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Thermal stress analysis of defect formation in fused-cast alumina refractories Au, Dominic Ka Man 1999

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T H E R M A L STRESS A N A L Y S I S OF DEFECT FORMATION IN FUSEDCAST A L U M I N A REFRACTORIES by DOMINIC K A M A N A U B . A . S c , The University of British Columbia, 1997  A T H E S I S S U B M I T T E D FN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF  M A S T E R OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Metals and Materials Engineering)  W e accept this thesis as conforming tc^heTrojunfed standard  T H E UNIVERSITY OF BRITISH C O L U M B I A October 1999 © Dominic K a M a n A u , 1999  in  presenting  degree freely  at  this  the  thesis  in  partial  fulfilment  of  University  of  British  Columbia,  I agree  available for  copying  of  department publication  this or  reference  thesis by  for  his  of this thesis  and  study.  I further  scholarly purposes  or  her  for  gain shall  permission.  Department of  Metal?  The University of British Vancouver, Canada  Date  DE-6 (2/88)  QcToW  ahJL  MafetfcJb? £  Columbia  IW  requirements that  agree  may be  representatives.  financial  the  It not  is be  that  the  for  an  Library shall  permission for  granted  by  understood allowed  advanced  the that  make it extensive  head  of  my  copying  or  without my written  11  Abstract Mathematical models have been developed to predict the temperature, stress and strain evolution during the manufacture of fused-cast ap-alumina refractories approximately consistent with the process used at Monofrax Inc.. A n uncoupled thermal stress model, which consists of a heat transfer model to predict the temperature evolution and a mechanical model to predict the stress and strain evolution, has been formulated.  The  commercial finite element code A B A Q U S was employed in the thermal stress analysis. The thermal model was validated against industrial thermocouple and pyrometer measurements obtained at Monofrax Inc., located in Falconer, N Y .  The model  predictions were in good agreement with the thermocouple data obtained at several locations from within the graphite mold, during Stage I cooling, and in the alumina annealing ore, during Stage II cooling.  In Stage I cooling, it has been necessary to  augment the conductivity of the liquid alumina to account for convective heat transport. In Stage II cooling, it proved necessary to account for asymmetric placement of the block in the annealing bin. The temperatures obtained from the thermal model were utilized as input to the mechanical model. Elastic and elastic-plastic stress analyses were conducted to assess the evolution of stress and strain during the casting process. The strain rate independent inelastic behavior of the casting based on the flexural tests performed at Oak Ridge National Laboratory was incorporated into the elastic-plastic stress model. The results obtained from the elastic-plastic stress model were more realistic than those predicted by the elastic analyses. However, the plastic strain may be under-predicted during Stage II cooling owing to the strain rate independent plasticity employed in the analysis.  Ill  The preliminary stress/strain predictions indicate that the (3-alumina core plays an important role in the generation of tensile stresses and likely gives rise to the generation of cracks. Since high tensile stress/strain was found to develop within the refractory, it is likely that crack initiates subsurface and propagates outwards. Overall, the results of the present research show the importance and usefulness of developing the ability to predict temperature, stress and strain evolution in the fused-cast ap-alumina refractories during the manufacturing process.  IV  Table of Contents Abstract  ii  Table of Contents  iv  List of Tables  vi  List of Figures  vii  List of Symbols  xii  Acknowledgements  xiv  Chapter 1 Introduction and Overview  1  1.1 Introduction 1.2 Background  1 2  1.3 Fundamental Based Approach  4  Chapter 2 Literature Review  9  2.1 Casting Processes  9  2.2 Computer-Based Mathematical Models on Refractory Components 2.3 Computer-Based Mathematical Models on Fused-Cast Refractories 2.4 Constitutive Behavior of Refractories 2.5 Monofrax-M 2.6 Summary Chapter 3 Scope and Objectives  11 13 15 18 19 24  3.1. Scope of the Research Programme  24  3.2 Objectives of the Research Programme  26  Chapter 4 Industrial and Laboratory Measurements  27  4.1 Industrial Measurements  27  AAA Experimental Techniques 4.1.2 Results 4.1.2.1 Initial Conditions 4.1.2.2 In-Mold Temperature Responses 4.1.2.3 Refractory Autopsy 4.2 Laboratory Measurements 4.2.1 Experimental Procedures 4.2.2 Results  27 29 29 30 31 31 32 33  Chapter 5 Thermal Model 5.7 General Thermal Model Formulation 5.2 Geometry 5.3 Boundary Conditions 5.3.1 Symmetry Boundaries 5.3.2 Casting-Mold Boundary 5.3.3 M o l d Exterior Boundary  5.4 Initial Conditions 5.5 Thermo-Physical Properties 5.6 Thermal Predictions and Comparisons to Measured Data 5.6.1 Stage I - Graphite Cooling Model 5.6.2 Stage II - Alumina Ore Cooling M o d e l  Chapter 6 Stress Model 6.1 General Stress Model Formulation 6.2 Formation of (3-Alumina and Void Distribution 6.3 Constitutive Behavior V 6.3.1 Elastic Analysis 6.3.2 Elastic-Plastic Analysis  6.3.2.1 Manipulation ofORNL Data for Input to ABAQUS Plasticity Model 6.4 Geometry 6.5 Boundary Conditions 6.6 Predictions Obtained from the Elastic Stress Model 6.6.1 Benchmark Case - ap-Alumina Crown 6.6.2 Effect of (3-Alumina Core Formation 6.6.3 Effect of (3-Alumina Core Formation and V o i d Distribution  46 46 47 49 49 49 51  51 52 53 53 56  71 71 72 74 74 75  76 77 77  77  79 80 81  6.7 Preliminary Predictions Obtained from the Elastic-Plastic Stress Model.... 82 Chapter 7 Summary and Conclusions 7.1 Recommendations for Future Work Bibliography  98 100 101  List of Tables Page T A B L E 2.1 -  COMPARISON OF T H E CREEP CONSTANTS OBTAINED FOR ALUMINA REFRACTORIES  TABLE 5.1-  VARIOUS  [18-19]  T H E R M A L M O D E L L O A D STEPS m  21  THE MONOFRAX-M THERMOCOUPLE  TRIAL  60  T A B L E 5.2 -  T H E R M O - P H Y S I C A L PROPERTIES O F M O N O F R A X - M [25-27]  61  T A B L E 5.3 -  THERMO-PHYSICAL PROPERTIES O F T H E M O L D I N G MATERIALS  T A B L E 5.4 -  T H E R M O - P H Y S I C A L PROPERTIES O F T H E S T E E L A N N E A L I N G BIN  TABLE 5.5-  T H E R M A L C O N D U C T I V I T Y O F V A R I O U S A L U M I N A ORES [26]  [26] [28]  62 63  64  vii  List of Figures Page FIGURE 1 . 1 -  S C H E M A T I C I L L U S T R A T I O N O F A F U S E D - C A S T OC(3-ALUMINA REFRACTORY CASTING WITH A N OVERSIZED H E A D E R (NOT TO S C A L E ) . . . 6  F I G U R E 1.2 -  CROSS-SECTION OF A STANDARD OPERATING PRACTICE ( S O P ) M O N O F R A X - M B L O C K T A K E N A T T H E VERTICAL P L A N E BISECTING T H E N A R R O W F A C E OF T H E CASTING (REFER TO F I G U R E  1.1).  C R A C K S A N D VOIDS C A N B E OBSERVED A N D T H E P-ALUMINA R E G I O N IS B E I N G O U T L I N E D .  NOTE THAT THE CRACKS ARE VERTICAL  A N D PERPENDICULAR TO T H E B R O A D F A C E O F T H E B L O C K R E S U L T I N G F R O M T H E TENSILE STRESS A C T I N G P A R A L L E L T O T H E B R O A D F A C E ( T E N S I L E S T R E S S I N D I R E C T I O N - 2 I N F I G U R E 1.1) F I G U R E 1.3 -  7  T O P VIEW OF A S O P M O N O F R A X - M B L O C K T A K E N A T THE H O R I Z O N T A L P L A N E JUST B E L O W T H E H E A D E R .  T H E (3-ALUMINA  REGION C A N B E OBSERVED IN THE CENTER OF THE CASTING A N D T H E CRACKS APPEAR TO E M A N A T E F R O M OR TERMINATE A T THE OCP/P-ALUMINA INTERFACE F I G U R E 2.1 -  8  T H E R M A L CONDUCTIVITY, T H E R M A L EXPANSION A N D TYPICAL COMPOSITION OF M O N O F R A X - M [1]  22  FIGURE 2.2 -  PHASE DIAGRAM OF N A 0 - A L 0  23  F I G U R E 4.1 -  THERMOCOUPLE MEASUREMENTS A N D PYROMETER DATA FROM  2  2  3  [3]  M O N O F R A X - M INDUSTRIAL TRIAL FOR S T A G E I COOLING IN T H E GRAPHITE M O L D FIGURE 4.2 -  T H E R M A L RESPONSE OBTAINED F R O M M O N O F R A X - M INDUSTRIAL TRIAL FOR S T A G E II COOLING IN THE A L U M I N A ORE  F I G U R E 4.3 -  37  SCHEMATIC ILLUSTRATION OF TYPICAL CRACKS FOUND IN M O N O F R A X - M (NOT TO SCALE)  FIGURE 4.4 -  37  38  SCHEMATIC ILLUSTRATION OF TYPICAL CRACKS FOUND IN M O N O F R A X - M A T T H E HORIZONTAL P L A N E JUST B E L O W T H E H E A D E R (NOT TO S C A L E )  F I G U R E 4.5 -  39  C H E M I C A L ANALYSIS RESULTS OBTAINED F R O M M O N O F R A X - M CROWN B L O C K MEASURED A T THE VERTICAL CENTERLINE OF THE P L A N E BISECTING T H E B R O A D F A C E OF T H E CASTING  39  F I G U R E 4.6  -  PHOTOGRAPH OF FLEXURAL APPARATUS WITH T H E FURNACE OPENED  F I G U R E 4.7  -  40  SCHEMATIC OF T H E CONFIGURATIONS PERFORMED AT  FIGURE  4.8 -  4.9 -  TESTS  ORNL  41  STRESS-STRAIN C U R V E OF M O N O F R A X - M SPECIMENS A T A T A S T R A I N R A T E O F 1 X l 0"  FIGURE  OF THE FLEXURAL  S"  6  1100°C 41  1  STRESS-STRAIN C U R V E OF M O N O F R A X - M SPECIMENS A T  1100°C  ATASTRAINRATEOF5X10" S"'  42  7  FIGURE  4.10 -  STRESS-STRAIN C U R V E OF M O N O F R A X - M SPECIMENS A T A T A S T R A I N R A T E O F 2X1 0"  FIGURE  4.11-  4.12-  4.13-  4.14 -  FIGURE  4.15 -  4.16 -  F I G U R E 5.1-  F I G U R E 5.2  7  1  43 1350°C  2 X 1 0 " S"'  44  7  1500°C  STRAIN RATES  44  1600°C  STRAIN RATES  45  STRESS-STRAIN C U R V E OF M O N O F R A X - M SPECIMENS A T AT VARIOUS  1350°C  5x10" s"  STRESS-STRAIN C U R V E OF M O N O F R A X - M SPECIMENS A T AT VARIOUS  FIGURE  43  STRESS-STRAIN C U R V E O F M O N O F R A X - M SPECIMENS A T AT VARIOUS  1350°C  1  STRESS-STRAIN C U R V E OF M O N O F R A X - M SPECIMENS A T AT A STRAIN R A T E OF  FIGURE  42  STRESS-STRAIN C U R V E OF M O N O F R A X - M SPECIMENS A T AT A STRAIN R A T E OF  FIGURE  l x l O " S" 6  1100°C  1  STRESS-STRAIN C U R V E OF M O N O F R A X - M SPECIMENS A T AT A STRAIN R A T E OF  FIGURE  S"  7  1700°C  STRAIN RATES  45  MONOFRAX-M CROWN BLOCK MESH  -  STAGE I COOLING MOLDING MATERIALS:  65  GRAPHITE A N D SILICA  ANNEALING SAND PLATFORM MESH F I G U R E 5.3-  S T A G E II C O O L I N G M O L D I N G M A T E R I A L S :  65 ALUMINA  ANNEALING  O R E A N D S T E E L F L A S K I N G BIN M E S H  F I G U R E 5.4-  M E A S U R E D A N D MODIFIED T H E R M A L CONDUCTIVITY MONOFRAX-M  66  OF 66  ix  F I G U R E  F I G U R E  5.5-  P L O T  5.6-  C O M P A R I N G  P R E D I C T I O N S  S T A G E  I  P L O T  5.7-  P L O T  I  5.8  -  F O R  F I G U R E  5.9-  P L O T  O F  F I G U R E  5.10-  H E A T  T O  II  T H E  E X P A N S I O N  C O M P A R I S O N  V E R T I C A L  6.4  -  T H E  6.5-  C O N T O U R  U S I N G  W I T H  M O D I F I E D  S O L I D I F I C A T I O N  P A T H  C O O L I N G  68  D A T A  M O D E L  W I T H  F O R  D A T A  M O D E L T O  U S I N G  F R A C T I O N  /  M O D E L  S T A G E  W I T H  II  P R E D I C T I O N S  C O O L I N G  M O D E L  S H O W I N G  T H E  68  P R E D I C T I O N S  S E N S I T I V I T Y  T H E R M O C O U P L E  L O C A T I O N S  D A T A  T H E  W I T H  M O D E L  S I N T E R I N G  P R E D I C T I O N S  A L G O R I T H M  F O R  O F  A L U M I N A  C O N T R A C T I O N  O R E  S I N T E R E D  B E H A V I O R  O F  70  CX(3-ALUMINA  F R O M  86  M O D E L  I N P U T A N D  M O N O F R A X - M  C E N T E R L I N E  P L O T  O F  C H E M I C A L  C R O W N  T H E P L A N E  A N A L Y S I S  B L O C K  R E S U L T S  M E A S U R E D  B I S E C T I N G  T H E B R O A D  P L O T  a-50%  O F  A T  T H E  F A C E 86  O F  T H E  U S E D  O F  IN  OC(3-ALUMINA A N D T H E  S T R E S S  T H E V O I D  T H E  ( 3 - A L U M I N A  M O D E L  D I S T R I B U T I O N  87  U S E D  IN  T H E  M O D E L  V A R I A T I O N 50%  I  D A T A  C A S T I N G  C O N T O U R  S T R E S S  F I G U R E  O F  O F  D I S T R I B U T I O N S  F I G U R E  F O R  P-ALUMINA  O B T A I N E D  -  P A R A M E T E R S  69  T H E R M A L  6.3  4)  O F  C O O L I N G  6.1 -  F I G U R E  W I T H  M O D I F I E D  69  M O D E L  F I G U R E  O F  D U R I N G  D A T A  LCMF  (AN  M O D E L  C O R R E C T  T H E R M O C O U P L E  P L O T  -  W I T H  C O O L I N G  C O N T O U R  6.2  S T A G E  D I S T R I B U T I O N  T H E R M A L  II  M O R E  T H E R M A L  5.11-  F I G U R E  D A T A M O D E L  U S I N G  P Y R O M E T E R  T H E T H E R M A L  T H E R M A L  F I G U R E  A N D  M O D E L  T R A N S F E R  A N D  T H E R M O C O U P L E  P L O T C O M P A R I N G  S T A G E  H E A T  T H E R M O C O U P L E  C O M P A R I N G  S T A G E  F O R  F O R  C O M P A R I N G  T E M P E R A T U R E  F O R  P Y R O M E T E R  O F M O N O F R A X - M  R E F L E C T A  T H E B A S E - C A S E  F O R  A N D  T H E T H E R M A L  T H E R M O C O U P L E  T H E B A S E - C A S E  F O R  T H E R M A L  67  M O N O F R A X - M  P L O T  F O R  C A S T I N G / M O L D  P R E D I C T I O N S  L A T E N T  F I G U R E  P Y R O M E T E R  C O O L I N G  C O M P A R I N G  M O D E L  O F  T H E R M O C O U P L E  C O N D U C T I V I T Y  M O D I F I E D  S T A G E  A N D  T H E B A S E - C A S E  67  P R E D I C T I O N S  T H E R M A L A N D  F O R  C O O L I N G  C O M P A R I N G  M O D E L  F I G U R E  T H E R M O C O U P L E  M O D E L  87  E L A S T I C  M O D U L U S  (3-ALUMINA A N D  W I T H  T E M P E R A T U R E  p - A L U M I N A  F O R 88  X  F I G U R E  6.6 -  S T R E S S - P L A S T I C  S T R A I N  I N P U T T O A B  A Q U S  F O R  F I G U R E  6.7  -  C O N T O U R  P L O T  M O N O F R A X - M F O R  F I G U R E  6 . 8 -  S T A G E  F I G U R E  6.9  -  I  P L O T  6.10  S T R E S S  -  C R O W N  5  1  88  M O D E L  S H O W I N G  I N T H E •  T H E S E L E C T E D  F O RM O N O F R A X - M  F O R T H E E L A S T I C CC(3-ALUMINA  N O D E S  S T R E S S  S T R E S S  A N A L Y S I S  A T T H E E N D O F A N A L Y S I S  W I T H 89  C A S T I N G  O F M O N O F R A X - M  E V O L U T I O N  E L A S T I C  S T R E S S F O R  B L O C K  lxlO" s"  A T  89  O F S22  C O O L I N G  0 $ - A L U M I N A  F I G U R E  P L A S T I C I T Y  O F T H E ( 3 - A L U M I N A D I S T R I B U T I O N  H O M O G E N E O U S  T H E  O F M O N O F R A X - M  I N T E R P R E T A T I O N  C O N T O U R  A  C U R V E S  W I T H  A  S T A G E  D U R I N G  I C O O L I N G  F O R  H O M O G E N E O U S 90  C A S T I N G  O F M O N O F R A X - M  E V O L U T I O N  T H E E L A S T I C  S T R E S S  A N A L Y S I S  S T A G E  D U R I N G  W I T H  A  n  C O O L I N G  H O M O G E N E O U S 90  CC(3-ALUMINA C A S T I N G  F I G U R E  6.11  -  C O N T O U R C O O L I N G  P L O T  O F S22  F O R M O N O F R A X - M  F O R T H E E L A S T I C  S T R E S S  A N A L Y S I S  A T T H E E N D O F S T A G E W I T H  (3-ALUMINA  n  C O R E 91  F O R M A T I O N  F I G U R E  6 . 1 2 -  S T R E S S T H E  F I G U R E  6 . 1 3 -  F O R  F I G U R E  6 . 1 4 -  E L A S T I C  S T R E S S  T H E E L A S T I C  C O N T O U R  P L O T  C O N T O U R T H E  F I G U R E  6 . 1 6 -  A N D  F I G U R E  6.17  -  P L O T  II  V O I D  S T R E S S  A N A L Y S I S  JJ  W I T H  [3-ALUMINA  A N D V O I D  O F M O N O F R A X - M A N A L Y S I S  E V O L U T I O N - O F  F O R M A T I O N  W I T H  C R O W N  D U R I N G  S T R E S S  B L O C K  A T  A N A L Y S I S  D I S T R I B U T I O N  S T A G E  P - A L U M I N A  93  I C O O L I N G F O R  C O R E  F O R M A T I O N 93  M O N O F R A X - M  S T R E S S  A N D V O I D  n  C O R E  D I S T R I B U T I O N  T H E E L A S T I C  92  F O R M A T I O N  92  F O R T H E E L A S T I C  F O R M A T I O N  91  C O O L I N G  A T T H E E N D O F S T A G E  F O RT H E M O N O F R A X - M  C O O L I N G  C O R E  S T R E S S  (3-ALUMINA  F O RM O N O F R A X - M  F O R M A T I O N . . .  C O R E  S T A G E  D U R I N G  W I T H  I C O O L I N G F O R  D I S T R I B U T I O N  O F S22  E V O L U T I O N  E L A S T I C  S T R E S S F O R  A N D V O I D  p - A L U M I N A  S T R E S S T H E  O F S22  A N A L Y S I S  S T A G E  D U R I N G  p - A L U M I N A  W I T H  O F M O N O F R A X - M  S T R E S S  E N D O F S T A G E  W I T H  A N A L Y S I S  F O RT H E E L A S T I C  F O R M A T I O N  6 . 1 5 -  S T R E S S  E V O L U T I O N  C O O L I N G  F I G U R E  O F M O N O F R A X - M  E V O L U T I O N  A N A L Y S I S  D U R I N G  W I T H  D I S T R I B U T I O N  S T A G E  P - A L U M I N A  n  C O O L I N G  C O R E  94  xi  F I G U R E  6.18-  P L A S T I C P L A S T I C  S T R A I N S T R E S S  R A T E  E V O L U T I O N  M O D E L  O F  O B T A I N E D  M O N O F R A X - M  F R O M  D U R I N G  T H E  E L A S T I C -  S T A G E  I  C O O L I N G  F I G U R E  6.19-  P L A S T I C P L A S T I C  94  S T R A I N S T R E S S  R A T E  E V O L U T I O N  M O D E L  O F  O B T A I N E D  M O N O F R A X - M  F R O M  D U R I N G  T H E  E L A S T I C -  S T A G E  II  C O O L I N G  F I G U R E  6.20 -  95  C O N T O U R S T R E S S  P L O T  M O D E L  PE22  O F O F  O B T A I N E D  M O N O F R A X - M  F R O M  A T  T H E  T H E E N D  E L A S T I C - P L A S T I C O F  S T A G E  II  C O O L I N G  F I G U R E  6.21  -  S T R E S S E V O L U T I O N M O D E L  F I G U R E  6.22  -  6.23  -  6.24  -  O F  P L A S T I C S T R E S S  F I G U R E  O F  S T R E S S  S T R A I N  S T R A I N  M O D E L  S T A G E  II  96  S T R E S S  C O O L I N G  T H E  S T A G E  F R O M  D U R I N G  S T R E S S  C O O L I N G  F R O M  D U R I N G  O B T A I N E D  M O N O F R A X - M  I  T H E P L A S T I C  O B T A I N E D  M O N O F R A X - M  E V O L U T I O N  O F  T H E E L A S T I C - P L A S T I C  S T A G E  F R O M  D U R I N G  E V O L U T I O N  O F  F R O M  D U R I N G  O B T A I N E D  M O N O F R A X - M  M O D E L  P L A S T I C  O B T A I N E D  M O N O F R A X - M  S T R E S S E V O L U T I O N M O D E L  F I G U R E  95  E L A S T I C - P L A S T I C I  T H E  S T A G E  96  C O O L I N G  97  E L A S T I C - P L A S T I C II  C O O L I N G  97  Xll  List of Symbols Latin Symbols  Description  a  distance from the support to the load applicator when the specimen is straight material constant specimen width specific heat Young's modulus fraction of contact between the casting and mold surfaces a field variable used in the ufield subroutine reflecting the fraction to which the alumina annealing ore is sintered a field variable used in the ufield subroutine reflecting the phase distribution in the ascast material at each integration point a field variable used in the ufield subroutine reflecting the void distribution within the casting specimen height heat transfer coefficient due to contact conduction effective heat transfer coefficient due to radiation material constant thermal conductivity  A  b c E  P  fc fs  M  h  k k  Units  Page  m  34,41 16,17,21 34,41 46,61-63 34  m J kg K Pa 1  50,53-55  57,58  72,73  m Wm" K Wrn  Qc Q  volumetric heat source term  Wm"  S22  predicted stress which is parallel to the broad face of the refractory time current total model time  Pa s s  P  PE22 R  q q  t t  1  2  K"  1  m  n n n  2  Wm" K  span length between the lower supports material constant stress exponent normal to surface direction applied load predicted plastic strain which is parallel to the broad face of the refractory molar gas constant (8.3144) heat flux heat flux across the surface of the mold activation energy for creep  L  73 34,41 50,53,55 50 10,76 46,49, 61-63 34,41 10,76 16,17,21 49 34  1  1  N  J m o l " K" Wm" Wm" Jmol" 2  2  3  1  83,95,97 16 49 51 16,17,21 46 78-81, 89-94,96 49 50,55  xiii  teiapsed  T T  Tj T T Th 2  a  T T  s  s  Th t  U  Greek Symbols  ^Monofrax-M + load train  &load train  &SiC bend bar  £ £ £ £\ £2 £ Ptheoretical P  a <7 (7e do  elapsed time, or duration, of Stage I or II cooling temperature current temperature of the alumina ore evaluated at each material integration point within an element temperature of the casting temperature of the mold ambient temperature temperature above which the alumina ore becomes hard sintered temperature of the casting surface temperature above which the alumina ore becomes weakly sintered threshold temperature below which sintering does not occur span length between the upper supports  Description corrected load-point displacement of Monofrax-M displacement obtained from the flexural tests on the Monofrax-M specimens displacement gathered from the flexural tests on the S i C bend bar load-point displacement based on the elastic deflection of the S i C bend bar outer-fiber tensile strain effective emissivity emissivity of the mold (Equation 5.7) emissivity of Monofrax M emissivity of the mold total equivalent plastic strain creep strain rate theoretical density density stress Stefan-Boltzmann constant ( 5 . 6 7 x l 0 ) V o n Mises effective stress yield stress -8  s K  50,55 16,46,49  °C K K K  58 50 50 51  °C K  58 51  °C  58  °C m  58 34,41  Units  Page  m  33,34  m  33,34  m  33,34  m  33,34 34 50,51 51 51 51 10 16 17 17,46, 61-63 16,34 50,51 10 10  kg m" kg m"  3  3  Pa Wm" K" Pa Pa 2  4  xiv  Acknowledgements I would like to thank my advisor, Dr. Steve L . Cockcroft for his utmost support and encouragement throughout the research programme.  I am grateful to the Monofrax  related personnel, Dr. Steve M . Winder, M r . A m u l Gupta, and M r . Dennis Walrod, for their technical help. I wish to thank Monofrax Inc. for the support of research expenses, and the provision of valuable data.  I would also like to thank Oak Ridge National  Laboratory ( O R N L ) and its personnel in the high temperature materials laboratory, Dr. Andrew A . Wereszczak and Dr. Kristin Breder, for the support and technical help on the laboratory experiments. In addition, I have profited from numerous discussions with my colleagues and professors at the University of British Columbia.  I particularly thank D r . Daan M .  Maijer, Dr. M a r y A . Wells, Dr. T o m Troczynski and Dr. Warren J. Poole.  1  CHAPTER 1 INTRODUCTION AND OVERVIEW 1.1 Introduction Refractories are often used in high temperature industrial processes involving materials manufacturing because of their insulation capabilities and ability to contain reactive materials at elevated temperatures. They are essential in the production of glass, metals, cements and other materials.  One class of refractory, called fused-cast refractory, is  widely used in glass industry for furnace linings owing to its superior corrosion/erosion resistance to the environment present in float-glass furnaces. These include refractories such as fused-cast alumina-zirconia-silica ( A Z S ) , and a|3-alumina. During the solidification process, fused-cast refractories are subject to a variety of defects that are comparable to those found in the metal castings. These include cracks that arise due to excessive thermal stresses or strains and porosity due to volumetric contraction associated with solidification. Under extreme conditions existing in modern glass furnaces, these defects can lead to excessive chemical attack, which can have a negative effect on glass quality. On-going efforts to improve glass quality while at the same time reducing costs have led to a need for higher quality fused-cast refractories. T o achieve high quality products, these defects must be controlled and preferably eliminated. This thesis is focused on the development of a mathematical model capable of predicting the thermal and stress/strain history in fused-cast a|3-alumina refractories during the casting process.  The ultimate goal of this work is to understand  the  mechanism(s) leading to the formation of cracks, which are a common defect, found in  2 this type of refractory block. This work has been sponsored by Monofrax Inc., a leading manufacturer of fusion cast products.  1.2 Background Monofrax Inc., located in Falconer, N Y , manufactures  various types of  fused-cast  ceramic components for different applications. Fusion cast a(3-alumina, sold under the trademark  Monofrax-M,  is one class of product  and is used  in glass  furnace  superstructure, known as crown block. Monofrax-M is a dense, high alumina fused-cast refractory formed by first melting a fusion of oxides, typically 94.5 weight % AI2O3, 3.8 weight % N a 0 and other minor oxides, in an electric arc furnace and then pouring the 2  resulting liquid into a graphite mold [1]. After a short time, the graphite mold is removed once a suitable solid shell has formed and the refractory is placed into an annealing bin (box) and covered in an insulating annealing sand (alumina ore), where it may take a few weeks to cool. To avoid cavity formation within the refractory during solidification, components are cast with an oversized header (refer to Figure 1.1), which serves as a liquid reservoir. Typical melt densities are 2660 k g / m and solid densities are 3590 k g / m , representing a 3  3  - 3 5 % volume reduction [2]. The resulting void distribution in a cast block with header attached is shown in Figure 1.2. The header is removed from the refractory block using a diamond-saw prior to surface grinding. Typical block sizes for a glass furnace crown are 0.4 m by 0.5 m by 1.0 m (16 inches by 20 inches by 40 inches).  3  Under equilibrium solidification conditions [3], the mixture of oxides would solidify to form a material consisting of crystalline phases of alpha- (-38 vol.%) and beta-alumina (-62 vol.%) together with a small amount of interstitial glass, less than 1% [1].  However, in practice, refractory blocks of this composition solidify with an P-  alumina core surrounded by oc-p shell. The P core forms due to rejection of N a 0 , a P 2  stabilizer, at the solid/liquid interface - refer to Figures 1.2 and 1.3 in which the P core can be seen. In addition to void formation, the other major defect and the most problematic from a process viewpoint, is crack formation. Cracks have been found in all the castings that have been cut open for examination. Figures 1.2 and 1.3 show the orientations and locations of the cracks found in a typical refractory. The formation of the P core would appear to have important implications in terms of the formation of crack defects as the majority of cracks appear to emanate from or terminate at the a-p/p  boundary.  These  cracks can reduce the erosion-corrosion resistance of the refractory and can also lower the ability of the refractory to withstand thermal cycling.  Blocks that exhibit surface  cracks at the hot face of the block (the bottom horizontal plane of the casting that is in contact with the vapor phase species generated at the glass melt surface inside a glass furnace) must be rejected and recast. Moreover, if the customer is not willing to accept a cracked product, the refractory is recycled at a considerable cost. Hence, there is a strong desire to determine the origins of these cracks and to eliminate them.  1.3 Fundamental Based Approach Historically, fused-casting processes have been developed on the basis of trial-and-error optimization. In terms of solving specific problems, such as the formation of a particular crack, this approach is often time-consuming, costly and ineffective. The main objective of the present study is to develop a fundamental understanding of crack formation during fusion  casting of a(3-alumina refractories.  This is to be achieved through the  development of a finite element method ( F E M ) based model to simulate the thermal behavior of a solidifying Monofrax-M crown block and the resulting  stress/strain  evolution during different stages of the standard production process at Falconer. The ' M o n o f r a x - M ' model is broken into two components - the thermal model and the mechanical, or stress model - which, are run separately and are uncoupled. The geometry of the block necessitates a three-dimensional analysis in both the thermal and the mechanical models. The thermal model is based on the Fourier's law of heat conduction using the law of conservation of energy.  The various stages of the  casting process need to be properly accounted for in the thermal model to reflect the different steps in the manufacturing process.  It is essential that both the refractory and  the molding materials including those associated with the graphite mold and the annealing bin be included in the thermal model in order to properly account for the flow of heat in the block. Characterization of the various thermal boundary conditions and initial conditions is also necessary as is the quantification of the various thermo-physical properties.  Moreover, since these materials are subject to a large range in temperature,  properties such as the thermal conductivity and the specific heat need to be incorporated  5 into the model as temperature dependent. Once validated against thermocouple data, the resulting thermal history can then be input to the thermal stress model. The stress model is solved based on the differential equations of equilibrium and the compatibility conditions. The temperature predictions obtained from the heat transfer model serve as input (thermal loads) to the stress model. Unlike the thermal model, only the fused-cast alumina refractory is involved in the stress analysis as the casting can be assumed to be free of interaction with the mold (surface traction free).  Owing to the  broad range of temperature that the casting experiences, the material properties such as the elastic modulus and the thermal expansion coefficient need to be implemented into the model as a function of temperature.  In addition, the effect of the different thermal  expansion or contraction behavior of the ocp-alumina and the P-alumina has also to be addressed.  This entails the incorporation of an algorithm into the model in which the  regions of ap-alumina and P-alumina can be distinguished. Assessment of the effect of void formation w i l l also be required due to a decrease in elastic modulus or strength in the void region.  The resulting evolution of stress and strain in the refractory during  various stages of the casting process can be linked to the occurrence of crack defects. Mechanisms can then be formulated and remedial actions can be prescribed for their elimination.  6  Figure 1 . 1 - Schematic illustration of a fused-cast ap-alumina refractory casting with an oversized header (not to scale).  Figure 1.2 - Cross-section of a Standard Operating Practice (SOP) M o n o f r a x - M block taken at the vertical plane bisecting the narrow face of the casting (refer to Figure 1.1). Cracks and voids can be observed and the [3-alumina region is being outlined. Note that the cracks are vertical and perpendicular to the broad face of the block resulting from the tensile stress acting parallel to the broad face (tensile stress in direction-2 in Figure 1.1).  8  Figure 1.3 - Top view of a S O P Monofrax-M block taken at the horizontal plane just below the header. The (3-alumina region can be observed in the center of the casting and the cracks appear to emanate from or terminate at the ccpVR-alumina interface.  9  CHAPTER 2 LITERATURE REVIEW A review of the literature has indicated a scarcity of information on the processing of fused-cast refractories. The few studies that do exist are presented below together with some relevant literature on metal-casting processes.  The review of metal casting  processes focuses on those studies that apply fundamental based mathematical models to understand the development of crack defects.  In addition, the constitutive behavior of  fused-cast refractories is also examined.  2.1 Casting Processes Crack defect formation is found in many metal castings owing to thermal stresses [4-6]. In the continuous casting of steel billets, for example, crack formation arises due to the generation of thermal stresses in the solid shell owing to the high heat-extraction rates. Mathematical models of heat flow, together with the measured material properties of steel at various temperatures, have led to an understanding of the mechanisms responsible for defect formation and important improvements in billet quality [4]. In the manufacturing of thick parts of high-strength aluminum alloys, which are often rolled, forged, or extruded, significant distortion of the products arises from the generation of thermal stresses resulting from a drastic quenching of the component during casting [5].  T o help develop methodologies for control of such complex phenomena,  finite element (FE) calculations have been applied by leanmart and Bouvaist [5] to determine the temperature distribution, thermal strains, and residual stresses arising during manufacture.  In the investigation by leanmart and Bouvaist, a non-linear heat  10 transfer analysis was first performed followed by an un-coupled elastic-plastic stress analysis using the predicted temperature distribution as the 'thermal loads' input to the stress analysis.  The elastic-plastic stress analysis was based on the Ramberg-Osgood  relationship to model the stress-strain curve of the material using the formulation [5] (2.1) where, <j is the V o n Mises effective stress, <T„ is the yield stress, k and n are constants e  and £ is the total equivalent plastic strain. Since plasticity data is required in the elasticplastic stress analysis, tensile tests were carried out at various temperatures  to  characteristic the inelastic behavior of the material. The model predictions were shown to have good agreement with the measured values of residual stresses. Moreover, these computer simulations made considerable contributions to the physical understanding of the residual stress build-up during production. In gray iron castings, the problems are similar to those found in the fused-cast refractory processes. In a study by Wiese and Dantzig [6], finite element analysis ( F E A ) was used to simulate the heat transfer and thermal stress evolution as the casting solidifies and cools. In this approach, stresses resulting from thermal displacements in the cooling casting were computed in an elastic-viscoplastic stress analysis requiring that the creep behavior of the gray iron to be evaluated in addition to the 'plastic' response. Due to the mechanical behavior of the gray cast iron, different properties in tension and compression were also incorporated into the model.  11  2.2 Computer-Based Mathematical Models on Refractory Components In addition to metal castings, numerical techniques have also been applied to refractory components to address design problems [7-9].  According to Bradley et al. [7], who  examined the thermal stress fracture of refractory components in  high-temperature  industrial furnace linings and metallurgical process vessels, heat flow may be assumed to be one-dimensional (1-D). A F E A conducted by the authors for selected cases revealed that, not surprisingly, the maximum principal tensile stress is directly proportional to elastic modulus, heating or cooling rate, and coefficient of thermal expansion. In terms of applicability to the present study, there are several shortcomings to the Bradley study. Firstly, the study is too general in nature and thus limits its applicability to specific refractory lining structures. In general, the modeling of heat flow is a complex matter that requires a case-by-case analysis using a numerical method such as F E rather than assuming that the heat flow is always one-dimensional. The assumption, therefore, eliminates the possibility of multi-dimensional stresses and strains that may be generated in the refractory.  Secondly, the material properties were assumed to be independent of  temperature in Bradley's model. In fact, thermal conductivity is often non-linear and temperature dependent in many refractory materials.  The analysis of two-dimensional  stress distribution may therefore be somewhat limiting.  In addition, the Bradley study  was only concerned with the thermo-elastic model and failed to address any hightemperature inelastic or plastic deformation during service. In another study by Chang et al. [8], segmented thick-walled refractory structures were analyzed.  A major simplification was made by assuming that the thermal  conductivity and thermal diffusivity were independent of temperature and position. The  12 resulting analytical temperature distribution was one-dimensional and dependent on time and position along the length of the refractory. In the Chang study, the stresses were analyzed using a two-dimensional finite element model.  A limited number of three-dimensional cases were also investigated.  The calculated stress profiles indicated that the maximum tensile thermal stresses were always found along the centerline of the refractory. It was also found that the magnitude of stress decreased with increasing thermal diffusivity and that changes in length of the refractory did not appear to reduce the probability of thermal stress fracture. The Chang study has a similar drawback as the Bradley's model in obtaining the temperature distribution.  The analytical thermal profile is simplified, being one-  dimensional, and caution should be exercised due to this limitation.  It should also be  noted that like the Bradley study, the temperature dependency of thermal conductivity was not addressed in the Chang study. Furthermore, many candidate materials for basic oxygen furnaces exhibit substantial softening and creeping at high temperatures.  The  model, however, only dealt with elastic behavior. Thus, the predicted thermal stresses might be over-estimated in this case, which would affect the predicted probability of failure and accuracy of the model. Unlike the previous two studies, Knauder and Rathner [9] evaluated both the temperature and stress distributions of the refractories based on F E M .  In their thermo-  mechanical analysis of basic refractories in a bottom blown converter, a quarter of the vessel bottom was modeled. A 2-D model assuming no circumferential heat flow was also developed for the 3-D quarter section of the bottom, which represented a symmetry plane with the 3-D model. A comparison of the 2-D and 3-D results confirmed that the  13 models were in agreement.  Moreover, Knauder and Rathner analyzed the stress  distribution based on the temperature results. A two-dimensional F E M elastic analysis was carried out to reduce computational effort.  Three-dimensional results were also  obtained for limited cases. In the Knauder and Rathner study [9], the reported stresses were way above the uniaxial compressive strength of the refractory materials.  This is mainly due to two  reasons: 1) the assumption of linear elastic behavior and 2) the homogeneous continuum at the bottom. In regard to the constitutive behavior, it was noticed that the stress-strain curves of carbon-bonded refractories are highly non-linear and thus, for a given strain, the linear elastic model over-predicts stress. Hence, a more quantitative analysis should address stress relaxation due to creeping of the material, which may be significant at high temperatures. In regard to the second shortcoming, joints often present in the refractory lines can significantly alter the thermal stress field depending on gap clearances and mortar material.  A proper analysis should therefore include  refractory-to-refractory  contact as part of the compliment of boundary conditions. Nonetheless, the analysis, as a whole, is good for qualitative comparison.  2.3 Computer-Based Mathematical Models on Fused-Cast Refractories Cockcroft and Brimacombe [10-11] performed thermal stress analysis on fused-cast A Z S refractories using a three-dimensional finite element model.  Uncoupled thermal and  stress models were employed to estimate the temperature distribution, as well as stresses and strains in the solidifying shell of the cast refractory.  The stress model assumed  simple thermo-elastic behavior but included the thermal dilatational expansion associated  14 with the tetragonal-to-monoclinic phase transformation of Zr0 . 2  The model predictions  revealed the critical role of the tetragonal-to-monoclinic phase transformation, which is predicted to generate tensile strains around the transformation zone. W i t h the aid of the mathematical model, it is evident that the major thermal resistance to heat extraction from the molten refractory lies in the mold and that changes to the mold design and the control of the differential cooling are the solution to the problem. Therefore, to eliminate the cracking problems found in the A Z S refractories, an understanding of the mechanisms responsible for crack formation is crucial. Similar to the other refractory studies that employed the use of numerical techniques, the stress model by Cockcroft and Brimacombe [10-11] is based on elastic behavior of the A Z S refractories. Owing to the glass constituents within the refractories, inelastic or irreversible deformation takes place at elevated temperature.  It is likely that  the stress predictions obtained from the model would be over-estimated and caution must be used in interpreting the results of this analysis. In an attempt to consider the inelastic deformation of the fused-cast A Z S refractories, L u et al. [12] developed a finite element model to study the stress and strain evolution in the casting. Unlike the stress analysis by Cockcroft and Brimacombe [1011], L u et al. incorporated the Drucker-Prager plasticity into the model, which assumed a spherical block casting rather than the standard rectangular block. Uniaxial stress-strain behaviors in tension and compression were evaluated for the model input and it was found that there is considerable difference between the tensile and compressive flow stress in the refractories. Although there are several limitations in this model in terms of the shape of the block and the thermal histories, the analysis was able to identify critical  15 parameters that affect the inelastic strains, stresses, and the cracking that develop in the A Z S castings. In a recent study, Wang [13] investigated the effects of the geometric and physical parameters of a cooling system on the temperature distributions in the solidifying fusedcast oc(3-alumina refractories.  A two-dimensional thermal analysis was conducted using  the finite difference method. Boundary conditions relating to heat transfer between the refractory and the mold, as well as the gap formation mechanism, were based on the fused-cast A Z S refractory analysis by Cockcroft and Brimacombe [10-11]. It is found that the thermal conductivity of the insulation material as well as the  geometric  parameters of the annealing system play an important role in the temperature gradient of the casting. The analysis by Wang can be used to begin to understand the complicated processes leading to crack formation in fused-cast a^-alumina refractories.  However,  Wang's analysis falls short in that it was not three-dimensional and does not describe key features of the casting process. Moreover, the computer simulation of heat transfer in the casting process was not verified against industrial measurements and many of the required material properties were not characterized.  2.4 Constitutive Behavior of Refractories In most of the above studies on refractories, inelastic behavior of the material is often neglected to simplify the analysis.  However, most ceramics exhibit some degree of  permanent deformation when under load at elevated temperatures.  During the casting  process, the cd3-alumina may be expected to be subject to a broad range of thermally  16 induced stresses as the block solidifies and cools to near room temperature.  The two  most common approaches (constitutive models) used to describe inelastic or permanent deformation in materials are plasticity and creep. Physically, the strain resulting from either plastic or creep deformation are indistinguishable from one another [14]. Thus, either approach could be used to quantify stresses and strains arising in the casting process. Typically, time-dependent deformation or creep in ceramics takes place at high temperatures under modest stress levels [15]. This time-dependent deformation usually exhibits three distinct regimes: a primary creep stage during which the creep rate rapidly decreases, a secondary or steady-state stage with a nearly constant creep rate, and finally the tertiary stage during which the creep rate increases in an unstable manner just before rupture [16]. The power-law creep equation is one of the most widely used constitutive creep models. In this model, the creep strain rate is given by the relationship  £ = A<r"exp V  RT  (2.2)  J  where, e is the creep strain rate, A is a constant, o is the stress, n is the stress exponent, Q is the activation energy for creep in J mol" , R is the molar gas constant (= 8.3144 J 1  c  mol" K " ) , and T is the temperature in K e l v i n [17]. This power-law creep equation is 1  1  applicable to the steady-state (secondary) creep region exhibited by many materials including some ceramics. While the power law creep equation could address sensitivity to temperature and stress, its applicability for the quantification of the stress and strain field in the fused-casting process is unclear and w i l l depend on the extent to which steady state flow conditions prevail.  17 In terms of available data, Munro [18] determined the steady-state creep constants, A and n, as well as the activation energy Q , for a sintered a-alumina (mass c  fraction of AI2O3 > 0.995, relative density pm).  (p/ptheoreticai)  ^ 0-98 and nominal grain size is 5  In another creep study, Wereszczak et al. [19] evaluated the steady-state constants  of  100% (3-alumina and  5 0 % a / 50% (3-alumina  compressive  creep  refractories.  Table 2.1 compares the creep characteristics and test conditions of the  various alumina refractories. It can be observed from Table 2.1 that the creep characteristics are sensitive to the various grades of alumina refractories and only the steady-state creep information is evaluated. Deformation associated with primary creep may also be important in transient loading applications such as those occurs in a casting process or in the early stages of thermal cycling. Unlike creep equations based on steady-state creep, the plasticity approach has the ability to easily address the early stages of deformation. Analyses of casting processes that adopted the plasticity approach include the study by leanmart and Bouvaist [5] on an aluminum alloy and L u et al. [12] on fused-cast A Z S refractories. The well-known V o n Mises yield criterion commonly used for metal plasticity can be used for materials exhibiting isotropic behavior. Temperature dependencies in the yield stress as well as the hardening behavior can be easily accommodated in most commercial F E packages. Some packages also permit strain rate dependent plastic behavior needed to address the varying loading conditions existing in the casting process. The one potential drawback of the plasticity approaches is that they fail to accumulate inelastic strain at a constant load -  18 i.e. exhibit time-dependent deformation, which may be expected to occur to some extent in the casting process.  2.5 Monofrax-M Thermo-physical and Thermo-mechanical Behavior - In addition to the constitutive behavior of fused-cast  ap-alumina refractories,  quantification of the thermal  and  stress/strain fields during solidification processing requires data describing the thermophysical and thermo-mechanical behavior of this material over the temperature range experienced in the casting process. Some of these data are found in the product brochure [1].  For instance, typical data for the thermal conductivity, thermal expansion and  compositions of Monofrax-M are shown in Figure 2.1.  Since very little additional  information can be obtained from the literature, most has had to be measured. Volumetric Latent Heat of Fusion - The volumetric latent heat of fusion is another thermo-physical property that needs to be quantified for the heat transfer analysis. According to the N a 0 - A l 2 0 3 phase diagram (refer to Figure 2.2) [3], a melt containing 2  between 90 and 100 mole % AI2O3, would solidify by first forming primary a-alumina until a temperature of approximately 2000°C is reached at which point the (3-alumina would form to complete the solidification process. Assuming equilibrium solidification, Monofrax-M, which consists of roughly 93.2 mole % A 1 0 , starts solidifying as primary 2  3  a-alumina at 2026°C, the liquidus, and reaches the solidus at 2000°C at which point the volume percent of a-alumina and |3-alumina would be expected to be approximately 38 and 62, respectively. The latent heat of fusion for a-alumina is 111.086 kJ m o f [20] but 1  no value for (3-alumina is reported in the literature.  19 a/3-Alumina and (3-Alumina - A s discussed in Section 1.3, the effect of the different thermal expansion or contraction behavior of the aP-alumina and the P-alumina needs to be examined.  The investigation of Cockcroft and Brimacombe [10-11] on fused-cast  A Z S refractories, which included the thermal dilatational expansion associated with the tetragonal-to-monoclinic phase transformation, showed the critical role of the phase transformation on the generation of stresses and strains on the casting. Void Distribution - In addition to addressing the aP-alumina and the  P-alumina  distribution and the associated expansion and contraction behavior, the void formation occurring in the casting process also needs to be evaluated.  N o reported literature has  been found on the assessment of the effect of void formation in the  fused-cast  refractories. However, the void distribution within the casting may have an effect on the generation of stresses and strains due to a decrease in elastic modulus or strength in the void region.  2.6 Summary Based on the literature review, fundamental based mathematical models can be used to predict the temperature,  stress and strain evolution of the fused-cast  refractories to understand the development of crack defects.  ap-alumina  Due to the nature of the  casting process, the analyses are required to be three-dimensional and should incorporate the actual geometry of the casting and the molding materials.  The temperature  predictions obtained from the thermal analysis can serve as 'thermal loads' input to the stress analysis. validation  of  In order to formulate the heat transfer and the mechanical models, the  thermal  model  through  industrial  measurements  and  proper  20 quantification of the boundary conditions, as well as the thermo-physical and themomechanical behavior of this material over the temperature range experienced in the casting process are required. The characterization of the a(3-alumina and the R-alumina regions and the void distribution needs to be addressed. Furthermore, strain rate sensitive inelastic behavior of the refractory needs to be examined and implemented into the stress model to properly reflect the deformation behavior taking place in the casting process.  21 Table 2 . 1 - Comparison of the creep constants obtained for various alumina refractories [18-19]. Refractories  A (s" )  n  Q (kJ-mol" )  Temperature (°C)  Applied Stress (MPa)  i~08  323  1200-1800  100-200  2  0.9  151  1400-1593  0.17-1.03  1 0  0.9  544  1400-1593  0.17-1.03  c  1  Sintered a-alumina  3.6xlO  100% [3-alumina  5.6xl0'  50% a/50% (3-alumina  l.lxlO'  u  Technical Data  Typical Thermal Conductivity 60 (8.65) O  °  40  E  £ (2.88)  .c  53  I  I  °C 400 °F 752 Mean Temperature  I  I  600 1112  800 1472  I  1000 1832  .  1200 2192  Typical Thermal Expansion  °C 400 °F 752 Temperature  600 1112  800 1472  1000 1832  Typical Chemical Analysis: Al 0 Na 0 Si0 Fe 0 CaO Other 2  3  2  2  2  3  Figure 2.1 - Thermal conductivity, thermal expansion M o n o f r a x - M [1].  1200 2192  1400 2552  % 94.5 3.8 0.8 0.1 0.2 0.6  and typical  composition  23  liquid FROM REF. 2 ^.—'f^— + Li(J.  /  /  II !|  —-J  +' a-AI Oj 2  :l 1S85«*10° fcNoAJO + 0  j? ss  :  "V4O0F SO0A-. RICH 0"  1200|  1000  FROM ^ NoF + AljOj  r  50 >"Nfl0  60  70  80  90  100 Al 0 2  2  Figure 2.2 - Phase diagram of N a 0 - A l 0 3 [3]. 2  2  3  24  CHAPTER 3 SCOPE AND OBJECTIVES 3.1 Scope of the Research Programme The purpose of this research programme is to develop a mathematical model capable of predicting the temperature, stress and strain distribution during the solidification and subsequent cooling of the Monofrax-M castings. The ultimate goal is to understand the mechanism(s) leading to the formation of cracks found in this type of refractory block. To achieve this goal, fundamental based mathematical models were developed based on the commercial finite element code, A B A Q U S ^  The non-linear solution  capabilities in A B A Q U S are well-developed and robust making it well suited to solve the heat transfer and stress problem.  Additional features of the code relevant to the  Monofrax-M crown block include the ability to handle multiple thermal load steps as well as the addition and removal of materials such as is required in the simulation of graphite stripping, floating and annealing in alumina ore. Also, the recently introduced sparse matrix solver enables larger geometrically more complex 3-D models to be tackled with reasonable execution times. Other features of the code such as the ability to define field variables and specify field variable dependencies permit additional features of the Monofrax-M casting processes to be tackled including the P-alumina core and the solidification shrinkage void distribution. Owing to a scarcity of the information in the literature, most of the input parameters for the heat transfer and mechanical models were evaluated by Monofrax Inc. or independent research laboratories. A n assessment of the sensitivity of the models to  25  this data was performed to determine the effect of this data on the analyses.  Once  formulated, the thermal model was then validated by comparison to industrial plant data obtained from Monofrax Inc.  Thermocouples embedded in the molds (both graphite  mold and alumina ore annealing bin) were used to obtain local thermal histories for the purpose of verifying the assumptions made in formulating the model. A good understanding of the high-temperature behavior of this refractory is essential to formulate crack formation mechanisms.  The preliminary elastic stress  analysis was formulated based on the elastic modulus values measured by a research laboratory.  Additional information or the inelastic stress-strain behavior at elevated  temperatures was also needed.  Experiments were performed at O R N L (Oak Ridge  National Laboratory in Oak Ridge, Tennessee) in an attempt to evaluate the inelastic behavior of this material. Preliminary results obtained from the elastic and elastic-plastic stress analyses were compared to determine the influence of assumptions relating to the constitutive behavior on the prediction of crack formation during solidification and cooling.  ABAQUS is a registered trademark of Hibbitt, Karlsson & Sorensen, Inc.  26 3.2 Objectives of the Research  Programme  The objectives of the present study are as follows: [1]  T o formulate, develop and verify mathematical models capable of predicting the temperature and stress/strain field evolution in solidifying M o n o f r a x - M blocks.  [2]  T o calculate the temperature and stress/strain distributions in solidifying blocks under typical process conditions which have led to the formation of cracks.  [3]  To understand the mechanism(s) of the crack formation based on the links between the model predictions and the occurrence of crack defects.  27  CHAPTER 4 INDUSTRIAL A N D L A B O R A T O R Y M E A S U R E M E N T S A s mentioned earlier, in-mold temperature  measurements  were obtained from an  industrial trial performed at Monofrax Inc. These thermocouple measurements serve to validate the heat transfer model as well as characterize the boundary conditions at the refractory/mold interface. In addition, experiments were conducted at O R N L to evaluate the inelastic behavior of the refractory at elevated temperatures and provide data for input to the mechanical model.  4.1 Industrial Measurements To understand the solidification of the fused-cast ap-alumina refractories, temperature data was obtained from thermocouples embedded at various locations in the molds employed in the Monofrax-M casting process. A pyrometer was employed to measure the pour stream as well as the surface temperature of the refractory block. In the same industrial study, chemical analysis was also performed in which both the pour stream and the block chemistries were evaluated. This section discusses the experimental techniques employed in the industrial study and the corresponding results.  4.1.1 Experimental Techniques The M o n o f r a x - M casting process can be divided into two regimes of behavior in terms of the heat transfer. Stage I cooling in the graphite mold, characterized by high cooling rates lasting a short duration; and Stage II cooling in the annealing ore, characterized by  28 relatively slow cooling rates extending over a long period of time (of the order of weeks). In the instrumented test casting, thermocouples were inserted into the graphite mold to record the variation in temperature with time during Stage I cooling and in the alumina ore to record the evolution in temperature during Stage II cooling in the bin. For Stage I cooling, three thermocouples were placed in the graphite mold. The thermocouples were located adjacent to the center of the broad face of the block at a vertical height of 0.3556 m (14 inches), measured from the base of the block. One was located approximately at the casting/graphite interface and the other two were located in the graphite 0.0254 m (1 inch) and 0.0968 m (3.8125 inches) from the interface.  The  casting/graphite interface thermocouple was located at the center of the broad face and a Type-R (Platinum-13% Rhodium vs. Platinum) thermocouple was employed.  The  0.0254-m (1-inch) and 0.0968-m (3.8125-inches) thermocouples were offset from the center toward the block narrow face by 0.0762 m (3 inches) and were T y p e - K (NickelChromium vs. Nickel-Aluminum) thermocouples. In addition, a pyrometer was used to obtain an estimate of the pour stream as well as the block surface temperature.  A two-  color optical pyrometer was employed and calibrated at Monofrax Inc. The pyrometer was calibrated against heated samples of Monofrax-M of a known temperature. For Stage II cooling, a total of 6 thermocouples were inserted in the alumina annealing ore.  The thermocouples were located approximately at the casting/ore  interface, and at 0.0508, 0.0523, 0.1016, 0.2032, and 0.2921 m (2, 2.06, 4, 8 and 11.5 inches) from the refractory/ore interface adjacent to the broad face of the block. The 0.0508 and 0.1016-m (2 and 4-inches) thermocouples were offset from the center of the broad face toward the narrow face by 0.0762 m (3 inches), whereas the 0.2032 and  29 0.2921-m (8 and 11.5-inches), and the 0.0523-m (2.06-inches) thermocouples were offset from the center by 0.1524 (6 inches) and 0.2286 m (9 inches), respectively.  Type-B  (Platinum-30% Rhodium vs. Platinum-6% Rhodium) thermocouples were used for the casting/ore interface, and 0.0508 and 0.0523 m (2 and 2.06 inches, respectively) from the interface. For the remaining locations, T y p e - K thermocouples were selected.  4.1.2 Results The results obtained from the industrial trial are presented below in three subsections: initial conditions, thermal responses and post-cast refractory autopsy.  4.1.2.1 Initial  Conditions  Based on the pyrometer measurements on the pour stream, the molten liquid temperature was approximately 2049°C, which serves as the initial condition for the block in the model.  The ambient air temperature was roughly 25°C, which serves as the initial  condition for the various components of the mold and the annealing bin (including the graphite, the silica annealing sand platform, the alumina annealing ore and the steel bin) used in the model. During the 'pouring' of the molten alumina into the graphite mold, a small sample was taken for chemical analysis. The results showed that the pour stream chemistries contained approximately 94.5 weight % AI2O3, 4.2 weight % NajO and other minor oxides.  30 4.1.2.2 In-Mold Temperature Responses A m o n g the three thermocouples that were inserted into the graphite mold, the one located at 0.0968 m (3.8125 inches) from the casting/graphite interface failed completely. Figure 4.1 shows the thermocouple measurements obtained from the graphite mold during Stage I cooling along with the pyrometer data of the surface temperature of the casting.  It  should be noted that here and in the remaining plots to follow, the frequency of the measured data is adjusted to distinguish various curves in the plots. A s can be seen in Figure 4.1, there is initially a gradual increase in the temperature  for the first  approximately 40 seconds followed by a relatively rapid increase associated with the liquid level reaching 0.3556 m (14 inches) height.  The graphite temperature at the  interface is observed to reach a peak temperature just in excess of 1000°C, whereas the surface of the block is between 1539 and 1737°C as recorded by the pyrometer.  The  greater scatter in the second set of pyrometer data shown in Figure 4.1 is a result of the motion of the casting and the difficulty of maintaining focus on the center of the broad face of the M o n o f r a x - M crown block. Unlike the transient Stage I cooling, Stage II cooling involves slower cooling rates over a longer time in the alumina ore annealing medium. O f the 6 thermocouples that were installed in the alumina ore, 1 failed completely and the thermocouples that were placed 0.0508 and 0.0523 m (2 and 2.06 inches) from the casting/ore interface produced similar results.  Figure 4.2 presents the thermal response obtained from the  thermocouples that were inserted 0.0508, 0.1016, 0.2032, and 0.2921 m (2, 4, 8 and 11.5 inches) from the interface.  31 4.1.2.3 Refractory Autopsy The results of the refractory autopsy are illustrated schematically in Figures 4.3 and 4.4. A s shown in Figure 4.3, the cracks are vertical and perpendicular to the broad face of the block. Tensile stresses acting parallel to the broad face (tensile stresses in direction-2) likely give rise to such cracks.  Moreover, these cracks appear below the p-alumina  region. For the horizontal plane just below the header, Figure 4.4 shows that the cracks appear to emanate from or terminate at the a(3/(3-alumina interface. morphologies  indicate that the cracks  are intergranular  Furthermore, crack  and formed  at  elevated  temperatures. In addition to examining the crack defects found in the fused-cast ap-alumina refractories, chemical analysis was also performed on the casting along the vertical centerline of the plane bisecting the broad face. In the analysis, the amount of AI2O3 and Na20 was measured to determine the distribution of P-alumina within the block. Figure 4.5 presents the plot of the amount of P-alumina in mole % evaluated from the base to the top of the block. The data can be used in the model to distinguish between the  aP-  alumina and the P-alumina regions.  4.2 Laboratory Measurements A s discussed earlier, an understanding of the high-temperature mechanical behavior of Monofrax-M is needed for the stress analysis. Four-point flexural tests were carried out at O R N L to evaluate the stress-strain behavior of the casting at elevated temperatures.  32 This section presents the experimental procedures used to perform the flexural tests and the corresponding results.  4.2.1 Experimental Procedures The specimens used in the flexural tests at O R N L were machined from the same casting as the thermocouple trial discussed in Section 4.1.1. The nominal size of the rectangular specimens is 80 m m by 10 m m by 10 mm.  The test frame for the flexural tests was  comprised of components capable of operating at high temperatures and under either the load-control or displacement-control mode.  The four-point bending tests in this study  were performed using the displacement-control function owing to the need to examine the strain rate sensitive inelastic behavior of the material. The test conditions such as the strain rates or the displacement rates, as well as the temperatures, were selected based on the temperature evolution of the refractory predicted with the thermal model during casting. Figure 4.6 shows the photograph of flexural test apparatus with the  furnace  opened. A s can be seen, the load train consisted of a silicon carbon (SiC) load rod and a S i C support rod. fixtures.  During the experiments, the specimens were supported by the S i C  Figure 4.7 displays the configurations of the four-point bending tests on the  M o n o f r a x - M specimen with the S i C fixtures. Test temperatures were 1100, 1350, 1500, 1600, and 1700°C and test strain rates were l x l O " , l x l O " , l x l O " , 5xl0" , and 2 x l 0 " 4  5  6  7  7  s" . 1  The strain rates that the casting experiences during Stage I cooling are covered by the test conditions whereas for Stage II cooling the strain rates are in the range of l x l O " to 2x10" 7  9  33 s" . 1  The limitation of strain rates for Stage II cooling is due to the flexural testing  machine at O R N L , which is not capable of loading the system below 2xl0" s" . 7  1  For each test, the experimental set-up was arranged as shown in Figures 4.6 and 4.7.  Prior to commencing the test, the furnace  temperatures for 30 to 90 minutes.  was pre-heated  to the  selected  After the temperature was stabilized, the specimen  was loaded at a constant strain rate. The tests were conducted at a constant temperature and were terminated after the maximum force was reached.  Results obtained from the  experiments were the force and corresponding displacement. It should be noticed that the flexural test results also included displacements of the load train. A s a result, flexural tests of S i C bend bar, which was made of the same material as the load train and same size as the M o n o f r a x - M specimens, were also conducted at various test temperatures in order to make corrections to the displacements to reflect the actual displacements of the fused-cast alumina refractories.  4.2.2 Results A total of 28 bending tests were carried out and the initial results were gathered in loaddisplacement format. A s mentioned earlier, the displacements of the flexural tests on the M o n o f r a x - M specimens consist of the displacements of both the testing samples and the S i C load train. Additional bending tests on S i C bend bar were conducted at all the test temperatures to evaluate the displacements of the load train.  Corrections of the  displacements were made using the relationship 8 — §Moruifrax-M  + load train '~ (b\ d train ~ 8$iC bend bar) im  (4.1)  34  where,  8 nofrax-M Mo  + load train  M o n o f r a x - M specimens,  was the displacement obtained from the flexural tests on the  8u, d train a  was the displacement gathered from the flexural tests  on the S i C bend bar, and Ssic bend bar was the load-point displacement based on the elastic deflection of the S i C bend bar according to the following equation  ^ c  f  c  * ^ A ^ - 1 6 « 2Ebn  2  )  (4.2)  where, P is the applied load, a is the distance from the support to the load applicator when the specimen is straight, E is the modulus of the S i C bend bar, b is the specimen width, h is the specimen height, and L is the span length between the lower supports (refer to Figure 4.7) [21]. After the corrected displacements were evaluated using Equation 4.1, stress-strain behavior of M o n o f r a x - M  was calculated based  on the corresponding  load-point  displacement and applied load. The stress and strain calculations shown in the following equations were derived based on simple beam theory assuming that the material is isotropic and homogeneous, the moduli of elasticity in tension and compression are identical, and the material is linearly elastic [21-22] 3 cr •  e  P  2 bh'  =--8 2 a  \L-U)  3L-4a  (4.3)  (4.4)  where, U is the span length between the upper supports, e is the outer-fiber tensile strain and Sis the corrected load-point displacement computed based on Equation 4.1. The resulted stress-strain curves of Monofrax-M at various strain rates and temperatures are displayed in Figures 4.8 - 4.16. A s a comparison, the plots also include  35 the elastic response of the refractory based on the interpolated values of the elastic modulus measured acoustically at various temperatures at United Technologies Research Center [23]. Figures 4.8 to 4.10 show the stress-strain response at 1100°C and indicate that the material exhibits largely elastic behavior up to the maximum stress. This is evident since the slopes on loading are in reasonable agreement with the stress-strain response based on perfectly elastic behavior using the measured elastic modulus.  The departure from  linearity that is observed near the peak stress likely reflects micro-cracking and damage accumulation and not significant plastic or creep deformation.  Hence, the behavior at  1100°C may be assumed to be perfectly elastic up to a failure load or between roughly 12 and 17 M P a . It should be noticed that the unreasonable behavior of Sample 2-8 in Figure 4.10 could be attributed to experimental artifacts such as thermal instability, poor articulation, and wedging. A t higher temperatures, it can be seen from Figures 4.11 to 4.16 that the stressstrain curves deviate from the elastic behavior or linearity well before the maximum stresses have been attained indicating plastic or creep deformation. temperature between  1100 and  Thus, at some  1350°C inelastic deformation mechanisms  become  important. The results for the different strain rates tests at 1350°C indicate little sensitivity to strain rate. However, for the sample tests at 1500°C and above, this is not the case and the stress-strain response is sensitive to the deformation rate. A t higher strain rates, the material is generally 'stiffer' as would be expected.  36  Overall, the stress-strain or constitutive behavior of M o n o f r a x - M over the range of temperatures experienced in the casting process is complex, exhibiting at elevated temperature, strain rate sensitive inelastic behavior. The approach adopted in the stress analysis is to greatly simplify this behavior by assuming that the oc(3-alumina is an elasticplastic material that does not exhibit any strain rate sensitivity. T o prepare that data for input to A B A Q U S (to be discussed later in Chapter 6), the stress-strain data was further processed.  37  2000 1800 1600 _  1400  u o  ^  3  1200  2 IOOO  I" S  H  A  A  • O  i  u  800  Measured  Location  600 A  AD  400  Block Surface (Pyrometer data) Graphite Interface 0.0254 m (1 inch) from Interface  •  _l  200  100  200  I  I  I  300  I  I  i_  _i  400  I  I  I  500  I  '  '  600  700  800  Time (s)  0  1  Figure 4.1 - Thermocouple measurements and pyrometer data from industrial trial for Stage I cooling in the graphite mold.  Monofrax-M  1800 1600  Measured  -++ •+ +  Location 0.0508 0.1016 0.2032 0.2921  + A  1400  •  &  O  +  1200 fcFA  A  (2 inches) from Interface (4 inches) from Interface (8 inches) from Interface (11.5 inches) from Interface  +  K  1000  m m m m  + +  A  +  +  •A  800  A  600 A  •  •XT**  • •  A  400 f ? o Q  m  o  0 tir°. 0  A  -4-  A  + + +  *  A  -*A> A  n •  200  A  D  o O O o o o  0  I I I II 50  (  o o o Oo  o o oo  100  0  Q  ° O n  A  A  A  A  ..A? + +  • • •  o OOOOOO p p o O O  150  200  • • o p o o Oo Oo o 250  300  350  Time (hr)  Figure 4.2 - Thermal response obtained from Monofrax-M industrial trial for Stage II cooling in the alumina ore.  38  Face  Figure 4.3 - Schematic illustration of typical cracks found in M o n o f r a x - M (not to scale).  39  )  )  [3-alumina  Figure 4.4 - Schematic illustration of typical cracks found in M o n o f r a x - M at the horizontal plane just below the header (not to scale).  Distance f r o m the base o f the b l o c k (inches)  30 -I 0  1  1  1  1  1  1  1  0.05  0.1  0.15  0.2  0.25  0.3  0.35  H  0.4  Distance f r o m the base of the b l o c k (m)  Figure 4.5 - Chemical analysis results obtained from Monofrax-M crown block measured at the vertical centerline of the plane bisecting the broad face of the casting.  Figure 4 . 6 - Photograph of flexural apparatus with the furnace opened.  41  Figure 4.7 - Schematic of the configurations of the flexural tests performed at O R N L .  Sample 1-3 Sample  1-10  Sample 2-9 Elastic response at 1100°C  0.0002  0.0004  0.0006  0.0008  0.0012  0.0014  Strain Figure 4.8 - Stress-strain curve of Monofrax-M specimens at 1100°C at a strain rate of l x l O ' s" . 6  1  42  Sample 1-6 Sample 2-5 Sample 2-7 Elastic response at 1100°C  ta  CL.  CO  0.0002  0.0004  0.0006  0.0008  0.0012  0.0014  0.0016  Strain  Figure 4.9 - Stress-strain curve of Monofrax-M specimens at 1100°C at a strain rate of 5 x l 0 " s" . 7  1  18 16 14 h 12 ca CL.  10  CO  -0°.0001  Sample 1-1 Sample 1-9 Sample 2-8 — Elastic response at 1100°C i I i i i i I i i i i I i i i i i i i i i 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 Strain  Figure 4.10 - Stress-strain curve of Monofrax-M specimens at 1100°C at a strain rate of 2xl0" s" . 7  1  43  A^ rsrf3 rPOc.] n a  1  a  A c  cP a  A A  rf ,A J  Pi  a  AA A  *A&  AA  S a m p l e 1-4 S a m p l e 2-3 E l a s t i c r e s p o n s e at 1 3 5 0 ° C J 0.0005  0.001  I  _l  L.  0.0015  I  1  I  0.002  I  I I 0.0025  I  I  I  I  0.003  Strain  Figure 4.11 - Stress-strain curve of M o n o f r a x - M specimens at 1350°C at a strain rate of  lxlO- s"'. 6  18  16  aotn*  14  r  12  pD  cr• •A ^ A A A ^ ^ ^ A A f i n . . . , A  A  A  4A  D  *A A  10  A a  A^  -A' <Z2  A  J  A  4?  i  4A  A  1  if  S a m p l e 1-8  B A i  S a m p l e 2-4  A  i  i  i I 0.0005  _i i  i  I i 0.001  i_  _i  I 0.0015  i i  i  i  E l a s t i c r e s p o n s e at 1 3 5 0 ° C  I i 0.002  i  i  I i i i i_  i 0.0025  0.003  Strain  Figure 4.12 - Stress-strain curve of M o n o f r a x - M specimens at 1350°C at a strain rate of  5xl(X s"'. 7  44  4* BH  S a m p l e 1-5 S a m p l e 2-1 E l a s t i c r e s p o n s e at 1 3 5 0 ° C 0.0015  0.0005  j  0.002  i i  0.0025  i _  0.003  Strain Figure 4.13 - Stress-strain curve of Monofrax-M specimens at 1350°C at a strain rate of 2x10" s" . 7  1  18 S a m p l e 1-7, l x l O ' V 16 H  A  S a m p l e 2-6, l x l O " s"'  o  S a m p l e 3-5, l x l O " s '  5  5  1  S a m p l e 3-8, 2 x l 0 " s"' 7  14  E l a s t i c r e s p o n s e at 1 5 0 0 ° C  12 OH  10  O  n  A  D  A A  at  C  •  A  •A  A A A  :o  °  % o  A o  I  0  I  I  I  I  I  0.001  I  l _  0.002  0.003  0.004  0.005  0.006  0.007  Strain Figure 4.14 - Stress-strain curve of Monofrax-M specimens at 1500°C at various strain rates.  45  18 Sample 2-2, 1x10" s"  1  16  A  Sample 3-1, l x l O " s"  o  Sample 3-7, l x l O " s"'  5  1  6  Elastic response at 1 6 0 0 ° C  14 12 10  •  A  A  O o ° °o  O  0 < >  o  •A.  o  :o  A i i i i I i ' ' i I ' i 0.001 0.002 0.003  i  i I ' i 0.004  A i A  i  LJ  ,-,  n  i A i'-I i ,-t I i i i i I i i 0.005 0.006 0.007  i  i I i i 0.008  i  i I i i i0 i.01 0.009  Strain Figure 4.15 - Stress-strain curve of Monofrax-M specimens at 1600°C at various strain rates. :  -  •  Sample 3-6, l x l O ' V  A  Sample 3-2, l x l O " s" 5  1  Elastic response at 1 7 0 0 ° C  f.  • r-i  C O  •  \ ^ ^ ^ ^ ,  0  1  1  1  ! 1 1 0.001  1  1  1  1  0.002  n  q  A  AA  ^  1  , , i i , 0.003  1  , 'i fcjr;, , 0.004 a  c  LJ  ,  A  AA  A  , i i AA4AAI 0.005 0.006  I I I M I 0.007 0.008  Strain Figure 4.16 - Stress-strain curve of Monofrax-M specimens at 1700°C at various strain rates.  46  CHAPTER 5 THERMAL MODEL Based on the modeling work reviewed in Chapter 2, the finite element method seems best suited to develop a mathematical model capable of predicting the evolution of temperature, stress and strain in a Monofrax-M crown block casting [4-12].  The  approach adopted in the present study is a thermal analysis followed by an un-coupled elastic-plastic stress analysis similar to that applied by Jeanmart and Bouvaist [5] on an aluminum alloy and L u et al. [12] on fused-cast A Z S refractories. The commercial finite element package A B A Q U S has been used to develop the thermal model and the thermalstress (mechanical) model of the crown block. This section focuses on the development of the thermal model, and its validation.  5.7 General Thermal Model  Formulation  Due to the time-dependent nature of the Monofrax-M casting process, the heat transfer model of the Monofrax-M crown block must be transient and must also account for the variation in temperature with time in the molding materials including the graphite mold, the silica annealing sand platform (on which the graphite mold was placed), the alumina ore, and the steel bin. In the thermal model, the governing partial differential equation is shown in Equation 5.1  / ? c ( r ) ^ = v[^(r)vr]+e ot / )  (5.1)  where, p is the density, C is the specific heat, T is the temperature, k is the thermal p  conductivity, and Q is a volumetric source term associated with latent heat evolution.  47 Fluid flow in the liquid during filling of the mold and during the early stages of the casting process has been ignored and heat is assumed to be transferred by diffusion only.  Temperature dependencies in the specific heat and thermal conductivity of the  materials present have been included to the extent to which they are available. The density has been assumed to be temperature independent to conserve mass with a fixed mesh.  Latent heat evolution associated with the solidification of the a/(3 alumina has  been included.  The casting and the molding materials have been assumed to exhibit  isotropic behavior.  5.2 Geometry A s previously mentioned, the geometric aspects of the casting process dictate a threedimensional analysis. In addition, the simulation of the Monofrax-M casting process requires the removal and addition of different mold materials during the various stages of the process. The various changes to geometry and the associated load steps utilized in the model to simulate the casting process are presented in Table 5.1. In Stage I cooling, the model geometry encompasses the crown block, graphite mold and silica annealing sand platform. Four-fold symmetry in heat flow about the vertical center planes bisecting the broad and narrow faces can be assumed during Stage I cooling. Likewise, under ideal conditions, the block would be expected to be centrally located in the bin, and four-fold symmetry in the flow of heat would also be expected in Stage II cooling. The original model developed in the present study took advantage of this symmetry and analyzed only VA section of the crown block, Stage I mold and Stage II annealing bin.  However, numerous runs with the VA section model failed to obtain  48 agreement between the predictions and the measured data taken from thermocouples located in the annealing ore. On close inspection of the bin geometry, it was realized that the crown block was not placed inside the annealing bin symmetrically. A t a height of 0.3556 m (14 inches) up the block face (measured from the block bottom), it was found that there was approximately 0.2921 m (11.5 inches) of alumina ore adjacent to one side of the broad face and roughly 0.4128 m (16.25 inches) of alumina ore on the other. Adjacent to the narrow face, the asymmetry was less, with approximately 0.2616 m (10.3 inches) and 0.2350 m (9.25 inches) of alumina ore on either side of the bin. A s a result, it was necessary to adopt only two-fold symmetry in heat transfer about a vertical center plane bisecting the broad face of the block. Thus, both Stage I and II geometry in the final model adopted only two-fold symmetry. First-order, 8-node brick linear heat transfer elements ( A B A Q U S designation D C 3 D 8 ) were employed in the current research. The resulting three-dimensional mesh employed in Stages I and II contained a total of 45619 elements comprising 54003 nodes. O f these, 6400 elements and 7497 nodes were located in the block. Figure 5.1 shows the M o n o f r a x - M crown block mesh adopted for the thermal model. Figure 5.2 shows the mesh utilized for the graphite mold and the silica annealing sand platform in Stage I cooling and Figure 5.3 shows the mesh used for Stage II cooling in the annealing bin. The inclusion of the steel flasking segments and fins in the Stage II mesh was determined to be necessary as they were determined to have a small but not insignificant effect on block cooling.  49 5.3 Boundary  Conditions  A key aspect in the development of the thermal model is characterization of the thermal boundary conditions, which govern the flow of heat between the various components in the mold and to the surrounding environment. M o v i n g from the crown block outward, heat is transferred between the block and the graphite mold (Stage I) and annealing ore (Stage II) via a combination of contact conduction and radiation. Between the ore and steel flasking, heat is assumed to flow without any interface resistance.  A t the graphite  mold exterior (Stage I) and flasking exterior (Stage II), heat transferred  to  the  surrounding environment by a combination of natural convection and radiation.  5.3.1 Symmetry Boundaries A s previously outlined, conditions of symmetry have been applied with respect to heat flow perpendicular to a vertical plane bisecting the center of the broad face. A l o n g this plane, a zero heat-flow, or adiabatic boundary condition has been applied, as in Equation 5.2  q = ~k — = 0fort>0 dn  (5.2)  where, q is the heat flux in W m " , k is the thermal conductivity in Wm' K'', T is the 2  l  temperature in K , n is the normal to surface direction and t is the time in seconds.  5.3.2 Casting-Mold Boundary A t the interface between the casting and the mold(s) (graphite in Stage I cooling and alumina ore in Stage II cooling), heat transfer is described by a Cauchy type boundary  50 condition.  Across the interface, heat is transferred  conduction and radiation.  by a combination of contact  Initially, contact conduction dominates and the rate of heat  transport is high. A s time proceeds, a gap is formed between the casting and the mold leading to a gradual decrease in contact conduction, a gradual increase in the radiation component and an overall reduction in heat transfer between the two surfaces. Equation (5.3) has been used in A B A Q U S to quantify heat transfer at the interface. Gap Conductance = f h c  In Equation 5.3, f  c  + (1 - f )h c  (5.3)  md  is the fraction of contact between the two surfaces, h is the heat  c  c  transfer coefficient due to contact conduction in W m ' K " ' , and h d is the effective heat 2  m  transfer coefficient due to radiation in W m " K " ' . 2  To account for gap formation, f  c  was calculated based on the  following  expression: f= e  t  —  +  l  (5.4)  ^elapsed  where, t i d is the elapsed time, or duration, of Stage I or II cooling, t is the current total e apse  model time, f  c  = 1 at the beginning of Stage I or II cooling, and f  c  = 0.2 at the end of  Stage I and II cooling. The radiative component of heat transfer, quantified by the effective heat transfer coefficient, h d, was calculated as follows: m  K =oe^—f-  (5-5)  d  where, CT is the Stefan-Boltzmann constant (5.67x10" W m " K " ) , £ is the effective 8  2  4  emissivity, Tj is the temperature of the casting in K e l v i n and T is the temperature of the 2  51 mold in K e l v i n at the location being processed. The effective emissivity is a function of the emissivities of both the casting and the mold as given below in Equation (5.6)  e =—  (5.6)  —+ — - 1  where, £[ is the emissivity of Monofrax-M as a function of temperature and £i is the emissivity of the mold.  5.3.3 M o l d Exterior Boundary Heat transfer from the mold exterior to its surroundings occurs by a combination of natural convection and radiation. The natural convection component is characterized by a Cauchy type boundary condition. The heat transfer coefficient for natural convection calculated for the mold exterior has been determined to be approximately 6 W m ' K ' ' , 2  based on an empirical expression for free-convection for vertical plates [24].  For the  radiation component, a boundary radiation was prescribed in A B A Q U S and the heat flux across the surface of the mold was evaluated as in Equation 5.7 q = <J£(T:-T:)  (5.7)  where, o is the Stefan-Boltzmann constant, 8 is the emissivity of the mold, T is the s  temperature of the casting surface in K e l v i n and T is the ambient temperature in K e l v i n . a  5.4 Initial  Conditions  The initial temperature of the casting was set to 2049°C, the approximate pour stream temperature measured at Monofrax Inc.  The various components of the mold and  52 annealing bin (including the graphite, the silica sand, the alumina ore and the steel bin) were initially assumed to be at a uniform temperature of 25°C.  5.5 Thermo-Physical  Properties  The thermo-physical data needed for Monofrax-M and the molding materials was obtained from various sources including Monofrax Inc., Virginia Polytechnic Institute, Orton Refractory Testing and Research Center, O R N L , and the Metals Handbook [2, 2528] and is summarized in Tables 5.2 to 5.4. Because the behavior of these materials is modeled over a large range of temperature (25 - 2049°C), temperature dependencies must be accounted for in the thermal model. This has been done to the extent possible with the available data. In some cases, to be discussed below, it has been necessary to extrapolate this data beyond the upper limit in the empirical data. The volumetric latent heat of fusion is another thermo-physical data that is required in the heat transfer model when phase transformations are involved.  As  discussed in Chapter 2.5, the liquidus and solidus of Monofrax-M were found to be 2026°C and 2000°C, respectively, assuming equilibrium solidification.  In the thermal  model, latent heat of 1089506 I kg" (111.086 k i mol" , [20]) was released linearly 1  between  the  transformation  start and finish  1  temperatures,  2026°C  and  2000°C,  respectively. In addition, a sensitivity analysis was performed to investigate the effect of releasing the latent heat non-linearly according to a more realistic solidification path.  53 5.6 Thermal Predictions and Comparisons to Measured  Data  During solidification, the Monofrax-M crown block is subject to a broad range of cooling conditions. T o validate the thermal model, thermocouple data has been obtained at Monofrax Inc. for both Stage I and Stage II cooling as discussed in Chapter 4.1.  5.6.1 Stage I - Graphite Cooling Model A base-case F E M thermal model was developed and run prior to conducting an investigation of model sensitivity to variations in some of the key input parameters and formulation assumptions. The following conditions were adopted in the base-case model: •  The heat transfer coefficient due to contact conduction, h  in Equation 5.3, was  c  assumed to be 1000 W m " K " . 2  •  1  The fraction of contact between the refractory and the mold, f , was evaluated as in c  Equation 5.4. •  The Virginia Polytechnic Institute thermal conductivity data has been used and has been linearly extrapolated to the pour temperature.  (Note: as both sets of  measurements in Table 5.2 indicate a general trend to increasing conductivity at elevated temperature (refer to Figure 5.4), the trend in empirical conductivity data was used as basis for extrapolation. Moreover, according to M o n o f r a x - M personnel [2], the thermal conductivity data obtained by laser flash tends to be more reliable than that by calorimeter.) The temperature predictions obtained from the base-case thermal model are presented together with the thermocouple data in Figure 5.5. A s can be seen, the agreement is poor.  One obvious problem with the base-case model is that the model  54 predicts an initial rate of temperature increase that is too high. In the thermal model, the mold is assumed to be filled instantaneously, whereas in the process this takes a finite amount of time. Hence, the model initially over-predicts the heat transfer to the mold at the height of the thermocouples. A second obvious shortcoming of the model is that the peak  temperatures  obtained in the mold are too low. A n extensive parameter investigation with the model encompassing 18 runs, revealed that this problem was related to a failure to transfer enough heat from within the block to the block/mold interface, which is controlled principally by the thermal conductivity of Monofrax-M. A s can be seen in Figure 5.4, the laser flash data is relatively low at the upper range of temperature required in the model whereas the calorimeter data leads to comparatively high conductivities at the same range.  In an attempt to improve model  agreement, the average of the regression lines obtained from both data was adopted. In addition, the conductivity of Monofrax M at 2026°C was increased above the value obtained from extrapolation at 2000°C to take into account the effect of convective heat transport in the liquid pool. This is a common approximation used in the simulation of heat transfer in the liquid phase in casting processes and involves the use of a L i q u i d Conductivity Multiplication Factor ( L C M F ) [29]. The model was also found to be quite sensitive to the parameters quantifying casting/mold interface heat transfer - i.e. the gap conductance defined in Equation 5.3 andf in Equation 5.4.. To 'fine-tune' the thermal model, the two parameters were varied c  in a series of runs. The 'best' results appear to have been achieved with h set to 600 c  W m " K " ' and/ . evaluated based on Equation 5.8 2  6  55  /.=^1« +1  (5.8)  ^elapsed  where, f  c  = 1 at the beginning of Stage I cooling, and f  c  = 0.4 at the end of Stage I  cooling. Figure 5.6 shows the model results obtained with the modified M o n o f r a x - M thermal conductivity with an L C M F of 4 and the modified h andf . A s can be seen from c  (:  the comparison, the results of the thermal model are now in relatively good agreement with the thermocouple data.  The temperature predictions of the block surface are also  improved, but remain too low (assuming the pyrometer data is correct). Finally, a sensitivity analysis was conducted to assess the influence of rate of volumetric latent heat release on the temperature predictions.  In contrast to a linear  release of latent heat, assumed in the base-case model, the phase diagram indicates that approximately 38% of the latent heat should be released from 2026°C to 2002°C and the balance from 2002°C to 2000°C. Figure 5.7 compares the thermocouple measurements with the results obtained from the thermal model using the more correct solidification path. A s can be seen in the figure, the revised model predictions deviate further from the measured data. It is believed that the thermal conductivity could be further adjusted to refit the heat transfer model. However, since accounting for the non-linear heat release doubles the computational time from 19 hours to 40 hours, it was felt that this change to the model was not justified. A s a result, the current thermal model assumes linear release of latent heat.  Future work should be performed to better reflect the nature of the  solidification path of Monofrax-M.  56 5.6.2 Stage II - Alumina Ore Cooling Model During Stage II cooling, the alumina ore represents the major resistance to heat transfer. Thus, factors that influence the conductivity of the ore are likely to have a significant impact on the predictions of the model - such as for example, process variability in the ore composition, different ore sources (suppliers) and extended exposure of the ore to high temperatures near the block face (sintering). Prior to conducting a sensitivity analysis, a base-case model was run to serve as a benchmark for comparison. The base-case model used the thermo-physical properties for system alumina ore with a density of 1270 kg m" (79.3 pcf), which is consistent with the ore used in the instrumented test casting.  In addition, it has been observed that  approximately 0.0381 m (1.5 inches) of ore is 'hard' sintered immediately next to the surface of M o n o f r a x - M crown block and another 0.0381 m (1.5 inches) of ore is 'weakly' or soft sintered adjacent to hard sintered ore (measured at the center of the broad face). Table 5.5 compares the measured thermal conductivity values for the various states of alumina ore. A s can be seen, the thermal conductivity data of both hard and soft sintered ores is much higher than the system ore. Thus, sintering in the bin adjacent to the block face may be expected to have a significant effect on heat transport. Initially, however, no sintering behavior was incorporated in the base-case model, so that the effect of ore sintering could be assessed. Figure 5.8 shows a comparison between the results obtained from the base-case thermal model and the thermocouple measurements during Stage II cooling. The model results were obtained at the corresponding thermocouple locations to within the limits in mesh resolution. A s can be seen, the model predictions were in good agreement with the  57 thermocouple measurements.  The peak temperature predicted at 0.0508 m (2 inches)  from the refractory/ore interface, however, was slightly too low. A n important point to make is that the temperature predictions are very sensitive to thermocouple location.  Figure 5.9 shows the model results obtained at locations  0.0254, 0.0762, 0.1524, and 0.2540 m (1, 3, 6 and 10 inches) from the refractory/ore interface.  A s can be seen, the peak temperature obtained from the thermocouple at  0.0508 m (2 inches) from the interface lies between the predicted peak temperatures at 0.0254 and 0.0508 m (1 and 2 inches) from the interface. Thus, any error in the location of the thermocouple will substantially alter the result. A s errors of the order of ±0.0127 m (±0.5 inches) in the location of the thermocouples could be expected, the fit presented in Figure 5.9 seems reasonable. In view of the sintering occurring in the ore and its effect on conductivity, the next step was to introduce this into the model. The sintering behavior in the ore is introduced into A B A Q U S through a subroutine called ufield in which a field variable,/^, can be defined reflecting the fraction, or degree, to which the ore is sintered.  In the  present model, whenf = 1, the properties of the hard sintered ore are used, w h e n / i = 0.5, s  the properties of the weakly sintered ore are used, and when f = 0, the properties of the s  system alumina ore are used. The fraction of alumina ore sintered is calculated based on the peak temperature the ore experiences at a particular location in the bin. The criteria adopted uses 3 threshold temperatures as given in Equation 5.9  58  [0 i f T < T, + 0.5 i f T  <T<T  lh  S  (5-9) if  T <T<T S  h  [1 i f T > T  h  where, T is the current temperature of the alumina ore (evaluated at each material integration point temperature within an element), T  tn  is the threshold temperature below  which sintering does not occur, T is the temperature above which the alumina ore s  becomes weakly sintered and Th is the temperature above which the alumina ore becomes hard sintered.  Linear interpolation is used for temperatures  between the threshold  temperatures. The temperatures, T, , T , and Th, were determined through trial and error to be n  s  500, 1700 and 1900°C, respectively. Using these values, -0.0381 m (-1.5 inches) of hard sintered ore was predicted to form next to the block surface at the center of the broad face and -0.0381 m (-1.5 inches) of weakly sintered ore was predicted adjacent to the hard sintered ore. Figure 5.10 shows a comparison between the model results using sintering and the thermocouple measurements.  A s can be seen, the agreement for the  peak temperatures at the various thermocouple locations are if anything worse. Figure 5.11 presents the contour plot of the fraction of alumina ore sintered at the end of the casting process.  A s can be seen, there is considerable variability in the  amount  (thickness) of material sintered, as would be expected based on variability in peak temperatures obtained at the various locations within the bin. A sensitivity analysis was also conducted on parameters controlling heat transfer from the exterior steel bin to the environment and across the interface between the  59 refractory and the alumina ore.  Varying these parameters within the model failed to  improve agreement and decrease the peak temperatures predicted by the model. One possible explanation for the excess peak temperature is that the sintering actually occurs over time rather than instantaneously at a given temperature, as was the case in the preliminary sintering model. To explore this, a second parameter,/;, could be introduced into the sintering model to affect a time delay in sintering. Delaying sintering could have the desired effect of initially reducing conductive heat transfer close to the block face resulting in a decrease in the peak temperatures predicted by the model. It has been found, in general, that decreasing the amount of ore sintered had the effect of improving the agreement between the model and the measured temperature response. A s a result, it is observed that although a sintering algorithm was introduced to account for the increase in conductivity due to densification of ore on exposure to elevated temperature in the bin, optimal agreement was achieved with the sintering algorithm switched off. In addition, similar to Stage I cooling, the influence of latent heat on the model predictions was assessed. The results obtained with the Monofrax-M solidification path of 38 and 62 volume percent of a-alumina and (3-alumina, respectively, showed limited effect in Stage II cooling.  Hence, the use of linear release of latent heat is not  unreasonable because the major resistance to heat transfer during Stage II cooling is the thermal conductivity of the alumina ore.  Table 5.1 - Thermal model load steps in the Monofrax-M thermocouple trial. Step #  Simulation Description  Step 1  Remove alumina ore and steel bin. Ramp Step. Stage I Cooling. Remove upper graphite section. Ramp Step. Strip upper graphite section. Remove graphite mold and silica annealing sand platform. Ramp Step. Float (crown block). A d d 0.3048 m (12 inches) of alumina ore on the bin bottom and 1.2192 m (48 inches) of steel flasking segments. Ramp Step. Block remains inside the steel bin with 0.3048 m (12 inches) of alumina ore below it. A d d alumina ore to top of bin to give total height of 1.2192 m (48 inches). Ramp Step. Block remains inside the annealing bin. A d d additional 0.2286 m (9 inches) of alumina ore and 0.3048 m (12 inches) high flasking segment. Ramp Step. Complete Stage II Cooling.  Step Step Step Step  2 3 4 5  Step 6 Step 7  Step 8  Step 9 Step 10 Step 11  Step 12  Note: Elapsed times/duration's for the various load steps have not been included to avoid disclosure of proprietary data.  61  Table 5.2 - Thermo-physical properties of Monofrax-M [25-27]. Material  Temperature (°C)  Thermal Conductivity k (Wm-'K- ) 1  Monofrax-M  Temperature (°C)  Heat Capacity C„ (J k g ' )  From Virginia Polytechnic Institute (using laser 20 6.50 250 5.18 500 3.75 750 3.45 1000 3.57 1250 3.76 1500 4.08 1600 4.30  flash)  From Orton Refractory Testing and Research Center (using calorimeter) 22 787 101 4.49 4.24 942 195 127 394 3.99 206 1020 304 4.03 1100 597 402 1134 797 4.28 4.2 868 518 1177 602 982 4.41 1198 4.44 1220 987 717 1000 4.62 811 1235 1094 4.87 917 1246 4.83 1003 1253 1099 1130 1262 1207 5.57 1324 1275 1208 5.53 1282 1305 6.57 1440 1613 1292  Density p(kgm" ) 3  From ORNL 3598  62  Table 5.3 - Thermo-physical properties of the molding materials [26]. Material  Temperature (°C)  Thermal Conductivity k (Wirf'lC )  Temperature (°C) 17 129 207 304 449 521 631 720 829 1652 0 100 200 300 400 500 600 700 750 2000  838 1048 1257 1383 1508 1592 1676 1718 1760 1760 821 900 969 1027 1075 1111 1137 1152 1155 1264  1690  0 400 800 1200 1600 2000  782 1100 1208 1289 1363 1436  1270  1  Graphite  25 500 1000 1500 2000 2500  110.8 60.3 42.0 31.3 25.9 24.1  Silica Sand  0 100 403 803 1200  1.0 0.48 0.50 0.64 0.87  A l u m i n a Ore  24 400 800.5 1002 1205  0.23 0.29 0.36 0.39 0.37  Heat Capacity C (Jkg')  Density P (kg n-f ) 3  p  1620  63  Table 5.4 - Thermo-physical properties of the steel annealing bin [28]. Temperature (°C)  Thermal Conductivity k (Wra-'K" )  Temperature (°C)  Heat Capacity C„(Jkg')  51.9 51.5 51.0 49.8 48.5 46.4 44.4 43.5 42.7 41.0 38.1 37.7 35.6 33.9 31.8 28.5 25.9 25.9 26.4 26.8 27.2 28.0  50 100 101 150 151 200 201 250 251 300 301 350 351 400 401 450 451 500 501 550 551 600 601 650 651 700 701 750 751 800 801 850 851 900 901 950 951 1000 1001 1050  477.0 477.0 493.7 493.7 510.4 510.4 527.2 527.2 543.9 543.9 564.8 564.8 589.9 589.9 615.0 615.0 648.5 648.5 694.5 694.5 740.6 740.6 778.2 778.2 836.8 836.8 1447.7 1447.7 820.1 820.1 556.5 556.5 535.6 535.6 589.9 589.9 598.3 598.3 606.7 606.7  1  0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050  Temperature (°Q 0 15 50 100 150 • 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050  Density P (kg " ) m  7863 7859 7840 7831 7819 7803 7787 7770 7753 7736 7718 7699 7679 7689 7635 7617 7620 4621 7616 7600 7574 7548 7522  3  64  Table 5.5 - Thermal conductivity of various alumina ores [26]. System Alumina Ore Temp. Thermal Conductivity (°C) (Wrrf'K" ) 0.23 24 400 0.29 800.5 0.36 1002 0.39 1205 0.37 1  Weakly Soft Sintered Ore Temp. Thermal Conductivity (°C) (Wm"'K"') 20 398 798 1100 1202  0.84 0.61 0.55 0.58 0.52  Hard Sintered Ore Temp. Thermal Conductivity (°C) (Wm" K"') l  20 398 798 1100 1202  2.64 1.8 1.8 2.41 2.52  65  Figure 5.2 - Stage I cooling molding materials: graphite and silica annealing sand platform mesh.  66  Plane 3-1: Narrow Face Plane 3-2: Broad Face  Vi Section (Center of Block Facing D o w n to Left)  Figure 5.3 - Stage II cooling molding materials: alumina annealing ore and steel flasking bin mesh. 50  • 45 40  Ui  i  (A) Virginia Polytechnic Institute (laser flash)  * (B) Orton Refractory Testing and Research Center (calorimeter) (A) - extrapolation - Base-Case Model - - (B) - extrapolation  35  Average of regressions of (A) & (B) with an LCMF of 4  30  >  § 25 9  T3  C O  u rt s u a  20 15 10 5  •-^2=r**»0~~  1  a g p ^ j f e - r O r - -r.Hf-  0 500  1000  1500  Temperature (C)  Figure 5.4 - Measured and modified thermal conductivity of M o n o f r a x - M .  2000  67  2000 1800 1600 1400 1200  2  1000  I  .A  , AA A A  &  800 t  A"  „A A" •" n Y-l n A' A ^ ' A /A •  600  1&  1  1  Predicted  100  _i  i  I  200  i  i i  i  I  300  i  '^/n,,..  Measured Location + Block Surface A Graphite Interface a 0.0254 m (1 inch) from Interface  200 00  u^  •a  IP-'  n  400  A A A A  i i  i  Ii i  400  i  i500 Ii  i  i  i  I  600  i  i  i 700 i_  800  Time (s)  Figure 5.5 - Plot comparing thermocouple and pyrometer data with model predictions for the base-case thermal model during Stage I cooling. 2000  1200 h  g 1000 t a  I  H  800 600  200  m,  -rCT  33 400  A  /v—A  A  •A  --A  A  w <  /A  Measured  Location Block Surface Graphite Interface 0.0254 m (1 inch) from Interface  a  ptlf^ I I I I 0  Predicted  100  200  I  I  I  L_  300  _l  1  I  I 1 400  I  L_  _1  500  I  I  I  I  600  I  1_  _l_j_  700  800  Time (s)  Figure 5.6 - Plot comparing thermocouple and pyrometer data with model predictions for the thermal model using modified thermal conductivity of M o n o f r a x - M (an L C M F of 4) and modified casting/mold heat transfer parameters for Stage I cooling.  68  1200  $  ^AAAAAAAAAAAA  S 1000  „ A  I 800 H O,  ..A <  "  A_f  r  A  A  • f  o-  ; • • 5'd'D ^  0  ^  0  '-  r J  '' /DA n  1  A  Predicted  Measured  Location B l o c k Surface Graphite Interface 0.0254 m (1 inch) from Interface  400 200  a  Ll .LJ. - —  13"  /  600  A  A  Is  100  200  300  i 1 iiiiI iiiiI  400  500  600  i  i  i  700  800  Time (s)  Figure 5.7 - Plot comparing thermocouple and pyrometer data with model predictions for the thermal model using modified latent heat to reflect a more correct solidification path of M o n o f r a x - M for Stage I cooling. 1800 Predicted  Measured  1600  Location 0.0508 0.1016 0.2032 0.2921  m m m m  (2 inches) from Interface (4 inches) from Interface (8 inches) from Interface (11.5 inches) from Interface  ^ A £ f > ^ ° O o - r 3D  D a  a a a „ a a  fori, i  o i 3 - L T c r a trn-DTJn i  p -e-e o a-p. .Q-. v.. - ..pffOo"'0'0' o o;o'D-0'«-o«'p'  oooo 50  ,  100  150  ,  p  200  0  ;  |  p  250  e  0 ;e  300  Q  .....  t  350  Time (hr) Figure 5.8 - Plot comparing thermocouple data with model predictions for the base-case thermal model for Stage II cooling.  70  Field Variable F r a c t i o n  Sintered  Plane 3-1: Narrow Face  0.000  Plane 3-2: Broad Face  Vi Section (Center of Ore Facing D o w n to Left)  Figure 5 . 1 1 - Contour plot of fraction of alumina ore sintered.  71  CHAPTER 6 STRESS M O D E L  The heat transfer analysis, presented in Chapter 5, provides part of the input necessary to perform a stress analysis and predict the evolution of stress/strain during the solidification and subsequent cooling of Monofrax-M crown blocks. Unlike the thermal model, which included the influence of the molding materials on the temperature evolution, the effect of the molding materials has been assumed to be negligible in the stress model. This section discusses the development of the Monofrax-M mechanical model and the preliminary stress/strain predictions. The resulting evolution of stress and strain in the refractory during various stages of the casting process is then linked to the occurrence of crack defects.  6.1 General Stress Model  Formulation  The stress model is solved using the differential equations of equilibrium based on a force balance on an elemental volume and the compatibility conditions based on  the  displacement field and its relationship to strain in the body. The F E M equations can be developed through the minimization of virtual work within the element [30].  The  temperature predictions obtained from the thermal model serve as 'thermal loads' input to the stress model in place of mechanical loads normally appearing in a structural analysis.  72 6.2 Formation of /3-Alumina and Void  Distribution  A s previously discussed, Monofrax-M solidifies to form a multi-crystalline a-(3 solid comprised of roughly 38 % by volume of a-alumina and 62 % by volume of P-alumina at 2000°C assuming equilibrium solidification.  In practice, however, the  solidifies with a P-alumina core surrounded by a-(3 shell.  refractory  The P core forms due to  segregation and rejection of Na20, a P stabilizer, at the solid/liquid interface, which accumulates in the center of the casting in the last section to freeze (refer to Figures 1.2 and 1.3 in which the p core can be seen).  According to the study by Cockcroft and  Brimacombe [10-11] on A Z S refractories, the differential thermal dilatation associated with phase transformations can play a critical role on the generation of stresses and strains in castings.  Similarly, for Monofrax-M, differences in the thermal contraction  behavior of the aP-alumina and the P-alumina can be expected to contribute to the generation of stresses and strains. Figure 6.1 shows the thermal expansion (heating) or contraction (cooling) behavior of the two phases.  Adopting a reference temperature of  2 0 0 0 ° C (solidus) for the onset of contraction, it is clear that the ap-alumina contracts significantly more than the P-alumina. The difference in mechanical properties associated with the P-alumina core can be introduced into A B A Q U S through a subroutine called ufield.  In ufield, a field variable,  fvl, is defined at each integration point in the mesh reflecting the phase distribution in the as-cast material. Based on Figures 1.2 and 1.3, the shape of the p-alumiria core is treated as a simplified ellipsoid in the stress model. Moreover, the P-alumina core is assumed to be present throughout the entire casting process.  In the model, when fvl  = 1, the  73  properties of the p-alumina are used and when fvl  = 0.5, the properties of the 50% a-  50%p-alumina are used. The 50% cc-50%P-alumina is selected based on the average of the chemical compositions measured in the industrial trial (discussed in Section 4.1.2.3). Figure 6.2 shows the comparison between the measured p-alumina profile in mole % traversing a vertical line up the center of the casting and profile input to the model based on the simplified distribution. Figure 6.3 shows the P-alumina distribution in the form of a contour plot. In addition to forming a P-alumina core, Monofrax-M crown blocks typically solidify with a substantial amount of shrinkage void in the header. It is not unreasonable to expect that the lower effective elastic modulus or stiffness within the void may influence the predicted stress or strain evolution in the mechanical model. The void has been included in the same afield subroutine in A B A Q U S as the ap-alumina and P~ alumina distributions.  This has entailed defining a second field variable, fv2, in the  subroutine to reflect the void distribution within the casting. The void region, based on Figure 1.2, is simplified as another smaller ellipsoid similar to the P-alumina core and is also assumed to form instantaneously and be present throughout the entire casting process.  In the algorithm, when fvl  = 1, the elastic modulus of the void region is  assumed to be 20 Pa (since numerical solutions may not be achieved with a 0 Pa modulus) and when fv2 = 0, the properties of either the ap-alumina or P-alumina are used, depending on the first field variable, fvl.  The contour plot of the void distribution  used in the stress analysis is presented in Figure 6.4.  74 6.3 Constitutive  Behavior  To explore the influence of constitutive behavior on the predictions of stress and strain, two material models have been employed to simulate the mechanical behavior of Monofrax-M.  The two approaches are presented in two subsections entitled: Elastic  Analysis and Elastic-Plastic Analysis.  6.3.1 Elastic Analysis In the elastic stress model, the elastic strain can be calculated using the elastic modulus and Poisson's ratio of the casting.  Similar to the thermo-physical properties in the  thermal model, a temperature dependent elastic modulus is required due to the large temperature range the casting experiences.  The elastic moduli of ap-alumina and P~  alumina are plotted in Figure 6.5 versus temperature (data was measured at the United Technologies Research Center [23] and was linearly interpolated). Based on the data of a 96% alumina [31], the Poisson's ratio of the a(3-alumina and the P-alumina is assumed to be 0.21. A s can be seen in Figure 6.5, in both cases, the elastic modulus at 2 0 4 9 ° C has been ramped to a low value of 1% of the modulus at 2000°C. The elastic modulus has been drastically reduced to minimize the influence of the liquid on the results of the stress analysis.  Ideally, the liquid would be removed from the analysis as it is incapable of  supporting shear. However, in a fixed domain analysis, such as the one applied here, specifying a small but finite elastic modulus approximates its removal. Based on the literature review, most mechanical analyses of refractory structures adopt elastic material behavior due to the limited ductility associated with ceramics at low temperature and because of ease of implementation. In the present study, an elastic  75 analysis has also been performed to serve as a benchmark against which the influence of adopting plasticity can be assessed.  The shortcoming of this analysis is that inelastic  deformation, which likely takes place in the refractory at elevated temperature, is not considered. Consequently, the evolution of stress predicted by the elastic model w i l l be only qualitative at best. While this does not preclude an attempt to develop correlation between the model predictions and typical cracks found in as-cast blocks, the limitations of the approach should be borne in mind while drawing conclusions.  6.3.2 Elastic-Plastic Analysis Plasticity and creep are the two mathematical constructs commonly used to describe inelastic or permanent deformation in materials. O f the plasticity approaches available, the V o n Mises yield criterion, also called 'Metal Plasticity' (discussed in Chapter 2), is probably the most widely used.  O f the creep equations available, most are based on  steady state creep behavior. A s discussed in Chapter 4.2, stress-strain data was measured at O R N L employing flexural tests.  This data has been adopted for use in the present' model assuming  symmetric tensile and compressive behavior - see the following section. After careful considerations of the options available, an elastic-plastic model was developed for analysis of the Monofrax-M crown blocks. Note: While plasticity can be used to quantify both stresses and inelastic strains  arising  in the material at elevated temperatures, it has certain limitations in application  to the  present problem.  One problem with the elastic-plastic approach is that it fails to address  creep - e.g. continued accumulation  of strain at constant load, a condition that can be  76  approached in casting processes. Since time-dependent deformation or creep in ceramics often takes place at high temperatures, this could result in error. relation  in Equation  deformation.  2.2, for  example,  The power-law  could be used to address  creep  time-dependent  However, its applicability depends on the extent to which steady state flow  conditions prevail. It should be noticed that during the transient stage loading  conditions  that occur in a casting process, deformation associated with primary creep may also be important.  As a result, it is essential to have proper mechanical data measuring both the  primary and secondary creep behavior for a large range of temperatures for M.  Monofrax-  Due to the limited availability of the creep data, an elastic-creep analysis has not  been undertaken in the present research.  6.3.2.1 Manipulation ofORNL  Data for Input to ABAQUS  Plasticity Model  To prepare the data for input to the stress model in A B A Q U S , the stress-strain data presented in Section 4.2.2 was further processed to evaluate separately the plastic strain from the total strain.  This was done by subtracting the elastic strain, based on the  measured elastic moduli, from the total strain obtained from the flexural tests. Since the results for the different strain rates tests at 1100 and 1350°C indicate little sensitivity to strain rate, the average of the measured data were adopted. Due to the limited number of tests performed, data at the low strain rate (2xl0" s' ) for temperatures of 1600°C and 7  higher was not available.  1  A n attempt to fit a simple plasticity hardening law (the  Ramberg-Osgood relationship in Equation 2.1) was made to allow extrapolation to elevated temperatures. Unfortunately, the k and n constants evaluated at each strain rate and test temperature could not be correlated in a logical and consistent manner.  As a  77 result, the data at l x l O " s" , which was the only data available at a consistent strain rate at 5  1  all the test temperatures, was employed to generate the data for input to the stress model. Thus, strain rate sensitivity could not be addressed. Figure 6.6 summarizes the resulting stress-plastic strain curves input to the model. It should be noticed that the material was assumed to be elastic-perfectly plastic at 1100°C.  6.4 Geometry The same 3-dimensional Monofrax-M refractory casting mesh utilized in the thermal model, shown in Figure 5.1, was employed in the stress analysis. The mesh consists of 8node linear brick stress/displacement elements available within A B A Q U S .  6.5 Boundary  Conditions  A s previously discussed, the stress analysis has been limited to the fused-cast refractory. Since interaction with the surrounding molding materials has been neglected, only the symmetry boundary needs quantification.  The symmetry boundary used in the stress  model follows directly from the assumption of symmetry in the heat transfer model. On the vertical plane which bisects the broad face of the casting, displacement in the direction normal to the plane of symmetry has been suppressed, as the thermal loads are assumed to be equal and opposite.  6.6 Predictions  Obtained from the Elastic Stress Model  In the elastic stress analysis, the stress evolution predicted by the model can be observed via plots of stress versus time or contour plots of the stress distribution in the casting.  78 Referring to the coordinate system defined in Figure 1.1, the three normal components of stress in the 1,2 and 3 directions and three components of strain can be output from A B A Q U S , together with other quantities such as the principal stresses and V o n Mises stress, if desired.  One of the challenges in interpretation is selecting the appropriate  output so as to avoid being overwhelmed by the large volume of data available for output. Fortunately, as previously discussed, tensile stresses acting parallel to the broad face, S22, are consistent with the orientation of the cracks that have been found in the crown blocks.  Interpretation of the data is thus focused on the tensile stresses and strains in  direction-2. Further, based on the P-alumina distribution and the location of the cracks formed - refer to Figures 1.2 and 1.3 - a detailed analysis of the data can be limited to only a few nodes to further reduce the volume of data.  Figure 6.7 shows the three  selected nodes (A, B , and C) located in the Monofrax-M crown block. A s can be seen, node A is at the transition in phase from P to oq3-alumina, whereas, nodes B and C are in the a(3-alumina region.  Furthermore, comparisons are also made with the measured  modulus of rupture ( M O R ) of Monofrax-M to evaluate the severity of the tensile stresses predicted by the model. To determine the influence of the P-alumina formation and the void distribution on the stress evolution during the casting process, three stress analyses have been completed: one, based on a homogeneous aP-alumina casting, a second including the palumina core formation, and a third with both the p-alumina formation and the void distribution. Further, the preliminary sensitivity analysis was limited to elastic material behavior only, to avoid the complication of plastic strain generation.  79  6.6.1 Benchmark Case - ap-Alumina Crown  A n elastic stress analysis performed based on the properties of aP-alumina only serves as a benchmark case. The stress evolution developed in this model is mainly driven by the temperature gradient predicted by the thermal analysis.  When the refractory  has  completely solidified and cooled, there should be no residual stresses" remaining at the end of the casting process. During Stage I cooling, the shell of the casting has solidified while the core remains molten. Since the core of the refractory is liquid, it is in a near zero state of stress. The resulting force balance on the shell results in the outer cooler shell in tension and inner hotter shell in compression.  The S22 stress (parallel to the  broad face) is illustrated in Figure 6.8. The large tensile stress on the base edge of the block due to high heat extraction can also be observed in the figure. Figure 6.9 shows the stress evolution at discrete points within the refractory during Stage I cooling, and for the early stages of Stage II cooling in the alumina ore. The temperature evolution is also included in the plot for comparison. A s can be seen, the stresses at the surface  node C are tensile and reach a peak magnitude of  approximately 425 M P a at roughly 800 seconds. A t the location of nodes A and B , the material is in a state of compression and peaks between 800 and 900 seconds. The stress at 0.1524 m (6 inches) from the block center (node B ) is higher in magnitude than the stress at 0.0762 m (3 inches) from the block center (node A ) consistent with a decrease in stress toward the block center.  It should be noted that the measured M O R for this  refractory is approximately 26 and 23 M P a at room temperature and 1510°C, respectively [2]. The unrealistic high tension and compression stresses obtained in this analysis are due to the assumption of pure elastic behavior of the material.  80 Figure 6.10 shows the stress evolution within the refractory during Stage II cooling.  A s can be seen, the S22 stresses rapidly decrease during the early stages of  Stage II cooling, which is a result of thermal rebounding of the block surface temperature - see also Figure 6.9.  The 'plateau' in temperature at nodes A , B and C , indicates  continued latent heat evolution in the block associated with solidification.  Shortly after  solidification is completed, roughly at 20 hours of cooling, the stresses at nodes A , B and C rise to a second peak associated with the completion of solidification and the increase in stiffness in the center of the block.  A s cooling in Stage II proceeds, the stresses  gradually moderate approaching zero as the block cools toward room temperature and gradients decrease.  6.6.2 Effect of (3-Alumina Core Formation  To assess the influence of the P-alumina formation on the stress development within Monofrax-M crown block, the model was run with the P-alumina core - refer to Section 6.2. Unlike the previous analysis, both the temperature gradient as well as the different thermal dilatational behavior of aP-alumina and p-alumina drive the stress evolution in the case of the refractory with the p-alumina core. The contour plot of S22 at the end of the casting process, shown in Figure 6.11, clearly illustrates this as it shows the Palumina core in the high state of compression.  Compression in the p-alumina zone  develops because it has lower thermal contraction than the aP-alumina. Figure 6.12 shows the stress evolution obtained from the model during Stage I cooling. Comparing the results with the previous analysis, it can be observed that the stress development follows similar trend during Stage I cooling. Figure 6.13 presents the  81  Stage II cooling stress profile obtained from the analysis incorporating the P-alumina distribution.  A s can be seen, during Stage II cooling, the P-alumina formation has a  pronounced effect on the stress evolution predicted by the model. Since the p-alumina core contracts less than the aP-alumina, the material at node A is subjected to increasing compressive stresses as the refractory block cools. Similarly, nodes B and C , which are composed of ap-alumina, increase in tension toward the end of Stage II cooling. Clearly, the P-alumina core has a major influence on the generation of tensile hoop stresses (S22) in the crown blocks and likely also plays a role in the generation of cracks.  6.6.3 Effect of p-Alumina Core Formation and V o i d Distribution  In addition to the p-alumina formation, the void distribution discussed i n Section 6.2 was also introduced into the model and run to evaluate the effect of the void distribution on the model predictions. The contour plots of S22 for the whole casting (with header) and the crown block (header removed) at the end of Stage II cooling are shown in Figures 6.14 and 6.15, respectively. The void region, which has a reduced capacity to support any thermal load can be observed in Figure 6.14. A s shown in Figure 6.15, high tensile stresses have developed adjacent to the ap/p-alumina interface. This high tensile region correlates well with the occurrence of cracks. Figures 6.16 and 6.17 present the stress evolution of Monofrax-M during Stages I and II cooling, respectively.  A s can be seen from both Figures, the stress profiles  predicted by the model including the void are slightly modified (the magnitude of the stresses are smaller) when comparing to stresses predicted without consideration of the  82 void, Figures 6.12 and 6.13. This indicates that the stress distribution predicted by the model is only slightly sensitive to the void distribution.  6.7 Preliminary  Predictions  Obtained from the Elastic-Plastic  Stress Model  Unlike the previous three elastic stress analyses, permanent or inelastic deformation is considered in the plastic stress model, which would occur predominantly at elevated temperatures in Monofrax-M.  Both stresses and plastic strains developed in the casting  process can be quantified in the plastic stress analysis. Since the cracks were found to be intergranular and formed at elevated temperature (refer to Section 4.1.2.3), the predicted plastic strain, in addition to the stress predictions, can be correlated against cracks in order to propose mechanisms for crack formation in Monofrax-M.  Similarly, the  interpretation of the results is focused on the stress and plastic strain in the direction-2, adjacent to the center of the broad face of the block consistent with the orientation and location of production cracks. The drawback to the present plastic stress analysis is the lack of rate sensitivity in the plasticity data employed in the model - recall only the data for l x l O "  5  s"  1  was  employed in the elastic-plastic stress analysis. Figures 6.18 and 6.19 show the absolute values of the predicted plastic strain rate for Stages I and II cooling, respectively.  The  temperature evolution has been included in the plots for comparison. A s can be seen, the l x l O " s" rate used in the model is applicable only at the surface of the block during 5  1  Stage I cooling whereas it is too high for application to the material in the center. Moreover, looking at Stage II cooling, Figure 6.19, it is somewhere between 4 and 5 orders of magnitude too high.  Consequently, the results of the elastic-plastic model  83 during Stage I cooling are probably not unreasonable. However, during Stage II cooling, the plastic strain may well be significantly under-predicted by the model - e.g. the constitutive model used will be too stiff as it is valid at a much higher strain rate. The contour plot of P E 2 2 (plastic strain in the direction-2) predicted by the plastic stress model incorporating the (3-alumina core and the void distribution at the end of the casting process is shown in Figure 6.20. Similar to the previous elastic stress predictions, the (3-alumina region, which is under compression due to thermal dilatational behavior of the two phases, can be seen in the plot as a region that has accumulated significant compressive plastic strain.  Also shown in the plot, is the region below the (3-alumina  zone in which high tensile strains can be observed consistent with the formation of cracks found in that location. The evolution of stress and temperature predicted by the model for the refractory during Stage I cooling are shown in Figure 6.21. Comparing with the previous elastic stress predictions, the results obtained by the plastic stress model are more realistic as the values are slightly below the M O R of Monofrax-M.  A s shown in the figure, during  graphite and air cooling, tensile stresses gradually develop on the surface of the casting. However, there is a peak in tensile stress at the block surface (location C) associated with the rapid drop in temperature caused by accelerated heat transfer.  The temperature then  rebounds - e.g. rapidly heats up. This rebound in temperature results in a change in state of stress at the surface of the refractory from tensile to compressive. Likewise, node B shifts from compression to tension. This sub-surface tension has the potential to give rise to the formation of subsurface cracks particularly i f the inner, hotter material has less ductility.  84 Figure 6.22 shows the stress predictions for the crown during Stage II cooling. A s can be seen, there is once again a stress reversal in that the surface and interior nodes are predicted to change stress states from compressive back to tensile (at the surface) and from tensile to compressive (subsurface).  This behavior is due to both a moderation in  temperature gradients experienced by the casting as well as the differential thermal contraction behavior of the a(3-alumina and the P-alumina.  A s the rate of the cooling  decreases, the stresses at both the center and the surface related to temperature gradient decrease.  However, the stresses associated with the p-alumina core (compressive in the  p-alumina region and tensile in the aP-alumina region) increase gradually placing the interior of the crown in compression and exterior in tension. A s discussed, the strain is an important parameter that may also be related to fracture or tearing in the fused-cast refractory. The plastic strain evolution predicted by the model during Stage I cooling is shown in Figure 6.23 together with the temperature variation. A s can been seen, high tensile strain develops at the block surface. The strain at which damage begins to occur in Monofrax-M (the critical strain at which the peak stress is reached) observed from the flexural tests at 1350 and 1500°C (refer to Figures 4.11 to 4.14) is approximately 0.0015 to 0.002. A s a result, the surface of the casting is susceptible to the formation of damage during Stage I cooling. Figure 6.24 shows the plastic strain evolution of the refractory during Stage II cooling. A s can be seen, the strains developed at node B change from compression to tension. The strains at nodes B and C are close to the critical strain values obtained from the flexural tests. Bearing in mind that the strains are likely underestimated during Stage II cooling because of the use of the l x l O " s" data, this suggests continued potential for 5  1  85 damage in Stage II cooling throughout the outer regions of the block extending outward from the (3-alumina core. Based on the stress/strain predictions, it is clear that the (3-alumina core plays a major role in the generation of tensile stresses. The current process has the potential to generate large subsurface tensile stresses.  Overall, it is likely that crack initiates  subsurface and propagates outwards. However, without proper accounting for the effect of strain rate on the constitutive behavior of this material, the exact direction of propagation is unclear.  86  -2 -I 0  1  1  1  1  1  1  1  1  1  1  200  400  600  800  1000  1200  1400  1600  1800  2000  Temperature (°C)  Figure 6.1 - Thermal expansion / contraction behavior of aP-alumina and p-alumina. Distance from the base of the block (inches)  30  -I  1  1  1  1  1  1  1  1  1  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35 '  0.4  0.45  Distance from the base of the block (m)  Figure 6.2 - Comparison of model input and chemical analysis results obtained from Monofrax-M crown block measured at the vertical centerline of the plane bisecting the broad face of the casting.  87  m mm  Plane 3-1: Narrow Face Plane 3-2: Broad Face  VA Section (Center of B l o c k Facing D o w n to Left and Right)  Figure 6.3 - Contour plot o f the aP-alumina and the P-alumina distributions used in the stress model.  m 1  fv 2 — u.uuu  — 0.0769 -  0.154  -  0.231  - 0.308 - 0.385 -0.462 —  0.538 0.615 0.692  Plane 3-1: Narrow Face Plane 3-2: Broad Face  0.769 - 0.846 — 0.923 X . uuu  A Section (Center of B l o c k Facing D o w n to Left and Right) L  Figure 6.4 - Contour plot of the void distribution used in the stress model.  88 250 »  5 0 % alpha - 5 0 % beta-alumina  - o - - beta-alumina  200 +  P. O 150  •~-  •••  s  •  a  -  •  i  O  .a  ioo +  i  VI  5  1  i  i i  50  0  1  1  1  200  400  600  1,,  , ,  800  ,  1000  1—J9  1  1  1  1  1200  1400  1600  1800  2000  2200  Temperature (°C) Figure 6.5 - Variation of elastic modulus with temperature for 50% a-50% P-alumina and P-alumina.  0.0005  0.001  0.0015  0.002  0.0025  0.003  Plastic Strain Figure 6.6 - Stress-plastic strain curves of Monofrax-M at 1x10" A B A Q U S plasticity model.  s"  for input to  89  Plane 3-1: Narrow Face  •HP i mm:  Plane 3-2: Broad Face  Yi Section (Center of B l o c k Facing D o w n to Left)  Figure 6.7 - Contour plot of the P-alumina distribution in the M o n o f r a x - M crown block showing the selected nodes for interpretation.  1  Plane 3-1: Narrow Face  Plane 3-2: Broad Face  Vi Section (Center of B l o c k Facing D o w n to Left)  :::::  Figure 6.8 - Contour plot of S22 for Monofrax-M at the end of Stage I cooling for the elastic stress analysis with a homogeneous aP-alumina casting.  90  500 2000 400 +  300  «  PM  1500  200 +  a  100  o -m  + IOOO 2  •A-  100  =fc  ±  2lro~---aoji__  a. E 01  400  -•A-  500-  IOOO  -100 +  A  Stress  Temperature  500  Location  A [ 0 . 0 7 6 2 m (3 i n c h e s ) f r o m C e n t e r ]  -200  B [ 0 . 1 5 2 4 m (6 i n c h e s ) f r o m C e n t e r ] C [Surface]  -300  Time (s)  Figure 6.9 - Stress evolution of Monofrax-M during Stage I cooling for the elastic stress analysis with a homogeneous od3-alumina casting. 500 S tress  400 +  Temperature  Location  + 2000  A [ 0 . 0 7 6 2 m (3 i n c h e s ) f r o m C e n t e r ] — A -  B [ 0 . 1 5 2 4 m (6 i n c h e s ) f r o m C e n t e r ] C [Surface]  300  PM  WJ  1500  U  200 +  o  100 4-  4- 1000  a E Ol  y.Y.V.>l.».V.V.V.V.V|».V.».V«  W3  C3 Ul  250 -100 +  300  350  H  500  -200  -300  Time (hr)  Figure 6.10 - Stress evolution of Monofrax-M during Stage II cooling for the elastic stress analysis with a homogeneous aP-alumina casting.  91  1 1P  Plane 3-1: Narrow Face  li:  IS  psig^i'iiiii  1  219 249 280  3  Plane 3-2: Broad Face  Vz Section (Center o f Block Facing D o w n to Left)  Figure 6.11 - Contour plot o f S22 for Monofrax-M at the end of Stage II cooling for the elastic stress analysis with p-alumina core formation. 500  4- 2000 400 + 300 +  I N  + 1500  V.--  200 +  •-  3  .&  100 +  IOOO 2  a E V  •4-  100  200-—-3.Q0_  400  500  600  700  800  -100 4-  +  Stress  -200  900  Temperature  lopo -+50I  Location  A [0.0762 m (3 inches) f r o m Center] — A  —  B [0.1524 m (6 inches) f r o m Center] C [Surface]  -300  Time (s) Figure 6.12 - Stress evolution of Monofrax-M during Stage I cooling for the elastic stress analysis with P-alumina core formation.  92  500  Time (hr) Figure 6.13 - Stress evolution of Monofrax-M during Stage II cooling for the elastic stress analysis with P-alumina formation.  •t • i  •i • r  Li  Plane 3-1: Narrow Face Plane 3-2: Broad Face  Yi Section (Center of B l o c k Facing D o w n to Left)  Figure 6.14 - Contour plot of S22 for Monofrax-M at the end of Stage II cooling for the elastic stress analysis with P-alumina core formation and void distribution.  93  Plane 3-1: Narrow Face  mil  ••fiiilSl!!  imi  Plane 3-2: Broad Face  V2 Section (Center of B l o c k Facing D o w n to Left)  um  Figure 6.15 - Contour plot of S22 for the Monofrax-M crown block at the end of Stage II cooling for the elastic stress analysis with P-alumina core formation and void distribution. 500  400  +  2000  +  1500  +  3 0 0 4-  «  <n  V,  2 0 0 4-  100  a  + +  1000  fa  0 -K*-  II  100  200-  .3.00  400  500  600  700  800  - 1 0 0 4-  4-  900  . Stress  -200  41  a E  in  Temperature  .  H  iopo _4-  500  Location  A [0.0762 m (3 inches) f r o m Center]  —A —  B [0.1524 m (6 inches) f r o m Center] C [Surface]  -300  Time (s) Figure 6.16 - Stress evolution of Monofrax-M during Stage I cooling for the elastic stress analysis with P-alumina core formation and void distribution.  94  500  Stress -A —  400  Temperature Location + A [0.0762 m (3 inches) from Center] B [0.1524 m (6 inches) from Center] C [Surface]  2000  300  1500  rt &. § 200  U  [»  g  IOOO 2  a. £  100 +  bl <Z3  H  . .KXXXX-X-X->>&X^-X-XXX-XX.X-XX-»<-XX-X-X-XX-XXX-X-X  •x-x x-x-x  X  •4-  HA A  ^^Tso  200  A-'f  250  350500  300  -100  -200  Time (hr) Figure 6.17 - Stress evolution of Monofrax-M during Stage II cooling for the elastic stress analysis with p-alumina core formation and void distribution. 2100 d»  ~\ 100  200  300  400  500  600  700  800  900  IOOO  + 1600  u + 1100 P ,  - A [0.0762 m (3 inches) from Center] - B [0.1524 m (6 inches) from Center] -- C [Surface] Model Input Time (s) Figure 6.18 - Plastic strain rate evolution obtained from the elastic-plastic stress model of M o n o f r a x - M during Stage I cooling.  95  -t-  100  Rate  150  200  H  1—  250  300  2100 350  Temperature Location A [0.0762 m (3 inches) from Center] B [0.1524 m (6 inches) from Center] + 1600 C [Surface] Model Input U  + 1100 °_  •3 cs hi u Q.  + 600  | H  -400  Time (hr) Figure 6.19 - Plastic strain rate evolution obtained from the elastic-plastic stress model of M o n o f r a x - M during Stage II cooling.  -0.00465  Plane 3-1: Narrow Face  ' -0.00400 -0.00336 ' -0.00271 -0.00206 -0.00141  Plane 3-2: Broad Face  -0.000765 -0.000117 +0.000530 +0.00118 +0.00183 +0.00247 +0.00312 +0.00377  Vi Section (Center of B l o c k Facing D o w n to Left)  Figure 6.20 - Contour plot of P E 2 2 obtained from the elastic-plastic stress model o f M o n o f r a x - M at the end of Stage II cooling.  96  20 2000 15 10  1500  u  -A-—A-  o  4)  u  o trjO--- 20Q. A  l!  300  400  9  +  9-U-  -«  500  600  800  900  IOOO 2  qoo  Cu  E <u H + 500  Stress -15  - A  —  Location Temperature A [0.0762 m (3 inches) from Center] B [0.1524 m (6 inches) from Center] C [Surface]  -20  Time (s) Figure 6.21 - Stress evolution obtained from the elastic-plastic Monofrax-M during Stage I cooling.  stress model of  Time (hr) Figure 6.22 - Stress evolution obtained from the elastic-plastic Monofrax-M during Stage II cooling.  stress model of  97  0.005 SSXXX^  + 2000  0.004  1500 <M 0.003 U  W BH  2  0.002  IOOO 2 CU  u  Plastic Strain  J5  Temperature  c CU  Location  A [0.0762 m (3 inches) f r o m Center]  S o.ooi  H  B [0.1524 m (6 inches) from Center] C [Surface]  500  X —9 6—0 " " ^ - - A - A —  —A—A-A--A-A  -AA--A---A-  A-  ;  +  -0.001 0  100  200  300  400  500  600  0 700  800  900  1000  Time (s) Figure 6.23 - Plastic strain evolution obtained from the elastic-plastic stress model of M o n o f r a x - M during Stage I cooling. 0.005 Plastic Strain  Temperature  Location  + 2000  A [0.0762 m (3 inches) from Center] B [0.1524 m (6 inches) from Center] C [Surface] + 1500  CU u 2  1000  _ ^ A - A - A - A - A - A - A - A - A - A - * ^ * .X-XX-X  X - X - X X X X - X X K X X - X K - X K X X X - X - X  2  Ol CM  E  CU  H 4- 500  -0.001 50  100  150  200  250  300  350  Time (hr) Figure 6.24 - Plastic strain evolution obtained from the elastic-plastic stress model of M o n o f r a x - M during Stage II cooling.  98  CHAPTER 7 SUMMARY AND CONCLUSIONS The present study has focused on developing mathematical models capable of predicting the temperature, stress and strain evolution in Monofrax-M crown blocks during the casting process. The finite element method has been adopted and utilized to analyze the manufacturing process. Industrial thermocouple and pyrometer measurements have been obtained to validate the heat transfer model. Experiments have also been performed to characterize the inelastic behavior of the fused-cast ap-alumina refractory at elevated temperature. The commercial finite element code A B A Q U S was employed to develop an uncoupled thermal stress model.  The thermal model was 'fine-tuned' and validated  against the industrial thermocouple and pyrometer measurements obtained at Monofrax Inc. This has been accomplished based on a comparison between the predictions of the model and thermocouple data obtained at several locations from within the graphite mold, during Stage I cooling, and in the alumina annealing ore, during Stage II cooling. The 'fine-tuning' involved adjustment  of the parameters that describe the thermal  boundary conditions between the refractory and the molding materials.  In Stage I  cooling, it has been necessary to adopt a liquid conductivity multiplication factor in the thermal conductivity of the refractory to account for convective heat transport in the liquid. A sintering algorithm was also introduced in Stage II cooling to account for the increase in the thermal conductivity of the alumina annealing ore due to densification of ore on exposure to elevated temperature in the bin. However, optimal agreement was achieved with the sintering algorithm switched off.  99 The temperature predictions obtained from the thermal model were employed as input to the stress model. The a(3/(3-alumina distribution as well as the void distribution were introduced into the A B A Q U S stress model through a user subroutine. Elastic and elastic-plastic stress analyses were performed to investigate the stress and strain evolution developed during the casting process. The preliminary results predicted by the elasticplastic stress model were more realistic than those obtained from the elastic analyses. Due to the limited available stress-strain data, the influence of strain rate dependent plasticity has not been investigated.  Hence, the results of the plastic stress analysis  should be considered only semi-qualitative. The plastic strain obtained employing data generated at a strain rate of l x l O " s" may be under-predicted during Stage II cooling. 5  1  Nevertheless, the predictions of the model have been used to propose a mechanism for formation of the intergranular crack found in the Monofrax-M crown blocks. The results of the stress analysis indicate that the temperature gradient that the casting experiences and the different dilatational behavior of the ocpVp-alumina are the main drivers of the stress/strain evolution, whereas the void distribution appears to have only a small effect.  The P-alumina core, in particular, plays an important role in the  generation of tensile stresses and likely gives rise to the generation of cracks. However, some cracking may also be associated with some of the processing steps. Based on the high tensile stress/strain developed within the refractory, it is likely that crack initiates subsurface and propagates outwards. Overall, the results of the present study show the importance and usefulness of developing the ability to predict stress and strain evolution in the fused-cast aP-alumina  100  refractories casting process. Based on an understanding of the mechanism of crack defect formation, procedures may be developed to reduce the likelihood of crack formation.  7.1 Recommendations for Future Work Owing to the computational efficiency, the latent heat of Monofrax-M implemented in the present model is simplified such that the volumetric latent heat is released linearly over the liquidus and solidus of the refractory.  Work should be undertaken to more  correctly account for primary alpha and peritectic beta formation. The (3-alumina core distribution incorporated in the stress model was shown to be important in the stress and strain predictions.  The (3-alumina core, however, was  assumed to be simplified ellipsoid and be present throughout the entire casting process. In reality, the P-alumina core is formed gradually when Na20 is being rejected at the ap/p-alumina interface.  A n algorithm based on mass diffusion should be implemented  into the thermal model to attempt to predict its formation. Finally, further investigation into time-dependent  plasticity/creep relations is  essential. It is likely that the rate-dependent plasticity has an influence on the inelastic deformation during both Stages I and II cooling. Also, creep tends to occur in the fusedcast  refractory  at elevated  temperatures.  Both  primary  and secondary  creep  characteristics should be assessed over a large temperature range that the casting experiences and implemented into the model to enhance the capability of the stress analysis.  101  Bibliography 1.  Monofrax Inc. Engineered Materials Product Advertisement Brochure.  2.  Private conversations with Monofrax Inc. personnel, 1997-1999.  3.  A . E . M c H a l e , R . S . Roth, eds. Phase Equilibria American Ceramic Society, 1996.  4.  J . K . Brimacombe, I.V. Samarasekera, S.L. Cockcroft. "Computer Simulation of Solidification and Casting Processes." International Conference on Computerassisted Materials Design and Process Simulation, (1993).  5.  P. Jeanmart, J. Bouvaist. "Finite Element Calculation and Measurement of thermal stresses in quenched plates of high-strength 7075 aluminium alloy." Materials Science and Technology, 1 (October 1985): 765-769.  6.  I . W . Wiese, I . A . Dantzig. "Modeling Stress Development during Solidification of Gray Iron Castings." Met. Trans., 2 1 A (1990): 489-497.  7.  F. Bradley, A . C . D . Chaklader, A . Mitchell. "Thermal Stress Fracture of Refractory Lining Components; Part I, Thermoelastic Analysis." Metall. Trans. B, 18 (1987): 355-363.  8.  W . S . Chang, C E . Knight, D . P . H . Hasselman, R . G . Mitchiner. "Analysis of Thermal Stress Failure of Segmented Thick-Walled Refractory Structures." I. Am.Ceram.Soc, 66, no. 10 (1983):708-713.  9.  I. Knauder, R. Rathner. "Thermomechanical Analysis of Basic Refractories in a Bottom B l o w i n g Converter." Rader Rundschau, 4 (December 1990): 354-365.  10.  S.L. Cockcroft, I . K . Brimacombe. "Thermal Stress Analysis of Fused-Cast A Z S Refractories during Production: Part I, Industrial Study." / . Am. Ceram. Soc, 11, no. 6 (1994): 1505-1511.  11.  S.L. Cockcroft, I . K . Brimacombe. "Thermal Stress Analysis of Fused-Cast A Z S Refractories during Production: Part II, Development of Thermo-elastic Stress M o d e l . " / . Am. Ceram. Soc, 11, no. 6 (1994): 1512-1521.  12.  T J . L u , A . G . Evans, I.W. Hutchinson, G . V . Srinivasan, S . M . Winder. "Stress and Strain Evolution in Cast Refractory Blocks during Cooling." J. Am. Ceram. Soc, 81, no. 4 (1998): 917-925.  13.  T.J. Wang. "Modelling of the Cooling of the Fused Cast a, P - A 1 0 Refractory Mouldings." Glass Technology, 40, no. 1 (February 1999): 33-40.  Diagrams.  V o l . 12, F i g . 9883,  2  3  the  102  14.  A . M o , E . J . H o l m . "On the Use of Constitutive Internal Variable Equations for Thermal Stress Predictions in A l u m i n u m Casting." Journal of Thermal Stresses (USA), 14, no. 4 (1991): 571-587.  15.  J . B . Wachtman. Mechanical 1996.  Properties  16.  N . E . Dowling. Mechanical International, Inc., 1999.  Behavior  17.  D . W . Richerson. Modern Ceramic Engineering,  18.  R . G . Munro. "Evaluated Material Properties for a Sintered a - A l u m i n a . " J. Am. Ceram. Soc, 80, no. 8 (1997): 1919-1928.  19.  A . A . Wereszczak, T.P. Kirkland, G . A . Pecoraro, R . A . New. "Compressive Creep Behavior of Fusion-Cast Alumina Refractories." Advances in the Fusion and Processing of Glass, Transactions of the American Ceramic Society, (1997): in press.  20.  JANAF thermochemical  21.  J . M . Gere, S.P. Timoshenko. Mechanics Company, 1997.  22.  Annual Book of ASTM Standards, C l 161. American Society for Testing and Materials, 1994.  23.  United Technologies Research Center, Unpublished, 1997-1998.  24.  F. Kreith, W . Z . Black. Basic Heat Transfer. Harper & R o w Publishers, 1980.  25.  Materials Science and Engineering Department, Virginia Polytechnic Institute, Unpublished, 1998.  26.  Orton Refractory Testing and Research Center, Unpublished, 1998-1999.  27.  Oak Ridge National Laboratory, Unpublished, March, 1999.  28.  Metals Handbook, V o l . 1, 10 ed. A S M International, 1990.  29.  J . - M . Drezet, M . Rappaz. "Modeling of Ingot Distortions During Direct C h i l l Casting of A l u m i n u m A l l o y s . " Metallurgical and Materials Transactions A, 2 7 A (1996): 3214-3225.  tables. 3  rd  of Ceramics, John Wiley & Sons, Inc.,  of Materials,  2  n d  ed.  Prentice  Hall  Marcel Dekker, Inc., 1992.  ed. U . S . National Bureau of Standards, 1985. of Materials,  4  th  ed. P W S Publishing  th  103  O . C . Zienkiewicz, R . L . Taylor. The Finite Element Method, 4 Book C o . , 1994.  ed. M c G r a w - H i l l  "Alumina, alpha A1203, 96%," in MatWeb: The Online Materials Information Resource [database on-line by Automation Creations, Inc.]; available from http://www.matls.com; Internet; accessed 20 August 1998.  

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