@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Materials Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Iyer, Jayaraman Rajagopalan"@en ; dcterms:issued "2010-04-22T22:52:27Z"@en, "1983"@en ; vivo:relatedDegree "Master of Applied Science - MASc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description "With the ultimate objective of quantitatively predicting the mechanical properties of steels, a mathematical model has been developed to compute the transient temperature distribution and austenite-pearlite transformation in an eutectoid steel rod during controlled cooling. The model is based on one-dimensional, unsteady-state heat conduction and incorporates empirical TTT data in the form of the parameters n and b(T) from the Avrami equation and the CCT start time, t[sub AV-CCT]. This data was obtained using a diametral dilatometer for an eutectoid steel of composition 0.82% C -0.82% Mn - 0.26% Si and a grain size of 5-7 ASTM. CCT kinetics are predicted from the TTT data by the additivity principle originally proposed by Scheil. The adequacy of the model was cheeked by comparing model1 predictions of the centre-line temperature of 9 and 10 mm diameter rods to measurements made during air cooling from an initial temperature between 840 and 870°C. The agreement obtained was good. Also the conditions determined by Avrami and Cahn for the additivity principle to hold were checked. Even though model predictions of CCT from TTT data generally were good, the application restrictions were not satisfied. Thus a new sufficient condition has been proposed which holds for the steel under study and establishes a firm theoretical foundation for model calculations. The condition, termed \"effective site saturation\", indicates that for growth dominated reactions, wherein the rate of reaction is governed by the growth of nuclei nucleated very early in the reaction, the kinetics can be considered additive due to the relative unimportance of subsequent nucleation. This condition suggests that the additivity rule may have a much broader range of applicability than was originally supposed. The calculation of TTT from CCT has been studied and a new method, involving an interative procedure using the additivity rule, has been derived. Agreement between calculated and measured TTT data is good. Finally the model has been employed to study the effect of centre segregation of manganese on the transformation behaviour of eutectoid steel rods and also to predict the mechanical properties of the same steel. Calculations indicate that segregation can lead to the formation of martensite at the centre of the rods with faster cooling rates. The calculation of mechanical properties is based on published relationships between pearlite spacing, undercooling and mechanical properties."@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/24080?expand=metadata"@en ; skos:note "MATHEMATICAL MODELLING OF PHASE TRANSFORMATION IN A P L A I N CARBON EUTECTOID STEEL By ^IYER JAYARAMAN RAJAGOPALAN B. T e c h . ( M e t a l l u r g i c a l E n g i n e e r i n g ) , I n d i a n I n s t i t u t e o f T e c h n o l o g y , B o m b a y , I n d i a , 1974 M . B . A . , I n d i a n I n s t i t u t e o f M a n a g e m e n t , A h m e d a b a d , I n d i a , 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE i n THE FACULTY OF GRADUATE STUDIES D e p a r t m e n t o f M e t a l l u r g i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA O c t o b e r 1983 © I y e r J a y a r a m a n R a j a g o p a l a n , 1983 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 fo r 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 a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or p u b l i c a t i o n 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 The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 M £ T A U L O £ G ( d A L_ Date DE-6 (3/81) i i A B S T R A C T W i t h t h e u l t i m a t e o b j e c t i v e o f q u a n t i t a t i v e l y p r e d i c t i n g t h e m e c h a n i c a l p r o p e r t i e s o f s t e e l s , a m a t h e m a t i c a l m o d e l h a s b e e n d e v e l o p e d t o c o m p u t e t h e t r a n s i e n t t e m p e r a t u r e d i s t r i -b u t i o n a n d a u s t e n i t e - p e a r l i t e t r a n s f o r m a t i o n i n a n e u t e c t o i d s t e e l r o d d u r i n g c o n t r o l l e d c o o l i n g . T h e m o d e l i s b a s e d o n o n e - d i m e n s i o n a l , u n s t e a d y - s t a t e h e a t c o n d u c t i o n a n d i n c o r -p o r a t e s e m p i r i c a l T T T d a t a i n t h e f o r m o f t h e p a r a m e t e r s n a n d b ( T ) f r o m t h e A v r a m i e q u a t i o n a n d t h e C C T s t a r t t i m e , t A V - C C T ' T n \" \" s d a t a w a s o b t a i n e d u s i n g a d i a m e t r a l d i l a t o -m e t e r f o r a n e u t e c t o i d s t e e l o f c o m p o s i t i o n 0 . 8 2 % C -0 . 8 2 % Mn - 0 . 2 6 % S i a n d a g r a i n s i z e o f 5 - 7 A S T M . C C T k i n e t i c s a r e p r e d i c t e d f r o m t h e T T T d a t a b y t h e a d d i t i v i t y p r i n c i p l e o r i g i n a l l y p r o p o s e d b y S c h e i l . T h e a d e q u a c y o f t h e m o d e l w a s c h e e k e d b y c o m p a r i n g m o d e l 1 p r e d i c t i o n s o f t h e c e n t r e - l i n e t e m p e r a t u r e o f 9 a n d 1 0 mm d i a m e t e r r o d s t o m e a s u r e m e n t s m a d e d u r i n g a i r c o o l i n g f r o m a n i n i t i a l t e m p e r a t u r e b e t w e e n 8 4 0 a n d 8 7 0 ° C . T h e a g r e e m e n t o b t a i n e d w a s g o o d . A l s o t h e c o n d i t i o n s d e t e r m i n e d b y A v r a m i a n d C a h n f o r t h e a d d i t i v i t y p r i n c i p l e t o h o l d w e r e c h e c k e d . E v e n t h o u g h m o d e l p r e d i c t i o n s o f C C T f r o m T T T d a t a g e n e r a l l y w e r e g o o d , t h e a p p l i c a t i o n r e s t r i c t i o n s w e r e n o t s a t i s f i e d . T h u s a n e w s u f f i c i e n t c o n d i t i o n h a s b e e n p r o p o s e d w h i c h h o l d s i i i f o r t h e s t e e l u n d e r s t u d y a n d e s t a b l i s h e s a f i r m t h e o r e t i c a l f o u n d a t i o n f o r m o d e l c a l c u l a t i o n s . T h e c o n d i t i o n , t e r m e d \" e f f e c t i v e s i t e s a t u r a t i o n \" , i n d i c a t e s t h a t f o r g r o w t h d o m i n a t e d r e a c t i o n s , w h e r e i n t h e r a t e o f r e a c t i o n i s g o v e r n e d b y t h e g r o w t h o f n u c l e i n u c l e a t e d v e r y e a r l y i n t h e r e a c t i o n , t h e k i n e t i c s c a n b e c o n s i d e r e d a d d i t i v e d u e t o t h e r e l a t i v e u n i m p o r t a n c e o f s u b s e q u e n t n u c l e a t i o n . T h i s c o n d i t i o n s u g -g e s t s t h a t t h e a d d i t i v i t y r u l e m a y h a v e a m u c h b r o a d e r r a n g e o f a p p l i c a b i l i t y t h a n w a s o r i g i n a l l y s u p p o s e d . T h e c a l c u l a t i o n o f T T T f r o m C C T h a s b e e n s t u d i e d a n d a n e w m e t h o d , i n v o l v i n g a n i n t e r a t i v e p r o c e d u r e u s i n g t h e a d d i -t i v i t y r u l e , h a s b e e n d e r i v e d . A g r e e m e n t b e t w e e n c a l c u l a t e d a n d m e a s u r e d T T T d a t a i s g o o d . F i n a l l y t h e m o d e l h a s b e e n e m p l o y e d t o s t u d y t h e e f f e c t o f c e n t r e s e g r e g a t i o n o f m a n g a n e s e o n t h e t r a n s f o r m a t i o n b e -h a v i o u r o f e u t e c t o i d s t e e l r o d s a n d a l s o t o p r e d i c t t h e m e c h a n i c a l p r o p e r t i e s o f t h e s a m e s t e e l . C a l c u l a t i o n s i n d i -c a t e t h a t s e g r e g a t i o n c a n l e a d t o t h e f o r m a t i o n o f m a r t e n s i t e a t t h e c e n t r e o f t h e r o d s w i t h f a s t e r c o o l i n g r a t e s . T h e c a l c u l a t i o n o f m e c h a n i c a l p r o p e r t i e s i s b a s e d o n p u b l i s h e d r e l a t i o n s h i p s b e t w e e n p e a r l i t e s p a c i n g , u n d e r c o o l i n g a n d m e c h a n i c a l p r o p e r t i e s . i v TABLE OF CONTENTS P a g e A b s t r a c t \"i\" 1 ' T a b l e o f C o n t e n t s i v L i s t o f T a b l e s v i i i L i s t o f F i g u r e s x L i s t o f S y m b o l s X 1 1 A c k n o w l e d g e m e n t x n i C h a p t e r 1 INTRODUCTION . 1 2 LITERATURE SURVEY 5 2 . 1 K i n e t i c s o f t h e A u s t e n i t e - P e a r l i t e R e a c t i o n . . . 5 2 . 2 The A d d i t i v i t y R u l e . . . . . 7 2 . 3 F o r m u l a t i o n o f N u c l e a t i o n a n d G r o w t h 9 Q 2 . 4 K i n e t i c s o f A d d i t i v e R e a c t i o n s 2 . 5 A l t e r n a t i v e A p p r o a c h e s t o t h e S t u d y o f N o n - / i s o t h e r m a l R e a c t i o n K i n e t i c s ^ 2 . 6 M a t h e m a t i c a l M o d e l l i n g o f P h a s e T r a n s f o r m a t i o n s 14 2 . 7 S c o p e o f P r e s e n t Work 15 2 . 8 O b j e c t i v e s 17 3 THEORY OF ADDITIVE REACTIONS 19 3 . 1 R e a c t i o n K i n e t i c s i n t h e A d d i t i v i t y Range . . . . . 21) 3 . 1 . 1 D e f i n i t i o n o f A d d i t i v i t y Range 27 3 . 2 K i n e t i c s o f N u c l e a t i o n a n d G r o w t h R e a c t i o n s a n d ?j t h e C r i t e r i o n o f E f f e c t i v e S i t e S a t u r a t i o n 27 3 . 2 . 1 K i n e t i c s o f I s o t h e r m a l Homogeneous N u c l e a t i o n a n d G r o w t h R e a c t i o n s 28 C h a p t e r P a g e 3 . 2 . 2 E f f e c t i v e S i t e S a t u r a t i o n C r i t e r i o n f o r V a r i a b l e N u c l e a t i o n R a t e I s o t h e r m a l R e a c t i o n s 37 3 . 2 . 3 E f f e c t i v e S i t e S a t u r a t i o n C r i t e r i o n f o r H e t e r o g e n e o u s I s o t h e r m a l R e a c t i o n s . . 40 3 . 3 V a l i d a t i o n o f t h e E f f e c t i v e S i t e S a t u r a t i o n C r i t e r i o n b y E x p e r i m e n t a l R e s u l t s 4 6 3 . 4 A p p l i c a t i o n o f A d d i t i v i t y t o D e r i v e TTT f r o m CCT by t h e A d d i t i v i t y M e t h o d 4 6 3 . 5 D e r i v a t i o n o f TTT f r o m CCT b y t h e A d d i t i v i t y M e t h o d 5 1 4 DEVELOPMENT OF A MATHEMATICAL MODEL TO STUDY PHASE TRANSFORMATION 58 4 . 1 I n t r o d u c t i o n 58 4 . 2 M o d e l F o r m u l a t i o n 58 4 . 3 C o m p u t e r P r o g r a m 62 4 . 4 P r o g r a m L o g i c . . — 65 5 EXPERIMENTAL 72 5 . 1 O b j e c t i v e s o f E x p e r i m e n t s 7 2 5 . 2 E x p e r i m e n t a l P r o c e d u r e s 72 5 . 2 . 1 TTT T e s t s 7 2 5 . 2 . 2 CCT T e s t s 7 6 5 . 2 . 3 C e n t r e - l i n e T e m p e r a t u r e M e a s u r e m e n t s i n A i r - c o o l i n g T e s t s 7 7 6 RESULTS AND DISCUSSION 8 2 6 . 1 TTT T e s t R e s u l t s 8 2 v i C h a p t e r P a g e 6 . 2 CCT T e s t R e s u l t s 92 6 . 3 C o m p a r i s o n o f M o d e l - p r e d i c t e d a n d E x p e r i m e n t a l R e s u l t s o f C e n t r e - l i n e T e m p e r a t u r e M e a s u r e m e n t s 96 6 . 4 M o d e l P r e d i c t i o n a n d V a l i d a t i o n w i t h M e a s u r e d T e m p e r a t u r e D a t a 99 6 . 5 D i s c u s s i o n ^ 2 6 . 6 S c o p e o f A p p l i c a t i o n o f t h e M a t h e m a t i c a l M o d e l . 1 1 9 6 . 7 E f f e c t o f S e g r e g a t i o n o n P h a s e T r a n s f o r m a t i o n . . 120 6 . 8 C a l c u l a t i o n o f M e c h a n i c a l P r o p e r t i e s o f W i r e Rod 1 3 4 7 SUMMARY AND CONCLUSIONS 1 4 1 BIBLIOGRAPHY 1 4 5 APPENDICES 1 The P r i n c i p l e o f A d d i t i v i t y . 1 4 9 2 A d d i t i v i t y o f t h e A v r a m i E q u a t i o n K i n e t i c s ^ 4 3 D e m o n s t r a t i n g t h e I n d e p e n d e n c e o f N ( T ) a n d G ( T ) W i t h R e s p e c t t o T i m e 1 5 6 4 I t e r a t i o n r e s u l t s f o r CCT t o TTT C a l c u l a t i o n s b y t h e A d d i t i v i t y M e t h o d 1 5 9 5 L i s t i n g o f C o m p u t e r P r o g r a m t o C a l c u l a t e TTT D a t a f r o m CCT b y t h e A d d i t i v i t y M e t h o d 1 6 2 1 6 3 6 T r i - d i a g o n a l S y s t e m o f E q u a t i o n s 7 C o m p a r i s o n o f M o d e l P r e d i c t e d a n d A n a l y i c a l S o l u t i o n 1 6 8 APPENDICES v i i Page 8 L i s t i n g o f C o m p u t e r P r o g r a m t o C a l c u l a t e T e m p e r a t u r e R e s p o n s e o f a S t e e l Rod U n d e r -g o i n g C o o l i n g 9 L i s t i n g o f C o m p u t e r P r o g r a m t o C a l c u l a t e t h e T e m p e r a t u r e R e s p o n s e o f a C e n t r e - s e g r e g a t e d S t e e l Rod U n d e r g o i n g C o o l i n g — v i i i L I S T OF TABLES C h a p t e r 3 P a g e 3 . 1 Summary o f t h e h e t e r o g e n e i t y c o - e f f i c i e n t c a l c u l a t i o n s 4 4 3 . 2 S u m m a r y . o f v o l u m e c o n t r i b u t i o n c a l c u l a t i o n s 4 7 3 . 3 E f f e c t i v e s i t e s a t u r a t i o n r a t i o c a l c u l a t i o n s f o r a u s t e n i t e - p e a r l i t e r e a c t i o n i n a p l a i n c a r b o n e u t e c t o i d s t e e l 4 8 3 . 4 E f f e c t i v e s i t e s a t u r a t i o n r a t i o c a l c u l a t i o n s f o r some e u t e c t o i d s t e e l s 4 9 3 . 5 C o m p a r i s o n o f e x p e r i m e n t a l a n d c a l c u l a t e d v a l u e s o f t^y_-j-i~j. f o r a p l a i n c a r b o n e u t e c t o i d s t e e l 57 C h a p t e r 4 4 . 1 M o d e l p r e d i c t e d r e c a l e s c e n c e c a l c u l a t i o n s f o r CO a p l a i n c a r b o n e u t e c t o i d s t e e l C h a p t e r 5 71 5 . 1 S t e e l c o m p o s i t i o n C h a p t e r 6 P a g e 6 . 1 E r r o r s i n n a n d b f o r 0 . 8 2 C e u t e c t o i d s t e e l ( 5 - 7 ASTM) 93 6 . 2 . ^ V - C C T f 0 r 0 , 8 2 C S t e e 1 ^ 5 \" 7 A S ™ ) 9 5 6 . 3 M o d e l p r e d i c t e d t i m e - t e m p e r a t u r e r e s p o n s e s f o r 0 . 8 2 C s t e e l r o d s u n d e r d i f f e r e n t c o o l i n g c o n d i -t i o n s e n c o u n t e r e d i n t h e e x p e r i m e n t s o f c e n t r e -l i n e t e m p e r a t u r e m e a s u r e m e n t 1 0 ° 6 . 4 n a n d b f o r 0 . 8 2 C s t e e l ( 5 - 7 ASTM) f o r t = 0 a t T 114 6 . 5 C o m p a r i s o n o f m o d e l p r e d i c t i o n s o f t i m e - t e m p e r a t u r e r e s p o n s e s w i t h 5 = 0 a t a n d t = 0 a t T^-j f o r 0 . 8 2 C s t e e l ( 5 - 7 ASTM) 1 1 6 6 . 6 t A V - T T T f o r 0 , 8 C ~ 1 - 8 8 M n s t e e l ( 5 - 8 ASTM) 1 2 1 6 . 7 n a n d b f o r 0 . 8 C - 1 . 8 8 Mn s t e e l ( 5 - 8 ASTM) 1 2 3 6 . 8 tM-ZCl f 0 r 0 , 8 C \" 1 - 8 8 M n S t e e l ^5\"8 AS™^ 1 2 4 6 . 9 t o M o d e l p r e d i c t e d c e n t r e - l i n e t e m p e r a t u r e s f o r a \\ ' .>•'<. 6 . 1 4 s e g r e g a t e d s t e e l r o d 125-130 6 . 1 5 t o C a l c u l a t i o n o f m e c h a n i c a l p r o p e r t i e s f o r 0 . 8 2 C s t e e l ( 5 - 7 ASTM) 1 3 7 - 1 3 9 C h a p t e r 7 7 . 1 C o m p a r i s o n o f t h e s c o p e o f a d d i t i v i t y c r i t e r i a 144 X L I S T OF FIGURES C h a p t e r 3 P a g e F i g u r e 3 . T R e l a t i o n s h i p b e t w e e n t h e r e a l v o l u m e a n d e x t e n d e d v o l u m e r a t i o s 35 3 . 2 R e l a t i o n s h i p b e t w e e n t h e e f f e c t i v e s i t e s a t u r a t i o n r a t i o ( x l ^ - ) a n d t h e r e a l a n d e x t e n d e d v o l u m e r a t i o s . . 3 6 3 . 3 , 3 . 4 I n h o m o g e n e i t y c o - e f f i c i e n t f o r a u s t e n i t e - p e a r l i t e r e a c t i o n s i n a p l a i n c a r b o n e u t e c t o i d s t e e l 4 1 - 4 2 3 . 5 I l l u s t r a t i n g t h e p r i n c i p l e o f a d d i t i v i t y i n c a l c u l a -t i n g t A V _ T T T f r o m t A V _ C C T 55 C h a p t e r 4 F i g u r e 4 . 1 C o m p u t e r p r o g r a m f l o w c h a r t 6 1 4 . 2 , 4 . 3 T y p i c a l m o d e l - p r e d i c t e d t i m e - t e m p e r a t u r e c h a r t s 6 6 - 6 7 C h a p t e r 5 F i g u r e 5 . 1 E x p e r i m e n t a l a p p a r a t u s u s e d i n TTT a n d CCT t e s t s 75 5 . 2 S p e c i m e n a s s e m b l y u s e d f o r e x p e r i m e n t s i n c e n t r e -l i n e t e m p e r a t u r e m e a s u r e m e n t . 7 8 5 . 3 A r r a n g e m e n t o f a i r b l o w e r a n d s p e c i m e n u s e d i n c e n t r e - l i n e t e m p e r a t u r e m e a s u r e m e n t e x p e r i m e n t s 8 0 X T C h a p t e r 6 P a g e F i g u r e 6 . 1 T y p i c a l t i m e - t e m p e r a t u r e - d i l a t i o n r e c o r d o f a TTT t e s t 8 3 6 . 2 T y p i c a l t i m e - t e m p e r a t u r e - d i l a t i o n r e c o r d o f a CCT t e s t 8 4 6 . 3 , 6 . 4 I n I n ( , — ) v e r s u s t i m e 8 7 - 8 8 6 - 5 ^ V - T T T f o r 0 , 8 2 c a r b o n e u t e c t o i d s t e e l ( 5 - 7 ASTM) . . . 8 9 6 . 6 n a n d b v a l u e s i n t h e A v r a m i e q u a t i o n f o r 0 . 8 2 c a r b o n e u t e c t o i d s t e e l ( 5 - 7 ASTM) 9 0 - 9 1 6 . 7 ^ v - C C T f o r ° ' 8 2 e a r b o n e u t e c t o i d s t e e l ( 5 - 7 ASTM) . . . 9 7 6 . 8 I l l u s t r a t i n g t h e c o n s i s t e n c y o f r e s u l t s i n t h e CCT QQ a n d t h e c e n t r e - l i n e t e m p e r a t u r e m e a s u r e m e n t t e s t s 6 . 9 t o M o d e l p r e d i c t e d a n d e x p e r i m e n t a l r e s u l t s o f c e n t r e -^ * 1 9 101-111 l i n e t e m p e r a t u r e m e a s u r e m e n t t e s t s 6 . 2 0 N o m e n c l a t u r e o f t e r m s u s e d i n T a b l e s 6 . 3 a n d 6 . 5 115 6 . 2 1 T y p i c a l t i m e - t e m p e r a t u r e m o d e l p r e d i c t i o n s u s i n g t = 0 a t T A ] 117 6 . 2 2 Amount o f m a r t e n s i t e f o r m e d a t t h e c e n t r e o f a c o o l -i n g s t e e l r o d a s a f u n c t i o n o f c o o l i n g r a t e s 132 6 . 2 3 P h o t o g r a p h s h o w i n g s e g r e g a t i o n a t t h e c e n t r e o f a w i r e r o d 133 D e s c r i p t i o n n \\ N u c l e a t i o n r a t e ( c o n s t a n t ) N u c l e a t i o n r a t e ( f u n c t i o n o f t e m p e r a t u r e ) G r o w t h r a t e ( c o n s t a n t ) G r o w t h r a t e ( f u n c t i o n o f t e m p e r a t u r e ) T i m e Time f o r x% v o l u m e t r a n s f o r m e d E x t e n d e d v o l u m e t r a n s f o r m e d R e a l v o l u m e t r a n s f o r m e d ( e x t e n d e d v o l u m e c o r r e c t e d f o r i m p i n g e m e n t ) E x t e n d e d v o l u m e t r a n s f o r m e d a t . t A E x t e n d e d v o l u m e t r a n s f o r m e d a t t i m e t o f n u c l e i A n u c l e a t i n g b e t w e e n t i m e =0 a n d t i m e = t V o l u m e f r a c t i o n t r a n s f o r m e d D e n s i t y S p e c i f i c h e a t T h e r m a l c o n d u c t i v i t y T e m p e r a t u r e Dummy v a r i a b l e r e p r e s e n t i n g t i m e x i i i A C K N O W L E D G E M E N T T w o u l d l i k e t o t h a n k P r o f e s s o r s J . K . B r i m a c o m b e a n d E . B . H a w b o l t f o r t h e i r h e l p a n d g u i d a n c e d u r i n g t h e c o u r s e o f t h e p r o j e c t . T h a n k s a r e a l s o d u e t o M r . B i n h C h a u , M r . B a h a K u b a n , M r . S . C h a t t o p a d h y a y , M r . R a m a p r a s a d a n d M r . N e i l W a l k e r f o r a l l t h e h e l p r e n d e r e d . F i n a n c i a l a s s i s t a n c e w a s r e c e i v e d i n t h e f o r m o f a r e s e a r c h g r a n t f r o m t h e A m e r i c a n I r o n a n d S t e e l I n s t i t u t e . 1 C h a p t e r 1 1 . 1 I N T R O D U C T I O N T h e m e c h a n i c a l p r o p e r t i e s o f s t e e l s d e p e n d o n t h e i r c o m p o s i t i o n , g r a i n s i z e a n d s t r u c t u r e . T h e l a t t e r i s n o r m a l l y c o n t r o l l e d b y a p p l y i n g s p e c i f i c c o o l i n g c o n d i t i o n s i n t h e l a s t s t a g e o f p r o c e s s i n g . A n e x a m p l e i s t h e p r o -d u c t i o n o f s t e e l r o d s i n w h i c h a f t e r t h e l a s t r o l l i n g p a s s , t h e r o d s a r e c o n t r o l c o o l e d f r o m a b o u t 9 0 0 ° C b y f o r c e d a i r o n a S t e l m o r l i n e . B y a d j u s t i n g t h e r e s i d e n c e t i m e o f r o d s a n d t h e a i r v e l o c i t y i n i n d i v i d u a l - c o o l i n g z o n e s , t h e d e s i r e d s t r u c t u r e , e . g . f r a c t i o n p e a r l i t e a n d f e r r i t e , c a n b e o b t a i n e d . B e c a u s e t h e s t r u c t u r e h a s a s t r o n g i n f l u e n c e o n t h e m e c h a n i c a l p r o p e r t i e s , i t i s i m -p o r t a n t t h a t t h e l i n k b e t w e e n s t r u c t u r e f o r e a c h s t e e l a n d p r o c e s s v a r i a b l e s s u c h a s , i n t h e c a s e o f t h e S t e l m o r p r o c e s s , r o d d i a m e t e r , a i r v e l o c i t y a n d l i n e s p e e d i s w e l l e s t a b l i s h e d . Up t o t h e p r e s e n t t i m e , s u c h l i n k s h a v e b e e n d e t e r m i n e d e m p i r i c a l l y . P r a c t i c e s h a v e b e e n d e v e l o p e d i n t h i s w a y , f o r e x a m p l e , t o c o n t r o l r o d c o o l i n g a n d a c h i e v e s p e c i f i c p e a r l i t e s p a e i n g s w h i c h g o v e r n t h e m e c h a n i c a l p r o p e r t i e s . ^ H o w e v e r t h i s a p p r o a c h , o n a p l a n t s c a l e , i s t i m e c o n s u m i n g a n d e x p e n s i v e . T h e r e i s 2 c o n s i d e r a b l e i n c e n t i v e , t h e r e f o r e , f o r t h e d e v e l o p m e n t o f a p r e d i c t i v e c a p a b i l i t y , s u c h a s a m a t h e m a t i c a l m o d e l , w h i c h c a n p r e d i c t m e c h a n i c a l p r o p e r t i e s o f a g i v e n s t e e l a s a f u n c t i o n o f p r o c e s s v a r i a b l e s . T h i s i s t h e s u b j e c t o f t h e p r e s e n t s t u d y . D e v e l o p m e n t o f a m a t h e m a t i c a l m o d e l t o p r e d i c t m e c h a n i -c a l p r o p e r t i e s , h o w e v e r , i s a d i f f i c u l t t a s k o w i n g t o t h e c o m p l e x i t y o f t h e p r o c e s s e s w h i c h d e t e r m i n e s t r u c t u r e i n s t e e l s . O n e m a j o r p r o b l e m i s t h a t h e a t f l o w a n d p h a s e t r a n s f o r m a t i o n k i n e t i c s a r e c o u p l e d . I n a c o n t r o l l e d c o o l i n g p r o c e s s , t h e s t e e l u n d e r g o e s a c o n t i n u o u s c h a n g e , o f t e m p e r a -t u r e , t h e r a t e o f w h i c h d e p e n d s o n t h e l o c a t i o n w i t h i n t h e s t e e l . A t t h e s a m e t i m e , a s p h a s e t r a n s f o r m a t i o n t a k e s p l a c e h e a t i s e v o l v e d w h i c h f r e q u e n t l y c a u s e s r e c a l e s e e n c e . T h u s t h e c h a n g i n g t e m p e r a t u r e f i e l d i s a f f e c t e d b y h e a t e x t r a c t i o n f r o m , a n d c o n d u c t i o n w i t h i n , t h e r o d a s w e l l a s h e a t g e n e r a t i o n w h i c h d e p e n d s o n t h e k i n e t i c s o f t h e p h a s e t r a n s f o r m a t i o n . T h e t r a n s f o r m a t i o n k i n e t i c s , i n t u r n a r e d e p e n d e n t o n t e m p e r a t u r e . A s e c o n d p r o b l e m i s t h a t t h e t r a n s f o r m a t i o n k i n e t i c s h a v e b e e n c h a r a c t e r i z e d e m p i r i c a l l y i n i s o t h e r m a l t e s t s ( T T T ) a n d e x p e r i m e n t s i n w h i c h t h e c o o l i n g r a t e i s c o n s t a n t ( C C T ) . B u t n e i t h e r c o n d i t i o n h o l d s a t a g i v e n l o c a t i o n w i t h i n t h e s t e e l s h a p e d u r i n g c o o l i n g . T h u s t h e q u e s t i o n 3 b e c o m e s h o w d a t a o b t a i n e d i n t h e 1 a b o r a t o r y c a n b e a p p l i e d t o t h e c o m p l e x n o n - i s o t h e r m a l s i t u a t i o n o f c o n t r o l l e d c o o l i n g . T h u s t h e m a t h e m a t i c a l m o d e l m u s t i n c o r p o r a t e h e a t c o n d u c t i o n w i t h i n t h e s t e e l , h e a t e x t r a c t i o n f r o m t h e s u r -f a c e o f t h e s t e e l a n d r e c a l e s c e n c e w h i c h i s d e p e n d e n t o n t h e c o u p l e d p h a s e - t r a n s f o r m a t i o n k i n e t i c s . T h e h e a t - e x t r a c -t i o n p a r t o f t h e m o d e l i s r e l a t i v e l y s t r a i g h t f o r w a r d c o m -p a r e d t o t h e t r a n s f o r m a t i o n k i n e t i c s ; t h e l a t t e r m u s t b e d e t e r m i n e d e m p i r i c a l l y f o r e a c h s t e e l c o m p o s i t i o n a n d a u s t e n -i t e g r a i n s i z e . M o r e o v e r , a s m e n t i o n e d a b o v e , o n c e t h e t r a n s f o r m a t i o n d a t a h a v e b e e n m e a s u r e d , u s u a l l y u n d e r i s o -t h e r m a l c o n d i t i o n s , a v a l i d p r o c e d u r e m u s t b e d e v e l o p e d t o a p p l y t h e d a t a i n t h e p r e d i c t i o n o f n o n - i s o t h e r m a l t r a n s -f o r m a t i o n . T h e f u n d a m e n t a l i n t e r - r e l a t i o n s h i p s b e t w e e n t h e p r o c e s s v a r i a b l e s a n d t h e c o o l i n g r a t e a r e d e v e l o p e d i n C h a p t e r 3. I n t h e p r e s e n t s t u d y i t w a s d e c i d e d t o m o d e l t h e a u s t e n i t e - p e a r l i t e r e a c t i o n i n a p l a i n - c a r b o n e u t e c t o i d s t e e l . T h i s m a t e r i a l t r a n s f o r m s f r o m a u s t e n i t e t o p e a r l i t e a t t h e AC-j t e m p e r a t u r e u n d e r e q u i l i b r i u m c o n d i t i o n s a n d d o e s n o t e x h i b i t a n y o t h e r p h a s e s , l i k e f e r r i t e o r c e m e n t l t e , a n d h e n c e i s s i m p l e s t t o m o d e l . T h e m o d e l , w h i c h i s d e -s c r i b e d i n C h a p t e r 4 , h a s b e e n w r i t t e n t o p r e d i c t t h e 4 t e m p e r a t u r e r e s p o n s e o f a c y l i n d r i c a l r o d , s i n c e t h e h e a t t r a n s f e r a n d b o u n d a r y c o n d i t i o n s a r e w e l l d e f i n e d f o r s u c h a s h a p e . S u c h a s h a p e a l s o h a s w i d e i n d u s t r i a l a p p l i c a b i l i t y a n d i s s i m p l e t o u s e i n e x p e r i m e n t s u n d e r c o n t r o l l e d c o n -d i t i o n s . T h e m o d e l i n t e g r a t e s C C T a n d T T T d a t a f o r t h e s t e e l a s m e a s u r e d i n e x p e r i m e n t s d e s c r i b e d i n C h a p t e r . 5 a n d e a s i l y m e a s u r a b l e p r o c e s s v a r i a b l e s s u c h a s i n i t i a l t e m p e r a t u r e a n d c o o l i n g p a r a m e t e r s . T h e m o d e l h a s b e e n v a l i d a t e d b y c o m p a r i n g p r e d i c t i o n s o f c e n t r e l i n e t e m p e r a t u r e t o m e a s u r e -m e n t s d e s c r i b e d i n C h a p t e r 6 . T h e e f f e c t o f c e n t r e s e g r e g a -t i o n o n t r a n s f o r m a t i o n , h a s b e e n s t u d i e d b y m o d i f y i n g t h e m o d e l . C a l c u l a t i o n s w e r e d o n e t o p r e d i c t t h e e f f e c t o f c e n t r e s e g r e g a t i o n ( o f c o m p o s i t i o n 0 . 8 0 % C - 1 . 8 8 % Mn i n a m a t r i x o f c o m p o s i t i o n 0 . 8 2 % C - 0 . 8 2 % Mn.) i n a n , a i r c o o l e d r o d , d e s c r i b e d i n C h a p t e r 6 . F i n a l l y , c a l c u l a t i o n s w e r e d o n e t o d e r i v e m e c h a n i c a l p r o p e r t i e s o f s t e e l r o d s u n d e r d i f f e r e n t c o o l i n g c o n d i t i o n s u s i n g t h e m o d e l g e n e r a t e d d a t a a n d a r e d e s c r i b e d i n C h a p t e r 6 . 5 C h a p t e r 2 L I T E R A T U R E S U R V E Y 2 . 1 K i n e t i c s o f t h e A u s t e n i t e - P e a r l i t e R e a c t i o n . A s y s t e m a t i c s t u d y o f t h e a u s t e n i t e - p e a r l i t e r e a c t i o n w a s m a d e b y B a i n . ^ S u b s e q u e n t l y s e v e r a l o t h e r s t u d i e s , b a s e d o n m e a s u r e m e n t s o f t e m p e r a t u r e , d i l a t i o n a n d h a r d n e s s 1 2 - 1 8 a s w e l l a s m e t a l l o g r a p h i c t e c h n i q u e s , w e r e c o n d u c t e d . T h e m o s t i m p o r t a n t w o r k s , w h i c h a d v a n c e d t h e u n d e r -1 9 s t a n d i n g o f r e a c t i o n k i n e t i c s , a r e t h o s e o f J o h n s o n - M e h l , 2 0 - 2 2 1 7 A v r a m i a n d S c h e i l . J o h n s o n - M e h l g a v e a c o m p r e h e n s i v e m a t h e m a t i c a l t r e a t m e n t o f t h e a u s t e n i t e - p e a r ! i t e r e a c t i o n k i n e t i c s . T h e y d e r i v e d a n e q u a t i o n f o r k i n e t i c s o f n u c l e a -t i o n a n d g r o w t h r e a c t i o n s u n d e r t h e f o l l o w i n g a s s u m p t i o n s : i ) C o n s t a n t n u c l e a t i o n a n d g r o w t h r a t e s i i ) R a n d o m n u c l e a t i o n i i i ) T h e r e a c t i o n p r o d u c t f o r m s t r u e s p h e r e s e x c e p t w h e n d u r i n g g r o w t h , i m p i n g e m e n t o n o t h e r g r o w i n g s p h e r e s o c c u r . W i t h t h e a b o v e a s s u m p t i o n s t h e v o l u m e f r a c t i o n t r a n s -f o r m e d , X , i s r e l a t e d t o t h e n u c l e a t i o n r a t e , N , a n d t h e g r o w t h r a t e , G , b y t h e f o l l o w i n g r e l a t i o n s h i p 1 - e x p ( - | N G 3 t 4 ) ( 2 . 1 ) H o w e v e r t h e r e i s s o m e q u e s t i o n : t h a t t h e a s s u m p t i o n s a r e v a l i d . E v e n f o r a n i s o t h e r m a l r e a c t i o n i t i s d o u b t f u l t h a t t h e n u c l e a t i o n r a t e r e m a i n s c o n s t a n t . U n d e r c o o l i n g i s t h e d r i v i n g f o r c e f o r t h e n u c l e a t i o n p r o c e s s , a n d f o r a n i s o t h e r m a l r e a c t i o n , i t m a y s e e m p o s s i b l e t h a t N m a y r e -m a i n c o n s t a n t . B u t t h i s i s t o o s i m p l i s t i c a v i e w w h i c h n e g l e c t s t h e e f f e c t o f c o m p o s i t i o n , s t r u c t u r e a n d t h e t r a n s -f o r m a t i o n p r o d u c t o n t h e p h e n o m e n o n o f n u c l e a t i o n . B r o w n 3 6 a n d R i d l e y h a v e s h o w n t h a t i t i s p o s s i b l e t o h a v e a d e -c r e a s i n g n u c l e a t i o n r a t e a f t e r a b o u t 2 0 % t r a n s f o r m a t i o n . O t h e r e v i d e n c e a l s o e x i s t s t o s u g g e s t t h a t n u c l e a t i o n m a y 1 9 d e c r e a s e a s t h e r e a c t i o n p r o c e e d s . H o w e v e r , t h e g r o w t h r a t e o f p e a r l i t e i s c o n s t a n t a t a g i v e n t e m p e r a t u r e . T h e a s s u m p t i o n o f r a n d o m n u c l e a t i o n i s a l s o . q u e s t i o n a b l e , e s p e c i a l l y i n c o m m e r c i a l s t e e l s w h i c h a r e p r o n e t o s o m e d e g r e e o f s e g r e g a t i o n o f e l e m e n t s l i k e Mn. a n d . P . A l s o n o n -u n i f o r m i t y i n g r a i n s i z e m a y h a v e a n e f f e c t . T h e a s s u m p t i o n o f c o m p l e t e l y s p h e r i c a l g r o w t h i s , l i k e w i s e , q u e s t i o n a b l e 3 7 i n t h e l i g h t o f m i c r o g r a p h i c s t u d i e s c o n d u c t e d b y K u b a n . A f u r t h e r d i f f i c u l t y i n u s i n g E q . ( 2 . 1 ) i s t h e r e q u i r e d d e t e r m i n a t i o n o f N a n d G b y c o n d u c t i n g c o n t r o l l e d e x p e r i -m e n t s . M o r e o v e r , r e a c t i o n s o f i n d u s t r i a l i m p o r t a n c e a r e 7 u s u a l l y n o n - i s o t h e r m a l . S i n c e E q . ( 2 . 1 ) c a n n o t b e u s e d f o r n o n - i s o t h e r m a l r e a c t i o n s , i t s u s e i s v e r y r e s t r i c t e d . A v r a m i ' s f o r m u l a t i o n i s m o r e u s e f u l i n t h i s r e g a r d . L i k e J o h n s o n - M e h l , A v r a m i d e r i v e d a n e q u a t i o n f o r a u s t e n i t e -p e a r l i t e r e a c t i o n s a s : X = 1 - e x p ( - b t n ) ( 2 . 2 ) w h e r e n i s a c o n s t a n t a n d b i s a t e m p e r a t u r e d e p e n d e n t p a r a -m e t e r . C l e a r l y E q . ( 2 . 2 ) i s a m o r e g e n e r a l f o r m o f E q . ( 2 . 1 ) . T h o u g h b a n d n a r e e m p i r i c a l c o n s t a n t s , A v r a m i u s e d s o u n d t h e o r e t i c a l p r i n c i p l e s t o d e r i v e E q . ( 2 . 2 ) . H i s t r e a t m e n t o f n u c l e a t i o n r a t e i s s u p e r i o r t o t h a t o f J o h n s o n a n d M e h l a n d : h e ; ; j d e r i v e d a s i m p l e r f o r m u l a f o r i n c l u d i n g t h e e f f e c t o f i m p i n g e m e n t d u r i n g g r o w t h , i n t h e v o l u m e c a l c u l a t i o n . A v r a m i a l s o s h o w e d t h a t h i s e q u a t i o n i n c l u d e d t h o s e o f p r e -v i o u s a u t h o r s , l i k e A u s t i n a n d R i c k e t t , 1 8 Z e n e r , 1 4 J o h n s o n 1 9 a n d M e h l a s s p e c i a l c a s e s . 2 . 2 T h e A d d i t i v i t y R u l e S c h e i l 1 7 f i r s t e n u n c i a t e d t h e a d d i t i v i t y r u l e , w h i c h l i n k s t h e i s o t h e r m a l k i n e t i c s t o n o n - i s o t h e r m a l r e a c t i o n s . T h i s r u l e s i m p l i f i e d t h e p r o b l e m o f s t u d y i n g n o n - i s o t h e r m a l 2 5 r e a c t i o n k i n e t i c s . C h r i s t i a n g a v e a n u p - t o - d a t e v e r s i o n o f t h i s r u l e , w h i c h s t a t e s t h a t t w h e r e : J t d t = . 1 ( 2 . 3 ) 0 a T f J • ( S e e A p p e n d i x 1 f o r d e r i v a t i o n . ) t = t i m e f o r a n o n - i s o t h e r m a l r e a c t i o n t o r e a c h a s p e c i f i c a m o u n t o f t r a n s f o r m a t i o n . t ( T ) = t i m e t o r e a c h t h e s a m e t r a n s f o r m a t i o n i s o -a t h e r m a l l y a t t e m p e r a t u r e T . T h i s r u l e h o l d s t r u e f o r r e a c t i o n s f o r w h i c h t h e i n -s t a n t a n e o u s r e a c t i o n r a t e i s o n l y a f u n c t i o n o f t h e t e m p e r a -t u r e a n d t h e a m o u n t t r a n s f o r m e d , i r r e s p e c t i v e o f t h e p r e v i o u s t h e r m a l h i s t o r y . A v r a m i ' s d e r i v a t i o n i s v e r y i m p o r t a n t i n t h i s r e g a r d . H e s h o w e d t h a t E q . ( 2 . 2 ) d e s c r i b e s t h e k i n e t i c s N o f a d d i t i v e r e a c t i o n s p r o v i d e d t h e r a t i o ^ r e m a i n s c o n s t a n t o v e r t h e t e m p e r a t u r e r a n g e o f t h e r e a c t i o n . He d e f i n e d t h i s t e m p e r a t u r e r a n g e a s t h e \" I s o k i n e t i c R a n g e \" . H o w e v e r , b e -c a u s e t h e c h a n g e i n N w i t h t e m p e r a t u r e i s m u c h m o r e r a p i d t h a n t h a t o f G f o r m a n y a u s t e n i t e - p e a r l i t e t r a n s f o r m a t i o n s i n s t e e l , t h e e x i s t e n c e o f s u c h a r a n g e i s d o u b t f u l . ' T h e a d v a n t a g e o f E q . ( 2 . 2 ) o v e r E q . ( 2 . 1 ) i s t h a t A v r a m i d i r e c t l y a d d r e s s e d t h e p r o b l e m o f a d d i t i v i t y a n d p r o v i d e d 9 a t l e a s t o n e s u f f i c i e n t c o n d i t i o n f o r a d d i t i v i t y t o h o l d . D e s p i t e t h e q u e s t i o n a b i 1 i t y o f t h e i s o k i n e t i c r a n g e , t h e A v r a m i E q u a t i o n , ( 2 . 2 ) w i t h e m p i r i c a l l y d e t e r m i n e d v a l u e s o f b a n d n , p r e d i c t s t h e n a t u r e o f t h e a u s t e n i t e -p e a r l i t e r e a c t i o n k i n e t i c s q u i t e a c c u r a t e l y . T h e d i f f i c u l t y l i e s i n d e t e r m i n i n g t h e a p p r o p r i a t e v a l u e s o f b a n d n . 2 . 3 F o r m u l a t i o n o f N u c l e a t i o n a n d G r o w t h I n o r d e r t o d e r i v e w a y s o f f i n d i n g b a n d n , a n d t o d e s c r i b e t h e t h e o r e t i c a l i m p o r t a n c e a n d b a s i s o f t h e s e , s e v e r a l a t t e m p t s h a v e b e e n m a d e t o f o r m u l a t e t h e n u c l e a -t i o n a n d g r o w t h p h e n o m e n a i n f u n d a m e n t a l t e r m s . E q u a t i o n s h a v e b e e n d e r i v e d f o r p l a t e - l i k e g r o w t h , n e e d l e - l i k e g r o w t h , g r a i n - b r o u n d a r y n u c l e a t e d g r o w t h e t c . b y s e v e r a l w o r k e r s ; a c o m p r e h e n s i v e t r e a t m e n t o f a l l o f t h e s e c a n b e f o u n d i n r e f e r e n c e ( 3 5 ) . T h e r e s u l t i n g e q u a t i o n s , e s s e n t i a l l y , a r e e x t e n s i o n s o f t h e J o h n s o n - M e h l t y p e o f c a l c u l a t i o n s a n d a r e s u b j e c t t o s i m i l a r a s s u m p t i o n s . T h e s e m e t h o d s u l t i -m a t e l y r e s u l t i n t h e f o r m u l a t i o n o f a n e q u a t i o n l i k e E q . ( 2 . 2 ) w i t h d i f f e r e n t v a l u e s f o r t h e c o n s t a n t n . S i n c e t h e v a l i -d i t y o f t h e a s s u m p t i o n s m a d e a r e q u e s t i o n a b l e , a c l o s e r e x a m i n a t i o n o f t h e r e a c t i o n k i n e t i c s i s i n o r d e r . 2 . 4 K i n e t i c s o f A d d i t i v e R e a c t i o n s 2 3 ?& I n 1 9 5 6 , C a h n ' p r o p o s e d t h a t r e a c t i o n k i n e t i c s w h i c h 10 c a n b e d e s c r i b e d b y : ( 2 . 4 ) w h e r e : X = v o l u m e f r a c t i o n t r a n s f o r m e d t = t i m e h ( T ) = a f u n c t i o n o f t e m p e r a t u r e g ( X ) = a f u n c t i o n o f v o l u m e f r a c t i o n t r a n s f o r m e d , c a n b e e x p e c t e d t o b e a d d i t i v e . I t c a n b e s h o w n t h a t t h e A v r a m i e q u a t i o n c a n b e m o d i f i e d t o b e o f t h e s a m e f o r m a s E q . ( 2 . 4 ) ( p r o v i d e d ' n ' i s a c o n s t a n t i n d e p e n d e n t o f T a n d X ) s u c h t h a t 1 h ( T ) = n ( - b ) n ( 2 . 5 ) a n d n-1 n g(x) l 4 x { l o g \\ l - X ) } ( 2 ' 6 ) ( S e e A p p e n d i x 2 f o r d e r i v a t i o n . ) H e n c e t h e A v r a m i e q u a t i o n d e s c r i b e s t h e k i n e t i c s o f a d d i t i v e r e a c t i o n s . S e v e r a l a u t h o r s h a v e s h o w n t h a t , d e s p i t e t h e q u e s t i o n a b l e a s s u m p t i o n s , t h e a d d i t i v i t y r u l e h o l d s 26 29 39 44 t r u e f o r a u s t e n i t e - p e a r l i t e a n d b a i n r t e . r e a c t i o n s . ' ' ' A n i m p o r t a n t f e a t u r e o f a l l t h e s e w o r k s i s t h e a s s u m p t i o n t h a t , i r r e s p e c t i v e o f t h e r e a c t i o n c o n d i t i o n s ( i . e . i s o t h e r m a l 11 o r n o n - i s o t h e r m a l ) , t h e t r a n s f o r m a t i o n o f a u s t e n i t e t o p e a r l i t e b e g i n s a t t h e e q u i l i b r i u m t r a n s f o r m a t i o n t e m p e r a -35 t u r e . H a w b o l t e t a l . , i n a v e r y r e c e n t w o r k h a v e d e r i v e d a d i f f e r e n t m e t h o d f o r d e t e r m i n i n g t h e . \" s t a r t \" o f t h e t r a n s -f o r m a t i o n u n d e r n o n - e q u i l i b r i u m r e a c t i o n c o n d i t i o n s . T h i s p r o c e d u r e i s d i s c u s s e d i n d e t a i l i n C h a p t e r 6 , a n d m a y c o n -t r a s t w i t h t h e p u b l i s h e d T T T o r C C T d i a g r a m s w h i c h s h o w t h e \" s t a r t \" l i n e a s 0 . 1 % o r 1% t r a n s f o r m e d . I n t h e n e w p r o -c e d u r e , f o r a s s e s s i n g t h e k i n e t i c d a t a , t h e i n c u b a t i o n t i m e i s n e g l e c t e d a n d t h e A v r a m i e q u a t i o n i s a p p l i e d o n l y t o d e s c r i b e t h e n u c l e a t i o n a n d g r o w t h p h e n o m e n o n . T h e s t a r t o f t h e t r a n s f o r m a t i o n o c c u r s a f t e r a n i n c u b a t i o n t i m e t ^ y ( f o r v RAM I ^ ' I n t h e p r e s e n t w o r k , t h i s t i m e h a s b e e n u s e d a s t h e \" s t a r t \" t i m e . T h i s i s a m a j o r d e p a r t u r e f r o m t h e c o n v e n t i o n a l m e t h o d s o f s t u d y i n g t h e k i n e t i c s . T h e r e -s u l t s f r o m t h e p r e s e n t w o r k c o n f i r m t h a t t h e u s e o f t ^ y f o r a d d i t i v i t y c a l c u l a t i o n s g i v e s b e t t e r a g r e e m e n t w i t h e x p e r i m e n t a l o b s e r v a t i o n s . 2.5 A l t e r n a t i v e A p p r o a c h e s t o t h e S t u d y o f N o n - i s o t h e r m a l R e a c t i o n K i n e t i c s 2 8 I n 1 9 4 1 , G r a n g e a n d K e i f e r d e s c r i b e d a s i m p l e a n d e l e g a n t m e t h o d o f d e r i v i n g C C T f r o m T T T d a t a . T h i s m e t h o d w a s e m p i r i c a l i n n a t u r e a n d i n v o l v e d a s s u m p t i o n s r e g a r d i n g t h e k i n e t i c s . T h o u g h t h e s e w e r e s i m p l i s t i c a s s u m p t i o n s , 1 2 a n d h e n c e t h e r e s u l t s a p p r o x i m a t e , t h e m e t h o d i s e a s y t o u s e . B u t i t d i d n o t e m p l o y a d d i t i v i t y . H o w e v e r , s i n c e t h i s m e t h o d c o u l d n o t b e j u s t i f i e d o n s o u n d t h e o r e t i c a l g r o u n d s , i t h a s n o t f o u n d m u c h a p p l i c a t i o n . •I c A n o t h e r m e t h o d , e m p l o y e d b y M a n n i n g a n d L o r i g , u s e d t h e e x p e r i m e n t a l d e t e r m i n a t i o n o f t h e \" s t a r t \" o f t r a n s f o r m a -t i o n d u r i n g c o n t i n u o u s c o o l i n g b y c o n d u c t i n g c o n t r o l 1 e d -c o o l i n g e x p e r i m e n t s . T h e s e a r e t i m e c o n s u m i n g p r o c e d u r e s a n d t h e r e s u l t s g e n e r a t e d d o n o t l e n d t h e m s e l v e s u s e f u l f o r g e n e r a l a p p l i c a t i o n . A n e w a p p r o a c h t o t h e p r o b l e m o f n o n - i s o t h e r m a l k i n e t i c s 2 3 2 4 w a s g i v e n b y C a h n ' i n 1 9 5 6 . He s h o w e d t h a t s i n c e t h e i s o k i n e t i c r a n g e i s o n l y a s u f f i c i e n t c o n d i t i o n f o r a d d i -t i v i t y , t h e r u l e o f a d d i t i v i t y c o u l d a l s o b e a p p l i e d t o r e a c t i o n s u n d e r a c o n d i t i o n c a l l e d \" s i t e s a t u r a t i o n \" . He d e d u c e d t h i s f r o m m i c r o g r a p h s f r o m e x p e r i m e n t s o n e u t e c t o i d a l l o y s t e e l s h a v i n g a l a r g e a u s t e n i t e g r a i n s i z e . I n s u c h s t e e l s , t h e r e a c t i o n i s i n i t i a t e d a t g r a i n b o u n d a r i e s w h e r e t h e n u c l e a t i o n e v e n t i s s o r a p i d t h a t i n t h e v e r y e a r l y s t a g e s o f t r a n s f o r m a t i o n ( 1 0 - 2 0 % ) t h e g r a i n b o u n d a r i e s a r e s a t u r a t e d w i t h t h e g r o w i n g n e w p h a s e . N u c l e a t i o n i s t h e r e -f o r e c o m p l e t e i n t h e e a r l y s t a g e s o f t h e r e a c t i o n a n d p l a y s n o f u r t h e r r o l e . T h e e n s u i n g t r a n s f o r m a t i o n i s t h e n c o n -t r o l l e d b y t h e g r o w t h r a t e . S i n c e t h e g r o w t h r a t e i s a 1 3 t e m p e r a t u r e d e p e n d e n t p a r a m e t e r , a d d i t i v i t y m u s t b e e x p e c t e d t o h o l d . T h i s w a s a d e f i n i t e n e w d i r e c t i o n i n t h e w o r k o n k i n e t i c s o f r e a c t i o n s . B y e l i m i n a t i n g n u c l e a t i o n a s a v a r i a b l e , C a h n s i m p l i f i e d t h e p r o c e s s o f c h a r a c t e r i s i n g t h e k i n e t i c s b y t h e g r o w t h r a t e a l o n e , t h e r e b y e l i m i n a t i n g t h e a s s u m p t i o n s r e g a r d i n g t h e n u c l e a t i o n r a t e . T h o u g h t h e c o n d i t i o n o f \" s i t e s a t u r a t i o n \" i n c r e a s e d t h e n u m b e r o f t r a n s f o r m a t i o n s f o r w h i c h t h e a d d i t i v i t y p r i n c i p l e e o u l d b e a p p l i e d , t h i s i s n o t a u n i v e r s a l p h e n o -3 7 m e n o n . I n t h e w o r k b y K u b a n , w h e r e a p l a i n c a r b o n e u t e c -t o i d s t e e l w a s s t u d i e d , m i c r o g r a p h s - u n a m b i g u o u s l y r e v e a l t h e a b s e n c e o f g r a i n b o u n d a r y s a t u r a t i o n . I t i s p o s s i b l e , h o w e v e r , t h a t s i t e s a t u r a t i o n : i s m o r e p r o b a b l e i n t h e c a s e o f a l l o y s t e e l s d u e t o t h e p r e s e n c e o f a l l o y i n g e l e m e n t s w h i c h m a y e n c o u r a g e g r a i n b o u n d a r y n u c l e a t i o n . A l s o t h e e f f e c t o f g r a i n s i z e o n s i t e s a t u r a t i o n n e e d s t o b e e x a m i n e d . I n t u i t i v e l y , i t w o u l d a p p e a r g r a i n b o u n d a r y s i t e s a t u r a t i o n i s m o r e p r o b a b l e i n l a r g e r g r a i n s i z e m a t e r i a l d u e t o t h e r e d u c e d a m o u n t o f g r a i n b o u n d a r y a r e a p e r u n i t v o l u m e . C a h n ' s m e t h o d d i f f e r s f r o m o t h e r e m p i r i c a l m e t h o d s i n t h a t i t h a s a f i r m t h e o r e t i c a l b a s i s . T h i s i s e v i d e n t 2 7 3 9 w h e n c o m p a r i n g C a h n ' s w o r k w i t h S a k a m o t o a n d T z i t z e l k o v . T h e s e a n d o t h e r w o r k e r s 2 6 ' 2 9 ' 3 0 h a v e e m p l o y e d c u r v e - f i t t i n g m e t h o d s , a i d e d b y c o m p u t e r - b a s e d c a l c u l a t i o n s , t o d e r i v e C C T f r o m T T T . S i n c e t h e s e a r e n o t b a s e d o n t h e o r e t i c a l c o n -s i d e r a t i o n s t h e y c a n n o t b e c o n s i d e r e d a s c o n t r i b u t i o n s t o t h e u n d e r s t a n d i n g o f t h e r e a c t i o n k i n e t i c s . T h e y f i n d t h e i r u s e i n s p e c i f i c s i t u a t i o n s . 2 . 6 M a t h e m a t i c a l M o d e l l i n g o f P h a s e T r a n s f o r m a t i o n s C a l c u l a t i o n o f C C T f r o m T T T b y u s i n g t h e a d d i t i v i t y p r i n c i p l e i s c o m p l e x a n d l a b o r i o u s . ^ 5 ' ^ ' 2 8 S u c h c a l c u l a -t i o n s a r e n o r m a l l y d o n e t o p l o t t h e C C T d i a g r a m f o r a ; m a t e r i a l o f a g i v e n c h e m i s t r y a n d g r a i n s i z e . T h e y c a n a l s o b e u s e d t o c a l c u l a t e r e a c t i o n k i n e t i c s i n a m a t e r i a l u n d e r g o i n g p r o c e s s i n g i n i n d u s t r i a l s i t u a t i o n s , e . g . , a n i n f i n i t e l y l o n g r o d o f c i r c u l a r c r o s s - s e c t i o n i n a w i r e r o d m i l l . T o s t u d y r e a c t i o n s t a k i n g p l a c e i n s u c h s h a p e s , u n d e r d i f f e r e n t p r o c e s s i n g c o n d i t i o n s , e . g . , c o o l i n g r a t e s , t h e k i n e t i c s m u s t b e r e l a t e d t o t h e t h e r m a l h i s t o r y , w h i c h i s i n t u r n g o v e r n e d b y t h e m a t e r i a l p r o p e r t i e s a n d p r o c e s s c o n d i t i o n s . D e v e l o p m e n t s i n t h e f i e l d s o f h e a t t r a n s f e r , s o l u t i o n o f d i f f e r e n t i a l e q u a t i o n s b y f i n i t e - d i f f e r e n c e m e t h o d s a n d r a p i d c o m p u t e r - a i d e d c a l c u l a t i o n s h a v e r e s u l t e d i n t h e f o r m u l a t i o n o f m a t h e m a t i c a l m o d e l s t o s t u d y s u c h p h o n e o m e n a a s p h a s e t r a n s f o r m a t i o n s , s t r e s s f i e l d s , 31 3 2 3 3 t e m p e r a t u r e f i e l d s , e t c . ' ' A m a t h e m a t i c a l m o d e l t o s t u d y r e a c t i o n k i n e t i c s i n a p l a i n c a r b o n e u t e c t o i d s t e e l r o d w a s f i r s t a t t e m p t e d b y 1 5 31 A g a r w a l a n d B r i m a c r o m b e . T h e y s o l v e d t h e s e c o n d - o r d e r d i f f e r e n t i a l e q u a t i o n f o r h e a t t r a n s f e r i n a n i n f i n i t e l y l o n g c i r c u l a r c r o s s - s e c t i o n r o d b y u s i n g a 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 p r o c e d u r e . T h e k i n e t i c s o f t r a n s f o r m a t i o n w e r e i n c o r p o r a t e d i n t o t h e m o d e l b y u s i n g p u b l i s h e d T T T d a t a a n d a d d i t i v i t y . T h e y c o m p a r e d t h e i r m o d e l - p r e d i c t e d r e -s u l t s w i t h t h e e x p e r i m e n t a l w o r k o f T a k e o e t a l . ^ a n d f o u n d t h a t a g r e e m e n t w a s r e l a t i v e l y p o o r . I t w a s t h o u g h t t h a t t h i s m i g h t h a v e b e e n d u e t o t h e T T T d a t a u s e d i n t h e m o d e l a n d a l s o t h e a s s u m p t i o n t h a t t h e t r a n s f o r m a t i o n , u n d e r n o n - e q u i l i b r i u m c o n d i t i o n s , b e g a n a t t h e e q u i l i -b r i u m 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 . T h o u g h t h e a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s w a s n o t g o o d , t h e w o r k d e m o n -s t r a t e d t h e f e a s i b i l i t y o f t h e u s e o f m o d e l s t o s t u d y p h a s e t r a n s f o r m a t i o n s . T h e i r s t u d y i n d i c a t e d t h e n e e d f o r t h e u s e o f a p p r o p r i a t e t r a n s f o r m a t i o n d a t a f o r s u c c e s s f u l m o d e l a p p l i c a t i o n . 2 . 7 S c o p e o f P r e s e n t W o r k T h e p r e s e n t w o r k w a s u n d e r t a k e n p r i m a r i l y t o d e v e l o p a m a t h e m a t i c a l m o d e l o f h e a t f l o w a n d t r a n s f o r m a t i o n i n e u t e c t o i d w i r e r o d s u s i n g c a r e f u l l y m e a s u r e d T T T a n d C C T d a t a . A l s o , i t w a s d e c i d e d t o c o n d u c t e x p e r i m e n t s t o v a l i -d a t e m o d e l c a l c u l a t i o n s . T h e s e w e r e t o b e a c c o m p l i s h e d b y m e a s u r i n g c e n t r e - l i n e t e m p e r a t u r e o f a i r - c o o l e d s t e e l r o d s 16 u n d e r c o n t r o l l e d c o n d i t i o n s o f c h e m i s t r y , g r a i n s i z e a n d c o o l i n g r a t e . S i n c e m o d e l c a l c u l a t i o n s w o u l d i n v o l v e t h e u s e o f a d d i t i v i t y , i t w a s d e c i d e d t o e x a m i n e t h e f u n d a m e n t a l q u a n t i t i e s i n v o l v e d , l i k e t h e n u c l e a t i o n a n d g r o w t h r a t e s . 3 7 T h i s w o r k w a s c a r r i e d o u t i n d e p e n d e n t l y , b y K u b a n , o n a m a t e r i a l v e r y s i m i l a r t o t h e o n e u s e d i n t h e p r e s e n t w o r k . T h e r e s u l t s f r o m t h e s t u d y b y K u b a n w e r e e x a m i n e d t o c h e c k w h e t h e r t h e c o n d i t i o n s n e e d e d f o r a d d i t i v i t y , l i k e i s o -k i n e t i c r a n g e , s i t e s a t u r a t i o n e t c . , d i d e x i s t i n t h e m a t e r i a l u n d e r s t u d y . A s m e n t i o n e d e a r l i e r , f r o m K u b a n ' s w o r k , t h e f o l l o w i n g o b s e r v a t i o n s w e r e m a d e : 1 ) N a n d G , f o r i s o t h e r m a l r e a c t i o n s , a r e c o n s t a n t u p t o a b o u t 2 0 % t r a n s f o r m a t i o n . i i ) N a n d G v a r y w i t h t e m p e r a t u r e . A n i s o k i n e t i c r a n g e , a s d e f i n e d b y A v r a m i , d o e s n o t e x i s t . i i i ) T h e r e i s n o e v i d e n c e o f s i t e s a t u r a t i o n , a s r e v e a l e d b y m e t a l 1 o g r a p h s . i v ) C a l c u l a t i o n s , u s i n g N a n d G v a l u e s a s f o u n d e x p e r i m e n t a l l y , r e v e a l t h a t t h e i s o t h e r m a l r e a c t i o n k i n e t i c s a r e m u c h s l o w e r t h a n w o u l d b e p r e d i c t e d b y t h e J o h n s o n - M e h l e q u a t i o n . v ) G r o w t h o f g r a i n s i s n o t t r u l y s p h e r i c a l , e s p e c i a l l y f o r T a r g e g r a i n s i z e , n o r i s n u c l e a t i o n r a n d o m . 1 7 S i n c e t h e s t e e l s u s e d i n . t h e p r e s e n t w o r k a n d t h a t o f K u b a n a r e v i r t u a l l y t h e s a m e ( p l a i n c a r b o n e u t e c t o i d ) , t h e s a m e k i n e t i c c o n d i t i o n s a p p l y i n b o t h c a s e s . S i n c e i t w a s f o u n d t h a t t h e c o n d i t i o n s r e q u i r e d f o r t h e a d d i t i v i t y r u l e t o h o l d , t h e i s o k i n e t i c r a n g e a n d s i t e s a t u r a t i o n , d i d n o t e x i s t , b u t t h a t t h e m o d e l c a l c u l a t i o n s u s i n g a d d i -t i v i t y a g r e e d w e l l w i t h e x p e r i m e n t a l r e s u l t s , i t w a s d e -c i d e d t o r e - e x a m i n e t h e c o n d i t i o n s n e e d e d f o r a d d i t i v i t y . I n a d d i t i o n , a l t e r n a t i v e c o n d i t i o n s s a t i s f y i n g t h e a d d i t i v i t y r u l e w e r e a l s o i n v e s t i g a t e d . A s a r e s u l t , a n e w c o n d i t i o n f o r a d d i t i v i t y , a s u f f i c i e n t c o n d i t i o n , h a s b e e n p r o p o s e d t o e x p l a i n t h e s u c c e s s f u l a p p l i c a t i o n o f a d d i t i v i t y i n t h e p r e s e n t c o n t e x t . F i n a l l y s i n c e i t i s e x p e r i m e n t a l l y m o r e d i f f i c u l t t o o b t a i n T T T d a t a t h a n C C T , d u e t o l i m i t a t i o n s o f t h e e x p e r i m e n t a l a p p a r a t u s , i t w a s d e c i d e d t o i n v e s t i g a t e t h e p o s s i b i l i t y o f d e v i s i n g a s i m p l e m a t h e m a t i c a l p r o c e d u r e t o d e r i v e T T T f r o m C C T . 2.8 O b j e c t i v e s T h e o b j e c t i v e s o f t h i s s t u d y c a n b e s u m m a r i z e d a s f o l l o w s : i ) T o d e v e l o p a p r o c e d u r e f o r q u a n t i t a t i v e p r e -d i c t i o n o f m e c h a n i c a l p r o p e r t i e s o f p l a i n c a r b o n s t e e l r o d s c o n t r o l c o o l e d i n a S t e l m o r - t y p e p r o c e s s . 1 8 i i ) T o d e v e l o p a c o m p u t e r - b a s e d m a t h e m a t i c a l m o d e l t o c a l c u l a t e t h e h e a t f l o w s a n d a u s t e n i t e -p e a r l i t e r e a c t i o n k i n e t i c s i n a p l a i n c a r b o n e u t e c t o i d s t e e l r o d . i i i ) T o t e s t t h e a d e q u a c y o f t h e m a t h e m a t i c a l m o d e l b y c o m p a r i n g m o d e l p r e d i c t i o n s w i t h m e a s u r e -m e n t s o f t h e c e n t r e - l i n e t e m p e r a t u r e o f r o d s u n d e r d i f f e r e n t c o o l i n g c o n d i t i o n s . i v ) T o e x a m i n e t h e t h e o r y o f a d d i t i v i t y a n d e x p l a i n i t s a p p l i c a b i l i t y t o t h e k i n e t i c s o f t h e e x p e r i -m e n t a l l y d e t e r m i n e d a u s t e n i t e - p e a r l i t e r e a c t i o n . v ) T o d e v i s e a p r o c e d u r e f o r c a l c u l a t i n g T T T f r o m C C T d a t a . v i ) T o u s e t h e m o d e l t o e x a m i n e t h e e f f e c t o f s e g r e -g a t i o n o n r e a c t i o n k i n e t i c s a n d t r a n s f o r m a t i o n b e h a v i o u r i n p l a i n c a r b o n e u t e e t o i d s t e e l r o d s . v i i ) T o p r e d i c t m e c h a n i c a l p r o p e r t i e s b y u s i n g t h e i n f o r m a t i o n g e n e r a t e d b y t h e . m o d e l . 1 9 C h a p t e r 3 T H E O R Y OF A D D I T I V E R E A C T I O N S A s d e s c r i b e d i n C h a p t e r 2 , t h e c o n d i t i o n s n e c e s s a r y f o r a d d i t i v i t y t o h o l d f o r t h e a u s t e n i t e - p e a r l i t e r e a c t i o n , i . e . A v r a m i ' s i s o k i n e t i c r a n g e a n d C a h n ' s e a r l y s i t e s a t u r a -t i o n , d o n o t o b t a i n f o r t h e r e a c t i o n s o b s e r v e d i n t h e p r e -s e n t w o r k . H o w e v e r m o d e l c a l c u l a t i o n s u s i n g a d d i t i v i t y ( d e s c r i b e d i n C h a p t e r s 5 a n d 6 ) , s h o w g o o d a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s . T h e r e f o r e i t i s n e c e s s a r y t o e x a m i n e a l t e r n a t e c o n d i t i o n s u n d e r w h i c h t h e a d d i t i v i t y p r i n c i p l e w i l l b e c o m e a p p l i c a b l e . I n t h i s c h a p t e r , t w o n e w s u f f i c i e n t c o n d i t i o n s f o r a d d i t i v i t y a r e e x a m i n e d . T h e s e a r e : i ) t h e a d d i t i v i t y r a n g e i i ) e f f e c t i v e s i t e s a t u r a t i o n . T h e s e t w o c o n d i t i o n s h a v e b e e n d e r i v e d a f t e r a c a r e f u l t h e o r e t i c a l a n a l y s i s o f t h e f u n d a m e n t a l a s p e c t s o f r e a c t i o n k i n e t i c s - t h e n u c l e a t i o n r a t e a n d t h e g r o w t h r a t e . T h e a d d i t i v i t y r a n g e , a s w i l l b e s h o w n i n S e c t i o n 3 . 1 , i s a n e x t e n s i o n o f t h e i s o k i n e t i c r a n g e . T h e e f f e c t i v e s i t e s a t u r a t i o n c r i t e r i o n , d e s c r i b e d i n S e c t i o n 3 . 2 , i s s i m i l a r t o C a h n ' s e a r l y s i t e s a t u r a t i o n p r i n c i p l e . T h e d e r i v a t i o n 2 0 o f t h e e f f e c t i v e s i t e s a t u r a t i o n c r i t e r i o n h a s b e e n d o n e i n t h r e e s t a g e s : i ) I t i s f i r s t d e r i v e d f o r a h o m o g e n e o u s r e a c t i o n w i t h c o n s t a n t N a n d 6 ( s e c t i o n 3 . 2 . 1 ) . i i ) I t i s e x t e n d e d t o c o v e r h o m o g e n e o u s r e a c t i o n s w i t h a c o n s t a n t G a n d a v a r i a b l e n u c l e a t i o n r a t e , w h i c h i n c l u d e s ( i ) a s a s p e c i a l c a s e ( s e c t i o n 3 . 2 . 2 ) . i i i ) I t i s t h e n d e r i v e d f o r a h e t e r o g e n e o u s r e a c t i o n w i t h c o n s t a n t N a n d G . I t i s a l s o i n d i c a t e d t h a t t h e r e s u l t s f o r a h e t e r o g e n e o u s r e a c t i o n w i t h a v a r i a b l e n u c l e a t i o n r a t e a n d a c o n s t a n t G w o u l d y i e l d t h e s a m e r e s u l t s a s o b t a i n e d i n ( i i ) ( s e c t i o n 3 . 2 . 3 ) . T h e a p p l i c a b i l i t y o f t h e c r i t e r i o n t o t h e p r e s e n t w o r k i s d e s c r i b e d a n d b y u s i n g e x p e r i m e n t a l r e s u l t s , i t h a s b e e n s h o w n t h a t t h e r e a c t i o n s e n c o u n t e r e d i n t h e p r e s e n t w o r k a r e a d d i t i v e ( s e c t i o n 3 . 3 ) . F i n a l l y , a n e w m e t h o d h a s b e e n d e v i s e d , u s i n g a d d i t i v -i t y , t o d e r i v e T T T d a t a f r o m C C T ( s e c t i o n 3 . 4 ) . A s h o r t s u m m a r y o f t h e w o r k d o n e i n t h i s r e g a r d i s a l s o i n c l u d e d i n t h i s s e c t i o n t o d e s c r i b e t h e c o n t e x t a n d r e l e v a n c e o f t h i s p r o c e d u r e . I t m u s t b e p o i n t e d o u t t h a t t h e d e r i v a t i o n s l e a d i n g t o t h e a d d i t i v i t y r a n g e , e f f e c t i v e s i t e s a t u r a t i o n a n d t h e m e t h o d o f o b t a i n i n g t ^ y _ J J J f r o m a r e o r i g i n a l a n d m u s t b e c o n s i d e r e d a s f r e s h c o n t r i b u t i o n s t o k n o w l e d g e i n t h i s f i e l d . 3 . 1 R e a c t i o n K i n e t i c s i n t h e A d d i t i v i t y R a n g e 21 A s p e r A v r a m i , t h e e x t e n d e d v o l u m e f r a c t i o n t r a n s -f o r m e d , V , f o r a n u c l e a t i o n a n d g r o w t h r e a c t i o n i s : w h e r e V e x = 0 / N 4 ( T \" z ) 3 e _ Z d z ( 3 - 1 ) 0 4TT s h a p e f a c t o r (= - y f o r s p h e r i c a l p a r t i c l e g r o w t h ) H = n u m b e r o f g e r m n u c l e i a t t h e s t a r t o f t r a n s f o r m a t i o n g = G / N x = c h a r a c t e r i s t i c t i m e S i n c e N i s i n d e p e n d e n t o f t i m e , T V e x = a N V ^ ( T \" z ) 3 e \" Z d z ( 3 - 2 ) 0 I f N a n d G a r e f u n c t i o n s o f t e m p e r a t u r e a l o n e , t h e n V e x = 0 N 4 ®3f (x_z)3 e\"Z dz (3'3) 0 • 2 2 On i n t e g r a t i n g E q . ( 3 . 3 ) a n d c o r r e c t i n g f o r i m p i n g e m e n t , V = i - e x p ( - V e x ) 2 1 ( 3 . 4 ) w h e r e V = r e a l v o l u m e f r a c t i o n t r a n s f o r m e d , we o b t a i n , V = 1 - e x p ( - b t n ) 2 1 ( 3 . 5 ) w h e r e ri = c o n s t a n t b = f u n c t i o n o f t e m p e r a t u r e . E q . ( 3 . 5 ) i s t h e A v r a m i e q u a t i o n w h i c h h a s b e e n s h o w n i n ( A p p e n d i x 1 ) t o d e s c r i b e t h e k i n e t i c s o f a d d i t i v e r e a c t i o n s . T h e t e m p e r a t u r e r a n g e o v e r w h i c h E q . ( 3 . 5 ) h o l d s i s t h e \" a d d i t i v i t y r a n g e \" . T h e J o h n s o n - M e h l e q u a t i o n ( E q . 2 . 1 ) i s a s p e c i f i c c a s e o f t h e m o r e g e n e r a l A v r a m i e q u a t i o n o f r e a c t i o n k i n e t i c s . F o r a n i s o t h e r m a l r e a c t i o n , f o r w h i c h t h e r e i s e v i d e n c e t h a t N a n d G a r e c o n s t a n t f o r s o m e n u c l e a t i o n a n d g r o w t h 36 r e a c t i o n s , , t h e J o h n s o n - M e h l e q u a t i o n c a n b e a p p l i e d t o s t u d y t h e k i n e t i c s . B u t b y u s i n g t h e c o n c e p t o f t h e a d d i - ' t i v i t y r a n g e , i t c a n b e s h o w n t h a t t h e J o h n s o n - M e h l e q u a -t i o n , w i t h s l i g h t m o d i f i c a t i o n s , c a n b e u s e d t o d e s c r i b e t h e k i n e t i c s o f n o n - i s o t h e r m a l r e a c t i o n s w h i c h a r e a d d i t i v e . 2 3 C o n s i d e r a n o n - i s o t h e r m a l r e a c t i o n ( a n u c l e a t i o n a n d g r o w t h r e a c t i o n ) o c c u r r i n g b e t w e e n t h e t e m p e r a t u r e s TQ a n d T^QQ. I n t h i s t e m p e r a t u r e r a n g e , l e t N(T) = n u c l e a t i o n r a t e ( a f u n c t i o n o f t e m p e r a t u r e a l o n e ) G(T) = g r o w t h r a t e ( a f u n c t i o n o f t e m p e r a t u r e a l o n e ) = r e a l v o l u m e f r a c t i o n t r a n s f o r m e d a t t i m e ' t ' . T • To v - 0 T - T100 V e x \" 0 T t T t + d t v - 1 I 1 t = t D u r i n g t h e i n f i n i t e s i m a l l y s m a l l t i m e s t e p ' d t ' , t h e n u m b e r o f n u c l e i t h a t n u c l e a t e = N ' ( T ) d t . N ' ( T ) = t e m p e r a t u r e a v e r a g e d n u c l e a t i o n r a t e o v e r t h e t i m e i n t e r v a l d t C o n s i d e r t h e g r o w t h o f o n e n u c l e u s w h i c h s t a r t s i t s g r o w t h d u r i n g d t . T h e e x t e n d e d v o l u m e o f t h i s n u c l e u s a t t i m e ' t - j i s : T l 0 0 3 2 4 V e x = { 1 f G ( T ) d T ( t 1 0 Q - t)j ( 3 . 6 ) 1 1 0 0 \" lt.J J 1 ' t T t w h e r e : ^ 1 0 0 Tioo - T G ( T ) d T T . i s t h e t e m p e r a t u r e a v e r a g e d g r o w t h r a t e b e t w e e n t i m e s t a n d t l 0 0 * H e n c e t h e v o l u m e o f a l l n u c l e i n u c l e a t i n g b e t w e e n t a n d t + d t a t t - | QQ . i s : t + d t T l 0 0 3 T t + d t \" / f e f 7 ^ / « T ) d T ( . 1 0 0 - . } { J F / . O l , d T j ( . t ( 3 . 7 ) t \" T t T t T h e r e f o r e , t h e t o t a l e x t e n d e d v o l u m e o f g r o w t h o f a l l n u c l e i n u c l e a t i n g b e t w e e n t Q a n d t ^ g a t t i m e t ^ Q 0 i s : ^ 0 0 T 1 0 0 3 T 1 0 0 ^° = a f l ^ r G ( T ) d T ( tioo - 4 f r — - f ^ ( T ) d i d t ; ( 3 . 8 ) J u 'ioo\"'t / J 1 1ioo\"'o^ J to To To L e t T 1 Q 0 G ^ T ) = ]- J G ( T ) d T ( 3 . 9 ) T l 0 0 \" T 0 T 0 1 , 0 0 2 5 N,(T) = 5 f N(T)dT (3 .10) T 1 0 0 \" T 0 * T 0 N - j ( T ) a n d G - j ( T ) a r e t h e t e m p e r a t u r e a v e r a g e d g r o w t h r a t e s f o r t h e r e a c t i o n . T h e n , E q . ( 3 . 8 ) c a n b e w r i t t e n a s t 1 0 0 V e x ° ° = a J G 1 ( T ) ' V T ) * ( t 1 0 0 _ t ) 3 d t { 3 J 1 ) G-^ CT) . . a n d , N^'(T.) a r e : f u n c t i o n s i n d e p e n d e n t o f t i m e . T h i s i s i l l u s t r a t e d i n A p p e n d i x 3 . B e c a u s e o f t h i s i n d e p e n d e n c e , G ^ ( T ) a n d N - j ( T ) c a n b e t a k e n o u t o f t h e i n t e g r a l i n E q . ( 3 . 1 1 ) t l 0 0 •'• V Ix°° = a j G l ( T ) N l ( T ) ^ ( t 1 0 0 \" t ) 3 d t ( 3 ' 1 2 ) C o n s i d e r i n g t h i s t r a n s f o r m a t i o n t o h a v e o c c u r r e d i n a u n i t v o l u m e , a t a n y t i m e \" T \" d u r i n g t h e r e a c t i o n , V e T x = a - G 3 ( T ) - N T ( T ) - f l*~*)3 d t <3-13) w h e r e T T G ( T ) = - f G(T) dT ( 3 . 1 4 ) T - T n J T 0 To 2 6 T T N ( T ) = — ! f N ( T ) d T ( 3 . 1 5 ) T - T n J T 0 • • Vj[ = a G 3 ( T ) N 3 ( T ) ^ - ( 3 . 1 6 ) F o r s p h e r i c a l p a r t i c l e g r o w t h , V e x = ~ N T ( T ) G ? ( T ) t 4 ( 3 ' 1 7 > 3 S i n c e V * = 1 - e x p | - V * x J . ( 3 . 1 8 ) . \" . V * = 1 - e x p ^ N t ( T ) • G j j ( T ) t 4 j . ( 3 . 1 9 ) F o r a n i s o t h e r m a l r e a c t i o n , E q . ( 3 . 1 9 ) c a n b e w r i t t e n a s V * = 1 - e x p (- - N G 3 t 4 ) ( 3 . 2 0 ) 3 w h i c h i s t h e J o h n s o n - M e h l e q u a t i o n . E q . ( 3 . 2 0 ) i s o f t h e s a m e f o r m a s t h e A v r a m i e q u a t i o n w i t h b = ^ N t ( T ) G 3 ( T ) ( 3 . 2 1 ) n = 4 ( 3 . 2 2 ) H e n c e E q . ( 3 . 1 9 ) a l s o d e s c r i b e s t h e k i n e t i c s o f a d d i t i v e c > 2 7 r e a c t i o n s . I t h o l d s u n d e r t h e f o l l o w i n g a s s u m p t i o n s : i ) N a n d G a r e f u n c t i o n s o f t e m p e r a t u r e a l o n e f o r n o n - i s o t h e r m a l r e a c t i o n s . i i ) R a n d o m n u c l e a t i o n . i i i ) S p h e r i c a l p a r t i c l e g r o w t h u n t i l i m p i n g e m e n t . T h e t e m p e r a t u r e r a n g e f o r w h i c h E q . ( 3 . 1 9 ) h o l d s i s t h e \" a d d i t i v i t y r a n g e \" . 3 . 1 . 1 D e f i n i t i o n o f A d d i t i v i t y R a n g e T h e a d d i t i v i t y r a n g e i s t h e t e m p e r a t u r e r a n g e , f o r a n u c l e a t i o n a n d g r o w t h r e a c t i o n , i n w h i c h t h e n u c l e a -t i o n a n d g r o w t h r a t e s a r e d e p e n d e n t o n l y o n t h e t e m p e r a t u r e . R e a c t i o n s o c c u r r i n g i n s u c h a r a n g e o b e y S c h e i l ' s a d d i t i v i t y p r i n c i p l e . 3 . 2 K i n e t i c s o f N u c l e a t i o n a n d G r o w t h R e a c t i o n s a n d t h e C r i t e r i o n o f E f f e c t i v e S i t e S a t u r a t i o n I s o t h e r m a l r e a c t i o n s w h i c h d o n o t f o l l o w t h e J o h n s o n -M e h l e q u a t i o n d u e t o t h e v i o l a t i o n o f a n y o n e o r a l l o f t h e a s s u m p t i o n s m a d e i n d e r i v i n g t h e e q u a t i o n c a n b e t e r m e d n o n -h o m o g e n e o u s r e a c t i o n s . S u c h r e a c t i o n s a r e n o t c o v e r e d b y E q . ( 3 . 1 9 ) . H e n c e i t i s n e c e s s a r y t o d e r i v e a s u f f i c i e n t c o n d i t i o n o f a d d i t i v i t y f o r s u c h r e a c t i o n s . S i n c e t h e 2 8 a u s t e n i t e - p e a r l i t e r e a c t i o n , s t u d i e d i n t h e p r e s e n t w o r k , i s h e t e r o g e n e o u s , i t b e c o m e s a l l t h e m o r e e s s e n t i a l . 1 I n o r d e r t o s t u d y h e t e r o g e n e o u s r e a c t i o n k i n e t i c s , i t i s i m p o r t a n t t o u n d e r s t a n d t h e k i n e t i c s o f h o m o g e n e o u s r e a c t i o n s 3 . 2 . 1 K i n e t i c s o f I s o t h e r m a l H o m o g e n e o u s N u c l e a t i o n a n d G r o w t h R e a c t i o n s d't t = 0 t = t t = t ] t = t 2 A s s u m p t i o n s : i ) C o n s t a n t N a n d G i i ) S p h e r i c a l p a r t i c l e g r o w t h i i i ) R a n d o m n u c l e a t i o n . N u m b e r o f n u c l e i f o r m e d u p t o t i m e t = N t N u m b e r o f n u c l e i n u c l e a t i n g i n a n i n f i n i t e s i m a l l y s m a l l t i m e i n t e r v a l d t T h e e x t e n d e d g r o w t h v o l u m e o f t h e s e ^ N 3 n u c l e i a t t = t N d t N d t v i •' 1 \"t o / | ( t r t ) ) ' T h e . e x t e n d e d ' g r o w t h : v o l u m e : ' o f a l l n u c l e i ' , n u c l e a t i n g d u r i n g t h e t i m e i n t e r v a l 0 t o t - j i s • 3 . , / , j . \\ 3 I OJG°N ( t , - t r dt 3 4 i i ^ ^ i ' 2 9 T h e t o t a l n u m b e r o f s i t e s c o n s u m e d d u e t o n u c l e a t i o n a n d t h e v o l u m e t r i c g r o w t h b y t h e e n d o f t i m e t = t - j i s : 3 4 t , a G N t ? = . l i t , + . _ _ L , , I 0 ( 3 . 2 4 ) w h e r e I Q = I n i t i a l n u m b e r o f a v a i l a b l e s i t e s f o r n u c l e a t i o n . T h e r a t e o f s i t e c o n s u m p t i o n i s ( o b t a i n e d b y d i f f e r e n t i a t i n g E q . ( 3 . 2 4 ) ) S * 1 = N + a N G 3 t 3 I n ( 3 . 2 5 ) e x . 1 0 C o n s i d e r i n g a u n i t v o l u m e o f m a t e r i a l , f r o m E q . ( 3 . 2 3 ) , v t = a N G 3 . ; t 4 ( 3 ^ 2 6 ) e x A l s o , S * = N t + I n - V * ( 3 . 2 7 ) e x 0 e x • t S = N + I n • V * ( 3 . 2 8 ) e x 0 e x F o r s p h e r i c a l g r o w t h (a = , v t = I N G 3 t 4 ( 3 . 2 9 ) e x 3 S i n e e V t = 1 - e x p ( - V ^ ) ( 3 . 3 0 ) S * = 1 - e x p ( - S * x ) ( 3 . 3 1 ) 3 0 We h a v e , 1 - e x p ( - | N G 3 t 4 ) ( . 3 . 3 2 ) S * o , l - e x p - ( N t + I n • V t ) r 0 e x ( 3 . 3 3 ) N + ! n — N G 3 t 3 • e x p - ( N t + I • V * ) ( 3 . 3 4 ) E q . ( 3 . 3 2 ) t o E q . ( 3 . 3 4 ) a r e t h e t h r e e b a s i c e q u a t i o n s o f h o m o g e n e o u s i s o t h e r m a l r e a c t i o n k i n e t i c s . E q . ( 3 . 3 2 ) i s t h e J o h n s o n - M e h l e q u a t i o n , w h i c h i s t h e v o l u m e f r a c t i o n t r a n s f o r m e d a t a n y t i m e t d u r i n g t h e r e a c t i o n . T h i s v o l u m e g r o w t h i s d u e t o t h e g r o w t h o f n u c l e i n u c l e a t e d t h r o u g h o u t t h e r e a c t i o n . I t i s a l s o p o s s i b l e t o c a l c u l a t e t h e v o l u m e -t r i c g r o w t h o f n u c l e i n u c l e a t i n g d u r i n g a s p e c i f i c t i m e i n -t e r v a l i n t h e r e a c t i o n . I t i s u s e f u l t o c a l c u l a t e t h i s q u a n t i t y b e c a u s e i t w i l l i n d i c a t e t h e r e l a t i v e i m p o r t a n c e o f t h e g r o w t h o f t h e s e n u c l e i t o t h e t o t a l g r o w t h v o l u m e . I f t h e v o l u m e c o n t r i b u t i o n d u e t o n u c l e i n u c l e a t i n g v e r y e a r l y i n t h e r e a c t i o n t o t h e t o t a l v o l u m e t r i c g r o w t h a t a v e r y l a t e s t a g e i n t h e r e a c t i o n i s s i g n i f i c a n t l y h i g h , t h e n i t c a n b e p o s t u l a t e d t h a t t h e r e a c t i o n i s e s s e n t i a l l y g r o w t h d o m i n a t e d . A s b e f o r e , t h e e x t e n d e d v o l u m e o f g r o w t h f r o m n u c l e i n u c l e a t i n g b e t w e e n t i m e s 0 a n d t - , a t t i m e t 0 i s : t 1 31 V!x ! = I 0 G 3 ( t ? - t ) 3 N d t (3.35) e x o / t 1 J d 0 = °—: A - ( t 2 - t l ) 4 (3.36) A l s o , t h e t o t a l e x t e n d e d v o l u m e o f g r o w t h a t t ^ i s : \\ \\ ^a^lA. ( 3 . 3 7 ) A m o r e g e n e r a l f o r m o f E q . (3.35) i s t . J _ / _p3,,/ i\\3 - J a la. V v = J G, N(x-1) d t (3.38) e x t . / t . i J t . l = 0 N S ! ( t _ t , * . ( T . t ,4 ( 3 . 3 9 ) H e n c e , a t a n y t i m e t d u r i n g t h e r e a c t i o n , t h e f r a c t i o n a l v o l u m e c o n t r i b u t e d b y n u c l e i n u c l e a t i n g b e t w e e n t h e t i m e s t ^ a n d t . t o t h e t o t a l e x t e n d e d v o l u m e t r a n s f o r m e d i s : J e X t . / t . ( t - t . ) 4 - ( t - t ) 4 t 1 J = ] r ^ — ( 3 . 4 0 ) t e x I f 3 2 t h e n , f r o m E q . ( 3 . 4 1 ) t ' 2 0 . _ 9 0 v 9 0 2 0 ' 9 0 e x 0 / t t 4 - f t - f ) 4 ^ 9 0 t 4 e x r 9 0 ( 3 . 4 1 ) Now i f , f o r t h i s r e a c t i o n , t h e g r o w t h r a t e i s t h e d o m i n -a n t f e a t u r e , ( a n d c o n s e q u e n t l y , t h e n u c l e a t i o n r a t e i s r e -l a t i v e l y u n i m p o r t a n t ) t h e n i t c a n b e e x p e c t e d t h a t t h e f r a c -i o n a l v o l u m e c o n t r i b u t e d b y t h e n u c l e i n u c l e a t i n g i n t h e v e r y e a r l y s t a g e s o f t h e r e a c t i o n ( s a y , u p t o 2 0 % t r a n s f o r m a t i o n ) t o t h e v o l u m e t r a n s f o r m e d a t t h e f i n a l s t a g e s o f t h e r e a c t i o n ( s a y , 9 0 % t r a n s f o r m a t i o n ) m u s t b e c l o s e t o u n i t y . I f t h e r a t i o i s c l o s e t o u n i t y , s a y 0 . 8 5 , t h e n t h e n u c l e a t i o n e v e n t a f t e r 2 0 % t r a n s f o r m a t i o n b e c o m e s u n i m p o r t a n t a n d t h e r e a c t i o n i s g r o w t h r a t e d o m i n a t e d . S i n c e i t i s k n o w n t h a t t h e g r o w t h r a t e i s a f u n c t i o n o f t e m p e r a t u r e o n l y f o r a u s t e n i t e - p e a r 1 i t e r e a c t i o n s , t h e s e r e a c t i o n s c a n b e c o n s i d e r e d a d d i t i v e . H e n c e , i f y ^ 9 0 e \" ° / t 2 0 = ' 9 0 - ( ^ 9 Q - t 2 0 ) : > _ Q > 8 5 ( 3 . 4 2 ) V 9 0 *9Q e x t 2 Q = > 0 . 3 8 t g Q ( 3 . 4 3 ) 3 3 h o l d s f o r t h e r e a c t i o n , i t s h o u l d b e a d d i t i v e . I t s h o u l d b e n o t e d t h a t E q . ( 3 . 4 2 ) i n v o l v e s t h e r a t i o o f e x t e n d e d v o l u m e s I t a l s o c a n b e s h o w n , a s d e s c r i b e d b e l o w , t h a t *90 /go 0 / t 2 0 e x 0 / t S i n c e a n d A l s o , > — ^ - ( 3 . 4 4 ) v t 9 0 V* 9 0 e x V * 9 0 = 1 - e x p ( - V ^ 9 0 ) ( 3 . 4 5 ) V 0 / t = 1 - e x p - V ^ 9 0 ( 3 . 4 6 ) U / t 2 0 ° / t 2 0 U / t 2 0 0 / t 9 n - t — 7. t 2 0 ( 3 . 4 7 ) 9 0 , , , 9 0 V 1 - e x p - V e x V * 9 0 = 0 . 9 0 ( 3 . 4 8 ) 1 - e x p ( - V g 9 0 ) = 0 . 9 0 ( 3 . 4 9 ) V ^ 9 Q = I n 1 0 ( 3 . 5 0 e x 3 4 F r o m E q . ( . 3 . 4 3 ) , V I x ° > Q . 8 5 \\ltg0 ( 3 . 5 1 ) e x o / t 2 Q e x . . E q . ( 3 . 4 8 ) , c o m b i n e d w i t h E q . ( 3 . 4 4 ) , c a n be w r i t t e n a s : 0 / t 2 0 > 1 - exp ( - 0 . 8 5 I n 1 0 ) V * 9 0 \" 0 . 9 1 ° - 9 5 ( 3 . 5 2 ) T h e r e l a t i o n s h i p b e t w e e n t h e r a t i o s o f e x t e n d e d v o l u m e s a n d r e a l v o l u m e s i s s h o w n i n F i g . ( 3 . 1 ) a n d F i g . ( 3 . 2 ) . T h e r e -f o r e , f r o m E q . ( 3 . 5 2 ) , i t c a n b e c o n c l u d e d t h a t t h e r a t i o o f t h e r e a l v o l u m e s i s g r e a t e r t h a n t h e r a t i o o f t h e e x t e n d e d v o l u m e s . T h u s , E q . ( 3 . 4 3 ) , w h i c h i s t h e \" E f f e c t i v e S i t e S a t u r a t i o n \" c o n d i t i o n f o r c o n s t a n t n u c l e a t i o n a n d g r o w t h r a t e h o m o g e n e o u s r e a c t i o n s , c a n b e u s e d t o s t u d y t h e a p p l i c a b i l i t y o f a d d i t i v i t y t o s u c h r e a c t i o n s . T h e c r i t e r i o n i s s o c a l l e d b e c a u s e t h e n a t u r e o f c o n c l u s i o n s d e r i v e d b y i t a r e e s s e n t i -a l l y s i m i l a r t o C a h n ' s s i t e s a t u r a t i o n . B u t i t d o e s n o t r e q u i r e t h a t n u c l e a t i o n s i t e s s a t u r a t e p h y s i c a l l y d u r i n g t h e r e a c t i o n . A l s o , t h e c r i t e r i o n r e q u i r e s o n l y a k n o w l e d g e o f t ^ Q a n d t g Q f o r s u c h r e a c t i o n s . H e n c e i t i s v e r y e a s y t o u s e . T h e o n l y r e s t r i c t i o n i n u s i n g t h e e f f e c t i v e s i t e F i g . 3 . 1 R e l a t i o n s h i p b e t w e e n t h e r a t i o s o f e x t e n d e d v o l u m e s a n d r e a l v o l u m e s . F i g u r e 3 . 2 R e l a t i o n s h i p b e t w e e n t h e s a t u r a t i o n r a t i o a n d t h e e x t e n d e d v o l u m e s r a t i o s . e f f e c t i v e s i t e r e a l v o l u m e s a n d 3 7 s a t u r a t i o n c o n d i t i o n , a s d e r i v e d i n E q . ( 3 . 4 3 ) , i s t h a t N a n d G b e c o n s t a n t . H o w e v e r , i t c a n b e e x t e n d e d t o i n c l u d e r e -a c t i o n s i n w h i c h N v a r i e s w i t h t i m e . 3 . 2 . 2 E f f e c t i v e S i t e S a t u r a t i o n C r i t e r i o n f o r V a r i a b l e N u c l e a t i o n R a t e I s o t h e r m a l R e a c t i o n s I f we c o n s i d e r a r e a c t i o n s u c h t h a t , N = N ( t ) ( t > t . . ) ( 3 . 5 3 ) A s b e f o r e , u p t o t t V ( 3 . 5 4 ) A l s o , ( 3 . 5 5 ) t . ( 3 . 5 6 ) 3 8 C o n s i d e r a d e c r e a s i n g n u c l e a t i o n r a t e , m N ( t ) N 0 ( 3 . 5 7 ) w h i c h i s t h e m o s t p r o b a b l e f o r m o f e q u a t i o n b a s e d o n e v i d e n c e i n t h e l i t e r a t u r e . M o r e o v e r , t h e a p p l i c a t i o n o f L e C h a t e l i e r ' s p r i n c i p l e p o i n t s t o t h e p o s s i b i l i t y t h a t t h e n u c l e a t i o n r a t e s h o u l d d e c r e a s e w i t h t i m e d u e t o t h e \" b a c k p r e s s u r e \" e x e r t e d b y t h e t r a n s f o r m e d p r o c u c t . E q . ( 3 . 5 7 ) a l s o i m p l i e s t h a t t h e n u c l e a t i o n r a t e i s n e g l i g i b l y s m a l l a f t e r t g g , w h i c h a g a i n i s a r e a s o n a b l e a s s u m p t i o n , ' m ' i s a c o - e f f i c i e n t w h i c h w i l l d e p e n d o n f a c t o r s l i k e c o m p o s i t i o n , t e m p e r a t u r e e t c . S u b s t i t u t i n g E q . ( 3 . 5 7 ) f o r N ( t ) i n E q . ( 3 . 5 0 ) , we o b t a i n N ( 3 . 5 8 ) m+4 (3.59.) H e n c e , V e x 0 / t N ( 3 . 6 0 ) e x 3 9 t 4 - f t - t ) 4 * 9 0 ^ 9 0 V ( 3 . 6 : 1 ) 4 ( t . t ) m + 4 ^ : ,0nt4) : . 9 0 ( 9 0 V T h e e f f e c t i v e s i t e s a t u r a t i o n c r i t e r i o n i s ' 9 0 e x 0 / t . 0 . 8 5 ' 9 0 e x ( 3 . 6 2 ) ( 3 . 6 3 ) E q . ( 3 . 6 3 ) i s t h e g e n e r a l s t a t e m e n t o f t h e e f f e c t i v e s i t e s a t u r a t i o n c r i t e r i o n w h i c h i n c l u d e s E q . ( 3 . 4 3 ) . E q . ( 3 . 6 3 ) c o n s i d e r a b l y e x p a n d s t h e s c o p e o f r e a c t i o n s c o v e r e d b y t h e e f f e c t i v e s i t e s a t u r a t i o n c o n d i t i o n . H o w e v e r , i n p r a c t i c e , h o m o g e n e o u s r e a c t i o n s a r e n o t c o m m o n . T h i s i s e s p e c i a l l y t r u e o f t h e a u s t e n i t e - p e a r l i t e r e a c t i o n , w h i c h i s h e t e r o g e n e o u s . M i c r o g r a p h i c e v i d e n c e a l s o 3 7 e x i s t s t o s h o w t h a t g r a i n g r o w t h m a y n o t b e s p h e r i c a l . I n 3 7 t h e w o r k b y K u b a n , t h e r e a c t i o n k i n e t i c s c a l c u l a t e d b y u s i n g e x p e r i m e n t a l l y d e t e r m i n e d N a n d 6 i n t h e J o h n s o n - M e h l 4 0 e q u a t i o n f o r i s o t h e r m a l r e a c t i o n s , p r e d i c t e d m u c h f a s t e r r a t e s o f t r a n s f o r m a t i o n ! , t h a n t h o s e e x p e r i m e n t a l l y o b s e r v e d . T h u s , f o r t h e e f f e c t i v e s i t e s a t u r a t i o n c r i t e r i o n t o b e u s e -f u l f o r h e t e r o g e n e o u s r e a c t i o n s , i t m u s t b e e x t e n d e d t o c o v e r t h e c o n d i t i o n s o b t a i n e d i n s u c h r e a c t i o n s . 3 . 2 . 3 E f f e c t i v e S i t e S a t u r a t i o n C r i t e r i o n f o r H e t e r o g e n e o u s I s o t h e r m a l R e a c t i o n s T h e J o h n s o n - M e h l e q u a t i o n r e p r e s e n t s t h e k i n e t i c s o f h o m o g e n e o u s r e a c t i o n s . V H o m = 1 \" e x p ( \" 3 N g 3 t 4 ) ( 3 - 6 4 ) D u e t o h e t e r o g e n e i t y , t h e r e a c t i o n k i n e t i c s a r e s l o w e r t h a n t h a t p r e d i c t e d b y E q . ( 3 . 6 4 ) . T h e s e k i n e t i c s c a n b e r e p r e s e n t e d a s V H e t = 1 \" e x p ( \" b t l 1 ) ( 3 * 6 5 ) w h e r e b a n d n r e p r e s e n t t h e e f f e c t o f h e t e r o g e n e i t y . W e : c a n d e f i n e a f a c t o r , c a l l e d t h e \" I n h o m o g e n e i t y C o - e f f i c i e n t \" , a s V * I . = ( 3 . 6 6 ) v H e t I t i s n o t p o s s i b l e , a t t h i s s t a g e , t o s p e c u l a t e u p o n t h e p r e c i s e n a t u r e o f 1 ^ . V a l u e s o f 1^ c a l c u l a t e d f r o m 3 7 t h e w o r k o f K u b a n a r e s h o w n i n F i g s . 3 . 3 a n d 3 . 4 . Volume Fraction Transformed (%) F i g u r e 3 . 3 E f f e c t o f g r a i n s i z e a n d r e a c t i o n t e m p e r a -t u r e o n t h e I n h o m o g e n e i t y c o - e f f i c i e n t . 4 2 Volume Fraction Transformed (%) F i g . 3 . 4 E f f e c t o f g r a i n s i z e a n d r e a c t i o n t e m p e r a -t u r e o n t h e I n h o m o g e n e i t y c o - e f f i c i e n t . 4 3 T h e i n h o m o g e n e i t y e o - e f f i c i e n t , l^, h e l p s t o c h a r a c t e r -i z e t h e d e g r e e o f i n h o m o g e n e i t y o f a r e a c t i o n , b u t c o m p l i -c a t e s t h e d e r i v a t i o n o f a c r i t e r i o n o f t h e f o r m o f E q . ( 3 . 4 3 ) , I n t r o d u c i n g .1 i n a s l i g h t l y m o d i f i e d f o r m , a s h \" t , t h e \" h e t e r -o g e n e i t y c o - e f f i c i e n t \" , h e l p s a c c o m p l i s h t h i s o b j e c t i v e . v t 1 - . e x p - V * t = v H o m = - H o r n V H e t = 1 - e x p - H • V * ( 3 . 6 7 ) z e x H o m S i n c e • V V L = H . • - N G 3 t 4 = b t n ( 3 . 6 8 ) z e x H o m z 3 . ' • H = — — t \" \" 4 ( 3 . 6 9 ) | N G 3 A s f o r 1 ^ , i t i s d i f f i c u l t t o p r e d i c t t h e n a t u r e o f f o r d i f f e r e n t r e a c t i o n s . V a l u e s o f f o r t h e r e a c t i o n s s t u d i e d b y K u b a n a r e . s u m m a r i z e d i n T a b l e 3 . 1 . A s c a n b e s e e n f r o m t h i s t a b l e , t h e v a l u e o f i s c o n s t a n t f o r s o m e r e a c -t i o n s a n d v a r y i n g f o r o t h e r s . F o r a h e t e r o g e n e o u s r e a c t i o n w i t h c o n s t a n t N a n d G , we h a v e , 4 4 T a b l e 3.1 Summary o f H e t e r o g e n e i t y C o - e f f i c i e n t C a l c u l a t i o n s R e a c t i o n A u s t e n i t i s i n g Range o f A v e r a g e T e m p e r a t u r e T e m p e r a t u r e H t H t ( ° C ) ( ° ' C ) 640 800 0 : 0 4 t o 0 . 2 6 0 . 1 0 640 840 0 . 0 4 t o 0 . 4 8 0 . 1 5 640 9 5 0 0 . 0 7 t o 0 . 0 9 0 . 0 8 640 1100 0 . 0 3 t o 0 . 0 3 5 0 . 0 3 3 6 9 0 800 0 . 1 3 t o 0 . 2 3 0 . 1 7 6 9 0 840 0 . 0 0 1 8 t o 0 . 0 0 3 4 0 . 0 0 2 4 6 9 0 9 0 0 0 . 0 5 t o 0 . 0 2 7 0 . 1 7 6 9 0 950 1 . 4 t o 1 2 . 3 4 . 4 2 4 5 V«L = H t „ „ T N s 3 t 9 0 ( 3 - 7 0 ) ' H e t \" 9 0 3 V e x ° = H t 1 N ^ ^ g o - ^ O ^ 4 ( 3 ' 7 1 ) H e t . / t z90 3 a u ^ u ZZ0f 9 0 V « ! L \" H t o n 7 ^ . t S o - ^ W 4 ] ( 3 - 7 2 ) ' H e t n / . \" 9 0 3 u / t 2 0 S i n c e t h e e f f e c t i v e s i t e s a t u r a t i o n c o n d i t i o n i s : / g o e x o / t - > 0 . 8 5 / g o e x f o r a h e t e r o g e n e o u s r e a c t i o n , t 4 - ( t - t ) 4 - ™ [ 9 0 2 ° J > . . 0 . 8 5 ( 3 . 7 3 ) t 9 0 t 2 0 > 0 . 3 8 t g o ( 3 . 7 4 ) E q . ( 3 . 7 4 ) i s t h e s a m e a s E q . ( 3 . 4 3 ) . T h e s a m e d e r i v a -t i o n s c a n b e d o n e f o r a v a r i a b l e n u c l e a t i o n r a t e r e a c t i o n w i t h t h e s a m e r e s u l t s a s o b t a i n e d i n E q . ( 3 . 6 3 ) . T h u s , t h e e f f e c t i v e s i t e s a t u r a t i o n c r i t e r i o n r e m a i n s u n a l t e r e d f o r h o m o g e n e o u s , a n d h e t e r o g e n e o u s r e a c t i o n s . 4 6 3 . 3 V a l i d a t i o n o f t h e E f f e c t i v e S i t e S a t u r a t i o n C r i t e r i o n b y E x p e r i m e n t a l R e s u l t s T h e e f f e c t i v e s i t e s a t u r a t i o n c r i t e r i o n w a s a p p l i e d t o e x p e r i m e n t a l r e s u l t s t o c h e c k i t s v a l i d i t y . T h e m o d e l p r e -d i c t i o n s , u s i n g a d d i t i v i t y , s h o w v e r y g o o d a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s i n t h e p r e s e n t w o r k , i n d i c a t i n g t h a t a d d i t i v i t y h o l d s u n d e r t h e e x p e r i m e n t a l c o n d i t i o n s e n c o u n t e r -e d . T h e m a t e r i a l u s e d a n d e x p e r i m e n t a l c o n d i t i o n s a r e 3 7 v i r t u a l l y t h e s a m e i n t h e p r e s e n t w o r k a n d t h a t o f K u b a n a n d h e n c e c o m p a r a b l e i n t e r m s o f k i n e t i c b e h a v i o u r . T h e r e -s u l t s o f v o l u m e c o n t r i b u t i o n c a l c u l a t i o n s f o r e x p e r i m e n t s 3 7 c o n d u c t e d b y K u b a n a r e s u m m a r i z e d i n T a b l e 3 . 2 a n d s h o w v e r y g o o d a g r e e m e n t w i t h t h e r e s u l t e x p e c t e d f r o m E q . ( 3 . 7 4 ) . T a b l e 3 . 3 s h o w s t h e c a l c u l a t i o n o f t h e r a t i o f o r t h e 3 7 9 0 e x p e r i m e n t c o n d u c t e d b y K u b a n . T a b l e 3 . 4 s h o w s t h e s a m e r a t i o c a l c u l a t e d f o r e x p e r i m e n t s c o n d u c t e d i n t h e p r e s e n t 3 0 4 3 5 0 s t u d y a n d o t h e r s . ' ' T h e v a l u e s i n d i c a t e v e r y g o o d a g r e e m e n t w i t h t h e e f f e c t i v e s i t e s a t u r a t i o n c r i t e r i o n . T h e a b o v e r e s u l t s v a l i d a t e t h e u s e o f t h e e f f e c t i v e s i t e s a t u r a -t i o n c r i t e r i o n t o e n s u r e a d d i t i v i t y i n r e a c t i o n s . T h e y a l s o e s t a b l i s h a f i r m t h e o r e t i c a l f r a m e w o r k f o r u s i n g a d d i t i v i t y i n t h e m o d e l c a l c u l a t i o n s . 3 . 4 A p p l i c a t i o n o f A d d i t i v i t y t o D e r i v e T T T f r o m C C T b y t h e A d d i t i v i t y M e t h o d A n i m p o r t a n t a p p l i c a t i o n o f t h e a d d i t i v i t y r u l e i s t h e 4 7 T a b l e 3 . 2 : Summary o f E f f e c t i v e S i t e S a t u r a t i o n C a l c u l a t i o n s R e a c t i o n T e m p e r a t u r e ( ° C ) A u s t e n i t i s i n g T e m p e r a t u r e ( ° C ) V o l u m e C o n t r i b u t i o n * (%) E x t e n d e d V o l u m e R e a l V o l u m e 6 4 0 8 0 0 86 95 640 840 82 9 4 640 9 5 0 - 96 9 8 6 4 0 1100 97 99 6 9 0 8 0 0 94 9 8 6 9 0 840 93 97 6 9 0 8 9 9 88 96 6 9 0 9 5 0 85 95 * V o l u m e c o n t r i b u t e d by n u c l e i n u c l e a t i n g b e t w e e n 0 a n d 20% t r a n s f o r m a t i o n t o t h e t o t a l v o l u m e t r a n s f o r m e d a t 90% t r a n s f o r m a t i o n . T a b l e 3 . 3 . . E f f e c t i v e S i t e S a t u r a t i o n C r i t e r i o n V a l u e s 4 8 S t e e l C h e m i s t r y : 0 . 7 8 C E u t e c t o i d ( p l a i n c a r b o n ) R e a c t i o n A u s t e n i t i s i n g T e m p e r a t u r e T e m p e r a t u r e t 2 0 ^ 9 0 t 2 0 ( ° C ) ( ° C ) ( s ) ( s ) ^ 0 640 800 ( 9 . 1 ASTM) 3 . 2 2 8 . 3 8 0 . 3 8 640 840 ( 7 . 8 ASTM) 3 . 0 8 8 . 9 3 0 . 3 4 640 9 5 0 ( 7 . 3 ASTM) 6 . 8 1 2 . 7 0 . 5 3 6 4 0 1100 ( 3 ASTM) 3 1 . 7 6 5 5 . 1 4 0 . 5 8 6 9 0 800 ( 9 . 1 ASTM) 5 1 . 1 1 0 1 . 4 0 . 5 1 6 9 0 840 ( 7 . 8 ASTM) 119 2 4 3 0 . 4 9 6 9 0 9 0 0 ( 7 . 5 ASTM) 918 2 2 7 5 0 . 4 0 6 9 0 9 5 0 ( 7 . 3 ASTM) 847 2301 0 . 3 7 ( D a t a f r o m R e f e r e n c e 3 7 . ) 4 9 Table 3.jf Effective Site Saturation Criterion Values Experimental Conditions Reaction Temperature ( 6C) So (s) So (s) So So Reference No. 0.82 C Eutectoid Steel Austenitised at850°C for 5 mts 5-7 ASTM 660 650 630 615 603 32.4 13.5 5.6 3.25 3.15 72 25.9 9.8 5.4 5.4 0.45 0.52 0.57 0.62 0.58 * 0.78 C Eutectoid Steel Austenitised at 875°C for 30 mts 5.25 ASTM 500 540 600 4.20 4.8 6.4 5.5 6.5 10.0 0.76 0.74 0.64 43 0.80 C Eutectoid Steel Austenitised at 875°C for 30 mts 4.25 ASTM 630 650 690 8.0 23.0 700.0 20.0 42.0 1100.0 0.40 0.54 0.63 43 1.10 C Eutectoid 5 ASTM 0.57 C Eutectoid 5 ASTM 0.93 C Eutectoid 1 ASTM Al l Steels Austenitised at 875°C for 30 mts 662 691 689 4.7 80.0 35.0 6.5 200.0 46.0 0.72 0.40 0.75 43 SKD-6 715 670 95 340 200 830 0.475 0.410 30 SKS-5 Austenitised at 1100°C for 15 mts 632 622 612 601 31 24 19 16.5 55 42 34 30 0.56 0.57 0.56 0.55 50 * Present work. 5 0 d e t e r m i n a t i o n o f C C T f r o m T T T . I n t h e l i t e r a t u r e , s e v e r a l 15 16 a t t e m p t s h a v e b e e n m a d e i n t h i s d i r e c t i o n . ' T h e d e r i v a -t i o n o f C C T f r o m T T T i s p o s s i b l e o n l y a f t e r t h e e x p e r i m e n t a l d e t e r m i n a t i o n o f t h e T T T d a t a . B u t t h e e x p e r i m e n t a l d e t e r -m i n a t i o n o f T T T i s v e r y d i f f i c u l t . T o d e t e r m i n e t h e T T T , t h e s p e c i m e n m u s t b e c o o l e d f r o m . t h e a u s t e n i t i s i n g t e m p e r a -t u r e ( u s u a l l y a r o u n d 8 5 0 - 9 0 0 ° C ) . t o t h e i s o t h e r m a l t e s t t e m p e r a t u r e ( u s u a l l y a r o u n d 6 5 0 ° C ) i n a v e r y s h o r t p e r i o d o f t i m e , u s u a l l y a s e c o n d o r t w o . T h i s m u s t b e d o n e i n o r d e r t o e n s u r e t h a t t h e t r a n s f o r m a t i o n d o e s n o t b e g i n b e -f o r e t h e t e s t t e m p e r a t u r e i s r e a c h e d . T h i s c a l l s f o r c o o l -i n g r a t e s o f t h e o r d e r o f 1 0 0 t o 1 5 0 ° C / s o r m o r e w h i c h a r e v e r y d i f f i c u l t t o a c h i e v e . I n a s a l t p o t , w h i c h h a s b e e n t h e m e d i u m u s e d b y s e v e r a l w o r k e r s i n t h e l i t e r a t u r e t o a c h i e v e s u c h c o o l i n g r a t e s , i t i s i m p o s s i b l e t o d o s o . H e n c e t h i s i n t r o d u c e s a n . e r r o r i n t h e e x p e r i m e n t a l m e a s u r e m e n t s . A l s o i t i s i m p o s s i b l e t o e n s u r e e q u a l c o o l i n g r a t e s a t a l l l o c a t i o n s e v e n i n a t h i n d i s c - s h a p e d s p e c i m e n i n a s a l t p o t q u e n c h . On t h e o t h e r h a n d , C C T d a t a i s m u c h s i m p l e r t o d e t e r m i n e e x p e r i m e n t a l l y a n d m o r e a c c u r a t e . T h u s a m e t h o d f o r d e t e r m i n i n g T T T f r o m C C T i s v e r y m u c h n e e d e d t o c h e c k e x p e r i m e n t a l ; T T T r e s u l t s . T h e p r e s e n t s t u d y e n u m e r a t e s a s i m p l e i t e r a t i v e p r o c e d u r e c a l l e d t h e \" a d d i t i v i t y m e t h o d \" , f o r d e r i v i n g t h e 111 f r o m t h e C C T . T h i s p r o c e d u r e i s e a s y t o u s e a n d m u c h l e s s t i m e c o n s u m i n g w h e n c o m p a r e d t o t h e 1 5 1 6 l e n g t h y c a l c u l a t i o n s s u g g e s t e d b y o t h e r s . ' 51 3 . 5 D e r i v a t i o n o f T T T f r o m C C T b y t h e A d d i t i v i t y M e t h o d T h e T T T c u r v e c a n b e e x p r e s s e d b y a m a t h e m a t i c a l e q u a t i o n t A V - T T T s a b * ( 3 ' 7 5 ) w h e r e ^ A V - T T T = s t a r t t i m e a t t h e t e m p e r a t u r e T a , b , c = c o n s t a n t s X = T A T - T T^.| = e q u i l i b r i u m t r a n s f o r m a t i o n t e r m p e r a -t u r e o f t h e m a t e r i a l T h e C C T s t a r t c a n a l s o b e e x p r e s s e d i n a s i m i l a r f o r m a s x T A V - C C T = a l b l e ( 3 - 7 6 ) T h e c o n s t a n t s a ^ , b ^ , c . j i n E q . ( 3 . 7 6 ) c a n b e c a l c u l a t e d b y u s i n g a m u l t i p l e - r e g r e s s i o n p r o c e d u r e b y u s i n g e x p e r i m e n t -a l l y d e t e r m i n e d t ^ y r r T v a l u e s . B y u s i n g a d d i t i v i t y a n d t h e i t e r a t i v e p r o c e d u r e t o b e d e s c r i b e d , t h e c o n s t a n t s a , b , c i n E q . ( 3 . 7 5 ) c a n b e d e t e r m i n e d . T o s t a r t t h e i t e r a t i v e p r o c e d u r e , a n e s t i m a t e o f a , b a n d c m u s t b e m a d e . T h i s c a n b e d o n e b y u s i n g p u b l i s h e d T T T d i a g r a m s . a s . a g u i d e o r c a n b e d e t e r m i n e d o n a n y a r b i t r a r y b a s i s . A p r o p e r e s t i m a t i o n o f t h e s e c o n s t a n t s r e s u l t s i n a r e d u c t i o n o f t h e n u m b e r o f i t e r a t i o n s n e e d e d t o d e t e r m i n e 5 2 t h e c o r r e c t T T T s t a r t c u r v e . H e n c e t h e e s t i m a t i o n o f t h e s e c o n s t a n t s i s n o t c r u c i a l t o t h e s u c c e s s o f t h e m e t h o d . T h e T T T s t a r t c u r v e d e t e r m i n e d b y t h e e s t i m a t e d c o n s t a n t s a , b a n d c i s t h e f i r s t a p p r o x i m a t i o n t o t h e c o r r e c t t ^ y _ - r - p - r . C o n s i d e r a c o o l i n g r a t e A ° C / s . T h e p r i n c i p l e o f a d d i t i v i t y s t a t e s t h a t t A V - C C T J d t = 1 { 3 7 7 ) v A V - T T T 0 f o r a c o n s t a n t c o o l i n g r a t e , E q . ( 3 . 7 7 ) c a n b e w r i t t e n a s : t , ' A V - C C T d T 1 f ^ = 1 ( 3 . 7 8 ) A t A V - T T T t A V - C C T f d J _ * A ( 3 ^ 7 - 9 - ) -•» t n w -r-r-r ' A V - T T T T h e E q . ( 3 . 7 9 ) c a n b e a p p r o x i m a t e d b y t A V - C C T ' ' AT t A V - T T T 0 A ( 3 . 8 0 ) 5 3 A s A T - > 0 , E q . ( 3 . 8 0 ) b e c o m e s t h e s a m e a s E q . ( 3 . 7 9 ) . F o r t h e c o o l i n g r a t e A, t h e c a n b e f o u n d f r o m t h e C C T c u r v e . U s i n g t ^ v T T T C a l c u l a t e d a t e a c h t e m p e r a t u r e , A f r o m T^-j t o t h e T ^ y _ r r j i n s t e p s o f A T , b y u s i n g t h e E q . ( 3 . 7 5 ) , t h e L H S o f E q . ( 3 . 8 0 ) c a n b e c a l c u l a t e d . I f t h i s i s g r e a t e r t h a n t h e R H S , t h e f i r s t a p p r o x i m a t i o n o f t h e T T T u p t o T/\\y_crT i s z0 t f i e l e f t °^ t n e c o r r e c t t ^ v _ T T T . I f t h e L H S v a l u e i s A^ , t h e n b y m u l t i p l y i n g a l l t h e t ^ v _ T T T A v a l u e s u p t o T^^.QQJ ^ y A^/A, t h e i d e n t i t y e x p r e s s e d i n E q . ( 3 . 8 0 ) w i l l h o l d t r u e u p t o T^ C C T ' N o w ' u s \" ' n 9 t h e s e v a l u e s o f t A V _ r r j u p t o T A V _ C C j » t h e c o n s t a n t s a , b a n d c i n E q . ( 3 . 7 5 ) a r e r e c a l c u l a t e d b y u s i n g a m u l t i p l e r e g r e s s i o n p r o -c e d u r e . T h i s e q u a t i o n i s u s e d i n t h e n e x t i t e r a t i o n f o r c a l c u l a t i n g t h e L H S i n E q . ( 3 . 8 1 ) . A c o o l i n g r a t e A-j ° C / s i s n o w c h o s e n s u c h t h a t A. >A. T h e c a l c u l a t i o n s , a s d o n e d u r i n g A l t h e f i r s t i t e r a t i o n , a r e r e p e a t e d , n o w u p t o a n e w C C T * ° T h i s p r o c e d u r e i s c a r r i e d o n u n t i l t h e c o r r e c t t ^ ^ _ T j j i s o b t a i n e d . T h e i t e r a t i o n s c a n b e s t o p p e d , w h e n r e q u i r e d , d e p e n d i n g u p o n t h e n e e d e d a c c u r a c y . T h e t h e o r e t i c a l j u s t i f i c a t i o n f o r t h e i t e r a t i v e p r o -c e d u r e i s a s f o l l o w s . F o r a c o o l i n g r a t e A, f o r a n a d d i t i v e r e a c t i o n , E q . ( 3 . 8 0 ) m u s t h o l d . I f t h e L H S o f E q . ( 3 . 8 0 ) , c a l c u l a t e d u s i n g t h e f i r s t a p p r o x i m a t i o n f o r t ^ T T T ' ^ s A.j , t h e n 5 4 ' A V - C C T •S-T A V - T T T A ' A V - C T T AT A V - T T T A ' A V - C C T AT A V - T T T ( 3 . 8 1 ) ( 3 . 8 2 ) ( 3 . 8 3 ) \" • A V - C C T . . \" . Z r - ^ = A ( 3 . 8 4 ) _ 1 . . t A A V - T T T A ^1 . . b y m u l t i p l y i n g t A V _ T J T u p t o T A V _ C C T b y , we c a n e n s u r e t h a t E q . ( 3 . 7 9 ) w i l l h o l d . F i g . 3 . 5 i s a p i c t o r i a l r e p r e s e n t a -t i o n o f t h i s p r o c e d u r e . T h i s i t e r a t i v e m e t h o d w a s u s e d t o p r e d i c t t h e t A y JJJ f o r t h e m a t e r i a l u s e d i n t h e p r e s e n t w o r k . T h e t A V T T T a n d t A V c c a r e , r e s p e c t i v e l y , t h e i n c u b a t i o n p e r i o d s f o r i s o t h e r m a l a n d c o n t i n u o u s c o o l i n g r r a n s f o r m a t i o n s . T h e d e t a i l s o f t h e 5 5 F i g u r e 3 . 5 N o m e n c l a t u r e o f p a r a m e t e r s u s e d t o c h a r a c t e r i z e i s o t h e r m a l a n d c o n t i n u o u s c o o l i n g r e a c t i o n s . 5 6 c a l c u l a t i o n s a r e s h o w n i n A p p e n d i x 4 . T h e t ^ y _ T T T v a l u e s c a l c u l a t e d b y t h e i t e r a t i v e p r o c e d u r e a n d t h a t f o u n d b y e x p e r i m e n t s a r e c o m p a r e d i n T a b l e 3 . 5 . A s c a n b e s e e n f r o m t h i s t a b l e , t h e a g r e e m e n t b e t w e e n p r e d i c t e d a n d 5 c a l c u l a t e d v a l u e s i s q u i t e g o o d . B . H a w b o l t e t a l . h a v e r e p o r t e d t h a t , i n t h e i r e x p e r i m e n t s , t h e a d d i t i v i t y r u l e d i d n o t w o r k w e l l i n t h e i n c u b a t i o n p e r i o d . T h i s i s c o n t r a r y t o t h e o b s e r v a t i o n s i n t h e p r e s e n t w o r k . S i n c e t h e n a t u r e o f t h e r e a c t i o n s i n t h e i n c u b a t i o n p e r i o d ( g r o w t h o f e m b r y o t o n u c l e u s ) i s e s s e n t i a l l y t h e s a m e a s i n t h e r e g i o n b e y o n d t h e i n c u b a t i o n p e r i o d , i t i s p o s s i b l e t o e x p e c t t h e a d d i -t i v i t y r u l e t o w o r k i n t h e i n c u b a t i o n p e r i o d . a s w e l l . C a l c u l a t i o n s t o f i n d t h e v a l u e s o f t h e L H S o f E q . ( 3 . 8 0 ) h a v e b e e n a c c o m p l i s h e d b y a F O R T R A N c o m p u t e r p r o g r a m . A l i s t i n g o f t h i s p r o g r a m i s s h o w n i n A p p e n d i x 5 . 5 7 T a b l e 3 . 8 V C o m p a r i s o n o f E x p e r i m e n t a l l y D e t e r m i n e d a n d C a l c u l a t e d ( b y t h e A d d i t i v i t y M e t h o d ) , t . . . T T T T e m p e r a t u r e ^AV-TTT ( ° C ) ( s ) E x p e r i m e n t a l C a l c u l a t e d 6 8 0 4 3 4 1 . 3 670 5 . 6 1 2 . 9 6 6 0 6 . 2 5 . 7 6 5 0 3 . 0 3 . 2 6 3 0 1 . 8 1 . 8 6 2 3 1 . 6 1 . 7 615 1 . 5 1 . 6 6 0 3 1 . 9 1 . 8 5 8 C h a p t e r 4 D E V E L O P M E N T OF A M A T H E M A T I C A L M O D E L TO S T U D Y P H A S E T R A N S F O R M A T I O N 4 . 1 I n t r o d u c t i o n I n t h i s c h a p t e r , a d e s c r i p t i o n i s g i v e n o f t h e m a t h e -m a t i c a l m o d e l d e v e l o p e d t o p r e d i c t t h e p h a s e t r a n s f o r m a t i o n o f a u s t e n i t e t o p e a r l i t e i n a p l a i n c a r b o n e u t e c t o i d s t e e l . T h e m o d e l i s b a s e d o n e x p e r i m e n t a l l y m e a s u r e d T T T a n d C C T d a t a . T o c h a r a c t e r i z e t h e n o n - i s o t h e r m a l a u s t e n i t e - p e a r l i t e r e a c t i o n k i n e t i c s , t h e p r i n c i p l e o f a d d i t i v i t y h a s b e e n u s e d , t h e t h e o r e t i c a l j u s t i f i c a t i o n f o r w h i c h h a s b e e n g i v e n i n C h a p t e r 3 . T h e m o d e l , i n i t s p r e s e n t f o r m , c a n b e u s e d t o p r e d i c t t h e t e m p e r a t u r e r e s p o n s e i n a n \" i n f i n i t e l y \" l o n g p l a i n c a r b o n e u t e c t o i d s t e e l r o d o f c i r c u l a r c r o s s - s e c t i o n b e i n g c o o l e d i n a i r . 4 . 2 M o d e l F o r m u l a t i o n F o r a c y l i n d r i c a l r o d c o o l i n g i n a i r , h e a t f l o w t o t h e s u r r o u n d i n g s i s g o v e r n e d b y h e a t c o n d u c t i o n w i t h i n t h e r o d a n d h e a t t r a n s f e r f r o m t h e s u r f a c e o f t h e r o d . T h e h e a t c o n d u c t i o n i n a r o d u n d e r g o i n g c o o l i n g i n a m e d i u m i s g i v e n i n q u a n t i t a t i v e t e r m s b y t h e f o l l o w i n g e q u a t i o n : 5 9 w h e r e : K = t h e r m a l c o n d u c t i v i t y o f t h e r o d m a t e r i a l r = r a d i a l d i s t a n c e f r o m t h e c e n t r e o f t h e r o d T = t e m p e r a t u r e o f t h e r o d Cp = s p e c i f i c h e a t o f t h e r o d m a t e r i a l p - d e n s i t y o f t h e r o d m a t e r i a l t = t i m e qA p = v o l u m e t r i c r a t e o f l a t e n t h e a t l i b e r a t e d d u e t o t h e a u s t e n i t e - p e a r l i t e t r a n s f o r m a t i o n . T h i s t e r m i s z e r o w h e n t h e r e i s n o t r a n s f o r m a -t i o n t a k i n g p i a c e . E q . ( 4 i l ) i s v a l i d u n d e r t h e f o l l o w i n g a s s u m p t i o n s : i ) I n f i n i t e l y l o n g r o d . i i ) N e g l i g i b l e a x i a l h e a t f l o w . i i i ) R a d i a l s y m m e t r y o f t e m p e r a t u r e i v ) U n i f o r m i n i t i a l t e m p e r a t u r e v ) U n i f o r m c i r c u l a r c r o s s - s e c t i o n . T h e s e c o n d i t i o n s a p p l y t o a w i r e r o d u n d e r g o i n g c o o l i n g i n a c o n t r o l l e d c o o l i n g p r o c e s s s u c h a s S t e l . m o r c o o l i n g . T h e b o u n d a r y c o n d i t i o n s a r e : i ) t > 0 , r = 0 , - K — - = 0 i i ) t > 0 , r = r a , K — = h ( T r - T a ) 60 w h e r e r = r a d i u s o f t h e r o d a h =• s u r f a c e h e a t t r a n s f e r c o - e f f i c i e n t T = a t m o s p h e r i c t e m p e r a t u r e = s u r f a c e t e m p e r a t u r e o f t h e r o d . T h e i n i t i a l c o n d i t i o n i s : • t = 0 , 0 < r < r . , T- = T . a l T.. = u n i f o r m i n i t i a l t e m p e r a t u r e . S o l u t i o n o f E q . ( 4 . 1 ) w i t h t h e p r o p e r c o n s t a n t s w i l l g i v e t h e t h e r m a l h i s t o r y o f t h e r o d u n d e r a g i v e n c o o l i n g c o n d i t i o n . I n c o r p o r a t i o n o f t h e l a t e n t - h e a t l i b e r a t i o n t e r m i n t h e h e a t - t r a n s f e r e q u a t i o n r e q u i r e s a k n o w l e d g e o f t h e k i n e t i c s o f t h e a u s t e n i t e - p e a r l i t e r e a c t i o n . T h i s t e r m c a n b e c a l c u l a t e d a s : = p H. ( 4 . 2 ) q A P At w h e r e H = e n t h a l p y o f t h e a u s t e n i t e - p e a r l i t e r e a c t i o n AF A P = v o l u m e f r a c t i o n t r a n s f o r m e d d u r i n g t h e time At. C a l c u l a t i o n o f A F ^ p m u s t b e d o n e b y u s i n g t A V _ r r T a n d t h e p r i n c i p l e o f a d d i t i v i t y , a s d e s c r i b e d i n S e c t i o n 4 . 4 . O w i n g t o t h e c o m p l e x i t y a n d t h e a m o u n t o f c a l c u l a t i o n s 61 S i m p l i f i e d Computer Program Flow Chart Read Input Parameters Specify I n i t i a l Conds. No. of Nodes of Heat Transfer Coefficient Tine steps • 1,2 ... n I Calculate Thermo-Physical Properties Ves Record the Time Solve Tridiagonal Systen. Calculate New Nodal Temperatures. Write Time am) Temperature of Centre and Surface Nodes Stop Calculate Fraction Transformed Recalculate Thermo-Physical Properties Solve Tridiagonal System. Recalculate New Nodal Temperatures. F i g . 4 . 1 C o m p u t e r p r o g r a m f l o w - c h a r t . 6 2 i n v o l v e d , i t i s a p p r o p r i a t e t o s o l v e E q . ( 4 . 1 ) s u b j e c t t o t h e b o u n d a r y a n d i n i t i a l c o n d i t i o n s w i t h t h e a i d o f a d i g i t a l c o m p u t e r . 4 . 3 C o m p u t e r P r o g r a m S o l u t i o n o f E q . ( 4 . 1 ) c a n b e a c c o m p l i s h e d b y u s i n g a o n e - d i m e n s i o n a l i m p l i c i t f i n i t e - d i f f e r e n c e a p p r o x i m a t i o n . ( T h e d i a g o n a l s y s t e m o f e q u a t i o n s i s s h o w n i n A p p e n d i x 6 . ) T h e n o d e a r r a n g e m e n t , t h e f i n i t e d i f f e r e n c e e q u a t i o n s a n d t h e t r i - d i a g o n a l s y s t e m o f e q u a t i o n s w a s s o l v e d b y u s i n g t h e G a u s s i a n e l i m i n a t i o n m e t h o d . A f l o w c h a r t o f t h e c o m p u t e r p r o g r a m i s s h o w n i n F i g . 4 . 1 . S o m e i m p o r t a n t f e a t u r e s o f t h e p r o g r a m a r e : i ) T h e r m a l c o n d u c t i v i t y a n d s p e c i f i c h e a t h a v e 4 6 b e e n c o n s i d e r e d a s f u n c t i o n s o f t e m p e r a t u r e . i i ) D e n s i t y i s a s s u m e d c o n s t a n t t o k e e p t h e n o d e s i z e c o n s t a n t . i i i ) T h e - t ^ y . ^ Q - p - h a s b e e n u s e d a s a . . . f u n c t i o n o f t e m p e r a t u r e . T h e r e l a t i o n s h i p b e t w e e n ^ A V - C C T a n c ' t i m e c a n b e f o u n d f r o m t h e e q u a t i o n t A V - C C T = A x B e ° X ^ 4 ' 3 ^ w h e r e A , B , C = c o n s t a n t s \" ' T A 1 \" T T = T e m p e r a t u r e . 6 3 T h e c o n s t a n t s A , B a n d C w e r e f o u n d b y a m u l t i p l e - r e g r e s s i o n p r o c e d u r e . i v ) B e c a u s e o f t h e r e l a t i o n s h i p b e t w e e n q ^ p a n d A F ^ p , E q . ( 4 . 2 ) , a n i t e r a t i v e p r o c e d u r e w a s r e q u i r e d a t e a c h t i m e s t e p , a f t e r t h e s t a r t o f t r a n s f o r m a -t i o n , t o c h e c k f o r c o n v e r g e n c e . I t w a s f o u n d t h a t w i t h i n 3 t o 4 i t e r a t i o n s , t h e t e m p e r a t u r e v a l u e s a t t h e n o d e s c o n v e r g e d t o g i v e a d i f f e r e n c e l e s s - 4 t h a n 1 0 ° C f o r s u c c e s s i v e t e m p e r a t u r e a p p r o x i -m a t i o n s . v ) T h e n u m b e r o f n o d e s a n d t h e t i m e s t e p m u s t b e c h o s e n c a r e f u l l y , d e p e n d i n g o n t h e r o d s i z e a n d t h e c o o l i n g r a t e . T h e n o d e s i z e m u s t b e n o t m o r e t h a n 0 . 2 5 mm. T h e t i m e - s t e p i n t e r v a l c o u l d b e l . s f o r s l o w c o o l i n g r a t e s ( l e s s t h a n 1 0 ° C / s ) a n d s h o u l d b e a p p r o x i m a t e l y 0 . 1 s f o r f a s t e r c o o l -i n g r a t e s . v i ) n a n d b i n t h e A v r a m i e q u a t i o n , E q . ( 2 . 2 ) h a v e b e e n i n c o r p o r a t e d a s f u n c t i o n s o f t e m p e r a t u r e . T h e s e f u n c t i o n s a r e c a l c u l a t e d i n s e p a r a t e s u b -r o u t i n e s . T h e p r o g r a m w a s c h e c k e d f o r i n t e r n a l c o n s i s t e n c y b y c o m p a r i n g t h e s o l u t i o n g e n e r a t e d b y a s s u m i n g q ^ p = 0 a n d c o n s t a n t t h e r m o p h y s i c a l p r o p e r t i e s f o r a n e u t e c t o i d s t e e l w i t h a n a n a l y t i c a l s o l u t i o n o f E q . ( 4 . 1 ) . T h e r e s u l t s a g r e e d t o w i t h i n 2% o f e r r o r a n d a r e s h o w n i n A p p e n d i x 7 . A c o m p a r i s o n w a s a l s o m a d e f o r t h e c a s e o f s m a l l d i a m e t e r r o d s w i t h n e g l i g i b l e i n t e r n a l r e s i s t a n c e . T h e a b o v e c h e c k s c o n f i r m e d t h a t t h e p r o g r a m i s f r e e o f l o g i c a l a n d o t h e r e r r o r s . A c o m p l e t e l i s t i n g o f t h e p r o g r a m i s g i v e n i n A p p e n d i x 8 . T h e m o d e l c a l c u l a t e s t h e t e m p e r a t u r e p r o -f i l e s a t a l l l o c a t i o n s i n s i d e t h e r o d u n d e r g o i n g c o o l i n g . T h e m o d e l o u t p u t c o n s i s t s o f : i ) T e m p e r a t u r e o f s u r f a c e a n d c e n t r a l n o d e s a n d t h e c o r r e s p o n d i n g t i m e . . i i ) S t a r t t i m e a n d t e m p e r a t u r e o f t r a n s f o r m a t i o n a t s u r f a c e a n d c e n t r a l n o d e s . i i i ) V o l u m e f r a c t i o n t r a n s f o r m e d a t s u r f a c e a n d c e n t r a l n o d e s a t e a c h t i m e s t e p . B y p l o t t i n g t h e t i m e - t e m p e r a t u r e d a t a g e n e r a t e d b y m o d e l , f o r a g i v e n c o o l i n g c o n d i t i o n , t h e a m o u n t o f r e -c a l e s c e n c e c a n b e c a l c u l a t e d a n d t h e t e m p e r a t u r e r a n g e o v e r w h i c h t h e t r a n s f o r m a t i o n t a k e s p l a c e c a n b e d e t e r m i n e d . B y p l o t t i n g t h e v o l u m e f r a c t i o n t r a n s f o r m e d - t i m e p r e d i c t i o n s , t h e c o u r s e o f t h e r e a c t i o n d u r i n g t h e p r o c e s s o f c o o l i n g c a n b e c h a r t e d . T y p i c a l t i m e - t e m p e r a t u r e p l o t s f r o m m o d e l p r e d i c t i o n s f o r a s t e e l r o d s o f c o m p o s i t i o n 0 . 8 2 % 0 . 8 2 % Mn - 0 . 2 6 % S i ( G r a i n s i z e 5 - 7 A S T M ) a n d d i a m e t e r s 6 5 5 . 5 m m , 2 0 mm:s a n d 2 5 mm a t d i f f e r e n t c o o l i n g r a t e s a r e s h o w n i n F i g s . 4 . 2 a n d 4 . 3 . T y p i c a l c a l c u l a t i o n s f r o m m o d e l p r e d i c t i o n s f o r t h e s a m e m a t e r i a l a r e s h o w n i n T a b l e 4 . 1 . 4 . 4 P r o g r a m L o g i c T h e r o d d i a m e t e r , n u m b e r o f n o d e s , i n i t i a l t e m p e r a t u r e o f t h e r o d , a m b i e n t t e m p e r a t u r e ( o r t h e t e m p e r a t u r e o f t h e c o o l i n g m e d i u m ) , t h e s u r f a c e h e a t t r a n s f e r c o - e f f i c i e n t a n d t h e t i m e - s t e p a r e i n p u t t o t h e p r o g r a m . I n t h e p r e -t r a n s f o r m a t i o n p e r i o d , s a y u p t o 7 0 0 ° C ( t h i s d e p e n d s u p o n t h e r o d d i a m e t e r a n d t h e c o o l i n g r a t e f o r a - g i v e n • m a t e r i a l ) t h e t r i - d i a g o n a l s y s t e m o f e q u a t i o n s i s s o l v e d a t e a c h t i m e s t e p , i n t h e a b s e n c e o f t r a n s f o r m a t i o n a l h e a t , t o d e t e r m i n e t h e n e w t e m p e r a t u r e s a t a l l n o d e s . A f t e r t h e s e t e m p e r a t u r e s a r e d e t e r m i n e d , e a c h n o d e i s c h e c k e d , a g a i n s t t ^ y _ Q P j , t o d e t e r -m i n e w h e t h e r t h e t r a n s f o r m a t i o n h a s b e g u n a t t h a t n o d e . I f t h e t r a n s f o r m a t i o n \" s t a r t s \" a t a n y n o d e , t h e f r a c t i o n t r a n s f o r m e d i s c a l c u l a t e d , f o r t h i s n o d e , b y u s i n g t h e p r i n c i p l e o f a d d i t i v i t y . I n g e n e r a l , a t t h e j - l t ' 1 t i m e s t e p , f o r t h e i n o d e u n d e r g o i n g t r a n s f o r m a t i o n , F j - l exp -b (TJ -_ 1 ) e: ( 4 . 4 ) w h e r e F j - l v o l u m e f r a c t i o n t r a n s f o r m e d a t n o d e i d u r i n g t h e t i m e s t e p j - l 66 T 1 1 — r Time (s) F i g . 4 . 2 T y p i c a l m o d e l - p r e d i c t e d c e n t r e - l i n e t e m p e r a -t u r e r e s p o n s e f o r a n a i r - c o o l e d s t e e l r o d . ( R o d d i a m e t e r = 5 . 5 mm) ( F i g u r e s o n t h e c u r v e s i n d i c a t e t h e a i r v e l o c i t y i n m / s . ) 67 F i g . 4 . 3 T y p i c a l m o d e l - p r e d i c t e d t u r e a n d t r a n s f o r m a t i o n c o o l e d s t e e l r o d s . c e n t r e - l i n e t e m p e r a -p r o f i l e s f o r a i r -T a b l e 4 . 1 T y p i c a l M o d e l P r e d i c t i o n s o f U n d e r c o o l i n g a n d R e c a l e s e e n c e a t t h e C e n t r e - l i n e o f A i r C o o l e d S t e e l Rods S t e e l : 0 . 8 2 % C - 0 . 8 2 % G r a i n S i z e : 5 - 7 ASTM Mn - 0 . 26% S i Rod D i a m e t e r : A i r V e l o c i t y ( m / s ) (mm) 0 20 i 40 Minimum ( ° C ) 69 92 X 5 Maximum ( ° C ) 83 135 -R e c a l e s e e n c e ( ° C ) 14 4 3 -C o o l i n g R a t e ( ° C / s ) 4 . 3 8 2 4 . 3 2 -Minimum ( ° C ) 60 76 81 10 Maximum ( ° C ) 81 103 109 R e c a l e s e e n c e ( ° C ) 21 23 2 8 C o o l i n g R a t e ( ° C / s ) 1 . 9 8 . 5 1 2 . 6 Minimum ( ° C ) 54 65 69 20 Minimum ( ° C ) 72 84 86 R e c a l e s e e n c e ( ° C ) 18 19 17 C o o l i n g R a t e ( ° C / s ) 0 . 9 3 . 4 5 . 0 N o t e : Minimum = M i n i m u m u n d e r c o o l i n g (= T ^ - T) C o o l i n g r a t e a t T f l 1 6 9 e1. . j - l b ( T l , ) = v a l u e o f c o - e f f i c i e n t b i n t h e A v r a m i e q u a t i o n a t t h e t e m p e r a t u r e o f t h e i n o d e a t t i m e s t e p j - l - t i m e t a k e n t o t r a n s f o r m t h e c u m u l a t i v e f r a c t i o n o a t t l i e t e m p e r a t u r e J.\\ •, n ( T \" ! , ) = v a l u e o f t h e c o - e f f i c i e n t n i n t h e A v r a m i J I j_ L e q u a t i o n a t t h e i n o d e a t t e m p e r a t u r e F o r t h e j t i m e s t e p , w h e r e e1. 1 - e x p - b ( T J ) ) T Q = T ( 0 ) - T ( » ) T ( t ) = t e m p e r a t u r e o f b o d y a t t i m e ' t ' ( ° C ) T ( 0 ) = i n i t i a l t e m p e r a t u r e o f b o d y ( ° C ) T(c°) = t e m p e r a t u r e o f t h e c o o l i n g m e d i u m ( ° C ) h = c o n v e c t i v e h e a t t r a n s f e r c o - e f f i c i e n t ( W / m 2 ° C ) 3 p = d e n s i t y o f t h e b o d y ( K g / m ) 9 3 T a b l e 6 . 1 E r r o r s i n n a n d b V a l u e s T e m p e r a t u r e ( ° C ) n l n b E x p e r i m e n t a l P r e d i c t e d E x p e r i m e n t a l P r e d i c t e d 6 8 0 2 . 1 2 5 2 . 1 0 2 - 9 . 8 1 - 9 . 8 3 6 7 0 1 . 6 1 9 1 . 6 1 9 - 9 1 4 4 - 9 . 4 3 6 6 0 2 . 4 6 7 2 . 5 3 6 - ' 9 . 4 7 - 9 . 4 5 6 5 0 2 . 9 4 6 2 . 8 8 5 - 8 . 3 9 - 8 . 4 8 6 3 0 3 . 1 6 6 3 . 1 9 0 - 5 . 7 3 - 5 . 5 7 6 2 3 3 . 1 4 8 3 . 1 3 1 - 4 . 5 1 - 4 . 4 8 6 1 5 2 . 9 2 2 2 . 9 3 4 - 3 . 1 4 - 3 . 3 2 6 0 3 2 . 3 4 6 2 . 3 4 1 - 2 . 0 4 - 1 . 9 8 9 4 V = v o l u m e o f t h e b o d y (m ) C = s p e c i f i c h e a t o f t h e b o d y ( W / K g ° C S e c ) t = t i m e ( S e c ) w h e r e F r o m E q . ( 6 . 3 ) , t a k i n g t h e n a t u r a l l o g o f b o t h s i d e s , T t I n = - H t ( 6 . 4 ) 'o H = y A ( 6 . 5 ) P H = 5 - ( 6 . 6 ) S i n c e t h e c o o l i n g t e s t s w e r e c o n d u c t e d i n a u n i f o r m n i t r o g e n f l o w , t h e c o n v e c t i v e h e a t - t r a n s f e r c o - e f f i c i e n t c a n b e e x p e c t e d t o b e n e a r l y c o n s t a n t . A t t h e t i m e t h e t r a n s -f o r m a t i o n b e g i n s , t h e v a l u e o f ' H ' s h o u l d c h a n g e m a r k e d l y , p r i m a r i l y d u e t o t h e r e e a l e s c e n c e c a u s e d b y t h e r e l e a s e o f l a t e n t h e a t o f t r a n s f o r m a t i o n . F o r a n y g i v e n C C T t e s t , t h e v a l u e s o f H w e r e c a l c u l a t e d f o r p r o g r e s s i v e l y i n c r e a s i n g v a l u e s o f t . T h e t i m e a t w h i c h t h e r e i s a s u d d e n c h a n g e i n t h e v a l u e o f H i s t h e n t h e t A y _ r r T . T h e s e v a l u e s w e r e a l s o c h e c k e d w i t h t h e d i l a t o m e t e r d a t a . T n e ^ A V - C C T t n u s c a l c u l a t e d f r o m e x p e r i m e n t a l d a t a a r e s h o w n i n T a b l e 6 . 2 . B y u s i n g a m u l t i p l e r e g r e s s i o n p r o c e d u r e , a c u r v e f o r d e t e r m i n i n g t A V _ r r j a t v a r i o u s t e m p e r a t u r e s , f o r u s e i n T a b l e 6 . 2 C o n t i n u o u s C o o l i n g D a t a ( t „ . , r r T ) E x p e r i m e n t t A V T A V # ( s ) (°c) 1 3 . 2 5 8 5 . 5 0 2 3 . 7 6 0 6 . 7 5 3 4 . 1 6 0 5 . 5 0 4 4 . 6 6 0 7 . 5 0 5 4 . 9 6 1 1 . 5 0 6 5 . 1 6 1 2 . 7 5 7 3 . 1 5 8 9 . 5 0 8 4 . 0 5 9 4 . 0 0 9 5 . 1 6 1 3 . 7 5 10 3 . 8 5 7 0 . 5 0 11 3 . 3 5 9 1 . 5 0 12 4 . 3 6 0 7 . 0 0 13 1 2 . 0 6 2 9 . 0 0 14 1 4 . 0 6 3 6 . 0 0 15 1 9 . 8 6 4 2 . 0 0 16 3 8 . 0 6 4 9 . 5 0 9 6 m o d e l c a l c u l a t i o n s , w a s o b t a i n e d . . T h e e q u a t i o n o f t h e c u r v e s o o b t a i n e d i s I n ( t A V _ C C T ) = ( 6 2 . 7 ) ( 7 2 8 - T ) \" 1 5 - 4 - e x p 0 . 1 ( 7 2 8 - T ) ( 6 . 7 ) T h e t A ^ _ r r j c a l c u l a t e d b y E q . ( 6 . 7 ) f i t s t h e e x p e r i m e n t a l d a t a q u i t e w e l l . A p l o t o f t A y _ r r T i s s h o w n i n F i g . 6 . 7 . I n o r d e r t o c h e c k f o r c o n s i s t e n c y i n t h e C C T t e s t s , s o m e o f t h e t e s t s w e r e r e p e a t e d . T h e c o n s i s t e n c y w a s f o u n d t o b e g o o d a s d e t e r m i n e d f r o m t h e r r T c a l c u l a t i o n s a n d t h e d i l a t o m e t e r r e s p o n s e s ( F i g . 6 . 8 ) 6 . 3 C o m p a r i s o n o f M o d e l P r e d i c t e d a n d E x p e r i m e n t a l R e s u l t s o f C e n t r e - 1 i n e T e m p e r a t u r e M e a s u r e m e n t s F o r a r o d o f a s p e c i f i c d i a m e t e r u n d e r g o i n g a i r - c o o l i n g , t h e t e m p e r a t u r e - t i m e r e s p o n s e c a n b e c a l c u l a t e d f r o m t h e m o d e l b y i n p u t t i n g t h e i n i t i a l t e m p e r a t u r e o f t h e r o d a n d t h e h e a t - t r a n s f e r c o - e f f i c i e n t w h i c h d e p e n d s o n t h e c o o l i n g c o n d i t i o n s . I n t h e p r e s e n t s t u d y , t h e a p p r o p r i a t e h e a t -t r a n s f e r c o - e f f i c i e n t h a s b e e n c a l c u l a t e d b y t h e f o l l o w i n g p r o c e d u r e . F o r a g i v e n r o d d i a m e t e r a n d i n i t i a l t e m p e r a t u r e , d i f -f e r e n t c o o l i n g p r o f i l e s w e r e g e n e r a t e d w i t h t h e m o d e l b y u s i n g d i f f e r e n t h e a t - t r a n s f e r c o - e f f i c i e n t s . . T h e t i m e -t e m p e r a t u r e p r o f i l e f r o m a n e x p e r i m e n t w a s t h e n c o m p a r e d w i t h 9 7 1 1 1 T 6 5 0 -6 3 0 ( J 0 610 Steel: 0-82 % C , 0 - 8 2 % M n 3 0 - 2 6 % Si -O ^ . Groin Size: 5 - 7 (ASTM) O) CL O Experimentol — E cu 5 9 0 c / ° Best Fit Curve of 5 7 0 -5 5 0 — I - 1 1 1 10 20 3 0 4 0 50 Time (s) F i g . 6 . 7 t A V _ C C T 4 0 0 1 '— 1 > 1 — » 0 2 0 4 0 6 0 80 100 Time For Temperature Tests (s) 0 2 4 6 8 10 Time For CCT Curves (s) F i g . 6 . 8 I l l u s t r a t i n g t h e c o n s i s t e n c y o f r e s u l t s o b s e r v e d d u r i n g C C T a n d c e n t r e - l i n e t e m p e r a t u r e m e a s u r e m e n t t e s t s . t h e m o d e l g e n e r a t e d t i m e - t e m p e r a t u r e p r o f i l e s t o f i n d t h e v a l u e o f t h e h e a t - t r a n s f e r c o - e f f i c i e n t p r i o r t o t r a n s -f o r m a t i o n t h a t b e s t f i t s t h e e x p e r i m e n t a l r e s u l t . U s i n g t h i s v a l u e o f t h e h e a t - t r a n s f e r c o - e f f i c i e n t a s t h e f i r s t a p p r o x i m a t i o n , t h e m o d e l c a l c u l t i o n s w e r e r e p e a t e d u n t i l t h e m o d e l - p r e d i c t e d v a l u e s a g r e e d w i t h t h e e x p e r i m e n t a l r e s u l t f o r a g i v e n v a l u e o f t h e h e a t - t r a n s f e r c o - e f f i c i e n t . T h e r e s u l t s o f t h e m o d e l - p r e d i c t e d a n d e x p e r i m e n t a l t i m e -t e m p e r a t u r e p r o f i l e s a r e s h o w n i n F i g s . 6 . 9 t o 6 . 1 9 a n d T a b l e 6 . 3 . A s c a n b e s e e n f r o m t h e s e , t h e m o d e l p r e d i c t e d v a l u e s a g r e e v e r y w e l l w i t h t h e e x p e r i m e n t a l r e s u l t s . 6 . 4 M o d e l P r e d i c t i o n a n d V a l i d & t i o n w i t h M e a s u r e d T e m p e r a t u r e D a t a F r o m t h e c o m p a r i s o n o f m o d e l - p r e d i c t e d a n d e x p e r i -m e n t a l r e s u l t s o f c e n t r e - l i n e t e m p e r a t u r e m e a s u r e m e n t s d u r i n g a i r c o o l i n g o f a s t e e l r o d , i t i s e v i d e n t t h a t t h e m o d e l c a l c u l a t i o n s a r e s u f f i c i e n t l y a c c u r a t e f o r t h e e x p e r i m e n t a l s i t u a t i o n s i n t h e p r e s e n t w o r k , p a r t i c u l a r l y s o , i n t h e l i g h t o f t h e n u m b e r o f d i f f e r e n t i n p u t s . T h e a c c u r a c y o f m o d e l c a l c u l a t i o n s i s g o v e r n e d b y t h e a c c u r a c y o f t h e i n p u t s . I n t h i s r e g a r d , t h e h e a t - t r a n s f e r c o -e f f i c i e n t , h , w h i c h h a s b e e n a s s u m e d c o n s t a n t d u r i n g c o o l i n g , m e r i t s s p e c i a l c o n s i d e r a t i o n . T h i s a s s u m p t i o n s e e m s r e a s o n -a b l e i n t h e t i m e p e r i o d p r i o r , t o t r a n s f o r m a t i o n b e c a u s e a g o o d f i t b e t w e e n p r e d i c t e d a n d m e a s u r e d t e m p e r a t u r e s w a s T a b l e 6 . 3 M o d e l P r e d i c t e d T i m e - t e m p e r a t u r e R e s p o n s e a t C e n t r e - l i n e o f A i r - c o o l e d S t e e l Rods C o o l i n g R a t e Amount o f a t T A 1 T . m i n T max t . m i n R e c a l e s e e n c e T m a x \" T m i n ( ° C / s ) (°C) ( ° C ) ( s ) (°c) 4 . 3 636 6 6 0 57 24 5 . 0 6 3 8 6 6 0 5 3 22 5 . 5 636 6 6 0 5 3 . 5 24 5 . 0 637 6 6 0 49 2 3 6 . 2 631 656 2 5 25 1 0 . 5 624 652 26 28 1 0 . 0 624 651 25 27 1 0 . 6 624 652 27 2 8 o o 0 101 F i g . 6 . 9 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . 1 0 2 9 0 0 840 o o O 780 CD 1^ 20 660 6 0 0 - 1 1 1 1 ~T 1 i I i Steel : 0 - 8 2 % C , 0 - 8 2 % M n , 0 - 2 6 % S i Groin Size: 5 - 7 (ASTM) \\ o Cooling Rote at T A i = 5 ° C / s O Experimental Model Prediction - --mm \\ o o 0 -1 1 1 1 I I I 1 I 0 20 4 0 6 0 Time (s) 80 100 F i g . 6 . 1 0 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . 103 9 0 0 o 8 4 0 CD o 7 8 0 CD CL E CD 7 2 0 h 6 6 0 h 6 0 0 0 2 0 4 0 60 Time (s) 8 0 100 F i g . 6 . 1 1 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . 9 0 0 8 4 0 o o OJ Z5 \"o 780 h cu £ 7 2 0 CD 6 6 0 6 0 0 4 0 6 0 Time (s) F i g . 6 . 1 2 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - 1 i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . 0 20 4 0 6 0 Time (s) F i g . 6 . 1 3 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . 1 0 6 F i g . 6 . 1 4 T e m p e r a t u r e r e . s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . F i g . 6 . 1 5 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . 9 0 0 8 4 0 o o £ 780 D O CU C L I 720 6 6 0 h 6 0 0 0 20 40 Time (s) 60 F i g . 6 . 1 6 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . F i g . 6 . 1 7 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . Steel:0-82 % C , 0 8 2 % M n f 10 Time (s) F i g . 6 . 1 8 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . T Time (s) F i g . 6 . 1 9 T e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a i r c o o l e d s t e e l r o d ( 1 0 mm d i a m e t e r ) . 1 o b t a i n e d , F i g s . 6 . 9 t o 6 . 1 9 . T h i s i s n o t s u r p r i s i n g s i n c e t h e d o m i n a n t m o d e o f h e a t t r a n s f e r i s t e m p e r a t u r e i n d e p e n -d e n t c o n v e c t i o n , e s p e c i a l l y a t h i g h e r c o o l i n g r a t e s . T h e t e m p e r a t u r e d e p e n d e n t t e r m , t h e r a d i a t i v e h e a t t r a n s f e r , i s a s m a l l p o r t i o n o f t h e t o t a l h e a t t r a n s f e r ( u s u a l l y l e s s t h a n 5 - 1 0 % a t h i g h e r c o o l i n g r a t e s ) . H e n c e i t i s t o b e e x p e c t e d t h a t ' h ' w i l l b e r o u g h l y c o n s t a n t i n t h e p r e -t r a n s f o r m a t i o n p e r i o d a n d t h e u s e o f a c o n s t a n t ' h ' i s t h u s j u s t i f i e d . 6 . 5 D i s c u s s i o n T h e g o o d a g r e e m e n t o f t h e m o d e l p r e d i c t i o n s w i t h t h e e x p e r i m e n t a l r e s u l t s i s p r i m a r i l y d u e t o t h e f o l l o w i n g f a c t o r s : i ) A c c u r a t e i n p u t s , l i k e t h e r m a l c o n d u c t i v i t y , s p e c i f i c h e a t e t c . i i ) U s e o f t A V _ T T T a n d t A y _ C C T i i i ) V a l i d i t y o f t h e a d d i t i v i t y p r i n c i p l e f o r t h e e x p e r i m e n t a l c o n d i t i o n s . A s m e n t i o n e d e a r l i e r , t h e t r a n s f o r m a t i o n i s a s s u m e d 3 5 t o \" s t a r t \" a t t A V _ C C T d u r i n g c o n t i n u o u s c o o l i n g . T h e c o n v e n t i o n a l a p p r o a c h i s t o a s s u m e t h a t t h e t r a n s f o r m a t i o n s t a r t s a t T A - | , e v e n d u r i n g c o n t i n u o u s c o o l i n g . I n t h e p r e s e n t w o r k , i f n a n d b a r e c a l c u l a t e d u s i n g t = 0 a t T f l , 1 1 3 t h e v a r i a t i o n i n n w i t h t e m p e r a t u r e i s m u c h h i g h e r t h a n f o r t h e c a s e o f t = 0 a t T ^ y ( s e e T a b l e 6 . 4 ) . T h e r a n g e o f v a l u e s o f n ( 1 . 9 t o 4 . 8 ) i s i n c o n t r a s t w i t h t h e w o r k o f 2 9 T a m u r a e t a l . a n d o t h e r s , w h o p r o p o s e a v a l u e o f 4 f o r t h e a u s t e n i t e - p e a r l i t e . r e a c t i o n . A l s o t h e r a n g e i s t o o w i d e t o e x p e c t a c o n s t a n t n , a s r e q u i r e d b y a d d i t i v i t y . I n a d d i t i o n , m o d e l c a l c u l a t i o n s u s i n g t = 0 a t a r e n o t i n a g r e e m e n t w i t h t h e t i m e - t e m p e r a t u r e r e s p o n s e o f t h e s t e e l r o d s m e a s u r e d e x p e r i m e n t a l l y , a s s h o w n i n T a b l e 6 . 5 a n d F i g . 6 . 2 1 . ( N o m e n c l a t u r e o f t e r m s u s e d i n T a b l e 6 . 5 i s shown i n F i g . 6 . 2 0 . ) , T h e v a l u e o f n c a l c u l a t e d b y u s i n g t = 0 a t v a r i e s w i t h i n a n a r r o w r a n g e ( 1 . 7 t o 3 ) , w i t h a n a v e r a g e c l o s e t o 2 . 5 . T h i s i s l o w e r t h a n t h e v a l u e e x p e c t e d b y T a m u r a 2 9 e t a l . T h i s d i s c r e p a n c y c a n b e p a r t l y e x p l a i n e d b y t h e e f f e c t o f i n h o m o g e n e i t y o f t h e r e a c t i o n i n t h e e x p e r i -m e n t a l s i t u a t i o n s e n c o u n t e r e d i n t h i s w o r k . D u e t o i n -h o m o g e n e i t y , t h e v o l u m e t r i c g r o w t h o f t h e t r a n s f o r m e d p r o -d u c t d o e s n o t f o l l o w t h e J o h n s o n - M e h l e q u a t i o n ( w h i c h w o u l d m e a n n = 4 ) . T h e v a l u e o f n a n d b w o u l d b e d e t e r m i n e d b y t h e d e g r e e o f i n h o m o g e n e i t y p r e s e n t i n t h e r e a c t i o n . F u r t h e r w o r k i s n e e d e d t o s t u d y t h e n a t u r e o f t h i s i n h o m o g e n e i t y a n d i t s e f f e c t o n t h e r e a c t i o n k i n e t i c s . T h e m o d e l c a l c u l a t i o n s a l s o d e m o n s t r a t e t h e v a l i d i t y T a b l e 6 . 4 V a l u e s o f n a n d b f o r t = 0 a t T T e m p e r a t u r e ( ° C ) n I n b 6 8 0 1 . 9 4 . - 1 1 . 5 7 6 7 0 2 . 2 9 - 1 0 . 7 2 6 6 0 2 . 9 1 - 1 1 . 5 4 6 5 0 3 . 6 0 - 1 0 . 8 3 6 3 0 4 . 3 2 - 8 . 9 6 6 2 3 4 . 6 3 - 8 . 1 6 615 4 . 7 9 - 7 . 1 6 6 0 3 4 . 7 6 - 6 . 9 8 1 1 5 min { max Time (s) F i g . 6 . 2 0 N o m e n c l a t u r e o f p a r a m e t e r s d e s c r i b i n g a c o n t i n u o u s c o o l i n g - r e a c t i o n . T a b l e 6 . 5 C o m p a r i s o n o f M o d e l P r e d i c t e d T i m e - t e m p e r a t u r e P r o f i l e s a t C e n t r e - l i n e o f A i r - c o o l e d S t e e l Rods U s i n g t = 0 a t t . . a n d t = 0 a t Jn w i t h E x p e r i m e n t a l R e s u l t s . T tl • (°c) T (°C) max v ' m i n E x p t . M o d e l 1 M o d e l 2 E x p t . M o d e l 1 M o d e l 2 E x p t . M o d e l 1 M o d e l 2 6 3 3 637 645 668 661 661 57 57 56 629 6 3 8 645 660 661 661 53 53 5 3 6 3 3 636 646 670 661 661 54 5 3 . 5 5 4 629 6 3 7 644 660 660 6 6 0 49 49 46 635 631 637 6 6 3 656 6 5 5 25 25 26 620 6 2 4 6 3 0 646 652 651 2 6 . 5 2 5 . 6 26 614 624 629 640 651 6 4 9 25 25 24 614 624 631 640 652 6 5 0 27 27 26 Model 1 : t = 0 a t t\" M o d e l 2 : t = 0 a t T 1 1 7 9 0 0 8 0 0 o o 700 CD 13 \"5 k_ CD §- 6 0 0 CD h-5 0 0 4 0 0 O A Experimental Model Predicted S t e e l : 0 - 8 2 % C , 0 - 8 2 % M n 0 - 2 6 % Si Grain S i z e : 5 - 7 (ASTM) t=0 at T AI _L 1 0 20 4 0 60 Time (s) 80 100 F i g . 6 . 2 V C o m p a r i s o n o f t y p i c a l m o d e l p r e d i c t e d ( w i t h t = 0 a t ) a n d e x p e r i m e n t a l r e s u l t s o f c e n t r e - l i n e t e m p e r a t u r e s o f a i r - c o o l e d s t e e l r o d s . ( R o d d i a m e t e r = 10 mm) 1 1 8 o f t h e a d d i t i v i t y p r i n c i p l e . T h i s p r i n c i p l e i s d e r i v e d f r o m a c o n s i d e r a t i o n o f t h e f u n d a m e n t a l a s p e c t s o f t h e n u c l e a t i o n a n d g r o w t h r e a c t i o n s , c h a r a c t e r i z e d b y n u c l e a -t i o n a n d g r o w t h r a t e s . I t i s i m p o r t a n t t o r e a l i s e t h a t t h e c o n c e p t o f n u c l e a t i o n e n v i s i o n e d b y A v r a m i i n d e r i v i n g h i s e q u a t i o n i s d i f f e r e n t f r o m t h a t o f J o h n s o n a n d M e h l . A v r a m i t r e a t e d n u c l e a t i o n i n t w o s t e p s : i ) a p r o b a b i l i t y ' P ' o f a g e r m n u c l e u s b e c o m i n g a g r o w t h n u c l e u s i i ) a g e r m n u c l e u s b e c o m i n g u n a v a i l a b l e f o r g r o w t h d u e t o c o n s u m p t i o n b y t h e g r o w i n g p h a s e . I n c o n t r a s t , J o h n s o n a n d M e h l d e f i n e d t h e n u c l e a t i o n r a t e , N ( a c o n s t a n t ) a s t h e n u m b e r o f n u c l e i n u c l e a t i n g a n d g r o w i n g . E v i d e n t l y A v r a m i ' s c o n c e p t o f n u c l e a t i o n i s c l o s e r t o t h e r e a l s i t u a t i o n . A v r a m i t h e n s t a t e d t h a t P / G b e c o n s t a n t i n t h e i s o k i n e t i c r a n g e . T h e A v r a m i P i s t h e e q u i v a l e n t o f t h e J o h n s o n - M e h l N . A v r a m i s t a r t e d h i s d e r i v a t i o n b y a s s u m i n g t h a t t h e r e a r e H g e r m n u c l e i p r e -s e n t b e f o r e t h e t r a n s f o r m a t i o n b e g a n . ( H e n c e , PN = N . ) T h i s a s s u m p t i o n i s e q u i v a l e n t t o s t a t i n g t h a t t h e r e i s a n i n c u b a t i o n t i m e f o r e a c h r e a c t i o n , b e f o r e t h e t r a n s f o r m a -t i o n s t a r t s , w h i c h c o r r e s p o n d s t o t A V _ T T T a n d t A V _ c c - r u s e d i n t h e p r e s e n t w o r k . T h e n u c l e a t i o n r a t e N , u s e d i n d e r i v -i n g t h e e f f e c t i v e s i t e s a t u r a t i o n c o n d i t i o n , i s P N \" . I n 1 1 9 t e r m s o f C a h n ' s s i t e s a t u r a t i o n p r i n c i p l e , ' P ' i s v e r y c l o s e t o u n i t y . T h e b e h a v i o u r o f ' P ' i s , i n d e e d , d i f f i c u l t t o d e t e r m i n e . I t i s a f f e c t e d b y t h e t h e r m o d y n a m i c s o f t h e a u s t e n i t e - p e a r l i t e r e a c t i o n , t h e g r a i n s i z e , t h e s i t e e n e r g y e t c . I t i s c l o s e l y r e l a t e d t o t h e f o r c e s g o v e r n i n g a t o m i c b e h a v i o u r . B u t t h e p r i n c i p l e o f e f f e c t i v e s i t e s a t u r a t i o n i s e a s i e r t o d e a l w i t h , a n d s i n c e i t i s b a s e d o n t h e f u n d a m e n t a l a s p e c t s o f r e a c t i o n m e c h a n i s m b e c o m e s a u s e f u l t o o l t o s t u d y t h e e f f e c t o f t h e s e v a r i a b l e s o n t h e m o r e e a s i l y m e a s u r a b l e a n d i m p o r t a n t p r o c e s s p a r a m e t e r s s u c h a s t e m p e r a t u r e , c o o l i n g r a t e e t c . 6 . 6 S c o p e o f A p p l i c a t i o n o f t h e M a t h e m a t i c a l M o d e l B y s i i g h t l y m o d i f y i n g t h e m o d e l i n i t s p r e s e n t f o r m , i t i s p o s s i b l e t o s t u d y t h e e f f e c t o f s e g r e g a t i o n o n t h e t r a n s f o r m a t i o n b e h a v i o u r o f w i r e r o d s u n d e r g o i n g c o o l i n g i n a S t e l m o r - 1 i k e p r o c e s s . D u e t o t h e s e g r e g a t i o n o f e l e m e n t s l i k e m a n g a n e s e a n d p h o s p h o r u s i n c a s t i n g o t s o r b i l l e t s t h e f i n a l p r o d u c t m a y c o n t a i n s e g r e g a t e d r e g i o n s a t t h e c e n t r e . D u e t o t h e h i g h e r h a r d e n a b i 1 i t y a s s o c i a t e d w i t h t h e h i g h Mn ( a n d P ) c o n t e n t s , t h e c e n t r a l r e g i o n i n t h e f i n i s h e d p r o d u c t m a y h a v e t r a n s f o r m e d t o ' . m a r - t e n s i t e i f t h e p o s t - r o l l i n g c o n t r o l l e d c o o l i n g p a r a m e t e r s a r e d e s i g n e d f o r t h e m a t r i x m a t e r i a l . T h e b r i t t l e m a r t e n s i t e m a y f r a c t u r e d u r i n g s u b s e q u e n t p r o c e s s i n g , e . g . w h e n w i r e r o d s 1 2 0 a r e d r a w n i n t o w i r e s . A n e x a m p l e o f Mn s e g r e g a t i o n i n a w i r e r o d a n d t h e e f f e c t o n t r a n s f o r m a t i o n i s d e s c r i b e d i n s e c t i o n 6 . 7 . F i n a l l y , b u t m o s t i m p o r t a n t l y , t h e m o d e l c a l c u l a t i o n s c a n b e u s e d f o r e v a l u a t i n g t h e a v e r a g e m e c h a n i c a l p r o -p e r t i e s o f f i n i s h e d s t e e l r o d s . A n e x a m p l e o f s u c h c a l c u -l a t i o n s i s s h o w n i n s e c t i o n 6 . 8 . 6 . 7 E f f e c t o f S e g r e g a t i o n o n P h a s e T r a n s f o r m a t i o n T o s t u d y t h e e f f e c t o f s e g r e g a t i o n o n p h a s e t r a n s -f o r m a t i o n , a p l a i n c a r b o n e u t e c t o i d s t e e l o f 0 . 8 % C - 1 . 8 8 % Mn ( G r a i n S i z e = 5 t o 8 A S T M ) w a s c h o s e n a s t h e c o m p o s i -t i o n i n t h e s e g r e g a t e d r e g i o n i n a m a t r i x c o m p o s i t i o n o f 0 . 8 2 % C - 0 . 8 2 % Mn ( G r a i n S i z e = 5 t o 7 A S T M ) . N o r m a l l y t h e r a n g e o f s e g r e g a t i o n f o r m a n g a n e s e i s a b o u t 1 . 2 t o 1 . 3 t i m e s t h e m a t r i x c o m p o s i t i o n . B u t n o p u b l i s h e d T T T d i a -g r a m o f a s t e e l w i t h 0 . 8 % C a n d a b o u t 1% Mn c o u l d b e o b t a i n e d f r o m t h e l i t e r a t u r e . H e n c e , t h o u g h t h e m a n g a n e s e c o n t e n t o f 1 . 8 8 % i s t o o h i g h , t h e c a l c u l a t i o n s w e r e d o n e w i t h t h e a i m o f d e m o n s t r a t i n g t h e t r e n d s t h a t c o u l d b e e x p e c t e d i n t h e t r a n s f o r m a t i o n b e h a v i o u r d u e t o s e g r e g a t i o n . T h e T A V _ T T T f o r t h e s e g r e g a t e d r e g i o n h a s b e e n c a l c u -4 5 l a t e d f r o m p u b l i s h e d d a t a a n d i s s h o w n i n T a b l e 6 . 6 . T h e T a b l e 6 . 6 TTT D a t a f o r S e g r e g a t e d S t e e l T e m p e r a t u r e * A V ( ° C ) ( s ) 675 93 650 29 625 13 6 0 0 6 . 2 575 5 . 4 ' 5 5 0 3 . 5 525 3 . 4 5 0 0 2 . 9 475 1 . 9 4 5 0 2 . 5 425 4 . 2 4 0 0 7 . 5 n a n d b v a l u e s i n t h e A v r a m i e q u a t i o n h a v e b e e n c a l c u l a t e d f r o m t h e t A V _ T T T a n d a r e s h o w n i n T a b l e 6 . 7 . T h e t f t v C C T h a s b e e n c a l c u l a t e d f r o m t f t V _ T T T b y u s i n g a d d i t i v i t y ( T a b l e 6 . 8 ) . T h e t h e r m a l c o n d u c t i v i t y , s p e c i f i c h e a t a n d d e n s i t y h a v e b e e n a s s u m e d t o b e t h e s a m e a s t h a t o f t h e m a t r i x m a t e r i a l s i n c e t h e c a r b o n c o n t e n t s a r e t h e s a m e . T h e 4 5 M g t e m p e r a t u r e f o r t h e s e g r e g a t e d s t e e l i s 1 8 0 ° C . T h e p r o g r a m , u s e d f o r c a l c u l a t i o n s o n t h e 0 . 8 2 % C s t e e l , h a s b e e n m o d i f i e d t o i n c l u d e t h e p a r a m e t e r s f o r t h e s e g r e -g a t e d c o m p o s i t i o n . T h e p r o g r a m l i s t i n g i s s h o w n i n t h e A p p e n d i x 9 . T h e p r o g r a m l o g i c i s t h e s a m e a s b e f o r e . T h e o n l y c h a n g e s a r e i n t h e t ^ y - . ^ y ' a n d t h e n a n d b v a l u e s u s e d f o r c a l c u l a t i n g t h e s t a r t o f t r a n s f o r m a t i o n a n d i t s f u r t h e r c o u r s e i n t h e s e g r e g a t e d r e g i o n s . N o r m a l l y , t h e s e g r e g a t e d r e g i o n a t t h e c e n t r e o c c u p i e s l e s s t h a n 5% o f t h e c r o s s -s e c t i o n a l a r e a o f t h e w i r e r o d . F o r m o d e l l i n g p u r p o s e s , t w o s i t u a t i o n s h a v e b e e n c o n s i d e r e d , T% a n d 4% o f c r o s s -s e c t i o n a l a r e a o c c u p i e d b y s e g r e g a t e d m a t e r i a l . D u e t o t h e p r e s e n c e o f t h e h i g h h a r d e n a b i 1 i t y m a t e r i a l a t t h e c o r e , m a r t e n s i t e m a y b e e x p e c t e d t o f o r m a t t h e c o r e , d e p e n d i n g o n t h e c o o l i n g c o n d i t i o n s a n d r o d s i z e . T h e e f f e c t o f s e g r e g a t i o n h a s b e e n i n v e s t i g a t e d f o r t h r e e d i f f e r e n t r o d d i a m e t e r s - 5 , 1 0 a n d 1 5 mm. T h e r e s u l t s o f t h e m o d e l c a l c u l a t i o n s a r e s h o w n i n T a b l e s 6 . 9 t o 6 . 1 4 . T h e a m o u n t o f m a r t e n s i t e t h a t c a n b e e x p e c t e d t o f o r m a t t h e c o r e f o r t h e d i f f e r e n t r o d s i z e s a t d i f f e r e n t c o o l i n g r a t e s a r e T a b l e 6 . 7 n a n d b V a l u e s f o r S e g r e g a t e d S t e e l T e m p e r a t u r e (°C) n I n b 675 0 . 7 7 - 6 . 0 9 6 5 0 0 . 8 3 - 4 . 6 0 6 2 4 1 . 2 8 - 5 . 0 8 6 0 0 1 . 6 5 - 5 . 5 7 575 1 . 2 1 - 4 . 0 0 5 5 0 1 . 1 0 - 4 . 0 2 5 2 5 0 . 7 4 - 2 . 9 2 500 0 . 7 2 - 2 . 9 8 475 0 . 8 5 - 3 . 7 8 450 0 . 9 1 - 3 . 9 7 4 2 5 0 . 9 9 - 4 . 3 8 400 1 . 0 6 - 5 . 0 3 T a b l e 6 . 8 CCT D a t a f o r S e g r e g a t e d S t e e l T e m p e r a t u r e l A V - C C T ( ° C ) (S) 696 3 0 6 0 675 4 7 3 6 6 7 286 641 86 591 23 545 1 1 . 4 512 8 . 3 4 5 8 5 . 9 4 4 3 5 . 6 4 2 3 5 . 4 383 5 . 6 364 5 . 9 1 2 5 T a b l e 6 . 9 E f f e c t o f C o o l i n g R a t e on M a r t e n s i t e F o r m a t i o n a t C e n t r e o f Rod \\ I n i t i a l T e m p e r a t u r e = 8 5 0 ° C Rod D i a m e t e r = 5 mm Amount o f S e g r e g a t e d A r e a = 1 % C o o l i n g R a t e a t T A 1 ( ° C / s ) \\ i i n (°0 ^max (°0 t • m i n % M a r t e n s i t e f o r m e d 6 629 6 5 4 36 <34 9 6 2 0 6 5 0 25 <53 13 6 1 4 646 20 <62 61 - - - 100 1 2 6 T a b l e 6 . 1 0 E f f e c t o f C o o l i n g R a t e on M a r t e n s i t e F o r m a t i o n a t C e n t r e o f Rod I n i t i a l T e m p e r a t u r e = 8 0 5 ° C Rod D i a m e t e r = 5 mm Amount o f S e g r e g a t e d A r e a = 4 % . C o o l i n g R a t e a t T A 1 T m i n T max t . . . m i n % M a r t e n s i t e f o r m e d ( ° C / s ) ( ° C ) ( ° C ) ( s ) 6 6 2 8 656 36 <33 9 622 6 4 6 25 <52 13 6 1 3 642 20 <62 61 - - - TOO 9 1 2 7 T a b l e 6 . 1 1 E f f e c t o f C o o l i n g R a t e o n M a r t e n s i t e F o r m a t i o n a t C e n t r e o f Rod I n i t i a l T e m p e r a t u r e = 8 5 0 ° C Rod D i a m e t e r = 1 0 mm Amount o f S e g r e g a t e d A r e a = 1 % C o o l i n g R a t e a t T A 1 ( ° C / s ) 3 4 . 5 61 T • mm ( ° C ) 6 4 0 631 max ( ° C ) 6 6 2 6 5 6 Vin ( s ) 78 38 \"% M a r t e n s i t e f o r m e d < 5 < 3 0 100 1 2 8 T a b l e 6 . 1 2 E f f e c t o f C o o l i n g R a t e o n M a r t e n s i t e F o r m a t i o n a t C e n t r e o f Rod I n i t i a l T e m p e r a t u r e = 8 5 0 ° C Rod D i a m e t e r = 10 nm Amount o f S e g r e g a t e d A r e a = 16 % C o o l i n g R a t e a t T A 1 ( ° C / s ) T m i n ( ° C ) T m a x ( ° C ) t . m i n ( s ) % M a r t e n s i t e Formed 2 . 5 6 3 9 6 6 2 8 5 < 5 4 . 0 6 3 4 6 5 7 57 <16 6 . 0 631 6 5 5 45 <30 5 8 . 0 - - - 100 1 2 9 T a b l e 6 . 1 3 E f f e c t o f C o o l i n g R a t e o n M a r t e n s i t e F o r m a t i o n a t C e n t r e o f Rod I n i t i a l T e m p e r a t u r e = 8 5 0 ° C Rod D i a m e t e r = 15 mm Amount o f S e g r e g a t e d A r e a - 1 % C o o l i n g R a t e a t T A 1 ( ° C / s ) T . m m ( ° C ) T m a x ( ° C ) t • m m ( s ) % M a r t e n s i t e F o r m e d 2 . 0 646 6 6 6 99 < 1 3 . 0 642 6 6 3 69 < 4 4 . 5 6 3 8 6 6 0 53 <12 5 2 . 0 - - - 100 1 3 0 T a b l e 6 . 1 4 E f f e c t o f C o o l i n g R a t e o n M a r t e n s i t e F o r m a t i o n a t C e n t r e o f Rod I n i t i a l T e m p e r a t u r e = 8 5 0 ° C Rod D i a m e t e r = 15 mm Amount o f S e g r e g a t e d A r e a = 16 % C o o l i n g R a t e a t T A 1 ( ° C / s ) T . m m ( ° C ) T m a x ( ° C ) t • m i n ( s ) % M a r t e n s i t e Formed 1 . 5 6 4 6 664 100 < 0 . 1 2 . 5 641 6 6 0 69 < 1 4 . 0 6 3 8 6 5 7 53 < 5 4 8 . 0 - - - 100 1 31 s h o w n i n F i g . 6 . 2 2 . A s e x p e c t e d , t h e p e r c e n t a g e m a r t e n s i t e f o r m e d i n t h e s e g r e g a t e d r e g i o n . i s h i g h e r f o r s m a l l e r r o d d i a m e t e r a n d f a s t e r c o o l i n g r a t e s . T h e p r o b l e m o f s e g r e g a t i o n i s n o t u n u s u a l i n a w i r e r o d m i l l . A n e x a m p l e o f s e g r e g a t i o n i s s h o w n i n F i g . 6 . 2 3 . M a n y m e t h o d s h a v e b e e n s t u d i e d t o m i n i m i s e o r e l i m i n a t e t h e m a r t e n s i t e f o r m a t i o n a t t h e c e n t r e . O n e s u c h m e t h o d , w h i c h h a s b e e n t r i e d i n a n i n d u s t r i a l s i t u a t i o n , i s d e s c r i b e d b y 4 7 V a n V u u r e n . He r e p o r t s t h a t t h e c o o l i n g r a t e w a s r e d u c e d t o e n s u r e t r a n s f o r m a t i o n t o p e a r l i t e a t t h e s e g r e g a t e d c e n t r e r e g i o n . B u t t h i s r e s u l t e d i n a w i d e s c a t t e r i n t h e t e n s i l e v a l u e s a n d h e n c e t h e p r o c e d u r e w a s u n a c c e p t a b l e . T h e p r o -c e d u r e w h i c h w o r k e d s u c c e s s f u l l y w a s t o h a v e a h i g h c o o l i n g r a t e i n i t i a l l y f o l l o w e d b y s l o w e r c o o l i n g . T h i s r e s u l t e d i n a n i n i t i a l s h a r p d r o p i n t e m p e r a t u r e , b u t l a t e r , t h e d r o p w a s v e r y s l o w . T h i s p r o c e d u r e a l l o w s t i m e f o r h o m o -g e n i s a t i o n o f t e m p e r a t u r e . I t i s c l a i m e d t h a t t h i s p r o -c e d u r e h e l p e d h o l d t h e s c a t t e r i n t h e t e n s i l e s t r e n g t h v a l u e s t o a m i n i m u m w h i 1 e a v o i d i n g t h e f o r m a t i o n o f m a r t e n -s i t e . O b v i o u s l y , t h i s p r o c e d u r e r e q u i r e s ! g o o d c o n t r o l o v e r t h e p r o c e s s c o o l i n g . c o n d i t i o n s . T h e m o d e l c a l c u l a t i o n s c a n b e u s e d t o s t u d y t h e e f f e c t o f s u c h c h a n g e s i n c o o l i n g c o n d i t i o n s a n d t h e i r e f f e c t o n 1 3 2 F i g . 6 . 2 2 E f f e c t o f c o o l i n g r a t e o n c e n t r e - l i n e m a r t e n s i t e f o r m a t i o n d u e t o s e g r e g a t i o n . 1 3 3 F i g . 6 . 2 3 L o n g i t u d i n a l s e c t i o n t h r o u g h a c o l d d r a w n r o d s h o w i n g a w h i t e c e n t r e l i n e m a r t e n s i t i c p h a s e f r a c t u r e d d u r i n g s u b s e q u e n t c o l d d r a w i n g . 1 3 4 t h e t r a n s f o r m a t i o n . T h e s e t h e n c a n b e u s e d a s a g u i d e f o r d e s i g n i n g t h e p r o c e s s p a r a m e t e r s . 6 . 8 C a l c u l a t i o n o f M e c h a n i c a l P r o p e r t i e s o f a W i r e R o d T h e m e c h a n i c a l p r o p e r t i e s o f p l a i n c a r b o n s t e e l r o d s d e p e n d p r i m a r i l y o n : i ) c o m p o s i t i o n ( e s p . N , S i , Mn a n d C ) i i ) a m o u n t o f t h e f e r r i t e p h a s e i i i ) p e a r l i t e s p a c i n g . F o r a g i v e n s t e e l c o m p o s i t i o n , t h e a m o u n t o f f e r r i t e , a n d m o r e i m p o r t a n t l y , t h e p e a r l i t e s p a c i n g d e p e n d s o n t h e u n d e r c o o l i n g d u r i n g t r a n s f o r m a t i o n . F r e q u e n t l y , a n d e s p e c i a l l y d u r i n g s l o w c o o l i n g , t h e s t e e l u n d e r g o e s r e -c a l e s c e n c e a f t e r t h e s t a r t o f t h e a u s t e n i t e - p e a r l i t e t r a n s -f o r m a t i o n i n t h e c a s e o f . a e u t e c t o i d s t e e l . T h i s i m p l i e s t h a t t h e t r a n s f o r m a t i o n t a k e s p l a c e o v e r a r a n g e o f t e m p e r a -t u r e . I f t h i s r a n g e i s n a r r o w , t h e n t h e r e a c t i o n c a n b e c o n s i d e r e d i s o t h e r m a l f r o m a p r a c t i c a l s t a n d p o i n t . I f t h e r o d s i z e a n d c o o l i n g c o n d i t i o n s a r e s u c h t h a t t h e t e m p e r a -t u r e s a t d i f f e r e n t l o c a t i o n s a r e s i g n i f i c a n t l y d i f f e r e n t , t h e n i t i s p o s s i b l e t h a t t r a n s f o r m a t i o n s a t t h e s e l o c a t i o n s t a k e p l a c e o v e r d i f f e r e n t t e m p e r a t u r e r a n g e s . T h i s i n t r o -d u c e s a n o t h e r d i f f i c u l t y i n c a l c u l a t i n g a n a v e r a g e p e a r l i t e s p a c i n g v a l u e f o r t h e r o d . A t t h i s s t a g e , i t i s n o t p o s s i b l e 1 3 5 t o c a l c u l a t e a c c u r a t e l y t h e e f f e c t o f s u c h a t e m p e r a t u r e r a n g e o n p e a r l i t e s p a c i n g . H o w e v e r , f o r i l l u s t r a t i v e p u r p o s e s , we c a n a s s u m e t h a t t h e a v e r a g e u n d e r c o o l i n g a t t h e c e n t r e o f t h e r o d d e t e r m i n e s t h e p e a r l i t e s p a c i n g i n t h e r o d . T h e m o d e l c a l c u l a t i o n s a r e v e r y u s e f u l i n t h i s r e g a r d . T h e m o d e l p r e d i c t i o n o f t h e t i m e - t e m p e r a t u r e r e s p o n s e a t t h e c e n t r e - l i n e o f a r o d b e i n g c o o l e d c a n b e u s e d t o d e t e r -m i n e t h e a v e r a g e u n d e r c o o l i n g . S i n c e t h e m o d e l c a l c u l a t e s t h e t e m p e r a t u r e a n d t h e f r a c t i o n t r a n s f o r m e d , i t i s p o s -s i b l e t o d e t e r m i n e t h e s t a r t a n d e n d 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 s a s w e l l a s t h e m a x i m u m a n d m i n i m u m u n d e r c o o l i n g v a l u e s . U s i n g t h e s e v a l u e s , t h e p e a r l i t e s p a c i n g f o r t h e s t e e l c a n 5 4 b e d e t e r m i n e d f r o m p u b l i s h e d e m p i r i c a l r e l a t i o n s h i p s . T h e m e c h a n i c a l p r o p e r t i e s a r e r e l a t e d t o t h e p e a r l i t e 5 4 s p a c i n g a n d s t e e l . . c h e m i s t r y a s f o l l o w s : i ) a v c . (MPa) 1 _ 1 a 3 J2 . 3 + 3 . 8 (%Mn) + 1 . 1 3 d 2*l+ UTS(MPa) a v o l u m e f r a c t i o n o f f e r r i t e s P d p e a r l i t e s p a c i n g (mm) g r a i n d i a m e t e r (mm) 1 3 6 F o r i l l u s t r a t i v e p u r p o s e s , t h e m e c h a n i c a l p r o p e r t i e s f o r t h e s t e e l u s e d i n t h e p r e s e n t s t u d y h a v e b e e n c a l c u l a -t e d , f o r t w o d i f f e r e n t r o d s i z e s a n d t w o c o o l i n g c o n d i t i o n s . P e a r l i t e s p a c i n g s h a v e b e e n c a l c u l a t e d b y u s i n g t h e p r o -c e d u r e m e n t i o n e d a b o v e a n d a r e s h o w n i n T a b l e s 6 . 1 7 a n d 6 . 1 8 . F o r a e u t e c t o i d s t e e l , E q . ( 6 . 8 ) a n d ( 6 . 9 ) c a n b e r e -w r i t t e n a s : i ) a v s (MPa) = 1 5 . 4 ^ 1 1 . 6 +.• Q . . 2 5 5 - s p * + 4 . 1 {% S i ) + 2 7 . 6 (%N)j- ( 6 i i ) UTS. (MPa) =-.l-5..4--J46.7 + 0 . . 2 3 s p 2 + - 6 . 3 - ( % . S i ) j . (6 T h e m e c h a n i c a l p r o p e r t i e s c a l c u l a t e d u s i n g E q . (\"6. 'TO) a n d ( 6 . 1 1 ) a r e s h o w n i n T a b l e s 6 . 1 5 t o 6 . 1 7 . T h i s e x a m p l e i l l u s -t r a t e s a p p l i c a b i l i t y o f t h e m o d e l t o c a l c u l a t i n g t h e m e c h a n i -c a l p r o p e r t i e s o f t h e w i r e r o d s f o r d i f f e r e n t c o o l i n g r a t e s . T h i s a p p l i c a t i o n c a n b e e x t e n d e d t o o t h e r m a t e r i a l s b y s u i t a b l y m o d i f y i n g t h e m o d e l . I t i s t h u s d e m o n s t r a t e d t h a t t h e m e c h a n i c a l p r o p e r t i e s c a n b e c a l c u l a t e d f r o m a k n o w l e d g e o f t h e r e a c t i o n k i n e t i c s o f t h e t r a n s f o r m a t i o n . A d m i t t e d l y , t h e c a l c u l a t i o n s h a v e b e e n d o n e w i t h s i m p l i f i e d a s s u m p t i o n s . F o r e x a m p l e , t h e e f f e c t o f t h e d i f f e r e n t t e m p e r a t u r e r a n g e s o f t r a n s f o r m a t i o n a t d i f f e r e n t l o c a t i o n s i n t h e r o d o n t h e p e a r l i t e s p a c i n g n e e d s t o b e s t u d i e d f u r t h e r . A l s o t h e t i m e t a k e n f o r t h e t r a n s f o r m a t i o n a t d i f f e r e n t l o c a t i o n s i n t h e r o d m a y a l s o a f f e c t p e a r l i t e s p a c i n g . N e v e r t h e l e s s , 1 3 7 T a b l e 6 . 1 5 P e a r l i t e S p a c i n g C a l c u l a t i o n s Rod D i a m e t e r = 5 mm I n i t i a l T e m p e r a t u r e o f Rod = 8 5 0 ° C A i r V e l o c i t y ( m / s ) 0 10 C o o l i n g r a t e a t T f t l - ( ° C / s ) 4 . 4 1 8 . 1 Maximum u n d e r c o o l i n g ( ° C ) 8 3 127 Minimum u n d e r c o o l i n g ( ° C ) 69 82 A v e r a g e u n d e r c o o l i n g ( ° C ) 76 1 0 4 . 5 Minimum i n t e r l a m e l l a r o p e a r l i t e s p a c i n g ( A ) 1000 6 7 5 T a b l e 6 . 1 6 P e a r l i t e S p a c i n g C a l c u l a t i o n s Rod D i a m e t e r = 15 mm I n i t i a l T e m p e r a t u r e o f Rod = 8 5 0 ° C A i r V e l o c i t y ( m / s ) 0 10 C o o l i n g r a t e a t T A 1 ( ° C / s ) 1 . 2 3 . 7 Maximum u n d e r c o o l i n g ( ° C ) 76 8 8 Minimum u n d e r c o o l i n g ( ° C ) 57 66 A v e r a g e u n d e r c o o l i n g ( ° C ) 6 7 . 5 77 M i n i m u m i n t e r l a m e l l a r o p e a r l i t e s p a c i n g (A) 1050 1000 1 3 9 T a b l e 6 . 1 7 M e c h a n i c a l P r o p e r t i e s C o o l i n g C o n d i t i o n s a Y S (MPa) UTS (MPa) RA ( X ) Rod d i a = 5 mm C R T = 4 . 3 8 ° C / s ' A I A i r V e l o c i t y = 0 m/s 599 1 0 9 8 37 Rod d i a = 5 mm C R T = 1 8 . 1 3 ° C / s ' A I A i r V e l o c i t y = 10 m/s 6 7 5 1175 4 0 Rod d i a = 15 mm C R T = 1 . 2 ° C / s ' A l A i r V e l o c i t y = 0 m/s 5 8 0 1090 35 Rod d i a = 15 mm C R T = 3 . 7 ° C / s ' A I A i r V e l o c i t y = 10 m/s 599 1 0 9 8 36 140 t h e s e c o n s i d e r a t i o n s o n l y i m p l y t h a t t h e c a l c u l a t i o n s n e e d t o b e m o r e s o p h i s t i c a t e d t h a n t h e o n e a t t e m p t e d i n t h e p r e -s e n t s t u d y . B u t t h e y d o n o t i n a n y w a y b r i n g i n t o d o u b t t h e v a l i d i t y o f u s i n g t h e m o d e l t o d o s u c h c a l c u l a t i o n s . I t h a s b e e n d e m o n s t r a t e d i n t h e p r e s e n t s t u d y , a l b e i t i n p r i n c i p l e , t h a t t h e m o d e l c a n b e u s e d t o i n t e g r a t e t h e r e a c t i o n k i n e t i c s w i t h t h e m a c r o l e v e l v a r i a b l e s s u c h a s t e m p e r a t u r e a n d m e c h a n i c a l p r o p e r t i e s . 1 4 1 C h a p t e r 7 S U M M A R Y A N D C O N C L U S I O N S W i t h t h e u l t i m a t e o b j e c t i v e o f c a l c u l a t i n g m e c h a n i c a l p r o p e r t i e s o f w i r e r o d s i n t h e S t e l m o r p r o c e s s u s i n g p h a s e -t r a n s f o r m a t i o n d a t a , a m a t h e m a t i c a l m o d e l h a s b e e n d e v e l o p e d . T h e p r e d i c t i o n s , m a d e b y t h e m o d e l , o f t h e c e n t r e - l i n e t e m p e r a t u r e p r o f i l e s h a v e b e e n s h o w n t o b e i n g o o d a g r e e -m e n t w i t h e x p e r i m e n t a l r e s u l t s f o r a p l a i n c a r b o n e u t e c t o i d s t e e l . C a l c u l a t i o n s h a v e a l s o b e e n d o n e t o d e r i v e t h e m e c h a n i c a l p r o p e r t i e s o f w i r e r o d s u s i n g m o d e l - p r e d i c t e d d a t a . H o w e v e r , t h e s e c a l c u l a t i o n s a r e o n l y i l l u s t r a t i v e a n d n e e d t o b e r e f i n e d f u r t h e r w i t h t h e h e l p o f e x p e r i m e n t a l d a t a . T h e m o d e l h a s b e e n m o d i f i e d t o s t u d y t h e e f f e c t o f c e n t r e - s e g r e g a t i o n o n t r a n s f o r m a t i o n i n a i r - c o o l e d s t e e l r o d s . C a l c u l a t i o n s h a v e b e e n c a r r i e d o u t w i t h a c e n t r e s e g r e g a t i o n c o m p o s i t i o n o f 0 . 8 % C - 1 . 8 8 % Mn i n a m a t r i x o f 0 . 8 2 % C - 0 . 8 2 % M n . T h o u g h t h e m a n g a n e s e i n t h e s e g r e -g a t e d r e g i o n i s h i g h e r t h a n w o u l d b e n o r m a l l y e x p e c t e d , t h e c a l c u l a t i o n s , n e v e r t h e l e s s , g i v e a q u a n t i t a t i v e e s t i m a t e o f t h e a m o u n t o f m a r t e n s i t e t h a t c a n b e e x p e c t e d t o f o r m a t t h e c e n t r e o f t h e r o d . A l s o t h e e f f e c t o f i n c r e a s i n g t h e c o o l i n g r a t e , t h e r o d d i a m e t e r a n d t h e a m o u n t o f s e g r e -g a t i o n o n t h e a m o u n t o f m a r t e n s i t e f o r m e d a t t h e c e n t r e c a n b e p r e d i c t e d q u a n t i t a t i v e l y b y t h e m o d e l . E x p e r i m e n t s a r e n e e d e d t o v a l i d a t e t h e m o d e l p r e d i c t i o n s . H o w e v e r t h e c a p a b i l i t y o f t h e m o d e l t o p e r f o r m t h e c a l c u l a t i o n s a n d g i v e m e a n i n g f u l p r e d i c t i o n s h a s b e e n d e m o n s t r a t e d . T h i s s h o u l d h e l p r e d u c e t h e a m o u n t o f e m p i r i c a l e x p e r i m e n t a t i o n t o d e s i g n p r o c e s s p a r a m e t e r s , s u c h a s c o o l i n g r a t e s f o r d i f f e r e n t r o d s i z e s , i n a S t e l m o r - 1 i n e . T h e s u c c e s s o f t h e m o d e l i s p r i m a r i l y d u e t o t w o f a c t o r s : i ) u s e o f t A V _ j T T a n d t A V _ C C T f o r c a l c u l a t i o n o f n a n d b i n t h e A v r a m i e q u a t i o n a n d t h e s t a r t o f t r a n s f o r m a t i o n d u r i n g c o o l i n g , r e s p e c t i v e l y . i i ) a d d i t i v i t y . T h e u s e o f t A y _ - r - r . r a n d t / ^ . Q Q j w a s p r o p o s e d b y B . 3 5 H a w b o l t e t a l . a n d h a s w o r k e d w e l 1 i n t h e p r e s e n t s t u d y . T h e c o n d i t i o n s f o r a d d i t i v i t y p r o p o s e d b y A v r a m i a n d C a h n w e r e n o t o b t a i n e d i n t h e p r e s e n t w o r k , t h e r e b y n e c e s s i t a -t i n g f u r t h e r e x p l o r a t i o n i n t o t h e m e c h a n i c s o f a d d i t i v e r e a c t i o n s . A s a r e s u l t , t w o n e w s u f f i c i e n t c o n d i t i o n s f o r a d d i t i v i t y h a v e b e e n d e r i v e d . T h e s e a r e : 1 4 3 i ) a d d i t i v i t y r a n g e i i ) e f f e c t i v e s i t e s a t u r a t i o n . T h e s e t w o s u f f i c i e n t c o n d i t i o n s i n c r e a s e t h e s c o p e o f r e a c t i o n s t o w h i c h t h e a d d i t i v i t y r u l e b e c o m e s a p p l i c -a b l e ( s e e T a b l e 7 . 1 ) , . T h e e f f e c t i v e s i t e s a t u r a t i o n r a t i o i s a s i m p l e b u t e f f e c t i v e m e t h o d o f d e t e r m i n i n g a d d i t i v i t y f o r n u c l e a t i o n a n d g r o w t h r e a c t i o n s a n d c a n b e u s e d f o r h o m o g e n e o u s a n d h e t e r o g e n e o u s r e a c t i o n s . T h e r e a s o n s f o r t h e h e t e r o g e n e i t y e n c o u n t e r e d i n t h e r e a c t i o n s i n t h e p r e s e n t s t u d y n e e d t o b e s t u d i e d f u r t h e r . T h e e f f e c t i v e l y s i t e s a t u r a t e d r e a c t i o n s a r e g r o w t h d o m i n a t e d . T o s t u d y t h e d e v i a t i o n s o f h e t e r o g e n e o u s r e a c t i o n k i n e t i c s f r o m t h e h o m o g e n e o u s , t h e I n h o m o g e n e i t y C o - e f f i c i e n t ( I t ) a n d t h e H e t e r o g e n e i t y C o - e f f i c i e n t ( H ^ ) h a v e b e e n p r o p o s e d . F u r t h e r e x p e r i m e n t s a r e n e e d e d t o s t u d y t h e n a t u r e o f t h e h e t e r o -g e n e i t y a n d i t s e f f e c t o n t h e c o e f f i c i e n t s b a n d n i n t h e A v r a m i e q u a t i o n . U s i n g t h e a d d i t i v i t y r u l e , a n e w i t e r a t i v e p r o c e d u r e , c a l l e d t h e \" a d d i t i v i t y m e t h o d \" , h a s b e e n d e r i v e d t o c a l c u -l a t e T T T d a t a f r o m C C T . T h e m e t h o d h a s b e e n s h o w n t o w o r k s u c c e s s f u l l y f o r t h e d a t a i n t h e p r e s e n t s t u d y . Table 7.1 Scope of Reactions Covered by D i f f e r e n t A d d i t i v i t y C r i t e r i a 144 C r i t e r i o n Remarks Avrami's I s o k i n e t i c Range N 1. must be constant i n the temperature range. Also implies t h a t N and G must be constant f o r an isothermal r e a c t i o n . 2. Normally not encountered f o r a u s t e n i t e - p e a r l i t e r e a c t i o n s i n s t e e l . Cahn's S i t e S a t u r a t i o n 1. The n u c l e a t i o n event i s n e a r l y complete i n the e a r l y stages o f the r e a c t i o n . The growth r a t e must be a f u n c t i o n of temperature alone (thereby implying that i t be constant f o r an isothermal r e a c t i o n ) . 2. May hold true f o r r e a c t i o n s i n a l l o y s t e e l s , e s p e c i a l l y grain-boundary nucleated r e a c t i o n s . A d d i t i v i t y Range 1. N and G are f u n c t i o n s o f temperature alone. N 2. Does not r e q u i r e t h a t ^ be constant. Thus i t i s l e s s r e s t r i c t i v e than the Avrami c r i t e r i o n . Implies t h a t N and G be constant f o r an isothermal r e a c t i o n . 3. There i s evidence i n the l i t e r a t u r e to support N and G are f u n c t i o n s of temperature alone. A l s o , the c r i t e r i o n does not c a l l f o r p h y s i c a l s a t u r a t i o n of n u c l e a t i o n s i t e s . Thus i t i s l e s s r e s t r i c t i v e than Cahn's c r i t e r i o n and includes r e a c t i o n s which are not covered by the Avrami c r i t e r i o n . A p p l i c a b l e to a l l s t e e l s . E f f e c t i v e S i t e S a t u r a t i o n 1. G i s a f u n c t i o n o f temperature alone, and hence constant f o r an isothermal r e a c t i o n . 2. Includes a l l r e a c t i o n s covered by the Avrami, Cahn and A d d i t i v i t y Range c r i t e r i a f o r a u s t e n i t e - p e a r l i t e r e a c t i o n s i n s t e e l f o r growth dominated r e a c t i o n s , which i s u s u a l l y the case. 3. Also includes r e a c t i o n s with decreasing n u c l e a t i o n rates and heterogeneous r e a c t i o n s . 4. The most f l e x i b l e and p r a c t i c a l l y u s e f u l c r i t e r i o n . B I B L I O G R A P H Y 1 4 5 1 . M o r g a n C o n s t r u c t i o n C o m p a n y , \" T h e S t e l m o r P r o c e s s \" , A u g u s t ( 1 9 7 8 ) . 2 . F e l d m a n , U . \" C o n t r o l l e d C o o l i n g o f W i r e S u r f a c e f r o m t h e R o l l i n g H e a t \" , I r o n a n d S t e e l E n g i n e e r , J a n . , p p . 6 2 - 6 7 : ( 1 9 8 0 ) . 3 . A m m e r l i n g , W . J . \" C o n t r o l l e d C o o l i n g o f W i r e R o d f r o m R o l l i n g T e m p e r a t u r e \" , I r o n a n d S t e e l E n g i n e e r , D e c , p p . 9 9 - 1 0 7 ( 1 9 7 0 ) . 4 . S c h u m m e r , A . \" C o n t r o l l e d C o o l i n g P r o c e s s e s f o r W i r e R o d a n d S t r u c t u r a l S h a p e s \" , S E A I S I Q u a r t e r l y , O c t . , p p . 6 - 1 2 : ( 1 9 7 8 ) . 5 . M c L e a n , D . W . \" T h e H i s t o r y o f C o n t r o l l e d C o o l i n g o f R o d a t S t e l c o \" , W i r e , V o l . 3 9 , p p . 1 6 0 6 - 1 6 0 9 ( 1 9 6 4 ) . 6 . D o v e , A . B . \" W i r e M a n u f a c t u r i n g U s i n g S t e l m o r C o n t r o l -l e d C o o l e d R o d \" , W i r e , V o l . 3 9 , p p . 1 6 1 0 - 1 6 1 5 ( 1 9 6 4 ) . 7 . H i t c h c o c k , J . H . \" M e c h a n . i c a l A s p e c t s \" , W i r e , V o l . 3 9 , p . 1 6 2 2 ( 1 9 6 4 ) . 8 . G r a t t a n , E . ; T w i g g , G . M . ; B e n s o n , P . \" S - E D - C : A n A d v a n c e i n t h e C o n t r o l l e d C o o l i n g o f C a r b o n S t e e l R o d \" . I r o n a n d S t e e l I n t e r n a t i o n a l , O c t . , p p . 2 7 7 - 2 8 0 ( 1 9 7 9 ) . 9 . M a l m g r e n , N . G . ; T a r n b l o m , S . G . \" D - p a t e n t e d W i r e R o d \" , W i r e , V o l . 2 5 , p p . 2 1 1 - 2 1 8 ( 1 9 7 5 ) . 1 0 . B e a u j e a n , R . ; G o d a r t , F . ; L a m b e r t , M . ; E c o n o m o p o u l o s , M . \" R e s e a r c h f o r O b t a i n i n g t h e L e a d P a t e n t i n g S t r u c t u r e b y T r e a t m e n t o f t h e W i r e R o d D i r e c t l y a f t e r t h e R o l l i n g M i l l \" , C e n t r e d e R e s e r c h e s M e t a l l u r g i q u e s , V o l . 3 2 , p p . 1 0 - 3 3 ( 1 9 7 2 ) . 1 1 . B a i n , E . C . \" O n t h e R a t e s o f R e a c t i o n s i n S o l i d S t e e l \" , T r a n s . A . I . M . E . , V o l . 1 0 0 , p p . 1 3 - 4 6 ( 1 9 3 2 ) . 1 2 . B a i n , E . C . ; D a v e n p o r t , E . S . , T r a n s . A . I . M . E . , V o l . 9 0 , p . , 1 1 7 ( 1 9 3 0 ) . 1 3 . B r a n d t , H . T r a n s . A . I . M . E . , V o l . 1 6 7 , p . 4 0 5 ( 1 9 4 6 ) . 1 4 . Z e n e r , C . \" K i n e t i c s o f t h e D e c o m p o s i t i o n o f A u s t e n i t e \" , T r a n s . A . I . M . E . , V o l . 1 6 7 , p . 5 5 0 ( 1 9 4 6 ) . 1 4 6 1 5 . H o l l o m a n , J . H . ; J a f f e , L . D . \" A n i s o t h e r m a l D e c o m p o s i -t i o n o f A u s t e n i t e \" , T r a n s . A . I . M . E . , V o l . . 1 6 7 , p . 4 1 9 ( 1 9 4 6 ) . 1 6 . M a n n i n g , G . K . ; L o r i g , C H . \" T h e R e l a t i o n s h i p B e t w e e n T r a n s f o r m a t i o n a t C o n s t a n t T e m p e r a t u r e a n d T r a n s f o r m a -t i o n D u r i n g C o o l i n g \" , T r a n s . A . I . M . E . , V o l . 1 6 7 , p . 4 4 2 ( 1 9 4 6 ) . 1 7 . S c h e i l , E . \" I n i t i a t i o n T i m e o f t h e A u s t e n i t e T r a n s -f o r m a t i o n \" , A r e h . i v . E i s e n h u t t e n w e s e n , V o l . 8 , p p . 5 6 5 -5 6 7 ( 1 9 3 5 ) . 1 8 . A u s t i n , J . B . ; R i c k e t t , R . L . M e t a l s T e c h n o l o g y , T . P . # 9 6 4 , S e p t . ( 1 9 3 8 ) . 1 9 . J o h n s o n , W . A . ; M e h l , R . F . \" R e a c t i o n K i n e t i c s i n P r o c e s s e s o f N u c l e a t i o n a n d G r o w t h \" , T r a n s . A . I . M . E . , V o l . 1 3 5 , p p . 4 1 6 - 4 4 2 ( 1 9 3 9 ) . 2 0 . A v r a m i , M . \" K i n e t i c s o f P h a s e C h a n g e I \" , J . o f C h e m . P h y s i c s , V o l . 7 , p p . 1 1 0 3 - 1 1 1 2 ( 1 9 3 9 ) . 2 1 . A v r a m i , M . \" K i n e t i c s o f P h a s e C h a n g e T I \" , J . o f C h e m . P h y s i c s , V o l . 8 , p p . 2 1 2 - 2 2 4 ( 1 9 4 0 ) . 2 2 . A v r a m i , M . \" K i n e t i c s o f P h a s e C h a n g e I I I \" , J . o f C h e m . P h y s i c s , V o l . 9 , p p . 1 7 7 - 1 8 3 ( 1 9 4 0 ) . 2 3 . C a h n , J . W . \" T h e K i n e t i c s o f G r a i n B o u n d a r y N u c l e a t e d R e a c t i o n s \" , A c t a M e t . , V o l . 4 , p , 4 4 9 ( 1 9 5 6 ) . 2 4 . C a h n , J . W . J - o f M e t a l s , J a n . , p . 1 4 6 ( 1 9 5 6 ) . 2 5 . C h r i s t i a n , J . W . \" T h e T h e o r y o f T r a n s f o r m a t i o n s i n M e t a l s a n d A l l o y s \" , P e r g a m o n P r e s s , C h a p t e r s 1 , 1 2 ( 1 9 6 5 ) . 2 6 . S h i m i z u , N . ; T a m u r a , I . T r a n s . I S I J , V o l . 1 8 , p . 4 4 5 ( 1 9 7 8 ) . 2 7 . S a k a m o t o , Y . e t a l . Y o I t o R e s e a r c h L a b s . , K a w a s a k i S t e e l C o r p n . , C h i b a , J a p a n 2 8 0 . 2 8 . G r a n g e , R . A . ; K e i f e r , J . M . \" T r a n s f o r m a t i o n o f A u s t e n i t e o n C o n t i n u o u s C o o l i n g a n d i t s R e l a t i o n t o T r a n s f o r m a t i o n a t C o n s t a n t T e m p e r a t u r e \" , T r a n s . A . S . M . , V o l . 2 9 , p . 2 9 ( 1 9 4 1 ) . 1 4 7 2 9 . S h i m i z u , N . ; T a m u r a , I . T r a n s . I S I J , V o l . 1 7 , p . 1 7 ( 1 9 7 7 ) . 3 0 . U m e m o t o , M . ; T a m u r a , I . \" C o n t i n u o u s C o o l i n g T r a n s f o r m a -t i o n K i n e t i c s o f S t e e l s \" , T r a n s . I S I J , V o l . 2 1 , p p . 3 8 3 -3 9 2 ( 1 9 8 1 ) . 3 1 . A g a r w a l , P . K . ; B r i m a c o m b e , J . K . \" M a t h e m a t i c a l M o d e l o f H e a t F l o w a n d A u s t e n i t e - P e a r l i t e T r a n s f o r m a t i o n i n E u t e c t o i d C a r b o n S t e e l R o d s f o r W i r e \" , M e t . T r a n s . ' B ' , V o l . 1 2 B , p p . 1 2 1 - 1 3 2 ( 1 9 8 1 ) . 3 2 . H i 1 d e n w a l 1 , B . \" P r e d i c t i o n o f t h e R e s i d u a l S t r e s s e s C r e a t e d D u r i n g Q u e n c h i n g \" , L i n k o p i n g S t u d i e s i n S c i e n c e a n d T e c h n o l o g y , D i s s e r t a t i o n # 3 9 ( 1 9 7 9 ) . 3 3 . H i 1 d e n w a l 1 , B . ; E r i c s s o n , T . \" H a r d e n a b i 1 i t y C o n c e p t s w i t h A p p l i c a t i o n t o S t e e l \" , T h e M e t a l l u r g i c a l S o c i e t y o f A . I . M . E . , W a r r e n d a l e , P a . ( 1 9 7 8 ) . 3 4 . K i r k a l d y , J . S . ; S h a r m a , R . C . \" A New P h e n o m e n o l o g y f o r S t e e l I T a n d CCT. C u r v e s \" , S c r i p t a M e t . , V o l . 1 6 , p p . 1 1 9 3 - 1 1 9 8 ( 1 9 8 2 ) . 3 5 . P r i v a t e c o m m u n i c a t i o n w i t h D r . H a w b o l t , E . B.. . a n d B r i m a e o m b e , J . K . 3 6 . B r o w n , D. ; R i d l e y , N . \" K i n e t i c s o f t h e P e a r l i t e R e a c t i o n i n H i g h - p u r i t y N i c k e l E u t e c t o i d S t e e l s \" , 0 . o f t h e I r o n a n d S t e e l I n s t i t u t e , S e p t . , p p . 1 2 3 2 -1 2 4 0 ( 1 9 6 9 ) . 3 7 . K u b a n , B . M . A . S c . T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , . V a n c o u v e r , B . C . , C a n a d a ( 1 9 8 3 ) . 3 8 . M a r k o w i t z , L . M . ; R i c h m a n , M . H . \" T h e C o m p u t a t i o n o f C o n t i n u o u s T r a n s f o r m a t i o n D i a g r a m s f r o m I s o t h e r m a l D a t a \" , T r a n s . A . I . M . E . , V o l . 2 3 9 , p p . 1 3 1 - 1 3 2 ( 1 9 6 7 ) . 3 9 . T z i t z e l k o v , I . ; H o g a r d y , H . P . ; R o s e , A . \" M a t h e m a t i c a l D e s c r i p t i o n o f t h e T T T D i a g r a m f o r I s o t h e r m a l T r a n s -f o r m a t i o n a n d C o n t i n u o u s C o o l i n g \" , R e p o r t # 1 8 0 8 o f t h e M a t e r i a l s C o m m i t t e e o f t h e A s s o c i a t i o n o f G e r m a n I r o n a n d S t e e l E n g i n e e r s ( 1 9 7 4 ) . 4 0 . T a k e o , K . e t a l . \" T h e D i r e c t P a t e n t i n g o f H i g h C a r b o n S t e e l W i r e R o d b y F i l m B o i l i n g \" , T r a n s . 1 S T J , V o l . 1 5 , p p . 4 2 2 - 4 2 7 ( 1 9 7 5 ) . 1 4 8 4 1 . C a h n , J . W . ; H a g e l , W . C . \" T h e o r y o f t h e P e a r l i t e R e a c t i o n \" , D e c o m p o s i t i o n o f A u s t e n i t e b y D i f f u s i o n a l P r o c e s s e s . E d . V . F . Z a c k a y a n d H . I . A a r o n s o n , A . I . M . E . P u b l i c a t i o n ( 1 9 6 2 ) . 4 2 . S c h e i l , E . ; L a n g e - W e i s e , A . \" S t a t i s t i e h e Ge f u g e u n t e r s u c h u n g e r n \" , V o l . 2 , p . 9 3 ( 1 9 3 7 - 1 9 3 8 ) . 4 3 . H u l l , F . C . ; C o l t o n , R . . A . ; M e h l , R . F . \" R a t e o f N u c l e a -t i o n a n d R a t e o f G r o w t h o f P e a r l i t e \" , T r a n s . A . I . M . E . , V o l . 1 5 0 , p p . 1 8 5 - 2 0 7 ( 1 9 4 2 ) . 4 4 . P r i v a t e C o m m u n i c a t i o n s w i t h D r . I . T a m u r a . 4 5 . A t l a s o f I s o t h e r m a l T r a n s f o r m a t i o n a n d C o o l i n g T r a n s -f o r m a t i o n D i a g r a m s . A S M , M e t a l s P a r k , O h i o ( 1 9 7 7 ) . 4 6 . \" P h y s i c a l C o n s t a n t s o f S o m e C o m m e r c i a l S t e e l s a t E l e v a t e d T e m p e r a t u r e s \" , B I S R A , S c i e n t i f i c P u b l i c a -t i o n s , L o n d o n ( 1 9 7 8 ) . 4 7 . V a n V u u r e n , C . J . \" O p e r a t i n g a n d Q u a l i t y C o n t r o l A s p e c t s o f t h e P r o d u c t i o n o f C r i t i c a l L o w a n d H i g h C a r b o n P r o d u c t s f r o m C o n t i n u o u s l y C a s t B l o o m s \" , S o u t h A f r i c a n I r o n a n d S t e e l I n d u s t r i a l C o r p o r a t i o n L t d . , I s c o r , N e w c a s t l e W o r k s , S o u t h A f r i c a . 4 8 M a r d e r , A . R . ; B r a m f i t t , B . L . \" E f f e c t o f C o n t i n u o u s C o o l i n g o n t h e M o r p h o l o g y a n d K i n e t i c s o f P e a r l i t e \" , M e t . T r a n s . , V o l . 6 A , p p . 2 0 0 9 - 2 0 1 4 ( 1 9 7 5 ) . 4 9 . \" G r a i n S i z e D e t e r m i n a t i o n \" - M e t a l l o g r a p h y , S t r u c t u r e s a n d P h a s e D i a g r a m s , A . S . M . M e t a l s H a n d b o o k , V o l . 8 , 8 t h E d i t i o n , M e t a l s P a r k , O h i o . 5 0 . U m e m o t o , M . ; H o r i u c h i , K . ; T a m u r a , I . \" T r a n s f o r m a t i o n K i n e t i c s o f B a i n i t e D u r i n g I s o t h e r m a l C o o l i n g a n d C o n t i n u o u s C o o l i n g \" , T r a n s . I . S . I . J . , V o l . 2 2 , p p . 8 5 4 -8 6 1 ( 1 9 8 2 ) . 5 1 . C a r n a h a n , B . e t a l . \" A p p l i e d N u m e r i c a l M e t h o d s \" , J o h n W i l e y ( 1 9 6 9 ) . 5 2 . K r e i t h , F . \" ' P r i n c i p i e s , o f H e a t T r a n s f e r \" , 3 r d e d . , I n t e x t P u b l i s h e r s , New Y o r k ( 1 9 7 3 ) . 5 3 . B . H a w b o l t e t a l P a p e r t o b e p u b l i s h e d i n M e t . T r a n s . 5 4 . L e s l i e , W . C . \" T h e P h y s i c a l M e t a l l u r g y o f S t e e l s \" , M c G r a w - H i l l B o o k C o . , p p . 1 6 4 - 1 6 5 ( 1 9 8 1 ) . 1 4 9 A p p e n d i x 1 T H E P R I N C I P L E OF A D D I T I V I T Y [ R e f e r e n c e : \" T h e T h e o r y o f T r a n s f o r m a t i o n s i n M e t a l s a n d A l l o y s \" , J . W . C h r i s t i a n , 1 9 6 5 , P e r g a m o n , L o n d o n , p p . 5 4 5 - 5 4 6 . ] C o n s i d e r t h e s i m p l e s t t y p e - o f n o n - i s o t h e r m a l t r a n s -f o r m a t i o n , o b t a i n e d b y c o m b i n i n g t w o i s o t h e r m a l t r a n s f o r m a -t i o n s . T h e a s s e m b l y i s t r a n s f o r m e d a t t e m p e r a t u r e T-j , w h e r e t h e k i n e t i c l a w i s f = f n ( t ) f o r a t i m e t - j ( f = v o l u m e f r a c t i o n t r a n s f o r m e d . ) a n d i s t h e n s u d d e n l y t r a n s -f e r r e d t o a s e c o n d t e m p e r a t u r e T , , . I f t h e r e a c t i o n i s a d d i t i v e , t h e c o u r s e o f t h e r e a c t i o n a t 1^ 1 S e x a c t l y t h e s a m e a s i f t h e f r a c t i o n t r a n s f o r m e d f ^ ( t - j ) h a d a l l b e e n f o r m e d a t 1^. T h u s i f t ^ i s t h e t i m e t a k e n a t T^ t o p r o d u c e t h e s a m e a m o u n t o f t r a n s f o r m a t i o n a s i s p r o d u c e d a t T.| i n a t i m e t - j , we h a v e a n d t h e c o u r s e o f t h e w h o l e r e a c t i o n i s f = f ] ( t ) t < t 1 = f 2 ( t + t 2 - t 1 ) t > t ] 1 5 0 S u p p o s e t h a t t a l i s t h e t i m e t a k e n t o p r o d u c e a f i x e d a m o u n t o f t r a n s f o r m a t i o n f a a t T-j a n d t a 2 i s t h e c o r r e s p o n d -i n g t i m e t o p r o d u c e t h e s a m e a m o u n t o f t r a n s f o r m a t i o n a t T , , . T h e n i n t h e c o m p o s i t e p r o c e s s a b o v e , a n a m o u n t f o f t r a n s -a f o r m a t i o n w i l l b e p r o d u c e d i n a t i m e t = t a 2 - t 2 + t ] i f t h e r e a c t i o n i s a d d i t i v e ( s e e F i g u r e A . l ) t - t 1 + t 2 t a 2 P± + ^ - * 1 r a 2 za2 t 1 t I f T~ = ~ - t h e n • a l ZaZ t - t , t , T a 2 r a l F o r a n a d d i t i v e r e a c t i o n t h e t o t a l t i m e t o r e a c h a s p e c i f i e d s t a g e o f t r a n s f o r m a t i o n i s o b t a i n e d b y a d d i n g t h e f r a c t i o n s o f t h e t i m e t o r e a c h t h i s s t a g e i s o t h e r m a l l y u n t i l t h e s u m b e c o m e s E q u a t i o n A l . 1 c a n b e w r i t t e n i n t h e m o r e g e n e r a l i s e d f o r m a s 151 t r d t J i ( A l . 2 ) 0 W h e r e t ( T ) a r e t h e i s o t h e r m a l t r a n s f o r m a t i o n t i m e s a f o r a f r a c t i o n t r a n s f o r m e d ' a ' a t a t e m p e r a t u r e T a n d ' t ' i s t h e t i m e t a k e n t o r e a c h a f r a c t i o n t r a n s f o m r e d ' a ' f o r a n a i s o t h e r m a l r e a c t i o n p a t h . T h e a b o v e r e l a t i o n s h i p h o l d s i f , f o r a r e a c t i o n , t h e r e a c t i o n r a t e d e p e n d s o n l y o n t h e f r a c t i o n t r a n s f o r m e d a n d t e m p e r a t u r e . C o n s i d e r a t r a n s f o r m a t i o n f o r w h i c h t h e i n s t a n t a n e o u s r e a c t i o n r a t e m a y b e w r i t t e n T h e d e r i v a t i o n o f E q . A l . ; 2 i s b a s e d o n t h e r e l a t i o n -s h i p t h a t , f o r a n a d d i t i v e r e a c t i o n , t ( A l . 3 ) d X d t h ( T ) ( A 1 . 4 ) gTx) w h e r e : X v o l u m e f r a c t i o n t r a n s f o r m e d t t i m e h ( T ) f u n c t i o n o f t e m p e r a t u r e g ( x ) f u n c t i o n o f - v o l u m e ' f r a c t i o n t r a n s f o r m e d . 1 5 2 T h e n , / h ( T ) d t = X ) d X ( A 1 . 5 ) f o r a n y t r a n s f o r m a t i o n p a t h . ' . X = F y h ( T - ) d t ( A l . 6 ) a n d f o r a n i s o t h e r m a l r e a c t i o n X = F j h ( T ) t j ( A l . 7 ) A c c o r d i n g t o A 1 . 7 , t h e v o l u m e f r a c t i o n t r a n s f o r m e d a t a n y t e m p e r a t u r e i s a f u n c t i o n o f t i m e a n d t e m p e r a t u r e . F r o m A l . 5 A l s o h ( T ) = g ( X a ) ( A l . 8 ) t^TT d X = hir dt \" 9TXT t ( T ) d X - 9 ( X a ) V T ) dt ITXT d t _ g X . + i ^ r y - ttrj dt • • JtjT) ~ g - n r y j 9 ( x ) d x - tttry a % ' J V a ' ' a 0 1 5 3 T h i s i s t h e g e n e r a l a d d i t i v i t y r u l e ( E q . A l . 9 ) . I n p a r t i c u l a r , i f X = X , , t h e n a r dt 9 ( x a> F i g . A 1 . 1 . A d d i t i v i t y P r i n c i p l e . A p p e n d i x 2 A D D I T I V I T Y OF THE AVRAMI EQUATION K I N E T I C S T h e A v r a m i e q u a t i o n i s : X = 1 - e x p ( - b t n ) . ' . l o g ( l - X ) = - b t n D i f f e r e n t i a t i n g A 2 . 1 w . r . t . ' t ' , t n i o q n-x) - b d X d t = e x p ( - b tn ) ( - n b t n - 1 = ( 1 - X ) ( - b n ) f i o g d - x j •n n = ( l - X ) ( n ) ( - b ) n n ( - b ) n n - 1 I h ( T ) g(x) w h e r e 1 h(T) = n - ( - . b ) n 1 5 6 A p p e n d i x 3 I N D E P E N D E N C E OF N ( T ) A N D G ( T ) w . r . t . T I M E C o n s i d e r t h e e q u a t i o n t T 3 T T X N ( T ; ) d f - ( x - t ) 3 d t 0 [ 0 T 0 ( A 3 : : l ) S i n c e G ( t ) a n d N ( T ; ) a r e f u n c t i o n s of-', t e m p e r a t u r e a l o n e E q . ( A 4 . 1 ) c a n b e w r i t t e n a s : T 3 T = J G ( T ) d T J • — L — j N ( T ) d T -J ( x - t ) 3 d t e x \" ' -'0 \" \" ' x - ' O T Q T Q 0 ( A 3 . 2 ) I n t h e c a s e o f a c o n t i n u o u s c o o l i n g r e a c t i o n , s i n c e t h e t e m p e r a t u r e i s a f u n c t i o n o f t i m e ( w h i c h i s g i v e n b y t h e c o o l i n g r a t e ) , i t w o u l d s e e m t h a t T = f ( t ) ( A 3 . 3 ) . ' . N ( T ) = N{ f ( t ) } = •N1 ( t ) ( A 3 . 4 ) G ( T ) = G { f ( t ) } = G ^ t ) ( A 3 . 5 ) B u t t h i s i s n o t t r u e , a s c a n b e s e e n f r o m F i g . A $ . 1 . T h e f i g u r e i s a t h r e e - d i m e n s i o n a l r e p r e s e n t a t i o n o f t , G ( T ) a n d T . A s c a n b e s e e n , t h e G ( T ) f o r t h r e e d i f f e r e n t r e a c t i o n s c a n b e 1 5 7 o b t a i n e d , i n d e p e n d e n t o f t h e t i m e . T h e G ( T ) f o r a r e a c t i o n d e p e n d s o n l y o n t h e t e m p e r a t u r e p a t h o f t h e r e a c t i o n . T h e t i m e o f t h e r e a c t i o n c a n t h e n , b e b r o u g h t i n . , a n d w h e n m u l t i -p l i e d b y t h e t e m p e r a t u r e a v e r a g e d g r o w t h r a t e f o r t h e r e -a c t i o n , g i v e s t h e t o t a l g r o w t h o f o n e p a r t i c l e d u r i n g t h e t i m e i n t e r v a l . T h i s s h o u l d b e m u l t i p i i e d b y t h e n u m b e r o f p a r t i c l e s g r o w i n g t o g i v e t h e t o t a l g r o w t h . T h u s , i t c a n b e d e m o n s t r a t e d t h a t i f G i s a f u n c t i o n o f t e m p e r a t u r e a l o n e , i t c a n b e c a l c u l a t e d , i n d e p e n d e n t o f t i m e , f o r a n y r e a c t i o n p a t h . T h e s a m e i s t r u e f o r N . F i g . A 3 . 1 D e m o n s t r a t i n g t h e i n d e p e n d e n c e o f G ( T ) N ( T ) w i t h r e s p e c t t o t i m e . 1 5 9 A p p e n d i x 4 C A L C U L A T I O N OF t A V _ T T T FROM t A V - C C T B Y T H E A D D I T I V I T Y M E T H O D T h e a d d i t i v i t y m e t h o d h a s b e e n u s e d t o c a l c u l a t e t h e t^_jjj f o r t h e s t e e l u s e d i n t h e p r e s e n t w o r k f r o m t h e e x p e r i m e n t a l l y d e t e r m i n e d t A ^ _ Q ^ j . B y u s i n g a m u l t i p l e r e -g r e s s i o n p r o c e d u r e , a f i r s t a p p r o x i m a t i o n f o r t A ^ _ y j j w a s d e t e r m i n e d f r o m t h e e x p e r i m e n t a l l y d e t e r m i n e d t^^-ryy d a t a . T h i s i s s h o w n i n T a b l e A 4 . 1 . B y s u c c e s s i v e i t e r a t i o n s , t h e b e s t a p p r o x i m a t i o n f o r t A^_y^ T t o t h e e x p e r i m e n t a l d a t a w a s c a l c u l a t e d b y t h e a d d i t i v i t y m e t h o d . T h e r e s u l t s o f t h e i t e r a t i o n s a r e s h o w n i n T a b l e A 4 . 2 . T h e c o m p a r i s o n o f t h e t ^ y JJJ v a l u e s p r e d i c t e d b y t h e a d d i t i v i t y m e t h o d a n d t h e e x p e r i m e n t a l d a t a i s s h o w n i n T a b l e / . 3 . ; . S 1 -1 60 T a b l e A 4 . T C o m p a r i s o n o f E x p e r i m e n t a l a n d F i r s t A p p r o x i m a t i o n o f T A V - T T T T e m p e r a t u r e * A V -TTT ( s ) ( ° C ) E x p e r i m e n t a l . F i r s t i i i A p p r o x i m a t i o n 6 8 0 4 3 58 6 7 0 5 . 6 12 6 6 0 6 . 2 6 . 3 6 5 0 3 . 0 4 . 0 6 3 0 1 . 8 2 . 1 6 2 3 1 . 6 2 . 0 615 1 . 5 1 . 9 6 0 3 1 . 9 1 . 7 161 T a b l e A 4 . 2 I t e r a t i o n R e s u l t s A p p r o x i m a t i o n Number C o - e f f i c i e n t i n t h e ; E q u a t i o n f o r t ^ _ j j j * l o g a b c 1 4 0 . 0 0 - 1 0 . 00 0 . 1 0 0 2 39 8 5 - 1 0 39 0 . 0 8 7 3 45 75 - 1 2 32 0 . 1 1 7 4 4 3 66 -11 6 0 0 . 1 0 3 5 44 12 -11 74 0 . 1 0 5 6 4 3 . 86 -11 66 0 . 1 0 4 * l o g t A V _ T T T =• l o g a + b l o g x + c x x = 7 2 8 - T 1 6 2 A p p e n d i x 5 L I S T I N G OF C O M P U T E R P R O G R A M TO C A L C U L A T E T T T D A T A FROM C C T BY T H E A D D I T I V I T Y M E T H O D 1 6 2 a 1 C ***************************.***************************** 2 C This program c a l c u l a t e s the a d d i t i v i t y i n t e g r a l value 3 , C upto a time t(AV-CCT)for a given c o o l i n g rate.The f i r s t 4 C approximation for t(AV-TTT) i s to be input by r e p l a c i n g 5 C the values in statement no.34.The c o o l i n g r a t e i s to be 6 C s p e c i f i e d by r e p l a c i n g the value in statement no.23 by an 7 C appropriate temperature value.The t(AV-CCT) for the s t e e l 8 C must be input by r e p l a c i n g the values in statement no.24. 9 C The time step value i s s p e c i f i e d in statement no.29 and the 10 C T(A1) temperature i s s p e c i f i e d by statement no.30.The program 11 C c a l c u l a t e s the a d d i t i v i t y i n t e g r a l value f o r the given 12 C c o o l i n g rate and prepares the input data f o r the next 13 C i t e r a t i o n . T h e s e values are then used as input to the 14 C m u l t i p l e r e g r e s s i o n package a v a i l a b l e in the general MTS 15 C system as *STRP.The *STRP program gives the next 16 C approximation f o r the t(AV-TTT).Thais i s input to t h i s .17 C program for the next run.Thus the program must be run 18 C r e p e t i t i v e l y , a l o n g with * S T R P , t i l l the r e q u i r e d t(AV-TTT) 19 C i s obtained. 20 C ************************************************************** 21 IMPLICIT REAL*8(A-H,0-Z) 22 DIMENSION A(100),B(100),C2(100) 23 A1=600.0D0 24- TAVCCT=41.4594D0+0.0698528D0*(728.0D0-A1)-l0.096lD0* 25 1(DLOG(728.0D0-A1)) 26 TAVCCT=DEXP(TAVCCT) 27 CR=(728.0D0-A1)/TAVCCT 28 C=0.0D0 29 DT=0.10D0 30 T=727.0D0 31 C1=1.0D0/CR 32 DO 100 1=1,30000 33 IF (T.LE.A1) GO TO 500 34 T1=40.48l7D0-l0.6052D0*DLOG(728.0D0-T)+0.088608 DO* 35 1(728.0D0-T) 36 T1=DEXP (T1) 37 C=C1*(DT/T1)+C 38 100 T=T-DT 39 500 T=700.0D0 4 0 PRINT,C 41 DO 10 1=1,30 42 T2=40.478 D0-10.6132 D0*DLOG(728.ODO-T)+0.0889042 DO* 43 1(728.0D0-T) 44 T2=DEXP(T2) 45 IF (T.GE.A1) T2=T2*C 46 A(I)=DLOG(T2) 47 B(I)=728.0D0-T 48 C2(I)=DLOG(B(I)) 49 WRITE(6,200) A(I ) , B ( I ) , C 2 ( I ) 50 200 FORMAT(F20.10,F20.10,F20.10) 51 10 T=T-5.0D0 52 STOP 53 END End of f i l e 1 6 3 A p p e n d i x 6 DERIVATION OF I M P L I C I T F I N I T E DIFFERENCE EQUATIONS i ) Node A r r a n g e m e n t .Central node S u r f a c e node G e n e r a ! internal node i i ) H e a t F l o w E q u a t i o n s a ) C e n t r a l Node ,,1 I ID H e a t f l o w a c r o s s A'B'C'D' i s : 1 6 4 A R T . i , i + l A e . 1 i + 1 , h i ,ri A R R a t e o f h e a t a c c u m u l a t i o n i s n r T i » n + I \" T i , n 1 , A R * 2 , p Cp ~ A t 2 (.T ) A e 1 i , 1 + 1 T i , h + 1„ \" T i , n C AR*\" 8 A t 1 ' n + 1 1 ' n ' l . n V C A R ' T , „ V + p—2 -4 K i , i + l A t } \" T i + l , n C A R p • T . + . 4 K . i + 1 A t ^ T h e r a t e o f l a t e n t h e a t g e n e r a t e d d u e t o t r a n s f o r m a -t i o n i s : „ A F 1. / ARx H ' p A t 2 ( T ) C A R ' T , . II + P - 2 — i , n £ 4 K i j i = 1 A t J i+lvo 1 6 5 b ) G e n e r a l I n t e r n a l N o d e I n f l o w a c r o s s AD i s i - 1 , n i , n j R _ AJR 2 1 2 1 T - - T . , •'LAS ! ^ 1 * H J A R O u t f l o w a c r o s s BC i s : K i + l , n + K i , n + M | A e i + l , n i , n AR R a t e o f a c c u m u l a t i o n i s : T i , n + 1 ~ T i , n R M A R P A t R a t e o f o u t f l o w - R a t e o f I n f l o w R a t e o f a c c u m u l a t i o n + R a t e o f g e n e r a t i o n . R a t e o f h e a t g e n e r a t i o n ( d u e t o l a t e n t h e a t l i b e r a t i o n d u r i n g t r a n s f o r m a t i o n ) i s : Hp R AG AR i - l yn i - l , i 0 T i , n [ | K i + l , i 2R + AR + K 2R_ i+1 -1 CnRAR?1 A t J jf „ AR + 2R 1 ,nY S-+1..1 2 J C R A R A P = T i n+1 P - 2 + H P — R A R 1 , n ' t A t S u r f a c e Node I n f l o w a c r o s s AB i s : O u t f l o w a c r o s s CD i s : ' [ T . - T i . » ] R Q A B R a t e o f h e a t g e n e r a t i o n d u e t o l a t e n t h e a t o f t r a n s f o r m a -t i o n i s : U A F D A R I , A R q A - p p A T R 0 - T r A e , 1-T R a t e o f h e a t a c c u m u l a t i o n i s T ~ i , n + l T i , n D AR . n , AR p C p — * u > - R Q - -j- te-l.-j-T K AR - 2 R + i - l , n 2 A R T . K • , , 2 R . , \" A R •+ h R n i » n i - l 5 i 2 A R 0 ; C p A R ( 4 R . Q - A R ) 8 A t h R 0 T a + ARHp H ( 4 R 0 \" A R > + C A R p — P _ ( 4 R - A R ) . T 8 A t U . 1 , n 1 6 8 A p p e n d i x 7 C O M P A R I S O N OF M O D E L P R E D I C T I O N W I T H A N A L Y T I C A L S O L U T I O N OF E Q . (4.1) F o r a b o d y w i t h n e g l i g i b l e i n t e r n a l r e s i s t a n c e , E q . (4.1) c a n b e r e - w r i t t e n a s : V p C = h A ( T - T ) ( A 7 . 1 ) p d t a hA t w h e r e ^ 1 = e ' P V G P ( A 7 . 2 ) G 0 e ( t ) = T ( t ) - T S 0 n T N - T 0 0 0 0 T ( t ) = t e m p e r a t u r e o f b o d y a t a n y t i m e t T^ = t e m p e r a t u r e o f b o d y a t t i m e = 0 0 TQ = i n i t i a l t e m p e r a t u r e o f t h e b o d y h = c o n v e c t i v e h e a t - t r a n s f e r c o - e f f i c i e n t A = s u r f a c e a r e a p = d e n s i t y V = v o l u m e C = s p e c i f i c h e a t P t = t i m e 1 6 9 U s i n g t h e a b o v e e q u a t i o n , t h e v a l u e s o f w e r e c a l c u l a t e d f o r r o d s o f d i a 5 . 5 , 8 . 5 , 1 3 . 5 a n d 2 5 mm w i t h a v a l u e o f 2 h = 2 5 0 W/m ° C . T h e m o d e l p r e d i c t i o n s u s i n g t h e s a m e h v a l u e w e r e c a l c u l a t e d . T h e v a l u e s o f T^ p r e d i c t e d b y t h e m o d e l f o r t h e s u r f a c e o f t h e r o d w e r e t h e n c o m p a r e d w i t h t h e a n a l y t i c a l s o l u t i o n v a l u e s . A s a m p l e c o m p u t e r o u t p u t s h o w -i n g t h e c o m p a r i s o n i s s h o w n i n F i g . A ? . 1 . F o r a 5 . 5 mm d i a r o d , t h e m a x i m u m d i f f e r e n c e b e t w e e n t h e t w o s o l u t i o n s i s l e s s t h a n 1% i n t h e p r e - t r a n s f o r m a t i o n p e r i o d , t h e r e g i o n w h e r e t h e c o m p a r i s o n i s m e a n i n g f u l . S u m m a r y o f C o m p a r i s o n R e s u l t s ( A l l c o m p a r i s o n s m a d e b e f o r e t h e S t a r t o f t h e T r a n s -f o r m a t i o n . ) R o d d i a m e t e r (mm) M a x i m u m d i f f e r e n c e b e t w e e n m o d e l p r e d i c t e d a n d a n a l y t i c a l s o l u t i o n v a l u e s o f t e m p e r a t u r e {%) 5 . 5 1 . 0 8 . 5 0 . 9 1 3 . 5 1 . 4 2 5 2 . 6 COMPARISON OF ANALYTICAL AND CALCULATED VALUES OF TEMPERATURE FOR 0.8%C STEEL ROD *********************************************** DIAMETER OF ROD= 5.5 MM TIME(SECONDS) % DIFFERENCE ********************************** 0. 10 0. 395115344578634 0. 20 0 .505964762365217 0. 30 0. 550808718128153 0. 40 0. 569784433443964 0. 50 0. 577463027588642 0. 60 0. 57831 1-603873660 0. 70 0. 576850442366463 0. 80 0. 574224223074626 0. 90 - :- 0. 569262*48676523 1. 00 •0. 566578789140192 1. 10 0. 561-543548630365 1. 20 0. 555303321342505 1. 30 0. 550182225960195 1. 40 0. 543847722637660 1. 50 0. 537470009879105 1. 60 0. 531036110455511 1. 70 0. 524558328991617 1. 80 0. 516841917661739 1 90 0. 510258825212505 2 00 0 502428403952435 2 10 0 493338134677705 - 2 20 - = .:o 486594560740012 - 2 30 0 -477376003093878 2 40 0 468094394994059 2 50 0 458737923146185 2 60 0 449317476984690 2 70 :: 0 438601864670495 -2 80 0 42780748663V018 -2 .90 • - : . -0 418172622645464 3 .00 : 0 407223576883559 3 .10 0 .396199465938024 - -3 .20 : = .\"0 .383843670941287 3 .30 • ~ 6 372653719659355 ^ 3 .40 r -: \"6 .360127487640224 - 3 .50 - 0 .348774253660875 -3 .60 0 .336070751737211 - 3 .70 0 .322008113590259 - --3 .80 0 .309120976974933 - -3 .90 o 1296141228730804 4 .00 • :0 ;281787557798469 4 .10 '- % ' 6 ;267331568259515 4 .20 0 .252772762445194 - -4 .30 0 .239407821518229 ; .4 .40 •\"\" \"0 :223347647192569 : 4 .50 - - \" • • \" : • £ ) =208479920777633 4 .60 0 .192199573761493 4 .70 0 .175804568294831 4 .80 0 .160612962255957 4 .90 0 .143991334099522 5 .00 0 .127255133286501 5 .10 0 . 1 10401702580471 5 .20 0 .093430501054428 5 .30 0 .076340989754242 5 .40 0 .057788708355173 169b 5. 50 0. 040456893180318' 5. 60 0. 021651172546309 5. 70 0. 002715730871076 5. 80 -0. 014988009592328 5. 90 -0. 032814230437934 6. 00 -0. 052134782131489 6. 10 -0. 071588859672065 6. 20 -0.089795128959617 6. 30 -0. 109512323601988 6. 40 -0. 129364306579489 • 6. 50 -0. 149351646357633 6. 60 -0. 169477281640425 6. 70 -0. 189737319222494 6. 80 -0. 210137364962064 6. 90 -0. 229263667369554 7. 00 -0. 248516004430685 7. 10 -0. 270747834017325 7. 20 -0. 290273686618808 7. 30 -0. 309930696052692 7 . 40 -0. 329719378572555 7. 50 -0. 351085040598724 7. ,60 -0. ,369699166364627 7. ,70 -0. ,391341034071401 7. ,80 -0. ,41 1672822294564 7 . ,90 -0. , 430674577016040 B. ,00 -0. ,451271497868586 8. ,10 -0. ,470536175234161 8. .20 -0. ,489920369438442 8, .30 -0. .509438726254661 8. .40 -0. . 529084757687836 8, .50 -0, . 547363383483388 6 .60 -0, .565768740151578 8 .70 -0, .587287484753110 6 .80 -0, .607434270172263 6 .90 -0 .629216890304000 9 .00 -0 .655682838148267 9 .10 -0 .683823304942449 9 .20 -0 .718223583460949 9 .30 -0 .760462115160079 9 .40 -0 .810602558175372 9 .50 -0 .873288200562619 9 .60 -0 .9517019034038,12. 9 .70 -1 .04B983651395537 9 .80 -1 . 171394301914426 9 .90 -1 .322094747522043 10 .00 -1 .505746227816433 10 .10 -1 .723840987020306 10 .20 -1 .977839422006990 10 .30 -2 .270624578570190 10 .40 -2 .595793403077311 10 .50 -2 .949940405244339 10 .60 -3 .326696662329319 10 .70 -3 .722737528202555 10 .80 -4 .131765590017377 10 .90 -4 .544530137822069 1 1 .00 -4 .961002829053284 1 1 .10 -5 .375185445459516 1 1 .20 -5 .784053674520205 1 1 .30 -6 .186274204109633 1 1 .40 -6 .583374875373876 A p p e n d i x 8 L I S T I N G OF C O M P U T E R P R O G R A M TO C A L C U L A T E T E M P E R A T U R E R E S P O N S E OF A S T E E L ROD U N D E R -G O I N G C O O L I N G . S t e e l : 0 . 8 2 % C - 0 . 8 2 % Mn - 0 . 2 6 % S i ( P l a i n c a r b o n e u t e c t o i d ) G r a i n S i z e : 5 - 7 A S T M A u s t e n i t i s i n g C o n d i t i o n s : 8 5 0 ° C - 5 m i n u t e s 1 7 0 a j Q ****************************************************************** 2 C T h i s program c a l c u l a t e s the temperature ( i n *C) i n s i d e a 3 C c y l i n d r i c a l rod of 0 .82%C-0 .82%Mn-0 .26%Si s t e e l undergo ing 4 C c o o l i n g . l t t a k e s . into account the e f f e c t of the l a t e n t heat 5 C of t r a n s f o r m a t i o n of a u s t e n i t e to p e a r l i t e g e n e r a t e d , d u r i n g 6 C c o o l i n g , o n the temperature of the r o d . C a l c u l a t i o n s are 7 C based on a 1-D I m p l i c i t F i n i t e D i f f e r e n c e Unsteady S t a t e 8 C Heat T r a n s f e r Model .The model has been w r i t t e n f o r a c o n s t a n t 9 C node d i s t a n c e . D e n s i t y i s c o n s i d e r e d c o n s t a n t . V a r i a t i o n s of Thermal-10 C C o n d u c t i v i t y and S p e c i f i c Heat have been i n c o r p o r a t e d i n t o the 11 C model by u s i n g BISRA d a t a . J2 C *************************************************************: 13 14 15 16 17 IMPLICIT REAL*8 ( A - H , 0 - Z ) 18 REAL*8 KK 19 20 21 22 Q ****************************************************** 23 C Data f o r t h i s program i s : . 24 C ****************************** 25 C DX = Node d i s t a n c e , M e t e r s 26 C DT = Time inc rement ,Seconds 27 C T0TTIM = T o t a l t ime c o u n t e r , S e c o n d s ( I n i t i a l Value=0) 28 C H = C o n v e c t i v e Heat t r a n s f e r c o e f f i c i e n t a t the rod 29 C surface,W/m *C 30 C R = Rad ius of r o d , M e t e r s 31 C TATMOS = Atmospher ic temperature at rod s u r f a c e , * C 32 C XX = D i s t a n c e of node from the rod c e n t r e , M e t e r s 33 C M = Number of nodes+1 34 C DD = D e n s i t y of s t e e l , K g / c u b i c Meter 35 C T1 = Maximum t ime upto which c a l c u l a t i o n s a re t o 36 C be done 37 C FRMAX = Maximum F r a c t i o n t r a n s f o r m e d a f t e r which 38 C check f o r t r a n s f o r m a t i o n at the node i s 39 . C t e r m i n a t e d . 40 Q ************************************************************ 41 42 43 44 C ********************************************************** 4 5 C DATA FOR THIS PROGRAM IS ENTERED BY INPUTTING APPROPRIATE .46 C VALUES OF THE VARIABLES IN THE STATEMENT NO.62 . IF THE 4 7 C PROGRAM IS TO BE RUN WITH A CONSTANT HEAT TRANSFER 48 C COEFFICIENT,THEN INSERT THE FOLLOWING AFTER THE 4 9 C STATEMENT NO.167: 50 C GO TO 220 51 C INSERT AFTER STATEMENT NO.221 52 C H=PRESELECTED VALUE. 53 C BY THE ABOVE PROCEDURE THE CALCULATION OF THE HEAT 54 C TRANSFER COEFFICIENT FOR A GIVEN AIR VELOCITY,EMI SSIVITY, 55 C ETC. IS BYPASSED.(THE PROGRAM DOES NOT EXECUTE STATEMENT 56 C NOS.175 TO 221) 57 Q ***************************************************** 58 59 60 170b 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 ' 101 1 02 103 104 105 106 107 108 109 110 1 1 1 112 113 1 14 115 1 16 1 17 118 119 120 C C C C C C C C C C C C C C C C C C c c c c c c c c c c c c c c c c c c c c c c c c c c c c DATA DX,DT,T1 ,DD,TATMOS, R , XX , M ,FRMAX/ 0 . 0 0 7 5 D 0 , 0 . 1 DO,100 .0D0, 17 6 5 0 . 0 D 0 , 2 0 . 0 D 0 , 0 . 0 1 5 D 0 , 0 . 0 D 0 , 1 1 , 0 . 9 9 D 0 / VEL=20.0D0 EMISS=0.3D0 TOTTIM=0.0D0 J1=0 K9=0 K1=0 ************************************** M a t r i x I d e n t i f i c a t i o n AA , BB , C D T KK,CP THERM 1,THERM2 CP1,CP2 DF AN,ALB 33 TAVRAM BETA,GAMMA C o n t a i n c o e f f i c i e n t s of the T r i d i a g o n a l system at each t ime s t e p C o n t a i n s the RHS of the t r i d i a g o n a l system C o n t a i n s the temperature of each node at each t ime s t e p C o n t a i n Thermal C o n d u c t i v i t y and S p e c i f i c Heat of each node at each t ime s t e p C o n t a i n c o e f f i c i e n t s of the P o l y n o m i a l used to c a l c u l a t e Thermal C o n d u c t i v i t y as a f u n c t i o n of temperature f o r F e r r i t e and A u s t e n i t e r e s p e c t i v e l y C o n t a i n c o e f f i c i e n t s of the p o l y n o m i a l used to c a l c u l a t e S p e c i f i c Heat as a f u n c t i o n of temperature f o r A u s t e n i t e and P e a r l i t e r e s p l y C o n t a i n s t o t a l f r a c t i o n t r a n s f o r m e d f o r each node at each t ime s t e p C o n t a i n s the i n c r e m e n t a l f r a c t i o n t rans fo rmed f o r each node at each t ime s t e p C o n t a i n s the c o e f f i c i e n t s of the P o l y n o m i a l used t o c a l c u l a t e ' n ' and 'Log b' as a . f u n c t i o n of temperature .The data f o r these was genera ted a t the M e t a l l u r g y D e p t . o f UBC. C o n t a i n s the i n f o r m a t i o n on whether a node has s t a r t e d t r a n s f o r m a t i o n . I f J J ( i ) = i , then the ' i ' t h node has s t a r t e d t r a n s f o r m i n g . I f _ n o t , J J ( i ) = 0 C o n t a i n s the Avrami t ime f o r each node at each t ime s tep .The data f o r t h i s has been generated at the M e t a l l u r g y Dept . of UBC Are dummy m a t r i c e s used f o r c a l c u l a t i n g the temperature of each node at each t ime s tep ******************************************* * UNITS * ************* Time D i s t a n c e S p e c i f i c Heat -Thermal C o n d u c t i v i t y C o n v e c t i o n Heat T r a n s f e r C o e f f i c i e n t Dens i ty Note on D i m e n s i o n i n g ***** of M a t r i c e s Seconds Meters W/Kg *C W/M *C W/m2 *C Kg/m3 C M a t r i c e s AA ,BB ,C ,D ,T ,TT ,DF ,KK ,CP , J J ,F ,TAVRAM,BETA ,GAMMA shou ld C be d imens ioned at l e a s t M (=no. of nodes+i) C 1 7 0 c 121 1 22 123 1 24 125 DIMENSION A A ( 3 0 ) , B B ( 3 0 ) , C ( 3 0 ) , D ( 3 0 ) , T ( 3 0 ) , T T ( 3 0 ) , D F ( 3 0 ) , 126 1 K K ( 3 0 ) , C P ( 3 0 ) , A L B ( l 0 ) , A N ( 5 ) , T H E R M 1 (10) ,THERM2(10) ,F (50 ,2 ) , J J (31 127 1 CP 1(10) ,CP2(10) ,TAVRAM(30) ,H1(20) ,BETA(30) ,GAMMA(30) 128 DIMENSION K 5 ( 1 0 0 ) , T A V ( 1 0 0 ) , T A 1 ( 1 0 0 ) , T I ( 1 0 0 ) 129 CALL THER(THERM1) 130 CALL THERMA(THERM2) 131 CALL CEEPE2(CP2) 132 CALL CEEPE1(CP1) 133 CALL LOGB(ALB) 134 CALL EXPONT(AN) 136 DO 3001 1=1,M 137 3001 K5(I )=0 138 139 140 C ******************************************************** 141 C I n i t i a l i s a t i o n . o f Temperature at a l l nodes at TOTTIM=0.The 142 C f r a c t i o n t r a n s f o r m e d m a t r i x F and the J J m a t r i x a re a l s o 143 C i n i t i a l i s e d to 0 . 144 C ******************************************************************' 145 146 147 148 DO 10 1=1,M 149 F ( I , 1 ) = 0 . 0 D 0 150 F ( I , 2 ) = 0 . 0 D 0 151 J J ( I )=0 152 10 T( I )=850.0D0 153 154 155 156 C ************************************************************* 157 C S t a r t i n g w i t h TOTTIM=0 the t ime i s incremented i n s t e p s of 158 C DT.The c a l c u l a t i o n s w i l l s t o p when TOTTIM v a l u e i s G r e a t e r 159 C than or Equa l to a p r e s p e c i f i e d v a l u e 160 C *********************************************************** 161 162 163 164 165 25 TOTTIM=TOTTIM+DT 166 IF (TOTTIM.GE.T1) GO TO 1001 167 K7=0 168 169 170 171 C ******************************************************** 172 C C a l c u l a t i o n of c o n v e c t i v e heat t r a n s f e r c o e f f i c i e n t by 173 C u s i n g a i r v e l o c i t y 175 TEM=((T(M)+TATMOS)/2.0D0)* 1.8DO+32.0DO 176 IF (TEM.GT.900.0D0) AIRK=(1.617D-05*TEM+1.575D-02)/2419.0D0 177 IF (TEM.LE.900.0D0) AlRK=(1.860D-05*TEM+1.372D-02)/2419.0D0 178 IF ( V E L . L E . ( 0 . 1 D - 8 ) . A N D . V E L . G E . ( - 0 . 1 D - 8 ) ) GOTO 130 179 IF (TEM.GT.1000.0D0) A lRNU=(1 .233D-06*TEM-3 .060D-04)*9 .2894D-0 ; 180 IF (TEM.LE .1000.0D0.AND.TEM.GT.800.0D0) 181 1AIRNU=(1.0D-06*TEM-8.3D-05)*9.2894D-02 170d 182 163 1 84 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 21 1 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 IF (TEM.LE.800.0D0) AIRNU=(6.475D-07*TEM+3 . 9D- 05)*9.2894D-02 RE= (VEL*2.0D0*R)/AIRNU IF ( R E . G E . 0 . 4 D 0 . A N D . R E . L T . 4 . 0 D 0 ) VC = 0 . 891D0 IF ( R E . G T . 0 . 4 D 0 . A N D . R E . L T . 4 . 0 D 0 ) VN = 0. 33D0 IF ( R E . G T . 4 . 0 D 0 . A N D . R E . L T . 4 0.0D0) VC = 0. 821 DO IF ( R E . G T . 4 . 0 D 0 . A N D . R E . L T . 4 0.0D0) VN = 0. 385D0 IF (RE .GT .40 .0D0 .AND.RE .LT .4 0 00.0D0) VC = 0. 61 5D0 IF (RE.GT.4 0 .0D0.AND.RE .LT .4 0 0 0.0D0) VN= 0. 466D0 IF (RE.GT.4000.ODO.AND.RE.LT .40000.0D0) VC = 0. 174D0 IF (RE.GT.4000.ODO.AND.RE.LT .40000.0D0) VN = 0. 618D0 IF (RE.GT.40000.ODO.AND.RE.LT.400000.ODO) VC = 0 . 0239D0 IF (RE.GT.40000.ODO.AND.RE.LT .400000.0D0) VN= 0 . 805D0 H=(VC*AIRK/(2.0D0*R) )*(RE**VN) + 1 .366D-1 1 *EMISS* ('( (T(M) 1+273.0D0)**4)-((TATMOS+273.0D0)**4))/(T(M)-TATMOS) GO TO 230 Q *************************************************** C C a l c u l a t i o n of the r a d i a t i v e heat t r a n s f e r c o e f f i c i e n t Q *********************************************************** 130 D2=(T(M)-TATMOS)*1.8D0 IF (TEM.GT.10O0.OD0) GR=(335.29345D0*(10.0D0**(TEM / * ( - 0 . 0 0 1 1 0 2 1 8 D 0 ) ) ) ) * 1 0 0 0 . 0 D 0 IF (TEM.LE.1000.0D0.AND.TEM.GT.800.0D0) GR=(621.10241 DO *(10.0DO**(TEM*(-0.00136992DO)) ) )*1000.0D0 IF (TEM.LE.800.0D0.AND.TEM.GT.500.0D0) GR=(1100.7197D0 *(10.DO**(TEM*(-0.00168056DO)) ) )*1000.0D0 IF (TEM.LE.500.0D0.AND.TEM.GT.300.ODO) GR=(2071.8664D0 *(10.D0**(TEM*( -0 .00222993D0) ) ) )*1000.0D0 IF (TEM.LE.300.ODO) GR=(4200.0D0*(10.DO**(TEM*(-0.00325289D0) ) ) )*1000.0D0 D1=(2.0DO*R)*3.2808399DO NUS=0.53D0*((GR*(D1**3.0D0)*D2*0.7D0)**0.25D0) H=((AIRK*2419.0D0*NUS/D1)/737.3D0)+ 1.366D-11*EMISS*(((T(M)+273.ODO)**4 ) -((TATMOS+273.0D0)**4))/(T(M)-TATMOS) 230 H=H*1000.ODO H=4.18*D0*H-220 H=50.0D0 Q ************************************************************** C Loop 60 C ******* C In t h i s l o o p , t h e v a l u e s of Thermal C o n d u c t i v i t y and S p e c i f i c C Heat f o r A u s t e n i t e a re c a l c u l a t e d f o r each node a t each t ime C s t e p C Loop 70 Q ******* C In t h i s l o o p , v a l u e s of Thermal C o n d u c t i v i t y and S p e c i f i c C Heat of F e r r i t e are c a l c u l a t e d f o r each node at each C t ime s t e p . C Loop 55 C ******* C In t h i s l o o p the v a l u e s of F r a c t i o n Transformed f o r each node C at each t ime s t e p are checked to f i n d whether t r a n s f o r m a t i o n C of A u s t e n i t e t o P e a r l i t e i s complete at a l l n o d e s . I f s o , C c o n t r o l i s d i r e c t e d t o Loop 70 where in the v a l u e s of C Thermal C o n d u c t i v i t y and S p e c i f i c Heat a re c a l c u l a t e d . O n 170e 242 C e n t e r i n g Loop 70 .a counter J1 i s set equa l to l . F o r a l l 243 C f u t u r e t ime s teps c o n t r o l i s d i r e c t e d to Loop 70 wi thout 244 C go ing through Loop 55.On complete t r a n s f o r m a t i o n at a l l 245 C nodes, from Loop 70 c o n t r o l i s d i r e c t e d t o s tatement no.820 246 C where in the a p p r o p r i a t e T r i d i a g o n a l System c o e f f i c i e n t s 247 C are c a l c u l a t e d , b y p a s s i n g Loop 501. 248 C **************************************************** 249 250 251 252 IF ( J1 .EQ .1 ) GO TO 70 253 L=0 254 DO 55 1=1,M 255 IF ( F ( I , 1 ) . L T . F R M A X ) GO TO 55 256 L=L+1 2 57 55 CONTINUE 258 IF (L.EQ.M) GO TO 70 259 DO 60 1 = 1 ,M 260 KK(I)=THERM2(1)+THERM2(2)*T(I) 261 KK(I)=4 18 .6D0*KK(I) 262 CP( I )=CP1 (1 )+CP1 (2)*T( I )+CP1 (3)*(T( I )**2)+CP1 (4)*(T( I )**3) 263 1+CP1(5)*(T(I)**4)+CP1(6)*(T(I)**5) 264 CP(I )=4186.0D0*CP(I ) 265 60 CONTINUE 266 GO TO 700 267 70 DO 600 1 = 1 ,M 268 KK(I)=THERM 1{1)+THERM1(2)*T(I)+THERM1(3)*(T(I)**2 ) 269 1+THERM1(4)*(T(I)**3)+THERM1(5)*(T(I)**4) 270 KK( I )=KK( I )*418.6D0 271 CP(I )=CP2(1)+CP2(2)*T( I )+CP2(3)*(T( I )**2) 272 CP( I )=CP( I )*4186.0D0 273 600 CONTINUE 274 J1=1 275 GO TO 820 27 6 C ************************************************************** 277 278 279 • 280 C 'Loop 501 281 C ********* 282 C In t h i s l o o p , e a c h node i s c h e c k e d , a t each t i m e s t e p f o r 283 C t r a n s f o r m a t i o n s t a r t . A t each t ime s t e p , f o r each node, the 284 C Avrami .t ime i s c a l c u l a t e d . I f the TOTTIM v a l u e i s G r e a t e r 2B5 C than or Equal t o the Avrami t ime the node w i l l s t a r t 286 C t r a n s f o r m i n g . O n c e a node s t a r t s t r a n s f o r m i n g , J J ( n o d e ) i s 287 C set e q u a l to node number.For a l l f u t u r e t i m e s t e p s t h i s 288 C node w i l l not be checked a g a i n f o r t r a n s f o r m a t i o n s t a r t . 289 C When a node s t a r t s t r a n s f o r m i n g , c o n t r o l i s d i r e c t e d t o 290 C s tatement no .500 where in ' n ' , ' l o g b' f o r t h a t node 291 C temperature are c a l c u l a t e d . T h e f r a c t i o n t r a n s f o r m e d i s 292 C then c a l c u l a t e d . T h e S p e c i f i c Heat and Thermal C o n d u c t i v i t y 293 C of the t r a n s f o r m i n g node i s then c a l c u l a t e d by u s i n g the 294 C fo rmula 295 C S p e c i f i c Heat=%Transformed*Speci f ic Heat of F e r r i t e + 296 C If 1 - % t r a n s f o r m e d ) * S p e c i f i c Heat o f A u s t e n i t e 297 C at the node temperature 298 C When the f r a c t i o n t r a n s f o r m e d of the node i s e q u a l t o 1 299 C c o n t r o l i s t r a n s f e r r e d to statement no . 8000 where the 300 C S p e c i f i c Heat and Thermal C o n d u c t i v i t y v a l u e s of F e r r i t e 301 C a re used f o r f u r t h e r c a l c u l a t i o n s . 1 7 0 f 302 C ************************************************** 303 304 305 306 700 DO 501 1=1,M 307 I F ( T ( I ) . G T . 7 2 8 . 0 D 0 ) GO TO 501 308 IF (F ( I ,1 ) .GE .FRMAX) GO TO 501 309 IF ( J J ( I ) . N E . O ) GO TO 500 310 I F ( K 5 ( I ) . E Q . 1 ) GO TO 229 311 TAV(I)=TOTTIM 312 K5(I)=1 313 229 IF ( T ( I ) . G E . 7 0 0 . 0 D 0 ) GO TO 501 314 TAVRAM(I )=62.7348D0+0.105339D0*(728.0D0-T ( I ) ) -15.4 325D0 315 1*(DLOG(728.0D0-T( I ) ) ) 316 TAVRAM(I)=DEXP(TAVRAM(I)) 317, TA1(I)=TOTTIM-TAV(I) 318 IF (TA1( I ) .LT .TAVRAM(I ) ) GO TO 501 319 J J ( I )= I 320 500 EN=AN(1)+AN(2)*T(I)+AN(3)*(T(I)**2)+AN(4 ) * (T (I)**3) 321 TT ( I )=728.0D0-T ( I ) 322 ALOGB=ALB(1)+ALB(2)*(TT(I) )+ALB(3)*((TT(I ) )**2 ) + 323 1ALB(4)*((TT( I ) )**3) 3 24 ALOGB=DEXP(ALOGB) 325 THETA=DT+(DLOG(1.0D0/(1.0D0-F(I ,1)))/ALOGB)**(1 .0D0/EN) 326 F(I ,2)=1.0D0-DEXP(-ALOGB*(THETA**EN)) 327 D F ( I ) = F ( I , 2 ) - F ( I , 1 ) 328 FK=THERM1(1)+THERM1(2)*T(I)+THERM1(3)*(T(I)**2)+THERM1(4)* 329 1(T(I)**3)+THERM1(5)*(T(I)**4) 330 FK=FK*418.6D0 331 FCP=CP2(1)+CP2(2)*T(I)+CP2(3)*(T(I )**2) 332 FCP=FCP*4186.0D0 333 ' K K ( I ) = ( F ( I , 2 ) * F K + ( 1 . O D 0 - F ( I , 2 ) ) * K K ( I ) ) 334 C P ( I ) = ( F ( I , 2 ) * F C P + ( 1 . 0 D 0 - F ( I , 2 ) ) * C P ( I ) ) 335 501 CONTINUE 336 337 338 339 C *********************************************************** 340 C Loop 7000 341 Q * * * * * * * * * 342 C T h i s l o o p c a l c u l a t e s the a p p r o p r i a t e v a l u e s of the T r i d i a g o n a l 343 C System C o e f f i c i e n t s - A A , B B , C , D 345 346 347 348 349 820 XX=-DX 350 DO 7000 1 = 1 ,M 351 XX=XX+DX 352 7050 I F ( I . G T . I ) GO TO 7051 353 AK1=(KK(1)+KK(2))/2.0D0 354 BB(1)=1.0D0+((DD*CP(1)*(DX**2))/(4.OD0*AK1*DT)) 355 C(1)=-1.0D0 356 IF ( ( F ( I , 1 ) . L E . 0 . 0 D 0 ) . O R . ( F ( I , 1 ) . G E . F R M A X ) ) GO TO 450 357 D(1)=((DX**2)/(4.0D0*AK1))* 358 1(DD*80000.0D0*DF(1)/DT)+((DD*CP(1)*(DX**2)*T(I))/ 359 1(4.0D0*AK1*DT)) 360 GO TO 7000 361 • 450 D(1)=(BB(1) -1 .0D0)*T(1) 1 7 0 g 362 GO TO 7000 363 705! IF ( I .EQ.M) GO TO 7052 364 AK2=(KK( I -1 )+KK( I ) )/ (2 .0D0) 365 AA(I )=AK2*((DX-2.0D0*XX)/(2.0D0*DX)) 3 66 AK3=AK2*(2.0D0*XX-DX)/(2.0D0*DX) 367 AK4=(KK(I )+KK(I+1))/(2.0D0) 368 AK4=AK4*(2.0D0*XX+DX)/(2.0D0*DX) 369 BB(I)=AK3+AK4+((DD*CP(I)*XX*DX)/(DT)) 370 AK4=(KK'(I )+KK(I + 1 ) )/(2.0D0) 371 C(I )=-AK4*((2.0D0*XX+DX)/(2.0D0*DX)) 372 IF ( ( F ( I , 1 ) . L E . 0 . 0 D 0 ) . O R . ( F ( I , 1 ) . G E . F R M A X ) ) GO TO 451 373 D(I)=(XX*DX*(DD*80000.0D0*DF(I)/DT))+(DD*CP(I)*XX*DX*T(I)/DT) 374 GO TO 7000 375 451 D(I)=(DD*CP(I)*XX*DX*T(I))/DT 376 GO TO 7000 377 7052 IF ( ( F ( I , 1 ) . L E . 0 . 0 D 0 ) . O R . ( F ( I , 1 ) . G E . F R M A X ) ) GO TO 350 378 D(M)=((DX)*(4.0D0*R-DX)* 37 9 1(80000.0D0*DD*DF(M)/DT))/(8.0D0)+(H*TATMOS*R)+ 380 1((DD*CP(M)*DX*(4.0D0*R-DX)*T(M))/(8.0D0*DT)) 381 GO TO 351 382 350 D(M)=(H*R*TATMOS)+((DD*CP(M)*DX*(4.0D0*R-DX)*T(M))/ 383 1(8.0D0*DT)) 384 351 AK5=(KK(M-1)+KK(M))/(2.0D0) 385 AA(M)=AK5*((DX-2.0D0*R)/(2.0D0*DX)) 386 BB(M)= AK5*((2.0D0*R-DX)/(2.0D0*DX))+(H*R)+((DD*CP(M)*DX* 387 1(4.0D0*R-DX))/(8.0D0*DT)) 388 7000 CONTINUE 389 390 391 Q *************************************************** 392 C Th i s p a r t of the program c a l c u l a t e s the temperature of each node 393 C at each t ime s t e p . T h e a l g o r i t h m used i s the s o l u t i o n of a 394 C T r i d i a g o n a l System of S i m u l t a n e o u s E q u a t i o n s d e s c r i b e d i n the 395 C book ' A p p l i e d N u m e r i c a l Methods ' by Carnahan ,Lu ther and W i l k e s . 396 C T(M) i s the temperature of the s u r f a c e node and T ( l ) i s the 397 C temperature of the Cent re of the r o d . A f t e r the c a l c u l a t i o n s of 398 C temperature f o r one t ime s t e p are completed c o n t r o l i s 399 C t r a n s f e r r e d t o s tatement no. 25 where the TOTTIM i s incremented 400 C by DT and the c a l c u l a t i o n of the T r i d i a g o n a l System c o e f f i c i e n t s 401 C e t c . i s r e p e a t e d . 4 02 C ****************************************************************** 403 404 405 406 407 BETA(1)=BB(1) 408 GAMMA(1)=D(1)/BETA(1) 409 DO 110 1=2,M 410 B E T A ( I ) = B B ( I ) - A A ( I ) * C ( I - 1)/BETA( I -1) 411 110 GAMMA(I)=(D(I ) -AA(I )*GAMMA(1-1))/BETA(I) 412 TI(M)=GAMMA(M) 413 MAST=M-1 414 DO 120 J=1,MAST 4 15 I=M-J 416 120 TI ( I )=GAMMA(I) -C( I )*TI (1+1)/BETA(I ) 417 418 419 420 C ************************************************************* 421 C I f t r a n s f o r m a t i o n s t a r t s at any node , the l a t e n t heat 170h 422 C l i b e r a t e d due t o t h e t r a n s f o r m a t i o n i s c a l c u l a t e d by an 423 C i t e r a t i v e p r o c e d u r e . K 7 i s c o n t r o l s t h e number of i t e r a t i o n s 424 C t o be pe r fo rmed . 4 2 5 C ************************************************ 426 427 428 IF (K7.GE.3) GO TO 6002 429 IF ( JJ (M) .EQ.O) GO TO 6002 430 IF ( J1 .EQ .1 ) GO TO 6002 431 DO 6003 11=1,M 432 T l ( I I ) = ( T I ( I I ) + T ( I I ) ) / 2 . 0 D 0 433 IF ( J J ( I I ) . E Q . 0 ) GO TO 6600 434 IF ( F ( I I , 2 ) . G E . F R M A X ) GO TO 6005 435 EN=AN(1)+AN(2)*TI(11)+AN(3)*(Tl(11)**2)+AN(4)*(Tl(11 )**3) 436 T T ( I I ) = 7 2 8 . 0 D 0 - T ( I I ) 437 ALOGB=ALB(1)+ALB(2)*(TT(11))+ALB(3)*((TT(11))**2) + 438 1ALB(4)*( (TT( I I ) )**3) 439 ALOGB=DEXP(ALOGB) 440 THETA=DT+(DLOG(1.0D0/(1.0D0-F(I I ,1)))/ALOGB)**(1.0D0/EN) 441 F(I I ,2)=1.0D0-DEXP(-ALOGB*(THETA**EN)) 442 D F ( I I ) = F ( I I , 2 ) - F ( I I , 1 ) 44 3 6005 FK=THERM1(1)+THERM1(2)*T(II)+THERM1(3)*(T(II)**2)+THERM1(4)* 44 4 1(T(I I)**3)+THERM1(5)*(T(II)**4) 445 FK=FK*418.6D0 4 46 FCP=CP2(1)+CP2(2)*T(II)+CP2(3)*(T(11)**2) 447 FCP=FCP*4186.0D0 4 48 K K ( I I ) = (F ( I I ,2 )*FK+(1 .0D0-F (11 ,2 ) )*KK(11)) 44 9 C P ( I I ) = (F ( I I ,2 )*FCP+(1 .0D0-F (11 ,2 ) )*CP(11)) 450 GO TO 6003 451 6600 KK(II)=THERM2(1)+THERM2(2)*TI(II) 452 KK( I I )=418.6D0*KK( I I ) 4 53 CP( I I )=CP1 (1 )+CP1 (2)*T(II\")+CPr(3)*(T(II )**2)+CP1 (4)*(T( I I )**3) 454 1+CP1(5)*(T(II)**4)+CP1(6)*(T(11)**5) 455 CP( I I )=4186.0D0*CP( I I ) 456 6003 CONTINUE 457 K7=K7+1 458 GO TO 820 459 6002 DO 6004 I 1 = 1,M 460 F ( I 1 , 1 ) = F ( I 1 , 2 ) 461 6004 T ( I1 )=T I ( I1 ) 462 K9=K9+1 463 IF ( K 9 . L E . 9 ) GOTO 325 464 K9=0 465 WRITE(6,730) TOTTIM,T(M),T(1) 466 730 F O R M A T ( 5 X , F 8 . 2 , 5 X , F 6 . 2 , 5 X , F 6 . 2 ) 467 325 GO TO 25 468 469 470 471 c. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 472 C C a l c u l a t i o n of temperature f o r a l l nodes f o r the c u r r e n t t ime 473 C s t e p i s c o m p l e t e . C o n t r o l ' i s now t r a n s f e r r e d to statement no .25 474 C f o r i n c r e m e n t i n g the TOTTIM v a l u e by DT and f u r t h e r c a l c u l a t i o n s . 475 Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 476 477 478 479 1001 STOP 480 END 481 Ti70i 482 £83 C ************************************************** 464 C End of main p r o g r a m . S t a r t of s u b r o u t i n e s . 4 g 5 Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 486 487 488 SUBROUTINE THER(THERM 1) 489 490 491 C ******************************************************************* 492 C In t h i s s u b r o u t i n e a P o l y n o m i a l of the 4th degree i s f i t t e d to 493 C Thermal C o n d u c t i v i t y v a l u e s of F e r r i t e i n the temperature range 494 C 50*C to 750*C,the data fo r which has been o b t a i n e d from the 495 C BISRA r e p o r t . T h i s P o l y n o m i a l i s the best f i t f o r the d a t a used 496 C and c a l c u l a t e s Thermal C o n d u c t i v i t y v a l u e s in the temperature 497 C range w i t h i n 0.5% of the e x p e r i m e n t a l v a l u e s . 498 C ******************************************************************* 499 500 501 502 IMPLICIT REAL*8 ( A - H , 0 - Z ) 503 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) , 504 1SIGMA(20),P(2 0),THERM1(10) 505 DATA K , N / 4 , 1 4 / 506 X ( l )=50 .0D0 507 DO 3300 1=2,14 508 3300 X ( I )=X( I -1J+50.0D0 509 DATA ( Y d ),1= 1 , 1 4 ) / 0 . 1 1 8 D 0 , 0 . 1 1 5 D 0 . 0 . 1 1 2 D 0 , 0 . 1 0 8 D 0 , 0 . 1 0 3 D 0 , 510 1 0 . 0 9 9 D 0 , 0 . 0 9 6 D 0 , 0 . 0 9 1 DO,0 .087D0, 0 .084D0,0 .081 DO,0 .078D0, 511 10 .075D0,0 .072D0/ 512 LOGICAL LK 513 LK= .TRUE. 514 NWT=0 515 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 516 DO 3400 1=1,5 517 3400 THERM 1( I )=P( I ) 518 RETURN 5 1 9 END 520 C ******************************************************** 521 522 523 524 SUBROUTINE THERMA(THERM2) 525 C ******************************************************************* 526 C In t h i s s u b r o u t i n e a P o l y n o m i a l of 1st degree i s f i t t e d t o the 527 C Thermal C o n d u c t i v i t y data of A u s t e n i t e . T h e p r e d i c t i o n e r r o r i s 528 C l e s s than 0 . 8 % . 529 C ******************************************************************* 530 531 532 533 IMPLICIT REAL*8(A -H ,0 -Z ) 534 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) 535 1 ,S IGMA(20) ,P(20) 536 1,THERM2(4) 537 DATA K , N / 1 , 1 1 / 538 X(1)=700.0DO 539 DO 2100 1=2,11 540 2100 X( I )=X( I -1)+50.0DO 541 DATA (Y(I),1 = 1 , 1 1 ) / 0 . 0 5 3 D 0 , 0 . 0 5 5 D 0 , 0 . 0 5 7 D 0 , 0 . 059D0,0.061 DO, 1 7 0 j b 4 2 1 U . U 6 3 D U , 54 3 10.064D0,0.066D0,0.068D0,0.07ODO,0.07 2D0/ 54 4 LOGICAL LK 54 5 LK= .TRUE. 54 6 NWT=0 54 7 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 548 DO 2110 1=1,2 54 9 2110 THERM2(I)=P(I) 550 RETURN • 551 END 552 C ***************************************************** 553 554 555 SUBROUTINE CEEPE2(CP2) 55g Q ************************************************* 557 C In t h i s s u b r o u t i n e a P o l y n o m i a l of 2nd degree i s f i t t e d t o . t h e 558 C S p e c i f i c Heat d a t a . o f F e r r i t e . T h e p r e d i c t i o n e r r o r i s l e s s than '559 C 1 .0%. 5£Q Q ********************************************* 561 562 563 564 IMPLICIT REAL*8(A -H ,0 -Z ) 565 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , SIGMA ( 2 0 ) , 566 1 A ( 2 0 ) , B ( 2 0 ) , P ( 2 0 ) , C P 2 ( 3 ) 567 DATA K , N / 2 , 1 3 / 568 X(1)=75.0D0 569 DO 4000 1= 2 ,13 570 4000 X ( I )=X( I -1 )+50.0D0 571 DATA (Y ( I ) ,1=1,13)/0 .117D0,0 .124D0,0.127D0,0 .131D0,0 .135D0, 572 1 0 . 1 4 0 D 0 , 0 . 1 4 5 D 0 , 0 . 1 5 0 D 0 , 0 . 1 6 0 D 0 , 0 . 1 6 6 D 0 ,0 . 1 7 2 D 0 , 0 . 1 7 2 D 0 , 573 10.184D0/ 574 LOGICAL LK 575 LK= .TRUE. 576 NWT=0 577 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 578 DO 4010 1=1,3 579 4010 CP2( I )=P( I ) 580 RETURN 581 END 582 C ******************************************************* 583 584 585 586 SUBROUTINE CEEPE1(CP1) 587 C *********************************************************** 588 C In t h i s s u b r o u t i n e a P o l y n o m i a l of 5th degree i s f i t t e d to 589 C the S p e c i f i c Heat v a l u e s of A u s t e n i t e . T h e p r e d i c t i o n e r r o r 590 C i s l e s s then 0 . 9 % . 55^ \" Q ************************************************************** 592 593 594 595 IMPLICIT REAL*8(A -H ,0 -Z ) 596 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , S (20 ) ,WT(25) ,S IGMA(20) , 597 1 A ( 2 0 ) , B ( 2 0 ) , P ( 2 0 ) , C P 1 ( 1 0 ) 598 DATA K , N / 5 , 1 3 / 599 X(1)=675.0D0 600 DO 5000 1=2,13 601 5000 X ( I )=X( I -1 )+50.0D0 1i70k 60 2 DATA (Y(I),1=1,13)/0.139D0,0.141D0,0.14 3D0,0.14 5D0,0.14 8D0, 60 3 1 0. 1 4 9D0,0.151 DO,0.154D0 ,0 .156D0 ,0 .158D0 ,0 .160D0,0.162D0, 604 10.162D0/ 605 LOGICAL LK 606 LK= .TRUE. 607 NWT=0 608 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 609 DO 5010 1=1,6 610 5010 C P l ( I ) = P ( I ) 6 1 1 RETURN 612 END 613 Q **************************************************** 614 615 616 SUBROUTINE LOGB(ALB) 617 Q ************************************************************* .618 C In t h i s s u b r o u t i n e a p o l y n o m i a l of 6th degree i s f i t t e d t o 619 C the data of \"Log b' v a l u e s o b t a i n e d fo r 0 .82%C-0 .82%Mn-620 C 0 .26%Si s t e e l in the Department of M e t a l l u r g y at UBC. g2i C ************************************************************ 622 623 624 625 IMPLICIT REAL*8(A -H ,0 -Z ) 626 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) 627 1 ,S IGMA(20) ,P (20) ,ALB(10) 628 DATA K , N / 3 , 8 / 62 9 D A T A ( X ( I ) , I = 1 , 8 ) / 5 8 . 0 D 0 , 4 8 . 0 D 0 , 6 8 . 0 D 0 , 7 8 . 0 D 0 , 9 8 . 0 D 0 , 1 0 5 . 0 D 0 , 630 1113.0D0,125.0D0/ 631 DATA(Y( I ) ,1 = 1 , 8 ) / - 9 . 8 1 3 3 D 0 , - 9 . 4 4 4 6 4 D 0 , - 9 . 4 7 1 1 5 D 0 , - 8 . 3 921 DO, 632 1 - 5 . 7 2 7 9 D 0 , - 4 . 5 l 3 2 7 D 0 , - 3 . 1 3 8 5 D 0 , - 2 . 0 4 1 0 3 D 0 / 63 3 LOGICAL LK 634 LK= .TRUE. 635 NWT=0 636 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 637 DO 6100 1=1,4 638 6100 ALB( I )=P( I ) 63 9 RETURN 64 0 END 541 Q ********************************************************** 642 643 644 64 5 SUBROUTINE EXPONT(AN) 646 C *************************************************************** 647 C In t h i s s u b r o u t i n e a P o l y n o m i a l of 1st degree i s f i t t e d t o the 648 C d a t a ' o f ' n ' v a l u e s o b t a i n e d fo r 0 .82%C-0 .82%Mn-0 .26%Si s t e e l i n 649 C the Department of M e t a l l u r g y at UBC. 650 C **************************************************************** 651 652 653 654 IMPLICIT REAL*8 (A -H .O -Z ) 655 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) 656 1 ,S IGMA(20) ,P (20) ,AN(5) 657 DATA K , N / 3 , 8 / 658 D A T A ( X ( I ) , 1 = 1 , 8 ) / 6 7 0 . 0 D 0 , 6 8 0 . 0 D 0 , 6 6 0 . 0 D 0 , 6 5 0 . 0 D 0 , 6 3 0 . 0 D 0 , 659 1 6 2 3 . 0 D 0 , 6 1 5 . 0 D 0 , 6 0 3 . 0 D 0 / 660 D A T A ( Y ( I ) , 1 = 1 , 8 ) / 2 . 1 2 5 1 0 9 D 0 , 1 . 6 1 8 9 5 6 D 0 , 2 . 4 6 7 576D0,2 .946133D0, 661 13.1 66861 D O , 3 . 1 4 7 9 4 5 D 0 , 2 . 922434D0,2.346407D0/ 11701 662 LOGICAL LK 663 LK= .TRUE. 664 NWT=0 6 6 5 CALL DOLSF (K ,N ,X ,Y ,YF ,YD ,WT ,NWT,S ,S IGMA,A ,B ,SS ,LK ,P ) 666 DO 6200 1=1,4 667 6200 AN( I )=P( I ) 668 RETURN 669 END 670 C ************************************************** End of f i l e A p p e n d i x 9 L I S T I N G OF C O M P U T E R P R O G R A M TO C A L C U L A T E T H E T E M P E R A T U R E R E S P O N S E OF A C E N T R E - S E G R E G A T E D S T E E L ROD U N D E R G O I N G C O O L I N G M a t r i x S t e e l : 0 . 8 2 % C - 0 . 8 2 % Mn - 0 . 2 6 % S i G r a i n S i z e : 5 - 7 A S T M S e g r e g a t e d S t e e l : 0 . 8 % C - 1 . 8 8 % Mn G r a i n S i z e : 5 - 8 A S T M A u s t e n i t i s i n g C o n d i t i o n s : 8 5 0 ° C - 5 m i n u t e s 171a 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 1 0 c 1 1 c 1 2 c 1 3 c 1 4 c 1 5 c 1 6 c 1 7 c 18 c 19 c 20 21 22 23 24 25 26 27 28 29 c 30 c 31 c 32 c 33 c 34 c 35 c 36 c 37 c 38 c 39 c 40 c 41 c 42 c 43 c 44 c 45 46 47 48 49 50 c 51 c 52 c 53 c 54 c 55 c 56 c 57 c 58 c 59 60 ************************** *-* ************************************** T h i s program c a l c u l a t e s the temperature ( i n *C) i n s i d e a c y l i n d r i c a l rod of 0 .82%C-0 .82%Mn-0 .26%Si s t e e l w i t h a c e n t r e s e g r e g a t i o n of compos i t ion ' 0 . 8 % C - 1 . 6 8 % K n . I t takes i n t o account the e f f e c t of the t r a n s f o r m a t i o n a l heat 1 i b e r a t e d , d u r i n g c o o l i n g , o n the temperature of the r o d . I t i s assumed t h a t the segregated r e g i o n t r a n s f o r m s to e i t h e r p e a r l i t e or m a r t e n s i t e , d e p e n d i n g on the c o o l i n g r a t e . l t i s f u r t h e r assumed t h a t no heat i s l i b e r a t e d d u r i n g the a u s t e n i t e to m a r t e n s i t e t r a n s f o r m a t i o n . C a l c u l a t i o n s are ************************************************************** IMPLICIT REAL*8 (A -H .O -Z ) REAL*8 KK ******************************************************** Data f o r t h i s program i s : ****************************** DX = Node d i s t a n c e , M e t e r s DT = Time inc rement ,Seconds TOTTIM = T o t a l t ime c o u n t e r , S e c o n d s H = C o n v e c t i v e Heat t r a n s f e r c o e f f i c i e n t at the rod surface,W/m *C R = Rad ius of r o d , M e t e r s TATMOS \" Atmospher ic temperature at rod s u r f a c e , * C XX = D i s t a n c e of node from the rod c e n t r e . M e t e r s M = Number of nodes+1 DD = D e n s i t y of s t e e l , K g / c u b i c Meter T1 = Maximum t ime upto which c a l c u l a t i o n s are to be done ************************************************************ Q ********************************************************* heat t r a n s f e r c o e f f i c i e n t va lue must be input by r e p l a c i n g statement no .63 ********************************************************** DATA DX,DT,TOTTIM,T1,DD,TATMOS,R,XX,M/0.00075D0,1.ODO,0.ODO, 1 7 1 b 61 1400.0DO,7650.0DO,20.0DO,0 .007 5DO,O.ODO,11/ 62 Ki=0 63 H=1200.0D0 64 IS=2 65 66 67 6g Q ************************************************** 69 C M a t r i x I d e n t i f i c a t i o n 70 C *********************** 71 C AA ,BB,C * C o n t a i n c o e f f i c i e n t s of the T r i d i a g o n a l 72 C system at each t ime s t e p 73 C D * C o n t a i n s the RHS of the t r i d i a g o n a l system 74 C T * C o n t a i n s the temperature of each node at 75 C each t ime s t e p 76 C KK,CP * C o n t a i n Thermal C o n d u c t i v i t y and S p e c i f i c 77 C Heat of each node at each t ime s t e p 78 C THERM 1,THERM2 * C o n t a i n c o e f f i c i e n t s of the P o l y n o m i a l used 79 C to c a l c u l a t e Thermal C o n d u c t i v i t y as a f u n c t i o n 80 C of temperature f o r F e r r i t e and A u s t e n i t e 81 C r e s p e c t i v e l y 82 C CP1,CP2 * C o n t a i n c o e f f i c i e n t s of the p o l y n o m i a l used 83 C to c a l c u l a t e S p e c i f i c Heat as a f u n c t i o n of 84 C temperature f o r A u s t e n i t e and P e a r l i t e r e s p l y 85 C F * C o n t a i n s t o t a l f r a c t i o n t r a n s f o r m e d fo r each 86 C node at each t ime s t e p 87 C DF * C o n t a i n s the i n c r e m e n t a l f r a c t i o n t rans fo rmed 88 C f o r each node at each t ime s t e p 89 C AN,ALB * C o n t a i n s the c o e f f i c i e n t s of the P o l y n o m i a l 90 C used to c a l c u l a t e ' n ' and 'Log b' as a 91 C f u n c t i o n of temperature .The data f o r these 92 C was generated at the M e t a l l u r g y D e p t . o f UBC. 93 C JO * C o n t a i n s the i n f o r m a t i o n on whether a node 94 C has s t a r t e d t r a n s f o r m a t i o n . I f J J ( i ) = i , then 95 C the ' i ' t h node has s t a r t e d t r a n s f o r m i n g . I f 96 C n o t , J J ( i ) = 0 97 ' C TAVRAM * C o n t a i n s the Avrami t ime f o r each node at 98 C each t ime s t e p . T h e data f o r t h i s has been 99 C generated at the M e t a l l u r g y Dept . of UBC 100 C BETA,GAMMA * Are dummy m a t r i c e s used f o r c a l c u l a t i n g the 101 C temperature of each node at each t ime s t e p 102 C ************************************************************** 103 C * UNITS * 104 C ************* 105 C Time * Seconds 106 C D i s t a n c e * Meters 107 C S p e c i f i c Heat * W/Kg *C 108 C Thermal C o n d u c t i v i t y ' * W/M *C 109 C C o n v e c t i o n Heat 110 C T r a n s f e r C o e f f i c i e n t * W/m2 *C 111 C D e n s i t y * Kg/m3 112 C Note on D i m e n s i o n i n g of M a t r i c e s 1)3 rj ******************************** 114 C M a t r i c e s AA ,BB,C ,D ,T ,TT ,DF ,KK,CP , J J ,F ,TAVRAM,BETA,GAMMA shou ld 115 C be d imens ioned at l e a s t M (=no. of nodes+1) 1 16 £ ******************************************************************* 1 1 7 118 1 19 120 1 7 t i c 1 2 1 2 1 2 2 1 7 9 C In t h i s l o o p , v a l u e s of Thermal C o n d u c t i v i t y and S p e c i f i c DIMENSION A A ( 3 0 ) , B B ( 3 0 ) \" , C ( 3 0 ) , D ( 3 0 ) , T ( 3 0 ) , T T ( 3 0 ) DF(30) 1 K K ( 3 0 ) , C P ( 3 0 ) , A L B ( 1 0 ) , A N ( 5 ) , T H E R M 1 ( 10),THERM2 ( i0) , F ( :>C , 2 ) , J J ( 1 C P 1 ( 1 0 ) , C P 2 ( 1 0 ) , T A V R A M ( 3 0 ) , H 1 ( 2 0 ) , B E T A ( J 0 ) , G A M M A ( 3 0 ) 24 DIMENSION K 5 ( 1 0 0 ) , T A V ( 1 0 0 ) , T A 1 ( 1 00) , 125 1 ALB 1 ( 1 0 ) , A L B 2 ( 1 0 ),AL1 ( 1 0 ) ,AL2(10) 1 2 6 CALL THER(THERM 1 ) . 127 CALL THERMA(THERM2) 128 CALL CEEPE2(CP2) 129 CALL CEEPE1(CP1) 130 CALL LOGB(ALB) 131 CALL EXPONT(AN) 132 CALL LOB(ALB2) 1 33 CALL EXPO(ALB 1) 1 34 CALL EXP(AL1) 135 CALL EX(AL2) 1 3 6 J1=0 •1 37 DO 3001 1 = 1 ,M 138 3001 K5(I)=0 r ********************************************************** 1 39 1 40 14 1 <_ -142 C I n i t i a l i s a t i o n of Temperature at a l l nodes a t TOTTIM=0.The 143 C f r a c t i o n t r a n s f o r m e d m a t r i x F and the J J m a t r i x are a l s o 144 C i n i t i a l i s e d to 0 . 1 4 5 Q *************************************************** 1 46 147 1 48 14 9 DO 10 I = 1 , M 150 F ( I , 1 ) = 0 . 0 D 0 151 F ( I , 2 ) = 0 . 0 D 0 152 J J ( I )=0 153 10 T( I )=850.0D0 154 155 1 56 157 Q ************************************************************* 158 C S t a r t i n g w i t h TOTTIM=0 the t ime i s inc remented i n s t e p s of 159 C DT.The c a l c u l a t i o n s w i l l s t o p when TOTTIM v a l u e i s G r e a t e r 160 C than or Equa l t o a p r e s p e c i f i e d v a l u e •j g •) Q *********************************************************** 162 163 1 64 165 166 25 TOTTIM=TOTTIM+DT 167 IF (TOTTIM.GE.Tl) GO TO 1001 1 6 8 1 6 9 1 70 •-\"---'-->••'-••-•\"*»**** + * * * * * ************************ 172 C Loop 60 \\ll C In t h i s l o o p , t h e v a l u e s of Thermal C o n d u c t i v i t y and S p e c i f i c 175 C Heat fo r A u s t e n i t e are c a l c u l a t e d fo r each node at each t ime 176 C s t e p 177 C Loop 70 ******* 180 Heat of F e r r i t e are c a l c u l a t e d f o r each noae a t . e a c h 171 d 161 C t ime s t e p . 182 C Loop 55 ig2 c ******* 184 C In t h i s l o o p the v a l u e s of F r a c t i o n Transformed f o r each node 185 C at each t ime s t e p are checked to f i n d whether t r a n s f o r m a t i o n 186 C of A u s t e n i t e to P e a r l i t e i s complete at a l l n o d e s . I f s o , 187 C c o n t r o l i s d i r e c t e d to Loop 70 wherein the v a l u e s of .183 C Thermal C o n d u c t i v i t y and S p e c i f i c Heat are c a l c u l a t e d . O n 189 C e n t e r i n g Loop 70 a counte r J1 i s set equa l to 1.For a l l 190 C f u t u r e t ime s teps c o n t r o l i s d i r e c t e d to Loop 70 w i t h o u t 191 C go ing through Loop 55.On complete t r a n s f o r m a t i o n at a l l 192 C nodes, from Loop 70 c o n t r o l i s d i r e c t e d to s tatement no .820 193 C wherein the a p p r o p r i a t e T r i d i a g o n a l System c o e f f i c i e n t s 194 C are c a l c u l a t e d , b y p a s s i n g Loop 5 0 1 . 195 C *************************************************************** 196 .197 198 199 IF ( J 1 . E Q . 1 ) GO TO 7 0 20'0 L=0 201 DO 55 1=1,M 202 IF ( F ( I , 1 ) . L T . 1 . 0 D 0 ) GO TO 55 203 L=L+1 204 55 CONTINUE 205 IF (L .EQ.M) GO TO 70 206 DO 60 1=1,M 207 KK(I )=THERM2(1)+THERM2(2)*T(I) 208 KK(I )=418.6D0*KK(I ) 209 CP(I )=CP1(1)+CP1(2)*T(I)+CP1(3)*(T(I)**2)+CP1(4)*(T(I)**3) 210 1+CP1(5)*(T(I)**4)+CP1(6)*(T(I)**5) 211 CP( I )=4186.0D0*CP(I ) 212 60 CONTINUE 213 GO TO 700 214 70 DO 600 1=1,M 215 KK(I)=THERM1(1)+THERM1(2)*T(I)+THERM 1(3)*(T( I )**2) 216 1+THERM1(4)*(T(I)**3)+THERM1(5)*(T(I)**4) 217 KK( I )=KK( I )*418.6D0 218 CP(I )=CP2(1)+CP2(2)*T( I )+CP2(3)*(T( I )**2) 219 CP( I )=CP( I )*4186.0D0 220 600 CONTINUE 221 J1=1 222 GO TO 820 223 C ************************************************************** 224 225 226 227 C Loop 501 228 C ********* 229 C In t h i s l o o p , e a c h node i s c h e c k e d , a t each t ime s t e p f o r 230 C t r a n s f o r m a t i o n s t a r t . A t each t ime s t e p , f o r each node, the 231 C Avrami t ime i s c a l c u l a t e d . I f the TOTTIM v a l u e i s G r e a t e r 232 C than or Equa l to the Avrami t ime the node w i l l s t a r t 233 C t r a n s f o r m i n g . O n c e a node s t a r t s t r a n s f o r m i n g , J J ( n o d e ) i s 234 C set equa l t o node number.For a l l f u t u r e t ime s t e p s t h i s 235 C node w i l l not be checked a g a i n fo r t r a n s f o r m a t i o n s t a r t . 236 C 'when a node s t a r t s t r a n s f o r m i n g , c o n t r o l i s d i r e c t e d to 237 C statement no .500 wherein ' n ' . ' l o g b' fo r t h a t node 238 C temperature are c a l c u l a t e d . T h e f r a c t i o n t r a n s f o r m e d i s 239 C then c a l c u l a t e d . T h e S p e c i f i c Heat and Thermal C o n d u c t i v i t y 240 C of the t r a n s f o r m i n g node i s then c a l c u l a t e d by u s i n g the 171e 24 I C formula 242 C S p e c i f i c Heat=%Transformed*Soeci f ic Heat of F e r r i t e + 243 C (1-%trans'f ormed ) *Speci f i c Heat of A u s t e n i t e 244 C at the node temperature 245 C When the f r a c t i o n t r a n s f o r m e d of the node i s equa l to 1 246 C c o n t r o l i s t r a n s f e r r e d . t o statement no. 8000 where the 247 C S p e c i f i c Heat and Thermal C o n d u c t i v i t y v a l u e s of F e r r i t e 248 C are used f o r f u r t h e r c a l c u l a t i o n s . 249 Q ******************************************************** 250 251 252 253 700 DO 501 1=1,M IF ( F ( I , 1 ) . G E . 0 . 9 9 9 9 9 D 0 ) GO TO 501 254 255 I F ( T ( I ) . G T . 7 2 8 . 0 D 0 ) GO'TO 501 256 IF ( I . G T . I S ) GO TO 240 257 IF ( J J ( I ) . N E . 0 ) GO TO 400 258 IF ( K 5 ( I ) . E Q . 1 ) GO TO 230 259 TAV(I)=TOTTIM 260 K5(I)=1 261 230 IF ( T ( I ) . G E . 7 0 0 . 0 D 0 ) GO TO 501 262 IF ( T ( I ) . G E . 4 7 5 . 0 D 0 ) GO TO 4001 263 TAVRAM(I)=35.2807D0-7.07259D0*(DLOG(728.0D0-T( I ) ) )+0.0225313D0* 264 1 ( 7 2 8 . 0 D 0 - T ( I ) ) 265 GO TO 4002 266 4001 TAVRAM(I )=22.4126D0+0.0123409D0*(728.0D0-T( I ) ) -267 14.27 29lD0*(DLOG(7 2 8 . 0 D 0 - T ( I ) ) ) 268 4002 TAVRAM(I)=DEXP(TAVRAM(I)) 269 TA1(I)=TOTTIM-TAV(I) 270 IF (TA1( I ) .LT .TAVRAM(I ) ) GO TO 501 271 J J ( I ) = 1 272 400 IF ( T ( I ) . G E . 6 2 5 . 0 D 0 ) GO TO 4003 273 IF ( T ( I ) . LT .500 .0DO) GO TO 4004 274 EN=AL1(1)+AL1(2)*T(I)+AL1(3)*(T(I )**2) 275 GO TO 4005 276 4003 'EN=ALB1(1)+ALB1(2)*T(I) 277 GO TO 4005 278 4004 EN=AL2(1)+AL2(2)*T(I) 279 4005 ALOGB=ALB2(1)+ALB2(2)*(728.0D0-T( I ) )+ALB2(3)* 280 1 ( ( 7 2 8 . 0 D 0 - T U ) ) * * 2 ) + A L B 2 ( 4 ) * ( ( 7 2 8 . 0 D 0 - T ( I ) ) * * 3 ) 281 ALOGB=DEXP(ALOGB) 282 GO TO 410 283 240 IF ( J J ( I ) . N E . 0 ) GO TO 500 284 I F ( K 5 ( I ) . E Q . 1 ) GO TO 229 285 TAV(I)=TOTTIM 286 K5(I)=1 287 229 IF (T(I ) .GE.700.0D.0) GO TO 501 \"288 TAVRAM(I )=62.7348D0+0.105339D0*(728.0D0-T( I ) ) -15.4 325D0 289 1*(DLOG(728.0D0-T( I ) ) ) 290 TAVRAM(I)=DEXP(TAVRAM(I)) 291 TA1(I)=TOTTIM-TAV(I) 292 IF (TA1( I ) .LT .TAVRAM(I ) ) GO TO 501 293 J J ( I ) = I 294 500 EN=AN(1)+AN(2)*T(I)+AN(3)*(T(I)**2)+AN(4)*(T(I)**3) 295 TT ( I )=728.0D0-T ( I ) 296 ALOGB=ALB(1)+ALB(2)*(TT(I) )+AL3(3)*((TT(I) )**2)+ 297 1ALB(4)*((TT( I ) )**3) 298 ALOGB=DEXP(ALOGB) 299 4 1 0 THETA=DT+(DLOG(1.0D0/(1.0D0-FO,1)))/ALOGB)**(1.0D0/EN) 300 F( I ,2 ) = 1 . 0D0-DEXP(-ALOGB*(THETA**EN)) 1 7 t f 301 D F ( I ) = F ( I , 2 ) - F ( l , 1 ) 302 F(I ,1 ) = F(I , 2 ) 303 FK=THERM1(1)+THERM1(2)*T(I)+THERM 1(3)*(T(I )**2)+THERM1(4)* 3 04 1(T ( I )**3)+THERMl(5)*(T(I)**4) 305 FK=FK*418.6D0 306 FCP=CP2(1)+CP2(2)*T(I )+CP2(3)*(T(I )**2) 307 FCP=FCP*4 186.0D0 308 K K ( I ) = ( F ( I , 1 ) * F K + ( 1 . 0 D 0 - F(I , 1 ) ) * K K ( 1 ) ) 309 CP (I ) = (F(1 , 1 )*FCP+( 1 . 0 D 0 - F C , 1 ) )*CPd ) ) 310 501 CONTINUE 311 312 313 314 Q ************************************************ 315 C Loop 7000 316 Q ********* .317 C T h i s l o o p c a l c u l a t e s the a p p r o p r i a t e v a l u e s of the T r i d i a g o n a l 316 C System C o e f f i c i e n t s A A , B B , C , D 31 g Q ******************************************************************* 320 321 322 323 324 820 XX=-DX 325 DO 7000 1=1,M 326 XX=XX+DX 327 7050 I F ( I . G T . I ) GO TO 7051 328 AK1=(KK(1)+KK(2))/2.0D0 329 BB(1}=1.0D0+((DD*CP(1)*(DX**2))/(4.0D0*AK1*DT)) 330 C(1)= -1 .0D0 331 IF ( ( F ( I , 1 ) . L E . 0 . O D O ) . O R . ( F ( I , 1 ) . G E . 0 . 9 9 9 9 9 D 0 ) ) GO TO 450 332 D(1)=((DX**2)/(4.0D0*AK1))* 333 1(DD*80000.0DO*DF(1)/DT)+((DD*CP(1)*(DX**2)*T(I))/ 334 1(4.0DO*AK1*DT)) 335 GO TO 7000 336 450 D(1)=(BB(1) -1 .0D0)*T(1) 337 GO TO 7000 338 7051 IF ( I .EQ.M) GO TO 7052 339 AK2=(KK( I -1 )+KK( I ) )/ (2 .0D0) 340 AA(I )=AK2*((DX-2.0D0*XX)/(2.0D0*DX)) 341 AK3=AK2*(2.0D0*XX-DX)/(2.0D0*DX) 342 AK4=(KR(I )+KK(I+1))/(2.0D0) 34 3 AK4=AK4*(2.0D0*XX+DX)/(2.0D0*DX) 344 BB(I)=AK3+AK4+((DD*CP(I)*XX*DX)/(DT)) 345 AK4=(KK(I )+KK(I+1))/(2.0D0) 346 C(I)=-AK4*((2.0DO*XX+DX)/(2.0DO*DX)) 347 IF ( ( F ( I , 1 ) . L E . 0 . O D O ) . O R . ( F ( I , 1 ) . G E . 0 . 9 9 9 9 9 D 0 ) ) GO TO 451 348 D(I)=(XX*DX*(DD*80000.0D0*DF(I)/DT))+(DD*CP(I)*XX*DX*T(I)/DT) 349 GO TO 7000 350 451 D(I)=(DD*CP(I)*XX*DX*T(I))/DT 351 GO TO 7000 352 7.052 IF ( (F (I , 1 ) . L E . 0 . ODO ) .OR. (F (I , 1 ) .GE. 0 . 99999D0 ) ) GO TO 350 353 D(M)=((DX)*(4.0D0*R-DX)* 354 1(80000.0D0*DD*DF(M)/DT))/(8.ODO)+(H*TATMOS*R)+ 355 1 ((DD*CP(M)*DX*(4.0D0*R-DX)*T(M))/(8.0D0*DT)) 356 GO TO 351 357 350 D(M)=(H*R*TATMOS)+((DD*CP(M)*DX*(4.0D0*R-DX)*T(M))/ 358 1(8.0D0*DT)) 359 351 AK5=(KK(M-1)+KK(M))/(2.0D0) 360 AA(M)=AK5*((DX-2.0D0*R)/(2.0D0*DX)) 1 7 1 g J c ^ 3 6 3 3 64 396 399 400 401 402 403 ] (4.0D0*R-DX) )/(6.0D0*DT.; 7 000 CONTINUE job 366 C ******************************************+*********************** 367 C T h i s p a r t of the program c a l c u l a t e s t h e temperature of ea c h node 368 C et each t ime s t e p . T h e a l g o r i t h m used i s t h e s o l u t i o n of a 369 C T r i d i a g o n a l System of S i m u l t a n e o u s E q u a t i o n s d e s c r i b e d i n the 370 C book ' A p p l i e d N u m e r i c a l Methods ' by C a r n a h a n , L u t h e r and W i l k e s . 371 C T(M) i s the temperatu re of the s u r f a c e node and T ( l ) i s the 372 C t e m p e r a t u r e of the C e n t r e of the r o d . A f t e r the c a l c u l a t i o n s of 373 C t e m p e r a t u r e f o r one t ime s t e p are completed c o n t r o l i s 374 C t r a n s f e r r e d to s ta tement no . 25 where the TOTTIM i s inc remented 375 C by DT and the c a l c u l a t i o n of the T r i d i a g o n a l System c o e f f i c i e n t s 376 C e t c . i s r e p e a t e d . 377 C ******************************************************** 378 379 380 381 382 BETA(1)=BB(1) 383 GAMMA(1)=D(1)/BETA(1) 384 DO 110 1=2,M 385 BETA(I)=BB(I) -AA(I )*C(I-1 )/BETA(1-1) 366 110 GAMMA(I) = (D(1)-AA(I)*GAMMA(1 - 1))/BETA(I ) 387 T(M)=GAMMA(M) 388 WRITE(6,63 01)TOTTIM,T(M) 389 6301 F O R M A T ( 1 0 X , F 8 . 2 , ' , ' , F 6 . 2 ) 3 90 MAST=M-1 391 DO 120 J=1,MAST 392 I=M-J 393 1 20 T(I)=GAMMA ( I ) - C ( I ) *T(I + 1)/BETA(I) 394 WRITE(6,6001)TOTTIM,T(1) 395 6001 FORMAT(1 O X , F 8 . 2 , ' , ' , F 6 . 2 ) 396 P R I N T , F ( 1 , 1 ) , F ( 1 0 , 1 ) 7 GO TO 25 £ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C a l c u l a t i o n of temperature f o r a l l nodes f o r the c u r r e n t t ime uJ C s t e p i s c o m p l e t e . C o n t r o l i s now t r a n s f e r r e d to s tatement n o . 2 5 404 C f o r i n c r e m e n t i n g the TOTTIM v a l u e by DT and f u r t h e r c a l c u l a t i o n s . 405 C ******************************************************************* 406 407 408 409 1001 STOP 410 END 4 11 412 413 Q ************************************************************* 414 C End of main p r o g r a m . S t a r t of s u b r o u t i n e s . 4 15 4 1 6 4 17 416 C ************************************************************ i i 9 C The S u b r o u t i n e s T h e r , T h e r m a , C e e p e 2 , C e e p e l , L o g b , E x p o n t are 420 C d e s c r i b e d i n the program used fo r c a l c u l a t i o n Ox the 1 7 1 h 4 2 1 4 2 2 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 temperature wi thout segrega't ior . . Subrout i ne Lob c a l c u l a t e s the c o e f f i c i e n t s of the p o l y n o m i a l used t o f i n d the v a l u e of ' b ' a t d i f f e r e n t t e m p e r a t u r e s . S u b r o u t i n e s Expo ,Exp and Ex c a l c u l a t e the c o e f f i c i e n t s cf the p o l y n o m i a l s used to f i n d the v a l u e s of ' n ' i n the temperature ranges 700 to 625*C, 625 to 500*C,<500*C r e s p e c t i v e l y fo r the seg regated s t e e l . **************************************************** SUBROUTINE THER(THERM 1 ) IMPLICIT REAL*8 ( A - H , 0 - Z ) DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) 1 SIGMA(20),P(20),THERM1 (10) DATA K , N / 4 , 1 4 / X( 1) = 50.ODO DO 3300 1=2,14 3300 X ( I ) = X ( I - 1)+50.0D0 DATA ( Y ( I ) , 1 = 1 , 1 4 ) / 0 . 1 1 8 D 0 , 0 . 1 1 5 D 0 , 0 . 1 1 2 D 0 , 0 . 1 0 8 D 0 , 0 . 1 0 3 D 0 , 10.099D0, 0. 096D0,0.091 D O , 0 . 0 8 7 D 0 , 0 . 0 8 4 D 0 , 0 . 0 8 1 DO,0 .078D0, 10 .075D0,0 .072D0/ LOGICAL LK LK= .TRUE. NWT=0 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) DO 3400 1=1,5 3400 THERM1(I)=P(I) RETURN END Q ******************************************************** SUBROUTINE THERMA(THERM2) IMPLICIT REAL*8(A -H ,0 -Z ) DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) 1 ,S IGMA(20) ,P(20) 1,THERM2(4) DATA K , N / 1 , 1 1 / X(1)=700.0D0 DO 2100 1=2,11 2100 X ( I )=X( I -1 )+50.0D0 DATA ( Y ( I ) , 1 = 1,11 ) / 0 . 0 5 3 D 0 , 0 . 0 5 5 D 0 , 0 . O 5 7 D 0 , O . 0 5 9 D 0 , 0 . 0 6 1 DO, 10.063D0, 1 0 . 0 6 4 D 0 , 0 . 0 6 6 D 0 , 0 . 0 6 8 D 0 , 0 . 0 7 O D O , 0 . 0 7 2 D 0 / LOGICAL LK LK= .TRUE. HWT= 0 CALL D O L S F ( K ,N , X , Y , Y F , Y D ,WT,NWT , S , S I G M A , A , B , S S , L K , P ) DO 2110 1=1,2 2110 THERM2(I)=P(I) RETURN END 1 7 t i Q ************************************************ 482 483 484 SUBROUTINE CEEPE2(CP2) 485 486 487 488 • IMPLICIT REAL*8 (A-H ,0 -Z) 489 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , S I G M A ( 20 ) , 490 1 A ( 2 0 ) , B ( 2 0 ) , P ( 2 0 ) , C P 2 ( 3 ) 491 DATA K,N/2,13/ 492 X(1)=75.0D0 493 DO 4000 1= 2 ,13 494 4000 X( I )=X( I -1 )+50.0D0 495 DATA ( Y ( I ) , I = 1,13)/0.117D0,0.124D0,0.127D0,0.131 DO,0.135D0, 496 10.14 0 D 0 , 0 . 1 4 5 D 0 , 0 . 1 5 0 D 0 , 0 . 1 6 0 D 0 , 0 . 1 6 6 D 0 , 0 . 1 7 2 D 0 , 0 . 1 7 2 D 0 , 497 10.184D0/ 4 98 LOGICAL LK 499 LK= .TRUE. 500 NWT=0 501 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 502 DO 4010 1=1,3 503 4010 CP2( I )=P( I ) 504 RETURN 505 END 5Qg C ******************************************************* 507 508 509 510 SUBROUTINE CEEPE1(CP1) 51 1 512 513 514 IMPLICIT REAL*8 (A-H ,0 -Z) 515 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , S ( 2 0 ) , W T ( 2 5 ) , S I G M A ( 2 0 ) , 516 1 A ( 2 0 ) , B ( 2 0 ) , P ( 2 0 ) , C P 1 ( 1 0 ) 517 DATA K,N/5,13/ 518 X(1)=675.0D0 519 DO 5000 1=2,13 520 5000 X( I )=X( I -1 )+50.0D0 521 DATA ( Y ( I ),1= 1 , l 3 ) / 0 . 1 3 9 D 0 , 0 . 1 4 1 D 0 , 0 . 1 4 3 D 0 . 0 . 1 4 5 D 0 , O . 1 4 8 D 0 , 522 10. 14 9D0,0.151 D O , 0 . 1 5 4 D 0 , 0 . 1 5 6 D 0 , 0 . 1 5 8 D 0 , 0 . 1 6 0 D 0 , 0 . 1 6 2 D 0 , 523 10.162D0/ 524 LOGICAL LK 52 5 LK= .TRUE. 526- NWT=0 527 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 528 DO 5010 1=1,6 529 5010 CP1( I )=P( I ) 530 RETURN 531 END 532 C ************************************************** 533 534 535 SUBROUTINE LOGB(ALB) 536 537 538 539 IMPLICIT REAL*8 (A-H ,0 -Z) 540 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) 171 j 541 1 ,SIGMA ( 2 0) ,P(2 0 ) ,ALB(10) 54 2 DATA K , N / 3 , 8 / 54 3 DATA(X ( I ) ,1 = 1,8)/58.ODO,45 .ODO,68.ODO,78.ODO,96.ODO, 1 05. ODO , 544 1113.ODO,125.ODO/ 54 5 DATA(Y( I ) ,1 = 1 , 8 ) / - 9 . 8 1 3 3 D 0 , - 9 . 44464D0, - 9 . 4 71 1 5 D 0 , - 8 . 3 9 2 1 D 0 , 54 6 1-5.7 27 9 D 0 , - 4 . 5 1 3 2 7D0,\"3 .1385D0, -2 .04 1 03DO/ 54 7 LOGICAL LK 54 8 LK= .TRUE. 54 9 NWT=0 550 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 551 DO 6100 1=1,4 552 6100 ALB( I )=P( I ) 553 RETURN 554 END 555 C *************************************************** 556 557 558 559 SUBROUTINE EXPONT(AN) 560 561 562 563 IMPLICIT REAL*8 ( A - H , 0 - Z ) 564 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) 565 1 ,S IGMA(20) ,P (20) ,AN(5) 566 DATA K , N / 3 , 8 / 567 DATA(X( I ) , 1=1,8)/67 0.ODO,680.ODO,660.ODO,650.ODO,630.ODO, 568 1623.ODO,615.ODO,603.ODO/ 569 DATA(Y ( I ) , 1=1,8 )/2 .125109D0,1 .6 1 8956D0,2 .467576D0,2 .946133D0, 57 0 13. 166861 DO, 3 . 1 47 945D0 , 2 . 922 4 34D0 ,'2 . 34 64 07D0/ 571 LOGICAL LK 572 LK= .TRUE. 573 NWT=0 574 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 575 DO 6200 1=1,4 576 6200 AN( I )=P( I ) 577 RETURN 578 END 579 SUBROUTINE HEAT(H1) 580 IMPLICIT REAL*8(A -H ,0 -Z ) 581 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , 582 1 B ( 2 0 ) , S I G M A ( 2 0 ) , P ( 2 0 ) , H 1 ( 2 0 ) 583 DATA K , N / 6 , 1 0 / 584 DATA (Y (I ) , I = 1,10)/8 1 .94D0,165.67D0,1 95. 4 8D0,200.91 DO,223.82D0, 585 1231.82DO,236.57DO,252.67DO,2 4 4.06DO,238.85DO/ 586 DATA (X( I ) , I=1,10)/890.ODO,860.ODO,830.ODO,B05.ODO,770.ODO, 587 1740. ODO , 7 1 2 . ODO , 67.5 . ODO , 658 . ODO , 640 . ODO/ 588 LOGICAL LK 589 LK=.TRUE. 590 NWT=0 591 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 592 DO 6300 1=1,7 593 6300 H 1 ( I ) = P ( I ) 594 RETURN 595 END 556 C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 597 598 599 SUBROUTINE LOB(ALB2) 600 .171k 601 602 603 IMPLICIT REAL*8(A-H-,0-Z) 604 DIMENSION1 X(25),Y(25),YF(2s),YD(25),WT(25),S(20),A(20),B(20) 605 1,SIGMA(20),P(20),ALB2(10) 606 DATA K.N/3,12/ 607 DATA(X(I),1=1,12)/53.0D0,78.0D0,10 3.0D0,128.0D0,153.0D0, 608 1178.0D0,203.0D0,228.0D0,253.0D0,2 76.0D0,303.0D0,328.0D0/ 609 DATA(Y(I),1=1,12)/-6.09954D0,-4.60211D0,-5.08256D0, 610 1-5.57 406D0,-4.00138D0,-4.024 53D0,-2.9204D0,-2.97626D0, 611 1-3.78062D0,-3.967 8D0,-4.37844D0,-5.02969D0/ 612 LOGICAL LK 613 LK= .TRUE. 614 NWT=0 615 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 616 DO 6100 1=1,4 .617 6100 ALB2(I)=P(I) 6 1 8 RETURN 6 1 9 END 620 C *********************************************** 621 622 623 SUBROUTINE EXPO(ALB 1) 624 625 626 627 IMPLICIT REAL*8(A-H,0-Z) 628 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 629 1,SIGMA(20),P(2 0),ALB 1(10) 630 DATA K,N/1,3/ 631 DATA(X(I),1=1,3)/675.0D0,650.0D0,625.0D0/ 632 DATA(Y(I),1=1,3)/0.77 0414D0,0.825272D0,1.276907D0/ 633 LOGICAL LK 634 LK= .TRUE. 635 NWT=0 636 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 637 DO 6100 1=1,2 638 6100 ALB1(I)=P(I) 639 RETURN 640 END 641 c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 642 643 644 SUBROUTINE EXP(AL1) 645 646 647 648 IMPLICIT REAL*8(A-H,0-Z) 64 9 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 650 1,SIGMA(20),P(20),AL1(10) 651 DATA K,N/2,5/ 652 DATA(X(I),1=1,5)/600.0D0,575.0D0,550.0D0,52 5.0D0,500.0D0/ 653 DATA(Y(I),I=1,5)/l.646218D0,1.205357D0,1.103574D0,0.74475D0, 654 10.715708D0/ 655 LOGICAL LK 656 LK= .TRUE. 657 NV?T=0 658 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 659 DO 6100 1=1,3 660 6100 AL1(I)=P(I) 1 7 1 1 661 RETURN 662 END ggj Q ****************************************************** 664 665 666 SUBROUTINE EX(AL2) 667 668 669 670 IMPLICIT REAL*8(A -H ,0 -Z ) 67 1 DIMENSION X ( 2 5 ) , Y ( 2 5 ) , Y F ( 2 5 ) , Y D ( 2 5 ) , W T ( 2 5 ) , S ( 2 0 ) , A ( 2 0 ) , B ( 2 0 ) ' 6 7 2 1 ,S IGMA(20) ,P (20) ,AL2(10) 673 DATA K , N / l , 4 / 67 4 DATA(X(I),1=1,4)/475.ODO,450.ODO,425.ODO,400.ODO/ 67 5 DATA(Y ( I ),1=1 ,4 )/0 .851409D0,0 .905354D0,0 .987884D0,1 .057 34 7D0/ 67 6 LOGICAL LK •677 LK= .TRUE. 678 NWT=0 679 CALL D O L S F ( K , N , X , Y , Y F , Y D , W T , N W T , S , S I G M A , A , B , S S , L K , P ) 680 DO 6100 1=1,2 681 6100 AL2( I )=P( I ) 68 2 RETURN 683 END End of f i l e "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0078974"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Metals and Materials Engineering"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Mathematical modelling of phase transformation in a plain carbon eutectoid steel"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/24080"@en .