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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 TRANSFER IN THE VACUUM INVESTMENT CASTING OF SUPERALLOY 1N718 by BARBARA EVA DOMINIK B.A.Sc., The University of British Columbia, 1986 A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Metals and Materials Engineering) We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December 1993 © Barbara Eva Dominik, 1993  In  presenting this thesis  in  partial  fulfilment of the  requirements for degree at the University of British Columbia, I agree that the Library freely available for reference and study. I further agree that permission copying of this thesis for scholarly purposes may be granted by the department  or  by  his  or  her  representatives.  an advanced shall make it for extensive head of my  It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  4  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  Abstract Heat transfer in vacuum investment casting of a nickel-based superalloy, 1N718, was studied using a finite-element-based solidification heat transfer code. For the external radiation boundary, general 2- and 3-dimensional viewfactor calculation codes based on a ray tracing approach were developed and verified. Heat transfer at the mould-metal interface may occur by contact conduction between the metal and the mould, and by radiation across the interface gap areas. A simple, time-dependent model was developed to simulate the decreasing contact conduction as solidification progresses. Temperature measurements were made on casting moulds in a series of experiments done in collaboration with Deloro Stellite Inc. of Belleville, Ontario. The model was applied to the experimental casting configurations. The model results were most influenced by the value of the mould thermal conductivity, the interface contact function and the radiation environment surrounding the mould. The mould thermal conductivity which resulted in the best fit to the data ranged from 0.9 to 1.1 W/m-deg C for the cylindrical castings and 0.8 0.9 W/m-deg C for the finned castings. The interface contact conduction function decreased from 1400 W/m -deg C at 2 -  time t  0, to a value of 0 at t 700 seconds and t 200 seconds for the cylindrical and finned castings respectively. The model was used to simulate casting conditions for a tensile test bar which had been analyzed experimentally by Deloro Stellite Inc. Although the 2dimensional model used gave results that were in qualitative agreement with the experiments =  in terms of predicting effects on microporosity and secondary dendrite arm spacing, a 3dimensional model altered the solidification pattern and time scale for solidification by an order of magnitude. A 2-dimensional approximation, although requiring less model input time and computational time, may thus be misleading and result in incorrect conclusions being drawn. The model developed in this work provides a strong tool which can be used in conjunction with experiments to develop relationships between heat flow and the micro structural development of investment castings.  11  Table of Contents  Abstract Table of Contents List of Tables List of Figures Nomenclature Acknowledgments CHAPTER 1 Introduction 1.1 Background of the investment casting process 1.2 Steps in the investment casting process CHAPTER 2 Literature review CHAPTER 3 Mathematical formulation of boundary conditions 3.1.1 Farfield radiation 3.1.2 Gap interface radiation 3.1.3 Enclosure radiation 3.1.4 Viewfactor calculation 3.2 Finite element implementation of boundary conditions CHAPTER 4 Verification of computer code 4.1 Viewfactor calculation verification 4.2 Radiation boundary condition verification CHAPTER 5 Experiments 5.1 Experimental procedures 5.2 Discussion of experimental results 5.2.1 Cylindrical castings 5.2.2 Finned castings CHAPTER 6 Sensitivity analysis and analysis of the casting process 6.1 Sensitivity analysis cylindrical configuration 6.1.1 Interface heat transfer assumed contact time 6.1.2 Metal initial temperature 6.1.3 Mould thermal conductivity 6.1.4 Metal latent heat of solidification 6.1.5 Interface contact initial heat transfer coefficient 6.1.6 Fibrepaper shielding 6.1.7 Variables having minor effects 6.1.8 Analysis of heat flow resistances 6.2 Sensitivity analysis finned configuration 6.2.1 Temperature variation with radial position 6.2.2 Temperature variation with differing faces 6.2.3 Effect of fibrepaper shielding 6.3 Comparison of experiments with model results 6.3.1 Cylindrical castings 6.3.2 Finned castings CHAPTER 7 Industrial application of casting model to testbar -  -  111  Page ii iii v vi x xiv 1 2 3 5 18 18 19 23 27 29 39 39 39 46 46 50 51 53 71 71 72 72 73 74 74 74 75 76 79 79 80 80 81 81 83 105  Table of Contents (cont’d)  7.1 Background 7.2 Results of analysis 2-dimensional testbar model 7.3 Results of analysis 3-dimensional testbar model 7.4 Theoretical considerations in casting quality analysis 7.4.1 Cavity porosity 7.4.2 Microporosity 7.4.3 Secondary dendrite arm spacing 7.5 Correlation of model predictions with casting quality CHAPTER 8 Summary and recommendations 8.1 Summary and conclusions 8.2 Recommendations for further work Bibliography Appendix A Analytical solutions for verification of computer code -  -  iv  Page 105 105 107 108 108 109 110 111 120 120 123 124 127  List of Tables  Table 4.1  Table Table Table Table  5.1 5.11 5.111 5.IV  Table 6.1 Table 6.11 Table 6.111 Table 6.IV Table 6.V Table 6.VI Table 7.1 Table 7.11  Mean percent error and standard deviation for comparison of Monte Carlo viewfactor results with analytically calculated values (n=20) Location of thermocouples in cylindrical castings Location of thermocouples in finned castings Vacuum furnace pressures during melting and cooling Cylindrical mould initial cooling rates after removal from preheat furnace Sensitivity analysis parameters Constant material properties Effect of parameter variation on total solidification time Values of parameters used in fitting cylindrical experimental data Constant values used in analysis of finned experimental data Values of parameters used in fitting finned experimental cata Parameters used in testbar model analysis Effect of model casting parameters on local solidification time for testbar castings  V  Page 41  55 56 57 58 85 86 86 87 87 87 114 114  List of Figures  Figure 1.1 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4a Figure 4.4b Figure 4.4c Figure 4.4d Figure 4.4e Figure 4.4f Figure Figure 5. lb Figure 52 Figure 5.3 Figure Figure Figure  5 .4a 5.4b 5.4c  Figure 5.4d Figure 5.5 Figure 5.6 Figure 5.7  Steps in the investment casting process Schematic of interface contact resistance model Nomenclature for radiative exchange between two surfaces Irradiation and radiosity for a grey surface Nomenclature for emission direction of a ray emitted by a surface Heat balance on the boundary of a conducting medium Configurations for viewfactor code verification Comparison of Monte Carlo viewfactor calculation with analytical results Configurations for radiation boundary condition verification of THERMAL code Cooling of a 2D semi-infmite slab radiating to farfield environment at OK Cooling of a 3D semi-infinite slab radiating to farfield environment at OK Cooling of a 2D semi-infinite slab radiating to enclosure at OK Cooling of a 3D semi-infinite slab radiating to enclosure at OK Cooling of a 2D semi-infinite slab radiating across gap interface to solid at OK Cooling of a 3D semi-infinite slab radiating across gap interface to solid at OK Dimension drawing cylindrical test casting Dimension drawing finned test casting Balzers VSGO5O Vacuum Induction Furnace (Deloro Stellite mc) Schematic of thermocouple installation in ceramic mould Typical cylindrical mould instrumented for testing Typical finned mould instrumented for testing Cylindrical mould in furnace after casting, showing manifold shielding Finned mould in furnace after casting, showing manifold shielding Thermocouple location in finned castings (Ref. Table 5.11) Pt-Ptl3Rh equivalent thermocouple circuit Experimental data: casting C5 -  -  vi  Page 4 36 36 37 37 38 41 42 42 43 43 44 44 45 45 59 59 60 60 61 61 62 62 63 63 64  List of Figures (cont’d Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Figure 5.16 Figure 5.17 Figure 5.18 Figure 5.19 Figure Figure Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Figure 6.7 Figure 6.8 Figure 6.9 Figure 6.10 Figure 6.11 Figure 6.12 Figure 6.13 Figure 6.14  Figure 6.15 Figure 6.16 Figure 6.17 Figure 6.18  Experimental data: casting C6 Experimental data: casting C7 Experimental data: casting C8 Experimental data: casting C9 Experimental data: casting ClO Experimental data: casting Fl Experimental data: casting F4 Experimental data: casting F5 Experimental data: casting F6 Experimental data: casting F7 Experimental data: casting F8 Experimental data: casting F9 2D cylindrical finite element model unshielded 2D cylindrical finite element model shielded Sensitivity analysis effect of mould thermal conductivity Sensitivity analysis effect of mould heat capacity Sensitivity analysis effect of mould initial temperature Sensitivity analysis effect of metal initial temperature Sensitivity analysis effect of mould emissivity Sensitivity analysis effect of metal emissivity Sensitivity analysis effect of ambient temj,erature Sensitivity analysis effect of mould thickness Sensitivity analysis effect of gap contact initial area fraction Sensitivity analysis effect of gap contact time Sensitivity analysis effect of metal latent heat of solidification Sensitivity analysis effect of fiberpaper shielding 2D cylindrical model circumferential effect of fibrepaper shielding on temperature profiles Comparison of model results with experimental data casting C5 Comparison of model results with experimental data casting C6 Comparison of model results with experimental data casting C7 Comparison of model results with experimental data casting C8  Page 64 65 65 66 66 67 67 68 68 69 69 70 88 88 89  -  -  -  89 89  -  -  89  -  90 90 90 90 91  -  -  -  -  -  91 91  -  -  91 92  -  -  vii  -  -  -  -  93 93 94 94  List of Figures (cont’d)  Figure 6.19 Figure 6.20 Figure 6.21 a Figure 6.2 lb Figure 6.22 Figure 6.23 Figure 6.24 Figure 6.25 Figure 6.26 Figure 6.27 Figure 6.28 Figure 6.29  Figure 6.30  Figure 6.31  Figure &32  Figure 6.33  Figure 6.34  Figure 6.35  Comparison of model results with experimental data casting C9 Comparison of model results with experimental data casting Cl 0 2D finned fmite element model unshielded 2D finned finite element model shielded Sensitivity analysis variation of temperature with radial position: Face A Sensitivity analysis variation of temperature with radial position: Face B Sensitivity analysis variation of temperature with radial position: Face C Temperature difference between 90 deg and 135 deg face at same radial location: Position a Temperature difference between 90 deg and 135 deg face at same radial location: Position b Temperature difference between 90 deg and 135 deg face at same radial location: Position c Temperature difference between 90 deg and 135 deg face at same radial location: Position d Sensitivity analysis (shielded model) variation of temperature with radial position: Face A Sensitivity analysis (shielded model) variation of temperature with radial position: Face B Sensitivity analysis (shielded model) variation of temperature with radial position: Face C Temperature difference between 90 deg and 135 deg face at same radial location (shielded model): Position a Temperature difference between 90 deg and 135 deg face at same radial location (shielded model): Position b Temperature difference between 90 deg and 135 deg face at same radial location (shielded model): Position c Temperature difference between 90 deg and 135 deg face at same radial location (shielded model): Position d -  -  -  -  -  Page 95  95 96 96 97  -  97  -  97  -  -  -  VII’  98 98 98 98 99  99  99  100  100  100  100  Figure 6.36  Figure 6.37 Figure 6.38 Figure 6.39 Figure 6.40 Figure 6.41 Figure 6.42 Figure 6.43 Figure 7.1 Figure 7.2 Figure 7.3 Figure 7.4 Figure 7.5 Figure 7.6 Figure 7.7 Figure 7.8 Figure 79 Figure 7.10 Figure 7.11  List of Figures (cont’d) Variation in temperature with radial location on Face A: shielded model vs. unshielded model Comparison of model results with experimental data: casting Fl Comparison of model results with experimental data: casting F4 Comparison of model results with experimental data: casting F5 Comparison of model results with experimental data: casting F6 Comparison of model results with experimental data: casting F7 Comparison of model results with experimental data: casting F8 Comparison of model results with experimental data: casting F9 2D testbar finite element mesh 2D model results base case; time 800 seconds 2D model results high initial mould temperature; time 900 seconds 2D model results high initial metal temperature; time 700 seconds 2D model results thick mould; time 1000 seconds 2D model results mould wrap on upper half; time 1300 seconds 2D model results mould wrap on sides and top; time 2000 seconds 2D model results feeding to test section removed; time 400 seconds 3D testbar geometric representation 3D model results base case; time 25 seconds 3D model results base case; time 50 seconds -  -  -  -  -  -  -  -  -  -  -  ix  101  101 102 102 103 103 104 104 115 115 115 115 116 116 116 116 117 118 119  Nomenclature Chapter 2  A, B, C Aca Atotal c D E F  H h h he/i he/rn hcornb hcond t 0 h hr hrad h ht ka ICC kg km km 1 P qcond grad Rca F Ar R ,T 1 T 2  -  Constants obtained from fitting data to exponential model of interface heat transfer Contact area at gap interface Total interface area, = Aca + Void area at gap interface Roughness wavelength of rougher surface of contact interface Thermal conductivity of air Roughness of metal-mould interface Dimensionless constant in local air gap conduction heat transfer model Viewfactor from integration point (x,y,z) on differential area dA 1 to isothermal planar surface segment A 2 Viewfactor from elemental area dA to elemental area dA Hardness of softer surface of contact interface Overall gap interface heat transfer coefficient Air gap conduction component of interface heat transfer coefficient Air gap local conduction heat transfer coefficient Overall equivalent heat transfer coefficient Combined contact and void conduction interface heat transfer coefficient Air gap conduction component of interface heat transfer coefficient Surface contact conduction component of interface heat transfer coefficient Void radiation component of interface heat transfer coefficient Void radiation component of interface heat transfer coefficient Surface contact conduction component of interface heat transfer coefficient Overall gap interface heat transfer coefficient Thermal conductivity of gas in interface void spaces Thermal conductivity of casting Thermal conductivity of gas in interface void spaces Mean thermal conductivity of materials contacting on interface Thermal conductivity of mould Width of gap at contact interface Heat transfer gap interface normal vector Contact pressure at interface Conduction heat flux at interface Radiation heat flux at interface Interface contact conduction resistance Vector from casting geometric center to casting-mould interface Air gap width (time- and space-dependant) distance between surface area elements dA 1 and dA 1 Interface void conduction resistance Temperatures of surfaces on either side of a contact interface Average temperatuer over volume of solidified metal x  Nomenclature (cont’d)  V1x,Vly,vlz Wg Wmg Wrm c i’ 9, Oj  Solidus temperature Direction cosines of A 2 surface normal Total width of gap at contact interface Width of gap formed due to solidification shrinkage Initial gap width at contact interface Linear thermal expansion coefficient of metal Emissivities of surfaces on either side of a contact interface Angle bewtween ru and surface normals Stefan Boltzman constant (5.669x10 84 -K 2 W/m )  Chapter 3  ao, AA , 1 2 Ac/Atotal Ai/Atotai Cp , dA 1 dA 2 dq i -dA2 FAB G hgap hr hrad 11,, 1 1 b i (2k) J ka, kb k 1 km n n N PQ,f3) q  Coefficients of Polynomial describing gap interface conduction heat transfer coefficient as a function of time Areas of isothermal surfaces exchanging heat by radiation Contact area fraction in gap interface Void area fraction in gap interface Specific heat of a material Differential areas in radiation exchange calculation Radiation energy flux from dA 1 to dA 2 Emissive power of a blackbody Viewfactor of surface A to B; the fraction of energy leaving surface A which strikes surface B Radiosity of a surface; total radiation flux leaving the surface Gap interface conduction heat transfer coefficient Radiation heat transfer coefficient Gap interface radiation heat transfer coefficient Energy emitted by surface element dA 1 per unit time per unit projected area per unit solid angle subtended by surface element dA 2 Emissive intensity of a blackbody Spectral intensity Irradiation of a surface; total radiation flux striking a surface Thermal conductivities of equally rough materials on either side of a gap interface Thermal conductivity of gas in void spaces of gap interface Thermal conductivity of mould material Number of isothermal surfaces in an enclosure (Viewfactor calculation) Number of nodes in an element (Finite element method) Shape function for node i in Finite element Galerkin weighted residual method Probability density function of total radiation energy Internal heat source in heat conduction equation  xi  Nomenclature (cont’d)  qa qA i -A2 qe qh qr qr r/cond qr/load r 13 R 9 R S T ,T 1 T 2 T T 1 1 T V Wg 1 W cx  13 i ,  (‘ 13)  O 01, 02 p p a  Rate of radiation absorption per unit area of a surface Total radiation energy flux from A 1 to A 2 Constant heat flux to Finite element boundary Rate of radiation emission per unit area of a surface Convective heat flux to fininte element boundary Net incident radiation flux per unit area of radiating surface Radiative heat flux to fmite element boundary conductance matrix contribution of radiation heat flux boundary condition to finite element formulation Load vectyor contribution of radiation heat flux boundary condition to finite element formulation Distance between differential areas dA 1 and dA 2 Cumulative distribution function relating the ray emission direction variable b to a random number uniformly distributed between 0 and 1 Cumulative distribution function relating the ray emission direction variable q to a random number uniformly distributed between 0 and 1 Surface domain of element Approximation of temperature in a conducting medium Temperature of a surface Temperatures of surfaces on either side of a contact interface Average surface temperature of an element surface based on the corner nodal temperatures Temperature at node i in Finite element method Ambient or surrounding temperature Volume domain of element Total interface gap width Weighting function for node 1, N 1 in weighted residual method Absorptivity of a surface Elevation angle of radiation emission Emissivity of a surface Emissjvjties of surfaces on either side of a contact interface Directional spectral emissivity Circumferential angle of radiation emission Angles between ray connecting areas dA 1 and dA , and surface normals at 2 1 and dA dA 2 Density of a material (heat conduction equation) Reflectivity of a surface Stefan Boltzman Constant (5.669x10 84 -K 2 W/m ) Solid angle  xii  Nomenclature (cont’d) Chapter 4  Cp H k m r 1 t Tamb Tenc Topposing v 0 v v p  Specific heat of a medium Boundary heat transfer coefficient Thermal conductivity of a medium Exponent of temperature at boundary Number of rays traced per surface Number of surfaces in enclosure for viewfactor calculation Computational run time Ambient temperature Temperature of enclosure walls surrounding the medium Temperature of opposing face at gap interface temperature Initial temperature Surface temperature density of a medium  Chapter 7  G k 1 2 nrrR 1 P Al? SDAS t  ATf V /XVfreezing  Thermal gradient Constant relating secondary dendrite arm spacing to local solidification time Channel (dendrite) length Total channel area for n channels of radius R Critical pressure below which bubbles nucleate from gases dissolved in melt Pressure at dendrite root Pressure in the bulk melt Pressure drop across dendrite array Secondary dendrite arm spacing Tortuosity factor Local solidification time Freezing range of solidifying metal Liquidus isotherm velocity Volume change associated with freezing  xlii  Acknowledgements I would like to thank my advisers, Dr. Alec Mitchell and Dr. Steve Cockcroft for their support and guidance, and Mr. Bob Dawson of Deloro Stellite Inc. for providing the opportunity to do experiments in an industrial setting. I would also like to acknowledge the Natural 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, Al Schmaltz for his technical help, and his unfailing good spirits during our plant trials; Serge Milaire for helping out with things electronic; Brock Barlow of Deloro Stellite for operating the furnace for us and for putting up with the engineers; Dave Tripp, the computer Guru; and Steve 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 their friendship, and especially my husband, Steve, for his example as an engineer, and for the encouragement which he provided when it was needed the most.  xiv  Chapter 1 Introduction  The investment casting process for the production of complex components has many advantages over other more conventional casting methods, as well as over manufacturing routes such as machining and assembly. It is a well-established near-net shape technology allowing versatility of design, close tolerances and good surface quality, requiring little or no additional machining.  For these reasons, the process has become widely used in the  aerospace industry, where a further advantage of design for weight savings can often be realized. Casting in vacuum extends the range of materials which may be successfully cast to include reactive materials such as titanium and alloys containing reactive elements such as superalloys. 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, the  experience of the foundryman has been combined with extensive testing of finished parts to establish the casting design and parameters. This method, however, can be costly, as several design iterations are often necessary for intricate or thin-walled parts, where adequate feeding of 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 tools which will enable the prediction and location of possible macroscopic defects and correction of 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 microstructural features such as the location of microsegregation, and grain size, which ultimately determine the mechanical behaviour of the cast part. The ability of computer-based models to predict correctly the flow of heat in casting processes hinges on the quantitative characterization of heat transfer in a casting from the  1  time molten metal is poured into the mold through solidification and cooling to ambient temperature. Numerical formulations describe the phenomena of conduction through the continuous media (metal, ceramic mold) and heat transfer at the boundaries (mold external surface and mold-metal interface).  Thus it is essential to have access to the relevant  thermophysical properties at elevated temperatures and relationships describing the various boundary conditions. Heat transfer at the boundaries is by convection, gap conduction and radiation. In vacuum casting the primary mechanism of heat transfer at the boundaries is by ,  radiation. This work concentrates on the development of boundary conditions for the case of vacuum investment casting of 1N7 18, a nickel-based superalloy. The work has proceeded in two stages:  first, modifications have been made to a finite element-based mathematical  model of heat flow, developed in the Department of Metals and Materials Engineering at the University of British Columbia [1], to include farfield radiation, enclosure radiation and interface gap radiation in two and three dimensions.  Second, the modified mathematical  model has been verified against industrial measurements made at Deloro Stellite Inc., Belleville Ontario.  1.1 Background of the investment casting process  Investment casting, also known as precision casting or the lost wax process, is one of the 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 and Aztec cultures, where it was primarily used in the production of intricate jewelry.  The  process was discovered in Europe during the Italian Renaissance for casting large statues, and was first used commercially in 1867 for dental fillings. It came to industrial prominence during the second world war, with the huge demand for aircraft parts. The near-net shape technology offered by investment casting allowed costly machining and assembly steps to be bypassed.  In the past four decades investment casting has become a widely-used, well  2  established manufacturing technology.  It is used extensively not only in aircraft and  aerospace, but computers, electronic equipment, food processing machinery, gas turbines, machine tools, automotive applications, medical and dental uses, weapons systems, and many other applications [2]. The advantages of investment casting include excellent dimensional control, the ability to cast thin sections and good surface finish with little or no fmishing required. Moreover, through the use of sophisticated coring, component geometry is virtually unlimited, allowing internal passages, undercuts, and features which cannot be obtained by other casting methods.  1.2 Steps in the investment casting process  A flow diagram of the investment casting process is shown in Figure 1.1. First, a pattern (positive) is made to the dimensions of the finished part. From this, a wax injection mold is made, and replicas of the part produced from wax. Depending on component size and 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 and coated with a face coat, consisting of very fine refractory flour or sand.  The face coat  material is chosen to give the desired surface finish and to be as chemically unreactive with the metal being poured as possible. It may also contain grain nucleating agents. The coat is allowed 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 fully invested assembly is next dewaxed, leaving a hollow ceramic shell, which is fired to increase its strength. This step may also be the preheat for pouring of the metal. The shell is removed from the preheat furnace, placed under the melting crucible pour spout (in vacuum, if the material requires it) and the metal is poured into the shell. The casting is allowed to cool and  3  the ceramic is knocked off. Further treatment may include surface cleaning and finishing operations such as drilling of holes or facing of surfaces.  a: Produce wax paftem  d: Coat with stucco  b: Assemble mold tree  c: Dip in ceramic slurry  e: Dewax, tire, preheat  f: Pour metal  I U g: Break off mold  h: Remove gating, clean  Figure 1.1 Steps in the Investment Casting Process  4  i: Finished part  Chapter 2 Literature Review  A large quantity of material has been published on the mathematical modelling of solidification processes, the majority of which has been devoted to primary casting processes such as ingot casting and continuous casting. No attempt will be made to review this work. Rather, this review will focus on the modelling of boundary conditions at the mould-metal interface and at the mould surface in vacuum investment casting. In the vacuum investment casting process heat flows between the casting and the mould by conduction and/or radiation and from the exterior of the shell by radiation. The complex geometry of a typical shell requires that radiation exchange between different parts of the shell, where temperatures are changing spatially and with time, be accounted for. The interface heat transfer is also dependant on location and time due to the formation of a gap between the mould and the solid shell as the metal solidifies. The literature was reviewed for details on previously applied techniques in modelling radiation boundary conditions, viewfactor calculation and metal-mould interface heat transfer for use in the current work. Hamar {3J used a finite volume method to model microporosity formation in investment castings. The model included a mould filling algorithm, latent heatevolution and feeding of the mould due to density differences in the liquid and solid phases. In their model the ceramic shell was encased in a cylinder of foundry sand. This technique allowed a simple farfield radiation boundary condition to be used; the approach is non typical to the investment casting process, however. Duffy et. al. [4] applied the finite element method to the analysis of single crystal turbine blades. In this production method, the radiation boundary condition is changing with time as the blade is withdrawn from the furnace. Several blades are typically cast in a cluster arrangement. Thus a mould may exchange heat with the furnace hot and cold  5  zones, radiation baffles, copper chill and with other moulds. This results in a modelling problem similar to that of investment casting. The authors used a commercially available finite element package (MARC, MARC Analysis Inc.). This package did not incorporate radiation boundary conditions, which were added by the authors. View factors for the radiation exchange were calculated from an integration point on an element to a planar surface segment. Contour integration was used to obtain the view factors. Both view factors 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 dA 1  ,  to an  isothermal planar surface segment is calculated as  1 21[  Fdl—2— Vlxy  (z2—zl)dy2—(y2—yl)dz2  C2  f (x2—xl)dz2—(z2—zl)&2  £  +V1y  +vlzy  C2  C2  (y2—y1)d2—(x2—X1)dy2  2.1  where  vix, Vly  and  Viz  are the direction cosines of the  2 surface normal. This A  method does not account for partial views (shading of one surface by another), i.e. either one surface sees another surface entirely, or not at all. This limitation requires that the subdivision of surface segments be carefully considered beforehand, and may require a finer surface mesh than the temperature field requires, increasing the computational size of the problem. Huang and Berry [5] dealt with solidification heat transfer in aluminum investment castings. They investigated the effects of metal superheat, mould preheat, casting thickness and mould thickness on solidification time, both experimentally and computationally. A finite difference scheme was used in the numerical model.  The  mould-metal interface heat transfer is discussed in some detail. Details of the boundary conditions implementation are described in a Masters thesis by Huang.  6  Liu et. al. [6] discuss a finite difference model for unidirectional solidification of single crystal high temperature alloys. Enclosure radiation with a coarse view factor grid is incorporated. The enclosure is divided in the horizontal plane into angles subtending other moulds in the enclosure, and the furnace wall. Then in the wall zone, the vertical range of exchange with the hot zone, baffle, cold zone and chill plate are determined. In the range of exchange with other moulds, the castings are divided into five exchange zones in the vertical direction. This approach, while it attempts to account for varying enclosure 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 blade solidification using a finite element formulation and including enclosure radiation with shadowing effects.  For 3-dimensional geometries view factors are calculated by the  relation for elemental area dA 1 to elemental area dA : 1  Fy=  cos0icosOj dAj ltry 2  2.2  where ru is the distance between surface area elements dA 1 and dA 1 and O and 0 are the angles between  and the surface normals.  Implicit in this method is the  assumption that, over the areas involved, cosOj, cos0j and ru are constant. This is approximately true when rU is sufficiently large, or dA 1 and dA 1 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 so that the view factor from i to each segment is approximately equal. Each segment is associated 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  7  j in the enclosure.  A final technique for calculation of radiation viewfactors, and the one which was chosen for use in this work, is a Monte Carlo ray tracing approach. This technique was adopted for its ease of implementation and integration with the fmite element code and PATRAN preprocessor.  It is generally applicable for all complex geometries, easily  incorporating shading effects, and requiring no numerical integration.  The level of  accuracy for a given run time is good, and computation time for large, complex problems is comparable to the other numerical methods described above. The Monte Carlo method as it is used in engineering and the sciences today was first formulated by Von Neumann and Ulam at Los Alamos National Labs in the 1940’s to investigate neutron transport phenomena [8]. The first applications to radiation heat transfer problems were done by Howell and Perimutter in 1964 [9,10]. They used the method to determine heat transfer between grey walls through non-isothermal, absorbingemitting media, a problem which, due to the gas attenuation, rapidly becomes extremely complex 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, investigating radiation exchange between surfaces separated by non-participating media. Whereas the previous researchers calculated the total energy exchange between elements of their systems, several studies have used the Monte Carlo method primarily to calculate exchange areas, or view factors, as in the present work. Modest [13] examined 3-dimensional configurations, and curved surfaces, and allowed for arbitrary emission, absorption and reflection characteristics, as well as radiation through openings. Maitby and Bums [14] allowed for 3-dimensional enclosures filled with a non-participating medium. They approximated curved surfaces by planar elements, and allowed for non zero transmittance through the enclosure surfaces.  Hoff and Janni [15] calculated  radiation shape factors and included intersection criteria for planar, cylindrical and spherical surfaces.  Ikushima et. al. [16] compared computation time and accuracy  between area integration and Monte Carlo simulation to obtain the radiation view factors. 8  Plehiers and Froment [17] calculated exchange factors for the gas and furnace in steam reforming chemical process models. It is important to recognize that initially, in vacuum investment casting, heat transfer across the mould-metal interface may occur by more than one mechanism. On a microscopic level, the surface of the mould is rough. When the metal flows into the mould, there will be contact between the mould and the metal only at the asperities of the mould surface. At a microscopic level, this contact will depend on the relative roughness of 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 in the interface heat transfer. These are (1) conduction through solid contact between the metal and mould, (2) conduction in the gap between the metal and mould through the intervening gas (if present), and (3) radiation across the gap. A fourth possibility is heat transfer by convection in a gas across the gap. Initially, the solid contact conduction component of heat transfer may be large. As the metal solidifies, it contracts away from the mould surface, reducing the effective area over which solid contact conduction occurs, and radiation or gas conduction become more significant.  Modelling this  condition rigourously is a very complex undertaking, as the gap width is dependent on the geometry of the part, as well as the solidification. If solidification shrinkage is. restricted due to geometric constraints, the conduction component will be active longer than in unconstrained regions. Ho and Pehlke [18] investigated the first three mechanisms by solving an inverse heat conduction problem from thermocouple data using a permanent mould or chill rather than a sand mould.  Interfacial gap size over time was measured to derive air gap  coefficients across the gas-filled gap. The overall gap heat transfer coefficient ht is  2.3  hi=hs+hc+hr  9  where  3 = surface contact conduction coefficient h air gap conduction coefficient hr = radiation coefficient  The air gap conduction coefficient, h is proposed to depend on a mean conductivity of the contacting materials, km, the roughness wavelength of the rougher surface, c, the contact pressure F, and the hardness of the softer surface, H.  h!-(FH)  2.4  The air gap conduction coefficient is  h  =g  2.5  Cl  where kg is the conductivity of the gas and 1 is the gap width.  The radiation coefficient is given as  (T2+T2)(T+T)  hr_  26  (i 1 I —+—-—1 1 62 ‘. E  10  where T 1 and T 2 are the interface temperatures of the metal and mould, and are the interface emissivities of the metal and mould.  and E2  The authors reference several  works 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 transfer coefficient 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 heat transfer coefficient consisting of three terms:  h  2.7  + hCOfld + hmd  =  where h t is the conduction component due to solid contact, hcond is conduction 0 across the interface gap, and hrad is radiation across the interface gap. They assume that no solid contact conduction occurs, i.e. 0 h = 0. The gap conduction term t  2.8  hCOfld=!c  where Ka is the conductivity of the gas (air) and Wg is the gap width. The gap width consists of two components:  =  2.9  +  where Wrm is the initial gap width, determined in the paper from average particle size measurements 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 mm -  is used in the work.  This model, due to the assumption of negligible solid contact 11  conduction, is not applicable to a vacuum casting situation, where gap conduction is negligible, and solid contact conduction may be significant at initial times. The second model of Kanetkar uses an equivalent heat transfer coefficient described by a constant (steady state) plus and exponential (transient) term:  hcimA+Be_Ct  2.10  The constants A, B and C are adjusted to fit experimental data. The original work by Kanetkar et. al. was not available at the time of writing, and no details are given in [5] as to the assumptions and validity of the model. Nishida et. al. [19] conducted experiments to determine the dominant mechanism of heat transfer in the formation of the air gap. The relative movement of the casting and mould were measured, and temperatures recorded at several locations in aluminum alloy castings. They concluded that air gap conduction dominated over radiation heat transfer, and that, as the air gap became larger, convection effects became significant.  These  conclusions are not applicable in the case of vacuum investment casting, where conduction through the gas phase is negligible. Huang  Ct.  al. [20] and Huang [21] address the importance of estimation of the gap  width 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 air gap width, dependant on casting geometry, and thus enable estimation of the interface heat transfer coefficient. This method estimates the thermal contraction at an interface location as:  2.11  =c(F.ñ)(u_i)  12  where Ar F  =  vector from casting geometric center to casting-mould interface  =  ñ a u  air gap width  interface normal vector metal linear thermal expansion coefficient  =  =  =  solidus temperature average temperature over solidified metal  Geometric restraints to contraction from the mould are considered as, for example, where the casting rests on the mould due to gravity or where contraction is resisted by an internal core.  The authors account for this by initially computing an unconstrained  displacement, then adjusting the displacement at the constrained location and all locations contracting toward the constrained location by the unconstrained amount. Finally, the air local gap conduction heat transfer coefficient is given by:  D E+FAr  2.12  where  D  thermal conductivity of air  E = roughness of metal-mould interface F = dimensionless constant accounting for other effects such as mould dilatation  This model tries to account for geometry and time dependance of the air gap conduction component.  However, it is not applicable in vacuum situations, where the air gap  conduction is negligible relative to the solid contact conduction and radiation heat  13  transfer components. No mention is made of solid contact conduction component in this work. Liu et. al. [6] discuss mould-metal interface heat transfer accounting for a solid conduction 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 to heat flow is then  WI 1 WI Rca =—-I I+—-I I 2 KcAca) 2 KmAca)  2.13  The gap conduction resistance is  R=W  1  2.14  gv  where  Wg K  gap width =  Km Kg  thermal conductivity of casting  =  thermal conductivity of mould thermal conductivity of gas (air)  Aca = area of solid contact A = void area  The total resistance is  14  1  114ca(2KmKc)AK V IJ’ Km + K  Rtotai  2.15  The combined conduction heat transfer coefficient is thus  i(2K K 111 A total K, + K  hcomb  j47  +  A K Atotal  217  and the total heat transfer coefficient is  2.18  hiotat = hCOfl,b + hrad  It should be noted, however, that the two components  and hrad are based on  different surface areas:  qcond  =  hcombAtoialAT  2.19 hradAt4T  grad  Thus, the direct addition of the heat transfer coefficients is not strictly correct. The radiation heat transfer coefficient must be corrected by the factor Av/Atotai:  grad  =  hrad  qrad  =  h,JdA,OlOlAT  Atotal  2.20  AioiaiAT  2.21  15  had  =  2.22 Aioiai  kotai  =  hcomb + h  2.23  This 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 how these are determined in the paper are given. The thermal properties of investment casting ceramic shells are not well documented in the literature. These properties depend strongly on the shell composition and structure.  A layer of the shell contains a flour or stucco and the silica binder  component. Each layer may have a different proportion of stucco to binder, as well as different stucco grain size. Further, the shell is not completely dense, but contains voids which may account for up to 40% volume fraction of the shell. At any temperature, the shell properties are a function of composition and processing. Properties may also be a function of temperature. At higher temperatures (>500 deg C) radiation across the void spaces increases the apparent conductivity of the shell. Some researchers have further postulated a radiation transmission mechanism through the fused silica at temperatures greater than 1000 deg C, as the silica becomes partially transparent. Some work has been done 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 diffusivities for various shell systems. Values of thermal conductivities ranging from approximately 0.550 to 0.750 W/m-deg C were reported for shells of primarily silica materials. Slightly lower 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 2 cm / sec are reported. Huang et. al. [27] performed experiments to determine shell conductivity as a function of particle size, binder composition, and temperature, and simulated the effect of  16  porosity. Lower porosity volume fraction resulted in a higher thermal conductivity. For various shell systems, thermal conductivities ranging from 0.75 to 1.5 W/m-deg C were reported.  A linear variation of conductivity with temperature was indicated by the  experimental 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 100 deg C to 0.86 W/m-deg C at 1000 deg C, and a thermal diffusivity of approximately 0.003 cm /sec. 2  17  Chapter 3 Mathematical Formulation of Boundary Conditions  The solidification heat conduction code THERMAL, developed in the Department of Metals and Materials Engineering at the University of British Columbia [1], was modified to incorporate the boundary conditions necessary to model the vacuum investment casting process. In the vacuum investment casting process, heat transfer from the molten pool to the surrounding furnace takes place by the following five mechanisms:  i) conduction and  convection in the liquid pool, ii) conduction through the solid shell, iii) conduction and/or radiation 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 detail the 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 Radiation  A surface at temperature T, exchanging heat with its environment at temperature Tf emits radiation at a rate of  4 q=saT  3.1  where q is the rate of energy emission per unit area of the surface, a is the Stefan Boltzman constant with a value of 5.669x10 8 Wm 4 and K 2  is the total emittance of the  surface. The surface absorbs radiation from the environment at a rate of  q=cLa7  3.2  18  where q is the rate of energy absorption per unit area of the surface, a is the absorptivity of the surface (fraction of incident radiation absorbed) and aTjn/ is the blackbody emissive power of the environment. The net incident radiation per unit area is the difference between incoming and outgoing radiation flux:  3.3  If Kirchoffs law is assumed to be valid, that is, a  =  E,  then  3.4  The radiation heat flux is non-linear in the dependant variable,  T.  A linearization is  performed as follows. It is desireable to express qr as  q=h(2flf—T)  3.5  Comparing equations 3.4 and 3.5 results in:  =  hr  =Ea(f+T2)(flf+T) (if-T)  3.6  3.1.2 Gap interface radiation  The metal-mould interface heat transfer is complex. Referring to Figure 3.1 a, two surfaces in contact will, in general, not be in perfect contact due to the fact that the surfaces  19  are not perfectly smooth, but rough, and actually contact only at projections and asperities on the surfaces. Referring to Figure 3. ib, where it is assumed that both surfaces are equally rough  ,  an overall heat transfer coefficient, accounting for conduction across the contact  points and across the gaps may be expressed as [22]:  hgap  i(A 2KaKbAvK 1011 Ka + Kb A total f) tA  37  where  -deg C) 2 hgap = interface heat transfer coefficient (W/m Wg = total interface gap width (m) =  total contact area (m ) 2  =  total void area (m ) 2  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 modifications should be made. First, the conduction component of the gap is neglected: when the mean free path of the gas molecules approaches the width of the interface, the effective conductance of the gas decreases as the pressure is decreased [221. Second, it is assumed that the roughness is associated only with the mould, and that the metal surface is smooth. Since surface 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 reasonable  20  assumption.  The resulting interface is depicted schematically in Figure 3.1 c.  Thus, the  conduction heat transfer coefficient is  4Km  hgap  “total Vg  where Km is the thermal conductivity of the mould. In addition to conduction, radiation heat transfer across the gap must also be considered. This may be modelled as radiation exchange between infinite parallel plates if the gap is sufficiently small. The analytical expression in this case is  4  —  rad  a(T2+T2)(T+T)  Atotai  3.9  /  where -deg C) 2 hrad = effective radiation heat transfer coefficient (W/m =  total void area (m ) 2  Tj, T 2 = 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,  +  Atotai  4  3.10  =1  Atotat  21  As the casting solidifies, it contracts, and the solid contact between the metal and the mould is reduced. In this model, this effect is incorporated in a time-dependent function for the contact area-fraction Ac/Atotal =f(t). Thus, the solid contact conduction component of heat transfer decreases with time and the radiation heat transfer component becomes correspondingly larger. Due to the complexity of investment cast components, the time for gap formation and the gap width will not, in general, be uniform across the mould/metal interface. If geometric and temporal variations in gap width are to be accounted for (e.g. smaller gap due to the casting resting on a portion of the mould or being restrained from contraction through an integral core), the functions A/At i and Wg may become spatially variable as well as t 0 time dependent. These effects are not addressed in the present work. Implementation of the model described above requires that the gap conduction and radiation coefficients be linked through the contact area fraction 0 A/At i t . The source code allows input of a time-dependent ‘convection’ coefficient as a user-specified polynomial:  3.11  hgap=ao+ait + 2 t ...+a,it +a  If the factor Km/Wg is assumed to be constant, then  A  W  1 W  Atotai  Km  Km 3.12  O  A 1.O Atotai  22  From equation 3.10, the void fraction is then calculated for use in the radiation heat transfer component. The assumption of constant K,/Wg is not rigourously correct, as discussed previously. Investigations of the time and spatial variations of this parameter present good scope for future work.  3.1.3 Enclosure radiation  One of the major advantages of the investment casting process is its ability to deal with extremely complex geometries. In many cases this additional complexity means that modelling the process with only far-field radiation is a poor approximation and may not give the desired accuracy; the mould may be radiating to itself due to shielding from other parts of the mould. Thus, non-uniform, non-constant T 1 values arise. In addition, a farfield model may not be suitable as the vacuum furnace may not be sufficiently large relative to the casting size, and the radiative exchange with the furnace must thus be accounted for. To deal with this phenomenon, the mould geometry, enclosure geometry and surface radiative properties must be taken into consideration. Consider two differential areas dAj and dA , separated by distance rj., located 2 in isothermal surfaces A 1 and A 2 (Figure 3.2) [23]. The total energy per unit time leaving 1 and reaching dA dA 2 is  2 =ib,ldAlcosOldG)l _ 1 dq  3.13  1 per unit time per unit projected area per unit solid where b t, 1 is the energy emitted by dA 1 is: angle subtended by dA . The solid angle d0 2  23  do = 1 2’°°2  3.14  Also,  ebcYT t  3.15  It  Thus, the total energy leaving dA 1 which is incident upon dA 2 is  2 =a2 4 dq  ‘42  1 1 d 2 cosO A cosO  3.16  Similarly, the total energy leaving dA 1 is 2 arriving at dA  1 _ 2 dq  12 dA 2 dA cosO 1 cosO =  3.17  The net energy transferred is  12 dq  If  —  dq  =  —  T)  12 cosO cosO d 1 A  3.18  and A 2 are approximately isothermal (but not at the same temperature) then this  expression can be integrated over the areas to give the total energy exchange between areas 1 and A A . 2  24  r cosO cosO dAdA  —A 1 qA — 2  ijJ 1A A 2  2  cosO d 1 2 A  4 T  qA,_Aa2JJ  .0  2  A1A2  The geometric configuration factor (also known as the shape factor or view factor) is the fraction of the total energy emitted by surface A 1 which reaches A : 2  4 T  12 cosO d 1 A j $ cosO itr  2 1 A  1 A 4 a1  4 T  j$  dAidA 2 1$JcosOicosO 1A A i-2 2 1  321  cosO ljjcosO 2 d 1 A 2 A 2 AA 1—2 2  322  12 cosO cosO d 1 A  ALA,  It can be seen from (3.21) and (3.22) that  Pj_ 1 A 2  =  1 _ F 2 A  3.23  This is called the reciprocity relation, and is true for all pairs of isothermal surfaces exchanging heat by radiation. From the definition of the view factor it can be seen that F depends only on the relative geometries of the exchanging surfaces; it is independant of temperature and surface properties. It represents an energy fraction leaving one surface and arriving at another, and  25  provides the link for setting up equations of radiation exchange between surfaces in an enclosure exchange network. In order to describe the boundary conditions in such a radiative network, the total incident energy flux to a surface is required. The net radiant heat flux to the surface is then the difference between that emitted by the surface and the total incident flux. Two quantities are 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, which is the total incident flux to a surface, including all direct exchange components and indirect exchange through intermediate reflections.  The desired quantity for boundary condition  calculation is the irradiation I Referring to Figure 3.3, the radiosity G is:  Gjtzqj+pjJ  3.24  or in vector format, for all surfaces, where [p] is the diagonal matrix of reflectivities,  {G} = {q}+[p]{J}  3.25  The irradiation J, where n is the total number of surfaces in the enclosure, is:  J 1 A  =  =  j=1  3.26  or in matrix form:  26  {J} = [F]{G}  3.27  Substituting for {G} from 3.25 into 3.27 and rearranging gives  ([i] [F][p]){J} = [F]{q —  }  3.28  This set of equations can be solved for the irradiation vector, {J}, which is used directly in the finite element formulation of the incident radiation flux boundary condition.  3.1.4 View Factor Calculation  The following assumptions are made for calculation of the viewfactors: 1. No energy is transmitted through surfaces 2. Surfaces are separated by a non-participating medium 3. Surfaces are black (perfect absorbers/emitters) 4. Surfaces are planar, with four straight edges, but of otherwise arbitrary shape 5. The radiative enclosure is filly closed  Assumption (3) is made only for viewfactor calculation purposes, and is relaxed in the FE implementation. Mathematical modelling of the investment casting process with enclosure radiation involves 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 integral equations axe not available, and numerical integration can become difficult and time consuming, especially in cases involving partial shielding of one surface by another. An alternative approach is the Monte Carlo ray tracing method. This statistically-based method consists of tracing the history of an energy bundle from its point of emission to its final 27  absorption. Over a large number of energy bundles, the interactions between emitting and absorbing surfaces converge to the exchange factors for a given set of surfaces. This method can be readily adapted to non-uniform radiative surface properties, and complicated geometries. Both 2- and 3-dimensional models for viewfactor calcuation were developed for this work. In the ray tracing approach used to calculate viewfactors, two directions must be defined 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 angle df3,andover OO2n is[23]:  dq = 2rc(, 13)i, (2k) cos j3 sin f3ddX  3.29  where  (, p)  =  directional spectral emissivity  i (?) = spectral intensity of a blackbody,  =  emissive power per unit projected area  per unit wavelength per unit solid angle  A probability density function is defined by normalizing the function by the total energy:  330  4 saT  where c is the total emissivity. It can be shown [23] that a function Rp uniform over the interval 0 integrating equation 3.30 over all wavelengths 0  28  R  1 is obtained by  ) <cand from zero to :  14=  27t$i 00  (?, P)b (2) cos 13 sin 13d(3d2. 4 EcYT  3.31  For blackbodies and grey, diffuse surfaces equation 3.31 gives  f3 2 R=sin  3.32  13 is called the cumulative distribution function. In order to assign directions to an emitted R ray, a random number is chosen between zero and one, and equation 3.32 is solved for 0 is uniform over the interval 0  0  2it  13. As  , generation of a circumferential angle is simply  given by  0  3.33  2’t  3.2 Finite element implementation of boundary conditions  Neglecting convection in the liquid pool, the general 3-dimensional, time-dependent heat conduction equation in cartesian coordinates, with internal heat source q and temperature dependant material thermophysical properties may be expressed as:  =  ox)  If k and  Oy  Oy)  8z  Oz)  are assumed to be constant, this reduces to:  29  3.34  ( T 2 2 öx  T 2 6  6 ‘1 T 2 6T +q=pC öt 2 ) 6z  ki —+—-—+---—--+ 2  3.35  —  In 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 possibly mid-edge, mid-face or body-center locations [24]. Within each element, the temperature at  any 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)  3.36  =  where n is the number of nodes in the element, and the N 1 are weighting functions dependant upon (xy,z). Thus, when T is substituted into equation 3.35, the equality will, in general, not hold at all locations but will have a small error R, called the residual, such that  k  ö2 —-+--——+-—-+  ox  3))2  Oz  +q—pC —=R  3.37  The 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$wRdv=o  ,  i=l...n  30  3.38  or,  339  The Galerkin method of weighted residuals assumes that the weighting functions W 1 are taken to be the same as the nodal temperature weighting functions N . 1  Equation 3.39  becomes  JJ$Nk[2  dV  +  0 3.40  The first integral can be integrated by parts to give  jfk 1fl +-ny +-nz N.dS_ffjk I  6x x  öy 6y  i=1...n  .5z öz 3.41  The double integral over the surface S represents the boundary condition, as can be seen by an energy balance on a surface element (Figure 3.5). Heat conducted into the boundary must be equal to heat lost from the boundary by imposition of a heat flux q (positive when heat is added to the boundary), by convection qj, and by radiation q,..  31  (8T kI n —  T 8y  + —ny +  öT öz  —  \\  I q+  + q,.  3.42  ,j  The surface integral is used to incorporate the boundary conditions by substitution of 3.42 into 3.41:  JJk[nx +ny+n,dS=JJ(q  +q  +q)dS  ,  =  l...3.43  For the farfield radiation condition of equation 3.5, the radiative flux q is now inserted into equation 3.43, the surface integral expression:  JfqrI1ds=  H hr(linf  —  Fi’)N d 1 s  3.44  So the load vector contribution is  -  qrload  =  HhrlnfM  3.45  and the conductance matrix contribution is  q,COfld=JJTNdS  3.46  32  The radiative heat transfer coefficient hr is highly non-linear, since it depends on the local temperature to the third power. It must therefore be updated for each time step. A value of the local temperature at the time step being evaluated is required. One method of evaluating the radiative heat transfer coefficient is to iterate within the time step. Evaluate hr on the basis of an initial value (say, the previous time step temperature), solve the current  time step, then reevaluate hr based on the new solution. Compare successive solutions until a convergence criterion is attained. This method is computationally intensive, however. The method used in this work is simply to base the evaluation of the radiative heat transfer coefficient on the previous time step. Restricting the maximum temperature change at any node within one time step to a sufficiently small value (as implemented by the time step optimization 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 is  JJqCOfldJvdS = JJhOfld 0 TIVdS—JJh f ld1flfNjdS C  ,  i  =  l...n  3.47  where T -ij is the temperature at the corresponding integration point on the opposing 1 element. Similarly, the radiation component is  JqradNjdS=  H hrad TNidS_HhradlinfNidS  3.48  In each of equations 3.47 and 3.48, the first term on the right hand side contributes to the conductance matrix, while the second term contributes to the load vector.  33  The assumption of constant  Km/Wg  is not rigourously correct.  The mould  conductivity 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 the procedure. A more rigourous approach, addressing both geometric dependency and variation of K,n/Wg presents good scope for future work. In the enclosure radiation model, the finite element formulation of the net heat flux to surface becomes:  4 q =cxJ—EoT  3.49  So the boundary condition is  Jj(aJ_saT4)JVdS  3.50  Here again, there are two terms, one of which contributes to the conductance matrix, the other 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 load vector contribution, this appears in the vector {q} on the right hand side. As in the case of farfield radiation, {q} is evaluated based on the temperatures at the previous time step. An average temperature over the surface of the element is computed:  3.51  34  where k  2 for 2-dimensional elements and k  =  4 for 3-dimensional elements. A  tolerance flag in the program tracks the maximum temperature differe nce between the corner nodes and issues a warning if this difference exceeds a user-defined isother mal tolerance. The conductance matrix term is linearized:  4 $JEcY T I\1jdShr TIVdS  3.52  hrEGT  3.53  where  and, for k = 2 for 2-dimensional problems or k  4 for 3-dimensional problems, and the T 1  are the temperatures of the ends (2-d) or corners (3 -d) of the emitting segme nt,  =  3.54  35  C C/)  CD  CD  C  Cl) -  -  OCD  CD  C  CD  I.  C)  CD  CD  CD  I  C)  CD  C  CD  —  )  —  0) C,  0  C, CD  — 0) C)  -I  0) C)  z  0  C)  -  CD  -  — 0)  CD  CD  C,  -h  CD  CD 0  —  CD  1 z 3  CD 0  C)  0  g 0  CD—  00  -  ‘<CD  CDD .CD  C/)  CD  L)  C)  C  Ambient temperature Tinf  qh  -kdT/dx  =  qr =  —x  hconv(Tinf- T) -  ) 4 T  Surface at temperature T  Figure 3.5 Heat Balance on the Boundary of a Conducting Medium  38  Chapter 4 Verification of Computer Code  4.1 Viewfactor calculation verification  The analytical solutions for the calculation of viewfactors for 2-dimensional surfaces oriented as shown in Figures 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 the  analytical values and standard deviation of the %error (for n =20 observations) as a function of the number of rays traced from each surface. As expected, the mean error and standard deviation both decrease significantly as the number of rays traced is increased. Figure 4.2 plots the 95% confidence interval (mean plus two standard deviations) of the %error in viewfactors vs the number of rays traced. The view factor algorithm is computationally intensive, with the run time increasing with the number of rays, and the square of the number of surfaces:  4.1  n 2 tcxn  The desired level of accuracy in view factor calculation can thus be obtained at a cost in run time. For most of the work described here, view factors were calculated using r = 10000.  4.2 Radiation boundary condition verification  In order to verifr the boundary conditions, the code was compared with data published by other researchers obtained by semi-analytical and other numerical methods for specific problems. Jaeger [25] obtained solutions to the equation 39  2 ox  4.2  pC Ot  subject to the boundary condition  _k’_=HVSm=FS Odx  4.3  This corresponds to a one-dimensional heat flow case of the semi-infinite region x  <  0  having constant conductivity k, density p, and specific heat C. When the exponent m the boundary condition corresponds to a radiative heat exchange where a black surface radiates to a medium at zero K. Jaeger gives a graph of the resulting temperature at the surface of the slab as a function of time, in dimensionless form, as  vs. log T 10  The 2- and 3-dimensional models used to simulate the conditions of Jaeger are shown in Figure 4.3 for the far-field, enclosure and interface type boundary conditions. For the semi-infinite slab, l=4m was used, and the run terminated when a temperature change was observed at x=-4.0. Tamb, Tenc and Topposing were at OK, and all emissivities were set equal to 1. Constant values of k = 11.4 W/m-K, p =8190 kg/m 3 and C, =435 J/kg-K were used.  -  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 on the solution. All meshes show good agreement at longer times. At short times the model predicts 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 temperatures for simple heat flow cases in agreement with data obtained by other methods. Some care  40  must be taken to choose a mesh size of sufficient density to obtain a desired level of accuracy. Table 4.1 Mean %error and standard deviation for comparison of Monte Carlo viewfactor results with analytically calculated values (n=20) Analytical Number of Mean Solution rays 0.4142 1000 1.35 10000 0.03 100000 -0.01 0.2929 1000 -1.16 10000 0.44 100000 0.11 0.19982 1000 0.69 10000 -0.34 100000 0.06 0.20004 1000 -3.67 10000 0.14 100000 0.11  Figure  4.ld  Standard deviation 3.23 1.21 0.43 5.75 1.51 0.59 5.78 2.21 0.66 7.65 2.15 0.62  T  T -_  Fb  b: 2D Perpendicular  a: 2D Parallel  rrz C:  d: 3D Perpendicular  3D Parallel  Figure 4.1 Configurations for View Factor Code Verification  41  Monte Carlo Viewfactor calculation Comparison with analytical solutions for 2- and 3-dimensional geometries 95% Confidence Level, n20  ---El---1-.-  -.--.-  0 LU  20 2D 3D 3D  parallel perpendicular parallel perpendicular  7.0 6.0 5.0 4.0 3.0 2.0 1.0 1o  13 Number of rays traced  Figure 4.2 Comparison of Monte Carlo View Factor Calculation with Analytical Results  Tinfinity  =  OK  a: Semi-infinite solid cooling to environment at OK  V b: Farfield radiation model  Tim  4.Orn  i  Tambient  Ts.(N. T 0.05 ra’////////////////,4 . T H’ 4.Om Ts 1__ 0.0 5nV/////////////////)- T 4.On  Ti”  =  C: =  OK Enclosure radiation model  OK  d: Gap interface model =  OK interface  Figure 4.3 Configurations for Radiation Boundary Condition Verification of Thermal Code  42  Comparison of model results with data from Jaeger (1) Cooling of 2D semi-infinite slab radiating to environment at OK Model results for various mesh sizes 1.00,  -  -  -0.lOm f  rr...  =O.025m =0.Olm  S  0.80  0.60  N  0.20  (1) Jaeger, J.C., roceedirigs, Cambridge I  0.00 -2.0  -1.0  1950. pp. 634-641  ilosophical Society, vol.  0.0 Log T 10  1.0  2.0  Figure 4.4a Cooling of 2D semi-infinite slab radiating to Farfield Environment at OK Comparison of model results with data from Jaeger (1) Cooling of 3D semi-infinite slab radiating to environment at OK Mesh size 0.01 m 1.00  =model =(1) .  0.80  0.60  0.40  (1) Jaeger, J.C., roceedings, Cambridge  0.00 -2.0  -1.0  Ilosophical Society, vol.  0.0 Log T 10  1950, pp. 634-641  1.0  2.0  Figure 4.4b Cooling of 3D semi-infinite slab radiating to Farfield Environment at OK 43  Comparison of model results with data from Jaeger (1) Cooling of 2D semi-infinite slab radiating to enclosure at OK Mesh size 0.01 m 1.c =model  •  =(1) n oi  0.60  N SA  0.40  0.20  (1) Jaeger, J.C., roedinga, Cambridge I hilosophlcal Society, vol.  0.00 -2.0  -1.0  0.0 Log T 10  1950, pp. 634-641  1.0  2.0  Figure 4.4c Cooling of 2D semi-infinite slab radiating to Enclosure at OK Comparison of model results with data from Jaeger (1) Cooling of 3D semi-infinite slab radiating to enclosure at OK Mesh size 0.01 m lCD, =model  -.----------.--  =(1) 0.80  0.60  u.40  (1> Jaeger, j.c., roceedings, Cambridge hilosophiosi Society, vol.  0.00 -2.0  1950, pp. 634-641  —  -1.0  0.0 Log T 10  1.0  2.0  Figure 4.4d Cooling of 3D semi-infmite slab radiating to Enclosure at OK 44  Comparison of model results with data from Jaeger (1) Cooling of 2D semi-infinite slab radiating across gap interface to solid at OK Mesh size 0.01 m 1.0o  \  =model  —.-.----..  =(1) NA  0.80  NA  0.60 &  3 “A  0.40  0.20  (1) Jaeger, J.c., roceedings, Cambridge  0.00 -2.0  -1.0  ilosophlcal Society, vol.  0.0 Log T 10  1950, pp. 634-641  1.0  2.0  Figure 4.4e Cooling of 2D semi-infinite slab radiating across Gap Interface to Solid at OK Comparison of model results with data from Jaeger (1> Cooling of 3D semi-infinite slab radiating across gap interface to solid at OK Mesh size 0.01 m ‘  =model =(1) 0.80  -  N  0.60  A  0. 40  0.20  (1) Jaeger, ,i.c., roceedings, Cambridge hilosophical Society, vol.  0.00 -2.0  1950, pp. 634-641  —  -1.0  0.0 Log T 10  1.0  2.0  Figure 4.4f Cooling of 3D semi-infinite slab radiating across Gap Interface to Solid at OK 45  Chapter 5 Experiments  Experimental trials were conducted in collaboration with Deloro Stellite Inc. at their plant in Belleville, Ontario. Ceramic moulds of two configurations were instrumented with thermocouples and time-temperature histories recorded during casting and cooling. The data collected were used to verify the mathematical model formulation.  5.1 Experimental Procedures  The two casting configurations are shown Figures and b. The simple cylindrical mould was used because the design is easily modelled (to a first approximation) by a 2dimensional or axisymmetric model, due to the radial symmetry and the relatively large height-to-diameter ratio. Thus, this model could be used to obtain data on the mould-metal interface heat transfer, and to verify data on the thermophysical properties of the mould material.  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 production procedures [33,34]. The moulds were cemented to a refractory brick base, which also held a ceramic 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 high temperature capability, as well as its ability to be used in oxidizing and vacuum environments. Compensating lead type Cu-Alloy 11, 24 gauge extension wire was used to connect the thermocouple manifold through the thermocouple port in the furnace to the data acquisition 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 operating  46  pressure of 1  mbar.  The vacuum furnace mould chamber door was modified with a  thermocouple feed-through port. The data acquisition system consisted of an Advantech PCL8 18 data board with a PCLD789 MUXICJC/screw terminal. The system was controlled using LabTech Notebook installed on a 486-33MHz IBM compatible computer. For installation of the thermocouples, narrow grooves were ground into the mould using a disc-shaped grinder. Bare wire, twisted junction thermocouples were embedded in the moulds at three depths: near the surface (.050 .080”), at midthickness (.120 .180”) and -  at the facecoat (.260  -  -  .310”) of the mould (Figure 5.3a). As well, several thermocouples  were placed at the interface in the melt. These thermocouples were alumina-sheathed, and the bare junction was given a thin coating of ceramic slurry. The location and depth of each thermocouple was recorded. Thermocouples were cemented in place using Sairset, a mould repair ceramic  [35]1.  The thermocouples were then placed in alumina sheathing, and  threaded through the thermocouple manifold.  Figures 5.4a and b show examples of  instrumented moulds before casting. The cylindrical moulds were sectioned after casting to verify the thermocouple locations (Figure 5.3a).  Table 5.1 summarizes the locations of the thermocouples in the  moulds for the cylindrical castings. The final depth measured was somewhat less than the initial groove depth in most cases, due to displacement of the thermocouple junction in the embedding and drying stages. It was also observed that, upon drying, the Sairset ceramic on several thermocouples had contracted away from the bottom of the groove in which the thermocouple was installed; the thermocouple adhered to the Sairset, and was pulled away from the mould ceramic at the measured depth; a void was thus formed between the Sairset and thermocouple, and the mould (Figure 5.3b). Remarks on these observations are included in 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. This presented 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. -  47  well. 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 finned casting. The mould configuration was such that the mould tended to break apart during cooling. It was thus not possible to measure the thermocouple depth by sectioning the mould after casting, as was done with the cylindrical moulds. Based on the difference in final measured depth vs. initial depth as determined from the cylindrical moulds, corrections could perhaps have been applied to the finned casting thermocouple locations, on the assumption that the installation method used was the same. However, the errors associated with the mould non-uniform thickness and surface roughness, as well as the subsequent approximations of constant thermophysical properties for modelling runs, were judged to be large relative to the error in exact location of the thermocouples, and thus the upper bound depths are shown. The thermocouples were calibrated to within one degree Celsius using the boiling point 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 by connecting 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 induction furnace power supply and coils, attention was paid to the possibility of noise inducement in the 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 between the melt chamber and pour chamber. In two instances the mould lock was inadvertently opened before the power was turned off. Both these data runs exhibit slight noise prior to the pour. The entire mould-base-manifold assembly was placed into the preheat furnace at 1093 deg C (2000 deg F) for one to two hours. The charge was melted under vacuum (see Table 5.111), and superheated to 1565 deg C (2850 deg F). The mould was transferred from the preheat furnace to the furnace mould box.  The extension leads were connected to the 48  manifold. The chamber door was closed, and the mould chamber evacuated to between 10-2 and 1 ü mbar (see Table 5.111). Power to the induction coils was shut off just before the mould lock was opened. The mould box was positioned under the crucible, and the metal cast. 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 molten metal could occur, damaging the thermocouple manifold,  and shorting out the  thermocouples. To protect the manifold, a sheet of fiber paper was placed over the entire manifold in the furnace (Figures 5.4c and d). The procedure was used for the finned castings as well. It became apparent that the presence and location of the fiber paper with respect to the thennocouples had an effect on the cooling behaviour of the mould surface; some thermocouples were shielded from the water-cooled furnace wall by the fiber paper, which thus acted as an insulator. Others radiated directly to the cooled furnace wall. This effect is considered in the analysis of the data and is included in the sensitivity analysis to assess the magnitude of possible variations in the recorded temperatures. Data sampling was commenced at the moment that the mould was taken from the preheat furnace, and continued at a sampling rate of 1 Hz for one hour.  As each  thermocouple was connected, its response was checked on the display terminal to verify that it was functioning. After pouring, short-checks were conducted at the screw terminals. This was found to be necessary to establish that thermocouples were reading independent signals. During a pour, molten metal occasionally splashed over the mould onto the thermocouple manifold, causing shorting between thermocouples. By placing a jumper wire between the high and low terminal junctions, a signal could be reduced to zero (reading the temperature of the screw terminal junctions). If two or more signals reacted to shorting at one thermocouple connection, the thermocouples were assumed to be electrically coupled in the furnace. These thermocouples were subsequently disregarded in data analysis.  49  The time at which short  checks were conducted was recorded, and the data subsequently edited to remove these spikes by averaging between the nearest smooth data. The compensating leads were connected to copper alligator clips. The high and low clips were mounted together on a small ceramic base. This enabled the two leads to be connected 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 error in the thermocouple readings. With reference to Figure 5.6, it was judged that, due to the high conductivity and thermal diffusivity of the copper, and the small size of the clips, the PtCu and Cu-Alloy 11 junctions in the negative half of the circuit could be assumed to be at the same temperature and thus formed an isothermal block junction, having little or no effect on the net voltage recorded at the screw terminal.  5.2 Discussion of experimental results  Plots of the resulting thermal histories for the various runs are shown in Figures 5.7 to 5.12 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 the following behaviour: 1) cooling of the mould after connection of the thermocouples before pouring, 2) rapid heating of the mould after pouring, 3) slower cooling during solidification, corresponding to the latent heat arrest, and 4) rapid cooling asymptotically approaching ambient temperature. The finned configurations do not show a pronounced cooling plateau associated with the release of latent heat of solidification (stage 3 above).  This may be  explained by the larger surface area-to-volume ratio of the fin over the cylinder, which allows the 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.  50  A parameter of interest in process control of casting procedures is the mould temperature at time of pour. Efforts center on maintaining the preheat furnace for the mould at a sufficiently high temperature to enhance flow of liquid metal through the mould. The cylindrical casting data indicated that, although the moulds were initially preheated to ‘-jl 000 deg C, there was significant heat lost in the transfer of the mould from the preheat furnace to the 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 mould temperature at the time of pour, the mould cool time must be kept to a minimum. The beneficial effect of a higher preheat furnace temperature will be lost very rapidly as time to pour increases.  5.2.1 Cylindrical Castings  The thermocouples on the cylindrical castings were spaced 2  -  3 cm apart measured  along the mould outer circumference. It was expected, from the synmietry of the cylindrical casting geometry, that the heat flow would be radially symmetric, and that there should be no difference in thermal profiles measured at the same depth and height, but at different locations on the circumference. The validity of this assumption may be influenced by several factors.  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 sensitivity analysis in Chapter 6. Asymmetry in the mould-metal interface condition as solidification and 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 some characteristics which could not be explained when the assumptions of radial symmetry and  51  uniform cooling environment were applied. 2 In particular, the temperature at longer times was somewhat anomalous in some instances, as some of the temperatures at deeper locations fell below that at shallower locations. TC1 and TC2 of casting C7 fall below TC4, for example (Figure 5.9), and TC6 of casting C9 falls below TC4 (Figure 5.11). TC7 of casting C6 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 with attention to the position of the thermocouples relative to the fibrepaper shielding (Figure 5.4c). Over short time scales, the insulating effect of the fibrepaper is most significant on surface temperature measurements, resulting in a higher maximum temperature for thermocouples located near the surface of the mould.  At longer time, the enclosure  temperature directly ‘seen’ by the shielded thermocouples is higher. The apparent terminal temperature of shielded thermocouples (both near the surface and deeper within the mould) is therefore higher than for the unshielded thermocouples, which ‘see’ an enclosure at a much lower temperature.  Finally, at long times, the temperature drop across the mould will  decrease (the mould interior temperature approaches the mould exterior temperature), as the mould metal interface heat transfer rate becomes limiting. The position of the fibrepaper may result in temperature profiles differing with height location of the thermocouple (above or below the level of shielding) and with placement around the circumference (thermocouples directly facing the fibrepaper may be more  For casting C7 (Figure 5.9), position “e” was recorded as being monitored by TC4, and 2 position “b” by TC3. Position “e” was at a depth of 0.145” below the mould surface, and slightly shielded by the fibrepaper, while position “b” was at a depth of 0.090”. It would thus be expected that the temperature at “e” be higher than that at location ‘b’; the reverse was observed in the data, however. The embedded thermocouple junctions were examined for 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 incorrectly when the mould was removed from the furnace, with TC3 monitoring position “e”, and TC4 monitoring 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 the thermocouple junctions. -  52  shielded than those placed further around the sides of the mould). In Figures 5.7 to 5.12, the circumferential location is indicated in the diagram, while the designation “s” (shielded), “ps”  (thermocouple approximately level with upper edge of fibrepaper) and  ns” 11  (not  shielded) indicate vertical placement of the thermocouple relative to the fibrepaper. The localized differences in cooling due to the conditions described above were considered to be a possible explanation for the observed temperature behaviour. The possible magnitude of these 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 C8 drops more rapidly than at TC1 and TC5 of casting C6. discontinuity at approximately time t  =  It also shows a temperature  350 seconds. These observations may be the result  of imperfect contact between the casting surface and the thermocouples such that there is a slight surface resistance resulting in a temperature drop between the thermocouple and the metal surface. Thermocouple TC5 of casting C8 (Figure 5.10) was not consistent with any of the other temperature profiles. Based on the preceding discussion of shielding effects, TC3 could be expected to have a higher temperature throughout. The response of TC5 seemed to indicate that the material surrounding TC4 had a significantly higher thermal diffusivity than that 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 sensitivity analysis.  5.2.2 Finned Castings  The experimental data for the finned castings are shown in Figures 5.13 to 5.19. The curves differ from those of the cylindrical castings in the absence of a latent heat arrest plateau.  All show a fairly sharp maximum at approximately 100 seconds after casting, 53  followed by a continuously decreasing cooling rate. Except for castings Fl and F9, which had thermocouples near the face coat of the ceramic, the thermocouples in the finned castings were placed near the surface to detect variations in the cooling conditions, that is, maximize the sensitivity to the radiation boundary condition. It was attempted to measure temperature variations with respect to radial position along the fin, vertical position on a face, and corresponding location on the differently-oriented faces. The data of the finned castings follow self-consistent profiles, in that cooling rates between thermocouples of any single casting are consistent within reasonable variation due to the difference in location on the casting. Thermocouples which were shielded by the fiber paper are indicated in Figure 5.13 to 5.19 with the designation ‘s’. Examining the temperature profiles for castings F4, F5, F6 and F7 (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 in the mould is similar for all. Casting F9 (Figure 5.19), having all thermocouples located at the same height, and thus with approximately the same degree of shielding, shows very little temperature spread in the profiles, as does casting F8 (Figure 5.18), having all the thermocouples 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 were shielded by the fibre paper, show similar traces. The deep traces exhibit a discontinuity at time t = 1000 which makes subsequent data suspect.  54  Table 5.1 Location of Thermocouples in Cylindrical Castings Casting TC# Position height depth Mould (in) (in) thickness (in) C5 3 b 4.6 0.135 0.315 (large void) 4 c 4.6 0.040 0.335 6 a 4.6 0.175 .0335 C6 1 a 4.6 Metal Surface 7 b 4.6 0.265 0.320 4 c 4.6 0.030 0.325 6 e 4.6 0.030 0.320 3 f 4.6 0.295 0.340 (void) 5 4.6 g Metal surface C7 1 a 4.6 0.285 0.340 (void) 2 d 1.9 0.260 0.300 3 e 1.9 0.145 0.310 4 b 4.6 0.090 0.305 6 f 1.9 0.035 0.335 C8 1 a 4.6 0.290 0.320 (void) 3 c 4.6 0.035 0.340 4 d 4.6 Metal Surface 5 f 4.6 0.030 0.300 C9 3 e 1.9 0.235 0.280 4 f 1.9 0.030 5 c 4.6 0.050 0.315 6 d 4.6 0.135 0.335 ClO 3 e 1.9 0.240 0.300 4 f 1.9 0.035 0.295 5 4.6 c 0.040 0.335 6 d 4.6 0.135 355 shielded, ps = partially Shielded -  -  -  -  55  Shielding  Ps  Ps Ps ps Ps ps Ps Ps ps s s s s s Ps ps ps Ps s s p ps s Ps  ON C-ti -  L3  Tj  U ON -  Ci.)  ‘J  i i—’  CD  —  CD  4 4  i-+  .1  -  CD  C  CD  L) L) ..  CD  C-ti  b b  -  .  c  k)  C)  t’J  k) C.)  -  ON  CD  C-’I  ) )  -  ON  Ui  C-ti  ON  Tj  -  C-ti  —  Li.) L’J  -‘  ---  I—’  CD  -  C)  CD  —  -t  -‘  —‘  C CD  —  ON  CD  bc  C)  i—i  ------------i  —  -------------E CJ  ---—  be b ccc c be be bce  ‘J  -  -  —  CO  CO CO  CO CO  CO  Cl)  I I I I I I I I I  Cl)  —.  C/)  C/)  I  CO  C/)  I I  CO  Cl)  Cl)  CO  Cl)  CO  CO  Cl)  CO  CO  C/)  CO  CO  I I I I I I I I I I I I I I  C/)  Cd)  CO  I I  Cd)  Cl)  CO  —.  cccceceececpeppcpcpppppppp9pppc, j’j b c e c c e e e c c c c c c c c c c c c c t’ i c — oo 0000  E  Cf)  C)  NQ 00000000O\O,  c  ——----—————-  t’J  CD  CD — CD-———----—  p-  CD  CD CD-  CO  i—l  00  ----—  i.----------  CD  -  CO  —  CD  CD-—  I  CD  C/)  C/)  C)  CD  .  0  0  C CD  Casting C2a  C4 C5 C6  C7  C8 C9  ClO Fl F2 F4 F5  Table 5.111 Vacuum Furnace Pressures during Melting and Cooling Melt Vacuum Cooling Vacuum/time (mbar) (mbar/sec) 4 5.0x10 /190 2 5.0x10 2 / 1000 1.1x10 _/_1300 3 8.3x10 4 8.5x10 /265 2 1.8x10 2 5.0x10 /l00 0 5.8x10 3.4xl0 /180 2 1.8x10 2 / 550 1.8x10 3 / 1600 6.3x10 3 /2250 4.2x10 3 / 3350 2.6x10 2 l.5x10 /210 2 3.7x10 2 / 550 l.9x10 3 / 1420 7.1x10 3 / 1980 5.6x10 3 / 3360 2.7x10 3.5x10 /650 2 1.9x10 _/_1930 3 7.1x10 3 2.0x10 /240 2 2.5x10 3 /2020 6.0x10 3 / 2320 5.2x10 3 / 3000 3.5x10 3 / 3600 2.7x10 nld* /900 2 2.0x10 _/2830 3 4.7x10 4 1.1x10 /170 2 1.3x10 3 1.0x10 1 / 160 1.0x10 /2510 4 l.0x10 nld* /1370 3 8.0x10 _/_1776 3 6.9x10 4.7x10 /180 2 2.0x10 2 / 840 1.9x10 3 / 1800 9.8x10 3 /2400 8.2x10 3 / 2700 7.5x10 _/2100 3 7.1x10  *mssing value  57  Casting F6  F7  F8  F9  Table 5J (cont’d) Vacuum Furnace Pressures during Melting and Cooling Melt Vacuum Cooling Vacuumltime (mbar) (mbar/sec) nld* /150 2 7.5x10 2 / 420 2.9x10 2 / 1100 1.3x10 3 / 1700 8.7x10 _/_3000 3 5.8x10 3 5.0x10 /230 2 5.0x10 2 /440 2.8x10 / 1050 2 1.3x10 2 / 1350 1.0x10 3 / 1700 8.3x10 _/ 2100 3 7.1x10 4 1.6x10 2.5x10/310 2 / 390 5.8x10 2 / 750 2.1x10 3 / 1600 8.8x10 3 / 2100 6.8x10 3 / 3250 4.5x10 _/_3600 3 3.9x10 4 3.0x10 /140 2 2.0x10 2 / 500 2.6x10 / 1300 2 1.0x10 3 / 1550 8.3x10 3 / 1850 7.0x10 3 / 2850 4.8x10 3 / 3600 2.4x10  *missing value Table 5.IV Cylindrical mould initial cooling rates after removal from preheat furnace Casting Average initial cooling rate (deg_C/mm) C5 144.6 C6 147.0 C7 141.0 C8 160.2 C9 148.2 dO 130.2  58  5000L13  •38R MAX (TYP) 20000±38  I. DIHESIUS i K 2. SURFACE FINISH 3.2 RHR 3. ATER1AL SFRALLUY 718  CASTI, TEST, [YLINDRI[L ak  LQ  I nm  .  Ifjfl1iA IIuI. Lt. U8I T  Figure Cylindrical Test Casting  2iWLI3  I. DIdfli 2. II3.2 2. IITRJ4L: 9Fkia 218  CASTI, FIIfU, TEST AtE GT  I  o IflStS •‘- SH8 .  (&1LiLS l8I LISItY LC E8A 6I 114  Figure Finned Test Casting 59  l1I8 md1A  Figure 5.2 Balzers VSGO5O Vacuum Furnace (Deloro Stellite Inc.) Mould exterior surface  a:  Mould Interior surface  Shrinkage void in Sairset  Figure 5.3 Schematic of thermocouple installation in ceramic moulds a: Typical installation showing initial and final measured depth b: Example of shrinkage void observed in a few of the thermocouple locations  60  Figure 5.4a Typical cylindrical mould instrumented for testing 61  Figure 5.4b Typical finned mould instrumented for testing  Figure 5.4c Cylindrical mould in furnace after casting, showing manifold shielding  62  Figure 5.4d Finned mould in furnace after casting, showing manifold shielding  ::•:::,j::  Figure 5.5 Thermocouple location in finned castings (Ref. Table 5.11)  Cu clip Ptl3Rh÷  Cu  Tmef  Cu clip Connected circuit  Tm  Isothermal junction Equivalent circuit  Figure 5.6 Pt-Ptl3Rh equivalent thermocouple circuit  63  Trial RunC5 1400  1200  jl000  D  800  E  ,!600  400  Da wnpe 10t27192  200  Ddo S4ee. 8ee,Ie Ont. 8.Donii  500  1000  1500  2000  3000  lime (seconds)  Figure 5.7  Trial Run C6 1400  1200 0  1000  Q 0  800 E I—  600  400  DaasdnW4cd 1cv28’92 Doo Sce, Beevie t. B. Dommik  200  0 500  1000  1500 Time (seconds)  Figure 5.8  64  2000  2500  3000  Trial Run C7  1200  o  1000  800  E I-  600  400  Da STPI  200  Dio S*ete. Bdc’.4e O’d. B. DaninUc  0 500  1000  1500 Time (seconds)  2000  Figure 5.9  Figure 5.10 65  2500  3000  Figure 5.11  Figure 5.12 66  Figure 5.13  Figure 5.14 67  Trial Run F5 1400  1200 0  -g  1000  800  a,c,e 600  400  200  a: 0.075° top b: 0080” top C: 0.080° mid-height d: 0.075” mid-height e: 0.070” bottom f: 0.070° bottom  Datasampled 1112/92 Detom Steflke Inc.. Belleville Ont. B.DonIk  0  500  1000  1500 2000 Time (seconds)  2500  3000  Figure 5.15  Trial Run F6  1400  e,f  1200  1000  a, 0  E F—  b  800  a,b 600  400  200  a: b: e: 1:  0.055° inside face, top 0.060” inside face, mid-height (s) 0.060 outside face, mid-height 0.065° outside face, bottom  Data sarrIed 11/3/92 Doro Stellhte Inc., Behievihle Ont. B.Dominhk  0 500  1000  1500 2000 Time (seconds)  Figure 5.16 68  2500  3000  1400  d,e  Trial Run F7  1200  1000 C)  800  600  400  200  a: 0.055” top inside b: 0.055’ mid-height inside (s) C: 0.055” bottom inside (s) d: 0.045” top outside e: 0.050” mid-height outside  Data sanpled 11/3/92 Deloro Stellite Inc.. Belleville Ont. BDoninik  0 500  1000  1500 2000 lime (seconds)  2500  3000  Figure 5.17 Trial Run F8  /  1400  d,e,f  a,b,o  1200 0 .  1000  0 Co  0  800  a  E 0  600  400  a c,d Data sarrpled 11/4/92 Deloro Stellile Inc.. Bellevite Ont. B.Dominik  200  0 500  1000  1500 2000 Time (seconds)  Figure 5.18 69  2500  3000  Trial Run F9 1400  1200 C)  1000  a,  a,  800  f  a)  0 0  E a,  F.  600  d 0  a:  400  200  0.235” 0.070” d: 0.075” e: 0.150” f: 0.240” C:  Datasampled 1114192 Debra Stebthe Inc., Bebbedlie Ont. B.Dombnbk  0 500  1000  1500 2000 Time (seconds)  Figure 5.19  70  2500  3000  e  Chapter 6 Sensitivity Analysis and Analysis of the Casting Process  6.1 Sensitivity analysis -cylindrical configuration  A simple, 2-dimensional model of the cylindrical casting was analyzed for the purpose of conducting a sensitivity analysis. The sensitivity analysis was performed with  two aims: 1) to determine the effect of each parameter on the solidification and cooling of the metal, leading to an understanding of how they may influence casting quality, and 2) to determine the effect of each parameter on the temperature distribution in the mould, for adjustment in fitting the model results to the experimental data, and for assessment of the data. The casting system is potentially influenced by a large number of parameters for the metal, the mould, the interface conditions and the surroundings. The parameters chosen for the sensitivity analysis were those associated with the greatest uncertainty, or for which little data were available. These are summarized in Table 6.1. A number of parameters were obtained from the literature; it was assumed that these parameters were sufficiently well established, that their inclusion in the sensitivity analysis was unnecessary. These parameters and their values are summarized in Table 6.11. The results of the sensitivity analysis are shown in Figures 6.2 to 6.13, and summarized in Table 6.111.  In the figures, the four curves shown for each condition  correspond 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 total solidification time (center of casting), interface heat transfer (casting exterior and mould interior) and heat transfer to the surroundings (mould exterior) can be readily assessed. The total solidification time  (  t  in Table 6.111) is obtained from the temperature trace at the  casting 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 of  71  temperature. The parameters in Table 6.111 are ranked in the order of decreasing effect on total solidification time (%At)  6.1.1 Interface heat transfer assumed contact time  Figure 6.11 shows the effect of variation in the time over which interface contact is assumed to exist. It can be seen that the time temperature response depends on whether the -  contact time is increased or decreased over the base value. A decrease in the contact time by 50% results in a 14% increase in total solidification time, and results in a “double plateau” in the 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 the model. The “double plateau” in the mould cooling curves which resulted when the contact time was less than the solidification time was not observed in the experimental data. The gap formation was assumed to be due to solidification shrinkage only; after solidification was complete 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 solidification time. The magnitude of the effect of the interface contact function indicates that the gap formation characterization is very important in the implementation of an investment casting simulation model.  6.1.2 Metal initial temperature  Figure 6.5 shows the effect of varying the metal initial temperature. The metal initial temperature had a strong effect on total solidification time, and a slight effect on the mould temperature and metal surface temperature profile. This parameter was known from the 72  experiments 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 occurs very quickly during the pouring and mould filling stages, and the initial temperature profile may 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 metal  temperature was assumed, at a value adjusted for the best fit to the data, but between the bounds given above. The metal pour temperature is used in the foundry industry as a process control parameter, and is known to influence mould filling. In intricate parts with thin sections, a significant superheat may be required to ensure complete mould filling.  6.1.3 Mould thermal conductivity  Figure 6.2 shows the effect of the assumed mould thermal conductivity. The effect on total solidification time is significant (5.9% increase in solidification time for a 10% decrease in thermal conductivity). The mould thermal conductivity is associated with a large degree of uncertainty. The ceramic shell system is inhomogeneous, consisting of layers of varying composition. Each layer, as well, has several components, including the stucco Or flour, the binder and a porosity fraction. The thermal conductivity varies with composition, and also with temperature; intergranular radiation in the pores and transgranular radiation through the solid 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 thermophysical properties.  Because of the uncertainty associated with the mould conductivity and its  relatively large effect on the temperature profiles, this parameter was one of those used to adjust the model fit to the experimental data.  73  6.1.4 Metal latent heat of solidification  Figure 6.12 shows the effect of variation in the value of metal latent heat of solidification. 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 a 4.7% reduction in total solidification time. The effect on the mould temperature profiles was not significant. Due to this small effect, and the relatively good characterization of this value in the literature, the latent heat was not varied in fitting the model.  6.1.5 Interface contact initial heat transfer coefficient  Figure 6.10 demonstrates that, although there is a negligible effect on total solidification time, the assumed initial interface contact heat transfer coefficient strongly influences the mould temperature profiles. It is also associated with a large uncertainty. This parameter was therefore one of those adjusted in fitting the model to the data.  6.1.6 Fibrepaper shielding  Figure 6.13 assesses the effect of the presence or absence of the fibrepaper manifold shield on the temperature (see Figures 5.4c/d and  Although the effect on total  solidification time is negligible, there is a pronounced effect on the surface temperature over the whole time range, and on the entire mould profile at longer times. Although it was difficult to incorporate this effect for each individual thermocouple location, in particular with a 2-dimensional model, the possibility and magnitude of the error due to this effect was assessed in the analysis of the experimental data. As discussed in chapter 5, the circumferential placement of thermocouples with respect to the fibrepaper in the cylindrical castings may account for the temperature profiles 74  intersecting in the experimental data. A model run was done to assess the magnitude of circumferential temperature differences due to the fibrepaper shielding. Figure 6.14 shows the temperature profile at four points in the mould: two at 0 degree orientation with respect to the fibrepaper, and two at 90 degree orientation (Figure 6.ib). The depths shown are typical 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 drop consistent with the difference in depth location. Similarly, the temperatures at N1415 and N1405 maintain a consistent temperature drop. However, the position of the 0 degree vs. 90 degree traces changes, as the unshielded traces decrease in temperature more rapidly than the shielded 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 confirm that the circumferential position of the thermocouples may result in the intersecting temperature traces observed in the experimental data.  6.1.7 Variables having minor effects  Figure 6.4, showing the effect of mould initial temperature on the temperature profiles, 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 temperature is used in the foundry industry as a quality control parameter. This results may be due to a number of factors.  The cylinder is a very simple geometry, with no complex feeding  requirements, and of relatively large section (0.025m radius) somewhat atypical for an investment casting. Filling of the mould requires no flow through thin sections or convoluted geometries, which results in rapid loss of superheat, premature freezing, and non-fill of castings. 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 the  75  system. 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 mould temperature profile at shallow depths.  The mould thickness was seen to vary not only  between 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 the model to the data, a constant mould thickness was used throughout. Figure 6.6 shows the effect of varying the mould emissivity. A negligible effect on total solidification time is observed; however, the mould temperature is slightly affected, near the exterior surface and at long time.  This parameter was poorly documented in the  literature, 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 total solidification time and the mould temperature profiles.  Values from the literature were  therefore used and maintained constant. The assumed ambient temperature (Figure 6.8) and assumed metal emissivity (Figure 6.7) are shown to have negligible effects on both total solidification time and the mould temperature profiles, and were thus held constant.  6.1.8 Analysis of heat flow resistances  An analysis of the relative magnitude of heat flow through the metal and mould versus heat transfer at the surfaces can explain several important aspects of the mould metal -  cooling curves. As solidification progresses, the interface heat transfer conditions become increasingly more important. The heat transfer Biot number represents the ratio of resistance to heat transfer at the surface of a medium to resistance to heat flow through the medium: 76  Bi  h /Ax  where h is the surface heat transfer coefficient, k is the medium thermal conductivity and Ax is a representative dimension for the conducting medium. For Bi> 1.0, heat can be removed from the surface more rapidly than it can be conducted to the surface through the medium. For the mould, using a thermal conductivity of 1.0 W/m-deg C, a thickness of 0.008m and an initial surface heat transfer coefficient at the mould-metal interface of 1500 W/m 2 deg C results in  Bz,OUfd_Ifl  1500x0.008 1.0  =  12.0  Thus, at the mould interior surface, the mould conduction limits the heat flow.  This  conclusion is in agreement with the indication from the sensitivity analysis that, at short times, 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 2 W/m deg C), so  Bifliouldex  40x0.008 1.0  =  0.32  Thus, at the mould exterior, the rate of heat removal from the surface of the mould limits heat flow in the system. In the metal, for kmetal 0.025 m,  77  =  11.4 W/m-deg C, and a cylinder radius of  Blnieiai_ex =  1500x0.025 114  Here, conduction through the metal initially limits heat flow. As the metal solidifies and contracts, the interface contact component of heat transfer decreases, and the area fraction of radiation heat transfer increases.  When the contact  component of the heat transfer coefficient approaches zero, the radiation heat transfer coefficient is calculated to be approximately 157 2 W/m deg C:  Blfl,OUld_Ifl  157x0.008 = 1.25 = 10  Biniejat_ex  157x0.025 =0.34 11.4  Thus, at short times after pour, heat removal from the metal is limited by conduction through the so1idifying shell and conduction through the mould. At longer time, metal cooling is limited by the rate at which heat is removed from the metal surface. Mould conductivity limits 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 after pouring, the temperature drop between the metal center and metal surface and the mould interior and exterior surfaces is large, and the temperature drop across the interface is small reflecting the conduction resistance limitation.  As the interface heat transfer coefficient  decreases, the metal attains an approximately uniform temperature, and the temperature difference between the interior and exterior mould surfaces decreases.  The widening  interface temperature drop reflects the decreasing gap heat transfer coefficient, while the  78  decreasing mould temperature drop reflects the limitation of heat removal from the mould exterior surface.  6.2 Sensitivity analysis finned configuration -  A 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 shielding and self-irradiation on the predicted temperatures. In addition, the results of this analysis were used to aid in interpretation of the experimental data. As most of the thermocouples on the finned castings were near the mould surface, the locations used in the sensitivity analysis are nodes on the mould surface.  6.2.1 Temperature variation with radial position  With reference to Figure 6.21a for placement of the various nodes, Figures 6.22 to 6.24 show the variation in temperature on each face due to the radial location.  The  temperature near the casting center is highest in all cases, decreasing with increasing radial distance from the center. The position ‘d’ traces show an “end effect” reflecting the greater heat loss, due to proximity to the end of the fin. A temperature difference of approximately 70 deg C is indicated between the extreme radial positions (“a” and “d”) on face A. On faces B and C, the temperature difference is smaller (30 deg C). This smaller range reflects the fact that faces B and C experience less self-irradiation than face A.  79  6.2.2 Temperature variation for differing faces  Figures 6.25 to 6.28 show the data replotted to emphasize the difference in temperature due to the self-irradiation effects at each radial location. Faces B and C are at the same temperature at corresponding radial locations. Face A is at a higher temperature, with the largest temperature difference at the interior position (position “a”, maximum temperature difference of —P40 deg C).  The self-irradiation effect decreases at positions  further out along the fin.  6.2.3 Effect of fibrepaper shielding  Figure 6.2 lb shows the position of the fibrepaper manifold with respect to the fin faces. Figures 6.29 through 6.35 summarize the results of model analysis in a similar manner to that of the unshielded model.  As seen in Figures 6.29 to 6.31, the magnitude of  temperature difference between the radial extremes does not change significantly over the unshielded model (70 deg C for face A, 30 deg C for faces B and C).  However, the  temperature difference between the faces shows a dramatic increase (Figures 6.32 to 6.35). A maximum temperature difference of-45O deg C is observed between face A and faces B and C. A secondary effect of the shielding is shown on face B at longer times, as the traces of B and 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 unshielded models. The results of this analysis of the finned configuration indicate that, in analyzing the data from the finned casting runs, the fibrepaper shielding and physical location of the thermocouples are dominating factors in explaining the spread in temperature profiles seen in the data.  80  6.3 Comparison of experiments with model results  The 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 castings  Figures 6.15 through 6.20 show the best fit of the thermal model results with experimental data. As indicated by the sensitivity analysis, the mould conductivity and the interface heat transfer conditions had the most significant influence in adjusting the model predictions. For the interface heat transfer, a linearly decreasing function was used. As discussed in section 6.1.1, the time over which the interface contact was assumed to exist was set approximately equal to the total solidification time, as determined from preliminary model runs. The value of the initial contact coefficient was adjusted based on fitting the model to the interface data of castings C6 and C8. Initial values of the mould thermal conductivity and mould emissivity were obtained from the literature. Based on the model sensitivity analysis response, the parameters were varied to align the model with the experimental data.  Mould initial temperatures were set to those indicated by the  experimental data for each model run. The values of the parameters used are summarized in Table 6.IV. Average mould thermal conductivities range from 0.9 to 1.1 W/m-deg C. This is somewhat higher than values reported in the literature.  Thermal diffusivities range from 0.00375 to 0.00458  cm / 2 sec, which is in agreement with reported values. Mould emissivity values range from 0.55 to 0.65. Reasonably good fits to the experimental data can be obtained using the constant values of the mould thermophysical properties.  81  In particular, the cooling rate at mould  temperatures below 750 deg C is well reproduced. At this lower temperature, the thermal conductivity of the mould will likely be less temperature dependent as the influence of void radiation will be reduced. During solidification, the larger uncertainties associated with the heat transfer at the mould-metal interface and possibly the constant value of thermal conductivity of the mould resulted in slightly poorer fit of the model to the data.  Casting C5 (Figure 6.15) shows good  agreement of the model with the temperature profile at 0.040” and 0.175” depth. The model temperature 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 the experimental data, in particular over the first 700 seconds.  This data set, having a  thermocouple in the metal, was used to establish more closely the interface heat transfer contact conditions. Casting C8 (Figure 6.18) shows a reasonably good fit in the mould temperatures. The metal surface temperature is over-predicted by up to 8%. The model did not reproduce the temperature discontinuity at time t = 350 seconds. The effect of the fibre paper shielding on the mould temperature distribution has particular 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 not significantly affected until long times (after solidification is complete). The predicted surface temperature is higher by up to 50 deg C due to the presence of the shielding. As castings C9 and ClO combined shielded and unshielded thermocouples, the model was run without a shielding condition, and the temperatures for thermocouples TC4 (near the surface of the mould), were adjusted down by 50 deg C to compensate for this condition, based on the magnitude of the effect indicated in the sensitivity analysis.  This is indicated by the  designation TC4’ in Figures 6.14 and 6.15. Casting C9 shows good agreement between the model results and the experimental data. For casting Cl 0, the model overpredicts the surface  82  temperature, possibly due to non-uniform cooling effects in the casting run which are not included in the model.  6.3.2 Finned Castings  The results of the finned configuration sensitivity analysis can be used to help interpret the experimental data and to provide insight into the differences in temperature observed at thermocouples located at similar radial position and depth, but on different faces and at different heights on the mould.  On casting F8 (Figure 6.41), which shows a  comparison between predicted and measured temperature, the thermocouples were all located at similar radial locations and depths, and were not shielded from the furnace by the insulating fibre paper. Thus, the difference in temperatures observed in the experimental data must be due to combined effects of varying height on the casting and non-uniform cooling environment due to the presence of the mould box, as well as slight differences in depth location of the thermocouples. The experimental data plot of casting F8 indicates that the above conditions may account for a temperature difference of up to 50 degrees C at the mould surface. Therefore, when modelling the finned casting as a 2-dimensional model with only the furnace and fibre paper influencing the cooling, results within approximately 50 degrees of the experimental data represent a reasonably good fit. Castings F5, F6 and F7 (Figures 6.38 to 6.40), with thermocouples located on various faces near the mould surface, show obvious effects of shielding from the fiber paper. For casting F5, having thermocouples at shielded, partially shielded and unshielded positions on Face A, the magnitude of temperature difference between the shielded and unshielded positions is similar to that predicted by the model; the maximum temperature difference is approximately 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 due to a radial difference in location (temperature of ‘b’ higher than that of ‘a’ temperature of ‘d’ 83  higher than that of ‘c’); traces ‘e’ and ‘f show a reverse difference from that expected, possibly due to combined experimental error as discussed above. For casting F6, all thermocouples are at the same radial location. Traces e and f, both on Face B (unshielded), show similar temperatures. Trace ‘a’, on face A (unshielded), is at a higher temperature (60  -  80 degrees) relative to ‘e’ and ‘f’, consistent with sensitivity analysis  results. Position ‘b’, shielded by the fiber paper, is at a significantly higher temperature than position ‘a’ over the whole time range.  Casting  F7 also shows data consistent with the  sensitivity analysis with regard to shielded and unshielded temperatures for all thermocouples at 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 and 6.VI summarize the model parameters, and Figures 6.36 to 6.42 show the resulting temperature profiles.  For all of the castings except casting F8, while using a mould  emissivity of 0.60, an average mould conductivity of 0.8 W/m-deg C and thermal diffusivities ranging from 0.0026 to 0.003 8 cm /sec gave the best fit to the data. These 2 values are slightly lower than for the cylindrical castings, possibly due to the fact that the cylindrical mould remained at temperatures where void space radiation increases the apparent thermal conductivity for a longer time. Casting F8 required a mould conductivity of 0.5 W/m-deg C. This lower value is possible due to the fact that casting F8 was poured at ‘—350 seconds after removal from the furnace, whereas the others were poured at between 150 and 250 seconds.  This resulted in a lower initial mould temperature for F8;  the mould  conductivity would thus have less of a radiation conduction component, yielding a lower effective thermal conductivity.  84  m  1  h i  f g  e  b c d  a  Lmetai  tCOntact  h°gap  (mould)  Thick  Tamb  1 Tmeta  metal 8  pC  Kmould  Base  Varied Param.  -  xlO 2.4  1.0  Tliquidus  2.667  (J/m -deg 6 C)  (W/mdeg C)  0.90  Pp  K  fIberpaper 1 Superheat = Tmetal  Run  0.00375 0.00375  .0.00417  /sec) 2 (cm  0.65  0.60  mould 6  0.30  0.35  meral 6  85  650  600  1350 (10)1  1400 (60)1  150  30  Tmould Tmetai Tamb (deg C) (deg C) (deg C)  Table 6.1 Sensitivity analysis parameters  0.00762 5000  hap (mould) 2 (W/m (mm) degC) 0.00838 1400  Thick  350/ 1050  700  (see)  tcontact  258  272  yes  no  Ilberpa (kJ/kg) per Lmetai  Table 6.11 Constant Material Properties Kmetal (W/m-deg C) ) 3 Pmetal (kg/rn CPmetal (J/kg-deg C) Tsolidus (deg C) Tliguidus (deg C) Efurnace  11.4 8190 435 1260 1340 0.60  Table 6.111 Effect of Parameter Variation on Total Solidification Time Run Parameter t %Param. variation %At Varied (seconds) a Base 597 k 681 -50.0 +14.1 548 g 3.5 8.2 Tmetai (83)1 b 632 -10.0 +5.9 Kmould m 569 -5.0 -4.7 Lmetai f 610 + 8.0 + 2.2 Tmould i 584 -9.0 -2.2 ThlCkmould d 585 + 8.0 2.0 mould c pCi, 586 +11.0 -1.8 587 j +257.0 -1.7 h;ap 1 tcongact 588 +50.0 -1.7 1 fiberpaper 606 +1.5 h 600 + 400.0 + 0.5 Tamb e 598 -14.0 + 0.2 Emetal Number in brackets inaicates %variation in superheat -  -  -  I  86  Table 6.IV Value of Parameters used in fitting Cylindrical Experimental Data Run No. Kmould Tmould mould 6 (W/m-deg !sec) 2 (cm (deg C) C) C5 0.9 0.0038 0.55 650 C6 1.1 0.0046 0.60 600 C7 1.0 0.0042 0.60 600 C8 1.1 0.0046 0.65 550 C9 1.1 0.0046 0.65 550 1.1 ClO 0.0046 0.65 550 Table 6.V Constant values used in Analysis of Finned Castings Parameter Value 0.30 Emetal 1.00 EfIbre 0.60 Efurnace 0.60 mould 1400 degC Tmetal 30 deg C Tfurnace 200 deg C Tfibre  Run No. Fl F4 F5 F6 F7 F8 F9  Table 6.VI Value of Parameters used in fitting Finned Experimental Data Kmould Tmould (W/m-deg C) /sec) 2 (cm (deg C) 0.8 0.0033 650 0.8 0.0026 750 0.8 0.0026 750/800 0.8 0.0026 700 0.8 0.0026 550 0.0021 0.5 500 0.9 0.0038 800  87  Figure 6.ia 2D Cylinder Model: Casting O.D. =50 mm, Mould Thickness 0 deg orientation  90 deg orientation Figure 2D Cylinder Model with fiberfrax shielding  88  =  8.3 mm  I—  60)  (0  02 D  ()  ..  600  0000  1250  600  aiso  1b000  5.  1250  0  500  1600 (annrf  Figure 6.2  0000  T,m  2000  2500  O —Cazdngcit• 2- CasOn9 00 3- Mold lnn.r .wfaca 4—Moldo4jlIt,urlao.  Solid: Kld = 1.0 W/m-deg C Dashed: KmJld = 0.90 W/m..deg C  0  600  1600  Figure 6.4  Timeisecondsl  1000  2000  2500  - Castk,g oeM. 2-CusrigOD 3-Mold Inner .urt.c. 4-Mold outer Oixlaoe  Solid: Tmwia = 600 dog C Dashed: Tm=ld 650 deg C  Model results for 2-dimensional cylindrical casting Effect of mould Initial temperature on cooling and solidification  2  Model results for 2-dimensional cylindrical casting Effect of mould thermal conductivity on cooling and solidification  89  02  C-)  I-  I  260  600  1000  1250  600  750  000 e-l  0  1$0  600 -  1000  Figure 6.3  1600  Timo (snnnd1  2000  2500  -  0  2’  500  1000  Figure 6.5  1600  Tim (ttinnr4e  200)  4-Mold  2500  ereurfac.  3-Mold toter enofac.  I .Cadncentr 2-C..SngOO  Solid: Tm. = 1400 deg C Dashed: Tm_ = 1350 dog C  Model results for 2-dimensional cylindrical casting Effect of metal initial temperature on coolIng and solidification  0  I — CaV109 cens. 00 0 2-Cas5n 3- Mold iner swlac. 4- Mold cut suda  Solid: pC = 2.4x1 6 J/m -deg C 3 Dashed: pCi, = 2.664x1 o J/m -deg C 3  Model results for 2-dimensional cylindrical casting Effect of mould heat capadty on cooling and solidification  1250  1250  600  700  E0)  0.  600  750  ) 1000  00  E  0)  I000 00  C)  2  0  Figure 6.6  2000  2500  600  1600  Figure 6.8  1000 ri.,,  2000 2500  I -CasSn5OacS• 2. ae5n5 OD 3- MoM terle .wfsce 4— Mold outer surface  Solid: Tb =30 deg C Dashed:T=150de9C  Model results for 2-dimensional cylindrical casting Effect of ambient temperature on cooling and solidification  500  1 -Cas5rrgcac6• 2- Caidn OD 3—Mold Incel 50r1501 4. Mold outer .wfaoe  Solid: £mOUd = 0.60 Dashed: £moold = 0.65  Model results for 2-dimensional cylindrical casting Effect of mould emissivity on cooling and solidification  90  1250  1260  1:  I000  -  0  I:  1000 e.  C)  0  0  2  Figure 6.7  1Q00 1600 rr.... ,__..J_.’.  2000  2600  600  iSo  Figure 6.9  1500  Time (seconds)  2000  2500  S - Mold mac wrface 4-Mold ouferlurface  1 -CasSog c5r00 2Cae5rrgOD  Solid: Thlckld = 0.330” Dashed: Thick,,d = 0.300”  Model results for 2-dimensional cylindrical casting Effect of mould thickness on cooling end solidification  500  I — Cas5n cqn6. 2-CasSogOD 3—Mold lrwr swtac. 4-Mold Outer surface  Solid: £m.a = 0.35 Dashed: m.(iI = 0.30  Model results for 2-dimensional cylindrical casting Effect of metal emissivity on cooling and solidification  I-  600  Th0  D (15  0  500  1000 1500  Figure 6.10  Tim (,rvkI  2000  2600  I .Cas6ngo.nO. 2.CasSngOD 3— Mold nn& a4XfaC. 4-Mold outal surface  -  0  600  1000  Figure 6.12  1600  Tim (rldI  2000  2600  I — Cas6ng canS. 2- Ca.Sng 00 3—Moldhm.wfac. 4-Moldou,urfaoe  Solid: L , = 272 k.J/kg 0 Dashed: L = 258 kJikg  Model results for 2-dimensional cylindrical casting Effect of metal latent of solidification on cooling and solidification  -1260  260  600  760  ‘1000  I—  E 0)  1O00  5.1260  Model results for 2-dimensional cylindrical casting Effect of interface contact heat transfer coefficient Constant gap opening time , = 1400- 2.Ot 9 Solid: h Dashed: h 5 = 5000 7.1 4t  91  -.  1260  g’ 1000  1260  250  :  .  ()  S750  C)  600  1600 flJO  Figure 6.11  1000 Tln, ,IIII OWI  2000  2500  0  506  1666  Figure 6.13  1600  Time (seconds)  2000  2500  2.Ca.fngOO 3—Moldinssijte 4-Udjlr1ac.  —  Solid: No fiber paper shielding Dashed: With fiber paper shielding nOog I  Model results for 2-dimensional cylindrical casting Effect of fiber paper shielding on cooling and solidification  0  I ..0OthfllN 2— CasOng 00 3- Mold nnao ,urtao. 4-Mold te i1sce  -  Solid: hgp = 1400- 2.Ot Dashed: = 1400 4.Ot 6 = 14000- 1 .33t Dot-Dash: h  Model results for 2-dimensional cylindrical casting Effect of Interfacecontact heat transfer coefficient Constant initial contact area  1000  0  200  400  600  -  -  -  -  0  —  -  — —  —————  I  1000  0 dog) 90 dog) 0 dog) 90 dog)  1500 Time (seconds)  orientation orientation orientation orientation  I  0.030”, 0.135”, 0.1 50w, 0.285”,  500  N882 (depth N1415(depth N866 (depth N1405 (depth  2000  2500  3000  •  92  Figure 6.14 2D Cylindrical model circumferential effect of fibrepaper shielding on temperatures at various locations  0) I.  E  0)  CO  -  a) 800  0)  c,)  C)  1200  1400  2D Cylindrical Casting Model Effect of circumferential position relative to fibrepaper shielding  Trial Run CS Comparison of model results with experimental data Mod K. Me(aI c.nt.  1250  o4r O.135 O175  IE  750  0 500  1000  1500  Time (seconds)  Figure 6.15  Figure 6.16 93  2000  2500  3000  Trial Run C7 Comparison of experimental data with modd results  Ni N2 N5 N9 N13 N20 N22  1000  CasIn9 ceitec Cangudace  IAo depth O.030 Mo4d depth O.090 Uo4d depth O.145 o4d depth O.255 Uo4d depth O.285  500  0 500  1000  1500  2000  2500  3000  Time (seconds)  Figure 6.17  Trial Run C8 Comparison of experimental data with model results Idode suIts: Castingoenter Castingwiiace olddep(h0030 depth O.285  0 0)1000  750  500  250  0  500  1000  1500 Time (seconds)  Figure 6.18  94  2000  2500  3000  Trial Run C9  Comparison of model results with experimental data Model raeuff: Ca*mgoenae CWkwdac. O.030  1250  0  0235  1000 0  750 E 500  250  0  500  0  1000  1500 lime (seconds)  2000  2500  3000  Figure 6.19  Trial Run Gb results with experimental data model of rison Compa Model resulte: Castkigcenter Ckigwiiace O.O3O O.235  0  750 0.  500  250  0 0  500  1000  1500  lime (seconds)  Figure 6.20  95  2000  2500  3000  Face C  Position a Position b Position c Position d Face B  Figure 6.21a 2D Fin Model: Nodes indicated are referenced in Sensitivity Analysis  Figure 6.21b 2D Fin model with fiberfrax shielding  96  Ywiation in Tonpeca5z. with kadiat Poai9on Finned Casitng Face A  1000  -  800 800 0  N969 N025  700  N881  S  800  N837  Ponitiona Positionb Positionc Positiond  500  I  400 300  100  500  0  1000  1500  2000  2500  3000  Thneonconds)  Figure 6.22 Sensitivity analysis variation of temperaturewithradial position: Face A VwiaitoninTenpecatur.with Radisi Position Finned Cacting Face B  1000  -  900 800  N566 N614 N660 N702  700 600 S  Positiona Position b Positionc Positiond  500  200 tOO 0 0  500  1500  1000  2000  2500  3000  lime (seconds)  Figure 6.23 Sensitivity analysis vañationof temperature with radial position: Face B -  Vanation in Temperature with Radsi Posilion 1000  Finned Casting Face C -  900 800 0  N672 N713 N757 N805  700 600  ps  psajonb Positionc d  500  I  400 300 200 100 500  0  1000  1500  2000  2500  3000  Figure 6.24 Sensitivity analysis variation of temperature with radial position: Face C -  97  201  300  0  j  I  500  \.  1000  2000  Thrn (oni i 4  Figure 6.25  1000  2500  N969 Face A N566 Face B N672 Face C  3000  -o  &  300  400  500  000  900  0  I V  f \  1  500  ‘-  \  5500  2000  TImA (nroHo\  -.  Figure 6.27  1000  .1..,  - -  .,  2500  .1.,,  N881 Face A N60 FaceB N757 FaceC  3000  .1  Difference in Temperature between 90 degree Face and 135 degree Face 1000 at the same radial location (position c)  E  800  °°  900  Difference in Temperature between 90 degree Face and 135 degree Face 1000 at the same radial location (position a)  98  a  0  500  2000  2500  ..  FaceC  ..I....,.  Time (seconds  9500  Figure 6.26  5000  .l....,..  N713  3000  .1  E a)  o  200  300  400  600  °°  000  0  500  f \  I \  1500  2000  Timn (n4-nntk  Figure 6.28  1000  2000  I  N837 FaceA N702 Face B N805 Face C  3000  Difference in Temperature between 90 degree Face and 135 degree Face 1000 at the same radial location (position d)  200  300  400  asco E  --  FaceA  N614 FaceB  N925  Difference in Temparature between 90 degree Face and 135 degree Face at the same radial location (position b)  Vwiation in T.nasture with Radisi Posiiton Finned Casting Face A Flbr*ax shielded model  1000  -  800 630  0  N969 N825 p4881  €00  N837  Position. Poailionb Poaltionc Positiond  50o  0  500  1000  1500  2000  2500  3000  Thnes.conds)  Figure 6.29 variation of temperature with radial position: Face A (shielded) Sensitivity analysis Vanadon in T.mpatxl with RatiI Position Rnd Casting Face B  1000  -  Flitrefrax shielded model  900 800  N566 Position. N614 Position b  100  N660 Positionc N702 Position d  600  50o  200 100 0 0  500  1000  1500  2000  2500  3000  1111w (seconds)  Figure 6.30 variation of temperature with radial position: Face B Sensitivity analysis (shielded)VadaSon in Tenipenature with Radial Posihon Finned Casting-Face C  1000  Flbcefrax shielded model  900 800  N672 N713  700  N757 N805  600  Positionb Position c Positiond  500  k0 0  500  1000  1500  2000  2500  3000  Tinw(seconds  Figure 6.31 of temperature with radial position: Face C variation (shielded)analysis Sensitivity 99  900  400  I—  0  500  2000 2500  N969 Face A N566 FaceB N672 Face C  Time (seconds  1000  Figure 6.32  1000  S.”  “5’  3000  200  200  F-aoo  ?00  800  V  0  j  ‘  \,  500  5  ‘5’  1000  -  -  2000  V  - - - - -  TIme(seconds)  1500  I..  -  Figure 6.34  ‘N  -  2500  N881 FaceA N660 Face B N757 Face C  Shielded model  3000  Difference in Temperature between 90 degree Face and 135 degree Face 1000 at the same radial location (position c)  200  300  500  a’Q E  600  0  Difference in Temperature between 90 degree Face and 135 degree Face 1000 at the same radial location (position a) Shielded modal  100  5  (\  800 1004  V.  2000  2800  N925 FaceA N614 FaceB N713 FaceC  Time Cseconds  1800  --  Figure 6.33  N  S.  ‘  2000  .r  .  E  200  300  400  800  800  ‘°°  ,  .  0  500  r\ 1 \ 0800.’  800  “s  1000  2000  2860  .j....I.  Time (sends  1004  Figure 6.35  5  N837 Face A N702 Face B N805 FaceC  Shielded model  2000  .1  Difference in Temperature between 90 degree Face and 135 degree Face 1000 at the same radial location (position d)  200  ?00  Difference in Temperature between 90 degree Face and 135 degree Face 1000 at the same radial location (position b) Shielded model  Variation in temperature with radial position Face A: Shielded vs. Unshielded model  1000 900  Noshielding  800  --------Shieiding  o 700 600 a 500 C.  E  !400  300 200  -  -  100 0  500  1000  1500  2000  2500  3000  lime (seconds)  Figure 6.36 Variation in temperature with radial location on Face A: shielded vs. unshielded model  Trial Run Fl Comparison of model results wfth experimental data  1400  1200  N1980 N1977 N1933 N1930  J1000  Z  (0.055 (0.110 (O.22O (O.275  800  a 600  400  b: 0.100 mid-height (ps) 0.090 bottom (s)  C:  200  d: 0.250w top (ns) e: O.280 mid-height (ps)  0 500  1000  1500 Time (seconds)  Figure 6.37 101  2000  2500  3000  Trial Run F4 Comparison of model resilts with experimental data  1400  1200  N904  0.050” (shielded)  N904  0.060” (unshielded)  41000  600  400 C: 0.075” top d: 0.065” bottom (s) f: 0.075” bottom (s)  200  0  I  500  1000  1500 Time (seconds)  2000  2500  I  .  3000  Figure 6.38  Trial Run F5 Comparison of model results with experimental data  1400  N904 N904  1200  Face A. 0.060” (unsie4ded) Face A. 0.050” (shielded)  0 1000  1800 a,o,e  600  400  a: b: c: d: e:  0.075” 0.080” 0.080” 0.075” 0.070  200  f: 0.070” bottom (s)  top top mid-height (ps) mid-height (ps) bottom (s)  0 500  1000  1500 2000 Time (seconds)  Figure 6.39 102  2500  3000  Trial Run F6 Comparison of model results with experimental data 1400  e,t 1200 —  —  —  —  N639 N904  FaceBO.O5O Face A, O050 (siaIded)  0 1000  b  800  E  a,b  600  400  200  a: 0.055w lx 0.060w e: 0.060 t 0.065  top mid-height (s) mid-height bottom  0 500  1000  1500 2000 Time (seconds)  2500  3000  Figure 6.40  Trial Run F7 Comparison of model results with experimental data  1400  N736 N904  1200  d,e  Face C. O.050 Face A, 0.050 (shi&d€d)  1000 0 C,  .800 a,  200  a: b: c: d: e:  0.055 0.055 0.055 0.045 0.050  top mid-height(s) bottom (s) top mid-height  ‘C  d,e  0 500  1000  1500 2000 Time (seconds)  Figure 6.41 103  2500  3000  Trial Run F8 Comparison of model results with experimental data  1400  1200  N639 N736  C)  d,e,t  FaeeB,O.050” FaceCO.050  a,b,o  1000 0 D  800 E  0 I—  600  400  c,d  200  0 500  1000  1500 Time (seconds)  2000  2500  3000  Figure 6.42  F9 Trial Run F9 Comparison of model results with experimental data  1400  N94a N906 N865  1200  0.250” 0.150” 0050”  1000  800  400  200  f  aJ  E 600  e  a: 0.235” 0.070” d: 0.075” e: 0.150” f: 0.240” C:  0 500  1000  1500 Time (seconds)  Figure 6.43 104  2500  3000  Chapter 7 Industrial Application of Casting Model to Testbar  7.1 Background  Having completed the sensitivity analysis and achieved confidence in the predictive capability of the model, an analysis was undertaken with a view to providing qualitative guidance in optimizing the design of a specific casting.  The goal was to reduce the  propensity to form macrovoids while seeking to optimize microstructure with respect to morphology 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 a Taguchi test series conducted by Deloro Stellite, in order to evaluate the influence of various casting parameters on casting quality [36  -  381. The compiled results provide an excellent  data base for further verification of the predictive capability of the model. The parameters included in the Taguchi study were: mould preheat temperature, mould cool time, mould thickness, use of mould wrap, type of mould primary coat, metal molten time, and pour temperature. In the model analysis, the effects of mould preheat and mould cool time cannot be addressed separately.  Thus they were combined into the effect of the mould initial  temperature. Mould shell thickness, metal pour temperature and the presence of mould wrap were modelled individually. The effects of mould face coat type and metal molten time are not readily assessed by the current model, as their influence is manifested through the grain nucleation phenomenon.  7.2 Results of analysis  -  2-dimensional test bar model  The 2-dimensional finite element model used to analyze the test bar is shown in Figure 7.1.  Figures 7.2 through 7.8 show example contour plots of the 2-dimensional 105  analysis 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 the piots 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 mould  temperature (Figure 7.3), lower initial metal temperature (Figure 7.4) and increased mould thickness (Figure 7.5) all show a solidification progression similar to that of the base case. A solid skin forms rapidly on the feeder cup, and the test bar end buttons solidify on both the upper 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 test bar 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 to the top of the feeder, as well as to the upper half of the mould. The most significant effect is that 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. The time 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 gating system has been redesigned, with the feeding to the center section removed. No mould wrap is used. Skin formation is still observed on the top of the feeder cup, but the progression of solidification is changed in the testbar; the center portion solidifies well before the end buttons (“J” isotherm passes through the center section before it passes through the ends).  106  7.3 Results of analysis 3-dimensional test bar model -  Finally, a 3-dimensional, farfield radiation cooling model was analyzed. Although the 2-dimensional model could be used to assess the relative effects of casting parameters qualitatively, the 2-dimensional assumption may be expected to give results in which the solidification time is increased significantly over a 3-dimensional model. The metal surface area-to-volume ratio of the 2-dimensional case is much lower, and the system loses heat from one face adjacent to the testbar only. In the 3-dimensional case, the surface area-to-volume ratio is much higher, and heat is extracted from the mould surrounding the testbar on three sides. The 3-dimensional geometry is shown in Figure 7.9. A quarter section only has been modelled to reduce the problem size.  Figures 7.10 and 7.11 show the temperature  distribution in the cross-sectional plane of the testbar at various times. The time scale for solidification of the testbar is reduced by an order of magnitude (-400 seconds vs. —1000 seconds). The testbar gauge length solidifies before the end buttons, in contrast to the 2dimensional 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 for computational time, the results may be only qualitative at best, and can be misleading. The assumption of 2-dimensional heat flow must be examined carefully and the results interpreted in the light of this assumption. A 3-dimensional model with the model size reduced would perhaps yield more quantitative results, within the limits of the reduced model size. For example, the test bar center solidifies so rapidly that the temperature in the pour cup, at some distance away, has little effect on the center of the bar. The downsprue has not cooled significantly 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 the  107  nodes 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 boundary condition (zero heat flux) is applied to the center of the downsprue. This model could be used for sensitivity analysis of the casting parameters on the test bar gauge length. The limiting 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 quantitative results than those obtained using the 2-dimensional model.  7.4 Theoretical considerations in casting quality analysis  The quality criteria examined with the model that are directly influenced by the thermal history and which were also examined in the Taguchi experiments are: i) presence of cavity porosity ii) presence of microporosity iii) secondary dendrite arm spacing (SDAS)  7.4.1 Cavity porosity  Porosity in èastings can arise due to several mechanisms. Cavity porosity is due to the premature freezing off of feeding channels preventing feeding to solidifying areas. The last 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 irregular in shape.  108  7.4.2 Microporosity  Reference [36] reported that the test bar casting was susceptible to severe microporosity in the area of the feeding gate adjacent to the test length, and that “Microporosity morphology was interdendritic microshrinkage.”  distinct  and  elongated;  characteristic  of  Microporosity is formed when the feeding to the dendrite root is inhibited by the solidifying metal.  To feed the solidification shrinkage flow, a pressure drop must exist  across the dendrite array between the melt and the dendrite root.  This pressure drop is  described by [39]:  p  —  p  =  Al’  AT”freezmg  7.1  2 nmR  where AVfreezing is the volume change associated with freezing 1 is the channel (dendrite) length is a tortuosity factor, straight  t  >  1, accounting for the fact that the flow channels are not  2 is the total channel area for n channels of radius K mtR  The pressure at the dendrite root is  7.2  The volume change on freezing causes the local pressure at the dendrite root to decrease below the vapour pressure of the melt constituents. When P 1 falls below a critical pressure  109  microvoids will nucleate. Relation (7.1) above states that microporosity is a function of the pressure of the melt, the volume contraction on freezing, the square of the length of the mushy zone and the tortuosity. The main effect of the temperature distribution on microporosity is through the value of the channel length, 12. This length is a function of the solidification range, /Tf and the thermal gradient G in the 2-phase region:  1=AT/G  7.3  Regions of low thermal gradient during solidification will be subject to more microporosity than those of a higher thermal gradient. In the Taguchi experiments, the parameters found to have 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 pouring and cooling without a mould wrap around the mould. These conditions both lead to a higher gradient in the casting during solidification, and a faster cooling rate.  7.4.3 Secondary dendrite arm spacing  Secondary dendrite arm spacing (SDAS) is a function of local solidification time [40]:  7•5  3 SDAScct  where local solidification time is defined by the relation  7.6  t Iv  110  where 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 secondary dendrite arm spacing. As well, higher thermal gradients in the dendrite array result in a shorter array length 1, shorter  and smaller SDAS. In the Taguchi experiments, fmer  SDAS was achieved using a lower mould preheat, longer mould cool time and no mould wrap.  Each of these conditions contributes to a higher thermal gradient and more rapid  cooling, which is consistent with the theoretical considerations.  7.5 Correlation of model predictions with casting quality  In the following comparison, the results of the 2-dimensional model sensitivity analysis are applied qualitatively, in terms of the relative effect of the parameters on the casting thermal history.  A 3-dimensional model, while reducing the time scale for  solidification and altering the progression of solidification, can be expected to show similar trends with variation in the parameters. Cavity porosity in the testbar castings can be expected in the model conditions of Figures 7.2 to 7.6, as the feeding from the pour cup is inhibited by formation of the solid skin. The cavities will be located in the last volume of metal to solidify. The casting volume in the mushy zone at the beginning of solidification of the gauge length of the testbar is quite large, requiring a large volume of metal to feed the shrinkage, and also resulting in a large value 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 mould initial 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 solidification time only slightly. Adding external mould wrap to the casting sides (Figure 7.6) results in a more directional solidification pattern in the test bar gauge length. Feeding to the gauge  111  length can occur not only through the center gate, but also through the top end of the test bar itself. 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 thus some microporosity may be present. In Figure 7.7, insulation has been added to the top of the feeder cup, as well as to the top outer sides. This retards the formation of a solid skin on the feeder top, reducing the incidence of cavity porosity. Some shrinkage microporosity may still be expected, due to the large volume in the mushy zone at the onset of solidification of the gauge length. Solidification will be more directional, with enhanced feeding through the testbar 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 from the melt within seconds after pour under the conditions examined in Table 7.1. A measure of the testbar solidification time,  ,  for the cases under consideration is obtained from the time  at which the solidus (J isotherm) has passed through the gauge length of the test bar. This information, 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, thinner mould, no mould wrap, and no center feeding. These conclusions agree with the results of the Taguchi experiments. The fourth column of Table 7.11 summarizes the results of the Taguchi experiments with respect to the variation of each parameter [36 38]. -  As discussed above, the parameter having the largest effect on porosity and SDAS is the presence of a mould wrap on the casting during solidification. Whereas the mould wrap slightly improves conditions for reduced microporosity, the significant increase in overall solidification time leads to a coarser SDAS, which has a detrimental effect on the mechanical properties of the casting [39]. In order to reduce microporosity without increasing overall solidification time, the gating system was redesigned to remove the feeding gate to the test bar gauge length.  Figure 7.8 shows the casting with the feeding system redesigned to  eliminate microporosity and reduce solidification time. The gauge length section solidifies 112  first, while a large volume of metal at a higher temperature is available to feed the test length from the top and bottom.  Although cavity and microporosity may be expected in the  downsprue, which solidifies slowly, the testbar itself will be sound. Overall solidification time is reduced by 60%; a finer SDAS may be expected, with improved mechanical properties as a result. The 3-dimensional testbar with the center feeding shows a solidification progression similar to the 2-dimensional model with the center feeding removed. The solidification time is 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 practical application of the model to casting design.  The process control parameters of mould  temperature, metal temperature and mould thickness have a relatively small influence on the solidification progression. Referring to Table 7.11, varying the mould temperature, metal temperature 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 of process control, invariably results in an increase in total solidification time. In the cases analyzed, adding mould wrap insulation increases total solidification time by 130  -  150%.  Adding mould wrap to reduce microporosity will thus lead to reduced casting quality in terms of material properties. The analysis of the redesigned feeding system indicates that a sound casting can be obtained without a mould wrap, and greatly reduced total solidification time. Removing the center feeding reduces solidification time by 60%. The above points illustrate the importance of optimizing the casting through correct design of the mould and feeding system, rather than through the use of the various parameters at pour time.  113  Table 7.1 Model parameters used in testbar analysis Parameter Base value Varied value (W/m-deg C) 1.0 kmould (cm2/sec) 042 0.0 mould 272 Lmetal (kJ/kg) Mould initial temperature 600 800 Metal initial temperature 1400 1360 Mould thickness (inches) 0.300 0.330 -  -  -  *  Table 7.11 Effect of Model Casting Parameters on Local Solidification Time for Testbar Castings Parameter tf(seconds) SDAS effect * 1 %At Base 1000 high Tmould 1100 +10 Coarser low Tmetal 900 -10 Finer Thick mould 1150 +15 Coarser Mould wrap (sides) 2300 +130 Coarser Mould wrap (sides 2500 +150 Coarser &_top) No center feeding 400 -60 Finer Refs [36) [38] -  114  Deg C 1500=A 1340=13 1330=C 1320 = D 1310=E 1300=F 1290=G 1280=H 1270=1 1260 = J  Figure 7.1 2-dimensional testbar fmite element mesh  Figure 7.2 Base case, time 800 seconds  Deg C 1500=A 1340 = B 1330=C 1320=D 1310=E 1300=F 1290=G 1280 = H 1270 = 1260 = J  Deg C 1500=A 1340 = B 1330=C 1320 = D 1310=E 1300=F 1290 = G 1280 = H 1270=1 1260=J  Figure 7.3 High initial mould temperature time 900 seconds  Figure 7.4 Low initial metal temperature time 700 seconds  115  Deg C 1500=A 1340=B 1330=C 1320=D 1310=E 1300=F 1290 = G 1280=H 1270=1 1260 = J  Deg C 1500=A 1340 = 13 1330=C 1320 = D 1310=E 1300 = F 1290 = G 1280 = H 1270 = 1260 = J  Figure 7.5 Thick mould time 1000 seconds  Figure 7.6 Mould wrap on upper half time 1300 seconds  Deg C 1500=A 1340 = B 1330=C 1320 = D 1310=E 1300 = F 1290=G 1280 =11 1270=1 1260 = J  Deg C 1S0O=A 1340=B 1330 =C 1320 = D 1310=E 1300=F 1290 = G 1280=H 1270 = I 1260 =  Figure 7.7 Mould wrap sides and top time 2000 seconds  Figure 7.8 Feeding to test section removed time 400 seconds  116  Figure 7.9 3-dimensional testbar model a: casting outer view; b: casting inner view; c: mould outer view; d: mould inner view  117  118  Figure 7.10 3-dimensional testbar model results; time 25 seconds  119  3-dimensional testbar model results; time 50 seconds  Chapter 8 Summary and Recommendations  8.1 Summary and conclusions  This work has studied the heat transfer processes in vacuum investment casting of nickel-based superalloy 1N718. Boundary conditions were developed for a finite element based solidification model for radiation heat transfer at the mould exterior and the mouldmetal interface. The investment casting process cannot be modelled adequately with a simple farfield radiation condition, as there is significant self-irradiation due to the complex mould geometries. General 2- and 3-dimensional viewfactor codes were developed for use in the radiation heat transfer network. The Monte Carlo ray tracing approach used in this code allowed complete generality of geometry, and was found to have an error of less than 5% at a 95% confidence level when as few as 10000 rays were traced for each surface in the enclosure. The boundary condition algorithms were verified against a series solution from the literature 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 thermophysical properties and the contact area function were conducted in collaboration with Deloro Stellite of Belleville, Ontario. A total of thirteen data sets in two configurations was obtained, giving thermal histories for the mould and metal at various locations. The casting configurations were modelled using the developed code.  Sensitivity  analyses determined that the critical parameters were the mould conductivity, contact area function, metal initial temperature, mould thickness and the radiation environment of the castings during cooling and solidification.  In particular, the presence of an insulating  120  fibrepaper used to shield the thermocouple manifold significantly influenced the mould surface temperature profile. Thermophysical properties of the mould were obtained by fitting the model results to the experimental data. Constant values of the properties were used throughout. The mould thermal conductivity which resulted in the best fit to the data was in the range of 0.9 W/m-deg C for the cylindrical castings, and 0.8  -  -  1.1  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 over which the mould remained at higher temperatures. As the mould conductivity is expected to have 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 for a longer time.  The mould thermal diffusivity ranged from 0.0026  -  0.0046 2 cm / sec, in  agreement with data from the literature. A linear function decreasing from an initial value h 0 to zero at some time  was  used for the interface contact conduction heat transfer coefficient. An initial value h 0 of 1400 2 W/m deg C was found to result in the best fit overall to the data. This corresponded to an initial contact area fraction of 0.28, assuming a gap constant (kmould/Ax) of 5000 W/m 2 deg C. The time over which the contact reduced to zero was 700 seconds for the cylindrical moulds, and 200 seconds for the finned moulds, corresponding with the overall solidification time of each casting. The results of a model analysis of a tensile test bar are in qualitative agreement with those from a set of Taguchi experiments performed at Deloro Stellite Inc. A low mould initial temperature, low metal initial temperature, thinner mould, and absence of mould wrap result in a shorter overall solidification time, and therefore in a finer secondary dendrite arm spacing. Feeding to the testbar center section will result in possible microporosity in the test gauge 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. This will reduce the incidence of microporosity in the gauge length. A preliminary 3-dimensional 121  model demonstrates the need for a 3-dimensional modelling capability;  the total  solidification time of the test bar is reduced by an order of magnitude over the 2-dimensional model, 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 from a computational aspect, and from the time required for geometry and mesh preparation, the results may be highly misleading.  The assumption of 2-dimensional heat flow must be  closely 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 a slight influence on microporosity and secondary dendrite arm spacing. Use of a mould wrap for process control may result in slightly less microporosity, but will increase the solidification time, with a resulting decrease in casting material properties. These points and the analysis of the redesigned casting model emphasize the importance of casting optimization in the initial design of the feeding system, rather than through process control in the pouring stage. The experimental data yielded information on the cooling rate of a mould immediately before pouring. calculated to be between 130  The average cooling rate for the cylindrical moulds was -  160 deg C/minute.  This very high rate indicates that in  casting, the metal must be poured as quickly as possible if better fill characteristics due to the higher mould preheat are to be obtained. The benefits of a preheat higher by 200 deg C will be lost by a delay in pouring of approximately 1-1/2 minutes.  122  8.2 Recommendations for further work  The parameters having the most significant effect on model output are the mould thermal conductivity and the interface contact heat transfer formulation. These parameters are as yet not well-characterized. The temperature dependence of the mould thermal conductivity is an area requiring further work. Due to the complex shell structure, a highly non-linear temperature response is expected, with void radiation becoming significant at higher temperatures. Theoretical work on conduction in porous media and 2-phase continuous media can serve as a basis for developing 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 mould may restrict shrinkage of the casting in certain locations. A temperature-based approach incorporating solidification shrinkage effects and thermal contraction due to cooling of the solid will be more generally applicable than the time-based, linear function applied in this study.  Such an approach, while adding significant computational complexity, will also  provide 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 for 1N7 18 investment castings.  123  Bibliography  [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 and Experimental 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 Investment Castings”, ASME HTD vol. 104 pp. 93-99, 1988 [6] Lui, Jia-Chin, Lee, Tsang-Sheao and Huang, Weng-Sing; “Computer Model of Unidirectional 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.; “The Simulation of Single Crystal Turbine Blade Solidification”, in Proceedings, ASM International Conference, Synthesis, Processing and Modelling of Advanced Materials, September 1991 [8] Ulam, S., Richtmeyer, R.D. and von Neumann, J.; “Statistical Methods in Neutron Diffusion”, LAMS-551, Los Alamos National Laboratory, 1947 [9] Howell, J.R. and Perlmutter, M.; “Monte Carlo Solution of Thermal Transfer through Radiant 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 between Concentric 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 by the 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, NonDiffuse Surfaces” Nurn. Heat Transfer, vol. 1 pp.403-416, 1978  124  [14] Maitby, J.D. and Burns, P.J.; Performance, Accuracy and Convergence in a 3Dimensional Monte Carlo Radiative Heat Transfer Simulation” Num. Heat Transfer B, vol. 19, pp. , 1991 209 l19 [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.548556, 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 the Casting-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 Thermal Contact and Phase Change in Solidification Modelling”, in Heat and Mass Transfer in Solidification Modelling, ASME-HTD vol. 175, pp.53-60, 1991 [21] Huang, H.; An Investigation of Thermal Contact, Phase Change and Computational Efficiency in Modelling Shaped Casting Solidification, Ph.D. Thesis, University of Alabama, 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 Book Company, New York, 1972) [24] Huebner, K.H. and Thornton, E.A.; The Finite Element Method for Engineers (John Wiley and Sons, New York, 1982) [25] Jaeger, J.C.; “Conduction of Heat in a Solid with a Power Law of Heat Transfer at its Surface”, Proceedings, Cambridge Philosophical Society. vol. 26, pp.634-641, 1950 [26] Heames, K. and Geiger, G.H.; “I. The Thermal Conductivity of Shell Investment Materials, II. Heat Transfer in Investment Shell Molds”, Proceedings, 26th Annual Meeting of the Investment Casting Institute, 1973 [27] Huang, H., Berry, J.T., Zheng, X.Z. and Piwonka, T.S.; “Thermal Conductivity of Investment Casting Ceramics”, Proceedings, 37th Annual Technical Meeting, Investment Casting Institute, 1989 [28] Atterton, D.V.; “Apparent Thermal Conductivies of Moulding Materials at High Temperatures”, I Iron Steel Inst. vol. 174 pp 201-211, 1953  125  [29] Godbee, H.W. and Ziegler, W.T.; “Thermal Conductivities of MgO, A1 , and Zr0 3 0 2 2 Powders 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 of Measurement and Factors Affecting Thermal Conductivity of Ceramic Materials”, J Amer. Ceram. Soc. vol. 37 no. 2 pp 67-72, 1954  [311 Loeb, Al.; “Thermal Conductivity: VII, A Theory of Thermal Conductivity of Porous Materials”, 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 Use of Cobalt Aluminate “Blue” Primary Coat [34] Deloro Stellite Engineering Specification No. 4523B, January 1993, Preparation and Use of FascoteTM Back-up Coatings [35] Sairset Material Safety Data Sheet, A.P. Green Refractories Co., Mexico, Missouri USA 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, Deloro Stellite 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, New York, 1975) [40] Kurz, W. and Fisher, D.J.; Fundamentals of Solidification (TransTech Publications, Switzerland, 1989)  126  Appendix A Analytical solutions for verification of computer code Viewfactor for 2-dimensional, parallel plates of width-to-separation ration H (Ref. [23]):  F12=Jl+H2 —H  Viewfactor for 2-dimensional, perpendicular plates having a common edge and length ratio  of H (Ref. [23]):  F12  Viewfactor for 3-dimensional, parallel plates of length-to-width ratio X and length-to separation ratio Y (Ref. [23]):  2  F12=—  itXY  F(1+x2)(1+Y2)1 +X,J1+Y2tan1 ml (1+X2+Y2) j [  Viewfactor from plate  1  to plate 2  common edge, where for plate 1, X  =  +Y-J1+X t 2 an  Y 1 —Xtari’X—Ytan  for 3-dimensional, perpendicular plates having a  length-to-width ratio, for plate 2, Y  width ratio, and the width dimension forms a common edge (Ref. [23]):  127  =  height-to-  1  1  i Wtan —+Htan 2 +W 2 tan ——IH W H  F12*  (1+W2)(1+H2) 1 +—lfl (1+W2+H2)  1  1  H2+W2  W2(1+W2+H2) (1+W2)(1+H2)  2 W  H2(1+W2+H2)  2 H  (1+W)(1+H2)  Jaeger [25] developed a series solution to the 1-dimensional heat conduction equation for a semi-infinite medium where, on the boundary:  HVm  —k-dx  For m  =  Fs  4, corresponding to a radiation boundary condition, for short times, the first  terms are:  v  =  erfcX + l6T 1 erfcX + 2787TierfcX—..} i 3 erfcX 189.lT i 2 {i 2Ti 0 v —  —  where  )/Kv 0 T=(Kt)](v  X= (1ct)  The ifleifcX is the n-pie repeated integral of the error function [25]:  ierfc  =  128  


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