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Determination of the influence of growth front angle on freckle formation in superalloys Auburtin, Philippe Bernard Lucien 1998

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D E T E R M I N A T I O N O F T H E I N F L U E N C E O F G R O W T H F R O N T A N G L E O N F R E C K L E F O R M A T I O N I N S U P E R A L L O Y S by PHILIPPE BERNARD LUCIEN AUBURTIN B.A.Sc., Ecole Centrale Paris (FRANCE), 1992 M A . S c , University of British Columbia (CANADA), 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Metals and Materials Engineering) We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA luly 1998 © Philippe Bernard Lucien Auburtin, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract Freckles are macrosegregation defects which are usually found in nickel-base superalloys or specialty steels and which occur during solidification. They are presently one of the major defects encountered in advanced casting technology of superalloys. The evaluation of a numerical criterion able to provide quantitative insight on the conditions of freckle formation is now recognized as a major key toward the successful manufacture of large diameter VAR/ESR ingots and large DS/SX castings. This thesis summarizes the main features of freckle formation and gathers most of the criteria proposed in the literature to date, along with a brief critical description. Emphasis is put on the Rayleigh number which seems best adapted to freckling prediction. The various parameters of the Rayleigh criterion are numerically evaluated. Some of the hmitations in the ability of the Rayleigh criterion to predict freckling are also described. It is suggested that the angle of the solidification front with respect to the direction of gravity plays a critical role on freckle formation. An experimental vacuum induction furnace was built in order to directionally solidify freckle-prone superalloys (Waspaloy, Mar-M247) at various angles to the vertical under typical industrial conditions (thermal gradients ranging from 500 to 4000°C/m (5 to 40°C/cm) and growth rates ranging from 1.6xl0"5 to 10xl0~5m/s (1 to 6mm/min)). A comprehensive thermal modeling of this furnace was carried out with the finite element package ProCAST in order to accurately evaluate the solidification conditions (G and R) for each sample. Based on the obtained experimental results, two possible "modified" Rayleigh criteria are suggested, incorporating the anisotropy in the mushy zone permeability and the directionality of gravity, and are shown to accurately describe the effect of the casting conditions and growth front angle on the presence or absence of freckles. Finally, the direct application of these new criteria to actual industrial processes is illustrated. ii Table of Contents Abstract i i Table of Contents iii List of Tables v List of Figures vi List of Symbols x Acknowledgments xii 1. INTRODUCTION 1 2. LITERATURE REVIEW 4 2.1. T H E N A T U R E O F F R E C K L E S 4 2.1.1. Physical characteristics 4 2.1.2. Mechanism of freckle formation : Density inversion theory 8 2.2. T H E P R E D I C T I O N O F F R E C K L E F O R M A T I O N 11 2.2.7. Various freckling criteria '. 11 2.2.2. The Rayleigh criterion 14 2.2.3. Rayleigh criterion : Range and sensitivity of the parameters 16 2.3. L I M I T A T I O N S O F T H E R A Y L E I G H N U M B E R A S A F R E C K L I N G C R I T E R I O N 23 2.3.1. Numerical evaluation of the Rayleigh number 23 2.3.2. Other limitations of the Rayleigh criterion 24 2.3.2.1. The effect of casting size 24 2.3.2.2. Surface-only and mid-radius-only freckles 25 2.4. I N F L U E N C E O F T H E G R O W T H F R O N T A N G L E 26 3. RESEARCH OBJECTIVES 33 3.1. R E S E A R C H F O C U S 33 3.2. O B I E C T I V E S 34 4. METHODOLOGY 35 4.1. C H O I C E O F I N D U S T R I A L A L L O Y S 35 4.2. E X P E R I M E N T A L A P P A R A T U S 36 4.2.1. Tillable Bridgman furnace 36 4.2.2. Typical experiment 39 4.3. N U M E R I C A L M O D E L I N G O F T H E F U R N A C E 41 4.3.1. ProCAST. 42 4.3.2. Mesh geometry 43 4.3.3. Typical simulation 45 4.3.4. Model parameters, calibration and sensitivity analysis 47 iii 5. R E S U L T S 5 0 5 . 1 . M O D E L R E S U L T S 5 0 5.1.1. Isotherm geometry 50 5.1.2. Thermal gradients and growth rates 57 5.1.3. Primary dendrite arm spacing measurements 64 5 .2 . E X P E R I M E N T A L R E S U L T S 6 5 5.2.1. Typical appearance of the castings 65 5.2.2. Tilted castings summary 71 6. D I S C U S S I O N : D E V E L O P M E N T O F A N E W F R E C K L I N G C R I T E R I O N 7 2 6 . 1 . A P P L I C A T I O N O F T H E O R I G I N A L R A Y L E I G H N U M B E R C R I T E R I O N .'. 7 3 6 . 2 . E F F E C T O F T H E G R O W T H F R O N T A N G L E O N F R E C K L E F O R M A T I O N : M O D I F I C A T I O N S T O T H E R A Y L E I G H C R I T E R I O N 7 7 6.2.1. First theory 7 7 (5.2.2. Second theory 82 6 . 3 . A P P L I C A T I O N O F T H E M O D I F I E D R A Y L E I G H C R I T E R I A T O T H E R E S U L T S O F T H E P R E S E N T W O R K 8 5 6.3.1. Application of the first theory 85 6.3.2. Application of the second theory 88 6.3.3. Application of the modified Rayleigh formulations to process optimization 91 6 .4 . D I R E C T A P P L I C A B I L I T Y O F T H E M O D I F I E D R A Y L E I G H C R I T E R I A T O I N D U S T R I A L S I T U A T I O N S 9 2 6.4.1. The case ofVAR IN718 93 6.4.2. The case of DS/SX castings 96 6A.l. 1. Special modification of the Rayleigh criterion 9 6 6 . 4 . 2 . 2 . Potential growth front angles in industrial D S / S X castings 9 7 7. C O N C L U S I O N S A N D F U T U R E W O R K 1 0 6 7 . 1 . C O N C L U S I O N S 1 0 6 7 . 2 . R E C O M M E N D A T I O N S F O R F U T U R E W O R K 1 0 7 References ; 1 1 0 A P P E N D I X A : Furnace modeling calibration 1 1 4 A . 1. N U M E R I C A L D A T A 1 1 4 A.1.1. Material properties 114 A.1.2. Heat transfer coefficients 116 A.l.3. Enclosure parameters 116 A.l. C A L I B R A T I O N V A L I D A T I O N 1 1 8 A . 3 . S E N S I T I V I T Y A N A L Y S I S 121 A P P E N D I X B : Numerical data of the experimental castings (tiltable furnace) 1 2 3 A P P E N D I X C : Use of the experimental furnace to cast D S turbine blades 1 2 4 A P P E N D I X D : Heat Transfer Coefficients estimation 1 2 5 iv List of Tables Table I : Liquid density, viscosity and thermal diffusivity for the four superalloys investigated in this thesis at their liquidus temperature and 100°C above their liquidus temperature 17 Table I I : Melting range, freckle initiation temperature and nominal composition of the alloys considered in this thesis (except U B C 1 ) 36 Table III : Initial conditions of the materials at the start of the numerical simulation 46 Table IV : Thermal gradients at the liquidus and at the freckle initiation location (Guquidus and Gfreckie respectively) and solidification rate at the growth front (R) calculated with the P roCAST model for all the alloys studied in this work for various experimental casting conditions (control temperature and withdrawal speed in the furnace) 53 Table V : Chemical analysis (measured by microprobe) of the matrix and freckles from the experimental samples 68 Table V I : Chemical analysis by microprobe of bulk and freckle compositions in industrially cast large IGT blades 69 Table V I I : Estimated liquid fraction at the freckle initiation location in the mushy zone 70 Table V I I I : Calculated partition coefficients for Aluminum in Waspaloy, Mar-M247 and U B C 1 assuming a liquid fraction of 0.5 70 Table I X : Summary of some significant experimental results 71 Table X : Material properties used in modeling the DS furnace with P r o C A S T 115 Table X I : Heat transfer coefficients used in modeling the DS furnace with P r o C A S T 116 Table X I I : Enclosure parameters for P roCAST simulation 117 Table X I I I : Various parameters used in the P roCAST model and their deviations from the base values used to perform a sensitivity analysis 121 Table X I V : Summary of the directional solidification experiments carried out on the tiltable Bridgman furnace 123 v List of Figures Figure 1: Various appearance of freckles in industrial castings 7 Figure 2 : Schematic diagram of directional solidification and associated thermal (pT), solutal (pc) and thermosolutal (pT+c) density profiles illustrating the density inversion theory 9 Figure 3 : Freckle formation and associated fluid flow pattern 10 Figure 4 : Freckle plumes rising above the growth front in DS NH4CI/H2O [18] 10 Figure 5 : Relationship between cooling rate at the dendrite tips and primary/secondary dendrite arms spacing in various superalloys [28] 18 Figure 6 : Typical liquid fraction profile along the mushy zone of nickel base superalloys.(experimental measurements : (a) from [1] and (b) from [29].) 19 Figure 7 : Freckles at mid-radius only in V A R IN706 ingot. [41] 25 Figure 8 : Schematic description of Giamei's experiment with curved growth fronts 26 Figure 9 : Freckle flow path and growth front angle in various industrial castings 28 Figure 10 : Freckles on opposite sides of the same root section of a large S X blade casting (UBC1 alloy)(Freckles size : about l-2mm diameter) 29 Figure 11 : Schematic diagram illustrating vertical and slanted surface freckles in D S / S X castings 30 Figure 12 : V A R IN706 ingot, exhibiting freckle flow following the shape of the liquid pool (i.e. perpendicular to the primary dendrites) (see longitudinal plane section)[41] 30 Figure 13 : Relationship between " A " segregate angle to the horizontal and solidification rate [4] [42] .31 Figure 14 : Case of a variable undercooling, where the growth front angle cannot be identified by the TFreckle isotherm [43] 32 Figure 15 : Schematic diagram of the Bridgman-type furnace used in this study 37 Figure 16 : Tiltable directional solidification furnace 38 Figure 17 : Typical data acquisition record for each experiment 40 Figure 18 : Thermocouple locations for various measurements performed on the experimental furnace 41 Figure 19 : Axisymetric mesh geometry used in P roCAST to model the furnace 44 Figure 20 : Schematic heat flow diagram in the P roCAST model of the experimental tiltable furnace 45 Figure 21 : Comparison between measured (bold lines) and modeled (thin lines) cooling curves at five locations inside the casting (130, 122, 117, 110 and 100 mm above the top of the chill). 48 vi Figure 22 : Typical thermal profiles in the casting during solidification (ProCAST simulation). (control temperature : 1435°C , withdrawal speed : 3.3xlO" 5 m/s (2mm/min)) 55 Figure 23 : Influence of the withdrawal speed on the isotherm profiles in the casting during solidification (ProCAST simulation), (control temperature : 1435°C, z = 66 mm) 56 Figure 24 : Influence of the furnace control temperature on the isotherm profiles in the casting during solidification (ProCAST simulation), (withdrawal speed : 3.3xl0" 5 m/s (2mm/min), z = 66 mm) 57 Figure 25 : Thermal gradients at the growth front (liquidus temperature) in LN718 for various casting conditions 58 Figure 26 : Solidification rate at the growth front (liquidus temperature) in LN718 for various casting conditions 59 Figure 27 : Relationship at various casting speeds between the furnace control temperature and the thermal gradients in the sample for superalloy LN718 (ProCAST simulation) 60 Figure 28 : Relationship at various casting speeds between the furnace control temperature and the thermal gradients in the sample for superalloy Waspaloy (ProCAST simulation) 61 Figure 29 : Relationship at various casting speeds between the furnace control temperature and the thermal gradients in the sample for superalloy Mar-M247 (ProCAST simulation) 62 Figure 30 : Relationship at various casting speeds between the furnace control temperature and the thermal gradients in the sample for superalloy U B C 1 (ProCAST simulation) 63 Figure 31 : Primary dendrite arm spacing measured in U B C 1 experimental samples as a function of calculated cooling rate (ProCAST simulation) 64 Figure 32 : Typical longitudinal sections of directionally solidified experimental samples 66 Figure 33 : Appearance of freckles on the surface of tilted castings (castings diameter : 25mm) 66 Figure 34 : Micrographs of cross-sections of the surface freckles observed in the experimental castings (average diameter of each freckle : about l-2mm) 67 Figure 35 : Original Rayleigh number versus growth front angle for alloy Waspaloy 75 Figure 36 : Original Rayleigh number versus growth front angle for alloy Mar-M247 76 Figure 37 : Original Rayleigh number versus growth front angle for alloy U B C 1 76 Figure 38 : Schematic geometry for an angled growth front (general casting conditions) 77 Figure 39 : Modified Rayleigh numbers Rax and Raz (and maximum value Ral in bold) versus growth front angle for various thermal gradients Guquidus and GFreckk = Guquidus + 10°C/cm (constant growth rate #=3.3x10 "5 m/s (2mm/min)) 80 Figure 40 : Modified Rayleigh numbers Rax and Raz (and maximum value Ral in bold) versus growth front angle for various growth rates R (constant thermal gradients Guquidus = 15°C/cm and GFmkie =25°C/cm) 81 vii Figure 41 : Modified Rayleigh number Ra2 versus growth front angle for various thermal gradients GLiquid™ and GFreckie = GUquidus + 10°C/cm (constant growth rate /?=3.3xlCV5 m/s (2mm/min)) 83 Figure 42 : Modified Rayleigh number Ra2 versus growth front angle for various growth rates R (constant thermal gradients Guquidus = l 5°C/cm and GFrec«e =25°C/cm) 84 Figure 43 : Modified Rayleigh number Ral versus growth front angle for alloy Waspaloy 86 Figure 44 : Modified Rayleigh number Ral versus growth front angle for alloy Mar-M247 87 Figure 45 : Modified Rayleigh number Ral versus growth front angle for alloy U B C 1 87 Figure 46 : Modified Rayleigh number Ral versus growth front angle for alloy Waspaloy 89 Figure 47 : Modified Rayleigh number Ra2 versus growth front angle for alloy Mar-M247 90 Figure 48 : Modified Rayleigh number Ra2 versus growth front angle for alloy U B C 1 90 Figure 49 : Illustration of the potential application of the generic graphs based on the modified Rayleigh criteria 91 Figure 50 : Pool depth vs. Radius in V A R LN718 shown by the isotherms for the liquidus, freckle and solidus temperatures 94 Figure 51 : Thermal gradients at the liquidus, freckle and solidus temperatures and solidification rate in V A R LN718 (melt rate : 260kg/hr) (after [47]) 94 Figure 52 : Original and modified Rayleigh criteria profiles along the radius of V A R LN718 (melt rate : 260kg/hr) 95 Figure 53 : Schematic geometry for an angled growth front (DS/SX casting conditions) 96 Figure 54 : Curvature of the isotherms in the mushy zone in industrial D S / S X blade casting (numerical simulation) [48] 98 Figure 55 : Axisymetric mesh geometries used to investigate the influence of casting cross-sections on growth front angle 102 Figure 56 : Numerical simulation of the influence of casting cross-sections and withdrawal rates on the growth front angle found in D S / S X castings. (Control temperature 1435°C , after 130mm withdrawal) (ProCAST simulation) 103 Figure 57 : Axisymetric mesh geometries used to investigate the following effects on growth front angle in D S / S X blade castings 104 Figure 58 : Numerical simulation of the influence of withdrawal rates on the growth front angle found in the airfoil in D S / S X blade castings. (Control temperature 1435°C , after 130mm withdrawal) (ProCAST simulation) 105 Figure 59 : Numerical simulation of the influence of withdrawal rates on the growth front angle found in large sections in D S / S X blade castings (such as the root). (Control temperature 1435°C , after 130mm withdrawal) (ProCAST simulation) 105 Figure 60 : Susceptor temperature as a function of height above the baffle (for various control temperatures 1400°C, 1435°C, 1465°C and 1500°C). (black markers : measured values, white markers : interpolated values used in the P r o C A S T model) 117 viii Figure 61 : Comparison between measured (bold lines) and modeled (thin lines) cooling curves at five locations inside the casting (130, 122, 117, 110 and 100 mm above the top of the chill) 118 Figure 62 : Comparison between measured (bold lines) and modeled (thin lines) cooling curves at five locations inside the casting (130, 122, 117, 110 and 100 mm above the top of the chill) 119 Figure 63 : Comparison between measured (bold lines) and modeled (thin lines) cooling curves at five locations inside the casting (130, 122, 117, 110 and 100 mm above the top of the chill) 119 Figure 64 : Comparison between measured (bold lines) and modeled (thin lines) cooling curves at five locations inside the casting (130, 122, 117, 110 and 100 mm above the top of the chill) 120 Figure 65 : Comparison between measured (bold lines) and modeled (thin lines) cooling curves at five locations inside the casting (130, 122, 117, 110 and 100 mm above the top of the chill) 120 Figure 66 : Deviations from the base temperature for variations of various parameters in the calibrated P roCAST model of the experimental furnace 122 Figure 67 : Casting of small DS turbine blades with the experimental Bridgman furnace built in this work 124 ix List of Symbols Symbols SI Unit Meaning C Co DT II F F g G, G Guquidus Gsolidus G'Freckle Vertical ^Freckle h k K K0 Keffective Kx Kz M r R, R Ra Ral Ral Rax Raz Ra wt% wt% m2/s % variable N/m 4 m/s2 °C/m °C/m °C/m °C/m °C/m m m 2 m 2 m 2 m 2 m 2 kg/mol J.mof'.fC1 m/s Solute concentration Reference solute concentration Thermal diffusivity Liquid fraction Threshold value for various freckling criteria Driving force for freckle formation Gravitational acceleration (=9.81 m/s2) Thermal gradient Thermal gradient at the liquidus temperature of the alloy Thermal gradient at the solidus temperature of the alloy Thermal gradient at the temperature of freckle initiation Vertical thermal gradient at 7>retl«e Characteristic linear dimension Partition coefficient Permeability Reference permeability Effective permeability Permeability perpendicular to the primary dendrites Permeability parallel to the primary dendrites Molar weight Gas constant (=8.3144 I/mol.K) Solidification rate Rayleigh number Modified Rayleigh number (first theory) Modified Rayleigh number (second theory) Rayleigh number along the x axis Rayleigh number along the z axis Critical Rayleigh number (threshold value for freckling) t s Time T °C Temperature To °C Reference temperature T- Liquidus} T^Liq °C ' Liquidus temperature of the alloy Tsolidusi Tsol °C Solidus temperature of the alloy TFreckle °C Freckle initiation temperature of the alloy V, V m/s Fluid flow velocity X m First horizontal coordinate y m Second horizontal coordinate z m Vertical coordinate X Greek Symbols SI Unit Meaning a d e g . A n g l e w i t h the v e r t i c a l d i r e c t i o n P l/wt% S o l u t a l e x p a n s i o n c o e f f i c i e n t Y 1 /°C T h e r m a l e x p a n s i o n c o e f f i c i e n t £ °C/s C o o l i n g rate k g . m " 1 . s " 1 D y n a m i c v i s c o s i t y h m P r i m a r y d e n d r i t e a r m s p a c i n g m S e c o n d a r y d e n d r i t e a r m s p a c i n g P k g / m ' D e n s i t y Po k g / m 3 R e f e r e n c e d e n s i t y A p k g / m 3 D e n s i t y d i f f e r e n c e T - T o r t u o s i t y f a c t o r Abbreviations Meaning C F D C o m p u t a t i o n a l F l u i d D y n a m i c s D S D i r e c t i o n a l l y S o l i d i f i e d D S Q D i r e c t i o n a l S o l i d i f i c a t i o n a n d Q u e n c h E S R E l e c t r o - S l a g R e m e l t i n g E D X E n e r g y D i s p e r s i o n S p e c t r o m e t r y F E M F i n i t e E l e m e n t s M o d e l i n g H T C H e a t T r a n s f e r C o e f f i c i e n t I G T I n d u s t r i a l G a s T u r b i n e L S T L o c a l S o l i d i f i c a t i o n T i m e N / A N o t A p p l i c a b l e P D A S P r i m a r y D e n d r i t e A r m S p a c i n g S E M S c a n n i n g E l e c t r o n M i c r o s c o p e S D A S S e c o n d a r y D e n d r i t e A r m S p a c i n g S X S i n g l e C r y s t a l T C T h e r m o c o u p l e V A R V a c u u m A r c R e m e l t i n g U B C U n i v e r s i t y o f B r i t i s h C o l u m b i a W D W i t h d r a w a l N o t e : T h r o u g h o u t th is thes i s , t h e r m a l g r a d i e n t s w i l l be e x p r e s s e d m a i n l y i n °C/cm u n i t s , m o r e c o m m o n l y u s e d i n the i n d u s t r y , ra ther t h a n i n °C /m S I u n i t s . x i Acknowledgments The author would like to thank first and foremost his supervisor, Dr. A . Mitchell, for his invaluable guidance and encouragement throughout this Ph.D.. Discussions with Dr. S.L. Cockcroft, particularly about the numerical modeling aspect of this work, were greatly appreciated. Many thanks to A . J . Schmalz for all his help during the design, building and operating of the experimental equipment. A l l the support staff in the department of Metals & Materials Engineering at the University of British Columbia (UBC) (Vancouver, Canada) was also most helpful. The contributions at various levels from the following companies was very much appreciated : Asea-Brown-Boveri, Aubert & Duval, Canon-Muskegon, Inco Alloys International, Sandia National Laboratories, Special Metals Corporation. Finally, the author is very thankful to M s . K . H . Roe for her much needed moral support throughout all his graduate studies. xii 1. INTRODUCTION Freckles are macrosegregation defects which are usually found in nickel-base superalloys or specialty steels and which occur during solidification. Virtually any freezing process involving a columnar dendritic solidification structure can be subject to freckling : vacuum arc remelted ( V A R ) and electro-slag remelted (ESR) ingots, directionally solidified (DS) and single crystal (SX) investment castings, as well as static castings and ingots. Freckles appear as long vertical linear trails of equiaxed grains, whose chemical composition is noticeably shifted toward the alloy's eutectic composition. Freckles are highly undesirable in any type of casting because they are unremovable by thermo-mechanical treatments. Since the 1960's, when they were linked to the failure of several military engines, freckles have been considered to be unacceptable defects in industrial aerospace castings. It is now generally agreed that freckles are the product of specific fluid flow patterns, known as thermosolutal convection, originating in the interdendritic liquid during solidification. This flow is driven by a density inversion occurring in the mushy zone as a result of interdendritic segregation. Freckle occurrence is known to be dependent on three factors : alloy chemistry, casting conditions and casting geometry. Although much research has been published about freckles over the past three decades, its focus has been centered mainly on studying or numerically simulating analog systems (aqueous solutions, lead-base binary alloys, etc.). Thus, most of the knowledge to date about these macro segregates in industrial alloys remains qualitative and empirical. Superalloy casters have been able to avoid freckle formation in ingots by keeping ingot diameters and melting rates below critical values, and to minimize the occurrence of freckles in DS and S X castings by maintaining a high thermal gradient at the solidification front. These are empirically found solutions and are perfectly valid, although not necessarily optimum, for most 1 current applications. However, the recent interest in larger aircraft engines and especially in large land-based industrial gas turbines (IGT) for power generation requires a considerable scale-up of the diameter of turbine disks and the size of S X blades. One of the main problems encountered for this scaling-up is the extensive freckling observed in these larger castings. Current rejection rates due to freckling are unacceptably high, the loss of added value for each rejected blade being of the order of several thousands of dollars. Moreover, non-destructive testing technology is not yet capable of detecting internal freckles (at the center of V A R / E S R ingots or along cooling channels inside blades). As a result, casters are still very reluctant to engage in the commercial production of larger castings, for fear of the potential for catastrophic failure due to the presence of freckles and the resulting high liability costs involved. Previous empirical knowledge and empirically determined casting parameters seem to be of little help in finding the new appropriate casting conditions. The most promising procedure lies in the determination of a mathematical criterion which could be used in computer simulations in order to predict accurately the location of freckle formation. Several potential criteria (based on various mathematical expressions involving the local thermal gradient G , local solidification rate R, and local solidification time LST) have already been suggested in the literature. However, these criteria usually seem to perform rather poorly at predicting freckle occurrence. Moreover, they do not provide sufficient insight regarding the actions to be taken to minimize freckling. To date, the most complete criterion available seems to be the Rayleigh criterion, which combines two of the three factors influencing freckle formation : alloy chemistry and casting conditions, but not casting geometry. The literature review of this thesis (Chapter 2) first summarizes the main features of freckle formation. It then lists a brief description of the various freckling criteria reported to date. Emphasis is put on the Rayleigh number, with a numerical evaluation of its parameters and an estimation of its current limitations when used as a freckling criterion. In view of these limitations, the necessity of taking into account the slope of the growth front of the casting in conjunction with the Rayleigh number is suggested in order to evaluate more accurately the conditions leading to freckling. This latter point is novel and new and its confirmation is the goal of this research program (research objectives in Chapter 3). In Chapter 4 (methodology), a set of experiments is suggested to confirm this theory and to evaluate the critical values associated with the Rayleigh criterion for some commercial alloys. The modeling of the experimental apparatus used in this study is also presented in Chapter 4 (the details of the model calibration procedure and sensitivity analysis are presented in Appendix A) . Chapter 5 gathers the main results from the model (G and R conditions for the range of casting conditions and alloy systems considered). It also presents the freckling results observed in the actual casting experiments. Discussion of these results, along with the description of two new potential theories and modified Rayleigh criteria, are presented in Chapter 6. The direct applicability of the modified Rayleigh criteria to real industrial V A R / E S R or D S / S X cases is also presented in Chapter 6. Chapter 7 summarizes the conclusions of this research program, as well as suggestions for future work. 3 2 . L I T E R A T U R E REVIEW 2.1. The nature of freckles This chapter is a brief summary of the state of knowledge on freckle formation. For further details about the research carried out over the past 30 years on freckling, the reader is invited to refer to the extensive literature survey previously published in reference [1] by the author of this thesis, where many experiments and fluid flow numerical simulations (particularly using analog systems) relevant to the study of freckling have already been described. 2.1.1. Physical characteristics Freckles, also known as "channel segregates" or " A segregates", are a particular form of macrosegregation, which can potentially develop in any casting process involving directional (as opposed to equiaxed) sohdification. They can be found in a wide variety of industrial castings such as V A R and E S R superalloy billets [2], DS and S X superalloy castings [3], and large killed steel ingots [4][5](see Figure 1). Freckles appear as linear trails of equiaxed grains and segregated material, of the order of l-2mm in diameter, running vertically along the casting. In the case of V A R / E S R ingots, freckles are usually located in the center to mid-radius of the billet. In unidirectional solidification (DS and S X ) of superalloys, freckle lines are normally located on the exterior surface of the casting. In killed steel ingots, freckles usually form in the middle of the solidification zone which grows perpendicularly to the side walls. Some porosity and feeding shrinkage may also be noted in and adjacent to the freckle line. The misoriented grains associated with freckles and nucleated during casting do not normally penetrate deeply into the casting because they cannot effectively compete with the more rapidly growing preferred orientation. 4 It is interesting to mention that freckles are able to develop parallel to the general solidification direction (DS/SX castings), perpendicular to it (killed steel ingots), or at various angles to it ( V A R / E S R ingots). Freckles have been observed to be enriched in those elements which segregate normally (i.e. rejected into the liquid during dendrite growth) and depleted in those elements that segregate inversely. Thus, freckles are shifted toward the eutectic composition [1][3]. Freckles are highly undesirable in any type of casting . They tend to reduce the ductility and yield strength of the alloy. Moreover, they cannot be removed by subsequent thermomechanical treatments : (a) the combination of slow solid-state diffusion rates and relatively large freckle size would require prohibitively long homogenization heat treatment times, (b) the high levels of microporosity and precipitates (eutectic, primary carbides, etc.) cannot be significantly reduced, and (c) misoriented grains in freckle trails cannot be eliminated, and require the scrapping of any affected D S / S X casting. The probability of occurrence of freckles has been observed to depend on several factors : (1) Alloy composition : Freckling is highly dependent on alloy composition [6] [7] [8] [9]. Experiments involving binary alloys showed that, below a certain alloying limit, no freckles develop. Above this limit, the number of freckles seems to increase with the solute concentration. For example, in N i - A l castings, no freckles were found at compositions of lw t% and 5wt%Al , a few freckle grains were noted at 8wt%Al , and there were numerous freckles at 10wt%Al [3]. The nature of alloying element is also important. Superalloys with high titanium (segregating normally) or tungsten (segregating inversely) are reported to be more freckle prone [3]. 5 (2) Solidification parameters and operating conditions : It is often suggested that freckling can be significantly reduced and even avoided by operating at larger thermal gradients and faster solidification rates [2]. (3) Shape and geometry : In the case of remelted superalloys ingots, E S R ingots have been observed to be more prone to freckling than V A R ingots. This observation has been related to the V-shaped pool and mushy zone profile in ESR, as opposed to a U-shaped profile in V A R [2] [10]. a) " A " segregate in a large killed steel ingot [4]. (Figure 1, continued) 6 b) Center to mid-radius freckles in V A R IN718 (quarter of a cross-section). c) Surface freckles in the root portion of a large S X IGT Mar-M247 blade. Figure 1: Various appearance of freckles in industrial castings. 7 2.1.2. Mechanism of freckle formation : Density inversion theory The vertical upward directional solidification of an alloy is presented schematically in Figure 2. The heat flow is vertical downward (for example, from the hot furnace at the top to the chill plate at the bottom), creating a vertical thermal gradient along the casting. In addition to a thermal gradient, there also exists a variable solute concentration gradient in the liquid between the bottom of the mushy zone and the top of the casting. Moreover, the density, p, of a liquid alloy can be considered to be dependent on its temperature and its solute concentration. In this case, p is usually expressed in the following form [11][12][13][14][15]: p = p 0 X [1 - |3x(C-C 0) - yx(T-T0)] [eq. 1] In the case of most metallic alloys and analog systems, y is always positive (i.e. the density decreases when the temperature increases). However, p can be either positive or negative, depending on the relative densities of the solute and the solvent in binary alloys, and on the segregation tendency (normal or inverse). In the cases where the rejected solute is lighter than the solvent, (3 is positive. The combined influence of the temperature and concentration profiles in the liquid and mushy zone can lead to a density profile similar to that in Figure 2. It can be seen that the solute gradient in the mushy zone has a driving effect on the density inversion whereas temperature has a stabilizing effect. Given such a density profile, it can be seen that the interdendritic liquid lower in the mushy zone (enriched in solute) is less dense than the liquid at the dendrites tip. This is a case of density inversion at the growth front. This system is inherently unstable, which can lead to some fluid convection in order to reduce its potential energy [12]. This phenomenon is known as "thermosolutal" or "double diffusive'' convection, since it arises from the influence of both thermal and solute concentration gradients. It 8 is now widely agreed that thermosolutal convection is the cause of freckling [12][14][16][17]. The complete fluid flow pattern associated with freckling is shown in Figure 3. Photographs of the freckle plumes appearing above the mushy zone can be taken in transparent systems, an example of which is shown in Figure 4. It is interesting to mention that freckling does not occur in zero gravity conditions, where density differences cannot drive fluid flow. Solid Figure 2 : Schematic diagram of directional solidification and associated thermal (p-p), solutal (pc) and thermosolutal (PT+C) density profiles illustrating the density inversion theory. 9 Figure 3 : Freckle formation and associated fluid flow pattern. Figure 4 : Freckle plumes rising above the growth front in D S N H 4 C 1 / H 2 0 [18]. KB L i q u i d M e l t x, Liquidus 11 T3 -T-s olidus Solid Casting Freckle P lume A A Equiaxed Grains and/or Eutectic-Enriched Material (l-2mm) Heavier Non-JSegregatecj Liquid Lighter ••' Segregated) Liquid 10 2.2. The prediction of freckle formation Industrial casters of superalloy components are presently faced with the challenge of increasing the size of their castings for use in bigger aircraft engines (such as on Boeing 777), and more specifically in land based turbines for power generation [19]. One of the main obstacles to this scaling-up is the very high rejection rates due to freckle formation. The recent advances in memory and speed of computers in the past few years make numerical simulations increasingly practical and affordable over trial and error experimentation in order to modify an existing process or develop new process technology. Hence, the driving force is great to numerically characterize freckle formation, thus enabling casting techniques to be optimized by computer simulation rather than furnace experimentation. This chapter focuses on various criteria suggested in the literature to describe freckle formation. The Rayleigh number is selected as the most complete of these criteria. It is presented in greater details along with the numerical data necessary to its evaluation. 2.2.1. Various freckling criteria The mathematical expressions presented below have all been reported in the literature as criterion suggestions for freckle formation. The following expressions are compared to critical threshold values F, indicating freckling when the comparison is verified. F assumes different values (and units) depending on the criterion considered. G • R (a) The Gradient Acceleration Parameter [20]: GAP = < F [eq. 2] LST Initially used to relate micro structure and ultimate strength, this criterion performed poorly in numerical models to predict freckle formation. 11 c (b) The Xue porosity function [20]: XUE = js"lidus <F [eq. 3] V LST This criterion was developed without any relation to freckle formation. Although it has been reported to predict freckling locations with some degree of accuracy, it does not provide any insight on correction steps to be taken to decrease freckle occurrence. V G (c) Fleming's criterion [21]: < — 1 [eq. 4] Since e=GxR, this equation simply states that the formation of a freckle occurs when the interdendritic fluid velocity is greater than the solidification rate. This criterion is based on the consequences rather than the causes of density inversion. In any event, unless the density inversion parameters are known and taken into account in a fluid flow computation, this criterion is incomplete. Moreover, most mathematical casting models used in the industry are diffusive heat transport models (sometimes coupled with macroscopic fluid flow), which are not designed to handle microscopic (dendrite scale) fluid flow. (d) Thermal gradient criterion [6]: G<F [eq. 5] Due to the very nature of the phenomenon, thermosolutal convection will not occur in large vertical thermal gradients. Thus, it is now common knowledge that a high thermal gradient wil l prevent freckling. However, this criterion is probably very conservative. Moreover, the required high gradients cannot currently be achieved for large castings by the traditional Bridgman furnaces, due to a combination of Biot and Fourier criteria for the larger thermal sections. 12 (e) G/R criterion [6]: [eq. 6] G/R has been used to characterize the solidification structure (plane front, cells, D S or equiaxed) of a casting. Although it is a definite boundary for operating conditions in DS and S X castings, this criterion does not appear to be linked closely to freckle formation. (f) Primary dendrite arm spacing criterion [6]: A, > F [eq. 7] It has been shown that A,i is proportional to G'mxR'm. Thus, this criterion is sometimes written G'U2xR'[l4>F. It has been reported to predict relatively accurately the formation of spurious grains and freckles in S X castings. However, this criterion does not explain the fact that freckles tend to form repeatedly at preferential locations in castings, even though, in the case of D S and S X castings, the primary dendrite arm spacing is virtually constant throughout the entire casting. (g) Cooling rate criterion [11]: GxR <F [eq. 8] This criterion derives from the observation that freckle formation requires a certain time necessary for the density inversion induced fluid flow to develop, and erode and/or melt a freckle channel. Thus, freckles may appear if the local solidification time (LSI) is greater than a critical value. Where LST = (TUquidus-TSoiidus)l{GxR) [eq. 9] Hence the cooling rate criterion as it is presented above. This criterion has been shown to predict the presence of freckles with reasonable precision. However, it takes into account only the casting conditions, which is only one of the three factors influencing freckle formation. 13 2.2.2. The Rayleigh criterion The Rayleigh number is the ratio of the dimensionless Prandtl and Schmidt numbers and g • Ap • / i 3 can be written in its most general form as : Ra — [eq. 10] 77 • DT Ra is one of the dimensionless parameters appearing in the Navier-Stokes equations used to describe fluid flow mechanics. It represents the ratio between gravity and viscosity effects, and it is now widely agreed that the Rayleigh number characterizes the onset of fluid flow in unstable systems [18] [22] [23]. In order to characterize fluid flow associated with freckle formation, Sarrazin & Hellawell [24] suggested to use the following form for the Rayleigh number : „ dp rjD7 Ra = -&- [eq. 11] h4 The Rayleigh criterion can then be written in the form [24]: Ra>F [eq. 12] The parameter h is a characteristic linear dimension of the system. It has been linked to the dendritic array with the following expressions: ii=V [24] [eq. 13] or hA=Kxk{2 [6] [eq. 14] The Rayleigh criterion described above is based on the "thermal" Rayleigh number (expressed with the thermal diffusivity DT rather than the solute diffusivity). The reason for this choice lies in the fact that the diffusion of heat is much faster (by a factor 103) than the diffusion of solute in conventional metallic systems, and therefore the diffusion of solute can be considered negligible during the evolution of freckle flow in the mushy zone. This is confirmed by the observation in analog systems that freckle plumes rising above the mushy zone are usually in 14 thermal equilibrium with the surrounding bulk liquid, but still retain their high solute concentration [12]. The numerator in the Rayleigh number can be considered as depicting the driving force (due to density inversion) in the interdendritic liquid to produce freckling flow. The denominator represents the restriction opposed to this flow by viscosity and diffusivity of the melt, and permeability of the mushy zone. Sarrazin & Hellawell estimated that, considering hA=\C, then the critical threshold value for the Rayleigh criterion is Ra*=F=l. dp dp dT dp Since — = —x—— = —— x G , it is also possible to write the Rayleigh number in a dz dT dz dT slightly different manner from [eq. 11]: dp 8 XT Ra=—^-xGX\ [eq. 15] 7]DT The term g is a constant ; dp/dT is dictated by the micro-segregation profile of the melt [1], which is mostly dependent on alloy chemistry for typical industrial casting conditions ; r\ and DT are dependent on alloy chemistry but not on local solidification conditions. Moreover, if A,i is considered to be proportional to R'U4G'U2, then for a given alloy system, Ra becomes proportional to MGR. Therefore, the Rayleigh criterion and the cooling rate criterion can be considered similar. It is worth mentioning that Ra is then also proportional to the local solidification time LST. The Rayleigh number as presented above seems to predict freckling with relatively good precision (at least as accurately as the cooling rate criterion [eq.8]). Moreover, since it combines alloy properties/chemistry, and solidification conditions, the Rayleigh number currently appears to be the best basis for a freckling criterion and to provide the most insight on freckling conditions. 15 Like most of the other criteria suggested in the literature and presented in Chapter 2.2.1, the Rayleigh number describes the critical conditions for the initiation of freckle flow rather than its steady state. However, since freckling results from the breakdown of a metastable equilibrium state (heavier liquid atop a lighter one), it is assumed that freckle initiation will always produce fully grown freckles. The subsequent path of the resulting freckle is however not characterized by the criterion. 2.2.3. Rayleigh criterion : Range and sensitivity of the parameters Many variables in the Rayleigh number are either known or can be numerically evaluated from literature data: (a) Temperature T and thermal gradient G are typical outputs of a casting model. It should be mentioned that in most of the equations presented above (including the Rayleigh criterion), there is no clear indication regarding the temperature at which G should be evaluated (instead, it seems to be considered one of the process constants). However, in most casting processes, G is strongly dependent on the temperature at which it is measured or estimated (e.g. Guquidus and Gs<>udus are quite different). This should influence greatly any results given by a freckling criterion. Therefore, in the rest of this thesis, care wil l be taken to associate thermal gradients with their relevant temperature. 16 (b) Viscosity rj : measured values for the alloys of interest (and their composition variations due to segregation) are probably unavailable. However, r\ can be numerically evaluated for any alloy composition at any temperature with the following equations [25] [26]: r/= Aexp(BIrT) (±10%) (in kg/m.s) [eq. 16] i . 7 x i o - 7 - p 2 / 3 - r ^ - M - , / 6 where A- (in kg/m.s) [eq. 17] uidus 1 and 5 = 2 . 6 5 - 7 ^ (in J/mol) [eq. 18] (with r = 8.3144 J /mol.K (gas constant), p in kg/m 3 , T and Tuquidus in K and M in kg/mol) Typical values of viscosity for nickeLbase superalloys can be found below in Table I. (c) Thermal diffusivity DT : as for T), measured values of DT are probably also unavailable. Nevertheless, DT can be estimated numerically with reasonable precision (±10%) as well. The following values in Table I have been calculated by the program "Metals" developed by National Physical Laboratories (NPL, England) [26] [27]. Table I : Liquid density, viscosity and thermal diffusivity for the four superalloys investigated in this thesis at their liquidus temperature and 100°C above their liquidus temperature. Superalloy Temperature (°C) Liquid density (xlO"3 kg/m3) Viscosity (xlO3 Pa.s) Thermal diffusivity (xlO"6 m2/s) IN718 1336 7.45 4.18 7.61 IN718 1436 7.36 3.62 7.66 Waspaloy 1355 7.34 4.18 7.70 Waspaloy 1455 7.24 3.61 8.20 Mar-M247 1360 7.30 4.13 9.41 Mar-M247 1460 7.20 3.58 9.82 UBC1 7Liquidus 7.33 4.16 9.68 UBC1 Tnquidus+^00°C 7.23 3.61 10.12 (p calculated by weighted average [27], n calculated with the formula in [eq. 16] and DT calculated by the software "Metals" [27]. The nominal compositions of these superalloys can be found in Chapter 4.1.)(For proprietary reasons, the liquidus temperature of UBC1 cannot be printed.) 17 In view of the data in Table I, values of r\ and DT are expected to be relatively constant within the range of alloy compositions, temperatures and segregation patterns usually involved in freckle formation in superalloys. (d) Primary dendrite arm spacing (PDAS) A-i and secondary dendrite arm spacing (SDAS) %i can be evaluated from the cooling rate (product of the thermal gradient Guquidus and the solidification rate R at the dendrite tips) as shown on Figure 5 [28]. It can be seen that P D A S and S D A S are relatively independent of the alloy considered among various superalloys. 10 100 1000 10000 1 — i 1 — n q 1 1 — r r j 1 1—rrq d I N - / I B V IN-738LC O IN-792 O MAR-M-246 & NI-9AL-4CR 0 NASAIR-100 i ' i I I I I | I 111 I I I I I I I L j J 0.01 0.1 1 10 100 COOLING RATE (G x R), "C/SEC Figure 5 : Relationship between cooling rate at the dendrite tips and primary/secondary dendrite arms spacing in various superalloys [28]. The linear interpolations in Figure 5 can be translated to the following equations (the precision range reflects the scatter of the data) : 18 and 150x10" 0.33 {G Liquidus X R) 40X10" 6 v0.42 m (±30%) m (±30%) {G Liquidus X R) (k\ and X2 in m, cooling rate GuquidusXR in °C/s) [eq. 19] [eq. 20] (e) Liquid fraction fL : fL is usually one of the parameters in a casting model. It governs the latent heat release on solidification. In the case on nickel based superalloys, liquid fraction profiles are relatively independent of the alloy system considered, as can be seen in Figure 6. 1 0.9 -I 0.8 0.7 H § 0.6 ' o £ 0.5 • f 0.4H 0.3 0.2 0.1 0 •Model o FN100 (a) & LN718 (a) A LN718 (b) o Mar-M002 (b) • Mar-M247(a) • Mar-M247 (b) 0.1 0.2 0.3 , 0.4 0.5 0.6 0.7 Melting Range Fraction (TSolidus=0, T U q u U l u = l ) 0.8 0.9 Figure 6 : Typical liquid fraction profile along the mushy zone of nickel base superalloys.(experimental measurements : (a) from [1] and (b) from [29].) (The bold curve is the generic profile used in the numerical model used in this thesis and described in Chapter 4.3). 19 Of particular interest is the liquid fraction at which freckling originates. It has been shown that freckles initiate at a liquid fraction between 0.4 and 0.6 in superalloys [1][30]. A value of 0.5 (±20%) can therefore be considered a good approximation for most superalloys. Given the profile shown in Figure 6,/L=0.5 corresponds to a melting range fraction of about 0.8. Consequently, it is possible to define the temperature for freckle initiation TFreckie as : O.lTsoiidus+O&Tuquidus [eq. 21] (f) Permeability K : there are numerous of publications reporting expressions of the permeability of the mushy zone (considered as a porous medium) as a function of X\, %i and liquid fraction/I- Most of the research published to date (especially numerical simulations [14][31][32]) usually consider permeability as an isotropic parameter depending only on liquid fraction. One of the best known examples is the Kozeni-Carman equation [33][34] : K = K0{ffl{\-fl)2) [eq.22] Other expressions include : ln(y?) = ln(0.10e a 8 1) + 19.6e 0 3 3 l n ( / L ) [35] [eq. 23] or K = K0xfL2 [15][21][36] [eq. 24] (Where e is the cooling rate and Kb is a constant depending on the mushy zone structure (usually a function of the primary dendrite arm spacing X\ and some tortuosity factor %).) However, it is now accepted that the mushy zone in directionally solidified metallic systems is highly anisotropic. Poirier et al. [37] [38] [39] studied the permeability for fluid flow parallel and perpendicular to a DS mushy zone in analog systems. He also conducted numerical 20 simulations for perpendicular flow based on actual dendrites geometry. The permeability in a DS mushy zone was found to be best described by the following expressions : For flow parallel to the dendritic array : Kparaiui = 3.75 x 10"4 • fl • X] (±31 %) (for 0.17<ft<0.61) [eq. 25] For flow perpendicular to the dendritic array : Kperpendicuiar = 3.62 x 10 3 • flM • A 0 , 6 9 9 • A 2 2 7 3 (±30%) (for 0.19</ t<0.66) [eq. 26] (A-i and X2 in m and Kparaiid, Kperptnd\cuiar in m 2) N o data on the permeability in real superalloy systems has been published to date. Nevertheless, because the mushy zone structure is relatively similar between analog and industrial systems, equations [eq. 25] and [eq.26] have been chosen as the best available approximations. They wil l be used to estimate permeability in superalloys in the rest of this thesis. It is interesting to examine numerical values of the permeability for castings where freckles are usually encountered. Values of A-i=0.35mm (350Lim) and A,2=0.12mm (120Lim) are typical of both V A R / E S R ingots and small D S / S X castings, and/L=0.5 is the liquid fraction where freckles initiation takes place [1]. In this case : Kz = l . l x l O " 1 1 m 2 and Kx = 2.7x10"" m 2 In the case of large D S / S X castings, the dendrite arm spacing increases to A,i=0.5mm (500Lim) and A,2=0.175mm (175Lim), yielding : Kz = 2.3x10-" m 2 and Kx = 9.8x10"" m 2 Thus it can be seen that flow perpendicular to the direction of the dendrites should be noticeably easier than that parallel to the dendrites due to a permeability 2.5 to 4 times greater. 21 These expressions of permeability are probably quite sensitive to the details of dendrite geometry, particularly at large values of Xi and X2. Moreover, superalloy mushy zones may exhibit noticeable localized deviations from the average P D A S and S D A S values, possibly creating preferential paths of increased permeability and interdendritic fluid flow channeling (at the grain boundaries for example). This fact is best illustrated by considering the degree of scatter of the measured data in Figure 5. However, further work is required in order to obtain better estimates of the actual permeability in the mushy zone of industrial alloys. (g) Density inversion term dp/dT : The evaluation of the density inversion leading to freckling has been the main focus of previous research by the author of this thesis [1]. It was found that, independently of the freckle-prone system considered ( N H 4 C 1 / H 2 0 , lead-tin, lead-antimony, and nickel base superalloys), the density inversion factor leading to freckle formation was of the order of : dp/dT = 30 (kg/m 3 )/°C (±25%) This result was obtained by measuring by microprobe analysis on quenched samples the chemical composition of the interdendritic liquid during directional solidification. The resulting composition profiles could then be mathematically translated into density profiles by means of a mathematical model based on weighted averages of pure elements. The reader is invited to refer to [1][40] for further details on the experimental procedure. (h) Critical threshold value Ra* for the Rayleigh criterion : Sarrazin & Hellawell [24] reported that when the Rayleigh number is written with h4=X4, then Ra*=l for Pb-2wt%Sb, Pb-10wt%Sn and N H 4 C l - 3 0 w t % H 2 O . Moreover, it has been established in [1][40] that for a density 22 gradient of about 30(kg/m 3 ) /°C, Ra=l is also the critical threshold value for nickel-base alloys. Thus it is expected that the critical threshold for the Rayleigh criterion (with h4=X4) is Ra=l and is independent of the alloy considered. 2.3. Limitations of the Rayleigh number as a freckling criterion 2.3.1. Numerical evaluation of the Rayleigh number The precision associated with the numerical evaluation of most factors of the Rayleigh criterion is still relatively poor. r| and DT are known to ±10%. Expressions for the permeability K are estimated at ±30%. Moreover, as shown in [1], values of the density gradient dp/dT can be relatively sensitive to the combined accuracy of microprobe measurements and of the density calculation method, especially regarding minor alloying elements such as C, B , Zr or S i : (a) Microprobe analysis of low level alloying elements (<lwt%) shows relatively poor accuracy. (b) Current liquid density calculations are based on weighted averages of pure elements, and thus do not recognize the very different effects substitutional and interstitial elements have on density. In any event, the Rayleigh number shown in equation [eq. 15], could be written : Ra = ®(r\,DT,dp/dT) x T(G,R,h,LST) [eq. 27] Thus, it can be seen that these coefficients with limited precision are used only in the proportionality factor (O function) (which could also be named "alloy chemistry factor"), 23 describing the various freckling behaviors between identical castings of different alloys. Since this proportionality coefficient is not known with great precision, it may be necessary to recalibrate some of its terms to account for a given critical threshold value Ra*. Alternatively, it is also possible to calibrate Ra* instead. This calibration could be carried out by comparing freckle defect maps for actual castings to the numerical maps predicted by the criterion in a computer model. However, the more interesting factor is probably the second term, or "casting conditions factor" (T function). Indeed, it is this factor which will provide some insight, for a given alloy chemistry, about location and extent of freckling for various casting conditions and various casting procedures. 2.3.2. Other limitations of the Rayleigh criterion The Rayleigh criterion, as described earlier, is incomplete. Indeed, there are at least two features in freckled castings that are not currently accounted for by the terms in the Rayleigh number. 2.3.2.1 .The effect of casting size When several freckles appear in a given region of a casting, they are usually "equi-spaced". In other words, it is possible to attribute same-size cross-section areas to each freckle [3]. This size is considered to be the minimum cross-section area necessary for the fluid flow pattern associated with freckling to develop. Thus, it is believed that freckles should not develop in casting areas that are thinner than the minimum freckling area. This phenomenon is verified in blade casting, where freckles are usually found in the root rather than the airfoil section. This factor, however, is not currently included in the Rayleigh criterion. 24 2.3.2.2.Surface-only and mid-radius-only freckles The Rayleigh criterion does not presently include any factor taking into account the fact that virtually all freckles found in DS and. S X castings occur at the surface of the casting (mold/casting interface) whereas none are found inside the bulk of the metal. Moreover, in many V A R / E S R ingots, freckles are found mostly at mid-radius but not in the center (see Figure 7), even though the Rayleigh number is expected to be maximum at the ingot centerline, where the P D A S A,i and local solidification times LST are the greater. ^ l u l - l . k l l l l s - i M l K l lCikl lt Figure 7 : Freckles at mid-radius only in V A R IN706 ingot. [41] (Freckles graphically enhanced for purpose of presentation) 25 2.4. Influence of the growth front angle The key to explaining surface-only and mid-radius-only freckling may be found among early experiments conducted by Giamei et al. [11]. Experiments on the transparent analog system ammonium-chloride / water (NH4CI /H2O) yielded the following observations (schematically depicted in Figure 8): (1) In straightforward vertical experiments (Figure 8 a)), the growth front was horizontal flat, and freckles seemed to be randomly distributed across the casting. (2) Some thermal insulation was then introduced at the bottom of the casting to artificially curve the growth front. When the center of the casting was colder and higher than the edges (convex front) (Figure 8 b)), freckles appeared preferentially at the center. Conversely, when the edges were colder and higher than the center (concave front) (Figure 8 c)), freckles formed preferentially at the surface of the casting. (a) Flat growth front (b) Convex growth front (c) Concave growth front Figure 8 : Schematic description of Giamei's experiment with curved growth fronts. 26 This suggests that the segregated interdendritic liquid may first flow perpendicularly to the dendrites (path of higher permeability) at a certain depth in the mushy zone (probably along the iso-liquid fraction//=0.5 at which it initiates). It is then diverted upward, either by the mold wall (concave front case) or because the growth front angle diminishes to zero and cannot sustain radial flow at the center (convex front case). In another field of study, Hart [23] conducted experiments on the evolution of convection patterns in water with respect to the direction of gravity. Water was contained in a rectangular box which could be tilted at various angles to the vertical. The broad faces of the box were maintained at two different temperatures to induce thermal convection. Hart found that the onset of unstable convection (akin to freckle initiation) was best described by the value of the Rayleigh number of the system and that the critical threshold value above which the system was turbulent (i.e. similar to exhibiting freckles) depended on the tilt angle of the box with the following relationship : Unstable flow for: Ra > 77067cos(a) [eq. 28] This equation can be rewritten : Unstable flow for : Ra x cos(a) / 1708 > Ra*= 1 [eq. 29] This could be considered a "modified Rayleigh criterion", in which the original Rayleigh expression is amended to include a function of the angle, and thus including the missing geometry factor. The influence of an angle on freckle formation is probably best understood by considering that at low angles, the system is metastable and may not evolve. At higher angles however, the system becomes unstable and must evolve, generating fluid flow and possibly freckles. 27 It is therefore suggested that the angle of the growth front plays an important role on freckle formation and that some angles should favor freckling. Furthermore, this factor (which can be considered as a "casting geometry factor") provides a plausible explanation for the occurrence of surface-only or mid-radius-only freckles. This is described schematically in Figure 9. Freckle flow tends to follow the path of least resistance at first, which is perpendicular to the dendrites (as shown in Chapter 2.2.3, the permeability perpendicular to the dendrites is about 2 to 4 times greater than that parallel to the dendrites). Large DS/SX Casting ESR/VAR Ingot Large killed steel ingot (Freckles at the surface) (Freckles at mid-radius) ("A" segregates) Figure 9 : Freckle flow path and growth front angle in various industrial castings. In the case of D S / S X castings, this flow will reach the edge of the casting and then develop vertically upward forming a surface freckle (Figure 10 (a)). Freckles can also develop at a certain angle to the vertical, even at the surface of the casting (Figure 10 (b)). The latter probably occurs when the growth front is angled in the third direction and offers greater sideways permeability (see Figure 11). 28 The propensity of D S / S X castings to form surface-only freckles may also be enhanced by a potentially higher permeability to fluid flow at the casting/mold interface, thus favoring the transition from a flow perpendicular to the dendrites to a flow parallel to the dendrites in the upward direction. The permeability at the casting/mold interface is expected to depend on the ceramic face coat and alloy chemistry (wetting or non-wetting properties) as well as on mold roughness and possibly on dendritic geometry at the surface of the casting. However, no data on the permeability at the casting/mold interface for industrial situations is available, and comparison with permeabilities inside the bulk of the mushy zone are only speculative. In any event, more work is required to determine whether the interface permeability has any effect on freckle formation at the surface of D S / S X castings. (a) Straight vertical freckles (b) Slanted freckles. Figure 10 : Freckles on opposite sides of the same root section of a large S X blade casting (UBC1 alloy)(Freckles size : about l-2mm diameter). 29 Figure 11 : Schematic diagram illustrating vertical and slanted surface freckles in D S / S X castings. In the case of V A R / E S R ingots, as the flow gets farther from the centerline, the permeability of the mushy zone decreases, diverting the freckling flow upward at mid-radius. Figure 12 shows a V A R IN706 ingot exhibiting freckles which follow the shape of the liquid pool and the mushy zone. This is a clear indication of the greater permeability and preferential path for freckle flow perpendicular rather than parallel to the primary dendrites. Figure 12 : V A R IN706 ingot, exhibiting freckle flow following the shape of the liquid pool (i.e. perpendicular to the primary dendrites) (see longitudinal plane section)[41]. 30 In the case of large killed steel ingots, freckles develop on the side walls rather than the bottom of the ingot, also probably because of the higher permeability for flow perpendicular to the primary dendrites. Moreover, as the side walls growth front advances, permeability ahead of the freckle flow is reduced, progressively diverting the freckle flow towards the center of the ingot. Indeed, it has been observed that the angle of the " A segregates" is directly proportional to the solidification rate (Figure 13)[4] [42]. The faster the side walls solidify, the more the " A segregates" have to deviate from the vertical to follow the path of higher permeability perpendicular to the primary dendrites. SOLIDIFICATION R A T E R .mmmm Figure 13 : Relationship between " A " segregate angle to the horizontal and solidification rate [4] [42]. Note : The term "growth front angle" used in the above discussion actually refers to the angle between the horizontal (perpendicular to the gravitational direction) and the iso-liquid fraction contour at fu=0.5 in the mushy zone (corresponding to the location of freckle flow initiation), 31 rather than the actual dendrite tips (iso-liquid fraction of 1.0) This definition of the "growth front angle" is the one used in the remainder of this thesis. Given the above definition, the growth front angle can also be viewed as the angle between the horizontal and the isotherm for the temperature of freckle initiation 7V r e CHe. This is true for the processes where undercooling can be neglected. This is also true for the processes where the undercooling is constant throughout the casting. Indeed, an undercooling of 5°C in IN718 ingot casting, for example, means that the dendrite tips are located at 1331°C (actual liquidus temperature on cooling) instead of the traditional liquidus temperature of 1336°C. It is therefore possible to neglect the undercooling by shifting all the temperatures accordingly (especially TFreckle)- This applies to most casting situations. Therefore, in the rest of this thesis, the growth front angle will also refer to the angle between the horizontal and the Tpreckie isotherm, and undercooling wi l l be neglected. However, in processes where the degree of undercooling is variable in the casting (for example, locally at a sharp section change in D S / S X castings, such as a blade shroud)(see Figure 14), only the original definition of the growth front angle (based on iso-fraction liquid) would be valid. Figure 14 : Case of a variable undercooling, where the growth front angle cannot be identified by the TFreckle isotherm [43]. ^ N T U Q U I D U S A T 5 G ATp V P G 32 3. R E S E A R C H OBJECTIVES 3.1. Research focus Past research showed that freckle formation is influenced by three factors (alloy chemistry, casting conditions, casting geometry), the first two of which are currently already incorporated in the Rayleigh criterion. Moreover, the casting conditions data required for the evaluation of the Rayleigh criterion can be easily obtained in typical outputs from models now commonly found in the industry. Calculation of the Rayleigh number does not require computational fluid dynamics (CFD) modeling, which can be more accurate than diffusive heat transport modeling, but also much more computation intensive and less convenient in industrial settings. Thus, among, all the criteria suggested to numerically describe freckle formation, the Rayleigh criterion seems to be the most appropriate and the most promising for on-line application in an industrial environment. The numerical uncertainties mentioned in Chapter 2.3.1 can be fine-tuned to specific alloy systems thanks to high temperature physical measurements such as those conducted by National Physical Laboratories (UK) [26] [27], but also by comparing real freckling patterns with model-calculated threshold values. The focus of this research is therefore to incorporate the currently missing casting geometry factor into the Rayleigh criterion, in such a way as to provide for the limitations mentioned in Chapter 2.3.2. Among these limitations, the effect of minimum casting size is a minor issue in the current casting scale-up trend. Indeed, the thinnest sections in big D S / S X blade castings for IGT applications are usually much larger than the estimated minimum 25-50 mm 2 area required per freckle [1]. 33 Hence, the focus of this research program is to provide an explanation for the fact that freckles tend to form only at the surface of D S / S X superalloys castings and sometimes only at mid-radius of V A R / E S R ingots. This explanation should be quantifiable in order to be incorporated into the Rayleigh criterion. 3.2. Objectives The objectives of this research program are the following : (1) Demonstrate the influence of growth front angle on freckle formation in superalloys, (2) Develop a theory to explain the physical basis behind this influence, (3) Incorporate this theory numerically into a "modified" Rayleigh criterion, (4) Estimate quantitatively the threshold values Ra* of this modified Rayleigh criterion for several freckle-prone industrial superalloys (used in V A R / E S R ingot or D S / S X blade castings). (5) Illustrate the way to readily apply this modified Rayleigh criterion to actual casting processes such as ingot remelting and directional solidification. Although working with low temperature aqueous solutions or lead-base binaries may be easier than with high melting point nickel-base alloys, there is a great need for quantitative data relevant to actual industrial alloys. Thus, it was decided that this research program should concentrate on gathering data for superalloys currently in use rather than for analog systems, which have been the focus of much research already. 34 4. M E T H O D O L O G Y In order to fulfill these objectives, a modified Bridgman-type furnace has been built to specifically investigate the effect of growth front angle on freckling. For each sample, casting conditions (thermal gradient Guauidus and solidification rate R at the solidification front) were accurately determined by numerical simulation using a commercial F E M package named ProCAST, calibrated with thermocouple measurements in the furnace. 4.1. Choice of industrial alloys Three industrial alloys were selected for this experimental investigation : (a) Waspaloy : a typical V A R / E S R forging alloy , (b) Mar-M247 : a typical DS alloy, (c) U B C 1 : aNi-base superalloy. A l l three alloys are nickel-base superalloys prone to freckle formation. In addition, superalloy IN718 was used for the furnace calibration experiments. The nominal compositions and melting range of these alloys are presented in Table II. Note : The actual name and composition of alloy U B C 1 could not be given in this thesis for industrial proprietary reasons. 35 Table II : Melting range, freckle initiation temperature and nominal composition of the alloys considered in this thesis (except U B C 1 ) Alloy IN718 Waspaloy Mar-M247 Melting Range 1260-1336 1330-1355 1280-1360 TFreckle 1320 1350 1344 N i Bal . Bal . Ba l . Fe 18.0 - -Co 0.4 12.3 10.0 Cr 19.0 19.0 8.5 A l 0.5 1.2 5.5 T i 1.0 3.0 1.0 H f - - 1.4 M o 3.0 3.8 0.6 Nb 5.5 - -Ta - - 3.0 W - - 10.0 (Note : TFreckie=0.2Tsoiidwl+0.STutlUidus (see Chapter 2.2.3)) 4.2. Experimental apparatus 4.2.1. Tiltable Bridgman furnace A vacuum induction furnace has been built in order to melt and directionally solidify 25mm diameter x 150mm long rods ( l in . diameter x 6in. long). The basic design is similar to that of the classical Bridgman furnace used in the industry, and a schematic diagram is presented in Figure 15. The induction coil is connected to a 50kW (adjustable) / 4.5kHz power supply. The top lid and walls of the graphite susceptor are 30mm thick in order to absorb over 99% of the magnetic field and prevent any electro-magnetic stirring of the melt. 36 Ceramic Insulation-Graphite Susceptor-Induction Coi l" Graphite Felt • Copper Baffle Copper Chi l l Water Cooling O o o o o 3 . o o o o -Alumina M o l d O O • Metal Casting Steel Spacer 50mm Withdrawal Direction i -Figure 15 : Schematic diagram of the Bridgman-type furnace used in this study. In order to determine the casting conditions for each heat as accurately as possible, this furnace has been modeled with a commercial finite element modeling ( F E M ) package named P r o C A S T (Chapter 4.3 presents the main outline of this model). The calibration of the model was carried out with various thermocouple measurements (especially inside the melt) in the furnace and is described in further details in Appendix A , along with the model's sensitivity analysis. The bulk of the results is presented in Chapter 5. Except for transient zones at the start and the end of solidification (bottom and top of the casting respectively), the solidification conditions in this experimental furnace are essentially steady-state and relatively constant throughout the entire sample. In the remainder of this thesis, all the results (such as the presence or absence of freckles) will pertain to this steady-state middle 37 section of the castings, where gradients and solidification rates can be ascertained with reasonable precision. The thermal gradient at the growth front (Guquidwd can be adjusted between 5 and 40°C/cm and depends essentially on the withdrawal speed and on the temperature of the susceptor (called "control temperature" of the furnace in the rest of this thesis) relative to the liquidus temperature of the alloy. The solidification rate R in steady-state operation is governed by the withdrawal speed, which can be adjusted by the operator between 1.6xl0" 5 and lOxlO" 5 m/s (1 and 6 mm/min). These gradients and solidification rates are typical of the casting conditions commonly found in the industry. Under these conditions, the growth front is essentially flat and perpendicular to the length of the casting. The main feature of this furnace is its ability to be tilted from 0° (conventional vertical operation) to 40° to the vertical. The whole chamber and all its contents are tilted together as shown in Figure 16. This presents the great advantage of allowing the study of the influence of the growth front angle on freckle formation, while everything else remains unchanged from one run to the next, since the inside of the chamber is not modified. Figure 16 : Tiltable directional solidification furnace. 38 4.2.2. Typical experiment A typical experimental run follows the standard procedure outlined below : (1) The alumina crucible is filled with solid blocks of superalloy and raised in the top position inside the graphite susceptor ; (2) The furnace chamber is closed, evacuated and tilted to the desired angle ; (3) The experiment starts (t = Os) with a heat-up period to bring the graphite susceptor to the desired operation temperature (="control temperature") (about one hour). The power is then reduced to the proper level to maintain this control temperature constant throughout the remainder of the experiment. This control temperature is chosen as a function of the desired solidification gradient (see Chapter 5.1.2). (4) The casting is allowed to reach its equilibrium temperature for an additional 2000s before the withdrawal mechanism is started. Calibration experiments with thermocouples inside the melt (as reported in Appendix A ) showed that this period of 2000s was more than sufficient. (5) Withdrawal is started at t = 5500s and is carried out at a constant speed for the entire length of the casting. During the entire experiment, the furnace operating conditions (susceptor temperature, temperature alongside the casting, chill position, withdrawal speed and vacuum) are continuously monitored and recorded by a computer data acquisition system. A n example of the record associated with each experiment is shown in Figure 17. T C I , T C 2 and T C 3 are three thermocouples located outside the alumina tube alongside the casting at 152mm, 134mm, 120mm respectively above the chill plate (dots 1, 2 and 3 respectively in Figure 18). They are withdrawn from the hot zone along with the casting. The "Control T C s " are three static control 39 thermocouples located inside the hot zone, just below the center of the top part of the susceptor (dots 4, 5 and 6 in Figure 18). Also in Figure 18, dots 7 through 12 represent the thermocouple locations used to measure the susceptor thermal profiles (one of the parameters in the modeling of the furnace) (dots 7, 8, 9, 10, 11 and 12 respectively at 3mm, 10mm, 35mm, 75mm, 95mm and 130mm above the baffle). Thermocouple locations at dots 13, 14 and 15 were used to estimate the temperatures of the bottom of the casting, steel spacer and top of the copper chill for heat transfer coefficients calculations as well as partial validation of the numerical model. Finally, dots 16 through 20 depict the location of the thermocouples placed in the superalloy melt to measure the casting conditions inside the sample during steady-state directional solidification and thus validate the numerical model (dots 16, 17, 18, 19, and 20 respectively at 100mm, 110mm, 117mm, 122mm and 130mm above the top of the chill). A l l the thermocouples used were type D tungsten-rhenium (W-3%Re/W-25%Re), which have an estimated accuracy of about ±10°C in the range of temperatures considered (1000°C-1500°C). A l l thermocouples were sheathed in 2mm diameter alumina tubing. 1600 160.0 < ( Control TCs") 1400 ] 1200 i C1000 CD <5 800 CD Q . E £ 600 400 200 0 0.0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (s) Figure 17 : Typical data acquisition record for each experiment. 40 16 17 18 19 20 O o o o o o x 7*1 8 9 10. 1 1 . 1 2 * 4 * b T 6 13» 14* o o o o o o 15* Scale : 50mm Figure 18 : Thermocouple locations for various measurements performed on the experimental furnace. 4.3. Numerical modeling of the furnace Each experiment with the tiltable Bridgman furnace could be characterized with its corresponding operation parameters : control temperature of the susceptor and withdrawal speed of the casting. However, it was necessary in this research to estimate the actual sohdification conditions (Guquidus, Gprecku, R) at various locations inside the samples. It was therefore decided to develop a numerical model of the experimental furnace. The goal of this model, was to enable, after proper calibration, the desired translation from furnace operation parameters to actual solidification conditions in the casting, for each experiment. 41 4.3.1. ProCAST P r o C A S T is a general finite element heat transfer and fluid flow package developed by U E S , Inc., specifically designed to simulate mold casting processes. Several modules modeling fluid flow (mold filling and thermal convection), stress/strain analysis and/or electro-magnetic effects can be coupled to the main thermal conduction module. P r o C A S T is particularly well suited to simulate vacuum directional solidification since it can easily account for objects which are moving during the process (e.g. a casting being slowly withdrawn from the hot zone of a furnace) : view factors are automatically updated by the model in order to accurately compute the radiation heat transfers, which depend on the relative positions of the various furnace parts. Moreover, P r o C A S T readily accepts temperature dependent properties (thermal conductivity, density, heat capacity, fraction solid in the melting range, etc.) which contributes to a much more accurate high temperature solidification simulation. In particular, as stated in P roCAST ' s user's manual: "The enthalpy formulation employed in ProCAST accurately models the phase transformation of an alloy or pure metal, to the extent that the fraction solidified function has been quantified. The latent heat evolution cannot be bypassed with a large time step". [44]. Thus, in P roCAST, the release of latent heat on solidification (which is a critical parameter for modelers concerned with casting conditions at the solidification front, as is the case in the present work) is governed by the fraction solid profile in the melting range of the alloy, an example of which has been shown in Figure 6, rather than by a linear release between Tuqutdus and Ts,,iidus. P r o C A S T also has the ability to account for temperature-dependent heat transfer coefficients (HTC) . Heat transfer coefficients can then vary along a given interface, and the local H T C is governed by the local temperature of one of the two surfaces forming the interface. 42 4.3.2. Mesh geometry The mesh used to model the furnace was generated by the software P A T R A N . The geometry is axisymetric and corresponds accurately to the geometry and dimensions of the furnace, as it has been presented in Figure 15 and Figure 16. The axisymetric mesh of the whole furnace, as well as an enlargement of the hot zone, can be seen in Figure 19. The top section of the vacuum chamber (above the baffle level, and surrounding the susceptor) has not been included in the meshing of the radiation enclosure. This simplification is justified since most of the top part of the chamber is masked off by the baffle/susceptor assembly and wil l therefore play only a minimal role in the view factor calculations. It should be noted that the susceptor, baffle and furnace walls are not considered as bulk solid materials. Instead, they are considered to form a "radiation enclosure". This enclosure is divided into "bar 2" elements, each of which is assigned a given emissivity and temperature, which remain constant during the simulation. The copper chill, copper water cooling pipes, steel spacer, alumina crucible and superalloy casting are modeled as independent bulk materials and meshed with "quad 4" elements. In each individual material, the thermal profile is governed by thermal conduction. The temperature fields in these five materials interact with the radiation enclosure by radiative heat transfer between the exposed surfaces. They also interact with each other through coincident nodes interfaces. Given the geometry in Figure 19, six interfaces have been defined : copper chill / water pipes, copper chill / steel spacer, copper chill / alumina crucible, steel spacer / alumina crucible, steel spacer / superalloy casting, and superalloy casting / alumina crucible. Each of these six interfaces can be assigned a specific heat transfer coefficient (HTC), which can be either constant or temperature-dependent (see Appendix A for values). 43 A schematic heat flow diagram describing the interaction of the various components of the model is given in Figure 20. Hot Zone Furnace chamber 100mm 20mm Legend Casting Alumina tube Steel spacer Copper chill Water cooling pipes (a) Entire furnace. (b) Detail of the hot zone. Figure 19 : Axisymetric mesh geometry used in P roCAST to model the furnace. 44 PIPES 15°C \ COT D J3 J3 u (C = Conduction, I = Interface heat transfer, R = Radiation) Figure 20 : Schematic heat flow diagram in the P r o C A S T model of the experimental tiltable furnace. 4.3.3. Typical simulation In an actual experiment, the furnace and the casting are initially at room temperature. The susceptor is then heated up to the desired "control temperature" (maintained throughout the rest of the experiment). At this point, the casting is allowed an extra 2000s to reach its equilibrium temperature before the withdrawal is started, refer to Chapter 4.2.2. 45 However, in order to simplify the model parameters and to save computation time, a typical simulation initially begins with the furnace radiation enclosure already at its desired control temperature (see enclose/susceptor profiles in Appendix A ) . The various modeled materials (casting, crucible, chill, spacer and pipes) are initially set at the arbitrary temperatures given in Table III. The simulation is then started, with the casting in its topmost position, and the system is left to evolve until it reaches a steady-state condition (i.e. all the temperatures in the modeled materials are constant). This numerical steady-state condition corresponds to the thermal steady-state physically reached in the experimental furnace at 5500s after the start of the experiment. A t this point in the numerical simulation, and similarly to an actual experiment, the casting/chill assembly is withdrawn from the hot zone at a predetermined speed and calculations are carried out until solidification of the casting is complete. Table I I I : Initial conditions of the materials at the start of the numerical simulation. Material Initial temperature Superalloy casting Tuquidus of the superalloy Alumina tube TLiquid™ of the superalloy Steel spacer 600°C Water cooled copper chill 20°C The results from P r o C A S T can then be displayed with the post-processing modules Pos tCAST and V i e w C A S T as temperature versus time curves or as isotherm evolution animations. It is also possible to output specific maps, such as thermal gradients or solidification rates at the solidification front. 46 4.3.4. Model parameters, calibration and sensitivity analysis The model parameters have been calibrated with thermocouple measurements in the experimental furnace. A typical illustration of the calibrated model can be seen in Figure 21. It can be seen that the model-predicted temperatures inside the sample show satisfactory agreement with the thermocouple data measured in the furnace. The temperature difference between measured and predicted values is of the order of 15°C or less (which is comparable to the estimated precision of the thermocouple measurements). The maximum deviation occurs at about 1000-2000s, just ahead of the solidification front. This may be linked to the latent heat release formulation used in this model. Nevertheless, it can be seen that the thermal gradients (whose values are the most important result from the model) are very similar for both measured and calculated data (the thermal gradients are directly proportional to the temperature difference between two curves at a given time). The actual numerical comparison between measured and calculated thermal gradients was not necessary in view of the results mentioned above : since the model can predict the temperature field, it can therefore evaluate the thermal gradients and solidification rates. Moreover, such a comparison is rather difficult to carry out : whereas G and R can be calculated at discrete times and positions by the model, their measurement requires data from at least two thermocouples. If these two thermocouples are far apart (10mm or more), the measurements only represent some average value over the gap between the thermocouples, which may not be representative of the instantaneous local value. On the other hand, if the thermocouples are close together (a few millimeters), the uncertainty of the measurement of G and R then lies in the potential influence of the thermocouples on the thermal field, as well as in the precision of the determination of the distance separating the two thermocouples (the tip of the thermocouple is about 1mm in size and is sheathed in a 2mm diameter alumina tube). Furthermore, these effects 47 are compounded with the inherent uncertainty in the temperature measurements from these thermocouples. It should also be mentioned that these simulations show relatively little sensitivity to most of the parameters in the model which had to be estimated (emmissivities, surface temperatures, heat transfer coefficients). The parameters with the most influence are the heat transfer coefficient between the casting and the crucible, and the temperature profile of the susceptor at the exit of the hot zone. 1600 1500 1200 H 1100 1000 Control Temperature : 1435°C Withdrawal speed : 2 mm/min 500 1000 1500 2000 2500 Time (s) 3000 3500 4000 Figure 21 : Comparison between measured (bold lines) and modeled (thin lines) cooling curves at five locations inside the casting (130, 122, 117, 110 and 100 mm above the top of the chill), (control temperature : 1435°C, withdrawal rate (started at t=0s): 3 .3x l0 - 5 m/s (2mm/min)) 48 Note : Since this research is specifically aimed at studying and numerically evaluating the influence of growth front angle on freckle formation and not at developing a model of the particular experimental furnace used for this work, the author decided not to include the details of the model in the main body of this thesis. Nevertheless, the complete model parameters (material properties, heat transfer coefficients, enclosure parameters), the model validation with thermocouple measurements in the furnace, as well as a sensitivity analysis of the model, are all reported at the end of this thesis in Appendix A , which the reader is invited to refer to for further details on this numerical model. 4 9 5. RESULTS 5.1. Model results 5.1.1. Isotherm geometry A typical output from the model is the evolution of isotherms within the sample and alumina tube during the casting withdrawal. An example for IN718 is shown in Figure 22 for a control temperature of 1435°C and a withdrawal speed of 3.3xl0"5 m/s (2mm/min). It can be seen that the position of the growth front relative to the baffle lip varies only a small amount during most of the withdrawal (Figure 22 b) and c)). The height of the mushy zone is also constant. This indicates steady-state solidification conditions. Figure 23 presents the influence of withdrawal speed on the isotherms in the casting after 66mm withdrawal for three different simulations (same control temperature of 1435°C and three withdrawal speeds of 1.6xl0"5, 3.3xl0"5 and 7.5xl0~5 m/s (1, 2 and 4.5mm/min respectively)). It can be seen that, at the same withdrawal distance, the amount of solidified metal is greater for smaller withdrawal speeds, as was expected, since more time was available to dissipate heat from the casting. Figure 24 shows the influence of the furnace control temperature on the isotherms in the casting after 66mm withdrawal (three different control temperatures of 1400°C , 1435°C and 1465°C respectively, same withdrawal speed of 3.3xl0"5 m/s (2mm/min)). It can be seen that the higher the susceptor temperature, the lower the growth front. Also, it can be seen that the mushy zone thickness is smaller for higher susceptor temperatures, indicating larger thermal gradients. In Figure 22, Figure 23, and Figure 24, it is interesting to note that, independently of the control temperature, the isotherms in the metal are essentially flat and perpendicular to the length 50 of the casting for speeds of 1.6xl0" 5 and 3 . 3 x l 0 3 m/s (1 and 2 mm/min). Especially, this is true at the liquidus and freckle temperatures of the alloy. Thus, within the present range of operation, the thermal conditions in the furnace yield a growth front which is flat and perpendicular to the vertical axis of the casting and the axis of withdrawal. Therefore, at low withdrawal speeds, the tilt angle of the furnace corresponds to the angle of the growth front with the direction of gravity : O W = OCfumace (± 1°) [eq. 30] However, at withdrawal speeds of 7.5xl0* 5 m/s (4.5mm/min), a small curvature of the isotherms occurs (from 0° at the center to about 7° to the horizontal at the edge of the casting). Thus, for high withdrawal rates, the actual growth front angle should be slightly adjusted as follows : OCfront = OCfurnace + 4° (± 3°) [eq. 31 ] 5.1.2. Thermal gradients and growth rates Results from the model can be post-processed to produce maps of thermal gradients and of isotherm velocities. These values are calculated for each node in the casting when they reach a specific temperature, such as the alloy liquidus temperature or the freckle initiation temperature. In the present case, the global thermal gradient is defined as : 2 far} 2 (dT) + + \dx) [eq. 32] The solidification rate is evaluated in the following manner, as quoted in the P r o C A S T user's manual [44]: "When each node reaches a specified temperature, a point is located along the temperature gradient some distance away, and the time that it takes for the isotherm to reach that point is determined. R is then calculated as that distance divided by the difference in time". In simpler terms, R is the local isotherm velocity for the specified temperature. 51 Figure 25 and Figure 26 show the thermal gradient and isotherm velocity maps for IN718 at its liquidus temperatureT/j 9 l,,y u,=1336°C , for various casting conditions. It can be seen that the thermal gradients show relatively little variation throughout the casting except for the bottom and top transients, and can be approximated to an average value known with a precision of about ± 1 5 % . It can also be seen that the solidification rate is similar to the withdrawal rate (±20%) throughout the casting (top and bottom transients excluded). It should be mentioned that R is not dependent on the temperature (Juqmdus or Tsoiidus for example) at which it is evaluated, unlike the thermal gradient G. Thus, as initially expected, it is possible to consider the middle section of the experimental sample to be directionally solidified under quasi steady-state conditions. When looking at the presence or absence of freckles in these samples, as reported in the next chapter, only this steady-state middle section of the castings wi l l be considered. Similar simulations for Waspaloy, Mar-M247 and U B C 1 yield the results presented in Table I V , showing the effect of the control temperature and withdrawal speed on the solidification rate and thermal gradients at the liquidus temperature and at the freckle initiation temperature for each alloy system considered in this study. The numerical results in Table I V are presented graphically in Figure 27, Figure 28, Figure 29 and Figure 30. It can be seen in these Figures that, knowing the control temperature of the furnace and the withdrawal speed of the casting (as recorded during each experiment by data acquisition analysis), gradients and growth rates in the melt in the steady-state portion of the castings can be evaluated for each experimental sample. The main observation to be drawn from Table I V and Figure 27, Figure 28, Figure 29 and Figure 30 is that higher control temperatures and/or lower withdrawal speeds result in larger thermal gradients, as is conventionally expected. 52 Table I V : Thermal gradients at the liquidus and at the freckle initiation location (Guquidus and Gprtckit respectively) and sohdification rate at the growth front (R) calculated with the P r o C A S T model for all the alloys studied in this work for various experimental casting conditions (control temperature and withdrawal speed in the furnace). A l l o y TSolidus'TLiquidus 1 Freckle C o n t r o l T W D rate G Liquidus GFrccklc R CO (°C) (°C) (XlfJ-Ws) (°C/cm) (°C/cm) (xlfJ5m/s) 1400 1.6 16 23 1.6 1400 3.3 13 22 3.3 1400 7.5 8 20 7.5 1435 1.6 23 30 1.6 IN718 1260-1336 1321 1435 3.3 19 28 3.3 1435 7.5 13 24 7.5 1465 1.6 28 34 1.6 1465 3.3 24 32 3.3 1465 7.5 16 28 7.5 1400 1.6 12 18 1.6 1400 3.3 11 17 3.3 1400 7.5 6 16 7.5 1435 1.6 20 26 1.6 Waspaloy 1330-1355 1350 1435 3.3 18 26 3.3 1435 7.5 8 19 7.5 1465 1.6 26 32 1.6 1465 3.3 22 28 3.3 1465 7.5 10 23 7.5 1435 1.6 18 26 1.6 1435 3.3 16 26 3.3 1435 7.5 10 24 7.5 1465 1.6 24 31 1.6 M a r - M 2 4 7 1280-1360 1344 1465 3.3 22 30 3.3 1465 7.5 14 27 7.5 1500 1.6 31 36 1.6 1500 3.3 26 34 3.3 1500 7.5 17 30 7.5 1435 1.6 14 22 1.6 1435 3.3 13 22 3.3 1435 7.5 8 20 7.5 1465 1.6 22 27 1.6 U B C 1 - - 1465 3.3 18 26 3.3 1465 7.5 10 24 7.5 1500 1.6 28 33 1.6 1500 3.3 24 30 3.3 1500 7.5 14 26 7.5 5 3 The precision of the measurements for the data in Table IV is approximately the following (as can be seen in Figure 25 and Figure 26) : - Thermal gradients Gz.,-9„,y„.v and GFreckle '• ± 1 5 % (about ±3-4°C/cm) - Solidification rates R : ± 2 0 % (about ±0 .7x l0" 5 m/s) 54 Tsoi T(°C) '• Uq • Sol (a) (b) (c) (d) Figure 22 : Typical thermal profiles in the casting during solidification (ProCAST simulation), (control temperature : 1435°C , withdrawal speed : 3.3xl0" 5 m/s (2mm/min)) (a) t= 0 s, z - 0 mm (steady-state before withdrawal) (b) t=1000s, z = 33 mm (c) t=2000s, z = 6 6 m m (d) t=3000s, z = 9 9 m m 5 5 'I ><t ' -Hi 1500.0 C Liquidus Tsolidus 'I T(°C) (a) (b) (c) Figure 23 : Influence of the withdrawal speed on the isotherm profiles in the casting during solidification (ProCAST simulation), (control temperature : 1435°C, z = 66 mm) (a) withdrawal: 1.6x10 5 m/s (1 mm/min) (b) withdrawal : 3.3x10 5 m/s (2 mm/min) (c) withdrawal : 7 .5x l0 - 5 m/s (4.5 mm/min) 56 T(°C) (a) (b) (c) Figure 24 : Influence of the furnace control temperature on the isotherm profiles in the casting during solidification (ProCAST simulation), (withdrawal speed : 3.3xl0" 5 m/s (2mm/min), z = 66 mm) (a) Control temperature : 1465°C (b) Control temperature : 1435°C (c) Control temperature : 1400°C 57 (a) (b) (c) (d) (e) GLiquidus (°C/cm) Figure 25 : Thermal gradients at the growth front (liquidus temperature) in IN718 for various casting conditions. (a) Control temperature : 1465°C ; withdrawal speed : 3 .3x10 5 m/s (2.0mm/min) (b) Control temperature : 1400°C ; withdrawal speed : 3 .3xl0 ' 5 m/s (2.0mm/min) (c) Control temperature : 1435°C ; withdrawal speed : 3 .3x l0 - 5 m/s (2.0mm/min) (d) Control temperature : 1435°C ; withdrawal speed : 7.5xl0" 5 m/s (4.5mm/min) (e) Control temperature : 1435°C ; withdrawal speed : 1.6xl0 - 5 m/s (l.Omm/min) 58 1.0S000E-02 (a) (b) (c) (d) (e) 7.70000E-03 6.30000F-03 1 .-WOOOE-03 O.OOOOOE+00 RuquUIus (cm/s) Figure 26 : Solidification rate at the growth front (liquidus temperature) in IN718 for various casting conditions. 3 .3x10° m/s (2.0mm/min) -5 (a) Control temperature : 1465°C ; withdrawal speed : (b) Control temperature : 1400°C ; withdrawal speed : (c) Control temperature : 1435°C ; withdrawal speed : (d) Control temperature : 1435°C ; withdrawal speed : (e) Control temperature : 1435°C ; withdrawal speed : 1.6xl0" 5 m/s (l.Omm/min) 3 .3x10° m/s (2.0mm/min) 3.3xl0" 5 m/s (2.0mm/min) 7.5xl0" 5 m/s (4.5mm/min) 59 Thermal Gradient at the Liquidus Temperature of Alloy IN718 40 H 10 "R=1.0mm/min -R=2.0mm/min • R=4.5mm/min 0 • 1380 1400 (a) 1420 1440 1460 Furnace Control Temperature (°C) 1480 1500 Thermal Gradient at the Freckling Temperature of Alloy IN718 Figure 27 : Relationship at various casting speeds between the furnace control temperature and the thermal gradients in the sample for superalloy LN718 (ProCAST simulation). (a) GLiquid™ vs. Control temperature. (b) GFreckie vs. Control temperature. 60 Thermal Gradient at the Liquidus Temperature of Alloy Waspaloy Thermal Gradient at the Freckling Temperature of Alloy Waspaloy "R=1.0mm/min •R=2.0mm/min •R=4.5mm/min (b) 1420 1440 1460 Furnace Control Temperature (°C) 1480 1500 Figure 28 : Relationship at various casting speeds between the furnace control temperature and the thermal gradients in the sample for superalloy Waspaloy (ProCAST simulation). (a) GLiquidus vs. Control temperature. (b) GFreckie vs. Control temperature. 61 Thermal Gradient at the Liquidus Temperature of Alloy Mar-M247 40 • s R=1 .Omm/min R=2.0mm/min —°— R=4.5mm/min 1400 1420 1440 1460 1480 1500 1520 Furnace Control Temperature (°C) (b) Figure 29 : Relationship at various casting speeds between the furnace control temperature the thermal gradients in the sample for superalloy Mar-M247 (ProCAST simulation). (a) Guquidus vs. Control temperature. (b) GFreckle vs. Control temperature. 62 Thermal Gradient at the Liquidus Temperature of Alloy UBC1 40 0-| , , , , , 1 1400 1420 1440 1460 1480 1500 1520 Furnace Control Temperature (°C) (b) Figure 30 : Relationship at various casting speeds between the furnace control temperature and the thermal gradients in the sample for superalloy U B C 1 (ProCAST simulation). (a) GLiquid™ vs. Control temperature. (b) GFreckk vs. Control temperature. 63 5.1.3. Primary dendrite arm spacing measurements It is now well known that primary dendrite arm spacing in superalloys are directly linked to the cooling rate GuauidusXR in the casting (see Figure 5 in Chapter 2.2.3). As a further confirmation of the above model results, A-i was measured on some of the UBC1 castings for various operating conditions of the furnace. The polished samples were etched with Kallings II reagent and the standard procedure of counting the number N of dendrites in a given cross-section area A was used (X\=(A/N)V2). The results are presented in Figure 31 versus the GuquidusXR cooling rate for each sample estimated with ProCAST (the axes in Figure 31 are shown at the same scale as in Figure 5 for measurement precision comparison). It can be seen that there is reasonable agreement between the expected (straight line identical to that in Figure 5) and measured dendrite arms spacing (the scatter of the measured data is similar to that found in Figure 5). This validates furthermore the good calibration of the model and the accurate prediction of G and R from the numerical simulations. 1000 <0 2 g 10 : CO I CL 1 I 0.01 0.1 1 10 100 Cooling Rate G U q u i d u s xff (°C /s) Figure 31 : Primary dendrite arm spacing measured in UBC1 experimental samples as a function of calculated cooling rate (ProCAST simulation). (The solid line corresponds to that in Figure 5 and is described by [eq. 19].) 64 5.2. Experimental results 5.2.1. Typical appearance of the castings Longitudinal sections of typical castings are shown in Figure 32 at 0° and 35°. The melting limit of the initial charge (which was already DS) is shown by the dashed lines. A t 0°, the solidification front is flat and perpendicular to the casting, the grains growing parallel to the withdrawal direction throughout the entire casting. A t 35° however, the initial solidification front is not perpendicular to the casting, due to the prolonged effects of thermal convection in the tilted casting before withdrawal. However, it can be seen that soon after the start of solidification, the preferred orientation of all remaining grains is parallel to the length of the casting. Because grain growth occurs perpendicularly to the solidification front, it is possible to conclude that the solidification front is always perpendicular to the casting withdrawal axis (bottom transient zone excluded). Therefore, the tilt of the whole furnace is indeed representative of the angle of the growth front in the casting with the direction of gravity (despite any potential effects from the melt convection). The middle section of each sample (steady-state solidification zone) was visually inspected (as-cast and etched surface as well as etched cross-sections) in order to determine whether the corresponding set of casting conditions was subject to freckling or not. Etching was done by immersion (up to 1 minute) in a fresh solution of 2 parts HC1 + 1 part H 2 0 2 . The typical appearance of freckles in a casting is shown in Figure 33. It can be seen that freckles appear on the top portion of inclined samples. It is very similar to what had been observed by Giamei et al. [3]. A l l the freckles observed on the experimental samples in this study were surface freckles. None was detected on the cross-sections etched to investigate the inside of the castings. This is comparable to the total absence of freckles inside larger industrial D S / S X castings such as those 65 shown in Figure lc) or Figure 10. Micrographs of the surface freckles observed in experimental samples are shown in Figure 34. (a) UBC1 alloy (0° angle) (b) UBC1 alloy (35° angle) Figure 32 : Typical longitudinal sections of directionally solidified experimental samples. (a) Waspaloy (b) Mar-M247 (c) UBC1 Figure 33 : Appearance of freckles on the surface of tilted castings (castings diameter : 25mm). 66 (a )Waspaloy (b) Mar-M247 (c) U B C 1 Figure 34 : Micrographs of cross-sections of the surface freckles observed in the experimental castings (average diameter of each freckle : about l-2mm). 67 Microprobe analysis was carried out on some samples with a Scanning Electron Microscope (SEM) coupled with an Energy Dispersion Spectrometer ( E D X ) . The results of the microprobe analyses of the observed freckles are presented in Table V and Table V I . For each alloying element, the difference in composition between the matrix and the freckle is consistent with the general segregation patterns observed in Ni-base superalloys. This indicates that the observed macrosegregates are shifted toward the eutectic composition of the alloy, confirming the fact that they are indeed freckles. It can be noticed that freckle compositions in Table V and Table V I show similar shifts toward the eutectic composition of the alloy. In Table V I however, this shift is not as pronounced as in Table V . This may due to a conventional solutioning heat treatment on the industrial samples, which is aimed at homogenizing the bulk composition at the dendrite scale by solid state diffusion. Table V : Chemical analysis (measured by microprobe) of the matrix and freckles from the experimental samples. Alloy Waspaloy Mar-M247 Matrix Freckle Matrix Freckle (wt%) (wt%) (wt%) (wt%) A l 1.64 1.59 5.79 5.91 T i 3.03 4.12 0.78 1.01 Cr 19.95 19.04 8.30 8.25 C o 13.90 13.30 10.35 9.79 N i 56.89 56.96 57.42 56.58 M o 4.59 4.99 0.62 0.76 H f - - 1.73 2.71 Ta - - 4.08 5.16 W - - 10.93 9.83 (Note : Matrix/Freckle compositions for UBC1 have also been measured but cannot be shown for proprietary reasons.) 68 Table V I : Chemical analysis by microprobe of bulk and freckle compositions in industrially cast large IGT blades. Alloy Mar-M247 Matrix Freckle (wt%) (wt%) A l 6.78 7.38 T i 0.59 0.78 Cr 8.09 8.14 Co 9.46 8.80 N i 59.55 58.91 M o 0.76 0.89 H f 1.86 2.13 Ta 3.25 4.67 W 9.66 8.30 (Note : Matrix/Freckle compositions for U B C 1 have also been measured but cannot be shown for proprietary reasons.) Given the data in Table V , and knowing the average segregation coefficients for each alloying element, it is possible to estimate from the Scheil equation the liquid fraction in the mushy zone corresponding to the freckle composition. The Scheil equation is usually written in the following form [45]: Q i i.e. fL <C ^ k-l [eq. 33] [eq. 34] where CL is the composition of the liquid in the freckle plume (also corresponding to the average freckle composition), Co is the bulk composition of the alloy and k is the segregation coefficient of the considered alloying element. With this method, estimates of the liquid fraction at the freckle initiation location are given in Table VII . It can be seen that a liquid fraction of /I=0.5 is a very good overall estimate of the freckle initiation location. 69 Although aluminum segregation plays a significant role on the density inversion and freckle formation in superalloys, it was not included in Table V I I because its segregation coefficient cannot be known a priori since it is strongly dependent on the alloy system, leading to normal segregation in some superalloys and inverse segregation in others [1]. Other elements such as Cr, N i and Fe were not included in Table V I I because of their very weak segregation (k~l). In the case of aluminum, it is nevertheless possible to apply the reverse calculation to estimate the partition coefficient kA\ for each alloy, assuming a liquid fraction of 0.5 at the freckle initiation location. The results are given in Table VIII. The obtained partition coefficients are well within the range of partition coefficients reported in the literature for aluminum [1]. This is a further confirmation that/z. is of the order of 0.5 at the freckle initiation location. Table V I I : Estimated liquid fraction at the freckle initiation location in the mushy zone. (Partition coefficients values from reference [1].) Element Partition coefficient Liquid fraction at the freckle initiation location Waspaloy Mar-M247 U B C 1 Ti 0.60 0.46 0.52 0.26 Co 1.07 0.53 0.45 0.28 M o 0.90 0.43 - -H f 0.12 - 0.6 -Ta 0.66 - 0.5 0.39 W 1.34 - 0.73 0.40 Table VIII : Calculated partition coefficients for Aluminum in Waspaloy, Mar-M247 and U B C 1 assuming a liquid fraction of 0.5. Liquid fraction Pi Waspaloy irtition coefficieri Mar-M247 ts U B C 1 A l 0.5 1.04 0.97 0.81 70 5.2.2. Tilted castings summary Because it is better suited for graphical representation (see Chapter 6), the numerical data relevant to all of the experimental results for the tilted castings is gathered in Appendix B . However, the most significant results observed for the tilted experiments of the three superalloys Waspaloy, Mar-M247 and U B C 1 are presented in Table I X . For each experimental casting, the growth front angle and the casting conditions Guquidus, Gfreckie and R (estimated with the P r o C A S T model from the furnace operating conditions as shown in 5.1.2) and the presence or absence of freckles are reported. It can be readily seen from Table I X that, as already reported in the literature, low solidification rates and low thermal gradients favor freckle formation. However, the most significant result in Table I X is that, for a given alloy, identical casting conditions may or may not result in freckle formation, depending on the angle of the growth front. This in itself confirms the first objective of this research : the growth front angle plays an important role on the occurrence of freckling in superalloy castings. In view of these results, the main goal of the next chapter is to develop a representative freckling criterion, based on the Rayleigh number, and taking into account the growth front angle in order to quantitatively predict freckling. Table I X : Summary of some significant experimental results. Observation Alloy Angle Guquidus GFreckle R Freckling ? Effect of the withdrawal rate .. UBC1 17 12 21 1.30 Yes 17 12 21 3.00 No Effect of the withdrawal rate Waspaloy 34 12 19 1.30 Yes 34 12 19 3.00 No Effect of the thermal gradient UBC1 35 14 23 2.00 Yes 35 27 32 2.00 No Effect of the growth front angle UBC1 0 14 23 2.00 No 35 14 23 2.00 Yes 71 6. DISCUSSION ; DEVELOPMENT OF A NEW FRECKLING CRITERION It has been discussed earlier in this thesis that the main reason for the limited ability of the original Rayleigh criterion (as presented in [eq. 15]) to predict freckle formation lies in the fact that it does not take into account the growth front angle. In this chapter, this limited ability of the original Rayleigh criterion to predict freckling will be illustrated with the present experimental results. Two possible modifications of the formulation of the Rayleigh number wil l then be suggested, based on the more general theory of a curved growth front, forming a certain angle with the vertical direction (rather than the particular case of upward solidification with a horizontal front). These modified Rayleigh criteria will be shown to be better suited than the original Rayleigh criterion to the prediction of freckling for the present results, as well as for estimates of industrial casting conditions. The general theory stating that the Rayleigh number must be greater than some critical threshold value for freckling to occur is not contested. Only the formulation of the Rayleigh number (left hand side in equation [eq. 15]) will be subject to modification. Then freckling is expected to occur whenever the modified Rayleigh value exceeds some critical limit. The threshold limits for the modified Rayleigh criteria are however expected to be quite different from that for the original Rayleigh number (Ra*=l). Indeed, the threshold value depends mostly on the choice of the characteristic length h of the system : for example, studies of thermosolutal convection in geophysical systems (ocean currents, earth's molten core) consider characteristic lengths of the order of several kilometers, yielding critical threshold values Ra* of 10 6 or higher. 72 6.1. Application of the original Rayleigh number criterion The aim of this chapter is to translate the results of the present experimental and numerical work into a graphical representation based on the original Rayleigh criterion as it has been reported in Chapter 2.2.2. For each casting, it is possible to calculate the corresponding original Rayleigh number based on the casting conditions. The original Rayleigh number, as presented earlier ([eq. 15]) is : dp Ra^-^xGl^K [eq.35] 150-IO" 6 Apart from the primary dendrite arm spacing (A, = o~33~m)' m e other (G Liquidus X R ) parameters in the Rayleigh number can be considered independent of the casting conditions or of the alloy composition and to assume the values given below already presented in Chapter 2.2.3 : g=9.81m/s2 ^ = 30(kg /m 3 ) / °C dT r\= 0.004 kg/ms DT= 9X10" 6 m2/s Thus, for the present conditions and alloys considered, the original Rayleigh number can be reduced to : Ra = 4A 6xlO" 6 x G ™ x (Guquidus x Ry132 (±15%) [eq. 36] ( G % ! £ ! and Guquidus in °C/m, R in m/s) 73 As explained in Chapter 2.3.1, the possible inaccuracies in the numerical values of the physical properties are all combined in the alloy chemistry factor ("4.16xlfT 6" proportionality coefficient). This term may need to be refined for each alloy system considered, but this is, however, beyond the scope of this thesis. Instead of considering variable proportionality coefficients (based on specific alloy chemistry) and comparing Rayleigh numbers to a unique threshold value Ra* valid for all alloys, it is simpler and more practical to consider one given proportionality coefficient (valid for all alloys) and compare Rayleigh numbers to alloy-dependent threshold values. This is the case in this chapter, where "4.16xl0~ 6" is an estimated proportionality coefficient considered valid for all three superalloys studied in this work. The actual uncertainty on the calculation of Rayleigh numbers then lies on the estimation of the solidification conditions. Given the range of variations of thermal gradients and solidification rates observed in the quasi steady-state sections of each sample (see Figure 25 and Figure 26) (R varying from 2.8xl0" 5 m/s to 4.2x10"5 m/s about its average value of 3 .5xl0 ' 3 m/s for example), the Rayleigh numbers can be calculated with a precision of about ± 1 5 % . Using the experimental data gathered in Appendix B , it is possible to plot the original Rayleigh number versus angle of the growth front for all the experimental castings, which represent a variety of solidification conditions. This yields the graphs shown in Figure 35, Figure 36, and Figure 37 for the three superalloys Waspaloy, Mar-M247 and U B C 1 respectively. In each of these graphs, it is then possible to plot a line delineating the freckled and freckle-free regions. It can be clearly seen in Figure 35, Figure 36, and Figure 37 that in all cases, it is not possible to draw these lines horizontally. This indicates that the critical threshold value Ra* (above which freckling occurs and below which there is no freckling) is indeed dependent on 74 the growth front angle. Moreover, the negative slope of these lines also clearly indicates that higher angles favor freckle formation, as it has been mentioned earlier in this work (Chapter 2.4). In any event, the original formulation of the Rayleigh criterion, although possibly valid in the case of a horizontal front growing vertically upward, cannot be considered to be representative of freckle formation in the more general case of industrial castings exhibiting a certain angle between the growth front and the horizontal direction. The goal of the following chapter is to develop a modified Rayleigh criterion which could be compared to a constant threshold value, independent of the growth front angle. s c (50 1.4 1.2 H 1.0 0.8 § 0.6 H :g> 0.4 ^ 0.2 0.0 o No Freckles • Freckles - - Ra* (Waspaloy) (Freckles) (No Freckles) 10 15 20 25 Growth front angle (degree) 30 35 40 Figure 35 : Original Rayleigh number versus growth front angle for alloy Waspaloy. 75 CD X> 6 3 C Xi 1.4 1.2 4 1.0 0.8 A IT 0.6 C3 c ;g> 0.4 0.2 0.0 o No Freckles • Freckles - - Ra* (Freckles) (No Freckles) 10 15 20 25 Growth front angle (degree) (Mar-M247) 30 35 40 Figure 36 : Original Rayleigh number versus growth front angle for alloy Mar-M247. X> 3 a 1.4 1.2 H i.o H 0.8 A | °-6 a ;g> 0.4 0.2 H 0.0 o No Freckles • Freckles - - Ra* (Freckles) (No Freckles) 10 15 20 25 Growth front angle (degree) 30 (UBC1) 'V"i 35 40 Figure 37 : Original Rayleigh number versus growth front angle for alloy U B C 1 . 76 6.2. Effect of the growth front angle on freckle formation : modifications to the Rayleigh criterion This chapter suggests two possible modifications to the Rayleigh criterion to account for the effect of the growth front angle on freckle formation. The only difference between these two theories lies on the assumption regarding the initial direction of the freckling flow : along the main axes of the mushy zone (parallel or perpendicular to the primary dendrites), or along the vertical direction. 6.2.1. First theory Let us consider the following geometry, where the mushy zone (and the corresponding isotherms) are curved and form an angle a with the horizontal direction (see Figure 38) at the location of the freckle flow initiation. This geometry applies to static ingots, V A R / E S R processes as well as the experimental samples of this work. Figure 38 : Schematic geometry for an angled growth front (general casting conditions). 77 In the local projection axes (x,z) (perpendicular and parallel to the primary dendrites), the two components of the driving force F for freckle formation are (F is based on the numerator of the Rayleigh number): F^S-^-GFreMe-^n(a) [eq. 37] F z = 8 • ^ ' G ™ - c o s ( a ) [ e ( l - 3 8 ] Let us now assume that the initial flow due to density inversion and leading to freckling wi l l be either parallel or perpendicular to the primary dendrites in the mushy zone. It is then possible to define two separate Rayleigh numbers : For flow perpendicular to the dendrites : Rax = -—% [ e c l - 39] 1 d T / K X For flow parallel to the dendrites : Raz = -,—j [eq. 40] (Where ^ a n d Kzcm be calculated by expressions given in 2.2.3) In the original Ra expression, the term h4 (where h is a characteristic distance in the system) was taken to be equal to %4 [24]. Other authors suggested the value KX2 [6]. Given the geometry in Figure 38, it appears that K X 2 and K2 are the most logical choices for h4 (h = 4K is indeed a characteristic linear dimension of the mushy zone linked to the average "pore" diameter). This choice is consistent with the fact that permeability is expressed in units of m 2 . 78 Note : This theory (and the corresponding assumption of flow along the x or z direction) is in effect an extension of the original Rayleigh criterion to a discretized 2D geometry. Extension to a 3D geometry could be easily done in a similar manner by incorporating a third Rayleigh number Ray along the y axis. It is then suggested that, for given G and R conditions, freckling wil l occur when at least one of Rax or Raz is greater than some critical value Ra*. Depending on which Rayleigh number is greater, freckle flow will then originally initiate parallel or perpendicularly to the dendrites. The modified Rayleigh criterion for this first theory could therefore be written : Freckling occurs when : Ral = max (Rax, Raz) > Ral* [eq. 41] It is possible to plot these modified Rayleigh numbers as a function of growth front angle for various casting conditions (using the numerical data already presented in Chapter 2.2.3). The results are presented in Figure 39 and Figure 40. It can be seen from Figure 39 and Figure 40 that Ral* is much more likely to be overcome by Rax, even at relatively low growth front angles. This is due to the fact that the permeability perpendicular to the dendrites (Kx) is estimated to be 2 to 4 times greater than that parallel to the dendrite (Kz) (see Chapter 2.2.3). This result diverges drastically from the conventional assumption that freckle flow always initiates parallel to the dendrites (as is suggested in the classic diagram shown in Figure 3). 79 0 10 20 30 40 50 60 70 80 90 Angle (deg.) Figure 39 : Modified Rayleigh numbers Rax and Raz (and maximum value Ral in bold) versus growth front angle for various thermal gradients Guquidus and GFnckie = Guquidus + 10°C/cm (constant growth rate i?=3.3xl0"5 m/s (2mm/min)). 80 Figure 40 : Modified Rayleigh numbers Rax and Raz (and maximum value Ral in bold) versus growth front angle for various growth rates R (constant thermal gradients Guquidu=15°C/cm and =25°C/cm). 81 6.2.2. Second theory Instead of considering that the flow associated with freckle formation will start either parallel or perpendicular to the dendritic direction, it is also possible to assume that this flow may always start vertically, following the direction of the density inversion driving force. In this case, let us consider the same geometry of a tilted growth front already shown in Figure 38. The effective permeability of the mushy zone for fluid flow in the vertical direction is defined as [46] : 1 Keffective = ~ T ~ 2~T I ^ - 4 2 ] 11 sin a cos a + Kx Kz (Where KX and KZ can be calculated by expressions given in 2.2.3) Considering the characteristic linear dimension to be h = -JK similarly to the first theory in 6.2.1, it is then possible to define the corresponding Rayleigh number as : $ irp ^Freckle Ral = a i . 2 [eq. 43] 77 • DT j KEFFF.CTJW The modified Rayleigh criterion for this second theory can then be written : Freckling occurs when : Ra2 > Ral* [eq. 44] It is also possible to plot this modified Rayleigh number as a function of growth front angle for various casting conditions. The results are presented in Figure 41 and Figure 42. Once again, it can be clearly seen that the increased permeability perpendicular to the primary dendrites can dramatically increase the value of the modified Rayleigh number (and therefore increase the propensity for freckle formation), even at relatively low angles between the growth front and the horizontal. 82 0 10 20 30 40 50 60 70 80 90 Angle (degree) Figure 41 : Modified Rayleigh number Ra2 versus growth front angle for various thermal gradients Guquidus and GFreckie = Guquidus + 10°C/cm (constant growth rate /?=3.3xl0" 5 m/s (2mm/min)). 83 0 T 1 1 1 1 1 1 1 1 1 0 10 20 30 40 50 60 70 80 90 Angle (degree) Figure 42 : Modified Rayleigh number Ra2 versus growth front angle for various growth rates (constant thermal gradients Guquidus=15°C7cm and GFreckie =25°C/cm). 84 6.3. Application of the modified Rayleigh criteria to the results of the present work 6.3.1. Application of the first theory It is now possible to apply the modified Rayleigh number developed in Chapter 6.2.1 (first theory) to the experimental results of the present work. A similar numerical evaluation as in Chapter 6.1 is carried out. Basic values of viscosity, diffusivity etc. remain identical. Permeability is calculated using the expressions previously presented in section 2.2.3. Kz = 3.75 x K T 4 - fl -l\ m 2 Kx = 3 . 6 2 * 1 0 ? • -V™ -X™ m 2 where/z=0.5 (estimated liquid fraction at the freckle initiation site [1]) 150 IO" 6 , „ 40 • 10~6 a n d \ = ~ ( ^ T m a n d ^=7 ^ 2 ~ m -(GLiqUiduS X R) (GLiquiduS X R) The modified Rayleigh numbers of the first theory can then be reduced to : Rax = 4 .62xl0" 1 5 x GFmku x (Guquidus x R)'215 x sin(a) (±15%) [eq. 45] Raz= 3.64xl0" 1 4 x GFrecUe x (Guquidus x Ryl32x cos(a) (±15%)[eq. 46] and Ral = max(Rax, Raz) (±15%)[eq. 47] (Gpreckie and Guquidus in °C/m, R in m/s) The discussion regarding the precision of the proportionality coefficients and of the Rayleigh calculations (already presented in Chapter 6.1) is also valid in this case. The various casting conditions in Appendix B can then be plotted in the form of the modified Rayleigh number Ral versus growth front angle for the three superalloys considered in this study. The results are depicted in Figure 43, Figure 44 and Figure 45 along with the appropriate horizontal threshold lines delineating the freckled and freckle-free regions. It can be seen that this time, in all three cases, it is possible to evaluate for each alloy a critical threshold 85 value Ral* independent of growth front angle, below which freckling does not occur and above which freckling always occurs. Estimates from the present experimental results are : i?a7*(Waspaloy)= 15.0xl0" 9 #a7*(Mar-M247)= 7.5xl0" 9 /?a7*(UBCT)« lOxlO" 9 It can be seen that the critical threshold value Ral* is very similar for all three alloys and is of the order of 10~8. Given the fixed proportionality coefficients, the actual threshold values may be indicative of the relative propensity of each alloy to form freckles. In the present case, Waspaloy appears to be the least freckle-prone of the three alloys because it has the highest critical threshold value Ral *. The minor variations between all the alloys are probably due to slight differences in the physical properties (viscosity, diffusivity and density inversion factor, all parameters of the "alloy chemistry" O function in equation [eq.27]). 1.2E-07 es Oi <D X) s a 00 Oi T3 <U T3 O 8.0E-08 4.0E-08 A 0.0E+00 ° No Freckles • Freckles - - Ra* (Waspaloy) (Freckles) (No Freckles) 10 15 20 25 30 Growth front angle (degree) 35 40 Figure 43 : Modified Rayleigh number Ral versus growth front angle for alloy Waspaloy. 86 cd Oi Ui (D E 3 C J= T 3 O .2E-07 8.0E-08 4.0E-08 A 0.0E+00 o N o Freckles • Freckles - • R a * (Freckles) (No Freckles) (Mar-M247) 10 15 20 25 30 Growth front angle (degree) 35 40 Figure 44 : Modified Rayleigh number Ral versus growth front angle for alloy Mar-M247. 1.2E-07 cd Oi fe 8.0E-08 s 3 C SQ "53 -a 4.0E-08 CU TJ O 0.0E+00 o N o Freckles • Freckles - • R a * (Freckles) i 10 15 20 25 30 Growth front angle (degree) (UB2D (No Freckles) a = 35 40 Figure 45 : Modified Rayleigh number Ral versus growth front angle for alloy U B C 1 . 87 6.3.2. Application of the second theory A similar numerical evaluation as in Chapter 6.3.1 is carried put for the modified Rayleigh criterion of the second theory. Basic values of viscosity, diffusivity, etc. remain identical. P D A S , S D A S and permeability are calculated using the expressions previously presented in Chapter 2.2.3. The modified Rayleigh number (second theory) then reduces to : Ra2 = - : — T 1.48 x 10 7 • [GLiquidus x RfM • sin2(a) + 5.24x 10 6 • (GLiquillus x R)°M • c o s 2 ( a ) ] (±15%) [eq. 48] (Gpreckie and Guquidus in °C/m, R in m/s) The various casting conditions in Appendix B can then be plotted in the form of the modified Rayleigh number Ral versus growth front angle for the three superalloys considered in this study. The results are depicted in Figure 46, Figure 47, and Figure 48. Horizontal line threshold values are also plotted in these Figures. In the case of U B C 1, although two points are very slightly off-side with respect to the horizontal threshold value, they are still well within the expected range of imprecision in the experimental measurements or numerical estimates, as shown with the error bars. Thus it is possible to evaluate for each alloy a critical threshold value Ral*, independent of growth front angle, below which freckling does not occur and above which freckling always occurs. Estimates from the present experimental results are the following : ita2*(Waspaloy)= 7.80xl0" 9 2?a2*(MarM247)= 6.70xlfJ 9 i?a2*(UBCl)=5.40xl(T 9 88 It can be seen that the critical threshold value Ral* is very similar for all three alloys and is of the order of 6.50xllT 9 . Like for Ral*, it is also possible to consider the minor variations between alloys as being indicative of the relative propensity of various alloys to form freckles. Once again, Waspaloy appears to be the least freckle-prone (highest critical threshold value). These variations between the alloys are due to slight differences in the physical properties. The ranking of Mar-M247 and U B C 1 is however different with Ral and Ra2. This is probably due to the fact that one of the modified Rayleigh numbers is more accurate than the other. The results in this thesis cannot however point to the better formulation. Nevertheless, comparison of actual freckle maps and numerically predicted maps for industrial castings is the ultimate validation of these modified criteria, and should enable the selection of the most suitable criterion of the two. <3 D2 CD 3 2.0E-08 1.5E-08 J •Sb 1.0E-08 D i T3 M 5.0E-09 o 0.0E+00 ° No Freckles • Freckles - • Ra* (Freckles) (No Freckles) (Waspaloy) 10 15 20 25 30 Growth front angle (degree) 35 40 Figure 46 : Modified Rayleigh number Ral versus growth front angle for alloy Waspaloy. 89 2.0E-08 $ 1.5E-08 Pi <U x> E 3 C fb l.OE-08 cs Pi -o cu i 5.0E-09 o 0.0E+00 o No Freckles • Freckles - - Ra* (Mar-M247) (Freckles) (No Freckles) ~i 1 1 r -1 10 15 20 25 30 35 Growth front angle (degree) 40 Figure 47 : Modified Rayleigh number Ral versus growth front angle for alloy Mar-M247. 2.0E-08 <^  1.5E-08 Pi •~ x> B 3 C JS cd Pi -a *£ O 1.0E-08 5.0E-09 0.0E+00 o No Freckles • Freckles - - Ra* (UBCl) (Freckles) .j II i I (No Freckles) i r 10 15 20 25 30 Growth front angle (degree) 35 40 Figure 48 : Modified Rayleigh number Ra2 versus growth front angle for alloy U B C l . 90 6.3.3. Application of the modified Rayleigh formulations to process optimization Knowing the critical threshold value Ra* for a given alloy (for example, from the previous results), it is then possible to use the generic graphs presented in Figure 39, Figure 40, Figure 41 and Figure 42 for the modified Rayleigh criteria in order to estimate the required process conditions which will lead to freckle-free castings. This is illustrated in Figure 49, based on Figure 39, where the solidification rate R is known. For example, given a threshold value Ral* = 10 - 8, a thermal gradient at the liquidus of 20°C/cm wil l not yield freckles if the growth front angle is lower than 25°. Alternately, if the growth front angle is about.25°, then freckles will not develop if Guquidus is greater than 20°C/cm. Thus, knowing two of the three parameters G, R and a , it is possible with these generic graphs and the appropriate threshold value to estimate the range of values of the third parameter required to avoid freckling. Figure 49 : Illustration of the potential application of the generic graphs based on the modified Rayleigh criteria. 0 10 20 30 40 50 60 70 80 90 Angle (deg.) 91 6.4. Direct applicability of the modified Rayleigh criteria to industrial situations It was shown in the previous chapters that the original Rayleigh criterion was not suitable for the prediction of freckle formation in the experimental castings of the present work. It was then shown that modifications of the original Rayleigh number to account for the angle of the growth front proved to be very successful at predicting freckling in these same experimental castings. Consequently, the aim of the following two sub-chapters is the following : (1) First, the original, modified Ral and modified Ra2 Rayleigh numbers will be applied to some given casting conditions in V A R ingot which have been previously modeled [47]. It wil l be shown that Ral and Ral can indeed predict mid-radius-only freckles whereas the original Rayleigh criterion cannot. (2) Second, it will be shown by numerical modeling with P r o C A S T that, depending on the casting geometry and process operation conditions, growth front angles in D S / S X castings can range from 0° (flat horizontal front) up to 45° or more, thus demonstrating the potential applicability of the present research results to industrial situations. The following examples of the direct applicability of the modified Rayleigh criteria to industrial situations illustrate how the results of the present work are useful to casters to predict freckle formation in castings for some given process conditions. These criteria can also be used to numerically evaluate the actual effectiveness of various potential process modifications aimed at reducing freckle occurrence. In their present form, the modified criteria do not however point directly to these modifications. To date, the conventional suggestions of reducing local solidification times and increasing gradients, as well as the novel suggestion of this work of minimizing growth front angle, are still the best available guidelines to process design modifications. 92 6.4.1. The case of V A R IN718 The casting conditions in V A R / E S R processes have been the object of several investigations and numerical simulations [47]. A typical pool profile (i.e. temperature profiles of the mushy zone) for V A R IN718 (500mm diameter ingot, 260kg/hr melt rate) is presented in Figure 50 (profiles constructed from numerical model data reported in [47]). From this same data, it is possible to evaluate the angle of the growth front, the thermal gradients at the liquidus and freckle temperatures, as well as the solidification rates at various positions from the centerline of the ingot. These results are presented in Figure 51. Calculation of the original, modified Ral and modified Ral Rayleigh numbers from equations [eq. 36], [eq. 49] and [eq. 50] respectively is then very straightforward. The results are plotted in Figure 52. It can be seen that the original Rayleigh number is maximum at the centerline of the ingot. Thus, if the ingot is subject to freckling, it will be most heavily freckled at the centerline. Therefore the original Rayleigh criterion cannot illustrate the case of an ingot exhibiting freckles at mid-radius only. However, it can be seen that both modified Rayleigh numbers are at their maxima some distance away from the centerline. Thus, it is possible with these criteria to predict freckle formation at mid-radius but not at the center of the ingot. This is a further confirmation of the greater ability of the modified Rayleigh numbers compared to the original formulation at predicting freckle formation. Moreover, E S R ingots have been reported to be more prone to freckling than V A R ingot, allegedly due to their V-shaped (as opposed to U-shaped for V A R ) pool profiles. This is confirmed by the results of the present work. At similar melting rates and thermal gradients, the growth front angles in E S R are greater than in V A R (especially at the centerline, where the thermal gradients are lower), thus increasing the values of the modified Rayleigh numbers Ral 93 and Ra2, and thus indeed indicating a greater propensity to form freckles. The same argument is also valid for higher melting rates and greater power inputs in V A R / E S R ingots, resulting in deeper pools, steeper angles and thus increased freckling. Figure 50 : Pool depth vs. Radius in V A R IN718 shown by the isotherms for the liquidus, freckle and solidus temperatures. (260kg/hr melt rate) (after [47]). 600 50 Radius (mm) 100 150 200 -Liquidus - Freckle -Solidus 250 Figure 51 : Thermal gradients at the liquidus, freckle and solidus temperatures and solidification rate in V A R IN718 (melt rate : 260kg/hr) (after [47]). 50 100 150 Radius (mm) 200 5.0E-05 4.0E-05 3.0E-05 h 2.0E-05 .2 1.0E-05 0.0E+00 250 94 Figure 52 : Original and modified Rayleigh criteria profiles along the radius of V A R IN718 (melt rate : 260kg/hr). 95 6.4.2. The case of DS/SX castings 6.4.2.1 .Special modification of the Rayleigh criterion The two theories described in chapters 6.2.1 and 6.2.2 and the corresponding geometry shown in Figure 38 are representative of most directional solidification conditions (and in particular of the present experimental work), where the dendrites are perpendicular to the isotherms ( V A R / E S R ingots, killed steel ingots, etc.). However, when the dendrite direction is not perpendicular to the growth front, the expressions in the modified Rayleigh numbers should be rewritten accordingly. In particular, in the case of D S / S X castings, the growth front could exhibit a localized and temporary curvature (due to specific thermal conditions) which would not last long enough to alter the vertical orientation of the dendrites and compromise the D S / S X structure. The mushy zone structure in a vertical D S / S X casting can be described by the diagram shown in Figure 53, very similar to Figure 38 except for the dendrites orientation. Figure 53 : Schematic geometry for an angled growth front (DS/SX casting conditions). 96 In this case, similarly to the first theory, let us assume the freckle flow to initiate either parallel to the primary dendrites (vertically), or parallel to the growth front. The two corresponding Rayleigh numbers can then be written.: dp F g- — - GFrcMe • sin(a) Rax= 7-— = — -j— [eq. 49] 1 nDTK2x rpTKl dp ' G Freckle Ravertical = = ^ ^ and Ra = max (Rax, Radical) [eq. 51] (Kx and Kz : permeabilities for flow perpendicular and parallel to the primary dendrites) The corresponding generic graphs are very similar to those in Figure 39 and Figure 40, with the only difference being that Raz(a) is replaced with Raverticai, independent of the growth front angle a (and thus appearing as straight horizontal lines). The result remains that the value of the modified Rayleigh number increases with the growth front angle, indicating a greater propensity to freckle formation at higher angles. 6.4.2.2.Potential growth front angles in industrial D S / S X castings The aim of this chapter is to evaluate the growth front angles which can potentially be present in D S / S X castings. Most of the research to date on freckle formation focused on vertical upward directional solidification of analog systems, with a horizontal flat growth front. Similarly, virtually all numerical models developed so far to try and understand freckle formation consider a 97 fully insulated system where heat is extracted from the bottom only, in effect resulting in a horizontal flat front growing vertically. However, other numerical simulations of actual industrial D S / S X blade castings [48] indicate a definite curvature of the isotherms in the mushy zone, as seen on Figure 54. This curvature results from the fact that heat extraction from the casting occurs laterally at its surface. In bigger castings, which have a larger volume to surface ratio, the limiting step for heat extraction becomes the thermal conduction inside the metal rather than through the ceramic/melt interface or through the investment shell, resulting in more pronounced isotherm curvatures under similar casting conditions. Figure 54 : Curvature of the isotherms in the mushy zone in industrial D S / S X blade casting (numerical simulation) [48]. A prolonged exposure of the casting to such a curvature may yield an alteration of the dendrites direction (since they tend to grow perpendicular to the isotherms), thus destroying the vertical directional orientation. Therefore, such curvatures are probably localized in the D S / S X castings, occurring mostly at changes of cross-sections, or changes in the solidification conditions during the casting process (e.g. slower withdrawal speed at the transition from the airfoil to the 98 blade shroud). Nevertheless, local and temporary curvatures could be sufficient to favor freckle formation. In addition, once this curvature disappears, so do the conditions allowing freckling. This could explain the fact that freckles do not normally run along the entire length of a D S / S X casting, but instead tend to disappear about 20 to 30mm after initiating (see Figure 1). The following numerical work has been carried out to qualitatively illustrate the degree of curvature of the growth front a D S / S X casting could be subject to under industrial conditions, and to evaluate the dependence of this curvature on the withdrawal speed and casting size. A l l the relevant modeling was executed with P r o C A S T . A l l the parameters (casting and mold physical properties, metal/ceramic heat transfer coefficient, enclosure emmissivities and temperatures (susceptor thermal profile), etc.) are identical to those used for the calibrated model of the tiltable experimental furnace described in Chapter 4.3. The model running parameters (initial temperature of the materials, time to reach thermal equilibrium before withdrawal) are also identical. A l l these parameters can be found in Appendix A or Chapter 4.3. The numerical casting conditions, much like the experimental furnace, should therefore be fairly representative of actual casting conditions in industrial furnaces. The only differences with the original calibrated model of the experimental furnace are in the dimensions and the geometry. For simplification as well, the copper chil l , steel spacer and cooling pipes have been omitted. The first effect to be investigated in this chapter is the effect of size and withdrawal speed on the growth front angle. Three geometries have been selected, and the corresponding axisymetric meshes are shown in Figure 55. The crucible is 5mm thick and the casting is 240mm long and 10mm, 20mm and 50mm in diameter respectively. The effect of withdrawal speed has been modeled on each of the three sizes. The resulting mushy zone isotherms in the middle of the steady-state solidification portion of the casting (not the top and bottom transients) are presented in Figure 56. 99 Several qualitative observations can be drawn from Figure 56. As expected, faster withdrawal speeds result in shallower thermal gradients and lower growth front height relative to the baffle position. A n increase in the size of the casting produces similar effects. The most interesting result however is the fact that the growth front can be noticeably more sensitive to curvature for larger casting sizes, and it is not unreasonable to expect angles of up to 45° with the horizontal for some casting conditions. Since higher angles result in higher modified Rayleigh values (see Figure 45, Figure 46, Figure 47 and Figure 48), this observation is consistent with the fact that large IGT D S / S X blade castings are significantly more prone to freckling than smaller castings. Moreover, the highest angles are found right at the surface of the castings. Thus, the modified Rayleigh numbers will be maximum at the mold/ceramic interface, therefore possibly accounting for surface-only freckles in D S / S X castings. Similar numerical simulations have also been carried out with the axisymetric meshes presented in Figure 57. The objective of these geometries was to qualitatively evaluate the behavior, in a cluster fed by a central sprue, of : (a) airfoil section (with a ceramic insert at the cooling channel location); and (b) large root section. The results for these two geometries are depicted in Figure 58 and Figure 59 respectively. In Figure 58, it is interesting to notice that, in the case of the sprue and inner airfoil wall, the various withdrawal speeds only affect the position of the growth front relative to the baffle. Thermal gradients are identical for all three withdrawal speeds, and the growth fronts are horizontal flat. However, the outer airfoil wall is affected in manner similar to that described in Figure 56 : higher withdrawal speeds result in lower thermal gradients and increased growth front angles. 100 This situation is to the advantage of the industrial casters since it indicates that freckles should occur predominantly on the outside surface of the outer wall of the airfoil, rather than on the inside of the cooling channel, where they cannot be detected. Thus, a freckle-free outer airfoil wall most likely guarantees a freckle-free cooling channel. In Figure 59, it can be seen again that higher withdrawal speeds result in lower growth front positions and lower thermal gradients. B y comparing these results to those in Figure 56, it can be seen that the presence of a feeding sprue may tend to accentuate the growth front angle, thus possibly promoting more freckling in the blade root. Casters should therefore try to minimize the size of the sprue in a cluster, and possibly relocate it to the edges of the cluster rather than its center. Also, since larger sizes seem to be favorable to freckle formation, a suggestion could be to solidify only one casting at a time in a smaller furnace, rather than a larger cluster in a large furnace, and possibly eliminating the sprue altogether. This latter option may however render the cost of manufacturing large freckle-free D S / S X castings prohibitive. In any event, since the casting of several blades in a cluster cannot be modeled accurately with an axisymetric geometry, the above results are mostly qualitative and more extensive 3D modeling is required. 101 Casting Ceramic shell 30mm Figure 55 : Axisymetric mesh geometries used to investigate the influence of casting cross-sections on growth front angle. 102 10mm diameter casting 1 £ 20mm diameter casting 50mm diameter casting Ul —ppp-J 2mm/min 5mm/min 8mm/min Figure 56 : Numerical simulation of the influence of casting cross-sections and withdrawal rates on the growth front angle found in D S / S X castings. (Control temperature 1435°C , after 130mm withdrawal) (ProCAST simulation) 103 Casting Ceramic shell 30mm Figure 57 : Axisymetric mesh geometries used to investigate the following effects on growth front angle in D S / S X blade castings. (a) Feeding sprue and internal ceramic insert (simulating the casting of the airfoil) (b) Feeding sprue and large casting cross-section (simulating the casting of the root) 104 2mm/rnin 5 mm/min 8mm/min Figure 58 : Numerical simulation of the influence of withdrawal rates on the growth front angle found in the airfoil in D S / S X blade castings. (Control temperature 1435°C , after 130mm withdrawal) (ProCAST simulation) 2mm/min 5 mm/min 8mm/min Figure 59 : Numerical simulation of the influence of withdrawal rates on the growth front angle found in large sections in D S / S X blade castings (such as the root). (Control temperature 1435°C , after 130mm withdrawal) (ProCAST simulation) 105 7. CONCLUSIONS A N D F U T U R E W O R K 7.1. Conclusions Freckles are presently one of the major defects encountered in advanced casting technology of superalloys. The evaluation of a numerical criterion able to provide quantitative insight on the conditions of freckle formation is now recognized as a major key toward the successful manufacture of large diameter V A R / E S R ingots and large D S / S X castings. A n experimental furnace was built to directionally solidify cylindrical castings at various angles, thus simulating a tilted mushy zone. The casting conditions in the samples were determined accurately by a complete thermal modeling (calibrated with thermocouple measurements) of the furnace. The following conclusions were drawn in this thesis : (1) Among all the criteria suggested in the literature to date to predict freckle formation in industrial castings, the Rayleigh criterion is probably the most promising, because it incorporates two out of the three factors influencing freckle formation. Moreover, it was shown that the Rayleigh number can be relatively easily calculated. (2) The Rayleigh criterion, in its original form, is incomplete and cannot account for the presence of surface-only freckles in D S / S X castings or mid-radius-only freckles in V A R / E S R ingots. (3) The missing factor in the Rayleigh criterion is the geometry aspect of the casting. More specifically, this thesis suggested to take into account the angle of the growth front with respect to the direction of gravity. 106 (4) It was found that, under similar solidification conditions, tilted castings exhibited freckles whereas vertical castings were freckle-free. This confirms the influence of the growth front angle on freckle formation. (5) Two modified Rayleigh criteria were developed to account for the growth front angle. They were shown to describe the casting conditions leading to freckling with much better accuracy than the original Rayleigh number. These modified Rayleigh criteria were applied to typical superalloy casting conditions and were shown to be suitable to account for the occurrence of freckles only at the surface of D S / S X blade castings or only at mid-radius in E S R / V A R ingots. 1.2. Recommendations for future work The following are suggestions for future work to be done on the determination of a freckling criterion following the bases laid by this thesis : (1) This thesis suggested two possible modifications of the original Rayleigh criterion. Both these modified Rayleigh criteria showed excellent accuracy at predicting freckling in the experimental samples of the present work and at explaining mid-radius-only freckling in V A R / E S R ingot casting or surface-only freckles in D S / S X castings. However, the ultimate validation of these criteria (and possibly the choice of one modified Rayleigh criterion over the other) can only be done by comparing freckle defect-maps for actual industrial castings with the associated freckle defect-maps generated by the criteria in an accurately calibrated numerical simulation of these same castings. These were unfortunately not available in the scope of this research. 107 (2) Numerous C F D models simulating freckle formation have already been developed. Unfortunately, they all focus on vertical upward solidification with a horizontal flat growth front and an isotropic permeability in the mushy zone. It is suggested that these models be modified to incorporate anisotropic permeability of the mushy zone and allow for variations of the angle between the growth front and the direction of gravity. Conditions of initiation of the freckle flow could then be accurately observed, and comparisons with the modified Rayleigh criteria presented in this thesis could be made. Moreover, such simulations could not only characterize the onset of freckling flow, but also its subsequent path in the mushy zone (possibly accounting for angled freckles). (3) The critical threshold values for the modified Rayleigh criteria have been evaluated for three superalloys in this thesis. They were found to be very close to each other. However, more experiments with the tiltable furnace could confirm these values for numerous other freckle-prone alloys. Moreover, modification of such a furnace from DS to S X casting would make freckle characterization more straightforward since the exact orientation of the mushy zone could be determined. In addition, it would also be possible to quantify the extent (overall number, length, average spacing) of freckling for each casting condition. 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Flemings : "Segregation in castings and ingots", 1990 Elliott Symposium Proceedings /pp.253-272 [46] M . Muskat : "The flow of homogeneous fluids through porous media", McGraw-Hi l l Book Cie, 1937 [47] A . S . Ballantyne, A . Mitchell, J.F. Wader : "The prediction of ingot structure in VAR/ESR Inconel 718", Proc. 6 t h Int. Vac. Met. Conf., Apr i l 1979, G . K . Bhat & R. Schlatter ed., pp. 599-623 [48] M . Meyer ter Vehn, D . Dedecke, U . Paul, P.R. Sahm : "Undercooling related casting defects in single crystal turbine blades", T M S Superalloys 1996, pp.471-479 [49] M . C . Schneider, J.P. Gu , C. Beckermann, W.J . Boettinger, U . R . Kattner : "Modeling of micro- and macrosegregation and freckle formation in single-crystal nickel-bas superalloy directional solidification", Met. Trans. A , vol 28A, July 1997 / pp. 1517-1531 [50] N . S . Stoloff : "Wrought and P/M superalloys", A S M International Metals Handbook, "Properties and selection : irons, steels and high performance alloys", A S M International, V o l . 1, 10th edition, J.R. Davis et al. Ed. , 1990 / pp.950-1006 [51] F . M . White : "Heat and mass transfer", Addisson-Wesley publishing co., 1991 113 APPENDIX A ; Furnace modeling calibration A . l . Numerical data A . l . l . Material properties The material properties used in P r o C A S T for the modeling of the experimental Bridgman furnace are presented in Table X along with the relevant references. The physical properties shown in Table X are all based on the properties of superalloy IN718. This is because IN718 is one of the rare superalloys (if not the only) whose properties have been measured with relative accuracy at high temperature in both liquid and solid phases. Numerical simulations for other superalloys in this work were approximated by assuming the same physical properties as for IN718, except for the melting range. This approximation is currently widely used by most models published in the literature [49]. Moreover, a look at the physical properties of various superalloys (such as reported in [50]) confirms that variations are minor from one nickef-base alloy system to the next, thus validating the present approximation. 114 T a b l e X : M a t e r i a l p r o p e r t i e s u s e d i n m o d e l i n g the D S f u r n a c e w i t h P r o C A S T . ( N / A : N o t A p p l i c a b l e ) P r o p e r t i e s N i - b a s e d s u p e r a l l o y ( c a s t i n g ) F u s e d A l u m i n a ( c r u c i b l e ) S t a i n l e s s S t e e l ( s p a c e r ) C o p p e r ( c h i l l ) D e n s i t y ( k g / m 3 ) 21°C 8 1 9 0 TSolidus 7 8 0 0 Tuguidus 7 4 0 0 1600°C 7 3 0 0 [1] 3 9 7 0 7 9 0 0 8 9 0 0 S p e c i f i c H e a t ( k J / k g . K ) 21°C 0 . 4 3 538°C 0 . 5 6 8 7 F C 0 . 6 5 1600°C ( 0 . 8 4 ) [ 4 4 ] [ 5 0 ] 0 . 7 6 0 . 5 0 0 . 4 0 C o n d u c t i v i t y ( W / m . K ) 21°C 11 .4 538°C 19 .6 871°C 2 4 . 9 1600°C ( 3 6 . 0 ) [44] [50] 200°C 2 2 . 0 400°C 1 3 . 0 600°C 9 .3 800°C 7 .3 1000°C 6 .2 1600°C (3 .0 ) [51] 2 0 3 0 0 L a t e n t H e a t ( k J / k g ) 2 7 7 N / A N / A N / A F r a c t i o n s o l i d (-) Tsoiidus (1260°C) 1 .00 0 .3 Tuquidus+0.77Solidus 0 . 9 5 0.6Tuquidus+0.4Tsoiidm 0 . 8 0 0 .8 Tuquidus+0.2 TSolidus 0 . 5 0 r « „ , ( 1 3 3 6 ° C ) 0 . 0 0 N / A N / A N / A E m m i s s i v i t y (-) 0 . 4 0 . 4 N / A 0 . 3 N o t e : T e m p e r a t u r e d e p e n d e n t p r o p e r t i e s f o l l o w l i n e a r i n t e r p o l a t i o n s e g m e n t s b e t w e e n t h e p o i n t s p r e s e n t e d i n T a b l e X . T h e v a l u e s i n b r a c k e t s are e x t r a p o l a t i o n s f r o m l o w e r t e m p e r a t u r e d a t a . 115 A.l.2. Heat transfer coefficients H e a t t r a n s f e r c o e f f i c i e n t s are i m p o r t a n t p a r a m e t e r s g o v e r n i n g t h e ra te o f heat t r a n s f e r b e t w e e n t w o m a t e r i a l s w h i c h f a c e e a c h o t h e r but a re no t i n i n t i m a t e c o n t a c t . I n the p r e s e n t g e o m e t r y , t he re a re s i x heat t r a n s f e r c o e f f i c i e n t s a n d the i r e s t i m a t e d v a l u e s a re p r e s e n t e d i n T a b l e X I . T h e p r o c e d u r e o f e v a l u a t i o n o f these heat t r a n s f e r c o e f f i c i e n t s i s d e s c r i b e d i n A p p e n d i x D . T h e m o s t c r i t i c a l heat t r a n s f e r c o e f f i c i e n t is b e t w e e n the c a s t i n g a n d the c r u c i b l e t u b e . T h i s is the p a r a m e t e r Which w a s fined-tuned i n o r d e r t o c a l i b r a t e t h e m o d e l t o the e x p e r i m e n t a l m e a s u r e m e n t s . T a b l e X I : H e a t t r a n s f e r c o e f f i c i e n t s u s e d i n m o d e l i n g the D S f u r n a c e w i t h P r o C A S T . C o n t a c t F a c e s H e a t T r a n s f e r C o e f f i c i e n t ( W / m 2 K ) P i p e s / C h i l l 1 5 0 0 S p a c e r / C h i l l 1 5 0 A l u m i n a / C h i l l 3 0 S p a c e r / A l u m i n a 7 5 S p a c e r / I N 7 1 8 c a s t i n g 2 0 0 I N 7 1 8 c a s t i n g / A l u m i n a T u b e 20°C 5 Tsolidus 5 0 Tuquidus 5 0 0 1600°C 5 0 0 A.1.3. Enclosure parameters T h e t e m p e r a t u r e p r o f i l e o f the s u s c e p t o r w a s m e a s u r e d w i t h t h e r m o c o u p l e s at v a r i o u s l o c a t i o n s (see d o t s 7 t o 12 i n F i g u r e 18) , f o r v a r i o u s c o n t r o l t e m p e r a t u r e s . C o m p l e t e t h e r m a l p r o f i l e s ( u s e d i n P r o C A S T ) w e r e e x t r a p o l a t e d f r o m these m e a s u r e m e n t s . T e m p e r a t u r e m e a s u r e m e n t s a n d p r o f i l e s u s e d i n the n u m e r i c a l s i m u l a t i o n s are s h o w n i n F i g u r e 6 0 . T h e r e m a i n i n g d a t a u s e d to d e f i n e the e n c l o s u r e c o n d i t i o n s c a n b e f o u n d i n T a b l e X I I . 116 1600 1500 1400 °w1300 <D ID 1200 Q. E 1100 t— 1000 -\ 900 800 Temperature profiles of the graphite susceptor for various control temperatures -j-(l500°c)-5 10 15 Distance above copper baffle (cm) 20 F i g u r e 6 0 : S u s c e p t o r t e m p e r a t u r e as a f u n c t i o n o f h e i g h t a b o v e the b a f f l e ( f o r v a r i o u s c o n t r o l t e m p e r a t u r e s 1400°C, 1435°C, 1465°C a n d 1500°C). ( b l a c k m a r k e r s : m e a s u r e d v a l u e s , w h i t e m a r k e r s : i n t e r p o l a t e d v a l u e s u s e d i n the P r o C A S T m o d e l ) . T a b l e X I I : E n c l o s u r e p a r a m e t e r s f o r P r o C A S T s i m u l a t i o n . E n c l o s u r e s e c t i o n T e m p e r a t u r e E m i s s i v i t y C h a m b e r w a l l ( s ta in less steel ) 1 0 0 0.1 B a f f l e u n d e r s i d e ( s o l d e r a n d c o p p e r ) 1 0 0 0.1 B a f f l e l i p ( c o p p e r ) 3 0 0 0 . 3 B a f f l e t o p ( c o p p e r ) 8 0 0 0 . 3 S u s c e p t o r ( g r a p h i t e ) S e e F i g u r e 6 0 0 . 9 117 A.2. Calibration validation G i v e n the base p a r a m e t e r s d e s c r i b e d a b o v e , v a r i o u s r u n s w i t h the m o d e l w e r e c a r r i e d o u t t o s i m u l a t e a v a r i e t y o f o p e r a t i n g c o n d i t i o n s ( c o n t r o l t e m p e r a t u r e s o f 1400°C, 1435°C a n d 1465°C a n d w i t h d r a w a l s p e e d s o f 1 . 6 x l f J 5 , 3 . 3 x l 0 " 5 a n d 7 . 5 x l O " 5 m / s ( 1 , 2 a n d 4 . 5 m m / m i n r e s p e c t i v e l y ) ) . T h e s e s i m u l a t i o n r e s u l t s a re c o m p a r e d t o a c t u a l t h e r m o c o u p l e m e a s u r e m e n t s i n the m e l t f o r the s a m e o p e r a t i o n c o n d i t i o n s (see F i g u r e 6 1 , F i g u r e 6 2 , F i g u r e 6 3 , F i g u r e 6 4 a n d F i g u r e 6 5 ) . It c a n be seen that t he re is s a t i s f a c t o r y a g r e e m e n t b e t w e e n the s i m u l a t i o n a n d the p h y s i c a l m e a s u r e m e n t s (as a l r e a d y d i s c u s s e d i n C h a p t e r 4 . 3 . 4 ) . T h e m o d e l is t h e r e f o r e c o n s i d e r e d t o b e a p p r o p r i a t e l y c a l i b r a t e d a n d to d e s c r i b e a c c u r a t e l y the c a s t i n g c o n d i t i o n s i n t h e e x p e r i m e n t a l f u r n a c e u s e d i n t h i s w o r k . 1600 1500 j Time (s) F i g u r e 61 : C o m p a r i s o n b e t w e e n m e a s u r e d ( b o l d l ines ) a n d m o d e l e d ( th in l i nes ) c o o l i n g c u r v e s at f i v e l o c a t i o n s i n s i d e the c a s t i n g ( 1 3 0 , 122 , 117 , 1 1 0 a n d 1 0 0 m m a b o v e the t o p o f the c h i l l ) . ( C o n t r o l t e m p e r a t u r e : 1465°C , W i t h d r a w a l s p e e d : 3 . 3 x l 0 " 5 m / s ( 2 m m / m i n ) ) 118 1600 1500 4 .1400 1000 Control Temperature: 1400°C Withdrawal speed : 2 mm/min 500 1000 1500 2000 Time (s) 2500 3000 3500 4000 F i g u r e 6 2 : C o m p a r i s o n b e t w e e n m e a s u r e d ( b o l d l i nes ) a n d m o d e l e d ( th in l i nes ) c o o l i n g c u r v e s at f i v e l o c a t i o n s i n s i d e the c a s t i n g ( 1 3 0 , 122 , 117 , 1 1 0 a n d 1 0 0 m m a b o v e the t o p o f the c h i l l ) . ( C o n t r o l t e m p e r a t u r e : 1400°C , W i t h d r a w a l s p e e d : 3 . 3 x l 0 ' 5 m / s ( 2 m m / m i n ) ) 1600 1500 1100 H 1000 Control Temperature: 1435°C Withdrawal speed : 2 mm/min 500 1000 1500 2000 Time (s) 2500 3000 3500 4000 F i g u r e 6 3 : C o m p a r i s o n b e t w e e n m e a s u r e d ( b o l d l ines ) a n d m o d e l e d ( t h i n l i nes ) c o o l i n g c u r v e s at f i v e l o c a t i o n s i n s i d e the c a s t i n g ( 1 3 0 , 122 , 117 , 1 1 0 a n d 1 0 0 m m a b o v e the t o p o f the c h i l l ) . ( C o n t r o l t e m p e r a t u r e : 1435°C , W i t h d r a w a l s p e e d : 3 . 3 x l 0 " 5 m / s ( 2 m m / m i n ) ) 1 1 9 1600 F i g u r e 6 4 : C o m p a r i s o n b e t w e e n m e a s u r e d ( b o l d l ines ) a n d m o d e l e d ( th in l i nes ) c o o l i n g c u r v e s at f i v e l o c a t i o n s i n s i d e the c a s t i n g ( 1 3 0 , 122 , 117 , 1 1 0 a n d 1 0 0 m m a b o v e the t o p o f the c h i l l ) . ( C o n t r o l t e m p e r a t u r e : 1435°C , W i t h d r a w a l s p e e d : 7 . 5 x l 0 " 5 m / s ( 4 . 5 m m / m i n ) ) 1600 1500 1400 4 o o CD I 1300 . CO Q . £ CD H 1200 H Control Temperature : 1435°C Withdrawal speed : 1 mm/min 1100 1000 500 1000 1500 2000 2500 Time (s) 3000 3500 4000 F i g u r e 6 5 : C o m p a r i s o n b e t w e e n m e a s u r e d ( b o l d l ines ) a n d m o d e l e d ( t h i n l i nes ) c o o l i n g c u r v e s at f i v e l o c a t i o n s i n s i d e the c a s t i n g ( 1 3 0 , 122 , 117 , 1 1 0 a n d 1 0 0 m m a b o v e the t o p o f t h e c h i l l ) . ( C o n t r o l t e m p e r a t u r e : 1435°C , W i t h d r a w a l s p e e d : 1 . 6 x l 0 " 5 m / s ( l m m / m i n ) ) 1 2 0 A .3. Sensitivity analysis T h e m o d e l resu l t s w e r e a l so a n a l y z e d f o r the i r s e n s i t i v i t y t o v a r i o u s p a r a m e t e r s w h i c h c o u l d n o t be m e a s u r e d a c c u r a t e l y (heat t rans fe r c o e f f i c i e n t s , e m m i s s i v i t i e s a n d s u r f a c e t e m p e r a t u r e s ) . E a c h p a r a m e t e r w a s v a r i e d i n d i v i d u a l l y ( h i g h o r l o w v a r i a t i o n ) w h i l e a l l the o t h e r s r e t a i n e d t h e i r b a s e v a l u e . T h e v a l u e s o f the v a r i a t i o n s f r o m the b a s e f o r e a c h p a r a m e t e r a re s h o w n i n T a b l e X I I I . T h e c o r r e s p o n d i n g n u m e r i c a l " r u n " l e t te r is a l so s h o w n i n the s a m e t a b l e ( l o w e r c a s e f o r l o w v a r i a t i o n , c a p i t a l f o r h i g h v a r i a t i o n ) . T h e r e s u l t s o f these s e n s i t i v i t y a n a l y s i s r u n s a re a l l g a t h e r e d i n F i g u r e 6 6 . O n e n o d e i n the m i d d l e o f the s t e a d y - s t a t e s o l i d i f i c a t i o n s e c t i o n o f t h e c a s t i n g w a s c h o s e n . I ts " b a s e " t h e r m a l p r o f i l e is s h o w n o n the s e c o n d a r y a x i s i n F i g u r e 6 6 . T h e d e v i a t i o n f r o m th is b a s e p r o f i l e f o r e a c h i n d i v i d u a l v a r i a t i o n w a s p l o t t e d o n F i g u r e 6 6 . It c a n be s e e n that , d e s p i t e v a r i a t i o n s o f the p a r a m e t e r s o f u p to 1 0 0 % , the d e v i a t i o n o f t h e r e s u l t s is o f the o r d e r o f 4 % o r less . T h e s e d e v i a t i o n s are e v e n l o w e r ( abou t ±1%) i n the s o l i d i f i c a t i o n r a n g e ( a r o u n d r = 1 3 0 0 ° C a n d r=2000s) . It is t h e r e f o r e c o n c l u d e d that the c a l i b r a t e d m o d e l is l i t t l e s e n s i t i v e t o i ts p a r a m e t e r s . T a b l e X I I I : V a r i o u s p a r a m e t e r s u s e d i n the P r o C A S T m o d e l a n d t h e i r d e v i a t i o n s f r o m the b a s e v a l u e s u s e d t o p e r f o r m a s e n s i t i v i t y a n a l y s i s . P a r a m e t e r L o c a t i o n B a s e L o w H i g h R u n s Chill / Pipes 1500 1000 2000 a, A Heat Chill / Spacer 150 100 300 b,B Transfer Chill / Alumina tube 30 20 50 c,C Coefficient Spacer / Alumina tube 75 50 100 d,D Spacer / Casting 200 100 350 e,E (W/m2K) Casting / Crucible (TS(,iidus) 50 40 70 f , F Casting / Crucible (TLimi,ius) 500 300 800 g,G Emmissivity Copper (baffle, chill) 0.3 0.2 0.4 h, H Chamber walls 0.1 0.05 0.2 U (-) Alumina tube 0.4 0.3 0.5 J,J Superalloy casting 0.4 0.3 0.5 k, K Graphite 0.9 0.8 1.0 1,L Temperature Baffle top 800 600 1000 m, M Baffle side 300 200 500 n, N (°Q Chamber walls 100 50 150 o, O Water pipes 15 10 25 P.P 121 0 1000 2000 3000 4000 Time after start of withdrawal (s) F i g u r e 6 6 : D e v i a t i o n s f r o m the base t e m p e r a t u r e f o r v a r i a t i o n s o f v a r i o u s p a r a m e t e r s i n c a l i b r a t e d P r o C A S T m o d e l o f the e x p e r i m e n t a l f u r n a c e . N o t e : A t t i m e t, the t h e r m a l " d e v i a t i o n f r o m b a s e " i n F i g u r e 6 6 is d e f i n e d as f o l l o w s : % D e v i a t i o n ( 0 = (Tvarialil)n(t) - ThaSe(t)) I ThaSe(t) [eq . 5 2 ] 1 2 2 APPENDIX B : Numerical data of the experimental castings (tiltable furnace) T a b l e X I V : S u m m a r y o f the d i r e c t i o n a l s o l i d i f i c a t i o n e x p e r i m e n t s c a r r i e d o u t o n the t i l t a b l e B r i d g m a n f u r n a c e . A l l o y A n g l e Guquidus GFreckle R F r e c k l i n g ? (deg ree ) (°C/cm) (°C/cm) ( m m / m i n ) 0 7 14 1.98 N o 0 2 3 2 9 1.92 N o 18 14 2 0 1.32 Y e s W a s p a l o y 2 5 14 2 0 1 .20 Y e s 3 4 11 18 3 . 0 0 N o 3 4 14 2 0 1.32 Y e s 3 4 31 38 1.26 Y e s 0 13 2 2 1 .30 Y e s 0 12 2 2 1.86 N o M a r - M 2 4 7 18 13 2 2 1 .32 Y e s 2 4 13 2 2 1.32 Y e s 3 5 11 2 2 2 . 7 6 Y e s 3 5 13 2 2 1.32 Y e s 0 13 2 2 2 . 1 0 N o 0 14 2 3 1.86 N o 0 14 2 3 1.92 N o 0 3 3 3 7 2 . 1 0 N o 10 2 6 3 2 1.98 N o 17 11 21 3 . 0 0 N o 17 14 2 2 1.32 Y e s U B C l 18 17 2 5 1.86 Y e s 18 18 2 6 1 .74 Y e s 2 7 17 2 5 1 .80 Y e s 31 18 2 7 2 . 7 0 N o 3 4 14 2 3 2 . 2 2 Y e s 3 4 2 7 3 2 2 . 0 4 N o 3 5 14 2 3 2 . 0 4 Y e s 3 6 12 21 1 .92 Y e s 3 6 18 2 7 1.98 Y e s N o t e : T h e c a s t i n g c o n d i t i o n s r e p o r t e d i n the tab le a b o v e d o n o t c o v e r the f u l l e x t e n t o f the (ang le , G, R) s p a c e f o r e a c h a l l o y . T h e y d o h o w e v e r c o v e r a w i d e e n o u g h r a n g e o f i n d u s t r i a l c a s t i n g c o n d i t i o n s to e n a b l e the d e t e r m i n a t i o n o f f r e c k l i n g t h r e s h o l d s , as d i s c u s s e d i n C h a p t e r 6 . A P P E N D I X C ; Use of the experimental furnace to cast DS turbine blades. T h e e x p e r i m e n t a l f u r n a c e bu i l t f o r th i s r e s e a r c h p r o g r a m has b e e n s u c c e s s f u l l y u s e d t o d i r e c t i o n a l l y c a s t s m a l l p o w e r g e n e r a t i o n t u r b i n e b lades . T y p i c a l e x a m p l e s o f the i n v e s t m e n t s h e l l ( a l s o m a n u f a c t u r e d at U B C ) a n d cas t b l a d e are s h o w n i n F i g u r e 6 7 . (a) I n v e s t m e n t s h e l l (b) D S t u r b i n e b l a d e F i g u r e 6 7 : C a s t i n g o f s m a l l D S t u r b i n e b l a d e s w i t h the e x p e r i m e n t a l B r i d g m a n f u r n a c e bu i l t i n t h i s w o r k . 12.* APPENDIX D : Heat Transfer Coefficients estimation. T h e heat t r a n s f e r c o e f f i c i e n t H o f a n i n t e r f a c e v e r i f i e s the f o l l o w i n g e q u a t i o n : Q = H(Tl-T2) [ e q . 5 3 ] w h e r e Q = hea t f l o w a c r o s s the i n t e r f a c e ( i n W / m 2 ) , a n d Ti , T2 = t e m p e r a t u r e s o f the t w o s u r f a c e s o f the i n t e r f a c e ( i n K ) . H o w e v e r , the t o t a l heat t r a n s f e r a c r o s s the i n t e r f a c e c a n b e c o n s i d e r e d t o b e a c o m b i n a t i o n o f c o n t a c t heat t r a n s f e r a n d r a d i a t i o n heat t r a n s f e r : Q = f >< Qa,n,ac, + (1 ~ / > * QradiaUon [eq. 5 4 ] w h e r e / = a r e a f r a c t i o n o f c o n t a c t b e t w e e n the t w o s u r f a c e s f o r m i n g the i n t e r f a c e , a n d Qcim!acl = Hcmtact • (7/, - T2) [eq . 5 5 ] fi™tetoI=OE-(7'i4-r24) [ e q - 5 6 ] W h e r e Hc„, !mc,==1000 W / m 2 K i s a n e s t i m a t e o f the H T C f o r t w o s u r f a c e s i n g o o d c o n t a c t , o = 5 . 6 7 x l 0 " 8 W / m 2 K 4 i s the S t e f a n - B o l t z m a n c o n s t a n t , 8 = e m m i s s i v i t y o f the s u r f a c e s o f the i n t e r f a c e . C o m b i n i n g a l l the e q u a t i o n s p r e s e n t e d a b o v e , i t i s f i n a l l y p o s s i b l e to c a l c u l a t e the g l o b a l heat t r a n s f e r c o e f f i c i e n t H o f a n y g i v e n i n t e r f a c e w i t h the f o l l o w i n g e x p r e s s i o n : H = f x Hcmma + (1 - / ) x cr • e • ^ ' ~ ! ? ) [eq . 5 7 ] 125 

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