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Mathematical modelling of phase transformation in a plain carbon eutectoid steel Iyer, Jayaraman Rajagopalan 1983

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MATHEMATICAL MODELLING OF PHASE TRANSFORMATION IN A PLAIN CARBON EUTECTOID STEEL By ^IYER JAYARAMAN RAJAGOPALAN B. Tech. (Metallurgical Engineering), Indian Institute of Technology, Bombay, India, 1974 M.B.A., Indian Institute of Management, Ahmedabad, India, 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Metallurgical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1983 © Iyer Jayaraman Rajagopalan, 1983 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 M £ TA U L O £ G ( d A L_ Date DE-6 (3/81) i i ABSTRACT With the ultimate objective of quantitatively predicting the mechanical properties of steels, a mathematical model has been developed to compute the transient temperature distri bution and austenite-pearlite transformation in an eutectoid steel rod during controlled cooling. The model is based on one-dimensional, unsteady-state heat conduction and incor porates empirical TTT data in the form of the parameters n and b(T) from the Avrami equation and the CCT start time, tAV-CCT' Tn""s data was obtained using a diametral dilato meter for an eutectoid steel of composition 0.82% C -0.82% Mn - 0.26% Si and a grain size of 5-7 ASTM. CCT kinetics are predicted from the TTT data by the additivity principle originally proposed by Scheil. The adequacy of the model was cheeked by comparing model1 predictions of the centre-line temperature of 9 and 10 mm diameter rods to measurements made during air cooling from an initial temperature between 840 and 870°C. The agreement obtained was good. Also the conditions determined by Avrami and Cahn for the additivity principle to hold were checked. Even though model predictions of CCT from TTT data generally were good, the application restrictions were not satisfied. Thus a new sufficient condition has been proposed which holds i i i for the steel under study and establishes a firm theoretical foundation for model calculations. The condition, termed "effective site saturation", indicates that for growth dominated reactions, wherein the rate of reaction is governed by the growth of nuclei nucleated very early in the reaction, the kinetics can be considered additive due to the relative unimportance of subsequent nucleation. This condition sug gests that the additivity rule may have a much broader range of applicability than was originally supposed. The calculation of TTT from CCT has been studied and a new method, involving an interative procedure using the addi tivity rule, has been derived. Agreement between calculated and measured TTT data is good. Finally the model has been employed to study the effect of centre segregation of manganese on the transformation be haviour of eutectoid steel rods and also to predict the mechanical properties of the same steel. Calculations indi cate that segregation can lead to the formation of martensite at the centre of the rods with faster cooling rates. The calculation of mechanical properties is based on published relationships between pearlite spacing, undercooling and mechanical properties. i v TABLE OF CONTENTS Page Abstract "i"1' Table of Contents iv List of Tables viii List of Figures x List of Symbols X11 Acknowledgement xni Chapter 1 INTRODUCTION . 1 2 LITERATURE SURVEY 5 2.1 Kinetics of the Austenite-Pearlite Reaction ... 5 2.2 The Additivity Rule ..... 7 2.3 Formulation of Nucleation and Growth 9 Q 2.4 Kinetics of Additive Reactions 2.5 Alternative Approaches to the Study of Non- / isothermal Reaction Kinetics ^ 2.6 Mathematical Modelling of Phase Transformations 14 2.7 Scope of Present Work 15 2.8 Objectives 17 3 THEORY OF ADDITIVE REACTIONS 19 3.1 Reaction Kinetics in the Additivity Range ..... 21) 3.1.1 Definition of Additivity Range 27 3.2 Kinetics of Nucleation and Growth Reactions and ?j the Criterion of Effective Site Saturation 27 3.2.1 Kinetics of Isothermal Homogeneous Nucleation and Growth Reactions 28 Chapter Page 3.2.2 Effective Site Saturation Criterion for Variable Nucleation Rate Isothermal Reactions 37 3.2.3 Effective Site Saturation Criterion for Heterogeneous Isothermal Reactions.. 40 3.3 Validation of the Effective Site Saturation Criterion by Experimental Results 46 3.4 Application of Additivity to Derive TTT from CCT by the Additivity Method 46 3.5 Derivation of TTT from CCT by the Additivity Method 51 4 DEVELOPMENT OF A MATHEMATICAL MODEL TO STUDY PHASE TRANSFORMATION 58 4.1 Introduction4.2 Model Formulation 58 4.3 Computer Program 62 4.4 Program Logic ..— 65 5 EXPERIMENTAL 75.1 Objectives of Experiments 72 5.2 Experimental Procedures 75.2.1 TTT Tests 72 5.2.2 CCT Tests 76 5.2.3 Centre-line Temperature Measurements in Air-cooling Tests 77 6 RESULTS AND DISCUSSION 82 6.1 TTT Test Results 8v i Chapter Page 6.2 CCT Test Results 92 6.3 Comparison of Model-predicted and Experimental Results of Centre-line Temperature Measurements 96 6.4 Model Prediction and Validation with Measured Temperature Data 99 6.5 Discussion ^2 6.6 Scope of Application of the Mathematical Model. 119 6.7 Effect of Segregation on Phase Transformation.. 120 6.8 Calculation of Mechanical Properties of Wire Rod 134 7 SUMMARY AND CONCLUSIONS 141 BIBLIOGRAPHY 145 APPENDICES 1 The Principle of Additivity . 149 2 Additivity of the Avrami Equation Kinetics ^4 3 Demonstrating the Independence of N(T) and G(T) With Respect to Time 156 4 Iteration results for CCT to TTT Calculations by the Additivity Method 159 5 Listing of Computer Program to Calculate TTT Data from CCT by the Additivity Method 162 163 6 Tri-diagonal System of Equations 7 Comparison of Model Predicted and Analyical Solution 168 APPENDICES vi i Page 8 Listing of Computer Program to Calculate Temperature Response of a Steel Rod Under going Cooling 9 Listing of Computer Program to Calculate the Temperature Response of a Centre-segregated Steel Rod Undergoing Cooling — v i i i LIST OF TABLES Chapter 3 Page 3.1 Summary of the heterogeneity co-efficient calculations 44 3.2 Summary.of volume contribution calculations 47 3.3 Effective site saturation ratio calculations for austenite-pearlite reaction in a plain carbon eutectoid steel 48 3.4 Effective site saturation ratio calculations for some eutectoid steels 49 3.5 Comparison of experimental and calculated values of t^y_-j-i~j. for a plain carbon eutectoid steel 57 Chapter 4 4.1 Model predicted recalescence calculations for CO a plain carbon eutectoid steel Chapter 5 71 5.1 Steel composition Chapter 6 Page 6.1 Errors in n and b for 0.82 C eutectoid steel (5-7 ASTM) 93 6.2. ^V-CCT f0r 0,82 C Stee1 ^5"7 AS™) 95 6.3 Model predicted time-temperature responses for 0.82 C steel rods under different cooling condi tions encountered in the experiments of centre line temperature measurement 10° 6.4 n and b for 0.82 C steel (5-7 ASTM) for t = 0 at T 114 6.5 Comparison of model predictions of time-temperature responses with 5 = 0 at and t = 0 at T^-j for 0.82 C steel (5-7 ASTM) 116 6.6 tAV-TTT for 0,8 C ~ 1-88Mn steel (5-8 ASTM) 121 6.7 n and b for 0.8 C - 1.88 Mn steel (5-8 ASTM) 123 6.8 tM-ZCl f0r 0,8 C " 1-88 Mn Steel ^5"8 AS™^ 124 6.9 to Model predicted centre-line temperatures for a \ '.>•'<. 6.14 segregated steel rod 125-130 6.15 to Calculation of mechanical properties for 0.82 C steel (5-7 ASTM) 137-139 Chapter 7 7.1 Comparison of the scope of additivity criteria 144 X LIST OF FIGURES Chapter 3 Page  Figure 3.T Relationship between the real volume and extended volume ratios 35 3.2 Relationship between the effective site saturation ratio (xl^-) and the real and extended volume ratios .. 36 3.3,3.4 Inhomogeneity co-efficient for austenite-pearlite reactions in a plain carbon eutectoid steel 41-42 3.5 Illustrating the principle of additivity in calcula-ting tAV_TTT from tAV_CCT 55 Chapter 4  Figure 4.1 Computer program flow chart 61 4.2,4.3 Typical model-predicted time-temperature charts 66-67 Chapter 5  Figure 5.1 Experimental apparatus used in TTT and CCT tests 75 5.2 Specimen assembly used for experiments in centre line temperature measurement . 78 5.3 Arrangement of air blower and specimen used in centre-line temperature measurement experiments 80 XT Chapter 6 Page  Figure 6.1 Typical time-temperature-dilation record of a TTT test 83 6.2 Typical time-temperature-dilation record of a CCT test 84 6.3,6.4 In In (,—) versus time 87-88 6-5 ^V-TTT for 0,82 carbon eutectoid steel (5-7 ASTM) ... 89 6.6 n and b values in the Avrami equation for 0.82 carbon eutectoid steel (5-7 ASTM) 90-91 6.7 ^v-CCT for °'82 earbon eutectoid steel (5-7 ASTM) ... 97 6.8 Illustrating the consistency of results in the CCT QQ and the centre-line temperature measurement tests 6.9 to Model predicted and experimental results of centre-^*19 101-111 line temperature measurement tests 6.20 Nomenclature of terms used in Tables 6.3 and 6.5 115 6.21 Typical time-temperature model predictions using t = 0 at TA] 117 6.22 Amount of martensite formed at the centre of a cool ing steel rod as a function of cooling rates 132 6.23 Photograph showing segregation at the centre of a wire rod 133 Description n\ Nucleation rate (constant) Nucleation rate (function of temperature) Growth rate (constant) Growth rate (function of temperature) Time Time for x% volume transformed Extended volume transformed Real volume transformed (extended volume corrected for impingement) Extended volume transformed at. t A Extended volume transformed at time t of nuclei A nucleating between time =0 and time = t Volume fraction transformed Density Specific heat Thermal conductivity Temperature Dummy variable representing time x i i i ACKNOWLEDGEMENT T would like to thank Professors J. K. Brimacombe and E. B. Hawbolt for their help and guidance during the course of the project. Thanks are also due to Mr. Binh Chau, Mr. Baha Kuban, Mr. S. Chattopadhyay, Mr. Ramaprasad and Mr. Neil Walker for all the help rendered. Financial assistance was received in the form of a research grant from the American Iron and Steel Institute. 1 Chapter 1 1.1 INTRODUCTION The mechanical properties of steels depend on their composition, grain size and structure. The latter is normally controlled by applying specific cooling conditions in the last stage of processing. An example is the pro duction of steel rods in which after the last rolling pass, the rods are control cooled from about 900°C by forced air on a Stelmor line. By adjusting the residence time of rods and the air velocity in individual - cooling zones, the desired structure, e.g. fraction pearlite and ferrite, can be obtained. Because the structure has a strong influence on the mechanical properties, it is im portant that the link between structure for each steel and process variables such as, in the case of the Stelmor process, rod diameter, air velocity and line speed is well established. Up to the present time, such links have been determined empirically. Practices have been developed in this way, for example, to control rod cooling and achieve specific pearlite spaeings which govern the mechanical properties.^However this approach, on a plant scale, is time consuming and expensive. There is 2 considerable incentive, therefore, for the development of a predictive capability, such as a mathematical model, which can predict mechanical properties of a given steel as a function of process variables. This is the subject of the present study. Development of a mathematical model to predict mechani cal properties, however, is a difficult task owing to the complexity of the processes which determine structure in steels. One major problem is that heat flow and phase transformation kinetics are coupled. In a controlled cooling process, the steel undergoes a continuous change,of tempera ture, the rate of which depends on the location within the steel. At the same time, as phase transformation takes place heat is evolved which frequently causes recaleseence. Thus the changing temperature field is affected by heat extraction from, and conduction within, the rod as well as heat generation which depends on the kinetics of the phase transformation. The transformation kinetics, in turn are dependent on temperature. A second problem is that the transformation kinetics have been characterized empirically in isothermal tests (TTT) and experiments in which the cooling rate is constant (CCT). But neither condition holds at a given location within the steel shape during cooling. Thus the question 3 becomes how data obtained in the 1aboratory can be applied to the complex non-isothermal situation of controlled cooling. Thus the mathematical model must incorporate heat conduction within the steel, heat extraction from the sur face of the steel and recalescence which is dependent on the coupled phase-transformation kinetics. The heat-extrac tion part of the model is relatively straightforward com pared to the transformation kinetics; the latter must be determined empirically for each steel composition and austen-ite grain size. Moreover, as mentioned above, once the transformation data have been measured, usually under iso thermal conditions, a valid procedure must be developed to apply the data in the prediction of non-isothermal trans formation. The fundamental inter-relationships between the process variables and the cooling rate are developed in Chapter 3. In the present study it was decided to model the austenite-pearlite reaction in a plain-carbon eutectoid steel. This material transforms from austenite to pearlite at the AC-j temperature under equilibrium conditions and does not exhibit any other phases, like ferrite or cementlte, and hence is simplest to model. The model, which is de scribed in Chapter 4, has been written to predict the 4 temperature response of a cylindrical rod, since the heat transfer and boundary conditions are well defined for such a shape. Such a shape also has wide industrial applicability and is simple to use in experiments under controlled con ditions. The model integrates CCT and TTT data for the steel as measured in experiments described in Chapter.5 and easily measurable process variables such as initial temperature and cooling parameters. The model has been validated by comparing predictions of centreline temperature to measure ments described in Chapter 6. The effect of centre segrega tion on transformation, has been studied by modifying the model. Calculations were done to predict the effect of centre segregation (of composition 0.80% C - 1.88% Mn in a matrix of composition 0.82% C - 0.82% Mn.) in an, air cooled rod, described in Chapter 6. Finally, calculations were done to derive mechanical properties of steel rods under different cooling conditions using the model generated data and are described in Chapter 6. 5 Chapter 2 LITERATURE SURVEY 2.1 Kinetics of the Austeni te-Pearlite Reaction. A systematic study of the austenite-pearlite reaction was made by Bain.^ Subsequently several other studies, based on measurements of temperature, dilation and hardness 12-18 as well as metallographic techniques, were conducted. The most important works, which advanced the under-1 9 standing of reaction kinetics, are those of Johnson-Mehl , 20-22 17 Avrami and Scheil. Johnson-Mehl gave a comprehensive mathematical treatment of the austenite-pear!ite reaction kinetics. They derived an equation for kinetics of nuclea tion and growth reactions under the following assumptions: i) Constant nucleation and growth rates ii) Random nucleation iii) The reaction product forms true spheres except when during growth, impingement on other growing spheres occur. With the above assumptions the volume fraction trans formed, X, is related to the nucleation rate, N, and the growth rate, G, by the following relationship 1 - exp (- | NG3t4) (2.1) However there is some question: that the assumptions are valid. Even for an isothermal reaction it is doubtful that the nucleation rate remains constant. Undercooling is the driving force for the nucleation process, and for an isothermal reaction, it may seem possible that N may re main constant. But this is too simplistic a view which neglects the effect of composition, structure and the trans formation product on the phenomenon of nucleation. Brown 36 and Ridley have shown that it is possible to have a de creasing nucleation rate after about 20% transformation. Other evidence also exists to suggest that nucleation may 19 decrease as the reaction proceeds. However, the growth rate of pearlite is constant at a given temperature. The assumption of random nucleation is also.questionable, especially in commercial steels which are prone to some degree of segregation of elements like Mn. and. P. Also non-uniformity in grain size may have an effect. The assumption of completely spherical growth is, likewise, questionable 37 in the light of micrographic studies conducted by Kuban. A further difficulty in using Eq. (2.1) is the required determination of N and G by conducting controlled experi ments. Moreover, reactions of industrial importance are 7 usually non-isothermal. Since Eq. (2.1) cannot be used for non-isothermal reactions, its use is very restricted. Avrami's formulation is more useful in this regard. Like Johnson-Mehl, Avrami derived an equation for austenite-pearlite reactions as: X = 1 - exp (- btn) (2.2) where n is a constant and b is a temperature dependent para meter. Clearly Eq. (2.2) is a more general form of Eq. (2.1). Though b and n are empirical constants, Avrami used sound theoretical principles to derive Eq. (2.2). His treatment of nucleation rate is superior to that of Johnson and Mehl and:he;; j derived a simpler formula for including the effect of impingement during growth, in the volume calculation. Avrami also showed that his equation included those of pre vious authors, like Austin and Rickett,18 Zener,14 Johnson 19 and Mehl as special cases. 2.2 The Additivity Rule Scheil17 first enunciated the additivity rule, which links the isothermal kinetics to non-isothermal reactions. This rule simplified the problem of studying non-isothermal 2 5 reaction kinetics. Christian gave an up-to-date version of this rule, which states that t where: J t dt =. 1 (2.3) 0 a TfJ • (See Appendix 1 for derivation.) t = time for a non-isothermal reaction to reach a specific amount of transformation. t (T) = time to reach the same transformation iso-a thermally at temperature T. This rule holds true for reactions for which the in stantaneous reaction rate is only a function of the tempera ture and the amount transformed, irrespective of the previous thermal history. Avrami's derivation is very important in this regard. He showed that Eq. (2.2) describes the kinetics N of additive reactions provided the ratio ^ remains constant over the temperature range of the reaction. He defined this temperature range as the "Isokinetic Range". However, be cause the change in N with temperature is much more rapid than that of G for many austenite-pearlite transformations in steel, the existence of such a range is doubtful. ' The advantage of Eq. (2.2) over Eq. (2.1) is that Avrami directly addressed the problem of additivity and provided 9 at least one sufficient condition for additivity to hold. Despite the questionabi1ity of the isokinetic range, the Avrami Equation, (2.2) with empirically determined values of b and n, predicts the nature of the austenite-pearlite reaction kinetics quite accurately. The difficulty lies in determining the appropriate values of b and n. 2.3 Formulation of Nucleation and Growth In order to derive ways of finding b and n, and to describe the theoretical importance and basis of these, several attempts have been made to formulate the nuclea tion and growth phenomena in fundamental terms. Equations have been derived for plate-like growth, needle-like growth, grain-broundary nucleated growth etc. by several workers; a comprehensive treatment of all of these can be found in reference (35). The resulting equations, essentially, are extensions of the Johnson-Mehl type of calculations and are subject to similar assumptions. These methods ulti mately result in the formulation of an equation like Eq. (2.2) with different values for the constant n. Since the vali dity of the assumptions made are questionable, a closer examination of the reaction kinetics is in order. 2.4 Kinetics of Additive Reactions 2 3 ?& In 1956, Cahn ' proposed that reaction kinetics which 10 can be described by: (2.4) where: X = volume fraction transformed t = time h(T) = a function of temperature g(X) = a function of volume fraction transformed, can be expected to be additive. It can be shown that the Avrami equation can be modified to be of the same form as Eq. (2.4) (provided 'n' is a constant independent of T and X) such that 1 h(T) = n(-b)n (2.5) and n-1 n g(x) l4x {log \l-X)} (2'6) (See Appendix 2 for derivation.) Hence the Avrami equation describes the kinetics of additive reactions. Several authors have shown that, despite the questionable assumptions, the additivity rule holds 26 29 39 44 true for austenite-pearl ite and bainrte. reactions. ' ' ' An important feature of all these works is the assumption that, irrespective of the reaction conditions (i.e. isothermal 11 or non-isothermal), the transformation of austenite to pearlite begins at the equilibrium transformation tempera-35 ture. Hawbolt et al . , in a very recent work have derived a different method for determining the. "start" of the trans formation under non-equilibrium reaction conditions. This procedure is discussed in detail in Chapter 6, and may con trast with the published TTT or CCT diagrams which show the "start" line as 0.1% or 1% transformed. In the new pro cedure, for assessing the kinetic data, the incubation time is neglected and the Avrami equation is applied only to describe the nucleation and growth phenomenon. The start of the transformation occurs after an incubation time t^y (for v RAM I ^' In the present work, this time has been used as the "start" time. This is a major departure from the conventional methods of studying the kinetics. The re sults from the present work confirm that the use of t^y for additivity calculations gives better agreement with experimental observations. 2.5 Alternative Approaches to the Study of Non-isothermal Reaction Kinetics 2 8 In 1941, Grange and Keifer described a simple and elegant method of deriving CCT from TTT data. This method was empirical in nature and involved assumptions regarding the kinetics. Though these were simplistic assumptions, 12 and hence the results approximate, the method is easy to use. But it did not employ additivity. However, since this method could not be justified on sound theoretical grounds, it has not found much application. •I c Another method, employed by Manning and Lorig, used the experimental determination of the "start" of transforma tion during continuous cooling by conducting control 1ed-cooling experiments. These are time consuming procedures and the results generated do not lend themselves useful for general application. A new approach to the problem of non-isothermal kinetics 23 24 was given by Cahn ' in 1956. He showed that since the isokinetic range is only a sufficient condition for addi tivity, the rule of additivity could also be applied to reactions under a condition called "site saturation". He deduced this from micrographs from experiments on eutectoid alloy steels having a large austenite grain size. In such steels, the reaction is initiated at grain boundaries where the nucleation event is so rapid that in the very early stages of transformation (10-20%) the grain boundaries are saturated with the growing new phase. Nucleation is there fore complete in the early stages of the reaction and plays no further role. The ensuing transformation is then con trolled by the growth rate. Since the growth rate is a 1 3 temperature dependent parameter, additivity must be expected to hold. This was a definite new direction in the work on kinetics of reactions. By eliminating nucleation as a variable, Cahn simplified the process of characterising the kinetics by the growth rate alone, thereby eliminating the assumptions regarding the nucleation rate. Though the condition of "site saturation" increased the number of transformations for which the additivity principle eould be applied, this is not a universal pheno-37 menon. In the work by Kuban, where a plain carbon eutec toid steel was studied, micrographs- unambiguously reveal the absence of grain boundary saturation. It is possible, however, that site saturation: is more probable in the case of alloy steels due to the presence of alloying elements which may encourage grain boundary nucleation. Also the effect of grain size on site saturation needs to be examined. Intuitively, it would appear grain boundary site saturation is more probable in larger grain size material due to the reduced amount of grain boundary area per unit volume. Cahn's method differs from other empirical methods in that it has a firm theoretical basis. This is evident 27 39 when comparing Cahn's work with Sakamoto and Tzitzelkov. These and other workers26'29'30 have employed curve-fitting methods, aided by computer-based calculations, to derive CCT from TTT. Since these are not based on theoretical con siderations they cannot be considered as contributions to the understanding of the reaction kinetics. They find their use in specific situations. 2.6 Mathematical Modelling of Phase Transformations Calculation of CCT from TTT by using the additivity principle is complex and laborious.^5'^'28 Such calcula tions are normally done to plot the CCT diagram for a; material of a given chemistry and grain size. They can also be used to calculate reaction kinetics in a material undergoing processing in industrial situations, e.g., an infinitely long rod of circular cross-section in a wire rod mill. To study reactions taking place in such shapes, under different processing conditions, e.g., cooling rates, the kinetics must be related to the thermal history, which is in turn governed by the material properties and process conditions. Developments in the fields of heat transfer, solution of differential equations by finite-difference methods and rapid computer-aided calculations have resulted in the formulation of mathematical models to study such phoneomena as phase transformations, stress fields, 31 32 33 temperature fields, etc. ' ' A mathematical model to study reaction kinetics in a plain carbon eutectoid steel rod was first attempted by 15 31 Agarwal and Brimacrombe. They solved the second-order differential equation for heat transfer in an infinitely long circular cross-section rod by using an implicit finite-difference procedure. The kinetics of transformation were incorporated into the model by using published TTT data and additivity. They compared their model-predicted re sults with the experimental work of Takeo et al.^ and found that agreement was relatively poor. It was thought that this might have been due to the TTT data used in the model and also the assumption that the transformation, under non-equilibrium conditions, began at the equili brium transformation temperature. Though the agreement with experimental results was not good, the work demon strated the feasibility of the use of models to study phase transformations. Their study indicated the need for the use of appropriate transformation data for successful model appli cati on. 2.7 Scope of Present Work The present work was undertaken primarily to develop a mathematical model of heat flow and transformation in eutectoid wire rods using carefully measured TTT and CCT data. Also, it was decided to conduct experiments to vali date model calculations. These were to be accomplished by measuring centre-line temperature of air-cooled steel rods 16 under controlled conditions of chemistry, grain size and cooling rate. Since model calculations would involve the use of additivity, it was decided to examine the fundamental quantities involved, like the nucleation and growth rates. 37 This work was carried out independently, by Kuban, on a material very similar to the one used in the present work. The results from the study by Kuban were examined to check whether the conditions needed for additivity, like iso kinetic range, site saturation etc., did exist in the material under study. As mentioned earlier, from Kuban's work, the following observations were made: 1) N and G, for isothermal reactions, are constant upto about 20% transformation. ii) N and G vary with temperature. An isokinetic range, as defined by Avrami, does not exist. iii) There is no evidence of site saturation, as revealed by metal 1ographs. iv) Calculations, using N and G values as found experimentally, reveal that the isothermal reaction kinetics are much slower than would be predicted by the Johnson-Mehl equation. v) Growth of grains is not truly spherical, especially for Targe grain size, nor is nucleation random. 1 7 Since the steels used in.the present work and that of Kuban are virtually the same (plain carbon eutectoid), the same kinetic conditions apply in both cases. Since it was found that the conditions required for the additivity rule to hold, the isokinetic range and site saturation, did not exist, but that the model calculations using addi tivity agreed well with experimental results, it was de cided to re-examine the conditions needed for additivity. In addition, alternative conditions satisfying the additivity rule were also investigated. As a result, a new condition for additivity, a sufficient condition, has been proposed to explain the successful application of additivity in the present context. Finally since it is experimentally more difficult to obtain TTT data than CCT, due to limitations of the experimental apparatus, it was decided to investigate the possibility of devising a simple mathematical procedure to derive TTT from CCT. 2.8 Objecti ves The objectives of this study can be summarized as follows: i) To develop a procedure for quantitative pre diction of mechanical properties of plain carbon steel rods control cooled in a Stelmor-type process. 18 ii) To develop a computer-based mathematical model to calculate the heat flows and austenite-pearlite reaction kinetics in a plain carbon eutecto id steel rod. iii) To test the adequacy of the mathematical model by comparing model predictions with measure ments of the centre-line temperature of rods under different cooling conditions. iv) To examine the theory of additivity and explain its applicability to the kinetics of the experi mentally determined austenite-pearlite reaction. v) To devise a procedure for calculating TTT from CCT data. vi) To use the model to examine the effect of segre gation on reaction kinetics and transformation behaviour in plain carbon euteetoid steel rods. vii) To predict mechanical properties by using the information generated by the.model. 19 Chapter 3 THEORY OF ADDITIVE REACTIONS As described in Chapter 2, the conditions necessary for additivity to hold for the austenite-pearlite reaction, i.e. Avrami's isokinetic range and Cahn's early site satura tion, do not obtain for the reactions observed in the pre sent work. However model calculations using additivity (described in Chapters 5 and 6), show good agreement with experimental results. Therefore it is necessary to examine alternate conditions under which the additivity principle will become applicable. In this chapter, two new sufficient conditions for additivity are examined. These are: i) the additivity range ii) effective site saturation. These two conditions have been derived after a careful theoretical analysis of the fundamental aspects of reaction kinetics - the nucleation rate and the growth rate. The additivity range, as will be shown in Section 3.1, is an extension of the isokinetic range. The effective site saturation criterion, described in Section 3.2, is similar to Cahn's early site saturation principle. The derivation 20 of the effective site saturation criterion has been done in three stages: i) It is first derived for a homogeneous reaction with constant N and 6 (section 3.2.1). ii) It is extended to cover homogeneous reactions with a constant G and a variable nucleation rate, which includes (i) as a special case (section 3.2.2). iii) It is then derived for a heterogeneous reaction with constant N and G. It is also indicated that the results for a heterogeneous reaction with a variable nucleation rate and a constant G would yield the same results as obtained in (ii) (section 3.2.3). The applicability of the criterion to the present work is described and by using experimental results, it has been shown that the reactions encountered in the present work are additive (section 3.3). Finally, a new method has been devised, using additiv ity, to derive TTT data from CCT (section 3.4). A short summary of the work done in this regard is also included in this section to describe the context and relevance of this procedure. It must be pointed out that the derivations leading to the additivity range, effective site saturation and the method of obtaining t^y_JJJ from are original and must be considered as fresh contributions to knowledge in this field. 3.1 Reaction Kinetics in the Additivity Range 21 As per Avrami, the extended volume fraction trans formed, V , for a nucleation and growth reaction is: where Vex = 0/ N4 (T"z)3 e_Z dz (3-1) 0 4TT shape factor (= -y for spherical particle growth) H = number of germ nuclei at the start of transformation g = G/N x = characteristic time Since N is independent of time, T Vex = a N V ^ (T"z)3 e"Z dz (3-2) 0 If N and G are functions of temperature alone, then Vex = 0 N 4 ®3f (x_z)3 e"Z dz (3'3) 0 • 22 On integrating Eq. (3.3) and correcting for impingement, V = i - exp (-Vex)21 (3.4) where V = real volume fraction transformed, we obtain, V = 1 - exp (-btn)21 (3.5) where ri = constant b = function of temperature. Eq. (3.5) is the Avrami equation which has been shown in (Appendix 1) to describe the kinetics of additive reactions. The temperature range over which Eq. (3.5) holds is the "additivity range". The Johnson-Mehl equation (Eq. 2.1) is a specific case of the more general Avrami equation of reaction kinetics. For an isothermal reaction, for which there is evidence that N and G are constant for some nucleation and growth 36 reactions,, the Johnson-Mehl equation can be applied to study the kinetics. But by using the concept of the addi-' tivity range, it can be shown that the Johnson-Mehl equa tion, with slight modifications, can be used to describe the kinetics of non-isothermal reactions which are additive. 23 Consider a non-isothermal reaction (a nucleation and growth reaction) occurring between the temperatures TQ and T^QQ. In this temperature range, let N(T) = nucleation rate (a function of temperature alone) G(T) = growth rate (a function of temperature alone) = real volume fraction transformed at time 't'. T • To v - 0 T - T100 Vex " 0 t Tt+dt v - 1 I 1 t = t During the infinitesimally small time step 'dt', the number of nuclei that nucleate = N'(T)dt. N'(T) = temperature averaged nucleation rate over the time interval dt Consider the growth of one nucleus which starts its growth during dt. The extended volume of this nucleus at time 't-j is: Tl 00 3 24 Vex = { 1 f G(T) dT (t10Q - t)j (3.6) 1100 " lt.J J  't T t where: ^100 Tioo - T G(T)dT T. is the temperature averaged growth rate between times t and tl 00 * Hence the volume of all nuclei nucleating between t and t+dt at t-| QQ . i s : t+dt Tl 00 3 Tt+dt "/fef7^ / «T)dT(.100-.}{JF/.Ol,dTj(.t(3.7) t "Tt Tt Therefore, the total extended volume of growth of all nuclei nucleating between tQ and t^g at time t^Q0 is: ^00 T100 3 T100 ^°  = afl^ rG(T)dT(tioo-4 fr—-f^(T)di dt; (3.8) J u 'ioo"'t /  J 11ioo"'o^ J to To To Let T1Q0 G^T) = ]- J G(T)dT (3.9) Tl00 " T0 T0 1,00 25 N,(T) = 5 f N(T)dT (3.10) T100" T0 * T0 N-j(T) and G-j(T) are the temperature averaged growth rates for the reaction. Then, Eq. (3.8) can be written as t100 Vex°° = a J G1(T) ' VT) * (t100_t)3 dt {3J1) G-^CT) ..and, N^'(T.) are:functions independent of time. This is illustrated in Appendix 3. Because of this independence, G^(T) and N-j(T) can be taken out of the integral in Eq.(3.11) tl 00 •'• VIx°° = ajGl(T)Nl(T) ^ (t100"t)3 dt (3'12) Considering this transformation to have occurred in a unit volume, at any time "T" during the reaction, VeTx = a-G3(T) -NT(T)- f l*~*)3 dt <3-13) where T T G (T) = - f G(T) dT (3.14) T - Tn J T 0 To 26 T T N(T) = —! f N(T) dT (3.15) T - Tn J T 0 • • Vj[ = a G3(T) N3(T) ^- (3.16) For spherical particle growth, Vex = ~ NT(T) G?(T) t4 (3'17> 3 Since V* = 1 - exp|-V*xJ. (3.18) .". V* = 1 -exp ^ Nt(T) • Gjj(T) t4j. (3.19) For an isothermal reaction, Eq. (3.19) can be written as V* = 1 - exp (- - NG3 t4) (3.20) 3 which is the Johnson-Mehl equation. Eq. (3.20) is of the same form as the Avrami equation with b = ^ Nt(T) G3(T) (3.21) n = 4 (3.22Hence Eq. (3.19) also describes the kinetics of additive c > 27 reactions. It holds under the following assumptions: i) N and G are functions of temperature alone for non-isothermal reactions. ii) Random nucleation. iii) Spherical particle growth until impingement. The temperature range for which Eq. (3.19) holds is the "additivity range". 3.1.1 Definition of Additivity Range The additivity range is the temperature range, for a nucleation and growth reaction, in which the nuclea tion and growth rates are dependent only on the temperature. Reactions occurring in such a range obey Scheil's additivity principle. 3.2 Kinetics of Nucleation and Growth Reactions and the Criterion of Effective Site Saturation Isothermal reactions which do not follow the Johnson-Mehl equation due to the violation of any one or all of the assumptions made in deriving the equation can be termed non-homogeneous reactions. Such reactions are not covered by Eq. (3.19). Hence it is necessary to derive a sufficient condition of additivity for such reactions. Since the 28 austenite-pearlite reaction, studied in the present work, is heterogeneous, it becomes all the more essential.1 In order to study heterogeneous reaction kinetics, it is important to understand the kinetics of homogeneous reactions 3.2.1 Kinetics of Isothermal Homogeneous Nucleation  and Growth Reactions d't t = 0 t=t t=t] t=t2 Assumptions: i) Constant N and G ii) Spherical particle growth iii) Random nucleation. Number of nuclei formed upto time t = Nt Number of nuclei nucleating in an infinitesimally small time interval dt The extended growth volume of these ^ N3 nuclei at t = t Ndt Ndt vi •' 1 "t o/|(trt))' The .extended' growth :volume: 'of all nuclei', nucleating during the time interval 0 to t-j is • 3., / , j. \ 3 I OJG°N (t,-tr dt 3 4 ii ^ ^ i ' 29 The total number of sites consumed due to nucleation and the volumetric growth by the end of time t = t-j is: 3 4 t, aG N t? = .lit, +.__L,,I0 (3.24) where IQ = Initial number of available sites for nucleati on. The rate of site consumption is (obtained by differentiating Eq. (3.24)) S*1 = N + a NG3 t3 In (3.25) ex .10 Considering a unit volume of material, from Eq. (3.23), vt = aNG3.;t4 (3^26) ex Al so, S* = Nt + In-V* (3.27) ex 0 ex • t S = N + In • V* (3.28) ex 0 ex For spherical growth (a = , vt = I NG3 t4 (3.29) ex 3 Si nee Vt = 1 - exp (- V^) (3.30) S* = 1 - exp (- S*x) (3.31) 30 We have, 1 - exp (- | NG3 t4) (.3.32) S*o , l - exp -(Nt + In • Vt ) r 0 ex (3.33) N+!n— NG3t3 • exp -(Nt+I • V* ) (3.34) Eq. (3.32) to Eq. (3.34) are the three basic equations of homogeneous isothermal reaction kinetics. Eq. (3.32) is the Johnson-Mehl equation, which is the volume fraction transformed at any time t during the reaction. This volume growth is due to the growth of nuclei nucleated throughout the reaction. It is also possible to calculate the volume tric growth of nuclei nucleating during a specific time in terval in the reaction. It is useful to calculate this quantity because it will indicate the relative importance of the growth of these nuclei to the total growth volume. If the volume contribution due to nuclei nucleating very early in the reaction to the total volumetric growth at a very late stage in the reaction is significantly high, then it can be postulated that the reaction is essentially growth dominated. As before, the extended volume of growth from nuclei nucleating between times 0 and t-, at time t0 is: t 1 31 V!x != I 0 G3(t?-t)3N dt (3.35) exo/t1 J d 0 = °—: A - (t2-tl)4 (3.36) Al so, the total extended volume of growth at t^ is: \\ ^a^lA. (3.37) A more general form of Eq. (3.35) is t. J _ / _p3,,/i\3 - J a la. V v = J  G, N(x-1) dt (3.38) ext./t. i J t. l = 0NS! (t_t ,* . (T.t ,4 (3.39) Hence, at any time t during the reaction, the fractional volume contributed by nuclei nucleating between the times t^ and t. to the total extended volume transformed is: J eXt./t. (t-t.)4- (t-t )4 t1 J = ] r ^— (3.40) t ex If 32 then, from Eq. (3.41) t '20. _ 90 v 90 20 '90 ex0/t t4 - ft - f )4 ^90 t4 ex r90 (3.41) Now if, for this reaction, the growth rate is the domin ant feature, (and consequently, the nucleation rate is re latively unimportant) then it can be expected that the frac-ional volume contributed by the nuclei nucleating in the very early stages of the reaction (say, up to 20% transformation) to the volume transformed at the final stages of the reaction (say, 90% transformation) must be close to unity. If the ratio is close to unity, say 0.85, then the nucleation event after 20% transformation becomes unimportant and the reaction is growth rate dominated. Since it is known that the growth rate is a function of temperature only for austenite-pear1ite reactions, these reactions can be considered additive. Hence, if y ^90 e"°/t20 = '90 - (^9Q-t20) :>_Q>85 (3.42) V 90 *9Q ex t2Q = > 0.38 tgQ (3.43) 33 holds for the reaction, it should be additive. It should be noted that Eq. (3.42) involves the ratio of extended volumes It also can be shown, as described below, that *90 /go 0/t20 ex0/t Since and Al so, > — ^- (3.44) vt90 V*90 ex V*90 = 1 - exp (-V^90) (3.45) V0/t = 1 - exp -V^90 (3.46) U/t20 °/t20 U/t20 0/t9n -t— 7. t 20 (3.47) 90 , ,,90 V 1 - exp -Vex V*90 = 0.90 (3.48) 1 - exp (-Vg90) = 0.90 (3.49) V^9Q = In 10 (3.50 ex 34 From Eq. (.3. 43) , VIx° > Q.85 \ltg0 (3.51) exo/t2Q ex . . Eq. (3.48), combined with Eq. (3.44), can be written as: 0/t20 > 1 - exp (-0.85 In 10) V*90 " 0.9 1 °-95 (3.52) The relationship between the ratios of extended volumes and real volumes is shown in Fig. (3.1) and Fig. (3.2). There fore, from Eq. (3.52), it can be concluded that the ratio of the real volumes is greater than the ratio of the extended volumes. Thus, Eq. (3.43), which is the "Effective Site Saturation" condition for constant nucleation and growth rate homogeneous reactions, can be used to study the applicability of additivity to such reactions. The criterion is so called because the nature of conclusions derived by it are essenti ally similar to Cahn's site saturation. But it does not require that nucleation sites saturate physically during the reaction. Also, the criterion requires only a knowledge of t^Q and tgQ for such reactions. Hence it is very easy to use. The only restriction in using the effective site Fig. 3.1 Relationship between the ratios of extended volumes and real volumes. Figure 3.2 Relationship between the saturation ratio and the extended volumes ratios. effective site real volumes and 37 saturation condition, as derived in Eq. (3.43), is that N and G be constant. However, it can be extended to include re actions in which N varies with time. 3.2.2 Effective Site Saturation Criterion for Variable  Nucleation Rate Isothermal Reactions If we consider a reaction such that, N = N(t) (t > t..) (3.53) As before, upto t t V (3.54) Al so, (3.55) t. (3.56) 38 Consider a decreasing nucleation rate, m N(t) N 0 (3.57) which is the most probable form of equation based on evidence in the literature. Moreover, the application of Le Chatelier's principle points to the possibility that the nucleation rate should decrease with time due to the "back pressure" exerted by the transformed procuct. Eq. (3.57) also implies that the nucleation rate is negligibly small after tgg, which again is a reasonable assumption, 'm' is a co-efficient which will depend on factors like composition, temperature etc. Substituting Eq. (3.57) for N(t) in Eq. (3.50), we obtain N (3.58) m+4 (3.59.) Hence, V ex 0/t N (3. 60) ex 39 t4 - ft - t )4 *90 ^90 V (3.6:1) 4 (t . t )m+4 ^ :,0nt4):. 90 ( 90 V The effective site saturation criterion is '90 ex 0/t. 0.85 '90 ex (3.62) (3.63) Eq. (3.63) is the general statement of the effective site saturation criterion which includes Eq. (3.43). Eq. (3.63) considerably expands the scope of reactions covered by the effective site saturation condition. However, in practice, homogeneous reactions are not common. This is especially true of the austenite-pearlite reaction, which is heterogeneous. Micrographic evidence also 37 exists to show that grain growth may not be spherical. In 37 the work by Kuban, the reaction kinetics calculated by using experimentally determined N and 6 in the Johnson-Mehl 40 equation for isothermal reactions, predicted much faster rates of transformation!, than those experimentally observed. Thus, for the effective site saturation criterion to be use ful for heterogeneous reactions, it must be extended to cover the conditions obtained in such reactions. 3.2.3 Effective Site Saturation Criterion for  Heterogeneous Isothermal Reactions The Johnson-Mehl equation represents the kinetics of homogeneous reactions. VHom = 1 " exp (" 3 Ng3 t4) (3-64) Due to heterogeneity, the reaction kinetics are slower than that predicted by Eq. (3.64). These kinetics can be represented as VHet = 1 " exp (" btl1) (3*65) where b and n represent the effect of heterogeneity. We:can define a factor, called the "Inhomogeneity Co-efficient", as V* I. = (3.66) vHet It is not possible, at this stage, to speculate upon the precise nature of 1^. Values of 1^ calculated from 37 the work of Kuban are shown in Figs. 3.3 and 3.4. Volume Fraction Transformed (%) Figure 3.3 Effect of grain size and reaction tempera ture on the Inhomogeneity co-efficient. 42 Volume Fraction Transformed (%) Fig. 3.4 Effect of grain size and reaction tempera ture on the Inhomogeneity co-efficient. 43 The inhomogeneity eo-efficient, l^, helps to character ize the degree of inhomogeneity of a reaction, but compli cates the derivation of a criterion of the form of Eq. (3.43), Introducing .1 in a slightly modified form, as h"t, the "heter ogeneity co-efficient", helps accomplish this objective. vt 1 -.exp - V* t = vHom = -Horn VHet = 1 - exp - H • V* (3.67) z exHom Since •VVL = H. • - NG3 t4 = btn (3.68) z exHom z 3 .'• H = —— t""4 (3.69) |NG3 As for 1^, it is difficult to predict the nature of for different reactions. Values of for the reactions studied by Kuban are. summarized in Table 3.1. As can be seen from this table, the value of is constant for some reac tions and varying for others. For a heterogeneous reaction with constant N and G, we have, 44 Table 3.1 Summary of Heterogeneity Co-efficient Calculations Reaction Austenitising Range of Average Temperature Temperature Ht Ht (°C) (°'C) 640 800 0:04 to 0.26 0.10 640 840 0.04 to 0.48 0.15 640 950 0.07 to 0.09 0.08 640 1100 0.03 to 0.035 0.033 690 800 0.13 to 0.23 0.17 690 840 0.0018 to 0.0034 0.0024 690 900 0.05 to 0.027 0.17 690 950 1.4 to 12.3 4.42 45 V«L = Ht„„ TNs3t90 (3-70) 'Het "90 3 Vex° =Ht 1 N^^go-^O^4 (3'71) Het. /t z90 3 au ^u ZZ0f 90 V «!L " Hton 7 ^.tSo-^W4] (3-72) 'Hetn/. "90 3 u/t20 Since the effective site saturation condition is: /go exo/t - > 0.85 /go ex for a heterogeneous reaction, t4 - (t -t )4 -™ [ 90 2°J >.. 0.85 (3.73) t90 t20 > 0.38 tgo (3.74) Eq. (3.74) is the same as Eq. (3.43). The same deriva tions can be done for a variable nucleation rate reaction with the same results as obtained in Eq. (3.63). Thus, the effective site saturation criterion remains unaltered for homogeneous, and heterogeneous reactions. 46 3.3 Validation of the Effective Site Saturation  Criterion by Experimental Results The effective site saturation criterion was applied to experimental results to check its validity. The model pre dictions, using additivity, show very good agreement with experimental results in the present work, indicating that additivity holds under the experimental conditions encounter ed. The material used and experimental conditions are 3 7 virtually the same in the present work and that of Kuban and hence comparable in terms of kinetic behaviour. The re sults of volume contribution calculations for experiments 37 conducted by Kuban are summarized in Table 3.2 and show very good agreement with the result expected from Eq. (3.74). Table 3.3 shows the calculation of the ratio for the 37 90 experiment conducted by Kuban. Table 3.4 shows the same ratio calculated for experiments conducted in the present 30 43 50 study and others. ' ' The values indicate very good agreement with the effective site saturation criterion. The above results validate the use of the effective site satura tion criterion to ensure additivity in reactions. They also establish a firm theoretical framework for using additivity in the model calculations. 3.4 Application of Additivity to Derive TTT from CCT  by the Additivity Method An important application of the additivity rule is the 47 Table 3.2: Summary of Effective Site Saturation Calculations Reaction Temperature (°C) Austenitising Temperature (°C) Volume Contribution* (%) Extended Volume Real Volume 640 800 86 95 640 840 82 94 640 950 - 96 98 640 1100 97 99 690 800 94 98 690 840 93 97 690 899 88 96 690 950 85 95 *Volume contributed by nuclei nucleating between 0 and 20% transformation to the total volume transformed at 90% transformation. Table 3.3.. Effective Site Saturation Criterion Values 48 Steel Chemistry: 0.78 C Eutectoid (plain carbon) Reaction Austenitising Temperature Temperature t20 ^90 t20 (°C) (°C) (s) (s) ^0 640 800 (9.1 ASTM) 3.22 8.38 0.38 640 840 (7.8 ASTM) 3.08 8.93 0.34 640 950 (7.3 ASTM) 6.8 12.7 0.53 640 1100 ( 3 ASTM) 31.76 55.14 0.58 690 800 (9.1 ASTM) 51.1 101.4 0.51 690 840 (7.8 ASTM) 119 243 0.49 690 900 (7.5 ASTM) 918 2275 0.40 690 950 (7.3 ASTM) 847 2301 0.37 (Data from Reference 37.) 49 Table 3.jf Effective Site Saturation Criterion Values Experimental Conditions Reaction Temperature (6C) So (s) So (s) So So Reference No. 0.82 C Eutectoid Steel Austenitised at850°C for 5 mts 5-7 ASTM 660 650 630 615 603 32.4 13.5 5.6 3.25 3.15 72 25.9 9.8 5.4 5.4 0.45 0.52 0.57 0.62 0.58 * 0.78 C Eutectoid Steel Austenitised at 875°C for 30 mts 5.25 ASTM 500 540 600 4.20 4.8 6.4 5.5 6.5 10.0 0.76 0.74 0.64 43 0.80 C Eutectoid Steel Austenitised at 875°C for 30 mts 4.25 ASTM 630 650 690 8.0 23.0 700.0 20.0 42.0 1100.0 0.40 0.54 0.63 43 1.10 C Eutectoid 5 ASTM 0.57 C Eutectoid 5 ASTM 0.93 C Eutectoid 1 ASTM All Steels Austenitised at 875°C for 30 mts 662 691 689 4.7 80.0 35.0 6.5 200.0 46.0 0.72 0.40 0.75 43 SKD-6 715 670 95 340 200 830 0.475 0.410 30 SKS-5 Austenitised at 1100°C for 15 mts 632 622 612 601 31 24 19 16.5 55 42 34 30 0.56 0.57 0.56 0.55 50 * Present work. 50 determination of CCT from TTT. In the literature, several 15 16 attempts have been made in this direction. ' The deriva tion of CCT from TTT is possible only after the experimental determination of the TTT data. But the experimental deter mination of TTT is very difficult. To determine the TTT, the specimen must be cooled from.the austenitising tempera ture (usually around 850-900 °C).to the isothermal test temperature (usually around 650 °C) in a very short period of time, usually a second or two. This must be done in order to ensure that the transformation does not begin be fore the test temperature is reached. This calls for cool ing rates of the order of 100 to 150 °C/s or more which are very difficult to achieve. In a salt pot, which has been the medium used by several workers in the literature to achieve such cooling rates, it is impossible to do so. Hence this introduces an. error in the experimental measurements. Also it is impossible to ensure equal cooling rates at all locations even in a thin disc-shaped specimen in a salt pot quench. On the other hand, CCT data is much simpler to determine experimentally and more accurate. Thus a method for determining TTT from CCT is very much needed to check experimental; TTT results. The present study enumerates a simple iterative procedure called the "additivity method", for deriving the 111 from the CCT. This procedure is easy to use and much less time consuming when compared to the 15 16 lengthy calculations suggested by others. ' 51 3.5 Derivation of TTT from CCT by the Additivity Method The TTT curve can be expressed by a mathematical equation tAV-TTT s a b* (3'75) where ^AV-TTT = start time at the temperature T a, b, c = constants X = TAT- T T^.| = equilibrium transformation termpera-ture of the material The CCT start can also be expressed in a similar form as x TAV-CCT = al bl e (3-76) The constants a^, b^, c.j in Eq. (3.76) can be calculated by using a multiple-regression procedure by using experiment ally determined t^y rrT values. By using additivity and the iterative procedure to be described, the constants a,b,c in Eq. (3.75) can be determined. To start the iterative procedure, an estimate of a,b and c must be made. This can be done by using published TTT diagrams .as. a guide or can be determined on any arbitrary basis. A proper estimation of these constants results in a reduction of the number of iterations needed to determine 52 the correct TTT start curve. Hence the estimation of these constants is not crucial to the success of the method. The TTT start curve determined by the estimated constants a, b and c is the first approximation to the correct t^y_-r-p-r. Consider a cooling rate A °C/s. The principle of additivity states that tAV-CCT J dt = 1 {377) vAV-TTT 0 for a constant cooling rate, Eq. (3.77) can be written as: t, 'AV-CCT dT 1 f ^ = 1 (3.78) A tAV-TTT tAV-CCT f dJ_ * A (3^7-9-)-•» t n w -r-r-r 'AV-TTT The Eq. (3.79) can be approximated by tAV-CCT ' ' AT tAV-TTT 0 A (3.80) 53 As AT->0, Eq. (3.80) becomes the same as Eq. (3.79). For the cooling rate A, the can be found from the CCT curve. Using t^v TTT Calculated at each temperature, A from T^-j to the T^y_rrj in steps of AT, by using the Eq. (3.75), the LHS of Eq. (3.80) can be calculated. If this is greater than the RHS, the first approximation of the TTT upto T/\y_crT is z0 tfie left °^ tne correct t^v_TTT. If the LHS value is A^, then by multiplying all the t^v_TTT A values upto T^^.QQJ ^y A^/A, the identity expressed in Eq. (3.80) will hold true upto ^T CCT' Now' us"'n9 these values of tAV_rrj upto TAV_CCj» the constants a, b and c in Eq. (3.75) are recalculated by using a multiple regression pro cedure. This equation is used in the next iteration for calculating the LHS in Eq. (3.81). A cool i ng rate A-j °C/s is now chosen such that A. >A. The calculations, as done during Al the first iteration, are repeated, now upto a new CCT*° This procedure is carried on until the correct t^^_Tjj is obtained. The iterations can be stopped, when required, depending upon the needed accuracy. The theoretical justification for the iterative pro cedure is as follows. For a cooling rate A, for an additive reaction, Eq. (3.80) must hold. If the LHS of Eq. (3.80), calculated using the first approximation for t^ TTT' ^s A.j , then 54 'AV-CCT •S-T AV-TTT A 'AV-CTT AT AV-TTT A 'AV-CCT AT AV-TTT (3.81 ) (3.82) (3.83) "•AV-CCT . .". Z r-^ = A (3.84) _1 .. t A AV-TTT A ^1 . . by multiplying tAV_TJT upto TAV_CCT by , we can ensure that Eq. (3.79) will hold. Fig. 3.5 is a pictorial representa tion of this procedure. This iterative method was used to predict the tAy JJJ for the material used in the present work. The tAV TTT and tAV cc are, respectively, the incubation periods for isothermal and continuous cooling rransformations. The details of the 55 Figure 3.5 Nomenclature of parameters used to characterize isothermal and continuous cooling reactions. 56 calculations are shown in Appendix 4. The t^y_TTT values calculated by the iterative procedure and that found by experiments are compared in Table 3.5. As can be seen from this table, the agreement between predicted and 5 calculated values is quite good. B. Hawbolt et al. have reported that, in their experiments, the additivity rule did not work well in the incubation period. This is contrary to the observations in the present work. Since the nature of the reactions in the incubation period (growth of embryo to nucleus) is essentially the same as in the region beyond the incubation period, it is possible to expect the addi tivity rule to work in the incubation period.as well. Calculations to find the values of the LHS of Eq. (3.80) have been accomplished by a FORTRAN computer program. A listing of this program is shown in Appendix 5. 57 Table 3.8V Comparison of Experimentally Determined and Calculated (by the Additivity Method), t... TTT Temperature ^AV-TTT (°C) (s) Experimental Calculated 680 43 41.3 670 5.6 12.9 660 6.2 5.7 650 3.0 3.2 630 1.8 1.8 623 1.6 1.7 615 1.5 1.6 603 1.9 1.8 58 Chapter 4 DEVELOPMENT OF A MATHEMATICAL MODEL  TO STUDY PHASE TRANSFORMATION 4.1 Introducti on In this chapter, a description is given of the mathe matical model developed to predict the phase transformation of austenite to pearlite in a plain carbon eutectoid steel. The model is based on experimentally measured TTT and CCT data. To characterize the non-isothermal austenite-pearlite reaction kinetics, the principle of additivity has been used, the theoretical justification for which has been given in Chapter 3. The model, in its present form, can be used to predict the temperature response in an "infinitely" long plain carbon eutectoid steel rod of circular cross-section being cooled in air. 4.2 Model Formulation For a cylindrical rod cooling in air, heat flow to the surroundings is governed by heat conduction within the rod and heat transfer from the surface of the rod. The heat conduction in a rod undergoing cooling in a medium is given in quantitative terms by the following equation: 59 where: K = thermal conductivity of the rod material r = radial distance from the centre of the rod T = temperature of the rod Cp = specific heat of the rod material p - density of the rod material t = time qAp = volumetric rate of latent heat liberated due to the austenite-pearlite transformation. This term is zero when there is no transforma tion taking pi ace. Eq. (4il) is valid under the following assumptions: i) Infinitely long rod. ii) Negligible axial heat flow. iii) Radial symmetry of temperature iv) Uniform initial temperature v) Uniform circular cross-section. These conditions apply to a wire rod undergoing cooling in a controlled cooling process such as Stel.mor cooling. The boundary conditions are: i) t > 0, r = 0, - K —- = 0 ii) t>0, r = ra, K — = h (Tr - Ta) 60 where r = radius of the rod a h =• surface heat transfer co-efficient T = atmospheric temperature = surface temperature of the rod. The initial condition is: • t = 0, 0 < r < r ., T- = T. a l T.. = uniform initial temperature. Solution of Eq. (4.1) with the proper constants will give the thermal history of the rod under a given cooling condition. Incorporation of the latent-heat liberation term in the heat-transfer equation requires a knowledge of the kinetics of the austenite-pearlite reaction. This term can be calculated as: = p H. (4.2) qAP At where H = enthalpy of the austenite-pearlite reaction AF AP = volume fraction transformed during the time At. Calculation of AF^p must be done by using tAV_rrT and the principle of additivity, as described in Section 4.4. Owing to the complexity and the amount of calculations 61 Simplified Computer Program Flow Chart Read Input Parameters Specify Initial Conds. No. of Nodes of Heat Transfer Coefficient Tine steps • 1,2 ... n I Calculate Thermo-Physical Properties Ves Record the Time Solve Tridiagonal Systen. Calculate New Nodal Temperatures. Write Time am) Temperature of Centre and Surface Nodes Stop Calculate Fraction Transformed Recalculate Thermo-Physical Properties Solve Tridiagonal System. Recalculate New Nodal Temperatures. Fig. 4.1 Computer program flow-chart. 62 involved, it is appropriate to solve Eq. (4.1) subject to the boundary and initial conditions with the aid of a digital computer. 4.3 Computer Program Solution of Eq. (4.1) can be accomplished by using a one-dimensional implicit finite-difference approximation. (The diagonal system of equations is shown in Appendix 6.) The node arrangement, the finite difference equations and the tri-diagonal system of equations was solved by using the Gaussian elimination method. A flow chart of the computer program is shown in Fig. 4.1. Some important features of the program are: i) Thermal conductivity and specific heat have 46 been considered as functions of temperature. ii) Density is assumed constant to keep the node size constant. iii) The- t^y . ^Q-p-has been used as a...f uncti on of temperature. The relationship between ^AV-CCT anc' time can be found from the equation tAV-CCT = A xB e°X ^4'3^ where A,B,C = constants " ' TA1 " T T = Temperature. 63 The constants A, B and C were found by a multiple-regression procedure. iv) Because of the relationship between q^p and AF^p, Eq. (4.2), an iterative procedure was required at each time step, after the start of transforma tion, to check for convergence. It was found that within 3 to 4 iterations, the temperature values at the nodes converged to give a difference less -4 than 10 °C for successive temperature approxi mations. v) The number of nodes and the time step must be chosen carefully, depending on the rod size and the cooling rate. The node size must be not more than 0.25 mm. The time-step interval could be l.s for slow cooling rates (less than 10 °C/s) and should be approximately 0.1s for faster cool ing rates. vi) n and b in the Avrami equation, Eq. (2.2) have been incorporated as functions of temperature. These functions are calculated in separate sub routines. The program was checked for internal consistency by comparing the solution generated by assuming q^p = 0 and constant thermophysical properties for an eutectoid steel with an analytical solution of Eq. (4.1). The results agreed to within 2% of error and are shown in Appendix 7. A comparison was also made for the case of small diameter rods with negligible internal resistance. The above checks confirmed that the program is free of logical and other errors. A complete listing of the program is given in Appendix 8. The model calculates the temperature pro files at all locations inside the rod undergoing cooling. The model output consists of: i) Temperature of surface and central nodes and the corresponding time.. ii) Start time and temperature of transformation at surface and central nodes. iii) Volume fraction transformed at surface and central nodes at each time step. By plotting the time-temperature data generated by model, for a given cooling condition, the amount of re-calescence can be calculated and the temperature range over which the transformation takes place can be determined. By plotting the volume fraction transformed-time predictions, the course of the reaction during the process of cooling can be charted. Typical time-temperature plots from model predictions for a steel rods of composition 0.82% 0.82% Mn - 0.26% Si (Grain size 5-7 ASTM) and diameters 65 5.5 mm, 20 mm:s and 25 mm at different cooling rates are shown in Figs. 4.2 and 4.3. Typical calculations from model predictions for the same material are shown in Table 4.1. 4.4 Program Logic The rod diameter, number of nodes, initial temperature of the rod, ambient temperature (or the temperature of the cooling medium), the surface heat transfer co-efficient and the time-step are input to the program. In the pre-transformation period, say upto 700°C (this depends upon the rod diameter and the cooling rate for a -given•material) the tri-diagonal system of equations is solved at each time step, in the absence of transformational heat, to determine the new temperatures at all nodes. After these temperatures are determined, each node is checked, against t^y_QPj, to deter mine whether the transformation has begun at that node. If the transformation "starts" at any node, the fraction transformed is calculated, for this node, by using the principle of additivity. In general, at the j-lt'1 time step, for the i node undergoing transformation, F j-l exp -b(TJ-_1) e: (4.4) where F j-l volume fraction transformed at node i during the time step j-l 66 T 1 1 —r Time (s) Fig. 4.2 Typical model-predicted centre-line tempera ture response for an air-cooled steel rod. (Rod diameter = 5.5 mm) (Figures on the curves indicate the air velocity in m/s.) 67 Fig. 4.3 Typical model-predicted ture and transformation cooled steel rods. centre-line tempera-profiles for air-Table 4.1 Typical Model Predictions of Undercooling and Recaleseence at the Centre-line of Air Cooled Steel Rods Steel: 0.82% C - 0.82% Grain Size: 5-7 ASTM Mn - 0. 26% Si Rod Diameter :Air Velocity (m/s) (mm) 0 20 i 40 Minimum (°C) 69 92 X 5 Maximum (°C) 83 135 -Recaleseence (°C) 14 43 -Cooling Rate (°C/s) 4.38 24.32 -Minimum (°C) 60 76 81 10 Maximum (°C) 81 103 109 Recaleseence (°C) 21 23 28 Cooling Rate (°C/s) 1.9 8.5 12.6 Minimum (°C) 54 65 69 20 Minimum (°C) 72 84 86 Recaleseence (°C) 18 19 17 Cooling Rate (°C/s) 0.9 3.4 5.0 Note: Minimum = Minimum undercooling (= T^ - T) Cooling rate at Tfl1 69 e1. . j-l b(Tl ,) = value of co-efficient b in the Avrami equation at the temperature of the i node at time step j-l - time taken to transform the cumulative fraction o at tlie temperature J.\ •, n(T"! ,) = value of the co-efficient n in the Avrami J I j_ L equation at the i node at temperature For the j time step, where e1. 1 - exp -b(TJ) <ej 1 At In 1 - F j-l b(Tj) n(TJ) (4.5) (4.6) Then the additional fraction transformed during the time interval j to j-l is: AF F1. - Fl , J J-l (4.7) and the corresponding latent heat liberated is qAP,j " pH T. (4.8) 70 Equation (4.6) is the mathematical statement of the addi tivity principle. This calculation, when ultimately used in Eq. (4.8) and subsequently as a term in thetri-diagonal system of equations, provides for the effect of reaction kinetics on the time-temperature response of the rod. Thus, the difficulties encountered in the direct measurement of reaction kinetics by means of the nuclea tion and growth rates have been overcome by linking the kinetics to a more easily and accurately measurable variable, like the temperature. Once the transformation is detected and the amount of transformation calculated, the co-efficients of the tri-diagonal system are calculated. Then the new temperatures at all nodes are calculated for that time step, including the effect of the latent heat of transformation. Since the rate of the latent heat liberation is of the form given in Eq. (4.7), an iterative procedure is necessary to check for stability of temperature. This is done and it has been established that four iterations are needed to give stable temperatures. This procedure is repeated for subsequent time steps until transformation is complete at all nodes. The program execution is terminated after a predetermined total time value is reached, which depends on the diameter and cooling rate. At each time step, for each node, the temperature dependent properties (thermal conductivity and specific heat) are calculated. Since these also depend 71 upon the transformation, the properties for austenite and pearlite for eutectoid plain carbon steel have been in corporated into the program as temperature functions. During transformation, the program adjusts the values of these variables at all nodes, according to the amount of transformation and temperature. In order to make the program efficient, several counters have been employed. These restrict the number of times a loop is used, thereby reducing program execution time. Si nee the ^/\\/_QQJ» thermal conductivity and specific heat are material specific properties, the program, in its pre sent form, can be used only for the 0.82% C - 0.82% Mn -0.26% Si eutectoid steel of grain size 5 to 7 ASTM. 72 Chapter 5 EXPERIMENTAL All experiments have been conducted on a plain carbon eutectoid steel having the composition shown in Table 5.1. 5.1 Objectives of Experiments Experiments have been conducted in pursuit of the fol lowing objectives: i) To determine the TTT "start" line and to charactize the kinetics of the austenite-pearlite reaction under isothermal conditions. ii) To determine the CCT "start" line for a range of continuous cooling rates. iii) To measure the time-temperature response at the centre-line of air cooled steel rods under dif ferent cooling conditions. 5.2 Experimental Procedures 5.2.1 TTT Tests The isothermal kinetics of a eutectoid plain car bon steel were obtained in a nitrogen atmosphere using a diametral dilatometer and a thermocouple attached to the Table 5.1 Steel Composition c 0.82% Mn 0.82% Si 0.26% S 0.01% P 0.016% Cu Nil Ni < 0.004% Cr 0.02% (Plain Carbon Eutectoid) 74 centre of the outside surface of the specimen. The testing chamber was evacuated with a mechanical forepump, after which nitrogen was introduced to obtain a positive pressure. This was repeated 3 times to ensure a low oxygen content in the atmosphere. A complete description of the apparatus used can be found in reference 35. The tests were conducted on tubular specimens of 8mm OD x 6 mm ID x 75 mm long. These specimens were machined out of rods of 10 mm OD x 150 mm length. The solid rods were used in the centre-line temperature tests. This was done to ensure that the material characteristics in all the tests (i.e. the TTT, CCT and centre-line temperature tests) were identical. To ensure uniform wall thickness of the tubular speci mens throughout the length, a procedure developed by Hawbolt 35 et al. was employed. A Chromel-Alumel thermocouple was spot-welded to the surface of the specimen at the centre. A quartz tipped dilatometer was also attached at this same diametral plane on the specimen. The temperature and dilational history of the specimens were recorded on a con tinuous chart recorder, during each test. Each specimen was mounted in the apparatus (Fig. 5.1) TEMPERATURE a DIAMETER RECORDER A Diometrol Dilotometer B Inlet for internal gos flow C Outlet for internal gas flow D Inlet for external gas flow E Thermocouple on sample F Support Structure Si PHASE SH I FTER TEMPERATURE CONTROL Fig. 5.1 Experimental apparatus for TTT and CCT tests. 76 and austenitised by resistance heating to 850°C and held for 5 minutes at this temperature. The specimen was then cooled to 740°G and held for 1 minute. Then it was cooled to the desired isothermal transformation temperature. A controlled flow of nitrogen gas over the external surface and through the internal core was used to cool the specimen to the test temperature. The maintenance of a nitrogen atmo sphere in the specimen chamber tends, to minimize the decar-burization, though it does not eliminate it. Metal 1ographic examinations revealed that decarb was less than 10% of the wall thickness and hence did not seriously affect the results. The diameter of the specimen during the test was measured by the dilatometer. Due to the dimensional changes accompanying the austenite-pearlite transformation, the dilatqmeter measurements, in conjunction with the thermal history recorded by the thermocouple, could be used to chart the course of transformation during the tests. In all, eight TTT tests were conducted between 680°G and 603°C. 5.2.2 CCT Tests The details of heating and cooling of the speci mens are the same as for the TTT tests. The difference is only in the continuous cooling conditions employed. After heating each specimen to 850°C and holding for 5 minutes, 77 then cooling to 740°C and holding for an additional 1 minute, it was then cooled down at different cooling rates (varying from 2°C/s to 50°C/s). The different cooling rates were achieved by varying the amount of : flow of nitrogen passed over the inside and the external surface of the speci men. As in the case of the TTT tests, the temperature and dilational history were recorded continuously during the tests. Sixteen CCT tests were conducted, some of which were duplicated. The reproducibility observed in these also reinforces the conclusion that the decarburization was mini mal and did not seriously affect the test results. 5.2.3 Centre-line Temperature Measurements in Air- Coo 1 i ng Tests Solid rod specimens of at least 150 mm length were machined to 10 mm diameter from the original hot rolled 12.5 mm diameter rod stock obtained from a local supplier. A centre hole of 3 mm diameter and 5 mm depth was drilled and tapped to the centre of each, rod along a radial line. A threaded plug, of the same steel, was made, with a 1.5 mm hole at its centre; this was screwed into the threaded hole in the rod. The specimen assembly is shown in Fig. 5.2. The hot junction of a 4 thou diameter Chrome!-Alumel thermocouple was held in contact with the rod centre-line by the plug. The thermocouple wires were encased in a ciramic sheath. The OD of the sheath was 1.5 mm, thus fitting r— A 78 A length: min. 150mm Thermocouple Two-hole Ceramic Tube Hole Steel Plug Steel Rod Fig. 5.2 Specimen assembly for centre-1ine temperature tests. 79 tightly into the hole at the centre of the plug. The cold junction of the thermocouple was maintained in an ice-water bath. The thermocouple leads were connected to a chart re corder to record the temperature of the centre-line of the specimen continuously during the cooling test. The specimen, with the thermocouple firmly embedded in place, was placed in an electrically heated furnace main tained at a temperature of 850°C. The specimen attained the furnace temperature, typically, within 3 minutes. It was held at this temperature for 5 minutes, then withdrawn from the furnace and allowed to cool in air. In order to obtain different cooling conditions air was blown across the surface of the rod transverse to the axis. A schematic diagram of this procedure is shown in Fig. 5.3. By using different air velocities, a range of cooling rates, from 4°G/s to 70°C/s, was obtained. Each rod specimen was used once for a test and then discarded, thereby preventing decarburization from affecting the test results. Some of the specimens, after the completion of the tests, were examined under the micro scope for grain size measurements and structure. The grain size measurements were done on specimens heated to 850°C, held for 5 minutes, and quenched in water. This procedure allows for some transformation of austenite to pearlite and the remainder transforms to martensite. On polishing and 80 5.3 Arrangement of air source and specimen during centre-line temperature measurement tests. 81 etching, the original austenite grain boundaries become visible. This micrograph was then compared with the stan dard ASTM grain size chart. The grain size so determined gave a range of 5 to 7 ASTM as the grain size for the material used in the tests. This was confirmed using another 49 method of measuring grain size. In this method, the number of grain intersections on a circle of.a specific diameter superimposed on the micrograph are counted. The grain size (ASTM) is then given by the formula: G(ASTM) = -10.0 - 6.64 In L3 where 4 • ^ 0 P-M P = number of grain intersections on the cir cumference of the superimposed circle M = magnification Lj = circumference of circle (cm) The austenitising treatment and hence the grain size, the test conditions and the constancy of material character istics obtained in the TTT, CCT and centre-!ine.temperature measurement tests ensure that the experimental conditions were the same for all specimens. 82 Chapter 6 RESULTS AND DISCUSSION During the TTT and CCT tests, the diametral response of the specimen and the temperature were recorded continuously. Typical results from these tests are shown in Figs. 6.1 and 6.2. Using these results, the tAy_T-r.r and the t^V-CCT nave been calculated, as described in sections 6.1 and 6.2. A comparison between model predicted and experimental time-temperature profiles measured at the centre-line of air-cooled steel rods is discussed in section 6.3, followed by analysis and discussion of the experimental results. 6.1 TTT Test Results To calculate the "start" of transformation, the data from the diametral dilatometer recordings was used. The fractional diameter change is equal to the fraction trans formed. From the dilatometer response, a graph of ln In (^py) versus. 1 n time (X = volume fraction transformed) was plotted for each isothermal test. The empirical Avrami equation, Eq. (2.1), was then used to find the "start" of the transformation by the following procedure. i) For values of X = 0.1, 0.2, 0.25, 0.30, 0.40, 0.50, 0.60, 0.70, 0.75, 0.80, and 0.90, the 83 Fig. 6.1 Ty pic-el diletometer end' temperature response during e TTT test. Fig. 6.2 Typical dilatometer during a 'CCT "test. and temperature response 85 corresponding 1t' values are obtained for each test from the di1atometer" data. The 11' values are adjusted to correspond to t = 0 at 728°C (T^-j temperature), ii) These values of X and t are used as data for a computer program which calculates the co efficients of a linear regression line fitted to the Avrami equation, Eq. (2.2). The program uses a variable 1t^y1, called "Avrami time", which is the "start" time of the transformation for a given test. During program execution, the value of 'tAV', starting with tAV = 0 at 1^ and incremented in predetermined steps, is subtracted from all values of 't'. A linear regression is then done and the re.gression coefficients printed out along with the sum of differences. This procedure is repeated for each value of "t^y". The regression fit which gives the least sum of differences of squares is the "best-fit" line for the data and the corresponding value of the "t^y" is the "start" time of transformation for the given test temperature. The TTT "start" time determined by the above procedure is different from the "start" times in conventional published TTT diagrams. Traditionally, TTT diagrams were plotted by conducting TTT tests using salt pots and subsequent metal-lographic examinations of the transformed specimens. These "start" lines for either 0.1% or 1 % transformed were deter mined assuming that the transformation started as soon as the temperature fell below the "T^-j" temperature; only the limitations of visual observations and experimental dif ficulties prevented the exact determination of the trans formation product. The method adopted in the present work, 35 first developed by Hawbolt et al., does not make this assumption. The "best-fit". 1 ine gives the "t^y" for an iso thermal temperature and it is assumed that the transforma tion starts at the time; the incubation period is not in cluded in the determination of Avrami's co-efficients (during the test). The plots of X versus t are shown in Figs. 6.3 and 6.4. The tAy_jyj plot, obtained from calculations described above, is shown in Fig. 6.5. The values of the co-efficients n and b from the best-fit line are plotted in Fig. 6.6. In the model calculations, to find the appropriate values of n and b at various temperatures, curve-fit routines are used. These routines calculate the co-efficients of a polynomial of the form, n(t) = a + bT + CT2 +. dT3 (6.1) b(T) = a1 + b^ + t^T2 + d^3 (6.2) where a, b, c etc. are co-efficients. In using this 87 ;0 0-8 "S 0-6 E CO c O ^_ h-c o 0-4 g 0-2 0 • i • • / / h • A / A / A A I h • A u • x • A M O' 0 / o o Steel: 0-82% C, 0-82 %Mn, 0 26% Si Groin Size: 5 -7 (ASTM) O 680°C A 670°C • 660°C -— Best Fit Curves 100 200 300 Time (s) 400 500 600 Fig. 6.3 Isothermal transformation kinetics. 88 • ii I / • A si i I O O S tee 1:0- 82 % C, 0- 8 2 % Mn 0-26% Si Grain Size :5-7 (ASTM) O 650°C A 630°C • 603°C Best Fit Curve _L -L 0 10 20 30 Time (s) 40 50 60 Fig. 6.4 Isothermal transformation kinetics. Steel: 0-82% C, 0-82%Mn 0-26% Si Groin Size: 5-7 (ASTM) O t=Oatt. 600 650 Temperature (°C) 700 Fig.6.;6(a) Variation of n with temperature Steel 0-82% C 0-82% Mn 0-26% Si Grain Size: 5-7 (ASTM) O t = 0 at t AV • t =0CTA| 600 620 640 660 Temperature (°C) 680 6.6(b) Variation of b with temperature. 92 procedure, a small error may be introduced in the values of n and b as seen by comparing the polynomial calculated and the experimentally determined values of n and b (Table 6.1). However, values of n(T) and b(T) calculated from Eq/s,. (6.1) and (6.2) are within 3% of experimentally measured values. Even though the variation in n with temperature is small (between 2 and 3), it has been used as a function of temperature in model calculations. 6.2 CCT Test Results A simple heat-transfer equation was used to find the "start" of transformation during continuous cooling, For a body with negligib1e internal resistance undergoing cooling by convection, where Tt = T(t) -.T(«>) TQ = T(0) - T(») T(t) = temperature of body at time 't' (°C) T(0) = initial temperature of body (°C) T(c°) = temperature of the cooling medium (°C) h = convective heat transfer co-efficient (W/m2°C) 3 p = density of the body (Kg/m ) 93 Table 6.1 Errors in n and b Values Temperature (°C) n ln b Experimental Predicted Experimental Predicted 680 2.125 2.102 - 9.81 - 9.83 670 1.619 1.619 - 9144 - 9.43 660 2.467 2.536 - '9.47 - 9.45 650 2.946 2.885 - 8.39 - 8.48 630 3.166 3.190 - 5.73 - 5.57 623 3.148 3.131 - 4.51 - 4.48 615 2.922 2.934 - 3.14 - 3.32 603 2.346 2.341 - 2.04 - 1.98 94 V = volume of the body (m ) C = specific heat of the body (W/Kg°C Sec) t = time (Sec) where From Eq. (6.3), taking the natural log of both sides, Tt In = -Ht (6.4) 'o H = yA (6.5P H = 5- (6.6) Since the cooling tests were conducted in a uniform nitrogen flow, the convective heat-transfer co-efficient can be expected to be nearly constant. At the time the trans formation begins, the value of 'H' should change markedly, primarily due to the reealescence caused by the release of latent heat of transformation. For any given CCT test, the values of H were calculated for progressively increasing values of t. The time at which there is a sudden change in the value of H is then the tAy_rrT. These values were also checked with the dilatometer data. Tne ^AV-CCT tnus calculated from experimental data are shown in Table 6.2. By using a multiple regression procedure, a curve for determining tAV_rrj at various temperatures, for use in Table 6.2 Continuous Cooling Data (t„., rrT) Experiment tAV TAV # (s) (°c) 1 3.2 585.50 2 3.7 606.75 3 4.1 605.50 4 4.6 607.50 5 4.9 611.50 6 5.1 612.75 7 3.1 589.50 8 4.0 594.00 9 5.1 613.75 10 3.8 570.50 11 3.3 591.50 12 4.3 607.00 13 12.0 629.00 14 14.0 636.00 15 19.8 642.00 16 38.0 649.50 96 model calculations, was obtained.. The equation of the curve so obtained is In (tAV_CCT) = (62.7) (728-T)"15-4 - exp 0.1 (728-T) (6.7) The tA^_rrj calculated by Eq. (6.7) fits the experimental data quite well. A plot of tAy_rrT is shown in Fig. 6.7. In order to check for consistency in the CCT tests, some of the tests were repeated. The consistency was found to be good as determined from the rrT calculations and the dilatometer responses (Fig. 6.8) 6.3 Comparison of Model Predicted and Experimental Results of Centre-1ine Temperature Measurements For a rod of a specific diameter undergoing air-cooling, the temperature-time response can be calculated from the model by inputting the initial temperature of the rod and the heat-transfer co-efficient which depends on the cooling conditions. In the present study, the appropriate heat-transfer co-efficient has been calculated by the following procedure. For a given rod diameter and initial temperature, dif ferent cooling profiles were generated with the model by using different heat-transfer co-efficients.. The time-temperature profile from an experiment was then compared with 97 1 1 1 T 650 -630 (J 0 610 Steel: 0-82 %C,0-82%Mn 3 0-26% Si -O ^. Groin Size: 5-7 (ASTM) O) CL O Experimentol — E cu 590 c/° Best Fit Curve of 570 -550 — I - 1 1 1 10 20 30 40 50 Time (s) Fig. 6.7 tAV_CCT 400 1 '— 1 > 1 —» 0 20 4 0 60 80 100 Time For Temperature Tests (s) 0 2 4 6 8 10 Time For CCT Curves (s) Fig. 6.8 Illustrating the consistency of results observed during CCT and centre-line temperature measurement tests. the model generated time-temperature profiles to find the value of the heat-transfer co-efficient prior to trans formation that best fits the experimental result. Using this value of the heat-transfer co-efficient as the first approximation, the model calcultions were repeated until the model-predicted values agreed with the experimental result for a given value of the heat-transfer co-efficient. The results of the model-predicted and experimental time-temperature profiles are shown in Figs. 6.9 to 6.19 and Table 6.3. As can be seen from these, the model predicted values agree very well with the experimental results. 6.4 Model Prediction and Valid&tion with Measured  Temperature Data From the comparison of model-predicted and experi mental results of centre-line temperature measurements during air cooling of a steel rod, it is evident that the model calculations are sufficiently accurate for the experimental situations in the present work, particularly so, in the light of the number of different inputs. The accuracy of model calculations is governed by the accuracy of the inputs. In this regard, the heat-transfer co efficient, h, which has been assumed constant during cooling, merits special consideration. This assumption seems reason able in the time period prior, to transformation because a good fit between predicted and measured temperatures was Table 6.3 Model Predicted Time-temperature Response at Centre-line of Air-cooled Steel Rods Cooling Rate Amount of at TA1 T . min T max t . min Recaleseence Tmax " Tmin (°C/s) (°C) (°C) (s) (°c) 4.3 636 660 57 24 5.0 638 660 53 22 5.5 636 660 53.5 24 5.0 637 660 49 23 6.2 631 656 25 25 10.5 624 652 26 28 10.0 624 651 25 27 10.6 624 652 27 28 o o 0 101 Fig. 6.9 Temperature response at the centre-line of air cooled steel rod (10 mm diameter). 102 900 840 o o O 780 CD 1^20 660 600 - 1 1 1 1 ~T 1 i I i Steel :0-82%C,0-82%Mn, 0-26%Si Groin Size: 5-7 (ASTM) \o Cooling Rote at TAi = 5°C/s O Experimental Model Prediction - --mm \ o o 0 -1 1 1 1 II I 1 I 0 20 40 60 Time (s) 80 100 Fig. 6.10 Temperature response at the centre-line of air cooled steel rod (10 mm diameter). 103 900 o 840 CD o 780 CD CL E CD 720 h 660 h 600 0 20 40 60 Time (s) 80 100 Fig. 6.11 Temperature response at the centre-line of air cooled steel rod (10 mm diameter). 900 840 o o OJ Z5 "o 780 h cu £720 CD 660 600 40 60 Time (s) Fig. 6.12 Temperature response at the centre-1ine of air cooled steel rod (10 mm diameter). 0 20 40 60 Time (s) Fig. 6.13 Temperature response at the centre-line of air cooled steel rod (10 mm diameter). 106 Fig. 6.14 Temperature re.sponse at the centre-line of air cooled steel rod (10 mm diameter). Fig. 6.15 Temperature response at the centre-line of air cooled steel rod (10 mm diameter). 900 840 o o £ 780 D O CU CL I 720 660 h 600 0 20 40 Time (s) 60 Fig. 6.16 Temperature response at the centre-line of air cooled steel rod (10 mm diameter). Fig. 6.17 Temperature response at the centre-line of air cooled steel rod (10 mm diameter). Steel:0-82 %C,082%Mnf 10 Time (s) Fig. 6.18 Temperature response at the centre-line of air cooled steel rod (10 mm diameter). T Time (s) Fig. 6.19 Temperature response at the centre-line of air cooled steel rod (10 mm diameter). 1 obtained, Figs. 6.9 to 6.19. This is not surprising since the dominant mode of heat transfer is temperature indepen dent convection, especially at higher cooling rates. The temperature dependent term, the radiative heat transfer, is a small portion of the total heat transfer (usually less than 5-10% at higher cooling rates). Hence it is to be expected that 'h' will be roughly constant in the pre-transformation period and the use of a constant 'h' is thus justified. 6.5 Discussion The good agreement of the model predictions with the experimental results is primarily due to the following factors: i) Accurate inputs, like thermal conductivity, specific heat etc. ii) Use of tAV_TTT and tAy_CCT iii) Validity of the additivity principle for the experimental conditions. As mentioned earlier, the transformation is assumed 3 5 to "start" at tAV_CCT during continuous cooling. The conventional approach is to assume that the transformation starts at TA-| , even during continuous cooling. In the present work, if n and b are calculated using t = 0 at Tfl, 113 the variation in n with temperature is much higher than for the case of t = 0 at T^y (see Table 6.4). The range of values of n (1.9 to 4.8) is in contrast with the work of 29 Tamura et al. and others, who propose a value of 4 for the austeni te-pearl i te. reaction. Also the range is too wide to expect a constant n, as required by additivity. In addition, model calculations using t = 0 at are not in agreement with the time-temperature response of the steel rods measured experimentally, as shown in Table 6.5 and Fig. 6.21. (Nomenclature of terms used in Table 6.5 is shown in Fig. 6.20.), The value of n calculated by using t = 0 at varies within a narrow range (1.7 to 3), with an average close to 2.5. This is lower than the value expected by Tamura 29 et al. This discrepancy can be partly explained by the effect of inhomogeneity of the reaction in the experi mental situations encountered in this work. Due to in homogeneity, the volumetric growth of the transformed pro duct does not follow the Johnson-Mehl equation (which would mean n = 4). The value of n and b would be determined by the degree of inhomogeneity present in the reaction. Further work is needed to study the nature of this inhomogeneity and its effect on the reaction kinetics. The model calculations also demonstrate the validity Table 6.4 Values of n and b for t = 0 at T Temperature (°C) n In b 680 1.94. -11.57 670 2.29 - 10.72 660 2.91 - 11.54 650 3.60 - 10.83 630 4.32 - 8.96 623 4.63 - 8.16 615 4.79 - 7.16 603 4.76 - 6.98 115 min { max Time (s) Fig. 6.20 Nomenclature of parameters describing a continuous cooling-reaction. Table 6.5 Comparison of Model Predicted Time-temperature Profiles at Centre-line of Air-cooled Steel Rods Using t = 0 at t.. and t = 0 at Jn with Experimental Results. T tl • (°c) T (°C) max v ' min <s> Expt. Model 1 Model 2 Expt. Model 1 Model 2 Expt. Model 1 Model 2 633 637 645 668 661 661 57 57 56 629 638 645 660 661 661 53 53 53 633 636 646 670 661 661 54 53.5 54 629 637 644 660 660 660 49 49 46 635 631 637 663 656 655 25 25 26 620 624 630 646 652 651 26.5 25.6 26 614 624 629 640 651 649 25 25 24 614 624 631 640 652 650 27 27 26 Model 1: t = 0 at t" Model 2: t = 0 at T 1 1 7 900 800 o o 700 CD 13 "5 k_ CD §- 600 CD h-500 400 OA Experimental Model Predicted Steel:0-82%C,0-82%Mn 0-26% Si Grain Size: 5-7 (ASTM) t=0 at T AI _L 1 0 20 40 60 Time (s) 80 100 Fig. 6.2V Comparison of typical model predicted (with t = 0 at ) and experimental results of centre-line temperatures of air-cooled steel rods. (Rod diameter = 10 mm) 118 of the additivity principle. This principle is derived from a consideration of the fundamental aspects of the nucleation and growth reactions, characterized by nuclea tion and growth rates. It is important to realise that the concept of nucleation envisioned by Avrami in deriving his equation is different from that of Johnson and Mehl. Avrami treated nucleation in two steps: i) a probability 'P' of a germ nucleus becoming a growth nucleus ii) a germ nucleus becoming unavailable for growth due to consumption by the growing phase. In contrast, Johnson and Mehl defined the nucleation rate, N (a constant) as the number of nuclei nucleating and growing. Evidently Avrami's concept of nucleation is closer to the real situation. Avrami then stated that P/G be constant in the isokinetic range. The Avrami P is the equivalent of the Johnson-Mehl N. Avrami started his derivation by assuming that there are H germ nuclei pre sent before the transformation began. (Hence, PN = N.) This assumption is equivalent to stating that there is an incubation time for each reaction, before the transforma tion starts, which corresponds to tAV_TTT and tAV_cc-r used in the present work. The nucleation rate N, used in deriv ing the effective site saturation condition, is PN". In 119 terms of Cahn's site saturation principle, 'P' is very close to unity. The behaviour of 'P' is, indeed, difficult to determine. It is affected by the thermodynamics of the austenite-pearlite reaction, the grain size, the site energy etc. It is closely related to the forces governing atomic behaviour. But the principle of effective site saturation is easier to deal with, and since it is based on the fundamental aspects of reaction mechanism becomes a useful tool to study the effect of these variables on the more easily measurable and important process parameters such as temperature, cooling rate etc. 6.6 Scope of Application of the Mathematical Model By siightly modifying the model in its present form, it is possible to study the effect of segregation on the transformation behaviour of wire rods undergoing cooling in a Stelmor-1ike process. Due to the segregation of elements like manganese and phosphorus in cast ingots or billets the final product may contain segregated regions at the centre. Due to the higher hardenabi1ity associated with the high Mn (and P) contents, the central region in the finished product may have transformed to'. mar-ten si te if the post-rolling controlled cooling parameters are designed for the matrix material. The brittle martensite may fracture during subsequent processing, e.g. when wire rods 1 20 are drawn into wires. An example of Mn segregation in a wire rod and the effect on transformation is described in section 6.7. Finally, but most importantly, the model calculations can be used for evaluating the average mechanical pro perties of finished steel rods. An example of such calcu lations is shown in section 6.8. 6.7 Effect of Segregation on Phase Transformation To study the effect of segregation on phase trans formation, a plain carbon eutectoid steel of 0.8% C - 1.88% Mn (Grain Size = 5 to 8 ASTM) was chosen as the composi tion in the segregated region in a matrix composition of 0.82% C - 0.82% Mn (Grain Size = 5 to 7 ASTM). Normally the range of segregation for manganese is about 1.2 to 1.3 times the matrix composition. But no published TTT dia gram of a steel with 0.8% C and about 1% Mn could be obtained from the literature. Hence, though the manganese content of 1.88% is too high, the calculations were done with the aim of demonstrating the trends that could be expected in the transformation behaviour due to segregation. The TAV_TTT for the segregated region has been calcu-45 lated from published data and is shown in Table 6.6. The Table 6.6 TTT Data for Segregated Steel Temperature *AV (°C) (s) 675 93 650 29 625 13 600 6.2 575 5.4 '550 3.5 525 3.4 500 2.9 475 1.9 450 2.5 425 4.2 400 7.5 n and b values in the Avrami equation have been calculated from the tAV_TTT and are shown in Table 6.7. The tftv CCT has been calculated from tftV_TTT by using additivity (Table 6.8). The thermal conductivity, specific heat and density have been assumed to be the same as that of the matrix material since the carbon contents are the same. The 45 Mg temperature for the segregated steel is 180°C. The program, used for calculations on the 0.82%C steel, has been modified to include the parameters for the segre gated composition. The program listing is shown in the Appendix 9. The program logic is the same as before. The only changes are in the t^y-.^y'and the n and b values used for calculating the start of transformation and its further course in the segregated regions. Normally, the segregated region at the centre occupies less than 5% of the cross-sectional area of the wire rod. For modelling purposes, two situations have been considered, T% and 4% of cross-sectional area occupied by segregated material. Due to the presence of the high hardenabi1ity material at the core, martensite may be expected to form at the core, depending on the cooling conditions and rod size. The effect of segregation has been investigated for three different rod diameters - 5, 10 and 15 mm. The results of the model calculations are shown in Tables 6.9 to 6.14. The amount of martensite that can be expected to form at the core for the different rod sizes at different cooling rates are Table 6.7 n and b Values for Segregated Steel Temperature (°C) n In b 675 0.77 - 6.09 650 0.83 - 4.60 624 1.28 - 5.08 600 1.65 - 5.57 575 1.21 - 4.00 550 1.10 - 4.02 525 0.74 - 2.92 500 0.72 - 2.98 475 0.85 - 3.78 450 0.91 - 3.97 425 0.99 - 4.38 400 1.06 - 5.03 Table 6.8 CCT Data for Segregated Steel Temperature lAV-CCT (°C) (S) 696 3060 675 473 667 286 641 86 591 23 545 11.4 512 8.3 458 5.9 443 5.6 423 5.4 383 5.6 364 5.9 125 Table 6.9 Effect of Cooling Rate on Martensite Formation at Centre  of Rod \ Initial Temperature = 850°C Rod Diameter = 5 mm Amount of Segregated Area = 1 % Cooling Rate at TA1 (°C/s) \iin (°0 ^max (°0 t • min % Martensite formed 6 629 654 36 <34 9 620 650 25 <53 13 614 646 20 <62 61 - - - 100 126 Table 6.10 Effect of Cooling Rate on Martensite Formation at Centre of Rod Initial Temperature = 805°C Rod Diameter = 5 mm Amount of Segregated Area = 4 % . Cooling Rate at TA1 Tmin T max t ... min % Martensite formed (°C/s) (°C) (°C) (s) 6 628 656 36 <33 9 622 646 25 <52 13 613 642 20 <62 61 - - - TOO 9 127 Table 6.11 Effect of Cooling Rate on Martensite Formation at Centre of Rod Initial Temperature = 850°C Rod Diameter =10 mm Amount of Segregated Area = 1 % Cooling Rate at TA1 (°C/s) 3 4.5 61 T • mm (°C) 640 631 max (°C) 662 656 Vin (s) 78 38 "% Martensite formed < 5 <30 100 128 Table 6.12 Effect of Cooling Rate on Martensite Formation at Centre of Rod Initial Temperature = 850°C Rod Diameter = 10 nm Amount of Segregated Area = 16 % Cooling Rate at TA1 (°C/s) Tmin (°C) Tmax (°C) t . min (s) % Martensite Formed 2.5 639 662 85 < 5 4.0 634 657 57 <16 6.0 631 655 45 <30 58.0 - - - 100 129 Table 6.13 Effect of Cooling Rate on Martensite Formation at Centre of Rod Initial Temperature = 850°C Rod Diameter = 15 mm Amount of Segregated Area - 1 % Cooling Rate at TA1 (°C/s) T . mm (°C) Tmax (°C) t • mm (s) % Martensite Formed 2.0 646 666 99 < 1 3.0 642 663 69 < 4 4.5 638 660 53 <12 52.0 - - - 100 1 30 Table 6.14 Effect of Cooling Rate on Martensite Formation at Centre of Rod Initial Temperature = 850°C Rod Diameter = 15 mm Amount of Segregated Area = 16 % Cooling Rate at TA1 (°C/s) T . mm (°C) Tmax (°C) t • min (s) % Martensite Formed 1.5 646 664 100 < 0.1 2.5 641 660 69 < 1 4.0 638 657 53 < 5 48.0 - - - 100 1 31 shown in Fig. 6.22. As expected, the percentage martensite formed in the segregated region.is higher for smaller rod diameter and faster cooling rates. The problem of segregation is not unusual in a wire rod mill. An example of segregation is shown in Fig. 6.23. Many methods have been studied to minimise or eliminate the martensite formation at the centre. One such method, which has been tried in an industrial situation, is described by 47 Van Vuuren. He reports that the cooling rate was reduced to ensure transformation to pearlite at the segregated centre region. But this resulted in a wide scatter in the tensile values and hence the procedure was unacceptable. The pro cedure which worked successfully was to have a high cooling rate initially followed by slower cooling. This resulted in an initial sharp drop in temperature, but later, the drop was very slow. This procedure allows time for homo-genisation of temperature. It is claimed that this pro cedure helped hold the scatter in the tensile strength values to a minimum whi1e avoiding the formation of marten site. Obviously, this procedure requires! good control over the process cooling.conditions. The model calculations can be used to study the effect of such changes in cooling conditions and their effect on 1 32 Fig. 6.22 Effect of cooling rate on centre-line martensite formation due to segregation. 133 Fig. 6.23 Longitudinal section through a cold drawn rod showing a white centre line martensitic phase fractured during subsequent cold drawing. 1 34 the transformation. These then can be used as a guide for designing the process parameters. 6.8 Calculation of Mechanical Properties of a Wire Rod The mechanical properties of plain carbon steel rods depend primarily on: i) composition (esp. N, Si, Mn and C) ii) amount of the ferrite phase iii) pearlite spacing. For a given steel composition, the amount of ferrite, and more importantly, the pearlite spacing depends on the undercooling during transformation. Frequently, and especially during slow cooling, the steel undergoes re-calescence after the start of the austenite-pearlite trans formation in the case of.a eutectoid steel. This implies that the transformation takes place over a range of tempera ture. If this range is narrow, then the reaction can be considered isothermal from a practical standpoint. If the rod size and cooling conditions are such that the tempera tures at different locations are significantly different, then it is possible that transformations at these locations take place over different temperature ranges. This intro duces another difficulty in calculating an average pearlite spacing value for the rod. At this stage, it is not possible 1 35 to calculate accurately the effect of such a temperature range on pearlite spacing. However, for illustrative purposes, we can assume that the average undercooling at the centre of the rod determines the pearl ite spacing in the rod. The model calculations are very useful in this regard. The model prediction of the time-temperature response at the centre-line of a rod being cooled can be used to deter mine the average undercooling. Since the model calculates the temperature and the fraction transformed, it is pos sible to determine the start and end transformation tempera tures as well as the maximum and minimum undercooling values. Using these values, the pearlite spacing for the steel can 54 be determined from published empirical relationships. The mechanical properties are related to the pearlite 54 spacing and steel ..chemi stry as follows: i) avc. (MPa) 1 _1 a3 J2.3 + 3.8 (%Mn) + 1.13 d 2*l+ UTS(MPa) a volume fraction of ferrite s P d pearlite spacing (mm) grain diameter (mm) 136 For illustrative purposes, the mechanical properties for the steel used in the present study have been calcula ted, for two different rod sizes and two cooling conditions. Pearlite spacings have been calculated by using the pro cedure mentioned above and are shown in Tables 6.17 and 6.18. For a eutectoid steel , Eq. (6.8) and (6.9) can be re written as: i) avs (MPa) = 15.4^11.6 +.• Q..255-sp * + 4.1 {% Si) + 27.6 (%N)j- (6 ii) UTS. (MPa) =-.l-5..4--J46.7 + 0..23 sp2+-6.3-(%.Si)j. (6 The mechanical properties calculated using Eq. ("6.'TO) and (6.11) are shown in Tables 6.15 to 6.17. This example illus trates applicability of the model to calculating the mechani cal properties of the wire rods for different cooling rates. This application can be extended to other materials by suitably modifying the model. It is thus demonstrated that the mechanical properties can be calculated from a knowledge of the reaction kinetics of the transformation. Admittedly, the calculations have been done with simplified assumptions. For example, the effect of the different temperature ranges of transformation at different locations in the rod on the pearlite spacing needs to be studied further. Also the time taken for the transformation at different locations in the rod may also affect pearlite spacing. Nevertheless, 137 Table 6.15 Pearlite Spacing Calculations Rod Diameter = 5 mm Initial Temperature of Rod = 850°C Air Velocity (m/s) 0 10 Cooling rate at Tftl-(°C/s) 4.4 18.1 Maximum undercooling (°C) 83 127 Minimum undercooling (°C) 69 82 Average undercooling (°C) 76 104.5 Minimum interlamellar o pearlite spacing (A) 1000 675 Table 6.16 Pearlite Spacing Calculations Rod Diameter = 15 mm Initial Temperature of Rod = 850°C Air Velocity (m/s) 0 10 Cooling rate at TA1 (°C/s) 1.2 3.7 Maximum undercooling (°C) 76 88 Minimum undercooling (°C) 57 66 Average undercooling (°C) 67.5 77 Minimum interlamellar o pearlite spacing (A) 1050 1000 1 39 Table 6.17 Mechanical Properties Cooling Conditions aYS (MPa) UTS (MPa) RA (X) Rod dia = 5 mm CRT = 4.38 °C/s 'AI Air Velocity = 0 m/s 599 1098 37 Rod dia = 5 mm CRT = 18.13 °C/s 'AI Air Velocity = 10 m/s 675 1175 40 Rod dia = 15 mm CRT = 1.2 °C/s 'Al Air Velocity = 0 m/s 580 1090 35 Rod dia = 15 mm CRT =3.7 °C/s 'AI Air Velocity = 10 m/s 599 1098 36 140 these considerations only imply that the calculations need to be more sophisticated than the one attempted in the pre sent study. But they do not in any way bring into doubt the validity of using the model to do such calculations. It has been demonstrated in the present study, albeit in principle, that the model can be used to integrate the reaction kinetics with the macro level variables such as temperature and mechanical properties. 141 Chapter 7 SUMMARY AND CONCLUSIONS With the ultimate objective of calculating mechanical properties of wire rods in the Stelmor process using phase-transformation data, a mathematical model has been developed. The predictions, made by the model, of the centre-line temperature profiles have been shown to be in good agree ment with experimental results for a plain carbon eutectoid steel. Calculations have also been done to derive the mechanical properties of wire rods using model-predicted data. However, these calculations are only illustrative and need to be refined further with the help of experimental data. The model has been modified to study the effect of centre-segregation on transformation in air-cooled steel rods. Calculations have been carried out with a centre segregation composition of 0.8% C - 1.88% Mn in a matrix of 0.82% C - 0.82% Mn. Though the manganese in the segre gated region is higher than would be normally expected, the calculations, nevertheless, give a quantitative estimate of the amount of martensite that can be expected to form at the centre of the rod. Also the effect of increasing the cooling rate, the rod diameter and the amount of segre gation on the amount of martensite formed at the centre can be predicted quantitatively by the model. Experiments are needed to validate the model predictions. However the capability of the model to perform the calculations and give meaningful predictions has been demonstrated. This should help reduce the amount of empirical experimentation to design process parameters, such as cooling rates for different rod sizes, in a Stelmor-1ine. The success of the model is primarily due to two factors: i) use of tAV_jTT and tAV_CCT for calculation of n and b in the Avrami equation and the start of transformation during cooling, respectively. ii) additivity. The use of tAy_-r-r.r and t/^.QQj was proposed by B. 35 Hawbolt et al. and has worked wel1 in the present study. The conditions for additivity proposed by Avrami and Cahn were not obtained in the present work, thereby necessita ting further exploration into the mechanics of additive reactions. As a result, two new sufficient conditions for additivity have been derived. These are: 143 i) additivity range ii) effective site saturation. These two sufficient conditions increase the scope of reactions to which the additivity rule becomes applic able (see Table 7.1),. The effective site saturation ratio is a simple but effective method of determining additivity for nucleation and growth reactions and can be used for homogeneous and heterogeneous reactions. The reasons for the heterogeneity encountered in the reactions in the present study need to be studied further. The effectively site saturated reactions are growth dominated. To study the deviations of heterogeneous reaction kinetics from the homogeneous, the Inhomogeneity Co-efficient (It) and the Heterogeneity Co-efficient (H^) have been proposed. Further experiments are needed to study the nature of the hetero geneity and its effect on the coefficients b and n in the Avrami equation. Using the additivity rule, a new iterative procedure, called the "additivity method", has been derived to calcu late TTT data from CCT. The method has been shown to work successfully for the data in the present study. Table 7.1 Scope of Reactions Covered by Different Additivity Criteria 144 Criterion Remarks Avrami's Isokinetic Range N 1. must be constant in the temperature range. Also implies that N and G must be constant for an isothermal reaction. 2. Normally not encountered for austenite-pearlite reactions in steel. Cahn's Site Saturation 1. The nucleation event is nearly complete in the early stages of the reaction. The growth rate must be a function of temperature alone (thereby implying that it be constant for an isothermal reaction). 2. May hold true for reactions in alloy steels, especially grain-boundary nucleated reactions. Additivity Range 1. N and G are functions of temperature alone. N 2. Does not require that ^ be constant. Thus it is less restrictive than the Avrami criterion. Implies that N and G be constant for an isothermal reaction. 3. There is evidence in the literature to support N and G are functions of temperature alone. Also, the criterion does not call for physical saturation of nucleation sites. Thus it is less restrictive than Cahn's criterion and includes reactions which are not covered by the Avrami criterion. Applicable to all steels. Effective Site Saturation 1. G is a function of temperature alone, and hence constant for an isothermal reaction. 2. Includes all reactions covered by the Avrami, Cahn and Additivity Range criteria for austenite-pearlite reactions in steel for growth dominated reactions, which is usually the case. 3. Also includes reactions with decreasing nucleation rates and heterogeneous reactions. 4. The most flexible and practically useful criterion. BIBLIOGRAPHY 145 1. Morgan Construction Company, "The Stelmor Process", August (1978). 2. Feldman, U. "Controlled Cooling of Wire Surface from the Rolling Heat", Iron and Steel Engineer, Jan., pp. 62-67: (1980). 3. Ammerling, W.J. "Controlled Cooling of Wire Rod from Rolling Temperature", Iron and Steel Engineer, Dec, pp. 99-107 (1970). 4. Schummer, A. "Controlled Cooling Processes for Wire Rod and Structural Shapes", SEAISI Quarterly, Oct., pp. 6-12: (1978). 5. McLean, D.W. "The History of Controlled Cooling of Rod at Stelco", Wire, Vol. 39, pp. 1606-1609 (1964). 6. Dove, A.B. "Wire Manufacturing Using Stelmor Control led Cooled Rod", Wire, Vol. 39, pp. 1610-1615 (1964). 7. Hitchcock, J.H. "Mechan.ical Aspects", Wire, Vol. 39, p.1622 (1964). 8. Grattan, E. ; Twigg, G.M. ; Benson, P. "S-ED-C: An Advance in the Controlled Cooling of Carbon Steel Rod". Iron and Steel International, Oct., pp. 277-280 (1979). 9. Malmgren, N.G.; Tarnblom, S.G. "D-patented Wire Rod", Wire, Vol. 25, pp. 211-218 (1975). 10. Beaujean, R. ; Godart, F. ; Lambert, M.; Economopoulos, M. "Research for Obtaining the Lead Patenting Structure by Treatment of the Wire Rod Directly after the Rolling Mill", Centre de Reserches Metallurgiques, Vol. 32, pp. 10-33 (1972). 11. Bain, E.C. "On the Rates of Reactions in Solid Steel", Trans. A.I.M.E., Vol. 100, pp. 13-46 (1932). 12. Bain, E.C. ; Davenport, E.S., Trans. A.I.M.E., Vol. 90, p.,117 (1930). 13. Brandt, H. Trans. A.I.M.E., Vol. 167, p. 405 (1946). 14. Zener, C. "Kinetics of the Decomposition of Austenite", Trans. A.I.M.E., Vol. 167, p. 550 (1946). 146 15. Holloman, J.H.; Jaffe, L.D. "Anisothermal Decomposi tion of Austenite", Trans. A.I .M. E. , Vol.. 167, p. 419 (1946). 16. Manning, G.K.; Lorig, CH. "The Relationship Between Transformation at Constant Temperature and Transforma tion During Cooling", Trans. A.I.M.E., Vol. 167, p. 442 (1946). 17. Scheil, E. "Initiation Time of the Austenite Trans formation", Areh.iv. E i senhuttenwesen , Vol . 8, pp. 565-567 (1935). 18. Austin, J.B.; Rickett, R.L. Metals Technology, T.P. #964, Sept. (1938). 19. Johnson, W.A.; Mehl, R.F. "Reaction Kinetics in Processes of Nucleation and Growth", Trans. A.I.M.E., Vol . 135, pp. 416-442 (1939). 20. Avrami, M. "Kinetics of Phase Change I", J. of Chem. Physics, Vol. 7, pp. 1103-1112 (1939). 21. Avrami, M. "Kinetics of Phase Change TI", J. of Chem. Physics, Vol. 8, pp. 212-224 (1940). 22. Avrami, M. "Kinetics of Phase Change III", J. of Chem. Physics, Vol. 9, pp. 177-183 (1940). 23. Cahn, J.W. "The Kinetics of Grain Boundary Nucleated Reactions", Acta Met., Vol. 4, p, 449 (1956). 24. Cahn, J.W. J- of Metals, Jan., p. 146 (1956). 25. Christian, J.W. "The Theory of Transformations in Metals and Alloys", Pergamon Press, Chapters 1,12 (1965). 26. Shimizu, N.; Tamura, I. Trans. ISIJ, Vol. 18, p. 445 (1978). 27. Sakamoto, Y. et al. Yo Ito Research Labs., Kawasaki Steel Corpn., Chiba, Japan 280. 28. Grange, R.A. ; Keifer, J.M. "Transformation of Austenite on Continuous Cooling and its Relation to Transformation at Constant Temperature", Trans. A.S.M., Vol. 29, p. 29 (1941). 147 29. Shimizu, N. ; Tamura, I. Trans. IS IJ, Vol . 17, p. 17 (1977). 30. Umemoto, M. ; Tamura, I. "Continuous Cooling Transforma tion Kinetics of Steels", Trans. ISIJ, Vol. 21, pp. 383-392 (1981). 31. Agarwal, P.K.; Brimacombe, J.K. "Mathematical Model of Heat Flow and Austenite-Pearlite Transformation in Eutectoid Carbon Steel Rods for Wire", Met. Trans. 'B', Vol. 12B, pp. 121-132 (1981). 32. Hi 1denwal1 , B. "Prediction of the Residual Stresses Created During Quenching", Linkoping Studies in Science and Technology, Dissertation #39 (1979). 33. Hi 1denwal1 , B. ; Ericsson, T. "Hardenabi1ity Concepts with Application to Steel", The Metallurgical Society of A.I.M.E. , Warrendale, Pa. (1978). 34. Kirkaldy, J.S. ; Sharma , R.C. "A New Phenomenology for Steel IT and CCT. Curves", Scripta Met. , Vol . 16, pp. 1193-1198 (1982). 35. Private communication with Dr. Hawbolt, E. B... and Brimaeombe , J.K. 36. Brown, D. ; Ridley, N. "Kinetics of the Pearlite Reaction in High-purity Nickel Eutectoid Steels", 0. of the Iron and Steel Institute, Sept., pp. 1232-1240 (1969). 37. Kuban, B. M.A.Sc. Thesis, University of British Columbia,. Vancouver, B.C., Canada (1983). 38. Markowi tz, L.M.; Richman, M.H. "The Computation of Continuous Transformation Diagrams from Isothermal Data" , Trans. A.I.M.E., Vol. 239 , pp. 1 31-1 32 (1967). 39. Tzi tzelkov , I. ; Hogardy, H.P. ; Rose, A. "Mathematical Description of the TTT Diagram for Isothermal Trans formation and Continuous Cooling", Report #1808 of the Materials Committee of the Association of German Iron and Steel Engineers (1974). 40. Takeo, K. et al. "The Direct Patenting of High Carbon Steel Wire Rod by Film Boiling", Trans. 1STJ, Vol. 15, pp. 422-427 (1975). 148 41. Cahn, J.W.; Hagel , W.C. "Theory of the Pearlite Reaction", Decomposition of Austenite by Diffusional Processes. Ed. V.F. Zackay and H.I. Aaronson , A.I.M.E. Publication (1962). 42. Scheil, E.; Lange-Weise, A. "Statistiehe Ge fugeuntersuchungern", Vol. 2, p. 93 (1937-1938). 43. Hull, F.C. ; Colton, R..A. ; Mehl, R.F. "Rate of Nuclea tion and Rate of Growth of Pearlite", Trans. A.I.M.E., Vol. 150, pp. 185-207 (1942). 44. Private Communications with Dr. I. Tamura. 45. Atlas of Isothermal Transformation and Cooling Trans formation Diagrams. ASM, Metals Park, Ohio (1977). 46. "Physical Constants of Some Commercial Steels at Elevated Temperatures", BISRA, Scientific Publica tions, London (1978). 47. VanVuuren, C.J. "Operating and Quality Control Aspects of the Production of Critical Low and High Carbon Products from Continuously Cast Blooms", South African Iron and Steel Industrial Corporation Ltd., Iscor, Newcastle Works, South Africa. 48 Marder, A.R. ; Bramfitt, B.L. "Effect of Continuous Cooling on the Morphology and Kinetics of Pearlite", Met. Trans., Vol. 6A, pp. 2009-2014 (1975). 49. "Grain Size Determination" - Metallography, Structures and Phase Diagrams, A.S.M. Metals Handbook, Vol. 8, 8th Edition, Metals Park, Ohio. 50. Umemoto, M. ; Horiuchi, K. ; Tamura, I. "Transformation Kinetics of Bainite During Isothermal Cooling and Continuous Cooling", Trans. I.S. I.J. , Vol. 22, pp. 854-861 (1982). 51. Carnahan, B. et al. "Applied Numerical Methods", John Wiley ( 1969) . 52. Kreith, F. "'Princi pies, of Heat Transfer", 3rd ed., Intext Publishers, New York (1973). 53. B. Hawbolt et al Paper to be published in Met. Trans. 54. Leslie, W.C. "The Physical Metallurgy of Steels", McGraw-Hill Book Co., pp. 164-165 (1981). 149 Appendix 1 THE PRINCIPLE OF ADDITIVITY [Reference: "The Theory of Transformations in Metals and Alloys", J.W. Christian, 1965, Pergamon, London, pp.545-546.] Consider the simplest type -of non-isothermal trans formation, obtained by combining two isothermal transforma tions. The assembly is transformed at temperature T-j , where the kinetic law is f = f n (t) for a time t-j (f = volume fraction transformed.) and is then suddenly trans ferred to a second temperature T,,. If the reaction is additive, the course of the reaction at 1^ 1S exactly the same as if the fraction transformed f^(t-j) had all been formed at 1^. Thus if t^ is the time taken at T^ to produce the same amount of transformation as is produced at T.| in a time t-j , we have and the course of the whole reaction is f = f](t) t < t1 = f2(t+t2-t1) t > t] 1 50 Suppose that tal is the time taken to produce a fixed amount of transformation fa at T-j and ta2 is the correspond ing time to produce the same amount of transformation at T,,. Then in the composite process above, an amount f of trans-a formation will be produced in a time t = ta2- t2 + t] if the reaction is additive (see Figure A.l) t-t1 + t2 ta2 P± + ^- * 1 ra2 za2 t1 t If T~ = ~- then •al ZaZ t-t, t, Ta2 ral For an additive reaction the total time to reach a specified stage of transformation is obtained by adding the fractions of the time to reach this stage isothermally until the sum becomes Equation Al.1 can be written in the more generalised form as 151 t r dt J i (Al .2) 0 Where t (T) are the isothermal transformation times a for a fraction transformed 'a' at a temperature T and 't' is the time taken to reach a fraction transfomred 'a' for an a isothermal reaction path. The above relationship holds if, for a reaction, the reaction rate depends only on the fraction transformed and temperature. Consider a transformation for which the instantaneous reaction rate may be written The derivation of Eq. Al.;2 is based on the relation ship that, for an additive reaction, t (Al.3) dX dt h(T) (A1.4) gTx) where: X volume fraction transformed t time h(T) function of temperature g(x) function of- volume'fraction transformed. 152 Then, /h(T) dt = X) dX (A1.5) for any transformation path . '. X = F yh(T-) dt (Al .6) and for an isothermal reaction X = F jh(T)tj (Al.7) According to A1.7, the volume fraction transformed at any temperature is a function of time and temperature. From Al.5 Al so h(T) = g(Xa) (Al.8) t^TT dX = hir dt " 9TXT t (T) dX - 9(Xa) VT) dt ITXT dt _ gX .+ i^ry - ttrj dt • • JtjT) ~ g-nryj 9(x)dx - tttry a% ' JV a' ' a 0 153 This is the general additivity rule (Eq. Al.9). In particular, if X = X,, then a r dt 9(xa> Fig. A 1.1. Additivity Principle. Appendix 2 ADDITIVITY OF THE AVRAMI EQUATION KINETICS The Avrami equation is: X = 1 - exp (-btn) .'. log(l-X) = -btn Differentiating A2.1 w.r.t. 't', t n ioq n-x) -b dX dt = exp(-btn) (-nb t n-1 = (1-X)(-bn) fiogd-xj •n n = (l-X)(n)(-b) n n (-b) n n-1 I h(T) g(x) where 1 h(T) = n-(-.b)n 1 56 Appendix 3 INDEPENDENCE OF N(T) AND G(T) w.r.t. TIME Consider the equation t T 3 T T X N(T;)df-(x-t)3dt 0 [0 T0 (A3::l) Since G(t) and N(T;) are functions of-', temperature alone Eq. (A4.1) can be written as: T 3 T = J G(T)dTJ • —L— j N(T)dT -J (x-t)3dt ex " ' -'0 " " 'x-'O TQ TQ 0 (A3.2) In the case of a continuous cooling reaction, since the temperature is a function of time (which is given by the cooling rate), it would seem that T = f(t) (A3.3) . ' . N (T) = N{ f (t) } = •N1 (t) (A3.4G(T) = G{f(t)} = G^t) (A3.5) But this is not true, as can be seen from Fig. A $.1. The figure is a three-dimensional representation of t, G(T) and T. As can be seen, the G(T) for three different reactions can be 157 obtained, independent of the time. The G(T) for a reaction depends only on the temperature path of the reaction. The time of the reaction can then, be brought in., and when multi plied by the temperature averaged growth rate for the re action, gives the total growth of one particle during the time interval. This should be multipiied by the number of particles growing to give the total growth. Thus, it can be demonstrated that if G is a function of temperature alone, it can be calculated, independent of time, for any reaction path. The same is true for N. Fig. A3.1 Demonstrating the independence of G(T) N(T) with respect to time. 1 59 Appendix 4 CALCULATION OF tAV_TTT FROM tAV-CCT BY THE ADDITIVITY METHOD The additivity method has been used to calculate the t^_jjj for the steel used in the present work from the experimentally determined tA^_Q^j. By using a multiple re gression procedure, a first approximation for tA^_yjj was determined from the experimentally determined t^^-ryy data. This is shown in Table A4.1. By successive iterations, the best approximation for tA^_y^T to the experimental data was calculated by the additivity method. The results of the iterations are shown in Table A4.2. The comparison of the t^y JJJ values predicted by the additivity method and the experimental data is shown in Table /.3.;.S1 -1 60 Table A4.T Comparison of Experimental and First Approximation  of TAV-TTT Temperature *AV -TTT ( s) (°C) Experimental . First iii Approximation 680 43 58 670 5.6 12 660 6.2 6.3 650 3.0 4.0 630 1.8 2.1 623 1.6 2.0 615 1.5 1.9 603 1.9 1.7 161 Table A4.2 Iteration Results Approximation Number Co-efficient in the; Equation for t^_jjj* log a b c 1 40. 00 -10. 00 0.100 2 39 85 -10 39 0.087 3 45 75 -12 32 0.117 4 43 66 -11 60 0.103 5 44 12 -11 74 0.105 6 43. 86 -11 66 0.104 *log tAV_TTT =• log a + b log x + cx x = 728-T 162 Appendix 5 LISTING OF COMPUTER PROGRAM TO CALCULATE TTT DATA FROM CCT BY THE ADDITIVITY METHOD 162a 1 C ***************************.***************************** 2 C This program calculates the additivity integral value 3 , C upto a time t(AV-CCT)for a given cooling rate.The first 4 C approximation for t(AV-TTT) is to be input by replacing 5 C the values in statement no.34.The cooling rate is to be 6 C specified by replacing the value in statement no.23 by an 7 C appropriate temperature value.The t(AV-CCT) for the steel 8 C must be input by replacing the values in statement no.24. 9 C The time step value is specified in statement no.29 and the 10 C T(A1) temperature is specified by statement no.30.The program 11 C calculates the additivity integral value for the given 12 C cooling rate and prepares the input data for the next 13 C iteration.These values are then used as input to the 14 C multiple regression package available in the general MTS 15 C system as *STRP.The *STRP program gives the next 16 C approximation for the t(AV-TTT).Thais is input to this .17 C program for the next run.Thus the program must be run 18 C repetitively,along with *STRP,till the required t(AV-TTT) 19 C is obtained. 20 C ************************************************************** 21 IMPLICIT REAL*8(A-H,0-Z) 22 DIMENSION A(100),B(100),C2(100) 23 A1=600.0D0 24- TAVCCT=41.4594D0+0.0698528D0*(728.0D0-A1)-l0.096lD0* 25 1(DLOG(728.0D0-A1)) 26 TAVCCT=DEXP(TAVCCT27 CR=(728.0D0-A1)/TAVCCT 28 C=0.0D0 29 DT=0.10D0 30 T=727.0D31 C1=1.0D0/CR 32 DO 100 1=1,30000 33 IF (T.LE.A1) GO TO 500 34 T1=40.48l7D0-l0.6052D0*DLOG(728.0D0-T)+0.088608 DO* 35 1(728.0D0-T) 36 T1=DEXP(T137 C=C1*(DT/T1)+C 38 100 T=T-DT 39 500 T=700.0D0 4 0 PRINT,C 41 DO 10 1=1,30 42 T2=40.478 D0-10.6132 D0*DLOG(728.ODO-T)+0.0889042 DO* 43 1(728.0D0-T) 44 T2=DEXP(T245 IF (T.GE.A1) T2=T2*C 46 A(I)=DLOG(T2) 47 B(I)=728.0D0-T 48 C2(I)=DLOG(B(I)) 49 WRITE(6,200) A(I),B(I),C2(I) 50 200 FORMAT(F20.10,F20.10,F20.10) 51 10 T=T-5.0D0 52 STOP 53 END End of file 1 63 Appendix 6 DERIVATION OF IMPLICIT FINITE DIFFERENCE EQUATIONS i) Node Arrangement .Central node Surface node Genera! internal node ii) Heat Flow Equations a) Central Node ,,1 I ID Heat flow across A'B'C'D' is: 164 AR T. i ,i+l Ae . 1 i+1 , h i ,ri AR Rate of heat accumulation is nr Ti »n+I " Ti,n 1 ,AR*2 , pCp ~At 2 (.T) Ae 1 i ,1+1 Ti,h + 1„ " Ti,n C AR*" 8At 1 'n + 1  'n 'l.n V C AR' T, „ V + p—2-4Ki,i+lAt }" Ti+l,n C A R p • T. +. 4K. i+1At ^ The rate of latent heat generated due to transforma tion is: „ AF 1. / ARx H'pAt 2 (T) CAR' T, . II +P-2— i,n £ 4Kiji = 1AtJ i+lvo 165 b) General Internal Node Inflow across AD is i-1 ,n i ,n jR _ AJR 2 1 2 1 T- - T. , •'LAS !^1*H J AR Outflow across BC is: Ki+l,n + Ki,n + M| Ae i + l ,n i ,n AR Rate of accumulation is: Ti,n + 1 ~ Ti,n R M AR P At Rate of outflow - Rate of Inflow Rate of accumulation + Rate of generation. Rate of heat generation (due to latent heat liberation during transformation) is: Hp R AG AR i-l yn i-l,i 0 Ti,n [|Ki+l,i 2R + AR + K 2R_ i+1 -1 CnRAR?1 At J jf „ AR + 2R 1 ,nY S-+1..1 2 J C RAR AP = Ti n+1 P-2 + HP — RAR 1,n ' t At Surface Node Inflow across AB is: Outflow across CD is: '[T.-Ti.»] RQAB Rate of heat generation due to latent heat of transforma tion i s : U AF D ARI , AR qA-p p AT R0 - TrAe,1-T Rate of heat accumulation is T ~ i,n + l Ti,n D AR .n , AR pCp —* u >- RQ - -j- te-l.-j-T K AR - 2R + i-l,n 2AR T. K • , , 2R.," AR •+ hRn i»n i-l5i 2 AR 0 ;CpAR (4R.Q -AR) 8At hR0Ta + ARHp H (4R0 " AR> + C AR p —P_ (4R - AR) .T 8At U . 1,n 168 Appendix 7 COMPARISON OF MODEL PREDICTION WITH ANALYTICAL SOLUTION OF EQ. (4.1) For a body with negligible internal resistance, Eq. (4.1) can be re-written as: VpC = hA(T-T ) (A7.1) p dt a hA t where ^1 = e ' PVGP (A7.2) G0 e(t) = T(t) - TS 0n TN - T 0 0 00 T(t) = temperature of body at any time t T^ = temperature of body at time = 00 TQ = initial temperature of the body h = convective heat-transfer co-efficient A = surface area p = density V = volume C = specific heat P t = time 169 Using the above equation, the values of were calculated for rods of dia 5.5, 8.5, 13.5 and 25 mm with a value of 2 h = 250 W/m °C. The model predictions using the same h value were calculated. The values of T^ predicted by the model for the surface of the rod were then compared with the analytical solution values. A sample computer output show ing the comparison is shown in Fig. A?.1. For a 5.5 mm dia rod, the maximum difference between the two solutions is less than 1% in the pre-transformation period, the region where the comparison is meaningful. Summary of Comparison Results (All comparisons made before the Start of the Trans formation .) Rod diameter (mm) Maximum difference between model predicted and analytical solution values of temperature {%) 5.5 1.0 8.5 0.9 13.5 1.4 25 2.6 COMPARISON OF ANALYTICAL AND CALCULATED VALUES OF TEMPERATURE FOR 0.8%C STEEL ROD *********************************************** DIAMETER OF ROD= 5.5 MM TIME(SECONDS) % DIFFERENCE ********************************** 0. 10 0. 395115344578634 0. 20 0.505964762365217 0. 30 0. 550808718128153 0. 40 0. 569784433443964 0. 50 0. 577463027588642 0. 60 0. 57831 1-603873660 0. 70 0. 576850442366463 0. 80 0. 574224223074626 0. 90 - :- 0. 569262*48676523 1. 00 •0. 566578789140192 1. 10 0. 561-543548630365 1. 20 0. 555303321342505 1. 30 0. 550182225960195 1. 40 0. 543847722637660 1. 50 0. 537470009879105 1. 60 0. 531036110455511 1. 70 0. 524558328991617 1. 80 0. 516841917661739 1 90 0. 510258825212505 2 00 0 502428403952435 2 10 0 493338134677705 - 2 20 - = .:o 486594560740012 - 2 30 0 -477376003093878 2 40 0 468094394994059 2 50 0 458737923146185 2 60 0 449317476984690 2 70 :: 0 438601864670495 -2 80 0 42780748663V018 -2 .90 • - : . -0 418172622645464 3 .00 : 0 407223576883559 3 .10 0 .396199465938024 - -3 .20 : = ."0 .383843670941287 3 .30 • ~ 6 372653719659355 ^ 3 .40 r -: "6 .360127487640224 - 3 .50 - 0 .348774253660875 -3 .60 0 .336070751737211 - 3 .70 0 .322008113590259 - --3 .80 0 .309120976974933 - -3 .90 o 1296141228730804 4 .00 • :0 ;281787557798469 4 .10 '- % ' 6 ;267331568259515 4 .20 0 .252772762445194 - -4 .30 0 .239407821518229 ; .4 .40 •"" "0 :223347647192569 : 4 .50 --"••":•£) =208479920777633 4 .60 0 .192199573761493 4 .70 0 .175804568294831 4 .80 0 .160612962255957 4 .90 0 .143991334099522 5 .00 0 .127255133286501 5 .10 0 . 1 10401702580471 5 .20 0 .093430501054428 5 .30 0 .076340989754242 5 .40 0 .057788708355173 169b 5. 50 0. 040456893180318' 5. 60 0. 021651172546309 5. 70 0. 002715730871076 5. 80 -0. 014988009592328 5. 90 -0. 032814230437934 6. 00 -0. 052134782131489 6. 10 -0. 071588859672065 6. 20 -0.089795128959617 6. 30 -0. 109512323601988 6. 40 -0. 129364306579489 • 6. 50 -0. 149351646357633 6. 60 -0. 169477281640425 6. 70 -0. 189737319222494 6. 80 -0. 210137364962064 6. 90 -0. 229263667369554 7. 00 -0. 248516004430685 7. 10 -0. 270747834017325 7. 20 -0. 290273686618808 7. 30 -0. 309930696052692 7 . 40 -0. 329719378572555 7. 50 -0. 351085040598724 7. ,60 -0. ,369699166364627 7. ,70 -0. ,391341034071401 7. ,80 -0. ,41 1672822294564 7 . ,90 -0. , 430674577016040 B. ,00 -0. ,451271497868586 8. ,10 -0. ,470536175234161 8. .20 -0. ,489920369438442 8, .30 -0. .509438726254661 8. .40 -0. . 529084757687836 8, .50 -0, . 547363383483388 6 .60 -0, .565768740151578 8 .70 -0, .587287484753110 6 .80 -0, .607434270172263 6 .90 -0 .629216890304000 9 .00 -0 .655682838148267 9 .10 -0 .683823304942449 9 .20 -0 .718223583460949 9 .30 -0 .760462115160079 9 .40 -0 .810602558175372 9 .50 -0 .873288200562619 9 .60 -0 .9517019034038,12. 9 .70 -1 .04B983651395537 9 .80 -1 . 171394301914426 9 .90 -1 .322094747522043 10 .00 -1 .505746227816433 10 .10 -1 .723840987020306 10 .20 -1 .977839422006990 10 .30 -2 .270624578570190 10 .40 -2 .595793403077311 10 .50 -2 .949940405244339 10 .60 -3 .326696662329319 10 .70 -3 .722737528202555 10 .80 -4 .131765590017377 10 .90 -4 .544530137822069 1 1 .00 -4 .961002829053284 1 1 .10 -5 .375185445459516 1 1 .20 -5 .784053674520205 1 1 .30 -6 .186274204109633 1 1 .40 -6 .583374875373876 Appendix 8 LISTING OF COMPUTER PROGRAM TO CALCULATE TEMPERATURE RESPONSE OF A STEEL ROD UNDER GOING COOLING. Steel: 0.82% C - 0.82% Mn - 0.26% Si (Plain carbon eutectoid) Grain Size: 5-7 ASTM Austenitising Conditions: 850°C - 5 minutes 170a j Q ****************************************************************** 2 C This program calculates the temperature (in *C) inside a 3 C cylindrical rod of 0.82%C-0.82%Mn-0.26%Si steel undergoing 4 C cooling.lt takes .into account the effect of the latent heat 5 C of transformation of austenite to pearlite generated,during 6 C cooling,on the temperature of the rod.Calculations are 7 C based on a 1-D Implicit Finite Difference Unsteady State 8 C Heat Transfer Model.The model has been written for a constant 9 C node distance.Density is considered constant.Variations of Thermal-10 C Conductivity and Specific Heat have been incorporated into the 11 C model by using BISRA data. J2 C *************************************************************: 13 14 15 16 17 IMPLICIT REAL*8 (A-H,0-Z) 18 REAL*8 KK 19 20 21 22 Q ****************************************************** 23 C Data for this program is:. 24 C ****************************** 25 C DX = Node distance,Meters 26 C DT  Time increment,Seconds 27 C T0TTIM = Total time counter,Seconds(Initial Value=0) 28 C H = Convective Heat transfer coefficient at the rod 29 C surface,W/m *C 30 C R = Radius of rod,Meters 31 C TATMOS = Atmospheric temperature at rod surface,*C 32 C XX = Distance of node from the rod centre,Meters 33 C M  Number of nodes+1 34 C DD = Density of steel,Kg/cubic Meter 35 C T1  Maximum time upto which calculations are to 36 C be done 37 C FRMAX = Maximum Fraction transformed after which 38 C check for transformation at the node is 39 . C terminated. 40 Q ************************************************************ 41 42 43 44 C ********************************************************** 4 5 C DATA FOR THIS PROGRAM IS ENTERED BY INPUTTING APPROPRIATE .46 C VALUES OF THE VARIABLES IN THE STATEMENT NO.62.IF THE 4 7 C PROGRAM IS TO BE RUN WITH A CONSTANT HEAT TRANSFER 48 C COEFFICIENT,THEN INSERT THE FOLLOWING AFTER THE 4 9 C STATEMENT NO.167: 50 C GO TO 220 51 C INSERT AFTER STATEMENT NO.221 52 C H=PRESELECTED VALUE. 53 C BY THE ABOVE PROCEDURE THE CALCULATION OF THE HEAT 54 C TRANSFER COEFFICIENT FOR A GIVEN AIR VELOCITY,EMI SSIVITY, 55 C ETC. IS BYPASSED.(THE PROGRAM DOES NOT EXECUTE STATEMENT 56 C NOS.175 TO 221) 57 Q ***************************************************** 58 59 60 170b 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 ' 101 1 02 103 104 105 106 107 108 109 110 1 1 1 112 113 1 14 115 1 16 1 17 118 119 120 C C C C C C C C C C C C C C C C C C c c c c c c c c c c c c c c c c c c c c c c c c c c c c DATA DX,DT,T1 ,DD,TATMOS, R , XX , M ,FRMAX/ 0.0075D0,0.1 DO,100.0D0, 17 650.0D0,20.0D0,0.015D0,0.0D0,11,0.99D0/ VEL=20.0D0 EMISS=0.3D0 TOTTIM=0.0D0 J1=0 K9=0 K1=0 ************************************** Matrix Identification AA , BB , C D T KK,CP THERM 1,THERM2 CP1,CP2 DF AN,ALB 33 TAVRAM BETA,GAMMA Contain coefficients of the Tridiagonal system at each time step Contains the RHS of the tridiagonal system Contains the temperature of each node at each time step Contain Thermal Conductivity and Specific Heat of each node at each time step Contain coefficients of the Polynomial used to calculate Thermal Conductivity as a function of temperature for Ferrite and Austenite respectively Contain coefficients of the polynomial used to calculate Specific Heat as a function of temperature for Austenite and Pearlite resply Contains total fraction transformed for each node at each time step Contains the incremental fraction transformed for each node at each time step Contains the coefficients of the Polynomial used to calculate 'n' and 'Log b' as a .function of temperature.The data for these was generated at the Metallurgy Dept.of UBC. Contains the information on whether a node has started transformation.If JJ(i)=i, then the 'i'th node has started transforming.If_ not,JJ(i)=0 Contains the Avrami time for each node at each time step.The data for this has been generated at the Metallurgy Dept. of UBC Are dummy matrices used for calculating the temperature of each node at each time step ******************************************* * UNITS * ************* Time Distance Specific Heat -Thermal Conductivity Convection Heat Transfer Coefficient Densi ty Note on Dimensioning ***** of Matrices Seconds Meters W/Kg *C W/M *C W/m2 *C Kg/m3 C Matrices AA,BB,C,D,T,TT,DF,KK,CP,JJ,F,TAVRAM,BETA,GAMMA should C be dimensioned at least M (=no. of nodes+i) C 170c 121 1 22 123 1 24 125 DIMENSION AA(30),BB(30),C(30),D(30),T(30),TT(30),DF(30), 126 1KK(30),CP(30),ALB(l0),AN(5),THERM 1(10),THERM2(10),F(50,2),JJ(31 127 1 CP 1(10),CP2(10),TAVRAM(30),H1(20),BETA(30),GAMMA(30) 128 DIMENSION K5(100),TAV(100),TA1(100),TI(100) 129 CALL THER(THERM1) 130 CALL THERMA(THERM2) 131 CALL CEEPE2(CP2) 132 CALL CEEPE1(CP1) 133 CALL LOGB(ALB) 134 CALL EXPONT(AN) 136 DO 3001 1=1,M 137 3001 K5(I)=0 138 139 140 C ******************************************************** 141 C Initialisation.of Temperature at all nodes at TOTTIM=0.The 142 C fraction transformed matrix F and the JJ matrix are also 143 C initialised to 0. 144 C ******************************************************************' 145 146 147 148 DO 10 1=1,M 149 F(I,1)=0.0D0 150 F(I,2)=0.0D151 JJ(I)=0 152 10 T(I)=850.0D0 153 154 155 156 C ************************************************************* 157 C Starting with TOTTIM=0 the time is incremented in steps of 158 C DT.The calculations will stop when TOTTIM value is Greater 159 C than or Equal to a prespecified value 160 C *********************************************************** 161 162 163 164 165 25 TOTTIM=TOTTIM+DT 166 IF (TOTTIM.GE.T1) GO TO 1001 167 K7=0 168 169 170 171 C ******************************************************** 172 C Calculation of convective heat transfer coefficient by 173 C using air velocity 175 TEM=((T(M)+TATMOS)/2.0D0)* 1.8DO+32.0DO 176 IF (TEM.GT.900.0D0) AIRK=(1.617D-05*TEM+1.575D-02)/2419.0D0 177 F (TEM.LE.900.0D0) AlRK=(1.860D-05*TEM+1.372D-02)/2419.0D0 178 IF (VEL.LE.(0.1D-8).AND.VEL.GE.(-0.1D-8)) GOTO 130 179 F (TEM.GT.1000.0D0) AlRNU=(1.233D-06*TEM-3.060D-04)*9.2894D-0; 180 IF (TEM.LE.1000.0D0.AND.TEM.GT.800.0D0) 181 1AIRNU=(1.0D-06*TEM-8.3D-05)*9.2894D-02 170d 182 163 1 84 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 21 1 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 IF (TEM.LE.800.0D0) AIRNU=(6.475D-07*TEM+3 . 9D- 05)*9.2894D-02 RE= (VEL*2.0D0*R)/AIRNU IF (RE.GE.0.4D0.AND.RE.LT.4.0D0) VC = 0. 891D0 IF (RE.GT.0.4D0.AND.RE.LT.4.0D0) VN = 0. 33D0 IF (RE.GT.4.0D0.AND.RE.LT.4 0.0D0) VC = 0. 821 DO IF (RE.GT.4.0D0.AND.RE.LT.4 0.0D0) VN = 0. 385D0 IF (RE.GT.40.0D0.AND.RE.LT.4 0 00.0D0) VC = 0. 61 5D0 IF (RE.GT.4 0.0D0.AND.RE.LT.4 0 0 0.0D0) VN= 0. 466D0 IF (RE.GT.4000.ODO.AND.RE.LT.40000.0D0) VC = 0. 174D0 IF (RE.GT.4000.ODO.AND.RE.LT.40000.0D0) VN = 0. 618D0 IF (RE.GT.40000.ODO.AND.RE.LT.400000.ODO) VC = 0. 0239D0 IF (RE.GT.40000.ODO.AND.RE.LT.400000.0D0) VN= 0. 805D0 H=(VC*AIRK/(2.0D0*R) )*(RE**VN) + 1 .366D-1 1 *EMISS* ('( (T(M) 1+273.0D0)**4)-((TATMOS+273.0D0)**4))/(T(M)-TATMOS) GO TO 230 Q *************************************************** C Calculation of the radiative heat transfer coefficient Q *********************************************************** 130 D2=(T(M)-TATMOS)*1.8D0 IF (TEM.GT.10O0.OD0) GR=(335.29345D0*(10.0D0**(TEM / *(-0.00110218D0))))*1000.0D0 IF (TEM.LE.1000.0D0.AND.TEM.GT.800.0D0) GR=(621.10241 DO *(10.0DO**(TEM*(-0.00136992DO))))*1000.0D0 IF (TEM.LE.800.0D0.AND.TEM.GT.500.0D0) GR=(1100.7197D0 *(10.DO**(TEM*(-0.00168056DO))))*1000.0D0 IF (TEM.LE.500.0D0.AND.TEM.GT.300.ODO) GR=(2071.8664D0 *(10.D0**(TEM*(-0.00222993D0))))*1000.0D0 IF (TEM.LE.300.ODO) GR=(4200.0D0*(10.DO**(TEM*(-0.00325289D0) )))*1000.0D0 D1=(2.0DO*R)*3.2808399DO NUS=0.53D0*((GR*(D1**3.0D0)*D2*0.7D0)**0.25D0) H=((AIRK*2419.0D0*NUS/D1)/737.3D0)+ 1.366D-11*EMISS*(((T(M)+273.ODO)**4 ) -((TATMOS+273.0D0)**4))/(T(M)-TATMOS) 230 H=H*1000.ODO H=4.18*D0*H-220 H=50.0D0 Q ************************************************************** C Loop 60 C ******* C In this loop ,the values of Thermal Conductivity and Specific C Heat for Austenite are calculated for each node at each time C step C Loop 70 Q ******* C In this loop,values of Thermal Conductivity and Specific C Heat of Ferrite are calculated for each node at each C time step. C Loop 55 C ******* C In this loop the values of Fraction Transformed for each node C at each time step are checked to find whether transformation C of Austenite to Pearlite is complete at all nodes.If so, C control is directed to Loop 70 wherein the values of C Thermal Conductivity and Specific Heat are calculated.On 170e 242 C entering Loop 70 .a counter J1 is set equal to l.For all 243 C future time steps control is directed to Loop 70 without 244 C going through Loop 55.On complete transformation at all 245 C nodes, from Loop 70 control is directed to statement no.820 246 C wherein the appropriate Tridiagonal System coefficients 247 C are calculated,bypassing Loop 501. 248 C **************************************************** 249 250 251 252 IF (J1.EQ.1) GO TO 70 253 L=0 254 DO 55 1=1,M 255 IF (F(I,1).LT.FRMAX) GO TO 55 256 L=L+1 2 57 55 CONTINUE 258 IF (L.EQ.M) GO TO 70 259 DO 60 1 = 1 ,M 260 KK(I)=THERM2(1)+THERM2(2)*T(I) 261 KK(I)=4 18.6D0*KK(I) 262 CP(I)=CP1(1)+CP1(2)*T(I)+CP1(3)*(T(I)**2)+CP1(4)*(T(I)**3) 263 1+CP1(5)*(T(I)**4)+CP1(6)*(T(I)**5) 264 CP(I)=4186.0D0*CP(I) 265 60 CONTINUE 266 GO TO 700 267 70 DO 600 1 = 1 ,M 268 KK(I)=THERM 1{1)+THERM1(2)*T(I)+THERM1(3)*(T(I)**2 ) 269 1+THERM1(4)*(T(I)**3)+THERM1(5)*(T(I)**4) 270 KK(I)=KK(I)*418.6D0 271 CP(I)=CP2(1)+CP2(2)*T(I)+CP2(3)*(T(I)**2) 272 CP(I)=CP(I)*4186.0D0 273 600 CONTINUE 274 J1=1 275 GO TO 820 27 6 C ************************************************************** 277 278 279 • 280 C'Loop 501 281 C ********* 282 C In this loop,each node is checked,at each time step for 283 C transformation start.At each time step,for each node, the 284 C Avrami .time is calculated.If the TOTTIM value is Greater 2B5 C than or Equal to the Avrami time the node will start 286 C transforming.Once a node starts transforming,JJ(node) is 287 C set equal to node number.For all future time steps this 288 C node will not be checked again for transformation start. 289 C When a node starts transforming,control is directed to 290 C statement no.500 wherein 'n','log b' for that node 291 C temperature are calculated.The fraction transformed is 292 C then calculated.The Specific Heat and Thermal Conductivity 293 C of the transforming node is then calculated by using the 294 C formula 295 C Specific Heat=%Transformed*Specific Heat of Ferrite + 296 C If 1-%transformed)*Specific Heat of Austenite 297 C at the node temperature 298 C When the fraction transformed of the node is equal to 1 299 C control is transferred to statement no. 8000 where the 300 C Specific Heat and Thermal Conductivity values of Ferrite 301 C are used for further calculations. 170f 302 C ************************************************** 303 304 305 306 700 DO 501 1=1,M 307 IF(T(I).GT.728.0D0) GO TO 501 308 IF (F(I,1).GE.FRMAX) GO TO 501 309 F (JJ(I).NE.O) GO TO 500 310 IF(K5(I).EQ.1) GO TO 229 311 TAV(I)=TOTTIM 312 K5(I)=1 313 229 IF (T(I).GE.700.0D0) GO TO 501 314 TAVRAM(I)=62.7348D0+0.105339D0*(728.0D0-T(I))-15.4 325D0 315 1*(DLOG(728.0D0-T(I))) 316 TAVRAM(I)=DEXP(TAVRAM(I)) 317, TA1(I)=TOTTIM-TAV(I) 318 IF (TA1(I).LT.TAVRAM(I)) GO TO 501 319 JJ(I)=I 320 500 EN=AN(1)+AN(2)*T(I)+AN(3)*(T(I)**2)+AN(4 ) * (T (I)**3) 321 TT(I)=728.0D0-T(I) 322 ALOGB=ALB(1)+ALB(2)*(TT(I))+ALB(3)*((TT(I))**2 ) + 323 1ALB(4)*((TT(I))**3) 3 24 ALOGB=DEXP(ALOGB325 THETA=DT+(DLOG(1.0D0/(1.0D0-F(I,1)))/ALOGB)**(1.0D0/EN) 326 F(I,2)=1.0D0-DEXP(-ALOGB*(THETA**EN)) 327 DF(I)=F(I,2)-F(I,1) 328 FK=THERM1(1)+THERM1(2)*T(I)+THERM1(3)*(T(I)**2)+THERM1(4)* 329 1(T(I)**3)+THERM1(5)*(T(I)**4) 330 FK=FK*418.6D0 331 FCP=CP2(1)+CP2(2)*T(I)+CP2(3)*(T(I)**2) 332 FCP=FCP*4186.0D0 333 ' KK(I)=(F(I,2)*FK+(1.OD0-F(I,2))*KK(I)) 334 CP(I)=(F(I,2)*FCP+(1.0D0-F(I,2))*CP(I)335 501 CONTINUE 336 337 338 339 C *********************************************************** 340 C Loop 7000 341 Q ********* 342 C This loop calculates the appropriate values of the Tridiagonal 343 C System Coefficients-AA,BB,C,D 345 346 347 348 349 820 XX=-DX 350 DO 7000 1 = 1 ,M 351 XX=XX+DX 352 7050 IF(I.GT.I) GO TO 7051 353 AK1=(KK(1)+KK(2))/2.0D0 354 BB(1)=1.0D0+((DD*CP(1)*(DX**2))/(4.OD0*AK1*DT)) 355 C(1)=-1.0D0 356 IF ((F(I,1).LE.0.0D0).OR. (F(I,1).GE.FRMAX)) GO TO 450 357 D(1)=((DX**2)/(4.0D0*AK1))* 358 1(DD*80000.0D0*DF(1)/DT)+((DD*CP(1)*(DX**2)*T(I))/ 359 1(4.0D0*AK1*DT)) 360 GO TO 7000 361 • 450 D(1)=(BB(1)-1.0D0)*T(1) 170g 362 GO TO 7000 363 705! IF (I.EQ.M) GO TO 7052 364 AK2=(KK(I-1)+KK(I))/(2.0D0) 365 AA(I)=AK2*((DX-2.0D0*XX)/(2.0D0*DX)) 3 66 AK3=AK2*(2.0D0*XX-DX)/(2.0D0*DX) 367 AK4=(KK(I)+KK(I+1))/(2.0D0) 368 AK4=AK4*(2.0D0*XX+DX)/(2.0D0*DX) 369 BB(I)=AK3+AK4+((DD*CP(I)*XX*DX)/(DT)) 370 AK4=(KK'(I )+KK(I + 1 ) )/(2.0D0) 371 C(I)=-AK4*((2.0D0*XX+DX)/(2.0D0*DX)) 372 IF ((F(I,1).LE.0.0D0).OR.(F(I,1).GE.FRMAX)) GO TO 451 373 D(I)=(XX*DX*(DD*80000.0D0*DF(I)/DT))+(DD*CP(I)*XX*DX*T(I)/DT) 374 GO TO 7000 375 451 D(I)=(DD*CP(I)*XX*DX*T(I))/DT 376 GO TO 7000 377 7052 IF ((F(I,1).LE.0.0D0).OR.(F(I,1).GE.FRMAX)) GO TO 350 378 D(M)=((DX)*(4.0D0*R-DX)* 37 9 1(80000.0D0*DD*DF(M)/DT))/(8.0D0)+(H*TATMOS*R)+ 380 1((DD*CP(M)*DX*(4.0D0*R-DX)*T(M))/(8.0D0*DT)) 381 GO TO 351 382 350 D(M)=(H*R*TATMOS)+((DD*CP(M)*DX*(4.0D0*R-DX)*T(M))/ 383 1(8.0D0*DT)) 384 351 AK5=(KK(M-1)+KK(M))/(2.0D0) 385 AA(M)=AK5*((DX-2.0D0*R)/(2.0D0*DX)) 386 BB(M)= AK5*((2.0D0*R-DX)/(2.0D0*DX))+(H*R)+((DD*CP(M)*DX* 387 1(4.0D0*R-DX))/(8.0D0*DT)) 388 7000 CONTINUE 389 390 391 Q *************************************************** 392 C This part of the program calculates the temperature of each node 393 C at each time step.The algorithm used is the solution of a 394 C Tridiagonal System of Simultaneous Equations described in the 395 C book 'Applied Numerical Methods' by Carnahan,Luther and Wilkes. 396 C T(M) is the temperature of the surface node and T(l) is the 397 C temperature of the Centre of the rod.After the calculations of 398 C temperature for one time step are completed control is 399 C transferred to statement no. 25 where the TOTTIM is incremented 400 C by DT and the calculation of the Tridiagonal System coefficients 401 C etc. is repeated. 4 02 C ****************************************************************** 403 404 405 406 407 BETA(1)=BB(1) 408 GAMMA(1)=D(1)/BETA(1) 409 DO 110 1=2,M 410 BETA(I)=BB(I)-AA(I)*C(I- 1)/BETA(I-1) 411 110 GAMMA(I)=(D(I)-AA(I)*GAMMA(1-1))/BETA(I) 412 TI(M)=GAMMA(M) 413 MAST=M-1 414 DO 120 J=1,MAST 4 15 I=M-J 416 120 TI(I)=GAMMA(I)-C(I)*TI(1+1)/BETA(I) 417 418 419 420 C ************************************************************* 421 C If transformation starts at any node,the latent heat 170h 422 C liberated due to the transformation is calculated by an 423 C iterative procedure.K7 is controls the number of iterations 424 C to be performed. 4 2 5 C ************************************************ 426 427 428 IF (K7.GE.3) GO TO 6002 429 F (JJ(M).EQ.O) GO TO 6002 430 IF (J1.EQ.1) GO TO 6002 431 DO 6003 11=1,M 432 Tl(II)=(TI(II)+T(II))/2.0D0 433 IF (JJ(II).EQ.0) GO TO 6600 434 F (F(II,2).GE.FRMAX) GO TO 6005 435 EN=AN(1)+AN(2)*TI(11)+AN(3)*(Tl(11)**2)+AN(4)*(Tl(11 )**3) 436 TT(II)=728.0D0-T(II) 437 ALOGB=ALB(1)+ALB(2)*(TT(11))+ALB(3)*((TT(11))**2) + 438 1ALB(4)*((TT(II))**3) 439 ALOGB=DEXP(ALOGB) 440 THETA=DT+(DLOG(1.0D0/(1.0D0-F(II,1)))/ALOGB)**(1.0D0/EN) 441 F(II,2)=1.0D0-DEXP(-ALOGB*(THETA**EN)) 442 DF(II)=F(II,2)-F(II,1) 44 3 6005 FK=THERM1(1)+THERM1(2)*T(II)+THERM1(3)*(T(II)**2)+THERM1(4)* 44 4 1(T(II)**3)+THERM1(5)*(T(II)**4) 445 FK=FK*418.6D0 4 46 FCP=CP2(1)+CP2(2)*T(II)+CP2(3)*(T(11)**2) 447 FCP=FCP*4186.0D0 4 48 KK(II) = (F(II,2)*FK+(1.0D0-F(11,2))*KK(11)) 44 9 CP(II) = (F(II,2)*FCP+(1.0D0-F(11,2))*CP(11)) 450 GO TO 6003 451 6600 KK(II)=THERM2(1)+THERM2(2)*TI(II) 452 KK(II)=418.6D0*KK(II) 4 53 CP(II )=CP1 (1 )+CP1 (2)*T(II")+CPr(3)*(T(II )**2)+CP1 (4)*(T(II )**3) 454 1+CP1(5)*(T(II)**4)+CP1(6)*(T(11)**5) 455 CP(II)=4186.0D0*CP(II) 456 6003 CONTINUE 457 K7=K7+1 458 GO TO 820 459 6002 DO 6004 I 1 = 1,M 460 F(I1,1)=F(I1,2) 461 6004 T(I1)=TI(I1) 462 K9=K9+1 463 IF (K9.LE.9) GOTO 325 464 K9=0 465 WRITE(6,730) TOTTIM,T(M),T(1) 466 730 FORMAT(5X,F8.2,5X,F6.2,5X,F6.2) 467 325 GO TO 25 468 469 470 471 c. ***************************************************************** 472 C Calculation of temperature for all nodes for the current time 473 C step is complete.Control' is now transferred to statement no.25 474 C for incrementing the TOTTIM value by DT and further calculations. 475 Q ****************************************************************** 476 477 478 479 1001 STOP 480 END 481 Ti70i 482 £83 C ************************************************** 464 C End of main program.Start of subroutines. 4 g 5 Q ************************************************************ 486 487 488 SUBROUTINE THER(THERM 1) 489 490 491 C ******************************************************************* 492 C In this subroutine a Polynomial of the 4th degree is fitted to 493 C Thermal Conductivity values of Ferrite in the temperature range 494 C 50*C to 750*C,the data for which has been obtained from the 495 C BISRA report.This Polynomial is the best fit for the data used 496 C and calculates Thermal Conductivity values in the temperature 497 C range within 0.5% of the experimental values. 498 C ******************************************************************* 499 500 501 502 IMPLICIT REAL*8 (A-H,0-Z) 503 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20), 504 1SIGMA(20),P(2 0),THERM1(10) 505 DATA K,N/4,14/ 506 X(l)=50.0D0 507 DO 3300 1=2,14 508 3300 X(I)=X(I-1J+50.0D0 509 DATA (Yd),1=1,14)/0.118D0,0.115D0.0.112D0,0.108D0,0.103D0, 510 10.099D0,0.096D0,0.091 DO,0.087D0, 0.084D0,0.081 DO,0.078D0, 511 10.075D0,0.072D0/ 512 LOGICAL LK 513 LK= .TRUE. 514 NWT=0 515 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 516 DO 3400 1=1,5 517 3400 THERM 1(I)=P(I) 518 RETURN 5 1 9 END 520 C ******************************************************** 521 522 523 524 SUBROUTINE THERMA(THERM2) 525 C ******************************************************************* 526 C In this subroutine a Polynomial of 1st degree is fitted to the 527 C Thermal Conductivity data of Austenite.The prediction error is 528 C less than 0.8%. 529 C ******************************************************************* 530 531 532 533 IMPLICIT REAL*8(A-H,0-Z) 534 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 535 1,SIGMA(20),P(20) 536 1,THERM2(4) 537 DATA K,N/1,11/ 538 X(1)=700.0DO 539 DO 2100 1=2,11 540 2100 X(I)=X(I-1)+50.0DO 541 DATA (Y(I),1 = 1 ,11)/0.053D0,0.055D0,0.057D0,0. 059D0,0.061 DO, 170j b42 1U.U63DU, 54 3 10.064D0,0.066D0,0.068D0,0.07ODO,0.07 2D0/ 54 4 LOGICAL LK 54 5 LK= .TRUE. 54 6 NWT=0 54 7 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 548 DO 2110 1=1,2 54 9 2110 THERM2(I)=P(I) 550 RETURN • 551 END 552 C ***************************************************** 553 554 555 SUBROUTINE CEEPE2(CP2) 55g Q ************************************************* 557 C In this subroutine a Polynomial of 2nd degree is fitted to.the 558 C Specific Heat data.of Ferrite.The prediction error is less than '559 C 1.0%. 5£Q Q ********************************************* 561 562 563 564 IMPLICIT REAL*8(A-H,0-Z) 565 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20) , SIGMA (20), 566 1A(20),B(20),P(20),CP2(3) 567 DATA K,N/2,13/ 568 X(1)=75.0D0 569 DO 4000 1= 2,13 570 4000 X(I)=X(I-1)+50.0D0 571 DATA (Y(I),1=1,13)/0.117D0,0.124D0,0.127D0,0.131D0,0.135D0, 572 10.140D0,0.14 5D0,0.150D0,0.160D0,0.166D0,0.172D0,0.172D0, 573 10.184D0/ 574 LOGICAL LK 575 LK= .TRUE. 576 NWT=0 577 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 578 DO 4010 1=1,3 579 4010 CP2(I)=P(I) 580 RETURN 581 END 582 C ******************************************************* 583 584 585 586 SUBROUTINE CEEPE1(CP1) 587 C *********************************************************** 588 C In this subroutine a Polynomial of 5th degree is fitted to 589 C the Specific Heat values of Austenite.The prediction error 590 C is less then 0.9%. 55^ " Q ************************************************************** 592 593 594 595 IMPLICIT REAL*8(A-H,0-Z) 596 DIMENSION X(25),Y(25),YF(25),YD(25) , S(20),WT(25),SIGMA(20), 597 1A(20),B(20),P(20),CP1(10) 598 DATA K,N/5,13/ 599 X(1)=675.0D0 600 DO 5000 1=2,13 601 5000 X(I)=X(I-1)+50.0D0 1i70k 60 2 DATA (Y(I),1=1,13)/0.139D0,0.141D0,0.14 3D0,0.14 5D0,0.14 8D0, 60 3 1 0. 1 4 9D0,0.151 DO,0.154D0,0.156D0,0.158D0,0.160D0,0.162D0, 604 10.162D0/ 605 LOGICAL LK 606 LK= .TRUE. 607 NWT=0 608 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 609 DO 5010 1=1,6 610 5010 CPl(I)=P(I) 6 1 1 RETURN 612 END 613 Q **************************************************** 614 615 616 SUBROUTINE LOGB(ALB) 617 Q ************************************************************* .618 C In this subroutine a polynomial of 6th degree is fitted to 619 C the data of "Log b' values obtained for 0.82%C-0.82%Mn-620 C 0.26%Si steel in the Department of Metallurgy at UBC. g2i C ************************************************************ 622 623 624 625 IMPLICIT REAL*8(A-H,0-Z) 626 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 627 1,SIGMA(20),P(20),ALB(10) 628 DATA K,N/3,8/ 62 9 DATA(X(I),I=1,8)/58.0D0,48.0D0,68.0D0,78.0D0,98.0D0,105.0D0, 630 1113.0D0,125.0D0/ 631 DATA(Y(I),1 = 1,8)/-9.8133D0,- 9.44464D0,-9.47115D0,-8.3 921 DO, 632 1-5.7279D0,-4.5l327D0,-3.1385D0,-2.04103D0/ 63 3 LOGICAL LK 634 LK= .TRUE. 635 NWT=0 636 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 637 DO 6100 1=1,4 638 6100 ALB(I)=P(I) 63 9 RETURN 64 0 END 541 Q ********************************************************** 642 643 644 64 5 SUBROUTINE EXPONT(AN) 646 C *************************************************************** 647 C In this subroutine a Polynomial of 1st degree is fitted to the 648 C data'of 'n' values obtained for 0.82%C-0.82%Mn-0.26%Si steel in 649 C the Department of Metallurgy at UBC. 650 C **************************************************************** 651 652 653 654 IMPLICIT REAL*8 (A-H.O-Z) 655 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 656 1,SIGMA(20),P(20),AN(5) 657 DATA K,N/3,8/ 658 DATA(X(I),1=1,8)/670.0D0,680.0D0,660.0D0,650.0D0,630.0D0, 659 1623.0D0,615.0D0,603.0D0/ 660 DATA(Y(I),1=1,8)/2.125109D0,1.618956D0,2.467 576D0,2.946133D0, 661 13.1 66861 DO,3.147945D0,2. 922434D0,2.346407D0/ 11701 662 LOGICAL LK 663 LK= .TRUE. 664 NWT=0 665 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 666 DO 6200 1=1,4 667 6200 AN(I)=P(I) 668 RETURN 669 END 670 C ************************************************** End of file Appendix 9 LISTING OF COMPUTER PROGRAM TO CALCULATE THE TEMPERATURE RESPONSE OF A CENTRE-SEGREGATED STEEL ROD UNDERGOING COOLING Matrix Steel: 0.82% C - 0.82% Mn - 0.26% Si Grain Size: 5-7 ASTM Segregated Steel: 0.8% C - 1.88% Mn Grain Size: 5-8 ASTM Austenitising Conditions: 850°C - 5 minutes 171a 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 1 0 c 1 1 c 1 2 c 1 3 c 1 4 c 1 5 c 1 6 c 1 7 c 18 c 19 c 20 21 22 23 24 25 26 27 28 29 c 30 c 31 c 32 c 33 c 34 c 35 c 36 c 37 c 38 c 39 c 40 c 41 c 42 c 43 c 44 c 45 46 47 48 49 50 c 51 c 52 c 53 c 54 c 55 c 56 c 57 c 58 c 59 60 ************************** *-* ************************************** This program calculates the temperature (in *C) inside a cylindrical rod of 0.82%C-0.82%Mn-0.26%Si steel with a centre segregation of composition' 0.8%C-1.68%Kn.It takes into account the effect of the transformational heat 1iberated,during cooling,on the temperature of the rod.It is assumed that the segregated region transforms to either pearlite or martensite,depending on the cooling rate.lt is further assumed that no heat is liberated during the austenite to martensite transformation. Calculations are ************************************************************** IMPLICIT REAL*8 (A-H.O-Z) REAL*8 KK ******************************************************** Data for this program is: ****************************** DX = Node distance,Meters DT  Time increment,Seconds TOTTIM = Total time counter,Seconds H = Convective Heat transfer coefficient at the rod surface,W/m *C R = Radius of rod,Meters TATMOS " Atmospheric temperature at rod surface,*C XX = Distance of node from the rod centre.Meters M  Number of nodes+1 DD = Density of steel,Kg/cubic Meter T1  Maximum time upto which calculations are to be done ************************************************************ Q ********************************************************* heat transfer coefficient value must be input by replacing statement no.63 ********************************************************** DATA DX,DT,TOTTIM,T1,DD,TATMOS,R,XX,M/0.00075D0,1.ODO,0.ODO, 171b 61 1400.0DO,7650.0DO,20.0DO,0.007 5DO,O.ODO,11/ 62 Ki=0 63 H=1200.0D0 64 IS=2 65 66 67 6g Q ************************************************** 69 C Matrix Identification 70 C *********************** 71 C AA,BB,C * Contain coefficients of the Tridiagonal 72 C system at each time step 73 C D * Contains the RHS of the tridiagonal system 74 C T  Contains the temperature of each node at 75 C each time step 76 C KK,CP * Contain Thermal Conductivity and Specific 77 C Heat of each node at each time step 78 C THERM 1,THERM2 * Contain coefficients of the Polynomial used 79 C to calculate Thermal Conductivity as a function 80 C of temperature for Ferrite and Austenite 81 C respectively 82 C CP1,CP2 * Contain coefficients of the polynomial used 83 C to calculate Specific Heat as a function of 84 C temperature for Austenite and Pearlite resply 85 C F * Contains total fraction transformed for each 86 C node at each time step 87 C DF * Contains the incremental fraction transformed 88 C for each node at each time step 89 C AN,ALB * Contains the coefficients of the Polynomial 90 C used to calculate 'n' and 'Log b' as a 91 C function of temperature.The data for these 92 C was generated at the Metallurgy Dept.of UBC. 93 C JO * Contains the information on whether a node 94 C has started transformation.If JJ(i)=i, then 95 C the 'i'th node has started transforming.If 96 C not,JJ(i)=0 97 ' C TAVRAM * Contains the Avrami time for each node at 98 C each time step.The data for this has been 99 C generated at the Metallurgy Dept. of UBC 100 C BETA,GAMMA * Are dummy matrices used for calculating the 101 C temperature of each node at each time step 102 C ************************************************************** 103 C * UNITS * 104 C ************* 105 C Time * Seconds 106 C Distance * Meter107 C Specific Heat * W/Kg *C 108 C Thermal Conductivity ' * W/M *C 109 C Convection Heat 110 C Transfer Coefficient * W/m2 *C 111 C Density * Kg/m3 112 C Note on Dimensioning of Matrices 1)3 rj ******************************** 114 C Matrices AA,BB,C,D,T,TT,DF,KK,CP,JJ,F,TAVRAM,BETA,GAMMA should 115 C be dimensioned at least M (=no. of nodes+1) 1 16 £ ******************************************************************* 1 1 7 118 1 19 120 17tic 1 2 1 2 122 179 C In this loop,values of Thermal Conductivity and Specific DIMENSION AA(30),BB(30)",C(30),D(30),T(30),TT(30) DF(30) 1KK(30),CP(30),ALB(10),AN(5),THERM 1 (10),THERM2(i0) , F ( :>C , 2 ) , J J ( 1CP1(10),CP2(10),TAVRAM(30),H1(20),BETA(J0),GAMMA(30) 24 DIMENSION K5(100),TAV(100),TA1 ( 1 00) , 125 1 ALB 1(10),ALB2(10),AL1 (10) ,AL2(10) 1 2 6 CALL THER(THERM 1 ) . 127 CALL THERMA(THERM2) 128 CALL CEEPE2(CP2) 129 CALL CEEPE1(CP1) 130 CALL LOGB(ALB) 131 CALL EXPONT(AN) 132 CALL LOB(ALB2) 1 33 CALL EXPO(ALB 1) 1 34 CALL EXP(AL1) 135 CALL EX(AL2) 136 J1=0 •1 37 DO 3001 1 = 1 ,M 138 3001 K5(I)=0 r ********************************************************** 1 39 1 40 14 1 <_ -142 C Initialisation of Temperature at all nodes at TOTTIM=0.The 143 C fraction transformed matrix F and the JJ matrix are also 144 C initialised to 0. 145 Q *************************************************** 1 46 147 1 48 14 9 DO 10 I = 1 , M 150 F(I,1)=0.0D0 151 F(I,2)=0.0D152 JJ(I)=0 153 10 T(I)=850.0D0 154 155 1 56 157 Q ************************************************************* 158 C Starting with TOTTIM=0 the time is incremented in steps of 159 C DT.The calculations will stop when TOTTIM value is Greater 160 C than or Equal to a prespecified value •j g •) Q *********************************************************** 162 163 1 64 165 166 25 TOTTIM=TOTTIM+DT 167 IF (TOTTIM.GE.Tl) GO TO 1001 168 169 1 70 •-"---'-->••'-••-•"*»**** + ***************************** 172 C Loop 60 \ll  In this loop ,the values of Thermal Conductivity and Specific 175 C Heat for Austenite are calculated for each node at each time 176 C step 177 C Loop 70 ******* 180 Heat of Ferrite are calculated for each noae at.each 171 d 161 C time step. 182 C Loop 55 ig2 c ******* 184 C In this loop the values of Fraction Transformed for each node 185 C at each time step are checked to find whether transformation 186 C of Austenite to Pearlite is complete at all nodes.If so, 187 C control is directed to Loop 70 wherein the values of .183 C Thermal Conductivity and Specific Heat are calculated.On 189 C entering Loop 70 a counter J1 is set equal to 1.For all 190 C future time steps control is directed to Loop 70 without 191 C going through Loop 55.On complete transformation at all 192 C nodes, from Loop 70 control is directed to statement no.820 193 C wherein the appropriate Tridiagonal System coefficients 194 C are calculated,bypassing Loop 501. 195 C *************************************************************** 196 .197 198 199 IF (J1.EQ.1) GO TO 7 0 20'0 L=0 201 DO 55 1=1,M 202 IF (F(I,1).LT.1.0D0) GO TO 55 203 L=L+1 204 55 CONTINUE 205 IF (L.EQ.M) GO TO 70 206 DO 60 1=1,M 207 KK(I )=THERM2(1)+THERM2(2)*T(I) 208 KK(I)=418.6D0*KK(I) 209 CP(I )=CP1(1)+CP1(2)*T(I)+CP1(3)*(T(I)**2)+CP1(4)*(T(I)**3) 210 1+CP1(5)*(T(I)**4)+CP1(6)*(T(I)**5) 211 CP(I)=4186.0D0*CP(I) 212 60 CONTINUE 213 GO TO 700 214 70 DO 600 1=1,M 215 KK(I)=THERM1(1)+THERM1(2)*T(I)+THERM 1(3)*(T(I)**2) 216 1+THERM1(4)*(T(I)**3)+THERM1(5)*(T(I)**4) 217 KK(I)=KK(I)*418.6D0 218 CP(I)=CP2(1)+CP2(2)*T(I)+CP2(3)*(T(I)**2) 219 CP(I)=CP(I)*4186.0D0 220 600 CONTINUE 221 J1=1 222 GO TO 820 223 C ************************************************************** 224 225 226 227 C Loop 501 228 C ********* 229 C In this loop,each node is checked,at each time step for 230 C transformation start.At each time step,for each node, the 231 C Avrami time is calculated.If the TOTTIM value is Greater 232 C than or Equal to the Avrami time the node will start 233 C transforming.Once a node starts transforming,JJ(node ) is 234 C set equal to node number.For all future time steps this 235 C node will not be checked again for transformation start. 236 C 'when a node starts transf orming,control is directed to 237 C statement no.500 wherein 'n'.'log b' for that node 238 C temperature are calculated.The fraction transformed is 239 C then calculated.The Specific Heat and Thermal Conductivity 240 C of the transforming node is then calculated by using the 171e 24 I C formula 242 C Specific Heat=%Transformed*Soecific Heat of Ferrite + 243 C (1-%trans'f ormed ) *Speci f ic Heat of Austenite 244 C at the node temperature 245 C When the fraction transformed of the node is equal to 1 246 C control is transferred.to statement no. 8000 where the 247 C Specific Heat and Thermal Conductivity values of Ferrite 248 C are used for further calculations. 249 Q ******************************************************** 250 251 252 253 700 DO 501 1=1,M IF (F(I,1).GE.0.99999D0) GO TO 501 254 255 IF(T(I).GT.728.0D0) GO'TO 501 256 IF (I.GT.IS) GO TO 240 257 IF (JJ(I).NE.0) GO TO 400 258 IF (K5(I).EQ.1) GO TO 230 259 TAV(I)=TOTTIM 260 K5(I)=1 261 230 IF (T(I).GE.700.0D0) GO TO 501 262 IF (T(I).GE.475.0D0) GO TO 4001 263 TAVRAM(I)=35.2807D0-7.07259D0*(DLOG(728.0D0-T(I)))+0.0225313D0* 264 1(728.0D0-T(I)) 265 GO TO 4002 266 4001 TAVRAM(I)=22.4126D0+0.0123409D0*(728.0D0-T(I))-267 14.27 29lD0*(DLOG(7 28.0D0-T(I))) 268 4002 TAVRAM(I)=DEXP(TAVRAM(I)) 269 TA1(I)=TOTTIM-TAV(I) 270 IF (TA1(I).LT.TAVRAM(I)) GO TO 501 271 JJ(I ) = 1 272 400 IF (T(I).GE.625.0D0) GO TO 4003 273 IF (T(I).LT.500.0DO) GO TO 4004 274 EN=AL1(1)+AL1(2)*T(I)+AL1(3)*(T(I )**2) 275 GO TO 4005 276 4003 'EN=ALB1(1)+ALB1(2)*T(I) 277 GO TO 4005 278 4004 EN=AL2(1)+AL2(2)*T(I) 279 4005 ALOGB=ALB2(1)+ALB2(2)*(728.0D0-T(I))+ALB2(3)* 280 1((728.0D0-TU))**2)+ALB2(4)*((728.0D0-T(I))**3) 281 ALOGB=DEXP(ALOGB) 282 GO TO 410 283 240 IF (JJ(I).NE.0) GO TO 500 284 IF(K5(I).EQ.1) GO TO 229 285 TAV(I)=TOTTIM 286 K5(I)=1 287 229 IF (T(I ) .GE.700.0D.0) GO TO 501 "288 TAVRAM(I)=62.7348D0+0.105339D0*(728.0D0-T(I))-15.4 325D0 289 1*(DLOG(728.0D0-T(I))) 290 TAVRAM(I)=DEXP(TAVRAM(I)) 291 TA1(I)=TOTTIM-TAV(I) 292 IF (TA1(I).LT.TAVRAM(I)) GO TO 501 293 JJ(I ) = I 294 500 EN=AN(1)+AN(2)*T(I)+AN(3)*(T(I)**2)+AN(4)*(T(I)**3) 295 TT(I)=728.0D0-T(I) 296 ALOGB=ALB(1)+ALB(2)*(TT(I))+AL3(3)*((TT(I))**2)+ 297 1ALB(4)*((TT(I))**3) 298 ALOGB=DEXP(ALOGB) 299 4 1 0 THETA=DT+(DLOG(1.0D0/(1.0D0-FO,1)))/ALOGB)**(1.0D0/EN) 300 F(I ,2) = 1 . 0D0-DEXP(-ALOGB*(THETA**EN)) 17tf 301 DF(I)=F(I,2)-F(l,1 ) 302 F(I ,1 )=F(I,2) 303 FK=THERM1(1)+THERM1(2)*T(I)+THERM 1(3)*(T(I)**2)+THERM1(4)* 3 04 1(T(I)**3)+THERMl(5)*(T(I)**4) 305 FK=FK*418.6D0 306 FCP=CP2(1)+CP2(2)*T(I)+CP2(3)*(T(I)**2) 307 FCP=FCP*4 186.0D0 308 KK(I)=(F(I,1)*FK+(1.0D0-F(I,1))*KK(1)) 309 CP (I ) = (F(1 , 1 )*FCP+( 1 . 0D0-FC , 1 ) )*CPd ) ) 310 501 CONTINUE 311 312 313 314 Q ************************************************ 315 C Loop 7000 316 Q ********* .317 C This loop calculates the appropriate values of the Tridiagonal 316 C System Coefficients AA,BB,C,D 31 g Q ******************************************************************* 320 321 322 323 324 820 XX=-DX 325 DO 7000 1=1,M 326 XX=XX+DX 327 7050 IF(I.GT.I) GO TO 7051 328 AK1=(KK(1)+KK(2))/2.0D0 329 BB(1}=1.0D0+((DD*CP(1)*(DX**2))/(4.0D0*AK1*DT)) 330 C(1)=-1.0D0 331 IF ((F(I,1).LE.0.ODO).OR. (F(I,1).GE.0.99999D0)) GO TO 450 332 D(1)=((DX**2)/(4.0D0*AK1))* 333 1(DD*80000.0DO*DF(1)/DT)+((DD*CP(1)*(DX**2)*T(I))/ 334 1(4.0DO*AK1*DT)) 335 GO TO 7000 336 450 D(1)=(BB(1)-1.0D0)*T(1) 337 GO TO 7000 338 7051 IF (I.EQ.M) GO TO 7052 339 AK2=(KK(I-1)+KK(I))/(2.0D0) 340 AA(I)=AK2*((DX-2.0D0*XX)/(2.0D0*DX)) 341 AK3=AK2*(2.0D0*XX-DX)/(2.0D0*DX) 342 AK4=(KR(I)+KK(I+1))/(2.0D0) 34 3 AK4=AK4*(2.0D0*XX+DX)/(2.0D0*DX) 344 BB(I)=AK3+AK4+((DD*CP(I)*XX*DX)/(DT)) 345 AK4=(KK(I)+KK(I+1))/(2.0D0) 346 C(I)=-AK4*((2.0DO*XX+DX)/(2.0DO*DX)) 347 IF ((F(I,1).LE.0.ODO).OR.(F(I,1).GE.0.99999D0)) GO TO 451 348 D(I)=(XX*DX*(DD*80000.0D0*DF(I)/DT))+(DD*CP(I)*XX*DX*T(I)/DT) 349 GO TO 7000 350 451 D(I)=(DD*CP(I)*XX*DX*T(I))/DT 351 GO TO 7000 352 7.052 IF ( (F (I , 1 ) . LE. 0 . ODO ) .OR. (F (I , 1 ) .GE. 0 . 99999D0 ) ) GO TO 350 353 D(M)=((DX)*(4.0D0*R-DX)* 354 1(80000.0D0*DD*DF(M)/DT))/(8.ODO)+(H*TATMOS*R)+ 355 1 ((DD*CP(M)*DX*(4.0D0*R-DX)*T(M))/(8.0D0*DT)) 356 GO TO 351 357 350 D(M)=(H*R*TATMOS)+((DD*CP(M)*DX*(4.0D0*R-DX)*T(M))/ 358 1(8.0D0*DT)) 359 351 AK5=(KK(M-1)+KK(M))/(2.0D0) 360 AA(M)=AK5*((DX-2.0D0*R)/(2.0D0*DX)) 171g J c ^ 3 6 3 3 64 396 399 400 401 402 403 ] (4.0D0*R-DX) )/(6.0D0*DT.; 7 000 CONTINUE job 366 C ******************************************+*********************** 367 C This part of the program calculates the temperature of each node 368 C et each time step.The algorithm used is the solution of a 369 C Tridiagonal System of Simultaneous Equations described in the 370 C book 'Applied Numerical Methods' by Carnahan,Luther and Wilkes. 371 C T(M) is the temperature of the surface node and T(l) is the 372 C temperature of the Centre of the rod.After the calculations of 373 C temperature for one time step are completed control is 374 C transferred to statement no. 25 where the TOTTIM is incremented 375 C by DT and the calculation of the Tridiagonal System coefficients 376 C etc. is repeated. 377 C ******************************************************** 378 379 380 381 382 BETA(1)=BB(1) 383 GAMMA(1)=D(1)/BETA(1) 384 DO 110 1=2,M 385 BETA(I)=BB(I)-AA(I)*C(I-1)/BETA(1-1) 366 110 GAMMA(I) = (D(1)-AA(I)*GAMMA(1 - 1))/BETA(I ) 387 T(M)=GAMMA(M) 388 WRITE(6,63 01)TOTTIM,T(M) 389 6301 FORMAT(10X,F8.2,',',F6.2) 3 90 MAST=M-1 391 DO 120 J=1,MAST 392 I=M-J 393 1 20 T(I)=GAMMA (I)-C(I) *T(I + 1)/BETA(I) 394 WRITE(6,6001)TOTTIM,T(1) 395 6001 FORMAT(1 OX,F8.2,' ,',F6.2) 396 PRINT,F(1,1),F(10,1) 7 GO TO 25 £ ***************************************************************** C Calculation of temperature for all nodes for the current time 4uJ C step is complete.Control is now transferred to statement no.25 404 C for incrementing the TOTTIM value by DT and further calculations. 405 C ******************************************************************* 406 407 408 409 1001 STOP 410 END 4 11 412 413 Q ************************************************************* 414 C End of main program.Start of subroutines. 4 15 4 1 6 4 17 416 C ************************************************************ ii9 C The Subroutines Ther,Therma,Ceepe2,Ceepel,Logb,Expont are 420 C described in the program used for calculation Ox the 171h 42 1 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 temperature without segrega't ior.. Subrout i ne Lob calculates the coefficients of the polynomial used to find the value of 'b'at different temperatures.Subroutines Expo,Exp and Ex calculate the coefficients cf the polynomials used to find the values of 'n' in the temperature ranges 700 to 625*C, 625 to 500*C,<500*C respectively for the segregated steel. **************************************************** SUBROUTINE THER(THERM 1 ) IMPLICIT REAL*8 (A-H,0-Z) DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 1 SIGMA(20),P(20),THERM1 (10) DATA K,N/4,14/ X( 1) = 50.ODO DO 3300 1=2,14 3300 X(I)=X(I- 1)+50.0D0 DATA (Y(I),1=1,14)/0.118D0,0.115D0,0.112D0,0.108D0,0.103D0, 10.099D0, 0. 096D0,0.091 DO,0.087D0,0.084D0,0.081 DO,0.078D0, 10.075D0,0.072D0/ LOGICAL LK LK= .TRUE. NWT=0 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) DO 3400 1=1,5 3400 THERM1(I)=P(I) RETURN END Q ******************************************************** SUBROUTINE THERMA(THERM2) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 1,SIGMA(20),P(20) 1,THERM2(4) DATA K,N/1,11/ X(1)=700.0D0 DO 2100 1=2,11 2100 X(I)=X(I-1)+50.0D0 DATA (Y(I),1 = 1,11 )/0.053D0,0.055D0,0.O57D0,O.059D0,0.061 DO, 10.063D0, 10.064D0,0.066D0,0.068D0,0.07ODO,0.072D0/ LOGICAL LK LK= .TRUE. HWT= 0 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) DO 2110 1=1,2 2110 THERM2(I)=P(I) RETURN END 17ti Q ************************************************ 482 483 484 SUBROUTINE CEEPE2(CP2) 485 486 487 488 • IMPLICIT REAL*8(A-H,0-Z) 489 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),SIGMA ( 20 ) , 490 1A(20),B(20),P(20),CP2(3) 491 DATA K,N/2,13/ 492 X(1)=75.0D0 493 DO 4000 1= 2,13 494 4000 X(I)=X(I-1)+50.0D0 495 DATA (Y(I),I = 1,13)/0.117D0,0.124D0,0.127D0,0.131 DO,0.135D0, 496 10.14 0D0,0.145D0,0.150D0,0.160D0,0.166D0,0.172D0,0.172D0, 497 10.184D0/ 4 98 LOGICAL LK 499 LK= .TRUE. 500 NWT=0 501 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 502 DO 4010 1=1,3 503 4010 CP2(I)=P(I) 504 RETURN 505 END 5Qg C ******************************************************* 507 508 509 510 SUBROUTINE CEEPE1(CP1) 51 1 512 513 514 IMPLICIT REAL*8(A-H,0-Z) 515 DIMENSION X(25),Y(25),YF(25),YD(25),S(20),WT(25),SIGMA(20), 516 1A(20),B(20),P(20),CP1(10) 517 DATA K,N/5,13/ 518 X(1)=675.0D0 519 DO 5000 1=2,13 520 5000 X(I)=X(I-1)+50.0D0 521 DATA (Y(I),1=1,l3)/0.139D0,0.141D0,0.143D0.0.145D0,O.148D0, 522 10. 14 9D0,0.151 DO,0.154D0,0.156D0,0.158D0,0.160D0,0.162D0, 523 10.162D0/ 524 LOGICAL LK 52 5 LK= .TRUE. 526- NWT=0 527 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 528 DO 5010 1=1,6 529 5010 CP1(I)=P(I) 530 RETURN 531 END 532 C ************************************************** 533 534 535 SUBROUTINE LOGB(ALB) 536 537 538 539 IMPLICIT REAL*8(A-H,0-Z) 540 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 171 j 541 1 ,SIGMA(2 0) ,P(2 0),ALB(10) 54 2 DATA K,N/3,8/ 54 3 DATA(X(I),1 = 1,8)/58.ODO,45.ODO,68.ODO,78.ODO,96.ODO, 1 05. ODO , 544 1113.ODO,125.ODO/ 54 5 DATA(Y(I),1 = 1,8)/-9.8133D0,- 9.44464D0,- 9.47115D0,-8.3921D0, 54 6 1-5.7 27 9D0,-4.5132 7D0,"3.1385D0,-2.04103DO/ 54 7 LOGICAL LK 54 8 LK= .TRUE. 54 9 NWT=0 550 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 551 DO 6100 1=1,4 552 6100 ALB(I)=P(I) 553 RETURN 554 END 555 C *************************************************** 556 557 558 559 SUBROUTINE EXPONT(AN) 560 561 562 563 IMPLICIT REAL*8 (A-H,0-Z) 564 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 565 1,SIGMA(20),P(20),AN(5) 566 DATA K,N/3,8/ 567 DATA(X(I),1=1,8)/67 0.ODO,680.ODO,660.ODO,650.ODO,630.ODO, 568 1623.ODO,615.ODO,603.ODO/ 569 DATA(Y(I),1=1,8)/2.125109D0,1.618956D0,2.467576D0,2.946133D0, 57 0 13. 166861 DO, 3. 1 47 945D0 , 2 . 922 4 34D0 ,'2 . 34 64 07D0/ 571 LOGICAL LK 572 LK= .TRUE. 573 NWT=0 574 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 575 DO 6200 1=1,4 576 6200 AN(I)=P(I) 577 RETURN 578 END 579 SUBROUTINE HEAT(H1) 580 IMPLICIT REAL*8(A-H,0-Z) 581 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20), 582 1B(20),SIGMA(20),P(20),H1(20) 583 DATA K,N/6,10/ 584 DATA (Y (I ) , I = 1,10)/8 1.94D0,165.67D0,1 95. 4 8D0,200.91 DO,223.82D0, 585 1231.82DO,236.57DO,252.67DO,2 4 4.06DO,238.85DO/ 586 DATA (X(I),I=1,10)/890.ODO,860.ODO,830.ODO,B05.ODO,770.ODO, 587 1740. ODO ,712. ODO , 67.5 . ODO , 658 . ODO , 640 . ODO/ 588 LOGICAL LK 589 LK=.TRUE. 590 NWT=0 591 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 592 DO 6300 1=1,7 593 6300 H1(I)=P(I) 594 RETURN 595 END 556 C ******************************************************** 597 598 599 SUBROUTINE LOB(ALB2) 600 .171k 601 602 603 IMPLICIT REAL*8(A-H-,0-Z) 604 DIMENSION1 X(25),Y(25),YF(2s),YD(25),WT(25),S(20),A(20),B(20) 605 1,SIGMA(20),P(20),ALB2(10) 606 DATA K.N/3,12/ 607 DATA(X(I),1=1,12)/53.0D0,78.0D0,10 3.0D0,128.0D0,153.0D0, 608 1178.0D0,203.0D0,228.0D0,253.0D0,2 76.0D0,303.0D0,328.0D0/ 609 DATA(Y(I),1=1,12)/-6.09954D0,-4.60211D0,-5.08256D0, 610 1-5.57 406D0,-4.00138D0,-4.024 53D0,-2.9204D0,-2.97626D0, 611 1-3.78062D0,-3.967 8D0,-4.37844D0,-5.02969D0/ 612 LOGICAL LK 613 LK= .TRUE. 614 NWT=0 615 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 616 DO 6100 1=1,4 .617 6100 ALB2(I)=P(I) 6 1 8 RETURN 6 1 9 END 620 C *********************************************** 621 622 623 SUBROUTINE EXPO(ALB 1) 624 625 626 627 IMPLICIT REAL*8(A-H,0-Z) 628 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 629 1,SIGMA(20),P(2 0),ALB 1(10) 630 DATA K,N/1,3/ 631 DATA(X(I),1=1,3)/675.0D0,650.0D0,625.0D0/ 632 DATA(Y(I),1=1,3)/0.77 0414D0,0.825272D0,1.276907D0/ 633 LOGICAL LK 634 LK= .TRUE. 635 NWT=0 636 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 637 DO 6100 1=1,2 638 6100 ALB1(I)=P(I) 639 RETURN 640 END 641 c ******************************************************** 642 643 644 SUBROUTINE EXP(AL1) 645 646 647 648 IMPLICIT REAL*8(A-H,0-Z) 64 9 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) 650 1,SIGMA(20),P(20),AL1(10) 651 DATA K,N/2,5/ 652 DATA(X(I),1=1,5)/600.0D0,575.0D0,550.0D0,52 5.0D0,500.0D0/ 653 DATA(Y(I),I=1,5)/l.646218D0,1.205357D0,1.103574D0,0.74475D0, 654 10.715708D0/ 655 LOGICAL LK 656 LK= .TRUE. 657 NV?T=0 658 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 659 DO 6100 1=1,3 660 6100 AL1(I)=P(I) 1711 661 RETURN 662 END ggj Q ****************************************************** 664 665 666 SUBROUTINE EX(AL2) 667 668 669 670 IMPLICIT REAL*8(A-H,0-Z) 67 1 DIMENSION X(25),Y(25),YF(25),YD(25),WT(25),S(20),A(20),B(20) '672 1,SIGMA(20),P(20),AL2(10) 673 DATA K,N/l,4/ 67 4 DATA(X(I),1=1,4)/475.ODO,450.ODO,425.ODO,400.ODO/ 67 5 DATA(Y(I),1=1,4)/0.851409D0,0.905354D0,0.987884D0,1.057 34 7D0/ 67 6 LOGICAL LK •677 LK= .TRUE. 678 NWT=0 679 CALL DOLSF(K,N,X,Y,YF,YD,WT,NWT,S,SIGMA,A,B,SS,LK,P) 680 DO 6100 1=1,2 681 6100 AL2(I)=P(I) 68 2 RETURN 683 END End of file 

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