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Investigation of panel crack formation in steel ingots using mathematical and physical models Thomas, Brian Gordon 1985

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INVESTIGATION OF PANEL CRACK FORMATION IN STEEL INGOTS USING MATHEMATICAL AND PHYSICAL MODELS By BRIAN GORDON THOMAS B. Eng., M c G i l l U n i v e r s i t y , 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of M e t a l l u r g i c a l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1985 © Brian Gordon Thomas, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of MEiAuu&GrxcAL £ M £ ) M £ £ / ? > N &-The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date APRIL 33^ 1 1 8 5" DE-6 £3/81) ABSTRACT An i n v e s t i g a t i o n of p a n e l c r a c k f o r m a t i o n i n s t e e l i n g o t s was u n d e r -t a k e n to improve u n d e r s t a n d i n g of the mechanisms by w h i c h they d e v e l o p and to e v a l u a t e p o s s i b l e s o l u t i o n s to the p r o b l e m . The i n v e s t i g a t i o n r e v e a l e d t h a t two d i s t i n c t types of p a n e l c r a c k s , b o t h of w h i c h a r e p a r t l y caused by i n t e r m e d i a t e - t e m p e r a t u r e e m b r i t t l e m e n t of s t e e l i n v o l v i n g aluminum n i t r i d e p r e c i p i t a t i o n , o p e r a t e under d i f f e r e n t mechanisms. I s o t h e r m a l , p h y s i c a l m o d e l l i n g e x p e r i m e n t s were c o n d u c t e d to d e t e r m i n e the f l o w p a t t e r n s , v e l o c i t y p r o f i l e s and flame geometry i n a b o t t o m - f i r e d s o a k i n g p i t and the r e s u l t a n t e f f e c t s on heat t r a n s f e r . An i n v e s t i g a t i o n i n v o l v i n g c o m p a r i s o n w i t h a n a l y t i c a l s o l u t i o n s d e t e r m i n e d the optimum n u m e r i c a l method to employ f o r the m a t h e m a t i c a l m o d e l l i n g of complex, t w o - d i m e n s i o n a l , t r a n s i e n t , h e a t -c o n d u c t i o n p r o b l e m s . T h i s method was f o r m u l a t e d to c a l c u l a t e the tempera-t u r e d i s t r i b u t i o n i n a s t e e l i n g o t d u r i n g the v a r i o u s p r o c e s s i n g s t a g e s from i n i t i a l c a s t i n g up to r o l l i n g and was v e r i f i e d w i t h i n d u s t r i a l measurements. A t r a n s i e n t , e l a s t o - v i s c o - p l a s t i c , t h e r m a l - s t r e s s model e m p l o y i n g the f i n i t e - e l e m e n t method was f o r m u l a t e d , d e v e l o p e d and v e r i f i e d u s i n g a n a l y -t i c a l s o l u t i o n s . Based on the t e m p e r a t u r e s c a l c u l a t e d by the f i n i t e -e l e m e n t , h e a t - t r a n s f e r model as i n p u t d a t a , the t r a n s i e n t , i n t e r n a l s t r e s s s t a t e o f the i n g o t was c a l c u l a t e d , t a k i n g i n t o a c c o u n t the e f f e c t s o f p h ase-t r a n s f o r m a t i o n volume changes and k i n e t i c s , c r e e p , and t e m perature-dependent m e c h a n i c a l p r o p e r t y b e h a v i o r . The s i m u l a t e d s t r e s s h i s t o r i e s were found to be d i r e c t l y l i n k e d to the p r o g r e s s of the p h a s e - t r a n s f o r m a t i o n f r o n t and were used to c l a r i f y the r o l e o f s t r e s s g e n e r a t i o n i n p a n e l c r a c k f o r m a t i o n . i i a F i n a l l y , the r e s u l t s of a m e t a l l u r g i c a l i n v e s t i g a t i o n of s t e e l ingot samples containing off-corner panel cracks were synthesized with the r e s u l t s of the p h y s i c a l and mathematical models to determine mechanisms and to suggest solutions for the formation of both mid-face and off-corner panel cracks. Mid-face panel cracks are apparently formed during a i r cooling when the mid-face s u r f a c e i s between the Ar^ and 500 °C. Off-corner panel cracks appear to i n i t i a t e i n t e r n a l l y during the early stages of reheating, but do not propagate to the surface u n t i l a i r cooling a f t e r removal from the soaking p i t . i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i i i L IST OF FIGURES x NOMENCLATURE x x i i i ACKNOWLEDGEMENTS x x v i i i 1 INTRODUCTION 1 2 PREVIOUS WORK 5 2.1 Other C r a c k Problems i n S t e e l 5 2.2 Hot D u c t i l i t y o f S t e e l 8 2.2.1 Zones of Reduced D u c t i l i t y 9 2.2.2 H i g h Temperature Zone 12 2.2.3 I n t e r m e d i a t e Temperature Zone..... 15 2.2.3.1 E f f e c t o f S t e e l C o m p o s i t i o n 16 2.2.3.2 E f f e c t of Thermal H i s t o r y 24 2.2.3.3 E f f e c t o f S t r a i n Rate 26 2.2.3.4 E f f e c t of G r a i n S i z e 29 2.2.3.5 Mechanisms 34 2.2.4 Lower Temperature Zone 43 2.2.4.1 Mechanisms 44 2.2.5 I m p l i c a t i o n s of D u c t i l i t y f o r P a n e l C r a c k i n g 50 2.3 P r e v i o u s P a n e l C r a c k i n g S t u d i e s 54 2.3.1 M i d - f a c e P a n e l C r a c k s 55 2.3.1.1 M e t a l l o g r a p h y 56 2.3.1.2 E f f e c t of C o m p o s i t i o n 59 2.3.1.3 E f f e c t o f I n g o t S i z e and Shape 60 2.3.1.4 E f f e c t of Thermal Treatment 61 2.3.2 O f f - c o r n e r P a n e l C r a c k s 62 2.3.2.1 M e t a l l o g r a p h y 66 2.3.2.2 E f f e c t o f C o m p o s i t i o n 68 2.3.2.3 E f f e c t of I n g o t S i z e and Shape 69 2.3.2.4 E f f e c t o f Thermal Treatment 69 2.4 Pr o p o s e d Mechanisms f o r P a n e l Crack F o r m a t i o n 72 2.4.1 Reduced H i g h Temperature D u c t i l i t y 72 2.4.1.1 M i d - f a c e P a n e l C r a c k s 73 2.4.1.2 O f f - c o r n e r P a n e l C r a c k s 74 i v Page 2.A.2 Stress Generation 77 2.4.2.1 Mid-face Panel Cracks 78 2.4.2.2 Off-corner Panel Cracks 80 2.5 Proposed Solutions 81 2.5.1 Steel Composition 82 2.5.2 Thermal Treatment 84 2.5.2.1 Mid-face Panel Cracks 85 2.5.2.2 Off-corner Panel Cracks 85 2.6 Conclusions 89 3 SCOPE OF THE PRESENT WORK 91 3.1 Objectives 91 3.2 Methodology 92 4 PHYSICAL MODELLING OF A BOTTOM-FIRED SOAKING PIT 100 4.1 Introduction 100 4.2 Soaking P i t Operation 101 4.3 Description of the Model 105 4.4 Physical Modelling C r i t e r i a 108 4.5 Flow Pattern Study I l l 4.5.1 Experimental Methodology 112 4.5.2 Results 114 4.5.2.1 E f f e c t of F i r i n g Conditions 118 4.5.2.2 E f f e c t of Ingot Arrangement 118 4.5.2.3 E f f e c t of Ingot Height 119 4.6 V e l o c i t y P r o f i l e Study 119 4.6.1 Experimental Methodology 120 4.6.2 Results 122 4.6.2.1 A i r I n l e t V e l o c i t i e s 122 4.6.2.2 Inside P i t V e l o c i t i e s 125 4.6.2.3 Wall V e l o c i t i e s 129 4.6.2.4 Recuperator Port V e l o c i t i e s 131 4.7 Mixing Study 135 4.7.1 Experimental Methodology 136 4.7.2 Results 139 4.7.2.1 E f f e c t of Excess A i r 141 4.7.2.2 E f f e c t of Ingot Arrangement 145 4.8 Comparison with I n d u s t r i a l Observations 145 4.9 Implications for Heat Transfer and Panel Cracking 148 4.10 Summary and Conclusions 151 5 COMPARISON OF NUMERICAL METHODS FOR HEAT TRANSFER MODELLING 154 5.1 Introduction 154 5.2 Problems Studied and A n a l y t i c a l Solutions 155 5.2.1 Reheating i n a Soaking P i t 155 5.2.2 S o l i d i f i c a t i o n i n the Mould 159 5.3 Description of Numerical Methods Tested 164 5.3.1 Formulation of F i n i t e Element Methods 166 5.3.2 Formulation of Boundary Conditions 170 V Page 5.3.3 Methods of L a t e n t Heat E v o l u t i o n 171 5.3.4 T i m e - S t e p p i n g T e c h n i q u e s 175 5.4 Comparison M e t h o d o l o g y 179 5.5 R e s u l t s and D i s c u s s i o n 181 5.5.1 S i z e of Mesh and Time S t e p 183 5.5.1.1 E f f e c t on A c c u r a c y 183 5.5.1.2 E f f e c t on S t a b i l i t y 185 5.5.1.3 G r a d i n g o f Mesh and Time Step 186 5.5.2 Comparison of N u m e r i c a l Methods 188 5.5.2.1 A c c u r a c y and S t a b i l i t y 188 5.5.2.2 C o s t s 195 5.6 C o n c l u s i o n s 197 6 HEAT TRANSFER MODEL FOR INGOT CASTING 199 6.1 Model F o r m u l a t i o n and A s s u m p t i o n s 200 6.2 Boundary C o n d i t i o n s 208 6.2.1 Mould C o o l i n g 211 6.2.2 A i r c o o l i n g 214 6.2.3 S o a k i n g P i t R e h e a t i n g 215 6.3 S o l u t i o n T e c h n i q u e 216 6.4 H e a t - t r a n s f e r Model V e r i f i c a t i o n 224 7 STRESS MODEL OF STATIC-CAST INGOT 227 7.1 Model F o r m u l a t i o n 228 7.1.1 Thermal S t r a i n 233 7.1.2 P l a s t i c S t r a i n 235 7.2 S o l u t i o n T echnique 239 7.3 Thermomechanical P r o p e r t y D a t a 251 7.3.1 P l a s t i c Creep F u n c t i o n 251 7.3.2 Thermal L i n e a r E x p a n s i o n F u n c t i o n 254 7.3.3 E l a s t i c Modulus 266 7.3.4 P o i s s o n ' s R a t i o 269 7.4 S t r e s s Model V e r i f i c a t i o n 269 7.4.1 F i r s t T e s t P r o b l e m 269 7.4.2 Second T e s t P r o b l e m 270 7.5 F r a c t u r e C r i t e r i a and P r e s e n t a t i o n o f R e s u l t s 277 7.5.1 P r i n c i p l e S t r e s s e s 278 7.5.2 E f f e c t i v e S t r e s s 280 7.5.3 P l a s t i c S t r a i n 281 8 MATHEMATICAL MODELLING RESULTS 283 8.1 Heat T r a n s f e r Model S e n s i t i v i t y A n a l y s i s 283 8.2 M i d - f a c e P a n e l C r a c k i n g 285 8.2.1 H e a t - t r a n s f e r Model R e s u l t s 285 8.2.2 S t r e s s Model R e s u l t s 294 8.3 O f f - c o r n e r P a n e l C r a c k i n g 317 8.3.1 H e a t - t r a n s f e r M odel R e s u l t s 324 8.3.1.1 E f f e c t of R e h e a t i n g S c h e d u l e 333 8.3.1.2 E f f e c t o f T r a c k Time 337 v i Page 8.3.2 Stress Model Results 342 8.3.2.1 E f f e c t of Reheating Schedule 363 8.3.2.2 E f f e c t of Track Time 372 8.3.2.3 E f f e c t of Mechanical Property Data Input 380 9 METALLURGICAL INVESTIGATION 385 9.1 O p t i c a l Metallography 385 9.2 Electron Microscopy 398 9.3 Segregation Analysis 401 9.4 C a l c u l a t i o n of Decarburized Zone Width 403 9.5 Summary 405 10 DISCUSSION 407 10.1 Mid-face Panel Cracks 408 10.2 Off-corner Panel Cracks 414 11 CONCLUSIONS 425 BIBLIOGRAPHY 432 APPENDIX I Establishment of Model Burner Dimensions (Using Thring-Newby S i m i l a r i t y C r i t e r i o n ) 455 APPENDIX II Reynolds Number Calculations 460 APPENDIX III C a l i b r a t i o n of O r i f i c e Plate Flow Meter 465 APPENDIX IV A i r - F u e l Flow Ratios and Excess A i r Calculations for D i f f e r e n t F i r i n g Conditions 469 APPENDIX V P i tot Tube C a l i b r a t i o n 480 APPENDIX VI C a l c u l a t i o n of Model Helium Concentrations f o r Flame Front Determination 485 APPENDIX VII Convective Heat Transfer i n the Soaking P i t 487 APPENDIX VIII Heat-transfer Model Computer Program 489 APPENDIX IX Accuracy of Heat-transfer Model i n P r e d i c t i n g Temperature Response for A n a l y t i c a l Solutions 510 APPENDIX X State Function to Account for Volume Change i n Thermal Stress Models 514 APPENDIX XI Stress Model Computer Program 516 APPENDIX XII The E f f e c t of Carbon Content on the Linear Expansion Accompanying the y •*• a Phase Transformation i n S t e e l . . . . 544 v i i Page APPENDIX XIII D i l a t o m e t e r Study to Determine T r a n s f o r m a t i o n Temperatures f o r Off-corner Panel-cracked Steel 551 APPENDIX XIV D e r i v a t i o n of A n a l y t i c a l S o l u t i o n to Time-Dependent Thermal Stress Problem 557 v i i i LIST OF TABLES Page 2.1 S o l u b i l i t y of Various N i t r i d e s i n A u s t e n t i t e 7 7 32 5.1 Conditions Assumed for F i r s t Problem of Ingot Reheating in a Soaking P i t 160 5.2 Conditions Assumed for Second Problem of Ingot S o l i d i f i c a -t i o n i n a Mould 160 5.3 Accuracy and S t a b i l i t y of Numerical Methods for F i r s t Problem 189 5.4 Accuracy and S t a b i l i t y of Numerical Method for Second Problem 190 5.5 Estimated Computer Costs per Time Step 196 6.1 S t e e l Compositions Used f o r Thermal P r o p e r t y Data Calculations 207 6.2 F i n i t e Element Meshes 220 7.1 Transformation Temperatures Used i n Model 261 7.2 Comparison Between A n a l y t i c a l and Numerical P l a s t i c - c r e e p S t r a i n Predictions for Second Test Problem 275 7.3 Comparison Between A n a l y t i c a l and Numerical S t r e s s Predictions f or Second Test Problem 275 8.1 Ingot Processing Conditions Simulated i n T y p i c a l Off-corner Panel Cracking Run #1 318 8.2 Ingot Processing Conditions Simulated for Off-corner Panel Cracking Study 320 9.1 Composition of Steel Samples Used i n M e t a l l u r g i c a l I nvestigation (weight percent) 387 9.2 Time Required to Produce I n t e r n a l l y Oxidized Zone of F e r r i t e 404 i x Page A9.1 Accuracy of Heat-transfer Model in P r e d i c t i n g Temperature Response for F i r s t Test Problem 511 A9.2 Accuracy of Heat-transfer Model in P r e d i c t i n g Temperature Response for Second Test Problem 512 X LIST OF FIGURES Page 1.1 T y p i c a l appearance of panel c r a c k running a l o n g corrugations of a 760 x 1520 mm, 25 ton st e e l i n g o t 1 4 3 2.1 Hot d u c t i l i t y of low-carbon s t e e l s containing manganese and aluminum as reported by various researchers 10 2.2 Schematic representation of temperature zones of reduced hot d u c t i l i t y of s t e e l related to embrittling mechanisms 13 2.3 R e l a t i o n s h i p between me c h a n i c a l p r o p e r t i e s i n the high-temperature zone of reduced d u c t i l i t y and: (A) corresponding schematic presentation of s o l i d / l i q u i d i n t e r f a c e during casting (B) carbon content (from S u z u k i 6 2 ) 14 2.4 E f f e c t of A l on the hot d u c t i l i t y of s t e e l 1 * 3 18 2.5 E f f e c t of Nb on the hot d u c t i l i t y of s t e e l 4 3 18 2.6 E f f e c t of Boron on the hot d u c t i l i t y of s t e e l 4 6 20 2.7 E f f e c t of Nitrogen on the hot d u c t i l i t y of C-Mn-Al-Nb s t e e l s 4 5 20 2.8 E f f e c t of Titanium on the hot d u c t i l i t y of s t e e l (from F u n n e l 4 2 ) 21 2 . 9 E f f e c t of S on hot d u c t i l i t y of low-carbon s t e e l (adapted from Weinberg 3 3) 23 2.10 E f f e c t of thermal treatment on intermediate-temperature d u c t i l i t y , i l l u s t r a t i n g the e f f e c t s of (A) hold time p r i o r to t e s t i n g 3 7 (B) dropping temperature into the two-phase region p r i o r to t e s t i n g 3 8 27 2.11 Dependence of intermediate-temperature d u c t i l i t y on s t r a i n r a t e 3 8 f o r : (A) low-carbon, Si-Mn s t e e l (B) Nb-bearing s t e e l 28 2.12 E f f e c t of various elements on the grain-coarsening temperature of plain-carbon s t e e l 33 x i Page 2.13 Possible fracture zones mapped for 0.2% C plain-carbon s t e e l i n s t r a i n - r a t e temperature s p a c e 6 6 35 2.14 Schematic r e p r e s e n t a t i i o n of the e f f e c t s of s t r a i n rate and temperature on the hot d u c t i l i t y of s t e e l 8 0 42 2.15 E f f e c t of hold time at 750°C on both f e r r i t e f i l m thickness and d u c t i l i t y 3 8 45 2.16 (A) K i n e t i c s of A1N p r e c i p i t a t i o n i n low-carbon s t e e l at various temperatures 7 9 (B) Contour l i n e s of 20% A1N p r e c i p i t a t i o n 7 1 47 2.17 Mechanism for embrittlement i n the low-temperature or two-phase z o n e 3 7 49 2.18 Schematic diagram showing the zones of embrittlement (see Figure 2.13) at intermediate s t r a i n rates for the Fe-C system 6 6 ; 51 2.19 Mid-face panel crack i n 350 x 350 mm square Enl8 (.4%C, 1.0%Cr) s t e e l i n g o t 1 57 2.20 T y p i c a l l o c a t i o n of mid-face panel cracks i n a transverse cross-section of a: (A) 370 mm square, .55% C, s t e e l I n g o t 1 ' 3 (B) 380 mm duodecagonal, .6% C s t e e l i n g o t 3 58 2.21 Off-corner panel cracks i n a 760 x 1520 mm, rectangular, corrugated (.14% C, 1.4% Mn, S i - k i l l e d , A l grain refined) s t e e l i n g o t 1 4 63 2.22 Relative l o c a t i o n of off-corner panel cracks found by Sussman 9 i n transverse cross-sections taken from the top of Ingots subjected to: (A) 1260 s ( 21 min) unjacketed time (B) 6480 s (108 min) unjacketed time 64 2.23 R e l a t i v e l o c a t i o n of o f f - c o r n e r panel c r a c k s i n transverse cross section taken from mid-height of a 760 x 1520 mm corrugated Stelco i n g o t 1 4 65 2.24 Close-up of off-corner panel crack showing associated f e r r i t e band and i n c l u s i o n s (etched i n 2% n i t o l , 120X) 67 2.25 The e f f e c t of cooling p r a c t i c e on the incidence of o f f -corner panel c r a c k i n g 1 4 70 2.26 Mechanism for off-corner panel crack formation involving reduced d u c t i l i t y 76 x i i Page 2.27 E f f e c t of cooling p r a c t i c e on panel c r a c k i n g 6 ' 1 4 86 2.28 Soaking p i t reheating p r a c t i c e recommended to reduce off-corner panel c r a c k i n g 1 0 88 3.1 Ingot section to be simulated with the mathematical heat transfer and stress models....... 99 4.1 Schematic cross section of prototype soaking p i t 103 4.2 Photograph of e n t i r e soaking p i t model 106 4.3 Model burner assembly • 107 4.4 D i s t o r t i o n of soaking p i t model dimensions using the Thring-Newby c r i t e r i o n 110 4.5 Model soaking p i t chamber with ingots charged i n : (A) standard 6 ingot arrangement and (B) standard 8 ingot arrangement 115 4.6 Photograph of helium bubble movement i l l u s t r a t i n g flow pattern (taken at high f i r e using standard 8 ingot arrangement) 116 4.7 Schematic diagram of o v e r a l l flow pattern 117 4.8 Upward v e r t i c a l v e l o c i t y ( i n m/s) versus p o s i t i o n i n the a i r i n l e t annulus 123 4.9 Schematic diagram of burner chamber and arched roof a i r i n l e t tunnels showing development of non symmetrical flow d i s t r i b u t i o n through a i r annulus 124 4.10 Horizontal v e l o c i t y p r o f i l e s i n lower p i t section (9% p i t height, standard 8 ingot arrangement) 126 4.11 Horizontal v e l o c i t y p r o f i l e i n upper section above ingots (96% p i t height, standard 8 ingot arrangement) 127 4.12 Upward v e r t i c a l v e l o c i t y at various locations i n the p i t (marked in Figure 4.11) 128 4.13 Downward v e r t i c a l v e l o c i t y along p i t walls 130 4.14 Recuperator port e x i t v e l o c i t i e s (west recuperator, standard 8 ingot arrangement) 132 4.15 E f f e c t of ingot positioning close to recuperator e x i t port on gas v e l o c i t y 134 x i i i Page 4.16 Mixing experiments and r e s u l t s (using standard 8 ingot arrangement) 138 4.17 Stoichiometric flame front contours at d i f f e r e n t p i t cross sections (high f i r e , 2% exhaust oxygen, standard 8 ingot arrangement).... 140 4.18 Dependence of flame height on excess a i r percentage for runs given in Figure 4.16 143 4.19 E f f e c t of ingot arrangement on flame geometry (high f i r e , 2% exhaust oxygen) 146 4.20 Photograph of the flames i n a Stelco soaking p i t with i t s roof removed 147 5.1 Schematic diagram of f i r s t problem 156 5.2 Temperature response of locations A, B, C, D, E (shown in Figure 5.1) from a n a l y t i c a l s o l u t i o n to f i r s t problem 158 5.3 Schematic diagram of second problem 161 5.4 P o s i t i o n of s o l i d i f i c a t i o n front at d i f f e r e n t times from a n a l y t i c a l s o l u t i o n to second problem 163 5.5 Meshes used for numerical methods 167 5.6 Examples of runs with varying s t a b i l i t y 182 5.7 E f f e c t of size of mesh and time step on accuracy for f i r s t problem 184 5.8 Accuracy improvement using v a r i a b l e time steps.... 187 5.9 E f f e c t of varying phase change temperature i n t e r v a l (PCTI) on accuracy for second problem 194 6.1 Thermal conductivity functions for s t e e l and cast i r o n used i n heat transfer model 203 6.2 Enthalpy functions for s t e e l and cast i r o n used i n heat transfer model 204 6.3 S p e c i f i c heat of s t e e l and cast i r o n assumed i n enthalpy functions i n Figure 6.2 205 6.4 Heat transfer model boundary condition regions for the 1/4 transverse section through a 355 x 355 mm mid-face panel-cracked ingot 209 x i v Page 6.5 Heat transfer model boundary condition regions for the 1/4 transverse section through a 760 x 1520 mm off-corner panel-cracked ingot 210 6.6 Airgap formation times h a l f way up mould for a 6 ton, 600 x 600 mm i n g o t 1 5 8 213 6.7 F i n i t e element mesh for a 355 x 355 mm (14 x 14") 2 ton, mid-face panel-cracked ingot and mould 218 6.8 F i n i t e element mesh for a 760 x 1520 mm (30 x 60"), 25 ton, off-corner panel-cracked ingot and mould 219 6.9 Flow chart of finite-element heat transfer model computer program 222 6.10 F i n i t e element mesh for a 230 x 405 mm (9 x 16") experimental ingot and mould used for comparison with i n d u s t r i a l temperature measurements 225 6.11 Comparison of measured and c a l c u l a t e d temperature responses for a 230 x 405 mm (9 x 16") s t e e l ingot 226 7.1 Stress model boundary conditions 230 7.2 F i n i t e element mesh for the 355 x 355 mm, (14 x 14"), 2 ton, mid-face panel-cracked ingot alone 240 7.3 F i n i t e element mesh for the 760 x 1520 mm (30 x 60"), 25 ton, corrugated, off-corner panel-cracked ingot alone 241 7.4 Flow chart of computer stress model 246 7.5 P l a s t i c creep s t r a i n rate function used f o r low carbon s t e e l 255 7.6 Stress-temperature curves at low s t r a i n rates for low carbon s t e e l assumed i n Figure 7.5 256 7.7 Thermal l i n e a r expansion of ir o n assumed i n model 257 7.8 Comparison between model predictions of transformation product, percentages and experimental measurements from analysis of dilatometer c u r v e 2 3 3 263 7.9 Thermal l i n e a r expansion curves f o r the cooling and heating of high carbon s t e e l assumed i n the model 264 7.10 Thermal l i n e a r expansion curves for the cooling and heating of low carbon s t e e l assumed i n the model 265 XV Page 7.11 E l a s t i c modulus functions used i n stress model 268 7.12 F i r s t test problem for stress model v e r i f i c a t i o n (A) Schematic diagram (B) Finite-element mesh 271 7.13 Second-test problem for stress model v e r i f i c a t i o n (A) Schematic diagram (B) Finite-element mesh 274 7.14 Comparison of a n a l y t i c a l and numerical stress predictions for v e r i f i c a t i o n of finite-element stress model 276 8.1 Temperature contours (°C) calculated by heat-transfer model for cooling of 355 x 355 mm ingot i n the mould (A) 240 s (4 min) 286 (B) 1440 s (0.40 h) 287 8.2 Temperature contours (°C) calculated from heat transfer model for 355 x 355 mm ingot during a i r cooling (A) 1800 s (0.5 h) (B) 4000 s (1.11 h) 288 (C) 4500 s (1.25 h) (D) 5400 s (1.5 h) 289 (E) 6300 s (1.75 h) (F) 7200 s (2.0 h) 290 (G) 9000 s (2.5 h) 291 8.3 Temperature p r o f i l e s for s l i c e through transverse section of 355 x 355 mm ingot calculated by heat-transfer model at various times 292 8.4 P r i n c i p a l stresses calculated by stress model for 355 x 355 mm ingot during a i r cooling (A) 1800 s (30 min) (B) 4000 s (1.11 h) 295 (C) 4500 s (1.25 h) (D) 5400 s (1-.5 h) 296 (E) 6300 s (1.75 h) (F) 7200 s (2.0 h) 297 (G) 9000 s (2.5 h) (H) 18000 s (5 h) 298 8.5 E f f e c t i v e stress contours (MPa) calculated by stress model fo r 355 x 355 mm ingot during a i r cooling (A) 1800 s (0.5 h) (B) 4000 s (1.11 h) 299 (C) 4600 s (1.28 h) (D) 5400 s (1.5 h) 300 (E) 6300 s (1.75 h) (F) 7200 s (2.0 h) 301 x v i Page 8.5 (G) 9000 s (2.5 h) (H) 18000 s (5 h) 302 8.6 Accumulated p l a s t i c s t r a i n contours (%) ca l c u l a t e d by stress model for 355 x 355 mm ingot during a i r cooling (A) 1800 s (0.5 h) (B) 5400 s (1.5 h) 306 (C) 7200 s (2.0 h) (D) 18000 s (5.0 h) 307 8.7 Stress h i s t o r i e s of two important locations i n 355 x 355 mm ingot during processing 309 8.8 S t r e s s - s t r a i n h i s t o r i e s f o r two important locati o n s i n 355 x 355 mm ingot during processing 310 8.9 Normal stress p r o f i l e s i n s l i c e through transverse section of 355 x 355 mm ingot at various times 312 8.10 Normal stress p r o f i l e s i n s l i c e through transverse section of 405 x 405 mm ingot at various times 313 8.11 E f f e c t of carbon content of the s t e e l on the thermal and stress h i s t o r i e s of the 405 x 405 mm ingot during processing (A) Mid-face surface l o c a t i o n 314 (B) Mid-face sub-surface l o c a t i o n 315 8.12 Soaking p i t reheating schedules simulated i n model 321 8.13 Terminology f o r s p e c i a l locations i n the 760 x 1520 mm off-corner panel cracked ingot 323 8.14 Temperature contours (°C) calculated by heat transfer model for cooling of 760 x 1520 mm ingot i n the mould (A) 2700 s (0.75 h) (B) 5400 s (1.5 h) 325 (C) 10800 s (3 h) (D) 14400 s (4 h) 326 8.15 Temperature contours (°C) calculated by heat transfer model for 760 x 1520 mm ingot during a i r cooling (A) 14400 s ( s t r i p a f t e r 4 h mould cooling) (B) 15000 s (10 min a f t e r s t r i p p i n g ) (C) 18900 s (1.25 h a f t e r s t r i p p i n g ) 327 (D) 20700 s (1.75 h a f t e r s t r i p p i n g ) (E) 25200 s (3 h a f t e r s t r i p p i n g ) (F) 32400 s (5 h a f t e r s t r i p p i n g ) 328 x v i i Page 8.16 Temperature contours (°C) cal c u l a t e d by heat transfer model for 760 x 1520 mm ingot during reheating i n the soaking p i t (run #1) (A) 20700 s (charge a f t e r 4 h mould cooling, 1.75 h a i r cooling) (B) 21300 s (10 min after charge) (C) 21900 s (20 min a f t e r charge) 330 (D) 23100 s (40 min aft e r charge) (E) 24300 s (1 h a f t e r charge) (F) 26100 s (1.5 h a f t e r charge) 331 (G) 27900 s (2 h a f t e r charge) (H) 29700 s (2.5 h a f t e r charge) (I) 38700 s (5 h a f t e r charge) 332 8.17 Temperature contours (°C) calculated by heat-transfer model for 760 x 1520 mm ingot during a i r cooling a f t e r removal from the soaking p i t (run #1) (A) 70900 s (15 min a f t e r draw) (B) 74500 s (1.25 h a f t e r draw) (C) 82600 s (3 hr a f t e r draw) 334 8.18 Temperature contours (°C) calculated by heat-transfer model for 760 x 1510 mm ingot during f a s t reheating i n an i n i t i a l l y cold soaking p i t (run #2) (A) 20700 s (charge a f t e r 4 h mould cooling, 1.75 h a i r cooling) (B) 21900 s (20 min aft e r charge) (C) 23100 s (40 min a f t e r charge) 335 (D) 24300 s (1 h aft e r charge) (E) 26100 s (1.5 h a f t e r charge) (F) 29700 s (2.5 h a f t e r charge) 336 8.19 Temperature contours (°C) cal c u l a t e d by heat transfer model for 760 x 1520 mm ingot during slow reheating i n an i n i t i a l l y cold soaking p i t (run #3) (A) 20700 s (charge) (B) 24300 s (1 h a f t e r charge) (C) 31500 s (3 h a f t e r charge) 338 8.20 Temperature contours (°C) cal c u l a t e d by heat transfer model for 760 x 1520 mm ingot during reheating i n the soaking p i t a f t e r very short track time (run #4) (A) 12600 s (charge a f t e r 3 h mould cooli n g , 0.5 h a i r cooling) (B) 14400 s (0. h a f t e r charge) (C) 18000 s (1.5 h a f t e r charge)... 339 x v i i i Page 8.21 Temperature contours (°C) cal c u l a t e d by heat-transfer model for 760 x 1520 mm ingot during reheating i n the soaking p i t a f t e r short track time (run #5) (A) 18900 s (charge a f t e r 4 h mould cooling, 1.25 h a i r cooling) (B) 20700 s (0.5 h a f t e r charge) (C) 22500 s (1 h aft e r charge) 340 8.22 Temperature contours (°C) calculated by heat-transfer model for 760 x 1520 mm ingot during reheating i n the soaking p i t aft e r long track time (run #6) (A) 25200 s (charge a f t e r 4 h mould cooling, 3 h a i r cooling) (B) 28800 s (1 h aft e r charge) (C) 30600 s (1.5 h aft e r charge) 341 8.23 P r i n c i p a l stresses calculated by stress model for 760 x 1520 mm ingot during a i r cooling (run #1) (A) 14400 s ( s t r i p a f t e r 4 h mould cooling) (B) 15000 s (10 min aft e r s t r i p p i n g ) (C) 18900 s (1.25 h a f t e r s t r i p p i n g ) 343 (D) 20700 s (1.75 h a f t e r s t r i p p i n g ) (E) 25200 s (3 h a f t e r s t r i p p i n g ) (F) 32400 s (5 h a f t e r s t r i p p i n g ) 344 8.24 P r i n c i p a l stresses calculated by stress model for 760 x 1520 mm ingot during fast reheating i n an i n i t i a l l y hot soaking p i t (run #1) (A) 20700 s (charge a f t e r 4 h mould cooling, 1.75 h a i r cooling) (B) 21300 s (10 min a f t e r charge) (C) 21900 s (20 min a f t e r charge) 346 (D) 23100 s (40 min a f t e r charge) (E) 24300 s (1 h a f t e r charge) (F) 26100 s (1.5 h a f t e r charge) 347 (G) 27900 s (2 h a f t e r charge) (H) 29700 s (2.5 h a f t e r charge) (I) 38700 s (5 h a f t e r charge) 348 8.25 E f f e c t i v e stress contours ca l c u l a t e d by stress model for 760 x 1520 mm ingot during f a s t reheating i n an i n i t i a l l y hot soaking p i t (run #1) (A) 20700 s (charge a f t e r 4 h mould cooling, 1.75 h a i r cooling) (B) 21000 s (5 min a f t e r charge) (C) 21300 s (10 min a f t e r charge) 349 (D) 21900 s (20 min a f t e r charge) (E) 22500 s (30 min a f t e r charge) (F) 23100 s (40 min a f t e r charge) 350 (G) 24900 s (70 min aft e r charge) (H) 27900 s (2 h a f t e r charge) (I) 38700 s (5 h a f t e r charge) 351 x i x Page 8.26 P r i n c i p a l stress crosses c a l c u l a t e d by stress model for 760 x 1520 mm Ingot during a i r cooling a f t e r removal from soaking p i t (run #1) (A) 70900 s (15 min aft e r draw) (B) 74500 s (1.25 h a f t e r draw) (C) 82600 s (3 h a f t e r draw) 354 8.27 Thermal and stress h i s t o r i e s of three important locati o n s i n 760 x 1520 mm ingot during processing under conditions i n run #1 (A) Off-corner surface l o c a t i o n 356 (B) Off-corner sub-surface l o c a t i o n 357 (C) Mid-face surface l o c a t i o n 358 8.28 Accumulated p l a s t i c s t r a i n contours (%) calculated by stress model for 760 x 1520 mm ingot during processing (run //l) (A) 14400 s ( s t r i p a f t e r 4 h mould cooling) (B) 20700 s (charge a f t e r 1.75 h a i r cooling) (C) 24300 s (1 h a f t e r charge) 361 (D) 29700 s (2.5 h a f t e r charge) (E) 38700 s (5 h a f t e r charge) (F) 74500 s (1.25 h a f t e r draw) 362 8.29 S t r e s s - s t r a i n h i s t o r i e s for 5 important locations i n 760 x 1520 mm ingot during processing (run #1) 364 8.30 P r i n c i p a l stresses calculated by stress model for 760 x 1520 mm ingot during f a s t reheating i n an i n i t i a l l y cold soaking p i t (run #2) (A) 20700 s (charge a f t e r 4 h mould cooling, 1.75 h a i r cooling) (B) 21900 s (20 min aft e r charge) (C) 23100 s (40 min a f t e r charge) 365 (D) 24300 s (1 h a f t e r charge) (E) 26100 s (1.5 h a f t e r charge) (F) 29700 s (2.5 h a f t e r charge) 366 8.31 P r i n c i p a l stresses calculated by stress model for 760 x 1520 mm ingot during slow reheating i n an i n i t i a l l y cold soaking p i t (run #3) (A) 20700 s (charge) (B) 24300 s (1 h a f t e r charge) (C) 31500 s (3 h a f t e r charge) 367 8.32 E f f e c t of d i f f e r e n t reheating schedules on the thermal and stress h i s t o r i e s of three important locations i n 760 x 1520 mm ingot during reheating a f t e r medium track time conditions (A) Off-corner surface l o c a t i o n . . 369 (B) Off-corner sub-surface l o c a t i o n 370 (C) Mid-wide-face surface l o c a t i o n 371 X X Page 8.33 P r i n c i p a l stresses calculated by str e s s model f o r 760 x 1520 mm ingot during reheating i n soaking p i t a f t e r very short track time (run #4) (A) 12600 s (charge a f t e r 3 h mould cool i n g , 0.5 h a i r cooling) (B) 14400 s (0.5 h af t e r charge) (C) 18000 s (1.5 h af t e r charge) 373 8.34 P r i n c i p a l stresses calculated by stress model for 760 x 1520 mm ingot during reheating i n soaking p i t a f t e r short track time (run #5) (A) 18900 s (charge a f t e r 4 h mould cooling, 1.25 h a i r cooling (B) 20700 s (0.5 h charge) (C) 22500 s (1 h a f t e r charge) 374 8.35 P r i n c i p a l stresses calculated by stress model for 760 x 1520 mm ingot during reheating i n soaking p i t a f t e r long track time (run #6) (A) 25200 s (charge a f t e r 4 h mould cooling, 3 h a i r cooling) (B) 28800 s (1 h a f t e r charge) (C) 30600 s (1.5 h a f t e r charge) 375 8.36 Thermal and stress h i s t o r i e s at off-corner l o c a t i o n during p r o c e s s i n g under v e r y s h o r t t r a c k time c o n d i t i o n s (run #4) 376 8.37 E f f e c t of a i r - c o o l i n g time p r i o r to reheating on the thermal and stress h i s t o r i e s of two important locati o n s i n the 760 x 1520 mm ingot during reheating i n soaking p i t (A) Off-corner surface l o c a t i o n 378 (B) Off-corner sub-surface l o c a t i o n 379 8.38 Influence of creep function on e f f e c t i v e stress contour d i s t r i b u t i o n at 18900 s (1.25 h a f t e r s t r i p p i n g ) (A) f u l l creep (Wray data with accelerated creep i n f e r r i t e ) (B) medium creep (Wray data with no extra creep i n f e r r i t e ) (C) low creep (Sakiu data) 382 8.39 E f f e c t i v e stress contours at 18900 s during a i r cooling c a l c u l a t e d assuming no creep 384 8.40 Maximum p r i n c i p a l t e n s i l e stress contours at 18900 s during a i r cooling calculated assuming no creep 384 9.1 Transverse section of 760 x 1520 mm off-corner panel-cracked ingot showing l o c a t i o n at lower l e f t corner where samples were obtained 386 xxi Page 9.2 Complete example of off-corner panel crack i n sectioned sample taken from ingot i n Figure 9.1, macroetched i n HC1, 1.2 X 388 9.3 Network of f e r r i t e bands and cracks at p r i o r austenite grain boundaries traced from macroetched sample i n Figure 9.2 390 9.4 Dendrites i n sectioned sample from a non-cracked ingot taken from same lo c a t i o n as cracked sample in Figure 9.2 macroetched i n HC1, 1.1 X 391 9.5 Panel-crack microstructures at 4 X magnification o u t l i n i n g the p r i o r austenite grain boundaries, obtained using three d i f f e r e n t etches 393 9.6 Decarburized f e r r i t e band containing p r e c i p i t a t e s etched i n 2% n i t a l (A) Including panel crack, 60 X (B) With no crack, 90 X 395 9.7 F e r r i t e / p e a r l i t e micros truetures etched i n 2% n i t a l from panel-cracked sample at 35 X magnification i l l u s t r a t i n g : (A) Intersection between decarburized panel crack and uncracked f e r r i t e band (B) Thin panel crack traversing f e r r i t e grains i n the absence of a c l e a r l y defined f e r r i t e zone 397 9.8 Section of panel-crack t i p e x i t i n g ingot surface and corresponding fracture surface showing curved facets of p r i o r austenite, columnar grain boundaries, 3 X 399 9.9 Sequence of S. E. M. micrographs Into seemingly smooth fr a c t u r e surface of panel crack, i l l u s t r a t i n g f i n e dimple structure on microscopic l e v e l 400 9.10 Sequence of S. E. M. micrographs i n d i c a t i n g p a r t i a l l y d u c t i l e fracture i n l a r g e s t panel crack near ingot surface 402 A3.1 C a l i b r a t i o n curve for 50.8 mm o r i f i c e plate flow meter ( 6 - 0.6) 468 A.51 P i t o t tubes used i n determining v e l o c i t y p r o f i l e s 484 A.13.1 Dilatometer data for slow cooling and heating tests f or off-corner panel-cracked s t e e l 552 A. 13.2 Length change curves for 0.8% C p l a i n carbon s t e e l 554 A.13.3 T y p i c a l set of length change curves from A t k i n s 2 3 0 p233 555 x x i i Page A.13.4 Dilatometer curves generated for off-corner panel-cracked low-carbon s t e e l specimen subjected to a s e r i e s of l°C/s heating and cooling sequences 556 x x i i i NOMENCLATURE A area of element (m 2) A^ nodal area (m 2) A constant in Eq. (7.48) o ASA acid soluble aluminum a^ parameter i n natural coordinate functions [ B ] 6 element strain-displacement matrix b superscript pertaining to external element boundary b^ parameter i n natural coordinate functions [c] global capacitance matrix [c ] e element capacitance matrix C p s p e c i f i c heat of s t e e l (J/kg°C) C e f f e c t i v e s p e c i f i c heat P c^ parameter i n natural coordinate functions {d} nodal displacement vector d distance along ingot side from corner (m) [ E ] e l a s t i c i t y matrix E e l a s t i c modulus of s t e e l (MPa) e superscript pertaining to element {F} thermal "force" vector in Eq. (5.49) {F } thermal force vector e o {F } p l a s t i c s t r a i n force vector (F^) boundary surface t r a c t i o n force vector H enthalpy (J/kg) xxiv H latent heat of s o l i d i f i c a t i o n (J/kg) s [h ]k heat-transfer c o e f f i c i e n t matrix 2 h o v e r a l l heat-transfer c o e f f i c i e n t (W/m °C) 2 h , conductive heat-transfer c o e f f i c i e n t (W/m °C) cond I subscript pertaining to ingot [K ] global conductivity matrix [K ] global s t i f f n e s s matrix [K ] E element conductivity matrix [K ] E element s t i f f n e s s matrix k s t e e l thermal conductivity (W/m °C) Z thickness of oxidized zone (mm) L length (m) [M ] global c o e f f i c i e n t matrix i n Eq. (5.49) M subscript pertaining to mould i n t e r p o l a t i n g shape functions NE number of elements i n mesh NN number of nodes in mesh NBE number of external boundaries n unit vector normal to surface PCTI phase change temperature i n t e r v a l (°C) {Q} global thermal force vector {Q}k boundary heat flux vector 2 q heat flux (W/m ) S p o s i t i o n of phase change boundary (function describing l o c a t i o n of s o l i d i f i c t i o n front) S t o t a l boundary enclosing region where stresses are to be calculated X X V U, V T T T c T O T w LIQ T SOL T PIT T PITI TSOAK TLE t t gap t s t r i p t t r a c k % f i r e tsoak portion of boundary where stress i s s p e c i f i e d portion of boundary where displacement i s s p e c i f i e d temperature (°C) time d e r i v a t i v e of temperature oT/dt i n i t i a l temperature surrounding temperature fixed wall temperature surface temperature fusion temperature dimensionless i n i t i a l temperature a n a l y t i c a l calculated temperature numerically calculated temperature temperature at which temperature-dependent terms i n [K ] , [c ] and {Q} are evaluated l i q u i d u s temperature (°C) solidus temperature (°C) i n t e r n a l soaking p i t temperature (°C) soaking p i t temperature at time of charging (°C) equilibrium soaking p i t temperature (°C) thermal l i n e a r expansion function f o r s t e e l (m/m) time (s) ingot/mould a i r gap formation time (s) time of s t r i p p i n g ingot from mould (s) time of charging ingot to soaking p i t (s) time when high f i r i n g rate begins (s) time when p i t temperature reaches T SOAK (s) xxv i 'draw time of drawing ingot from soaking p i t (s) At time step increment (s) U temperature displacement u, v displacement in x and y d i r e c t i o n s (m) x, y coordinate d i r e c t i o n s (m) x^, y^ centroid of t r i a n g l e a thermal expansion c o e f f i c i e n t (m/m °C) a ambient temperature, f e r r i t e / p e a r l i t e phase of s t e e l 6 l a t e n t to sensible heat r a t i o ; o r i f i c e diameter r a t i o y shear s t r a i n (m/m) y austenite phase of s t e e l 6 d e l t a - f e r r i t e phase of low carbon s t e e l 6 i j Kronecker d e l t a {J J " j} E j , E^ e m i s s i v i t y of ingot and mould surfaces e e f f e c t i v e e m i s s i v i t y e s t r a i n (m/m) • -1 £p p l a s t i c creep s t r a i n rate (s ) Ep p l a s t i c creep s t r a i n (m/m) e e l a s t i c s t r a i n (m/m) e E Q thermal s t r a i n (m/m) Ep p o s i t i v e l y accumulated t o t a l e f f e c t i v e p l a s t i c s t r a i n (m/m) p s t e e l density (kg/m ) 3? s p e c i f i e d surface t r a c t i o n (MPa) a stress (MPa) °I II p r i n c i p a l stresses a t o t a l e f f e c t i v e stress (MPa) 2 A a Steffan Boltzman r a d i a t i o n constant (W/m K ) shear stress (MPa) Poisson r a t i o square matrix column vector inverse of matrix transpose of matri x x v i i i ACKNOWLEDGEMENTS The author would l i k e to thank NSERC, Noranda and Stelco f o r scholarships and support of the project, and Indira Samarasekera, Keith Brimacombe, Bruce Hawbolt, Jim L a i t and the f a c u l t y , s t a f f and students of the M e t a l l u r g i c a l Engineering department at UBC for t h e i r assistance and valuable discussions. Special thanks are due to Binh Chau, Dominic Gr a n d i n e t t i , Steve Hsuing and e s p e c i a l l y Rodney Ng f o r t h e i r e f f o r t s i n performing experiments. Also g r e a t l y appreciated i s the work of Pat Wenman for the drawing of f i g u r e s , Eleanor Thomas for typing the manuscript, and Bruce Thomas f o r helping with photographs. F i n a l l y , thanks go to my wife, Sandi, and my family f o r t h e i r help and encouragement. 1 CHAPTER 1 INTRODUCTION Despite the advantages of the continuous casting process, over two t h i r d s of world s t e e l production currently follows the conventional ingot c a s t i n g route. Although the adoption of continuous casting i s a c c e l e r a t i n g , s t a t i c ingot casting w i l l continue to be an important mode of s t e e l production for decades to come. The q u a l i t y of ingots i s a matter of great concern, p a r t i c u l a r l y so because defects can d e l e t e r i o u s l y a f f e c t the y i e l d of t h i s highly energy intensive process. One serious q u a l i t y problem that has been nagging the s t e e l industry i n at le a s t seven countries f o r over 40 years Is the formation of panel cracks. The term "panel" describes the l o c a t i o n of the cracks which frequently appear i n the concave, panel areas on f l u t e d or corrugated ingots. However, t h i s defect has also been c a l l e d center face c r a c k i n g , 1 l o n g i t u d i n a l c r a c k i n g , 2 - 8 p e a r l i t i c c r a c k i n g , 4 cooling cracking, l o n g i t u d i n a l surface c r a c k i n g , 9 thermal stress c r a c k i n g , 1 0 reheating cracking, phase transformation cracking, h a i r l i n e c r a c k i n g , 6 - 7 t o r t o i s e s h e l l c r a c k i n g , 1 1 transverse c r a c k i n g , 7 o v a l l y arranged c r a c k i n g , 1 2 v e r t i c a l c r a c k i n g , 1 3 and even c r a z y c r a c k i n g . 1 ' 1 4 These d i f f e r e n t names give an i n d i c a t i o n of the many d i f f e r e n t manifestations of panel cracks and the v a r i e t y of mechanisms that have been proposed to explain t h e i r formation. 2 Panel cracks appear In a v a r i e t y of low and medium-carbon k i l l e d s t e e l s but are always associated with aluminum-treated grades and are g r e a t l y a f f e c t e d by the thermal treatment of the ingot. They have been found i n a wide range of ingot sizes and shapes, from 1.5 ton, square, f l a t b i l l e t s to 30 ton rectangular, corrugated ingots. Round, f l u t e d ingots also have been a f f e c t e d . 8 The defect i s characterized by one or more i r r e g u l a r , i n t e r g r a n u l a r , cracks which generally run l o n g i t u d i n a l l y down the face of the ingot as shown i n Figure 1.1. They extend to a considerable depth below the surface and t r a v e l along the austenite grain boundaries. The reasons f o r panel crack formation are not f u l l y understood and many complicated mechanisms have been proposed. However, i t i s generally agreed that the problem i s caused by a combination of reduced intermediate-temperature (600-900°C) d u c t i l i t y i n v o l v i n g the presence of aluminum n i t r i d e , or "A1N", p r e c i p i t a t e s and s t r e s s generation due to both thermal contraction and phase transforma-t i o n . Panel cracks usually are not discovered u n t i l a much l a t e r stage i n ingot processing, t y p i c a l l y during r o l l i n g . They present a serious and expensive problem because affected slabs cannot be salvaged and must be scrapped. Since the cost of scrapping an ingot i s about $200 per ton there i s a strong incentive to discover methods to eliminate panel crack forma-t i o n . Thus, the present work was undertaken to improve our understanding of panel cracking with the objective of determining the mechanisms behind i t s formation. The approach taken In t h i s thesis i s to focus mainly on the str e s s f i e l d generated i n an ingot during the d i f f e r e n t processing stages. 3 F i g u r e 1.1 T y p i c a l a ppearance of p a n e l c r a c k r u n n i n g a l o n g c o r r u g a t i o n s o f a 760 x 1520 mm, 25 t o n s t e e l i n g o t 1 4 4 This i s determined through the development of mathematical heat trans f e r and st r e s s models using f i n i t e element analysis and, i n t h i s respect, i s the f i r s t panel cracking study of i t s kind. In add i t i o n , the work includes a study using a physical model of a soaking p i t and, f i n a l l y , a m e t a l l u r g i c a l examination of the cracks themselves. A l l previous work on panel cracking has involved s t a t i s t i c a l a n alysis of i n d u s t r i a l data or observations from plant t r i a l s . Fundamental studies have concentrated on the reduced hot d u c t i l i t y aspects of the problem. The next chapter f i r s t summarizes the voluminous l i t e r a t u r e i n t h i s area, p r i o r to c o l l e c t i n g and reviewing information c u r r e n t l y a v a i l a b l e on panel cracking i t s e l f . 5 CHAPTER 2 PREVIOUS WORK 2.1 Other Crack Problems In Steel At the outset i t Is important to d i s t i n g u i s h between panel cracks and other types of cracks that form i n ingots by d i f f e r e n t mechanisms. This i s p a r t i c u l a r l y important when so many studies on panel cracking r e f e r to i t by a d i f f e r e n t name. One of these d i f f e r e n t mechanisms i s "hot tea r i n g " or "hot shortness" which i s responsible for transverse cracks i n s t a t i c a l l y cast ingots. I t also gives r i s e to v i r t u a l l y a l l of the crack defects i n continuously cast s t e e l with the exception of transverse surface cracks. Cracks r e s u l t i n g from hot tearing are i n t e r d e n d r i t i c and exhibit a smooth fra c t u r e surface, s i m i l a r i n appearance to panel cracks. They form during the early stages of ingot s o l i d i f i c a t i o n i n a zone of low d u c t i l i t y just below the solidus temperature. The stresses causing the cracks are usually generated by s t i c k i n g or bending i n the mould. Hot tearing i s r e l a t i v e l y i n s e n s i t i v e to subsequent thermal treatment but i s strongly influenced by the sulphur and phosphorus content and manganese/sulphur, or "Mn/S", r a t i o i n the s t e e l as well as conditions i n the mould such as metal temperature, f i l l rate, mould design, and s t i r r i n g . Another type of cracking, often c a l l e d " c l i n k i n g " because i t i s audible, has a d i s t i n c t l y d i f f e r e n t mechanism from both panel cracking and 6 hot t e a r i n g . Clinks appear only i n high-carbon or a l l o y s t e e l grades with high carbon equivalents and are generated at lower temperatures (about 300°C). Cooling c l i n k s occur a f t e r an ingot has been stripped e a r l y and exposed to a cold atmosphere and reheating c l i n k s are formed when a cold ingot i s charged into a hot p i t and ra p i d l y heated. Both types of c l i n k s are thought to be caused s o l e l y by the generation of high thermal stresses i n the outer ingot skin and are not associated with e i t h e r A1N or a d u c t i l -i t y loss at intermediate temperatures. A t h i r d type of cracking, "hydrogen f l a k i n g " , also a f f e c t s medium-carbon s t e e l ingots and Is c l o s e l y associated with thermal treatment and the hydrogen content of the s t e e l . Hydrogen f l a k i n g i s caused by hydrogen gas nucleation i n the s o l i d s t e e l and can be c o n t r o l l e d e i t h e r by lowering the hydrogen content or by holding at or slow cooling the ingot through about 650°C to f a c i l l i t a t e hydrogen d i f f u s i o n . Turning from ingot casting to other s t e e l treatment processes, many experience cracking problems with features s i m i l a r to panel cracks. Several examples can be c i t e d . Sand castings of carbon and low a l l o y s t e e l s with high aluminum and nitrogen contents occasionally e x h i b i t intergranular cracks known as "rock candy f r a c t u r e " . 1 5 - 1 7 L o r i g and E l s e a 1 5 concluded i n 1947 that A1N p r e c i p i t a t i o n at primary austenite grain boundaries was the p r i n c i p l e cause of th i s f r a c t u r e . Woodfine and Q u a r r e l l 1 6 subsequently confirmed t h i s mechanism and added that the problem was most severe i n large castings where a slow cooling rate took place a f t e r s o l i d i f i c a t i o n or the temperature was held between 800 and 1100°C. A s i m i l a r mechanism accounts for "surface break up" i n large, aluminum g r a i n - r e f i n e d , low a l l o y , n i c k e l 7 bearing, s t e e l f o r g i n g s . 1 B Nickel causes the s o l u t i o n temperature of A1N to increase by about 100°C which then allows the p r e c i p i t a t i o n of A1N at f o r g -ing temperatures. This reduces the hot work a b i l i t y of the s t e e l and r e s u l t s i n c r a c k i n g . 1 8 "Y" crack formation during r o l l i n g of low-carbon, a l l o y s t e e l s i s also associated with A1N p r e c i p i t a t i o n at austenite grain bound-a r i e s . Chuen 1 9 determined that s t e e l composition, teeming temperature, r o l l i n g v e l o c i t y , track time and reheating p r a c t i c e most influenced t h i s type of crack. Low carbon, silicon-manganese s t e e l s are subject to "temper-embrittlement" during heat treatment or reheating, i f n i t r i d e s such as A1N are allowed to p r e c i p i t a t e . 2 0 - 2 2 Inoue determined that nitrogen segregation to the austenite grain boundaries causes severe embrittlement i f the s t e e l i s tempered at 500 - 600°C and quenched. 2 1 Another example can be found i n the welding of low-carbon steels since cracks can form i n the heat-affected z o n e . 2 3 - 2 5 Crack formation i s s i g n i f i c a n t l y affected by cooling r a t e , 2 4 the A1N c o n t e n t 2 4 and a loss i n d u c t i l i t y around 600°C. 2 5 Transverse cracking i s experienced i n continuously cast s t e e l s l a b s 2 6 - 3 1 i f stresses generated during straightening occur when the surface temperature of the strand i s i n the intermediate-temperature range of reduced d u c t i l i t y between 700 and 900°C. Also, i t i s suspected that continuously cast blooms may be subject to panel cracking a f t e r e x i t i n g the caster i n the same manner as s t a t i c a l l y cast i n g o t s . 8 Each of these defects involves A1N embrittlement as well as i n t e r -granular cracking along p r i o r austenite grain boundaries, and i s greatly influenced by the thermal treatment. These facts strongly imply that a s i m i l a r mechanism i s operating i n a l l of these cracking problems as 8 w e l l as i n panel cracking, despite the major dif f e r e n c e s between the metal-l u r g i c a l processes involved. The one fa c t o r apparently i n common to each of these problems i s a loss i n the hot d u c t i l i t y of s t e e l , so the next s e c t i o n reviews studies made in this area. P a r t i c u l a r a t t e n t i o n i s given to A1N p r e c i p i t a t i o n and thermal h i s t o r y e f f e c t s In the intermediate-temperature range. Emphasis i s placed on those aspects which are most c l o s e l y related to the formation of panel cracks. 2.2 Hot D u c t i l i t y of S t e e l Several d i f f e r e n t methods have been applied to determine the hot duct-i l i t y of s t e e l . Tests have been made with an Instron machine and induction furnace but t h i s t r a d i t i o n a l method has problems associated with premature necking. A l t e r n a t i v e l y , researchers have employed t o r s i o n - t e s t i n g machines to achieve higher s t r a i n s before f r a c t u r e . However, the accuracy of both these methods has been questioned owing to the d i f f i c u l t y of reproducing an "as cast" structure by reheating s o l i d material from ambient temperature. To overcome t h i s , many workers have used a Gleeble machine, i n which a specimen can be melted and r e s o l i d i f i e d " i n s i t u " , slowly cooled, and then tested, possibly representing a better simulation of true casting condi-t i o n s . However, a disadvantage to t h i s technique i s that only a very small amount of material i s tested (about 1 cm 3) so l o c a l nonuniformities can play a large r o l e . The l o c a l nature of the test also makes i t impossible to record actual load and elongation so that mechanical behavior must be i n f e r r e d s o l e l y from reduction-in-area measurements and analysis of the cooled f r a c t u r e surface. 9 Notwithstanding these d i f f i c u l t i e s , numerous studies have been per-formed on the hot d u c t i l i t y of s t e e l which has been found to c o r r e l a t e remarkably well with a v a r i e t y of cracking problems. The following sections w i l l describe the d i f f e r e n t temperature zones of lowered d u c t i l i t y f o r plain-carbon and low-alloy s t e e l s . 2.2.1 Zones of reduced d u c t i l i t y In general, the d u c t i l i t y of s t e e l at elevated temperatures i s e x c e l -l e n t . However, there are at l e a s t two d i s t i n c t temperature regions i n which i t s d u c t i l i t y drops markedly. The f i r s t of these appears at high tempera-tures within about 50°C of the solidus temperature. The d u c t i l i t y i n t h i s zone i s v i r t u a l l y zero and as mentioned e a r l i e r , i s responsible f or hot t e a r i n g . The second range of reduced d u c t i l i t y extends from as high as 1200 o c 3 2 » 3 3 t Q a s l o w a s 6 0 0 ° C , 3 0 as shown i n F i g u r e 2.1. In t h i s broad i n t e r v a l , reduction i n area or "R.A." can have almost any value, ranging from a minimum of about 10% to almost 100%. While the t o t a l s t r a i n to f r a c t u r e can occ a s i o n a l l y approach zero, there i s always some l o c a l deforma-t i o n 3 4 which distinguishes low d u c t i l i t y cracking i n t h i s region from the high temperature zone. This second " d u c t i l i t y trough" can be divided further into at l e a s t two overlapping temperature zones i n which d i f f e r e n t e m b r i t t l i n g mechanisms operate. One of these a f f e c t s s t e e l while i t i s e n t i r e l y i n the austenite phase and l i e s i n an intermediate temperature range from the Ar temperature 10 500 7 0 0 9 0 0 1100 1300 1500 Temperature (°C) Figure 2 . 1 Hot ducti l i t y of low-carbon steels containing manganese and aluminum as reported by various researchers 11 ( u s u a l l y about 800°C) to as high as 1200°C. Because the s t a r t of the aus-t e n i t e - f e r r i t e or "y •*• a" phase transformation can be s u b s t a n t i a l l y delayed below equilibrium i n micro-alloy s t e e l s , t h i s temperature region can extend to below 700°C. While the mechanisms operating i n t h i s region are poorly understood, they are related to p r e c i p i t a t e pinning e f f e c t s , grain boundary s l i d i n g and delayed r e c r y s t a l l i z a t i o n . The other zone i s c l o s e l y associated with the y •*• a phase transforma-t i o n and l i e s i n a lower temperature range below the Ar^• S t r a i n concentra-t i o n at f e r r i t e networks surrounded by austenite, or p e a r l i t e at lower temp-eratures, are responsible for reduced d u c t i l i t y i n t h i s zone. This mechan-ism may extend to below the Ar^ temperature where f e r r i t e networks sur-rounded by p e a r l i t e may constitute a f i n a l zone of lower d u c t i l i t y . Although innumerable i n v e s t i g a t i o n s have been made on the hot d u c t i l i t y o f s t e e l below 1 2 0 0 ° C , 1 " 3 ' 7 ' 1 5~1 8> 2 4 » 29-51 f e w r e S e a r c h e r s have found two d i s t i n c t intermediate-temperature d u c t i l i t y troughs at the same time. E i t h e r one, the other, or a combination of the two usually i s observed, depending on the l o c a t i o n of the Ar^ phase transformation temperature with respect to the observed d u c t i l i t y trough. Although a d u c t i l i t y trough has been d i r e c t l y linked to the Ar^ temperature, 3 7' 4 8 ' 4 9 other studies report i t extends to much h i g h e r t e m p e r a t u r e s 4 3 ' 4 4 w h i l e s t i l l others f i n d a trough e n t i r e l y above the A r ^ . 2 9 ' 3 2 ' 3 4 Because the y •* a phase transform-ati o n i s influenced by a l l o y i n g elements, cooling rate, s t r a i n rate and p r e c i p i t a t e a c t i o n i t s e l f , r e l a t i n g the los s i n d u c t i l i t y to the Ar^ temper-ature i s often complicated. Thus, the d i v i s i o n of the second d u c t i l i t y trough into two further zones i s more due to a d i f f e r e n c e i n fr a c t u r e mech-12 anisms and s t e e l phases present than to separable ranges of temperature over which the mechanisms operate. These temperature zones of reduced d u c t i l i t y and t h e i r corresponding e m b r i t t l i n g mechanisms are i l l u s t r a t e d schematically i n Figure 2.2. The next sections w i l l elaborate on each i n turn. 2.2.2 High Temperature Zone At temperatures just below the s o l i d u s , the s t r a i n - t o - f r a c t u r e of s t e e l i s only about 1%. Many studies have been conducted on t h i s zone of reduced d u c t i l i t y 1 * 7 ' 5 2 - 6 2 and the mechanisms that are operative are probably the best understood. The d u c t i l i t y i s reduced by the microsegregation of S and P r e s i d u a l s at s o l i d i f y i n g dendrite i n t e r f a c e s which lowers the solidus temperature l o c a l l y i n the i n t e r d e n d r i t i c regions. The d u c t i l i t y remains e f f e c t i v e l y zero u n t i l the i n t e r d e n d r i t i c l i q u i d f i l m s begin to freeze. Severe embrittlement i s experienced at a l l temperatures above the "zero d u c t i l i t y temperature" which occurs within 30 - 70°C of the solidus as shown i n Figure 2.3. Any s t r a i n that i s applied to the s t e e l i n t h i s temperature region w i l l propagate cracks outward from the s o l i d i f i c a t i o n front between dendrites. The r e s u l t i n g f r a c t u r e surface e x h i b i t s a smooth, rounded appearance, c h a r a c t e r i s t i c of the presence of a l i q u i d f i l m at the time of f a i l u r e . The t r a n s i t i o n from b r i t t l e to d u c t i l e behavior occurs over a narrow temperature range on h e a t i n g . 5 2 However, on cool i n g , d u c t i l i t y may not approach 100% u n t i l somewhat lower temperatures are reached. In f a c t , 13 Crock Temperature (°C) Figure 2.2 Schematic representation of temperature zones of reduced hot d u c t i l i t y of s t e e l r e l a t e d to e m b r i t t l i n g mechanisms 14 Temperature Figure 2.3 Relationship between mechanical properties i n the high temperature zone of reduced d u c t i l i t y and: (A) corresponding schematic presentation of s o l i d / l i q u i d i n t e r f a c e during casting (B) carbon content (from S u z u k i 6 2 ) 15 Figure 2.3 indicates that some embrittlement may be encountered at tempera-tures as low as 1200°C. In theory, t h i s e m b r i t t l i n g mechanism can continue to operate to as low as the Fe-FeS e u t e c t i c temperature of 980°C. 3 2 > 3 1 0 6 3 Increased contents of S, P, S n , 5 9 C u , 7 ' 6 0 and S i 4 7 a l l worsen the d u c t i l i t y . Manganese i s b e n e f i c i a l since i t p r e f e r e n t i a l l y combines with S to form le s s harmful MnS p r e c i p i t a t e s , thereby preventing l i q u i d f i l m formation. A Mn/S r a t i o greater than 20 minimizes cracking by t h i s mechanism, 5 7 p a r t l y by r a i s i n g the d u c t i l e / b r i t t l e t r a n s i t i o n temperature. This zone of reduced d u c t i l i t y i s responsible f or the problem of hot tearing as previously discussed. The d u c t i l i t y i s r e l a t i v e l y i n s e n s i t i v e to subsequent thermal treatment 3** and s t r a i n r a t e . 5 2 In comparison, the fact that panel cracking i s s e n s i t i v e to the thermal h i s t o r y of the s t e e l ingot suggests, i n the absence of s t r a i n generation considerations, that the high-temperature zone of low d u c t i l i t y plays an i n s i g n i f i c a n t r o l e In the problem. 2.2.3 Intermediate-Temperature Zone With descending temperature, the second drop i n d u c t i l i t y experienced by s t e e l i s i n the single austenite phase and extends from the Ar^ tempera-ture to as high as 1 2 0 0 ° C . 3 3 , 3 7 Above t h i s temperature range, dynamic r e c r y s t a l l i z a t i o n occurs so r e a d i l y that high d u c t i l i t y i s assured, v i r t u a l l y unaffected by s t e e l composition and processing conditions. While a great deal of study has been done on the d u c t i l i t y of austenite below 1 2 0 0 ° C , 1 8 ' 2 4*' 2 9 - 5 0 e l u c i d a t i o n of the mechanisms o p e r a t i n g i n t h i s temperature region i s incomplete owing to t h e i r complexity. 16 The d u c t i l i t y of s t e e l specimens i n t h i s temperature range i s d i r e c t l y r e f l e c t e d i n the appearance of the frac t u r e surface. H i g h - d u c t i l i t y f r a c -tures are transgranular with c h a r a c t e r i s t i c dimples and a few large p r e c i p i -t a t e s , i n d i c a t i n g that fracture i n i t i a t e d at i s o l a t e d i n c l u s i o n s dispersed throughout the matrix. In contrast, tests made i n a temperature region of low d u c t i l i t y always exhibit an intergranular f r a c t u r e along austenite grain boundaries making large angles with the major stress a x i s . In f a c t , the v a r i a t i o n i n RA values correlates well with the f r a c t i o n of fr a c t u r e surface occurring on austenite grain b o u n d a r i e s . 3 8 The surface of specimens fractured i n the intermediate temperature zone of reduced d u c t i l i t y was intergranular and wavy with numerous p r e c i p i t a t e s of v a r y i n g types including sulphides (Mn, Fe, and possibly A l 8 1 ) , 1 ' 3 4 ' 3 7 ' 5 0 o x i d e s (Mn, Fe and A l ) 3 0 ' 3 7 and n i t r i d e s (A1N, 2 9' 1 + 1" l t l + niobium carbo-n i t r i d e compounds, or MNb(C,N)" 2 9' 3 7> 3 8 » 4 4 ' 4 5 ' 4 7 ' 6 4 » 6 5 and BN 2 3' 4 6 ) . Most of the f r a c t u r e surfaces i n d i c a t e a creep-type f a i l u r e due to the coa-lescence of c a v i t i e s nucleating at the grain-boundary p r e c i p i t a t e s . 2.2.3.1 E f f e c t of St e e l Composition S t e e l c o m p o s i t i o n i s extremely important i n d e t e r m i n i n g the intermediate-temperature d u c t i l i t y of low-alloy s t e e l s and has received the greatest a t t e n t i o n by researchers. While embrittlement i n th i s temperature range does not occur i n h i g h - p u r i t y i r o n 3 0 ' 3 7 ' 4 9 , i t has been found i n both plain-carbon s t e e l 4 5 and an Fe-0.24Si binary a l l o y . 6 6 These observa-tions suggest that embrittlement i s not possible without some p r e c i p i t a t e s and reveal the importance of even minor amounts of r e s i d u a l elements. 17 One of the most I n f l u e n t i a l elements a f f e c t i n g d u c t i l i t y i n t h i s region i s aluminum. As shown in Figure 2.4, increasing dissolved A l content, or "ASA" (acid soluble aluminum), within the range of 0.02 to 0.06% causes a marked drop i n hot d u c t i l i t y , p a r t i c u l a r l y below 900°C. 2 1** 3 8 ' k 3 t 4 5 ' 5 0 I t also extends the upper l i m i t of the d u c t i l i t y trough occurring i n p l a i n C s t e e l s to higher temperatures. 2 1*' 2 9 ' **3' 5 0 Further increases i n ASA above 0.07% recovers the d u c t i l i t y somewhat, presumably due to A1N p r e c i p i t a t e c o a r s e n i n g . 1 8 ' 21*' 2 9 The action of A l i n determining d u c t i l i t y i s undoubt-edly due to the p r e f e r e n t i a l p r e c i p i t a t i o n of A1N at the austenite grain boundaries. 1* 1* I t also r e f i n e s the austenite grain s i z e and retards austen-i t e r e c r y s t a l l i z a t i o n . Mintz & Arrowsmith 2 9' 4 5 report that increasing ASA also aggravates the e f f e c t of Nb(C,N) p r e c i p i t a t e s , causing them to become f i n e r , more c l o s e l y spaced and concentrated at the grain boundaries. The influence of niobium i s quite s i m i l a r to that of aluminum, both i n e f f e c t and s e v e r i t y as shown i n Figure 2.5. Increasing Nb content again causes a drop i n d u c t i l i t y v a l u e s 2 9 * 3 1 ' t f 3' 4 5 but i t i s even more i n f l u -e n t i a l than A l i n extending the trough to higher temperatures. 2 9' 3 1 ' 3 7 ' 3 8> 4 3 - 4 5 Researchers studying A l s t e e l s both with and without Nb observe that Nb(C,N) p r e c i p i t a t e s tend to predominate at higher temperatures while A1N i s more associated with the lower temperature 700 - 900°C ran g e . 3 8 ' 1*3' 4 5 Steels containing both A l and Nb have the deepest, widest d u c t i l i t y t r o u g h s . Niobium p r e c i p i t a t e s as NbC ,N i n h i g h N s t e e l s 3 8 ' 6 4 or NbC 7 C. i n low N s t e e l s 6 5 and i s r a t e c o n t r o l l e d by d i f f u s i o n of Nb i n a u s t e n i t e . 6 7 In s t e e l s where boron i s present, s i m i l a r observations to those wit-nessed f o r A l & Nb are r e p o r t e d 2 3 ' 4 6 (See Figure 2.6). I t i s presumably 1 8 500 900 1300 Temperature (°C) Figure 2.4 E f f e c t of A l on the hot d u c t i l i t y of s t e e l 43 100 o a> o 2 50 0) 25 o 3 0%Nb / — I j 0£»5%Nb V x /^a025%Nb V* V\ // ' \ x " / Increasing Nb \ %C 017-0.19 %Mn 1.5 %Si 0.36-050 % P 0.20 %S 0.01-Q02 %AI 0.035 e w 10 s _L 600 800 1000 1200 1400 Temperature (°C) Figure 2.5 E f f e c t of Nb on the hot d u c t i l i t y of s t e e l 1 9 due to the same mechanism with BN p r e c i p i t a t e s taking the place of or acting simultaneously with A1N and Nb(C,N). Knowing that n i t r i d e p r e c i p i t a t e s ( A l & Nb) are l a r g e l y responsible for lowering d u c t i l i t y , i t i s not s u r p r i s i n g that increasing nitrogen contents are associated with decreasing d u c t i l i t y 2 4 ' 3 0 ' 3 1 ' 3 8 ' 4 5 ' 5 0 and extension of the l o w - d u c t i l i t y trough to higher temperatures. 3 0 I t s e f f e c t i s not nearly as dramatic as that of A l and Nb, but unlike these elements, N i s also deleterious to the low temperature properties of s t e e l : strength and toughness. 6 8 Its e f f e c t i s i l l u s t r a t e d i n Figure 2.7. Vanadium, another common micro-alloy, and n i t r i d e former, i s not nearly as detrimental to d u c t i l i t y as A l and Nb and may even be b e n e f i c i a l . Some believe V acts i n a s i m i l a r manner to Nb but has a reduced e f f e c t because of the h i g h e r s o l u b i l i t y of VN i n a u s t e n i t e . 3 1 ' 4 5 Others claim that V may improve d u c t i l i t y 2 ' 1 3 > 2 0 p a r t i c u l a r l y at lower temperatures, since i t hinders A1N p r e c i p i t a t i o n . 6 9 The f i n a l n i t r i d e former, titanium, i s unique i n being the only element that i s unquestionably b e n e f i c i a l i n reducing the d u c t i l i t y problem. Figure 2.8 shows that although T i i s not e f f e c t i v e i n t o t a l l y removing the d u c t i l i t y t r o u g h 3 8 , i t can reduce i t to the same depth and extent as that observed f o r p l a i n C/Mn s t e e l s . 4 2 ' 4 5 Titanium appears able to eliminate the d e t r i m e n t a l e f f e c t s of A l 7 ' 4 5 by preventing A1N formation. I t does t h i s by p r e f e r e n t i a l l y combining with the a v a i l a b l e nitrogen and p r e c i p i t a -t i n g c o a r s e r , l e s s harmful, TIN p r e c i p i t a t e s 3 8 ' 4 1 * 4 2 ' 4 5 d i s t r i b u t e d throughout the matr i x . 7 I t also p r e c i p i t a t e s at higher temperatures due to 20 Figure 2.6 E f f e c t of Boron on the hot d u c t i l i t y of s t e e l 4 6 I O O I — T 0 0 0 8 % N O L O I I %N 1 1 1 Figure 2.7 7 0 0 8 0 0 9 0 0 1 0 0 0 Test temp (°C) E f f e c t of Nitrogen on the hot d u c t i l i t y of C-Mn-Al-Nb s t e e l s 4 5 2 1 Figure 2.8 E f f e c t of Titanium on the hot d u c t i l i t y of s t e e l (from F u n n e l 4 2 ) 22 i t s lower s o l u b i l i t y , leaving less N for the subsequent p r e c i p i t a t i o n of more detrimental n i t r i d e s . Titanium has a s i m i l a r but reduced e f f e c t on Nb b e a r i n g s t e e l s , 4 5 probably because Nb can a l s o form c a r b i d e s . 3 8 Apart from n i t r i d e s , which play t h e i r most important r o l e at lower temperatures, sulphides are seen to be detrimental.over the whole range of reduced d u c t i l i t y . They are p a r t i c u l a r l y damaging at the higher temperature 1000-1200°C range and at higher s t r a i n rates (above 10~ 3 s _ 1 ) . Increasing s u l p h u r content both deepens 3 2' 3 3 ' 3 4 ' 3 7 ' 6 4 and widens 3 0' 3 3 ' 3 7 the hot d u c t i l i t y trough, as shown i n Figure 2.9. The a d d i t i o n of manganese to a c h i e v e Mn/S r a t i o s greater than 20 g r e a t l y reduces the d u c t i l i t y t r o u g h 3 0 ' 32> 3"+i 36» 37 t ^ e s a m e mechanism t h a t i t a l l e v i a t e s the hot tearing problem. However, very high manganese l e v e l s (>1.6%) are reported to lower d u c t i l i t y , p o s s i b l y due to matrix h a r d e n i n g . 3 8 ' 5 1 Because the important e f f e c t s of these elements are themselves so com-p l i c a t e d , i t i s d i f f i c u l t to determine the possible influences of other minor elements. While P i s c l e a r l y detrimental at hot tearing temperatures, i t i s much less important i n the Intermediate temperature region. Some s t a t e i t has a n e g l i g i b l e e f f e c t 3 2 ' 3 8 while others even report a s l i g h t i n c r e a s e i n d u c t i l i t y w i t h i n c r e a s i n g P up to 0.3%. 2 9' 3 0 ' 4 5 This was a t t r i b u t e d to hindrance of Nb(C,N) p r e c i p i t a t i o n by P segregation to the austenite grain boundaries. Researchers hold mixed views on the influence of carbon, ranging from low-carbon s t e e l having lower d u c t i l i t y 3 1 to medium carbon s t e e l being w o r s e 1 8 ' 3 2 ' 4 9 to there being no e f f e c t at a l l . 3 6 ' 3 8 Although i t s a c t i o n 23 Temperature ( ° C ) Figure 2.9 E f f e c t of S on hot d u c t i l i t y of low-carbon s t e e l (adapted from Weinberg 3 3) 24 i s also unclear, molybdenum additions may be b e n e f i c i a l . Calcium additions may improve d u c t i l i t y by reducing sulphur and oxygen l e v e l s . 7 0 F i n a l l y , oxygen has been seen to have only a s l i g h t l y d eleterious e f f e c t , 2 8 ' 5 4 pre-sumably due to i t s contribution to (Fe, Mn, Al) i n c l u s i o n s and reduction i n i n t e r n a l c l e a n l i n e s s . Oxide p r e c i p i t a t e s are far less damaging than e i t h e r n i t r i d e s or sulphides. 2.2.3.2 E f f e c t of Thermal History The other major variable a f f e c t i n g hot d u c t i l i t y i s thermal h i s t o r y . Since time i s one of the basic parameters c o n t r o l l i n g the e m b r i t t l i n g pro-cesses, the e f f e c t s of thermal h i s t o r y are n a t u r a l l y linked to s t r a i n r a te. Thermal h i s t o r y e f f e c t s are r e v e r s i b l e since several researchers have observed that r e a u s t e n i t i z i n g a sample e x h i b i t i n g low d u c t i l i t y at a p a r t i c -u l a r test temperature, followed by a "favorable" thermal treatment and r e t e s t i n g at the same temperature, r e s t o r e s good d u c t i l i t y . 1 6 ' 1 7 ' 3 8 Unfortunately, there i s wide disagreement as to what constitutes a d e l e t e r -ious or favorable thermal treatment. The findings seem to depend, at l e a s t i n part, on the p r e c i p i t a t e species responsible for embrittlement. The time and temperature of annealing used to " i n i t i a l i z e " the sample has been found by several researchers to be very important. More severe embrittlement and extension of the d u c t i l i t y trough to higher temperatures occurs when the maximum heating temperature i s above the i n c i p i e n t g rain boundary m e l t i n g t e m p e r a t u r e . 3 3 ' 3 7 The same extended embrittlement was found when annealing for short times (60 s) at 1425°C 6 6 and to a l e s s e r 25 e x t e n t at temperatures between 1300°C and 1400°C. 3 4' 3 7 However, Wray also reports that annealing for long times at high temperature produces extensive g r a i n growth with no embrittlement when subsequently tested at 950°C. 6 6 Several studies report that annealing temperature has no e f f e c t on d u c t i l i t y provided that annealing i s done above the s o l u t i o n temperature of the various n i t r i d e s present for s u f f i c i e n t time to ensure r e d i s s o l u t i o n . 3 8 ' 4 3 This should take only 300 s at 1300°C for A1N which i s quick to d i s s o l v e when reheated above i t s s o l u t i o n temperature. However, longer times, i n excess of 1800 s may be needed for Nb(C,N) which i s much slower to d i s -s o l v e . 4 4 ' 7 1 Since many workers did not use a Gleeble apparatus, i t remains uncertain whether t h e i r experimental r e s u l t s r e f l e c t those of actual casting conditions, where the s t e e l i s i n i t i a l l y molten. S u b s t a n t i a l l y d i f f e r e n t behavior i s found i f A1N p r e c i p i t a t e s s t i l l remain a f t e r a n n e a l i n g . 4 0 Embrittlement i n t h i s temperature region i s s e n s i t i v e not only to annealing temperature but also to the subsequent thermal h i s t o r y . Slower cooling rates c o n s i s t e n t l y are found to be b e n e f i c i a l , both when sulphides were invo l v e d 3 0 ' 3 2 ' 3 7 (.05 vs 5 °C/s) and wh en A1N was the major embrit-t l i n g s p e c i e s . 2 4 ' 2 9 (1 vs 25 °C/s) . Many workers have found that short, (200-3000 s) isothermal periods before t e s t i n g also greatly improves d u c t i l i t y at temperatures above 900°C, when s u l p h i d e e m b r i t t l e m e n t i s i n v o l v e d . 3 2 ' 3 4 ' 3 7 F i g u r e 2.10 (A) i l l u s t r a t e s t h i s e f f e c t . However, when n i t r i d e s are responsible, holding 26 before t e s t i n g from 900 to 1800 s r e s u l t s i n s u b s t a n t i a l l y reduced d u c t i l i t y below 1 0 0 0 ° C . 3 8 ' 4 4 Even at higher temperatures, d u c t i l i t y decreases unless h o l d i n g i s done f o r much l o n g e r t i m e s . 3 8 ' 4 4 (more than 5400 s at 1 1 5 0 ° C ) 4 2 . This r e f l e c t s the slow k i n e t i c s of n i t r i d e p r e c i p i t a t i o n , p a r t i c u l a r l y Nb(C,N). C o o l i n g below the temperature and reheating to the a u s t e n i t i c range c o m p l e t e l y e l i m i n a t e s embrittlement, at l e a s t when MnS i s i n v o l v e d . 3 2 ' 3 4 ' 3 7 However, cooling e i t h e r into the two-phase region or just above the Ar^ r e s u l t s i n more severe n i t r i d e embrittlement. As shown i n Figure 2.10 (B), t h i s extends the zone of lower d u c t i l i t y to higher temperatures, p a r t i c u l a r l y when Nb i s p r e s e n t . 3 8 F i n a l l y , c y c l i n g the temperature a c r o s s the A^ temperature, such as occurs during continuous ca s t i n g , was found to be very d e t r i m e n t a l . 3 1 ' 4 2 2.2.3.3 E f f e c t of S t r a i n Rate Researchers who test at low s t r a i n rate and/or a t t r i b u t e d u c t i l i t y losses to n i t r i d e p r e c i p i t a t e s , unanimously concur that d u c t i l i t y i n t h i s temperature range decreases with decreasing s t r a i n r a te. Figure 2.11 shows that as s t r a i n rate i s lowered below about 10" 3 s - 1 , any observed d u c t i l i t y trough deepens d r a s t i c a l l y . This e f f e c t i s most prominent at temperatures below 9 0 0 ° C . 2 9 ' 3 0 ' 3 1 ' 3 8 ' 4 0 ' 4 3 While lowering the s t r a i n rate also has the e f f e c t of extending the d u c t i l i t y trough to higher temperatures, 7 2 the temperature of lowest d u c t i l i t y remains just above the A r , . 3 8 27 Time (min) 10 100 1000 Holding time (s) Thermal cycle 800 1000 Temperature (°C) Figure 2.10 E f f e c t of thermal treatment on intermediate-temperature d u c t i l i t y i l l u s t r a t i n g the e f f e c t s of: (A) hold time p r i o r to t e s t i n g 3 7 (B) dropping temperature i n t o the two-phase region p r i o r to t e s t i n g 3 8 28 O <U - c o u a> U a 100 80 60 40 20 100 80 60 40 20 0 i(s-') O V • A o 9 I 1X10 4X10"' 6X10"4 I XlO'4 7XI0"5 %C 0.15 % SI 0.29 %Mn 1.35 %S 0.006 % P 0.014 %N 0.0066 %AI Q022 + A O O — A ^ o -) — ° / • / A ' ' /%C0.I5_ / / %Si0.29 / / %Mn 1.35 O . - A ' / \ t %S Q006__| %P 0.014 %N 0.0056 %AI 0.022 %Nb 0.054 i 600 700 800 900 1000 Temperature (°C) Figure 2.11 Dependence of intermediate-temperature d u c t i l i t y on s t r a i n r a t e 3 8 f o r : (A) low-carbon, Si-Mn s t e e l (B) Nb-bearing s t e e l 29 At higher s t r a i n rates, (above 10" 3 s _ 1 ) the influence of this v a r i a b l e i s not as c l e a r . Researchers a t t r i b u t i n g embrittlement to n i t r i d e ( A l or Nb) p r e c i p i t a t e s s t i l l f i n d lower s t r a i n rates c o n s i s t e n t l y detrimental to d u c t i l i t y . 3 7 ' 4 0 ' 6 6 However, many of those a t t t r i b u t i n g embrittlement to sulphide (Fe or Mn) p r e c i p i t a t e s f i n d decreasing s t r a i n rate e i t h e r has no i n f l u e n c e 3 2 or increases d u c t i l i t y , 3 7 p a r t i c u l a r l y at higher temperatures (900-1200°C). 3 8 2.2.3.4 E f f e c t of Grain Size The grain boundaries are the weak l i n k i n s t e e l (and other metals) at elevated temperatures. Thus, coarse grained materials should exhibit lower d u c t i l i t y , p a r t i c u l a r l y at lower s t r a i n rates, where the grain-boundary weakening mechanisms have time to operate. This i s because s t r a i n concentration at the weakened grain boundaries i s enhanced when less grain boundary area i s present. Many studies indeed have determined coarse grain s i z e to be associated with lower d u c t i l i t y 3 0 * 3 1 ' 3 7 > 3 9 ' 4 3 > 4 5 but several others found f i n e grains to be worse 3 3' 4 1 ' 4 2 and another stated i t was not imp o r t a n t . 3 8 These contradictory findings are more understandable when one considers that grain s i z e i s i n t r i n s i c a l l y related to other variables such as grain r e f i n i n g agents and thermal h i s t o r y which themselves are highly i n f l u e n t i a l on hot d u c t i l i t y . The e f f e c t of A l additions on grain s i z e i s p a r t i c u l a r l y important to note. Some of the aluminum added to s t e e l acts as a deoxidant, being the most e f f e c t i v e and economical element to perform t h i s r o l e . More impor-t a n t l y , however, the remainder i s dissolved i n the s t e e l and i s very e f f e c -30 t i v e i n c o n t r o l l i n g the a u s t e n i t i c grain s i z e . The dissolved aluminum, or other grain r e f i n i n g element such as Nb, T i and to some extent Zr and V, accomplishes t h i s by preventing the grain growth that normally occurs at high temperature with increasing time. These a l l o y i n g elements act by form-ing "obstruction agents" which mechanically obstruct grain growth by pinning the austenite grain boundaries. The obstruction agents are f i n e , d i s c r e t e , p a r t i c l e s ( u s u a l l y carbides or n i t r i d e s ) that p r e c i p i t a t e during cooling at the austenite grain boundaries when t h e i r s o l u b i l i t y l i m i t has been exceeded. A1N p r e c i p i t a t e s are p a r t i c u l a r l y e f f e c t i v e 7 3 but niobium carbo-n i t r i d e s and titanium n i t r i d e s are also s u i t a b l e . 7 1 * With Increasing temperature during reheating, the f i n e p a r t i c l e s both coalesce (Ostwald ripening) and s t a r t to di s s o l v e back into s o l i d s o l u t i o n . As the p r e c i p i t a t e s coarsen and reduce i n number, the pinning e f f e c t i s reduced. When the rate of release of energy per unit displacement of gr a i n boundary (during grain growth) exceeds the rate of increase i n energy due to the unpinning process, grain growth o c c u r s . 7 5 At t h i s c r i t i c a l temperature, c a l l e d the grain coarsening temperature, detrimental secondary r e c r y -s t a l l i z a t i o n can begin. It i s manifested by the rapid growth of a few grains to large sizes which lowers f i n a l product q u a l i t y and consistency. Grain size control i s therefore very desirable during l a t e r reheating stages. The f i n e grain size r e s u l t i n g from A l addition has been c o r r e l a t e d with a decrease i n low temperature d u c t i l e / b r i t t l e t r a n s i t i o n temperature, an increase i n both low temperature strength and toughness and improved w e l d a b i l i t y , aging resistance and d i s t o r t i o n r e s i s t a n c e . 1 8 31 However, t h i s mechanism by which A l prevents grain coarsening Is the same one that contributes to a lowered d u c t i l i t y i n the Intermediate-temp-erature range. Indeed, the same factors that r e s u l t In lowered d u c t i l i t y have been found to achieve the f i n e s t g rain s i z e . For example, the grain coarsening temperature of s t e e l depends to a large extent on the s o l u t i o n temperature of A1N. 1 8' 7 6 The highest grain coarsening temperatures r e s u l t from large volume f r a c t i o n s of very f i n e p r e c i p i t a t e s made from i n t e r -m e t a l l i c compounds with low s o l u b i l i t y p r o d u c t s . 7 6 Because of i t s low solu-b i l i t y (see Table 2.1) and i t s readiness to form very fine p r e c i p i t a t e s (< 1 micron), A1N i s a very e f f e c t i v e obstruction agent. However, i f the alum-inum content of the s t e e l i s too high (above 0.08%), A1N can p r e c i p i t a t e at higher temperatures as coarse p a r t i c l e s which reduce i t s effectiveness and r e s u l t s i n a lower grain coarsening temperature as shown i n Figure 2.12. An optimum range exists at 0.015 - 0.05% A l . 3 The addition of titanium w i l l p r e f e r e n t i a l l y combine with the nitrogen to reduce A1N formation and again lower the grain coarsening temperature. Thus, when a fi n e grain s i z e i s observed associated with a lower duct-i l i t y , i t may simply r e f l e c t the effectiveness of n i t r i d e p r e c i p i t a t e s both i n preventing grain growth and i n weakening the austenite grain boundaries of that p a r t i c u l a r sample. 4 2 On the other hand, the action of A1N p r e c i p i -tates may prevent dynamic r e c r y s t a l l i z a t i o n which could r e s u l t i n a coarser grain s i z e . In t h i s case, lower d u c t i l i t y would appear associated with the coarse grain size for the a d d i t i o n a l reasons that A1N p r e c i p i t a t e a c t i o n has accelerated grain boundary cavity n u c l e a t i o n 3 9 and reduced grain boundary m o b i l i t y . 4 5 In conclusion, grain s i z e i t s e l f i s not always a major factor c o n t r o l l i n g the hot d u c t i l i t y of s t e e l . I t often simply r e f l e c t s the 3 2 Table 2.1 S o l u b i l i t y of various n i t r i d e s In austenite Compound S o l u b i l i t y (weight percent) VN l o g 1 0 [V] [N] = = -8300/T (°K) + 3.46 very soluble NbN l o g 1 0 [Nb] [N] = • -8500/T +2.80 A1N log [Al] 10 [N] = = -6770/T + 1.03 TiN log [Ti] 10 [N] £ 5 -7.0 @ 1100°C ZrN very stable 0 0.02 004 006 008 0J0 0.12 0.14 Element content (wt%) Figure 2.12 E f f e c t of various elements on the grain-coarsen temperature of plain-carbon s t e e l 34 influences of the processes which do co n t r o l d u c t i l i t y such as the e f f e c t -iveness of n i t r i d e p r e c i p i t a t e s i n grain boundary pinning. 2.2.3.5 Mechanisms I t i s f a i r l y well agreed that trends observed i n the intermediate-temperature zone of reduced d u c t i l i t y can be explained l a r g e l y by the action of p r e c i p i t a t e s at the austenite grain boundaries. However, theories d i f f e r as to how these p r e c i p i t a t e s operate to reduce d u c t i l i t y . While many of the apparently contradictory findings can be explained, the phenomena are f a r from being completely understood. There appear to be at l e a s t three separate mechanisms operating both simultaneously and independently to account for the complex behavior observed i n t h i s temperature region. Under any p a r t i c u l a r set of conditions, any one of these i s l i k e l y to predom-inate . Wray has represented these d i f f e r e n t fracture mechanisms i n terms of d i f f e r e n t s t r a i n - r a t e temperature zones v i a a fracture map shown i n Figure 2.13 6 6. Zone A i s the high-temperature zone responsible for hot t e a r i n g . Zones B, C, and D correspond to three mechanisms involving the a u s t e n i t i c phase: sulphide embrittlement at high s t r a i n rate, d u c t i l e intragranular f r a c t u r e , and intergranular creep f r a c t u r e , r e s p e c t i v e l y which operate i n t h i s intermediate-temperature zone. Zone E r e f e r s to the e m b r i t t l i n g mechanism operating i n the two-phase austenite and f e r r i t e region below the A^ temperature. These same lab e l s were used to i d e n t i f y the low d u c t i l i t y zones In Figure 2.2. 35 400 800 1200 1600 Temperature (°C) Figure 2.13 Possible fracture zones mapped for 0.2% C p l a i n -carbon s t e e l i n s t r a i n - r a t e temperature s p a c e 6 6 3 6 At low s t r a i n r a t e s (below 10~ 3 s - 1 ) , e m b r i t t l e m e n t o c c u r s i n a u s t e n i t e by t h e n u c l e a t i o n , growth and c o a l e s c e n c e of g r a i n boundary v o i d s . Under s t r e s s , s m a l l p r e c i p i t a t e p a r t i c l e s a s s i s t i n i n i t i a t i n g m i c r o - f i s s u r e s , o r e q u i a x e d , c r e e p c a v i t i e s w i t h f a c e t e d s u r f a c e s . W i t h i n c r e a s i n g s t r a i n , t h e c r e e p c a v i t y d e n s i t y i n c r e a s e s . 7 8 I n a d d i t i o n , growth o c c u r s by s p r e a d i n g a l o n g t h e g r a i n boundary p l a n e . 3 9 I f e i t h e r the r a t e of c a v i t y n u c l e a t i o n i s h i g h , 7 8 o r e x t e n s i v e c a v i t y growth o r c o a l e s c e n c e can o c c u r a l o n g t h e b o u n d a r i e s , t h e n low d u c t i l i t y , i n t e r g r a n u l a r f r a c t u r e r e s u l t s . The f i n a l morphology of the c r e e p c a v i t i e s changes w i t h i n c r e a s i n g s t r a i n r a t e ( o r s t r e s s ) from l e n s - s h a p e d t o wedge-shaped, accompanied by more p l a s t i c f l o w . 7 2 At h i g h s t r a i n r a t e s , v o i d s have i n s u f f i c i e n t t i m e t o n u c l e a t e and t r a n s g r a n u l a r r u p t u r e o c c u r s , i n the absence of o t h e r e m b r i t t l i n g phenomena. The work of F u n n e l l * * 1 ' 4 2 and o t h e r s 2 9 ' 3 7 ' 4 0 s u g g e s t s a mechanism f o r p r e c i p i t a t e a c t i o n i n t h i s i n t e r m e d i a t e - t e m p e r a t u r e r e g i o n a t low s t r a i n r a t e . I f g r a i n b o u n d a r i e s can m i g r a t e away from d e v e l o p i n g c a v i t i e s , t h e n c a v i t y g r o w t h s t o p s , s t r e s s c o n c e n t r a t i o n a t the g r a i n boundary i s r e l a x e d , and d u c t i l i t y i s p r e s e r v e d . However, i n a d d i t i o n t o p r o v i d i n g I n i t i a t i o n s i t e s f o r v o i d n u c l e a t i o n , p r e c i p i t a t e s a l s o tend t o h i n d e r or p r e v e n t g r a i n boundary m o b i l i t y . 4 0 - 4 2 T h i s e n c o u r a g e s c a v i t y c o a l e s c e n c e and i n t e r g r a n -u l a r f a i l u r e . A l t e r n a t i v e l y , the a c t i o n o f p r e c i p i t a t e s may enhance g r a i n b o u n d a r y s l i d i n g . 3 7 ' 4 1 ' 4 2 A c c o r d i n g t o t h i s m e c h a n i s m , t h e v o i d s n u c l e a t e , grow, and c o a l e s c e a l o n g t h e a u s t e n i t e g r a i n b o u n d a r i e s by t h e r e l a t i v e movement of n e i g h b o r i n g g r a i n s a l o n g t h e i r b o u n d a r i e s . 37 In e i t h e r case, the thermal-history s t r a i n - r a t e combinations that produce many f i n e p r e c i p i t a t e s at the austenite grain boundaries r e s u l t i n the lowest d u c t i l i t y . 1 + 0 _' t 2» 4 5 Mintz and Arrowsmith 4 5 suggest that a c r i t i -c a l p a r t i c l e s i z e e x i s t s , above which, grain boundary migration can occur. This s i z e would be a function of p a r t i c l e volume f r a c t i o n , i n i t i a l grain s i z e and the stored energy of deformation. Only a s u f f i c i e n t l y large number of p r e c i p i t a t e s smaller than t h i s c r i t i c a l s i z e causes grain boundary pinning and the resultant loss of d u c t i l i t y . Thus, d u c t i l i t y decreases for decreasing p r e c i p i t a t e s i z e and increasing volume f r a c t i o n . N i t r i d e s are the p r i n c i p a l p r e c i p i t a t e s responsible f o r enhancing t h i s mechanism. This Is because A1N and Nb(C,N) are slow to nucleate i n a u s t e n i t e 2 9 ' 1 | 0 ' 6 4 ' 7 1 ' 7 9 and under average cooling rates, produce very f i n e p r e c i p i t a t e s , a v e r a g i n g about 100 nm i n diameter for A1N 2 9' 4 1 and smaller than 50 nm for Nb(C,N). 2 9' 3 8 Because sulphide p r e c i p i t a t e s tend to be much larger (200-5000 nm), 5 4 as well as to nucleate e a s i l y and grow r a p i d l y , they are much less e f f e c t i v e at grain boundary pinning by t h i s mechanism. An a d d i t i o n a l a c t i o n of Nb(C,N) p r e c i p i t a t e s i n enhancing embrittlement i n t h i s zone i s the concentration of s t r a i n at the austenite grain bound-a r i e s . This occurs when networks of f i n e , intragranular, Nb(C,N) p a r t i c l e s cause p r e c i p i t a t e hardening i n the matrix. Many of the previously discussed observations can be explained by t h i s mechanism. A low s t r a i n rate allows time for the d i f f u s i o n - c o n t r o l l e d pro-cesses of ( A l , Nb, B) n i t r i d e p r e c i p i t a t i o n and grain boundary void 38 coalescence to take e f f e c t . As s t r a i n rate i s decreased, these e m b r i t t l i n g mechanisms are enhanced. Higher A l , Nb, B or N l e v e l s increase n i t r i d e p r e c i p i t a t e s o l u b i l i t y products which r e s u l t s i n increased p r e c i p i t a t i o n r a t e s . 6 8 Fast cooling rates or cooling to lower temperatures again encourages p r e c i p i t a t i o n 2 9 ' 3 8 ' 7 1 and r e s u l t s i n f i n e r , more c l o s e l y spaced p a r t i c l e s . 2 9 This i s because s o l u b i l i t y products decrease l o g a r i t h m i c a l l y with decreasing temperature (see Table 2.1) so supersaturation, the d r i v i n g force for nucleation, i s increased. Cooling to lower temperatures and reheating before testing promotes rapid n i t r i d e p r e c i p i t a t i o n f o r the same reason. Any of these factors causing increased n i t r i d e p r e c i p i t a t i o n rates tends to lower d u c t i l i t y and extend the d u c t i l i t y trough to higher temperatures. While u s u a l l y associated with s t r a i n rates below 10~ 3 s - 1 , g r a i n bound-ary s l i d i n g has been found to be at l e a s t p a r t i a l l y responsible f o r embrittlement at s t r a i n rates as high as 10" 1 s - 1 i n a 0.54Nb s t e e l at 9 00°C. 3 8 However, Ouchi & Matusumoto remark that because austenite g r a i n boundary s l i d i n g was not s e n s i t i v e to Nb, or N content or Increasing temperature above 900°C, i t cannot be the c o n t r o l l i n g factor f o r embrittle-ment. 3 8 The predominant mechanism e m b r i t t l i n g s t e e l at high s t r a i n rates involves (Fe, Mn) sulphide p r e c i p i t a t e s . Sulphur strongly and r a p i d l y seg-regates to the austenite grain boundaries to form weak sulphide films which can f a i l i n a manner reminiscent of high-temperature or hot-tearing e m b r i t t l e m e n t . 3 2 ' 3 3 Indeed, l i q u i d - f i l m f a i l u r e i t s e l f i s possible at 39 temperatures above the Fe-FeS e u t e c t i c i f l o c a l remelting of sulphur r i c h pockets can o c c u r . 3 3 Slow cooling or isothermal holding allows time for the slow d i f f u s i n g Mn to combine with S and form less harmful MnS p r e c i p i t a t e s which reduces FeS formation at the grain boundaries. In a d d i t i o n , high Mn/S r a t i o s encourage harmless MnS p r e c i p i t a t i o n i n s i d e the grains. This also explains the b e n e f i c i a l e f f e c t s of a high Mn/S r a t i o or low S l e v e l . However, l i q u i d - f i l m f a i l u r e alone cannot explain the problem since d u c t i l i t y losses occur even with Mn/S r a t i o s and temperatures well above those required to prevent l i q u i d FeS f i l m formation. Thus, i n a d d i t i o n , the p r e c i p i t a t e s themselves must be h a r m f u l , 3 4 presumably i n a manner s i m i l a r to that of n i t r i d e s . The c o n f l i c t i n g r e s u l t s obtained for varying thermal treatments and s t r a i n rate can be better understood by considering p r e c i p i t a t e thermo-dynamics. Above 1200°C, thermally activated processes such as d i s l o c a t i o n climb and r e c r y s t a l l i z a t i o n restore grain boundary m o b i l i t y . With i n c r e a s -ing time and temperature, p r e c i p i t a t e s both coarsen and d i s s o l v e , reducing t h e i r effectiveness at pinning grain boundaries, and recovering d u c t i l i t y . The aggravated embrittlement caused by high annealing temperatures i s presumably due to the complete d i s s o l u t i o n of MnS p r e c i p i t a t e s that occurs at 1420°C. 6 6 In a d d i t i on, i f l o c a l grain boundary melting can occur before c o o l i n g , t h i s detrimental action w i l l reduce d u c t i i t y at lower tempera-t u r e s . 3 3 ' 3 7 40 Between 1200°C and the temperature, d u c t i l i t y depends on the s i z e , number and l o c a t i o n of p r e c i p i t a t e s produced by the previous thermal t r e a t -ment. Slow cooling and isothermal holding induce both p r e c i p i t a t e nuclea-t i o n and growth. The same conditions of Increased time that allow large numbers of fin e n i t r i d e p r e c i p i t a t e s to form and lower d u c t i l i t y , cause sulphide p r e c i p i t a t e s to coarsen and thereby improve d u c t i l i t y . Cooling below the and r e h e a t i n g nucleates p r e c i p i t a t e s which grow r a p i d l y , and for the most part, harmlessly inside the grains. At higher s t r a i n r a t e s , l e s s time i s allowed for the coarsening of sulphide p r e c i p i t a t e s , r e s u l t i n g i n more e f f e c t i v e grain boundary pinning and thereby reducing d u c t i l i t y . Because they p r e c i p i t a t e and grow so much fa s t e r than n i t r i d e s , sulphides and oxides probably play t h e i r greatest r o l e i n the upper 900-1200°C temp-erature region at higher s t r a i n rates. At intermediate s t r a i n rates, a t h i r d mechanism comes into play.- This i s the competition between dynamic r e c r y s t a l l i z a t i o n and p l a s t i c t e n s i l e i n s t a b i l i t y . I f i t can occur, dynamic r e c r y s t a l l i z a t i o n completely relaxes any l o c a l stress concentrations and creates fresh grain boundaries that trap harmful p r e c i p i t a t e s and voids i n s i d e the grains. This r e s u l t s i n much improved d u c t i l i t y . In austenite above 1050°C, t h i s occurs so r e a d i l y that high d u c t i l i t y i s usually assured. However, i f the Considere s t r a i n (the s t r a i n at neck formation or the peak i n the o-e curve) i s l e s s than the s t r a i n required for dynamic r e c r y s t a l l i z a t i o n , then premature necking and f a i l u r e occurs. Thus, factors which either retard dynamic r e c r y s t a l l i z a t i o n or lower the work hardening parameter reduce d u c t i l i t y . ' Both A1N and Nb(C,N) retard austenite r e c r y s t a l l i z a t i o n , 3 7 ' **3' 4 5 which explains further the detrimental e f f e c t s of n i t r i d e s on d u c t i l i t y . Norstrom 8 0 explains the 41 d i f f e r e n t observed e f f e c t s of s t r a i n rate on d u c t i l i t y by t h i s mechanism, which i s i l l u s t r a t e d schematically i n Figure 2.14. At a lower s t r a i n rate, the recovery process i s ac t i v e which reduces the d r i v i n g force f o r dynamic r e c r y s t a l l i z a t i o n and thereby delays the d u c t i l i t y improving r e c r y s t a l l i z a t -ion process. At high s t r a i n r ate, the lack of time again prevents dynamic r e c r y s t a l l i z a t i o n , p o s s i b l y leading to an e a r l y transgranular, d u c t i l e f r a c t u r e . Thus, the best d u c t i l i t y should occur at intermediate s t r a i n r a t e s , e s p e c i a l l y i f the temperature i s high enough to allow r e c r y s t a l -l i z a t i o n . The absolute magnitude of the optimum s t r a i n rate w i l l depend on the creep, recovery and r e c r y s t a l l i z a t i o n c h a r a c t e r i s t i c s which depend on s t e e l composition. Since grain boundary pinning processes become l e s s e f f e c t i v e with i n c r e a s i n g s t r a i n r a t e , the prevention of r e c r y s t a l l i z a t i o n may be the main process responsible f o r continued low d u c t i l i t y at intermediate s t r a i n r a t e s . However, Ouchi and Matusumoto 3 8 point out that t h i s mechanism c l e a r l y i s not the only one operating i n the intermediate-temperature region since i t cannot explain: 1) the occurrence of f r a c t u r e with le s s than the Considere s t r a i n 2) the importance of thermal h i s t o r y 3) why Increasing s t r a i n rate sometimes Improves d u c t i l i t y while i t retards dynamic r e c r y s t a l l i z a t i o n as well as in c r e a s i n g the Considere s t r a i n . 4) why Nb c o n s i s t e n t l y lowers d u c t i l i t y at lower s t r a i n rates while i t suppresses recovery as well as r e c r y s t a l l i z a t i o n . 42 O Q Strain rate Figure 2.14 Schematic representation of the e f f e c t s of s t r a i n rate and temperature on the hot d u c t i l i t y of s t e e l 8 0 43 C l e a r l y , the mechanical behavior of s t e e l i n t h i s intermediate-temperature region i s not f u l l y understood. Further work needs to be done to unravel the complexities of the d i f f e r e n t e mbrittling mechanisms. 2.2.4 Lower Temperature Zone The t h i r d zone of low d u c t i l i t y i n s t e e l occurs i n the two-phase a u s t e n i t e - f e r r i t e region below the Ar^ temperature. I t corresponds to zone E i n Figures 2.2 and 2.13. In many respects, i t i s a continuation of the previously discussed intermediate-temperature zone i n v o l v i n g single-phase austen i t e . However, the presence of f e r r i t e appears to invoke another e m b r i t t l i n g mechanism i n the temperature range of 600-900°C. The fracture r e s u l t i n g from tests done i n the lower temperature, or "two-phase" zone, although appearing b r i t t l e due to i t s intergranular nature and lack of macroscopic s t r a i n to f r a c t u r e , i s thought to be a d u c t i l e f a i l u r e at the austenite grain boundaries on a microscopic s c a l e . 3 8 The fr a c t u r e surface was covered with dimples, many of which contained a p r e c i p i t a t e p a r t i c l e . 3 7 ' 3 8 > 5 0 These p r e c i p i t a t e s consisted mainly of A 1 N 1 8 , 24, 37, 43 , 44 , 5 0 b u t o t h e r n i t r i d e s , sulphides and a few oxides were also f o u n d . 3 0 ' 3 7 In st e e l s containing Nb or B, Nb(C,N) 2 9' 3 7 ' 3 8 ' 4 7 and B N 2 3 ' 1 + 6 p r e c i p i t a t e s were often found, both on the fr a c t u r e surface at the p r i o r a u s t e n i t i c grain boundaries and within the matrix. 44 The same detrimental e f f e c t s of n i t r i d e - p r e c i p i t a t e forming elements were found i n t h i s two-phase zone. 3 8 Increased Mn or S i also decreases d u c t i l i t y s l i g h t l y below 750°C. 3 8 In a d d i t i o n , r e s i d u a l (Cu, Sn, Sb, As) and impurity (S, P) elements segregate to the f e r r i t e grain boundaries to further lower d u c t i l i t y . 2 5 The e f f e c t s of thermal treatment also are generally the same except that increased holding time at 750°C was found to improve d u c t i l i t y 3 8 as shown i n Figure 2.15. Many researchers found that the temperature at the s t a r t of transformation was associated with the minimum d u c t i l i t y and that d u c t i l i t y r a p i d l y improved with decreasing temperature below the Ar^* 3**' 3 7 ' 4 3 ' 4 4 Most r e s e a r c h e r s agree that lowering s t r a i n rate i n the two-phase zone d r a s t i c a l l y reduces d u c t i l i t y , 3 7 p a r t i c u l a r l y near the A-j tempera-t u r e . 3 7 ' 3 8 However, Wray 6 6 suggests that embrittlement by t h i s mechanism may decline at low s t r a i n rate as shown i n zone E i n Figure 2.13. 2.2.4.1 Mechanisms The a d d i t i o n a l mechanism that has been proposed to co n t r o l e m b r i t t l e -ment In the two-phase zone places only secondary importance on the a c t i o n of p r e c i p i t a t e s . 7 ' 3 0 ' 3 8 ' 4 7 > 5 0 Grain boundary weakness i s instead a t t r i b u -ted mainly to s t r a i n concentration at the primary f e r r i t e f i l m forming along the austenite grain boundaries. This occurs because, at the same tempera-t u r e , f e r r i t e i s more d u c t i l e 3 5 ' 4 8 and has less s t r e n g t h 6 6 than austenite. This i s due p a r t l y to the higher atomic d i f f u s i v i t y of f e r r i t e and to the la r g e r number of s l i p systems i n bcc (48) compared with fee atomic s t r u c -tures ( 1 2 ) . 4 8 The austenite matrix can be hardened further by the add i t i o n Time (min) 5 10 50 T %c aos %P 0.014 %Si a305 % N 0.060 %Mn 1.56 %AI 0.022 %S 0.006 %Nb0.032 43 31 CL 7 10 O" € = 4XI0"3s' Thickness of ferrite (/x-m) 1 1 1 10 300 600 Holding time at 750°C (s) 3000 Figure 2.15 E f f e c t of hold time at 750°C on both f e r r i t e f i l m thickness and d u c t i l i t y 3 46 of elements such as C r " or by intragranular p r e c i p i t a t i o n i H such as Nb(C,N). 2 5' 2 9 ' ^ The presence of p r e c i p i t a t e s , ( p a r t i c u l a r l y n i t r i d e s ) further exacer-bates the problem by enhancing s t r a i n concentration and em b r i t t l i n g the grain-boundary f e r r i t e , each p r e c i p i t a t e nucleating a m i c r o v o i d . 5 0 > 5 1 In ad d i t i o n , the primary f e r r i t e encourages p r e f e r e n t i a l p r e c i p i t a t i o n at the gra i n boundaries because n i t r i d e s have a much lower s o l u b i l i t y i n f e r r i t e than i n austenite. ^' t , 7 1 For example, Figure 2.16 shows that A1N p r e c i p i t a -t i o n , which can take several hours i n single-phase a u s t e n i t e , 7 9 occurs within minutes once f e r r i t e i s p r e s e n t . 7 1 With continued s t r e s s , the microvoids multiply and the r e s u l t i s an in t e r g r a n u l a r , but microsc o p i c a l l y , d u c t i l e f r a c t u r e . D u c t i l i t y i s a minimum when the pockets of nucleating primary f e r r i t e f i r s t l i n k i n t o a continuous f i l m at the austenite grain boundaries. The thickness of t h i s pro-eutectoid, f e r r i t e f i l m i s the c o n t r o l l i n g factor f o r d u c t i l i t y accord-ing to th i s mechanism. 3 7 With lower temperatures or longer holding times, the accompanying increased thickness of the f e r r i t e f i l m (see Figure 2.15) i s believed to be responsible for the observed improvement i n d u c t i l i t y . 3 8 Yamanaka 5 0 s u c c e s s f u l l y correlated the d u c t i l i t y trough with the t h e o r e t i c a l f r a c t u r e s t r a i n calculated from the equations of Gurland & P l a t e a u 8 1 describing d u c t i l e fracture around i n c l u s i o n s . This mechanism, i l l u s t r a t e d in Figure 2.17 i s consistent with several of the previously discussed observations. I t explains the dimpled, d u c t i l e , Intergranular appearance of the frac t u r e surface and the as s o c i a t i o n of Time (hours) 0 5000 10000 Time (s) Time (hours) 1100 o o I 1000 0> E 900 r -800 0.1 1 1 0 1 1 1 0.0035-0.0045%N ASA (%) ~ 0.086 0.059 " A r 3 /° X / o o 1 1 1 1 I0 2 I 0 3 I0 4 I0 5 Time (s) Figure 2.16 (A) Ki n e t i c s of A1N p r e c i p i t a t i o n i n low-carbon s t e e l at various temperatures 79 (B) Contour l i n e s of 20% A1N p r e c i p i t a t i o n 48 minimum d u c t i l i t y with the Ar^ temperature. Increasing both grain boundary and intragranular p r e c i p i t a t e s , p a r t i c u l a r l y n i t r i d e s , n a t u r a l l y lowers d u c t i l i t y . The enhanced p r e c i p i t a t i o n rate of n i t r i d e s i n f e r r i t e also helps to explain the decrease i n the d u c t i l i t y of austenite when cooling i n t o the two-phase region and r e h e a t i n g . 3 8 Ouchi & Matusumoto 3 8 suggest that increasing s t r a i n rate might suppress s t r a i n concentration, thereby increasing d u c t i l i t y . However, s t r a i n concentration may occur even at high s t r a i n r ate, as evidenced by a d u c t i l i t y drop i n the two-phase region of Armco i r o n while t e s t i n g at 0.47 s - 1 . 4 8 Many researchers f e e l that s t r a i n concentration i n the f e r r i t e Is not necessary f o r embrittlement i n t h i s z o n e . 3 7 ' 4 1 The presence of a d u c t i l i t y drop even i n a single-phase a u s t e n i t i c s t a i n l e s s s t e e l over a s i m i l a r temp-erature region i s evidence of t h i s . 3 7 ' 4 1 ' 6 6 The continued loss i n d u c t i l -i t y below the Ar^ may simply r e f l e c t the continued operation of e m b r i t t l i n g mechanisms from the intermediate-temperature region such as grain boundary s l i d i n g . In many studies, the worst d u c t i l i t y was found at 750°C, regard-l e s s of whether the Ar^ was at 750°C or lower. This implies that d u c t i l i t y losses s t i l l may be c o n t r o l l e d by thermally activated processes i n v o l v i n g p r e c i p i t a t e s even If primary f e r r i t e i s p r e s e n t . 3 7 The improvement i n duct-i l i t y with continuing temperature reduction then would be due to the gradual decline i n importance of these mechanisms. 4 4 Wray states that the phase-transformation process may improve an inherent weakness i n austenite by t r a p p i n g harmful p r e c i p i t a t e s and voids i n s i d e new g r a i n s . 6 6 Below the A^ temperature, t h i s surely i s the case. The new, fully-transformed, f i n e -grained, f e r r i t e and p e a r l i t e structure generally has excel l e n t d u c t i l -i t y . 3 8 49 (a) (b) (c) (d) Ppt on "^ gb Void formation Nucleation of Coalescence of (Fe,Mn)S-0 (yg Dslip) proeutectoid- void AIN.BN ferrite NbCN, Figure 2.17 Mechanism for embrittlement i n the low-temperature or two-phase z o n e 3 7 50 However, a possible exception to t h i s e x i s t s for higher carbon s t e e l s where a fourth l o w - d u c t i l i t y zone may come Into play i f the cooling condi-tions and composition are such that a permanent, t h i n , f e r r i t e network r e s u l t s i n embrittlement below the A^ temperature. 3 Carbon contents near 0.8% would be the most susceptible, but a l l o y i n g elements such as Cr and Mn may also influence the amount of primary f e r r i t e that forms and produce large volume f r a c t i o n s of p e a r l i t e at lower carbon contents. This mechanism has been documented for hypereutectoid s t e e l s over a wide range of tempera-tures where cementite i s the b r i t t l e , grain-boundary phase. 6 6 Wray suggests that t h i s fracture zone begins at lower temperatures with decreasing carbon content for the f e r r i t e - p e a r l i t e c a s e . 6 6 This i s represented schematically i n zone F i n Figure 2.18, which shows the r e l a t i o n s h i p with carbon for zones A-E as w e l l . Indeed, several researchers studying medium-carbon s t e e l s found a drop i n d u c t i l i t y between 600 and 700°C that was also associated w i t h h i gh aluminum contents. 3' 1 6 ' 1 7 Titanium additions were found to be b e n e f i c i a l , 1 ' 3' 7 ' 1 6 presumably acting by the same mechanism that improves intermediate temperature d u c t i l i t y . In a d d i t i o n , Zr a d d i t i o n s 3 0 and poss-i b l y V as w e l l , were found to a l l e v i a t e the problem, although not as e f f e c t -i v e l y . 3 ' 1 6 ' 1 7 Higher N l e v e l s were also found to be detrimental i f accom-panied by aluminum. 1' 3 2.2.5 Implications of D u c t i l i t y f o r Panel Cracking The previous discussion of the zones of lowered d u c t i l i t y a f f e c t i n g s t e e l has several important implications for the panel cracking problem. Since thermal cracking can be prevented i f the material can accommodate as l i t t l e as 2% s t r a i n , then a severe d u c t i l i t y l o s s must be encountered before Figure 2.18 Schematic diagram showing the zones of embrittlement (see Figure 2.13) at intermediate s t r a i n rates f o r the Fe-C system. 6 6 52 panel cracking can occur. The d i f f e r e n t zones of reduced d u c t i l i t y are manifested under d i f f e r e n t and sometimes opposite processing conditions. I t i s therefore imperative to i d e n t i f y the p a r t i c u l a r e m b r i t t l i n g mechanism(s) responsible i f solutions to panel cracking are to be found. The cooling rates of a large ingot s o l i d i f y i n g i n i t s mould or even a i r cooling are very low. This r e s u l t s i n s t r a i n r a t e s , due to thermal contrac-t i o n , on the order of 2 x 10~ 6 s ~ 1 or l e s s . 7 Relative to the previously discussed d u c t i l i t y studies, t h i s s t r a i n rate i s extremely low. This f a c t and the importance of A1N together rule out several e m b r i t t l i n g mechanisms from being responsible for panel cracking. The f i r s t of these i s the high-temperature, hot-tearing zone of embrit-tlement. Panel cracking cannot be due to this mechanism for several addi-t i o n a l reasons. I t a f f e c t s s t e e l s even with high Mn/S r a t i o s and low S contents and the e f f e c t s of subsequent thermal h i s t o r y are too important. F i n a l l y , i t i s not responsive to changes In i n i t i a l casting conditions. The sulphide e m b r i t t l i n g mechanism a f f e c t i n g intermediate-temperature austenite i s also a very u n l i k e l y contributor to panel cracking. Besides the high Mn/S r a t i o s (usually greater than 50) and low S l e v e l s , the slow cooling rates and low s t r a i n rates encountered i n ingot casting would undoubtedly coarsen sulphide p r e c i p i t a t e s and eliminate embrittlement by this mechanism. 53 Since the t o t a l s t r a i n and s t r a i n rates involved i n panel cracking are very low and the cracks are intergranular, the e f f e c t s of r e c r y s t a l l i z a t i o n and t e n s i l e p l a s t i c i n s t a b i l i t y are also unimportant. This leads to the conclusion that the em b r i t t l i n g mechanism responsible f o r panel cracking must be either grain boundary void coalescence i n austen-i t e j u s t above the temperature or the lower temperature zone of embrit-tlement. In either case, improving the d u c t i l i t y of af f e c t e d grades near the A^ temperature would be b e n e f i c i a l . One method of achieving t h i s i s through a l t e r a t i o n of the s t e e l compo-s i t i o n to prevent the formation of detrimental, f i n e n i t r i d e p r e c i p i t a t e s . This can be achieved most e f f e c t i v e l y by lowering the a d d i t i o n of n i t r i d e -forming, elements such as A l , Nb, B, and e s p e c i a l l y N. A l t e r n a t i v e l y , r a i s i n g the ASA l e v e l m a rkedly 2 1 4' 4 2 or adding T i would produce coarse or harmless p r e c i p i t a t e s and again improve d u c t i l i t y . F i n a l l y , lowering S and 0 l e v e l s or adding Ca or Mn would only help to reduce sulphide e m b r i t t l e -ment, but at least would not do any harm. A second s o l u t i o n to avoid the production of f i n e , n i t r i d e p r e c i p i t a t e s i s through a l t e r a t i o n of the thermal treatment. Unfortunately the e f f e c t s of thermal h i s t o r y are s t i l l i n some dispute so i t i s not known what thermal treatment i s best. Suggestions have been made that temperature-time cycles to e i t h e r coarsen the p r e c i p i t a t e s 3 0 ' 3 7 ' 4 2 or keep them i n s o l u t i o n 4 2 w i l l both a l l e v i a t e the low d u c t i l i t y problem. However, the actual thermal h i s -t o r i e s to use are s t i l l unknown and the p o s s i b i l i t y of exacerbating the problem instead i s quite l i k e l y . Thus, solutions to n i t r i d e embrittlement 54 by a l t e r n a t e thermal treatment are not obvious given our present l e v e l of understanding. The f i n a l s o l u t i o n , i f i t can be c a l l e d that, i s simply to avoid s t r a i n i n g the s t e e l s i g n i f i c a n t l y while i t i s i n a region of low d u c t i l i t y . One way to achieve this might be to s t r i p the ingot from the mould e a r l y , keeping i t warm during transport and reheating i t quickly In an attempt to prevent the ingot surface from f a l l i n g into the low d u c t i l i t y temperature range. Then, immediate subsequent processing would be done using high s t r a i n rate operations such as r o l l i n g or forging where d u c t i l i t y problems r e l a t e d to n i t r i d e s are les s l i k e l y . 3 8 ' 4 2 However, we have seen that other e m b r i t t l i n g mechanisms operate at higher s t r a i n r a t e s . In ad d i t i o n , A1N p r e c i p i t a t i o n i t s e l f i s accelerated by deformation. 2 9' 4 0 ' 4 4 ' 6 4 Because of the wide range of temperatures a f f e c t e d by the d u c t i l i t y trough, i t i s v i r t u a l l y impossible to process s t e e l s only under conditions where good d u c t i l i t y e x i s t s . 4 2 Thus, the most p r a c t i c a l way to u t i l i z e t h i s s o l u t i o n i s by a l t e r i n g the thermal treatment to reduce the stresses acting i n the s o l i d i f y i n g and cooling ingot. 2.3 Previous Panel Cracking Studies In the l i g h t of an increased knowledge of the d u c t i l i t y of s t e e l at elevated temperatures, the studies made on panel cracking i n s t a t i c - c a s t , s t e e l ingots w i l l now be examined. During the l a t e 1950's, at l e a s t four s t u d i e s 1 - 4 were done on panel cracking, which was a well-known problem even at that time. Then, i n the l a t e 1970's, at least ten more studies into panel cracking were undertaken by major s t e e l companies from seven d i f f e r e n t 5 5 countries. 5-14 The cracking defects described i n these studies can be c l a s s i f i e d into two d i s t i n c t types. The f i r s t was experienced by a l l of the early workers 1-4 as well as i n several of the recent studies. 5-8 It i s found e x c l u s i v e l y i n small ingots, l e s s than s i x tons. I t a f f e c t s only medium carbon, aluminum treated s t e e l s with . 4 - . 7 % C or . 3 - . 6 % C i f e i t h e r 1% Cr, Ni or Mn i s present. This problem i s u s u a l l y manifested by a s i n g l e , continuous, l o n g i t u d i n a l crack down the center of one of the ingot faces so w i l l be referred to as "mid-face panel cracking". Most of the recent studies involved a d i s t i n c t l y d i f f e r e n t kind of panel cracking, although several companies have experienced d i f f i c u l t y w i t h both t y p e s . 6 ' 8 This second type of panel cracking i s mainly e x p e r i -enced i n much l a r g e r i n g o t s r a n g i n g i n s i z e from 2 0 - 3 5 t o n s . 9 - 1 1 ' 1 3 ' 1 4 Unlike mid-face panel cracking, i t only a f f e c t s low carbon s t e e l s ( . 1 - . 2 % C ) with manganese contents above . 7 % . 9 - 1 1 + The cracks are short and d i s c o n t i n u -ous, but quite deep ( 3 0 - 1 3 0 mm), having both transverse and l o n g i t u d i n a l components. 9' 1 2 ' 1 4 They most often occur i n bands near the edges of the wide face of the ingot and therefore w i l l be ref e r r e d to as "off-corner panel cracks." 2 . 3 . 1 Mid-face Panel Cracks Figure 2 . 1 9 presents a p i c t u r e of a mid-face panel cracked b i l l e t , showing the long, intergranular crack running down the center of one face. The cracks can extend to a depth of up to half the b i l l e t diameter and, 56 depending on the s e v e r i t y , as many as three of the four faces can be a f f e c t e d . 1 ' 3 ' 4 On duo-decagonal i n g o t s , the cracks run l o n g i t u d i n a l l y down the center of the f l u t e s . This i s i l l u s t r a t e d i n the transverse ingot cross sections i n Figure 2.20 which also show how cracks are sometimes found below the surface of the b i l l e t face that do not extend to the e x t e r i o r . This suggests that cracks may i n i t i a t e i n t e r n a l l y and then propagate out-wards. The defects are usually discovered either i n the melt shop a f t e r s t r i p p i n g or during subsequent r o l l i n g operations. The extent of the defect v a r i e s from only one or two ingots being a f f e c t e d to e n t i r e heats being scrapped. 2.3.1.1 Metallography A l l of the studies on mid-face panel cracking reported that the cracks appeared to follow the f e r r i t e network between the p e a r l i t e grains along the p r i o r - a u s t e n i t e grain boundaries. The f r a c t u r e surface had long, curved, intergranular f a c e t s . Only s l i g h t , s u p e r f i c i a l oxidation of the cracks was observed 1' 3' 4 ' 5 and no decarburization was p r e s e n t . 2 ' 3 Sulphur p r i n t i n g , macroetching, and chemical analysis revealed no macrosegregation associated with the d e f e c t s . 1 ' 3 Furthermore, microscopic examination could associate none of the v i s i b l e i n c l u s i o n s or s t r i n g e r s with the c r a c k s . 1 - 3 Rarely do researchers report the presence of p r e c i p i t a t e s associated with the f r a c t u r e i n the grain-boundary f e r r i t e . D e s a i 3 found occasional manganese sulphide and alumina i n c l u s i o n s but only E r i c s o n ' reported f i n d i n g A1N p r e c i p i t a t e s there. 5 7 58 Figure 2.20 T y p i c a l l o c a t i o n of mid-face panel cracks i n a transverse c r o s s - s e c t i o n of a: (A) 370 mm square, .55% C, s t e e l i n g o t 1 ' 3 (B) 380 mm duodecagonal, .6% C, s t e e l ingot 59 I t should be reemphasized here that p a r t i c l e s are most damaging when they f i r s t p r e c i p i t a t e and are very f i n e . 4 0 - 4 2 At t h i s stage, t h e i r detect-ion i s very d i f f i c u l t . By the time they are r e a d i l y i d e n t i f i a b l e , important events may have already occurred and the p r e c i p i t a t e s may have changed s i z e , shape, l o c a t i o n or e f f e c t i v e n e s s . Thus, grain boundary p r e c i p i t a t i o n i s always more severe than the examination methods suggest and many workers could not f i n d A1N p r e c i p i t a t e s even when ASA content c l e a r l y indicated high A l concentrations associated with c r a c k i n g . 1 ' 3' 1 + 5 2.3.1.2 E f f e c t of Composition As previously mentioned, mid-face panel cracking i s confined to medium-carbon, hypoeutectoid, p e a r l i t i c s t e e l s with carbon content between 0.4% and 0.7%. 1 - 8 However, c e r t a i n a l l o y s t e e l s containing e i t h e r 1% Cr, 1% NI or 1.5% Mn are p a r t i c u l a r l y prone to t h i s defect and are affected at s l i g h t l y lower carbon c o n t e n t s . 1 - 3 ' 5 There appears to be a lower l i m i t of 0.3%C. 8 The ad d i t i o n of 0.2 - 0.3% Mo to the high Cr or Mn s t e e l s eliminated t h e i r s u s c e p t i b i l i t y . 1 ' 5 Guerin and R o c c a t a g l i a t a 5 reported a detrimental influence of Pb, Cu and possibly Sn. Mid-face panel cracking only a f f e c t s aluminum treated s t e e l s and i t s i n c i d e n c e i n c r e a s e s w i t h i n c r e a s i n g ASA up to .06%. 1 - 5' 7 ' 8 Steels with l e s s than .015% 8 or more than .06% 1 experienced less problems. L i t t l e trouble was encountered with acid Open Hearth or "OH" s t e e l s while basic OH s t e e l s were susceptible to cracking and basic e l e c t r i c arc s t e e l s were worse s t i l l . 1 ' 8 B i g g s 1 a t t r i b u t e d the increased s u s c e p t i b i l i t y of e l e c t r i c arc s t e e l s to t h e i r higher nitrogen contents (.007-.012%) compared with OH 60 s t e e l s (.004-.006%). These fac t s imply that A1N p r e c i p i t a t i o n i s an import-ant f a c t o r , i f not the determining factor for mid-face panel cracking. F u r t h e r evidence of t h i s i s the b e n e f i c i a l influence of t i t a n i u m , 1 ' 3 - 5 ' 7 ' 8 y3» 5, 8 a n ( j p o s s i b l y Z r 5 i n r e d u c i n g the Incidence of mid-face panel cracking. Another i n t e r e s t i n g observation i s that while s t e e l s forming both bain-i t e and f e r r i t e ( i n the form of grain boundary networks) are prone to crack-i n g , f u l l y b a i n i t i c steels are n o t . 5 ' 7 2.3.1.3 E f f e c t of Ingot Size and Shape Mid - f a c e panel cracking i s only found i n small, 2-6 ton i n g o t s . 1 " " 5 ' 7 ' 8 Guerin and R o c c a t a g l i a t a 5 summarized the experiences of f i v e companies and agreed wi t h two other s t u d i e s 3 ' 4 i n c o n c l u d i n g that mid-face panel cracking does not a f f e c t ingots smaller than 2 tons. They a t t r i b u t e d t h i s to the i n a b i l i t y of very small ingots to generate i n t e r n a l thermal gradients during cooling s u f f i c i e n t i n magnitude to cause stresses that r e s u l t i n cracking. No researcher has reported f i n d i n g mid-face panel cracking i n ingots l a r g e r than 6 tons. Within t h i s range, there i s some disagreement as to the most suscept-i b l e ingot s i z e . E r i c s o n 7 found that two ton ingots were more prone to cracking than s i x ton ingots. Biggs stated that "intermediate sized" f o r g -ing ingots were most susceptible and that panel cracking was r a r e l y found i n "very large" forging i n g o t s . 1 However, Desai found that cracking tendency increased with increasing ingot s i z e above 9 inches square. 3 F i n a l l y , 61 others found cracking a f f e c t i n g the complete range of ingot sizes they produced 4' 5 Ingot shape or mould design appears to be unimportant. 1 Mid-face panel cracking has been found i n a wide v a r i e t y of ingot shapes ranging from square, f l a t - f a c e d b i l l e t s to f l u t e d , duodecagonal ingots. 2.3.1.A E f f e c t of Thermal Treatment Mid-face panel cracking i s independent of casting conditions such as teeming temperature and pouring r a t e . 1 ' 4 However, the subsequent cooling p r a c t i c e i s extremely i n f l u e n t i a l . Ingots that are stripped from the mould hot and transferred d i r e c t l y to the reheating furnaces or subsequent forging operations r a r e l y encounter c r a c k i n g . 1 - 3 Ingots allowed to cool excessively are the most susceptible to cracking. Mid-face panel cracks are generally a s s o c i a t e d w i t h l o n g j a c k e t e d times 6' 7 and/or long unjacketed times. 5' 6 The lack of s i g n i f i c a n t oxidation and decarburization i n these cracks i s evidence that they form at lower temperatures. Several researchers have proposed that mid-face panel cracks occur only a f t e r the ingot surface temp-erature has f a l l e n below a c r i t i c a l value. This c r i t i c a l temperature i s r e p o r t e d to be i n the range of 550-700°C 5' 6 with an upper l i m i t of 700°C 2 or 850°C. 3 I t i s suspected by some to be the Ar^ temperature at the comple-t i o n of the p e a r l i t e transformation. 8 The cooling rate and symmetry of cooling may also be important. Mid-face panel cracks are reported to occur p r e f e r e n t i a l l y on the i n s i d e , high 62 temperature faces between ingots that are too close to each other while cooling on the ingot buggy. 5' 6 In addition, stacking the ingots together or holding them i n a pre-heated furnace to allow slower cooling was found to prevent c r a c k i n g . 5 Guerin and Roccatagliata found that l a y i n g one face of the hot ingot on an i n s u l a t i n g bed of vermiculite eliminated panel cracking as w e l l . 5 Reheating p r a c t i c e appears to be much less important. Only one study suggested that cracking was affected by the reheating r a t e . 8 Thus, although s t i l l not conclusive, mid-face panel cracking probably occurs during c o o l -i n g . 5 ' 6 ' 8 2.3.2 Off-corner Panel Cracks A photograph of an ingot a f f e c t e d severely by off-corner panel cracking i s given i n Figure 2.21. The majority of the defects are l o n g i t u d i n a l , i n t e r g r a n u l a r , discontinuous cracks near the edges of the wide face of the ingot. However, near the ingot extremities, p a r t i c u l a r l y the bottom, they often begin to run i n a more transverse d i r e c t i o n , thus forming a rough oval p a t t e r n . 9 ' 1 2 ' 1 4 In a d d i t i o n , the i n g o t c r o s s s e c t i o n i n Figure 2.22 reveals a s i m i l a r r a d i a l pattern of cracks just beneath the surface of both the narrow and wide f a c e s . 9 Only those subsurface cracks that reach to the surface cause r e j e c t s . In a corrugated ingot, these damaging cracks are associated with the mould corrugations. Figure 2.23 shows how these cracks generally i n i t i a t e d i r e c t l y beneath the peak of a corrugation, often the one 63 Figure 2.21 Off-corner panel cracks i n a 760 x 1520 mm, rectangular, corrugated (.14% C, 1.4% Mn, S i - k i l l e d , A l grain refined) s t e e l i n g o t 1 4 64 Figure 2.22 Relative l o c a t i o n of off-corner panel cracks found by Sussman 9 i n transverse cross-sections taken from the top of ingots subjected to: (A) 1260 s ( 21 min.) unjacketed time (B) 6480 s (108 min.) unjacketed time 66 nearest the edge of the wide face. They then bend to reach the surface at a point between the corrugation peak and an adjacent t r o u g h . 1 4 The lack of cracking on the narrow face implies that r o l l i n g may close up subsurface cracks beneath the narrow face while i t contributes to opening up those on the wide f a c e . 9 ' 1 4 The exact time of cracking between i n i t i a l pouring and hot r o l l i n g i s not known as various theories have been pre-sented. However, the cracks are more associated with reheating since they are discovered, at the e a r l i e s t , a f t e r removal from the soaking p i t and u s u a l l y are not detected u n t i l e a r l y hot working stages. 8 Like mid-face panel cracking, the extent of the defect v a r i e s from one ingot to the e n t i r e heat and affected ingots must be scrapped. 2.3.2.1 Metallography Off-corner panel cracks are often found outlined with a t h i n f e r r i t e zone which can be seen i n Figure 2.24. Several researchers a t t r i b u t e t h i s to d e c a r b u r i z a t i o n . 6 ' 9 ' 1 2 ' 1 4 A l t e r n a t i v e l y , some of the f e r r i t e networks may have formed p r i o r to cracking as i n mid-face panel cracks. Figure 2.24 also shows that the f e r r i t e zone associated with the cracks may contain several types of i n c l u s i o ns. These were found to be mainly large oxides of a l l k i n d s (Fe, Mn, S i , A l ) 6 ' 9 ' 1 2 ' 1 4 which were at t r i b u t e d to high temp-e r a t u r e o x i d a t i o n a f t e r c r a c k i n g 9 ' 1 2 or p o s s i b l y during r o l l i n g . 1 2 An experiment done on two s t e e l rings t i g h t l y screwed together to simulate a crack, confirmed that high temperature oxidation can penetrate the crack and p r e f e r e n t i a l l y oxidize the grain boundaries, producing oxide p r e c i p i t a t e s quite s i m i l a r to those observed i n off-corner panel c r a c k s . 1 2 F i g u r e 2 . 2 4 C l o s e - u p of o f f - c o r n e r p a n e l crack, showing a s s o c i a t e d f e r r i t e band and i n c l u s i o n s ( e t c h e d i n 2% n i t o l , 1 2 0 X ) 68 In a d d i t i o n to the large oxides, very small (0.03-1.0 micron) p r e c i p i -t a t e s c o n t a i n i n g Mn, S i , or A l were a l s o found i n the f e r r i t e zone. 9' 1 2 However, only a few workers a c t u a l l y confirmed the presence of A1N p r e c i p i -t a t e s . 7 ' 9 N e g l i g i b l e segregation of S, Mo or Cr was found, 9' 1 2 although one study found the f e r r i t e zone depleted i n Mn and S i . 1 2 The crack may appear i n two portions with the top quite wide, oxidized, and open to the surface, while the bottom appears welded s h u t . 1 4 One researcher also found evidence that r e c r y s t a l l i z a t i o n takes place a f t e r c r a c k i n g . 6 2.3.2.2 E f f e c t of Composition As previously mentioned, off-corner panel cracking d i f f e r s from mid-face panel cracking i n that i t only a f f e c t s low carbon s t e e l s (between .1 and .2% C ) 6 ' 8 - 1 4 with high manganese contents (> .7% Mn). 6' 9 ' 1 0 ' 1 3 ' 1 4 However, l i k e mid-face panel cracking, i t only a f f e c t s A l k i l l e d or A l g r a i n - r e f i n e d s t e e l s c o n t a i n i n g .015-.6% ASA. 6' 8 ~ 1 0 ' 1 2 - 1 4 S o m e s t u d i e s found c r a c k i n g i n c i d e n c e more l i k e l y with increasing ASA content 9' 1 2 ' 1 3 but other studies reported no p a r t i c u l a r trend with increasing ASA, 6' 1 0 ' 1 4 so long as i t was s u f f i c i e n t for grain refinement (> .015% A S A 1 2 ) . Off-corner panel cracking was also found to be associated with high N c o n t e n t s (> .007% N ) . 9 ' 1 2 ' 1 4 Thus, n i t r i d e p r e c i p i t a t e s and A1N i n p a r t i c u l a r , are apparently detrimental to both types of panel cracking. 69 M i c r o - a l l o y s t e e l s , containing Nb or V, were generally reported to be e s p e c i a l l y p r o n e to o f f - c o r n e r p a n e l c r a c k i n g . 1 0 ' 1 4 However, one researcher reported a b e n e f i c i a l e f f e c t of both Nb and V . 1 2 Increasing Cu above .3% may increase cracking s u s c e p t i b i l i t y 1 2 while adding T i 1 2 , Z r 1 0 , or possibly Mo 1 0 a l l a l l e v i a t e i t . One company found off-corner panel cracking only i n s t e e l s that contained S i . 1 4 No c o n t r i b u t i n g e f f e c t s were found for r e s i d u a l l e v e l s of H, Sn, As, Pb, Sb, 0, 1 2 and most notably, S. 1 0' 1 2 Most of the s u s c e p t i b l e s t e e l grades had low sulphur contents. 6' 9 ' 1 0 ' 1 4 When combined with t h e i r high manganese contents, they consequently also had very h i g h Mn/S r a t i o s , u s u a l l y greater than 50 9' 1 0 ' 1 4 and o c c a s i o n a l l y exceed-ing 200. 1 4 2.3.2.3 E f f e c t of Ingot Size and Shape Off-corner panel cracks were mainly found i n very large, rectangular i n g o t s over 20 t o n s 9 - 1 1 ' 1 3 ' 1 4 such as the one pictured i n Figure 2.21. However, they have also been found i n ingots as small as only 10 t o n s . 9 ' 1 2  1 4 No p a r t i c u l a r trends with ingot si z e or shape have been noted except that cracking decreased with greater reduction r a t i o s to slabs, presumably due to t h e i r p a r t i a l l y s ealing up during r o l l i n g . 9 2.3.2.4 E f f e c t of Thermal Treatment Off-corner panel cracks are affected by cooling p r a c t i c e i n a d i f f e r e n t way than mid-face panel cracks. Some studies again f i n d that the incidence of cracking increases for long track times and occurs only when the track time exceeds 9000 to 21,600 seconds (or 2.5 8 to 6 6 hours). Others f i n d that 70 Figure 2.25 The e f f e c t of cooling practice on the incidence of off-corner panel c r a c k i n g 1 4 71 off-corner panel cracks only appear when there has been a short unjacketed time, l e s s than 7200 seconds (or 2 hours ) . 6 ' 1 4 This i s shown i n Figure 2.25. A l t e r n a t i v e l y , short track times, l e s s than 14,400 to 32,400 seconds (or 4 8 to 9 6 hours) have been found to be the most detrimental. Ingots which were allowed to cool to ambient temperature before reheating never experienced off-corner panel cracking. These c o n f l i c t i n g observations have been r a t i o n a l i z e d with the explana-t i o n that the most harmful treatment i s intermediate cooling that allows the , ingot surface to f a l l into some c r i t i c a l temperature range before reheat-i n g . 6 - 9 Many researchers believe t h i s c r i t i c a l temperature range to be the two-phase region between the Ar^ and Ar^ temperatures. 6' 8 > 9 ' 1 2 However, there i s wide disagreement as to the actual cooling p r a c t i c e to be avoided. Another i n t e r e s t i n g observation i s that the oval crack pattern observed by several researchers was found to displace towards the center of the ingot w i t h i n c r e a s i n g u n j a c k e t e d t i m e . 9 ' 1 4 The o v a l crack pattern coincided c l o s e l y with the l o c a t i o n of isothermal contours that o u t l i n e the boundary between the o r i g i n a l warmer i n t e r i o r and cooler e x t e r i o r . Thus, the cooling p r a c t i c e i s obviously highly important i n c o n t r o l l i n g the formation of o f f -corner panel cracks. The majority of researchers agree that off-corner panel cracking i s also g r e a t l y influenced by reheating p r a c t i c e , s p e c i f i c a l l y , the reheating r a t e 8 - 1 4 and time 9' 1 1 ' 1 3 ' 1 4 i n the soaking p i t . However, the same appa-rent contradictions and confusion about the e f f e c t s of cooling p r a c t i c e also e x i s t s f or reheating p r a c t i c e . Sussman found that cracking diminished for f a s t reheating rates and that the time spent between 650 and 1100°C should 72 be minimized. A l t e r n a t i v e l y , several other researchers believe that slow, c a r e f u l l y c o n t r o l l e d h e a t i n g conditions minimize c r a c k i n g . 1 0 - 1 2 ' 1 4 Long times i n the soaking p i t are r e p o r t e d by some to be d e t r i m e n t a l 9 ' 1 4 but Nashiwa 1 1 states that a longer time i n the soaking p i t i s b e n e f i c i a l . 2.4 Proposed Mechanisms f o r Panel Crack Formation Based on the findings of the numerous studies made on panel cracking, s e v e r a l attempts have been made at formulating mechanisms. There i s general agreement that both forms of panel cracking are caused by a combination of the two f a c t o r s : reduced hot d u c t i l i t y and stresses generated from thermal gradients and phase transformation. However, many researchers have only vaguely formulated mechanisms and the r e l a t i v e importance of these two f a c t -ors i s i n considerable dispute. 2.4.1 Reduced High Temperature D u c t i l i t y One group of researchers believe that panel cracking i s mainly due to a r e d u c t i o n i n the hot d u c t i l i t y of s t e e l . 1 ~ k ' 7 ' 8 ' 1 2 In order to f a i l i n an intergranular manner, the grain boundaries must be weakened at the temp-eratures where cracking occurs. The previous observations on the e f f e c t s of thermal treatment imply that the lower temperature zone of reduced duct-i l i t y , i n p a r t i c u l a r the zone of embrittlement e n t i r e l y below the A^ temp-erature, i s responsible for mid-face panel cracking. D u c t i l i t y experiments done by Desai confirmed that s t e e l from cracked ingots had lower d u c t i l i t y than uncracked ingots i n the 600-700°C range. 3 73 Although the Intermediate-temperature embrittlement zone of s t e e l i s known to extend much higher than 900°C, researchers a t t r i b u t e off-corner panel cracking s o l e l y to the low-temperature region between 700 and 900°C. Their mechanisms are formulated i n terms of A1N p r e c i p i t a t e pinning at the austenite grain boundaries. As previously deduced, t h i s f a l l s i n t o either the low s t r a i n - r a t e zone of embrittlement i n austenite inv o l v i n g the nuclea-t i o n , growth and coalescence of grain boundary voids or the lower-temperature embrittlement zone involving the two-phase region. 2.4.1.1 Mid-face Panel Cracks Two d i f f e r e n t explanations were postulated for the o r i g i n of the g r a i n -boundary weakness responsible for mid-face panel cracking. B i g g s 1 and Colombo and Cesar! 4 emphasized that the incidence of cracking increased with inc r e a s i n g aluminum and nitrogen contents. They argued that the grain boundaries were weakened mainly by the presence of mechanically weak, second-phase p a r t i c l e s . As the s o l i d i f i e d s t e e l cools, the p r e c i p i t a t i o n of f i n e A1N p a r t i c l e s w i l l occur at the austenite grain boundaries. These p r e c i p i t a t e s may p e r s i s t to lower the d u c t i l i t y of the grain-boundary f e r r i t e , e i t h e r before or after the p e a r l i t e transformation. Cracks would then tend to i n i t i a t e at the weakened grain boundaries under the a p p l i c a t i o n of s t r e s s . On the other hand, Irvine and P i c k e r i n g 2 argued that the f a i l u r e occurs at the g r a i n boundaries due to the presence of t h i n , f e r r i t e networks. Below the Ar^ temperature, austenite f i r s t transforms to primary f e r r i t e at nucleation s i t e s i n the grain boundaries. Then, as the s t e e l cools below 74 the Ar^ temperature, the remaining majority of the austenite transforms to p e a r l i t e , leaving a harder and less d u c t i l e phase surrounded by the f e r r i t e f i l m . Under s t r e s s , the weaker, more d u c t i l e f e r r i t e f i l m i s subjected to a disproportionate amount of s t r a i n and eventually f a i l s i n a d u c t i l e manner. This explains the presence of a f e r r i t e zone o u t l i n i n g an intergranular crack network and explains why the cracks only appear a f t e r longer cooling times when the s u r f a c e has dropped below the Ar^ temperature. I t suggests that a c r i t i c a l d i s t r i b u t i o n or thickness of grain boundary f e r r i t e might be i n v o l v e d , 1 ' 5 0 t h i s being c o n t r o l l e d by composition and cooling rate. This i s supported by the observation that only c e r t a i n carbon l e v e l s and thermal treatments were prone to panel cracking. This theory also accounts for the lack of cracking i n f u l l y b a i n i t i c s t e e l s or In Mo bearing s t e e l s , where the p e a r l i t i c transformation Is r e t a r d e d . 7 Since there i s some evidence supporting each of the two mechanisms, a l l of these researchers admit that a combination of both mechanisms most l i k e l y accounts for the intergranular nature of panel cracking. Mid-face panel cracking i s thereby explained by the slow nucleation of f i n e A1N p a r t i c l e s p r e f e r e n t i a l l y p r e c i p i t a t i n g at the grain boundaries. This further weakens the f e r r i t e f i l m network where subsequent stress and s t r a i n concentration r e s u l t s i n an intergranular f a i l u r e . 2.4.1.2 Off-Corner Panel Cracks Several studies a t t r i b u t e off-corner panel cracking to the e m b r i t t l i n g mechanisms that reduce the intermediate temperature d u c t i l i t y of s t e e l at low s t r a i n r a t e s . A mechanism has been developed to explain the r e l a t i o n -75 s h i p between o f f - c o r n e r panel c r a c k i n g and thermal treatment. and i s represented schematically i n Figure 2.26. When track times are very short, the surface i s prevented from f a l l i n g into the two-phase region. Research-ers subscribing to t h i s mechanism believe that cracking does not occur i n t h i s case because of the r e s u l t i n g lack of e m b r i t t l i n g p r e c i p i t a t e s . At intermediate track times, the surface f a l l s i nto the two-phase r e g i o n , but does not go below the Ar^ temperature. Thin f e r r i t e films form at the austenite grain boundaries along with fi n e n i t r i d e p r e c i p i t a t e s such as A1N. Cracking then occurs along the weakened, f e r r i t e networks during reheating or hot deformation. For long track times, the surface completely transforms to f e r r i t e and p e a r l i t e to a considerable depth. A1N p r e c i p i t a t e s again form at the grain boundaries. However, upon reheating, new austenite grains are formed when the surface retransforms and the dangerous chains of A1N p r e c i p i t a t e s are trapped harmlessly i n s i d e them. Thus, again there i s no cracking. E r i c s o n 7 has suggested an alternate explanation for the p a r t i c u l a r l y detrimental e f f e c t of A l on both types of panel cracking. As the ingot c o o l s , A l may segregate p r e f e r e n t i a l l y to the austenite grain boundaries, 7' 1 2 p a r t i c u l a r l y In the oval boundary region separating the warmer and cooler areas of the ingot's wide face. The segregation of other f e r r i t e formers (Cr, Mo, S, P, Si,) may be enhanced as w e l l . This segregation may be atten-uated by the delta-austenite p e r i t e c t i c phase transformation. 6' 8 Since an i n c r e a s e d concentration of these elements w i l l increase the Ar^ temperature l o c a l l y , t h i s would promote the e a r l i e r formation of primary f e r r i t e at the 76 Figure 2.26 Mechanism for off-corner panel crack formation inv o l v i n g reduced d u c t i l i t y 77 grain boundaries and i n the oval area. P r e c i p i t a t e s , p a r t i c u l a r l y n i t r i d e s , then r a p i d l y nucleate i n the grain boundary f e r r i t e . Combined with s t r e s s concentration i n the f e r r i t e f i l m , a c l a s s i c , i ntergranular, l o w - d u c t i l i t y f a i l u r e occurs. E r i c s o n 7 explains that slow, even cooling or long track times reduce the cracking tendency by lowering i n t e r n a l temperature gradients. This r e s u l t s i n less A l segregation and therefore no l o c a l f e r r i t e formation or p r e f e r e n t i a l p r e c i p i t a t i o n . Although the b e n e f i c i a l e f f e c t of T i has been explained previously i n terms of i t s p r e f e r e n t i a l formation of coarser, more evenly d i s t r i b u t e d p r e c i p i t a t e s , E r i c s o n 7 believes that T i helps prevent panel cracking i n two a d d i t i o n a l ways. F i r s t l y , i t promotes a f i n e a u s t e n i t i c structure with low microsegregation a f t e r s o l i d i f i c a t i o n by diminishing the s o l i d i f i c a t i o n temperature i n t e r v a l . Secondly, i t discourages f e r r i t e network formation even though T i i t s e l f i s a f e r r i t e former. I t does t h i s by reducing s o l i d state microsegregation of A l and by retarding the y •*• a transformation. This reduces mid-face panel cracking i n b a i n i t i c s t e e l s by encouraging bain-i t e formation, thereby preventing the formation of detrimental f e r r i t e networks. 2.4.2 Stress Generation Despite these convincing arguments for the hot d u c t i l i t y of s t e e l being the determining f a c t o r for panel cracking, most of the previous observations can also be explained i n terms of stress generation. A second group of r e s e a r c h e r s b e l i e v e that panel c r a c k i n g i s due m a i n l y , 9 ' 1 0 ' 1 3 i f not e n t i r e l y 1 1 to thermal and phase transformation s t r e s s . 78 2 . 4 . 2 . 1 Mid-face Panel Cracks On the b a s i s of two early studies, ' 3 a vaguely formulated mechanism for mid-face panel cracking due to stress generation has emerged. As the s t e e l s o l i d i f i e s and cools, the soft center seeks to contract within a r i g i d outer framework. Thus, i n t e r n a l t e n s i l e stresses develop due to changing thermal gradients and phase transformations. The stresses were thought to reach a maximum at the center of the b i l l e t face where the panel cracks were ul t i m a t e l y observed. Under t h i s s t r e s s , cracks i n i t i a t e below the ingot surface at the austenite grain boundaries. This mechanism requires that the temperature i s low enough at the time of crack formation that stresses are not r e l i e v e d by p l a s t i c deformation. The cracks then propagate outwards as the s t e e l continues to contract. Other r e s e a r c h e r s 5 ' 8 have r e f i n e d t h i s mechanism by considering the volume expansion that accompanies the y a phase transformation i n s t e e l . They emphasize the previously discussed observation that mid-face panel cracking occurs when the surface f a l l s below some c r i t i c a l temperature pre-sumably connected with the y •*• a phase transformation. According to t h e i r mechanism, the b i l l e t surface i s expanding during transformation while the a u s t e n i t i c i n t e r i o r i s s t i l l cooling and contracting. A subsurface t e n s i l e stress consequently develops. Cracking occurs at t h i s stage as the trans-formation front moves inward. However, t h i s mechanism i s incomplete, since the warm, a u s t e n i t i c i n t e r i o r may be able to deform s u f f i c i e n t l y through creep to avoid cracking u n t i l the surface drops below the Ar. temperature. As cooling proceeds, the 79 f u l l y transformed surface begins to contract again while the austenite transforming beneath the surface i s s t i l l expanding. This puts the surface i n t o tension while i t i s composed e n t i r e l y of f e r r i t e and p e a r l i t e . The zone of low d u c t i l i t y associated with p r e c i p i t a t e - e m b r i t t l e d f e r r i t e net-works 'surrounded by p e a r l i t e r e s u l t s i n cracking. This mechanism suggests that s t e e l s with a narrow two-phase region should be more prone to cracking since they are subjected to higher r e s u l t -ing stress gradients f o r a given expansion. This would imply l e s s cracking f o r low C s t e e l s , which have a wider two-phase region, and no cracking for b a i n i t i c s t e e l s , which experience no phase transformation expansion. Crack-ing would also be expected to occur i n the center of the panels where stresses are the highest. These stress generation mechanisms explain the improved r e s u l t s of slower c o o l i n g rates by the lowering of i n t e r n a l temperature gradients and the r e s u l t a n t decrease i n maximum stress l e v e l s . In a d d i t i o n , the absence of cracking i n ingots smaller than two tons can be a t t r i b u t e d to the i n a b i l i t y of very small ingots to generate i n t e r n a l thermal gradients during cooling s u f f i c i e n t to cause stresses that r e s u l t i n c r a c k i n g . 5 A f i n a l f a c t o r , already touched upon, i s the symmetry aspect i n c o o l -i n g . Asymmetrical cooling might set up higher, more unfavorable s t r e s s e s , 5 ' 6 but t h i s idea needs to be developed f u r t h e r . 80 2.4.2.2 Off-corner Panel Cracks The stress generation mechanism can also be used to explain the influence of thermal treatment on off-corner panel c r a c k i n g . 9 As cooling progresses a f t e r the ingot has been stripped, the surface transforms to f e r r i t e f i r s t . The transformation front then extends inward to a c e r t a i n depth. However, reheating i n the soaking p i t retransforms the surface back to austenite. The r e s u l t i s a thin zone of two-phase material undergoing expansion while transforming from austenite to f e r r i t e which i s surrounded by austenite on each side. As t h i s t h i n zone retransforms to austenite as w e l l , i t s contraction w i l l develop a large t e n s i l e stress beneath the surface and a compressive stress on the surface. Subsurface cracks w i l l form along the path of least r e s i s t a n c e : the p r e c i p i t a t e weakened g r a i n boundaries. Their p o s i t i o n should correspond to the maximum depth of the o r i g i n a l a u s t e n i t e - f e r r i t e transformation f r o n t . I t should be noted that the b r i t t l e , columnar, grain boundaries are in the most vulnerable o r i e n t a -t i o n , being perpendicular to any applied t e n s i l e s t r e s s . The compressive st r e s s on the surface causes i t to deform p l a s t i c a l l y by shrinking. As the i n t e r i o r continues to reheat, temperature gradients subside, causing the subsurface to expand. The r e s u l t i n g t e n s i l e stress at the surface causes the previously formed subsurface cracks to propagate through to the surface, again along the weakest grain boundary path. This mechanism of f e r s an alternate explanation for why the occurrence of off-corner panel cracking i s most l i k e l y when the surface f a l l s i n t o the 81 two-phase region before reheating. I t also predicts that cracking occurs during reheating and that a rapid soaking p i t reheating practice would be the most detrimental under these circumstances. Two a d d i t i o n a l factors have been suggested to contribute to stress generation i n the cooling ingot. F i r s t l y , the delta-austenite phase trans-formation occurring at high temperature i n low C s t e e l s i s accompanied by a cont r a c t i o n which may generate r e s i d u a l s t r e s s e s . Secondly, r e s i d u a l stresses are the highest when steep thermal gradients are present. Kawawa 1 1 determined that the thermal gradients i n a cooli n g , ingot reach a maximum 9000 seconds (or 2.5 hours) a f t e r s t r i p p i n g and correlated them with stress generation. 2.5 Proposed Solutions In an e f f o r t to eliminate panel cracking, various companies have pro-posed and t r i e d a number of d i f f e r e n t s o l u t i o n s , meeting with varied success. Since only a small percentage of s t e e l production s u f f e r s from panel cracks (about 1 % ) , 1 4 the f i r s t natural s o l u t i o n i s to simply remove the cracks from affected ingots by grinding or sc a r f i n g them a f t e r w a r d s . 1 4 Unfortunately, t h i s has proven unsuccessful. Grinding i s expensive and the lower, "closed" portion of the crack s t i l l opens up during r o l l i n g . Scarf-ing with an oxy-acetylene torch i t s e l f opens up the cracks and pushes them deeper. F i n a l l y , the detection of a l l remnants of cracks i n an affected ingot or slab i s v i r t u a l l y impossible and the consequences of allowing a 82 panel cracked product to go into service could be di s a s t r o u s . Thus, ingots a f f e c t e d by panel cracks are almost u n i v e r s a l l y rejected and sc r a p p e d . 1 2 ' 1 4 Another s o l u t i o n that has been suggested to reduce panel cracking i s to make smaller or thinner ingots that generate smaller thermal gradients and less r e s u l t i n g s t r e s s 5 but th i s has not been substantiated with any experi-mental evidence. As previously mentioned, higher ingot to slab reduction r a t i o s may reduce cracks by c l o s i n g them up during r o l l i n g , only i f they have not yet reached the surface. The remaining solutions to panel cracking f a l l i nto two categories: changing s t e e l composition to improve hot d u c t i l i t y and a l t e r i n g thermal treatment to either improve hot d u c t i l i t y or reduce stress generation. 2.5.1 S t e e l Composition Following the solutions to the n i t r i d e embrittlement problem p r e v i o u s l y discussed, the f i r s t chemistry s o l u t i o n to panel cracking i s to lower the nitrogen content of the s t e e l . Unfortunately, t h i s often proves to be d i f f i c u l t . Nitrogen enters s t e e l mainly by absorption and entrainment from the atmosphere so i t s content can be lowered, along with oxygen, by l i m i t i n g exposure to a i r . S p e c i f i c a l l y , rough spraying streams should be avoided and well designed shroud and nozzle systems should be u s e d . 7 7 Slag type Is also important as acid slags r e s u l t i n more N than basic slags and double slag systems r e s u l t i n more N than single ones. 5 One company found that blowing with high p u r i t y oxygen (>99.4%) reduced the incidence of panel c r a c k i n g . 1 2 I t i s also important to watch i n d i r e c t influences on N content such as the 83 a l l o y i n g agents and deoxidizers used. For example, N absorption from the a i r i s greater for desulphurized s t e e l . 7 7 F i n a l l y , furnace type i s very important as N content continuously Increases i n going from OH to BOF to e l e c t r i c furnace. 5 However, one company unexpectedly finds OH s t e e l s more prone to off-corner panel cracks than BOF s t e e l s despite t h e i r lower N con-t e n t . 1 4 I n t e r e s t i n g l y , t h e i r OH steels also have higher S l e v e l s . The second s o l u t i o n to improve d u c t i l i t y i s simply to lower the A l c o n t e n t 5 ' 7 ' 9 ' 1 2 and, i n m i c r o - a l l o y s t e e l s , the Nb and B contents as w e l l . However, the b e n e f i c i a l grain refinement and r e c r y s t a l l i z a t i o n r e t a r d i n g e f f e c t s of the n i t r i d e p r e c i p i t a t e s , so d e s i r a b l e i n l a t e r pro-cessing stages, w i l l also be reduced. The same mechanism of grain boundary pinning, by which these processes operate, i s responsible for reducing hot d u c t i l i t y as w e l l . Thus, In the words of Biggs, 1 " i t i s very much a com-promise between obtaining e f f e c t i v e grain size c o n t r o l and the avoidance of panel cracking." Several studies recommend the s o l u t i o n of lowered A l con-tent only i f lower notch toughness values are acceptable. 5, 1 2 Lowering A l content i s a d i f f i c u l t task i n i t s e l f since i t also acts as a deoxidizer. Several companies compromise by aiming for a composition window of . 0 1 - . 0 2 % A l . 8 However, because of the extremely v a r i a b l e recovery of A l , t h i s can be d i f f i c u l t to achieve. Adding A l d i r e c t l y to the l a d l e i s one way to make recoveries more consistent. In a d d i t i o n , f a c t o r s which c o n t r o l oxygen l e v e l , i n d i r e c t l y influence ASA so should be c a r e f u l l y con-t r o l l e d . These include a i r entrainment, S i and Mn additions and slag type. A l t e r n a t i v e l y , Si can be used as a deoxidizer although one company fi n d s off-corner panel cracks only i n Si k i l l e d , A l grain refi n e d s t e e l s . 1 4 On 84 top of t h i s , the lack of any d e f i n i t e trend with increasing Al content found by many studies of off-corner panel cracking makes t h i s second proposed s o l u t i o n very dubious. The t h i r d s t e e l composition based s o l u t i o n i s the use of other, l e s s detrimental, n i t r i d e formers. Various companies have claimed success i n r e d u c i n g the incidence of panel cracks through the addition of T i , 5 ' 7 ' 1 0 * 12 v > 5 » 8 . 12 Z r > 5 » 10 M o > 5 . 10 o r e y e n N b . 1 2 However, aluminum i s gener-a l l y considered to be the least c r i t i c a l grain growth i n h i b i t e r with respect to the introduction of undesirable non-metallic i n c l u s i o n s . Titanium pro-duces undesirable c a r b o - n i t r i d e s , r e s u l t i n g i n increased non-metallics and reduced m a c h i n a b i l i t y . 5 Zirconium i s a strong deoxidizer which r e s u l t s i n lower recoveries and increased non-metallics. Besides being very expensive, i t i s also less e f f e c t i v e than T i and again reduces machinability and notch toughness. F i n a l l y , the b e n e f i c i a l e f f e c t s of V and Nb are in dispute, at l e a s t for off-corner panel c r a c k i n g . 9 ' 1 0 ' 1 4 2.5.2 Thermal Treatment Although thermal treatment i s highly i n f l u e n t i a l on panel cracking, there i s l i t t l e agreement on the optimal thermal treatment. This i s due, i n part, to the dual r o l e i t plays i n i n f l u e n c i n g both high temperature d u c t i l -i t y and s t r e s s generation. 85 2.5.2.1 Mid-face Panel Cracks As previously discussed, the incidence of mid-face panel cracking has been reduced i n many s t e e l plants by avoiding very long track t i m e s . 5 - 8 Nashiwa 6 suggests that ingots be charged into the soaking p i t before a maxi-mum of 32,000-40,000 seconds (9-11 hours), depending on the jacketed time, as shown i n Figure 2.27. Other studies are vague as to the actual l i m i t s to set but the intent i s to keep the ingot surface above some c r i t i c a l tempera-t u r e . 3 ' 8 Another s o l u t i o n to mid-face panel cracking i s slow c o o l i n g 3 ' 5 ' 8 by e i t h e r stacking ingots together or keeping them i n a holding furnace. Cracking may increase i f ingots cool while standing too close together 5, 6 so symmetrical cooling may be the important feature here. F i n a l l y , as previously mentioned, Guerin & R o c c a t a g l i a t a 5 found that l a y i n g one face of the hot ingot on an i n s u l a t i n g bed of v e r m i c u l i t e , eliminated mid-face panel cracking. This was believed due to the asymetri-c a l cooling allowing the one hot face to absorb a l l of the generated stress by p l a s t i c creep deformation. 2.5.2.2 Off-corner Panel Cracks A v a r i e t y of d i f f e r e n t thermal solutions have met with some success i n reducing off-corner panel cracking. A t y p i c a l example of the cooling prac-t i c e s to avoid i s given i n Figure 2.27. Unfortunately, the exact times and reheating rates necessary are disputed, and vary between operations, as 86 (Hours) 15000 25000 Jacketed time(s) Figure 2.27 E f f e c t of cooling p r a c t i c e on panel c r a c k i n g 6 , 87 p r e v i o u s l y discussed. However, these solutions do f a l l into two general philosophies. The f i r s t i s to prevent the surface from f a l l i n g below the Ar^ temperature i n t o the two-phase r e g i o n . 8 ' 9 This i s achieved by s t r i p -ping the ingot early (short jacketed time) and quickly t r a n s f e r i n g the ingot to the soaking p i t (short unjacketed time). Sussman 9 suggests t h i s should be followed by a rapid reheat with a high gas f i r i n g rate i n a hot soaking p i t . This s o l u t i o n has the added advantage of thermal e f f i c i e n c y but i s often d i f f i c u l t to achieve due to l o g i s t i c s problems. 9' 1 4 The second s o l u -t i o n i s to ensure t h a t , i f the s u r f a c e does drop below the Ar^, that i t cools completely below the Ar^ before reheating. This i s achieved by s t r i p -p i n g l a t e , and c o o l i n g e x t e n s i v e l y b e f o r e charging to the soaking p i t . 6 ' 8-10' 1 2 ' 1 4 T h i s s o l u t i o n has the disadvantage of being expensive to implement due to lower production rates, l o g i s t i c problems and higher f u e l consumption. 9' 1 2 * 1 4 F i n a l l y , several companies have found success with c o n t r o l l e d reheating r a t e s i n the soaking p i t . 9 - 1 2 ' 1 4 such as shown i n Figure 2.28. Many com-panies achieve t h i s by lowering the soaking p i t temperature at charge for those s t e e l compositions designated as susceptible to off-corner panel c r a c k i n g 8 ' 1 0 ' 1 4 and/or by i n c r e a s i n g the time i n the soaking p i t . 1 1 Sussman 9 states that the slow cooling p r a c t i c e s o l u t i o n should be combined with slow reheating rates. 88 (Hours) 10 15 20 1400 I 2 0 0 o ~ 1000 <u w 3 o « 8 0 0 o. E 6 0 0 4 0 0 25 30 1 1 1 1 1 — — / Ingot Full heating / soaking ~~\Stripped ingot \ (air cooling) / — / Controlled / ingot reheating — Ingot/pit equalisation • 1 i • i i 1 0 50P00 Time after stripping (s) 100,000 Figure 2.28 Soaking p i t reheating p r a c t i c e recommended to reduce off-corner panel c r a c k i n g 1 0 89 2.6 Conclusions Panel cracking i s manifested as two d i s t i n c t l y d i f f e r e n t types of cracking problems: mid-face panel cracking and off-corner panel cracking. The e f f e c t s of s t e e l composition on both hot d u c t i l i t y and panel crack form-a t i o n are now f a i r l y well understood. However, solutions based on chemistry are often undesirable and many companies have adopted elaborate and expen-sive cooling and heating practices i n an attempt to reduce panel cracking. Due to the inconclusive feed back i n a d i v e r s i f i e d s t e e l plant, and the in t e r m i t t e n t nature of panel c r a c k i n g , 1 4 i t i s not known, i n the majority of cases, how b e n e f i c i a l these practices are. The section on reduced hot d u c t i l i t y revealed that the e f f e c t s of ther-mal h i s t o r y on d u c t i l i t y are not well understood. However, even If the best thermal treatment were known, i t i s impossible to give the same thermal treatment to every part of the ingot. While the surface i s heating r a p i d l y , the i n s i d e may be heating slowly or even c o o l i n g . When combined with the generation of thermal stress, and the Influences of phase transformation and creep, the e f f e c t s of thermal h i s t o r y on panel cracking are extremely com-plex . 90 In conclusion, there i s a large incentive to understand the influence of thermal treatment on panel cracking, but a great deal of confusion s t i l l e x i s t s . Most of the observations and r e s u l t s from previous cracking studies can be explained by eit h e r the reduced d u c t i l i t y or stress generation mech-anisms. Although they probably both contribute to some extent, i t i s not known, at the present, which i s the most important. Since the stress gen-e r a t i o n mechanism has received by far the least a t t e n t i o n , and consequently i s the most vaguely formulated, i t presents a very promising area for research using modelling techniques. 9 1 CHAPTER 3 SCOPE OF THE PRESENT WORK In view of the l i t e r a t u r e findings just presented, the present work was undertaken to increase our understanding of the panel cracking problem. The ultimate goal i s to determine the mechanism(s) behind both types of panel crack formation and to f i n d i n d u s t r i a l l y f e a s i b l e means for eli m i n a t i n g them. This would have tremendous s i g n i f i c a n c e , not only to s t a t i c - c a s t ingot production, but to continuous ca s t i n g as well, where the cracking of reheated blooms i n a manner reminiscent of panel cracking i s becoming an increasing concern. 3.1 Objectives In s t r i v i n g towards this ultimate goal, this research project pursued the following ob j e c t i v e s : 1) to construct a physical scale model of a soaking p i t and use i t to determine the e f f e c t s of process reheating v a r i a b l e s on flow pattern and flame structure and t h e i r possible r o l e i n panel crack formation. 2) to formulate, develop and v e r i f y , accurate mathematical computer models for c a l c u l a t i n g both heat transfer and stress generation i n a s o l i d i f y i n g , cooling and reheating, s t a t i c - c a s t s t e e l Ingot from the time of i n i t i a l pouring u n t i l the time of r o l l i n g . 92 3) using the models, to determine the i n t e r n a l state of stress under d i f f e r e n t processing conditions for two ingots representing t y p i c a l mid-face and off-corner panel cracking conditions. A) to conduct a b r i e f m e t a l l u r g i c a l i n v e s t i g a t i o n of actual panel-cracked samples. 5) to c l a r i f y the r o l e of stress generation i n both types of panel crack formation and i n so doing, to o f f e r mechanism(s) for panel crack formation that are consistent with the mathematical model p r e d i c t i o n s , physical model r e s u l t s , m e t a l l u r g i c a l i n v e s t i g a t i o n , and the observations made i n previous studies on d u c t i l i t y of s t e e l and panel cracking. 3.2 Methodology This i n v e s t i g a t i o n studied panel crack formation i n s t a t i c - c a s t s t e e l Ingots through phy s i c a l modeling, metallographic i n v e s t i g a t i o n , and mathematical heat-flow and s t r e s s - a n a l y s i s modeling. Because t h i s i n v e s t i g a t i o n i s the f i r s t of i t s kind i n several respects, a great deal of background work not d i r e c t l y related to panel cracking was required. The f i r s t was a comprehensive l i t e r a t u r e review on the pertinent subject of the hot d u c t i l i t y of s t e e l that was presented previously. In a d d i t i o n , the mathematical modeling techniques themselves are r e l a t i v e l y recent and comprise important areas for research i n t h e i r own r i g h t . F i n a l l y , input data on thermomechanical properties of s t e e l for the models was required, n e c e s s i t a t i n g both lab experiments and further l i t e r a t u r e i n v e s t i g a t i o n . 93 While t h i s work was concerned with e l u c i d a t i n g the mechanisms behind the general problem of panel cracking encountered i n t e r n a t i o n a l l y i n the s t e e l industry, i t focussed p r i m a r i l y on examining the nature of off-corner panel cracking experienced at the Stelco H i l t o n Works. This was due to the a v a i l a b i l i t y of ingot samples containing panel cracks, soaking p i t and ingot blue p r i n t s , d e t a i l e d studies on the incidence of panel cracking including s t e e l compositions and thermal processing schedules, and temperature measurement data for v e r i f i c a t i o n of the heat-transfer model. The s p e c i f i c methodology used i n the present i n v e s t i g a t i o n was as follows: 1 ) To better characterize heat trans f e r i n the soaking p i t , a p h y s i c a l , l / 8 t ^ scale model of a bottom-fired, Stelco soaking p i t was constructed based on isothermal modeling techniques. An i n v e s t i g a t i o n of the flow patterns, v e l o c i t y p r o f i l e s and flame geometry i n the model then was conducted. 2) The f i r s t step i n developing the mathematical heat-transfer model was a comparison of the a v a i l a b l e numerical (finite-element) methods f o r s o l v i n g t h i s type of complex, t r a n s i e n t , heat-conduction problem involving s o l i d i f i c a t i o n , v a r i a b l e thermophysical properties and i r r e g u l a r geometry (mould corruga-tions ). 3) The best method obtained, i n terms of accuracy, s t a b i l i t y and cost e f f i c i e n c y was formulated for the p a r t i c u l a r thermal property data and boundary conditions of the s t e e l ingot 94 s o l i d i f i c a t i o n , c o o l i n g , and reheating problem being considered. 4) The temperature predictions of the f i n a l heat-transfer model were v e r i f i e d against both a n a l y t i c a l solutions and i n d u s t r i a l measurements made at Stelco on a 230 x 405 mm ingot. 5) Finite-element meshes of nodes were constructed to simulate two d i f f e r e n t ingot s i z e s : a 350 mm x 350 mm, two-ton ingot representing the mid-face panel cracking case and a 760 x 1520 mm twenty-five ton ingot representing off-corner panel cracking. 6) The f i n a l heat-transfer model was run to construct temperature contour maps, p r o f i l e s and temperature-time pl o t s f o r several thermal processing h i s t o r i e s , c a r e f u l l y chosen to simulate conditions for mid-face panel cracking, off-corner panel cracking and non-cracking cases. 7) A finite-element stress model then was formulated to determine the stresses a r i s i n g from the previously calculated temperature data. This task was complicated by the involvement of several f a c t o r s , a l l of which had to be i n c l u d e d b e f o r e a c c u r a t e s t r e s s c a l c u l a t i o n s were achievable: (a) Since stress i s desired as a function of time, the model must be dynamic or t r a n s i e n t . (b) Thermo-mechanical properties such as e l a s t i c modulus and Poisson r a t i o change d r a s t i c a l l y over the large temperature ranges involved. 95 (c) S o l i d i f i c a t i o n involves l i q u i d material that i s not capable of withstanding load and i s further complicated by the movement of the s o l i d - l i q u i d i n t e r f a c e with time. (d) Stresses are caused by both thermal and phase-transformation volume changes. The volume change due to phase trans-formation i s further complicated by the k i n e t i c s of the y •*• a transformation which i s delayed on cooling s i g n i f i c a n t l y below the t r a n s f o r m a t i o n temperatures encountered on heating. (e) The s t r e s s - s t r a i n r e l a t i o n i s h i g h l y non-linear as s t r a i n contains both e l a s t i c and p l a s t i c components. The most important e f f e c t I n f l u e n c i n g high-temperature s t r e s s development i s creep which i s both temperature and st r e s s dependent. An a n a l y t i c a l s o l u t i o n for e l a s t o - p l a s t i c , time-dependent thermal stress development was derived and used to v e r i f y the stress model. The stress model then was formulated with appropriate boundary conditions and the best a v a i l a b l e thermo-mechanical property data i n c l u d i n g : (a) temperature-dependent e l a s t i c modulus. (b) temperature-, s t r e s s - , and structure-dependent p l a s t i c creep r a t e . 9 6 ( c ) t e m p e r a t u r e - , and c o m p o s i t i o n - d e p e n d e n t t h e r m a l e x p a n s i o n i n c l u d i n g phase t r a n s f o r m a t i o n . Due t o i n e x a c t l i t e r a t u r e d a t a , d i l a t o m e t e r e x p e r i m e n t s were c o n d u c t e d on samples o f p a n e l - c r a c k prone s t e e l t o d e t e r m i n e the e x a c t b e h a v i o r o f t h e y •*• a phase t r a n s f o r m a t i o n w i t h r e s p e c t t o t h e r m a l e x p a n s i o n . A major d e f i c i e n c y i n the m o d e l i n g o f s t r e s s g e n e r a t i o n i n s t e e l i s the l a c k of a g e n e r a l c o n s t i t u t i v e e q u a t i o n t o a d e q u a t e l y d e s c r i b e the v i s c o - p l a s t i c b e h a v i o r 8 2 . Thus, as p a r t o f t h e p r o j e c t , t h e a v a i l a b l e m e c h a n i c a l d a t a had t o be m a n i p u l a t e d t o d e v e l o p an adequate d e s c r i p t i o n o f t h e s t r e s s - s t r a i n b e h a v i o r o f s t e e l a t a p p r o p r i a t e t e m p e r a t u r e s and s t r a i n r a t e s and t o i n c o r -p o r a t e t h i s i n t o t h e s t r e s s m o d e l. The f i n a l s t r e s s model was r u n f o r the same c o n d i t i o n s as t h e h e a t - t r a n s f e r model t o r e l a t e s t r e s s g e n e r a t i o n t o the i m p o r t a n t i n f l u e n c i n g v a r i a b l e s : ( a ) i n g o t s i z e ( b ) s t r i p t i m e ( c ) t r a c k t i m e ( d ) s o a k i n g p i t r e h e a t i n g r a t e , as d e t e r m i n e d by s o a k i n g - p i t t e m p e r a t u r e a t c h a r g e , gas f i r i n g r a t e , and p o s s i b l y , i n g o t p o s i t i o n . ( e ) s o a k i n g - p i t r e s i d e n c e t i m e ( f ) p h a s e - t r a n s f o r m a t i o n t e m p e r a t u r e i n t e r v a l as a f f e c t e d by s t e e l c o m p o s i t i o n ( p a r t i c u l a r l y c a r b o n c o n t e n t ) . 97 (11) C r i t e r i a f o r t h i s type of high-temperature fracture then had to be found including conditions for panel-crack i n i t i a t i o n and propagation. These suggested the most e f f e c t i v e means of present-ing the output from the s t r e s s model. (12) No work on understanding a cracking problem would be complete without a m e t a l l u r g i c a l examination of the cracks themselves. Thus, the l a s t step was to conduct a metallographic i n v e s t i g a t i o n of off-corner panel-cracked samples supplied by Stelco i n c l u d i n g : (a) N i t a l , Oberhoffer, P i c r i c acid and HC1 macro-etches, a sulphfur p r i n t , microprobe and SIMS scans of polished specimens of the cracked regions to determine the r e l a t i o n of the cracks to the s t e e l structure and to detect the presence of any segregation. (b) an S.E.M. examination of the fr a c t u r e surface. (c) measurement and c a l c u l a t i o n of the t h i c k n e s s of the decarburized zone to Infer the approximate time of cracking. (13) F i n a l l y , the r e s u l t s of the mathematical heat-transfer and stress models were synthesized with the findings of the l i t e r a t u r e reviews, m e t a l l u r g i c a l i n v e s t i g a t i o n and ph y s i c a l modeling studies to o f f e r mechanism(s) for both types of panel crack formation. The accurate mathematical modeling of stresses i n a s o l i d i f y i n g ingot i s obviously a very complex problem. Fortunately, due to the p a r t i c u l a r way i n which panel cracking i s manifested, i t i s possible to make a few 98 s i m p l i f y i n g assumptions, while s t i l l r e t a i n i n g s u f f i c i e n t accuracy to ob-serve the determining thermal and stress related aspects of panel cracking. Both types of panel cracking have predominantly v e r t i c a l components over a large portion of the c e n t r a l region of the ingot. Indeed, cracks u s u a l l y appear f i r s t near the center and extend upwards and downwards only i n severe cases. I t should therefore be possible to understand panel cracking through examination of a two-dimensional, transverse s l i c e through the midplane of the ingot alone. Secondly, the two-fold symmetry of both the Ingot dimen-sions and panel cracks themselves requires that only 1/4 of the ingot s e c t i o n need be modeled. This region i s i l l u s t r a t e d i n Figure 3.1. F i n a l l y , the present thermo-mechanical problem being solved by the heat t r a n s f e r and stress models i s uncoupled. This means that any heat generated by i n e l a s t i c deformation has a n e g l i g i b l e e f f e c t on the temperature pro-f i l e s . Thus the heat-transfer model can be used Independently to generate temperature data f o r subsequent input into the stress model. 99 +- X Figure 3.1 Ingot section to be simulated with the mathematical heat-transfer and stress models 100 CHAPTER 4 PHYSICAL MODELLING OF A BOTTOM-FIRED SOAKING PIT 4 . 1 Introduction The reheating of s t e e l ingots i n a soaking p i t i s a c r u c i a l phase i n s t e e l processing where defects such as panel cracking can a r i s e . Poor flow patterns r e s u l t i n g i n uneven heating rates not only reduce heating e f f i c i e n c y ; they also aggravate and increase the l i k e l i h o o d of th i s and other types of cracking problems. Reheating i s achieved at Stelco using a bottom-fired Salem soaking p i t . Since t h i s type of p i t has received r e l a t i v e l y l i t t l e a t t e n t i o n , i t s flow patterns and heating c h a r a c t e r i s t i c s are not well understood. The main obje c t i v e of t h i s part of the work was to determine the e f f e c t s of process v a r i a b l e s such as f i r i n g c o n d i t i o n and ingot p o s i t i o n i n g on the flow pattern and flame structure i n t h i s bottom-fired soaking p i t . The r e s u l t s were then used to re l a t e panel cracking to soaking p i t operation and to a s s i s t i n c h a r a c t e r i z i n g heat transfer to the ingot during reheating f o r the mathematical model. An i n v e s t i g a t i o n of t h i s type could proceed along any of several d i f f e r e n t courses: i n - p l a n t measurements on an operating furnace, c o n t r o l l e d laboratory experiments using hot or cold models, or predictions using mathematical models. However, I n d u s t r i a l experiments are very e x p e n s i v e , time-consuming and sometimes i n c o n c l u s i v e owing to the 1 0 1 d i f f i c u l t i e s of taking measurements i n a h o s t i l e environment. Mathematical modelling i s also very d i f f i c u l t given the complex geometry of the burner nozzle and confined flow region in s i d e t h i s soaking p i t . Thus, f o r the present i n v e s t i g a t i o n , an isothermal physical model was constructed. Its advantages are low cost, r e l a t i v e s i m p l i c i t y and the ease and accuracy with which measurements can be made. This chapter describes the fundamental basis behind the model design and „ the r e s u l t s of three separate experimental studies made using t h i s model: o v e r a l l flow patterns, v e l o c i t y p r o f i l e s and the a i r - f u e l mixing behavior. To the author's knowledge, t h i s study i s the f i r s t to examine both flow patterns and mixing c h a r a c t e r i s t i c s within a bottom-fired soaking p i t . The r e s u l t s of t h i s study should also provide i n s i g h t s for improving soaking p i t operation and aid i n future furnace design. 4.2 Soaking P i t Operation The prototype i s a Salem, bottom-fired soaking p i t , representative of the 36 independently operated p i t s at the Stelco H i l t o n Works i n Hamilton, Ontario. Each p i t i s flanked by a "recuperator" to recover waste heat from the products of combustion or "exhaust gases" which leave the p i t at 1350°C. This heat i s transferred to the incoming a i r which r a i s e s i t s temperature to 800°C. 102 A schematic diagram of flow through the soaking p i t i s shown i n Figure A . l . The a i r flows from the recuperators, through two arch-roofed, b r i c k tunnels underneath the p i t f l o o r . These incoming ducts empty in t o a burner chamber located beneath the center of the p i t f l o o r , through which the f u e l i n l e t pipe protrudes. The a i r then d e f l e c t s upwards and enters the furnace through the annulus formed between the f u e l i n l e t pipe and a ceramic flame s t a b i l i z i n g r i n g . The unpreheated 50°C f u e l gases discharge from the ten equally-spaced, roughly-oval e x i t ports of a multiport burner nozzle situated atop the f u e l i n l e t pipe which i s concentric with the primary a i r duct. The burner applies no s w i r l to the incoming gases. The p i t i t s e l f measures 5.2 m x 5.2 m x 3.5 m deep. Zero gauge pressure i s maintained inside the p i t by balancing the incoming a i r and f u e l pressures with the external draft from the discharge stack. The combusted waste gases e x i t the p i t through twelve rectangular e x i t ports located along the bottom of each of the two opposing chamber walls adjacent to the recup-e r a t o r s . The other two opposing walls are r e f e r r e d to as the " b l i n d s i d e " w a l l s . Six to twelve ingots, varying i n si z e from 710 mm x 890 mm (28" x 35") ten ton ingots to 760 mm x 2000 mm (30" x 79") t h i r t y ton ingots, are charged, e i t h e r hot or cold, into the p i t with an overhead crane. Immed-i a t e l y a f t e r charging, the p i t i s run at "high f i r e " f o r 1-8 hours to qui c k l y heat the ingots to about 1250°C. Once a set p i t temperature i s achieved, i n d i c a t i n g that the ingot temperature has e q u i l i b r a t e d , the heat-103 Platform (ground level) t f t i / / > Flame stabilizing ring Z Z Z T J J J J J J J J Exit port to recuperator Waste gas Cold air InUt V / r r r r r -WM / Preheated air p u a | in arched - roof line ducts Waste gas ducts Figure 4.1 Schematic cross section of prototype soaking p i t 104 ing rate i s gradually dropped back to "soaking" or "low f i r e " conditions. This i s maintained f o r another 6 to 10 hours at which time the ingot i s removed for hot r o l l i n g . Aside from decreasing the absolute f u e l and a i r flow rates, the f u e l composition also varies from high to low f i r e . A 4:1 r a t i o mixture of blast-furnace gas and coke-oven gas i s used during h i g h - f i r i n g conditions. The c a l o r i f i c value of the f u e l i s reduced by decreasing the coke-oven gas f r a c t i o n and under low - f i r e conditions, pure blast-furnace gas i s combusted. Recently, a "new h i g h - f i r e " mixture of 6:1 blast-furnace to coke-oven gas has been used. For the purposes of t h i s study, the terms "high f i r e " , and "low f i r e " are used only to re f e r to the p a r t i c u l a r f u e l mixtures. The amount of unburned oxygen i n the waste combustion gases, or "exhaust oxygen percentage", i s monitored continuously and i s set at a standard of 2%. However, along with f u e l composition and mass flow rate, the exhaust oxygen percentage i s a process v a r i a b l e subject to change. 105 4.3 D e s c r i p t i o n of the Model A l / 8 t ^ - s c a l e p l e x i g l a s s model of the b o t t o m - f i r e d soaking p i t prototype was constructed at Stelco Research and the U n i v e r s i t y of B r i t i s h Columbia. Figure 4.2 shows a photograph of the model which consists of three main sections: f l o o r , mid-section, and roof. The f l o o r i s permanently attached to a portable table from which the a i r and f u e l I n l e t duct system Is suspended. The duct work includes the burner chamber, f u e l i n l e t pipe and arch-roofed tunnels f o r primary a i r . Separate l i n e s f or the simulated f u e l and a i r gas streams lead back to an a i r blower system ( i n c l u d i n g a Godfrey N 2250/3T Roots blower). Each l i n e has i t s own o r i f i c e plate flow meter attached to a U-tube manometer and adjustable gate valve f o r d e l i v e r i n g any desired flow rate of a i r and f u e l . Like the majority of previous p h y s i c a l models designed to run isothermally, a i r i s n a t u r a l l y used as the f l u i d f o r simulating the a i r , f u e l and combustion gases of the prototype. Although a i r moves through the model e x c l u s i v e l y from the blower pressure, the lack of flow resistence creates n e g l i g i b l e pressure build-up i n the p i t i t s e l f . The mid-section consists of the square soaking p i t area with 24 exhaust ports leading into the top sections of two i d e n t i c a l recuperators. I t i s constructed e n t i r e l y from transparent p l e x i g l a s s , as are the flame s t a b i l i z i n g r i n g , f u e l i n l e t tube and underlying duct work. A removable burner nozzle assembly was machined from brass to match the prototype and i s pictured i n Figure 4.3. Figure 4.2 Photograph of e n t i r e soaking p i t model Figure 4.3 Model burner assembly 108 The roof Is detachable from the mid-section to permit loading and unloading ingots. The ingots themselves are f l a t - b l a c k , wooden blocks, scaled to model Stelco's large 760 mm x 1520 mm (30" x 60") 25-ton s t e e l i n gots. To f a c i l i t a t e v i s u a l i z a t i o n of the p i t i n t e r i o r , a few model ingots were constructed from transparent p l e x i g l a s s . 4.4 P h y s i c a l Modelling C r i t e r i a The physical model was designed to duplicate both the flow and mixing c h a r a c t e r i s t i c s of the actual soaking p i t during operation. In order to ensure s i m i l i t u d e with the prototype, despite being an isothermal model, four separate modelling c r i t e r i a were adopted. Geometric s c a l i n g i s the f i r s t , most obvious requirement. With the exception of the f u e l burner nozzle, a i r i n l e t annulus, and the underlying duct work, a l l l i n e a r dimensions of the prototype were reduced by a f a c t o r of 8 f o r the model. A s u b s t a n t i a l expansion of gases accompanies combustion just beyond the burner nozzle In the prototype soaking p i t . To take t h i s i n t o account, and enable both the flow pattern and mixing behavior to be reproduced i n the isothermal ph y s i c a l model, I t i s necessary to d i s t o r t some of the model dimensions. Thring and Newby 8 3 developed a c r i t e r i o n for taking into account gas expansion from a s i n g l e , c i r c u l a r nozzle j e t t i n g flame into a much l a r g e r , c y l i n d r i c a l tunnel furnace. I t Is based on the p r i n c i p l e of maintaining constant burner j e t momentum. Simply stated, the area of the model burner nozzle must be d i s t o r t e d larger by the r a t i o between the 109 d e n s i t i e s of the cold Incoming gas and the hot combustion gases i n the f u r n a c e . 8 4 Beer and C h i g i e r 8 5 adapted the Thring-Newby c r i t e r i o n f o r modelling furnaces with double-concentric burners by applying t h i s c r i t e r i o n separately to d i s t o r t both the primary burner and the secondary burner annulus. I t was subsequently used to s u c c e s s f u l l y model a v a r i e t y of other systems, inc l u d i n g pulverized-coal flame p l a n t s 8 6 and U - f i r e d soaking p i t s . 8 7 ' 8 8 T h i s same procedure was used to d i s t o r t separately the burner nozzle and a i r annulus of the present model. Robertson 8 9 warns that i f the burner diameter i s too large i n r e l a t i o n to the chamber diameter, t h i s d i s t o r t i o n can modify the flow patterns somewhat. In the present model, the a i r duct work, burner chamber and flame s t a b i l i z i n g r i n g were a l l d i s t o r t e d by a factor of roughly 1.4. The e n t i r e burner assembly and f u e l pipe were d i s t o r t e d 2.IX. The a i r duct d i s t o r t i o n i s less since the a i r i n the prototype i s preheated. To maintain the same "view" for the Incoming a i r around the f u e l pipe, only the width of the arch-roofed a i r tunnel was d i s t o r t e d i n achieving the desired cross s e c t i o n a l area. Figure 4.4 i l l u s t r a t e s the extent of these d i s t o r t i o n s . D e t a i l s of the c a l c u l a t i o n s are presented i n Appendix I. The t h i r d parameter that must be kept the same i n the model and proto-type i s the mass flow rate r a t i o between the second and primary f l u i d s or the " a i r - f u e l r a t i o " . 8 h * 8 9 ' 9 0 This r a t i o varied between .8 and 1.8 for the d i f f e r e n t f u e l s used under d i f f e r e n t f i r i n g conditions. It was therefore treated as a process v a r i a b l e . — 102 mm • (•45 mm*) Fuel gas 144 mm 95 mm Air Air Geometrically scaled burner nozzle Actual burner nozzle with Thring-Newby distortion Geometrically scaled burner chamber Actual burner chamber with distorted width Figure 4 .4 Distortion of soaking pit model dimensions using the Thring-Newby criterion I l l The f i n a l s i m i l i t u d e c r i t e r i o n required to reproduce flow and mixing conditions from the prototype i n the model i s the Reynolds number. However, i n an Internal flow system such as the soaking p i t , where the Reynolds number exceeds 1 0 , 0 0 0 , transport phenomena are c o n t r o l l e d e n t i r e l y by i n e r -t i a l forces and the flow regime i s f u l l y turbulent. Several authors have found that flow patterns In the regime are r e l a t i v e l y i n s e n s i t i v e to changes i n Reynolds number. 8 4' 9 1 - 9 3 Thus, so long as the model a i r flow rates are above those necessary to ensure a f u l l y turbulent flow regime at the burner nozzle e x i t , they can be considerably l e s s than those necessary to achieve exact Reynolds number s i m i l a r i t y . 8 4 C a l c u l a t i o n s i n Appendix II show that both the f u e l and a i r j e t s for the prototype and model have Reynolds numbers well i n excess of 10,000 for a l l f i r i n g conditions. In any case, the e f f e c t of absolute mass flow rate on the flow pattern was determined as part of the i n v e s t i g a t i o n . Because of the dominance of the l n e r t i a l forces, i t i s unnecessary to consider Schmidt, Prandtl or Froude number s i m i l a r i t y . 8 4 4.5 Flow Pattern Study The f i r s t study using the soaking p i t model was to v i s u a l i z e the over-a l l gas flow and r e c i r c u l a t i o n patterns developed under a v a r i e t y of d i f f e r -ent process conditions. The aim was simply to gain a q u a l i t a t i v e under-standing of the flow patterns and to determine the most i n f l u e n t i a l process v a r i a b l e s . S p e c i f i c objectives of the study were: 112 1) to i n v e s t i g a t e the e f f e c t s of changing f i r i n g condition (as determined by a i r - f u e l mass flow r a t i o ) on the flow pattern. 2) to f i n d what changes i n flow pattern occur with d i f f e r e n t ingot arrangements. In p a r t i c u l a r , i t was desired to determine i f the flow pattern changes during charging or i s affected by blocking recuperator e x i t channels using non-standard ingot arrangements. 3) to explore the r e l a t i o n s h i p between flow pattern and ingot height, as determined both by d i f f e r e n t Ingot size and by scale build-up on the p i t bottom. 4.5.1 Experimental Methodology To obtain worthwhile observations of the flow patterns, i t i s extremely important to use a good flow v i s u a l i z a t i o n technique. Past researchers have used tracers varying from styrofoam beads or colored chemicals i n water m o d e l l i n g to b a l s a dust In a i r . 9 4 ' 9 5 However, to v i s u a l i z e flow patterns i n the present model, small (1-3 mm) h e l i u m - f i l l e d soap bubbles were i n j e c t e d into the incoming f u e l gas stream. Because of t h e i r excellent r e f l e c t i o n properties under proper l i g h t i n g conditions, they were found to be superior to other a i r stream t r a c e r s . Any bias due to buoyancy e f f e c t s was removed by balancing the soap-helium r a t i o to render the bubbles weightless. 113 Before experiments could begin, the o r i f i c e - p l a t e flow meters had to be c a l i b r a t e d . D e t a i l s of the c a l c u l a t i o n s and experiments for t h i s c a l i b r a -t i o n are given i n Appendix I I I . I n i t i a l experimental runs were video-taped but subsequently, 35 mm s t i l l photographs were used. For each experimental condition, top, front and side views of the soaking p i t were examined, with a photo flood lamp providing " l i g h t s e c t ioning". A i r - f u e l r a t i o s were then determined for the three d i f f e r e n t f i r i n g conditions c u r r e n t l y used at Stelco and are given In Appendix IV. Those r a t i o s , based on Stelco's normal operating practice of 2% exhaust oxygen, and t h e i r corresponding f i r i n g conditions are as follows: (1) 1.8 simulating the 4:1 r a t i o mixture of blast-furnace to coke oven gas used for the " h i g h - f i r e " condition. (2) 1.5 simulating the 6:1 r a t i o mixture of BF to CO gas used for the "new high f i r e " c o ndition. (3) 0.8 simulating pure BF gas used for the "low f i r e " or "soaking" c o n d i t i o n . I n i t i a l experiments were conducted to determine i f varying Reynolds number i n the f u l l y turbulent regime had any influence on the flow pattern. This was done by increasing the a i r and f u e l flow rates at a f i x e d a i r - f u e l r a t i o while leaving a l l other variables constant. The r e s u l t s of these i n i t i a l experiments showed no change in flow pattern with a 67% increase i n absolute flow rates for either burner. Thus, the assumption that s i m i l i t u d e between model and prototype can be maintained without achieving equally high 114 Reynolds number i n the model, so long as the flow regime i s f u l l y turbulent, was v e r i f i e d . In subsequent experiments, f i x e d a i r and f u e l flow rates were used: 45 1/s a i r : 25 1/s f u e l f o r high f i r e , 37:25 f or new high f i r e and 30:37 f or low f i r e . For each of the three heating conditions tested, the a i r and f u e l flow rates were f i r s t set at the desired l e v e l s . Then, ingots were changed two at a time and the flow patterns recorded photographically f o r each of the following ingot arrangements: empty p i t , 2 ingots, 4 ingots, 6 ingot standard, 8 ingot standard, and several non-standard 8 ingot arrangements. Figures 4.5A and B i l l u s t r a t e the standard 6 and 8 ingot arrangements r e s p e c t i v e l y . A f a l s e f l o o r was then inserted to rai s e the ingot height to simulate t a l l e r ingots and/or scale build-up on the p i t f l o o r . The standard ingot arrangements were re-tested. 4.5.2 Results Figure 4.6 i s a t y p i c a l example of the helium bubble photographs which shows the o v e r a l l flow pattern. A schematic diagram of t h i s flow pattern was constructed from the many photographs and observations that were made and i s presented i n Figure 4.7. The drawing i s divided down the center to show how the flow pattern appears both with and without the presence of ingots. Figure 4.5 Model soaking p i t chamber with ingots charged i n A) standard 6 ingot arrangement and B) standard 8 ingot arrangement 116 Figure 4.6 Photograph of helium bubble movement I l l u s t r a t i n g flow pattern (taken at high f i r e using standard 8 ingot arrangement). 117 Figure 4.7 Schematic diagram of o v e r a l l flow pattern 1 1 8 4.5.2.1 E f f e c t of F i r i n g Conditions Extensive experiments at several d i f f e r e n t a i r - f u e l r a t i o s have deter-mined that, over the normal range of f i r i n g conditions, the flow pattern i s independent of a i r - f u e l r a t i o . As i l l u s t r a t e d i n Figures 4.6 and 4.7 the i n i t i a l upward j e t i s f a i r l y s t r a i g h t with the upward flow appearing strongest at i t s center. Bubbles project upward to scatte r across the roof i n a l l d i r e c t i o n s . This flow i s strongly r a d i a l l y symmetric and i s reminiscent of a water jet from a garden hose splashing d i r e c t l y against a f l a t surface. The bubbles then c o l l e c t on the sides of the p i t and f i n a l l y e i t h e r rush toward the ex i t ports or slowly r e c i r c u l a t e towards the burner nozzle. Rapidly s w i r l i n g bubbles, Indicating turbulence, are observed In the corners near the top, between the ingots and on the f l o o r near the burner where the r e c i r c u l a t i n g gas i s re-entrained into the j e t . 4.5.2.2 E f f e c t of Ingot Arrangement The schematic drawing i n Figure 4.7 was divided down the center to show the d i f f e r e n c e s i n flow pattern r e s u l t i n g from the presence of in g o t s . In general, ingot p o s i t i o n i n g had an almost n e g l i g i b l e e f f e c t on flow pattern f o r a l l of the many ingot arrangements investigated. The same basic flow pattern was observed for 2, 4, 6 and 8 ingot charges, showing that the flow pattern i s not appreciably a l t e r e d during charging. Even when extremely non-standard ingot arrangements were tested, such as p h y s i c a l l y blocking most of the recuperator ports on both sides, the f a m i l i a r mushroom-like flow 119 pattern p r e v a i l e d . The only discernable e f f e c t of the ingots i s to force the gentle r e c i r c u l a t i n g regions to occur higher, above the ingots, or to move between them. 4.5.2.3 E f f e c t of Ingot Height The "headroom" above the Ingots can be changed eit h e r by scale build-up on the soaking p i t f l o o r or by charging d i f f e r e n t sized ingots. For the model experiments, t h i s was achieved by r a i s i n g the p i t f l o o r . I t was found to have l i t t l e e f f e c t on the flow pattern. The only change i s an increased turbulence i n the region above the ingots when bubble movement i s more con-f i n e d . The flow stream also elongates s l i g h t l y i n the v e r t i c a l d i r e c t i o n . 4.6 V e l o c i t y P r o f i l e Study The second stage of the i n v e s t i g a t i o n was to measure gas v e l o c i t i e s q u a n t i t a t i v e l y at various key lo c a t i o n s i n the model soaking p i t . The object i v e of t h i s study was to obtain a more precise understanding of flow i n these areas by detecting any s l i g h t aberrations i n the flow f i e l d not poss i b l e by v i s u a l means alone. The s p e c i f i c measurements made were: (1) upward v e r t i c a l v e l o c i t y through the a i r Inl e t annulus (2) h o r i z o n t a l and upward v e r t i c a l v e l o c i t i e s throughout the p i t (3) downward v e r t i c a l v e l o c i t y along the p i t walls (4) e x i t v e l o c i t y through the 24 ports to the recuperators 120 The e f f e c t s of both d i f f e r e n t ingot arrangements and ingot heights were again examined. In add i t i o n , the influence of p a r t i a l scale blockage of the burner was studied. 4.6.1 Experimental Methodology Gas v e l o c i t i e s i n the physical model were determined by the pressure drop between the s t a t i c holes and the dynamic hole i n a p i t o t tube using a type 504 i n c l i n e d manometer (Air Flow Developments). D e t a i l s of the c a l i -b r a t i o n of the system are given i n Appendix V. Measurements i n the four d i f f e r e n t areas of the p i t were made i n d i f f e r e n t ways. To measure a i r v e l o c i t i e s at the a i r i n l e t annulus, the t i p of a s t r a i g h t p i t o t - s t a t i c tube was positioned v e r t i c a l l y at f l o o r l e v e l over the midway point between the insi d e and outside burner ducts. Within the p i t i t s e l f , upward v e r t i c a l v e l o c i t i e s were determined at various distances from the c e i l i n g using the s t r a i g h t p i t o t - s t a t i c tube. At each p o s i t i o n , a h o r i z o n t a l a i r v e l o c i t y measurement also was taken by r o t a t i n g an L-shaped p i t o t - s t a t i c tube about i t s axis u n t i l the d i r e c t i o n of maximum v e l o c i t y was located. These measurements required d r i l l i n g 25 evenly-spaced holes through the roof of the soaking p i t . Since i t was evident from the previous study that a l l normal f i r i n g conditions f o r t h i s burner had a s i m i l a r behavior, only the high f i r e condition was studied. Each of the 18 conditions tested was repeated f o r both the standard 6 and 8 ingot arrangements. 121 The r e s u l t s have been displayed by: plan views i l l u s t r a t i n g the magni-tude and d i r e c t i o n of the h o r i z o n t a l v e l o c i t i e s and graphs of v e r t i c a l v e l o c i t y versus distance from the f l o o r of the p i t . To avoid confusion, these distances are expressed i n percentage of the p i t height which i s 440 mm i n the model and 3.5 m i n the prototype. I t i s important to note that the p i t o t tube must be positioned to point d i r e c t l y i n t o the flow f o r accurate v e l o c i t y measurements to be taken. Where the p i t o t tube i s badly misaligned with the flow d i r e c t i o n , v e l o c i t y readings are lower than actual and can even become negative. Since the flow i n large areas of the p i t i s neither predominantly v e r t i c a l nor h o r i z o n t a l , only a small sample of the t o t a l number of v e l o c i t y measurements made i s included i n t h i s chapter as fi g u r e s . The two h o r i z o n t a l v e l o c i t y p r o f i l e s were taken from near the bottom and top of the p i t r e s p e c t i v e l y where the flow i s predominantly h o r i -z o n t a l . In p l o t t i n g upward v e r t i c a l v e l o c i t i e s , dashed l i n e s were used to indicate v e l o c i t y components which were low because they were made i n regions of the p i t where flow was n o n - v e r t i c a l . Downward v e r t i c a l a i r v e l o c i t y measurements were made along the walls of the p i t at various distances from the c e i l i n g using a s p e c i a l l y -constructed, U-shaped p i t o t - s t a t i c tube. This required d r i l l i n g 24 new holes around the perimeter of the p i t , 30 mm from the walls. Runs were again repeated to investigate both the standard 6 and 8 ingot arrangements and graphs of v e l o c i t y versus percentage p i t height were p l o t t e d . F i n a l l y , a i r v e l o c i t i e s were measured at a l l 24 ports e x i t i n g to the l e f t and ri g h t recuperators, using the standard L-shaped p i t o t - s t a t i c tube. The t i p of the tube was placed at and perpendicular to the plane of the 122 recuperator entrance. This required the d r i l l i n g of holes i n the recup-erator roof d i r e c t l y behind and above each e x i t port. For each of the 71 experimental conditions examined, several a i r v e l o c i t y measurements were taken at each of the 24 ports and the r e s u l t s averaged s t a t i s t i c a l l y . The conditions were chosen to investigate the e f f e c t s of f i r i n g c ondition, standard and non-standard ingot p o s i t i o n i n g , f l o o r height and scale blockage (simulated by s t r a t e g i c a l l y positioned wood c h i p s ) . In ad d i t i o n , the e f f e c t s of both angle and distance of the ingot to the recuperator walls were inve s t i g a t e d . Graphs of e x i t v e l o c i t y versus port p o s i t i o n were drawn for each run to display the r e s u l t s . 4.6.2 Results 4.6.2.1 A i r In l e t V e l o c i t i e s The v a r i a t i o n i n v e r t i c a l a i r v e l o c i t y with p o s i t i o n around the a i r i n l e t annulus i s shown i n Figure 4.8. V e l o c i t i e s at the recuperator sides are generally much lower than at the b l i n d sides, i n d i c a t i n g a lack of r a d i a l symmetry that was not apparent by v i s u a l observation i n the f i r s t study. This implies that more a i r i s e x i t i n g through the bli n d sides. This d i f f e r e n c e i s at t r i b u t e d to a "tunnelling e f f e c t " caused by the underlying a i r duct work. Incoming a i r through the tunnel from each recuperator r e t a i n s i t s ho r i z o n t a l momentum even a f t e r h i t t i n g the f u e l pipe and d e f l e c t i n g upwards, as i l l u s t r a t e d i n Figure 4.9. It flows around the f u e l pipe to meet the stream from the opposite tunnel at the midway or b l i n d side" of the a i r annulus. There, the streams impinge and merge, r e s u l t i n g i n more upward flow on these sides of the a i r annulus. In addition, a i r 1 2 3 Figure 4.8 Upward v e r t i c a l v e l o c i t y ( i n m/s) versus p o s i t i o n i n the a i r i n l e t annulus 124 Figure 4.9 Schematic diagram of burner chamber and arched roof a i r i n l e t tunnels showing development of non symmetrical flow d i s t r i b u t i o n through a i r annulus 125 v e l o c i t i e s at the NW and SE d i r e c t i o n s appear p a r t i c u l a r l y high although there i s no immediately apparent reason f o r t h i s . 4.6.2.2 Inside P i t V e l o c i t i e s The horizontal v e l o c i t y measurements Indicated i n Figures 4.10 and 4.11 and the v e r t i c a l measurements graphed i n Figure 4.12 generally r e i n f o r c e the v i s u a l observations of flow patterns (documented i n section 4.5). Curve E i n Figure 4.12 shows the upward v e r t i c a l v e l o c i t y along the burner axis to be the highest at mid height of the p i t . Below 30% p i t height, the maximum upward v e r t i c a l v e l o c i t i e s are found at a radius of 75-100 mm from the burner axis, corresponding roughly to l o c a t i o n D. Lower v e l o c i t i e s are measured both farther away and closer to the burner a x i s . The low readings d i r e c t l y above the burner could be p a r t l y due to the high s t a t i c pressure but they i n d i c a t e that the a i r - f u e l j e t i s a c t u a l l y stronger on the outside i n the lower l e v e l s of the p i t . This was not apparent i n the v i s u a l obser-vations reported i n section 4.5. Figure 4.10 shows that h o r i z o n t a l v e l o -c i t i e s below mid-height, are n a t u r a l l y very low. Several positions have no measurement due to the presence of Ingots or the burner. The d i r e c t i o n of maximum hori z o n t a l v e l o c i t y constantly f l u c t u a t e s , r e f l e c t i n g the random s w i r l i n g nature of turbulent gas currents i n the burner j e t . The slow entrainment of surrounding a i r by the burner j e t i s indicated by v e l o c i t y vectors pointing generally inward towards the burner a x i s . 1 2 6 West recuperator X 1 1 1 1 1 1 1 0 1 2 3 4 5 6 Velocity (m/s) \ / X / \ 4 \ < \ East recuperator Figure 4.10 Horizontal v e l o c i t y p r o f i l e s i n lower p i t section (9% p i t height, standard 8 ingot arrangement) 1 2 7 O I I I I I I I 0 I 2 3 4 5 6 Velocity (m/s) / West recuperator \ East recuperator \ Figure 4.11 Horizontal v e l o c i t y p r o f i l e i n upper section above Ingots (96% p i t height, standard 8 ingot arrangement) 1 2 8 Figure 4.12 Upward v e r t i c a l v e l o c i t y at various locations in the p i t (marked i n Figure 4.11) 129 Above the mid-height of the p i t , the v e r t i c a l v e l o c i t y component along the burner axis gradually decreases while h o r i z o n t a l v e l o c i t i e s s t e a d i l y increase to maximum values near the roof. In Figure A.11, r a d i a l symmetry of the flow f i e l d i s r e f l e c t e d by the consistent d i r e c t i o n of the maximum ho r i z o n t a l v e l o c i t y away from the burner a x i s . This strong, general upward motion combined with s t e a d i l y increasing outward movement ind i c a t e s that the burner j e t fans out somewhat before h i t t i n g the roof. Near the f l o o r and between the burner j e t and the walls, both v e r t i c a l and h o r i z o n t a l v e l o c i t i e s are very low. This i n d i c a t e s that large "dead volumes" ex i s t where the only movement i s a gradual r e c i r c u l a t i o n of gases d r i f t i n g slowly towards the burner j e t to be entrained. No e f f e c t of changing ingot arrangement on the v e l o c i t y p r o f i l e s was found. This i s understandable since the ingots are positioned i n the pre-dominantly low v e l o c i t y "dead zone" of the p i t where obstructing the low v e l o c i t y flow would have l i t t l e e f f e c t . 4.6.2.3 Wall V e l o c i t i e s The graph of downward v e r t i c a l wall v e l o c i t y i n Figure 4.13 shows very low values near the c e i l i n g and f l o o r . This r e f l e c t s the non-vertical nature of flow i n these areas. A i r t r a v e l l i n g h o r i z o n t a l l y across the c e i l i n g over the ingots d e f l e c t s downwards upon reaching either the b l i n d side walls or recuperator walls. The greatest v e l o c i t i e s are found near the top of the p i t down the center of these s i d e s . While v e l o c i t i e s down the corners are r e l a t i v e l y low near the top, they gradually increase to become Middle of blind side " of recuperator side Near corner Figure 4.13 Downward v e r t i c a l v e l o c i t y along p i t walls 131 the highest below 50% p i t height. In a d d i t i o n , below 70% p i t height, velo-c i t y down the b l i n d sides i s s i g n i f i c a n t l y lower than down the recuperator sides. These r e s u l t s r e f l e c t the increasing a i r flow d e f l e c t i n g horizon-t a l l y towards the corners from the b l i n d sides. 4.6.2.4 Recuperator Port V e l o c i t i e s T y p i c a l recuperator port e x i t v e l o c i t i e s are shown i n Figure 4.14 where i t i s seen that the maximum v e l o c i t y i s at the corners. This trend p e r s i s t s f o r a l l f i r i n g conditions, ingot heights, and ingot arrangements, inc l u d i n g the empty p i t . Even when pieces of wood were inserted to block the spaces between ingots the " U " shaped v e l o c i t y p r o f i l e was observed. Thus, the phenomenon seems inherent to t h i s design of soaking p i t . The U-shaped v e l o c i t y p r o f i l e can be explained by considering that roughly h a l f of the o r i g i n a l burner j e t a i r h i t t i n g the roof t r a v e l s down the two b l i n d side walls and eventually d e f l e c t s toward the recuperators. The e x i t ports nearest the corners n a t u r a l l y must accept most of t h i s a i r i n a d d i t i o n to t h e i r share of a i r from the recuperator walls. The extent to which t h i s e f f e c t occurs can vary somewhat, depending on how close the ingots are against the recuperator walls. Moving c e n t r a l ingots c l o s e r to the walls r e s u l t s i n almost uniform a i r flow through a l l the ports. A l t e r n a t i v e l y , moving corner ingots closer to the wall or cover-ing both the front and back of the burner with wood chips simulating scale both exaggerate the U-shape v e l o c i t y p r o f i l e s . Another i n t e r e s t i n g observa-t i o n i s that covering one side of the burner simply increases v e l o c i t i e s 1 3 2 i—r~i—r 5.1 4.9r-i i i — m — r Low fire 3.9 3.7 i I I I I I I I I I L N 4 6 8 Port number 10 12 S Figure 4.14 Recuperator port e x i t v e l o c i t i e s (west recuperator, standard 8 ingot arrangement) 133 through the e x i t ports on the opposite side, as the e n t i r e flow pattern i s skewed. The r e s u l t s of experimental runs examining the importance of ingot p o s i t i o n i n g close to the recuperator walls are i l l u s t r a t e d i n Figure 4.15. I f the ingot face i s more than 35 mm from the port, ingot p o s i t i o n i n g has no e f f e c t on e x i t v e l o c i t y . Since t h i s corresponds to only .28 m (or about 12") i n the prototype, t h i s implies that e x i t port v e l o c i t y i s unaffected by ingot p o s i t i o n i n g unless an ingot p h y s i c a l l y touches the recuperator wall at some point. Moving the ingot face c l o s e r than 35 mm i n i t i a l l y causes a i r to accelerate s l i g h t l y through the p a r t i a l l y shielded e x i t ports. A maximum i n e x i t v e l o c i t y occurs when a c r i t i c a l distance of about 20 mm i s reached, corresponding to .16 m (or 6 inches) i n the prototype. Moving the ingot face even closer r e s u l t s i n a rapid decline i n e x i t v e l o c i t y as those ports e f f e c t i v e l y become blocked. While a i r v e l o c i t y through the blocked ports eventually decreases to zero, the o v e r a l l flow pattern i n the p i t remains unaffected. The same e f f e c t s were found when ingots touching the recup-erator wall along one edge were moved closer to reduce the angle between t h e i r mid-faces and the w a l l . A f i n a l observation i s that v e l o c i t i e s through the e x i t ports are always the highest at the top of i n d i v i d u a l ports. This e f f e c t i s exaggera-ted by r a i s i n g the f l o o r and indicates that a i r flowing down the recuperator wall exits at i t s e a r l i e s t opportunity. Ingot height (or scale build-up on the f l o o r ) had l i t t l e other e f f e c t on exit v e l o c i t y through the ports. 1 3 4 Figure 4.15 E f f e c t of ingot positioning close to recuperator exit port on gas v e l o c i t y 135 4.7 Mixing Study The f i n a l study using the soaking p i t model was to determine the height and shape of the burner flame, thereby gaining i n s i g h t into r a d i a t i v e heat transfer within the p i t . This ambitious undertaking using an isothermal model could only be done by assuming that the flames are d i f f u s i o n c o n t r o l l e d . Thus, the stoichiometric mixing front between the f u e l and a i r streams corresponds to the flame boundary. At the operating temperatures i n a soaking p i t , the chemical reactions of combustion are extremely rapid so th i s i s a good assumption. 9 6 The s p e c i f i c objectives of the mixing study were: (1) to determine the flame height and shape under standard operating conditions. (2) to find those operating variables most d i r e c t l y responsible for deter-mining flame geometry by i n v e s t i g a t i n g the e f f e c t s of changing: f i r i n g conditions, exhaust oxygen percentage and ingot arrangement. (3) to compare the resultant model flame height and shape predictions to observations and photographs of actual flames taken at Stelco. 136 4.7.1 Experimental Methodology In order to c a l c u l a t e the r e l a t i v e amounts of f u e l and a i r at location s throughout the model soaking p i t , helium gas was continuously injected as a tracer into the f u e l stream. An i n i t i a l experiment was performed to v e r i f y that s i m i l a r mixing r e s u l t s are obtained when i n j e c t i n g tracer into the a i r stream. Local helium concentrations were measured throughout the p i t using a 4 component gas analysis system con s i s t i n g of a sampling probe, gas chrom-atograph, d i g i t a l i n t e g r a t o r and 1/4 HP vacuum pump. The gas sample was c a r e f u l l y drawn at below 300 ml/s through a s p e c i a l l y tapered 45 mm x 1.5 mm D sampling probe i n order to minimize any d i s t o r t i o n of the flow pattern. The sample then flowed through 3 mm brass tubing to one of two gas analyzing columns i n the gas chromatograph. While one sample was being analyzed, another could be c o l l e c t e d . The d i g i t a l integrator connected to the gas chromatograph printed out helium counts which were converted to percent using the r e s u l t s from previous c a l i b r a t i o n s with a standard. Any problems of re s i d u a l gas accumulation were eliminated because the sampling system was capable of c o l l e c t i n g gas samples continuously. From the l o c a l helium concentrations, the p o s i t i o n of the flame front was determined by c a l c u l a t i o n s of the stoichiometric mixing r a t i o . Assuming that "mixed i s burned", the flame-front helium concentration i s a function only of i n i t i a l helium concentration, a i r - f u e l r a t i o and exhaust oxygen percentage. 137 These c a l c u l a t i o n s , given i n Appendix VI, show that, f o r an i n i t i a l h e l i u m concentration i n the f u e l stream of 2%, and an exhaust concentra-t i o n i n the prototype of 2%, the flame front occurs where helium concentra-tions drop to 0.78%, 0.87%, and 1.20% f o r the high f i r e , new high f i r e and low f i r e conditions r e s p e c t i v e l y . Complete mixing i n the p i t y i e l d s exhaust helium concentrations of 0.72%, 0.80% and 1.10%. The r e s u l t s were displayed by p l o t t i n g contour maps of helium concentration at each of 7 l e v e l s cut at d i f f e r e n t heights through the model soaking p i t . In add i t i o n , a side p r o f i l e of helium concentration versus height at a mid plane through the burner axis was made for each experimental run. I n i t i a l runs were made to check the system and to determine i f varying Reynolds number i n the f u l l y turbulent regime had any influence on mixing c h a r a c t e r i s t i c s . Raising the volumetric flow rates of f u e l and a i r at a fixed a i r - f u e l mass r a t i o was found to have no e f f e c t on either the flame height or shape. Thus, the previous assumption that Reynolds number was unimportant so long as flow was i n the f u l l y turbulent regime has been v e r i f i e d for mixing as well as for o v e r a l l flow pattern. Subsequent experimental runs were made using the three previously noted a i r and f u e l flow rates for the high f i r e , new high f i r e and soaking condi-t i o n s . Each run was repeated using three d i f f e r e n t exhaust oxygen concen-t r a t i o n s (1%, Stelco's standard set point of 2%, and 3%) and three d i f f e r e n t ingot arrangements (empty p i t , standard 6 ingot, and standard 8 i n g o t ) . Several of the experimental conditions tested are summarized i n Figure 4.16 along with t h e i r r e s u l t s . These runs include simulation of the conditions 1 3 8 Run f i r i n g c ondition exhaust a i r / f u e l excess s t o i c h i o m e t r i c % O2 mass flow a i r flame r a t i o (%) front high f i r e 1.0 high f i r e 2.0 high f i r e 3.0 new high f i r e 2.0 low f i r e (soak) 2.0 1.68 1.79 1.91 1.49 0.79 6.5 14.0 21.6 16.0 23.0 Mk Figure 4.16 Mixing experiments and r e s u l t s (using standard 8 ingot arrangement) 139 under which a picture was taken at Stelco of actual flames i n a soaking p i t with i t s roof removed. 4.7.2. Results The flame-front p r o f i l e s determined through stoichiometric concentra-t i o n c a l c u l a t i o n s for f i v e d i f f e r e n t experimental conditions are i l l u s t r a t e d i n Figure 4.16. Because the flame shape was found to be generally symme-t r i c , sampling for several runs was conducted i n only h a l f of the p i t to save time and to conserve helium. A t y p i c a l set of helium contours at d i f f -erent l e v e l s i n the p i t i s presented i n Figure 4.17. For c l a r i t y , only the flame front contour i t s e l f has been shown. The r e s u l t s reveal that the flame i n t h i s bottom-fired soaking p i t using low c a l o r i f i c fuels i s gener-a l l y quite short. Figure 4.17 describes the flame geometry under standard high f i r e con-d i t i o n s . The flame f i r s t appears j u s t above the burner and spreads to reach i t s maximum width by about 30% p i t height. I t then tapers o f f and event-u a l l y disappears at 60% p i t height. The presence of turbulent eddy currents causes the flame to move around slowly during the course of a run. This random f l u c t u a t i n g behavior makes i t very d i f f i c u l t to reproduce i d e n t i c a l r e s u l t s at every l e v e l . 1 4 0 percentage of p i t height: Figure 4.17 Stoichiometric flame front contours at d i f f e r e n t p i t cross sections (high f i r e , 2% exhaust oxygen, standard 8 ingot arrangement) 1 4 1 Figure 4.17 also shows that the flame has a s l i g h t tendency to elongate towards the recuperator walls. In addition, for runs where a i r v e l o c i t i e s through the i n l e t were greater In the NW and SE d i r e c t i o n s , the flame tended to lean toward the NE and SW d i r e c t i o n s . This i s due to the resistance to f u e l flow caused by increased a i r flow through the b l i n d sides of the a i r annulus. Combined with the h o r i z o n t a l momentum imparted to the a i r stream by the underlying ductwork, t h i s d e f l e c t s f u e l towards the recuperator si d e s . The r e s u l t i n g elongated flame i s consistent with the conclusion of previous w o r k e r s 9 6 that the flame i s longer where there i s an excess of f u e l or a shortage of a i r . 4.7.2.1 E f f e c t of Excess A i r An examination of Figure 4.16 reveals that the highest flames occur under high f i r e conditions. Decreasing the a i r - f u e l r a t i o to new high f i r e then low f i r e conditions r e s u l t s i n a continuously shortening flame. This i s understandable since the lower c a l o r i f i c - v a l u e f u e l mixtures, r e q u i r i n g more excess a i r , d i l u t e the f u e l stream f a s t e r . Thus, mixing to the s t o i c h -iometric percentage required f or combustion occurs c l o s e r to the burner. F i r i n g condition i s not the only important v a r i a b l e a f f e c t i n g flame l e n g t h , however. The p r e v i o u s tren d o n l y h o l d s at c o n s t a n t exhaust 0^ l e v e l s . Decreasing the percent oxygen i n the exhaust gas r e s u l t s i n drama-t i c increases i n flame length. In f a c t , I n j e c t i n g exactly the s t o i c h -iometric a i r requirement with no allowance for exhaust oxygen would r e s u l t 142 i n an i n f i n i t e flame length using the flame front c a l c u l a t i o n method i n Appendix VI. While i n p r a c t i c e , the r e s u l t would simply be incomplete com-bustion, Figure 4.16 shows that flames can h i t the roof and even l i c k down the p i t sides to the recuperator e x i t s i f the exhaust oxygen percentage f a l l s to 1%. This mushroom shaped flame i s never encountered for exhaust oxygen percentages of 2% or more as the flame length c o n t i n u a l l y decreases with increasing exhaust oxygen concentration. The e f f e c t s of these two i n f l u e n t i a l v a r i a b l e s ; a i r - f u e l r a t i o ( f i r i n g c ondition) and exhaust oxygen percentage, are not independent. In f a c t , the determining v a r i a b l e for flame height was found to be excess a i r , which was c a l c u l a t e d from given f u e l compositions and exhaust oxygen percentages i n Appendix IV. Figure 4.18 i l l u s t r a t e s the r e l a t i o n s h i p between excess a i r and flame length using data from the f i v e experimental runs given i n Figure 4.16. It i s of p a r t i c u l a r i n t e r e s t to note that nearly the same flame l e n g t h was a c h i e v e d i n run 3 (34%) using high f i r e and 3% exhaust as i n run 5 (28-32%) using low f i r e and 2% exhaust 0^. Since these two d i f f e r e n t combinations of c o n t r o l l a b l e process v a r i a b l e s both had nearly the same -21.5 vs 23% - excess a i r , t h i s r e s u l t indicates that percent excess a i r i s the more fundamental va r i a b l e determining flame height. However, the importance of excess a i r i s most pronounced when i t f a l l s below about 15%. Figure 4.18 shows that above t h i s l e v e l , the flame height i s f a i r l y short and consistent. Figure 4.18 Dependence of flame height on excess a i r percentage f o r runs given i n Figure 4.16 144 As the flame becomes shorter, i t also tends to become hollow i n the center, reducing in the extreme to a t o r r o i d or ring shape. Runs 4 and 5 i n Figure 4.16 i l l u s t r a t e t h i s phenomenon f or the new high f i r e and low f i r e cases. This happens because the Increased excess a i r promotes f a s t e r mix-in g . This a i r immediately d i l u t e s the f u e l stream to below stoi c h i o m e t r i c requirements before i t can move around the burner cap to the region d i r e c t l y above i t . While runs 3 and 5 had s i m i l a r flame heights and excess a i r percent-ages, Figure 4.16 shows that t h e i r flame shapes are quite d i f f e r e n t . Thus, excess a i r i s not the sole v a r i a b l e determining flame geometry. The d i f f e r -ences can be at t r i b u t e d to the d i f f e r e n t a i r - f u e l r a t i o s . Because the e x i t ports on the f u e l nozzle are sloped, increasing the r e l a t i v e flow through the f u e l l i n e (by decreasing the a i r - f u e l r a t i o ) imparts more h o r i z o n t a l momentum to the gases. This forces mixing to occur further away from the burner nozzle which r e s u l t s i n an increasing tendency f o r a torroid-shaped flame rather than a s o l i d one. The e f f e c t i s increased with increasing angle of the f u e l ports with respect to the burner a x i s . 1 4 5 4.7.2.2 E f f e c t of Ingot Arrangement Figure 4.19 shows that the flame geometry i s v i r t u a l l y independent of ingot arrangement regardless of whether zero, 6 or 8 ingots are charged. This lack of dependence of the flame on ingot arrangement i s consistent with the previous findings of the flow pattern and v e l o c i t y p r o f i l e studies. I t i s understandable since the areas of the p i t where ingots are placed are predominantly "dead zones" having l i t t l e influence on the i n i t i a l burner j e t . 4.8 Comparison with I n d u s t r i a l Observations A photograph of the flames i n a soaking p i t running with i t s roof removed i s presented i n Figure 4.20. The test was conducted under standard high f i r e conditions except that the exhaust oxygen percentage was unknown. The flame height i s estimated from t h i s photograph to be roughly 175% of p i t height. When compared with model run 2 i n Figure 4.16 done at high f i r e using the standard exhaust oxygen set point of 2%, t h i s observed height appears s u b s t a n t i a l l y higher. However, the model flame height recorded for run 1 in Figure 4.16 using 1% exhaust oxygen i s projected to have been 150-200% of p i t height i f the roof were not present. Considering that the model runs were performed with a closed roof, t h i s r e s u l t agrees with the i n d u s t r i a l observation, i f the exhaust oxygen percentage had f a l l e n below the 2% set point at the time the photograph was taken. In a d d i t i o n , mixing by molecular d i f f u s i o n , which i s required on the micro-scale before combust-Figure 4.19 E f f e c t of ingot arrangement on flame geometry (high f i r e , 2% exhaust oxygen) Photograph of the flames in a Stelco soaking p i t with I t s roof removed 1 4 8 ion takes place, can produce flames over 25% higher than those predicted by the macro-scale mixing assumption made i n the model. 9 6 This f a c t also could account for the longer flame observed i n the prototype. Besides the quantitative comparison of flame height, several q u a l i t a t -ive features of the model flames can also be compared with the prototype. I f examined c l o s e l y , one can see the s l i g h t l y lower In t e n s i t y of the i n t e r i o r of the flame. In a d d i t i o n , the flames i n the prototype were observed to wander and f l u c t u a t e , i n d i c a t i n g the presence of the same turbu-l e n t eddy currents that were encountered during the model simulations. The flame shape i t s e l f i s s l i g h t l y more v e r t i c a l l y elongated i n the prototype but t h i s i s most l i k e l y due to the lack of a roof over the soaking p i t . In conclusion, the model appears able to reproduce behavior i n the prototype i n at l e a s t a q u a l i t a t i v e sense. 4.9 Implications f o r Heat Transfer and Panel Cracking Heat i s transferred to the ingots i n a bottom f i r e d soaking p i t through a combination of both r a d i a t i o n and convection. Using an isothermal model, i t i s d i f f i c u l t to exactly quantify heat transfer i n the p i t . Nevertheless, several important conclusions regarding both r a d i a t i o n and convection can be drawn from these studies. 1 4 9 Because of the high operating temperatures, the majority of the heat tra n s f e r r e d to the ingots i s through r a d i a t i o n . C a l c u l a t i o n s which support t h i s conclusion are given i n Appendix VII. They are based on v e l o c i t y measurements i n the model soaking p i t and c a l c u l a t i o n s f o r forced convective heat t r a n s f e r . This heat can radiate from the flame i t s e l f , the hot com-bustion gas, or most importantly, the p i t walls and roof. Since the hot combustion gas flows across the roof and down the walls, i t s heat w i l l be transferred to the roof and walls as well as to the upper parts of the ingots. The walls, i n turn, w i l l radiate heat back to the tops of ingots and those sides facing walls. F i n a l l y , convective heat transfer i s highest where gas v e l o c i t i e s are highest. Thus, although convection plays only a minor role i n t r a n s f e r r i n g heat d i r e c t l y to the ingots, i t too w i l l be greatest on the outer sides and tops of the ingots. This w i l l r e s u l t i n ingot heating rates that are somewhat uneven as the sides f a c i n g the p i t walls heat f a s t e r . Cernoch reports t h i s same phenomemon i n other soaking p i t d e s i g n s . 9 7 This study has revealed very few i n d i c a t i o n s of an enhanced p o t e n t i a l f o r panel cracking i n ingots charged i n c e r t a i n p o s i t i o n s . In f a c t , the complete lack of dependence of e i t h e r the flow patterns or the mixing char-a c t e r i s t i c s with ingot arrangement i s s u r p r i s i n g , when one considers that 8 ingots comprise over 25% of the t o t a l p i t volume. The excel l e n t s t a b i l i t y of the flow and mixing patterns implies that ingot p o s i t i o n i n g i n the p i t i s not l i k e l y to be a fa c t o r with regard to cracking tendency. The only poss-i b l e exception i s for ingots charged very close (le s s than 1 foot) to ports i n the recuperator w a l l s . It Is therefore recommended that Ingots be kept away from the recuperator walls whenever p o s s i b l e . 1 5 0 A f a c t o r of much greater Importance i s the exhaust oxygen percentage. I f i t i s allowed to become very low ( l e s s than 1%), then the dramatic increase i n size of the flame could r e s u l t i n flame impingement on the ingots. The r e s u l t could be damage to those ingot faces by severe oxida-t i o n . Even i f d i r e c t flame impingement did not occur, the increased temp-eratures could r e s u l t i n extreme grain growth and perhaps even l o c a l grain boundary melting. This would enhance d u c t i l i t y related cracking problems l a t e r , p a r t i c u l a r l y for long times i n the soaking p i t . Thus, the c a r e f u l monitoring of exhaust oxygen l e v e l s and p i t temperatures i s recommended. A f i n a l f actor not previously discussed i s the alignment of the f u e l pipe i n the burner chamber. S l i g h t errors i n alignment are g r e a t l y magni-f i e d i n d i s t o r t i n g the shape of the a i r annulus. In the model studies, t h i s r e s u l t e d i n major d i s t o r t i o n s i n the symmetry of both the flow pattern, as in d i c a t e d by recuperator port e x i t v e l o c i t i e s and the flame shape. This v a r i a b l e i s therefore l i k e l y to influence heat transfer much more than ingot p o s i t i o n i n g . 151 4.10 Summary and Conclusions This chapter has described three experimental studies made using a l / 8 f c ^ scale model of a bottom-fired soaking p i t . The o v e r a l l flow patterns, v e l o c i t y p r o f i l e s and mixing c h a r a c t e r i s t i c s were determined under a v a r i e t y of d i f f e r e n t operating conditions. The combination of these three studies together can be applied to present a consistent d e s c r i p t i o n of flow and flame c h a r a c t e r i s t i c s i n the bottom-fired soaking p i t . The i n i t i a l j e t of f u e l and a i r discharging from the burner nozzle g e n e r a l l y mixes very quickly to produce a short, bushy flame. The height of t h i s flame i s c o n t r o l l e d by the percentage of excess a i r . For a given exha-ust oxygen percentage, i t becomes longer as the f u e l mixture becomes r i c h e r i n coke gas, increasing i n length continuously from low f i r e to new high f i r e to high f i r e . For a given f u e l , i t also becomes longer and can even touch the roof i f the percent exhaust oxygen f a l l s below the set point of 2%. The i n i t i a l a i r j e t i s stronger through the b l i n d side of the i n l e t annulus due to the configuration of the underlying duct work. This causes the flame to elongate towards the recuperators. The flame i s le a s t intense i n i t s center and, when excess a i r exceeds 15% causing very f a s t mixing, the flame reduces to a ring shape about the top of the burner. From the burner, the gas j e t sw i r l s upwards, fanning out s l i g h t l y before h i t t i n g the roof. The highest v e l o c i t i e s are found i n the outside of t h i s j e t , which slowly entrains surrounding a i r as i t r i s e s . A f t e r h i t t i n g the roof, the gas d e f l e c t s equally i n a l l d i r e c t i o n s to t r a v e l h o r i z o n t a l l y 152 across the roof, over the ingots towards the w a l l s . Gas flowing towards the recuperators d e f l e c t s downwards between the wall and the back, face of ingots and through the top of the e x i t ports. Gas flowing down the b l i n d side walls must s p l i t to flow towards either recuperator w a l l . This added flow increases gas v e l o c i t i e s through recuperator ports nearest the corners. V e l o c i t y decreases through ports next to ingots positioned very close to the recuperator w a l l . F i n a l l y , some of the gas moving down the walls d r i f t s inwards across the f l o o r and between the ingots to become re-entrained i n the i n i t i a l gas j e t . The large zones i n these lower regions of the p i t where the ingots are located exhibit very low v e l o c i t i e s . Because of them, the o v e r a l l flow pattern and flame geometry are almost completely unaffected by ingot arrangement (including number of ingots charged), ingot height and scale build-up. Within the range of standard operating conditions, the flow pattern i s also unaffected by the a i r - f u e l mass flow r a t i o . Several of the findings from t h i s study can be applied to enable a more comprehensive understanding of flow patterns and heat tran s f e r and a s s i s t i n the design of other types of furnaces. These general conclusions are as follows: (1) Flames produced by low c a l o r i f i c f u e l s , such as b l a s t furnace gas, are generally short and busy. In the bottom-fired soaking p i t , the flow of hot gases d e f l e c t s along the roof and down the walls to t r a n s f e r heat to the ingots more r a p i d l y from the outside inwards. 153 (2) The flow pattern i s i n s e n s i t i v e to obstructions i n low v e l o c i t y regions, even i f they are very l a r g e . Thus, ingot p o s i t i o n i n g i s not expected to be an important f a c t o r i n panel cracking. (3) Percent excess a i r i s the most important v a r i a b l e c o n t r o l l i n g flame length. Flame length increases dramatically with decreasing excess a i r which i s the fundamental v a r i a b l e c o n t r o l l i n g flame length. (4) Flame length increases with both increasing a i r - f u e l mass flow r a t i o ( f o r a fixed exhaust oxygen percentage) and decreasing exhaust oxygen percentage ( f o r a given f u e l ) . (5) The momentum imparted to gases moving through the duct work underlying the burner nozzle i s i n f l u e n t i a l i n the development of both flow pattern and flame shape. The importance of the p r i o r duct work there-fore must not be overlooked i n furnace design. (6) The a i r - f u e l r a t i o also influences flame shape when the burner ports are angled with respect to the burner a x i s . Increasing the angle or decreasing the a i r - f u e l r a t i o causes the flame to broaden. (7) Flow pattern and flame c h a r a c t e r i s t i c s are both independent of absolute a i r and f u e l flow rates so long as the gases e x i t the burner nozzle i n the f u l l y turbulent regime. (8) Isothermal physical models can be used e f f e c t i v e l y to study heat tr a n s f e r as well" as flow patterns In an i n d u s t r i a l furnace. 1 5 4 CHAPTER 5 COMPARISON OF NUMERICAL METHODS FOR HEAT TRANSFER MODELLING 5 . 1 Introduction Many numerical methods have been employed to solve f o r the temperature d i s t r i b u t i o n i n transient heat-conduction problems with or without change of phase. T r a d i t i o n a l l y , f i n i t e - d i f f e r e n c e techniques have been applied with considerable success; but as i n t e r e s t has grown i n complex shapes and combined heat flow/stress problems, an example of which i s the s o l i d i f i c a -t i o n of s t e e l ingots with corrugations, a t t e n t i o n has turned to f i n i t e -element methods developed o r i g i n a l l y for str e s s analysis of s t r u c t u r e s . As a r e s u l t , the number of numerical methods and versions of each, a v a i l a b l e f o r use i n tackl i n g a given heat-flow problem, has increased r a p i d l y . However, the comparative advantages of the d i f f e r e n t techniques with respect to accuracy, s t a b i l i t y and cost remain unclear. This chapter w i l l examine th i s question by comparing the temperature predictions of several d i f f e r e n t formulations of the standard finite-element method, 9 8 the matrix method of Ohnaka 9 9 and the a l t e r n a t i n g - d i r e c t i o n , i m p l i c i t f i n i t e - d i f f e r e n c e m e t h o d 1 0 0 against a n a l y t i c a l solutions for two problems. Because the immediate objective of th i s study Is to determine the best numerical method with which to model heat flow i n a corrugated s t e e l Ingot, the two problems have been chosen to approximate d i f f e r e n t stages i n ingot processing: reheating i n a soaking p i t and s o l i d i f i c a t i o n i n the mould. These problems also test the a b i l i t y of the numerical methods to handle 1 5 5 temperature-dependent boundary c o n d i t i o n s and the l a t e n t heat of s o l i d i f i c a t i o n r e s p e c t i v e l y . Two-dimensional heat flow i n the transverse mid-plane of the ingot has been considered. The f i r s t problem examined was the transient temperature d i s t r i b u t i o n i n the t r a n s v e r s e s e c t i o n of a s t e e l i n g o t (0.762 m x 1.524 m) "convectively" heated i n a soaking p i t , as shown i n Figure 5.1. Heat was assumed to transfer uniformly to a l l four sides of the ingot, g i v i n g r i s e to a temperature d i s t r i b u t i o n with two-fold symmetry; thus only one-quarter of the ingot section need be considered. The i n i t i a l temperature of the Ingot at c h a r g i n g , T , c o n v e c t i v e heat transfer c o e f f i c i e n t , h, surrounding p i t temperature, T o, and thermophysical properties, k, p and were a l l assumed to be constant. Heat conduction within the ingot then i s described mathematically by 5.2 Problems Studied and A n a l y t i c a l Solutions 5.2.1 Reheating i n a Soaking P i t the well-known p a r t i a l d i f f e r e n t i a l equation k ( + ay' (5 .1 ) * A l l symbols are defined in a Nomenclature sec t i o n at the end of t h i s chapter 1 5 6 y (m ) 0,762 Soaking pit atmosphere Tgj = II00°C q = h(Ts- T a )) Steel ingot / q = 0 t T (x.y.t) T0=600°C K • D y (0J905.Q3429) / • C (0.0762,0.1905) yf • B (0.0762 ,0.0762) / K V »• x (m ) T 0381 q^ds-T^) Figure 5.1 Schematic diagram of f i r s t problem applied over the two-dimensional, rectangular domain, 0 < x < 0.381 0 < y < 0.762 with the i n i t i a l condition, T = T o and boundary conditions: x = .381, - k = 0 5x y = .762, - k -g = 0 x = 0, - k | ^ = h (T - T ) ' ox 0 0 y = 0, - k | ^ = h (T - T ) J * by ro 1100 Time (s) Figure 5.2 Temperature response of locations A, B, C, D, E (Shown i n Figure 5.1) from a n a l y t i c a l s o l u t i o n to f i r s t problem 159 The a n a l y t i c a l s o l u t i o n to t h i s problem was o b t a i n e d by the superposition of two one-dimensional seri e s solutions from L u i k o v 1 0 1 . A s u f f i c i e n t number of terms were taken to ensure an estimated accuracy, with respect to the exact s o l u t i o n , of better than ± 0.01% for both short and long times. Values of the parameters used i n the c a l c u l a t i o n s are given i n Table 5.1 and Figure 5.2 shows sample temperature responses from the a n a l y t i c a l s o l u t i o n for various locations i n the ingot. 5.2.2 S o l i d i f i c a t i o n i n the Mould The second problem was concerned with the temperature d i s t r i b u t i o n i n a corner of the transverse section of the same s t e e l ingot during the early stages of s o l i d i f i c a t i o n i n the mould as shown i n Figure 5.3. Again symmetrical cooling was assumed so that only one quarter of the Ingot was c o n s i d e r e d . The temperature at the ingot/mould boundary, T , was taken to be f i x e d at 1150°C. The i n i t i a l temperature of the molten s t e e l was assigned a value of 1535°C which Included a 35°C superheat over the unique s o l i d i f i c a t i o n temperature, T^, of 1500°C. Thermophysical properties were again taken to be constant. The problem i s expressed mathematically by the same governing equation, Eq. (5.1), domain Eqs. (5.2), and (5.3), i n i t i a l condition, Eq. (5.4), and f i r s t two boundary conditions, Eqs. (5.5) and (5.6), as the f i r s t problem, but includes the boundary conditions, x = 0 and y = 0, T = T J ' w (5.9) 160 Table 5.1 Conditions Assumed for F i r s t Problem of Ingot Reheating i n a Soaking P i t Ingot dimensions 0.762 m x 1.524 m I n i t i a l s t e e l temperature, T Q 600°C (uniform) Thermal conductivity of s t e e l , k 30 W/m°C Density of s t e e l , p 7500 kg/m3 S p e c i f i c heat of s t e e l , C P 800 J/kg°C Convective heat-transfer c o e f f i c i e n t , h 394 W/m 2°C Surrounding p i t temperature, T CO 1100°C Table 5.2 Conditions Assumed for Ingot S o l i d i f i c a t i o n i n Second Problem of a Mould Ingot dimensions 0.762 m x 1.524 m I n i t i a l temperature of molten s t e e l , T o 1535°C S o l i d i f i c a t i o n temperature of s t e e l , T^ 1500°C * Thermal conductivity of s t e e l , k 30 W/m°C * Density of s t e e l , C P 7200 kg/m3 * S p e c i f i c heat of ste e l , C P 750 J/kg°C Latent heat of s o l i d i f i c a t i o n , H ' s 262.5 KJ/kg Temperature of ingot/mould boundary, T w 1150°C *Thermophysical properties of l i q u i d and s o l i d s t e e l are assumed to have the same values. y(m ) 4 0.762 V//////Y//////// Cast iron mould Ingot / mould boundary fixed at T w = 1150 °C L q = 0 I. Steel ingot T (x.y.t) T 0 = 1535 °C T w = I I50°C '/ / / I i ^—»•» x (m) 0.381 F i g u r e 5 . 3 S c h e m a t i c d i a g r a m of second p r o b l e m 162 and a d d i t i o n a l conditions regarding the p o s i t i o n of the s o l i d i f i c a t i o n f r o n t , S: y = S (x, t ) , T = T f ... (5.10) 5T |y = S 5y + J t 1 + <S)2]=^sf 5S \f - S ... (5.11) An approximate so l u t i o n for t h i s problem was obtained from the a n a l y t i c a l s o l u t i o n of Rathjen and J i j i 1 0 2 for s o l i d i f i c a t i o n i n an unbounded corner. The problem i s characterized, using t h e i r terms and parameters, by a dimensionless i n i t i a l temperature, * T - T * o r T. = r r =r- = 0.1 i T, - T f w ... (5.12) and a l a t e n t - t o - s e n s i b l e heat r a t i o , P = C (T - T ) p f w = 1.0 ... (5.13) To r e t a i n an estimated accuracy i n p r e d i c t i n g temperatures of better than ±0.3%, the a p p l i c a b i l i t y of t h i s s o l u t i o n i s r e s t r i c t e d to times less than 6000 s. The progress of the s o l i d i f i c a t i o n front with time, calculated from t h i s s o l u t i o n for the conditions summarized i n Table 5.2, i s i l l u s t r a t e d i n Figure 5.4. 163 y(m) 0.762 J x ( m ) 0.381 Figure 5 .4 Position of sol i d i f i c a t i o n front at different times from analytical solution to second problem 164 5.3 De s c r i p t i o n of Numerical Methods Tested The numerical methods were formulated, f o r th i s study, to solve the general heat-conduction problem, Eq. (5.1) with one of the following three boundary conditions s p e c i f i e d on each part of the boundary enclosing the region where temperatures are to be c a l c u l a t e d : i ) Neumann convective boundaries with s p e c i f i e d heat transfer c o e f f i c i e n t , h, -k | £ = h ( T o - T) ... (5.14) i i ) Neumann s p e c i f i e d heat f l u x , q, boundaries, on q ... (5.15) i i i ) D i r i c h l e t fixed temperature boundaries, T = T w ... (5.16) where n i s i n a d i r e c t i o n normal to the boundary. Thermophysical p r o p e r t i e s , k, p and are p o t e n t i a l l y functions of temperature and T^, may be functions of time, while q and h may be both temperature and time dependent. In addition, the general problem i s subject to the i n i t i a l condition given by Eq. (5.4) where T q may be a function of p o s i t i o n . 165 Three d i f f e r e n t numerical methods were studied f o r solving the two, previously described, s p e c i a l cases of t h i s problem. The a l t e r n a t i n g -d i r e c t i o n i m p l i c i t f i n i t e - d i f f e r e n c e , or "ADI" method of Peaceman and R a c h f o r d 1 0 0 was selected from the many f i n i t e - d i f f e r e n c e techniques a v a i l -able owing to i t s advantages (unconditional s t a b i l i t y , second-order accuracy and a c o s t - e f f i c i e n t s o l u t i o n algorithm in v o l v i n g t r i d i a g o n a l m a t r i c e s ) 1 0 3 f o r two and three-dimensional heat-conduction p r o b l e m s 1 0 4 . The ADI method then was used as a basis for comparison with the two finite-element methods studied i n t h i s i n v e s t i g a t i o n . F i n i t e element methods have advantages over f i n i t e d i f f e r e n c e schemes i n problems i n v o l v i n g complex geometry 1 0 5 and are more e a s i l y coupled with f i n i t e element thermal stress models. The most widely used f i n i t e element method, referred to here as the "Standard Method", formulates element matrix equations by evaluating terms from general i n t e g r a l equations using element Q O i n t e r p o l a t i o n functions . The Matrix method of Ohnaka" i s an alternate way to formulate the linear-temperature, three node tria n g u l a r element using a lumping procedure rather than the consistent d i s t r i b u t i o n of the Standard method. It was chosen over other lumping schemes because the element equations are derived i n a p h y s i c a l l y more l o g i c a l manner by applying heat balances to the i n d i v i -dual nodes. 166 5.3.1 Formulation of F i n i t e Element Methods For the Standard and Matrix methods, the s p a t i a l continuum was d i s c r e t i z e d into 3 node, l i n e a r temperature, tria n g u l a r elements as shown i n Figure 5.5. Triangles were chosen over other shapes such as rectangles owing to th e i r v e r s a t i l i t y i n d i s c r e t i z i n g regions of complex shape. Higher order elements were not considered since f o r the ingot s o l i d i f i c a t i o n problem i t was f e l t that the discontinuous temperature f i e l d across the s o l i d - l i q u i d boundary would be better approximated by a large number of elements than by fewer elements each having more degrees of freedom. By applying the finite-element method to the heat-conduction problem, v i z . Eqs. (5.1), (5.4) and (5.14)-(5.16), and summing the contributions from i n d i v i d u a l elements and boundaries, NE NBE [K] = E [K]J + Z [h]J ... (5.17) 1-1 1-1 NE [C] = E [C ]* ... (5.18) i-1 N B E K K K (Q) = E [ {Q}° - [h]J {TJI ] ... (5.19) Complete Coarse Mesh With Nodes Figure 5.5 Meshes used for numerical methods 168 the following global matrix equation i s obtained: [K] {T} + [ c ] {f} = {Q} ... (5.20) where {T} c o n t a i n s the time d e r i v a t i v e s of the unknown nodal temperatures, {T}. Equation (5.20) i s solved at each time step using one of the techniques discussed i n Section 5.3.4. The e l e m e n t c o n d u c t i v i t y m a t r i x , [K] 6, i s the same f o r both the Standard and Matrix methods, although the d e r i v a t i o n f or each i s d i f f e r e n t . Applying the Standard method to the linear-temperature t r i a n g u l a r element y i e l d s the following i n d i v i d u a l terms i n the 3 x 3 [K ] e matrix: K l j = 2A ( b i b j + C i C j } i , J = 1,2,3 ...(5.21) where D^ = 3^2 ~ v 3 c l = x3 x2 and the o t h e r b^ and c^ v a l u e s are obtained through c y c l i c permutation of the s u b s c r i p t s , 1,2,3, which represent the three nodes of the t r i a n g l e having c o o r d i n a t e s (x^, y ^ ) , (.^2' v2^ a n <* ^ x3' ^3^ r e s p e c t i v e l y . Although i t i s formulated by applying a f i n i t e - d i f f e r e n c e method to t r i a n g l e s , the element c o n d u c t i v i t y matrix obtained using the Matrix method i s i d e n t i c a l to that given by Eq. (5.21). 169 The only d i f f e r e n c e between the two methods i s the formulation of the c a p a c i t a n c e , or h e a t - s t o r a g e matrix, [c]e. Using the Standard method and assuming constant pC within the element r e s u l t s i n a consistent 3 x 3 [c ]e p matrix given by: pC A c i j " i f " t 1 + 6 i j l i , j = 1,2,3 ...(5.22) where 6 i s the Kronecker d e l t a . For the Matrix method, [c ] 6 i s calculated by unequally d i s t r i b u t i n g the c a p a c i t a n c e to each node i n p r o p o r t i o n to i t s nodal a r e a , A^, which i s defined by constructing perpendicular b i s e c t o r s on each side of the t r i a n g l e : A i = 2 ~ 4 ( a i + b i X c + C i y c ) 1 = 1 , 2 » 3 ( 5 ' 2 3 ) where ( x c » v c ) a r e t n e coordinates of the centroid of the t r i a n g u l a r element and a^ , i s defined i n a s i m i l a r manner to b^ and c^ . with a^ = x^y^ " X 3 V 2 * This r e s u l t s i n a capacitance matrix i n which terms are lumped along the main diagonal: C i j " p C p A i 6 i J 1 , j = 1 , 2 > 3 (5.24) 170 5.3.2 Formulation of Boundary Conditions Four d i f f e r e n t methods for formulating the boundary conditions f o r the f i r s t problem were compared for each version of both the Standard and Matrix methods. A Neumann convective boundary i s the n a t u r a l choice and Is accounted f o r In the finite-element methods by i n c l u d i n g the [h ] b matrix f o r each element side that forms part of the e x t e r i o r boundary where Eq. (5.14) a p p l i e s . Applying the Standard method assuming l i n e a r v a r i a t i o n f o r T and h along the boundary r e s u l t s In the " l i n e a r h formulation": L(h + h ) Lh hj\j 1 2 + - g ^ i . J = 1.2 ...(5.25) where L i s the length of the boundary segment connecting nodes a r b i t r a r i l y numbered 1 and 2. A l t e r n a t i v e l y , [h ] b may be defined by "lumping" h at the two boundary nodes, g i v i n g r i s e to: . h L h l j D = ^ - ( 6 l j ) i , j = 1,2 ...(5.26) This formulation, referred to as the "lumped h formulation", i s more t h e o r e t i c a l l y consistent with the Matrix method. 171 An a l t e r n a t i v e way to formulate the boundary conditions for the f i r s t problem i s through the use of Neumann heat-flux boundaries (Eq. 5.15). The finite-element methods account f o r these by incorporating a heat-flux vector, {Q}k, f o r each appropriate boundary segment. Applying the Standard method and assuming l i n e a r v a r i a t i o n of both q and T along the boundary leads to the " l i n e r q formulation": Q i = 6 ( q l + q2 + q i } 1 = 1 , 2 ...(5.27) which i s more compatible with the Standard method. The f i n a l option, termed the " l i n e a r q formulation", leads to: ... (5.28) For the second problem, D i r i c h l e t boundaries, Eq. (5.16), are required. The temperatures of nodes on these boundaries were e f f e c t i v e l y forced to assume desired values by employing the lumped h formulation with an 9 a r b i t r a r i l y . l a r g e h (eg. 10 ). 5.3.3 Methods of Latent-Heat E v o l u t i o n Accounting for latent-heat evolution i s a d i f f i c u l t task for numerical methods, p a r t i c u l a r l y when the phase-change temperature i n t e r v a l i s s m a l l . 1 0 6 The problem has received a great deal of a t t e n t i o n i n recent 172 l i t e r a t u r e . For problems dominated by latent-heat evolution, such as f r e e z i n g s o i l , several researchers have developed methods of dynamically deforming the element g r i d system to maintain the f i n e s t mesh i n the v i c i n i t y of the c r i t i c a l phase-change r e g i o n 1 0 7 . With these methods, the l o c a t i o n of the s o l i d i f i c a t i o n front i s c o n t i n u a l l y tracked. However, f o r the s t e e l s o l i d i f i c a t i o n problem i n v o l v i n g important boundary heat flows, which i s not dominated by l a t e n t heat e f f e c t s , the extra complication and expense of these methods i s not considered worthwhile. For f i x e d - g r i d systems, where the s o l i d i f i c a t i o n front i s generally at an unknown l o c a t i o n between nodes, two d i f f e r e n t classes of methods are a v a i l a b l e . The f i r s t t r e a t s l a t e n t heat as a temperature-dependent heat source term i n the o r i g i n a l heat-conduction equation, Eq. (5.1). The temperature of each n o d e 1 0 8 or e l e m e n t 1 0 9 undergoing phase change then i s f i x e d at the s o l i d i f i c a t i o n temperature u n t i l s u f f i c i e n t heat has been extracted to complete s o l i d i f i c a t i o n and allow the s o l i d i f i c a t i o n front to move on. These methods properly require i t e r a t i o n within a time step, however, and therefore were not considered further i n t h i s study. The second group of methods t r e a t s the latent heat i n terms of a temperature-dependent s p e c i f i c heat which, for finite-element methods, i s included i n the capacitance matrix [c ] e of the element. Two types of the s p e c i f i c - h e a t method were evaluated i n t h i s i n v e s t i g a t i o n . The f i r s t i s based on a temperature-dependent e f f e c t i v e s p e c i f i c heat, which i s a r t i f i c a l l y r aised above over the phase-change temperature i n t e r v a l , or "PCTI", to account for la t e n t heat: 173 H C = C + P P s for < T < T ( 5 . 2 9 ) - T sol l i q where T l i q and T so l are the l i q u i d u s and solidus temperatures r e s p e c t i v e l y . For problems i n v o l v i n g a unique s o l i d i f i c a t i o n temperature such as the second problem outlined e a r l i e r , t h i s method obviously requires the cr e a t i o n of an a r t i f i c i a l PCTI about the true s o l i d i f i c a t i o n temperature. To safe-guard against nodes that "jump" over the PCTI i n a si n g l e time step, and thus miss t h e i r latent-heat evolution, a p o s t - i t e r a t i v e c o r r e c t i o n technique i s used to readjust the temperatures of those nodes. This procedure, r e f e r r e d to as the "Specific-Heat method", i s commonly used i n f i n i t e -d i f f e r e n c e f o r m u l a t i o n s 1 0 4 and therefore was the only latent-heat evolution technique employed with the ADI method. Because t h i s method a t t r i b u t e s a d i f f e r e n t C to each i n d i v i d u a l node, rather than to the element, i t was not P considered applicable to the Standard finite-element method. However, i t was used i n conjunction with the Matrix method. The second s p e c i f i c - h e a t method investigated i s a c t u a l l y a sub-class of methods which c a l c u l a t e an e f f e c t i v e s p e c i f i c heat f o r an en t i r e element, through the use of an enthalpy function, H. The f i r s t of these methods, evaluated In t h i s i n v e s t i g a t i o n , was developed by Lemmon 1 1 0 i n which 1 7 4 c - c W a x ) 2 + W a y ) 2 1 / 2 _ ( 5 > 3 0 ) p (5T/oxr + (BT/ &y) where, f o r linear-temperature, t r i a n g u l a r elements, 5H/5x - b l H l + b 2 H 2 + b 3H 3 m ^ ( 5 > 3 1 ) 5H/ay = + c 2 H 2 + C3H3 _ ( 5 < 3 2 ) o/T5x - b l T l + b 2 T 2 + b 3 T 3 _ ( 5 > 3 3 ) oT/ a y = + c 2 T 2 + c 3 T 3 _ ( 5 > 3 4 ) The second method, reported by Del-Giudice et a l 1 1 1 , gives the following r e l a t i o n s h i p between and H: c = [OH/ox) (oT/ox) + (5H/oy) (oT/oy)j _ (5.35) p (aT/ax) 2 + O T / a y ) 2 In both methods, i f the denominator equals zero, the temperature i s constant throughout the element and the appropriate can e a s i l y be determined. The use of a PCTI i s optional for these methods. A t h i r d method developed by Comini et a l 1 1 2 uses . 1 aH/ax 8H/ a y P 2 laT/ax aT / a y J . . . ( 5 . 3 6 ) 175 but was not considered i n t h i s work because Del-Giudice et a l 1 1 1 found i t to be i n f e r i o r to that given by Eq. (5.35). 5.3.4 Time-Stepping Techniques The system of d i s c r e t e f i r s t - o r d e r , non-linear, d i f f e r e n t i a l equations obtained from the finite-element, s e m i - d i s c r e t i z a t i o n of space, given by the matrix equation, Eq. (5.20), was solved incrementally using a f i n i t e -d i f f e r e n c e approximation i n the time domain. Although several i n v e s t i g a t o r s have used f i n i t e elements In t i m e , 1 1 3 ' 1 1 4 f i n i t e - d i f f e r e n c e recurrance r e l a t i o n s h i p s , a great many of which are i n the l i t e r a t u r e , are u s u a l l y employed. Of these, three d i f f e r e n t time-stepping algorithms were i n v e s t i g a t e d . They are d i s t i n g u i s h e d by the way i n which {T} and {T} are evaluated i n terms of temperatures at known and unknown time l e v e l s . The f i r s t method investigated was the Dupont th r e e - l e v e l t e c h n i q u e 1 1 5 or "Dupont" method: 3T + T T T [ K ] { — - } + [ C ] { 3 2 ~ T 1 ) = {Q} . . . (5.37) Solving for the unknown temperatures at the t h i r d time l e v e l , y i e l d s {T3> - Xf-) 1 { T 2 } _ { T I } + { Q } ] . . . ( 5 . 3 8 ) 176 This method i s one of a c l a s s of second-order accurate, t h r e e - l e v e l techniques developed by Dupont et a l 1 1 5 and i s r e f e r r e d to by Hogge 1 1 6 as the "Dupont II scheme with a ° 1/4". I t was chosen because Hogge reported i t s o v e r a l l performance i n accuracy and s t a b i l i t y to be superior to other time-stepping methods i n solving the one dimensional, homogeneous equation: kT + f - 0 ... (5.39) where k had a s l i g h t l i n e a r temperature dependence. The Dupont method has u n c o n d i t i o n a l A q s t a b i l i t y 1 1 6 ' 1 1 7 ; but i n a t h e o r e t i c a l study on the s t a b i l i t y of various time-stepping techniques i n so l v i n g Eq. (5.39) with constant k, Wood 1 1 7 has demonstrated that any t h r e e - l e v e l method which i s A q stable cannot guarantee a l l r e a l eigenvalues, so i s termed " r e l a t i v e l y unstable". This means that the Dupont method could be prone to o s c i l l a t i o n under c e r t a i n circumstances. The second method tested was the Lees t h r e e - l e v e l t e c h n i q u e 1 1 8 or "Lees" method" T + T + T T T [K] {— ^ - } + [C] { ^ 1 } = (Q) ... (5.40) i Again s o l v i n g for the unknown temperatures at the t h i r d time l e v e l gives: <V = ^ + 2 f t - l " 1 I - I f i { T 2 } - I f I { T l } + i t - { T l } + ( 5- 4 l> 1 7 7 This method was chosen because i t has been used s u c c e s s f u l l y by several i n v e s t i g a t o r s 1 1 1 ' 1 1 2 ' 1 1 9 ' 1 2 0 i n modelling non-linear, m e t a l l u r g i c a l heat-t r a n s f e r problems. Like the Dupont method, i t has unconditional A q s t a b i l -i t y but i s r e l a t i v e l y unstable for both very small and very large time s t e p s 1 1 7 . The f i n a l method tested was the Crank-Nicolson two-level t e c h n i q u e 1 2 1 or "C-N" method: T + T T — T [K] f- 2—£—~) + [C] { 2 A t l} = {Q} ... (5.42) Solving f or the unknown temperatures at the second time l e v e l y i e l d s : {T 2} = [1^1+ l^)1 [ZL|L {Tl> +1^1 { T i} + { Q } ] ... (5.43) This well known, two-level technique was chosen f o r comparison with the t h r e e - l e v e l schemes, and a d d i t i o n a l l y was used to generate second-level temperatures to s t a r t the t h r e e - l e v e l methods. C-N i s the only second-order accurate, two-level scheme and has zero s t a b i l i t y , A q s t a b i l i t y and r e l a t i v e s t a b i l i t y 1 1 7 . However, i t s s t a b i l i t y i s only marginal so that t h i s method also i s prone to o s c i l l a t i o n i f large time steps are u s e d 1 1 7 ' 1 2 2 . For highly non-linear problems, several researchers have employed secant or tangent methods which involve i t e r a t i o n within each time s t e p 1 2 3 - 1 2 6 . These methods have been compared by Hogge 1 2 7 and are expensive, p a r t i c u l a r l y i n the case of the tangent method i n which new [K] 1 7 8 and [C] matrices are constructed every i t e r a t i o n . Since the s t e e l ingot s o l i d i f i c a t i o n problem i s only m i l d l y non-linear, these methods were not considered to be j u s t i f i e d economically. However, they may have us e f u l a p p l i c a t i o n f o r s o l i d i f i c a t i o n problems dominated by hi g h l y non-linear latent-heat e f f e c t s . No I t e r a t i o n was done within time steps i n t h i s i n v e s t i g a t i o n . Thus, the c a l c u l a t i o n of temperature dependent, or non-linear terms i n [K], [C] and {Q}, r e q u i r e s the use of temperatures, {T^}, evaluated at preceeding time l e v e l s . In the f i r s t problem, {Q} i s non-linear for the s p e c i f i e d q f o r m u l a t i o n owing to i t s dependence on the surface temperature, T . In the s second problem, [C] i s non-linear due to the temperature dependence of the s p e c i f i c heat, C . K P For the ADI and C-N methods, these c a l c u l a t i o n s must be performed using temperatures from the f i r s t - t i m e l e v e l , ( T^(t^)}. However, the Dupont and Lees methods may employ any l i n e a r combination of temperatures from the f i r s t and second time l e v e l s . In t h i s study, the following f i v e cases were in v e s t i g a t e d : V ^ ) = T ( t l ) (5.44) T * ( t 1 . 5 ) = °-5T<tl> + °-5T(V (5.45) T * ( t 2 ) = T ( t 2 ) ... (5.46) T ^ ( t 2 > 5 ) = 1.5T(t 2) - 0 . 5 1 ^ ) ... (5.47) V t 3 ) = 2.0T(t2) - T(t L ) (5.48) From a t h e o r e t i c a l s t a n d p o i n t , should be evaluated at for the Lees m e t h o d 1 1 6 ' 1 1 9 and at t- for the Dupont method 1 1 6. / . J Using these techniques, the matrix equations r e s u l t i n g from Eqs. (5.38), (5.41) and (5.43) can be written i n the form: [M] {T} = {F} ... (5.49) Since [M] i s a p o s i t i v e d e f i n i t e , symetric, banded matrix, the equation can be solved e f f i c i e n t l y at each time step using the Cholesky method 1 2 8. 5.4 Comparison Methodology To compare the finite-element and the ADI methods on an equal b a s i s , one fourth of the ingot section (0.381 m x 0.762 m) was d i s c r e t i z e d i n t o a rectangular g r i d of nodes. For the Standard and Matrix methods, the nodes were then connected to form a mesh of three-node, right-angled, t r i a n g u l a r elements. Three meshes, shown i n Figure 5.5, were employed: a coarse, 6 x 11 mesh with 66 nodes and 100 elements, a medium 11 x 21 mesh with 231 nodes and 400 elements and a fin e 21 x 42 mesh with 861 nodes and 1600 elements. I t should be noted that each refinement of the mesh completely contains a l l coarser meshes. Care was taken i n numbering the nodes to ensure that the band width of [M] was minimized. The numerical methods were coded as Fortran IV programs that were made to be as s i m i l a r and e f f i c i e n t as p o s s i b l e . The programs were run on an Amdahl 470 V/8, 12-megabyte computer 180 at the U n i v e r s i t y of B r i t i s h Columbia. The a n a l y t i c a l s o l u t i o n s were used to c a l c u l a t e n e a r l y e x a c t temperatures {T } for each node i n the mesh at 600 s time i n t e r v a l s f o r both a test problems investigated. Per cent dif f e r e n c e s between the "exact" temperatures and n u m e r i c a l l y generated values, {T }, were ca l c u l a t e d f o r n each node at these time i n t e r v a l s . Two c r i t e r i a were established to compare the r e l a t i v e accuracy of the various methods at each time step: i ) average absolute value of percent e r r o r : NN T - T 1/N I | ~ | x 100% ... (5.50) i=l a i i ) maximum percent error from the set of NN nodal errors i n Eq. (5.50). For problems involving s o l i d i f i c a t i o n such as the second problem, many previous investigators have based the comparison of numerical and a n a l y t i c a l solutions on the p o s i t i o n of the s o l i d i f i c a t i o n f r o n t . However, errors i n actual temperature predictions are more s e n s i t i v e , and therefore have been used as the c r i t e r i o n f o r comparison. Their accuracy i s also of p r a c t i c a l importance because the thermal stress c a l c u l a t i o n s require these temperatures. 181 S t a b i l i t y was estimated by v i s u a l l y examining the behavior of the maximum absolute percent error with increasing time, and assigning an " i n s t a b i l i t y index" of 0 to 3 i n order of increasing i n s t a b i l i t y as follows: 0 = completely stable, with maximum error decreasing monotonically with time. 1 = stable with a f l u c t u a t i n g maximum error that eventually decreases. 2 = unstable, with a f l u c t u a t i n g maximum error that gradually increases out of c o n t r o l . 3 = extremely unstable, with average error exceeding 100% a f t e r only a few time steps. Figure 5.6 shows examples of these defined forms of i n s t a b i l i t y f o r four t y p i c a l runs taken from the f i r s t problem. Costs for each method were estimated by considering both CPU time and core storage. 5.5 Results and Discussion Each numerical method generated many nodal temperatures at every time step of the simulation from which average and maximum percent errors were c a l c u l a t e d . Typical r e s u l t s are given i n Appendix IX. For comparative purposes, i t was desirable to characterize the o v e r a l l accuracy of each method using errors calculated at only a single time step. Extensive examination of the lo c a t i o n of the node with the maximum error has c o n s i s t e n t l y r e v e a l e d that e r r o r s i n temperature p r e d i c t i o n s are proportional to the temperature gradient, regardless of the method, mesh and time step used. Because temperatures are changing the most r a p i d l y e a r l y 0 3000 6000 9000 12000 T i m e ( s ) Figure 5 . 6 Examples of runs with varying s t a b i l i t y 1 8 3 i n the simulation, average and maximum errors at 600 s have been singled out to characterize accuracy for both problems. 5.5.1 Size of Mesh and Time Step 5.5.1.1 E f f e c t on Accuracy The influence of mesh refinement and time-step s i z e on accuracy can be seen by examining Figure 5.7. This f i g u r e shows the t y p i c a l e f f e c t s of these v a r i a b l e s on average error using equivalent formulations of the Standard and Matrix methods as examples. Unless s t a b i l i t y problems are encountered, accuracy generally improves with refinement of both mesh and time-step. However, u n t i l the time-step s i z e i s reduced below a c r i t i c a l value, mesh refinement does not lead to improvement, as accuracy i s s i m i l a r among a l l three meshes for large time steps. Once the time-step i s smaller than the c r i t i c a l value, f i n e r meshes g r e a t l y improve accuracy. This c r i t i -c a l time-step s i z e i s smaller for f i n e r meshes. Continued refinement of the time-step si z e ( i . e . increase i n the number of time-steps taken to reach 600 s) r e s u l t s i n l i t t l e further improvement and eventually the accuracy worsens. Thus, every given mesh has an inherent l i m i t i n achievable accur-acy and an optimum range of time-step size associated with i t . Time-step s i z e optima have been reported elsewhere i n previous studies comparing n u m e r i c a l m e t h o d s 1 1 3 ' 1 2 9 . Gray and S c h n u r r 1 2 9 found such optima only when using the finite-element method and postulated that any increase 1 8 4 Dupont -Matrix lumped q — Dupont-Standard linear q o Coarse Mesh • Medium Mesh A Fine Mesh o-o--o--o-2 5 10 20 50 100 200 Number of Time Steps to 600s Figure 5 . 7 E f f e c t of size of mesh and time step on accuracy for f i r s t problem 185 i n error with increasing degrees of freedom was due to simple computational round-off e r r o r . However, K e r a m i d a s 1 1 3 a t t r i b u t e d the optima to the space and time approximations inherent i n the numerical methods. He found sharp, d i s t i n c t optima to occur at a constant value of dimensionless At/Ax of 0.5 i n a study of a s i m p l i f i e d , one-dimensional, heat-conduction problem i n v o l v -ing a f i r s t - o r d e r p a r t i a l d i f f e r e n t i a l equation formulated i n terms of heat displacement, U, where T = SU/dx. I t was stated that for a more conven-t i o n a l formulation, the parameter for error and s t a b i l i t y c o n t r o l should instead be At/Ax 2. In the present study, time-step s i z e optima were found for every method tested, including the ADI. Unfortunately, the optimum values varied with the method, were problem dependent, and often were not d i s t i n c t . Thus, a precise r e l a t i o n s h i p between the mesh size and optimum time-step s i z e was d i f f i c u l t to e s t a b l i s h i n general. However, the optimum does occur at smaller time steps with f i n e r meshes; and the dimensionless parameter, k At (——) — appears to remain constant with a value of roughly 0.1 at the P C P ( A x ) 2 optimum of each of the three meshes examined. I t i s important to emphasize that increasing refinement of the time-step si z e for a given mesh does not continuously improve accuracy, but i t does s t e a d i l y r a i s e computing costs. 5.5.1.2 E f f e c t on S t a b i l i t y For the l i n e a r problem, i . e . the f i r s t problem formulated with h boundaries, a l l the numerical methods remained stable with time steps at le a s t up to 4800 s although the accuracy became unacceptably poor with higher values of At. In the case of the non-linear problems which included 186 a l l other formulations of the f i r s t problem and a l l formulations of the second problem {Q} and [C] are temperature dependent) every method examined eventually became unstable i f the time step was made s u f f i c i e n t l y l a r g e . However, i n s t a b i l i t y was much less severe with the second problem since I t was formulated with fixed-temperature boundaries. I n s t a b i l i t y was found f o r time steps i n excess of 300 s for both the Lees and C-N methods; f o r ADI, i n s t a b i l i t y occurred at time steps of 600 s or more. The Dupont method i s the most stable, as I t s s t a b i l i t y was retained for time-steps up to 1200 s. These values pose upper l i m i t s to s t a b i l i t y since they were determined for the optimum formulation of each method. They are independent of the mesh used and are s l i g h t l y lower for the Standard method, as compared with the Matrix method, which i s more stable. 5.5.1.3 Grading of Mesh and Time-Step The r e s u l t s i n the preceding sections have been obtained with constant time-steps and a uniform mesh. Accuracy may be improved, however, i f the mesh i s refined i n regions which experience rapid temperature changes or i f the time step i s reduced during periods when the temperature changes q u i c k l y . An example of improved accuracy achieved by using smaller time steps early i n the simulation of the f i r s t problem i s shown i n Figure 5.8. Other c a l c u l a t i o n s revealed that grading the time-steps i n t h i s way c o n s i s t e n t l y resulted i n improved accuracy over constant time-steps for an equal number of i t e r a t i o n s , and had no adverse e f f e c t on s t a b i l i t y provided that time-step l i m i t s were not exceeded. Thus s i g n i f i c a n t cost savings are possible i n solving r e a l p r o b l e m s 1 1 2 . However, grading the mesh by using f i n e r elements near the surface was found to be much less advantageous. 1 8 7 0 3 * 0 2 O > 3 01 o If) < a> o k_ a> > < Ox xx. Dupont-Matrix evaluated at t2 Lumped heat flux Medium mesh 120 time steps O Constant 120s step • Variable time step XL. • 0 _l_ 0 3000 6000 Time 9000 2000 (s) Figure 5 . 8 Accuracy improvement using v a r i a b l e time steps 1 8 8 While accuracy close to the surface improved, increased errors resulted i n the i n t e r i o r with no net s i g n i f i c a n t improvement i n accuracy f o r the two graded mesh configurations tested. In a d d i t i o n , the use of a graded mesh increased the tendency toward i n s t a b i l i t y . Although the benefits of mesh refinement should not be disregarded, these r e s u l t s i n d i c a t e that consider-able care must be taken i n s e l e c t i n g a graded mesh. In p a r t i c u l a r , elements with excessive aspect r a t i o s must be avoided. 5.5.2 Comparison of Numerical Methods 5.5.2.1 Accuracy and S t a b i l i t y The accuracy and s t a b i l i t y of the numerical methods tested are presented i n Tables 5.3 and 5.4. For comparative purposes, a constant time-step of 120 s was selected for the f i r s t problem and 30 s for the second problem i n conjunction with a regular, medium mesh. These conditions were chosen because they generally produced a reasonable accuracy of les s than 1% average error at 600 s for both problems. In addition, the r e s u l t s i n Tables 5.3 and 5.4 are representative of the comparisons made over the e n t i r e spectrum of mesh and time-step sizes examined. Although each of the seven methods compared can solve either problem e f f e c t i v e l y i f formulated optimally, some have d i s t i n c t advantages over others. Tables 5.3 and 5.4 and Figure 5.7 show that the Matrix method has better accuracy and s t a b i l i t y than the Standard finite-element method, although t h e i r performance i s generally quite s i m i l a r . Since i t represents an upper bound to the eigenvalues, the Standard method, with i t s consistent 189 Table 5.3 Accuracy and S t a b i l i t y of Numerical Methods for F i r s t Problem Boundary Condition Formulation Dupont Dupont Lees Lees C-N C-N Matrix Standard Matrix Standard Matrix Standard Lumped h -.43 .53 .52 .58 .37 .47 .08 1.6 1.8 2.1 1.9 1.6 1.4 0.3 0 0 0 0 0 0 0 Linear h -.43 .54 .52 .56 .38 .47 1.7 1.8 2.1 1.9 1.6 1.4 0 0 0 0 0 0 Lumped q ' l .87 .62 .69 1.51 .31 .44 .31 4.7 4.2 11.3 67.4 1.2 28.4 1.2 0 1 1 3 0 3 0 '1.5 .54 .52 .27 .31 2.8 5.6 1.1 14.6 0 0 0 2 H .23 .29 1.13 2.77 " Average Error (%) 1.1 1.5 38.9 189.4 -< Maximum Error (%) 0 1 3 3 •< Instability Index '2.5 .15 5.65 0.7 243.0 0 3 £3 .43 1.6 0 Linear q ll .85 .62 .68 1.42 .31 .34 4.6 4.1 8.1 44.1 1.9 11.4 0 0 1 3 0 1 h.5 .54 .51 .27 .25 2.7 4.3 1.3 5.4 0 0 0 1 C2 .23 .29 1.07 2.29 1.7 2.9 26.9 91.1 0 0 3 3 '2.5 .24 13.17 1.3 630.3 0 3 £3 .55 12.1 0 Evaluated for medium mesh with time step - 120 s Average and Maximum Errors recorded at 600 s Table 5 . 4 Accuracy and S t a b i l i t y of Numerical Methods for Second Problem S o l i d i f i c a t i o n Method Dupont Dupont Lees Lees C-N C-N ADI Matrix Standard Matrix Standard Matrix Standard Specific Heat l l .16 .36 1.2 2.8 3 1 1 '1.5 3 3 .15 1.1 1 3 ' 2 . 5 3 Lemmon .26 .63 .13 .39 5.1 5.7 3.1 2.9 1 1 1 1 .28 .38 .11 .24 3.8 2.6 2.8 1.9 1 1 1 1 l2 .12 .43 .17 .45 Average Error (%) 1.5 2.9 3.7 5.5 -> Maximum Error (%) 1 1 2 2 I n s t a b i l i t y Index ' 2 . 5 .21 .43 3.0 3.8 1 1 Del-Giudice ' l .26 .64 .24 .41 5.0 6.0 3.6 4.0 1 1 1 1 h.5 .28 .40 .12 .25 3.8 3.8 2.8 1.9 1 1 1 1 *2 .14 .46 .19 .45 2.9 3.8 4.6 5.6 1 1 2 . 2 £2.5 .22 .45 3.8 4.5 1 1 Phase Change Temperature Interval = 20°C for Specific Heat Technique = 2°C for Lemmon and Del-Giudice Evaluated for medium mesh with time step = 30 s Average and Maximum Errors recorded at 600 s 191 [C] matrix, has a greater tendency toward o s c i l l a t i o n f o r short time steps when steep temperature gradients are p r e s e n t 1 3 0 ' 1 3 1 . This i s seen i n Table 5.3 pertaining to the f i r s t problem where higher values of the i n s t a b i l i t y index are l i s t e d f o r the Standard method, p a r t i c u l a r l y f o r the Lees and C-N time-stepping techniques. In the second problem, Table 5.4, s t a b i l i t y of the Standard and Matrix methods i s s i m i l a r but the accuracy of the Matrix method i s c l e a r l y superior. Very l i t t l e d i f f e r e n c e i s seen between the lumped and l i n e a r boundary condition formulations i n Table 5.3. Average errors are the same but maxi-mum errors are s l i g h t l y higher for the l i n e a r formulation. Since there i s no d i f f e r e n c e In s t a b i l i t y between them for the Matrix method, i t would seem better to formulate the Matrix method using the lumped formulation. How-ever, the Standard method exhibits s l i g h t l y increased i n s t a b i l i t y with the lumped formulation and i s better with the l i n e a r formulation. These f i n d -ings are consistent with the t h e o r e t i c a l bases of these methods. The s i n g l e , most important v a r i a b l e a f f e c t i n g s t a b i l i t y and accuracy f o r both problems i s the choice of time l e v e l at which T^ i s evaluated for the t h r e e - l e v e l methods. Tables 5.3 and 5.4 i n d i c a t e that the optimum choice, although d i f f e r e n t for each time-stepping technique, i s the same for each problem. This i s e s p e c i a l l y s i g n i f i c a n t because the n o n - l i n e a r i t y occurs i n d i f f e r e n t places for each problem; temperature dependencies occur only i n {Q} f o r the f i r s t problem and i n [C] for the second. 1 9 2 The Lees method has a c c e p t a b l e s t a b i l i t y only at T ^ ( t ^ In f a c t , with the mesh and time-step s i z e used i n Table 5.3, i t became unstable ( i n s t a b i l i t y index > 2) for every choice other than T.(t c ) . In a d d i t i o n , optimum accuracy occurs at T ^ ( t ^ This f i n d i n g i s contrary to the theo-r e t i c a l p redictions of Bonacina et a l 1 1 9 that the best r e s u l t should be obtained at T A ( t 2 ) . The Dupont method, on the other hand, has excellent s t a b i l i t y f o r a l l ways of c a l c u l a t i n g T^. I t s accuracy i s at an optimum for T ^ ( t 2 ^ ) i n the f i r s t problem and i s best at T ^ ( t 2 ) f o r the second. However, when the time l e v e l exceeded t 2 f o r the f i r s t problem, the method became unstable beyond time-steps of 600 s. R e c a l l i n g that the optimum Dupont method, evaluated u s i n g T ^ ( t 2 ) , d i d not become unstable u n t i l greater than 1200 s, T ^ ( t 2 ) i s the safer o v e r a l l choice. This i s again contrary to the t h e o r e t i c a l analy-s i s that T should be evaluated at t_ c 1 1 6 . The r e s u l t s i n Tables 5.3 and 5.4 combined with the previous d i s c u s s i o n of upper l i m i t s to time steps, i n d i c a t e that the Dupont method has much better s t a b i l i t y than that of Lees. This agrees with the findings of Hogge 1 1 6 and Wood 1 1 7 that the Lees method i s more prone to o s c i l l a t i o n and i n s t a b i l i t y . In addition, Tables 5.3 and 5.4 show that the Dupont method i s s l i g h t l y more accurate than the Lees. The C-N two-level, time-stepping method has been used as a basis f o r comparison by several i n v e s t i g a t o r s 1 3 2 - 1 3 1 * . In t h i s i n v e s t i g a t i o n , i t s performance was s u r p r i s i n g l y good, p a r t i c u l a r l y with the Matrix method. 193 Tables 5.3 and 5.4 show i t s accuracy and s t a b i l i t y to be close to that of the best formulations of the Dupont and Lees methods. Three d i f f e r e n t methods for handling latent-heat evolution are compared i n Table 5.4. To enable implementation of the Specific-Heat method i n the second problem, inv o l v i n g a unique s o l i d i f i c a t i o n temperature, an a r t i f i c i a l phase change temperature i n t e r v a l (PCTI) was created. Although i t should t h e o r e t i c a l l y equal zero for the other methods, Figure 5.9 shows the e f f e c t on accuracy of varying the PCTI up to 50°C f o r a l l three methods. The Dupont-Matrix method with T^it^) was chosen for i l l u s t r a t i v e purposes. As one would expect, er r o r generally r i s e s as PCTI i s increased. The Del-Giudice method y i e l d s the lowest error at a PCTI of about 5°C while the Specific-Heat method has a minimum