area (m2)= total void area (m2)Ka = thermal conductivity of material a (W/m-deg C)Kb = thermal conductivity of material b (W/m-deg C)Kf= thermal conductivity of gas present in the voids (W/m-deg C)which is the relation used by Liu et. al. [6]. In the case of vacuum casting, two modificationsshould be made. First, the conduction component of the gap is neglected: when the meanfree path of the gas molecules approaches the width of the interface, the effectiveconductance of the gas decreases as the pressure is decreased [221. Second, it is assumed thatthe roughness is associated only with the mould, and that the metal surface is smooth. Sincesurface tension effects in the liquid metal will tend to reduce depressions in the metal surface,and there will be some flow of molten metal into the mould surface, this is a reasonable20assumption. The resulting interface is depicted schematically in Figure 3.1 c. Thus, theconduction heat transfer coefficient ish 4Kmgap“total Vgwhere Km is the thermal conductivity of the mould.In addition to conduction, radiation heat transfer across the gap must also beconsidered. This may be modelled as radiation exchange between infinite parallel plates ifthe gap is sufficiently small. The analytical expression in this case is— 4 a(T2+T2)(T+T)rad / 3.9Atotaiwherehrad = effective radiation heat transfer coefficient (W/m2-deg C)= total void area (m2)Tj, T2 = temperatures of the surfaces on either side of the interface (deg C)El, E2 = surface emissivities of the surfaces on either side of the interface.Also,+ 4=1 3.10Atotai Atotat21As the casting solidifies, it contracts, and the solid contact between the metal and themould is reduced. In this model, this effect is incorporated in a time-dependent function forthe contact area-fraction Ac/Atotal =f(t). Thus, the solid contact conduction component ofheat transfer decreases with time and the radiation heat transfer component becomescorrespondingly larger.Due to the complexity of investment cast components, the time for gap formation andthe gap width will not, in general, be uniform across the mould/metal interface. If geometricand temporal variations in gap width are to be accounted for (e.g. smaller gap due to thecasting resting on a portion of the mould or being restrained from contraction through anintegral core), the functions A/At0ti and Wg may become spatially variable as well astime dependent. These effects are not addressed in the present work.Implementation of the model described above requires that the gap conduction andradiation coefficients be linked through the contact area fraction A/At0ti. The source codeallows input of a time-dependent ‘convection’ coefficient as a user-specified polynomial:hgap=ao+ait+a2t+...+a,it 3.11If the factor Km/Wg is assumed to be constant, thenA W W1Atotai Km Km3.12O A 1.OAtotai22From equation 3.10, the void fraction is then calculated for use in the radiation heat transfercomponent. The assumption of constant K,/Wg is not rigourously correct, as discussedpreviously. Investigations of the time and spatial variations of this parameter present goodscope for future work.3.1.3 Enclosure radiationOne of the major advantages of the investment casting process is its ability to dealwith extremely complex geometries. In many cases this additional complexity means thatmodelling the process with only far-field radiation is a poor approximation and may not givethe desired accuracy; the mould may be radiating to itself due to shielding from other partsof the mould. Thus, non-uniform, non-constant T1 values arise. In addition, a farfieldmodel may not be suitable as the vacuum furnace may not be sufficiently large relative to thecasting size, and the radiative exchange with the furnace must thus be accounted for. To dealwith this phenomenon, the mould geometry, enclosure geometry and surface radiativeproperties must be taken into consideration.Consider two differential areas dAj and dA2, separated by distance rj., locatedin isothermal surfaces A1 and A2 (Figure 3.2) [23]. The total energy per unit time leavingdA1 and reaching dA2 isdq1_2=ib,ldAlcosOldG)l 3.13where tb, 1 is the energy emitted by dA1 per unit time per unit projected area per unit solidangle subtended by dA2. The solid angle d01 is:23do1=2’°°2 3.14Also,ebcYT 3.15t ItThus, the total energy leaving dA1 which is incident upon dA2 is4 ‘42 cosOdA1cosO1 3.16dq2 =a2Similarly, the total energy leaving dA2 arriving at dA1 isdA1 cosO1dA2cosO2 3.17dq2_1 =The net energy transferred isdq12 — dq = — T) cosO1cosO2dA1 3.18If and A2 are approximately isothermal (but not at the same temperature) then thisexpression can be integrated over the areas to give the total energy exchange between areasA1 and A2.24r cosO cosO dAdAqA1—A2 ijJ 2A1 A2T4 1cosO2dAqA,_Aa2JJ 2 .0A1A2The geometric configuration factor (also known as the shape factor or view factor) is thefraction of the total energy emitted by surface A1 which reaches A2:T4 j $ cosO1cosO2dA1A12 itr 1$JcosOicosO2dAidA 321a14A1 A1 A12 i-2T4 j$ cosO1cosO2dA1ALA, ljjcosO1cos2dA 322A 22 AA2 1—2It can be seen from (3.21) and (3.22) thatA1Pj_2 =A2F_1 3.23This is called the reciprocity relation, and is true for all pairs of isothermal surfacesexchanging heat by radiation.From the definition of the view factor it can be seen that F depends only on therelative geometries of the exchanging surfaces; it is independant of temperature and surfaceproperties. It represents an energy fraction leaving one surface and arriving at another, and25provides the link for setting up equations of radiation exchange between surfaces in anenclosure exchange network.In order to describe the boundary conditions in such a radiative network, the totalincident energy flux to a surface is required. The net radiant heat flux to the surface is thenthe difference between that emitted by the surface and the total incident flux. Two quantitiesare now defined [24]: the radiosity G, which is the total radiation leaving a surface,including the emitted component and the reflected component, and the irradiation J, whichis the total incident flux to a surface, including all direct exchange components and indirectexchange through intermediate reflections. The desired quantity for boundary conditioncalculation is the irradiation I Referring to Figure 3.3, the radiosity G is:Gjtzqj+pjJ 3.24or in vector format, for all surfaces, where [p] is the diagonal matrix of reflectivities,{G} = {q}+[p]{J} 3.25The irradiation J, where n is the total number of surfaces in the enclosure, is:A1J = =j=13.26or in matrix form:26{J} = [F]{G} 3.27Substituting for {G} from 3.25 into 3.27 and rearranging gives([i] — [F][p]){J} = [F]{q } 3.28This set of equations can be solved for the irradiation vector, {J}, which is used directly inthe finite element formulation of the incident radiation flux boundary condition.3.1.4 View Factor CalculationThe following assumptions are made for calculation of the viewfactors:1. No energy is transmitted through surfaces2. Surfaces are separated by a non-participating medium3. Surfaces are black (perfect absorbers/emitters)4. Surfaces are planar, with four straight edges, but of otherwise arbitrary shape5. The radiative enclosure is filly closedAssumption (3) is made only for viewfactor calculation purposes, and is relaxed in the FEimplementation.Mathematical modelling of the investment casting process with enclosure radiationinvolves the calculation of radiant exchange factors by multiple integration over the areas(equation 3.21, 3.22). For shapes of arbitrary complexity, analytical solutions to the integralequations axe not available, and numerical integration can become difficult and timeconsuming, especially in cases involving partial shielding of one surface by another. Analternative approach is the Monte Carlo ray tracing method. This statistically-based methodconsists of tracing the history of an energy bundle from its point of emission to its final27absorption. Over a large number of energy bundles, the interactions between emitting andabsorbing surfaces converge to the exchange factors for a given set of surfaces. This methodcan be readily adapted to non-uniform radiative surface properties, and complicatedgeometries. Both 2- and 3-dimensional models for viewfactor calcuation were developed forthis work.In the ray tracing approach used to calculate viewfactors, two directions must bedefined for an emitted ray (Figure 3.4). These are determined as follows.In general, the energy emitted by an area dA in wavelength interval d?. over angledf3,andover OO2n is[23]:dq = 2rc(, 13)i, (2k) cos j3 sin f3ddX 3.29where(, p) = directional spectral emissivityi (?) = spectral intensity of a blackbody, = emissive power per unit projected areaper unit wavelength per unit solid angleA probability density function is defined by normalizing the function by the total energy:330saT4where c is the total emissivity.It can be shown [23] that a function Rp uniform over the interval 0 R 1 is obtained byintegrating equation 3.30 over all wavelengths 0 ) <cand from zero to :2827t$i (?, P)b (2) cos 13 sin 13d(3d2.14= 00 3.31EcYT4For blackbodies and grey, diffuse surfaces equation 3.31 givesR=sin2f3 3.32R13 is called the cumulative distribution function. In order to assign directions to an emittedray, a random number is chosen between zero and one, and equation 3.32 is solved for 13. As0 is uniform over the interval 0 0 2it , generation of a circumferential angle is simplygiven by03.332’t3.2 Finite element implementation of boundary conditionsNeglecting convection in the liquid pool, the general 3-dimensional, time-dependentheat conduction equation in cartesian coordinates, with internal heat source q and temperaturedependant material thermophysical properties may be expressed as:= 3.34ox) Oy Oy) 8z Oz)If k and are assumed to be constant, this reduces to:29(2T 62T 62T ‘1 6Tki —+—-—+---—--+ +q=pC — 3.35öx2 2 6z2 ) ötIn the finite element method (FEM), a continuous domain is divided into subdomains,or elements, and a grid of nodes superimposed on the elements at the corners, and possiblymid-edge, mid-face or body-center locations [24]. Within each element, the temperature atany location is approximated by a sum of weighted temperatures at the node locations:T(x,y,z,t) T(x,y,z,t)=IV(x,y,z)7(t) 3.36where n is the number of nodes in the element, and the N1 are weighting functionsdependant upon (xy,z). Thus, when T is substituted into equation 3.35, the equality will, ingeneral, not hold at all locations but will have a small error R, called the residual, such thatö2k —-+--——+-—-+ +q—pC —=R 3.37ox 3))2 OzThe method of weighted residuals maintains that an approximate solution to equation(3.35) can be obtained by requiring that the sum over the subdomain of the residuals,weighted by a weighting factor W, must vanish:Jj$wRdv=o , i=l...n 3.3830or,339The Galerkin method of weighted residuals assumes that the weighting functions W1 aretaken to be the same as the nodal temperature weighting functions N1. Equation 3.39becomesJJ$Nk[2+ dV 03.40The first integral can be integrated by parts to givejfk 1fl +-n +-n N.dS_ffjky z I 6x x öy 6y .5z özi=1...n 3.41The double integral over the surface S represents the boundary condition, as can be seen byan energy balance on a surface element (Figure 3.5). Heat conducted into the boundary mustbe equal to heat lost from the boundary by imposition of a heat flux q (positive when heat isadded to the boundary), by convection qj, and by radiation q,..31(8T T öT \\kI — n + —ny +— I q + + q,. 3.428y öz ,jThe surface integral is used to incorporate the boundary conditions by substitution of 3.42into 3.41:JJk[nx +ny+n,dS=JJ(q +q +q)dS , = l...3.43For the farfield radiation condition of equation 3.5, the radiative flux q is nowinserted into equation 3.43, the surface integral expression:JfqrI1ds= H hr(linf — Fi’)N1ds 3.44So the load vector contribution is- qrload = HhrlnfM 3.45and the conductance matrix contribution isq,COfld=JJTNdS 3.4632The radiative heat transfer coefficient hr is highly non-linear, since it depends on thelocal temperature to the third power. It must therefore be updated for each time step. Avalue of the local temperature at the time step being evaluated is required. One method ofevaluating the radiative heat transfer coefficient is to iterate within the time step. Evaluatehr on the basis of an initial value (say, the previous time step temperature), solve the currenttime step, then reevaluate hr based on the new solution. Compare successive solutions untila convergence criterion is attained. This method is computationally intensive, however. Themethod used in this work is simply to base the evaluation of the radiative heat transfercoefficient on the previous time step. Restricting the maximum temperature change at anynode within one time step to a sufficiently small value (as implemented by the time stepoptimization algorithm of the code) minimizes the error introduced by this approximation.Overall computation time is reduced over an iterative approach.For the metal-mould interface, heat transfer is represented by equations 3.8 and 3.9.In the finite element implementation, the solid contact conduction component isJJqCOfldJvdS = JJhOfld TIVdS—JJhC0fld1flfNjdS , i = l...n 3.47where T1-ij is the temperature at the corresponding integration point on the opposingelement. Similarly, the radiation component isJqradNjdS= H hrad TNidS_HhradlinfNidS 3.48In each of equations 3.47 and 3.48, the first term on the right hand side contributes to theconductance matrix, while the second term contributes to the load vector.33The assumption of constant Km/Wg is not rigourously correct. The mouldconductivity may vary with temperature, and the gap width will vary over some time interval.These effects are not accounted for in this work, as they add considerable complexity to theprocedure. A more rigourous approach, addressing both geometric dependency and variationof K,n/Wg presents good scope for future work.In the enclosure radiation model, the finite element formulation of the net heat flux tosurface becomes:q =cxJ—EoT4 3.49So the boundary condition isJj(aJ_saT4)JVdS 3.50Here again, there are two terms, one of which contributes to the conductance matrix, theother which contributes to the load vector.In both terms of equation 3.50, the radiative heat transfer coefficient is non-linear,depending on the surface temperatures at all surfaces. In the calculation of J for the loadvector contribution, this appears in the vector {q} on the right hand side. As in the case offarfield radiation, {q} is evaluated based on the temperatures at the previous time step. Anaverage temperature over the surface of the element is computed:3.5134where k 2 for 2-dimensional elements and k = 4 for 3-dimensional elements. Atolerance flag in the program tracks the maximum temperature difference between the cornernodes and issues a warning if this difference exceeds a user-defined isothermal tolerance.The conductance matrix term is linearized:$JEcYT4I\1jdShr TIVdS 3.52wherehrEGT 3.53and, for k = 2 for 2-dimensional problems or k 4 for 3-dimensional problems, and the T1are the temperatures of the ends (2-d) or corners (3 -d) of the emitting segment,=3.5435CDDC.CDCDCD—C)‘<CDI CDCD-00C) CCD0g-hCD—CD0)C, CDCDCD—CDOCD—CD--0)CDCD-CDCl)- 0C)C)CzC/)0CD0)CC) -I— 03z1CDC)— 0)0C, CD0) C,I.C C) L)CD C/)Ambient temperature TinfFigure 3.5Heat Balance on the Boundary of a Conducting Medium-kdT/dx—xqh = hconv(Tinf- T)qr =- T4)Surface at temperature T38Chapter 4 Verification of Computer Code4.1 Viewfactor calculation verificationThe analytical solutions for the calculation of viewfactors for 2-dimensional surfacesoriented as shown in Figures 4.la to d are given in Appendix A. For verification purposes, 1= w = h = 1.0 and A = B = C = 1.0 were used. Table 4.1 shows the mean %error from theanalytical values and standard deviation of the %error (for n =20 observations) as a functionof the number of rays traced from each surface. As expected, the mean error and standarddeviation both decrease significantly as the number of rays traced is increased. Figure 4.2plots the 95% confidence interval (mean plus two standard deviations) of the %error inviewfactors vs the number of rays traced.The view factor algorithm is computationally intensive, with the run time increasingwith the number of rays, and the square of the number of surfaces:tcxn2n 4.1The desired level of accuracy in view factor calculation can thus be obtained at a cost in runtime. For most of the work described here, view factors were calculated using r = 10000.4.2 Radiation boundary condition verificationIn order to verifr the boundary conditions, the code was compared with datapublished by other researchers obtained by semi-analytical and other numerical methods forspecific problems.Jaeger [25] obtained solutions to the equation394.2ox2 pC Otsubject to the boundary condition_k’_=HVSm=FS 4.3OdxThis corresponds to a one-dimensional heat flow case of the semi-infinite region x < 0having constant conductivity k, density p, and specific heat C. When the exponent mthe boundary condition corresponds to a radiative heat exchange where a black surfaceradiates to a medium at zero K. Jaeger gives a graph of the resulting temperature at thesurface of the slab as a function of time, in dimensionless form, as vs. log10TThe 2- and 3-dimensional models used to simulate the conditions of Jaeger are shownin Figure 4.3 for the far-field, enclosure and interface type boundary conditions. For thesemi-infinite slab, l=4m was used, and the run terminated when a temperature change wasobserved at x=-4.0. Tamb, Tenc and Topposing were at OK, and all emissivities were setequal to 1. Constant values of k = 11.4 W/m-K, p =8190 kg/m3 and C, =435 J/kg-K wereused.-The results of the model are compared with Jaeger’s data in Figures 4.4a through f.Figure 4.4a also shows the effect of finite element mesh density at the radiating surface onthe solution. All meshes show good agreement at longer times. At short times the modelpredicts a higher temperature by up to 5% than the series solution for coarse mesh densities.For Figures 4.4b through f, a fine mesh of 0.Olm was used.The above comparisons verified that the code closely predicts transient temperaturesfor simple heat flow cases in agreement with data obtained by other methods. Some care40must be taken to choose a mesh size of sufficient density to obtain a desired level ofaccuracy.Table 4.1Mean %error and standard deviation forcomparison of Monte Carlo viewfactor results withanalytically calculated values (n=20)Figure Analytical Number of Mean StandardSolution rays deviation4.la 0.4142 1000 1.35 3.2310000 0.03 1.21100000 -0.01 0.434.lb 0.2929 1000 -1.16 5.7510000 0.44 1.51100000 0.11 0.594.lc 0.19982 1000 0.69 5.7810000 -0.34 2.21100000 0.06 0.664.ld 0.20004 1000 -3.67 7.6510000 0.14 2.15100000 0.11 0.62T T-_Fba: 2D Parallel b: 2D PerpendicularrrzC: 3D Parallel d: 3D PerpendicularFigure 4.1Configurations for View Factor Code Verification41Figure 4.2Comparison of Monte Carlo View Factor Calculation with Analytical ResultsTs.(N.0.05ra’////////////////,4 T = OKT H’ 4.Om.1__ Ts0.05nV/////////////////)- T = OKTi” 4.On interfaceFigure 4.3Configurations for Radiation Boundary Condition Verification of Thermal CodeMonte Carlo Viewfactor calculationComparison with analytical solutions for 2- and 3-dimensional geometries95% Confidence Level, n200LU---El----1-.--.--.-20 parallel2D perpendicular3D parallel3D perpendicular7.06.05.04.03.02.01.01 3Number of rays traced1 oVTinfinity = OKa: Semi-infinite solidcooling to environmentat OKb: Farfield radiation modelTim 4.Orn i Tambient = OKC: Enclosure radiation model__________________________d: Gap interface model42S=O.025m=0.OlmNFigure 4.4aCooling of 2D semi-infinite slab radiating to Farfield Environment at OK=model.=(1)Figure 4.4bCooling of 3D semi-infinite slab radiating to Farfield Environment at OKComparison of model results with data from Jaeger (1)Cooling of 2D semi-infinite slab radiating to environment at OKModel results for various mesh sizes1.00, -- -0.lOmf rr...0.800.600.200.00-2.0 -1.0 0.0 1.0 2.0Log10T(1) Jaeger, J.C., roceedirigs, Cambridge I ilosophical Society, vol. 1950. pp. 634-641Comparison of model results with data from Jaeger (1)Cooling of 3D semi-infinite slab radiating to environment at OKMesh size 0.01 m1.000.800.600.40(1) Jaeger, J.C., roceedings, Cambridge Ilosophical Society, vol. 1950, pp. 634-6410.00-2.0 -1.0 0.0 1.0 2.0Log10T43N• =model=(1)(1) Jaeger, J.C., roedinga, Cambridge I hilosophlcal Society, vol. 1950, pp. 634-641SAFigure 4.4cCooling of 2D semi-infinite slab radiating to Enclosure at OK-.----------.-- =model0.80=(1)0.60u.40Figure 4.4dCooling of 3D semi-infmite slab radiating to Enclosure at OKComparison of model results with data from Jaeger (1)Cooling of 2D semi-infinite slab radiating to enclosure at OKMesh size 0.01 m1.cn oi________________ ________________ ________________0.600.400.200.00-2.0 -1.0 0.0Log10T1.0 2.0lCD,Comparison of model results with data from Jaeger (1)Cooling of 3D semi-infinite slab radiating to enclosure at OKMesh size 0.01 m(1> Jaeger, j.c., roceedings, Cambridge hilosophiosi Society, vol. 1950, pp. 634-6410.00 —-2.0 -1.0 0.0 1.0 2.0Log10T44\—.-.----.. =modelNA=(1)NA“AFigure 4.4esemi-infinite slab radiating across Gap Interface to Solid at OKCooling of 2D‘=model=(1)NA40Figure 4.4fCooling of 3D semi-infinite slab radiating across Gap Interface to Solid at OKComparison of model results with data from Jaeger (1)Cooling of 2D semi-infinite slab radiating across gap interface to solid at OKMesh size 0.01 m1.0o0.800.60&30.400.20(1) Jaeger, J.c., roceedings, Cambridge ilosophlcal Society, vol. 1950, pp. 634-6410.00-2.0 -1.0 0.0 1.0 2.0Log10T0.80- 0.600.Comparison of model results with data from Jaeger (1>Cooling of 3D semi-infinite slab radiating across gap interface to solid at OKMesh size 0.01 m2.00.200.00 —-2.0 -1.0(1) Jaeger, ,i.c., roceedings, Cambridge hilosophical Society, vol. 1950, pp. 634-6410.0Log10T1.045Chapter 5 ExperimentsExperimental trials were conducted in collaboration with Deloro Stellite Inc. at theirplant in Belleville, Ontario. Ceramic moulds of two configurations were instrumented withthermocouples and time-temperature histories recorded during casting and cooling. The datacollected were used to verify the mathematical model formulation.5.1 Experimental ProceduresThe two casting configurations are shown Figures 5.la and b. The simple cylindricalmould was used because the design is easily modelled (to a first approximation) by a 2-dimensional or axisymmetric model, due to the radial symmetry and the relatively largeheight-to-diameter ratio. Thus, this model could be used to obtain data on the mould-metalinterface heat transfer, and to verify data on the thermophysical properties of the mouldmaterial. The more complex fmned configuration was employed to attempt to detect self-irradiation effects between the fins during cooling of the casting.Wax patterns and moulds were made by Deloro Stellite following normal productionprocedures [33,34]. The moulds were cemented to a refractory brick base, which also held aceramic thermocouple manifold. The moulds were instrumented with type R (Pt-Pt13%Rh)0.010” diameter thermocouple wire. This type of thermocouple was chosen for its hightemperature capability, as well as its ability to be used in oxidizing and vacuumenvironments. Compensating lead type Cu-Alloy 11, 24 gauge extension wire was used toconnect the thermocouple manifold through the thermocouple port in the furnace to the dataacquisition system.The vacuum furnace (Figure 5.2) was a Balzers VSGO50 horizontal loading unit,induction melting at 2 - 3 klzlz, 100 kW maximum power, with a minimum rated operating46pressure of 1 mbar. The vacuum furnace mould chamber door was modified with athermocouple feed-through port. The data acquisition system consisted of an AdvantechPCL8 18 data board with a PCLD789 MUXICJC/screw terminal. The system was controlledusing LabTech Notebook installed on a 486-33MHz IBM compatible computer.For installation of the thermocouples, narrow grooves were ground into the mouldusing a disc-shaped grinder. Bare wire, twisted junction thermocouples were embedded inthe moulds at three depths: near the surface (.050 - .080”), at midthickness (.120 - .180”) andat the facecoat (.260 - .310”) of the mould (Figure 5.3a). As well, several thermocoupleswere placed at the interface in the melt. These thermocouples were alumina-sheathed, andthe bare junction was given a thin coating of ceramic slurry. The location and depth of eachthermocouple was recorded. Thermocouples were cemented in place using Sairset, a mouldrepair ceramic [35]1. The thermocouples were then placed in alumina sheathing, andthreaded through the thermocouple manifold. Figures 5.4a and b show examples ofinstrumented moulds before casting.The cylindrical moulds were sectioned after casting to verify the thermocouplelocations (Figure 5.3a). Table 5.1 summarizes the locations of the thermocouples in themoulds for the cylindrical castings. The final depth measured was somewhat less than theinitial groove depth in most cases, due to displacement of the thermocouple junction in theembedding and drying stages. It was also observed that, upon drying, the Sairset ceramic onseveral thermocouples had contracted away from the bottom of the groove in which thethermocouple was installed; the thermocouple adhered to the Sairset, and was pulled awayfrom the mould ceramic at the measured depth; a void was thus formed between the Sairsetand thermocouple, and the mould (Figure 5.3b). Remarks on these observations are includedin Table 5.1. The mould thickness at each thermocouple location is recorded in Table 5.1 as‘The Sairset compound containes 2- 3% NaKO in the form of a liquid sodium silicate. Thispresented the possibility of thermocouple decalibration by dissolution of Na in the platinum.No experimental verification was done to try to extablish the magnitude of the possible error,but it was expected this would be small relative to the overall experimental error.47well. The average thickness was calculated to be 0.320”, with a standard deviation of 0.019”.Table 5.11 and Figure 5.5 summarize the location of thermocouples in the finnedcasting. The mould configuration was such that the mould tended to break apart duringcooling. It was thus not possible to measure the thermocouple depth by sectioning the mouldafter casting, as was done with the cylindrical moulds. Based on the difference in finalmeasured depth vs. initial depth as determined from the cylindrical moulds, corrections couldperhaps have been applied to the finned casting thermocouple locations, on the assumptionthat the installation method used was the same. However, the errors associated with themould non-uniform thickness and surface roughness, as well as the subsequentapproximations of constant thermophysical properties for modelling runs, were judged to belarge relative to the error in exact location of the thermocouples, and thus the upper bounddepths are shown.The thermocouples were calibrated to within one degree Celsius using the boilingpoint of water as a reference, using both a chart recorder and the data acquisition system.After cementing in place, each thermocouple was checked for continuity and polarity byconnecting it to a chart recorder and heating the surface with an acetylene torch.As the trials were being conducted in an industrial setting, and near the inductionfurnace power supply and coils, attention was paid to the possibility of noise inducement inthe thermocouples. Signals were observed to be stable to within ± one bit flip (± 2 deg C).As a precaution, power to the induction coils was turned off before opening the lock betweenthe melt chamber and pour chamber. In two instances the mould lock was inadvertentlyopened before the power was turned off. Both these data runs exhibit slight noise prior to thepour.The entire mould-base-manifold assembly was placed into the preheat furnace at 1093deg C (2000 deg F) for one to two hours. The charge was melted under vacuum (see Table5.111), and superheated to 1565 deg C (2850 deg F). The mould was transferred from thepreheat furnace to the furnace mould box. The extension leads were connected to the48manifold. The chamber door was closed, and the mould chamber evacuated to between 10-2and 1ü mbar (see Table 5.111). Power to the induction coils was shut off just before themould lock was opened. The mould box was positioned under the crucible, and the metalcast. The mould was then moved back to the mould chamber and the mould lock closed.During casting of the cylindrical runs, it was found that significant spillage of moltenmetal could occur, damaging the thermocouple manifold, and shorting out thethermocouples. To protect the manifold, a sheet of fiber paper was placed over the entiremanifold in the furnace (Figures 5.4c and d). The procedure was used for the finned castingsas well. It became apparent that the presence and location of the fiber paper with respect tothe thennocouples had an effect on the cooling behaviour of the mould surface; somethermocouples were shielded from the water-cooled furnace wall by the fiber paper, whichthus acted as an insulator. Others radiated directly to the cooled furnace wall. This effect isconsidered in the analysis of the data and is included in the sensitivity analysis to assess themagnitude of possible variations in the recorded temperatures.Data sampling was commenced at the moment that the mould was taken from thepreheat furnace, and continued at a sampling rate of 1 Hz for one hour. As eachthermocouple was connected, its response was checked on the display terminal to verify thatit was functioning.After pouring, short-checks were conducted at the screw terminals. This was found tobe necessary to establish that thermocouples were reading independent signals. During apour, molten metal occasionally splashed over the mould onto the thermocouple manifold,causing shorting between thermocouples. By placing a jumper wire between the high andlow terminal junctions, a signal could be reduced to zero (reading the temperature of thescrew terminal junctions). If two or more signals reacted to shorting at one thermocoupleconnection, the thermocouples were assumed to be electrically coupled in the furnace. Thesethermocouples were subsequently disregarded in data analysis. The time at which short49checks were conducted was recorded, and the data subsequently edited to remove thesespikes by averaging between the nearest smooth data.The compensating leads were connected to copper alligator clips. The high and lowclips were mounted together on a small ceramic base. This enabled the two leads to beconnected to the manifold quickly with one hand, and to be handled while wearing heavy,heat-resistant gloves. The use of copper clips in both junctions is a potential source of errorin the thermocouple readings. With reference to Figure 5.6, it was judged that, due to thehigh conductivity and thermal diffusivity of the copper, and the small size of the clips, the Pt-Cu and Cu-Alloy 11 junctions in the negative half of the circuit could be assumed to be at thesame temperature and thus formed an isothermal block junction, having little or no effect onthe net voltage recorded at the screw terminal.5.2 Discussion of experimental resultsPlots of the resulting thermal histories for the various runs are shown in Figures 5.7 to5.12 for the cylindrical configurations and Figures 5.13 to 5.19 for the finned configurations.The thermocouple responses recorded for the cylindrical castings generally showed thefollowing behaviour: 1) cooling of the mould after connection of the thermocouples beforepouring, 2) rapid heating of the mould after pouring, 3) slower cooling during solidification,corresponding to the latent heat arrest, and 4) rapid cooling asymptotically approachingambient temperature. The finned configurations do not show a pronounced cooling plateauassociated with the release of latent heat of solidification (stage 3 above). This may beexplained by the larger surface area-to-volume ratio of the fin over the cylinder, which allowsthe finned casting to solidify much more rapidly, due to the higher rate of heat removal.Subsequent model runs were able to recreate this observed phenomenon.50A parameter of interest in process control of casting procedures is the mouldtemperature at time of pour. Efforts center on maintaining the preheat furnace for the mouldat a sufficiently high temperature to enhance flow of liquid metal through the mould. Thecylindrical casting data indicated that, although the moulds were initially preheated to ‘-jl 000deg C, there was significant heat lost in the transfer of the mould from the preheat furnace tothe pour location. The recorded mould temperatures at time of pour were around 600 deg C.The average initial cooling rates of the moulds were calculated and are shown in Table 5.IV.These high rates (around 145 deg C/minute) indicate that, in order to obtain a high mouldtemperature at the time of pour, the mould cool time must be kept to a minimum. Thebeneficial effect of a higher preheat furnace temperature will be lost very rapidly as time topour increases.5.2.1 Cylindrical CastingsThe thermocouples on the cylindrical castings were spaced 2- 3 cm apart measuredalong the mould outer circumference. It was expected, from the synmietry of the cylindricalcasting geometry, that the heat flow would be radially symmetric, and that there should be nodifference in thermal profiles measured at the same depth and height, but at differentlocations on the circumference. The validity of this assumption may be influenced by severalfactors. The thickness of the mould at the thermocouple locations varied by up to 10%within a single casting (Table 5.1). The effect of this variation is assessed in the sensitivityanalysis in Chapter 6. Asymmetry in the mould-metal interface condition as solidificationand cooling progress is another factor which may result in non-symmetrical cooling. Finally,variation in the external cooling environment will affect the temperature history.On first examination, the test data from the cylindrical casting runs showed somecharacteristics which could not be explained when the assumptions of radial symmetry and51uniform cooling environment were applied.2 In particular, the temperature at longer timeswas somewhat anomalous in some instances, as some of the temperatures at deeper locationsfell below that at shallower locations. TC1 and TC2 of casting C7 fall below TC4, forexample (Figure 5.9), and TC6 of casting C9 falls below TC4 (Figure 5.11). TC7 of castingC6 approaches TC4 and TC6 more rapidly than does TC3 (Figure 5.8).In light of these apparent anomalies, it was necessary to consider the data withattention to the position of the thermocouples relative to the fibrepaper shielding (Figure5.4c). Over short time scales, the insulating effect of the fibrepaper is most significant onsurface temperature measurements, resulting in a higher maximum temperature forthermocouples located near the surface of the mould. At longer time, the enclosuretemperature directly ‘seen’ by the shielded thermocouples is higher. The apparent terminaltemperature of shielded thermocouples (both near the surface and deeper within the mould) istherefore higher than for the unshielded thermocouples, which ‘see’ an enclosure at a muchlower temperature. Finally, at long times, the temperature drop across the mould willdecrease (the mould interior temperature approaches the mould exterior temperature), as themould metal interface heat transfer rate becomes limiting.The position of the fibrepaper may result in temperature profiles differing with heightlocation of the thermocouple (above or below the level of shielding) and with placementaround the circumference (thermocouples directly facing the fibrepaper may be more2For casting C7 (Figure 5.9), position “e” was recorded as being monitored by TC4, andposition “b” by TC3. Position “e” was at a depth of 0.145” below the mould surface, andslightly shielded by the fibrepaper, while position “b” was at a depth of 0.090”. It wouldthus be expected that the temperature at “e” be higher than that at location ‘b’; the reversewas observed in the data, however. The embedded thermocouple junctions were examinedfor both locations, and good contact with the Sairset, as well as good Sairset - mould contact,were found. It was concluded that the thermocouple lead positions were recorded incorrectlywhen the mould was removed from the furnace, with TC3 monitoring position “e”, and TC4monitoring position “b”. This correction has been applied to the plotted data. Similarly,traces TC3 and TC4 of casting run C5 (Figure 5.7) were reversed upon examination of thethermocouple junctions.52shielded than those placed further around the sides of the mould). In Figures 5.7 to 5.12, thecircumferential location is indicated in the diagram, while the designation “s” (shielded),“ps” (thermocouple approximately level with upper edge of fibrepaper) and 11ns” (notshielded) indicate vertical placement of the thermocouple relative to the fibrepaper. Thelocalized differences in cooling due to the conditions described above were considered to bea possible explanation for the observed temperature behaviour. The possible magnitude ofthese effects is assessed further in the sensitivity analysis.Castings C6 and C8 had thermocouples located in the metal at the casting surface(TC1 and TC5 of casting C6, TC4 of casting C8). The temperature at TC4 of casting C8drops more rapidly than at TC1 and TC5 of casting C6. It also shows a temperaturediscontinuity at approximately time t = 350 seconds. These observations may be the resultof imperfect contact between the casting surface and the thermocouples such that there is aslight surface resistance resulting in a temperature drop between the thermocouple and themetal surface.Thermocouple TC5 of casting C8 (Figure 5.10) was not consistent with any of theother temperature profiles. Based on the preceding discussion of shielding effects, TC3could be expected to have a higher temperature throughout. The response of TC5 seemed toindicate that the material surrounding TC4 had a significantly higher thermal diffusivity thanthat at TC3; it showed more rapid heating, a higher maximum temperature and faster cooling.The possibility of inhomogeneities in the moulding ceramic are assessed in the sensitivityanalysis.5.2.2 Finned CastingsThe experimental data for the finned castings are shown in Figures 5.13 to 5.19. Thecurves differ from those of the cylindrical castings in the absence of a latent heat arrestplateau. All show a fairly sharp maximum at approximately 100 seconds after casting,53followed by a continuously decreasing cooling rate. Except for castings Fl and F9, whichhad thermocouples near the face coat of the ceramic, the thermocouples in the finned castingswere placed near the surface to detect variations in the cooling conditions, that is, maximizethe sensitivity to the radiation boundary condition. It was attempted to measure temperaturevariations with respect to radial position along the fin, vertical position on a face, andcorresponding location on the differently-oriented faces. The data of the finned castingsfollow self-consistent profiles, in that cooling rates between thermocouples of any singlecasting are consistent within reasonable variation due to the difference in location on thecasting.Thermocouples which were shielded by the fiber paper are indicated in Figure 5.13 to5.19 with the designation ‘s’. Examining the temperature profiles for castings F4, F5, F6 andF7 (Figures 5.14 through 5.17), it can be seen that the temperatures recorded by the ‘s’thermocouples are significantly higher (slower cooling) in all cases, even though the depth inthe mould is similar for all. Casting F9 (Figure 5.19), having all thermocouples located at thesame height, and thus with approximately the same degree of shielding, shows very littletemperature spread in the profiles, as does casting F8 (Figure 5.18), having all thethermocouples on the outside faces, and thus no shielding effect. The effect on casting Fl(Figure 5.13) is more difficult to assess. The shallow traces (b and c), both of which wereshielded by the fibre paper, show similar traces. The deep traces exhibit a discontinuity attime t = 1000 which makes subsequent data suspect.54Table 5.1Location of Thermocouples in Cylindrical CastingsCasting TC# Position height depth Mould Shielding(in) (in) thickness(in)C5 3 b 4.6 0.135 0.315 Ps(largevoid)4 c 4.6 0.040 0.335 Ps6 a 4.6 0.175 .0335 PsC6 1 a 4.6 Metal- psSurface7 b 4.6 0.265 0.320 Ps4 c 4.6 0.030 0.325 ps6 e 4.6 0.030 0.320 Ps3 f 4.6 0.295 0.340 Ps(void)5 g 4.6 Metal- pssurfaceC7 1 a 4.6 0.285 0.340 s(void)2 d 1.9 0.260 0.300 s3 e 1.9 0.145 0.310 s4 b 4.6 0.090 0.305 s6 f 1.9 0.035 0.335 sC8 1 a 4.6 0.290 0.320 Ps(void)3 c 4.6 0.035 0.340 ps4 d 4.6 Metal- psSurface5 f 4.6 0.030 0.300 PsC9 3 e 1.9 0.235 0.280 s4 f 1.9 0.030- s5 c 4.6 0.050 0.315 p6 d 4.6 0.135 0.335 psClO 3 e 1.9 0.240 0.300 s4 f 1.9 0.035 0.2955 c 4.6 0.040 0.335 Ps6 d 4.6 0.135 355shielded, ps = partially Shielded55—----——---——---COTjTj-00--‘CD--------------EONC-ti-L3UON-Ci.)‘JC-ti.k)C.)ONC-’ICJONUiC-tiONC-tiLi.)L’J—C)i.------------------------ii—lCDCDi-+CDCDCDC)CDCDCD-tCCDCDC)CO CD CD-CD p-t’Jii—’—44.1CL)L)..-k)t’J‘J))I—’—-‘—‘i—i——----—————--bbccbebccccbebebceONbcCD — CD-———----—-—CD -NQ00000000O\O,cccceceececpeppcpcpppppppp9pppc,j’j—.bcecceeeccccccccccccct’ic——.C)oo0000Cf) ECOCOCOCOCl)COCOCOCOCOCOCl)C/)C/)COC/)COCl)C/)Cl)COCl)COCl)COCOC/)Cd)COCd)Cl)IIIIIIIIIIIIIIIIIIIIIIIIIIIIC CD 0 0 .CDCD CD-—C) C/) C/) CD ITable 5.111Vacuum Furnace Pressures during Melting and CoolingCasting Melt Vacuum Cooling Vacuum/time(mbar) (mbar/sec)C2a 5.0x104 5.0x102/1901.1x102/ 10008.3x10_/_130C4 8.5x104 1.8x102/265C5 5.0x102 5.8x100/l00C6 3.4xl0 1.8x102/1801.8x102/ 5506.3x10 / 16004.2x103/22502.6x103/ 3350C7 l.5x102 3.7x10/210l.9x102/ 5507.1x103/ 14205.6x103 / 19802.7x103/ 3360C8 3.5x10 1.9x10/6507.1x103_/_193C9 2.0x103 2.5x10/2406.0x103 /20205.2x103/ 23203.5x10 / 30002.7x103 / 3600ClO nld* 2.0x10/9004.7x103_/283Fl 1.1x104 1.3x10/170F2 1.0x103 1.0x10 / 160l.0x104/25F4 nld* 8.0x103/1376.9x10_/_1776F5 4.7x10 2.0x10/1801.9x102/ 8409.8x103 / 18008.2x103 /24007.5x103 / 27007.1x10_/210*mssing value57Table 5J (cont’d)Vacuum Furnace Pressures during Melting and CoolingCasting Melt Vacuum Cooling Vacuumltime(mbar) (mbar/sec)F6 nld* 7.5x102/1502.9x10 / 4201.3x102/ 11008.7x103/ 17005.8x10_/_300F7 5.0x103 5.0x102/2302.8x10 /4401.3x10/10501.0x102/ 13508.3x10 / 17007.1x10_/2100F8 1.6x104 2.5x10/3105.8x102/ 3902.1x10 / 7508.8x103/ 16006.8x103 / 21004.5x103 / 32503.9x10_/_360F9 3.0x104 2.0x10/1402.6x10 / 5001.0x10/13008.3x10 / 15507.0x103 / 18504.8x103/ 28502.4x103/ 3600*missing valueTable 5.IVCylindrical mould initial cooling rates after removal from preheat furnaceCasting Average initial cooling rate(deg_C/mm)C5 144.6C6 147.0C7 141.0C8 160.2C9 148.2dO 130.258ak LQFigure 5.laCylindrical Test CastingCASTI, FIIfU, TESTAtEGTFigure 5.lbFinned Test Casting5000L13•38R MAX (TYP)20000±38I. DIHESIUS i K_________2. SURFACE FINISH 3.2 RHR3. ATER1AL SFRALLUY 718CASTI, TEST, [YLINDRI[LI Ifjfl1iAnm. IIuI. Lt. U8I T2iWLI3I. DIdfli2. II3.22. IITRJ4L: 9Fkia 218o IflStS (&1LiLS l8I LISItY l1I8 md1AI •‘- SH8 . LC E8A 6I 11459Figure 5.3Schematic of thermocouple installation in ceramic mouldsa: Typical installation showing initial and final measured depthb: Example of shrinkage void observed in a few of the thermocouple locationsFigure 5.2Balzers VSGO5O Vacuum Furnace (Deloro Stellite Inc.)Mould exterior surfacea:Mould Interior surfaceShrinkage void in Sairset60Figure5.4aTypicalcylindricalmouldinstrumentedfortesting61Figure5.4bTypicalfinnedmouldinstrumentedfortestingFigure5.4cCylindricalmouldinfurnaceaftercasting,showingmanifoldshieldingFigure5.4dFinnedmouldinfurnaceaftercasting,showingmanifoldshielding62Figure 5.6Pt-Ptl3Rh equivalent thermocouple circuit::•:::,j::Figure 5.5Thermocouple location in finned castings (Ref. Table 5.11)Cu clipPtl3Rh÷ CuTmefCu clipConnected circuitTmIsothermal junctionEquivalent circuit63Trial RunC5Figure 5.7Figure 5.814001200jl000D 800E,!600400200500 1000Da wnpe 10t27192Ddo S4ee. 8ee,Ie Ont.8.Donii1500lime (seconds)2000 3000Trial Run C61400120010000Q0EI—8006004002000DaasdnW4cd 1cv28’92Doo Sce, Beevie t.B. Dommik500 1000 1500 2000 2500Time (seconds)300064Trial Run C7Figure 5.91200800o 1000EI-4002006000Da STPIDio S*ete. Bdc’.4e O’d.B. DaninUc500 1000 1500 2000 2500Time (seconds)3000Figure 5.1065Figure 5.11Figure 5.1266Figure 5.13Figure 5.1467Trial Run F5Figure 5.15Figure 5.16140012000-g 10008006004002000a: 0.075° topb: 0080” topC: 0.080° mid-heightd: 0.075” mid-heighte: 0.070” bottomf: 0.070° bottoma,c,e500 1000Datasampled 1112/92Detom Steflke Inc.. Belleville Ont.B.DonIk1500 2000Time (seconds)2500 3000Trial Run F6be,f140012001000800a,0EF— 6004002000a: 0.055° inside face, topa,bb: 0.060” inside face, mid-height (s)e: 0.060 outside face, mid-height1: 0.065° outside face, bottom500 1000Data sarrIed 11/3/92Doro Stellhte Inc., Behievihle Ont.B.Dominhk1500Time (seconds)2000 2500 3000681400 Trial Run F7Figure 5.17Figure 5.18d,e12001000C)8006004002000a: 0.055” top insideb: 0.055’ mid-height inside (s)C: 0.055” bottom inside (s)d: 0.045” top outsidee: 0.050” mid-height outside500Data sanpled 11/3/92Deloro Stellite Inc.. Belleville Ont.BDoninik1000 1500lime (seconds)2000 2500 30001400Trial Run F81200/0. 10000Co0aE0800d,e,fa,b,o6004002000ac,dData sarrpled 11/4/92Deloro Stellile Inc.. Bellevite Ont.B.Dominik500 1000 1500 2000 2500Time (seconds)300069Trial Run F9Figure 5.19C)a,a,00Ea,F.1400120010008006004002000a)a: 0.235”C: 0.070”fd: 0.075”e: 0.150”f: 0.240”0d eDatasampled 1114192Debra Stebthe Inc., Bebbedlie Ont.B.Dombnbk500 1000 1500 2000 2500 3000Time (seconds)70Chapter 6 Sensitivity Analysis and Analysis of the Casting Process6.1 Sensitivity analysis -cylindrical configurationA simple, 2-dimensional model of the cylindrical casting was analyzed for thepurpose of conducting a sensitivity analysis. The sensitivity analysis was performed withtwo aims: 1) to determine the effect of each parameter on the solidification and cooling ofthe metal, leading to an understanding of how they may influence casting quality, and 2) todetermine the effect of each parameter on the temperature distribution in the mould, foradjustment in fitting the model results to the experimental data, and for assessment of thedata. The casting system is potentially influenced by a large number of parameters for themetal, the mould, the interface conditions and the surroundings. The parameters chosen forthe sensitivity analysis were those associated with the greatest uncertainty, or for which littledata were available. These are summarized in Table 6.1. A number of parameters wereobtained from the literature; it was assumed that these parameters were sufficiently wellestablished, that their inclusion in the sensitivity analysis was unnecessary. These parametersand their values are summarized in Table 6.11.The results of the sensitivity analysis are shown in Figures 6.2 to 6.13, andsummarized in Table 6.111. In the figures, the four curves shown for each conditioncorrespond to nodes at the casting center (trace 1), casting surface (trace 2), mould interior(trace 3) and mould exterior (trace 4) surfaces. Thus the effect of the parameters on totalsolidification time (center of casting), interface heat transfer (casting exterior and mouldinterior) and heat transfer to the surroundings (mould exterior) can be readily assessed. Thetotal solidification time ( t in Table 6.111) is obtained from the temperature trace at thecasting center (trace 1 in Figures 6.2 to 6.13) and is taken as the end of the latent heat“plateau”, at which the temperature trace shows a sharp increase in the rate of change of71temperature. The parameters in Table 6.111 are ranked in the order of decreasing effect ontotal solidification time (%At)6.1.1 Interface heat transfer assumed contact timeFigure 6.11 shows the effect of variation in the time over which interface contact isassumed to exist. It can be seen that the time - temperature response depends on whether thecontact time is increased or decreased over the base value. A decrease in the contact time by50% results in a 14% increase in total solidification time, and results in a “double plateau” inthe mould temperature. An increase in contact time by 50% has a very small effect (-1.7%)on total solidification time, and the shape of the curves is not significantly altered.This behaviour gave some insight into the value of contact time to be used in themodel. The “double plateau” in the mould cooling curves which resulted when the contacttime was less than the solidification time was not observed in the experimental data. The gapformation was assumed to be due to solidification shrinkage only; after solidification wascomplete the gap was assumed to remain constant. For the cylindrical and finned models,therefore, the gap formation time was set to be approximately equal to the total solidificationtime.The magnitude of the effect of the interface contact function indicates that the gapformation characterization is very important in the implementation of an investment castingsimulation model.6.1.2 Metal initial temperatureFigure 6.5 shows the effect of varying the metal initial temperature. The metal initialtemperature had a strong effect on total solidification time, and a slight effect on the mouldtemperature and metal surface temperature profile. This parameter was known from the72experiments only within a range having the last measured temperature as the upper bound(1565 deg C) and the melt liquidus (1340 deg C) as a lower bound. Loss of superheat occursvery quickly during the pouring and mould filling stages, and the initial temperature profilemay well be non-uniform. To represent this process, a coupled mould filling (fluid flow) -heat transfer model would be required. For the purpose of this work, a uniform metaltemperature was assumed, at a value adjusted for the best fit to the data, but between thebounds given above.The metal pour temperature is used in the foundry industry as a process controlparameter, and is known to influence mould filling. In intricate parts with thin sections, asignificant superheat may be required to ensure complete mould filling.6.1.3 Mould thermal conductivityFigure 6.2 shows the effect of the assumed mould thermal conductivity. The effect ontotal solidification time is significant (5.9% increase in solidification time for a 10% decreasein thermal conductivity). The mould thermal conductivity is associated with a large degreeof uncertainty. The ceramic shell system is inhomogeneous, consisting of layers of varyingcomposition. Each layer, as well, has several components, including the stucco Or flour, thebinder and a porosity fraction. The thermal conductivity varies with composition, and alsowith temperature; intergranular radiation in the pores and transgranular radiation through thesolid may contribute to an effective conductivity at temperatures greater than -500 deg C.Very little data are available in the literature for investment casting ceramic thermophysicalproperties. Because of the uncertainty associated with the mould conductivity and itsrelatively large effect on the temperature profiles, this parameter was one of those used toadjust the model fit to the experimental data.736.1.4 Metal latent heat of solidificationFigure 6.12 shows the effect of variation in the value of metal latent heat ofsolidification. This parameter was assumed to be fairly well characterized in the literature;hence the small parameter variation was used. A 5% reduction in latent heat resulted in a4.7% reduction in total solidification time. The effect on the mould temperature profiles wasnot significant. Due to this small effect, and the relatively good characterization of this valuein the literature, the latent heat was not varied in fitting the model.6.1.5 Interface contact initial heat transfer coefficientFigure 6.10 demonstrates that, although there is a negligible effect on totalsolidification time, the assumed initial interface contact heat transfer coefficient stronglyinfluences the mould temperature profiles. It is also associated with a large uncertainty. Thisparameter was therefore one of those adjusted in fitting the model to the data.6.1.6 Fibrepaper shieldingFigure 6.13 assesses the effect of the presence or absence of the fibrepaper manifoldshield on the temperature (see Figures 5.4c/d and 6.lb). Although the effect on totalsolidification time is negligible, there is a pronounced effect on the surface temperature overthe whole time range, and on the entire mould profile at longer times. Although it wasdifficult to incorporate this effect for each individual thermocouple location, in particularwith a 2-dimensional model, the possibility and magnitude of the error due to this effect wasassessed in the analysis of the experimental data.As discussed in chapter 5, the circumferential placement of thermocouples withrespect to the fibrepaper in the cylindrical castings may account for the temperature profiles74intersecting in the experimental data. A model run was done to assess the magnitude ofcircumferential temperature differences due to the fibrepaper shielding. Figure 6.14 showsthe temperature profile at four points in the mould: two at 0 degree orientation with respectto the fibrepaper, and two at 90 degree orientation (Figure 6.ib). The depths shown aretypical of those sampled in the cylindrical castings. The temperatures at N882 and N866,both at 0 degree orientation, although they converge somewhat, maintain a temperature dropconsistent with the difference in depth location. Similarly, the temperatures at N1415 andN1405 maintain a consistent temperature drop. However, the position of the 0 degree vs. 90degree traces changes, as the unshielded traces decrease in temperature more rapidly than theshielded ones. N866, at a depth of 0.150”, approaches Ni 405, at a depth of 0.285”. Ni 415,at a depth of 0.135”, actually drops below N882, at a depth of 0.030”. These results confirmthat the circumferential position of the thermocouples may result in the intersectingtemperature traces observed in the experimental data.6.1.7 Variables having minor effectsFigure 6.4, showing the effect of mould initial temperature on the temperatureprofiles, indicates that this parameter has almost no effect on the cooling or solidification.This conclusion may be somewhat misleading in light of the fact that the mould temperatureis used in the foundry industry as a quality control parameter. This results may be due to anumber of factors. The cylinder is a very simple geometry, with no complex feedingrequirements, and of relatively large section (0.025m radius) somewhat atypical for aninvestment casting. Filling of the mould requires no flow through thin sections or convolutedgeometries, which results in rapid loss of superheat, premature freezing, and non-fill ofcastings. The volume of metal in the cylinder is large compared to the volume of ceramic;thus the heat content of the ceramic contributes less to the temperature response of the75system. For thin sections, the mould heat content is large compared with that of the metal;the mould initial temperature may thus be a significant control parameter.Figure 6.9 shows the effect of varying mould thickness on the temperature profiles.The effect on total solidification time is negligible. A significant effect is seen on the mouldtemperature profile at shallow depths. The mould thickness was seen to vary not onlybetween castings, but within a single casting. It was not feasible to model this variation.Thus, while recognizing that variation in thickness may account for a small error in fitting themodel to the data, a constant mould thickness was used throughout.Figure 6.6 shows the effect of varying the mould emissivity. A negligible effect ontotal solidification time is observed; however, the mould temperature is slightly affected, nearthe exterior surface and at long time. This parameter was poorly documented in theliterature, and it was thus varied in the model runs to aid in fitting the results.Figure 6.3 shows that the mould heat capacity has a negligible effect on both the totalsolidification time and the mould temperature profiles. Values from the literature weretherefore used and maintained constant.The assumed ambient temperature (Figure 6.8) and assumed metal emissivity (Figure6.7) are shown to have negligible effects on both total solidification time and the mouldtemperature profiles, and were thus held constant.6.1.8 Analysis of heat flow resistancesAn analysis of the relative magnitude of heat flow through the metal and mouldversus heat transfer at the surfaces can explain several important aspects of the mould - metalcooling curves. As solidification progresses, the interface heat transfer conditions becomeincreasingly more important.The heat transfer Biot number represents the ratio of resistance to heat transfer at thesurface of a medium to resistance to heat flow through the medium:76hBi/Axwhere h is the surface heat transfer coefficient, k is the medium thermal conductivity andAx is a representative dimension for the conducting medium. For Bi> 1.0, heat can beremoved from the surface more rapidly than it can be conducted to the surface through themedium. For the mould, using a thermal conductivity of 1.0 W/m-deg C, a thickness of0.008m and an initial surface heat transfer coefficient at the mould-metal interface of 1500W/m2-deg C results in1500x0.008Bz,OUfd_Ifl 1.0= 12.0Thus, at the mould interior surface, the mould conduction limits the heat flow. Thisconclusion is in agreement with the indication from the sensitivity analysis that, at shorttimes, the mould conductivity is of prime importance in influencing the heat flow path.At the mould exterior surface, the radiation heat transfer coefficient is much lower(calculated to be approximately 40 W/m2-deg C), so40x0.008Bifliouldex 1.0= 0.32Thus, at the mould exterior, the rate of heat removal from the surface of the mould limits heatflow in the system. In the metal, for kmetal = 11.4 W/m-deg C, and a cylinder radius of0.025 m,771500x0.025Blnieiai_ex= 114Here, conduction through the metal initially limits heat flow.As the metal solidifies and contracts, the interface contact component of heat transferdecreases, and the area fraction of radiation heat transfer increases. When the contactcomponent of the heat transfer coefficient approaches zero, the radiation heat transfercoefficient is calculated to be approximately 157 W/m2-deg C:157x0.008Blfl,OUld_Ifl= 1 0= 1.25157x0.025Biniejat_ex 11.4=0.34Thus, at short times after pour, heat removal from the metal is limited by conduction throughthe so1idifying shell and conduction through the mould. At longer time, metal cooling islimited by the rate at which heat is removed from the metal surface. Mould conductivitylimits the heat transfer rate during the entire solidification and cooling process. At long time,the rate of heat removal from the mould exterior surface also becomes a limiting factor.The above conclusions are reflected in the sensitivity analysis curves. Shortly afterpouring, the temperature drop between the metal center and metal surface and the mouldinterior and exterior surfaces is large, and the temperature drop across the interface is smallreflecting the conduction resistance limitation. As the interface heat transfer coefficientdecreases, the metal attains an approximately uniform temperature, and the temperaturedifference between the interior and exterior mould surfaces decreases. The wideninginterface temperature drop reflects the decreasing gap heat transfer coefficient, while the78decreasing mould temperature drop reflects the limitation of heat removal from the mouldexterior surface.6.2 Sensitivity analysis- finned configurationA 2-dimensional model (Figure 6.21) was used to analyze the finned configuration.The finned sensitivity analysis was performed to ascertain the effect of radiation shieldingand self-irradiation on the predicted temperatures. In addition, the results of this analysiswere used to aid in interpretation of the experimental data. As most of the thermocouples onthe finned castings were near the mould surface, the locations used in the sensitivity analysisare nodes on the mould surface.6.2.1 Temperature variation with radial positionWith reference to Figure 6.21a for placement of the various nodes, Figures 6.22 to6.24 show the variation in temperature on each face due to the radial location. Thetemperature near the casting center is highest in all cases, decreasing with increasing radialdistance from the center. The position ‘d’ traces show an “end effect” reflecting the greaterheat loss, due to proximity to the end of the fin. A temperature difference of approximately70 deg C is indicated between the extreme radial positions (“a” and “d”) on face A. On facesB and C, the temperature difference is smaller (30 deg C). This smaller range reflects the factthat faces B and C experience less self-irradiation than face A.796.2.2 Temperature variation for differing facesFigures 6.25 to 6.28 show the data replotted to emphasize the difference intemperature due to the self-irradiation effects at each radial location. Faces B and C are atthe same temperature at corresponding radial locations. Face A is at a higher temperature,with the largest temperature difference at the interior position (position “a”, maximumtemperature difference of —P40 deg C). The self-irradiation effect decreases at positionsfurther out along the fin.6.2.3 Effect of fibrepaper shieldingFigure 6.2 lb shows the position of the fibrepaper manifold with respect to the finfaces. Figures 6.29 through 6.35 summarize the results of model analysis in a similar mannerto that of the unshielded model. As seen in Figures 6.29 to 6.31, the magnitude oftemperature difference between the radial extremes does not change significantly over theunshielded model (70 deg C for face A, 30 deg C for faces B and C). However, thetemperature difference between the faces shows a dramatic increase (Figures 6.32 to 6.35). Amaximum temperature difference of-45O deg C is observed between face A and faces B andC. A secondary effect of the shielding is shown on face B at longer times, as the traces of Band C diverge. This reflects the hotter opposing face of B vs. that of C. Figure 6.36.summarizes the temperature difference on face A between the shielded and unshieldedmodels.The results of this analysis of the finned configuration indicate that, in analyzing thedata from the finned casting runs, the fibrepaper shielding and physical location of thethermocouples are dominating factors in explaining the spread in temperature profiles seen inthe data.806.3 Comparison of experiments with model resultsThe finite element models shown in Figures 6.1 (cylindrical casting) and Figure 6.21(finned casting) were analyzed with a view to aligning the model to the industrial process.6.3.1 Cylindrical castingsFigures 6.15 through 6.20 show the best fit of the thermal model results withexperimental data. As indicated by the sensitivity analysis, the mould conductivity and theinterface heat transfer conditions had the most significant influence in adjusting the modelpredictions. For the interface heat transfer, a linearly decreasing function was used. Asdiscussed in section 6.1.1, the time over which the interface contact was assumed to exist wasset approximately equal to the total solidification time, as determined from preliminarymodel runs. The value of the initial contact coefficient was adjusted based on fitting themodel to the interface data of castings C6 and C8. Initial values of the mould thermalconductivity and mould emissivity were obtained from the literature. Based on the modelsensitivity analysis response, the parameters were varied to align the model with theexperimental data. Mould initial temperatures were set to those indicated by theexperimental data for each model run.The values of the parameters used are summarized in Table 6.IV. Average mouldthermal conductivities range from 0.9 to 1.1 W/m-deg C. This is somewhat higher thanvalues reported in the literature. Thermal diffusivities range from 0.00375 to 0.00458cm2/sec, which is in agreement with reported values. Mould emissivity values range from0.55 to 0.65.Reasonably good fits to the experimental data can be obtained using the constantvalues of the mould thermophysical properties. In particular, the cooling rate at mould81temperatures below 750 deg C is well reproduced. At this lower temperature, the thermalconductivity of the mould will likely be less temperature dependent as the influence of voidradiation will be reduced.During solidification, the larger uncertainties associated with the heat transfer at themould-metal interface and possibly the constant value of thermal conductivity of the mouldresulted in slightly poorer fit of the model to the data. Casting C5 (Figure 6.15) shows goodagreement of the model with the temperature profile at 0.040” and 0.175” depth. The modeltemperature profile at 0.135” is higher initially by up to 10% at t=300 seconds.Casting C6 (Figure 6.16) shows good agreement between the model results and theexperimental data, in particular over the first 700 seconds. This data set, having athermocouple in the metal, was used to establish more closely the interface heat transfercontact conditions.Casting C8 (Figure 6.18) shows a reasonably good fit in the mould temperatures. Themetal surface temperature is over-predicted by up to 8%. The model did not reproduce thetemperature discontinuity at time t = 350 seconds.The effect of the fibre paper shielding on the mould temperature distribution hasparticular significance in the analysis of casting runs C9 and ClO (Figures 6.19 and 6.20).The sensitivity analysis predicted that the temperature near the mould-metal interface is notsignificantly affected until long times (after solidification is complete). The predicted surfacetemperature is higher by up to 50 deg C due to the presence of the shielding. As castings C9and ClO combined shielded and unshielded thermocouples, the model was run without ashielding condition, and the temperatures for thermocouples TC4 (near the surface of themould), were adjusted down by 50 deg C to compensate for this condition, based on themagnitude of the effect indicated in the sensitivity analysis. This is indicated by thedesignation TC4’ in Figures 6.14 and 6.15. Casting C9 shows good agreement between themodel results and the experimental data. For casting Cl 0, the model overpredicts the surface82temperature, possibly due to non-uniform cooling effects in the casting run which are notincluded in the model.6.3.2 Finned CastingsThe results of the finned configuration sensitivity analysis can be used to helpinterpret the experimental data and to provide insight into the differences in temperatureobserved at thermocouples located at similar radial position and depth, but on different facesand at different heights on the mould. On casting F8 (Figure 6.41), which shows acomparison between predicted and measured temperature, the thermocouples were all locatedat similar radial locations and depths, and were not shielded from the furnace by theinsulating fibre paper. Thus, the difference in temperatures observed in the experimental datamust be due to combined effects of varying height on the casting and non-uniform coolingenvironment due to the presence of the mould box, as well as slight differences in depthlocation of the thermocouples. The experimental data plot of casting F8 indicates that theabove conditions may account for a temperature difference of up to 50 degrees C at themould surface. Therefore, when modelling the finned casting as a 2-dimensional model withonly the furnace and fibre paper influencing the cooling, results within approximately 50degrees of the experimental data represent a reasonably good fit.Castings F5, F6 and F7 (Figures 6.38 to 6.40), with thermocouples located on variousfaces near the mould surface, show obvious effects of shielding from the fiber paper. Forcasting F5, having thermocouples at shielded, partially shielded and unshielded positions onFace A, the magnitude of temperature difference between the shielded and unshieldedpositions is similar to that predicted by the model; the maximum temperature difference isapproximately 160 degrees, consistent with the sensitivity analysis predictions. Traces ‘a’ and‘b’, and ‘c’ and ‘d’ show a temperature difference consistent with what would be expected dueto a radial difference in location (temperature of ‘b’ higher than that of ‘a’ temperature of ‘d’83higher than that of ‘c’); traces ‘e’ and ‘f show a reverse difference from that expected, possiblydue to combined experimental error as discussed above.For casting F6, all thermocouples are at the same radial location. Traces e and f, bothon Face B (unshielded), show similar temperatures. Trace ‘a’, on face A (unshielded), is at ahigher temperature (60 - 80 degrees) relative to ‘e’ and ‘f’, consistent with sensitivity analysisresults. Position ‘b’, shielded by the fiber paper, is at a significantly higher temperature thanposition ‘a’ over the whole time range. Casting F7 also shows data consistent with thesensitivity analysis with regard to shielded and unshielded temperatures for all thermocouplesat the same radial location.Combining results from the shielded and unshielded two-dimensional model,computer results were fit to the experimental data for the finned castings. Tables 6.V and6.VI summarize the model parameters, and Figures 6.36 to 6.42 show the resultingtemperature profiles. For all of the castings except casting F8, while using a mouldemissivity of 0.60, an average mould conductivity of 0.8 W/m-deg C and thermaldiffusivities ranging from 0.0026 to 0.003 8 cm2/sec gave the best fit to the data. Thesevalues are slightly lower than for the cylindrical castings, possibly due to the fact that thecylindrical mould remained at temperatures where void space radiation increases the apparentthermal conductivity for a longer time. Casting F8 required a mould conductivity of 0.5W/m-deg C. This lower value is possible due to the fact that casting F8 was poured at ‘—350seconds after removal from the furnace, whereas the others were poured at between 150 and250 seconds. This resulted in a lower initial mould temperature for F8; the mouldconductivity would thus have less of a radiation conduction component, yielding a lowereffective thermal conductivity.84Table6.1SensitivityanalysisparametersRunVariedKPp6mould6meralTmouldTmetaiTambThicktcontactLmetaiIlberpaParam.(W/m-(J/m-deg(cm2/sec)(degC)(degC)(deg C)(mould)hap(see)(kJ/kg)perdegC)C)6(mm)(W/m2xlOdegC)aBase1.02.4.0.004170.600.356001400300.008381400700272no(60)1bKmould0.900.00375cpC2.6670.00375d0.65e8metal0.30f650gTmeta11350(10)1hTamb150iThick0.00762(mould)h°5000gaptCOntact350/10501Lmetai258mfIberpaperyes1Superheat =Tmetal-Tliquidus85Table 6.11Constant Material PropertiesKmetal (W/m-deg C) 11.4Pmetal (kg/rn3) 8190CPmetal (J/kg-deg C) 435Tsolidus (deg C) 1260Tliguidus (deg C) 1340Efurnace 0.60Table 6.111Effect of Parameter Variation on Total Solidification TimeRun Parameter t %Param. variation %AtVaried (seconds)a Base 597k 681 -50.0 +14.1g Tmetai 548 - 3.5 - 8.2(83)1b Kmould 632 -10.0 +5.9m Lmetai 569 -5.0 -4.7f Tmould 610 + 8.0 + 2.2i ThlCkmould 584 -9.0 -2.2d mould 585 + 8.0 - 2.0c pCi, 586 +11.0 -1.8j 587 +257.0 -1.7h;ap1 tcongact 588 +50.0 -1.71 fiberpaper 606 +1.5h Tamb 600 + 400.0 + 0.5e Emetal 598 -14.0 + 0.2I Number in brackets inaicates %variation in superheat86Table 6.IVValue of Parameters used in fitting Cylindrical Experimental DataRun No. Kmould 6mould Tmould(W/m-deg (cm2!sec) (deg C)C)C5 0.9 0.0038 0.55 650C6 1.1 0.0046 0.60 600C7 1.0 0.0042 0.60 600C8 1.1 0.0046 0.65 550C9 1.1 0.0046 0.65 550ClO 1.1 0.0046 0.65 550Table 6.VConstant values used in Analysis of Finned CastingsParameter ValueEmetal 0.30EfIbre 1.00Efurnace 0.60mould 0.60Tmetal 1400 degCTfurnace 30 deg CTfibre 200 deg CTable 6.VIValue of Parameters used in fitting Finned Experimental DataRun No. Kmould Tmould(W/m-deg C) (cm2/sec) (deg C)Fl 0.8 0.0033 650F4 0.8 0.0026 750F5 0.8 0.0026 750/800F6 0.8 0.0026 700F7 0.8 0.0026 550F8 0.5 0.0021 500F9 0.9 0.0038 80087Figure 6.ia2D Cylinder Model: Casting O.D. =50 mm, Mould Thickness = 8.3 mm0 deg orientation90 deg orientationFigure 6.lb2D Cylinder Model with fiberfrax shielding88Figure6.2Figure6.4Figure6.3Figure6.52Modelresultsfor2-dimensionalcylindricalcastingEffectofmouldthermalconductivityoncoolingandsolidificationSolid: Kld=1.0W/m-degC5.1250Dashed:KmJld=0.90W/m..degC1b000aiso 600O—Cazdngcit•2-CasOn9003-Moldlnn.r.wfaca4—Moldo4jlIt,urlao.Modelresultsfor2-dimensional cylindricalcastingEffect ofmouldheatcapadtyoncoolingandsolidification1$00 e-l000 I I-0500000016002000T,m(annrf750Solid: pC=2.4x16J/m3-degCDashed:pCi, =2.664x1oJ/m3-degCI—CaV109cens.2-Cas5n0003-Moldinerswlac.4-Moldcutsuda2500600Modelresultsfor2-dimensionalcylindricalcastingEffectofmouldInitialtemperatureoncoolingandsolidification0600-10001600_____________Timo(snnnd1000020002500..1250() 02 D (0 6 0) I—Solid: Tmwia=600dogCDashed:Tm=ld650degC-Castk,goeM.2-CusrigOD3-MoldInner.urt.c.4-MoldouterOixlaoe2’1250600Modelresultsfor 2-dimensional cylindricalcasting-Effect ofmetalinitial temperatureoncoolIngandsolidification1000C-) 02060010001600TimeisecondslSolid: Tm.=1400degCDashed:Tm_=1350dogCI.Cadncentr2-C..SngOO3-Moldtoterenofac.4-Moldereurfac.20002500600260050010001600Tim(ttinnr4e200)250089Figure6.6Figure6.8Figure6.7Figure6.9Modelresultsfor2-dimensionalcylindricalcastingEffectofmouldemissivityoncoolingandsolidification1250C) I00000 0)700E 00600Solid: £mOUd=0.60Dashed:£moold=0.651-Cas5rrgcac6•2-CaidnOD3—MoldIncel50r15014.Moldouter.wfaoeModelresultsfor2-dimensionalcylindricalcastingEffect ofmetalemissivityoncoolingandsolidification1250C) e.1000 I:2Solid: £m.a=0.35Dashed:m.(iI=0.30I—Cas5ncqn6.2-CasSogOD3—Moldlrwrswtac.4-MoldOutersurface50020002500Modelresultsfor2-dimensionalcylindricalcastingEffectofambienttemperatureoncoolingandsolidification205001Q00160020002600rr....,__..J_.’.1250)10000.750E 0)Solid: Tb=30degCDashed:T=150de9CI-CasSn5OacS•2.ae5n5OD3-MoMterle.wfsce4—MoldoutersurfaceModelresultsfor2-dimensionalcylindricalcastingEffectofmouldthicknessoncoolingendsolidification600-12600 I000 1:060010001600ri.,,Solid: Thlckld=0.330”Dashed:Thick,,d=0.300”1-CasSogc5r002Cae5rrgODS-Moldmacwrface4-Moldouferlurface200025000600iSo1500Time (seconds)2000250090Figure6.10Figure6.12-.1260C) S750Modelresultsfor2-dimensional cylindrical castingEffectofInterfacecontactheat transfer coefficientConstantinitial contactarea060010001600Tln,Solid: hgp=1400- 2.OtDashed:=1400-4.OtDot-Dash:h6=14000-1.33tI..0OthfllN2—CasOng003-Moldnnao,urtao.4-Moldtei1sce,IIIIOWIflJOFigure6.11Modelresultsfor2-dimensional cylindrical castingEffectoffiberpapershieldingoncoolingandsolidificationSolid:Nofiberpapershielding()1260Dashed:Withfiberpapershieldingg’I—nOog.2.Ca.fngOO10003—Moldinssijte4-Udjlr1ac.: 25005061666160020002500Time(seconds)Figure6.13Modelresultsfor2-dimensionalcylindricalcastingEffectofinterfacecontactheattransfercoefficientConstantgapopeningtimeSolid:h9,=1400-2.OtDashed:h5=5000-7.14t5.12601O00 760E 0) I—I.Cas6ngo.nO.2.CasSngOD3—Moldnn&a4XfaC.4-Moldoutal surface60026005001000150020002600Tim(,rvkI20002500Solid:L0,=272k.J/kgDashed:L=258kJikgI—Cas6ngcanS.2-Ca.Sng003—Moldhm.wfac.4-Moldou,urfaoeModelresultsfor2-dimensional cylindricalcastingEffectofmetallatentofsolidificationoncoolingandsolidification-1260‘1000D (15 Th0I-6000600100016002000Tim(rldI2600912DCylindricalCastingModel1400-Effectofcircumferentialpositionrelativetofibrepapershielding1200-1000C) c,) 0) -800a) CO 0)600E 0) I.400-N882(depth0.030”,orientation0dog)N1415(depth0.135”,orientation90dog)200-—————N866(depth0.150w,orientation0dog)———N1405(depth0.285”,orientation90dog)0II050010001500200025003000Time(seconds)•Figure6.142DCylindricalmodel-circumferentialeffectoffibrepapershieldingontemperaturesatvariouslocations92Figure 6.151250Trial Run CSComparison of model results with experimental dataMod K.Me(aI c.nt.o4rO.135O175750IE0500 1000 1500 2000 2500 3000Time (seconds)Figure 6.1693Figure 6.17Figure 6.18Trial Run C7Comparison of experimental data with modd results1000NiN2N5N9N13N20N22CasIn9 ceitecCangudaceIAo depth O.030Mo4d depth O.090Uo4d depth O.145o4d depth O.255Uo4d depth O.2855000500 1000 1500 2000 2500 3000Time (seconds)Trial Run C8Comparison of experimental data with model resultsIdode suIts:CastingoenterCastingwiiaceolddep(h0030Motddepth O.28500)10007505002500 500 1000 1500 2000 2500Time (seconds)300094Figure 6.19Figure 6.201250Trial Run C9Comparison of model results with experimental data010000E750Model raeuff:Ca*mgoenaeCWkwdac.O.030O.135023550025000 500 1000 1500 2000 2500 3000lime (seconds)Trial Run GbComparison of model results with experimental data00.750Model resulte:CastkigcenterCkigwiiaceO.O3OO.135O.23550025000 500 1000 1500 2000 2500 3000lime (seconds)95Face C2D Fin Model:Figure 6.21aNodes indicated are referenced in Sensitivity AnalysisPosition aPosition bPosition cPosition dFace BFigure 6.21b2D Fin model with fiberfrax shielding96Sensitivity analysisFigure 6.22variation of temperaturewithradial position: Face AFigure 6.23Sensitivity analysis - vañationof temperature with radial position: Face B200Figure 6.24Sensitivity analysis - variation of temperature with radial position: Face CYwiation in Tonpeca5z. with kadiat Poai9onFinned Casitng- Face A10008008007008005004000SIN969 PonitionaN025 PositionbN881 PositioncN837 Positiond3001000 500 1000 1500 2000 2500 3000Thneonconds)N566 PositionaN614 Position bN660 PositioncN702 PositiondVwiaitoninTenpecatur.with Radisi Position1000 Finned Cacting - Face B900800700600S500200tOO00 500 1000 1500 2000 2500 3000lime (seconds)Vanation in Temperature with Radsi PosilionFinned Casting - Face C10009008007006005004003000IN672 psN713 psajonbN757 PositioncN805 d1000 500 1000 1500 2000 2500 300097Figure6.25Figure6.26Figure6.27Figure6.28DifferenceinTemperaturebetween90degreeFaceand135degreeFace1000atthesameradiallocation(positiona)900°°I\.N969FaceAN566FaceBjN672FaceC800E300201050010001000200025003000Thrn(oni4iDifferenceinTemparaturebetween90degreeFaceand135degreeFaceatthesameradiallocation(positionb)N925FaceAN614FaceB--N713FaceCascoE a400300200.l....,....I....,....1050050009500200025003000Time(secondsDifferenceinTemperaturebetween90degreeFaceand135degreeFace1000atthesameradiallocation(positionc)900&000f\N881FaceA-o1 I\N60FaceBV‘-N757FaceC500400--300.1..,-..,.1.,,.1050010005500200025003000TImA(nroHo\DifferenceinTemperaturebetween90degreeFaceand135degreeFace1000atthesameradiallocation(positiond)000oI\N837FaceA°°f\N702FaceB600N805FaceCE a)400300200I050010001500200020003000Timn(n4-nntk98Sensitivity analysis (shielded)Figure 6.29variation of temperature with radial position: Face AFigure 6.30Sensitivity analysis (shielded)- variation of temperature with radial position: Face BFigure 6.31Sensitivity analysis (shielded)- variation of temperature with radial position: Face CVwiation in T.nasture with Radisi Posiiton1000 Finned Casting - Face AFlbr*ax shielded model8006300€0050oN969 Position.N825 Poailionbp4881 PoaltioncN837 Positiond0 500 1000 1500 2000 2500 3000Thnes.conds)Vanadon in T.mpatxl with RatiI PositionRnd Casting - Face BFlitrefrax shielded model100090080010060050oN566 Position.N614 Position bN660 PositioncN702 Position d20010000 500 1000 1500 2000 2500 30001111w (seconds)VadaSon in Tenipenature with Radial PosihonFinned Casting-Face CFlbcefrax shielded model1000900800700600500k0N672N713 PositionbN757 Position cN805 Positiond0 500 1000 1500 2000 2500 3000Tinw(seconds99Figure6.32Figure6.34Figure6.33Figure6.35“5’S.”DifferenceinTemperaturebetween90degreeFaceand135degreeFace1000atthesameradial location(positiona)Shieldedmodal9000 600a’ Q500E I—400300200N969FaceAN566FaceBN672FaceC050010001000Time(seconds200025003000Difference inTemperaturebetween90degreeFaceand135degreeFace1000atthesameradiallocation(positionb)Shieldedmodel(\‘N925FaceA?00N--N614FaceBS.N713FaceC200V.580010041800200028002000TimeCsecondsDifferenceinTemperaturebetween90degreeFaceand135degreeFace1000atthesameradiallocation(positionc)VShieldedmodel800‘\,‘5’N881FaceA?00j5‘NN660FaceBN757FaceC---------F-aoo 200V200I..050010001500200025003000TIme(seconds)DifferenceinTemperaturebetween90degreeFaceand135degreeFace1000atthesameradiallocation(positiond)800.,Shieldedmodel0800.’.rr\.N837FaceA‘°°1\“sN702FaceB8005N805FaceC800E400300200.j....I..1050010001004200028602000Time(sends100Figure 6.36Variation in temperature with radial location on Face A: shielded vs. unshielded modelFigure 6.37Noshielding--------ShieidingVariation in temperature with radial position1000 Face A: Shielded vs. Unshielded model900800o 700600a 500C.E!400300 -200-1000 500 1000 1500 2000 2500 3000lime (seconds)Trial Run FlComparison of model results wfth experimental dataN1980N1977N1933N1930(0.055(0.110(O.22O(O.27514001200J1000Z 800a6004002000b: 0.100 mid-height (ps)C: 0.090 bottom (s)d: 0.250w top (ns)e: O.280 mid-height (ps)500 1000 1500 2000 2500 3000Time (seconds)101Figure 6.38Figure 6.39Trial Run F4Comparison of model resilts with experimental dataN904N9040.050” (shielded)0.060” (unshielded)14001200410006004002000C: 0.075” topd: 0.065” bottom (s)f: 0.075” bottom (s)500 1000 1500 2000Time (seconds)2500I. I3000Trial Run F5N904N904Comparison of model results with experimental dataFace A. 0.060” (unsie4ded)Face A. 0.050” (shielded)14001200010001800600400a: 0.075” topb: 0.080” topc: 0.080” mid-height (ps)d: 0.075” mid-height (ps)e: 0.070 bottom (s)f: 0.070” bottom (s)200a,o,e0500 1000 1500 2000 2500 3000Time (seconds)102Trial Run F6Figure 6.40Figure 6.4114001200Comparison of model results with experimental dataN639 FaceBO.O5O— — —— N904 Face A, O050 (siaIded)b01000E800e,t600400200a: 0.055w toplx 0.060w mid-height (s)e: 0.060 mid-heightt 0.065 bottoma,b0500 1000 1500 2000 2500Time (seconds)3000Trial Run F7Comparison of model results with experimental dataN736N904Face C. O.050Face A, 0.050 (shi&d€d)d,e1400120010000C,.800a,2000a: 0.055 topb: 0.055 mid-height(s)c: 0.055 bottom (s)d: 0.045 tope: 0.050 mid-height‘Cd,e500 1000 1500 2000 2500 3000Time (seconds)103Figure 6.421400Trial Run F8Comparison of model results with experimental data1200 N639 FaeeB,O.050”N736 FaceCO.050C)10000DE0I—d,e,t800600a,b,o4002000c,d500 1000 1500 2000 2500 3000Time (seconds)Trial Run F9Comparison of model results with experimental dataN94a 0.250”N906 0.150”N865 0050”F9140012001000800E6004002000a: 0.235”C: 0.070”d: 0.075”aJe: 0.150”f: 0.240”fe500 1000 1500Time (seconds)Figure 6.432500 3000104Chapter 7 Industrial Application of Casting Model to Testbar7.1 BackgroundHaving completed the sensitivity analysis and achieved confidence in the predictivecapability of the model, an analysis was undertaken with a view to providing qualitativeguidance in optimizing the design of a specific casting. The goal was to reduce thepropensity to form macrovoids while seeking to optimize microstructure with respect tomorphology and microporosity. A casting of a tensile test bar cluster (Figures 7.1 and 7.9)was chosen for the analysis since this particular configuration had been used previously in aTaguchi test series conducted by Deloro Stellite, in order to evaluate the influence of variouscasting parameters on casting quality [36- 381. The compiled results provide an excellentdata base for further verification of the predictive capability of the model. The parametersincluded in the Taguchi study were: mould preheat temperature, mould cool time, mouldthickness, use of mould wrap, type of mould primary coat, metal molten time, and pourtemperature. In the model analysis, the effects of mould preheat and mould cool time cannotbe addressed separately. Thus they were combined into the effect of the mould initialtemperature. Mould shell thickness, metal pour temperature and the presence of mould wrapwere modelled individually. The effects of mould face coat type and metal molten time arenot readily assessed by the current model, as their influence is manifested through the grainnucleation phenomenon.7.2 Results of analysis - 2-dimensional test bar modelThe 2-dimensional finite element model used to analyze the test bar is shown inFigure 7.1. Figures 7.2 through 7.8 show example contour plots of the 2-dimensional105analysis for the varying casting conditions summarized in Table 7.1. In the figures, the “B”isotherm represents the liquidus while the “J” isotherm represents the solidus. Thus, in thepiots the contour lines encompass the solidification temperature range only.Figure 7.2 shows the temperature distribution for the base case described in Table 7.1,at 800 seconds after pouring. The figure shows the distribution at the time when the solidus(“J’ isotherm) begins to pass through the test bar gauge length. A higher initial mouldtemperature (Figure 7.3), lower initial metal temperature (Figure 7.4) and increased mouldthickness (Figure 7.5) all show a solidification progression similar to that of the base case. Asolid skin forms rapidly on the feeder cup, and the test bar end buttons solidify on both theupper and lower end of the bar before the center section of the bar.In Figure 7.6, insulating wrap has been added to the upper half of the mould.Whereas a solid skin still forms on the feeder cup, the progression of solidification in the testbar itself changes significantly, becoming more directional from the bottom toward the top.Solidification of the top end button is delayed. In Figure 7.7, insulation has been added tothe top of the feeder, as well as to the upper half of the mould. The most significant effect isthat the formation of the solid skin on the feeder cup is inhibited. Although the “J” (solidus)isotherm has reached the edges of the cup, the center is still in the mushy range.Solidification of the entire casting begins at the bottom and progresses toward the top. Thetime scale for solidification is significantly increased, from 800 seconds to 2000 seconds.Figure 7.8 shows the results of the final 2-dimensional test bar analysis. The gatingsystem has been redesigned, with the feeding to the center section removed. No mould wrapis used. Skin formation is still observed on the top of the feeder cup, but the progression ofsolidification is changed in the testbar; the center portion solidifies well before the endbuttons (“J” isotherm passes through the center section before it passes through the ends).1067.3 Results of analysis - 3-dimensional test bar modelFinally, a 3-dimensional, farfield radiation cooling model was analyzed. Althoughthe 2-dimensional model could be used to assess the relative effects of casting parametersqualitatively, the 2-dimensional assumption may be expected to give results in which thesolidification time is increased significantly over a 3-dimensional model. The metal surfacearea-to-volume ratio of the 2-dimensional case is much lower, and the system loses heat fromone face adjacent to the testbar only. In the 3-dimensional case, the surface area-to-volumeratio is much higher, and heat is extracted from the mould surrounding the testbar on threesides.The 3-dimensional geometry is shown in Figure 7.9. A quarter section only has beenmodelled to reduce the problem size. Figures 7.10 and 7.11 show the temperaturedistribution in the cross-sectional plane of the testbar at various times. The time scale forsolidification of the testbar is reduced by an order of magnitude (-400 seconds vs. —1000seconds). The testbar gauge length solidifies before the end buttons, in contrast to the 2-dimensional model predictions.These differences illustrate an important aspect of solidification computer modelling.Although a 2-dimensional model requires significantly less time for model input and forcomputational time, the results may be only qualitative at best, and can be misleading. Theassumption of 2-dimensional heat flow must be examined carefully and the results interpretedin the light of this assumption. A 3-dimensional model with the model size reduced wouldperhaps yield more quantitative results, within the limits of the reduced model size. Forexample, the test bar center solidifies so rapidly that the temperature in the pour cup, at somedistance away, has little effect on the center of the bar. The downsprue has not cooledsignificantly below the liquidus temperature before the test bar center section has solidified.The model size could be reduced, therefore, by removing the pour cup, while maintaining the107nodes at the pour cup neck at the liquidus temperature. Only one testbar need be modelled,using a farfield radiation condition on the mould exterior surface; a symmetry boundarycondition (zero heat flux) is applied to the center of the downsprue. This model could beused for sensitivity analysis of the casting parameters on the test bar gauge length. Thelimiting assumption would be the constant temperature assumption at the pour cup neck.Thus, the model would be valid for short times only, but could result in more quantitativeresults than those obtained using the 2-dimensional model.7.4 Theoretical considerations in casting quality analysisThe quality criteria examined with the model that are directly influenced by thethermal history and which were also examined in the Taguchi experiments are:i) presence of cavity porosityii) presence of microporosityiii) secondary dendrite arm spacing (SDAS)7.4.1 Cavity porosityPorosity in èastings can arise due to several mechanisms. Cavity porosity is due tothe premature freezing off of feeding channels preventing feeding to solidifying areas. Thelast volume of liquid to solidify will not have a supply of liquid to feed the entire volume.The resulting void is large (on the scale of the dimensions of the casting) and often irregularin shape.1087.4.2 MicroporosityReference [36] reported that the test bar casting was susceptible to severemicroporosity in the area of the feeding gate adjacent to the test length, and that“Microporosity morphology was distinct and elongated; characteristic ofinterdendritic microshrinkage.”Microporosity is formed when the feeding to the dendrite root is inhibited by thesolidifying metal. To feed the solidification shrinkage flow, a pressure drop must existacross the dendrite array between the melt and the dendrite root. This pressure drop isdescribed by [39]:AT”p — p= Al’ freezmg 7.1nmR2whereAVfreezing is the volume change associated with freezing1 is the channel (dendrite) lengtht is a tortuosity factor, > 1, accounting for the fact that the flow channels are notstraightmtR2 is the total channel area for n channels of radius KThe pressure at the dendrite root is7.2The volume change on freezing causes the local pressure at the dendrite root to decreasebelow the vapour pressure of the melt constituents. When P1 falls below a critical pressure109microvoids will nucleate. Relation (7.1) above states that microporosity is a function ofthe pressure of the melt, the volume contraction on freezing, the square of the length of themushy zone and the tortuosity.The main effect of the temperature distribution on microporosity is through the valueof the channel length, 12. This length is a function of the solidification range, /Tf and thethermal gradient G in the 2-phase region:1=AT/G 7.3Regions of low thermal gradient during solidification will be subject to more microporositythan those of a higher thermal gradient. In the Taguchi experiments, the parameters found tohave the most significant effect on porosity were mould preheat and the use of a mould wrap.A lower mould preheat (1500 deg C vs. 2000 deg C) resulted in less porosity, as did pouringand cooling without a mould wrap around the mould. These conditions both lead to a highergradient in the casting during solidification, and a faster cooling rate.7.4.3 Secondary dendrite arm spacingSecondary dendrite arm spacing (SDAS) is a function of local solidification time[40]:SDAScct3 7•5where local solidification time is defined by the relationt 7.6Iv110where 1 is the length of the dendrite array and V is the liquidus isotherm velocity. Thus,faster cooling rates will result in shorter local solidification times and smaller secondarydendrite arm spacing. As well, higher thermal gradients in the dendrite array result in ashorter array length 1, shorter and smaller SDAS. In the Taguchi experiments, fmerSDAS was achieved using a lower mould preheat, longer mould cool time and no mouldwrap. Each of these conditions contributes to a higher thermal gradient and more rapidcooling, which is consistent with the theoretical considerations.7.5 Correlation of model predictions with casting qualityIn the following comparison, the results of the 2-dimensional model sensitivityanalysis are applied qualitatively, in terms of the relative effect of the parameters on thecasting thermal history. A 3-dimensional model, while reducing the time scale forsolidification and altering the progression of solidification, can be expected to show similartrends with variation in the parameters.Cavity porosity in the testbar castings can be expected in the model conditions ofFigures 7.2 to 7.6, as the feeding from the pour cup is inhibited by formation of the solidskin. The cavities will be located in the last volume of metal to solidify. The casting volumein the mushy zone at the beginning of solidification of the gauge length of the testbar is quitelarge, requiring a large volume of metal to feed the shrinkage, and also resulting in a largevalue for the flow path 1 to the dendrite roots in the gauge length (equation 7.1).Interdendritic microporosity may thus be expected in the gauge length. Varying the mouldinitial temperature (Figure 7.3), metal initial temperature (Figure 7.4) and mould thickness(Figure 7.5) does not affect the solidification pattern, and affects the overall solidificationtime only slightly. Adding external mould wrap to the casting sides (Figure 7.6) results in amore directional solidification pattern in the test bar gauge length. Feeding to the gauge111length can occur not only through the center gate, but also through the top end of the test baritself. The feeder cup solidifies prematurely, however, well before the test bar gauge length,resulting in a large volume of metal in the mushy zone without adequate feeding, and thussome microporosity may be present. In Figure 7.7, insulation has been added to the top ofthe feeder cup, as well as to the top outer sides. This retards the formation of a solid skin onthe feeder top, reducing the incidence of cavity porosity. Some shrinkage microporosity maystill be expected, due to the large volume in the mushy zone at the onset of solidification ofthe gauge length. Solidification will be more directional, with enhanced feeding through thetestbar top half, due to the better feeding from the feeder cup.Solidification theory predicts that SDAS will be a function of local solidification time(equation 7.5). A review of the solidification sequence reveals that the superheat is lost fromthe melt within seconds after pour under the conditions examined in Table 7.1. A measure ofthe testbar solidification time, , for the cases under consideration is obtained from the timeat which the solidus (J isotherm) has passed through the gauge length of the test bar. Thisinformation, from the model data, is summarized in Table 7.11. With reference to Table 7.11,conditions for a finer SDAS include low mould temperature, low pour temperature, thinnermould, no mould wrap, and no center feeding. These conclusions agree with the results ofthe Taguchi experiments. The fourth column of Table 7.11 summarizes the results of theTaguchi experiments with respect to the variation of each parameter [36- 38].As discussed above, the parameter having the largest effect on porosity and SDAS isthe presence of a mould wrap on the casting during solidification. Whereas the mould wrapslightly improves conditions for reduced microporosity, the significant increase in overallsolidification time leads to a coarser SDAS, which has a detrimental effect on the mechanicalproperties of the casting [39]. In order to reduce microporosity without increasing overallsolidification time, the gating system was redesigned to remove the feeding gate to the testbar gauge length. Figure 7.8 shows the casting with the feeding system redesigned toeliminate microporosity and reduce solidification time. The gauge length section solidifies112first, while a large volume of metal at a higher temperature is available to feed the test lengthfrom the top and bottom. Although cavity and microporosity may be expected in thedownsprue, which solidifies slowly, the testbar itself will be sound. Overall solidificationtime is reduced by 60%; a finer SDAS may be expected, with improved mechanicalproperties as a result.The 3-dimensional testbar with the center feeding shows a solidification progressionsimilar to the 2-dimensional model with the center feeding removed. The solidification timeis much shorter, and thermal gradients in the bar center are much steeper.The results of the above analysis lead to an important conclusion in the practicalapplication of the model to casting design. The process control parameters of mouldtemperature, metal temperature and mould thickness have a relatively small influence on thesolidification progression. Referring to Table 7.11, varying the mould temperature, metaltemperature or mould thickness results in a change in total solidification time of 10 - 15%.The addition of mould wrap, although it has a much more pronounced effect in terms ofprocess control, invariably results in an increase in total solidification time. In the casesanalyzed, adding mould wrap insulation increases total solidification time by 130 - 150%.Adding mould wrap to reduce microporosity will thus lead to reduced casting quality interms of material properties. The analysis of the redesigned feeding system indicates that asound casting can be obtained without a mould wrap, and greatly reduced total solidificationtime. Removing the center feeding reduces solidification time by 60%. The above pointsillustrate the importance of optimizing the casting through correct design of the mould andfeeding system, rather than through the use of the various parameters at pour time.113Table 7.1Model parameters used in testbar analysisTable 7.11Effect of Model Casting Parameters on Local Solidification Time for Testbar CastingsParameter tf(seconds) %At1 SDAS effect *Base 1000high Tmould 1100 +10 Coarserlow Tmetal 900 -10 FinerThick mould 1150 +15 CoarserMould wrap (sides) 2300 +130 CoarserMould wrap (sides 2500 +150 Coarser&_top)No center feeding 400 -60 FinerParameter Base value Varied valuekmould (W/m-deg C) 1.0 -mould (cm2/sec) 0.0042 -Lmetal (kJ/kg) 272 -Mould initial temperature 600 800Metal initial temperature 1400 1360Mould thickness (inches) 0.300 0.330* Refs [36)- [38]114Figure 7.12-dimensional testbar fmite element meshFigure 7.3High initial mould temperaturetime 900 secondsFigure 7.2Base case, time 800 secondsFigure 7.4Low initial metal temperaturetime 700 secondsDeg C1500=A1340=131330=C1320 = D1310=E1300=F1290=G1280=H1270=11260 = JDeg C1500=A1340 = B1330=C1320 = D1310=E1300=F1290 = G1280 = H1270=11260=JDeg C1500=A1340 = B1330=C1320=D1310=E1300=F1290=G1280 = H1270 =1260 = J115Figure 7.5Thick mouldtime 1000 secondsFigure 7.6Mould wrap on upper halftime 1300 secondsFigure 7.7Mould wrap sides and toptime 2000 secondsFigure 7.8Feeding to test section removedtime 400 secondsDeg C1500=A1340 = 131330=C1320 = D1310=E1300 = F1290 = G1280 = H1270 =1260 = JDeg C1500=A1340=B1330=C1320=D1310=E1300=F1290 = G1280=H1270=11260 = JDeg C1500=A1340 = B1330=C1320 = D1310=E1300 = F1290=G1280 =111270=11260 = JDeg C1S0O=A1340=B1330 =C1320 = D1310=E1300=F1290 = G1280=H1270 = I1260 =116Figure 7.93-dimensional testbar modela: casting outer view; b: casting inner view; c: mould outer view; d: mould inner view117118Figure7.103-dimensionaltestbarmodelresults;time25seconds1193-dimensional testbarmodel results;time50secondsChapter 8 Summary and Recommendations8.1 Summary and conclusionsThis work has studied the heat transfer processes in vacuum investment casting ofnickel-based superalloy 1N718. Boundary conditions were developed for a finite elementbased solidification model for radiation heat transfer at the mould exterior and the mould-metal interface. The investment casting process cannot be modelled adequately with a simplefarfield radiation condition, as there is significant self-irradiation due to the complex mouldgeometries. General 2- and 3-dimensional viewfactor codes were developed for use in theradiation heat transfer network. The Monte Carlo ray tracing approach used in this codeallowed complete generality of geometry, and was found to have an error of less than 5% at a95% confidence level when as few as 10000 rays were traced for each surface in theenclosure.The boundary condition algorithms were verified against a series solution from theliterature for simple geometries.A simple model to account for initial mould-metal contact conduction was developed,based on a time-dependent contact area function.Experimental data to verif’ the model and obtain data on the mould thermophysicalproperties and the contact area function were conducted in collaboration with Deloro Stelliteof Belleville, Ontario. A total of thirteen data sets in two configurations was obtained, givingthermal histories for the mould and metal at various locations.The casting configurations were modelled using the developed code. Sensitivityanalyses determined that the critical parameters were the mould conductivity, contact areafunction, metal initial temperature, mould thickness and the radiation environment of thecastings during cooling and solidification. In particular, the presence of an insulating120fibrepaper used to shield the thermocouple manifold significantly influenced the mouldsurface temperature profile.Thermophysical properties of the mould were obtained by fitting the model results tothe experimental data. Constant values of the properties were used throughout. The mouldthermal conductivity which resulted in the best fit to the data was in the range of 0.9 - 1.1W/m-deg C for the cylindrical castings, and 0.8 - 0.9 W/m-deg C for the finned castings.This slightly lower value for the finned castings was possibly due to the shorter time overwhich the mould remained at higher temperatures. As the mould conductivity is expected tohave a pore radiation effective conductivity contribution at higher temperature (>500 deg C),the average thermal conductivity would be higher for a mould remaining above 500 deg C fora longer time. The mould thermal diffusivity ranged from 0.0026- 0.0046 cm2/sec, inagreement with data from the literature.A linear function decreasing from an initial value h0 to zero at some time wasused for the interface contact conduction heat transfer coefficient. An initial value h0 of1400 W/m2-deg C was found to result in the best fit overall to the data. This corresponded toan initial contact area fraction of 0.28, assuming a gap constant (kmould/Ax) of 5000 W/m2-deg C. The time over which the contact reduced to zero was 700 seconds for the cylindricalmoulds, and 200 seconds for the finned moulds, corresponding with the overall solidificationtime of each casting.The results of a model analysis of a tensile test bar are in qualitative agreement withthose from a set of Taguchi experiments performed at Deloro Stellite Inc. A low mouldinitial temperature, low metal initial temperature, thinner mould, and absence of mould wrapresult in a shorter overall solidification time, and therefore in a finer secondary dendrite armspacing. Feeding to the testbar center section will result in possible microporosity in the testgauge length. A redesigned feeding system, with no feeding to the center section,significantly alters the order of solidification, so that the gauge length solidifies first. Thiswill reduce the incidence of microporosity in the gauge length. A preliminary 3-dimensional121model demonstrates the need for a 3-dimensional modelling capability; the totalsolidification time of the test bar is reduced by an order of magnitude over the 2-dimensionalmodel, and the direction of solidification is altered significantly. It is concluded that,whereas results may be obtained much more rapidly using a 2-dimensional model, both froma computational aspect, and from the time required for geometry and mesh preparation, theresults may be highly misleading. The assumption of 2-dimensional heat flow must beclosely examined, and the results obtained must be interpreted accordingly.From the testbar analysis it was concluded that the use of metal initial temperature,mould initial temperature and mould thickness as process control parameters will have only aslight influence on microporosity and secondary dendrite arm spacing. Use of a mould wrapfor process control may result in slightly less microporosity, but will increase thesolidification time, with a resulting decrease in casting material properties. These points andthe analysis of the redesigned casting model emphasize the importance of castingoptimization in the initial design of the feeding system, rather than through process control inthe pouring stage.The experimental data yielded information on the cooling rate of a mouldimmediately before pouring. The average cooling rate for the cylindrical moulds wascalculated to be between 130- 160 deg C/minute. This very high rate indicates that incasting, the metal must be poured as quickly as possible if better fill characteristics due to thehigher mould preheat are to be obtained. The benefits of a preheat higher by 200 deg C willbe lost by a delay in pouring of approximately 1-1/2 minutes.1228.2 Recommendations for further workThe parameters having the most significant effect on model output are the mouldthermal conductivity and the interface contact heat transfer formulation. These parametersare as yet not well-characterized.The temperature dependence of the mould thermal conductivity is an area requiringfurther work. Due to the complex shell structure, a highly non-linear temperature response isexpected, with void radiation becoming significant at higher temperatures. Theoretical workon conduction in porous media and 2-phase continuous media can serve as a basis fordeveloping a model for the multi-component, porous shell.The contact conduction at the mould-metal interface is an area of active research.Part geometry will influence the contraction of metal away from the mould, in that the mouldmay restrict shrinkage of the casting in certain locations. A temperature-based approachincorporating solidification shrinkage effects and thermal contraction due to cooling of thesolid will be more generally applicable than the time-based, linear function applied in thisstudy. Such an approach, while adding significant computational complexity, will alsoprovide a more quantitative approach to the predicting of microporosity in castings.The castings produced for this work can be analyzed and microstructuraIly quantified.Such data would provide a base for developing heat flow vs. microstructure correlations for1N7 18 investment castings.123Bibliography[1] Cockcroft, S.L.; Thermal Stress Analysis of Fused-Cast Monofrax-S Refractories; Ph.D.Thesis, University of British Columbia, 1990[2] Investment Casting Handbook 1980 (Investment Casting Institute, 1979)[3] Hamar, R; “Numerical Simulation in Precision Casting”, mt. I Num. Meth. Eng. vol.24, pp219-229 (1978)[4] Duffy, M.O., Morris, P.M. and Mador, R.J.; “Casting Solidification Analysis andExperimental Verification” in Finite Elements in Analysis and Design vol. 4pp. 1-7(1988)[5] Huang, H. and Berry, J.T.; “Solidification Heat Transfer in Aluminum InvestmentCastings”, ASME HTD vol. 104 pp. 93-99, 1988[6] Lui, Jia-Chin, Lee, Tsang-Sheao and Huang, Weng-Sing; “Computer Model ofUnidirectional Solidification of Single Crystals of High-Temperature Alloys”,Materials Science and Technology, vol. 7, pp.954-964, 1991[7] Desbiolles, J.L., ImVinkelried, T., Rappaz, M., Rossman, S. and Thevoz, Ph.; “TheSimulation of Single Crystal Turbine Blade Solidification”, in Proceedings, ASMInternational Conference, Synthesis, Processing and Modelling of AdvancedMaterials, September 1991[8] Ulam, S., Richtmeyer, R.D. and von Neumann, J.; “Statistical Methods in NeutronDiffusion”, LAMS-551, Los Alamos National Laboratory, 1947[9] Howell, J.R. and Perlmutter, M.; “Monte Carlo Solution of Thermal Transfer throughRadiant Media between Grey Walls” I Heat Transfer, vol. 96 pp.1 16-122, 1964[10] Perimutter, M. and Howell, J.R.; “Radiant Transfer through a Grey Gas betweenConcentric Cylinders using Monte Carlo” I Heat Transfer, vol. 86, pp.169-179,1964[11] Corlett, R.C.; “Direct Monte Carlo Calculation of Radiative Heat Transfer in Vacuum”I Heat Transfer, vol.88, pp.3’76-382, 1966[12] Toor, J.S. and Viskanta, R.; A Numerical Experiment of Radiant Heat Interchange bythe Monte Carlo Method’t mt. I Heat Mass Transfer, vol. 11, pp883-897, 1968[13] Modest, M.F.; Three-Dimensional Radiative Exchange Factors for Non-Grey, Non-Diffuse Surfaces” Nurn. Heat Transfer, vol. 1 pp.403-416, 1978124[14] Maitby, J.D. and Burns, P.J.; Performance, Accuracy and Convergence in a 3-Dimensional Monte Carlo Radiative Heat Transfer Simulation” Num. Heat TransferB, vol. 19, pp.19l-209 1991[15] Hoff, S.J. and Janni, K.A.; Transactions ASME, vol. 32, pp.1023-1028, 1989[16] Ikushima, T., Suzuki, K., and Yoshida, H.; I Atom Energy Soc. Japan, vol. 30, pp.548-556, 1988[17] Plehiers, P.M. and Froment, G.F.; Chem. Eng. Technol., vol. 12, pp.20-26, 1989[18] Ho, K. and Pehlke, R.D.; “Mechanisms of Heat Transfer at a Mold-Metal Interface”,AFS Transactions vol. 92, pp.567-598, 1984[19] Nishida, Y., Droste, W. and Engler, S.; “The Air Gap Formation Process at theCasting-Mold Interface and the Heat Transfer Mechanism through the Gap”, Met.TransB,vol. 17B,pp.833-844, 1986[20] Huang, H., Sun, V.K., Hill, J.L. and Berry, J.T.; “Some Important Aspects of ThermalContact and Phase Change in Solidification Modelling”, in Heat and Mass Transfer inSolidification Modelling, ASME-HTD vol. 175, pp.53-60, 1991[21] Huang, H.; An Investigation of Thermal Contact, Phase Change and ComputationalEfficiency in Modelling Shaped Casting Solidification, Ph.D. Thesis, University ofAlabama, 1992[22] Holman, J.P.; Heat Transfer (McGraw-Hill Book Company, New York, 1981)[23] Siegel, R. and Howell, J.R.; Thermal Radiation Heat Transfer (McGraw-Hill BookCompany, New York, 1972)[24] Huebner, K.H. and Thornton, E.A.; The Finite Element Method for Engineers (JohnWiley and Sons, New York, 1982)[25] Jaeger, J.C.; “Conduction of Heat in a Solid with a Power Law of Heat Transfer at itsSurface”, Proceedings, Cambridge Philosophical Society. vol. 26, pp.634-641, 1950[26] Heames, K. and Geiger, G.H.; “I. The Thermal Conductivity of Shell InvestmentMaterials, II. Heat Transfer in Investment Shell Molds”, Proceedings, 26th AnnualMeeting of the Investment Casting Institute, 1973[27] Huang, H., Berry, J.T., Zheng, X.Z. and Piwonka, T.S.; “Thermal Conductivity ofInvestment Casting Ceramics”, Proceedings, 37th Annual Technical Meeting,Investment Casting Institute, 1989[28] Atterton, D.V.; “Apparent Thermal Conductivies of Moulding Materials at HighTemperatures”, I Iron Steel Inst. vol. 174 pp 201-211, 1953125[29] Godbee, H.W. and Ziegler, W.T.; “Thermal Conductivities of MgO, A1203and Zr02Powders to 850 deg C. II. Theoretical”, J Appi. Phys. vol. 37 no. 1 pp 56-65, 1966[30] Kingery, W.D. and McQuarrie, M.C.; “Thermal Conductivity: I, Concepts ofMeasurement and Factors Affecting Thermal Conductivity of Ceramic Materials”, JAmer. Ceram. Soc. vol. 37 no. 2 pp 67-72, 1954[311 Loeb, Al.; “Thermal Conductivity: VII, A Theory of Thermal Conductivity of PorousMaterials”, I Amer. Ceram. Soc. vol. 37 no. 2 pp 96-99, 1954[32] Russell, H.W.; “Principles of Heat Flow in Porous Insulators”, I Amer. Ceram. Soc.vol. 18 pp 1-5, 1935[33] Deloro Stellite Engineering Specification No. 4521A, June 1992, Preparation and Useof Cobalt Aluminate “Blue” Primary Coat[34] Deloro Stellite Engineering Specification No. 4523B, January 1993, Preparation andUse of FascoteTM Back-up Coatings[35] Sairset Material Safety Data Sheet, A.P. Green Refractories Co., Mexico, MissouriUSA May 1988[36] Ronan, K., Vacuum Melting and Casting of Inconel 718: Taguchi Analysis - Tier I,Deloro Stellite Internal Report, Sept. 1992[37] Ronan, K., Vacuum Melting and Casting of Inconel 718: Detailed Report, DeloroStellite Internal Report, Sept. 1992[38] Dawson, R.J. Vacuum Melting and Casting of Nickel Base Superalloys: Final Report,Deloro Stellite Internal Report, May 1993[39] Verhoeven, J.D.; Fundamentals of Physical Metallurgy (John Wiley and Sons, NewYork, 1975)[40] Kurz, W. and Fisher, D.J.; Fundamentals of Solidification (TransTech Publications,Switzerland, 1989)126Appendix AAnalytical solutions for verification of computer codeViewfactor for 2-dimensional, parallel plates of width-to-separation ration H (Ref. [23]):F12=Jl+H2—HViewfactor for 2-dimensional, perpendicular plates having a common edge and length ratioof H (Ref. [23]):F12Viewfactor for 3-dimensional, parallel plates of length-to-width ratio X and length-toseparation ratio Y (Ref. [23]):2 F(1+x2)(1+Y2)1__________F12=— ml +X,J1+Y2tan1 +Y-J1+X2tan —Xtari’X—Ytan1YitXY [ (1+X2+Y2) jViewfactor from plate 1 to plate 2 for 3-dimensional, perpendicular plates having acommon edge, where for plate 1, X = length-to-width ratio, for plate 2, Y = height-to-width ratio, and the width dimension forms a common edge (Ref. [23]):1271 1 i 1 1Wtan —+Htan ——IH2+ W2 tanW H H2+W2F12*1 (1+W2)(1+H2) W2(1+W2+H2)W2H2(1+W2+H2) H2+—lfl (1+W2+H2) (1+W2)(1+H2) (1+W)(1+H2)Jaeger [25] developed a series solution to the 1-dimensional heat conduction equation for asemi-infinite medium where, on the boundary:—k-- HVm FsdxFor m = 4, corresponding to a radiation boundary condition, for short times, the firstterms are:v = v0{i — 2Ti1erfcX +l6T2ierfcX — 189.lT3ierfcX+ 2787TierfcX—..}whereT=(Kt)](v0)/ vX= (1ct)The ifleifcX is the n-pie repeated integral of the error function [25]:ierfc =128

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