CHARACTERIZATION OF THE COOLING AND TRANSFORMATION OF STEELS ON A RUN-OUT

I

CHARACTERIZATION OF T H E COOLING AND

TRANSFORMATION OF STEELS O N A

RUN-OUT T A B L E OF A HOT-STRIP M I L L

By

CRAIG A L L E N M C C U L L O C H

B.A.Sc, The University of British Columbia, 1986

A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF

T H E REQUIREMENTS FOR T H E D E G R E E O F

MASTER OF APPLIED SCIENCE

in

T H E F A C U L T Y OF G R A D U A T E STUDIES

M E T A L S A N D MATERIALS ENGINEERING

We accept this thesis as confonning to the required standard

T H E UNIVERSITY OF BRITISH COLUMBIA August 1988

©Craig Allen McCulloch, 1988

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 M e t a l s a n d M a t e r i a l s E n q i n e e r i n g

The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3

DE-6(3/81)

A B S T R A C T

A mathematical model has been developed to predict the thermal history of strip

during cooling on the run-out table of a hot strip mill. The model incorporates phase

transformation kinetics and accounts for the heat of transformation. To characterize the

cooling by laminar water sprays, in-plant trials were conducted at the Stelco Lake Erie

Works hot strip mill. The temperature data was used in the thermal model to calculate

an overall heat transfer coefficient for a laminar water bank of 1 kW/m 2 , C. Isothermal

diametral dilatometer testing was used to generate phase transformation kinetics for a

0.34 weight percent plain carbon steel. Continuous cooling dilatometer testing was

used to calculate the transformation start time as a function of the cooling rate. The

high cooling rates of 40 *C/s to 50*C/s, experienced on the run-out table had the effect

of depressing the transformation start temperature by over 100'C.

The phase transformation kinetics were incorporated in a phase transformation model

and employed to predict thermal profiles for a 0.34 carbon plain-carbon steel. The

temperature predictions were within 25"C of the plant pyrometer readings using the

calculated overall heat transfer coefficient and within 35°C of the plant pyrometer

values using literature derived heat transfer coefficients.

A simulation of the model predicted cooling conditions on a Gleeble high

temperature testing machine showed that the transformation was occurring at

approximately 730*C. The empirical transformation start time, obtained from cooling

ii

rate versus transformation start time tests, which was used in the phase transformation

portion of the model, and the Gleeble simulation gave excellent agreement with the

model thermal profile predictions.

iii

T A B L E OF CONTENTS

Abstract ii

Table of Contents iv

List of Tables viii

List of Figures ix

Acknowledgment xvi

1.0 INTRODUCTION 1

2.0 LITERATURE REVIEW 3

2.1 Heat Transfer on the Run-out Table 3

2.1.1 Heat Transfer Coefficients for Water Bar and Water

Curtain Cooling from Plant Data 4

2.1.2 Heat Transfer Coefficients for Water Bar Cooling

from Experimental Measurements 5

2.1.3 Heat Transfer Coefficient for Roll Contact Cooling

from Experimental Measurements 9

2.2 Phase Transformation Kinetics 10

2.3 Review of Related Models , 13

2.4 Figures 16

3.0 SCOPE A N D OBJECTTVES 17

3.1 Scope 17

3.2 Objectives 18

4.0 PROCEDURE 19

iv

4.1 Mathematical Model 19

4.1.1 Formulation 20

4.1.2 Numerical Methods 24

4.2 Heat Transfer Coefficient 27

4.2.1 Calculation from Literature 28

4.2.2 Calculation from Plant Data 30

4.3 Phase Transformation Characterization 33

4.3.1 Material 33

4.3.2 Metallography 33

4.3.3 Dilatometer 34

4.3.3.1 Isothermal Tests 35

4.3.3.2 Continuous Cooling Tests 36

4.3.4 Phase Transformation Model Calculations 37

4.4 Tables and Figures 39

5.0 RESULTS A N D DISCUSSION 55

5.1 Heat Transfer Coefficient 55

5.1.1 Literature 55

5.1.1.1 Laminar Water Bar Cooling 55

5.1.1.2 Film Boiling Cooling 56

5.1.1.3 Support Roller Contact Cooling 56

5.1.1.4 Combined Cooling 57

5.1.2 Plant Trials 58

v

5.1.2.1 Overall Heat Transfer Coefficient 59

5.1.2.1.1 Calculation 59

5.1.2.1.2 Sensitivity 60

5.1.2.2 Individual Heat Transfer Coefficient 62

5.2 Phase transformation 63

5.2.1 Material 63

5.2.2 Isothermal Cooling Tests 64

5.2.3 Continuous Cooling Tests 65

5.2.3.1 Metallography 66

5.2.3.2 Coiling Temperature 67

5.2.4 Model Phase Transformation Calculations 69

5.3 Mathematical Model 71

5.3.1 Sensitivity 71

5.3.2 Validation 72

5.4 Tables and Figures 73

6.0 CONCLUSIONS 133

6.1 Summary 133

6.2 Conclusions 135

6.3 Future Considerations 138

7.0 BIBLIOGRAPHY 139

8.0 APPENDIX 142

8.1 Nomenclature 142

vi

8.2 Derivation of Finite Difference Equations 145

8.2.1 Top Surface Node 145

8.2.2 Interior Nodes 146

8.2.3 Bottom Surface Node 146

8.2.4 Solution , 147

8.3 Hatta et al. Thermal Boundary Layer Calculations 147

vii

LIST OF TABLES

Table I Composition for the three steel chemistries used. 39

Table IIa....Plant conditions for four runs 73

Table Ho....Plant conditions for four runs 74

Table IIc....Plant conditions for four runs 75

Table IJJ Industrial plant cooling conditions 76

Table IV Metaliographic data for the 0.34 carbon samples, for the

down-coiler sample.the continuous cooling samples, and the Gleeble

simulation sample; with tabulated values for, cooling rate, fraction ferrite,

undercooling, and average austenite grain size 77

Table V Comparison of the composition of the down-coiler and transfer bar

medium carbon samples 78

Table VI....Grain size versus coiling temperature for 0.054 weight percent

carbon grade steel 79

Table VH....Tabulated model predictions, for low (7'C/s) and high (45'C/s)

cooling rates, and for the literature heat transfer coefficients at an average

cooling rate, (26'C/s) 80

viii

LIST OF FIGURES

Figure 1 Specific Heat as a Function of Temperature for five carbon levels,

BISRA 16

Figure 2 Hot-strip geometry used for the model 40

Figure 3 Schematic of the STELCO Lake Erie Works Hot Strip Mill

Run-out Table 41

Figure 4 Specific Heat as a Function of Temperature for a 0.34 % carbon

steel, BISRA, w/o phase transformation 42

Figure 5 Thermal Conductivity as a Function of Temperature for a 0.06 %

plain carbon steel, BISRA 43







Figure 9 Flow chart for the basic program 47

Figure 10 The six types of cooling regime experienced by the steel strip 48

Figure 11 The various film boiling heat transfer coefficients from Kokada et

al.[6] for three cooling water temperatures with two values from the

Berensen[24] horizontal surface boiling equation 49

Figure 12 Experimental verification of TAC3 and TAC1 50

ix

Figure 13 A typical dilation versus time plot for an isothermal dilatometer

test 51

Figure 14 A typical dilation and temperature versus time plot showing

transformation start and finish times 52

Figure 15 Experimental dilation and thermal dilation plots, used with

divergence method (Campbell[27]) for calculation of transformation start 53

Figure 16 Flow sheet for the iterative solution of the Avrami fraction

transformed equation as a function of temperature 54

Figure 17 Black zone radius as a function of a constant steel surface

temperature 81

Figure 18 Hatta laminar water bar heat transfer coefficient as a function of

contact radius 82

Figure 19 Thermal profile model sensitivity to changes in the water

temperature for the Kokada film boiling heat transfer coefficient 83

Figure 20 Thermal profile model sensitivity to changes in the support roller

conduction cooling 84

Figure 21 Thermal profile model literature heat transfer coefficients 0.05%

carbon, 3.89 mm gauge, target coiling temperature 720*C 85

Figure 22 Thermal profile model literature heat transfer coefficients, 0.05%

carbon, 2.62 mm gauge, target coiling temperature 720*C 86


carbon, 0.024% Nb, 3.89 mm gauge, target coiling temperature 720° C 87

x


carbon, 0.024% Nb, 2.62 mm gauge, target coiling temperature 720°C 88


carbon, 2.62 mm gauge, target coiling temperature 620"C 89

Figure 26 A sample temperature profile from the plant data. 90

Figure 27 Thermal profile model, overall heat transfer coefficient calculated

from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target

coiling temperature 720"C 91


from plant pyrometer measurements, 0.07% carbon, 0.024% Nb, 3.89 mm

gauge, target coiling temperature 720'C 92


from plant pyrometer measurements, 0.07% carbon, 0.024% Nb, 2.62 mm

gauge, target coiling temperature 720*C 93



coiling temperature 720*C 94



coiling temperature 620"C 95



xi

coiling temperature 540°C 96

Figure 33 Thermal profile model sensitivity, overall heat transfer coefficient

calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge,

target coiling temperature 720°C 97

Figure 34 Thermal profile model sensitivity, individual laminar water bar

heat transfer coefficient, 10 kW/m 2 ,C with 20 kW/m2 ,C and 5 kW/m2 oC

deviations,target coiling temperature 720*C 98

Figure 35 Isothermal dilatometer results for 673*C test, dilation-time and

temperature-time 99

Figure 36 AD/AT as a function of time for the 673'C isothermal test 100

Figure 37 Isothermal dilatometer test sample plot lnln(l/(l-FX)) vs ln(t) for

673"C fraction ferrite transformed 101

Figure 38 ln(b) Avrami coefficient for the isothermal formation of ferrite in

the 0.34 carbon steel 102

Figure 39 ln(b) Avrami coefficient for the isothermal formation of pearlite in

the 0.34 carbon steel 103

Figure 40 Avrami coefficient, nf, for the austenite-to-ferrite transformation in

the 0.34 % C, plain carbon steel 104

Figure 41 Avrami coefficient, n,,, for the austenite-to-pearlite transformation

in the 0.34 % C, plain carbon steel 105

Figure 42 Calculated ln(b) values for the ferrite transformation assuming n<=

1.25, for 0.34% carbon steel 106

xii

Figure 43 Calculated ln(b) values for the pearlite transformation assuming Ap

= 1.14, for 0.34% carbon steel 107

Figure 44 Average Avrami coefficient V for 0.34% carbon compared to

other experimental values (Campbell[27]) 108

Figure 45 Comparison of the ln(b) Avrami coefficient for the

austenite-ferrite transformation in several plain-carbon steels(Campbell[27]) 109

Figure 46 Comparison of the ln(b) Avrami coefficient for the

austenite-pearlite transformation in several plain-carbon steels(Campbell[27])... 110

Figure 47 Temperature as a function of time for a continuous cooling rate of

27'C/s I l l

Figure 48 Thermal and Experimental dilatometer values as a function of

time for a cooling rate of 27*C/s 112

Figure 49 The undercooling for the austenite-to-ferrite start temperature as a

function of cooUng rate 113

Figure 50 Fraction ferrite as a function of cooling rate, from metallographic

examination 114

Figure 51 Continuously cooled dilatometer sample 115

Figure 52 Continuously cooled dilatometer sample showing banding 116

Figure 53 Medium carbon down-coiler sample 117

Figure 54 Surface thermal profile, using the overall heat transfer coefficient,

1 kW/m 2 , C, Table rn run one cooling conditions, and a transformation start

temperature of 732'C (dT/dt = 7°C/s) 118

xiii


1 kW/m 2 o C, Table in run two cooling conditions, and a transformation start

temperature of 732'C (dT/dt = TCJs) 119


1 kW/m 2 °C, Table LTI run three cooling conditions, and a transformation start

temperature of 732'C (dT/dt = 7'C/s) 120


1 kW/m 2*C, Table DI run one cooling conditions, and a transformation start



1 kW/m 2*C, Table in run two cooling conditions, and a transformation start



1 kW/m2"C, Table DI run three cooling conditions, and a transformation start


Figure 60 Surface thermal profile, using the literature heat transfer

coefficients, Table DI run one cooling conditions, and a transformation start

temperature of 710"C (dT/dt = 26'C/s) 124

Figure 61 Surface thermal profile, using the literature heat transfer

coefficients, Table DI run two cooling conditions, and a transformation start


Figure 62.....Surface thermal profile, using the literature heat transfer

xiv

coefficients, Table DJ run three cooling conditions, and a transformation start

temperature of 710*C (dT/dt = 26'C/s) 126

Figure 63 Effect on predicted center line temperature of changes in the

number of nodes through thickness 127

Figure 64 Effect on predicted center line temperature of changes in the step

size, where the step size equals the strip velocity times the time increment 128

Figure 65 Effect of the ±0.3m/s deviation of the strip velocity on the

predicted temperature profile 129

Figure 66 Industrial cooling profile simulated on the Gleeble high

temperature testing machine 130

Figure 67 Industrial cooling conditions, simulated on a Gleeble high

temperature testing machine 131

Figure 68 Microstructure of the Gleeble cooling simulation sample for the

Table HI run one cooling conditions 132

xv

Acknowledgment

I would like to acknowledge the support provided for this project by NSERC and

Stelco Inc. The guidance of my thesis supervisors I.V. Samarasekera and E.B. Hawbolt

is also much appreciated. On the experimental side, Keith Barnes, Henk Averink, and

Barbara Zbinden were of major assistance in securing the industrial temperature data as

was Bihn Chau with the dilatometer phase transformation kinetics.

xvi

INTRODUCTION 1

1 INTRODUCTION

Historically the production of flat rolled product has been based on previous experi

ence. New materials or new physical properties for existing materials, were produced by

a trial and error process by examining the effects of minor modifications to the

rolhng/ccoling conditions. The introduction of the computer to industrial applications

initially changed the experience reservoir from the on-line staff to the machine, with little

modification of the trial and error methods historically used.

Approximately one half of the finished steel in North America is in the form of

sheet or strip, with hot rolled strip being used in a wide variety of applications ranging

from auto body to the shells of stoves and refrigerators. Furthermore, there has been a

proliferation of non ferrous products in the market place of lightweight materials with

mechanical properties equivalent to those of steel. The steel industry has responded by

developing lighter gauge steel to lower the weight without an attendant loss of strength.

To produce a strip steel of improved physical properties with decreased weight the vari

ables in the process must be well controlled; the final overall physical properties are

affected by the chemical composition and thermo-mechanical history of the steel. Thus,

the initial composition of the steel, the casting and reheating processes, together with the

thermo-mechanical history of the steel during rolling and subsequent cooling profoundly

influences the final mechanical properties of the strip.

The economic down-tum at the start of this decade emphasized the need for tighter

control of the hot rolling process to minimize costs. The use of niobium-titanium-

1

INTRODUCTION 1

vanadium additives for increased strength became routine, although mill scheduling was

accomplished by trial and error. Hot strip mills world wide are now controlled by

empirical or semi-empirical computer models with some operator input, but, a better

thenno-mechanical understanding of the entire rolling mill is needed in order to produce

control models that decrease deviations from target physical properties.

The rolling mill and the run-out table have been modeled from both an experimen-

tal(empirical) and a theoretical(mathematical) point of view in an effort to provide better

mill operational control. Mathematical models of these processes are also being

developed in order to provide a better understanding of the theoretical aspects of thermo-

mechanical processing of steel. One area that is not well understood is the relationship

between the process variables and the transformation from austenite to ferrite/pearlite

during run-out table cooling. The phase transformation effects have historically been

included in models by incorporating a specific heat value that includes the recalescence

due to phase transformation, however, the cooling rate strongly influences the start of

transformation and thus kinetics must be considered in any modeling effort.

The object of this thesis was to model the temperature and microstructure of the

hot-strip as it passes along the run-out table from exit of the finishing stands until

entrance to the pinch roller of the down-coiler. A better understanding of the heat trans

fer to the cooling water sprays and of the kinetics of the phase transformation was sought

in an attempt to provide a realistic model of the process.

2

LrTERATURE REVIEW 2.1

2 LITERATURE REVIEW The current literature on hot strip run-out table cooling covers all aspects of the

process. These range from a description of new cooling techniques to composition

control for the production of steels with improved mechanical properties. They also

include reviews of models for process control. Of particular interest to this thesis are

measurements of or expressions quantifying the heat transfer between the strip and

cooling water on the run-out table, phase transformation kinetics under non-equilibrium

conditions and mathematical models coupling the two phenomena.

2.1 Heat Transfer on the Run-Out Table

On the run-out table the strip is cooled by a laminar water bar or water curtain. The

former technique has received its name because of the 'glassy' or 'rod-like' non-turbulent

appearance rather than due to a strict Reynolds number definition of laminar flow. A

water curtain has been described as a continuous water bar in that it resembles a laminar

sheet or 'curtain' of water. Cooling occurs by forced convection in the zone of direct

contact and by film boiling across a vapour barrier in the region surrounding the impact

zone. The heat transfer coefficient for laminar water bar, water curtain, and film boiling

may be determined empirically from in-plant temperature measurements or from

laboratory experiments. Heat is also transferred to the support rolls by conduction and to

the surrounding air by radiation.

3

LITERATURE REVIEW 2.1.1

2.1.1 Heat Transfer Coefficients for Water Bar and Water

Curtain Cooling from Plant Data

In this approach heat transfer coefficients are back calculated from in-plant strip

surface temperature measurements. Tacke et al.[l] studied the two types of water cooling

as well as the spray nozzle cooling previously employed on run-out tables. To a run-out

table with a standard spray nozzle header configuration, they added a bank of laminar

water bar headers and a bank of water curtain headers . Each cooling bank had a

blow-off and a pyrometer mounted before and after the bank to give an accurate reading

of surface temperature. Using the measured temperature changes across a cooling bank

in a finite element model they calculated an overall heat transfer coefficient for each type

of cooling bank as a function of the water application rate. From this they derived values

of 1800 W / m 2 , C ± 2 0 0 W / m 2 , C for the bank of water curtain cooling, 1300 W/m2*C ±200

W/m 2 o C for the bank of laminar water bar cooling, and 900 W/m 2 o C ±150 W / m 2 , C for the

bank of spray nozzle cooling. While the water curtain cooling had the highest heat

transfer coefficient, the water application was uneven over the strip. The laminar water

bar gave an even water application over the full strip width and so was chosen for plant

use even thoygh the resultant heat transfer coefficient was less than that for water curtain

cooling. They report cooling rates of 50°C/s up to 200°C/s depending on strip gauge with

laminar water bar cooling. They also report that front end loading the cooling, that is

using water sprays at the finish mill end of the run-out table resulted in an increase in

yield strength over tail end loading, cooling with the sprays at the down-coiler end of the

4


run-out table. They note that in a series of over seven hundred strips using the laminar

water bar cooling coupled with the calculated heat transfer coefficients in their plant

control model the two sigma deviation was reduced from an average value of 45°C down

to 20'C.

Colds and Sellars[2] have calculated a heat transfer coefficient for an individual

water curtain by using a finite difference heat transfer model with various values included

for water curtain and film boiling cooling. The resulting thermal profile is compared to

observed values to arrive at a result of 17 kW/m 2 , C for an individual water curtain; they

comment on the difficulty in producing an exact value due to the short residence time of

the strip under the water curtain contact area which they assume to have a diameter of

two to three times the water curtain diameter. They use a film boiling cooling heat

transfer value of 150 W/m2"C for the region outside the contact zone which compares

well with the Farber and Scorah[3] value of between 150 W / m 2 , C and 170 W/m 2 o C.

2.1.2 Heat Transfer Coefficients for Water Bar Cooling from

Experimental Measurements

Individual laminar water bar heat transfer coefficients and values for the associated

film boiling cooling in the surrounding region were presented in three articles by

Hatta[5,6], Kokada[7], et al. The Hatta et al. results were based on an examination of the

cooling associated with a single laminar water bar over a stationary 10 mm thick stainless

steel plate with low water flow rates. The plate was instrumented with five

thermocouples inserted from the back of the plate at a depth of 8 mm from the top surface

5


at 20 mm increments from the water bar contact center line. The plate was heated in a

reducing furnace to hot-strip temperatures, it was then removed and placed under a

laminar water bar header. The temperature change over time for various water

temperature and flow rates was recorded. This data was then used in a finite difference

model to derive an equation for a heat transfer coefficient. A heat transfer coefficient of

15.93 W/m 2"C was adopted for natural convection in air in the Hatta model.

The Hatta et al. experiments produced a number of general observations about the

water cooling under a laminar water bar. First, that there is a 'black zone' around the

area under the water bar which did not show any boiling phenomena. Second, around

this 'black zone' was an area of film boiling. Third, the transition between the boiling

and non-boiling areas appeared to be instantaneous; that is, there did not appear to be a

visible transition cooling regime. Using these observations and the data produced from

the thermocouples, a heat transfer coefficient equation was obtained,

. . . 1

where 0.063 is an experimentally derived constant

k is the thermal conductivity of the water, in W/mK

r is the laminar water bar contact radius, in meters

Re is the Reynolds number

and Pr is the Prandd number

The heat transfer coefficient for the film boiling region is,

6


a™ = 200* 2420-21.70V) . . . 2

,•8

where T w is the average water temperature, in Kelvin

T s is the steel temperature, in Kelvin

and T S A X is the saturation temperature of the water, in Kelvin.

The water saturation temperature under one atmosphere pressure is 100 °C. The

water temperature at which the transition from water contact cooling to film boiling

cooling occurs, Tom-, is described by the equation,

The value of ranges from 18.75 "C for a steel temperature of 1000 *C to 100'C

for a steel temperature of 350 *C. Hatta, Kokada, et al. noted that film boiling was not

observed for a water temperature lower than 68 °C. The cooling water temperature, T w ,

must therefore be between the minimum critical transition temperature, Tdm- = 68 *C,

and the saturation temperature, T S A X = 100 *C. The water temperature used in the model

is a simple average of these two values,

-(7^-1150)

8

. . . 3 cm ~

1 0 0 o C + 6 8 ° C 2

= 84°C . . . 4

which is used as T w for Eq.2.

7


A horizontal water velocity1 is needed to derive a water film thickness as well as for

computation of the Reynolds number used in Eq.l; this is calculated based on the

assumption that the horizontal water velocity is equal to the vertical water velocity which

is determined by the water flow rate. Hatta et al. used the heat transfer coefficients

calculated in Eq. 1 and Eq.2 to calculate a thermal profile for the plate. The calculated

profiles were then compared to the thermocouple data and it was found that greater

cooling was predicted by the calculated heat transfer coefficient than was observed

experimentally. To compensate for this over cooling Hatta et al.[6] postulated that there

is a 'thermal zone' in the water film layer, not all of the water film thickness was affected

by the heat flow. The thermal zone2, a boundary layer phenomena, is the thickness of

water above the plate that is heated in a finite time period. This was used in Hatta's

model with Eq. 1 for the area under the laminar water bar and Eq.2 for the film boiling

zone to calculate a new thermal profile which gave good agreement with the

experimental results. Eq. l is insensitive to the water flow rate but relatively sensitive to

the area of contact under the laminar water bar3.

1 Equation 1 in the appendix

2 Water film thickness, described in the appendix

3 From Eq.Al in the appendix

8


2.1.3 Heat Transfer Coefficient for Roll Contact Cooling from

Experimental Measurements

Diener and Drastik[7] examined heat flow between guide rolls4 and continuously

cast slab using instrumented rolls and developed heat flux profiles for various cooling

types. Using their 'quasi-stationary' heat flux value of 75 kW/m with an average

temperature difference of 900 *C, an average heat transfer coefficient of 83 W/m 2 o C can

be calculated.

4 On the inside of the curve above the slab

9

LITERATURE REVIEW 2.2

2.2 Phase T rans fo rmat ion

The phase transformation and its associated heat of transformation has been

characterized with a variety of methods, the predominant one being the use of a modified

specific heat value. The specific heat values tabulated by the British Iron and Steel

Research Association, BISRA[8], include the effects of the phase change by

incorporating the heat of transformation in the specific heat value to give a greatly

increased value at the phase transformation temperature, as can be seen in Figure 1. If

the specific heat is taken as a temperature dependent value which includes the heat of

transformation, then the thermal effects of the phase transformation can be accounted for

in this way. The BISRA specific heat values were obtained from plant measurements

made in the early 1950's and do not include the modem alloys. The data range is based

only on the weight percent carbon and a variation in carbon content of 0.06 to 0.40

weight percent; individual values are an average for a 50 °C temperature range. The

BISRA values are for equilibrium and any effects of cooling rate are ignored.

The use of the isothermal kinetics to describe the continuous cooling transformation

is based on the Avrami [10] formula,

X = l-expH>r") • • • 5

where X is the fraction transformed, t is time, and b and n are two coefficients called the

'Avrami' coefficients. This is based on the additivity concept first postulated by Scheil5

in 1935. An additive system is one in which the transformation is only a function of the

5 For incubation not for phase transformation as such

10


temperature and the fraction previously transformed. In an additive system a continuous

process can be approximated as the sum of a series of discrete steps; this is very useful in

mathematical modeling. The Avrami formula presents the fraction transformed ( X ) as a

function of time (t) and the two 'Avrami' coefficients 'b' and V . Avrami[9,10,ll], and

later Cahn[12] postulated separate criterion for determining if a system is additive.

Avrami described an ' isokinetic' condition in which the ratio of nucleation and growth

rate is constant. Cahn described a site saturation criterion based on preferential

nucleation sites. Agarwal and Brimacombe[13] used the additivity concept in their

model of rod cooling, noting that while the system being examined did not satisfy either

criterion, the results from the model based on the assumption of an additive system

agreed with experimental observations. Kuban et al.[14] examined the additivity of the

austenite to pearlite transformation to determine conditions over which additivity applied.

They postulated a criterion of 'effective site saturation' based on the concept that most of

the growth of the new phase, pearlite, is growth at the initially nucleated sites with the

sites nucleated near the end of the transformation contributing very little to the overall

volume change. The effective site saturation criterion was found to be valid if the time

for twenty percent of the transformation was experimentally greater than 0.28 times the

time for ninety percent of the transformation,

r 2 0> 0.28^ . . . 6

Hawbolt et al.[15,16] examined the austenite-to-pearlite transformation for a

eutectoid steel and the austenite-to-ferrite and pearlite transformations for a 1025 steel

using a dilatometer to determine phase transformation kinetics and start temperatures.

11


The Avrami coefficients, n, and, b, were determined from isothermal tests. The

transformation start time (or temperature) and the total fraction ferrite formed as a

function of cooling rate were determined using continuous cooling tests.

A different method of dealing with the phase transformations occurring on a run-out

table was examined by Morita et al.[17]. They used an on-line transformation detector

measuring the change in magnetic resistance of the strip to determine the fraction

transformed on the run-out table. The concept of an on-line transformation detector

under the strip is potentially very desirable. However, machine calibration and data

interpretation seem dependent on trial and error. Until a theoretical model capable of

interpreting the change in magnetic resistance in terms of the kinetics of the austenite to

ferrite and ferrite plus pearlite transformation is available, the on-line transformation

detector, while sophisticated, requires substantial experimental data to describe the

transformation behavior.

12


2.3 Review of Related Models

Of the various published models pertaining to run-out table cooling of hot strip,

most are intended for use in mill control. They range from the Hinrichsen[18] dynamic

systems approach to the Hurkmans et al.[19] experimentally produced deformation

transformation model. Hinrichsen[18] modeled the run-out table as a dynamic simulation

via a systems control approach used widely in chemical engineering applications. He

formulated a dynamic model of the run-out table including the gain and dead time of each

component, from the spray water valves to the run-out table pyrometer. He then used

experimental data to tune the response of the dynamic model. The entire process is

controlled with a proportional-integral controller using modified feedback compensated

feed-forward control to prevent cumulative errors from inducing increasing oscillation.

This is a widely used control system in areas where there is small variability in the

desired output product. With hot-strip, current production requires output of many

products with different properties from the same production line, which makes this type

of model of limited utility.

A basic model used in a wide variety of plants is the basic heat transfer model as

exemplified by Tacke et al.[l]. This model is described in the preceding section A. and is

an excellent example of the use of empirical data and mathematical modeling to control

run-out table output. The heat transfer models found in the literature vary in their levels

of sophistication. These range from the simple Longenberger[20] model to the

sophisticated Tacke et al.[l] model. The Longenberger[20] one dimensional model

13


discretizes the strip through thickness into three nodes and the resulting model is tuned

through statistical regression. Miyake[21] has produced a more sophisticated model

which mathematically characterizes the losses due to radiation and water cooling but is

still fine tuned with empirical data. The Tacke et al.[l] model, previously described, uses

a finite element approach and back calculated heat transfer coefficients to produce an

on-line control model and represents the most sophisticated of the purely heat transfer

models.

More complex still are the models that add microstructural considerations to the

basic thermal model. Yada[20] uses additivity and the assumption of a transformation

rate independent of time. The entire rolling mill is approximated as a series of

independent models; one for hot deformation, resistance to hot deformation, a

temperature profile model, a transformation model based on nucleation and growth, and a

structure versus properties model. The model outputs are combined to produce a

prediction of the final microstructure and the physical properties and are used as an

on-line mill control model. The model is used on-line to compensate strip cooling for

variations in strip velocity to maintain the consistency of the strip properties. Yada notes

that some form of on-line microstructural information would be useful during processing

to eliminate cumulative errors and to this end he suggests the use of the magnetic

transformation detector described by Morita et al.[17].

The most comprehensive approach is that of Hurkmans et al.[19] in which

dilatometry is used to characterize the phase transformation kinetics for a given

chemistry. The dilatometric data for a given test is reduced to a group of between thirty

14


to sixty points which are then fitted to a cubic spline interpolation. From the interpolated

data a set of thirty data values are produced and used for all future calculations. From the

fitted data set the rate of diametral change over time and the rate of temperature change

over time is calculated. This is similar to the method used by Hawbolt et al.[15,16] but,

with the interpolation of the raw data, variations due to experimental differences between

individual data runs should be minimized. The diametral change with time data is

integrated to produce fraction transformed data. This data is then fitted to an equation by

a least squares approximation to produce values for the constants A±, B k , and Q used in,

dt

where 'az Jk

4

e

= Ak(Zk + e)%>

is the rate of transformation for phase k,

is the fraction of phase k transformed,

is the fraction of y phase transformed,

is a small number needed in integration of the

differential equation.

The constants are derived for a given phase, composition, and austenitizing

condition. Hurkmans et al.[19] have used the model for ferrite, pearlite, bainite, and

martensite transformations. This data is used in an in-plant control model and has

resulted in a reduction of overall water consumption while maintaining the desired

microstructure.

15


2.4 Figures

c o "9 v $ v O v g v O v.e(

w co CM o a o 5 d ci d ci a O) CD

5

>«r>+

-Mac

-K>

O + <

+ •

- m a x • M I X

1 1 -

(fl m CO CM T -

I 1 1 1 1 1— i - O) oo s to in ^ d d d d d ci

Specific Heat W/kg C

(X 1000)

Figure 1 Specific Heat as a Function of Temperature for five carbon levels, BISRA

16

SCOPE A N D OBJECTIVES 3.1

3 SCOPE A N D OBJECTIVES

3.1 Scope

The impetus for this work lies in the need to link the microstructure and properties

of hot band to processing parameters in the hot strip mill. This requires the integration of

effects of composition, casting, reheat, rough and finish rolling, run-out table cooling,

and down-coiler cooling on the microstructure. This may be best accomplished by

developing mathematical models of the individual processes and linking them up to trace

the changes in the microstructure due to processing.

This project focuses on the cooling and phase transformations on the run-out table

of a hot strip mill subject to certain limitations. This examination is limited to the run-out

table without regard to the prior thermo-mechanical history, even though this is accepted

as having an effect on the microstructure. The model incorporates heat transfer and

phase transformation kinetics associated with the cooling and any thermally generated

run-out table stresses or strains are ignored. The model will examine only a medium

carbon (< 0.40% ) steel and the resulting austenite to ferrite and ferrite plus pearlite phase

transformations. The bainite and martensite transformation kinetics will be left to future

workers. Transformation and cooling in the down-coiler is also outside the scope of the

model.

17

SCOPE A N D OBJECTIVES 3.2

3.2 Objectives

(i) Production of a heat transfer model of the hot strip on the run-out table,

from exit from the final stand of the finish mill until entrance into the pinch roller of

the down-coiler.

(ii) Determination of phase transformation kinetics for a medium carbon,

plain carbon steel of 0.34 % C.

(iii) Determination of individual and overall heat transfer coefficients for

laminar water bar spray banks.

(iv) Integration of the heat transfer coefficients and phase transformation

kinetics in an overall heat transfer model to predict coiling temperatures.

(v) Microstructure prediction1 for the coiled steel from dilatometer data and

the integrated model.

1 ferrite-pearlite ratios

18

PROCEDURE 4.1

4 PROCEDURE

4.1 Mathematical Model

The strip geometry assumed for this model is shown in Figure 2. The model has

been formulated for the Stelco Lake Erie Works Hot Strip Mill Run-Out Table, which is

shown schematically in Figure 3. The cooling water for this run-out table is delivered by

laminar water bar sprays over the top of the strip and water curtain spray for the bottom

of the strip. The cooling system consists of five banks of sprays with six headers in each

bank. The five banks cover the first half of the run-out table with banks one, two, and

three used as the main cooling banks and the fifth bank used to trim the strip temperature

to the desired down-coiler temperature. Bank four was being installed and was not in use

for the duration of this work.

On the run-out table hot steel strip moves at high speed and undergoes rapid

cooling. The significant phenomena that occur as a result are internal heat flow, variable

external heat transfer, phase transformation and associated heat generation. To

mathematically model the hot-strip on the run-out table, the following is required:

(i) basic physical description of the strip and the layout of the run-out table,

(ii) ....heat flow equations,

(iii) ...boundary conditions,

(iv) ....phase transformation and recalesence equations.

The heat flow equations are well understood and will be described in section 4.1

along with the basics of the mathematical model. The external environment the strip sees

19

PROCEDURE 4.1.1

varies down the length of the run-out table. Heat transfer occurs by convection and

radiation to the air as well as by convection and film boiling to the cooling water. The

various heat transfer regimes, the resulting heat transfer coefficients, and the theoretical

and empirical formulae for their calculation will be examined in section 4.2. The phase

transformations and recalescence as well as the methods for their characterization will be

exarnined in section 4.3 with the figures for sections 4.1,4.2, and 4.3 following in section

4.4.

4.1.1 Formulation

The basic unsteady state equation for a three dimensional control volume is,

k{s?+B?+^r- + v p c ' l a 7 + a7 + 3 7 j = P C ' 3 T

where the first three terms account for internal heat conduction, qg is the heat generated

by the phase change, and the last three terms involving the velocity, v, of the strip are the

heat flow due to bulk motion; the right hand side is the energy change in the volume as a

function of time.

qfis calculated by taking the fraction transformed for a given time step (which is

detailed in 4.3.4) and multiplying by the volume of one node. The calculated volume

transformed is used with the Zacay and AAronson[23] values for the heat generated by

phase transformation per mole along with a density value to produce a heat flux for a

20

PROCEDURE 4.1.1

given fraction transformed.

In order to simplify Eq.8 a number of assumptions about the physical geometry of

the hot strip as it travels on the run-out table were made:

(i) The strip is continuous and no distinction is made between the head end,

tail end, or central portion of the strip.

(ii) The process is operating at steady state and the temperature profile at a

fixed location is invariant with time.

(iii) Since the width to thickness ratio is large1, a zero temperature gradient is

assumed across the strip width perpendicular to the direction of travel.

(iv) Although a Biot number calculation based on an overall heat transfer

coefficient indicates that there should be no gradient in the z direction through the strip

thickness, the local heat transfer coefficient beneath a water spray is sufficiendy high to

produce internal gradients. Therefore,

(v) The rate of heat transferred into a stationary control volume due to bulk

motion of the strip is much greater than the rate of heat transfer by conduction so the

latter term in the x direction will be assumed to be negligible,

Thus the governing equation simplifies to

1 1 meter wide to 0.004 meters thick

21

PROCEDURE 4.1.1

, ,1ft? ^fdr) . . . 1 0

Therefore while the through thickness nodes must be solved simultaneously, the

steps along the axis of travel may be solved sequentially, which greatly simplifies the

model calculations.

The boundary and initial conditions for Eq.8 for a strip of thickness'd' are given

below.

Boundary Conditions, x > 0 , z = 0, z = d

-k^ = h(x)(T-TA) " A l

Initial Conditions

x = 0 , 0 £ z £ d T = Tj . . . 1 2

As can be seen in Eq.l 1 the heat removed from the surface is a product of the

temperature difference between the strip and cooling medium and a heat transfer

coefficient, h(x); the heat transfer coefficient is a function of the type of cooling at the

particular location which will be examined in section B.

The basic physical properties for steel were derived from the British Iron and Steel

Research Association data tables[8]. BISRA compiled values for specific heat, thermal

conductivity, density, thermal expansion, thermal diffusivity, and resistivity. The data for

density, specific heat, and thermal conductivity were examined for temperature

22

PROCEDURE 4.1.1

dependence over the conditions of the run-out table and while the density was found to be

relatively temperature independent2, all three were included as variables for each grade of

steel. The BISRA specific heat and thermal conductivity are strongly temperature

dependent A cubic spline interpolation of the BISRA[8] specific heat and density data

was used to provide equations for the model. The temperature dependence of the specific

heat data can be seen in Figure 1. This specific heat data includes the effects of the heat

generated by the phase transformation; for this model a specific heat value that is

independent of the heat of transformation is required since the latter has been

incorporated separately. As the variation of specific heat with temperature for

non-equilibrium conditions is not known a simple linear approximation of the austenite

and ferrite regions in Figure 1 was used. Figure 4 shows the linear extrapolations of the

specific heat of the gamma and alpha regions for 0.343 weight percent carbon steel.

Initially a weighted average of the specific heat values was to have been used with the

proportion of the gamma and alpha phases determining the proportion of the austenite

and ferrite specific heats used. As the specific heat values are only linear approximations

of discrete data points, a weighted average was viewed as having greater precision than

2 7.615 gm/cm ±0.0105 between 700 °C and 950 ' C

3 The BISRA data is an interpolation of the 0.23 weight percent carbon value and the 0.40 weight percent carbon values for plain carbon steel. An interpolation of the low alloy values gave similar results.

23

PROCEDURE 4.1.2

the data would allow. The model, therefore, uses the austenite specific heat value at

temperatures greater than the transformation start temperature and the ferrite value for

temperatures at or below this temperature.

The thermal conductivity data from the BISRA tables is described as a pair of linear

equations with an inflection point at a temperature that varies according to the carbon

content. The values for 0.06,0.08,0.23, and 0.34 weight percent carbon are shown in

Figures 5, 6,7, and 8 respectively.

The values for the heat generated by the phase transformation were taken as 776

cal/mole[23] for the austenite/fenite transformation and 1000 cal/mole[23] for the

austenite/pearlite.

4.1.2 Numerical methods

Equation 10, subject to the boundary conditions given in Eq. l 1, was solved

numerically by an implicit finite difference method. The strip thickness was discretized

into a series of nodes and finite difference equations were formulated for each node; the

equations are derived in Appendix Eq.A. l to Eq.A.8. Figure 9 shows the flow chart of

the computation scheme. The physical data, such as strip gauge and speed, cooling water

flow rates, spray position, run-out table length, and steel composition are inputs to the

model together with an initial steel temperature. The program computes the position

along the run-out table and the heat transfer conditions for that location are determined.

The coefficients for the tridiagonal matrix are calculated, the matrix is then solved and

the node temperatures are altered. The data is then output and the position counter is

24

PROCEDURE 4.1.2

incremented; if the down-coiler position has not been reached the process starts over with

a new position calculation. For the second and subsequent calculations the temperature

of all the nodes at that location are examined to determine if any are less than the

transformation start temperature. When the node temperature is below the transformation

start temperature the model becomes slightly more complicated. The fraction

transformed, and subsequently the amount of heat generated, and the resulting

temperature increase are a function of the temperature at which the transformation takes

place. The calculation of recalescence is therefore an iterative process which is repeated

until the difference in two succeeding temperatures is below an error value. This process

is exarnined in greater detail in section 4.3.

The choice of a time step and through thickness node size for the model was based

on the diameter of the laminar water bar. The time step for this model is a distance along

the strip divided by the strip velocity. The laminar water bar diameter at the header

nozzle is slighdy less than 40 mm and so to ensure that the step size is capable of

resolving an individual laminar water bar, the step size had to be at least less than half the

laminar water bar diameter or just under 20 mm. A step size of 10 mm was chosen so

that each laminar water bar would be represented by at least three steps. 200 nodes

through thickness were chosen after running various values for the number of nodes

through thickness with the model and a 10 mm step size. The results of the model tests

with various combinations of step size and through thickness nodes will be shown in

section 5.3.

25

PROCEDURE 4.1.2

The model testing and validation is obtained through comparison of predicted

thermal history and microstructure with plant data and the microstructure in down-coiler

samples and will be exarnined in Chapter 5 sections 5.2 and 5.3.

The model was written in FORTRAN and run on the University of British

Columbia Amdahl V8 mainframe computer with approximately 250 seconds CPU time in

an elapsed time of one-half hour. The model was also run on a C O M P A Q portable Ll

personal computer with a 80286 CPU and an 80287 math co-processor, with an

approximate running time of 3 1/2 hours for a 200 node by 10mm step size configuration.

Due to the different floating point representations of the two machines, double precision

was necessary for the Amdahl while only single precision was needed for the Personal

Computer.

26

PROCEDURE 4.2

4.2 Heat Transfer Coefficient

The magnitude of the heat flow from the steel surface to

the surrounding fluid, which consists of air, water, or some combination of the two, is

deterrnined by the local heat transfer coefficient. The hot steel strip experiences six

different cooling regimes as it proceeds along the run-out table, as shown schematically

in Figure 10 and described as,

(1) air cooling on the top and bottom of the strip,

(2) air cooling on the top of the strip with roller contact below,

(3) cooling by film boiling on top and air cooling below,

(4) cooling by film boiling on top and roller contact below,

(5) laminar water bar cooling on top and roller contact below,

(6) cooling by film boiling on top and water curtain cooling below.

To describe the six cooling regimes, the following five heat transfer coefficients are

needed,

(a) convection and radiation cooling to air,

(b) conduction to the water cooled support rollers,

(c) convection to the vapour film surrounding the laminar water bar,

(d) convection to the laminar water bar,

(e) convection to the water curtain.

27

PROCEDURE 4.2.1

The five heat transfer coefficients are developed from theoretical relationships

found in the accelerated water cooling Uterature, examined in section 4.2.1, and by back

calculation from plant temperature measurements, examined in section 4.2.2.

4.2.1 Calculation from Literature

From an examination of the literature on accelerated cooling of hot steel strip it is

clear that relatively few studies have been performed for the determination of heat

transfer coefficients between the moving strip and the cooling water, either in plant or by

laboratory simulation. The plant trial-derived values are best illustrated by the Tacke et

al.fl] paper in which 1.8 ±0.3 kW/m 2 K is reported for an overall heat transfer coefficient

for a water curtain cooling bank and a value of 1.3 ±0.25 kW/m 2 K is given for an overall

heat transfer coefficient for a laminar water bar. These two values are for an entire bank

of water sprays and include convective cooling in the contact zone beneath a water

curtain or water bar and cooling by film boiling in the surrounding region.

The laminar water bar heat transfer coefficient can be calculated using the Hatta et

al.[5] Eq. l .

While the film boiling heat transfer coefficient is calculated using the Kokada et

al.[7] relationship Eq.2.

a, = 0.063* - *Re**Pr ,8) ... i •WB

2420-21.7(7V)

(Ts ~ TSAT)* . . . 2

a™ = 200*

28

PROCEDURE 4.2.1

The temperature at which a transition from Eq. 1 type cooling to Eq.2 type cooling

takes place is calculated with Eq.3.

r 5 -1150 . . . 3 T = — 1 cm _g

The Reynolds number, Prandd number, and k are temperature dependent and

calculated internally in the model. T w , the temperature of the water in the film boiling

section, is greater than or equal to 68"C by the definition of Tcxst- At one atmosphere

pressure T w can be assumed to have a maximum value of 100'C. Therefore, T w must

always have a value between 68*C and 100#C. An average of these two values was used

in Eq.2. Figure 11 plots the film boiling heat transfer coefficient as a function of the

difference in temperature between the water and the steel surface. The values are plotted

for 68*C, 100'C, and the average value 84*C. The two Berensen[24] values are for film

boiling on a horizontal surface for a water film-steel surface temperature difference of

816*C and 636°C; these represent the average temperature difference realized just before

and after the water cooling zones on the run-out table. The Berensen values agree with

the Kokada et al. Eq.2 values calculated with a water temperature of 68 °C.

A water curtain cooling heat transfer value of 17 kW/m 2 °C has been reported by

Colds and Sellars[2], assuming the existence of a surface oxide layer in order to produce

a 'black zone' that will appear black at the calculated temperatures. They have employed

a heat transfer coefficient for film boiling cooling of 150 W/m 2 °C. This value is less than

the Kokada et al.[7] value for a water temperature of 100 °C, is much less than the

Berensen[24] values, but, agrees quite well with the Farber and Scorah[3] values for

29

PROCEDURE 4.2.2

small diameter wires. Eq.2 predicts a value of 520 W/m 2 °C for a steel temperature of

1000 ' C which increases to 990 W/m 2 o C for a steel temperature of 500 'C. The water

curtain cooling for the hot strip is only on the underside of the strip; film boiling cooling

does not occur as the water immediately falls off of the strip. For this reason, the Colds

and Sellers film boiling heat transfer coefficient was ignored and the Kokada et al. film

boiling heat transfer coefficient was used for the top surface along with Eq. 1 for the

laminar water bar in this model.

There does not appear to be any literature describing heat transfer at the support

roller in a hot strip mill run-out table. However, Diener and Drastik[7] reported some

data on heat transfer to support rollers in the secondary cooling zone of a continuous slab

caster. For a water spray cooled roller4, a heat flow of 75 kW/m with an average roll/slab

temperature difference of 900 *C was given resulting in an 83 W/m 2 'C heat transfer

coefficient As there is no available data on the size of the roller/strip contact area, a

value of one model step size has been used.

4.2.2 Calculation from Plant Data

An alternative method of generating heat transfer coefficients is to use the

mathematical model to back calculate specific machine dependent values from in-plant

surface temperature measurements. To gather this data, a C O M P A Q portable computer

with a Data Translation DT2805/DT707T data acquisition board was connected to four

pyrometers positioned along the run-out table of the Stelco L E W Hot-Strip Mill. All the

4 a 0.3 meter diameter roll of 1.75 meters length, 16 Cr and 44 Mo.

30

PROCEDURE 4.2.2

pyrometer and plant engineering log data was stored on 5 1/4 inch, high density, floppy

diskettes. The four pyrometers, PI, P2, P3, and P4, are shown schematically in Figure 3

and were mounted to the hand rail of the walkway over the run-out table water cooling

bank section. The four walkway pyrometers were supplied and installed by Stelco

Research and Development specifically for trial data acquisition; calibration for these

units was done by Stelco with an IRCON portable black body. This device was also used

to calibrate the three plant pyrometers FEXT, ROT, and DC. Units PI and P2 were

IRCON R series two colour units with a range of 700*C to 1400'C, while units P3 and P4

are single colour IRCON 6000 units with a 500*C to 1500*C range. As these units were

only in place for the twelve runs during the trials, air and water blow-offs were not in

place at the strip locations measured.

Additional data in the form of the engineering logs for the trials was available from

the rolling mill computer. This data listed the speed, gauge, average number of cooling

sprays, finish mill exit temperature, and down-coiler temperature of the strip along with

the standard deviations of these values for one run. Temperature data was also available,

in the plant engineering log for the three permanently installed plant pyrometers, FEXT,

ROT, and D C which are shown schematically in Figure 3. The plant pyrometers were

IRCON 2000 series with a 700°C to 1100'C range for the F E X T pyrometer and a 500°C

to 800'C range for the ROT and D C pyrometers. The plant pyrometers were aimed at

areas with water and air blow-offs and were recorded in the engineering logs with all

water, speed, and physical data taken at one second intervals.

31

PROCEDURE 4.2.2

The heat transfer coefficients were calculated by using the plant pyrometer

temperature data for strips that were coiled at a high enough temperature that

recalescence effects did not occur until the down-coiler. A value for a heat transfer

coefficient was input into the model and the resulting thermal profile compared to the

pyrometer data. The best fit with the pyrometer data will be taken as the heat transfer

coefficient for that set of conditions. An overall heat transfer coefficient for an entire

cooling bank of six headers was calculated as was a value for individual header laminar

water bars.

32

PROCEDURE 4.3.1

4.3 Phase Transformation Characterization

4.3.1 Material

Three grades of steel were chosen for the phase transformation kinetics

characterization due to availability of test samples and plant temperature data. These

were a 0.054 weight percent carbon, a 0.074 weight percent carbon with 0.024 weight

percent niobium, and a 0.343 weight percent carbon. These steels will be referred to as

the 0.05 carbon, 0.07 carbon with niobium, and 0.34 carbon steels respectively for the

rest of this thesis. The chemical composition for all three steels is listed in Table L

4.3.2 Metallography

Down-coiler samples were obtained from Stelco for the various steel chemistries

examined. These were transversely sectioned, polished to a five micrometer diamond

surface, etched with 5% Picral etch and photographed on Polaroid type 55 positive

negative film.

The percentage ferrite for the down-coiler was determined with a Wild-Leitz Image

Analyzer, using five randomly selected sample areas per specimen.

It was necessary to determine the percentage ferrite and percentage pearlite in the

continuously cooled dilatometer test samples from metallographic studies; a visible

transition from ferrite to pearlite in the dilation-time plots was not observable at the high

cooling rates used.

33

PROCEDURE 4.3.3

4.3.3 D i la tometer

The diametral dilatometer, which measures the change in diameter of a tubular

sample divring isothermal or continuous cooling conditions has been previously described

by Hawbolt et al.[4,5] In this device, a thin walled tube is used as a specimen and the

diametral dilation is measured. A thin walled tube is used to minimize internal

temperature gradients and to provide the same cooling rate around the periphery of the

specimen. A control thermocouple is attached to the outside of the tube at the plane of

the dilation measurement The diameter change, as a function of time and temperature, is

recorded and is used to provide phase transformation kinetics and transformation start

times or temperatures as a function of time, temperature, and cooling rate. The AC3

temperature of 785*C and the AC1 temperature of 723*C, were calculated using the

Andrews[25] formula and checked using a very slow heating rate for the 0.34 weight

percent carbon sample. The experimental values of 800°C for AC3 and 733*C for AC1

are shown in the temperature-time plot of Figure 12. All samples were heated to 850" C

and held for 3 minutes. The samples were then air cooled to 820°C and held for 1

minute. The isothermal test samples were then rapidly cooled to the test temperature

while the continuous cooling samples were cooled at a constant rate for the duration of

the test.

As the down-coiler strip was too thin for preparation of dilatometer samples, the

tubular samples were machined from transfer bar taken at the end of the rougher rolling

stage, which precedes the finish rolling stage. The transfer bar samples were cut to

34

PROCEDURE 4.3.3

approximate sample dimensions and then fully annealed.5 It is recognized that these

samples do not duplicate the grain size and thermal history of the steel as it arrives at the

run-out table. However, the transfer bar does have the same chemistry. The isothermal

transformation kinetics obtained from the annealed transfer bar samples are characteristic

of a given austenitizing condition (grain size).

4.3.3.1 Isothermal Dilatometer Tests

The Avrami coefficients, b, and, n, are determined from data generated during

isothermal diametral dilation tests. The isothermal dilatometer tests measure diametral

dilation versus time at a constant temperature. From the dilation-time data the onset of

dilation change is taken as the transformation start time, or tA V» as shown in Figure 13.

The fraction transformed, for ferrite or pearlite, which is proportional to the diametral

dilation, is calculated by dividing the dilation value at time, t, by the dilation value

associated with completion of each transformation. The equilibrium fraction ferrite that

will form at a given temperature is calculated from the Fe-C phase diagram using a lever

law and an extrapolation of the y and lines to temperatures below the TAC1 using

the Kirkaldy et al.[30] equations. The fraction transformed that corresponds to this

equilibrium fraction ferrite (AD(ferrite) in Figure 13) is used as the ferrite stop, pearlite

start point. Thus, the total fraction pearlite that will form is one minus the total fraction

ferrite. For example, if the total fraction ferrite that forms at 680°C is 0.45, (AD/AD X =

5 30 minutes at TAc3 + 50°C, followed by furnace cooling.

35

PROCEDURE 4.3.3

0.45 ), then the fraction ferrite transformed at a given time, t, is the measured ferrite

dilation divided by 0.45 which gives the ferrite fraction transformed. The fraction

transformed for pearlite is obtained by dividing the measured pearlite dilation by 0.55.

The transformations can be described using the Avrami equation, Eq.5, in the form:

The Avrami coefficients, n, and, b, are calculated from the graph of lnln(l/(l-X)) versus

ln(t); with, n, as the slope and ln(b) as the intercept, where ln(t) = 0. This assumes that n

is a constant value during the isothermal test, as is indicated by the experimental data.

4.3.3.2 Continuous Cooling Tests

Continuous cooling tests were performed by passing a controlled flow of cooling

gas over the interior and exterior surface of the hollow tubular sample while measuring

dilation and temperature versus time. Typical data, shown in Figure 14, is used to

calculate the transformation start temperature as a function of cooling rate. The

transformation start temperature (or time) for each cooling rate was determined for a

range of cooling rates equivalent to those obtained on the run-out table. This temperature

was calculated using the diametral dilation versus time and the temperature versus time

data shown in Figure 14. AD/AT is calculated using six dilation values and the

corresponding six temperature values. The difference between the average of the first

three dilation values and the second three dilation values is divided by the difference

between the average of the first three temperature values and the second three

. . .13

36

PROCEDURE 4.3.4

temperature values. Thus at some time, t,

(AD/AT), = | - r „ 2 , r „ 1 + r , y d ^ J

...14

the point at which AD/AT changes slope is taken as the transformation start temperature,

as shown in Figure 15. This is an effective procedure for determining the transformation

start temperature (or time) because both dilation and temperature are affected by the

onset of transformation; the heat of transformation causes recalescence in the

temperature-time response.

4.3.4 Phase Transformation Model Calculations

The model incorporates relationships describing the calculated transformation start

temperature and the experimental percentage ferrite formed as a function of the cooling

rate.

The phase transformation rate at any time step in the model is assumed to be a

function of the fraction transformed and the temperature at which the transformation

takes place; this assumes that the phase transformation is additive. The fraction that

undergoes transformation during one time increment generates a finite amount of heat

which in turn raises the temperature of the node. This requires an iterative solution to

determine the temperature and amount transformed; this is detailed in the flow chart

shown in Figure 16.

37

PROCEDURE 4.3.4

The fraction transformed in the previous time step, Fx(k-1), and the current node

temperature, T(k), are used to calculate a virtual time, tv, the time that would be required

to produce Fx(k-l) at temperature T(k). Eq.5 is rearranged to deterrnine the virtual time,

, .1 . . . 15 [v - .

V ~ b )

The time step, dt, is added to t v and a new fraction transformed, Fx(k), is calculated

for temperature T(k). The difference between Fx(k-1) and Fx(k) is the fraction

transformed, dFx(k), for the time increment, dt, and is used to calculate a new

temperature T(k)' based on T(k) and the heat generated by the new fraction transformed,

dFx(k). Using T(k)' and Fx(k) a new virtual time t v' is calculated and in a similar

manner a new temperature T(k)". T(k)" - T(k)' is compared to an acceptable error value

(0.05 *C). If the difference is lower than O.OS'C the loop is exited. If the temperature

difference is greater than 0.05°C, T(k)' becomes T(k)" and the process repeats until the

0.05°C limit is satisfied.

It should be noted that the model uses through-thickness nodes to model an

observed rebound of surface temperature after a cooling spray. At strip velocities ranging

from 5 m/s to 7 m/s the re-heating times are too short for a reverse transformation to

austenite and the temperature is usually too low. For this reason, the model assumes that

no transformation will take place if the temperature difference between the current step

and the previous step is positive, that is if the node is increasing in temperature there is no

reverse transformation.

38

PROCEDURE 4.4

4.4 Tables and Figures

Low carbon Low carbon-

niobium

Medium carbon

c 0.054 0.074 0.343

Mn 0.270 0.540 0.700

P 0.006 0.005 0.008

S 0.011 0.008 0.009

Si 0.020 0.017 0.009

Cu 0.044 0.021 0.021

Ni 0.007 0.008 0.006

Cr 0.062 0.012 0.023

Mo 0.002 0.003 0.003

V 0.000 0.000 0.000

Nb 0.000 0.024 0.000

A l 0.030 0.047 0.043

Table I Composition of the three steel chemistries intended for use in this study.

39

40

PROCEDURE 4.4

03 CD k_

3

CO

03

c 'E 03

8 ©-

Q_ (5-V4

p—

o DC

®4 CO Q_ ©4 CM CL

CL ej ©-fc v — L U

o o O c 5 o Q

c CO m CD -•—» CO

-ee-oo oo olo 'c

Figure 3 Schematic of the STELCO Lake Erie Works Hot Strip Mill Run-out Table

41

PROCEDURE 4.4

Specific Heat W/kg C

Figure 4 Specific Heat as a Function of Temperature for a 0.34 % carbon steel, BISRA,

with out phase transformation.

42

PROCEDURE 4.4

o o

Thermal Conductivity W/mC

Figure 5 Thermal Conductivity as a Function of Temperature for a 0.06 % plain

carbon steel, BISRA

43

PROCEDURE 4.4

44

PROCEDURE 4.4

o o

o o CM

o o o

o

CD

o o co

o o

o o CM

CO

E


Figure 7 Thermal Conductivity as a Function of Temperature for a 0.23 % plain carbon

steel, BISRA

45

PROCEDURE 4.4

o


Figure 8 Thermal Conductivity as a Function of Temperature for a 0.34 % plain carbon

steel, BISRA

46

PROCEDURE 4.4

47

PROCEDURE 4.4

Film Boiling

Laminar: y Water

Bar

Film Boiling

Film Boiling

Water Curtain

Roller

Roller

Roller

Figure 10 The six types of cooling regime experienced by the steel strip

48

PROCEDURE 4.4

i i i i i i i i i i i i i i r Q

Heat Transfer Coefficient

Kw/mC 2

Figure 11 The various Film boiling heat transfer coefficients from Kokada et al.[6] for

three cooling water temperatures with two values from the Berensen[24] horizontal

surface boiling equation.

49

PROCEDURE 4.4

PROCEDURE 4.4

• 1 Dilation

Figure 13 A typical dilation versus time plot for an isothermal dilatometer test.

51

PROCEDURE 4.4

§

Dilation

^ Temperature ( Q

Figure 14....A typical dilation and temperature versus time plot showing transformation

start and finish times.

52

PROCEDURE 4.4

53

PROCEDURE 4.4

T(k) Fx(k-1) tv.Fx(k)

START

CALCULAT6 T(k)'

t' ,Fx(k) v CALCULATE

T(k)"

STOP Figure 16 Flow sheet for the iterative solution of the fraction transformed as a function

of temperature.

54

RESULTS & DISCUSSION 5.1.1

5 R E S U L T S & D I S C U S S I O N

5.1 Heat Transfer Coefficient

5.1.1 Literature

5.1.1.1 Laminar Water Bar Cool ing

The Hatta et al.[5] laminar water bar heat transfer coefficient calculated using

Eq. l is sensitive to the value of the contact radius, r. Colds and Sellars[2] in their

water curtain heat transfer calculation have noted that a value of two to three times the

water curtain width seemed reasonable for a contact diameter. To examine the effect

of steel surface temperature on the contact radius or 'black zone' diameter a simple one

dimensional model of laminar water bar cooling, using Equations 1 and 3 and the Hatta

et al.[5] heat flow and thermal layer calculations (in the appendix section 8.3) was used

to calculate the radius of the 'black zone' as a function of a constant steel surface tem

perature. Figure 17 shows the results of this model calculation for steel surface tem

peratures in the range from 400°C to 1100'C. For steel surface temperatures greater

than 600°C the 'black zone' radius changes slowly with temperature. An average

value, 33.7 mm, was chosen for the temperature range of 700°C to 900'C; this is the

range of interest on the run-out table. The heat transfer coefficient for various contact

radii between 0.1 mm and 100 mm was calculated and the results are presented in Fig

ure 18. The heat transfer coefficient values seen in Figure 18 are stable for any contact

radius greater than 20 mm with an average heat transfer coefficient value of 11

kW/m 2 o C calculated for a contact radius of 33.7 mm. The thermal profile model

55


combines the Colas and Sellars[2] water curtain heat transfer coefficient of 17

kW/m 2 o C, the laminar water bar heat transfer coefficient calculated with Eq. l , and a

film boiling heat transfer coefficient calculated with Eq.2.

5.1.1.2 Film B oiling Cooling

The film boiling heat transfer coefficient calculated with Eq.2 was shown, in Fig

ure 11, to be sensitive to the cooling water temperature, T w . To assess the sensitivity

of the thermal profile model predictions to this parameter, the through strip thermal

profile was modeled using film boiling heat transfer coefficients calculated with Eq.2.

The minimum, average, and maximum values for the water film temperature of 68'C,

84*C, and lOO'C respectively were used along with the laminar water bar heat transfer

coefficient calculated in E q . l 1 . The results of this model are shown in Figure 19, it is

evident that the model predictions of strip surface temperature are only mildly sensitive

to the water film temperature. The predicted surface temperatures at the down-coiler

location are 769*C, 774°C, and 778*C for the respective water film temperatures, T w , of

68°C, 84°C, and 100'C, while the measured pyrometer value at that location is 718*C

±12 'C.

5.1.1.3 Support Roller Contact Cooling

The Diener and Drastik[7] support roller, conduction cooling, heat-transfer coeffi

cient is an approximation for a different physical system; therefore to evaluate the sen

sitivity to the 83 W/m 2*C value the model was employed. A ±50% change in the

1 33.7 millimeter contact radius

56


support roller conduction heat transfer coefficient results in a top surface, down-coiler

location, temperature prediction of 772"C and 775*C respectively. The temperature

predictions are shown in Figure 20, as is the predicted value of 773*C for the Diener

and Drastik[7] heat transfer coefficient of 83 W/m 2 °C.

5.1.1.4 Combined Cooling

The literature derived heat transfer coefficient values were combined into one

model to test the predictions in relation to the plant pyrometer data from section 5.1.2.

The laminar water bar heat transfer coefficient from Eq. l , the film boiling heat transfer

coefficient from Eq.2, and the Diener and Drastik[7] support roller contact heat transfer

coefficient were employed as input to the thermal profile model and used to predict top

surface strip temperatures under a variety of cooling conditions from the plant data.

For the 0.05 carbon, 720*C target coiling temperature, the model predictions for

the strip surface are 774"C and 765"C at the down-coiler position for the 3.89 mm and

2.69 mm gauges respectively. These results are shown in Figures 21 and 23 and it is

seen that the corresponding in-plant temperature measurements at the same location are

considerably lower with values of 718*C ±12*C and 717"C ±8"C respectively. For the

0.07 carbon with niobium grade with gauges of 3.89 mm and 2.69 mm, the model pre

dicted surface temperatures at the down-coiler are 776*C and 758°C, the results are

shown in Figures 22 and 24. The pyrometer readings at the corresponding positions

in-plant are 710*C ±15°C and 714*C ±8"C. In all four cases the in-plant pyrometer

measured values are 40° C to 55" C lower than the model predicted values.

57


The Hatta et al.[5] laminar water bar and the Kokada et al.[6] film boiling heat

transfer coefficients were experimentally derived using a stationary stainless steel plate

under a water nozzle as opposed to the plant cooling conditions of a moving plain-

carbon strip under a water nozzle. The composition of the plate should have little or

no effect, while strip movement will elongate the water contact area and this may

increase cooling. The overall result would be lower pyrometer temperatures than pre

dicted by the model and these lower predictions are what has been observed with the

model calculations based on literature derived heat transfer coefficients.

The model predictions for the cooling conditions experienced by a 3.89 mm 0.05

carbon steel with a 630*C target coiling temperature are shown in Figure 25. These

cooling conditions were modeled to determine what effect the lower coiling tempera

ture would have on the model prediction-pyrometer temperature difference and the

model prediction of 725*C is much higher than the pyrometer value of 629*C ±14*C.

5.1.2 Plant Trials

The run-out table history for a total of twelve strips is listed in Table Ua, lib, and

lie. The variables for the twelve runs were; 0.05 percent carbon and 0.07 percent car

bon with 0.024 percent niobium, 3.89 millimeter and 2.69 millimeter gauges, and three

aim coiling temperatures of 720'C, 630*C, and 550°C. Basic run-out table parameters

for the trial coil runs, along with average temperature values and standard deviations

are also listed in Table II. An example set of temperature readings for each pyrometer,

for one trial, taken at one second intervals, are plotted in Figure 26. The readings for

58


the three permanent plant pyrometers, F E X T , ROT, and DC have been compensated at

the pyrometer for emissivity. PI and P2 being two colour units need no emissivity

compensation. P3 and P4 are recorded at an emissivity of 1.00 and therefore must be

compensated for a strip emissivity of 0.80.

5.1.2.1 Overall Heat Transfer Coefficient

5.1.2.1.1 Calculation

An overall heat transfer coefficient was calculated using the plant temperature

data. At the cooling rates experienced on the run-out table, between 40*C/s and 50*C/s,

the phase transformation starts at a lower temperature than in an air cooled sample2.

To avoid including phase transformation effects in the calculation of an overall heat

transfer coefficient, only those coils with an aim coiling temperature greater than 700*C

were used for the calculations. Using the thermal profile model with the cooling condi

tions experienced by the 0.05 carbon, 3.89 mm gauge strip, coiled at 718°C ±18*C vari

ous values for an overall laminar water bank heat transfer coefficient were evaluated.

A value of lkW/m 2 e C gave the best fit with a predicted temperature of 716"C for the

strip surface at the down-coiler pyrometer location. The plant pyrometer value of

7 1 8 * C ± 1 8 * C i s shown with the model predictions in Figure 27.

To evaluate the effect of using the lkW/m 2*C overall effective heat transfer coef

ficient with the cooling conditions experienced by other strips, the thermal profile

2 This is examined in the continuous cooling section of section 5.2.4

59


model was run for a number of other grade/gauge combinations. For a given strip,

depending on the finish mill exit temperature, gauge, strip speed, and desired coiling

temperature; the number of headers turned on in each bank is varied. In the model the

overall effective heat transfer coefficient is applied over the region spanning the head

ers that were turned on during that particular run.

Model predicted top surface temperatures have been compared with in-plant tem

perature measurements for a 3.89 mm gauge strip of 0.07 carbon with niobium, a 2.62

mm gauge strip of the same composition, and a 2.62 mm gauge strip with 0.05 carbon,

and the results are shown in Figures 28, 29 and 30. The 0.07 carbon with niobium

gives predicted values of 715'C for the 3.89 mm gauge and 715°C for the 2.62 mm

gauge. This is close agreement with the respective pyrometer values of 710°C ±15*C

and 714'C ±8*C. The 0.05 carbon, 2.62mm gauge prediction of 716*C is also in close

agreement with the pyrometer value of 718*C ±8'C. The lkW/m 2 °C heat transfer coef

ficient appears to be valid for the plant cooling data.

5.1.2.1.2 Sensitivity

To check the sensitivity of model predictions with the overall heat transfer coeffi

cient the thermal profile model was run using the 0.05 carbon, 3.89 mm gauge cooling

conditions for the two lower coiling temperatures of 630°C and 550°C, in which phase

transformation should have some effect. Figure 31 shows the predicted surface value at

the down-coiler position of 620° C as well as the pyrometer value of 620° C ±14°C for

the 630°C target coiling temperature sample. Figure 32 displays the model down-coiler

60


position temperature prediction of 656"C for the 550*C target coiling cooling conditions

as well as the measured pyrometer value of 559°C ±16"C. These two figures show no

discernible trend for the relation of the overall heat transfer coefficient predictions and

the coiling temperature.

The effect of changing the overall heat transfer coefficient on the model predic

tion is shown in Figure 33, with a ±0.2 kW/m 2 o C change resulting in predicted surface

temperature values of 736'C and 700°C for a respective decrease and increase of 0.2

kW/m 2*C in the overall heat transfer coefficient Both values are within the ±18*C

deviation of the 718°C pyrometer value.

For all six figures3 one trend emerges; the model consistently predicts a value

slightly less than the measured PI pyrometer temperature, slighdy greater than than the

measured P2 pyrometer value, and a much higher4 value than the P3 and P4 measured

pyrometer readings. The model predictions agree quite well, however, with the

readings for the three permanently installed plant pyrometers. The consistent deviation

of the readings from the pyrometers installed for the trials makes the data of doubtful

value. This deviation is possibly due to the lack of air and water blow-offs, which are

employed at the permanent pyrometer locations, resulting in an unclear view of the

strip surface for the temporary pyrometers.

3 Figures 27 through 32.

4 100°C to 200*C

61


5.1.2.2 Individual Heat Transfer Coefficient

for Laminar Water Cooling

Individual heat transfer coefficients for laminar water bar sprays were determined

based on an initial value of 10kW/m 2 ,C, approximately the same as that calculated by

Hatta et al.[5] for an average contact radius. This value, along with the Colds and Sel-

lars[2] water curtain heat transfer coefficient of 17kW/m2*C, and the Kokada et al.[7]

film boiling heat transfer calculation from Eq.2, was input to the thermal profile model.

The temperature distribution through the strip was predicted for a 0.05 weight percent

carbon steel of 3.89mm gauge and with a 720*C target coiling temperature. The

results are presented in Figure 34. The 10kW/m2*C heat transfer coefficient resulted in

a predicted down-coiler position surface temperature of 741°C. The same model, under

similar conditions, with a 20kW/m2*C laminar water bar heat transfer coefficient, yields

a prediction of 730*C while a 5kW/m 2 °C laminar water bar heat transfer coefficient

results in a prediction of 746'C as compared to the pyrometer reading of 718*C±12"C.

The large changes in the individual laminar water bar heat transfer coefficient result in

temperature prediction changes that are less than the deviation of the pyrometer

reading. Under these conditions an accurate individual heat transfer coefficient value

cannot be calculated.

62


5.2 Phase Transformation

5.2.1 Material

Of the three steels intended for use in this study, two were not amenable to dilato-

metric characterization with the current machine configuration. To date, the kinetics of

the isothermal and continuous cooling austenite decomposition to ferrite and ferrite plus

pearlite have been characterized only for the plain-carbon grade. Thus, all the model

phase transformation calculations pertain to the 0.34 carbon steel. The in-plant cooling

conditions available for this steel are listed in Table LTf.

Modifications are currently being made to incorporate the diametral dilation mea

suring capability on a Gleeble high temperature testing machine. The isothermal and

continuous cooling decomposition kinetics of the low carbon steel will be measured by

other workers when these modifications have been completed.

The 0.07 carbon with niobium grade presents a problem. Le Bon et al.[26], in a

study of the recrystallization of niobium and plain carbon steels present data showing a

50 percent static recrystallization time for a niobium-free grade of approximately 1 sec

ond with 100 percent recrystallization by 2 seconds at a temperature of 900" C. In the

temperature range of interest for run-out table cooling, 700*C to 900*C, a niobium

bearing HSLA steel can have a 50 percent recrystallization time two orders of magni

tude higher, or approximately 100 seconds. The strip velocity on the run-out table is

such that the total transit time is approximately 20 seconds. Thus, the 0.07 carbon with

niobium grade is entering the down-coiler in an unrecrystallized condition. The present

63


dilatometer setup cannot duplicate this unrecrystallized condition and therefore cannot

simulate the appropriate run-out table conditions for the HSLA steel. With the incorpo

ration of the dilatometer into the Gleeble system it is hoped that this problem can be

resolved.

5.2.2 Isothermal Cooling Tests

Figure 35 shows isothermal dilatometer results for a 673°C test The horizontal

line indicates the fraction of ferrite formed, as deterrnined from the extrapolated lines

of the phase diagram {Kirkaldy et al.[30}. The fraction of the equilibrium amount of

each phase formed is obtained by dividing the measured dilation by the maximum(equi-

librium) dilation for each product phase. The results are shown in Figure 36 for both

the ferrite and pearlite transformations. Figure 37 shows the ferrite data for a typical ln

ln ( 1 / (1-FX)) as a function of ln( t) plot for an isothermal temperature test of

673*C. The slope of a best fit line is the Avrami coefficient value, n, while the inter

cept at ln(t) = 0 is the natural logarithm of the Avrami coefficient, b. Figure 37 is for

the austenite to ferrite transformation at 673*C and is typical5 of the isothermal

transformation data for the formation of ferrite in plain-carbon steels. The ln(b) values

for the ferrite and pearlite transformation, for a range of isothermal temperatures are

plotted in Figures 38 and 39 respectively. Figures 40 and 41, show the Avrami time

exponent, n, for the ferrite and pearlite transformations respectively.

5 See Hawbolt et al.[16]

64


The additivity concept requires, n, to be a constant, independent of temperature.

Assuming that, n, is a constant and that the scatter in the test data is due to experimen

tal variation, an average, n, value for ferrite, n f, and pearlite, fip, is shown in Figures 40

and 41 respectively. Using the average values, n f and fip, new ln(bf) and ln(bp) values

were calculated for each isothermal test Figure 42 shows the temperature dependent

values of ln(bf) for fif equal to 1.25. Figure 43 shows the calculated values for ln(bp)

with fip equal to 1.14. The best fit equations for this new data are shown in the respec

tive figures and will be used in the model to characterize the Avrami coefficient, b.

The Avrami time exponent, n, and the ln b parameter for the ferrite and pearlite

transformations are in good agreement with data reported for other steel grades by

Campbell[27]. Figure 44 shows good agreement for the n f and fip values with various

grades of steel, while the ln(bf), and ln(bp) values in Figures 45 and 46 indicate that a

linear fit best describes the parameters.

5.2.3 Continuous Cooling Tests

Figure 47 shows the dilation-time and temperature-time data for a continuous

cooling dilatometer trial, with a cooling rate of 27°C/s. The AD/AT values produced

from this data are shown as a function of time in Figure 48. The temperature-time data

in Figure 47 gives a transformation start temperature of 704"C for the transformation

start time deterrnined in the AD/AT plot, Figure 48. The undercooling, the difference

between the continuous cooling rate transformation start temperature and the equilib

rium value of 785*C; is listed in Table l Y and shown in Figure 49 as a function of the

65


cooling rate.

5.2.3.1 Metallography

Figure 50 shows the average fraction ferrite as a function of cooling rate for the

sectioned and etched continuous cooling dilatometer samples; the linear best fit line is

also shown. The best fit equation for fraction ferrite was used in the model to calculate

the total fraction ferrite formed as a function of the cooling rate, which was input as an

initial model parameter. The fraction ferrite was determined at five random locations

on the cross section of each tubular sample using the quantitative analysis capability of

the Wild-Leitz image analyzer. A typical microstructure is shown in Figure 51. The

wide scatter is due to the inhomogenous nature of the samples which exhibit a micro-

structural banding, as shown in Figure 52. This is attributed to segregation associated

with solidification during continuous casting. It is apparent that rough rolling in the

plant and a full furnace anneal6 during dilatometer sample preparation was insufficient

to produce a homogenous product

An overall average austenite grain size of 29 um ±8 urn was calculated from the

individual values listed in Table IV. Counting the grains was very difficult as only in

the high cooling rate samples were the grains easily recognizable as large areas of pear

lite outlined by a small fraction ferrite. At slow cooling rates it was difficult to deter

mine if each pearlite cluster represented one prior austenite grain or if multiple clusters

6 30 minutes at a temperature 50'C above the A3 temperature followed by a furnace cool

66


were the result of a single austenite grain. For this reason while the local deviation

from the average ( 18um to 37um) is greater than expected for the one austenitizing

condition experienced by all of the samples, an average prior austenite grain size value

was calculated.

A sample of down-coiler material with the same nominal composition ( compo

sition listed in Table V ) as the transfer bar samples used in the dilatometer tests was

polished and etched for Wild-Leitz image analysis of percentage ferrite and prior

austenite grain size, with the results listed in Table V and the structure shown in Figure

53. However, the chemical analysis does indicate that the down-coiler sample contains

more C and Mn then does the transfer bar sample. This would encourage more pearlite

formation for a given cooling rate, consistent with the microstructure observed in Fig

ure 53. The percentage ferrite value of .16 ±.03 compares with the continuous cooling

test samples for a cooling rate of 90*C/s which is higher than that experienced by the

strip. The prior austenite grain size of 35 um ±9 um is in good agreement with the

continuous cooling sample value of 29 um ±8 um and supports the use of transfer bar

samples to characterize the conditions in a down-coiler sample for these experiments.

5.2.3.2 Coil ing Temperature

Samples from the 720*C and the 630*C target coiling temperature, 0.05 carbon

continuous cooling dilatometer tests were examined to determine if the coiling tempera

ture had an observable effect on the grain size. The four samples were polished to 5

67


(im diamond and etched for 20 seconds with 2 percent Nital. The results are listed in

Table VI and do not show an overall effect on grain size attributable to coiling temper

ature.

68


5.2.4 Model Phase Transformation Calculations

Using the industrial data, listed in Table HI, obtained from Devadas[28] for the

0.34 weight percent carbon grade, the austenite decomposition kinetics from section

5.2.3, the phase transformation model, and the calculated overall heat transfer coeffi

cient from section 5.1.2, a series of thermal profile predictions were made. Run one,

shown in Figure 54, predicted a down-coiler position, (DC), strip top surface

temperature of 706"C at the run-out table location that has a pyrometer reading of

686*C ±8*C. Figures 55 and 56, for runs two and three respectively, result in similar

agreement with predictions of 699'C and 700"C versus pyrometer readings of 680"C

±7°C and 676'C ±11 *C respectively. The percentage ferrite and transformation start

temperature for these model predictions were calculated for a cooling rate of 7"C/s,

which is the strip surface cooling rate prior to the transformation of the strip. From

Figures 54, 55, and 56 it was seen that the strip undergoes two basic cooling regimes

on the run-out table, one under the laminar water bar cooling banks with an average

cooling rate of 45"C/s, and a second of air cooling on the balance of the run-out table

at 7"C/s. Figures 57, 58, and 59 show the results of repeating the cooling conditions

used in Figures 54, 55, and 56 for a cooling rate of 45*C/s instead of 7*C/s. In Figure

57 the model prediction of 688*C is very close to the pyrometer temperature of 686*C

±8*C. The value for run two, shown in Figure 58, of 686*C predicted versus 680'C

± 7 ' C pyrometer reading, and the run three values shown in Figure 59 of a 676*C model

prediction for a pyrometer reading of 676'C ±12*C, show similar agreement. The

69


phase transformation model results are also listed in Table VLT.

The model incorporating phase transformation was also run with the industrial

cooling conditions from Table IV, those used for Figures 54 to 59, but, using the litera

ture derived heat transfer coefficients rather than the overall heat transfer coefficient

value of 1 kW/m 2 'C. Using the Hatta et al.[6] laminar water bar heat transfer

coefficient, the Kokada et al.[7] Film boiling heat transfer coefficient, the Colas and

Sellars[2] water curtain values, and the 0.34 phase transformation kinetics as input to

the phase transformation model, a thermal profile prediction was made. The industrial

cooling conditions as presented in Table DI, with a cooling rate input of 26*C/s, which

is the average cooling rate of the combined 45*C/s and 7*C/s sections were used. For

run one cooling conditions the model predicted a top surface temperature for the down-

coiler position of the run-out table of 711'C and the pyrometer reading is 686*C ±8*C,

these values are shown in Figure 60. The model predicted temperature for run two

cooling conditions was 705"C with a pyrometer reading of 680*C ±7*C, as shown in

Figure 61. For the run three cooling conditions the model top surface predicted tem

perature is 709*C with an equivalent pyrometer location value of 676*C ±12*C. The

results are listed in Table V H with the other phase transformation model results, all of

which show reasonable agreement with the pyrometer values.

70


5.3 Mathematical Model

5.3.1 Sensitivity

The time step for the model is the step size along the direction of strip travel

divided by the strip velocity. A 10 mm step size, which results in a time step of 1.3 to

2.0 milliseconds, was chosen based on the size of the laminar water bar, as previously

stated in section 4.1.2. The thermal profile model was run with various numbers of

nodes through the thickness of the strip and the results are plotted in Figure 63. The

predicted final temperature value increases as the number of nodes through the thick

ness of the strip increases or decreases towards the value of 200 nodes, so this value

was used in the model which results in an average node thickness of 15 urn to 20 urn.

The step size was varied from 10 mm to determine the effect on model predictions.

Using 200 nodes through thickness, the results are shown in Figure 64 and it can be

seen that varying the step size from 5mm to 40mm results in less than 10"C of varia

tion in the model temperature prediction.

The industrial data from Table Dl shows that the strip velocity deviation is ±0.3

m/s. Using the overall model incorporating phase transformation, cooling conditions

from run one Table Dl, with a 45 'C/s cooling rate input; the model was run three

times with the only variable being the strip velocity, with values of 5.54 m/s, 5.24 m/s,

and 4.94 m/s. Figure 65 shows that the ±0.3 m/s variation in strip velocity results in a

± H " C deviation in the predicted temperature, showing that the model is sensitive to the

strip velocity.

71


5.3.2 Validation

The overall model predicted temperatures for the down-coiler position on the

run-out table gave reasonable agreement with the pyrometer readings as shown in Fig

ures 54 to 62, and listed in Table VLL

To provide some external validation of the overall model incorporating phase

transformation, the cooling conditions experienced by the 0.34 carbon steel in run one

Table DI, were duplicated on a Gleeble high temperature testing machine, recently

installed in the Metals and Materials Engineering department. The test conditions,

shown in Figure 66, consisted of heating from room temperature to 800*C at 400*C/s,

heating at 65*C/s from 800°C to 930*C, holding at 930°C for 2 seconds, cooling to

900"C in 3 seconds (10°C/s), cooling to 770*C in 3 seconds (43"C/s), and then cooling

to room temperature at 7*C/s. The results of this simulation are shown in Figure 67

with some model predicted temperature values included to show the accuracy of the

duplication of the model predicted cooling rates. Examination of the test data showed

that transformation started at approximately 732*C, which is approximately the transfor

mation start temperature for a cooling rate of 7°C/s. The simulation test sample was

polished and etched in 5 percent Picral and is shown in the photomicrograph, Figure

68, with the percentage ferrite of 0.39 and a prior austenite grain size of 18 um ± 6 |im

in reasonable agreement with the transfer bar and down-coiler samples listed in Table

rv.

72

RESULTS & DISCUSSION 5.4

5.4 Tables and Figures 0.054 Carbon 720"C target coiling temperature Strip Velocity 359.6 m/min Strip Gauge 3.89 mm Strip Width 1.056 m

Pyrometer F E X T ROT DC PI P2 P3 P4

Tempera ture(C) 910 770 856 856 759 786 730 Deviation ± 6 6 12 17 11 5 11

0.054 Carbon 630*C target coiling temperature

3.89 mm Strip Width 1.052 m



0.054 Carbon 550*C target coiling temperature Strip Velocity 375.3 m/min Strip Gauge 3.89 mm Strip Width 1.052 m

Pyrometer F E X T ROT D C PI P2 P3 P4


0.07 Carbon w/niobium 550°C target coiling temperature Strip Velocity 377.9 m/min

3.89 mm Strip Width 1.053 m


Temperature(C) 895 637 539 847 700 645 625 Deviation ± 74 81 36 28 0 12 0

Table Ha Plant Conditions for Four Runs

73


0.07 Carbon w/niobium 720*C target coiling temperature Strip Velocity 329.9 nVmin Strip Gauge 3.89 mm Strip Width 1.053 m


Temperature(C) Deviation ±

917 36

757 5

710 15

863 16

716 7

773 9

716 13

0.07 Carbon w/niobium 630"C target coiling temperature Strip Velocity 348.0 m/min Strip Gauge 3.89 mm Strip Width 1.053 m

Pyrometer F E X T ROT DC PI P2 P3


921 29

721 10

629 10

819 20

700 0

704 25

P4

637 9

0.07 Carbon w/niobium 720*C target coiling temperature Strip Velocity 328.1 m/min Strip Gauge 3.89 mm Strip Width 1.056 m


Temperature(C) 924 752 711 865 704 767 723 Deviation ± 12 7 26 21 7 10 23

0.07 Carbon w/niobium 550*C target coiling temperature

Strip Gauge 2.62 mm Strip Width 1.049 m



Table LTb Plant Conditions for Four Runs

74


0.07 Carbon w/niobium 720°C target coiling temperature Strip Velocity 414.3 m/min Strip Gauge 2.62 mm Strip Width 1.050 m



917 7

753 4

714 8

831 18

711 7

769 6

724 17

0.054 Carta Strip Veloc Strip Gauge Strip Width

an 630°C target c ity 45' j 2.

1.1

oiling temperature 3.8 rn/min 62 mm 353 m

0.054 Carta Strip Veloc Strip Gauge Strip Width

oiling temperature 3.8 rn/min 62 mm 353 m



895 7

717 8

629 14

835 4

701 3

732 15

638 11

0.054 Carbon 550*C target coiling temperature Strip Velocity 457.9 m/min

Strip Width ...1.053 m



0.054 Carbon 720°C target coiling temperature Strip Velocity 458.2 rn/min Strip Gauge 2.62 mm Strip Width 1.050 m


Tempera ture(C) Deviation ±

897 6

779 4

718 8

864 15

784 6

795 11

735 9

Table lie Plant Conditions for Four Runs

75


Plant cooling conditions, 0.34 weight percent carbon.

Run

Number

Number of

Sprays On

F E X T

Temperature

C C )

D C

Temperature

C C )

Strip

Velocity

(m/s)

1 10 ±2 914 ± 9 686 ±8 5.24 ±.3

2 9 ±2 909 ±8 680 ±7 4.89 ±.3

3 11 ±3 924 ± 10 676 ±11 5.20 ± 3

Table m Industrial Plant Cooling Conditions.

76


Metallographic Data

Sample Fraction Ferrite

Undercooling C C )

Gamma Grain Size (um)

C C T @ 5'Os 0.53 42 19 ± 2 *

C C T @ 10 °C/s 0.51 63 28 ± 7 *

C C T @ 15 'as 0.56 66 22 ± 3 *

C C T @ 27 'as 0.65 81 18 ± 2 *

C C T @ 40 °c/s 0.15 90

C C T @ 51 *C/s 0.41 95 37 ± 7 *

C C T @ 55 °C/s 0.38 28 ± 3 *

C C T @ 65 'as 0.40 101 32 ± 6*

C C T @ 78 'as 0.07 114 37 ± 5 *

C C T @ 103 *C/s 0.13 118 36 ± 4 *

C C T average 29 ± 9 *

0.34 C down-coiler 0.16 35 ± 9

0.34 C Gleeble 0.39 18 ± 3

•....variation due to difficulties in counting technique as all C C T samples were the result of the same austenitizing conditions of 3 minutes at 850*C, 1 minute at 820*C, followed by continuous cooling rate. An overall average prior austenite grain diameter of 29 um ± 9 um will be used.

Table IV Metallographic data for the 0.34 carbon samples, for the down-coiler sam

ple, the continuous cooling test samples, and the Gleeble simulation sample; with tabu

lated values for, cooling rate, fraction ferrite, undercooling, and average austenite grain

size.

77


Composition

Transfer bar sample Down-coiler sample

C 0.343 0.370

Mn 0.700 0.840

P 0.008 0.018

S 0.009 0.010

Si 0.009 0.150

Cu 0.021 <0.02

Ni 0.006 <0.08

Cr 0.023 < 0.025

Mo 0.003 0.000

V 0.000 0.000

Nb 0.000 0.000

A l 0.043 0.023

Table V Comparison of the composition of the down-coiler and transfer bar medium

carbon test samples.

78


Grain size as a function of coiling temperature for a low carbon steel

0.054 % carbon, 3.89 mm gauge, 720 °C 17 ± 2 um

0.054 % carbon, 3.89 mm gauge, 630 ' C 14 ± 1 um

0.054 % carbon, 2.69 mm gauge, 720 "C 16 ± 2 um

0.054 % carbon, 2.69 mm gauge, 630 *C 16 ± 2 um

Table VI Grain size versus coiling temperature for a 0.054 weight percent carbon

grade steel.

79


Model temperature predictions for various cooling conditions

Model Temperature

Values

Pyrometer

Reading

Cooling rate = Tsan

Figure 54 706*C 686'C ± 8 ' C 7'C/s = 732'C

Figure 55 699'C 680'C ± 7 ' C 7'C/s = 732'C

Figure 56 700'C 676'C ±12 'C 7'C/s = 732'C

Figure 57 688'C 686'C ± 8 ' C 45'C/s = 688'C

Figure 58 685'C 680'C ±7 'C 45'C/s = 688'C

Figure 59 676'C 676°C ±12*C 45'C/s = 688'C

Figure 60 711'C 686*C ±8*C 26'C/s = 710'C

Figure 61 705'C 680'C ±7*C 26°as = 710'C

Figure 62 709'C 676'C ±12 'C 26'C/s = 710'C

Table VII.„..Tabulated model predictions, for low (7*C/s) and high (45*C/s) cooling

rates, and for the literature heat transfer coefficients at an average cooling rate,

(26'C/s).

80


O C O C D t ^ O C O ^ T f C M O C O t O ^ C M O ^

Black Zone Radius ( mm)

Figure 17 Black zone radius as a function of a constant steel surface temperature, per

Hatta et al.[4]

81


Heat Transfer Coefficient

Kw/m C

Figure 18 Hatta laminar water bar heat transfer coefficient as a function of contact

radius

82

O

§

i Distance from Finish Stand F4 2: c ? meters ?

RESULTS & DISCUSSION

o o

o o o o o o o o m o m o w o m o O J C D c o o o r ^ r ^ c o c o

Temperature (C)

Figure 21 Thermal profile model literature heat transfer coefficients 0.05% carbon,

3.89 mm gauge, target coiling temperature 720*C.

85


o o

o o o o o o o o m o w o w o m o O > o > c o o 5 f » * r ^ c o c o

Temperature (C)

Figure 22 Thermal profile model literature heat transfer coefficients, 0.07% carbon,

0.024% Nb, 3.89 mm gauge, target coiling temperature 720°C.

86


o o

c n o ) c o o o r ^ r > . c o c D

Temperature (C)


2.62 mm gauge, target coiling temperature 720'C.

87


o o

o o o o o o o o m o m o m o u D O c o c n c o c o r « » r ^ c o c o

Temperature (C)


0.024% Nb, 2.62 mm gauge, target coiling temperature 720 *C.

88


o o

o o o o o o o o i n o w o m o m o C O O J O O C O h - r ^ C O C O

Temperature (C)


3.89 mm gauge, target coiling temperature 630*C.

89


Temperature (C)

Figure 26 A sample temperature profile from the plant data.

90


Temperature (C)

Figure 27 Thermal profile model, overall heat transfer coefficient calculated from

plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target coiling

temperature 720"C

91


Temperature (C)


plant pyrometer measurements, 0.07% carbon, 0.024% Nb, 3.89 mm gauge, target

coiling temperature 720*C

92


o o

o o o o o o o o o o m o m o m o m o w o

Temperature (C)


plant pyrometer measurements, 0.07% carbon, 0.024% Nb, 2.62 mm gauge, target

coiling temperature 720'C

93


o o

o o o o o o o o o o i n o m o m o L o o m o o i c o o o o o r ^ r x - c o c o m i o

Temperature (C)



temperature 720°C

94


o o

o o o o o o o o o o L n o m o m o m o L o o c o c o o o o o r ^ r - c o c o m m

Temperature (C)



temperature 630* C

95


o o

o o o o o o o o o o L o o m o m o m o m o c 7 ) c o a o a o r « r ^ c o c o m m

Temperature (C)



temperature 550°C

96


o o

o o o o o o o o o o o o o fflcooiooooooooeoSsSNN

Temperature (C)

Figure 33 Thermal profile model sensitivity, Overall Heat Transfer Coefficient

Calculated from plant pyrometer measurements, 0.05% carbon, 3.89 mm gauge, target

coiling temperature 720° C

97


o o o o o o o o o o o o o o o o o o o o o o eo CM o cn oo h- co m ^ co CM t- o o> oo r*. co i n - r co cvj cjiO)0>oioococooocooococococor*»S-r^r»-r*»-S'r^r ,>.

Temperature (C)

Figure 34 Thermal profile model results for a 0.05 carbon, 3.89 mm gauge steel

using an individual laminar water bar heat transfer coefficient of 10 kW/m 2 °C and

showing the effect of using 20 kW/m2*C and 5 kW/m 2 o C on the target coiling

temperature of 720*C.

98


Fraction Transformed

Figure 35 Isothermal dilatometer results for 673*C test

99


Fraction Transformed

Figure 36 Fraction transformed as a function of time for constant temperature =

673#C.

100


• \ • \

• \

CO CD

\ if) CO \ C\J

> CD «—«

Q

CO

CO II • \

r -

• \

OJ

h CNJ

CO

CM

CO ©

T T

d

d CVJ d in d

T - IT) CM i If)

c\i

In(ln(1/(1-Fx)))

Figure 37 Isothermal 673*C austenite to ferrite kinetics plotted as lnln(l/(l-FX))

versus ln(t).

101


ln(b), ferrite

Figure 38 ln(b) Avrami coefficient for the isothermal formation of ferrite in the 0.34

carbon steel.

102


i i i i i i I i i i i i i i i i i r Tru3cor^coc»i-^CMcoTrir)cDr^coo>c\i-i-CMoo o o o o o o

i i i i i i I I I I I CM CM CM i i i

ln(b), pearlite

Figure 39 ln(b) Avrami coefficient for the isothermal formation of pearlite in the

0.34 carbon steel.

103


n (ferrite)

Figure 40 Avrami coefficient, n f, for the austenite-to-ferrite transformation in the 0.34

% C, plain carbon steel.

104


o

-O i _ C3 O o> CO

' c i I I i r

m co m CM m T -

C O ^ C M ^ to T -

o cn cn in oo

d

n (pearlite)

Figure 41 Avrami coefficient, n,,, for the austenite-to-pearlite transformation in the

0.34 % C, plain carbon steel.

105


ln(b), ferrite

Figure 42 Calculated ln(b) values for the ferrite transformation assuming ii f = 1.25,

for the 0.34% carbon steel.

106


ln(b), pearlite

Figure 43 Calculated ln(b) values for the pearlite transformation assuming ftp = 1.14,

for the 0.34% carbon steel.

107


LOTTOOCNJT—cxjo^cqr^couo CNJ C\i C\j C\i C\j T ^ T - ^ T - ^ T — i - ^ T - ^ - r ^ T - ^ - r ^ o o o o o

n v a l u e

Figure 44 Average Avrami coefficient V for 0.34% carbon compared to other

experimental values (Campbell[27])

108


+•

<

• x

x

r . r-. r*. r CM J CM I CM I C\l ) C\J <

CD CD CD CD CD CD CD CD CD CD To GO "55 CO GO Q

rr co CM T - 3 o o o o o n . T- y- -r- T - O

• + O < X «

o

o • +

+ +

CO

~ I I 1 I I

CM i - O i - CM CO r f i If)

O 00

o CO

o T t

o CM

o o

o oo

o CO

o rr

CM

CO

< h-o

s o CD

" O c 3

CO

ln(b), ferrite

Figure 45 Comparison of the ln(b) Avrami coefficient for the austenite-ferrite

transformation is several plain-carbon steels(Campbell[27]).

109


• 9 © <

O

co d

CD CD © CD CD CD CD CD CD CD CD CD

wcyScyjcy) c/5co com coo o o T -CMCOTT cor*, o o o o o o X < o + • >

CM,

"55 .O Q. E CO

O

o oo

o CO

_ o

o CM

CM T - O T - CM CO i i co 00

ln(b), pearlite

Figure 46 Comparison of the ln(b) Avrami coefficient for the austenite-ferrite

transformation is several plain-carbon steels(Campbell[27]).

110


Dilation - Temperature Figure 47 Dilatometer-time and temperature-time data for a continuous cooling rate

of 27'C/s*

111


112


Undercooling ( C ) (785 - Tst) Figure 49 The undercooling for the austenite-to-ferrite start temperature as a function

of the cooling rate.

113


I 1 1 1 1 1 1 i h o C O r ^ C O U O T T C O C N J - r - O o o o o o o o o

Percentage Ferrite

Figure 50....Fraction ferrite as a function of cooling rate, from metallographic

examination.

114


Figure 51 Continuously cooled dilatometer sample.

115




Temperature (C)

Figure 54 Surface thermal profile, using the overall heat transfer coefficient = 1

kW/m 2*C, Table DT-Run #1 cooling conditions, 3.94mm gauge, and a transformation

start temperature of 732'C ( dT/dt = 7'C/s).

118


o o

o o o o o o o o o o o o o o o o o C O r r c \ J O C O C D r f C \ J O O O C D T t C \ J O C O C O T t CnO5O)O)COCOCO00 0 0 S N N N N ( D t O ( O

Temperature (C)


kW/m 2*C, Table DJ-Run #2 cooling conditions, 3.94mm gauge, and a transformation

start temperature of 732'C ( dT/dt = 7'C/s).

119


o o

o o o o o o o o o o o o o o o o o C D ^ C N J O C O C D T f C N J O C O t O T r C N J O C O C O T f O l C I C n c n c O C O O O C O C O N N N N N O l D C D

Temperature (C)

Figure 5 6 Surface thermal profile, using the overall heat transfer coefficient = 1

kW/m 2*C, Table DI-Run #3 cooling conditions, 3.94mm gauge, and a transformation

start temperature of 732'C ( dt/dt = 7'C/s).

120


o o

O O O O Q O O O O O O O O O

O J O ) 0 > 0 0 0 0 0 0 0 0 0 3 N N N S N ( 0

Temperature (C)


kW/m 2*C, Table HI-Run #1 cooling conditions, 3.94mm gauge, and a transformation

start temperature of 688°C ( dt/dt = 45*C/s).

121


o o

o o o o o o o o o o o o o o t ( v J O 0 0 « 0 t C v | O 0 0 ( f i t C M O 0 0 O ) O ) O ) C O C O C O 0 0 0 0 N N N N N C O

Temperature (C)


kW/m 2 °C, Table Hi-Run #2 cooling conditions, 3.94mm gauge, and a transformation

start temperature of 688°C ( dt/dt = 45'C/s).

122

meters


o o

o o o o o o o o o o o o o o o o o c o r f o j o a j c D T r c N i o c o c o T r c M O c o c o r r cooJOTOioocooococo i^r^ .r^r^r^cococD

Temperature (C)

Figure 60 Surface thermal profile, using the literature heat transfer coefficients ,

Table DI-Run #1 cooling conditions, 3.94mm gauge, and a transformation start

temperature of 710*C ( dt/dt = 26*C/s).

124


Temperature (C)


Table LTI-Run #2 cooling conditions, 3.94mm gauge, and a transformation start

temperature of 710"C ( dt/dt = 26'C/s).

125


Temperature (C)


Table HI-Run #3 cooling conditions, 3.94mm gauge, and a transformation start

temperature of 710°C ( dt/dt = 26'C/s).

126


to in in in m U D m m m <<* -<t -t TJ- T T C O C O C O

Temperature (C)

Figure 63 Effect on predicted center line temperature of changes in the number of

nodes through thickness.

127


E E

co N

CO Q . ©

CO

Temperature (C)

Figure 64 Effect on predicted center line temperature of changes in the step size,

where the step size equals the strip velocity times the time increment.

128


o o

o o o o o o o o o o o o o o o T T C N O C O C O r f C N J O C O C O T f C N O C O C O C D O C n o O O O C O C O C O N N N N S C O C O

Temperature (C)

Figure 65 Effect of the ±0.3 m/s deviation of the strip velocity on the predicted

temperature profile.

129


Temperature (C)

Figure 66....Industrial cooling profile simulated on the Gleeble high temperature testing

machine.

130


Temperature (C)

Figure 67 Industrial cooling conditions, simulated on a Gleeble high temperature

testing machine.

131


Figure 68 Microstructure of the Gleeble cooling simulation sample of the Table UJ

run one cooling conditions.

132

CONCLUSIONS 6.1

6 CONCLUSIONS

6.1 Summary

A mathematical model has been formulated to predict the through thickness

temperature distribution of strip as it cools on the run-out table of a hot-strip mill. The

model incorporates the heat of transformation associated with the austenite to ferrite

and ferrite plus pearlite phase change.

The phase transformation has been characterized by an Avrami equation and the

coefficients, n and b, have been determined from isothermal diametral dilatometer tests.

The transformation start temperature is a function of the cooling rate experienced by

the steel and a relationship between the cooling rate and the temperature at which the

phase transformation starts has been determined from continuous cooling diametral

dilatometer tests.

The total fraction ferrite that will form is also a function of the cooling rate and

the fraction ferrite has been characterized as a function of cooling rate using the

quantitative analysis capability of the Wild-Leitz image analyzer with polished and

etched continuous cooling test samples.

In-plant temperature measurements for various strip cooling regimes were

obtained and used with the thermal profile model to calculate an overall effective heat

transfer coefficient for a laminar water bar cooling bank. Back calculation of a heat

transfer coefficient for individual laminar water bar headers using the plant temperature

data was unsuccessful due to the insensitivity of the strip temperature to individual

133

CONCLUSIONS 6.1

laminar water bars due to the short strip residence rime under any individual header.

Metallographic examination of strip samples from the in-plant temperature data

acquisition tests to determine a relationship between coiling temperature and grain size

was inconclusive as no obvious relationship was seen.

Heat transfer coefficients for various cooling regimes; laminar water bar, water

curtain, film boiling, support roll contact, and air cooling by convection/radiation, were

found in the literature. These values were used in the overall model incorporating

phase transformation along with industrial cooling conditions to predict strip thermal

profiles and the predicted temperatures were compared with plant pyrometer values.

The effect of variations in the various heat transfer coefficients on the model

thermal profile predictions was examined and the model was shown to be relatively

insensitive to small variations in the heat transfer coefficients obtained from literature

or calculated. The model did, however, show some sensitivity to variations in strip

velocity.

The cooling regime predicted by the overall model, incorporating phase

transformation, was simulated on a Gleeble 1500 testing machine. The resulting

transformation start temperature of 730*C is similar to that predicted by the emperical

transformation start versus constant cooling rate data for an average cooling rate of

7'C/s.

The dilatometer test samples were from transfer bar stock, due to specimen size

constraints, and had not received the finish stand rolling that a down_coiled sample

would. To determine the validity of using transfer bar stock to predict down-coiled

134

CONCLUSIONS 6.2

reactions a down-coiled sample was acquired and metallographic examination of the

transfer bar, down-coiled, and Gleeble samples gave reasonable agreement in prior

austenite grain size for the samples tested.

6.2 Conclusions

(i) For a 0.34 carbon plain-carbon steel the

Avrami equation coefficients were determined as:

n (ferrite) = 1.25 . . . 16

ln(b) (ferrite) = 23.86 - 0.038559 * ( T ) . . . 17

fi (pearlite) = 1.14 . . . 18

ln(b) (pearlite) = 25 - 0.040 * ( T ) . . . 19

(ii) The transformation start temperature as a

function of the cooling rate (dT/dt) is:

T (start) = 785 - [ 38.0933 + 2.1164(dT/dt)

- 0.0242(dT/dt)2 + 0.00011043(dT/dt)3] . . . 20

(iii) The total percentage ferrite formed as a function of

cooling rate (dT/dt) is:

Fraction ferrite = 0.59317 - 0.0048177(dT/dt) . . . 21

(iv) An overall average effective heat transfer coefficient for a

laminar water bar cooling bank of 1 kW/m2'C has been calculated and

shown to be effective for describing the plant data.

135

CONCLUSIONS 6.2

(v) An individual laminar water bar heat transfer coefficient

cannot be calculated with the present data due to the short residence time of

the strip under any one individual spray.

(vi) While no correlation between grain size and coiling

temperature was observed, a larger number of tests might separate trends

from experimental scatter.

(vii) The overall model, incorporating phase transformation,

calculated a strip thermal profile for various industrial cooling conditions, for

a 0.34 carbon plain-carbon steel. For the calculated overall heat transfer

coefficient of 1 kW/m 2 , C, using a transformation start temperature of 732"C

calculated for the cooling rate of 7*C/s experienced in the air cooling section

of the run-out table, the model predicted temperatures 19*C to 24*C greater

than the measured pyrometer values.

(viii) For the same conditions as in (vii) with a transformation start

temperature of 688'C calculated for the 45*C/s cooling rate experienced in

the water cooling section of the run-out table, the model predicted

temperatures of l ' C to 5*C greater than the pyrometer readings.

(ix) Using the literature derived heat transfer coefficient values

instead of the calculated overall heat transfer coefficient, with the same

cooling conditions as in (vii) and (viii), and a cooling rate of 26°C/s, which

is the average of the 45*C/s and 7*C/s values, model predictions were

compared with pyrometer values. For this cooling rate the transformation

136

CONCLUSIONS 6.2

start temperature is 710*C and the model predictions are 25*C to 33'C

greater than the plant pyrometer readings. These literature derived heat

transfer coefficients are useful but require fine tuning for the effects of strip

velocity which was not present in the initial experiments used to produce

these relationships.

(x) The cooling regime used for the model predictions was

simulated on a Gleeble testing machine and the model predictions were

within XX* C of the simulation temperatures. The simulation sample started

phase transformation at approximately 730" C which is also the

experimentally produced transformation start temperature for a cooling rate

of 7'C/s.

(xi) The continuous cooling and Gleeble test samples, made from

transfer bar stock, and a down-coiler sample, where examined to compare

the prior austenite grain sizes. The average prior austenite grain size for the

continuous cooling dilatometer samples of 29 um ± 8 um compares well

with the down-coiler value of 35 um ± 9 um. The Gleeble sample shows

some local microstructural variation due to banding, as shown in Figure 68,

and an average austenite grain size of 18 um ± 3 urn. This grain size is

finer than that obtained in the continuous cooling transformation samples but

is similar in magnitude. The closer austenite grain size similarity between

the continous cooling samples and the down-coiler sample supports the use

of transfer bar to measure the transformation kinetics.

137

CONCLUSIONS 6.3

6.3 Future Considerations

Modification of the existing hardware to allow characterization of other low

carbon and HSLA grades of steel as well as duplication of the prior thermo-mechanical

history is needed to increase the scope of the work. The addition of the Gleeble high

temperature testing machine means that the deformation due to the finishing stands may

be duplicated in the future or that a method of using test samples of smaller dimensions

will allow the use of down-coiler samples rather than the transfer bar samples currently

employed. Modifications to the model to include prior thermo-mechanical history and

grain growth effects prior to the run-out table should be added to allow grain growth

prediction on the run-out table as well as prediction of final grain size and thus some

mechanical properties prediction. As the grades of steel that can be characterized are

increased the bainite and martensite phase transformation kinetics should be

characterized to increase the scope of the model.

Finally the run-out table model could be integrated into an overall rolling mill

model for use in production forecasting.

138

BIBLIOGRAPHY 7

7 BIBLIOGRAPHY

1 G. Tacke, H. Litzke and E . Raquet, "Investigations into the Efficiency of cooling systems for Wide-Strip Hot Rolling Mills and Computer-aided Control of Strip Cooling", Accelerated Cooling of Steel, Proceedings of a symposium sponsored by The Metallurgical Society of AIME, Pittsburgh, Pennsylvania, August 19-21, 1985, pp 35-54.

2 R. Colas and C M . Sellars, "Computed Temperature Profiles of Hot Rolled Plate and Strip During Accelerated Cooling", (CTM convention , Winnipeg, Manitoba 1987)

3 E.A. Farber and R.L. Scorah, "Heat Transfer to Water Boiling Under Pressure", Transactions of the A S M E , 70 (1948), pp.369-384.

4 Natsuo Hatta, Jun-ichi Kokada and Koichi Hanasaki, "Numerical Analysis of Cooling Characteristics for Water Bar", Transactions of ISU, Volume 23, 1983, pp 555-564.

5 Natsuo Hatta, Jun-ichi Kokada, Hirohiko Takuda, Jun Harada and Keizo Hiraku, "Predictable Modeling for Cooling Process of a Hot Steel Plate by Laminar Water Bar", Archiv fur das Eisenhuttenwesen, 55(1984) Nr. 4 April, pp.143-148.

6 Jun-ichi Kokada, Natsuo Hatta, Hirohiko Takuda, Jun Harada and Nobuo Yasuhira, "An Analysis of Film Boiling Phenomena of Subcooled Water Spreading Radially on a Hot Steel Plate", Archiv fur das Eisenhiittenwessen, Nr. 55(1984) March

7 Arnulf Diener and Alfons Drastik, "Heat Exchange Between Strands and Guide Rollers in the Secondary Cooling Zone of a Slab Continuous Casting Machine", Fourth Japan-Germany Seminar, Nov. 1980, Tokyo, The Iron And Steel Institute of Japan

8 "Physical Constants of Some Commercial Steels at Elevated Temperatures", Edited by the British Iron and Steel Research Association, Metallurgy (General) Division Thermal Treatment Sub-Committee, London, 1953

9 M . A^rami: "Kinetics of Phase Change. I, General Theory", Journal of Chemical Physics, December 1939, pp.1103-12.

10 M . Avrami: "Kinetics of Phase Change. LI, Transformation-Time Relations for Random Distribution of Nuclei", Journal of Chemical Physics, February 1940, pp.212-224.

139

BIBLIOGRAPHY 7

11 M. Avrami: "Granulation, Phase Change, and Microstructure. Kinetics of Phase Change HI", Journal of Chemical Physics, February 1941, pp. 177-84.

12 J.W. Cahn, "The Kinetics of Grain Boundary Nucleated Reactions", Acta Metallurgies Vol. 4, September 1956, pp.449-459.

13 Prakash K. Agarwal and J.K. Brimacombe, "Mathematical Model of Heat Flow and Austerute-Pearhte Transformation in Eutectoid carbon Steel Rods for Wire", Metallurgical Transactions B, Vol 12B, March 1981, pp.121-133.

14 M.B. Kuban, R. Jayaraman, E.B. Hawbolt, and J.K. Brimacombe: "An Assessment of the Additivity Principle in Predicting Continuous-Cooling Austenite-to-Pearlite Transformation Kinetics Using Isothermal Transformation Data", Metallurgical Transactions A, Vol. 17A, September 1986, pp. 1493-1503.

15 E.B. Hawbolt, B. Chau, and J.K. Brimacombe, "Kinetics of Austenite-Pearlite Transformation in Eutectiod Carbon Steel", Metallurgical Transactions A, Vol. 14A, September 1983, pp.1803-15.

16 E.B. Hawbolt, B. Chau, and J.K. Brimacombe: "Kinetics of Austenite-Ferrite and Austenite-Pearlite Transformations in a 1025 Carbon Steel", Metallurgical Transactions A, Vol. 16A, April 1985, pp.565-77.

17 M . Morita, K. Hashiguchi, O. Hashimoto, M . Nishida, and S. Okano, "On-Line Transformation Detector for Property Control of Hot Rolled Steel", Accelerated Cooling of Steel, Proceedings of a symposium sponsored by The Metallurgical Society of AJME, Pittsburgh, Pennsylvania, August 19-21, 1985, pp.449-461

18 Eric N. Hinrichsen, "Hot Strip Mill Runout Table Cooling - A System View of Control, Operation and Equipment", Yearly Proceedings of the AISE, 1976, pp.403-408.

19 A. Hurkmans, G.A. Duit, Th.M. Hoogendoom and F. Hollander, "Accelerated cooling and the Transformation of Steel", Accelerated Cooling of Steel, Proceedings of a symposium sponsored by The Metallurgical Society of AIME, Pittsburgh, Pennsylvania, August 19-21, 1985, pp.481-499

20 Edward B. Longenberger, "Computer Modeling of a Hot Strip Mill Run out Table", Proceedings of the XXVIII Conference on Mechanical Working and Steel Processing, The Iron and Steel Society of the AIME, Pittsburgh, PA, October 26-28, 1986, pp. 169-172

140

BIBLIOGRAPHY 7

21 Yushi Miyake, Teruyuki Nishide and Shoichi Moriya, "Device and System for Controlled Cooling for Hot Strip Mill", Transactions of the ISIJ, vol 20, 1980 pp.496-503

22 H. Yada, "Prediction of Microstructural Changes and Mechanical Properties in Hot Strip Rolling", Presented at the Accelerated Cooling of Steel Symposium at the Conference of the Canadian Institute of Mining and Metallurgy, Winnipeg, Manitoba, August 24, 1987.

23 Zacay and AAronson editors, "Decomposition of Austenite by Diffusional Processes", 1962, pp. 336

24 P J . Berensen, "Film-Boiling Heat Transfer From a Horizontal Surface", Transactions of the A S M E , Journal of Heat Transfer, vol 83, August 1961, pp.351-358.

25 K.W. Andrews, "Empirical Formula for the Calculation of Some Transformation Temperatures", Journal of the Iron and Steel Institute, July 1965, pp. 721-727

26 A. le Bon, J. Rofes-Vemis, and C. Rossard, "Recrystallization and Precipitation during Hot Working of a Nb-Bearing HSLA Steel", Metal Science, volume 9, 1975, pp36-40.

27 P. Campbell, Private communication of unpublished work.

28 C. Devadas, Private communication of unpublished work.

29 Yukihisa Kuriyama, Matsuo Ataka, Masayuki Nakanishi, Masayuki Miyatake, Kouichiro Goto and Shuichi Hamauzu, "On-line Model for Prediction of Strip Temperature in Hot Rolling", Transactions of the ISIJ, vol. 23 number 9, 1983, p.B338.

30 J.S. Kirkaldy and E.A. Baganis, "Thermodynamic Prediction of the Ae3 Temperature of Steels with Additions of Mn, Si, Ni, Cr, Mo, Cu", Metallurgical Transactions A, Vol. 9A, April 1978, pp.495-501.

141

APPENDIX

APPENDIX

8.1 N O M E N C L A T U R E

*/ Avrami coefficient, 'b' for ferrite

K Avrami coefficient, 'b' for pearlite

c. Specific heat, (W/kg*C)

D Laminar water bar nozzle internal diameter, (m)

Fx(k -1) Fraction transformed in the previous time step

Fx(k) Fraction transformed in the current time step

8 Acceleration due to gravity, (m/s2)

h Heat transfer coefficient, (W/m2*C)

H Laminar water bar header height over run-out table, (m)

k Thermal conductivity, (W/m'C)

nf Avrami coefficient, V for ferrite

NP Avrami coefficient, V for pearlite

"/ average n(ferrite)

"P average n(pearlite)

Pr Prandtl number

4. Heat flux generated due to phase transformation, (W)

Q Water flow rate, (m3/s)

r Laminar water bar contact radius, (m)

Re Reynolds number

142

APPENDIX 8.1

t Time, (s)

tv Virtual time, (s)

/• Time for completion of 20% of the phase

transformation, (s)

r Time for completion of 90% of the phase

transformation, (s)

TCRTT Critical water temperature for film boiling

to water contact cooling transition, (*C)

T(k) Node temperature at this time step, (°C)

7(j y Internally calculated model temperature values, (*C)

T(k) Internally calcualted model temperature values, (*C)

Ts Steel surface temperature, (°C)

TSAJ. Water saturation temperature, (*C)

T Transformation start temperature, (*C)

Tw Water temperature, (*C)

x,y,z Spatial coordinates, (m)

X Fraction transformed

a Ferrite phases

a n Film boiling heat transfer coefficient, (W/m 2 ,C)

ot j Laminar water bar heat transfer coefficient, (W/m 2 #C)

y Austenite phase

5 Water film thickness, (m)

143

APPENDIX 8.1

6V Thermally affected water film thickness, (m)

AFx{k),dFx(k) Incremental fraction transformed for this time step

At,dt Model time step, (s)

Ax,etc Model step size, (m)

Az Model node size, (m)

e Emissivity

v Strip velocity, (m/s)

p Density, (kg/m3)

ej Stefan Boltzmann constant, (W/m 2K 4)

144

APPENDIX 8.2.1

8.2 Finite Difference Equation Derivation

Based on the assumptions in section 4 and using 200 nodes through the thickness

of the strip, the basic finite difference equations are;

8.2.1 The top surface node,

dUt p*C*A*bc^dTx

dt ~ 2 * dt

where

dUi -jfi- = <?2-1 + 4lT- 1 + Qrad + 4

4Fr-l=hc*A*<T'Fr"-T2'+")

.Al

....A2

....A3

....A4

....AS

Qnai = the heat generatedby the phase transformation ....A6

Due to the small size of the time step and the relativly large, compared to the step

size, fluid volume it will be assumed that the surrounding fluid ( air or water) is of

145

APPENDIX 8.2.2

constant temperature. To facilitate the solution of the radiation calculations the node

temperature from the current time step is used for the future time step calculations.

Which results in the equation :

2A/ 1 1 2

.A7

f p * C , * A ^ 1 2A/

8.2.2 Interior nodes

The interior nodes were less difficult to describe due to the absence of radiation

or convection effects in Eq.A.7. Through a similar set of assumptions to those in the

top surface node, Eq.A.8 was produced.

o*C *Ar

^ ....A8

= T'* * a

+4 rtcai 2*Ar

8.2.3 The bottom node

As with the formulation of the top surface node (1) equations and with similar

fluid temperature and radiation cooling simplifications the bottom node, node 'n' in this

example, could be described, after manipulation by Eq.A.9.

146

APPENDIX 8.3

-k*Ax*T. • -I ....A9

„JP*C,*Ax + hc*T'„ + a*f^(T'J-(T:)) + <i rtcai

8.2.4 Solution

The equations for the top and bottom surfacve nodes along with the

interior nodes can be gatherd together as a sparse tridiagonal matrix. A

number of simple subroutines exist to make this problem readily solvable by

computer.

8.3 Hatta et al. Model Thermal Boundary Layer Calculation

The thermal boundary layer concept used by Hatta et al. is effected by the water

velocity, which is calculated with

where Q is the water flow rate in m 3 / min, D is the diameter of the laminar water bar

nozzle in meters, H is the height above the strip in meters, and g is the acceleration due

to gravity. The thickness of the water film layer is then calculated by Eq.A. l 1.

....A10

. . .All k*dr 2KVW

147

APPENDIX 8.3

For Eq.A. l 1 k is the step number and dr is the incremental distance. V w is the

water velocity from Eq.A.10. The over prediction of cocooling by the model caused

Hatta et al.[5] to postulate that the entire water film thickness was not affected by the

heat flow. They hypothesize a thermally affected zone, a type of boundry layer in the

water film and calculate the thickness with Eq.A.12.

In Eq.A.l2 Re is the Reynolds number and Pr is the Prandd number.

If the water film from the laminar water bar was thought of as radially

symmetrical and divided into k rings of thickness r and area,

The calculation of the water temperature of any one cell, using the thermal zone

concept could then be formulated as,

....A12

dS = (7C*(2* - l)*dr2) ....A13

T(k) = rdS(k-l) , dS(k)

oy*-l)-cW*-l) dTik)

+ 8wr(*-l)' M i ....A14

148

Documents

CHARACTERIZATION OF THE COOLING AND TRANSFORMATION OF STEELS ON A RUN-OUT