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Measurement of quench heat transfer coefficients and their use in heat treatment design Gupta, Shashi Mohan 1977

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MEASUREMENT OF QUENCH HEAT TRANSFER COEFFICIENTS AND THEIR USE IN HEAT TREATMENT DESIGN SHASHI MOHAN GUPTA B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER -OF APPLIED^SCIENCE i n THE FACULTY OF GRADUATE STUDIES i n the Department of METALLURGY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 0 Shashi Mohan Gupta, 1977 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f i nanc ia l gain sha l l not be allowed without my writ ten permiss ion. Department of Metallurgy The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Oct. 4, 1977 ABSTRACT The heat-transfer phenomena i n the quenching process have been studied using s t a i n l e s s s t e e l and mild s t e e l specimens i n brine (3% by wt. NaCl), water, o i l and a i r , under c o n t r o l l e d conditions. The experimental data were analysed using a simple mathematical model of the quenching process to study the r e l a t i o n s h i p between the surface temperature of the specimen and the surface heat-transfer c o e f f i c i e n t . The influence on t h i s r e l a t i o n s h i p , of important v a r i a b l e s such as i n i t i a l specimen temperature, quenchant temperature, surface oxidation, etc. has been studied. The r e s u l t s from the experimental data are i n good agreement with the r e s u l t s of previous workers. The r e s u l t s obtained i n t h i s work, together with Jominy-test data and the mathematical model, were used to determine the necessary quenching conditions required to obtain a desired thermal h i s t o r y or mechanical property at a given p o s i t i o n i n a 4 inch diameter s t e e l g r i n d i n g - b a l l . A s t e e l b a l l was then quenched under the above determined conditions and an examination of the b a l l section showed that the desired property was indeed present at the given l o c a t i o n . i i TABLE OF CONTENTS Page ABSTRACT 1 1 LIST OF FIGURES v i LIST 'OF TABLES . . l x LEGEND OF EQUATIONS SYMBOLS . . x ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . x i i Chapter 1 INTRODUCTION . . . . . . . 1 1.1 General .. 1 1.2 Heat Treatment Design .. .. 2 1.2.1 Grossmann hardenability test .. . . . . 2 1.2.2 Hardenability from the chemical composition .. .. .. .. .. . . . . .. 3 1.2.3 The Jominy end-quench hardenability test .. . . . . . . . . .. .. . . . . 3 1.2.4 Ap p l i c a t i o n of C.C.T. diagrams .. .. .. 4 1.2.5 Proposed method 5 1.3 Review of Previous Work . . . . .. . . . . .. 6 1.4 Scope of the Present Work .. .. 8 2 MATHEMATICAL MODEL - THE INVERSE BOUNDARY VALUE PROBLEM 9 2.1 The Model . . . . . . . . . . . . .. .. . . . . 9 2.2 Assumptions Made i n the Model . . . . . . . . .. 9 i i i Chapter Page 2.3 Model Geometry . . . . .. .. 10 2.4 Heat Balance Equations 12 2.5 Solution of the Inverse Problem .. . . . . .. 17 2.6 Numerical Calculations .. .. .. . . . . .. .. 20 2.7 V a l i d i t y of the Model .. . . . . . . . . . . . . 21 2.7.1 A n a l y t i c a l s o l u t i o n .. .. .. 21 2.7.2 Optimum values of the parameters . . . . 23 3 EXPERIMENTAL .. . . . . . . 26 3 i l Apparatus . . . . . . . . . . . . . . . . . . . . 26 3.2 Procedure 30 4 RESULTS, DISCUSSIONS AND APPLICATION 33 4.1 Treatment of Experimental Data . . . . . ^  .. .. 33 4.2 R e p r o d u c i b i l i t y of Results . . . . . . . . . . . . 39 4.3 Quench Variables .. .. . . . . . . . . . . . . 39 4.3.1 I n i t i a l specimen temperature 39 4.3.2 I n i t i a l bath temperature .. .. . . . . 43 4.3.3 A g i t a t i o n of bath . . . . .. 43 4.3.4 Surface oxidation 48 4.3.5 Specimen material .. .. . . . . . . . . 52 4.3.6 Quench medium .. . . . . 55 4.4 Comparison with Previous Work .. 55 4.4.1 Quenching i n s t i l l a i r .. .. . . . . .. 58 4.4.2 Quenching i n s t i l l water . . . . . . .. 58 4.4.3 Quenching i n s t i l l brine (3% NaCl) so l u t i o n . . . . . . 58 i v Chapter Page 4.4.4 Quenching i n s t i l l o i l .. . . . . . . . . 61 4.5 A p p l i c a t i o n to Design Problem .. . . . . .. .. 63 4.6 Comments . . . . . . . . . ; . . 69 5 SUMMARY AND CONCLUSION .. . . . . .. .. . . . . 72 5.1 Summary of the E f f e c t of Quenching Variables . . . -. . . . . .. .. . . . . . . . . 72 5.2 Conclusion .. ; .. . . . . 73 6 SUGGESTIONS FOR FUTURE WORK .. .. . . . . . . . . 74 REFERENCES . . . . . . .. 75 APPENDIX A . . . . . . . . . . 76 APPENDIX B . . . . .. .. . . . . .. . . . . 83 v LIST OF FIGURES Figure Page 2.1 Model geometry showing d i v i s i o n of f l a t specimen into nodes . . . . . . . . . . . . . . .. 11 2.2 Influence of point of measurement on accuracy of r e s u l t s [10] .. . . . . . . . . . . . . . . . . .. 24 3.1 Schematic diagram of the quenching apparatus used . . . 27 3.2 Configuration of di s c specimen showing thermo-couple l o c a t i o n .. . . . . .; .. . . . . . . . . 28 4.1 Linear p l o t s , of q v s T g and h. vs T g f o r the r e s u l t s shown i n Table 4.2. Various b o i l i n g regions are shown . . .. . . . . . . . . . . . . . . . . . . . . 36 4.2 Semi-logarithmic plots of q vs T s and h vs T s for the r e s u l t s shown i n Table 4.2. Various b o i l i n g regions are shown .. .. .. . . . . .. .. .; .. 37 4.3 Plots of h vs T s for two s i m i l a r quenches of the s t a i n l e s s s t e e l specimen from 987 ± 5°C i n s t i l l water at 10 ± 2°C .. .. . . . . . . . . . . . . .. 40 4.4 Plots of q vs T s for the s t a i n l e s s s t e e l specimen i n s t i l l water at 20°C from two d i f f e r e n t i n i t i a l temperatures . . . . . . . . . . . . . . 41 4.5 Plots of h vs T s for the s t a i n l e s s s t e e l specimen i n s t i l l water at 20°C from two d i f f e r e n t i n i t i a l temperatures .. . . . . . . . . . . . . 41 4.6 P l o t s of q vs T g for the s t a i n l e s s s t e e l specimen i n s t i l l water at three bath temperatures .. .. .. 44 4.7 P l o t s of h vs T s for the s t a i n l e s s s t e e l specimen i n s t i l l water at three bath temperatures .. .. .. 44 4.8 Plots of q vs T s for the s t a i n l e s s s t e e l specimen i n 20°C water i n s t i l l and s t i r r e d conditions .. .. 46 v i Figure Page 4.9 Plots of h vs T s for the s t a i n l e s s s t e e l specimen i n 20°C water i n s t i l l and s t i r r e d conditions .. .. • 46 4.10 Plots of h vs T s for the s t a i n l e s s s t e e l specimen i n 40°C water i n s t i l l and s t i r r e d conditions .. .. 49 4.11 Plots of h vs T s for the s t a i n l e s s s t e e l specimen i n 60°C water i n s t i l l and s t i r r e d conditions . . . . 49 4.12 Plots of q vs T s for quenching into s t i l l water at 20°C, the p l a i n carbon s t e e l specimen i n the non-oxidized ( l i g h t oxide) and the oxidized (heavy oxide) conditions .. 50 4.13 Plots of h vs T s for quenching into s t i l l water at 20°C, the p l a i n carbon s t e e l specimen i n the non-oxidized ( l i g h t oxide) and the oxidized (heavy oxide) conditions . . .. .. .. 50 4.14 Plots of q vs T s for quenching into s t i l l water at 20°C, the s t a i n l e s s s t e e l specimen and the plain'carbon (AISI-1038) s t e e l specimen i n the non-oxidized ( l i g h t oxide) condition , .. .. .. .. 53 4.15 Plots of h vs T s for quenching into s t i l l water at 20°C, the s t a i n l e s s s t e e l specimen and the p l a i n carbon (AISI-1038) s t e e l specimen i n the non-oxidized ( l i g h t oxide) condition 53 4.16 P l o t s of h vs T s for the s t a i n l e s s s t e e l specimen quenched i n four d i f f e r e n t quench media, v i z . brine (3% by wt. NaCl), water, o i l (HOUGHTO-OUENCH-G) and A i r .. . . .. 56 4.17 Plots of h vs T s for the p l a i n carbon s t e e l specimen quenched i n four d i f f e r e n t quench media, v i z . brine (3% by wt. NaCl), water, o i l (HOUGHTO-QUENCH-G) and A i r 57 4.18 Comparison of h-curve determined i n t h i s work with that obtained by Lambert, et a l . [10] for non-oxidized s t e e l quenched i n s t i l l a i r under s i m i l a r conditions 59 4.19 Comparison of h-curve determined i n t h i s work, using a s t a i n l e s s s t e e l specimen, with those obtained by Lambert, et a l . [10] using a n i c k e l specimen, for a s t i l l water quench under s i m i l a r conditions . . .. 59 v i i Figure Page 4.20 Comparison of h-curve determined i n t h i s work, using a s t a i n l e s s s t e e l specimen, with that obtained by Lambert, et a l . [11] using a n i c k e l specimen of c y l i n d r i c a l shape, for a s t i l l brine (3% by wt. NaCl) s o l u t i o n quench under s i m i l a r conditions .. . . . . 60 4.21 Comparison of h-curve determined, i n t h i s work, using a s t a i n l e s s s t e e l specimen, with that obtained by Sto l z , et a l . [6] using a s i l v e r specimen of sp h e r i c a l shape, for a s t i l l o i l quench under s i m i l a r conditions . . . . 62 4.22 Distance vs hardness pl o t f o r the Jominy Bar 64 4.23 Distance vs hardness plot for the Commercial Grinding B a l l . . .. .. . . . . .. 65 4.24 Comparison of the required h-curve with the h-curve for the 20°C s t i r r e d water quench 67 4.25 Continuous-cooling-transformation diagram for the AISI-5160 s t e e l [17]. The measured thermal h i s t o r y i n the Jominy bar and the desired thermal h i s t o r y i n the s t e e l b a l l are pl o t t e d on i t . . . . . . . . .. . . 68 4.26 Distance vs hardness p l o t f o r the Heat-treated B a l l .. .. . . . . . . . . . . 70 B - l Model geometry showing d i v i s i o n of s p h e r i c a l specimen into nodes 84 v i i i LIST OF TABLES Table Page 2.1 Back calculated h-values f or an a n a l y t i c a l run .. . . . . . . . . . . . . . . . . .. 22 4.1 T y p i c a l computer input data . . . . . . . . . . . . .. 34 4.2 Computer output for data-in Table 4.1 .. . . . . .. 35 4.3 Analysis of F i g . 4.4 .. . . . . . . .. .. . . . . 42 4.4 Analysis of F i g . 4.6 .. . . . . . . . . .. .. .. 45 4.5 Analysis of F i g . 4.8 .. .. .. .. .. 47 4.6 Analysis of F i g . 4.12 . . . . . . .. 51 4.7 Analysis of F i g . 4.14 .. .. .. .. .. 54 4.8 Comparison of Quenching O i l s .. . . . . 61 4.9 Comparison of composition of commercial grinding b a l l s t e e l with AISI 5160 s t e e l .. . . . . . . . . . . . . 66 i x LEGEND OF EQUATIONS SYMBOLS ' 2 A^ Area between nodes i and i+1, normal to heat flow, (cm ) i Cp^ S p e c i f i c heat of the material at temperature T_^, (cal/gm.°G) F^ Symbol used to denote — — for 1 ^  i s p ± Cp i Ax L Half thickness of the f l a t specimen, (cm) M Symbol used to denote [12(i - l ) 2 + 1] for 1 < i < s 2 M g Symbol used to denote [3(4s - 5) +1] N. Symbol used to denote j for 1 ^  i < s 1 P ± Cp ± Ar R Radius of the sp h e r i c a l grinding b a l l , (cm) S Tot a l number of nodes T ^ Temperature of the quench medium, . (°C) T ^ Past temperature of the i t h node, (°C) T ^ Present temperature of the i t h node, (°C) T q Uniform i n i t i a l temperature of the specimen, (°C) 3 Volume surrounding the i t h node, associated with area A^, (cm ) 2 h Overall surface h e a t - t r a n s f e r - c o e f f i c i e n t , (cal/cm .sec.°C) k^ Thermal conductivity of the material at temperature T^, (cal/cm.sec.°C) 2 q Heat f l u x , (cal/cm .sec) s Subscript for the surface node t Time from the beginning of the quench, (sec) a k 2 Thermal d i f f u s i v i t y , — — , (cm./sec) p-Cp 6 Symbol used to denote (L-X ) n n x AT Temperature d i f f e r e n c e , (°C) Ar Distance between successive nodes for the sp h e r i c a l specimen,.(cm) At Time increment, (sec) Ax Distance between successive nodes for the f l a t specimen, (cm) X^ One of the c h a r a c t e r i s t i c values of X that s a t i s f y . t h e equation Cot (XL) = ^ h p. Density of the material at T' (gm/cm^) x i ACKNOWLEDGEMENTS The author wishes to s i n c e r e l y thank h i s research supervisors Drs. J.K. Brimacombe and E.B. Hawbolt for t h e i r assistance and guidance throughout the course of t h i s project. Thanks are'also extended to the other f a c u l t y members of the department and fellow graduate students f o r the h e l p f u l discussions, with s p e c i a l appreciation to Mr. B. Prabhakar and Mr. G. Bacon. The help received from a l l the s t a f f members of the department i s very much appreciated. In a d d i t i o n , thanks are extended to two s t e e l companies for providing necessary information and materials f o r the various t e s t s . F i n a n c i a l support from the National Research Council i n the form of a research a s s i s t a n t s h i p i s g r a t e f u l l y acknowledged. x i i CHAPTER 1 INTRODUCTION 1.1 General The cooling rates i n a piece of hot s t e e l during quenching depend p r i m a r i l y upon the h e a t - t r a n s f e r - c o e f f i c i e n t between the s t e e l surface and the cooling medium. The rates of the d i f f u s i o n c o n t r o l l e d s t r u c t u r a l transformations are very s e n s i t i v e to the cooling rates p r e v a i l i n g during these transformations. The austenite-to-martensite transformation occurs below the M g temperature during cooling, the amount of austenite a v a i l a b l e being strongly dependent on the thermal h i s t o r y . Continuous-cooling-transformation (C.C.T.) diagrams have been established to show the r e l a t i o n -ship between the thermal h i s t o r i e s and the structure f o r several selected s t e e l s . Empirical r e l a t i o n s h i p s have also been established r e l a t i n g the structure, hardness and the ultimate t e n s i l e strength f o r s t e e l s . There-fo r e , to e f f e c t i v e l y design the heat treatment of a s t e e l to a t t a i n a c e r t a i n desired structure or hardness at a given depth from the surface, i t i s necessary to have a p a r t i c u l a r thermal h i s t o r y at that l o c a t i o n during the heat treatment. To obtain the desired thermal h i s t o r y , heat transfer c o e f f i c i e n t vs. surface temperature data f o r a v a r i e t y of quenchants i s required. This data can then be used to s e l e c t the necessary quenching con-d i t i o n s which give the desired heat transfer c o e f f i c i e n t s and hence produce the desired thermal h i s t o r y at the given l o c a t i o n i n the part. 1 2 1.2 Heat Treatment Design There are b a s i c a l l y four d i f f e r e n t approaches used i n s e l e c t i n g heat treatment for a s p e c i f i c s t e e l [1,2,3]. A l l of these methods have a very l i m i t e d use i n the actual design of heat-treatment, but are very useful i n p r e d i c t i n g and comparing the depth of hardening of various s t e e l s . 1.2.1. Grossmann hardenability test The hardenability of a s t e e l i s defined as i t s a b i l i t y to harden by the formation of martensite and/or b a i n i t e f o r a given quenching condition. It i s used as a measure of the depth of hardening obtained on quenching and i s s p e c i f i e d as the distance below the surface where the structure i s 50% martensite and/or b a i n i t e . A c r i t i c a l diameter (Do) i s defined as the bar diameter which, when quenched i n a given medium under a given condition of a g i t a t i o n , w i l l y i e l d a centre-structure containing 50% martensite and/or b a i n i t e . An i d e a l diameter (Di) i s s i m i l a r l y defined for an i n f i n i t e l y rapid cooling condition, and i s independent of the cooling medium. Do, for a given cooling condition, i s determined by a metallographic examination of a number of c y l i n d r i c a l bars of the given s t e e l with d i f f e r e n t diameters and hardened under the given cooling condition. The Do-value thus obtained i s then converted to a Di-value using a set of standard graphical r e l a t i o n s h i p s . This method i s expensive because i t requires the t e s t i n g of a large number of specimens to determine Do for a p a r t i c u l a r grade of s t e e l . I t i s best applicable to c y l i n d r i c a l sections. I t only i d e n t i f i e s a zone with 50% martensite and/or b a i n i t e . D i f f e r e n t quenching mediums are described i n terms of a s e v e r i t y of quench which i s a parameter describing the average quenching 3 c h a r a c t e r i s t i c s f o r the whole duration of the quench. For these reasons t h i s method i s not well suited for the design of heat-treatments. 1.2.2 Hardenability from the chemical composition For low and medium-alloy s t e e l s , empirical r e l a t i o n s h i p s have been established r e l a t i n g the Grossmann Di-values, the grain s i z e and the carbon content of the s t e e l s . Standard graphs giving the multi p l y i n g factors f o r d i f f e r e n t a l l o y i n g elements based on the a l l o y content i n s o l u t i o n at the a u s t e n i t i z i n g temperature, are used to determine the e f f e c t of each a l l o y element on the hardenability. The Do-value i s calculated by multi p l y i n g Di-value, obtained above, by the i n d i v i d u a l m u l t i p l i e r s associated with each of the a l l o y i n g elements. Since i t . i s very d i f f i c u l t to assess the exact amount of each element i n s o l u t i o n at the conventional a u s t e n i t i z i n g temperatures, the Do-values c a l -culated by this, method can only be used for comparing two s t e e l s . To ensure complete d i s s o l u t i o n of a l l a l l o y i n g elements a higher temperature would have to be employed which might r e s u l t i n harmful grain growth e f f e c t s and a high retained austenite content i n the s t e e l . 1.2.3 The Jomlny. end-quench hardenability test A c y l i n d r i c a l bar specimen of standard dimensions i s made from the desired s t e e l . The specimen i s austenitized and end-quenched under a set of s p e c i f i e d conditions. I t i s then machined i n a prescribed manner and tested for hardness along i t s l o n g i t u d i n a l axis. A curve i s then prepared r e l a t i n g the hardness to the corresponding distance from the quenched end. The cooling rate decreases progressively from the quenched end along the length of the specimen. This gives a unique thermal h i s t o r y f o r each l o c a t i o n along the axis with a c e r t a i n hardness associated with i t . Empirical r e l a t i o n -4 ships are also a v a i l a b l e , r e l a t i n g the hardness obtainable at d i f f e r e n t locations on the Jominy specimen to the s t e e l chemistry and the grain s i z e , for low-alloy s t e e l s . Jominy test data can be manipulated to r e l a t e i t to several quenching conditions and to d i f f e r e n t specimen shapes, using the cooling curves as the basis for the comparison. Jominy t e s t i n g i s commonly used, due to i t s s i m p l i c i t y , r e p r o d u c i b i l i t y and f l e x i b i l i t y , as a method of q u a l i t y c o n t r o l i n s t e e l production. In order to make use of Jominy data, for the design of heat-treatment for a given s t e e l part, a c o r r e l a t i o n must be established between the Jominy cooling curve for a desired hardness and the quench conditions that w i l l produce that thermal h i s t o r y at a desired l o c a t i o n i n the given part. Also, since the s t e e l composition a f f e c t s i t s cooling r a t e , a set of standard Jominy curves cannot be used for d i f f e r e n t s t e e l s . A Jominy test must be performed for each composition and a u s t e n i t i c grain s i z e . 1.2.4 A p p l i c a t i o n of C.C.T. diagrams The structure associated with a s p e c i f i c thermal h i s t o r y i n a s t e e l of a given grain s i z e can be obtained from i t s C.C.T. diagram. A c o r r e l a t i o n between t h i s thermal h i s t o r y and the quench conditions that w i l l produce t h i s thermal, h i s t o r y at a desired l o c a t i o n i n the given part must s t i l l be established. The composition of the s t e e l and i t s grain s i z e are important parameters that modify the l o c a t i o n of the various transformation boundaries i n the C.C.T. diagram. A C.C.T. diagram would, therefore, be required for each s t e e l chemistry and for each grain s i z e . Since i t requires a considerable amount of time and e f f o r t to produce a C.C.T. diagram, only a few such d i a -grams are a v a i l a b l e for a few selected compositions having a s p e c i f i e d grain s i z e . 5 1.2.5 Proposed method The Jominy and the C.C.T. diagram methods are the only two methods that make use of s p e c i f i c thermal h i s t o r i e s to determine d i f f e r e n t structures i n s t e e l s . Grossmann method uses H-factors for average cooling rates during the quench. A l l these methods lack the c o r r e l a t i o n between the desired thermal h i s t o r y at a given l o c a t i o n and the required quenching conditions; t h i s i s needed for the design of e f f e c t i v e heat-treatment of a given s t e e l part. The proposed method provides t h i s c o r r e l a t i o n . The required thermal h i s t o r y for a desired structure or hardness i s obtained either from the C.C.T. diagram of the given s t e e l , i f a v a i l a b l e , or by Jominy t e s t method. This thermal h i s t o r y i s then imposed at the given location, i n a s i m p l i f i e d mathematical model of the part, to c a l c u l a t e the heat-transfer c o e f f i c i e n t curve needed during the quench. By comparing t h i s c a l -culated curve with various predetermined heat-transfer c o e f f i c i e n t curves f o r d i f f e r e n t quenching conditions, the required quenching conditions are determined. The advantages of t h i s method are: (a) i t can be applied to any s t e e l without any reservations, (b) i t can be applied to any geometrical shape for which the heat flow can be described mathematically and (c) i t can be applied to a small portion of or the whole range of the transformations. The major l i m i t a t i o n of t h i s method Is that a computer i s required to perform the large number of mathematical computations needed for the a n a l y s i s . Also, the accuracy of the method decreases as the chosen l o c a t i o n moves away from the surface of the part. 6 1.3 Review of Previous Work The procedures to determine the surface heat-transfer c o e f f i c i e n t as a function of the surface temperature of a s o l i d during quenching were f i r s t established by Paschkis, et a l . [4,5,6]. They used s i l v e r spheres, assuming that the surface heat-^transfer c o e f f i c i e n t at a given temperature i s dependent on the surface roughness only, and i s independent of the material below the surface. They examined the e f f e c t of quenchant temperature, depth of immersion, v e l o c i t y of i n s e r t i o n , tank s i z e , gas content ( i n water only), and a g i t a t i o n of the quench medium for various mediums. Comparison of various quenching conditions was made on the basis of the time i t took for the specimen to reach 250°F (121°C) from an i n i t i a l temperature of 1600°F (871°C) for each of the quenching conditions. Stolz [7],. In.his a n a l y s i s , used a numerical inversion of the a n a l y t i c a l s o l u t i o n of the s i m p l i f i e d d i r e c t problem of heat conduction with s p e c i a l reference to the sphere. This approach i s l i m i t e d i n that i t cannot be applied to the case of v a r i a b l e thermo-physical properties. He wrote the temperature s o l u t i o n at an i n t e r i o r point corresponding to a sequence of step changes i n surface heat f l u x , the f i r s t step being applied at time = 0, the second at time = At, the t h i r d at time = 2At, and so f o r t h . With such an expression for the temperature at hand, Stolz proposed to work backwards, u t i l i z i n g a prescribed temperature h i s t o r y at an I n t e r i o r point as the input and solving for the successive steps i n surface heat f l u x . In t h i s approach, i t was assumed that during the time i n t e r v a l from 0 to At, the prescribed temperature h i s t o r y was due to a s i n g l e step i n surface heat f l u x applied at time = 0; that during the time i n t e r v a l from At to 2At, the temperature h i s t o r y was due to a heat f l u x step applied at time = 0 and a second step 7 applied at time = At, and so f o r t h . The actual numerical work may be tedious inasmuch as i t i s necessary to know the value of an i n f i n i t e s e r i e s at every.stage of the calculation.. Although t h i s method can also be applied to the plane slab and to the cylinder as well as the sphere, the d e t a i l e d c a l c u l a t i o n s are d i f f e r e n t for the three geometries because the basic temperature solutions are d i f f e r e n t . Stolz's method [7] was r e s t r i c t e d to s i t u a t i o n s i n which the i n i t i a l temperature of the body was uniform. In contrast to the d i r e c t numerical approach of S t o l z , Sparrow et a l [8] took an e n t i r e l y d i f f e r e n t approach to the inverse problem of transient heat conduction. By u t i l i z i n g operational mathematics at an early stage of the a n a l y s i s , they devised a s o l u t i o n method which was more general and which may be computationally simpler. They processed the given temperature data the same way regardless of whether the body was a sphere, a c y l i n d e r , or plane slab. The major part of the computations involved i n t h i s method could be c a r r i e d out g r a p h i c a l l y u t i l i z i n g the standard heat-transfer charts r e a d i l y a v a i l a b l e i n various text books. Economopoulos, et a l . [9,10] proposed a purely numerical method for c a l c u l a t i n g the heat-transfer c o e f f i c i e n t s for f a s t cooling rates with v a r i a b l e thermo-physical properties. They used two types of probes, a long c y l i n d -r i c a l one and a f l a t disk, to measure the temperature at one point near the surface during cooling. An i m p l i c i t method was used to pass from the p a r t i a l d e r i v a t i v e s form to the f i n i t e d i fferences form of the numerical so l u t i o n . Nickel and mild s t e e l were used as probe materials. Lambert, et a l . [11] did a s i m i l a r study using the same probes and procedures. 8 1.4 Scope of the Present Work The aim of the present work was (a) to determine the surface heat-transfer c o e f f i c i e n t s p r e v a i l i n g during the quenching of s t e e l i n several commonly used quenching media, under d i f f e r e n t conditions, and (b) to devise a more precise method for the design of heat-treatment of s t e e l by quenching. To determine the heat-rtransfer c o e f f i c i e n t s , experimentally measured cooling curves were used i n numerical s o l u t i o n of the inverse problem of heat conduction. (These formed a set of standard heat-transfer c o e f f i c i e n t curves for d i f f e r e n t quenching conditions studied). A more precise method for the design of heat-treatment of s t e e l by quenching requires measuring of the thermal h i s t o r y at a l o c a t i o n during a Jominy t e s t . This thermal h i s t o r y provides the c o r r e l a t i o n between various structures and thermal h i s t o r i e s for the s t e e l . The thermal h i s t o r y for the desired structure i s imposed at the desired, l o c a t i o n , i n a s i m p l i f i e d mathematical model of the part, to c a l c u l a t e the heat-transfer curve needed during the quench. A comparison of t h i s curve with various standard curves y i e l d s the required quenching conditions for the desired structure. The v a l i d i t y of the method was checked by a c t u a l l y heat-treating a 4 inch diameter s t e e l b a l l . CHAPTER 2 MATHEMATICAL MODEL - THE INVERSE BOUNDARY VALUE PROBLEM To determine the temperature and heat-transfer c o e f f i c i e n t at the surface of the specimen, the temperature at an i n t e r i o r l o c a t i o n was experi-mentally measured as a function of time. This gives r i s e to an inverse problem of transient heat conduction. A mathematical model that quanti-t a t i v e l y describes the heat-transfer during quenching and a numerical method for c a l c u l a t i o n of r e s u l t s for the aforementioned inverse problem are outlined f i r s t . 2.1 The Model The mathematical model of heat-transfer during the quenching process i s based on fundamental p r i n c i p l e s of heat flow. While the development •presented here s p e c i f i c a l l y applies to a s e m i - i n f i n i t e slab, i t can be e a s i l y extended to cylinders and spheres. The model simulates the quenching of a s t e e l f l a t p l ate from a uniform i n i t i a l temperature into a large quenching bath at a lower uniform temperature. 2.2 Assumptions Made i n the Model In order to s i m p l i f y the complex mathematical problem of describing the quenching process a number of assumptions have been made and those are l i s t e d 9 10 below: i . I n i t i a l l y the specimen i s at a uniform temperature throughout i t s volume. i i . The thermo-physical properties of the specimen material are equal i n a l l d i r e c t i o n s at any given temperature. i i i . There i s no heat generation or consumption i n the specimen material due to any phase transformations within the temperature range of the model. i v . The bulk bath temperature of the quench medium remains uniform and constant, r e l a t i v e to the temperature changes i n the specimen, during the quench. v. One-dimensional heat flow i s symmetrical about the plane describing the centre of the thickness of the specimen. Edge e f f e c t s are to be neg-lected so that only the co-ordinate measured i n the d i r e c t i o n of the f i n i t e thickness i s needed to describe p o s i t i o n s . In the case of spheres and .cylinders symmetrical r a d i a l heat flow can be assumed and the r a d i a l distances w i l l describe p o s i t i o n s . v i . There i s no volume or shape change i n the specimen due to the temperature changes i n the specimen. 2.3 Model Geometry The f l a t p l ate i s considered to be made up of an odd number of t h i n s l i c e s , each of thickness. Ax, except for the surface s l i c e s which have a thickness of Ax/2 each, F i g . 2.1. The mid-point of the c e n t r a l s l i c e i s designated as node 1 and each successive node i s a distance of Ax from the previous node i n the outward d i r e c t i o n on both sides of the c e n t r a l s l i c e . This puts the i t h node i n the centre of the i t h s l i c e but the l a s t node (surface node) l i e s on the outer face of each outer s l i c e . Therefore the Figure 2.1 Model geometry showing d i v i s i o n of f l a t specimen into nodes. 12 distance of the ith node from the central node (node 1) equals (i-l)Ax. Since geometrical symmetry is assumed about the central node, only half the thickness of the plate (In the positive x direction, Fig. 2.1) is 2 considered for analysis. Also for simplicity, a unit area (1 cm ) normal to the x-direction, and a total S number of nodes (node s lies on outer surface of the surface slice) are considered for mathematical analysis. Therefore, for the ith node: 2 A^ = 1, cm (unit area) for 1 < i ^  S Ax 3 V = - r — , cm for i = 1 and i = S i 2 3 = Ax, cm for 1 < i < S 2.4 Heat Balance Equations An implicit finite difference method is used to derive the heat balance equations. The general equation used to describe the heat balance in any system is the following: Rate of _ Rate of + Rate of heat _ Rate of heat _ Rate of heat ,^  heat input heat output generated consumed accumulated (a) For the ith node, 1 < i < S: Heat-f I ow i - l 13 Heat i s supplied to the i t h node by the ( i - l ) t h node. The i t h node i n turn supplies heat to the ( i + l ) t h node. 3T Therefore Heat input = - k -A. , • — i-1 i -1 9x x=x i-1 Heat 9T output = - k. «A.*— i x 9x x=x, Heat generated = 0 Heat consumed = 0 dT Heat accumulated = p.'Cp.'V. - r — x x x 3t Substituting back into Eq. (2-1) the r e s u l t i s - k 9T i-1 3x x=x + k ^ fci 3x i-1 x=x. 3T - P i C p i A x 9 t -OR ( T. , - T. ) ( T - T ) (T. - T.) r i - 1 x y , - i i+l.. „ . x x - k^ , + k. = p . Cp. Ax • • • i - 1 Ax i Ax x r x • At Then multi p l y i n g throughout by (- Ax) the equation becomes , - , , N P. C p ^ A x ) 2 (I! k i - i ( T i - i " V " k i ( T i " W - — A l T ±) Rearranging the equation gives i t i t s f i n a l form - ( V i F i ) T i - i + [ 1 + W i + k i ^ T i - ( V i ) T i + i = T i ( 2 " 2 ) where F. = x At p ± C P i Ax for a l l i . 14 (b) Boundary condition, at the 1st node Helot-low 2 Heat input = 0 — 3T Heat output = - k. • — 1 1 3x x=x„ 3T Heat accumulated = Pi*Cp 1 ,V — 1 1 1 o t Substituting back into Eq. (2-1) the r e s u l t i s k 3T 9x x=x. Ax 3T P l C p l T "3T OR ^ ( .Tj_ - T 2 ) Ax o . Ax ( T l - V  P l C p l 2 A t -Then multip l y i n g throughout by (- Ax) the equation becomes 2 - k x ( T^ - T 2 ) = 2At Rearranging the equation gives i t i t s f i n a l form (1 + 2 1 ^ ) T.| - (2k1F]_> T 2 = T x (2-3) 15 (c) Boundary Condition, at the surface node At the surface of the p l a t e , the heat i s transferred to the quench medium. Heat-, f I o.w S-l To maintain mathematical continuity, a f i c t i t i o u s node s + 1, outside the plate surface, a distance Ax from node s i s considered. The heat flow across the f i c t i t i o u s unit area A associated with node s + 1 i s considered s+1 unaltered. Then between nodes S - l and S + 1 , 3T Heat l o s t by surface = - k 'A^-— s s 9x x=x s Heat gained by quench medium = Ag?h- ( A T ) ^ ^ u i ( j _ s u r f ace But Heat l o s t by surface = Heat gained by medium i i - k A ( T s - l " Ts+1 ) = A h (T, - T') s s — s f s 2Ax Rearranging the equation y i e l d s But also at the surface node Eq. (2-1) applies. 16 Therefore for the surface node s — 3T Heat input = - k, - -A ., •— r s-1 s-1 3x x=x s-1 3T Heat output = - k, «A • — s s 3x x=x, 3 T Heat accumulated = p, «Cp »V s s s 3t Substituting back into Eq. (2-1) the r e s u l t i s - k 3T s-1 3x x=x + k a s 3x s-1 x=x • Ax 3T = p s C p s — ¥ F OR (T* - T' ) ( T' - T' ) . (T - T') r s-1 s ' , , .••.s_s+l- „ Ax s s y - k i T + k -. = p. Cp — — s-1 Ax s Ax K s ^s 2 At (2-5) Substituting Eq. (2-4) into Eq. (2-5) and rearranging y i e l d s - 2F.(k | + k )T' + [1 + 2F (k , + k + 2hAx)]f' s s-1 s s-1 s s-1 s s = T s - < 4 F s h A x ) T f (2-6) The model describes the heat transfer aspect of the quenching process i n terms of the temperature f i e l d i n the specimen, the thermo-physical properties of the plate material, the surface heat-transfer c o e f f i c i e n t between the plate and the quench medium, the nodal distance and time increment by Eqs. (2-2, 2-3 and 2-6). The equations are mathematically stable for a l l time under the conditions of the model. 17 2.5 Solution of the Inverse Problem Any s o l u t i o n of the inverse problem of transient heat-conduction su f f e r s from inherent uncertainty. In a transient heat-conduction system, the e f f e c t of boundary conditions i s always damped at i n t e r i o r points, and the inverse problem involves b a s i c a l l y the extrapolation of the damped response to the surface, and sometimes d i f f e r e n t i a t i o n at the surface. The damped information cannot be completely recovered, but t h i s does give reasonable r e s u l t s [7]. The three equations used for the so l u t i o n are Eqs. (2-2, 2-3 and 2-6). Since the surface heat transfer c o e f f i c i e n t i s to be calcul a t e d , the Eq. (2-6) i s rearranged to the form - 2F (k . + k )T* . + [1 + 2F (k . + k )]T* s s - l s s - l s v s - l s J s + 4F Ax(T* - TJ.)h = T (2-7) s s f s again, Eqs. (2-2 and 2-3) are - (Vi Fi ) TI-i + [1 + W i + V ] T i - ( V i ) T i + i = T i ( 2 ~ 2 ) and (1 + Ik^^T^ - ( 2 ^ ^ ) ^ = T± (2-3) Eqs. (2-2, 2-3 and 2-7) can be written i n very general forms as follows: a,T. + b.T. + C.T' = T. (2-8) i i - l i i I l + l I and -a T' + b T' + c h = T (2-9) s s - l s s s s s J where For i = 1, from Eq. (2-3) & 1 = 0 b 1 = 1 + 2 k 1 F 1 C l - " 2^1 F1 • • 18 For 1 < i < S, from Eq. (2-2) a . = - k. . F. i i-1 l b i = 1 + F i ( k i - i + V c = - k F I I i For i = S, from Eq. (2-7) a = - 2F (k . + k ) s s s-1 s b = 1 + 2 F ( k . + k ) s s s-1 s c = 4F Ax(T - T.) S 8 8 f Let the node with known (experimentally measured) temperature time r e l a t i o n s h i p be the nth node, such that n i s close to but l e s s than S. Now Eq. (2-8) can be used to form a t r i - d i a g o n a l matrix of (n - 1) rows b l T l + C 1 T 2 a 2 T[ + b 2 T 2 + c z where d. l a, T*. , + b. T| + C . T'.,. i i - i i i I l + l a -1 T' + b T* n - i n-2 n-1 n-1 T, for 1 < i < (n-1) I T i " °i Ti+1 f o r 1 = ( n _ 1 ) — -d l d2 d. l n-1 Then the lower diagonal containing a l l ( a ) g can be eliminated to y i e l d 19 B B l T i + C 1 T 2 BB 2 T 2 + c 2 T3 B B i T l + C i T x + l BB T n-1 n-1 — — D D 1 DD 2 DD. X DD -n-1 where BB- = b. BB. = b. a. c . — x 1-1 BB i-1 for 1 < i ^ (n-1) DD, = d DD. = d -a. DD. . x 1-1 BB i-1 for 1 < i ^ (n-1) The upper diagonal containing a l l (c) can be eliminated to y i e l d a simple diagonal matrix as follows B B 1 T l BB 2 T 2 BB. T. 1 1 BB T n-1 n-1 DDD„ DDD, DDD. 1 DDD n-1 (2-10) where DDD - = DD , n-1 n-1 DDD. 1 DD - c. DDD. i 1 l + l BB i+1 for 1 £ i < (n-1) Rearranging matrix form of Eqs. (2-10) y i e l d s DDD. T' ^ i BB. for 1 < i < (n-1) (2-11) 20 Substituting back into Eq. (2-8), and rearranging, the r e s u l t i s T i+1 for n -$ i < s (2-12) Substituting back into Eq. (2-9) , and rearranging, the r e s u l t i s T -s a T s s - l - b T h = s s (2-13) c s The value of h can thus be calculated f or each step on the known temperature-time r e l a t i o n s h i p to give a r e l a t i o n s h i p between the surface temperature (T') and the surface heat-transfer c o e f f i c i e n t (h). 2.6 Numerical Calculations An IBM 370/168 computer was used for the numerical c a l c u l a t i o n s . The values of the thermophysical properties (k, p, Cp) as a function of temperature were given i n the form of subroutines. The experimentally measured cooling curve was d i g i t i z e d as d i s c r e t e points d i v i d i n g the curve into approxi-mately s t r a i g h t - l i n e segments. This data, together with the other quench data was read into the computer with the main program. The main program together with various subroutines i s given i n Appendix A. From.the beginning of the quench, the computer interpolates the experi-mental cooling curve l i n e a r l y at 10°C step reductions i n temperature to provide corresponding time steps. The temperature at each step i s then imposed on the nth node and the temperatures at a l l the nodes are calculated using equations 2-11 and 2-12. These nodal temperatures are then adjusted for the corres-ponding thermophysical properties using an i t e r a t i v e procedure. The cut o f f point for t h i s procedure was that the d i f f e r e n c e i n the surface temperature, 21 being most s e n s i t i v e , for the l a s t two i t e r a t i o n s must be l e s s than 10 °C. The corresponding value of h i s then calculated using equation 2-13. The r e s u l t s for the step are printed and the next temperature step i s performed. This continues u n t i l the end of the measured cooling curve i s reached. At the end the l a s t data point i s used for the c a l c u l a t i o n even though the temperature step i s l e s s than the prescribed 10°C. 2.7 V a l i d i t y of the Model To check the v a l i d i t y of the model used, i t s r e s u l t s were compared with values calculated a n a l y t i c a l l y f o r a s i m p l i f i e d case. The l a t t e r was chosen as s i m i l a r as possible to the model. 2.7.1 A n a l y t i c a l s o l u t i o n A s e m i - i n f i n i t e f l a t p l ate of h a l f thickness (L), having a uniform i n i t i a l temperature (T ) and constant thermophysical properties (k, p, Cp), quenched i n a f l u i d at i n i t i a l temperature T^ = 0°C, was considered for the a n a l y t i c a l case. The a n a l y t i c a l s o l u t i o n [12,13,14,15] f o r a,location x from the centre of thickness and at time, t from the beginning of the quench can be written as Sin 6n Cos (6n x/L)~ 6n + Sin 6n Cos 6n Several a n a l y t i c a l solutions were prepared f o r d i f f e r e n t values of h. An i n t e r n a l cooling curve from each of these runs was used as input i n the numerical s o l u t i o n to back c a l c u l a t e h. The following observations were made regarding the error i n the calculated value of h: = 2T . 2 , n=l exp ( - at Sn i . maximum error was observed i n the f i r s t step of the numerical solu t i o n . i i . the error reduced to less than 1% i n the f i r s t 0.25 seconds of the quench. i i i . the maximum error increased as the h-value increased, i v . the calculated h-values converged f a s t e r to the actual h-value fo r larger h-values. Table 2.1 shows the back calculated h-values for an a n a l y t i c a l run using h = 0.100., (cal/cm 2-sec. °C) , k = 0.05 (cal/cm. sec. °C) and a = 0.05 2 (cm /sec) . Quench Time (sec) Surface Temp. (°c) • h (cal/cm 2.sec.°C) Error CO 0.00 1000 - -0.01 951.5 0.1041 + 4.1 0.05 896.5 0.0981 - 1.9 0.10 858.5 0.0985 - 1.5 0.15 831.1 0.0988 - 1.2 0.20 809.0 0.0989 - 1.1 0.25 790.4 0.0991 - 0.9 0.50 723.6 0.0994 - 0.6 0.75 678.4 0.0995 - 0.5 1.0 643.8 0.0996 - 0.4 TABLE 2.1: Back calculated h-values f o r an a n a l y t i c a l run using h = 0.100 cal/cm 2.sec.°C; k = 0.05 cal/cm.sec.°C and a = 0.05 cm 2/sec. 23 2.7.2 Optimum values of the parameters Two types of parameters a f f e c t the model most, v i z : (i) those that a f f e c t the experimental measurement of the temperature at a l o c a t i o n near the surface i n the specimen. ( i i ) those r e l a t e d to the numerical c a l c u l a t i o n of h-values. (i) Experimental measurement: The parameters r e l a t e d to the experimental measurement of the temperature are the shape and s i z e of the specimen and the p o s i t i o n of the measuring thermocouple. Lambert, et a l . [10] found that i f a disc i s used to represent a s e m i - i n f i n i t e f l a t plate i t s diameter-to-thickness r a t i o must be kept above s i x to ensure n e g l i g i b l e edge e f f e c t s at i t s centre. Therefore, a diameter of 20 cm and a thickness of l e s s than 2.5 cm were used f o r a l l specimens. F i g . 2.2 shows the influence of the p o s i t i o n of the point of measurement on the accuracy of the r e s u l t s as established by Lambert, et a l . [10]. Due to machining l i m i t a t i o n s the point of measurement was kept at less than 2 mm from the surface. This, according to F i g . 2.2, gives a mean error of about 7% i n the r e s u l t s . ( I i ) Numerical c a l c u l a t i o n : The c a l c u l a t i o n method adopted involves the use of c e r t a i n parameters on which the s t a b i l i t y and the accuracy, of the c a l -c u l a t i o n procedure depend, v i z . the space increment or number of nodal sections and the time increment. The accuracy of the numerical methods based on the introduction of f i n i t e differences increases with the fineness of the space l a t t i c e adopted. However, to minimize the occupation of the core memory of the computer and to maintain reasonable accuracy, c a l c u l a t i o n s were performed using the r e s u l t s of an a n a l y t i c a l s o l u t i o n as input i n the numerical c a l c u l a t i o n s f o r various t o t a l number of nodes. It was found that no s i g n i f i c a n t increase i n accuracy 25 i s achieved by increasing the t o t a l number of nodes above 20. Since the point of temperature measurement must l i e on one of the nodes, two d i f f e r e n t but very close space increments were used i n the c a l c u l a t i o n s . One space increment was e f f e c t i v e between the centre of the specimen and the point of measurement and the other between the point of measurement and the specimen surface. Stolz [7] has pointed out that any change i n the surface temperature propagates into the specimen at a f i n i t e rate. As a r e s u l t , during the s o l u t i o n of the inverse problem, i f the time increment i s too small, the c a l c u l a t i o n at the measured point w i l l be c a r r i e d out before the e f f e c t of any surface perturbation has reached t h i s point. This w i l l cause o s c i l l a t i o n s of increasing magnitude. To avoid such a s i t u a t i o n , the c a l c u l a t i o n s were ca r r i e d out by imposing equal temperature steps at the measured point. CHAPTER 3 EXPERIMENTAL The apparatus needed and the procedure followed to obtain the temperature at an i n t e r i o r l o c a t i o n i n the specimen are described i n t h i s chapter. 3.1 Apparatus A schematic diagram of the apparatus i s shown i n F i g . 3.1. I t e s s e n t i a l l y consists of the following components: i . Furnace An induction furnace was constructed from t r a n s i t e . The chamber of the furnace was l i n e d with porcelain wool and then with low density f i r e b r i c k s . The f i n i s h e d dimensions of the chamber were approximately 25 cm high x 20 cm long x 3 cm wide, with induction c o i l s pressed against the outside of the larger sides. The furnace could be opened at the bottom through remote handles. The atmosphere i n the chamber could be c o n t r o l l e d by supplying the desired atmosphere through the i n l e t nozzle i n the door of the furnace. A hole i n the top of the furnace was provided to allow for the specimen holder to pass through, i i . Specimen The over a l l dimensions of the specimen were approximately 20 cm d i a . x 2 cm thick. The specimen was machined out of a one inch t h i c k plate to the shape shown i n F i g . 3.2. A 110 cm long, 13 mm outer diameter s t a i n l e s s 26 27 i Furnace f.i Specimen "ij-L-i Quench Tank . iv.,. Gverall_S . u p e r g S t r u c t u r e v- Specimen Movement Control Figure 3.1 Schematic diagram of the quenching apparatus used. Figure 3.2 Configuration of d i s c specimen showing thermocouple l o c a t i o n . 29 s t e e l pipe was fastened to one edge of the specimen using a s t a i n l e s s s t e e l bridge. The function of t h i s pipe was (a) to hold the specimen i n place f i r m l y , (b) to accommodate thermocouple wires from the specimen without obstructing i t s f a s t movement, and (c) to pass through a v e r t i c a l brass guide to cont r o l the d i r e c t i o n and or i e n t a t i o n of the specimen. At the opposite edge from the edge with the pipe, a small p i s t o n with a long rod was attached to the specimen. The length of t h i s rod was j u s t enough to locate the specimen i n the quench tank, i n the middle of the quench medium. When the specimen was dropped into the tank, the piston s l i d into a f i t t e d c y linder at the bottom of the tank, thus f o r c i n g a l l l i q u i d out of the cylin d e r through grooves provided. This absorbed the energy of the f a l l and provided a hold on the specimen to avoid any o s c i l l a t i o n s , i i i . Quench Tanks Two d i f f e r e n t tanks were used, one for o i l and the other for water and brine. The tank used for o i l was a simple b a r r e l 40 cm diameter x 55 cm deep with a small cylinder fastened on the in s i d e centre at the bottom to receive the piston at the bottom of the specimen. The tank used for water was approximately 61 cm x 61 cm x 76 cm deep with a p a i r of p l e x i g l a s s opposite walls. A small cylinder was fastened on the i n s i d e centre at the bottom to receive the piston at the bottom of the specimen. Provisions were made for (a) moving the f u l l tank by use of . casters, (b) emptying the tank by the use of a valve and a small pump near the bottom of the tank, (c) s t i r r i n g the medium by blowing cleaned a i r through a manifold bubbler at the bottom of the tank, and (d) heating the water by c i r c u l a t i n g i t through a c o i l of copper tubing, heated by gas burners, using the valve and pump mentioned above. 30 i v . Overall Super-structure The super-structure was made out of s t r u c t u r a l s t e e l and wood and was on heavy duty casters to provide mobility. The o v e r a l l dimensions were approximately 1 . 5 m x l . 5 m x 3 m high. I t was r i g i d l y made to avoid un-wanted v i b r a t i o n s i n the specimen and r e l a t i v e movement between d i f f e r e n t components. v. Specimen Movement Control By using a 15 cm long c y l i n d r i c a l brass guide located, j u s t above the furnace top, on a cross member of the super-structure, a smooth v e r t i c a l movement of the specimen was ensured. The s t a i n l e s s s t e e l tube on the specimen passed through the brass guide and was connected to a t h i n s t e e l cable. The cable i n turn, passed over three pulleys and came down at one corner of the main structure where a counter weight of about 500 gm was connected to i t s other end. There was a locking device that could lock the cable at any p o s i t i o n . The weight of the specimen kept the cable t i g h t on a l l the pulleys. To r a i s e the specimen into the furnace the counter weight was pulled down and the cable locked i n p o s i t i o n . To drop the specimen the cable lock was released and the specimen dropped due to i t s own weight. To minimize the drop time of the specimen, the furnace, the brass guide and the pulleys were properly aligned. Also the pulleys were equipped with double b a l l bearings to minimize f r i c t i o n . 3.2 Procedure Type K (Chromel-Alumel) bare thermocouple wires of 0.31 mm diameter each and about 2.5 m long were welded at the bottom of the thermocouple holes i n the specimen (Fig. 3.2). The wires were then insulated by threading them through t h i n ceramic tube i n s u l a t o r s of appropriate diameter and length. 31 The specimen was then assembled and mounted i n the apparatus. The output leads were plugged into the thermocouple wires. To obtain reasonably reproducible r e s u l t s the following steps were followed f or each experimental run on the specimens: i ) the resistance of the thermocouple was measured while r a i s i n g and lowering the specimen. If any f l u c t u a t i o n s i n the resistance were observed, It indicated a damaged thermocouple, which was repaired and retested. i i ) the specimen was lowered and rested on a removable bridge for r i g i d i t y . If the specimen was a used one, both i t s faces were prepared evenly using emery paper #120 on a rubber pad mounted on a v a r i a b l e speed power d r i l l . The thickness of the specimen was measured to keep a record of metal removel. i i i ) the specimen was raised into the cold furnace and the resistance of the thermocouple was rechecked for any damage during step i i . I f damage was found i t was repaired. iv) the quench tank was moved into p o s i t i o n ensuring good alignment between the cylinder at the bottom of the tank and the piston at the bottom of the specimen. v) the specimen was enclosed i n the furnace and i t s doors were closed. The desired atmosphere was supplied to the furnace. v i ) the quench tank was f i l l e d with the quench medium and i t s temp-erature was s t a b i l i z e d . v i i ) the thermocouple leads were connected to the chart recorder. The recorder was then allowed to s t a b i l i z e . The recorder was c a l i b r a t e d by apply-ing a constant coarse suppression and adjusting the d e f l e c t i o n . 32 v i i i ) the thermocouple leads were removed from the recorder and the f u r -nace was switched on. This was done to protect the recorder against damage due to induced currents i n the thermocouple wires crossing the e l e c t r o -magnetic f i e l d used i n the induction heating. ix) the temperature of the specimen was monitored using a potentio-meter. When the specimen reached the required temperature the power input to the furnace was.lowered to j u s t maintain the temperature. x) a f t e r the temperature was uniform i n the specimen, the atmosphere supply was shut o f f , the power to the furnace was shut o f f , the thermo-couple leads were connected to the recorder, the furnace doors were opened and the specimen cable lock was released. As the specimen dropped, i t turned on the recorder by a microswitch mounted at the top of the specimen tube. x i ) when the specimen temperature was low enough (as indicated by the thermocouple) the specimen was. ra i s e d out of the quench medium. This auto-m a t i c a l l y shut o f f the recorder. x i i ) the quench tank was emptied, i f the medium i n i t was not reusable, and moved out of the way to prepare for the next experimental run. A f t e r the required number of runs had been performed on a specimen, i t was then sectioned to determine the exact distance of the thermocouple wires from the surface. CHAPTER 4 RESULTS, DISCUSSIONS AND APPLICATION 4.1 Treatment of Experimental Data The thermal h i s t o r y of a point below the surface of the specimen was experimentally recorded during each quench as a time (sec.) vs. temperature ( m i l l i v o l t s ) p l o t . ' These plo t s were d i g i t i z e d and used as part of the input data for the computer program, given i n Appendix vA, to determine the surface temperature ( T g , °C) with the corresponding o v e r a l l heat-transfer co-2 . 2 e f f i c i e n t s (h, cal/sec.cm . °C) and heat fluxes (q, cal/sec.cm ). Other input data consisted of parameters describing the specimen and the quenching conditions. The computer output was plotted as q vs. T g and h vs. T g. Table 4.1 shows a t y p i c a l input .data and Table 4.2 shows i t s computer out-put. This output i s plotted i n F i g . 4.1 using l i n e a r scales f o r q and h. The data i s replotted i n F i g . 4.2 using logarithmic scales f o r q and h to examine the high temperature region. Fi g s . 4.1 and 4.2 can be divided into four temperature regions with t h e i r boundaries at T q (891°C), 575°C, 150°C, B.P. (^  100°C) and T f (60°C). These boundaries represent r e s p e c t i v e l y the specimen temperature at the beginning of the quench, end of stable f i l m b o i l i n g , end of t r a n s i t i o n b o i l i n g , end of nucleate b o i l i n g and the ambient temperature of water. These f i v e boundaries approximately separate the four stages of the cooling phenomena 33 TABLE 4.1: T y p i c a l computer input data Specimen Quench medium Location of thermocouple Thickness of specimen T o t a l number of nodes Thermocouple node # Number of data points Total quench time 304 s t a i n l e s s s t e e l S t i l l water at 60°C 0.140 cm from the surface 2.150 cm 23 20 39 52.00 sec. Time (sec) Temperature (m.v.) Time (sec) Temperature (m.v.) 0.00 37.0 45.70 22.0 0.30 36.8 46.15 21.6 4.20 36.0 46.50 21.2 10.00 34.4 47.25 20.0 14.50 34.0 47.43 19.6 17.80 33.0 47.70 18.8 21.00 32.0 48.10 17.2 24.28 31.1 48.30 16.4 24.60 29.8 48.50 15.4 24.83 29.2 48.80 13.6 25.20 28.8 49.10 12.4 27.95 27.6 49.23 12.0 31.80 26.4 49.40 11.6 34.90 26.0 49.56 11.2 36.70 25.8 50.03 10.4 39.20 25.2 50.40 10.0 41.50 24.4 50.80 9.6 42.40 24.0 51.40 9.2 44.60 22.8 52.00 9.0 45.20 22.4 TABLE 4.2: Computer output for data i n Table 4.1 Ts calculated surface temperature (°C) 2 q surface heat f l u x (cal/sec.cm ) 2 h o v e r a l l h e a t - t r a n s f e r - c o e f f i c i e n t (cal/sec.cm .°C) Ts h • q Ts h q Ts h h 874 .0047 3.83 592 .0150 7.98 184 .5041 62.51 863 .0050 4.02 582 .0151 7.88 165 .6181 64.90 852 .0056 4.44 572 .0152 7.78 149 .7481 66.58 841 .0063 4.92 562 .0160 8.03 141 .7846 63.55 830 .0069 5.31 549 .0193 9.44 135 .8088 60.66 819 . 0075 5.69 536 .0217 10.33 129 .8260 56.99 810 .0072 5.40 524 .0241 11.18 128 .7759 52.76 800 .0071 5.25 512 .0265 11.98 126 .7412 48.92 790 .0075 5.48 500 .0294 12.94 124 .4396 28.13 780 .0078 5.62 486 .0330 14.06 769 .0083 5.88 473 .0375 15.49 757 .0099 6.90 458 .0441 17.55 745 .0111 7.60 443 .0513 19.65 735 .0112 7.56 427 .0592 21.73 724 .0123 8.17 410 .0712 24.92 714 .0131 8.56 392 .0847 28.12 702 .0142 9.12 374 .0990 31.09 690 .0150 9.45 354 .1202 35.34 679 .0161 9.97 332 .1483 40.34 667 .0169 10.26 312 .1745 43.97 652 .0179 10.60 294 .1985 46.45 639 .0171 9.90 278 .2209 48.16 627 .0164 9.30 262 .2466 49.81 618 .0159 8.87 245 .2790 51.62 611 .0156 8.60 226 .3271 54.30 602 .0154 8.35 206 .3986 58.20 37 Figure 4.2 Semi-logarithmic p l o t s of q vs T s and h vs T g for the r e s u l t s shown i n Table 4.2. Various b o i l i n g regions are shown. 38 commonly observed during quenching, v i z . stable f i l m b o i l i n g , t r a n s i t i o n b o i l i n g , nucleate b o i l i n g and natural convection. A d e s c r i p t i o n of these phenomena follows. At. the beginning of the quench, the hot metal surface comes i n contact with r e l a t i v e l y cold l i q u i d causing a sudden heat l o s s at the surface i n a small f r a c t i o n of a second, a period f or which the numerical method does not provide a v a l i d s o l u t i o n [10]. A vapor f i l m begins to form around the specimen reducing the i n i t i a l l y large heat f l u x from the metal to the l i q u i d . Under agreeable conditions, t h i s f i l m becomes stable causing a persistant drop i n both q and h values. The break up of t h i s stable f i l m marks the onset of t r a n s i t i o n b o i l i n g [13,15,16] during which both q and h values con-tinuously increase. The end of t r a n s i t i o n b o i l i n g i s marked by the peak i n the q-value [13,15], which occurs at 150°C i n F i g s . 4.1 and 4.2. As the surface temperature drops further the h-value continues to increase, reaches a peak and then sharply decreases while the q-value decreases continuously. As the surface temperature approaches the b o i l i n g point of the quenchant, nucleate b o i l i n g ends and heat trans f e r occurs by natural convection u n t i l the whole specimen reaches the temperature of the surrounding quenchant. A l l quenches were discontinued 'before the surface temperature reached the b o i l i n g point of the quenchant to protect the thermocouples from any possible damage due to quenchant contact. The temperature range of each of these stages varies from quench to quench. The duration of each stage during a quench a f f e c t s the q-values and thus the h-values and the cooling rates i n the specimen. 39 4.2 Re p r o d u c i b i l i t y of Results To check the r e p r o d u c i b i l i t y of the r e s u l t s , two runs were made under s i m i l a r quench conditions. F i g . 4.3 shows the pl o t s of h-values for both runs and t h e i r quench conditions. Exact duplicate conditions were not achieved f o r the runs because of experimental l i m i t a t i o n s . However the two r e s u l t s are i n reasonable agreement with a maximum deviation of ± 17.9% occurring at 800°C. 4.3 Quench Variables To study the influence of a quench v a r i a b l e on the h-curve, a l l reasonable precautions were taken to keep the other v a r i a b l e s constant i n the corresponding experimental runs. Six quenching v a r i a b l e s were studied i n a l l . 4.3.1 I n i t i a l specimen, temperature Fi g s . 4.4 and 4.5 compare the q-values and h-values r e s p e c t i v e l y obtained by quenching the s t a i n l e s s s t e e l specimen i n s t i l l water at 20°C from two d i f f e r e n t i n i t i a l temperatures. Table 4.3 gives the analysis of Fi g . 4.4. No stable f i l m b o i l i n g stage i s observed i n the 842°C run, whereas i n the 891°C run the stable f i l m b o i l i n g stage may be present. A l a t e i o n s e t of nucleate b o i l i n g i n the 891°C run suggests that a higher i n i t i a l specimen temperature increases the s t a b i l i t y of the vapor f i l m during t r a n s i t i o n and/or f i l m b o i l i n g stages. The e f f e c t of the temperature range of the various cooling stages on the h-values can be seen i n F i g . 4.5. The maximum differe n c e i n the h-values fo r the two cases occurs at about 500°C where the h-value f o r the 842°C run 0 100 200 300 400 500 600 700 800 900 1000 Surface temperature, (°c) Figure 4.3 Plots of h vs T s for two s i m i l a r quenches of the s t a i n l e s s s t e e l specimen from 987 ± 5°C i n s t i l l water at 10 ± 2°C. 41 1000 cr u (j w o u 100 o a> jC <u o o H~ k. 3 10 Initial temperature 842"C JL JL JL JL JL 100 200 300 400 500 600 700 Surface temperature, (°c. 800 900 Figure 4.4 Plots of q vs T s for the stainless steel specimen in s t i l l water at 20°C from two different i n i t i a l temperatures. u u V o u 5 IC? in c o o X 1 1 1 1 1 1 1 1 — zz ZZ — — — — — Initial temperature — - V >^-842*C — — — zz «-— — — — - -- -_ — zz — _ — — — -- --1 1 1 I 1 1 1 1 100 200 300 400 500 600 700 Surface temperature, (°c) 800 900 Figure 4.5 Plots of h vs T s for the stainless steel specimen in s t i l l water at 20°C from two different i n i t i a l temperatures. 42 TABLE 4.3: Analysis of F i g . 4.4 I n i t i a l Specimen Temp., T Q (°C) STABLE FILM BOILING: Ends, T S B.(°C) Quench time, t ^ (sec) Temp, drop, ( T Q - T S B ) , (°C) Duration, t ^ (sec) TRANSITION BOILING: Ends, T T B (°C) Quench time, t 2 (sec) Temp, drop, ( T S B - T T B ) , (°C) Duration, (t2 - t ^ ) , (sec) NUCLEATE BOILING: Temp, at end of run, T E (°C) Quench time, t3 (sec) Temp. d r o p , . ( T T B - Tg), (°C) Duration, (t3 - t 2 ) , (sec) Run Low T; 842 410 7.12 432 7.12 150 9.40 260 2.28 High T Q 891 805* 4.46 330 10.72 150 12.18 86** 4.46 475 6.26 180 1.46 * Stable f i l m b o i l i n g may not be present i n t h i s case. I f so, ignore t h i s set of numbers. ** If stable f i l m b o i l i n g i s not present add t h i s set of numbers to the set of numbers for the immediately following stage. 43 i s about 59% greater than that for the 891°C run. Since our r e p r o d u c i b i l i t y error i s l e s s than ± 20%, the above difference may be considered s i g -n i f i c a n t . Lambert, et a l . [11] observed no s i g n i f i c a n t differences i n the heat fluxes measured on a n i c k e l specimen heated to 850, 900, 925 and 950°C, and water-quenched. Mitsutsuka, et a l . [16] found that a higher i n i t i a l specimen temperature tends to increase the s t a b i l i t y of the f i l m b o i l i n g phenomenon, i n agreement with the present fi n d i n g s . 4.3.2 I n i t i a l bath temperature Fi g s . 4.6 and 4.7 compare the q-values and h-values r e s p e c t i v e l y , obtained by quenching the s t a i n l e s s s t e e l specimen i n s t i l l water at three bath temperatures. Table 4.4 gives the analysis of F i g . 4.6. I t i s observed that as the bath temperature increases, the temperature range of the stable f i l m b o i l i n g stage increases. .r This suggests that an increase i n the bath temperature increases the s t a b i l i t y of the vapor f i l m . Correspondingly slower cooling rates during t r a n s i t i o n b o i l i n g and l a t e r onset of nucleate b o i l i n g at higher bath temperatures support the deduction above. The slower cooling rates at higher bath temperature during nucleate b o i l i n g are due to f corresponding decrease i n the d r i v i n g force (T - T^). The e f f e c t of the temperature range of the various cooling stages on the h-values can be seen i n F i g . 4.7. Lambert, et a l . [11], Mitsutsuka, et a l . [16] and Paschkis, et a l . [5] have reported s i m i l a r findings. 4.3.3 A g i t a t i o n of bath Figs. 4.8 and 4.9 compare the q-values and h-values r e s p e c t i v e l y , obtained by quenching the s t a i n l e s s s t e e l specimen i n 20°C water i n s t i l l and s t i r r e d conditions. Table 4.5 gives the analysis of F i g . 4.8. I t i s observed that X I I I 1 1 1 1 1 1 1 0 100 200 300 400 S00 600 700 800 900 Surface temperature, (°c) Figure 4.7 Plots of h vs T s for the stainless s t e e l specimen i n s t i l l water at three bath temperatures. 45 TABLE 4.4: Analysis of F i g . 4.6 I n i t i a l Specimen Temp., T q (°C) STABLE FILM BOILING: Ends, T g B (°C) Quench time, t ^ (sec) Temp, drop, (T q - • T g B ) , (°C) Duration, t ^ (sec) TRANSITION BOILING: Ends, T T B (°C) Quench time, (sec) Temp, drop, ( T g B - T ^ ) , (°C) Duration, (t^ - t ^ ) , (sec) NUCLEATE BOILING: Temp, at end of run, T„ (°C) Quench time, t ^ (sec) Temp, drop, ( T T B - T £ ) , (°C) Duration, (t-^ - t^), (sec) Run 20°C 40°C 60°C 891 884 891 805 777* 575* 4.46 10.89 40.75 86 107** 316** 4.46 10.89 40.75 330 190 150 10.72 26.32 48.82 475 587 166 6.26 15.43 8.07 150 150 125 12.18 28.50 52.00 180 40 25 1.46 2.18 3.18 * Stable f i l m b o i l i n g may not be present i n t h i s case. I f so ignore t h i s set of numbers. ** If stable f i l m b o i l i n g i s not present add t h i s set of numbers to the set of numbers f or the immediately following stage. t l I I I I L_ I I 1 1 1 0 100 200 300 400 300 600 700 800 900 1000 Surface temperature, (°c) Figure 4.8 Plots of q vs T g for the s t a i n l e s s s t e e l specimen i n 20°C water i n s t i l l and s t i r r e d conditions. Surface temperature, (°c) Figure 4.9 P l o t s of h vs T g for the s t a i n l e s s s t e e l specimen i n 20°C water i n s t i l l and s t i r r e d conditions. TABLE 4.5: Analysis.of F i g . 4.8 Run S t i l l Quench S t i r r e d Quench I n i t i a l Specimen Temp., T q (°C) 891 894 STABLE FILM BOILING: Ends, T g B (°C) 805* 825* Quench time, t ^ (sec) 4.46 2.70 Temp. drop,.(T q - T g B ) , (°C) 86** 69** Duration, t ^ (sec) 4.46 2.70 TRANSITION BOILING: Ends, T T B (°C) 330 375 Quench time, (sec) 10.72 5.17 Temp, drop, ( T g B - T ^ ) , (°C) 475 450 Duration,, (t^ - t ^ ) , (sec) 6.26 2.47 NUCLEATE BOILING: Temp, at end of run, T_ (°C) 150 155 Quench time, t ^ (sec) 12.18 11.16 Temp. drop,.(T T B.- T £ ) , (°C) 180 295 Duration, ( t ^ - t^), (sec) 1.46 5.99 * Stable f i l m b o i l i n g may not be present i n t h i s case. I f so ignore t h i s set of numbers. ** If stable f i l m b o i l i n g i s not present add t h i s set of numbers to the set of numbers for the immediately following stage. 48 the cooling rates during t r a n s i t i o n b o i l i n g and/or stable f i l m b o i l i n g are s i g n i f i c a n t l y higher i n the s t i r r e d case. An early onset of nucleate b o i l i n g i n the s t i r r e d case suggests that s t i r r i n g of the bath reduces the s t a b i l i t y of the vapor f i l m by p h y s i c a l l y disturbing i t s outer l a y e r . During nucleate b o i l i n g slower cooling rates are observed i n the s t i r r e d case. An explanation for t h i s may be that since a i r was bubbled to cause s t i r r i n g , these bubbles may have a blanketing e f f e c t on the specimen surface reducing the heat f l u x during nucleate b o i l i n g . Such an e f f e c t does not come into play during stable f i l m and t r a n s i t i o n b o i l i n g because a layer of vapor f i l m already exists, at the specimen surface. An o v e r a l l reduction i n quench time i s observed due to s t i r r i n g . Paschkis, et a l . [4] and Lambert, et a l . [10] have reported s i m i l a r findings except that they did not report slower cooling rates during nucleate b o i l i n g stage i n the s t i r r e d case, but then they did not use a i r bubbling as a means for s t i r r i n g . F i g s . 4.10 and 4.11 compare the h-values for the s t i l l and s t i r r e d quenches i n water at 40°C and 60°C re s p e c t i v e l y . I t i s observed that the e f f e c t of s t i r r i n g (air-bubbling) on h-values decreases as the bath tempera-ture i s increased. 4.3.4 Surface oxidation F i g s . 4.12 and 4.13 compare the q-values and h-values r e s p e c t i v e l y , obtained by quenching into s t i l l water at 20°C, the p l a i n carbon s t e e l specimen i n the non-oxidized ( l i g h t oxide) and the oxidized (heavy oxide) conditions. Table 4.6 gives the analysis of F i g . 4.12. I t i s observed that the drop i n surface temperature during stable f i l m b o i l i n g for the oxidized case i s s i g n i f i c a n t l y l e s s and i t occurs over a longer time period. Since the thermal resistance of the heavy oxide scale adds to that of the vapor in c o o w X ,-<?l i 1 1 1 1 1 1 1 — 0 100 200 300 400 500 600 700 800 900 Surface temperature, (°c) F i g u r e 4 . 1 0 P l o t s o f h v s T s f o r t h e s t a i n l e s s s t e e l s p e c i m e n i n 4 0 ° C w a t e r i n s t i l l a n d s t i r r e d c o n d i t i o n s -F i g u r e 4 . 1 1 P l o t s o f h v s T s f o r t h e s t a i n l e s s s t e e l s p e c i m e n i n 6 0 ° C w a t e r i n s t i l l a n d s t i r r e d c o n d i t i o n s . 50 •I ' i I I 1 I 1 1 1 0 100 200 300 400 500 600 700 800 900 Surface temperature, (°c) Figure 4.12 Plots of q vs T s for quenching into s t i l l water at 20°C, the p l a i n carbon s t e e l specimen i n the non-oxidized ( l i g h t oxide) and the oxidized (heavy oxide) conditions. Surface temperature, (°c) Figure 4.13 Plots of h vs T s for quenching into s t i l l water at 20°C, the p l a i n carbon s t e e l specimen i n the non-oxidized ( l i g h t oxide) and the oxidized (heavy oxide) conditions. 51 TABLE 4.6: Analysis of F i g . 4.12 Run Non-oxidized Oxidized I n i t i a l Specimen Temp.-, T q (°C) 891 891 STABLE FILM BOILING: Ends, T g B (°C) 700* 750 Quench time, t ^ (sec) 7.71 9.65 Temp, drop, (T q - T g B ) , (°C) 191** 141 Duration, t ^ (sec) 7.71 9.65 TRANSITION BOILING: Ends, T T B (°C) 235 307 Quench time, t 2 (sec) 19.62 17.88 Temp, drop, ( T G B - T ^ ) , (°C) 465 443 Duration, ( t 2 - t ^ ) , (sec) 11.91 8.23 NUCLEATE BOILING: Temp, at end of run, T_ (°C) 150 150 Quench time, t ^ (sec) 21.06 33.61 Temp, drop, 0*. - T £ ) , (°C) 41 157 Duration,.(t^ - t ^ ) , (sec) 1.44 15.73 * Stable f i l m b o i l i n g may not be present i n t h i s case. I f so ignore t h i s set of numbers. ** If stable f i l m b o i l i n g i s not present add t h i s set of numbers to the. set of numbers for the immediately following stage. 52 f i l m , i t reduces the cooling rates during stable f i l m b o i l i n g , thus i n -creasing the time duration of t h i s stage but reduces the s t a b i l i t y of the vapor f i l m , causing t r a n s i t i o n b o i l i n g to begin at a higher temperature. Faster cooling rates during t r a n s i t i o n b o i l i n g and an early onset of nucleate b o i l i n g . i n the oxidized case support the deduction above. S i g n i f i c a n t l y slower cooling rates during nucleate b o i l i n g are observed i n the oxidized case. I t i s thought that at the onset of nucleate b o i l i n g the bath water penetrates the thick oxide scale, comes into contact with the hot specimen surface, and bo i l s . . This phenomena causes the oxide scale to become lo o s e l y attached to the specimen at c e r t a i n l o c a t i o n s , forming various vapor pockets between the scale and the- specimen surface, thus reducing the cooling rates during .this stage. (Loose oxide scale was observed on the specimen at the end of quench i n the oxidized case). The o v e r a l l e f f e c t i s that heavy oxide f i l m on s t e e l increases the t o t a l quenching time considerably. 4.3.5 Specimen material Figs. 4.14 and 4.15 compare the q-values and h-values r e s p e c t i v e l y , obtained by quenching into s t i l l water at 20°C, the s t a i n l e s s s t e e l specimen and the p l a i n carbon s t e e l specimen i n the non-oxidized ( l i g h t oxide) con-d i t i o n . The general shapes of the q and h curves f o r both cases i s approxi-mately the same. Table 4.7 gives the analysis of F i g . 4.14. Stable f i l m b o i l i n g may be present i n both cases. The p l a i n carbon s t e e l specimen has a longer t r a n s i t i o n boiling-stage with lower cooling rates. I t suggests that the vapor; f i l m on the p l a i n carbon s t e e l specimen i s more stable than on the st a i n l e s s s t e e l specimen. This i s thought to be due to differences i n the surface oxides and t h e i r roughness. • The thermophysical properties of the oxide layer depend upon i t s chem-i s t r y , which i s a function of the specimen material. Since the thermophysical 53 F i g u r e 4 . 1 4 P l o t s o f q v s T s f o r q u e n c h i n g i n t o s t i l l w a t e r a t 2 0 ° C , t h e s t a i n l e s s s t e e l s p e c i m e n a n d t h e p l a i n c a r b o n ( A I S I - 1 0 3 8 ) s t e e l s p e c i m e n i n t h e n o n - o x i d i z e d ( l i g h t o x i d e ) c o n d i t i o n . Surface temperature, (°c) F i g u r e 4 . 1 5 P l o t s o f h v s T s f o r q u e n c h i n g i n t o s t i l l w a t e r a t 2 0 ° C , t h e s t a i n l e s s s t e e l s p e c i m e n a n d t h e p l a i n c a r b o n ( A I S I - 1 0 3 8 ) s t e e l s p e c i m e n i n t h e n o n - o x i d i z e d ( l i g h t o x i d e ) c o n d i t i o n . 54 TABLE 4.7: Analysis of F i g . 4.14 Run Stainless Steel P l a i n Carbon Steel I n i t i a l Specimen Temp.,.TQ (°C) 891 891 STABLE FILM BOILING: Ends, T g B (°C) ft 805 ft 700 Quench time, t- (sec) 4.46 7.71 Temp, drop, (T q - T g B ) , (°C) ** 86 ** 191 Duration, t ^ (sec) 4.46 7.71 TRANSITION BOILING: Ends, T T B (°C) 330 235 Quench time, (sec) 10.72 19.62 Temp, drop, ( T g B - T ^ ) , (°C) 475 465 Duration, (t^ ~ t ^ ) , (sec) 6.26 11.91 NUCLEATE BOILING: Temp, at end of run, T„ (°C) 150 150 Quench time, t ^ (sec) 12.18 21.06 Temp, drop, (T^ B - T £ ) , (°C) 180 41 Duration, ( t ^ - t^), (sec) 1.46 1.44 * Stable f i l m b o i l i n g may not be present .in t h i s case. I f so ignore t h i s set of numbers. ** I f stable f i l m i s not present add t h i s set of numbers to the set of numbers f or the immediately following stage. 55 properties of the specimen material were taken into account i n determining the surface temperature, the h-values should be independent of the specimen material i n the t r u l y non-oxidized surface condition. Since i t i s com-me r c i a l l y d i f f i c u l t to a t t a i n the t r u l y non-oxidized surface condition, the chemistry of the oxide layer a f f e c t s the h-values. 4.3.6 Quench medium Figs. 4.16 and 4.17 compare the h-values obtained by quenching into four d i f f e r e n t media the s t a i n l e s s s t e e l specimen and the non-oxidized ( l i g h t oxide) p l a i n carbon s t e e l specimen re s p e c t i v e l y . The four quench media used were water, brine (3% NaCl), o i l and a i r . The cooling phenomena i n the case of water, o i l and brine are e s s e n t i a l l y the same except that i n the case of brine the presence of dissolved s a l t i n the water has a twofold e f f e c t on the k i n e t i c s of heat-transfer. I t modifies the thermal con-d u c t i v i t y , of water and the thermodynamic s t a b i l i t y and adherence c h a r a c t e r i s t i c s of the vapor f i l m during f i l m b o i l i n g [11]. In the case of an a i r quench, the cooling i s by both r a d i a t i o n and natural convection. F i g s . 4.16 and 4.17 ind i c a t e that the o v e r a l l quenching e f f i c i e n c i e s i n the decreasing order were for the quenching media: brine, water, o i l and a i r r e s p e c t i v e l y . 4.4 Comparison with Previous Work Lambert, et a l . [11] state that the value of the heat-transfer co-e f f i c i e n t f o r a given process depends on phenomena which obey very complex and generally poorly defined ph y s i c a l laws, as well as on the surface con-d i t i o n of the product and the type of f a c i l i t y used. They also point out that i t i s l i k e l y to vary from one f a c i l i t y to another, even i f these 100 200 300 400 500 600 700 800 Surface temperature, (°c) Figure 4.16 Plots of h vs T s for the s t a i n l e s s s t e e l specimen quenched i n four d i f f e r e n t quench media, v i z . brine (3% by wt. NaCl), water, o i l (H0UGHT0-QUENCH-G) and Air. 100 200 300 400 500 600 700 Surface temperature, (°c) 800 Figure 4.17 Plots of h vs T s for the p l a i n carbon s t e e l specimen quenched i n four d i f f e r e n t quench media, v i z . brine (3% by wt. NaCl), water, o i l (HOUGHTO-QUENCH-G) and A i r . 58 f a c i l i t i e s are s i m i l a r . Precautions were taken to approximate the known conditions used i n the previous work. 4.4.1 Quenching i n s t i l l a i r Fig.. 4.18 compares the h-curve determined i n t h i s work to that obtained by Lambert, et a l . [10] for non-oxidized s t e e l quenched i n s t i l l a i r under si m i l a r . c o n d i t i o n s . The maximum deviation i n the two curves i s at 550°C, 2 where the.respective h-values are 0.00164 and 0.0013 cal/cm sec.°C, i . e . our value i s 26.2% higher than theirs.. For most of the temperature range the h-values from both experiments are i n good agreement. 4.4.2 Quenching i n s t i l l water F i g . 4.19 compares the h-curve determined i n t h i s work, using a s t a i n -l e s s s t e e l specimen, to those obtained by Lambert, et a l . [10] using a n i c k e l specimen., for a s t i l l water quench under s i m i l a r conditions. They report two curves -for apparently s i m i l a r conditions. Above 225°C the three curves have approximately the same shape but below t h i s temperature t h e i r curves show a continuous reduction i n the h-values. The curve obtained i n t h i s study, however, reaches a peak i n h-value at 157°C before showing a reduction. The observed differences i n the curves may be a t t r i b u t a b l e to the differences i n the surface roughness, oxide properties, i n i t i a l specimen temperature and bath temperature. 4.4.3. .Quenching i n s t i l l , brine (3% NaCl) s o l u t i o n Fig..4.20 compares the h-curve determined i n t h i s work using a s t a i n l e s s s t e e l specimen, to that obtained by Lambert, et a l . [11] using a n i c k e l specimen, of c y l i n d r i c a l shape, for a s t i l l brine (3% by wt. NaCl) s o l u t i o n quench under s i m i l a r conditions. The two curves i n t e r s e c t at several points 5 9 p . 6 2 3 </> o o c q io3 c o o X T 1 1 1 r T 1 r -26-2% • T -Ref.(IO) T ( ° C ) T £ ( ° C ) o f 8 7 5 8 9 1 X X X X 2 4 X X 100 200 300 400 S00 600 700 Surface temperature, (°c) 800 900 1000 F i g u r e 4 . 1 8 C o m p a r i s o n o f h - c u r v e d e t e r m i n e d i n t h i s w o r k w i t h t h a t o b t a i n e d b y L a m b e r t , e t a l . [ 1 0 ] f o r n o n - o x i d i z e d s t e e l q u e n c h e d i n s t i l l a i r u n d e r s i m i l a r c o n d i t i o n s . Surface temperature, (°c) F i g u r e 4 . 1 9 C o m p a r i s o n o f h - c u r v e d e t e r m i n e d i n t h i s w o r k , u s i n g a s t a i n l e s s s t e e l s p e c i m e n , w i t h t h o s e o b t a i n e d b y L a m b e r t , e t a l . [ 1 0 ] u s i n g a n i c k e l s p e c i m e n , f o r a s t i l l w a t e r q u e n c h u n d e r s i m i l a r c o n d i t i o n s . 0 100 200 300 400 500 600 700 800 900 1000 Surface temperature, (°c) Figure 4.20 Comparison of h-curve determined i n t h i s work, using a s t a i n l e s s s t e e l specimen, with that obtained by Lambert, et a l . [11] using a n i c k e l specimen of c y l i n d r i c a l shape, f o r a s t i l l b r i n e (3% by wt. NaCl) solution quench under s i m i l a r conditions. 61 above 210°C but are reasonably coincident. The maximum differe n c e i n the two curves occurs at 750°C where t h e i r values are within 25% of the mean value. Below 210°C the curve obtained i n t h i s work shows a slower reduction i n h-value than that of Lambert, et a l . ' s . The cause for these differences i n the two curves i s not very c l e a r . 4.4.4 Quenching, i n s t i l l o i l Fi g . 4.21 compares the h-curve determined i n t h i s work using a s t a i n l e s s s t e e l specimen, to that obtained by Stolz et a l . [6] using a s i l v e r specimen of s p h e r i c a l shape, for a s t i l l o i l quench. The o i l used i n the present study was a. commercial quench o i l known as 'Houghto-Quench-G', and was selected because i t s known properties were approximately the same as those of the slow o i l used by St o l z , et a l . Comparison of the two o i l s i s given i n Table 4.8. TABLE 4.8: Comparison of Quenching O i l s Property Slow O i l [6] HQ-G Flash Point (°F) 360 355 F i r e Point (°F) 410 410 V i s c o s i t y at 100°F 103 105-115 The peaks in. both curves occur at a common temperature of 485°C, where t h e i r h-values are within 10% of the mean value. It i s noted that i n the quench performed i n t h i s work the stable f i l m b o i l i n g was not present. Below 485°C the h-values are comparable i n the two cases. The differences i n the curves may be a t t r i b u t e d to the differences i n the quench o i l s and quenching conditions. 62 1.0 0 o d < cal. .1 10 * — * c <v o o o er c o 1 o a> X 100 200 300 400 500 600 700 Surface temperature, (°c) 800 900 Figure 4.21 Comparison of h-curve determined i n t h i s work, using a s t a i n l e s s s t e e l specimen, with that obtained by- S t o l z , et a l . [6] using a s i l v e r specimen of sp h e r i c a l shape, for a s t i l l o i l quench under s i m i l a r conditions. 63 4.5 A p p l i c a t i o n to Design Problem The r e s u l t s obtained i n t h i s work, together with the Jominy-test data and the mathematical model, were used to determine the necessary quenching conditions required to obtain a desired thermal h i s t o r y or mechanical property at a given p o s i t i o n i n a 4 inch diameter s t e e l g r i n d i n g - b a l l . STELCO manufactures these b a l l s and they were water quenched during t h e i r manufacture. The procedure employed i n t h i s a p p l i c a t i o n was as follows: 1. A Jominy bar specimen was prepared from an annealed b a l l with a small thermocouple placed r a d i a l l y at 3mm from the quenched end. i i . The standard Jominy t e s t was performed while recording the cooling curve at the thermocouple l o c a t i o n . i i i . The Jominy distance vs. hardness data was pl o t t e d as shown i n F i g . 4.22. i v . An examination of the STELGO-heat-treated b a l l gave a hardness d i s t r i b u t i o n i n the b a l l as shown i n F i g . 4.23. The p o s i t i o n 5mm from the surface of the b a l l had a hardness of HRC (61-64.5). I t was chosen to produce a hardness of HRC (61-64.5) at 5mm from the surface i n the test b a l l . v. An equivalent Jominy distance f or the above hardness i s 6mm from the quenched' end, having a hardness of HRC (60-64). v i . In a one dimensional mathematical simulation of the Jominy t e s t , the thermocouple data was used to c a l c u l a t e the thermal h i s t o r y of the 6mm Jominy p o s i t i o n . v i i . In the s i m p l i f i e d mathematical model of the grinding b a l l , the calculated thermal h i s t o r y of the 6mm Jominy p o s i t i o n was imposed at 5mm l o c a t i o n from the surface i n the b a l l to y i e l d the h-curve required. I i 0 20 40 60 80 100 Distance from quenched-end, (mm.) Figure 4.22 Distance vs hardness p l o t for the Jominy Bar. 65 Distance from surface, (mm.) Figure 4.23 Distance vs hardness plot for the Commercial Grinding Ball. 66 v i i i . This r e q u i r e d h-curve was compared w i t h the h-curves f o r d i f f e r e n t quenching c o n d i t i o n s to s e l e c t the appropriate quenching medium having a s i m i l a r curve. F i g . 4.24 shows that above 300°C the h—curve f o r the 20°C s t i r r e d water quench i s approximately the same as the r e q u i r e d h-curve. Below 300°C the r e q u i r e d h-value decreases where as the s t i r r e d water quench has e s s e n t i a l l y , a constant h-value. The composition of the g r i n d i n g b a l l s t e e l i s shown i n Table 4.9, along w i t h that of AISI 5160 s t e e l . Of the a v a i l a b l e CCT curves, the composition of the AISI 5160 s t e e l best approximated the b a l l composition. F i g . 4.25 shows a CCT diagram f o r the AISI 5160 s t e e l w i t h the measured and the d e s i r e d c o o l i n g curves p l o t t e d on i t . Using t h i s CCT TABLE 4.9: Comparison of composition of commercial g r i n d i n g b a l l s t e e l w i t h AISI 5160 s t e e l Element Commercial Grinding B a l l S t e e l CO AISI 5160 (%) C 0.65 0.63 Mn 0.61 0.86 S i 0.31 0.23 Cr 0.47 0.83 N i 0.36 Cu 0.39 Mo 0.05 V 0.008 diagram as a f i r s t approximation f o r the b a l l s t e e l , the l o c a t i o n of the d e s i r e d c o o l i n g curve, that r e p r e s e n t i n g the 5mm p o s i t i o n i n the b a l l , i n -d i c a t e s that l e s s than 1% a u s t e n i t e transforms i n t o f e r r i t e + b a i n i t e and the Surface temperature, (°c) Figure 4.24 Comparison of the required h-curve with the h-curve for the 20°C s t i r r e d water quench. 68 for AISI 5160 1600 Homer.Research Laboratories, Bethlehem Steal Corp. 20 50 Cooling Time, Sec 1000 Figure 4.25 Continuous-cooling-transformation diagram f or the AISI-5160 s t e e l . [ 1 7 ] , The measured thermal h i s t o r y i n the Jominy bar and the desired thermal h i s t o r y i n the s t e e l b a l l are plotted on i t . i measured thermal h i s t o r y (J-bar, 3mm) i i desired thermal h i s t o r y (J bar, 6mm -> b a l l , 5mm) 69 remainder transforms to martensite on cooling below about 560°F (293°C). If the b a l l was quenched i n 20°C s t i r r e d water, according to F i g . 4.24 the 5mm b a l l p o s i t i o n should follow the desired cooling curve shown i n F i g . 4.25 u n t i l the b a l l surface reaches about 300°C, i . e . for the f i r s t 7.5 seconds of the quench. The cooling rates w i l l then be f a s t e r than the ones given by the desired cooling curve. However, t h i s should not a f f e c t the desired property s i g n i f i c a n t l y since only l e s s than 1% austenite transforms into ferrite..+ b a i n i t e at the desired cooling rates. Therefore, 20°C s t i r r e d water quench was selected as the quench condition required to produce a hardness, of HRC (61-64.5) at a p o s i t i o n 5mm below the surface of the 4 inch diameter s t e e l grinding b a l l . i x . To t e s t the selected quench condition, an annealed b a l l was quenched i n 20°C s t i r r e d water from about 845°C. F i g . 4.26 shows the hardness d i s -t r i b u t i o n obtained i n the b a l l . At the 5mm p o s i t i o n below the surface the obtained hardness was HRC (61-63.5), comparing favorably with the desired hardness of HRC (61-64.5). 4.6 Comments The basic d i f f e r e n c e between the use of the r e s u l t s of t h i s work as suggested in.§ 4.5 and other methods discussed i n § 1.2 i s that the other methods are very e f f e c t i v e i n choosing the proper s t e e l for a given part whereas the method used here i s e f f e c t i v e i n choosing quench conditions to obtain c e r t a i n property at a given l o c a t i o n i n a given s t e e l part. The choice of quench conditions may also be l i m i t e d due to the part design, i f i t develops undesirable d i s t o r t i o n , quench cracks, r e s i d u a l stresses or surface f i n i s h . Our approach provides a more precise method for the design of 70 Figure 4.26 Distance vs hardness pl o t f o r the Heat-rtreated B a l l . 71 heat treatment of s t e e l by providing a c o r r e l a t i o n between the d e t a i l e d thermal h i s t o r y of a given l o c a t i o n i n the part and the quenching conditions. Our approach does not replace any of the e x i s t i n g methods but compliments them. CHAPTER 5 SUMMARY AND CONCLUSION 5.1 Summary of the e f f e c t of quenching v a r i a b l e s i . A higher i n i t i a l specimen temperature increases the s t a b i l i t y of the vapor f i l m , thus prolonging the range of the stable f i l m b o i l i n g and t r a n s i t i o n b o i l i n g stages. i i . ' A hotter quench bath also increases the s t a b i l i t y of the vapor f i l m . i i i . A g i t a t i o n of the bath reduces the s t a b i l i t y of the vapor f i l m by introducing ph y s i c a l disturbances. However, an increase i n the bath temperature reduces the effectiveness of s t i r r i n g . i v . Heavy oxide scale reduces the heat flow rate and the s t a b i l i t y of the vapor f i l m . The composition, thickness and adherence of the oxide scale a f f e c t the rate of heat flow through i t . v. D i f f e r e n t quench media have d i f f e r e n t quenching e f f i c i e n c i e s . In the order of decreasing e f f i c i e n c y the four media tested were brine (3% by wt. NaCl), tap water, o i l (Houghto-Quench-G) and a i r . v i . The r e s u l t s obtained i n t h i s work are reproducible to within/± 18% and are i n reasonable agreement with the r e s u l t s obtained by previous workers. 72 73 5.2 Conclusion The r e s u l t s obtained i n t h i s work, together with the Jominy test data and the mathematical model were e f f e c t i v e l y used to determine the quenching conditions required to obtain a desired, mechanical property at a given l o c a t i o n i n a s t e e l b a l l . CHAPTER 6 SUGGESTIONS FOR FUTURE WORK The discussion and the a p p l i c a t i o n of the r e s u l t s of the present work suggest that more quench media may be Investigated which are of i n t e r e s t to the Industry. The e f f e c t of the thickness, composition and adherence of the oxide scale on the heat transfer process requires a closer i n v e s t i g a t i o n . The degree of a g i t a t i o n of the quenchant may be quantified and i t s e f f e c t s categorized. Attempts were made to use the r e s u l t s of t h i s work together with a CCT diagram and the mathematical model to predict hardness obtainable i n a s t e e l by quenching i n water and o i l . These tests f a i l e d because the s t e e l being tested and the cooling rates obtained did not duplicate those used to obtain the CCT diagram. A CCT diagram can only be used, with confidence, f o r the given cooling rates, composition, grain s i z e and a u s t e n i t i z i n g temperature. Jominy test was chosen over CCT diagram because of i t s s i m p l i c i t y . Since the cooling rates i n a specimen are a function of i t s shape, i t would be i n t e r e s t i n g to study and standardize a test for r a d i a l cooling i n spheres and cy l i n d e r s . The data from such a test could be r e a d i l y used i n the a p p l i c a t i o n of the h e a t - t r a n s f e r - c o e f f i c i e n t curves. A better understanding of the quenching phenomenon may be achieved by employing high-speed photography to observe the specimen surface and syn-cronizing i t with i t s thermal h i s t o r y during a quench. 1 74 REFERENCES 1. Thelning, K.E., Steel and i t s Heat-treatment, Bofors Handbook, 1975. 2. The Making, Shaping and Treating of Stee l , U.S. Steel Corporation. 3. Metals Handbook, Vol. 2 (Heat-treating), 8th Ed., ASM, 1964. 4. Paschkis, V. and S t o l t z , G. J r . , J . Metals, 1956, 8, pp. 1074-75. 5. Paschkis, V. and S t o l t z , G. J r . , Iron Age, V o l . 178, No. 21, Nov. 22, 1956, pp. 95-97. 6. S t o l t z , G. J r . , Paschkis, V., B o n i l l a , C F . and Acevedo, G., J . ISI, Oct. 1959, pp. 116-123. 7. S t o l t z , G. J r . , Trans. ASME, V o l . 82, Series C, 1960, pp. 20-26. 8. Sparrow, E.M., Trans. ASME, Sept. 1964, pp. 369-375. 9. Economopoulos, M., C.N.R.M., No. 14, March 1968, pp. 45-58. 10. Lambert, N. and Economopoulos, M., J. ISI, 208, 8 (1970), pp. 917-928. 11. Lambert, N. and Greday, T., C.N.R.M. , No. 44, Sept. 1975, pp. 13-27. 12. Carslaw,.H.S. and Jaeger, J.C., Conduction of Heat i n Sol i d s , 2nd Ed., Oxford U n i v e r s i t y Press, 1959. 13. K r e i t h , F., P r i n c i p l e s of Heat Transfer, 2nd Ed., Int. Text Book Company, 1969. 14. Developments i n Heat Transfer, E d i t o r : W.M. Rohsenow, The M.I.T. Press, 1964. 15. Rohsenow, W.M. and Choi, H., Heat, Mass and Momentum Transfer, Prentice H a l l , 1961. 16. Mitsutsuka, M. and Fukuda, K., Trans. ISU, V o l . 16, 1976, pp. 46-50. 17. Metal Progress Materials and Process Engineering DATABOOK, Second E d i t i o n , pp.83. 75 c C 7 6 C C c c c c c C A P P E N D I X A C C c C S O U R C E L I S T I N G O F T H E C O M P U T E S P R O G B A H . C c c C * * T H I S P B O G B A H C A L C U L A T E S T H E Q U E N C H H E A T T R A N S F E R C O E F F I C I E N T S F O R C * * T H E F L A T P L A T E S P E C I M E N , T H E T E M P E B A T U B E I S M E A S U R E D E X P E 8 I M E N T A -C * * L L Y A T T H E N O D E ( N T ) W I T H B E S P E C T T O T H E Q U E N C H T I M E . C I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) . C O M M O N K B O C P N C = 0 J T = 2 0 0 0 E P S = 1 . 0 D - 0 6 D I M E N S I O N T O ( 3 0 ) , TJS ( 3 0 ) , T N ( 3 0 ) , A ( 3 0 ) , B ( 3 0 ) , C ( 3 0 ) , D ( 3 0 ) , B B ( 3 0 ) , 1 D D ( 3 0 ) , D D D ( 3 0 ) , A L F A ( 3 0 ) ,11 ( 7 0 ) , T M V ( 7 0 ) , T P ( 7 0 ) , P T ( 2 0 0 ) , P H ( 2 0 0 ) R E A L * 4 S S T ( 2 0 0 ) , S H T ( 2 0 0 ) C C * * * T H E F O L L O W I N G T W O C A L L S B E A D T O T A L N U M B E R O F N O D E S , T H E H U H B E B O F C * * * T H E N O D E A T W H I C H T H E T E M P E B A T U B E I S M E A S U R E D ( C E N T R E - L I N E I S #1) , C * * * A N D K B O C P ( I P K B O C P . N E . 1 , I T M E A N S T H A T T H E V A L U E S O F K , R O S C P C * * * V A B Y W I T H T E M P E R A T U S E ) ; A N D T H E D I S T A N C E O F T H E M E A S U R E D N O D E C * * * F R O M T H E C E N T R E - L I N E O F T H E P L A T E R E S P E C T I V E L Y . C C A L L F R E A D ( 5 , ' I N T E G E R . . . * , N N , N T , K B O C P ) C A L L F R E A D ( 5 , * R * 8 . . . * , D 1 N T ) C C * * # T H E F O L L O W I N G C A L L H E A D S T H E 1 S T . D A T A C A R D W H I C H P R O V I D E S T H E C * * * V A L U E S F O B Q U E N C H M E D I U M T E M P . ( T F I N ) , D I S T A N C E O F N O D E N T F B O M T H E C * * * S U R F A C E O F T H E P L A T E A N D T O T A L Q U E N C H T I M E . C 1 C A L L F B E A D (5 , ' S * 8 . , . * , T F T N , D N T N N ^ T Q T ) C C * * * T H E F O L L O W I N G C A L L B E A D S T H E 2 N D . D A T A C A B D W H I C H P R O V I D E S T H E C * * * N U M B E R O F D A T A C A R D S . C 6 C A L L F B E A D ( 5 , » I H T E G E B . . . » , N D C ) C D O 1 0 I = 1 , N D C C A L L F B E A D ( 5 , ' R * 8 * , T I ( I ) , T M V ( I ) ) P = T M V ( I ) 1 0 T P ( I ) = - 3 . 0 + ( 2 5 . 8 6 1 * P ) - ( 0 . 1 0 8 7 6 * ( P * * 2 ) ) • ( 0 . 0 0 1 7 0 1 4 * f P * * 3 ) ) T I N 1 = T P ( 1 ) D F = T I N L - T F I N N M = N N - 1 N S = N T - 1 D X C = D 1 N T / N S D X S = ( D N T N N / ( N N - N T ) ) c C 77 XX=D1NT+DNTNN TS1=10.0 TE=TINL QT=0.0 WBITE(6,18)TINL,TFIN,XX,DNTNN,TQT,DXC,DXS WRITE(6,19)NN,NT,NDC,KBOCP 18 FORMAT (/8F15.6) 19 FORMAT (/8115) DO 20 .1=1, NN TO(I)=TINL 20 TN(I)=TINL WBITE (6,86) QT, (TN (I) ,I=NT,NN) C C*** THE FOLLOWING PAST OF THE PBOGBAW INTERPOLATES FOB TIME (QT) FOR A C*** GIVEN TEMP. (TE) USING TEMP. STEP (TS1) . C*** THE FIRST 5 CARDS INITIALISE THE VARIABLES AND THEN #95 DO LOOP c*** PERFORMS ONE CALCULATION FOB EACH INTERPOLATION. C N1 = 1 N=2 QTQ=0.0 X 1=TI (N 1) Y1 = TP(N1) DO 95 J=1,JT NC=NC+1 TE=TE-TS1 22 ¥2=TP(N) IF (12.IE.TE) GO TO 25 N1 = N N=N + 1 IF{N1. GE.NDC) GO TO 26 XI = TI (N1) Y 1 = TP (N 1) GO TO 22 2 5 X2=TI(N) QT=X1- ( ( (X2-X1)/(Y2-Y1) ) *(Y1-TE) ) X 1=QT Y1 = TE GO TO 28 26 TE=TP(N1) QT=TI(N 1) 28 DT=QT-QTQ QTQ=QT 29 DO 30 1=1,NS 30 TM(I)=TN(I) TM (NT) =TE DO HQ 1=1, NS ALFA (I) = (DT/ (ROCP (TM(I) ) *DXC*DXC) ) IF (I.GT.1) GO TO 31 A(I)=0.0 C (I)=- (ALFA (I) * (AK (TM (I) ) + AK (TM (1+1) )) ) B(I) = 1.0-C(I) BB(I)=B(I) D (I) =TO (I) DD(I)=D(I) GO TO 40 31 A X=AK (TM (1-1)) AY=AK (TM (I)) AZ=AK (TM (1 + 1) ) A fl)=- ( (ALFA (I) ) * ( <AX + AY)/2. 0) ) C (I)=- { (ALFA (I) )*{(AZ+AY) /2.0) ) B <I) = 1.0~{A (I)+C (X) ) BB (I) = B (I) - ( (A (I) *C (I- 1) ) /BB (I- 1) ) IF (I.EQ. NS) GO TO 3 3 D (I)=TO (I) DD (I) =D (I) - ( {A (I) *D D (I- 1) ) /BB (I-1) ) GO TO 40 D(I)=TO (I) -|C(I) *TE) GO TO 32 CONTINUE DO 50 13=1, NS I=NT-M IF{I.IT.NS) GO TO 41 DDD (.I)=DD(I) GO TO 50 DDD (I) =DD (I) - ( (C (I) *DDD (I* 1) ) /BB (1+ 1) } CONTINUE DO 60 1=1,NS TN (I)=DDD(I)/BB(I) IF(KROCP.EQ. 1) GO TO 6 1 DIFS=DABS(TNpS) -TM (NS) ) IF(DIFS.GT.EPS) GO TO 29 TN (NT) =TE MM=NN-NT IF(MM.LT.I) GO TO 81 DO 70 I=NS,NN TM(I) = TN(I) DO 80 I=NT , N N AX=AK (TM (1-1) ) A Y=AK (TM (I) ) AZ=AK{TH{X + 1}) IF(I.GT.NT) GO TO 71 ALFA (I) = ( (2.0*DT) /{ROCP (TM (I) ) * (DXC + DXS) ) ) A CI)=-MALFA (I) ) * { { AX+AY)/ (2. G*DXC) ) ) C <!)=-{ (ALFA (I) )*{ (AZ+AY)/{2. 0*DXS) ) ) GO TO 72 ALFA (I)=|DT/{ROCP{TM (I) ) *DXS*DXS) ) IF (I.EQ. NN) GO TO 74 A (I) =- ( (ALFA (I) ) * ( (AX + AY) /2. 0) ) C (I)=- ( {ALFA (I) ) *( (AZ*AY)/2.0) ) B (I) = 1.0-{A(I) + C(I) ) D(I)=TO(I) TN (1 + 1 )= (D (I) - (A(I) *TN (1-1))- (B(I)*TN (I) ) )/C (I) GO TO 80 DRFOBC=TN(I)-TFIN A (I) =- ( ALF A (I) * (AX+ <3.0*AY) ) ) B (I) = 1.0-A(I) C{I)=4.0*ALFA(I)*DXS*DRFORC D (I)=TO (I) CONTINUE IF(KROCP.EQ. 1) GO TO 81 DIFS=DABS (TN (NN) —TM (NN) ) IF(DIES.GT.EPS) GO TO 6 2 I=NN HTC=<D(I)-(A(I) *TN(I-1))-(B{I)*TN (I) J )/C(I) PT (NC) =TN { NN) c C 7 9 P H ( N C ) = H T C W H I T E ( 6 , 8 6 ) O f , ( T N ( I ) , I = N T , N N ) , H T C , D T 8 6 F O R M A T ( F 1 0 . 3 , 4 F 9 . 0 , F 1 0 . 6 , F 1 0 . 3 , 2 F 1 0 . 1 ) I F < D B F O B C . L T . 0 . 0 ) G O T G 1 0 0 I F ( D B F O R C . G T . D F ) G O T O 1 0 0 I F ( Q T . G E . T Q T ) G O T O 1 0 0 I F ( N 1 . G E . N D C ) G O T O 1 0 0 D O 9 0 1 = 1 , N N 9 0 T O ( I ) = T N ( I ) 9 5 C O N T I N U E 1 0 0 D O 1 3 0 1 = 1 , N C T 1 = 0 . 0 0 D O 1 2 0 K = 1 , N C T 2 = P T ( K ) I F ( T 2 . L E . T 1 ) G O T O 1 2 0 T 1 = T 2 J = K 1 2 0 C O N T I N U E S S T ( I ) = T 1 S H T ( I ) = P H ( J ) P T ( J ) = 0 . 0 0 W R I T E ( 6 , 1 2 5 ) I , S S T ( I ) , S H T ( I ) , J 1 2 5 F O R M A T ( 1 1 0 , F 1 0 . 0 , F 1 0 . 4 , 1 1 0 ) 1 3 0 C O N T I N U E 2 5 0 C O N T I N U E M 1 = 1 D O 3 1 0 K = 1 , 1 0 0 0 M 2 = M l + 7 I F ( M 2 . G T . N D C ) M 2 = N D C W R I T E ( 6 , 3 0 1 ) ( T I ( I ) , T P ( I ) , I = M 1 , M 2 ) 3 0 1 F O R M A T {/8 ( F 1 0 . 3 , F 6 . 0 ) ) I F ( M 2 . G E . N D C ) G O T O 3 1 1 3 1 0 M 1 = M 2 + 1 3 1 1 S T O P E N D C C c C * * D E P E N D I N G U P O N T H E M A T E R I A L O F T H E S P E C I M E N O N E O F T H E F O L L O W I N G C * * T W O S U B - R O U T I N E S ' M A Y B E O S E D I N T H E P R O G R A M . C C C C * * T H E F O L L O W I N G S O B - R O U T I N E G I V E S T H E T H E R M O - P H Y S I C A L P R O P E R T I E S O F C * * 3 0 4 - S T A I N L E S S S T E E L A T A G I V E N T E M P E R A T U R E . C C F U N C T I O N A K ( T Z ) C * * * T H I S F U N C T I O N G I V E S T H E M E A N T H E R M A L C O N D U C T I V I T Y F O R 3 0 4 - S . S T E E L C * * * A T A G I V E N T E M P E B A T U H E . C C I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) C O M M O N K R O C P I F ( K R G C P . E Q . 1) G O T O 1 0 1 0 X = 0 . 0 0 0 0 2 X F | T Z . 1 T . 9 5 0 . 0 ) G O T O 1 0 0 1 A K = 0 . 0 4 7 + ( X * T Z ) C 8 0 G O T O 1 0 1 1 1 0 0 1 I F ( T Z . I T . 9 0 0 . 0 ) G O T O 1 0 0 2 A K = Q . 0 2 8 + ( 2 . 0 * X * T Z ) G O T O 1 0 1 1 1 0 0 2 I F ( T Z . I T . 8 5 0 . 0 ) G O T O 1 0 0 3 A K = 0 . 0 4 6 + ( X * T Z ) G O T O 1 0 1 1 1 0 0 3 I F ( T Z . L T . 8 0 0 . 0 ) G O T O 1 0 0 4 A K = 0 . 0 8 0 - ( X * T Z ) G O T O 1 0 1 1 1 0 0 4 I F ( T Z . L T . 7 5 0 . 0 ) G O T O 1 0 0 5 A K = 0 . 0 4 8 + ( X * T Z ) G O T O 1 0 1 1 1 0 0 5 I F ( T Z . L T . 5 5 0 . 0 ) G O T O 1 0 0 6 A K = 0 . 0 3 3 + ( 2 . 0 * X * T Z ) G O T O 1 0 1 1 1 0 0 6 I F ( T Z . L T . 5 0 0 . 0 ) G O T O 1 0 0 7 A K = 0 . 0 2 2 + ( 3 . 0 * X * T Z ) G O T O 1 0 1 1 1 0 0 7 I F ( T Z . L T . 2 5 0 . 0 ) G O T O 1 0 0 8 A K = 0 . 0 3 2 + ( 2 . 0 * X * T Z ) G O T O 1 0 1 1 1 0 0 8 I F ( T Z . L T . 5 0 . 0 ) G O T O 1 0 0 9 A K = 0 . 0 3 7 + ( X * T Z ) G O T O 1 0 1 1 1 0 0 9 A K = 0 . 0 3 8 G O T O 1 0 1 1 1 0 1 0 A K = 0 . 0 5 0 1 0 1 1 R E T U R N E N D F U N C T I O N R O C P ( T Z ) C * * * T H I S F D N C T I O N G I V E S T H E P R O D U C T O F M E A N D E N S I T Y A N D M E A N S P E C I F I C C * * * H E A T F O R 3 0 4 - S T A I N L E S S S T E E L A T A G I V E N T E M P E R A T U R E . I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) C O f l f l O N K R O C P I F ( K R O C P . E Q . 1 ) G O T O 2 0 1 0 I F ( T Z . L T . 1 5 0 . 0 ) G O T O 2 0 0 1 R O = 7 . 9 3 6 4 7 1 - ( 0 . 0 0 0 4 7 6 4 7 1 * T Z ) G O T O 2 0 0 3 2 0 0 1 I F ( T Z . L T . 1 0 0 . 0 ) G O T O 2 0 0 2 R O = 7 . 9 5 5 - ( 0 . 0 0 0 6 * T Z ) G O T O 2 0 0 3 2 0 0 2 R O = 7 . 9 2 - ( 0 . 0 0 0 2 5 * T Z ) 2 0 0 3 I F ( T Z . L T . 8 7 5 . 0 ) G O T O 2 0 0 4 C P = 0 . 1 3 1 1 2 5 + ( 0 . 0 0 0 0 2 5 * T Z ) G O T O 2 0 0 9 2 0 0 4 I F ( T Z . L T . 8 2 5 . 0 ) G O T O 2 0 0 5 C P = 0 . 1 8 8 - ( 0 . 0 0 0 0 4 * T Z ) G O T O 2 0 0 9 2 0 0 5 I F ( T Z . L T . 6 6 5 . 0 ) G O T O 2 0 0 6 C P = 0 . 1 2 1 4 8 4 3 7 5 + ( 0 . 0 0 0 0 4 0 6 2 5 * T Z ) G O T O 2 0 0 9 2 0 0 6 I F ( T Z . L T . 5 7 5 . 0 ) G O T O 2 0 0 7 C P = 0 . 1 9 6 5 2 7 7 7 6 - ( 0 . 0 0 0 0 7 2 2 2 2 * T Z ) G O T O 2 0 0 9 2 0 0 7 I F ( T Z . L T . 4 5 0 . 0 ) G O T O 2 0 0 8 C P = 0 . 0 8 6 + ( 0 . 0 0 0 1 2 * T Z ) G O T O 2 0 0 9 c C 81 2 0 0 8 C P = 0 . 1 1 8 4 + ( 0 . 0 G 0 0 4 8 * T Z ) 2 00 9 SOCP=BO*CP GO TO 2 0 1 1 2 0 1 0 R O C P - 1 . 0 2 0 1 1 BETDBH END C C *•* END OF S U B - B O U T I N E . C c c C * * T H E F O L L O W I N G S U B - E O U T I N E G I V E S T H E T H E B H O - P H Y S I C A L P R O P E R T I E S OF C * * P L A I N C A R B O N S T E E L ( A I S I 1 0 3 8 ) AT A G I V E N T E M P E R A T U R E . C C F U N C T I O N A K ( T Z ) C * * * T H I S F U N C T I O N G I V E S T H E MEAN T H E B H A L C O N D O C T I V I T Y FOR MED IUM C * * * C A B B O N S T E E L ( 0 . 4 % C) AT A G I V E N T E M P E R A T U R E . I M P L I C I T BEAL*8 ( A - H , 0 - Z ) COMMON K B O C P I F ( K B O C P . E Q . 1) G O T O 1 0 9 9 I F ( T Z . L T . 8 1 0 . ) GO TO 1 0 0 1 A K = 0 . 0 3 0 6 7 + ( 3 . 3 3 3 3 D - 0 5 * T Z ) GO TO 1 1 0 0 1 0 0 1 I F ( T Z . L T . 7 0 0 . ) GO TO 1 0 0 2 A K = 0 . 1 6 3 - ( 1 . 3 D - 0 4 * T Z ) GO TO 1 1 0 0 1 0 0 2 I F ( T Z . L T . 3 0 0 . ) GO TO 1 0 0 3 A K = 0 . 1 3 8 5 - ( 9 . 5 D - 0 5 * T Z ) GO TO 1 1 0 0 1 0 0 3 A K = 0 . 1 2 5 - ( 5 . 0 D - 0 5 * T Z ) GO TO 1 100 1 0 9 9 A K = 0 . 0 9 1 1 0 0 B E T U B N END F U N C T I O N R O C P ( T Z ) C * * * T H I S F U N C T I O N G I V E S T H E P R O D U C T O F MEAN D E N S I T Y AND MEAN S P E C I F I C C * * * H E A T FOB MED IUM CARBON S T E E L { 0 . 4 JSC) AT A G I V E N T E M P E B A T U B E . I M P L I C I T B E A L * 8 ( A - H , 0 - Z ) COMMON KROCP I F ( K B O C P . E Q . 1) GO TO 2 0 9 9 IF(TZ.LT.800.) GO TO 2 0 0 1 B 0 = 7 . 8 4 9 8 7 7 5 3 9 - ( 1 . 9 5 4 5 5 D - 0 7 * T Z ) GO TO 2 0 0 6 2 0 0 1 I F ( T Z . L T . 6 0 0 . ) GO TO 2 0 0 2 R 0 = 7 . 8 4 9 4 6 3 0 7 9 + ( 3 . 2 2 6 2 D - 0 7 * T Z ) GO TO 2 0 0 6 2 0 0 2 I F ( T Z . L T . 4 0 0 . ) GO TO 2 0 0 3 B O = 7 . 8 4 9 7 2 7 2 9 8 - ( 1 . 1 7 7 4 5 D - 0 7 * T Z ) GO TO 2 0 0 6 2 0 0 3 I F ( T Z . L T . 2 0 0 . ) GO TO 2 0 0 4 BO=7 . 8 4 9 7 4 8 0 2 - ( 1 . 6 9 5 5 D - 0 7 * T Z ) GO TO 2 0 0 6 2 0 0 4 I F ( T Z . L T . 1 0 0 . ) GO TO 2 0 0 5 B0=7. 849757912- ( 2 . 1 901 D - 0 7 * T Z ) , GO TO 2 0 0 6 2 0 0 5 B O = 7 . 8 5 O - ( 2 . 6 3 9 8 9 D - 0 6 * T Z ) 2006 I F ( T Z . L T . 7 7 5 . ) GO TO 2 0 0 7 c C 82 CP=0,1349+(1.5D-05*TZ) GO TO 2010 2007 IF(TZ.LT.750.) GO TO 2008 CP=1.65- (1.94D-03*TZ) GO TO 2010 2008 IF{TZ.IT.350.) GO TO 2009 CP=0.085+(1. 4667D-04*TZ) GO TO 2010 2009 CP=0. 1 1+<7. 636D-05*TZ) 2010 ROCP=RO*CP GO TO 2100 2099 ROCP=1.1775 2100 BETURN END C c C ** END OF SUB-BOUTINE. C ** DATA CARDS APPENDIX B MODEL FOR THE GRINDING BALL (SPHERE) Consider the grinding b a l l of radius R to be a s o l i d sphere, made up of several concentric s p h e r i c a l s h e l l s of wall thickness Ar on top of a cen t r a l sphere of diameter Ar. The centre of the sphere i s designated as nodel point 1 and successive nodal points 2, 3, ... are each a r a d i a l distance of Ar from the previous point. In other words, the i t h nodal point i s i n the middle of the i t h s h e l l thickness and i t s distance from the centre of the sphere (node 1) i s equal to ( i - 1) Ar. A^ refe r s to the surface area located halfway between nodal points 1 and 2, and to the volume of the s h e l l (here i t i s the c e n t r a l sphere) bounded by the nodal point 1 and the surface A^. S i m i l a r l y , the volume re f e r s to the volume of the s h e l l bounded by surfaces A. , and A., i . e . l - l l volume of the s h e l l housing i t h nodal point i n i t s wall thickness. Therefore, for the i t h node: A i-1 = 4TT(1 = 7T (21 A. I = 4TT(1 2 = IT (21 2 83 8 4 Figure B - l Model geometry showing d i v i s i o n of s p h e r i c a l specimen into nodes. Heat balance for the i t h node Heat i n - Heat out = Heat accumulated r . 9T " i - 1 V l 3 r i-1 + k. A. — i x 3 r T - T' T* - T* OR - k . _ 1 [ ^ ( 2 i - 3 ) 2 A r 2 ] ( ^ ) + k ± [ u ( 2 i - l ) 2 A r 2 ]( 1 A r 1 + 1 3 T - T OR N. k ± _ l ( 2 1 " 3 ) ' M~ V l + T. -l N. — t ^ ( 2 1 - 3 ) z + k . ( 2 i - i r i + 1 N. k. i l . ,.2 Ti+1 = T i where M = 12(i - l ) 2 + 1 and N. = 3- A^ 1 i „ . 2 for 1 < i < s p ± Cp ± Ar 85 Heat balance for the 1st node Heat i n - Heat out = Heat accumulated r) T . p i C p i v i -3T T - T OR k x (uAr 2) ( X A r 2 ) 3 T - T r t ^Ar . 1 1 . P i c P i ( —T- X TZ ) '1 ~*1 v 6 At OR (2N k± + 1)T^ - (2N 1 k ^ T ^ = (B-2) Heat balance for the surface node Surface node i s the l a s t node and i t l i e s on the surface of the sphere. Only half a nodal volume i s associated with t h i s node. Consider a f i c t i t i o u s node s + 1 outside the sphere surface at a r a d i a l distance Ar from the surface node. Assume that t h i s f i c t i t i o u s node has a temperature due to an unaltered heat flow from the surface node. But at the surface Heat l o s t by surface = Heat gained by quench f l u i d 3T OR - k A „ s s 3r = h A (AT) s s t t T - T OR - k ( S - 1 O A 8 + 1 )= h (T. - T') s 2Ar s f s OR T* = T' - ( T . _ j s+1 s - l k s f s But f o r the surface node Heat i n - Heat out = Heat accumulated r . _3T s - l s - l 3r s - l + k A ,, — s s+1 3r s+1 p Cp V ? s *s s 3t 86 > ' I T T - T T — - T OR - k . (2s - 3 ) 2 A r 2 ( - ^ 7 - ) + k (2s - l ) 2 A r 2 ( — — ) s-1 Ar s Ar P s C p s A r 2 24At [ 3 ( 4 S " 5 ) + 1 ] ( T s " V OR 8N M [ k g _ 1 ( 2 s - 3 )2 + k g (2s - l ) 2 ] T . + s-1 8N c [k .(2s - 3)' s-1 + k (2s - 1) ] + 1 s T + s 16N Ar , — ~ - (2s - l ) Z ( T s - T f) h = T (B-3) where M = 3(4s - 5) + 1 and N s s 3At 2 ' P i S i A r By replacing eqs.2-2, 2-3 and 2-7 i n Chapter 2 with eqs. B - l , B-2 and B-3 r e s p e c t i v e l y , and by adjusting the matrix v a r i a b l e s a_^ , b^, c_^ , ... etc. the mathematical model can be used for the sph e r i c a l geometry. 

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